id	sid	tid	token	lemma	pos
cana-5363	1	1	communications	communication	NOUN
cana-5363	1	2	on	on	ADP
cana-5363	1	3	applied	apply	VERB
cana-5363	1	4	nonlinear	nonlinear	ADJ
cana-5363	1	5	analysis	analysis	NOUN
cana-5363	1	6	issn	issn	NOUN
cana-5363	1	7	:	:	PUNCT
cana-5363	1	8	1074	1074	NUM
cana-5363	1	9	-	-	PUNCT
cana-5363	1	10	133x	133x	NUM
cana-5363	1	11	vol	vol	VERB
cana-5363	1	12	32	32	NUM
cana-5363	1	13	no	no	NOUN
cana-5363	1	14	.	.	PUNCT
cana-5363	2	1	10s	10	NOUN
cana-5363	2	2	(	(	PUNCT
cana-5363	2	3	2025	2025	NUM
cana-5363	2	4	)	)	PUNCT
cana-5363	2	5	2005	2005	NUM
cana-5363	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5363	2	7	generalization	generalization	NOUN
cana-5363	2	8	of	of	ADP
cana-5363	2	9	open	open	ADJ
cana-5363	2	10	sets	set	NOUN
cana-5363	2	11	in	in	ADP
cana-5363	2	12	pythagorean	pythagorean	PROPN
cana-5363	2	13	fuzzy	fuzzy	ADJ
cana-5363	2	14	nano	nano	PROPN
cana-5363	2	15	topological	topological	ADJ
cana-5363	2	16	spaces	space	NOUN
cana-5363	2	17	and	and	CCONJ
cana-5363	2	18	its	its	PRON
cana-5363	2	19	real	real	ADJ
cana-5363	2	20	application	application	NOUN
cana-5363	2	21	x.	x.	NOUN
cana-5363	2	22	arul	arul	PROPN
cana-5363	2	23	selvaraj1	selvaraj1	NOUN
cana-5363	2	24	and	and	CCONJ
cana-5363	2	25	n.	n.	NOUN
cana-5363	2	26	prabavathy2	prabavathy2	PROPN
cana-5363	2	27	1department	1department	NUM
cana-5363	2	28	of	of	ADP
cana-5363	2	29	mathematics	mathematics	PROPN
cana-5363	2	30	,	,	PUNCT
cana-5363	2	31	dde	dde	PROPN
cana-5363	2	32	,	,	PUNCT
cana-5363	2	33	annamalai	annamalai	PROPN
cana-5363	2	34	university	university	PROPN
cana-5363	2	35	,	,	PUNCT
cana-5363	2	36	annamalai	annamalai	PROPN
cana-5363	2	37	nagar	nagar	VERB
cana-5363	2	38	608	608	NUM
cana-5363	2	39	002	002	NUM
cana-5363	2	40	,	,	PUNCT
cana-5363	2	41	india	india	PROPN
cana-5363	2	42	;	;	PUNCT
cana-5363	2	43	(	(	PUNCT
cana-5363	2	44	deputed	depute	VERB
cana-5363	2	45	to	to	PART
cana-5363	2	46	)	)	PUNCT
cana-5363	2	47	periyar	periyar	PROPN
cana-5363	2	48	arts	art	NOUN
cana-5363	2	49	collge	collge	PROPN
cana-5363	2	50	,	,	PUNCT
cana-5363	2	51	cuddalore-607	cuddalore-607	ADJ
cana-5363	2	52	001	001	NUM
cana-5363	2	53	,	,	PUNCT
cana-5363	2	54	tamil	tamil	PROPN
cana-5363	2	55	nadu	nadu	PROPN
cana-5363	2	56	,	,	PUNCT
cana-5363	2	57	india	india	PROPN
cana-5363	2	58	.	.	PUNCT
cana-5363	3	1	xaselvarajmaths@gmail.com	xaselvarajmaths@gmail.com	X
cana-5363	3	2	2	2	NUM
cana-5363	3	3	department	department	NOUN
cana-5363	3	4	of	of	ADP
cana-5363	3	5	mathematics	mathematic	NOUN
cana-5363	3	6	,	,	PUNCT
cana-5363	3	7	arignar	arignar	PROPN
cana-5363	3	8	anna	anna	PROPN
cana-5363	3	9	gov	gov	PROPN
cana-5363	3	10	.	.	PROPN
cana-5363	3	11	arts	arts	PROPN
cana-5363	3	12	college	college	PROPN
cana-5363	3	13	,	,	PUNCT
cana-5363	3	14	musiri-621	musiri-621	ADJ
cana-5363	3	15	211	211	NUM
cana-5363	3	16	,	,	PUNCT
cana-5363	3	17	trichy(d.t	trichy(d.t	PROPN
cana-5363	3	18	.	.	PUNCT
cana-5363	3	19	)	)	PUNCT
cana-5363	3	20	,	,	PUNCT
cana-5363	3	21	tamil	tamil	PROPN
cana-5363	3	22	nadu	nadu	PROPN
cana-5363	3	23	,	,	PUNCT
cana-5363	3	24	india	india	PROPN
cana-5363	3	25	.	.	PUNCT
cana-5363	4	1	1,2	1,2	NUM
cana-5363	4	2	department	department	NOUN
cana-5363	4	3	of	of	ADP
cana-5363	4	4	mathematics	mathematics	PROPN
cana-5363	4	5	,	,	PUNCT
cana-5363	4	6	annamalai	annamalai	PROPN
cana-5363	4	7	university	university	PROPN
cana-5363	4	8	,	,	PUNCT
cana-5363	4	9	annamalai	annamalai	PROPN
cana-5363	4	10	nagar	nagar	VERB
cana-5363	4	11	608	608	NUM
cana-5363	4	12	002	002	NUM
cana-5363	4	13	,	,	PUNCT
cana-5363	4	14	india	india	PROPN
cana-5363	4	15	.	.	PUNCT
cana-5363	5	1	online2020av@gmail.com	online2020av@gmail.com	PROPN
cana-5363	5	2	,	,	PUNCT
cana-5363	5	3	prabamarch23@gmail.com	prabamarch23@gmail.com	PROPN
cana-5363	5	4	(	(	PUNCT
cana-5363	5	5	corresponding	correspond	VERB
cana-5363	5	6	author	author	NOUN
cana-5363	5	7	:	:	PUNCT
cana-5363	5	8	n.	n.	PROPN
cana-5363	5	9	prabavathy	prabavathy	ADJ
cana-5363	5	10	)	)	PUNCT
cana-5363	5	11	article	article	NOUN
cana-5363	5	12	history	history	NOUN
cana-5363	5	13	:	:	PUNCT
cana-5363	5	14	received	receive	VERB
cana-5363	5	15	:	:	PUNCT
cana-5363	5	16	12	12	NUM
cana-5363	5	17	-	-	SYM
cana-5363	5	18	01	01	NUM
cana-5363	5	19	-	-	PUNCT
cana-5363	5	20	2025	2025	NUM
cana-5363	5	21	revised	revise	VERB
cana-5363	5	22	:	:	PUNCT
cana-5363	5	23	15	15	NUM
cana-5363	5	24	-	-	NUM
cana-5363	5	25	02	02	NUM
cana-5363	5	26	-	-	PUNCT
cana-5363	5	27	2025	2025	NUM
cana-5363	5	28	accepted	accept	VERB
cana-5363	5	29	:	:	PUNCT
cana-5363	5	30	01	01	NUM
cana-5363	5	31	-	-	SYM
cana-5363	5	32	03	03	NUM
cana-5363	5	33	-	-	PUNCT
cana-5363	5	34	2025	2025	NUM
cana-5363	5	35	abstract	abstract	NOUN
cana-5363	5	36	:	:	PUNCT
cana-5363	5	37	the	the	DET
cana-5363	5	38	purpose	purpose	NOUN
cana-5363	5	39	of	of	ADP
cana-5363	5	40	this	this	DET
cana-5363	5	41	paper	paper	NOUN
cana-5363	5	42	is	be	AUX
cana-5363	5	43	to	to	PART
cana-5363	5	44	define	define	VERB
cana-5363	5	45	and	and	CCONJ
cana-5363	5	46	study	study	VERB
cana-5363	5	47	a	a	DET
cana-5363	5	48	new	new	ADJ
cana-5363	5	49	class	class	NOUN
cana-5363	5	50	of	of	ADP
cana-5363	5	51	sets	set	NOUN
cana-5363	5	52	called	call	VERB
cana-5363	5	53	pythagorean	pythagorean	PROPN
cana-5363	5	54	fuzzy	fuzzy	ADJ
cana-5363	5	55	nano	nano	NOUN
cana-5363	5	56	𝑍	𝑍	PROPN
cana-5363	5	57	(	(	PUNCT
cana-5363	5	58	resp	resp	NOUN
cana-5363	5	59	.	.	PUNCT
cana-5363	6	1	𝛿	𝛿	ADJ
cana-5363	6	2	,	,	PUNCT
cana-5363	6	3	𝛿𝒮	𝛿𝒮	NOUN
cana-5363	6	4	and	and	CCONJ
cana-5363	6	5	pre)-open	pre)-open	NOUN
cana-5363	6	6	sets	set	NOUN
cana-5363	6	7	in	in	ADP
cana-5363	6	8	pythagorean	pythagorean	PROPN
cana-5363	6	9	fuzzy	fuzzy	ADJ
cana-5363	6	10	nano	nano	PROPN
cana-5363	6	11	topological	topological	ADJ
cana-5363	6	12	spaces	space	NOUN
cana-5363	6	13	.	.	PUNCT
cana-5363	7	1	basic	basic	ADJ
cana-5363	7	2	properties	property	NOUN
cana-5363	7	3	of	of	ADP
cana-5363	7	4	pythagorean	pythagorean	PROPN
cana-5363	7	5	fuzzy	fuzzy	ADJ
cana-5363	7	6	nano	nano	NOUN
cana-5363	7	7	𝑍	𝑍	PROPN
cana-5363	7	8	(	(	PUNCT
cana-5363	7	9	resp	resp	NOUN
cana-5363	7	10	.	.	PUNCT
cana-5363	8	1	𝛿	𝛿	ADJ
cana-5363	8	2	,	,	PUNCT
cana-5363	8	3	𝛿𝒮	𝛿𝒮	NOUN
cana-5363	8	4	and	and	CCONJ
cana-5363	8	5	pre	pre	ADJ
cana-5363	8	6	)	)	PUNCT
cana-5363	8	7	-open	-open	ADJ
cana-5363	8	8	and	and	CCONJ
cana-5363	8	9	their	their	PRON
cana-5363	8	10	respective	respective	ADJ
cana-5363	8	11	closed	closed	ADJ
cana-5363	8	12	sets	set	NOUN
cana-5363	8	13	are	be	AUX
cana-5363	8	14	analysed	analyse	VERB
cana-5363	8	15	.	.	PUNCT
cana-5363	9	1	we	we	PRON
cana-5363	9	2	also	also	ADV
cana-5363	9	3	used	use	VERB
cana-5363	9	4	them	they	PRON
cana-5363	9	5	to	to	PART
cana-5363	9	6	introduce	introduce	VERB
cana-5363	9	7	the	the	DET
cana-5363	9	8	new	new	ADJ
cana-5363	9	9	notions	notion	NOUN
cana-5363	9	10	like	like	ADP
cana-5363	9	11	pythagorean	pythagorean	PROPN
cana-5363	9	12	fuzzy	fuzzy	ADJ
cana-5363	9	13	nano	nano	NOUN
cana-5363	9	14	𝑍	𝑍	PROPN
cana-5363	9	15	(	(	PUNCT
cana-5363	9	16	resp	resp	NOUN
cana-5363	9	17	.	.	PUNCT
cana-5363	10	1	𝛿	𝛿	ADJ
cana-5363	10	2	,	,	PUNCT
cana-5363	10	3	𝛿𝒮	𝛿𝒮	NOUN
cana-5363	10	4	and	and	CCONJ
cana-5363	10	5	pre)-closure	pre)-closure	PROPN
cana-5363	10	6	(	(	PUNCT
cana-5363	10	7	resp	resp	NOUN
cana-5363	10	8	.	.	PUNCT
cana-5363	11	1	interior	interior	PROPN
cana-5363	11	2	)	)	PUNCT
cana-5363	11	3	and	and	CCONJ
cana-5363	11	4	their	their	PRON
cana-5363	11	5	relations	relation	NOUN
cana-5363	11	6	with	with	ADP
cana-5363	11	7	already	already	ADV
cana-5363	11	8	existing	exist	VERB
cana-5363	11	9	well	well	ADV
cana-5363	11	10	known	know	VERB
cana-5363	11	11	sets	set	NOUN
cana-5363	11	12	are	be	AUX
cana-5363	11	13	also	also	ADV
cana-5363	11	14	investigated	investigate	VERB
cana-5363	11	15	.	.	PUNCT
cana-5363	12	1	keywords	keyword	NOUN
cana-5363	12	2	:	:	PUNCT
cana-5363	12	3	pythagorean	pythagorean	VERB
cana-5363	12	4	fuzzy	fuzzy	ADJ
cana-5363	12	5	nano	nano	NOUN
cana-5363	12	6	open	open	ADJ
cana-5363	12	7	set	set	NOUN
cana-5363	12	8	,	,	PUNCT
cana-5363	12	9	pythagorean	pythagorean	PROPN
cana-5363	12	10	fuzzy	fuzzy	ADJ
cana-5363	12	11	nano	nano	PROPN
cana-5363	12	12	pre	pre	X
cana-5363	12	13	open	open	ADJ
cana-5363	12	14	set	set	NOUN
cana-5363	12	15	,	,	PUNCT
cana-5363	12	16	pythagorean	pythagorean	PROPN
cana-5363	12	17	fuzzy	fuzzy	ADJ
cana-5363	12	18	nano	nano	PROPN
cana-5363	12	19	𝛿	𝛿	DET
cana-5363	12	20	semi	semi	ADJ
cana-5363	12	21	open	open	ADJ
cana-5363	12	22	set	set	NOUN
cana-5363	12	23	,	,	PUNCT
cana-5363	12	24	pythagorean	pythagorean	PROPN
cana-5363	12	25	fuzzy	fuzzy	ADJ
cana-5363	12	26	nano	nano	NOUN
cana-5363	12	27	𝑍	𝑍	PROPN
cana-5363	12	28	open	open	ADJ
cana-5363	12	29	set	set	NOUN
cana-5363	12	30	.	.	PUNCT
cana-5363	13	1	ams(2000	ams(2000	ADJ
cana-5363	13	2	)	)	PUNCT
cana-5363	13	3	subject	subject	ADJ
cana-5363	13	4	classification	classification	NOUN
cana-5363	13	5	:	:	PUNCT
cana-5363	13	6	03e72	03e72	NUM
cana-5363	13	7	,	,	PUNCT
cana-5363	13	8	54a05	54a05	NUM
cana-5363	13	9	,	,	PUNCT
cana-5363	13	10	54a40	54a40	NUM
cana-5363	13	11	.	.	NOUN
cana-5363	13	12	1	1	NUM
cana-5363	13	13	introduction	introduction	NOUN
cana-5363	13	14	zadeh	zadeh	PROPN
cana-5363	13	15	in	in	ADP
cana-5363	13	16	[	[	X
cana-5363	13	17	21	21	NUM
cana-5363	13	18	]	]	PUNCT
cana-5363	13	19	established	establish	VERB
cana-5363	13	20	the	the	DET
cana-5363	13	21	idea	idea	NOUN
cana-5363	13	22	of	of	ADP
cana-5363	13	23	fuzzy	fuzzy	ADJ
cana-5363	13	24	in	in	ADP
cana-5363	13	25	1965	1965	NUM
cana-5363	13	26	,	,	PUNCT
cana-5363	13	27	which	which	PRON
cana-5363	13	28	is	be	AUX
cana-5363	13	29	a	a	DET
cana-5363	13	30	generalization	generalization	NOUN
cana-5363	13	31	of	of	ADP
cana-5363	13	32	usual	usual	ADJ
cana-5363	13	33	set	set	NOUN
cana-5363	13	34	using	use	VERB
cana-5363	13	35	fuzzy	fuzzy	NOUN
cana-5363	13	36	where	where	SCONJ
cana-5363	13	37	each	each	DET
cana-5363	13	38	element	element	NOUN
cana-5363	13	39	has	have	VERB
cana-5363	13	40	a	a	DET
cana-5363	13	41	membership	membership	NOUN
cana-5363	13	42	degree	degree	NOUN
cana-5363	13	43	in	in	ADP
cana-5363	13	44	[	[	X
cana-5363	13	45	0,1	0,1	NUM
cana-5363	13	46	]	]	PUNCT
cana-5363	13	47	.	.	PUNCT
cana-5363	14	1	the	the	DET
cana-5363	14	2	subsequent	subsequent	ADJ
cana-5363	14	3	advanement	advanement	NOUN
cana-5363	14	4	of	of	ADP
cana-5363	14	5	fuzzy	fuzzy	ADJ
cana-5363	14	6	subsets	subset	NOUN
cana-5363	14	7	was	be	AUX
cana-5363	14	8	the	the	DET
cana-5363	14	9	intuitionistic	intuitionistic	ADJ
cana-5363	14	10	fuzzy	fuzzy	ADJ
cana-5363	14	11	set	set	NOUN
cana-5363	14	12	published	publish	VERB
cana-5363	14	13	by	by	ADP
cana-5363	14	14	atanassov	atanassov	NOUN
cana-5363	14	15	,	,	PUNCT
cana-5363	14	16	[	[	X
cana-5363	14	17	4	4	X
cana-5363	14	18	]	]	PUNCT
cana-5363	14	19	in	in	ADP
cana-5363	14	20	1983	1983	NUM
cana-5363	14	21	,	,	PUNCT
cana-5363	14	22	which	which	PRON
cana-5363	14	23	has	have	VERB
cana-5363	14	24	elements	element	NOUN
cana-5363	14	25	having	have	VERB
cana-5363	14	26	membership	membership	NOUN
cana-5363	14	27	and	and	CCONJ
cana-5363	14	28	non	non	ADJ
cana-5363	14	29	-	-	ADJ
cana-5363	14	30	membership	membership	ADJ
cana-5363	14	31	degree	degree	NOUN
cana-5363	14	32	.	.	PUNCT
cana-5363	15	1	in	in	ADP
cana-5363	15	2	1968	1968	NUM
cana-5363	15	3	,	,	PUNCT
cana-5363	15	4	chang	chang	PROPN
cana-5363	15	5	in	in	ADP
cana-5363	15	6	[	[	X
cana-5363	15	7	5	5	NUM
cana-5363	15	8	]	]	PUNCT
cana-5363	15	9	defined	define	VERB
cana-5363	15	10	fuzzy	fuzzy	ADJ
cana-5363	15	11	topological	topological	ADJ
cana-5363	15	12	space	space	NOUN
cana-5363	15	13	and	and	CCONJ
cana-5363	15	14	fundamental	fundamental	ADJ
cana-5363	15	15	results	result	NOUN
cana-5363	15	16	such	such	ADJ
cana-5363	15	17	as	as	ADP
cana-5363	15	18	continuity	continuity	NOUN
cana-5363	15	19	,	,	PUNCT
cana-5363	15	20	open	open	ADJ
cana-5363	15	21	and	and	CCONJ
cana-5363	15	22	closed	closed	ADJ
cana-5363	15	23	set	set	NOUN
cana-5363	15	24	.	.	PUNCT
cana-5363	16	1	following	follow	VERB
cana-5363	16	2	this	this	PRON
cana-5363	16	3	,	,	PUNCT
cana-5363	16	4	lowen	lowen	PROPN
cana-5363	16	5	in	in	ADP
cana-5363	16	6	[	[	X
cana-5363	16	7	7	7	NUM
cana-5363	16	8	]	]	PUNCT
cana-5363	16	9	defined	define	VERB
cana-5363	16	10	fuzzy	fuzzy	ADJ
cana-5363	16	11	topological	topological	ADJ
cana-5363	16	12	space	space	NOUN
cana-5363	16	13	in	in	ADP
cana-5363	16	14	other	other	ADJ
cana-5363	16	15	form	form	NOUN
cana-5363	16	16	.	.	PUNCT
cana-5363	17	1	coker	coker	NOUN
cana-5363	17	2	introduced	introduce	VERB
cana-5363	17	3	the	the	DET
cana-5363	17	4	idea	idea	NOUN
cana-5363	17	5	of	of	ADP
cana-5363	17	6	intuitionistic	intuitionistic	ADJ
cana-5363	17	7	fuzzy	fuzzy	ADJ
cana-5363	17	8	topological	topological	ADJ
cana-5363	17	9	space	space	NOUN
cana-5363	17	10	with	with	ADP
cana-5363	17	11	few	few	ADJ
cana-5363	17	12	properties	property	NOUN
cana-5363	17	13	,	,	PUNCT
cana-5363	17	14	see	see	VERB
cana-5363	17	15	[	[	X
cana-5363	17	16	6	6	NUM
cana-5363	17	17	]	]	PUNCT
cana-5363	17	18	.	.	PUNCT
cana-5363	18	1	the	the	DET
cana-5363	18	2	concept	concept	NOUN
cana-5363	18	3	of	of	ADP
cana-5363	18	4	pythagorean	pythagorean	PROPN
cana-5363	18	5	fuzzy	fuzzy	PROPN
cana-5363	18	6	subset	subset	NOUN
cana-5363	18	7	which	which	PRON
cana-5363	18	8	is	be	AUX
cana-5363	18	9	a	a	DET
cana-5363	18	10	typical	typical	ADJ
cana-5363	18	11	fuzzy	fuzzy	ADJ
cana-5363	18	12	subset	subset	NOUN
cana-5363	18	13	was	be	AUX
cana-5363	18	14	presented	present	VERB
cana-5363	18	15	by	by	ADP
cana-5363	18	16	yager	yager	NOUN
cana-5363	18	17	,	,	PUNCT
cana-5363	18	18	see	see	VERB
cana-5363	18	19	[	[	X
cana-5363	18	20	18	18	NUM
cana-5363	18	21	,	,	PUNCT
cana-5363	18	22	19	19	NUM
cana-5363	18	23	,	,	PUNCT
cana-5363	18	24	20	20	NUM
cana-5363	18	25	]	]	PUNCT
cana-5363	18	26	.	.	PUNCT
cana-5363	19	1	pythagorean	pythagorean	PROPN
cana-5363	19	2	fuzzy	fuzzy	ADJ
cana-5363	19	3	topological	topological	ADJ
cana-5363	19	4	space	space	NOUN
cana-5363	19	5	was	be	AUX
cana-5363	19	6	introduced	introduce	VERB
cana-5363	19	7	by	by	ADP
cana-5363	19	8	olgun	olgun	NOUN
cana-5363	19	9	in	in	ADP
cana-5363	19	10	[	[	X
cana-5363	19	11	11	11	NUM
cana-5363	19	12	]	]	PUNCT
cana-5363	19	13	by	by	ADP
cana-5363	19	14	taking	take	VERB
cana-5363	19	15	the	the	DET
cana-5363	19	16	lead	lead	NOUN
cana-5363	19	17	as	as	ADP
cana-5363	19	18	from	from	ADP
cana-5363	19	19	chang	chang	PROPN
cana-5363	19	20	.	.	PUNCT
cana-5363	20	1	in	in	ADP
cana-5363	20	2	2013	2013	NUM
cana-5363	20	3	,	,	PUNCT
cana-5363	20	4	a	a	DET
cana-5363	20	5	new	new	ADJ
cana-5363	20	6	topology	topology	NOUN
cana-5363	20	7	called	call	VERB
cana-5363	20	8	nano	nano	NOUN
cana-5363	20	9	topology	topology	NOUN
cana-5363	20	10	was	be	AUX
cana-5363	20	11	introduced	introduce	VERB
cana-5363	20	12	by	by	ADP
cana-5363	20	13	lellis	lellis	PROPN
cana-5363	20	14	thivagar	thivagar	NOUN
cana-5363	20	15	[	[	X
cana-5363	20	16	9	9	NUM
cana-5363	20	17	]	]	PUNCT
cana-5363	20	18	which	which	PRON
cana-5363	20	19	is	be	AUX
cana-5363	20	20	an	an	DET
cana-5363	20	21	extension	extension	NOUN
cana-5363	20	22	of	of	ADP
cana-5363	20	23	rough	rough	ADJ
cana-5363	20	24	set	set	NOUN
cana-5363	20	25	theory	theory	NOUN
cana-5363	20	26	.	.	PUNCT
cana-5363	21	1	he	he	PRON
cana-5363	21	2	also	also	ADV
cana-5363	21	3	introduced	introduce	VERB
cana-5363	21	4	nano	nano	NOUN
cana-5363	21	5	topological	topological	ADJ
cana-5363	21	6	spaces	space	NOUN
cana-5363	21	7	which	which	PRON
cana-5363	21	8	were	be	AUX
cana-5363	21	9	defined	define	VERB
cana-5363	21	10	in	in	ADP
cana-5363	21	11	terms	term	NOUN
cana-5363	21	12	of	of	ADP
cana-5363	21	13	approximations	approximation	NOUN
cana-5363	21	14	and	and	CCONJ
cana-5363	21	15	boundary	boundary	ADJ
cana-5363	21	16	region	region	NOUN
cana-5363	21	17	of	of	ADP
cana-5363	21	18	a	a	DET
cana-5363	21	19	subset	subset	NOUN
cana-5363	21	20	of	of	ADP
cana-5363	21	21	a	a	DET
cana-5363	21	22	universe	universe	NOUN
cana-5363	21	23	using	use	VERB
cana-5363	21	24	an	an	DET
cana-5363	21	25	equivalence	equivalence	NOUN
cana-5363	21	26	relation	relation	NOUN
cana-5363	21	27	on	on	ADP
cana-5363	21	28	it	it	PRON
cana-5363	21	29	.	.	PUNCT
cana-5363	22	1	the	the	DET
cana-5363	22	2	elements	element	NOUN
cana-5363	22	3	of	of	ADP
cana-5363	22	4	a	a	DET
cana-5363	22	5	nano	nano	ADJ
cana-5363	22	6	topological	topological	ADJ
cana-5363	22	7	space	space	NOUN
cana-5363	22	8	are	be	AUX
cana-5363	22	9	called	call	VERB
cana-5363	22	10	the	the	DET
cana-5363	22	11	nano	nano	NOUN
cana-5363	22	12	open	open	ADJ
cana-5363	22	13	sets	set	NOUN
cana-5363	22	14	and	and	CCONJ
cana-5363	22	15	its	its	PRON
cana-5363	22	16	complements	complement	NOUN
cana-5363	22	17	are	be	AUX
cana-5363	22	18	called	call	VERB
cana-5363	22	19	the	the	DET
cana-5363	22	20	nano	nano	NOUN
cana-5363	22	21	closed	close	VERB
cana-5363	22	22	sets	set	NOUN
cana-5363	22	23	.	.	PUNCT
cana-5363	23	1	nano	nano	NOUN
cana-5363	23	2	means	mean	VERB
cana-5363	23	3	something	something	PRON
cana-5363	23	4	very	very	ADV
cana-5363	23	5	small	small	ADJ
cana-5363	23	6	.	.	PUNCT
cana-5363	24	1	nano	nano	NOUN
cana-5363	24	2	topology	topology	NOUN
cana-5363	24	3	thus	thus	ADV
cana-5363	24	4	literally	literally	ADV
cana-5363	24	5	means	mean	VERB
cana-5363	24	6	the	the	DET
cana-5363	24	7	study	study	NOUN
cana-5363	24	8	of	of	ADP
cana-5363	24	9	very	very	ADV
cana-5363	24	10	small	small	ADJ
cana-5363	24	11	surface	surface	NOUN
cana-5363	24	12	.	.	PUNCT
cana-5363	25	1	the	the	DET
cana-5363	25	2	fundamental	fundamental	ADJ
cana-5363	25	3	ideas	idea	NOUN
cana-5363	25	4	in	in	ADP
cana-5363	25	5	nano	nano	NOUN
cana-5363	25	6	topology	topology	NOUN
cana-5363	25	7	are	be	AUX
cana-5363	25	8	those	those	PRON
cana-5363	25	9	of	of	ADP
cana-5363	25	10	approximations	approximation	NOUN
cana-5363	25	11	and	and	CCONJ
cana-5363	25	12	indiscernibility	indiscernibility	NOUN
cana-5363	25	13	relation	relation	NOUN
cana-5363	25	14	.	.	PUNCT
cana-5363	26	1	furthermore	furthermore	ADV
cana-5363	26	2	nano	nano	VERB
cana-5363	26	3	𝛿	𝛿	DET
cana-5363	26	4	open	open	ADJ
cana-5363	26	5	sets	set	NOUN
cana-5363	26	6	in	in	ADP
cana-5363	26	7	nano	nano	NOUN
cana-5363	26	8	topological	topological	ADJ
cana-5363	26	9	space	space	NOUN
cana-5363	26	10	was	be	AUX
cana-5363	26	11	studied	study	VERB
cana-5363	26	12	in	in	ADP
cana-5363	26	13	[	[	X
cana-5363	26	14	13	13	NUM
cana-5363	26	15	]	]	PUNCT
cana-5363	26	16	.	.	PUNCT
cana-5363	27	1	mailto:prabamarch23@gmail.com	mailto:prabamarch23@gmail.com	PROPN
cana-5363	27	2	communications	communication	NOUN
cana-5363	27	3	on	on	ADP
cana-5363	27	4	applied	apply	VERB
cana-5363	27	5	nonlinear	nonlinear	ADJ
cana-5363	27	6	analysis	analysis	NOUN
cana-5363	27	7	issn	issn	NOUN
cana-5363	27	8	:	:	PUNCT
cana-5363	27	9	1074	1074	NUM
cana-5363	27	10	-	-	PUNCT
cana-5363	27	11	133x	133x	NUM
cana-5363	27	12	vol	vol	VERB
cana-5363	27	13	32	32	NUM
cana-5363	27	14	no	no	NOUN
cana-5363	27	15	.	.	PUNCT
cana-5363	28	1	10s	10	NOUN
cana-5363	28	2	(	(	PUNCT
cana-5363	28	3	2025	2025	NUM
cana-5363	28	4	)	)	PUNCT
cana-5363	28	5	2006	2006	NUM
cana-5363	28	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5363	29	1	recently	recently	ADV
cana-5363	29	2	,	,	PUNCT
cana-5363	29	3	lellis	lellis	PROPN
cana-5363	29	4	thivagar	thivagar	PROPN
cana-5363	29	5	et	et	PROPN
cana-5363	29	6	.	.	PUNCT
cana-5363	30	1	al	al	PROPN
cana-5363	31	1	[	[	X
cana-5363	31	2	10	10	NUM
cana-5363	31	3	]	]	PUNCT
cana-5363	31	4	explored	explore	VERB
cana-5363	31	5	a	a	DET
cana-5363	31	6	new	new	ADJ
cana-5363	31	7	concept	concept	NOUN
cana-5363	31	8	of	of	ADP
cana-5363	31	9	neutrosophic	neutrosophic	ADJ
cana-5363	31	10	nano	nano	NOUN
cana-5363	31	11	topology	topology	NOUN
cana-5363	31	12	,	,	PUNCT
cana-5363	31	13	and	and	CCONJ
cana-5363	31	14	also	also	ADV
cana-5363	31	15	𝑍	𝑍	VERB
cana-5363	31	16	-open	-open	ADJ
cana-5363	31	17	sets	set	NOUN
cana-5363	31	18	in	in	ADP
cana-5363	31	19	topological	topological	ADJ
cana-5363	31	20	spaces	space	NOUN
cana-5363	31	21	by	by	ADP
cana-5363	31	22	el	el	PROPN
cana-5363	31	23	-	-	PUNCT
cana-5363	31	24	magharabi	magharabi	NOUN
cana-5363	31	25	and	and	CCONJ
cana-5363	31	26	mubarki	mubarki	NOUN
cana-5363	32	1	[	[	X
cana-5363	32	2	8	8	NUM
cana-5363	32	3	]	]	PUNCT
cana-5363	32	4	,	,	PUNCT
cana-5363	32	5	𝑀	𝑀	PROPN
cana-5363	32	6	-open	-open	NOUN
cana-5363	32	7	sets	set	NOUN
cana-5363	32	8	in	in	ADP
cana-5363	32	9	a	a	DET
cana-5363	32	10	nano	nano	NOUN
cana-5363	32	11	topological	topological	ADJ
cana-5363	32	12	spaces	space	NOUN
cana-5363	32	13	by	by	ADP
cana-5363	32	14	padma	padma	PROPN
cana-5363	32	15	et	et	PROPN
cana-5363	32	16	.	.	PUNCT
cana-5363	33	1	al	al	PROPN
cana-5363	34	1	[	[	X
cana-5363	34	2	12	12	NUM
cana-5363	34	3	]	]	PUNCT
cana-5363	34	4	,	,	PUNCT
cana-5363	34	5	𝑍-open	𝑍-open	ADJ
cana-5363	34	6	sets	set	NOUN
cana-5363	34	7	in	in	ADP
cana-5363	34	8	nano	nano	NOUN
cana-5363	34	9	topological	topological	ADJ
cana-5363	34	10	spaces	space	NOUN
cana-5363	34	11	by	by	ADP
cana-5363	34	12	selvaraj	selvaraj	ADJ
cana-5363	34	13	and	and	CCONJ
cana-5363	34	14	balakrishna	balakrishna	NOUN
cana-5363	34	15	[	[	X
cana-5363	34	16	3	3	NUM
cana-5363	34	17	]	]	PUNCT
cana-5363	34	18	and	and	CCONJ
cana-5363	34	19	fuzzy	fuzzy	ADJ
cana-5363	34	20	𝑍	𝑍	NOUN
cana-5363	34	21	-closed	-close	VERB
cana-5363	34	22	sets	set	NOUN
cana-5363	34	23	and	and	CCONJ
cana-5363	34	24	generalized	generalize	VERB
cana-5363	34	25	fuzzy	fuzzy	ADJ
cana-5363	34	26	𝑍	𝑍	NOUN
cana-5363	34	27	-closed	-close	VERB
cana-5363	34	28	sets	set	NOUN
cana-5363	34	29	in	in	ADP
cana-5363	34	30	double	double	ADJ
cana-5363	34	31	fuzzy	fuzzy	ADJ
cana-5363	34	32	topological	topological	ADJ
cana-5363	34	33	spaces	space	NOUN
cana-5363	34	34	by	by	ADP
cana-5363	34	35	shiventhiradevi	shiventhiradevi	ADJ
cana-5363	34	36	et	et	NOUN
cana-5363	34	37	.	.	PUNCT
cana-5363	35	1	al	al	PROPN
cana-5363	35	2	in	in	ADP
cana-5363	35	3	[	[	X
cana-5363	35	4	14	14	NUM
cana-5363	35	5	,	,	PUNCT
cana-5363	35	6	15	15	NUM
cana-5363	35	7	]	]	PUNCT
cana-5363	35	8	.	.	PUNCT
cana-5363	36	1	thangammal	thangammal	PROPN
cana-5363	36	2	et	et	PROPN
cana-5363	36	3	.	.	PUNCT
cana-5363	37	1	al	al	PROPN
cana-5363	38	1	[	[	X
cana-5363	38	2	16	16	NUM
cana-5363	38	3	]	]	PUNCT
cana-5363	38	4	introduced	introduce	VERB
cana-5363	38	5	fuzzy	fuzzy	ADJ
cana-5363	38	6	nano	nano	NOUN
cana-5363	38	7	𝑍-open	𝑍-open	ADJ
cana-5363	38	8	sets	set	NOUN
cana-5363	38	9	in	in	ADP
cana-5363	38	10	fuzzy	fuzzy	ADJ
cana-5363	38	11	nano	nano	NOUN
cana-5363	38	12	topological	topological	ADJ
cana-5363	38	13	spaces	space	NOUN
cana-5363	38	14	.	.	PUNCT
cana-5363	39	1	vadivel	vadivel	VERB
cana-5363	39	2	et	et	PROPN
cana-5363	39	3	al	al	PROPN
cana-5363	39	4	.	.	PUNCT
cana-5363	40	1	[	[	X
cana-5363	40	2	17	17	NUM
cana-5363	40	3	]	]	PUNCT
cana-5363	40	4	discussed	discuss	VERB
cana-5363	40	5	some	some	DET
cana-5363	40	6	open	open	ADJ
cana-5363	40	7	sets	set	NOUN
cana-5363	40	8	in	in	ADP
cana-5363	40	9	fuzzy	fuzzy	ADJ
cana-5363	40	10	nano	nano	NOUN
cana-5363	40	11	topological	topological	ADJ
cana-5363	40	12	spaces	space	NOUN
cana-5363	40	13	.	.	PUNCT
cana-5363	41	1	research	research	NOUN
cana-5363	41	2	gap	gap	NOUN
cana-5363	41	3	:	:	PUNCT
cana-5363	41	4	no	no	DET
cana-5363	41	5	investigation	investigation	NOUN
cana-5363	41	6	on	on	ADP
cana-5363	41	7	some	some	DET
cana-5363	41	8	stronger	strong	ADJ
cana-5363	41	9	and	and	CCONJ
cana-5363	41	10	weaker	weak	ADJ
cana-5363	41	11	forms	form	NOUN
cana-5363	41	12	of	of	ADP
cana-5363	41	13	pythagorean	pythagorean	PROPN
cana-5363	41	14	fuzzy	fuzzy	ADJ
cana-5363	41	15	nano	nano	NOUN
cana-5363	41	16	open	open	ADJ
cana-5363	41	17	sets	set	NOUN
cana-5363	41	18	such	such	ADJ
cana-5363	41	19	as	as	ADP
cana-5363	41	20	pythagorean	pythagorean	PROPN
cana-5363	41	21	fuzzy	fuzzy	ADJ
cana-5363	41	22	nano	nano	PROPN
cana-5363	41	23	𝛿	𝛿	DET
cana-5363	41	24	open	open	ADJ
cana-5363	41	25	set	set	NOUN
cana-5363	41	26	,	,	PUNCT
cana-5363	41	27	pythagorean	pythagorean	PROPN
cana-5363	41	28	fuzzy	fuzzy	ADJ
cana-5363	41	29	nano	nano	NOUN
cana-5363	41	30	𝛿-semi	𝛿-semi	PROPN
cana-5363	41	31	open	open	ADJ
cana-5363	41	32	set	set	PROPN
cana-5363	41	33	,	,	PUNCT
cana-5363	41	34	pythagorean	pythagorean	PROPN
cana-5363	41	35	fuzzy	fuzzy	ADJ
cana-5363	41	36	nano	nano	PROPN
cana-5363	41	37	pre	pre	X
cana-5363	41	38	open	open	ADJ
cana-5363	41	39	set	set	VERB
cana-5363	41	40	and	and	CCONJ
cana-5363	41	41	pythagorean	pythagorean	PROPN
cana-5363	41	42	fuzzy	fuzzy	ADJ
cana-5363	41	43	nano	nano	NOUN
cana-5363	41	44	𝑍	𝑍	VERB
cana-5363	41	45	open	open	ADJ
cana-5363	41	46	sets	set	NOUN
cana-5363	41	47	on	on	ADP
cana-5363	41	48	pythagorean	pythagorean	PROPN
cana-5363	41	49	fuzzy	fuzzy	ADJ
cana-5363	41	50	nano	nano	PROPN
cana-5363	41	51	topological	topological	ADJ
cana-5363	41	52	space	space	NOUN
cana-5363	41	53	has	have	AUX
cana-5363	41	54	been	be	AUX
cana-5363	41	55	reported	report	VERB
cana-5363	41	56	in	in	ADP
cana-5363	41	57	the	the	DET
cana-5363	41	58	pythagorean	pythagorean	ADJ
cana-5363	41	59	fuzzy	fuzzy	ADJ
cana-5363	41	60	literature	literature	NOUN
cana-5363	41	61	.	.	PUNCT
cana-5363	42	1	in	in	ADP
cana-5363	42	2	this	this	DET
cana-5363	42	3	paper	paper	NOUN
cana-5363	42	4	some	some	DET
cana-5363	42	5	preliminary	preliminary	ADJ
cana-5363	42	6	concepts	concept	NOUN
cana-5363	42	7	required	require	VERB
cana-5363	42	8	in	in	ADP
cana-5363	42	9	our	our	PRON
cana-5363	42	10	work	work	NOUN
cana-5363	42	11	are	be	AUX
cana-5363	42	12	briefly	briefly	ADV
cana-5363	42	13	recalled	recall	VERB
cana-5363	42	14	in	in	ADP
cana-5363	42	15	section	section	NOUN
cana-5363	42	16	2	2	NUM
cana-5363	42	17	.	.	PUNCT
cana-5363	43	1	in	in	ADP
cana-5363	43	2	section	section	NOUN
cana-5363	43	3	3	3	NUM
cana-5363	43	4	,	,	PUNCT
cana-5363	43	5	we	we	PRON
cana-5363	43	6	introduce	introduce	VERB
cana-5363	43	7	the	the	DET
cana-5363	43	8	concept	concept	NOUN
cana-5363	43	9	of	of	ADP
cana-5363	43	10	𝒫ℱ𝒩𝑜	𝒫ℱ𝒩𝑜	PROPN
cana-5363	43	11	,	,	PUNCT
cana-5363	43	12	𝒫ℱ𝒩𝛿𝑜	𝒫ℱ𝒩𝛿𝑜	PROPN
cana-5363	43	13	,	,	PUNCT
cana-5363	43	14	𝒫ℱ𝒩𝛿𝒮𝑜	𝒫ℱ𝒩𝛿𝒮𝑜	PROPN
cana-5363	43	15	,	,	PUNCT
cana-5363	43	16	𝒫ℱ𝒩𝒫𝑜	𝒫ℱ𝒩𝒫𝑜	NOUN
cana-5363	43	17	and	and	CCONJ
cana-5363	43	18	𝒫ℱ𝒩𝑍𝑜	𝒫ℱ𝒩𝑍𝑜	NOUN
cana-5363	43	19	sets	set	VERB
cana-5363	43	20	and	and	CCONJ
cana-5363	43	21	studied	study	VERB
cana-5363	43	22	some	some	PRON
cana-5363	43	23	of	of	ADP
cana-5363	43	24	their	their	PRON
cana-5363	43	25	properties	property	NOUN
cana-5363	43	26	.	.	PUNCT
cana-5363	44	1	also	also	ADV
cana-5363	44	2	,	,	PUNCT
cana-5363	44	3	we	we	PRON
cana-5363	44	4	discuss	discuss	VERB
cana-5363	44	5	on	on	ADP
cana-5363	44	6	pythagorean	pythagorean	PROPN
cana-5363	44	7	fuzzy	fuzzy	ADJ
cana-5363	44	8	nano	nano	NOUN
cana-5363	44	9	𝑍-interior	𝑍-interior	PROPN
cana-5363	44	10	and	and	CCONJ
cana-5363	44	11	pythagorean	pythagorean	PROPN
cana-5363	44	12	fuzzy	fuzzy	ADJ
cana-5363	44	13	nano	nano	PROPN
cana-5363	44	14	𝑍-closure	𝑍-closure	PROPN
cana-5363	44	15	operators	operator	NOUN
cana-5363	44	16	in	in	ADP
cana-5363	44	17	pythagorean	pythagorean	PROPN
cana-5363	44	18	fuzzy	fuzzy	ADJ
cana-5363	44	19	nano	nano	PROPN
cana-5363	44	20	topological	topological	ADJ
cana-5363	44	21	spaces	space	NOUN
cana-5363	44	22	.	.	PUNCT
cana-5363	45	1	2	2	NUM
cana-5363	45	2	preliminaries	preliminary	NOUN
cana-5363	45	3	definition	definition	NOUN
cana-5363	45	4	2.1	2.1	NUM
cana-5363	46	1	[	[	X
cana-5363	46	2	21	21	NUM
cana-5363	46	3	]	]	PUNCT
cana-5363	46	4	a	a	DET
cana-5363	46	5	function	function	NOUN
cana-5363	46	6	𝜆	𝜆	ADP
cana-5363	46	7	from	from	ADP
cana-5363	46	8	𝑋	𝑋	PROPN
cana-5363	46	9	into	into	ADP
cana-5363	46	10	the	the	DET
cana-5363	46	11	unit	unit	NOUN
cana-5363	46	12	interval	interval	NOUN
cana-5363	46	13	𝐼	𝐼	PROPN
cana-5363	46	14	is	be	AUX
cana-5363	46	15	called	call	VERB
cana-5363	46	16	a	a	DET
cana-5363	46	17	fuzzy	fuzzy	ADJ
cana-5363	46	18	set	set	NOUN
cana-5363	46	19	in	in	ADP
cana-5363	46	20	𝑋.	𝑋.	PROPN
cana-5363	46	21	for	for	ADP
cana-5363	46	22	every	every	DET
cana-5363	46	23	𝑥	𝑥	PRON
cana-5363	46	24	∈	∈	PROPN
cana-5363	46	25	𝑋	𝑋	PROPN
cana-5363	46	26	,	,	PUNCT
cana-5363	46	27	𝜆(𝑥	𝜆(𝑥	PROPN
cana-5363	46	28	)	)	PUNCT
cana-5363	46	29	∈	∈	PROPN
cana-5363	46	30	𝐼	𝐼	PROPN
cana-5363	46	31	is	be	AUX
cana-5363	46	32	called	call	VERB
cana-5363	46	33	the	the	DET
cana-5363	46	34	grade	grade	NOUN
cana-5363	46	35	of	of	ADP
cana-5363	46	36	membership	membership	NOUN
cana-5363	46	37	of	of	ADP
cana-5363	46	38	𝑥	𝑥	PROPN
cana-5363	46	39	in	in	ADP
cana-5363	46	40	𝜆.	𝜆.	NOUN
cana-5363	46	41	some	some	DET
cana-5363	46	42	authors	author	NOUN
cana-5363	46	43	say	say	VERB
cana-5363	46	44	that	that	SCONJ
cana-5363	46	45	𝜆	𝜆	PRON
cana-5363	46	46	is	be	AUX
cana-5363	46	47	a	a	DET
cana-5363	46	48	fuzzy	fuzzy	ADJ
cana-5363	46	49	subset	subset	NOUN
cana-5363	46	50	of	of	ADP
cana-5363	46	51	𝑋	𝑋	PROPN
cana-5363	46	52	instead	instead	ADV
cana-5363	46	53	of	of	ADP
cana-5363	46	54	saying	say	VERB
cana-5363	46	55	that	that	SCONJ
cana-5363	46	56	𝜆	𝜆	PRON
cana-5363	46	57	is	be	AUX
cana-5363	46	58	a	a	DET
cana-5363	46	59	fuzzy	fuzzy	ADJ
cana-5363	46	60	set	set	NOUN
cana-5363	46	61	in	in	ADP
cana-5363	46	62	𝑋.	𝑋.	PROPN
cana-5363	46	63	the	the	DET
cana-5363	46	64	class	class	NOUN
cana-5363	46	65	of	of	ADP
cana-5363	46	66	all	all	DET
cana-5363	46	67	fuzzy	fuzzy	ADJ
cana-5363	46	68	sets	set	NOUN
cana-5363	46	69	from	from	ADP
cana-5363	46	70	𝑋	𝑋	NOUN
cana-5363	46	71	into	into	ADP
cana-5363	46	72	the	the	DET
cana-5363	46	73	closed	closed	ADJ
cana-5363	46	74	unit	unit	NOUN
cana-5363	46	75	interval	interval	NOUN
cana-5363	46	76	𝐼	𝐼	PROPN
cana-5363	46	77	will	will	AUX
cana-5363	46	78	be	be	AUX
cana-5363	46	79	denoted	denote	VERB
cana-5363	46	80	by	by	ADP
cana-5363	46	81	𝐼𝑋.	𝐼𝑋.	NOUN
cana-5363	46	82	definition	definition	NOUN
cana-5363	46	83	2.2	2.2	NUM
cana-5363	46	84	[	[	X
cana-5363	46	85	21	21	NUM
cana-5363	46	86	]	]	X
cana-5363	46	87	if	if	SCONJ
cana-5363	46	88	𝜆	𝜆	NOUN
cana-5363	46	89	and	and	CCONJ
cana-5363	46	90	𝜉	𝜉	PROPN
cana-5363	46	91	are	be	AUX
cana-5363	46	92	any	any	DET
cana-5363	46	93	two	two	NUM
cana-5363	46	94	fuzzy	fuzzy	ADJ
cana-5363	46	95	subsets	subset	NOUN
cana-5363	46	96	of	of	ADP
cana-5363	46	97	a	a	DET
cana-5363	46	98	set	set	ADJ
cana-5363	46	99	𝑋	𝑋	NOUN
cana-5363	46	100	,	,	PUNCT
cana-5363	46	101	then	then	ADV
cana-5363	46	102	𝜆	𝜆	PRON
cana-5363	46	103	is	be	AUX
cana-5363	46	104	said	say	VERB
cana-5363	46	105	to	to	PART
cana-5363	46	106	be	be	AUX
cana-5363	46	107	included	include	VERB
cana-5363	46	108	in	in	ADP
cana-5363	46	109	𝜉	𝜉	NOUN
cana-5363	46	110	or	or	CCONJ
cana-5363	46	111	𝜆	𝜆	NOUN
cana-5363	46	112	is	be	AUX
cana-5363	46	113	contained	contain	VERB
cana-5363	46	114	in	in	ADP
cana-5363	46	115	𝜉	𝜉	NOUN
cana-5363	46	116	or	or	CCONJ
cana-5363	46	117	𝜆	𝜆	NOUN
cana-5363	46	118	is	be	AUX
cana-5363	46	119	less	less	ADJ
cana-5363	46	120	than	than	ADP
cana-5363	46	121	or	or	CCONJ
cana-5363	46	122	equal	equal	ADJ
cana-5363	46	123	to	to	ADP
cana-5363	46	124	𝜉	𝜉	PROPN
cana-5363	46	125	iff	iff	PROPN
cana-5363	46	126	𝜆(𝑥	𝜆(𝑥	PROPN
cana-5363	46	127	)	)	PUNCT
cana-5363	46	128	≤	≤	NOUN
cana-5363	47	1	𝜉(𝑥	𝜉(𝑥	PROPN
cana-5363	47	2	)	)	PUNCT
cana-5363	47	3	for	for	ADP
cana-5363	47	4	all	all	DET
cana-5363	47	5	𝑥	𝑥	NOUN
cana-5363	47	6	in	in	ADP
cana-5363	47	7	𝑋	𝑋	NOUN
cana-5363	47	8	and	and	CCONJ
cana-5363	47	9	is	be	AUX
cana-5363	47	10	denoted	denote	VERB
cana-5363	47	11	by	by	ADP
cana-5363	47	12	𝜆	𝜆	DET
cana-5363	47	13	≤	≤	PROPN
cana-5363	47	14	𝜉.	𝜉.	PROPN
cana-5363	47	15	equivalently	equivalently	PROPN
cana-5363	47	16	,	,	PUNCT
cana-5363	47	17	𝜆	𝜆	DET
cana-5363	47	18	≤	≤	PROPN
cana-5363	47	19	𝜉	𝜉	VERB
cana-5363	47	20	iff	iff	PROPN
cana-5363	47	21	𝜇𝜆(𝑥	𝜇𝜆(𝑥	NUM
cana-5363	47	22	)	)	PUNCT
cana-5363	47	23	≤	≤	NOUN
cana-5363	47	24	𝜇𝜉(𝑥	𝜇𝜉(𝑥	X
cana-5363	47	25	)	)	PUNCT
cana-5363	47	26	for	for	ADP
cana-5363	47	27	all	all	PRON
cana-5363	47	28	𝑥	𝑥	PROPN
cana-5363	47	29	in	in	ADP
cana-5363	47	30	𝑋.	𝑋.	PROPN
cana-5363	47	31	note	note	NOUN
cana-5363	47	32	that	that	SCONJ
cana-5363	47	33	every	every	DET
cana-5363	47	34	fuzzy	fuzzy	ADJ
cana-5363	47	35	subset	subset	NOUN
cana-5363	47	36	is	be	AUX
cana-5363	47	37	included	include	VERB
cana-5363	47	38	in	in	ADP
cana-5363	47	39	itself	itself	PRON
cana-5363	47	40	and	and	CCONJ
cana-5363	47	41	empty	empty	ADJ
cana-5363	47	42	fuzzy	fuzzy	ADJ
cana-5363	47	43	subset	subset	NOUN
cana-5363	47	44	is	be	AUX
cana-5363	47	45	included	include	VERB
cana-5363	47	46	in	in	ADP
cana-5363	47	47	every	every	DET
cana-5363	47	48	fuzzy	fuzzy	ADJ
cana-5363	47	49	subset	subset	NOUN
cana-5363	47	50	.	.	PUNCT
cana-5363	48	1	definition	definition	NOUN
cana-5363	48	2	2.3	2.3	NUM
cana-5363	48	3	[	[	X
cana-5363	48	4	21	21	NUM
cana-5363	48	5	]	]	SYM
cana-5363	48	6	two	two	NUM
cana-5363	48	7	fuzzy	fuzzy	ADJ
cana-5363	48	8	subsets	subset	NOUN
cana-5363	48	9	𝜆	𝜆	ADP
cana-5363	48	10	and	and	CCONJ
cana-5363	48	11	𝜇	𝜇	X
cana-5363	48	12	of	of	ADP
cana-5363	48	13	a	a	DET
cana-5363	48	14	set	set	NOUN
cana-5363	48	15	𝑋	𝑋	NOUN
cana-5363	48	16	are	be	AUX
cana-5363	48	17	said	say	VERB
cana-5363	48	18	to	to	PART
cana-5363	48	19	be	be	AUX
cana-5363	48	20	equal	equal	ADJ
cana-5363	48	21	,	,	PUNCT
cana-5363	48	22	written	write	VERB
cana-5363	48	23	𝜆	𝜆	ADP
cana-5363	48	24	=	=	PUNCT
cana-5363	48	25	𝜇	𝜇	ADP
cana-5363	48	26	,	,	PUNCT
cana-5363	48	27	if	if	SCONJ
cana-5363	48	28	𝜆(𝑥	𝜆(𝑥	NOUN
cana-5363	48	29	)	)	PUNCT
cana-5363	48	30	=	=	SYM
cana-5363	48	31	𝜇(𝑥	𝜇(𝑥	PROPN
cana-5363	48	32	)	)	PUNCT
cana-5363	48	33	for	for	ADP
cana-5363	48	34	every	every	DET
cana-5363	48	35	𝑥	𝑥	NOUN
cana-5363	48	36	in	in	ADP
cana-5363	48	37	𝑋.	𝑋.	PROPN
cana-5363	48	38	definition	definition	NOUN
cana-5363	48	39	2.4	2.4	NUM
cana-5363	48	40	[	[	X
cana-5363	48	41	21	21	NUM
cana-5363	48	42	]	]	PUNCT
cana-5363	48	43	the	the	DET
cana-5363	48	44	complement	complement	NOUN
cana-5363	48	45	of	of	ADP
cana-5363	48	46	a	a	DET
cana-5363	48	47	fuzzy	fuzzy	ADJ
cana-5363	48	48	subset	subset	NOUN
cana-5363	48	49	𝜆	𝜆	ADP
cana-5363	48	50	in	in	ADP
cana-5363	48	51	a	a	DET
cana-5363	48	52	set	set	ADJ
cana-5363	48	53	𝑋	𝑋	NOUN
cana-5363	48	54	,	,	PUNCT
cana-5363	48	55	denoted	denote	VERB
cana-5363	48	56	by	by	ADP
cana-5363	48	57	1	1	NUM
cana-5363	48	58	−	−	NOUN
cana-5363	48	59	𝜆	𝜆	NOUN
cana-5363	48	60	,	,	PUNCT
cana-5363	48	61	is	be	AUX
cana-5363	48	62	the	the	DET
cana-5363	48	63	fuzzy	fuzzy	ADJ
cana-5363	48	64	subset	subset	NOUN
cana-5363	48	65	of	of	ADP
cana-5363	48	66	𝑋	𝑋	PROPN
cana-5363	48	67	defined	define	VERB
cana-5363	48	68	by	by	ADP
cana-5363	48	69	1	1	NUM
cana-5363	48	70	−	−	NUM
cana-5363	48	71	𝜆(𝑥	𝜆(𝑥	NOUN
cana-5363	48	72	)	)	PUNCT
cana-5363	48	73	for	for	ADP
cana-5363	48	74	all	all	DET
cana-5363	48	75	𝑥	𝑥	PROPN
cana-5363	48	76	in	in	ADP
cana-5363	48	77	𝑋.	𝑋.	PROPN
cana-5363	48	78	note	note	NOUN
cana-5363	48	79	that	that	SCONJ
cana-5363	48	80	1	1	NUM
cana-5363	48	81	−	−	NOUN
cana-5363	48	82	(	(	PUNCT
cana-5363	48	83	1	1	NUM
cana-5363	48	84	−	−	NOUN
cana-5363	48	85	𝜆	𝜆	X
cana-5363	48	86	)	)	PUNCT
cana-5363	48	87	=	=	PUNCT
cana-5363	48	88	𝜆.	𝜆.	NOUN
cana-5363	48	89	definition	definition	NOUN
cana-5363	48	90	2.5	2.5	NUM
cana-5363	49	1	[	[	X
cana-5363	49	2	21	21	NUM
cana-5363	49	3	]	]	PUNCT
cana-5363	49	4	the	the	DET
cana-5363	49	5	union	union	NOUN
cana-5363	49	6	of	of	ADP
cana-5363	49	7	two	two	NUM
cana-5363	49	8	fuzzy	fuzzy	ADJ
cana-5363	49	9	subsets	subset	NOUN
cana-5363	49	10	𝜆	𝜆	ADP
cana-5363	49	11	and	and	CCONJ
cana-5363	49	12	𝜇	𝜇	X
cana-5363	49	13	in	in	ADP
cana-5363	49	14	a	a	DET
cana-5363	49	15	set	set	ADJ
cana-5363	49	16	𝑋	𝑋	NOUN
cana-5363	49	17	,	,	PUNCT
cana-5363	49	18	denoted	denote	VERB
cana-5363	49	19	by	by	ADP
cana-5363	49	20	𝜆	𝜆	DET
cana-5363	49	21	∨	∨	NUM
cana-5363	49	22	𝜇	𝜇	ADP
cana-5363	49	23	,	,	PUNCT
cana-5363	49	24	is	be	AUX
cana-5363	49	25	fuzzy	fuzzy	ADJ
cana-5363	49	26	subset	subset	NOUN
cana-5363	49	27	in	in	ADP
cana-5363	49	28	𝑋	𝑋	PROPN
cana-5363	49	29	defined	define	VERB
cana-5363	49	30	by	by	ADP
cana-5363	49	31	(	(	PUNCT
cana-5363	49	32	𝜆	𝜆	PROPN
cana-5363	49	33	∨	∨	NUM
cana-5363	49	34	𝜇)(𝑥	𝜇)(𝑥	NUM
cana-5363	49	35	)	)	PUNCT
cana-5363	49	36	=	=	SYM
cana-5363	49	37	𝑚𝑎𝑥{𝜆(𝑥	𝑚𝑎𝑥{𝜆(𝑥	PROPN
cana-5363	49	38	)	)	PUNCT
cana-5363	49	39	,	,	PUNCT
cana-5363	49	40	𝜇(𝑥	𝜇(𝑥	PROPN
cana-5363	49	41	)	)	PUNCT
cana-5363	49	42	}	}	PUNCT
cana-5363	49	43	,	,	PUNCT
cana-5363	49	44	for	for	ADP
cana-5363	49	45	all	all	DET
cana-5363	49	46	𝑥	𝑥	NOUN
cana-5363	49	47	in	in	ADP
cana-5363	49	48	𝑋.	𝑋.	PROPN
cana-5363	49	49	in	in	ADP
cana-5363	49	50	general	general	ADJ
cana-5363	49	51	,	,	PUNCT
cana-5363	49	52	the	the	DET
cana-5363	49	53	union	union	NOUN
cana-5363	49	54	of	of	ADP
cana-5363	49	55	a	a	DET
cana-5363	49	56	family	family	NOUN
cana-5363	49	57	of	of	ADP
cana-5363	49	58	fuzzy	fuzzy	ADJ
cana-5363	49	59	subsets	subset	NOUN
cana-5363	49	60	{	{	PUNCT
cana-5363	49	61	𝜉𝑖	𝜉𝑖	PROPN
cana-5363	49	62	:	:	PUNCT
cana-5363	49	63	𝑖	𝑖	SYM
cana-5363	49	64	∈	∈	PROPN
cana-5363	49	65	𝐼	𝐼	PROPN
cana-5363	49	66	}	}	PUNCT
cana-5363	49	67	is	be	AUX
cana-5363	49	68	a	a	DET
cana-5363	49	69	fuzzy	fuzzy	ADJ
cana-5363	49	70	subset	subset	NOUN
cana-5363	49	71	denoted	denote	VERB
cana-5363	49	72	by	by	ADP
cana-5363	49	73	∨𝑖∈𝐼	∨𝑖∈𝐼	PROPN
cana-5363	49	74	𝜉𝑖	𝜉𝑖	PROPN
cana-5363	49	75	and	and	CCONJ
cana-5363	49	76	defined	define	VERB
cana-5363	49	77	by	by	ADP
cana-5363	49	78	(	(	PUNCT
cana-5363	49	79	∨𝑖∈𝐼	∨𝑖∈𝐼	PROPN
cana-5363	49	80	𝜉𝑖)(𝑥	𝜉𝑖)(𝑥	PROPN
cana-5363	49	81	)	)	PUNCT
cana-5363	50	1	=	=	SYM
cana-5363	50	2	sup{𝜉𝑖(𝑥	sup{𝜉𝑖(𝑥	PROPN
cana-5363	50	3	):	):	PUNCT
cana-5363	50	4	𝑖	𝑖	SYM
cana-5363	50	5	∈	∈	PROPN
cana-5363	50	6	𝐼	𝐼	PROPN
cana-5363	50	7	}	}	PUNCT
cana-5363	50	8	,	,	PUNCT
cana-5363	50	9	for	for	SCONJ
cana-5363	50	10	all	all	DET
cana-5363	50	11	𝑥	𝑥	PROPN
cana-5363	50	12	in	in	ADP
cana-5363	50	13	𝑋.	𝑋.	PROPN
cana-5363	50	14	definition	definition	NOUN
cana-5363	50	15	2.6	2.6	NUM
cana-5363	50	16	[	[	X
cana-5363	50	17	21	21	NUM
cana-5363	50	18	]	]	PUNCT
cana-5363	50	19	the	the	DET
cana-5363	50	20	intersection	intersection	NOUN
cana-5363	50	21	of	of	ADP
cana-5363	50	22	two	two	NUM
cana-5363	50	23	fuzzy	fuzzy	ADJ
cana-5363	50	24	subsets	subset	NOUN
cana-5363	50	25	𝜆	𝜆	ADP
cana-5363	50	26	and	and	CCONJ
cana-5363	50	27	𝜇	𝜇	X
cana-5363	50	28	in	in	ADP
cana-5363	50	29	a	a	DET
cana-5363	50	30	set	set	ADJ
cana-5363	50	31	𝑋	𝑋	NOUN
cana-5363	50	32	,	,	PUNCT
cana-5363	50	33	denoted	denote	VERB
cana-5363	50	34	by	by	ADP
cana-5363	50	35	𝜆	𝜆	DET
cana-5363	50	36	∧	∧	PROPN
cana-5363	50	37	𝜇	𝜇	ADP
cana-5363	50	38	,	,	PUNCT
cana-5363	50	39	is	be	AUX
cana-5363	50	40	fuzzy	fuzzy	ADJ
cana-5363	50	41	subset	subset	NOUN
cana-5363	50	42	in	in	ADP
cana-5363	50	43	𝑋	𝑋	PROPN
cana-5363	50	44	defined	define	VERB
cana-5363	50	45	by	by	ADP
cana-5363	50	46	(	(	PUNCT
cana-5363	50	47	𝜆	𝜆	PROPN
cana-5363	50	48	∧	∧	PROPN
cana-5363	50	49	𝜇)(𝑥	𝜇)(𝑥	NOUN
cana-5363	50	50	)	)	PUNCT
cana-5363	50	51	=	=	SYM
cana-5363	51	1	𝑚𝑖𝑛{𝜆(𝑥	𝑚𝑖𝑛{𝜆(𝑥	PROPN
cana-5363	51	2	)	)	PUNCT
cana-5363	51	3	,	,	PUNCT
cana-5363	51	4	𝜇(𝑥	𝜇(𝑥	PROPN
cana-5363	51	5	)	)	PUNCT
cana-5363	51	6	}	}	PUNCT
cana-5363	51	7	,	,	PUNCT
cana-5363	51	8	for	for	ADP
cana-5363	51	9	all	all	DET
cana-5363	51	10	𝑥	𝑥	NOUN
cana-5363	51	11	in	in	ADP
cana-5363	51	12	𝑋.	𝑋.	PROPN
cana-5363	51	13	in	in	ADP
cana-5363	51	14	general	general	ADJ
cana-5363	51	15	,	,	PUNCT
cana-5363	51	16	the	the	DET
cana-5363	51	17	intersection	intersection	NOUN
cana-5363	51	18	of	of	ADP
cana-5363	51	19	a	a	DET
cana-5363	51	20	family	family	NOUN
cana-5363	51	21	of	of	ADP
cana-5363	51	22	fuzzy	fuzzy	ADJ
cana-5363	51	23	subsets	subset	NOUN
cana-5363	51	24	{	{	PUNCT
cana-5363	51	25	𝜉𝑖	𝜉𝑖	PROPN
cana-5363	51	26	:	:	PUNCT
cana-5363	51	27	𝑖	𝑖	SYM
cana-5363	51	28	∈	∈	PROPN
cana-5363	52	1	𝐼	𝐼	PROPN
cana-5363	52	2	}	}	PUNCT
cana-5363	52	3	is	be	AUX
cana-5363	52	4	a	a	DET
cana-5363	52	5	fuzzy	fuzzy	ADJ
cana-5363	52	6	subset	subset	NOUN
cana-5363	52	7	denoted	denote	VERB
cana-5363	52	8	by	by	ADP
cana-5363	52	9	∧𝑖∈𝐼	∧𝑖∈𝐼	NOUN
cana-5363	52	10	𝜉𝑖	𝜉𝑖	PROPN
cana-5363	52	11	and	and	CCONJ
cana-5363	52	12	defined	define	VERB
cana-5363	52	13	by	by	ADP
cana-5363	52	14	(	(	PUNCT
cana-5363	52	15	∧𝑖∈𝐼	∧𝑖∈𝐼	PROPN
cana-5363	52	16	𝜉𝑖)(𝑥	𝜉𝑖)(𝑥	PROPN
cana-5363	52	17	)	)	PUNCT
cana-5363	53	1	=	=	SYM
cana-5363	53	2	inf{𝜉𝑖(𝑥	inf{𝜉𝑖(𝑥	PROPN
cana-5363	53	3	):	):	PUNCT
cana-5363	53	4	𝑖	𝑖	PROPN
cana-5363	53	5	∈	∈	PROPN
cana-5363	53	6	𝐼	𝐼	PROPN
cana-5363	53	7	}	}	PUNCT
cana-5363	53	8	,	,	PUNCT
cana-5363	53	9	for	for	SCONJ
cana-5363	53	10	all	all	DET
cana-5363	53	11	𝑥	𝑥	PROPN
cana-5363	53	12	in	in	ADP
cana-5363	53	13	𝑋.	𝑋.	PROPN
cana-5363	53	14	communications	communication	NOUN
cana-5363	53	15	on	on	ADP
cana-5363	53	16	applied	apply	VERB
cana-5363	53	17	nonlinear	nonlinear	ADJ
cana-5363	53	18	analysis	analysis	NOUN
cana-5363	53	19	issn	issn	NOUN
cana-5363	53	20	:	:	PUNCT
cana-5363	53	21	1074	1074	NUM
cana-5363	53	22	-	-	PUNCT
cana-5363	53	23	133x	133x	NUM
cana-5363	53	24	vol	vol	VERB
cana-5363	53	25	32	32	NUM
cana-5363	53	26	no	no	NOUN
cana-5363	53	27	.	.	PUNCT
cana-5363	54	1	10s	10	NOUN
cana-5363	54	2	(	(	PUNCT
cana-5363	54	3	2025	2025	NUM
cana-5363	54	4	)	)	PUNCT
cana-5363	54	5	2007	2007	NUM
cana-5363	55	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5363	55	2	definition	definition	NOUN
cana-5363	55	3	2.7	2.7	NUM
cana-5363	55	4	[	[	SYM
cana-5363	55	5	18	18	NUM
cana-5363	55	6	,	,	PUNCT
cana-5363	55	7	19	19	NUM
cana-5363	55	8	,	,	PUNCT
cana-5363	55	9	20	20	NUM
cana-5363	55	10	]	]	PUNCT
cana-5363	55	11	let	let	VERB
cana-5363	55	12	𝑋	𝑋	NOUN
cana-5363	55	13	be	be	AUX
cana-5363	55	14	a	a	DET
cana-5363	55	15	universal	universal	ADJ
cana-5363	55	16	set	set	NOUN
cana-5363	55	17	.	.	PUNCT
cana-5363	56	1	then	then	ADV
cana-5363	56	2	,	,	PUNCT
cana-5363	56	3	a	a	DET
cana-5363	56	4	pythagorean	pythagorean	PROPN
cana-5363	56	5	fuzzy	fuzzy	ADJ
cana-5363	56	6	set	set	PROPN
cana-5363	56	7	𝐴	𝐴	PROPN
cana-5363	56	8	,	,	PUNCT
cana-5363	56	9	which	which	PRON
cana-5363	56	10	is	be	AUX
cana-5363	56	11	a	a	DET
cana-5363	56	12	set	set	NOUN
cana-5363	56	13	of	of	ADP
cana-5363	56	14	ordered	order	VERB
cana-5363	56	15	pairs	pair	NOUN
cana-5363	56	16	over	over	ADP
cana-5363	56	17	𝑋	𝑋	PROPN
cana-5363	56	18	,	,	PUNCT
cana-5363	56	19	is	be	AUX
cana-5363	56	20	defined	define	VERB
cana-5363	56	21	by	by	ADP
cana-5363	56	22	the	the	DET
cana-5363	56	23	following	following	NOUN
cana-5363	56	24	:	:	PUNCT
cana-5363	56	25	𝐴	𝐴	PROPN
cana-5363	56	26	=	=	PUNCT
cana-5363	56	27	{	{	PUNCT
cana-5363	56	28	<	<	X
cana-5363	56	29	𝑥	𝑥	X
cana-5363	56	30	,	,	PUNCT
cana-5363	56	31	𝜇𝐴(𝑥	𝜇𝐴(𝑥	NUM
cana-5363	56	32	)	)	PUNCT
cana-5363	56	33	,	,	PUNCT
cana-5363	56	34	𝜆𝐴(𝑥)|𝑥	𝜆𝐴(𝑥)|𝑥	NOUN
cana-5363	56	35	∈	∈	PROPN
cana-5363	56	36	𝑋	𝑋	PROPN
cana-5363	56	37	}	}	PUNCT
cana-5363	56	38	or	or	CCONJ
cana-5363	56	39	𝐴	𝐴	PROPN
cana-5363	56	40	=	=	PUNCT
cana-5363	56	41	{	{	PUNCT
cana-5363	56	42	⟨	⟨	NOUN
cana-5363	56	43	𝜇𝐴(𝑥),𝜆𝐴(𝑥	𝜇𝐴(𝑥),𝜆𝐴(𝑥	PUNCT
cana-5363	56	44	)	)	PUNCT
cana-5363	56	45	𝑥	𝑥	PRON
cana-5363	56	46	⟩	⟩	NOUN
cana-5363	56	47	|𝑥	|𝑥	NOUN
cana-5363	56	48	∈	∈	PROPN
cana-5363	56	49	𝑋	𝑋	PROPN
cana-5363	56	50	}	}	PUNCT
cana-5363	56	51	,	,	PUNCT
cana-5363	56	52	where	where	SCONJ
cana-5363	56	53	the	the	DET
cana-5363	56	54	functions	function	NOUN
cana-5363	56	55	𝜇𝐴(𝑥	𝜇𝐴(𝑥	VERB
cana-5363	56	56	):	):	PUNCT
cana-5363	56	57	𝑋	𝑋	PROPN
cana-5363	56	58	→	→	SYM
cana-5363	56	59	[	[	X
cana-5363	56	60	0,1	0,1	NUM
cana-5363	56	61	]	]	PUNCT
cana-5363	56	62	and	and	CCONJ
cana-5363	56	63	𝜆𝐴(𝑥	𝜆𝐴(𝑥	NUM
cana-5363	56	64	):	):	PUNCT
cana-5363	56	65	𝑋	𝑋	PROPN
cana-5363	56	66	→	→	SYM
cana-5363	56	67	[	[	X
cana-5363	56	68	0,1	0,1	NUM
cana-5363	56	69	]	]	PUNCT
cana-5363	56	70	define	define	VERB
cana-5363	56	71	the	the	DET
cana-5363	56	72	degree	degree	NOUN
cana-5363	56	73	of	of	ADP
cana-5363	56	74	membership	membership	NOUN
cana-5363	56	75	and	and	CCONJ
cana-5363	56	76	the	the	DET
cana-5363	56	77	degree	degree	NOUN
cana-5363	56	78	of	of	ADP
cana-5363	56	79	nonmembership	nonmembership	NOUN
cana-5363	56	80	,	,	PUNCT
cana-5363	56	81	respectively	respectively	ADV
cana-5363	56	82	,	,	PUNCT
cana-5363	56	83	of	of	ADP
cana-5363	56	84	the	the	DET
cana-5363	56	85	element	element	NOUN
cana-5363	56	86	𝑥	𝑥	PRON
cana-5363	56	87	∈	∈	PROPN
cana-5363	56	88	𝑋	𝑋	NOUN
cana-5363	56	89	to	to	ADP
cana-5363	56	90	𝐴	𝐴	PROPN
cana-5363	56	91	,	,	PUNCT
cana-5363	56	92	which	which	PRON
cana-5363	56	93	is	be	AUX
cana-5363	56	94	a	a	DET
cana-5363	56	95	subset	subset	NOUN
cana-5363	56	96	of	of	ADP
cana-5363	56	97	𝑋	𝑋	PROPN
cana-5363	56	98	,	,	PUNCT
cana-5363	56	99	and	and	CCONJ
cana-5363	56	100	for	for	ADP
cana-5363	56	101	every	every	DET
cana-5363	56	102	𝑥	𝑥	DET
cana-5363	56	103	∈	∈	PROPN
cana-5363	56	104	𝑋	𝑋	NOUN
cana-5363	56	105	,	,	PUNCT
cana-5363	56	106	0	0	NUM
cana-5363	56	107	≤	≤	NOUN
cana-5363	56	108	(	(	PUNCT
cana-5363	56	109	𝜇𝐴(𝑥))2	𝜇𝐴(𝑥))2	NOUN
cana-5363	56	110	+	+	CCONJ
cana-5363	56	111	(	(	PUNCT
cana-5363	56	112	𝜆𝐴(𝑥))2	𝜆𝐴(𝑥))2	NOUN
cana-5363	56	113	≤	≤	NUM
cana-5363	56	114	1	1	NUM
cana-5363	56	115	.	.	PUNCT
cana-5363	57	1	supposing	suppose	VERB
cana-5363	57	2	(	(	PUNCT
cana-5363	57	3	𝜇𝐴(𝑥))2	𝜇𝐴(𝑥))2	NOUN
cana-5363	57	4	+	+	CCONJ
cana-5363	57	5	(	(	PUNCT
cana-5363	57	6	𝜆𝐴(𝑥))2	𝜆𝐴(𝑥))2	NOUN
cana-5363	57	7	≤	≤	NUM
cana-5363	57	8	1	1	NUM
cana-5363	57	9	,	,	PUNCT
cana-5363	57	10	then	then	ADV
cana-5363	57	11	there	there	PRON
cana-5363	57	12	is	be	VERB
cana-5363	57	13	a	a	DET
cana-5363	57	14	degree	degree	NOUN
cana-5363	57	15	of	of	ADP
cana-5363	57	16	indeterminacy	indeterminacy	NOUN
cana-5363	57	17	of	of	ADP
cana-5363	57	18	𝑥	𝑥	DET
cana-5363	57	19	∈	∈	PROPN
cana-5363	57	20	𝑋	𝑋	NOUN
cana-5363	57	21	to	to	ADP
cana-5363	57	22	𝐴	𝐴	PROPN
cana-5363	57	23	defined	define	VERB
cana-5363	57	24	by	by	ADP
cana-5363	57	25	𝜋𝐴(𝑥	𝜋𝐴(𝑥	NUM
cana-5363	57	26	)	)	PUNCT
cana-5363	57	27	=	=	PUNCT
cana-5363	58	1	√1	√1	ADV
cana-5363	58	2	−	−	PROPN
cana-5363	59	1	[	[	X
cana-5363	59	2	(	(	PUNCT
cana-5363	59	3	𝜇𝐴(𝑥))2	𝜇𝐴(𝑥))2	NOUN
cana-5363	59	4	+	+	CCONJ
cana-5363	59	5	(	(	PUNCT
cana-5363	59	6	𝜆𝐴(𝑥))2	𝜆𝐴(𝑥))2	NOUN
cana-5363	59	7	]	]	PUNCT
cana-5363	59	8	and	and	CCONJ
cana-5363	59	9	𝜋𝐴(𝑥	𝜋𝐴(𝑥	NUM
cana-5363	59	10	)	)	PUNCT
cana-5363	59	11	∈	∈	NOUN
cana-5363	60	1	[	[	X
cana-5363	60	2	0,1	0,1	NUM
cana-5363	60	3	]	]	PUNCT
cana-5363	60	4	.	.	PUNCT
cana-5363	61	1	in	in	ADP
cana-5363	61	2	what	what	PRON
cana-5363	61	3	follows	follow	VERB
cana-5363	61	4	,	,	PUNCT
cana-5363	61	5	(	(	PUNCT
cana-5363	61	6	𝜇𝐴(𝑥))2	𝜇𝐴(𝑥))2	NOUN
cana-5363	61	7	+	+	CCONJ
cana-5363	61	8	(	(	PUNCT
cana-5363	61	9	𝜆𝐴(𝑥))2	𝜆𝐴(𝑥))2	NOUN
cana-5363	61	10	+	+	CCONJ
cana-5363	61	11	(	(	PUNCT
cana-5363	61	12	𝜋𝐴(𝑥))2	𝜋𝐴(𝑥))2	NOUN
cana-5363	61	13	=	=	SYM
cana-5363	61	14	1	1	X
cana-5363	61	15	.	.	PUNCT
cana-5363	61	16	otherwise	otherwise	ADV
cana-5363	61	17	,	,	PUNCT
cana-5363	61	18	𝜋𝐴(𝑥	𝜋𝐴(𝑥	NUM
cana-5363	61	19	)	)	PUNCT
cana-5363	61	20	=	=	SYM
cana-5363	61	21	0	0	PUNCT
cana-5363	61	22	whenever	whenever	SCONJ
cana-5363	61	23	(	(	PUNCT
cana-5363	61	24	𝜇𝐴(𝑥))2	𝜇𝐴(𝑥))2	NOUN
cana-5363	61	25	+	+	CCONJ
cana-5363	61	26	(	(	PUNCT
cana-5363	61	27	𝜆𝐴(𝑥))2	𝜆𝐴(𝑥))2	NOUN
cana-5363	61	28	=	=	SYM
cana-5363	61	29	1	1	X
cana-5363	61	30	.	.	X
cana-5363	62	1	we	we	PRON
cana-5363	62	2	denote	denote	VERB
cana-5363	62	3	the	the	DET
cana-5363	62	4	set	set	NOUN
cana-5363	62	5	of	of	ADP
cana-5363	62	6	all	all	DET
cana-5363	62	7	𝑃𝐹𝑆	𝑃𝐹𝑆	PROPN
cana-5363	62	8	’s	’s	NOUN
cana-5363	62	9	over	over	ADP
cana-5363	62	10	𝑋	𝑋	PROPN
cana-5363	62	11	by	by	ADP
cana-5363	62	12	𝑝𝑓𝑠(𝑋	𝑝𝑓𝑠(𝑋	PROPN
cana-5363	62	13	)	)	PUNCT
cana-5363	62	14	.	.	PUNCT
cana-5363	63	1	definition	definition	NOUN
cana-5363	63	2	2.8	2.8	NUM
cana-5363	63	3	[	[	SYM
cana-5363	63	4	20	20	NUM
cana-5363	63	5	]	]	PUNCT
cana-5363	63	6	let	let	VERB
cana-5363	63	7	𝐴	𝐴	PROPN
cana-5363	63	8	and	and	CCONJ
cana-5363	63	9	𝐵	𝐵	NOUN
cana-5363	63	10	be	be	AUX
cana-5363	64	1	𝑝𝑓𝑠	𝑝𝑓𝑠	PROPN
cana-5363	64	2	’s	’s	NOUN
cana-5363	64	3	of	of	ADP
cana-5363	64	4	the	the	DET
cana-5363	64	5	forms	form	NOUN
cana-5363	64	6	𝐴	𝐴	NOUN
cana-5363	64	7	=	=	PUNCT
cana-5363	64	8	{	{	PUNCT
cana-5363	64	9	<	<	X
cana-5363	64	10	𝑎	𝑎	NOUN
cana-5363	64	11	,	,	PUNCT
cana-5363	64	12	𝜇𝐴(𝑎	𝜇𝐴(𝑎	NOUN
cana-5363	64	13	)	)	PUNCT
cana-5363	64	14	,	,	PUNCT
cana-5363	64	15	𝜆𝐴(𝑎	𝜆𝐴(𝑎	PROPN
cana-5363	64	16	)	)	PUNCT
cana-5363	64	17	>	>	X
cana-5363	64	18	|𝑎	|𝑎	PROPN
cana-5363	65	1	∈	∈	PROPN
cana-5363	65	2	𝑋	𝑋	PROPN
cana-5363	65	3	}	}	PUNCT
cana-5363	65	4	and	and	CCONJ
cana-5363	65	5	𝐵	𝐵	NOUN
cana-5363	65	6	=	=	PUNCT
cana-5363	65	7	{	{	PUNCT
cana-5363	65	8	<	<	X
cana-5363	65	9	𝑎	𝑎	PROPN
cana-5363	65	10	,	,	PUNCT
cana-5363	65	11	𝜇𝐵(𝑎	𝜇𝐵(𝑎	NUM
cana-5363	65	12	)	)	PUNCT
cana-5363	65	13	,	,	PUNCT
cana-5363	65	14	𝜆𝐵(𝑎	𝜆𝐵(𝑎	PROPN
cana-5363	65	15	)	)	PUNCT
cana-5363	65	16	>	>	X
cana-5363	65	17	|𝑎	|𝑎	PROPN
cana-5363	66	1	∈	∈	PROPN
cana-5363	66	2	𝑋	𝑋	PROPN
cana-5363	66	3	}	}	PUNCT
cana-5363	66	4	.	.	PUNCT
cana-5363	67	1	then	then	ADV
cana-5363	67	2	1	1	X
cana-5363	67	3	.	.	PUNCT
cana-5363	67	4	𝐴	𝐴	PROPN
cana-5363	67	5	⊆	⊆	NUM
cana-5363	67	6	𝐵	𝐵	PROPN
cana-5363	67	7	if	if	SCONJ
cana-5363	67	8	and	and	CCONJ
cana-5363	67	9	only	only	ADV
cana-5363	67	10	if	if	SCONJ
cana-5363	67	11	𝜇𝐴(𝑎	𝜇𝐴(𝑎	NUM
cana-5363	67	12	)	)	PUNCT
cana-5363	67	13	≤	≤	NUM
cana-5363	67	14	𝜇𝐵(𝑎	𝜇𝐵(𝑎	NUM
cana-5363	67	15	)	)	PUNCT
cana-5363	67	16	and	and	CCONJ
cana-5363	67	17	𝜆𝐴(𝑎	𝜆𝐴(𝑎	PROPN
cana-5363	67	18	)	)	PUNCT
cana-5363	67	19	≥	≥	NOUN
cana-5363	67	20	𝜆𝐵(𝑎	𝜆𝐵(𝑎	INTJ
cana-5363	67	21	)	)	PUNCT
cana-5363	67	22	for	for	ADP
cana-5363	67	23	all	all	DET
cana-5363	67	24	𝑎	𝑎	PROPN
cana-5363	67	25	∈	∈	NOUN
cana-5363	67	26	𝑋.	𝑋.	PROPN
cana-5363	67	27	2	2	NUM
cana-5363	67	28	.	.	PUNCT
cana-5363	67	29	𝐴	𝐴	NOUN
cana-5363	67	30	=	=	PROPN
cana-5363	67	31	𝐵	𝐵	PROPN
cana-5363	67	32	if	if	SCONJ
cana-5363	67	33	and	and	CCONJ
cana-5363	67	34	only	only	ADV
cana-5363	67	35	if	if	SCONJ
cana-5363	67	36	𝐴	𝐴	PROPN
cana-5363	67	37	⊆	⊆	NUM
cana-5363	67	38	𝐵	𝐵	NOUN
cana-5363	67	39	and	and	CCONJ
cana-5363	67	40	𝐵	𝐵	NOUN
cana-5363	68	1	⊆	⊆	NUM
cana-5363	68	2	𝐴.	𝐴.	PROPN
cana-5363	68	3	3	3	NUM
cana-5363	68	4	.	.	PUNCT
cana-5363	68	5	𝐴̅	𝐴̅	NOUN
cana-5363	68	6	=	=	PUNCT
cana-5363	68	7	{	{	PUNCT
cana-5363	68	8	<	<	X
cana-5363	68	9	𝑎	𝑎	X
cana-5363	68	10	,	,	PUNCT
cana-5363	68	11	𝜆𝐴(𝑎	𝜆𝐴(𝑎	PROPN
cana-5363	68	12	)	)	PUNCT
cana-5363	68	13	,	,	PUNCT
cana-5363	68	14	𝜇𝐴(𝑎	𝜇𝐴(𝑎	PROPN
cana-5363	68	15	)	)	PUNCT
cana-5363	68	16	>	>	X
cana-5363	68	17	|𝑎	|𝑎	PROPN
cana-5363	69	1	∈	∈	PROPN
cana-5363	69	2	𝑋	𝑋	PROPN
cana-5363	69	3	}	}	PUNCT
cana-5363	69	4	.	.	PUNCT
cana-5363	70	1	4	4	X
cana-5363	70	2	.	.	X
cana-5363	70	3	𝐴	𝐴	PROPN
cana-5363	70	4	∩	∩	NOUN
cana-5363	70	5	𝐵	𝐵	NOUN
cana-5363	70	6	=	=	PUNCT
cana-5363	70	7	{	{	PUNCT
cana-5363	70	8	<	<	X
cana-5363	70	9	𝑎	𝑎	X
cana-5363	70	10	,	,	PUNCT
cana-5363	70	11	𝜇𝐴(𝑎	𝜇𝐴(𝑎	NOUN
cana-5363	70	12	)	)	PUNCT
cana-5363	70	13	∧	∧	PROPN
cana-5363	70	14	𝜇𝐵(𝑎	𝜇𝐵(𝑎	NUM
cana-5363	70	15	)	)	PUNCT
cana-5363	70	16	,	,	PUNCT
cana-5363	70	17	𝜆𝐴(𝑎	𝜆𝐴(𝑎	PROPN
cana-5363	70	18	)	)	PUNCT
cana-5363	70	19	∨	∨	NOUN
cana-5363	70	20	𝜆𝐵(𝑎	𝜆𝐵(𝑎	NUM
cana-5363	70	21	)	)	PUNCT
cana-5363	70	22	>	>	X
cana-5363	70	23	|𝑎	|𝑎	PROPN
cana-5363	70	24	∈	∈	PROPN
cana-5363	70	25	𝑋	𝑋	PROPN
cana-5363	70	26	}	}	PUNCT
cana-5363	70	27	.	.	PUNCT
cana-5363	71	1	5	5	X
cana-5363	71	2	.	.	X
cana-5363	71	3	𝐴	𝐴	PROPN
cana-5363	71	4	∪	∪	AUX
cana-5363	71	5	𝐵	𝐵	NOUN
cana-5363	71	6	=	=	PUNCT
cana-5363	71	7	{	{	PUNCT
cana-5363	71	8	<	<	X
cana-5363	71	9	𝑎	𝑎	NOUN
cana-5363	71	10	,	,	PUNCT
cana-5363	71	11	𝜇𝐴(𝑎	𝜇𝐴(𝑎	NOUN
cana-5363	71	12	)	)	PUNCT
cana-5363	71	13	∨	∨	NUM
cana-5363	71	14	𝜇𝐵(𝑎	𝜇𝐵(𝑎	NUM
cana-5363	71	15	)	)	PUNCT
cana-5363	71	16	,	,	PUNCT
cana-5363	71	17	𝜆𝐴(𝑎	𝜆𝐴(𝑎	PROPN
cana-5363	71	18	)	)	PUNCT
cana-5363	71	19	∧	∧	NOUN
cana-5363	71	20	𝜆𝐵(𝑎	𝜆𝐵(𝑎	PROPN
cana-5363	71	21	)	)	PUNCT
cana-5363	71	22	>	>	X
cana-5363	71	23	|𝑎	|𝑎	PROPN
cana-5363	71	24	∈	∈	PROPN
cana-5363	71	25	𝑋	𝑋	PROPN
cana-5363	71	26	}	}	PUNCT
cana-5363	71	27	.	.	PUNCT
cana-5363	72	1	6	6	NUM
cana-5363	72	2	.	.	X
cana-5363	72	3	0𝑃	0𝑃	NOUN
cana-5363	72	4	=	=	SYM
cana-5363	72	5	{	{	PUNCT
cana-5363	72	6	<	<	X
cana-5363	72	7	𝑎	𝑎	NOUN
cana-5363	72	8	,	,	PUNCT
cana-5363	72	9	0,1	0,1	NUM
cana-5363	72	10	>	>	SYM
cana-5363	72	11	|𝑎	|𝑎	NOUN
cana-5363	72	12	∈	∈	PROPN
cana-5363	72	13	𝑋	𝑋	PROPN
cana-5363	72	14	}	}	PUNCT
cana-5363	72	15	and	and	CCONJ
cana-5363	72	16	1𝑃	1𝑃	NOUN
cana-5363	72	17	=	=	SYM
cana-5363	72	18	{	{	PUNCT
cana-5363	72	19	<	<	X
cana-5363	72	20	𝑎	𝑎	PROPN
cana-5363	72	21	,	,	PUNCT
cana-5363	72	22	1,0	1,0	NUM
cana-5363	72	23	>	>	SYM
cana-5363	72	24	|𝑎	|𝑎	PROPN
cana-5363	72	25	∈	∈	PROPN
cana-5363	72	26	𝑋	𝑋	PROPN
cana-5363	72	27	}	}	PUNCT
cana-5363	72	28	.	.	PUNCT
cana-5363	73	1	7	7	X
cana-5363	73	2	.	.	X
cana-5363	73	3	1̅𝑃	1̅𝑃	NUM
cana-5363	73	4	=	=	NOUN
cana-5363	73	5	0𝑃	0𝑃	NOUN
cana-5363	73	6	and	and	CCONJ
cana-5363	73	7	0̅𝑃	0̅𝑃	NOUN
cana-5363	74	1	=	=	SYM
cana-5363	75	1	1𝑃.	1𝑃.	NUM
cana-5363	75	2	definition	definition	NOUN
cana-5363	75	3	2.9	2.9	NUM
cana-5363	75	4	[	[	X
cana-5363	75	5	1	1	NUM
cana-5363	75	6	]	]	PUNCT
cana-5363	75	7	let	let	VERB
cana-5363	75	8	𝑈	𝑈	PROPN
cana-5363	75	9	be	be	AUX
cana-5363	75	10	a	a	DET
cana-5363	75	11	non	non	ADJ
cana-5363	75	12	-	-	ADJ
cana-5363	75	13	empty	empty	ADJ
cana-5363	75	14	set	set	NOUN
cana-5363	75	15	and	and	CCONJ
cana-5363	75	16	𝑅	𝑅	PROPN
cana-5363	75	17	be	be	AUX
cana-5363	75	18	an	an	DET
cana-5363	75	19	equivalence	equivalence	NOUN
cana-5363	75	20	relation	relation	NOUN
cana-5363	75	21	on	on	ADP
cana-5363	75	22	𝑈.	𝑈.	PROPN
cana-5363	75	23	let	let	VERB
cana-5363	75	24	𝐴	𝐴	PROPN
cana-5363	75	25	be	be	AUX
cana-5363	75	26	a	a	DET
cana-5363	75	27	pythagorean	pythagorean	ADJ
cana-5363	75	28	fuzzy	fuzzy	NOUN
cana-5363	75	29	set	set	VERB
cana-5363	75	30	in	in	ADP
cana-5363	75	31	𝑈	𝑈	PROPN
cana-5363	75	32	with	with	ADP
cana-5363	75	33	the	the	DET
cana-5363	75	34	membership	membership	NOUN
cana-5363	75	35	function	function	VERB
cana-5363	75	36	𝜇𝐴(𝑥	𝜇𝐴(𝑥	NUM
cana-5363	75	37	)	)	PUNCT
cana-5363	75	38	and	and	CCONJ
cana-5363	75	39	non	non	ADJ
cana-5363	75	40	membership	membership	NOUN
cana-5363	75	41	function	function	NOUN
cana-5363	75	42	𝜆𝐴(𝑥	𝜆𝐴(𝑥	NOUN
cana-5363	75	43	)	)	PUNCT
cana-5363	75	44	,	,	PUNCT
cana-5363	75	45	∀	∀	PUNCT
cana-5363	75	46	𝑥	𝑥	DET
cana-5363	75	47	∈	∈	NOUN
cana-5363	75	48	𝑈.	𝑈.	PROPN
cana-5363	75	49	the	the	DET
cana-5363	75	50	pythagorean	pythagorean	PROPN
cana-5363	75	51	fuzzy	fuzzy	ADJ
cana-5363	75	52	nano	nano	NOUN
cana-5363	75	53	lower	low	ADJ
cana-5363	75	54	,	,	PUNCT
cana-5363	75	55	pythagorean	pythagorean	PROPN
cana-5363	75	56	fuzzy	fuzzy	ADJ
cana-5363	75	57	nano	nano	PROPN
cana-5363	75	58	upper	upper	ADJ
cana-5363	75	59	approximation	approximation	NOUN
cana-5363	75	60	and	and	CCONJ
cana-5363	75	61	pythagorean	pythagorean	PROPN
cana-5363	75	62	fuzzy	fuzzy	ADJ
cana-5363	75	63	nano	nano	NOUN
cana-5363	75	64	boundary	boundary	NOUN
cana-5363	75	65	of	of	ADP
cana-5363	75	66	𝐴	𝐴	PROPN
cana-5363	75	67	in	in	ADP
cana-5363	75	68	the	the	DET
cana-5363	75	69	approximation	approximation	NOUN
cana-5363	75	70	(	(	PUNCT
cana-5363	75	71	𝑈	𝑈	PROPN
cana-5363	75	72	,	,	PUNCT
cana-5363	75	73	𝑅	𝑅	PROPN
cana-5363	75	74	)	)	PUNCT
cana-5363	75	75	denoted	denote	VERB
cana-5363	75	76	by	by	ADP
cana-5363	75	77	𝒫ℱ𝒩(𝐴	𝒫ℱ𝒩(𝐴	NOUN
cana-5363	75	78	)	)	PUNCT
cana-5363	75	79	,	,	PUNCT
cana-5363	75	80	𝒫ℱ𝒩(𝐴	𝒫ℱ𝒩(𝐴	NOUN
cana-5363	75	81	)	)	PUNCT
cana-5363	75	82	and	and	CCONJ
cana-5363	75	83	𝐵𝒫ℱ𝒩(𝐴	𝐵𝒫ℱ𝒩(𝐴	NOUN
cana-5363	75	84	)	)	PUNCT
cana-5363	75	85	are	be	AUX
cana-5363	75	86	respectively	respectively	ADV
cana-5363	75	87	defined	define	VERB
cana-5363	75	88	as	as	SCONJ
cana-5363	75	89	follows	follow	VERB
cana-5363	75	90	:	:	PUNCT
cana-5363	76	1	1	1	X
cana-5363	76	2	.	.	X
cana-5363	76	3	𝒫ℱ𝒩(𝐴	𝒫ℱ𝒩(𝐴	NOUN
cana-5363	76	4	)	)	PUNCT
cana-5363	76	5	=	=	PRON
cana-5363	76	6	{	{	PUNCT
cana-5363	76	7	〈	〈	NOUN
cana-5363	76	8	𝑥	𝑥	NOUN
cana-5363	76	9	,	,	PUNCT
cana-5363	76	10	𝜇𝑅(𝐴)(𝑥	𝜇𝑅(𝐴)(𝑥	NOUN
cana-5363	76	11	)	)	PUNCT
cana-5363	76	12	,	,	PUNCT
cana-5363	76	13	𝜆𝑅(𝐴)(𝑥)〉/𝑦	𝜆𝑅(𝐴)(𝑥)〉/𝑦	PROPN
cana-5363	76	14	∈	∈	PROPN
cana-5363	77	1	[	[	X
cana-5363	77	2	𝑥]𝑅	𝑥]𝑅	NOUN
cana-5363	77	3	,	,	PUNCT
cana-5363	77	4	𝑥	𝑥	PRON
cana-5363	77	5	∈	∈	PROPN
cana-5363	77	6	𝑈	𝑈	PROPN
cana-5363	77	7	}	}	PUNCT
cana-5363	77	8	2	2	NUM
cana-5363	77	9	.	.	PUNCT
cana-5363	77	10	𝒫ℱ𝒩(𝐹	𝒫ℱ𝒩(𝐹	NOUN
cana-5363	77	11	)	)	PUNCT
cana-5363	77	12	=	=	PRON
cana-5363	77	13	{	{	PUNCT
cana-5363	77	14	〈	〈	NOUN
cana-5363	77	15	𝑥	𝑥	NOUN
cana-5363	77	16	,	,	PUNCT
cana-5363	77	17	𝜇𝑅(𝐴)(𝑥	𝜇𝑅(𝐴)(𝑥	NOUN
cana-5363	77	18	)	)	PUNCT
cana-5363	77	19	,	,	PUNCT
cana-5363	77	20	𝜆𝑅(𝐴)(𝑥)〉/𝑦	𝜆𝑅(𝐴)(𝑥)〉/𝑦	PROPN
cana-5363	77	21	∈	∈	PROPN
cana-5363	78	1	[	[	X
cana-5363	78	2	𝑥]𝑅	𝑥]𝑅	ADV
cana-5363	78	3	,	,	PUNCT
cana-5363	78	4	𝑥	𝑥	PRON
cana-5363	78	5	∈	∈	PROPN
cana-5363	78	6	𝑈	𝑈	PROPN
cana-5363	78	7	}	}	PUNCT
cana-5363	78	8	3	3	NUM
cana-5363	78	9	.	.	PUNCT
cana-5363	78	10	𝐵𝒫ℱ𝒩(𝐹	𝐵𝒫ℱ𝒩(𝐹	NUM
cana-5363	78	11	)	)	PUNCT
cana-5363	78	12	=	=	PUNCT
cana-5363	78	13	𝒫ℱ𝒩(𝐹	𝒫ℱ𝒩(𝐹	NOUN
cana-5363	78	14	)	)	PUNCT
cana-5363	78	15	−	−	PROPN
cana-5363	78	16	𝒫ℱ𝒩(𝐹	𝒫ℱ𝒩(𝐹	NOUN
cana-5363	78	17	)	)	PUNCT
cana-5363	78	18	where	where	SCONJ
cana-5363	78	19	𝜇𝑅(𝐴)(𝑥	𝜇𝑅(𝐴)(𝑥	NOUN
cana-5363	78	20	)	)	PUNCT
cana-5363	79	1	=	=	NOUN
cana-5363	79	2	∧𝑦∈[𝑥]𝑅	∧𝑦∈[𝑥]𝑅	NOUN
cana-5363	79	3	𝜇𝐴(𝑦	𝜇𝐴(𝑦	NUM
cana-5363	79	4	)	)	PUNCT
cana-5363	79	5	𝜆𝑅(𝐴)(𝑥	𝜆𝑅(𝐴)(𝑥	ADP
cana-5363	79	6	)	)	PUNCT
cana-5363	80	1	=	=	X
cana-5363	80	2	∧𝑦∈[𝑥]𝑅	∧𝑦∈[𝑥]𝑅	NOUN
cana-5363	80	3	𝜆𝐴(𝑦	𝜆𝐴(𝑦	NOUN
cana-5363	80	4	)	)	PUNCT
cana-5363	80	5	,	,	PUNCT
cana-5363	80	6	𝜇𝑅(𝐴)(𝑥	𝜇𝑅(𝐴)(𝑥	NOUN
cana-5363	80	7	)	)	PUNCT
cana-5363	81	1	=	=	NOUN
cana-5363	81	2	∨𝑦∈[𝑥]𝑅	∨𝑦∈[𝑥]𝑅	NOUN
cana-5363	81	3	𝜇𝐴(𝑦	𝜇𝐴(𝑦	NOUN
cana-5363	81	4	)	)	PUNCT
cana-5363	81	5	,	,	PUNCT
cana-5363	81	6	𝜆𝑅(𝐴)(𝑥	𝜆𝑅(𝐴)(𝑥	X
cana-5363	81	7	)	)	PUNCT
cana-5363	82	1	=	=	NOUN
cana-5363	82	2	∨𝑦∈[𝑥]𝑅	∨𝑦∈[𝑥]𝑅	NOUN
cana-5363	82	3	𝜆𝐴(𝑦	𝜆𝐴(𝑦	NOUN
cana-5363	82	4	)	)	PUNCT
cana-5363	82	5	.	.	PUNCT
cana-5363	83	1	definition	definition	NOUN
cana-5363	83	2	2.10	2.10	NUM
cana-5363	83	3	[	[	X
cana-5363	83	4	1	1	NUM
cana-5363	83	5	]	]	PUNCT
cana-5363	83	6	let	let	VERB
cana-5363	83	7	𝑈	𝑈	PROPN
cana-5363	83	8	be	be	AUX
cana-5363	83	9	an	an	DET
cana-5363	83	10	universe	universe	NOUN
cana-5363	83	11	of	of	ADP
cana-5363	83	12	discourse	discourse	NOUN
cana-5363	83	13	,	,	PUNCT
cana-5363	83	14	𝑅	𝑅	PROPN
cana-5363	83	15	be	be	AUX
cana-5363	83	16	an	an	DET
cana-5363	83	17	equivalence	equivalence	NOUN
cana-5363	83	18	relation	relation	NOUN
cana-5363	83	19	on	on	ADP
cana-5363	83	20	𝑈	𝑈	PROPN
cana-5363	83	21	and	and	CCONJ
cana-5363	83	22	𝐴	𝐴	PROPN
cana-5363	83	23	be	be	VERB
cana-5363	83	24	a	a	DET
cana-5363	83	25	pythagorean	pythagorean	ADJ
cana-5363	83	26	fuzzy	fuzzy	ADJ
cana-5363	83	27	set	set	VERB
cana-5363	83	28	in	in	ADP
cana-5363	83	29	𝑈	𝑈	PROPN
cana-5363	83	30	and	and	CCONJ
cana-5363	84	1	if	if	SCONJ
cana-5363	84	2	the	the	DET
cana-5363	84	3	collection	collection	NOUN
cana-5363	84	4	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	84	5	)	)	PUNCT
cana-5363	84	6	=	=	PRON
cana-5363	84	7	{	{	PUNCT
cana-5363	84	8	0𝒫	0𝒫	NOUN
cana-5363	84	9	,	,	PUNCT
cana-5363	84	10	1𝒫	1𝒫	INTJ
cana-5363	84	11	,	,	PUNCT
cana-5363	84	12	𝒫ℱ𝒩(𝐴	𝒫ℱ𝒩(𝐴	NOUN
cana-5363	84	13	)	)	PUNCT
cana-5363	84	14	,	,	PUNCT
cana-5363	84	15	communications	communication	NOUN
cana-5363	84	16	on	on	ADP
cana-5363	84	17	applied	apply	VERB
cana-5363	84	18	nonlinear	nonlinear	ADJ
cana-5363	84	19	analysis	analysis	NOUN
cana-5363	84	20	issn	issn	NOUN
cana-5363	84	21	:	:	PUNCT
cana-5363	84	22	1074	1074	NUM
cana-5363	84	23	-	-	PUNCT
cana-5363	84	24	133x	133x	NUM
cana-5363	84	25	vol	vol	VERB
cana-5363	84	26	32	32	NUM
cana-5363	84	27	no	no	NOUN
cana-5363	84	28	.	.	PUNCT
cana-5363	85	1	10s	10	NOUN
cana-5363	85	2	(	(	PUNCT
cana-5363	85	3	2025	2025	NUM
cana-5363	85	4	)	)	PUNCT
cana-5363	85	5	2008	2008	NUM
cana-5363	85	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5363	85	7	𝒫ℱ𝒩(𝐴	𝒫ℱ𝒩(𝐴	NOUN
cana-5363	85	8	)	)	PUNCT
cana-5363	85	9	,	,	PUNCT
cana-5363	85	10	𝐵𝒫ℱ𝒩(𝐴	𝐵𝒫ℱ𝒩(𝐴	NOUN
cana-5363	85	11	)	)	PUNCT
cana-5363	85	12	}	}	PUNCT
cana-5363	85	13	forms	form	VERB
cana-5363	85	14	a	a	DET
cana-5363	85	15	topology	topology	NOUN
cana-5363	85	16	then	then	ADV
cana-5363	85	17	it	it	PRON
cana-5363	85	18	is	be	AUX
cana-5363	85	19	said	say	VERB
cana-5363	85	20	to	to	PART
cana-5363	85	21	be	be	AUX
cana-5363	85	22	a	a	DET
cana-5363	85	23	pythagorean	pythagorean	ADJ
cana-5363	85	24	fuzzy	fuzzy	ADJ
cana-5363	85	25	nano	nano	NOUN
cana-5363	85	26	topology	topology	NOUN
cana-5363	85	27	.	.	PUNCT
cana-5363	86	1	we	we	PRON
cana-5363	86	2	call	call	VERB
cana-5363	86	3	(	(	PUNCT
cana-5363	86	4	𝑈	𝑈	PROPN
cana-5363	86	5	,	,	PUNCT
cana-5363	86	6	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	86	7	)	)	PUNCT
cana-5363	86	8	)	)	PUNCT
cana-5363	87	1	(	(	PUNCT
cana-5363	87	2	or	or	CCONJ
cana-5363	87	3	simply	simply	ADV
cana-5363	87	4	𝑈	𝑈	PROPN
cana-5363	87	5	)	)	PUNCT
cana-5363	87	6	as	as	ADP
cana-5363	87	7	the	the	DET
cana-5363	87	8	pythagorean	pythagorean	PROPN
cana-5363	87	9	fuzzy	fuzzy	ADJ
cana-5363	87	10	nano	nano	NOUN
cana-5363	87	11	topological	topological	ADJ
cana-5363	87	12	space	space	NOUN
cana-5363	87	13	.	.	PUNCT
cana-5363	88	1	the	the	DET
cana-5363	88	2	elements	element	NOUN
cana-5363	88	3	of	of	ADP
cana-5363	88	4	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	88	5	)	)	PUNCT
cana-5363	88	6	are	be	AUX
cana-5363	88	7	called	call	VERB
cana-5363	88	8	pythagorean	pythagorean	PROPN
cana-5363	88	9	fuzzy	fuzzy	ADJ
cana-5363	88	10	nano	nano	NOUN
cana-5363	88	11	open	open	ADJ
cana-5363	88	12	(	(	PUNCT
cana-5363	88	13	briefly	briefly	ADV
cana-5363	88	14	,	,	PUNCT
cana-5363	88	15	𝒫ℱ𝒩𝑜	𝒫ℱ𝒩𝑜	PROPN
cana-5363	88	16	)	)	PUNCT
cana-5363	88	17	sets	set	NOUN
cana-5363	88	18	.	.	PUNCT
cana-5363	89	1	remark	remark	VERB
cana-5363	89	2	2.1	2.1	NUM
cana-5363	89	3	[	[	X
cana-5363	89	4	1	1	NUM
cana-5363	89	5	]	]	X
cana-5363	89	6	[	[	X
cana-5363	89	7	𝜏ℛ(𝐴)]𝑐	𝜏ℛ(𝐴)]𝑐	NOUN
cana-5363	89	8	is	be	AUX
cana-5363	89	9	called	call	VERB
cana-5363	89	10	the	the	DET
cana-5363	89	11	dual	dual	ADJ
cana-5363	89	12	fuzzy	fuzzy	ADJ
cana-5363	89	13	nano	nano	NOUN
cana-5363	89	14	topology	topology	NOUN
cana-5363	89	15	of	of	ADP
cana-5363	89	16	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	89	17	)	)	PUNCT
cana-5363	89	18	.	.	PUNCT
cana-5363	90	1	elements	element	NOUN
cana-5363	90	2	of	of	ADP
cana-5363	90	3	[	[	X
cana-5363	90	4	𝜏ℛ(𝐴)]𝑐	𝜏ℛ(𝐴)]𝑐	NOUN
cana-5363	90	5	are	be	AUX
cana-5363	90	6	called	call	VERB
cana-5363	90	7	pythagorean	pythagorean	PROPN
cana-5363	90	8	fuzzy	fuzzy	PROPN
cana-5363	90	9	nano	nano	PROPN
cana-5363	90	10	closed	close	VERB
cana-5363	90	11	(	(	PUNCT
cana-5363	90	12	briefly	briefly	ADV
cana-5363	90	13	,	,	PUNCT
cana-5363	90	14	𝒫ℱ𝒩𝑐	𝒫ℱ𝒩𝑐	NOUN
cana-5363	90	15	)	)	PUNCT
cana-5363	90	16	sets	set	NOUN
cana-5363	90	17	.	.	PUNCT
cana-5363	91	1	thus	thus	ADV
cana-5363	91	2	,	,	PUNCT
cana-5363	91	3	we	we	PRON
cana-5363	91	4	note	note	VERB
cana-5363	91	5	that	that	SCONJ
cana-5363	91	6	a	a	DET
cana-5363	91	7	pythagorean	pythagorean	PROPN
cana-5363	91	8	fuzzy	fuzzy	NOUN
cana-5363	91	9	set	set	VERB
cana-5363	91	10	𝐺	𝐺	PROPN
cana-5363	91	11	of	of	ADP
cana-5363	91	12	𝑈	𝑈	PROPN
cana-5363	91	13	is	be	AUX
cana-5363	91	14	pythagorean	pythagorean	PROPN
cana-5363	91	15	fuzzy	fuzzy	ADJ
cana-5363	91	16	nano	nano	NOUN
cana-5363	91	17	closed	close	VERB
cana-5363	91	18	in	in	ADP
cana-5363	91	19	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	91	20	)	)	PUNCT
cana-5363	91	21	if	if	SCONJ
cana-5363	91	22	and	and	CCONJ
cana-5363	91	23	only	only	ADV
cana-5363	91	24	if	if	SCONJ
cana-5363	91	25	1𝑃	1𝑃	PROPN
cana-5363	91	26	−	−	PROPN
cana-5363	91	27	𝐺	𝐺	PROPN
cana-5363	91	28	is	be	AUX
cana-5363	91	29	pythagorean	pythagorean	ADJ
cana-5363	91	30	fuzzy	fuzzy	ADJ
cana-5363	91	31	nano	nano	NOUN
cana-5363	91	32	open	open	ADJ
cana-5363	91	33	in	in	ADP
cana-5363	91	34	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	91	35	)	)	PUNCT
cana-5363	91	36	.	.	PUNCT
cana-5363	92	1	definition	definition	NOUN
cana-5363	92	2	2.11	2.11	NUM
cana-5363	92	3	[	[	X
cana-5363	92	4	1	1	NUM
cana-5363	92	5	,	,	PUNCT
cana-5363	92	6	2	2	NUM
cana-5363	92	7	]	]	PUNCT
cana-5363	92	8	let	let	VERB
cana-5363	92	9	(	(	PUNCT
cana-5363	92	10	𝑈	𝑈	PROPN
cana-5363	92	11	,	,	PUNCT
cana-5363	92	12	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	92	13	)	)	PUNCT
cana-5363	92	14	)	)	PUNCT
cana-5363	92	15	be	be	AUX
cana-5363	92	16	a	a	DET
cana-5363	92	17	𝒫ℱ𝒩𝑡𝑠	𝒫ℱ𝒩𝑡𝑠	NOUN
cana-5363	92	18	with	with	ADP
cana-5363	92	19	respect	respect	NOUN
cana-5363	92	20	to	to	ADP
cana-5363	92	21	𝐴	𝐴	PROPN
cana-5363	92	22	where	where	SCONJ
cana-5363	92	23	𝐴	𝐴	PROPN
cana-5363	92	24	is	be	AUX
cana-5363	92	25	a	a	DET
cana-5363	92	26	pythagorean	pythagorean	ADJ
cana-5363	92	27	fuzzy	fuzzy	ADJ
cana-5363	92	28	subset	subset	NOUN
cana-5363	92	29	of	of	ADP
cana-5363	92	30	𝑈.	𝑈.	PROPN
cana-5363	92	31	let	let	VERB
cana-5363	92	32	𝑆	𝑆	PROPN
cana-5363	92	33	be	be	AUX
cana-5363	92	34	a	a	DET
cana-5363	92	35	pythagorean	pythagorean	ADJ
cana-5363	92	36	fuzzy	fuzzy	ADJ
cana-5363	92	37	subset	subset	NOUN
cana-5363	92	38	of	of	ADP
cana-5363	92	39	𝑈.	𝑈.	PROPN
cana-5363	92	40	then	then	ADV
cana-5363	92	41	pythagorean	pythagorean	VERB
cana-5363	92	42	fuzzy	fuzzy	ADJ
cana-5363	92	43	nano	nano	PROPN
cana-5363	92	44	1	1	NUM
cana-5363	92	45	.	.	PUNCT
cana-5363	92	46	interior	interior	NOUN
cana-5363	92	47	of	of	ADP
cana-5363	92	48	𝑆	𝑆	PROPN
cana-5363	92	49	(	(	PUNCT
cana-5363	92	50	briefly	briefly	ADV
cana-5363	92	51	,	,	PUNCT
cana-5363	92	52	𝒫ℱ𝒩𝑖𝑛𝑡(𝑆	𝒫ℱ𝒩𝑖𝑛𝑡(𝑆	X
cana-5363	92	53	)	)	PUNCT
cana-5363	92	54	)	)	PUNCT
cana-5363	92	55	is	be	AUX
cana-5363	92	56	defined	define	VERB
cana-5363	92	57	by	by	ADP
cana-5363	92	58	𝒫ℱ𝒩𝑖𝑛𝑡(𝑆	𝒫ℱ𝒩𝑖𝑛𝑡(𝑆	NOUN
cana-5363	92	59	)	)	PUNCT
cana-5363	92	60	=	=	NOUN
cana-5363	92	61	∪	∪	X
cana-5363	92	62	{	{	PUNCT
cana-5363	92	63	𝐼	𝐼	NOUN
cana-5363	92	64	:	:	PUNCT
cana-5363	92	65	𝐼	𝐼	PROPN
cana-5363	92	66	⊆	⊆	NUM
cana-5363	92	67	𝑆	𝑆	PROPN
cana-5363	92	68	&	&	CCONJ
cana-5363	92	69	𝐼isa𝒫ℱ𝒩𝑜set	𝐼isa𝒫ℱ𝒩𝑜set	PROPN
cana-5363	92	70	in𝑈	in𝑈	PROPN
cana-5363	92	71	}	}	PUNCT
cana-5363	92	72	.	.	PUNCT
cana-5363	93	1	2	2	X
cana-5363	93	2	.	.	X
cana-5363	93	3	closure	closure	NOUN
cana-5363	93	4	of	of	ADP
cana-5363	93	5	𝑆	𝑆	PROPN
cana-5363	93	6	(	(	PUNCT
cana-5363	93	7	briefly	briefly	ADV
cana-5363	93	8	,	,	PUNCT
cana-5363	93	9	𝒫ℱ𝒩𝑐𝑙(𝑆	𝒫ℱ𝒩𝑐𝑙(𝑆	PUNCT
cana-5363	93	10	)	)	PUNCT
cana-5363	93	11	)	)	PUNCT
cana-5363	93	12	is	be	AUX
cana-5363	93	13	defined	define	VERB
cana-5363	93	14	by	by	ADP
cana-5363	93	15	𝒫ℱ𝒩𝑐𝑙(𝑆	𝒫ℱ𝒩𝑐𝑙(𝑆	NOUN
cana-5363	93	16	)	)	PUNCT
cana-5363	93	17	=	=	NOUN
cana-5363	93	18	∩	∩	X
cana-5363	93	19	{	{	PUNCT
cana-5363	93	20	𝐴	𝐴	PROPN
cana-5363	93	21	:	:	PUNCT
cana-5363	93	22	𝑆	𝑆	PROPN
cana-5363	93	23	⊆	⊆	NUM
cana-5363	93	24	𝐴	𝐴	PROPN
cana-5363	93	25	&	&	CCONJ
cana-5363	93	26	𝐴isa𝒫ℱ𝒩𝑐set	𝐴isa𝒫ℱ𝒩𝑐set	PROPN
cana-5363	93	27	in𝑈	in𝑈	PROPN
cana-5363	93	28	}	}	PUNCT
cana-5363	93	29	.	.	PUNCT
cana-5363	94	1	3	3	X
cana-5363	94	2	.	.	X
cana-5363	94	3	regular	regular	ADJ
cana-5363	94	4	open	open	ADJ
cana-5363	94	5	(	(	PUNCT
cana-5363	94	6	briefly	briefly	ADV
cana-5363	94	7	,	,	PUNCT
cana-5363	94	8	𝒫ℱ𝒩𝑟𝑜	𝒫ℱ𝒩𝑟𝑜	NUM
cana-5363	94	9	)	)	PUNCT
cana-5363	94	10	set	set	VERB
cana-5363	94	11	if	if	SCONJ
cana-5363	94	12	𝑆	𝑆	PROPN
cana-5363	94	13	=	=	PUNCT
cana-5363	94	14	𝒫ℱ𝒩𝑖𝑛𝑡(𝒫ℱ𝒩𝑐𝑙(𝑆	𝒫ℱ𝒩𝑖𝑛𝑡(𝒫ℱ𝒩𝑐𝑙(𝑆	NOUN
cana-5363	94	15	)	)	PUNCT
cana-5363	94	16	)	)	PUNCT
cana-5363	94	17	.	.	PUNCT
cana-5363	95	1	4	4	X
cana-5363	95	2	.	.	X
cana-5363	95	3	regular	regular	ADJ
cana-5363	95	4	closed	close	VERB
cana-5363	95	5	(	(	PUNCT
cana-5363	95	6	briefly	briefly	ADV
cana-5363	95	7	,	,	PUNCT
cana-5363	95	8	𝒫ℱ𝒩𝑟𝑐	𝒫ℱ𝒩𝑟𝑐	PROPN
cana-5363	95	9	)	)	PUNCT
cana-5363	95	10	set	set	VERB
cana-5363	95	11	if	if	SCONJ
cana-5363	95	12	𝑆	𝑆	PROPN
cana-5363	95	13	=	=	SYM
cana-5363	95	14	𝒫ℱ𝒩𝑐𝑙(𝒫ℱ𝒩𝑖𝑛𝑡(𝑆	𝒫ℱ𝒩𝑐𝑙(𝒫ℱ𝒩𝑖𝑛𝑡(𝑆	PROPN
cana-5363	95	15	)	)	PUNCT
cana-5363	95	16	)	)	PUNCT
cana-5363	95	17	.	.	PUNCT
cana-5363	96	1	3	3	NUM
cana-5363	96	2	pythagorean	pythagorean	PROPN
cana-5363	96	3	fuzzy	fuzzy	ADJ
cana-5363	96	4	nano	nano	NOUN
cana-5363	96	5	𝒁	𝒁	PROPN
cana-5363	96	6	(	(	PUNCT
cana-5363	96	7	resp	resp	NOUN
cana-5363	96	8	.	.	PUNCT
cana-5363	97	1	𝜹	𝜹	X
cana-5363	97	2	,	,	PUNCT
cana-5363	97	3	𝜹𝓢	𝜹𝓢	ADJ
cana-5363	97	4	and	and	CCONJ
cana-5363	97	5	pre)-open	pre)-open	NOUN
cana-5363	97	6	sets	set	NOUN
cana-5363	97	7	definition	definition	NOUN
cana-5363	97	8	3.1	3.1	NUM
cana-5363	97	9	let	let	NOUN
cana-5363	97	10	(	(	PUNCT
cana-5363	97	11	𝑈	𝑈	PROPN
cana-5363	97	12	,	,	PUNCT
cana-5363	97	13	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	97	14	)	)	PUNCT
cana-5363	97	15	)	)	PUNCT
cana-5363	97	16	be	be	AUX
cana-5363	97	17	a	a	DET
cana-5363	97	18	𝒫ℱ𝒩𝑡𝑠	𝒫ℱ𝒩𝑡𝑠	NOUN
cana-5363	97	19	with	with	ADP
cana-5363	97	20	respect	respect	NOUN
cana-5363	97	21	to	to	ADP
cana-5363	97	22	𝐴	𝐴	PROPN
cana-5363	97	23	where	where	SCONJ
cana-5363	97	24	𝐴	𝐴	PROPN
cana-5363	97	25	is	be	AUX
cana-5363	97	26	a	a	DET
cana-5363	97	27	𝑝𝑓𝑠	𝑝𝑓𝑠	PROPN
cana-5363	97	28	of	of	ADP
cana-5363	97	29	𝑈.	𝑈.	PROPN
cana-5363	97	30	let	let	VERB
cana-5363	97	31	𝑆	𝑆	PROPN
cana-5363	97	32	be	be	AUX
cana-5363	97	33	a	a	DET
cana-5363	97	34	𝑝𝑓𝑠	𝑝𝑓𝑠	NOUN
cana-5363	97	35	of	of	ADP
cana-5363	97	36	𝑈.	𝑈.	PROPN
cana-5363	97	37	then	then	ADV
cana-5363	97	38	pythagorean	pythagorean	PROPN
cana-5363	97	39	1	1	NUM
cana-5363	97	40	.	.	PUNCT
cana-5363	97	41	fuzzy	fuzzy	ADJ
cana-5363	97	42	nano	nano	PROPN
cana-5363	97	43	𝛿	𝛿	DET
cana-5363	97	44	interior	interior	NOUN
cana-5363	97	45	of	of	ADP
cana-5363	97	46	𝑆	𝑆	PROPN
cana-5363	97	47	(	(	PUNCT
cana-5363	97	48	briefly	briefly	ADV
cana-5363	97	49	,	,	PUNCT
cana-5363	97	50	𝒫ℱ𝒩𝛿𝑖𝑛𝑡(𝑆	𝒫ℱ𝒩𝛿𝑖𝑛𝑡(𝑆	X
cana-5363	97	51	)	)	PUNCT
cana-5363	97	52	)	)	PUNCT
cana-5363	97	53	is	be	AUX
cana-5363	97	54	defined	define	VERB
cana-5363	97	55	by	by	ADP
cana-5363	97	56	𝒫ℱ𝒩𝛿𝑖𝑛𝑡(𝑆	𝒫ℱ𝒩𝛿𝑖𝑛𝑡(𝑆	NOUN
cana-5363	97	57	)	)	PUNCT
cana-5363	98	1	=	=	SYM
cana-5363	98	2	∪	∪	X
cana-5363	98	3	{	{	PUNCT
cana-5363	98	4	𝐼	𝐼	NOUN
cana-5363	98	5	:	:	PUNCT
cana-5363	98	6	𝐼	𝐼	PROPN
cana-5363	98	7	⊆	⊆	NUM
cana-5363	98	8	𝑆	𝑆	PROPN
cana-5363	98	9	&	&	CCONJ
cana-5363	98	10	𝐼isa𝒫ℱ𝒩𝑟𝑜	𝐼isa𝒫ℱ𝒩𝑟𝑜	ADV
cana-5363	98	11	set	set	VERB
cana-5363	98	12	in𝑈	in𝑈	NOUN
cana-5363	98	13	}	}	PUNCT
cana-5363	98	14	.	.	PUNCT
cana-5363	99	1	2	2	X
cana-5363	99	2	.	.	X
cana-5363	99	3	fuzzy	fuzzy	ADJ
cana-5363	99	4	nano	nano	NOUN
cana-5363	99	5	𝛿	𝛿	ADJ
cana-5363	99	6	closure	closure	NOUN
cana-5363	99	7	of	of	ADP
cana-5363	99	8	𝑆	𝑆	PROPN
cana-5363	99	9	(	(	PUNCT
cana-5363	99	10	briefly	briefly	ADV
cana-5363	99	11	,	,	PUNCT
cana-5363	99	12	𝒫ℱ𝒩𝛿𝑐𝑙(𝑆	𝒫ℱ𝒩𝛿𝑐𝑙(𝑆	ADV
cana-5363	99	13	)	)	PUNCT
cana-5363	99	14	)	)	PUNCT
cana-5363	99	15	is	be	AUX
cana-5363	99	16	defined	define	VERB
cana-5363	99	17	by	by	ADP
cana-5363	99	18	𝒫ℱ𝒩𝛿𝑐𝑙(𝑆	𝒫ℱ𝒩𝛿𝑐𝑙(𝑆	ADV
cana-5363	99	19	)	)	PUNCT
cana-5363	100	1	=	=	NOUN
cana-5363	100	2	∩	∩	NOUN
cana-5363	100	3	{	{	PUNCT
cana-5363	100	4	𝐴	𝐴	PROPN
cana-5363	100	5	:	:	PUNCT
cana-5363	100	6	𝑆	𝑆	PROPN
cana-5363	100	7	⊆	⊆	NUM
cana-5363	100	8	𝐴	𝐴	PROPN
cana-5363	100	9	&	&	CCONJ
cana-5363	100	10	𝐴isa𝒫ℱ𝒩𝑟𝑐set	𝐴isa𝒫ℱ𝒩𝑟𝑐set	PROPN
cana-5363	100	11	in	in	ADP
cana-5363	100	12	𝑈	𝑈	PROPN
cana-5363	100	13	}	}	PUNCT
cana-5363	100	14	.	.	PUNCT
cana-5363	101	1	definition	definition	NOUN
cana-5363	101	2	3.2	3.2	NUM
cana-5363	101	3	let	let	VERB
cana-5363	101	4	(	(	PUNCT
cana-5363	101	5	𝑈	𝑈	PROPN
cana-5363	101	6	,	,	PUNCT
cana-5363	101	7	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	101	8	)	)	PUNCT
cana-5363	101	9	)	)	PUNCT
cana-5363	101	10	be	be	AUX
cana-5363	101	11	a	a	DET
cana-5363	101	12	𝒫ℱ𝒩𝑡𝑠	𝒫ℱ𝒩𝑡𝑠	NOUN
cana-5363	101	13	with	with	ADP
cana-5363	101	14	respect	respect	NOUN
cana-5363	101	15	to	to	ADP
cana-5363	101	16	𝐴	𝐴	PROPN
cana-5363	101	17	where	where	SCONJ
cana-5363	101	18	𝐴	𝐴	PROPN
cana-5363	101	19	is	be	AUX
cana-5363	101	20	a	a	DET
cana-5363	101	21	𝑝𝑓𝑠	𝑝𝑓𝑠	NOUN
cana-5363	101	22	of	of	ADP
cana-5363	101	23	𝑈.	𝑈.	PROPN
cana-5363	101	24	then	then	ADV
cana-5363	101	25	a	a	DET
cana-5363	101	26	𝑝𝑓𝑠	𝑝𝑓𝑠	PROPN
cana-5363	101	27	𝑆	𝑆	PROPN
cana-5363	101	28	in	in	ADP
cana-5363	101	29	𝑈	𝑈	PROPN
cana-5363	101	30	is	be	AUX
cana-5363	101	31	said	say	VERB
cana-5363	101	32	to	to	PART
cana-5363	101	33	be	be	AUX
cana-5363	101	34	pythagorean	pythagorean	NOUN
cana-5363	101	35	:	:	PUNCT
cana-5363	101	36	1	1	X
cana-5363	101	37	.	.	X
cana-5363	101	38	fuzzy	fuzzy	ADJ
cana-5363	101	39	nano	nano	NOUN
cana-5363	101	40	𝛿-open	𝛿-open	NOUN
cana-5363	101	41	(	(	PUNCT
cana-5363	101	42	briefly	briefly	ADV
cana-5363	101	43	,	,	PUNCT
cana-5363	101	44	𝒫ℱ𝒩𝛿𝑜	𝒫ℱ𝒩𝛿𝑜	NUM
cana-5363	101	45	)	)	PUNCT
cana-5363	101	46	set	set	VERB
cana-5363	101	47	if	if	SCONJ
cana-5363	101	48	𝑆	𝑆	PROPN
cana-5363	101	49	=	=	SYM
cana-5363	101	50	𝒫ℱ𝒩𝛿𝑖𝑛𝑡(𝑆	𝒫ℱ𝒩𝛿𝑖𝑛𝑡(𝑆	PROPN
cana-5363	101	51	)	)	PUNCT
cana-5363	101	52	.	.	PUNCT
cana-5363	102	1	2	2	X
cana-5363	102	2	.	.	X
cana-5363	102	3	fuzzy	fuzzy	ADJ
cana-5363	102	4	nano	nano	NOUN
cana-5363	102	5	𝛿-𝛼-open	𝛿-𝛼-open	NOUN
cana-5363	102	6	(	(	PUNCT
cana-5363	102	7	or	or	CCONJ
cana-5363	102	8	)	)	PUNCT
cana-5363	102	9	fuzzy	fuzzy	ADJ
cana-5363	102	10	nano	nano	NOUN
cana-5363	102	11	𝑎-open	𝑎-open	PROPN
cana-5363	102	12	(	(	PUNCT
cana-5363	102	13	briefly	briefly	ADV
cana-5363	102	14	,	,	PUNCT
cana-5363	102	15	𝒫ℱ𝒩𝛿𝛼𝑜	𝒫ℱ𝒩𝛿𝛼𝑜	PROPN
cana-5363	102	16	(	(	PUNCT
cana-5363	102	17	or	or	CCONJ
cana-5363	102	18	)	)	PUNCT
cana-5363	102	19	𝒫ℱ𝒩𝑎𝑜	𝒫ℱ𝒩𝑎𝑜	NOUN
cana-5363	102	20	)	)	PUNCT
cana-5363	102	21	set	set	VERB
cana-5363	102	22	if	if	SCONJ
cana-5363	102	23	𝑆	𝑆	PROPN
cana-5363	102	24	⊆	⊆	NUM
cana-5363	102	25	𝒫ℱ𝒩𝑖𝑛𝑡(𝒫ℱ𝒩𝑐𝑙	𝒫ℱ𝒩𝑖𝑛𝑡(𝒫ℱ𝒩𝑐𝑙	PROPN
cana-5363	102	26	(	(	PUNCT
cana-5363	102	27	𝒫ℱ𝒩𝛿𝑖𝑛𝑡	𝒫ℱ𝒩𝛿𝑖𝑛𝑡	PROPN
cana-5363	102	28	(	(	PUNCT
cana-5363	102	29	𝑆	𝑆	PROPN
cana-5363	102	30	)	)	PUNCT
cana-5363	102	31	)	)	PUNCT
cana-5363	102	32	)	)	PUNCT
cana-5363	102	33	.	.	PUNCT
cana-5363	103	1	3	3	X
cana-5363	103	2	.	.	X
cana-5363	103	3	fuzzy	fuzzy	ADJ
cana-5363	103	4	nano	nano	NOUN
cana-5363	103	5	𝛿-semi	𝛿-semi	NOUN
cana-5363	103	6	open	open	ADJ
cana-5363	103	7	(	(	PUNCT
cana-5363	103	8	briefly	briefly	ADV
cana-5363	103	9	,	,	PUNCT
cana-5363	103	10	𝒫ℱ𝒩𝛿𝒮𝑜	𝒫ℱ𝒩𝛿𝒮𝑜	ADV
cana-5363	103	11	)	)	PUNCT
cana-5363	103	12	set	set	VERB
cana-5363	103	13	if	if	SCONJ
cana-5363	103	14	𝑆	𝑆	PROPN
cana-5363	103	15	⊆	⊆	NUM
cana-5363	103	16	𝒫ℱ𝒩𝑐𝑙(𝒫ℱ𝒩𝛿𝑖𝑛𝑡(𝑆	𝒫ℱ𝒩𝑐𝑙(𝒫ℱ𝒩𝛿𝑖𝑛𝑡(𝑆	PROPN
cana-5363	103	17	)	)	PUNCT
cana-5363	103	18	)	)	PUNCT
cana-5363	103	19	.	.	PUNCT
cana-5363	104	1	4	4	X
cana-5363	104	2	.	.	X
cana-5363	104	3	fuzzy	fuzzy	ADJ
cana-5363	104	4	nano	nano	NOUN
cana-5363	104	5	pre	pre	X
cana-5363	104	6	open	open	ADJ
cana-5363	104	7	(	(	PUNCT
cana-5363	104	8	briefly	briefly	ADV
cana-5363	104	9	,	,	PUNCT
cana-5363	104	10	𝒫ℱ𝒩𝒫𝑜	𝒫ℱ𝒩𝒫𝑜	PROPN
cana-5363	104	11	)	)	PUNCT
cana-5363	104	12	set	set	VERB
cana-5363	104	13	if	if	SCONJ
cana-5363	104	14	𝑆	𝑆	PROPN
cana-5363	104	15	⊆	⊆	NUM
cana-5363	104	16	𝒫ℱ𝒩𝑖𝑛𝑡(𝒫ℱ𝒩𝑐𝑙(𝑆	𝒫ℱ𝒩𝑖𝑛𝑡(𝒫ℱ𝒩𝑐𝑙(𝑆	NUM
cana-5363	104	17	)	)	PUNCT
cana-5363	104	18	)	)	PUNCT
cana-5363	104	19	.	.	PUNCT
cana-5363	105	1	the	the	DET
cana-5363	105	2	complement	complement	NOUN
cana-5363	105	3	of	of	ADP
cana-5363	105	4	an	an	DET
cana-5363	105	5	𝒫ℱ𝒩𝛿𝑜	𝒫ℱ𝒩𝛿𝑜	PROPN
cana-5363	105	6	(	(	PUNCT
cana-5363	105	7	resp	resp	NOUN
cana-5363	105	8	.	.	PUNCT
cana-5363	106	1	𝒫ℱ𝒩𝛿𝛼𝑜	𝒫ℱ𝒩𝛿𝛼𝑜	PROPN
cana-5363	106	2	,	,	PUNCT
cana-5363	106	3	𝒫ℱ𝒩𝛿𝒮𝑜	𝒫ℱ𝒩𝛿𝒮𝑜	PROPN
cana-5363	106	4	&	&	CCONJ
cana-5363	106	5	𝒫ℱ𝒩𝒫𝑜	𝒫ℱ𝒩𝒫𝑜	PROPN
cana-5363	106	6	)	)	PUNCT
cana-5363	106	7	set	set	NOUN
cana-5363	106	8	is	be	AUX
cana-5363	106	9	called	call	VERB
cana-5363	106	10	a	a	DET
cana-5363	106	11	pythagorean	pythagorean	ADJ
cana-5363	106	12	fuzzy	fuzzy	ADJ
cana-5363	106	13	nano	nano	PROPN
cana-5363	106	14	𝛿	𝛿	ADJ
cana-5363	106	15	(	(	PUNCT
cana-5363	106	16	resp	resp	NOUN
cana-5363	106	17	.	.	PUNCT
cana-5363	107	1	pythagorean	pythagorean	PROPN
cana-5363	107	2	fuzzy	fuzzy	ADJ
cana-5363	107	3	nano	nano	NOUN
cana-5363	107	4	𝛿-𝛼	𝛿-𝛼	NOUN
cana-5363	107	5	,	,	PUNCT
cana-5363	107	6	pythagorean	pythagorean	PROPN
cana-5363	107	7	fuzzy	fuzzy	ADJ
cana-5363	107	8	nano	nano	NOUN
cana-5363	107	9	𝛿-semi	𝛿-semi	PROPN
cana-5363	107	10	&	&	CCONJ
cana-5363	107	11	pythagorean	pythagorean	PROPN
cana-5363	107	12	fuzzy	fuzzy	ADJ
cana-5363	107	13	nano	nano	PROPN
cana-5363	107	14	pre	pre	NOUN
cana-5363	107	15	)	)	PUNCT
cana-5363	107	16	closed	closed	ADJ
cana-5363	107	17	(	(	PUNCT
cana-5363	107	18	briefly	briefly	ADV
cana-5363	107	19	,	,	PUNCT
cana-5363	107	20	𝒫ℱ𝒩𝛿𝑐	𝒫ℱ𝒩𝛿𝑐	NUM
cana-5363	107	21	(	(	PUNCT
cana-5363	107	22	resp	resp	NOUN
cana-5363	107	23	.	.	PUNCT
cana-5363	108	1	𝒫ℱ𝒩𝛿𝛼𝑐	𝒫ℱ𝒩𝛿𝛼𝑐	PROPN
cana-5363	108	2	,	,	PUNCT
cana-5363	108	3	𝒫ℱ𝒩𝛿𝒮𝑐	𝒫ℱ𝒩𝛿𝒮𝑐	PROPN
cana-5363	108	4	&	&	CCONJ
cana-5363	108	5	𝒫ℱ𝒩𝒫𝑐	𝒫ℱ𝒩𝒫𝑐	NOUN
cana-5363	108	6	)	)	PUNCT
cana-5363	108	7	)	)	PUNCT
cana-5363	108	8	in	in	ADP
cana-5363	108	9	𝑈.	𝑈.	PROPN
cana-5363	108	10	definition	definition	NOUN
cana-5363	108	11	3.3	3.3	NUM
cana-5363	108	12	let	let	VERB
cana-5363	108	13	(	(	PUNCT
cana-5363	108	14	𝑈	𝑈	PROPN
cana-5363	108	15	,	,	PUNCT
cana-5363	108	16	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	108	17	)	)	PUNCT
cana-5363	108	18	)	)	PUNCT
cana-5363	108	19	be	be	AUX
cana-5363	108	20	a	a	DET
cana-5363	108	21	𝒫ℱ𝒩𝑡𝑠	𝒫ℱ𝒩𝑡𝑠	NOUN
cana-5363	108	22	with	with	ADP
cana-5363	108	23	respect	respect	NOUN
cana-5363	108	24	to	to	ADP
cana-5363	108	25	𝐴	𝐴	PROPN
cana-5363	108	26	where	where	SCONJ
cana-5363	108	27	𝐴	𝐴	PROPN
cana-5363	108	28	is	be	AUX
cana-5363	108	29	a	a	DET
cana-5363	108	30	𝑝𝑓𝑠	𝑝𝑓𝑠	PROPN
cana-5363	108	31	of	of	ADP
cana-5363	108	32	𝑈.	𝑈.	PROPN
cana-5363	108	33	let	let	VERB
cana-5363	108	34	𝑆	𝑆	PROPN
cana-5363	108	35	be	be	AUX
cana-5363	108	36	a	a	DET
cana-5363	108	37	𝑝𝑓𝑠	𝑝𝑓𝑠	NOUN
cana-5363	108	38	of	of	ADP
cana-5363	108	39	𝑈.	𝑈.	PROPN
cana-5363	108	40	then	then	ADV
cana-5363	108	41	pythagorean	pythagorean	VERB
cana-5363	108	42	fuzzy	fuzzy	ADJ
cana-5363	108	43	nano	nano	NOUN
cana-5363	108	44	communications	communication	NOUN
cana-5363	108	45	on	on	ADP
cana-5363	108	46	applied	apply	VERB
cana-5363	108	47	nonlinear	nonlinear	ADJ
cana-5363	108	48	analysis	analysis	NOUN
cana-5363	108	49	issn	issn	NOUN
cana-5363	108	50	:	:	PUNCT
cana-5363	108	51	1074	1074	NUM
cana-5363	108	52	-	-	PUNCT
cana-5363	108	53	133x	133x	NUM
cana-5363	108	54	vol	vol	VERB
cana-5363	108	55	32	32	NUM
cana-5363	108	56	no	no	NOUN
cana-5363	108	57	.	.	PUNCT
cana-5363	109	1	10s	10	NOUN
cana-5363	109	2	(	(	PUNCT
cana-5363	109	3	2025	2025	NUM
cana-5363	109	4	)	)	PUNCT
cana-5363	109	5	2009	2009	NUM
cana-5363	109	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5363	109	7	1	1	X
cana-5363	109	8	.	.	PUNCT
cana-5363	110	1	𝛿	𝛿	DET
cana-5363	110	2	semi	semi	ADJ
cana-5363	110	3	interior	interior	NOUN
cana-5363	110	4	of	of	ADP
cana-5363	110	5	𝑆	𝑆	PROPN
cana-5363	110	6	(	(	PUNCT
cana-5363	110	7	briefly	briefly	ADV
cana-5363	110	8	,	,	PUNCT
cana-5363	110	9	𝒫ℱ𝒩𝛿𝒮𝑖𝑛𝑡(𝑆	𝒫ℱ𝒩𝛿𝒮𝑖𝑛𝑡(𝑆	PROPN
cana-5363	110	10	)	)	PUNCT
cana-5363	110	11	)	)	PUNCT
cana-5363	110	12	is	be	AUX
cana-5363	110	13	defined	define	VERB
cana-5363	110	14	by	by	ADP
cana-5363	110	15	𝒫ℱ𝒩𝛿𝒮𝑖𝑛𝑡(𝑆	𝒫ℱ𝒩𝛿𝒮𝑖𝑛𝑡(𝑆	PROPN
cana-5363	110	16	)	)	PUNCT
cana-5363	111	1	=	=	NOUN
cana-5363	111	2	∪	∪	X
cana-5363	111	3	{	{	PUNCT
cana-5363	111	4	𝐼	𝐼	NOUN
cana-5363	111	5	:	:	PUNCT
cana-5363	111	6	𝐼	𝐼	PROPN
cana-5363	111	7	⊆	⊆	NUM
cana-5363	111	8	𝑆	𝑆	PROPN
cana-5363	111	9	&	&	CCONJ
cana-5363	111	10	𝐼isa𝒫ℱ𝒩𝛿𝒮𝑜	𝐼isa𝒫ℱ𝒩𝛿𝒮𝑜	PROPN
cana-5363	111	11	set	set	VERB
cana-5363	111	12	in𝑈	in𝑈	NOUN
cana-5363	111	13	}	}	PUNCT
cana-5363	111	14	.	.	PUNCT
cana-5363	112	1	2	2	X
cana-5363	112	2	.	.	X
cana-5363	112	3	𝛿	𝛿	ADJ
cana-5363	112	4	semi	semi	ADJ
cana-5363	112	5	closure	closure	NOUN
cana-5363	112	6	of	of	ADP
cana-5363	112	7	𝑆	𝑆	PROPN
cana-5363	112	8	(	(	PUNCT
cana-5363	112	9	briefly	briefly	ADV
cana-5363	112	10	,	,	PUNCT
cana-5363	112	11	𝒫ℱ𝒩𝛿𝒮𝑐𝑙(𝑆	𝒫ℱ𝒩𝛿𝒮𝑐𝑙(𝑆	PROPN
cana-5363	112	12	)	)	PUNCT
cana-5363	112	13	)	)	PUNCT
cana-5363	112	14	is	be	AUX
cana-5363	112	15	defined	define	VERB
cana-5363	112	16	by	by	ADP
cana-5363	112	17	𝒫ℱ𝒩𝛿𝒮𝑐𝑙(𝑆	𝒫ℱ𝒩𝛿𝒮𝑐𝑙(𝑆	ADJ
cana-5363	112	18	)	)	PUNCT
cana-5363	112	19	=	=	NOUN
cana-5363	112	20	∩	∩	X
cana-5363	112	21	{	{	PUNCT
cana-5363	112	22	𝐴	𝐴	PROPN
cana-5363	112	23	:	:	PUNCT
cana-5363	112	24	𝑆	𝑆	PROPN
cana-5363	112	25	⊆	⊆	NUM
cana-5363	112	26	𝐴	𝐴	PROPN
cana-5363	112	27	&	&	CCONJ
cana-5363	112	28	𝐴isa𝒫ℱ𝒩𝛿𝒮𝑐set	𝐴isa𝒫ℱ𝒩𝛿𝒮𝑐set	NOUN
cana-5363	112	29	in	in	ADP
cana-5363	112	30	𝑈	𝑈	PROPN
cana-5363	112	31	}	}	PUNCT
cana-5363	112	32	.	.	PUNCT
cana-5363	113	1	3	3	X
cana-5363	113	2	.	.	X
cana-5363	113	3	pre	pre	ADJ
cana-5363	113	4	interior	interior	ADJ
cana-5363	113	5	of	of	ADP
cana-5363	113	6	𝑆	𝑆	PROPN
cana-5363	113	7	(	(	PUNCT
cana-5363	113	8	briefly	briefly	ADV
cana-5363	113	9	,	,	PUNCT
cana-5363	113	10	𝒫ℱ𝒩𝒫𝑖𝑛𝑡(𝑆	𝒫ℱ𝒩𝒫𝑖𝑛𝑡(𝑆	PROPN
cana-5363	113	11	)	)	PUNCT
cana-5363	113	12	)	)	PUNCT
cana-5363	113	13	is	be	AUX
cana-5363	113	14	defined	define	VERB
cana-5363	113	15	by	by	ADP
cana-5363	113	16	𝒫ℱ𝒩𝒫𝑖𝑛𝑡(𝑆	𝒫ℱ𝒩𝒫𝑖𝑛𝑡(𝑆	NOUN
cana-5363	113	17	)	)	PUNCT
cana-5363	113	18	=	=	NOUN
cana-5363	113	19	∪	∪	X
cana-5363	113	20	{	{	PUNCT
cana-5363	113	21	𝐼	𝐼	NOUN
cana-5363	113	22	:	:	PUNCT
cana-5363	113	23	𝐼	𝐼	PROPN
cana-5363	113	24	⊆	⊆	NUM
cana-5363	113	25	𝑆	𝑆	PROPN
cana-5363	113	26	&	&	CCONJ
cana-5363	113	27	𝐼isa𝒫ℱ𝒩𝒫𝑜	𝐼isa𝒫ℱ𝒩𝒫𝑜	PROPN
cana-5363	113	28	set	set	VERB
cana-5363	113	29	in𝑈	in𝑈	PROPN
cana-5363	113	30	}	}	PUNCT
cana-5363	113	31	.	.	PUNCT
cana-5363	114	1	4	4	X
cana-5363	114	2	.	.	X
cana-5363	114	3	pre	pre	VERB
cana-5363	114	4	closure	closure	NOUN
cana-5363	114	5	of	of	ADP
cana-5363	114	6	𝑆	𝑆	PROPN
cana-5363	114	7	(	(	PUNCT
cana-5363	114	8	briefly	briefly	ADV
cana-5363	114	9	,	,	PUNCT
cana-5363	114	10	𝒫ℱ𝒩𝒫𝑐𝑙(𝑆	𝒫ℱ𝒩𝒫𝑐𝑙(𝑆	PROPN
cana-5363	114	11	)	)	PUNCT
cana-5363	114	12	)	)	PUNCT
cana-5363	114	13	is	be	AUX
cana-5363	114	14	defined	define	VERB
cana-5363	114	15	by	by	ADP
cana-5363	114	16	𝒫ℱ𝒩𝒫𝑐𝑙(𝑆	𝒫ℱ𝒩𝒫𝑐𝑙(𝑆	PROPN
cana-5363	114	17	)	)	PUNCT
cana-5363	115	1	=	=	NOUN
cana-5363	115	2	∩	∩	NOUN
cana-5363	115	3	{	{	PUNCT
cana-5363	115	4	𝐴	𝐴	PROPN
cana-5363	115	5	:	:	PUNCT
cana-5363	115	6	𝑆	𝑆	PROPN
cana-5363	115	7	⊆	⊆	NUM
cana-5363	115	8	𝐴	𝐴	PROPN
cana-5363	115	9	&	&	CCONJ
cana-5363	115	10	𝐴isa𝒫ℱ𝒩𝒫𝑐set	𝐴isa𝒫ℱ𝒩𝒫𝑐set	PROPN
cana-5363	115	11	in	in	ADP
cana-5363	115	12	𝑈	𝑈	PROPN
cana-5363	115	13	}	}	PUNCT
cana-5363	115	14	.	.	PUNCT
cana-5363	116	1	definition	definition	NOUN
cana-5363	116	2	3.4	3.4	NUM
cana-5363	116	3	let	let	VERB
cana-5363	116	4	(	(	PUNCT
cana-5363	116	5	𝑈	𝑈	PROPN
cana-5363	116	6	,	,	PUNCT
cana-5363	116	7	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	116	8	)	)	PUNCT
cana-5363	116	9	)	)	PUNCT
cana-5363	116	10	be	be	AUX
cana-5363	116	11	a	a	DET
cana-5363	116	12	𝒫ℱ𝒩𝑡𝑠	𝒫ℱ𝒩𝑡𝑠	NOUN
cana-5363	116	13	with	with	ADP
cana-5363	116	14	respect	respect	NOUN
cana-5363	116	15	to	to	ADP
cana-5363	116	16	𝐴	𝐴	PROPN
cana-5363	116	17	where	where	SCONJ
cana-5363	116	18	𝐴	𝐴	PROPN
cana-5363	116	19	is	be	AUX
cana-5363	116	20	a	a	DET
cana-5363	116	21	𝑝𝑓𝑠	𝑝𝑓𝑠	NOUN
cana-5363	116	22	of	of	ADP
cana-5363	116	23	𝑈.	𝑈.	PROPN
cana-5363	116	24	then	then	ADV
cana-5363	116	25	a	a	DET
cana-5363	116	26	𝑝𝑓𝑠	𝑝𝑓𝑠	PROPN
cana-5363	116	27	𝑆	𝑆	PROPN
cana-5363	116	28	in	in	ADP
cana-5363	116	29	𝑈	𝑈	PROPN
cana-5363	116	30	is	be	AUX
cana-5363	116	31	said	say	VERB
cana-5363	116	32	to	to	PART
cana-5363	116	33	be	be	AUX
cana-5363	116	34	a	a	DET
cana-5363	116	35	pythagorean	pythagorean	ADJ
cana-5363	116	36	fuzzy	fuzzy	ADJ
cana-5363	116	37	nano	nano	NOUN
cana-5363	116	38	1	1	NUM
cana-5363	116	39	.	.	PUNCT
cana-5363	117	1	𝑍-open	𝑍-open	ADJ
cana-5363	117	2	(	(	PUNCT
cana-5363	117	3	briefly	briefly	ADV
cana-5363	117	4	,	,	PUNCT
cana-5363	117	5	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	PROPN
cana-5363	117	6	)	)	PUNCT
cana-5363	117	7	set	set	VERB
cana-5363	117	8	if	if	SCONJ
cana-5363	117	9	𝑆	𝑆	PROPN
cana-5363	117	10	⊆	⊆	NUM
cana-5363	117	11	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝑆	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝑆	PROPN
cana-5363	117	12	)	)	PUNCT
cana-5363	117	13	)	)	PUNCT
cana-5363	117	14	∩	∩	NOUN
cana-5363	117	15	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝑆	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝑆	ADJ
cana-5363	117	16	)	)	PUNCT
cana-5363	117	17	)	)	PUNCT
cana-5363	117	18	,	,	PUNCT
cana-5363	117	19	2	2	X
cana-5363	117	20	.	.	X
cana-5363	118	1	𝑍-closed	𝑍-close	VERB
cana-5363	118	2	(	(	PUNCT
cana-5363	118	3	briefly	briefly	ADV
cana-5363	118	4	,	,	PUNCT
cana-5363	118	5	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	PROPN
cana-5363	118	6	)	)	PUNCT
cana-5363	118	7	set	set	VERB
cana-5363	118	8	if	if	SCONJ
cana-5363	118	9	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝑆	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝑆	PROPN
cana-5363	118	10	)	)	PUNCT
cana-5363	118	11	)	)	PUNCT
cana-5363	118	12	∩	∩	X
cana-5363	118	13	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝑆	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝑆	NOUN
cana-5363	118	14	)	)	PUNCT
cana-5363	118	15	)	)	PUNCT
cana-5363	119	1	⊆	⊆	X
cana-5363	119	2	𝑆.	𝑆.	VERB
cana-5363	119	3	the	the	DET
cana-5363	119	4	family	family	NOUN
cana-5363	119	5	of	of	ADP
cana-5363	119	6	all	all	DET
cana-5363	119	7	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	119	8	(	(	PUNCT
cana-5363	119	9	resp	resp	NOUN
cana-5363	119	10	.	.	PUNCT
cana-5363	120	1	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	ADJ
cana-5363	120	2	)	)	PUNCT
cana-5363	120	3	sets	set	NOUN
cana-5363	120	4	of	of	ADP
cana-5363	120	5	a	a	DET
cana-5363	120	6	space	space	NOUN
cana-5363	120	7	(	(	PUNCT
cana-5363	120	8	𝑈	𝑈	PROPN
cana-5363	120	9	,	,	PUNCT
cana-5363	120	10	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	120	11	)	)	PUNCT
cana-5363	120	12	)	)	PUNCT
cana-5363	120	13	will	will	AUX
cana-5363	120	14	be	be	AUX
cana-5363	120	15	as	as	SCONJ
cana-5363	120	16	always	always	ADV
cana-5363	120	17	denoted	denote	VERB
cana-5363	120	18	by	by	ADP
cana-5363	120	19	𝒫ℱ𝔑𝑍𝑂(𝑈	𝒫ℱ𝔑𝑍𝑂(𝑈	NUM
cana-5363	120	20	,	,	PUNCT
cana-5363	120	21	𝐴	𝐴	PROPN
cana-5363	120	22	)	)	PUNCT
cana-5363	120	23	(	(	PUNCT
cana-5363	120	24	resp	resp	NOUN
cana-5363	120	25	.	.	PUNCT
cana-5363	121	1	𝒫ℱ𝔑𝑍𝐶(𝑈	𝒫ℱ𝔑𝑍𝐶(𝑈	NUM
cana-5363	121	2	,	,	PUNCT
cana-5363	121	3	𝐴	𝐴	PROPN
cana-5363	121	4	)	)	PUNCT
cana-5363	121	5	)	)	PUNCT
cana-5363	121	6	.	.	PUNCT
cana-5363	122	1	definition	definition	NOUN
cana-5363	122	2	3.5	3.5	NUM
cana-5363	122	3	let	let	VERB
cana-5363	122	4	(	(	PUNCT
cana-5363	122	5	𝑈	𝑈	PROPN
cana-5363	122	6	,	,	PUNCT
cana-5363	122	7	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	122	8	)	)	PUNCT
cana-5363	122	9	)	)	PUNCT
cana-5363	122	10	be	be	AUX
cana-5363	122	11	a	a	DET
cana-5363	122	12	𝒫ℱ𝒩𝑡𝑠	𝒫ℱ𝒩𝑡𝑠	NOUN
cana-5363	122	13	with	with	ADP
cana-5363	122	14	respect	respect	NOUN
cana-5363	122	15	to	to	ADP
cana-5363	122	16	𝐴	𝐴	PROPN
cana-5363	122	17	where	where	SCONJ
cana-5363	122	18	𝐴	𝐴	PROPN
cana-5363	122	19	is	be	AUX
cana-5363	122	20	a	a	DET
cana-5363	122	21	𝑝𝑓𝑠	𝑝𝑓𝑠	NOUN
cana-5363	122	22	of	of	ADP
cana-5363	122	23	𝑈.	𝑈.	PROPN
cana-5363	122	24	then	then	ADV
cana-5363	122	25	a	a	DET
cana-5363	122	26	𝑝𝑓𝑠	𝑝𝑓𝑠	PROPN
cana-5363	122	27	𝐾	𝐾	PROPN
cana-5363	122	28	in	in	ADP
cana-5363	122	29	𝑈	𝑈	PROPN
cana-5363	122	30	,	,	PUNCT
cana-5363	122	31	then	then	ADV
cana-5363	122	32	the	the	DET
cana-5363	122	33	pythagorean	pythagorean	PROPN
cana-5363	122	34	1	1	NUM
cana-5363	122	35	.	.	PUNCT
cana-5363	123	1	fuzzy	fuzzy	ADJ
cana-5363	123	2	nano	nano	NOUN
cana-5363	123	3	𝑍-interior	𝑍-interior	PROPN
cana-5363	123	4	of	of	ADP
cana-5363	123	5	𝐾	𝐾	PROPN
cana-5363	123	6	is	be	AUX
cana-5363	123	7	the	the	DET
cana-5363	123	8	union	union	NOUN
cana-5363	123	9	of	of	ADP
cana-5363	123	10	all	all	DET
cana-5363	123	11	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	123	12	sets	set	NOUN
cana-5363	123	13	contained	contain	VERB
cana-5363	123	14	in	in	ADP
cana-5363	123	15	𝐾	𝐾	PROPN
cana-5363	123	16	and	and	CCONJ
cana-5363	123	17	denoted	denote	VERB
cana-5363	123	18	by	by	ADP
cana-5363	123	19	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	PROPN
cana-5363	123	20	)	)	PUNCT
cana-5363	123	21	.	.	PUNCT
cana-5363	124	1	2	2	X
cana-5363	124	2	.	.	X
cana-5363	124	3	fuzzy	fuzzy	ADJ
cana-5363	124	4	nano	nano	PROPN
cana-5363	124	5	𝑍-closure	𝑍-closure	PROPN
cana-5363	124	6	of	of	ADP
cana-5363	124	7	𝐾	𝐾	PROPN
cana-5363	124	8	is	be	AUX
cana-5363	124	9	the	the	DET
cana-5363	124	10	intersection	intersection	NOUN
cana-5363	124	11	of	of	ADP
cana-5363	124	12	all	all	DET
cana-5363	124	13	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	ADJ
cana-5363	124	14	sets	set	NOUN
cana-5363	124	15	containing	contain	VERB
cana-5363	124	16	𝐾	𝐾	PROPN
cana-5363	124	17	and	and	CCONJ
cana-5363	124	18	denoted	denote	VERB
cana-5363	124	19	by	by	ADP
cana-5363	124	20	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	124	21	)	)	PUNCT
cana-5363	124	22	.	.	PUNCT
cana-5363	125	1	remark	remark	PROPN
cana-5363	125	2	3.1	3.1	NUM
cana-5363	125	3	let	let	VERB
cana-5363	125	4	𝐾	𝐾	PRON
cana-5363	125	5	be	be	AUX
cana-5363	125	6	a	a	DET
cana-5363	125	7	subset	subset	NOUN
cana-5363	125	8	of	of	ADP
cana-5363	125	9	a	a	DET
cana-5363	125	10	𝒫ℱ𝒩𝑡𝑠	𝒫ℱ𝒩𝑡𝑠	PROPN
cana-5363	125	11	(	(	PUNCT
cana-5363	125	12	𝑈	𝑈	PROPN
cana-5363	125	13	,	,	PUNCT
cana-5363	125	14	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	125	15	)	)	PUNCT
cana-5363	125	16	)	)	PUNCT
cana-5363	125	17	.	.	PUNCT
cana-5363	126	1	then	then	ADV
cana-5363	126	2	(	(	PUNCT
cana-5363	126	3	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾))𝑐	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾))𝑐	PROPN
cana-5363	126	4	=	=	SYM
cana-5363	126	5	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾𝑐	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾𝑐	PROPN
cana-5363	126	6	)	)	PUNCT
cana-5363	126	7	,	,	PUNCT
cana-5363	126	8	(	(	PUNCT
cana-5363	126	9	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾))𝑐	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾))𝑐	X
cana-5363	126	10	=	=	SYM
cana-5363	126	11	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾𝑐	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾𝑐	PROPN
cana-5363	126	12	)	)	PUNCT
cana-5363	126	13	.	.	PUNCT
cana-5363	127	1	theorem	theorem	VERB
cana-5363	127	2	3.1	3.1	NUM
cana-5363	127	3	let	let	NOUN
cana-5363	127	4	(	(	PUNCT
cana-5363	127	5	𝑈	𝑈	PROPN
cana-5363	127	6	,	,	PUNCT
cana-5363	127	7	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	127	8	)	)	PUNCT
cana-5363	127	9	)	)	PUNCT
cana-5363	127	10	be	be	AUX
cana-5363	127	11	a	a	DET
cana-5363	127	12	𝒫ℱ𝒩𝑡𝑠.	𝒫ℱ𝒩𝑡𝑠.	NOUN
cana-5363	127	13	then	then	ADV
cana-5363	127	14	,	,	PUNCT
cana-5363	127	15	(	(	PUNCT
cana-5363	127	16	i	i	NOUN
cana-5363	127	17	)	)	PUNCT
cana-5363	127	18	every	every	DET
cana-5363	127	19	𝒫ℱ𝔑𝛿𝑜	𝒫ℱ𝔑𝛿𝑜	PROPN
cana-5363	127	20	set	set	VERB
cana-5363	127	21	is	be	AUX
cana-5363	127	22	𝒫ℱ𝔑𝑜	𝒫ℱ𝔑𝑜	PROPN
cana-5363	127	23	set	set	NOUN
cana-5363	127	24	.	.	PUNCT
cana-5363	128	1	(	(	PUNCT
cana-5363	128	2	ii	ii	NOUN
cana-5363	128	3	)	)	PUNCT
cana-5363	128	4	every	every	DET
cana-5363	128	5	𝒫ℱ𝔑𝑜	𝒫ℱ𝔑𝑜	NOUN
cana-5363	128	6	set	set	VERB
cana-5363	128	7	is	be	AUX
cana-5363	128	8	𝒫ℱ𝔑𝒫𝑜	𝒫ℱ𝔑𝒫𝑜	NOUN
cana-5363	128	9	set	set	NOUN
cana-5363	128	10	.	.	PUNCT
cana-5363	129	1	(	(	PUNCT
cana-5363	129	2	iii	iii	X
cana-5363	129	3	)	)	PUNCT
cana-5363	129	4	every	every	DET
cana-5363	129	5	𝒫ℱ𝔑𝛿𝑜	𝒫ℱ𝔑𝛿𝑜	PROPN
cana-5363	129	6	set	set	VERB
cana-5363	129	7	is	be	AUX
cana-5363	129	8	𝒫ℱ𝔑𝛿𝒮𝑜	𝒫ℱ𝔑𝛿𝒮𝑜	PROPN
cana-5363	129	9	set	set	NOUN
cana-5363	129	10	.	.	PUNCT
cana-5363	130	1	(	(	PUNCT
cana-5363	130	2	iv	iv	X
cana-5363	130	3	)	)	PUNCT
cana-5363	130	4	every	every	DET
cana-5363	130	5	𝒫ℱ𝔑𝛿𝒮𝑜	𝒫ℱ𝔑𝛿𝒮𝑜	NOUN
cana-5363	130	6	set	set	NOUN
cana-5363	130	7	is	be	AUX
cana-5363	130	8	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	130	9	set	set	NOUN
cana-5363	130	10	.	.	PUNCT
cana-5363	131	1	(	(	PUNCT
cana-5363	131	2	v	v	NOUN
cana-5363	131	3	)	)	PUNCT
cana-5363	131	4	every	every	DET
cana-5363	131	5	𝒫ℱ𝔑𝒫𝑜	𝒫ℱ𝔑𝒫𝑜	NOUN
cana-5363	131	6	set	set	NOUN
cana-5363	131	7	is	be	AUX
cana-5363	131	8	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	131	9	set	set	NOUN
cana-5363	131	10	.	.	PUNCT
cana-5363	132	1	proof	proof	NOUN
cana-5363	132	2	.	.	PUNCT
cana-5363	133	1	(	(	PUNCT
cana-5363	133	2	i	i	NOUN
cana-5363	133	3	)	)	PUNCT
cana-5363	133	4	if	if	SCONJ
cana-5363	133	5	𝐾	𝐾	PROPN
cana-5363	133	6	is	be	AUX
cana-5363	133	7	a	a	DET
cana-5363	133	8	𝒫ℱ𝔑𝛿𝑜𝑠	𝒫ℱ𝔑𝛿𝑜𝑠	PROPN
cana-5363	133	9	in	in	ADP
cana-5363	133	10	𝑈	𝑈	PROPN
cana-5363	133	11	,	,	PUNCT
cana-5363	133	12	then	then	ADV
cana-5363	133	13	𝐾	𝐾	PROPN
cana-5363	133	14	=	=	PUNCT
cana-5363	133	15	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	X
cana-5363	133	16	)	)	PUNCT
cana-5363	133	17	⊆	⊆	NUM
cana-5363	133	18	𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	NOUN
cana-5363	133	19	)	)	PUNCT
cana-5363	133	20	.	.	PUNCT
cana-5363	134	1	therefore	therefore	ADV
cana-5363	134	2	,	,	PUNCT
cana-5363	134	3	𝐾	𝐾	PROPN
cana-5363	134	4	is	be	AUX
cana-5363	134	5	a	a	DET
cana-5363	134	6	𝒫ℱ𝔑𝑜𝑠.	𝒫ℱ𝔑𝑜𝑠.	PROPN
cana-5363	134	7	(	(	PUNCT
cana-5363	134	8	ii	ii	NOUN
cana-5363	134	9	)	)	PUNCT
cana-5363	134	10	if	if	SCONJ
cana-5363	134	11	𝐾	𝐾	PROPN
cana-5363	134	12	is	be	AUX
cana-5363	134	13	a	a	DET
cana-5363	134	14	𝒫ℱ𝔑𝑜𝑠	𝒫ℱ𝔑𝑜𝑠	PROPN
cana-5363	134	15	in	in	ADP
cana-5363	134	16	𝑈	𝑈	PROPN
cana-5363	134	17	,	,	PUNCT
cana-5363	134	18	then	then	ADV
cana-5363	134	19	𝐾	𝐾	PROPN
cana-5363	134	20	=	=	PUNCT
cana-5363	134	21	𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	PROPN
cana-5363	134	22	)	)	PUNCT
cana-5363	134	23	.	.	PUNCT
cana-5363	135	1	so	so	ADV
cana-5363	135	2	,	,	PUNCT
cana-5363	135	3	𝐾	𝐾	PROPN
cana-5363	135	4	=	=	SYM
cana-5363	135	5	𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	PROPN
cana-5363	135	6	)	)	PUNCT
cana-5363	135	7	⊆	⊆	NUM
cana-5363	135	8	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	NUM
cana-5363	135	9	)	)	PUNCT
cana-5363	135	10	)	)	PUNCT
cana-5363	135	11	.	.	PUNCT
cana-5363	136	1	therefore	therefore	ADV
cana-5363	136	2	,	,	PUNCT
cana-5363	136	3	𝐾	𝐾	PROPN
cana-5363	136	4	is	be	AUX
cana-5363	136	5	a	a	DET
cana-5363	136	6	𝒫ℱ𝔑𝒫𝑜𝑠.	𝒫ℱ𝔑𝒫𝑜𝑠.	ADJ
cana-5363	136	7	(	(	PUNCT
cana-5363	136	8	iii	iii	NOUN
cana-5363	136	9	)	)	PUNCT
cana-5363	136	10	if	if	SCONJ
cana-5363	136	11	𝐾	𝐾	PROPN
cana-5363	136	12	is	be	AUX
cana-5363	136	13	a	a	DET
cana-5363	136	14	𝒫ℱ𝔑𝛿𝑜𝑠	𝒫ℱ𝔑𝛿𝑜𝑠	PROPN
cana-5363	136	15	in	in	ADP
cana-5363	136	16	𝑈	𝑈	PROPN
cana-5363	136	17	,	,	PUNCT
cana-5363	136	18	then	then	ADV
cana-5363	136	19	𝐾	𝐾	PROPN
cana-5363	136	20	is	be	AUX
cana-5363	136	21	𝒫ℱ𝔑𝑜𝑠	𝒫ℱ𝔑𝑜𝑠	NUM
cana-5363	136	22	by	by	ADP
cana-5363	136	23	(	(	PUNCT
cana-5363	136	24	𝑖	𝑖	X
cana-5363	136	25	)	)	PUNCT
cana-5363	136	26	.	.	PUNCT
cana-5363	137	1	so	so	ADV
cana-5363	137	2	,	,	PUNCT
cana-5363	137	3	𝐾	𝐾	PROPN
cana-5363	137	4	⊆	⊆	NUM
cana-5363	137	5	𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	NOUN
cana-5363	137	6	)	)	PUNCT
cana-5363	137	7	⊆	⊆	NUM
cana-5363	137	8	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	NOUN
cana-5363	137	9	)	)	PUNCT
cana-5363	137	10	)	)	PUNCT
cana-5363	137	11	.	.	PUNCT
cana-5363	138	1	therefore	therefore	ADV
cana-5363	138	2	,	,	PUNCT
cana-5363	138	3	𝐾	𝐾	PROPN
cana-5363	138	4	is	be	AUX
cana-5363	138	5	a	a	DET
cana-5363	138	6	𝒫ℱ𝔑𝛿𝒫𝑜𝑠.	𝒫ℱ𝔑𝛿𝒫𝑜𝑠.	PROPN
cana-5363	138	7	communications	communication	NOUN
cana-5363	138	8	on	on	ADP
cana-5363	138	9	applied	apply	VERB
cana-5363	138	10	nonlinear	nonlinear	ADJ
cana-5363	138	11	analysis	analysis	NOUN
cana-5363	138	12	issn	issn	NOUN
cana-5363	138	13	:	:	PUNCT
cana-5363	138	14	1074	1074	NUM
cana-5363	138	15	-	-	PUNCT
cana-5363	138	16	133x	133x	NUM
cana-5363	138	17	vol	vol	VERB
cana-5363	138	18	32	32	NUM
cana-5363	138	19	no	no	NOUN
cana-5363	138	20	.	.	PUNCT
cana-5363	139	1	10s	10	NOUN
cana-5363	139	2	(	(	PUNCT
cana-5363	139	3	2025	2025	NUM
cana-5363	139	4	)	)	PUNCT
cana-5363	139	5	2010	2010	NUM
cana-5363	139	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5363	139	7	(	(	PUNCT
cana-5363	139	8	iv	iv	X
cana-5363	139	9	)	)	PUNCT
cana-5363	139	10	𝐾	𝐾	NOUN
cana-5363	139	11	is	be	AUX
cana-5363	139	12	a	a	DET
cana-5363	139	13	𝒫ℱ𝔑𝛿𝒮𝑜𝑠	𝒫ℱ𝔑𝛿𝒮𝑜𝑠	PROPN
cana-5363	139	14	,	,	PUNCT
cana-5363	139	15	then	then	ADV
cana-5363	139	16	𝐾	𝐾	PROPN
cana-5363	139	17	⊆	⊆	NUM
cana-5363	139	18	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	NUM
cana-5363	139	19	)	)	PUNCT
cana-5363	139	20	)	)	PUNCT
cana-5363	140	1	and	and	CCONJ
cana-5363	140	2	so	so	ADV
cana-5363	140	3	𝐾	𝐾	PROPN
cana-5363	140	4	⊆	⊆	NUM
cana-5363	140	5	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	NUM
cana-5363	140	6	)	)	PUNCT
cana-5363	140	7	)	)	PUNCT
cana-5363	141	1	⊆	⊆	NUM
cana-5363	141	2	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	NOUN
cana-5363	141	3	)	)	PUNCT
cana-5363	141	4	)	)	PUNCT
cana-5363	141	5	∪	∪	ADP
cana-5363	141	6	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	NOUN
cana-5363	141	7	)	)	PUNCT
cana-5363	141	8	)	)	PUNCT
cana-5363	141	9	.	.	PUNCT
cana-5363	142	1	∴	∴	PROPN
cana-5363	142	2	𝐾	𝐾	PROPN
cana-5363	142	3	is	be	AUX
cana-5363	142	4	a	a	DET
cana-5363	142	5	𝒫ℱ𝔑𝑍𝑜𝑠.	𝒫ℱ𝔑𝑍𝑜𝑠.	NOUN
cana-5363	142	6	(	(	PUNCT
cana-5363	142	7	v	v	NOUN
cana-5363	142	8	)	)	PUNCT
cana-5363	142	9	𝐾	𝐾	NOUN
cana-5363	142	10	is	be	AUX
cana-5363	142	11	a	a	DET
cana-5363	142	12	𝒫ℱ𝔑𝒫𝑜𝑠	𝒫ℱ𝔑𝒫𝑜𝑠	PROPN
cana-5363	142	13	,	,	PUNCT
cana-5363	142	14	then	then	ADV
cana-5363	142	15	𝐾	𝐾	PROPN
cana-5363	142	16	⊆	⊆	NUM
cana-5363	142	17	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	NUM
cana-5363	142	18	)	)	PUNCT
cana-5363	142	19	)	)	PUNCT
cana-5363	143	1	and	and	CCONJ
cana-5363	143	2	so	so	ADV
cana-5363	143	3	𝐾	𝐾	PROPN
cana-5363	143	4	⊆	⊆	NUM
cana-5363	143	5	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	NUM
cana-5363	143	6	)	)	PUNCT
cana-5363	143	7	)	)	PUNCT
cana-5363	144	1	⊆	⊆	NUM
cana-5363	144	2	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	NOUN
cana-5363	144	3	)	)	PUNCT
cana-5363	144	4	)	)	PUNCT
cana-5363	144	5	∪	∪	ADP
cana-5363	144	6	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	PROPN
cana-5363	144	7	)	)	PUNCT
cana-5363	144	8	)	)	PUNCT
cana-5363	144	9	.	.	PUNCT
cana-5363	145	1	∴	∴	PROPN
cana-5363	145	2	𝐾	𝐾	PROPN
cana-5363	145	3	is	be	AUX
cana-5363	145	4	a	a	DET
cana-5363	145	5	𝒫ℱ𝔑𝑍𝑜𝑠.	𝒫ℱ𝔑𝑍𝑜𝑠.	NOUN
cana-5363	145	6	the	the	DET
cana-5363	145	7	converse	converse	NOUN
cana-5363	145	8	of	of	ADP
cana-5363	145	9	the	the	DET
cana-5363	145	10	above	above	ADJ
cana-5363	145	11	propositions	proposition	NOUN
cana-5363	145	12	need	need	VERB
cana-5363	145	13	not	not	PART
cana-5363	145	14	to	to	PART
cana-5363	145	15	be	be	AUX
cana-5363	145	16	true	true	ADJ
cana-5363	145	17	.	.	PUNCT
cana-5363	146	1	the	the	DET
cana-5363	146	2	following	follow	VERB
cana-5363	146	3	examples	example	NOUN
cana-5363	146	4	show	show	VERB
cana-5363	146	5	it	it	PRON
cana-5363	146	6	.	.	PUNCT
cana-5363	147	1	example	example	NOUN
cana-5363	147	2	3.1	3.1	NUM
cana-5363	147	3	assume	assume	NOUN
cana-5363	147	4	𝑈	𝑈	PROPN
cana-5363	147	5	=	=	SYM
cana-5363	147	6	{	{	PUNCT
cana-5363	147	7	𝑠1	𝑠1	PROPN
cana-5363	147	8	,	,	PUNCT
cana-5363	147	9	𝑠2	𝑠2	NOUN
cana-5363	147	10	,	,	PUNCT
cana-5363	147	11	𝑠3	𝑠3	NOUN
cana-5363	147	12	,	,	PUNCT
cana-5363	147	13	𝑠4	𝑠4	PROPN
cana-5363	147	14	}	}	PUNCT
cana-5363	147	15	be	be	VERB
cana-5363	147	16	the	the	DET
cana-5363	147	17	universe	universe	NOUN
cana-5363	147	18	set	set	VERB
cana-5363	147	19	and	and	CCONJ
cana-5363	147	20	the	the	DET
cana-5363	147	21	equivalence	equivalence	NOUN
cana-5363	147	22	relation	relation	NOUN
cana-5363	147	23	is	be	AUX
cana-5363	147	24	𝑈/𝑅	𝑈/𝑅	ADJ
cana-5363	147	25	=	=	SYM
cana-5363	147	26	{	{	PUNCT
cana-5363	147	27	{	{	PUNCT
cana-5363	147	28	𝑠1	𝑠1	PROPN
cana-5363	147	29	,	,	PUNCT
cana-5363	147	30	𝑠4	𝑠4	PROPN
cana-5363	147	31	}	}	PUNCT
cana-5363	147	32	,	,	PUNCT
cana-5363	147	33	{	{	PUNCT
cana-5363	147	34	𝑠2	𝑠2	NOUN
cana-5363	147	35	}	}	PUNCT
cana-5363	147	36	,	,	PUNCT
cana-5363	147	37	{	{	PUNCT
cana-5363	147	38	𝑠3	𝑠3	NOUN
cana-5363	147	39	}	}	PUNCT
cana-5363	147	40	}	}	PUNCT
cana-5363	147	41	.	.	PUNCT
cana-5363	148	1	let	let	VERB
cana-5363	148	2	𝐴	𝐴	PROPN
cana-5363	148	3	=	=	PUNCT
cana-5363	148	4	{	{	PUNCT
cana-5363	148	5	⟨	⟨	X
cana-5363	148	6	𝑠1	𝑠1	PROPN
cana-5363	148	7	0.3,0.1	0.3,0.1	PROPN
cana-5363	148	8	⟩	⟩	NOUN
cana-5363	148	9	,	,	PUNCT
cana-5363	148	10	⟨	⟨	VERB
cana-5363	148	11	𝑠2	𝑠2	NOUN
cana-5363	148	12	0.1,0.5	0.1,0.5	PROPN
cana-5363	148	13	⟩	⟩	NOUN
cana-5363	148	14	,	,	PUNCT
cana-5363	148	15	⟨	⟨	VERB
cana-5363	148	16	𝑠3	𝑠3	PROPN
cana-5363	148	17	0.2,0.45	0.2,0.45	NUM
cana-5363	148	18	⟩	⟩	NOUN
cana-5363	148	19	,	,	PUNCT
cana-5363	148	20	⟨	⟨	VERB
cana-5363	148	21	𝑠4	𝑠4	PROPN
cana-5363	148	22	0.4,0.25	0.4,0.25	NUM
cana-5363	148	23	⟩	⟩	NOUN
cana-5363	148	24	}	}	PUNCT
cana-5363	148	25	be	be	AUX
cana-5363	148	26	a	a	DET
cana-5363	148	27	𝑝𝑓𝑠	𝑝𝑓𝑠	NOUN
cana-5363	148	28	of	of	ADP
cana-5363	148	29	𝑈.	𝑈.	PROPN
cana-5363	148	30	𝒫ℱ𝔑(𝐴	𝒫ℱ𝔑(𝐴	PROPN
cana-5363	148	31	)	)	PUNCT
cana-5363	148	32	=	=	NOUN
cana-5363	148	33	{	{	PUNCT
cana-5363	148	34	⟨	⟨	ADP
cana-5363	148	35	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	148	36	0.3,0.25	0.3,0.25	NUM
cana-5363	148	37	⟩	⟩	NOUN
cana-5363	148	38	,	,	PUNCT
cana-5363	148	39	⟨	⟨	VERB
cana-5363	148	40	𝑠2	𝑠2	NOUN
cana-5363	148	41	0.1,0.5	0.1,0.5	PROPN
cana-5363	148	42	⟩	⟩	NOUN
cana-5363	148	43	,	,	PUNCT
cana-5363	148	44	⟨	⟨	VERB
cana-5363	148	45	𝑠3	𝑠3	PROPN
cana-5363	148	46	0.2,0.45	0.2,0.45	NUM
cana-5363	148	47	⟩	⟩	NOUN
cana-5363	148	48	}	}	PUNCT
cana-5363	148	49	,	,	PUNCT
cana-5363	148	50	𝒫ℱ𝔑(𝐴	𝒫ℱ𝔑(𝐴	INTJ
cana-5363	148	51	)	)	PUNCT
cana-5363	148	52	=	=	NOUN
cana-5363	148	53	{	{	PUNCT
cana-5363	148	54	⟨	⟨	ADP
cana-5363	148	55	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	148	56	0.4,0.1	0.4,0.1	PROPN
cana-5363	148	57	⟩	⟩	NOUN
cana-5363	148	58	,	,	PUNCT
cana-5363	148	59	⟨	⟨	VERB
cana-5363	148	60	𝑠2	𝑠2	NOUN
cana-5363	148	61	0.1,0.5	0.1,0.5	PROPN
cana-5363	148	62	⟩	⟩	NOUN
cana-5363	148	63	,	,	PUNCT
cana-5363	148	64	⟨	⟨	VERB
cana-5363	148	65	𝑠3	𝑠3	PROPN
cana-5363	148	66	0.2,0.45	0.2,0.45	PUNCT
cana-5363	148	67	⟩	⟩	NOUN
cana-5363	148	68	}	}	PUNCT
cana-5363	148	69	,	,	PUNCT
cana-5363	148	70	𝐵𝒫ℱ𝔑(𝐴	𝐵𝒫ℱ𝔑(𝐴	PROPN
cana-5363	148	71	)	)	PUNCT
cana-5363	149	1	=	=	NOUN
cana-5363	149	2	{	{	PUNCT
cana-5363	149	3	⟨	⟨	ADP
cana-5363	149	4	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	149	5	0.25,0.3	0.25,0.3	NOUN
cana-5363	149	6	⟩	⟩	NOUN
cana-5363	149	7	,	,	PUNCT
cana-5363	149	8	⟨	⟨	VERB
cana-5363	149	9	𝑠2	𝑠2	NOUN
cana-5363	149	10	0.1,0.5	0.1,0.5	PROPN
cana-5363	149	11	⟩	⟩	NOUN
cana-5363	149	12	,	,	PUNCT
cana-5363	149	13	⟨	⟨	VERB
cana-5363	149	14	𝑠3	𝑠3	PROPN
cana-5363	149	15	0.2,0.45	0.2,0.45	NUM
cana-5363	149	16	⟩	⟩	NOUN
cana-5363	149	17	}	}	PUNCT
cana-5363	149	18	.	.	PUNCT
cana-5363	150	1	thus	thus	ADV
cana-5363	150	2	𝜏ℛ(𝐴	𝜏ℛ(𝐴	VERB
cana-5363	150	3	)	)	PUNCT
cana-5363	150	4	=	=	PRON
cana-5363	150	5	{	{	PUNCT
cana-5363	150	6	0𝒫	0𝒫	NOUN
cana-5363	150	7	,	,	PUNCT
cana-5363	150	8	1𝒫	1𝒫	INTJ
cana-5363	150	9	,	,	PUNCT
cana-5363	150	10	𝒫ℱ𝔑(𝐴	𝒫ℱ𝔑(𝐴	NOUN
cana-5363	150	11	)	)	PUNCT
cana-5363	150	12	,	,	PUNCT
cana-5363	150	13	𝒫ℱ𝔑(𝐴	𝒫ℱ𝔑(𝐴	NOUN
cana-5363	150	14	)	)	PUNCT
cana-5363	150	15	,	,	PUNCT
cana-5363	150	16	𝐵𝒫ℱ𝔑(𝐴	𝐵𝒫ℱ𝔑(𝐴	PROPN
cana-5363	150	17	)	)	PUNCT
cana-5363	150	18	}	}	PUNCT
cana-5363	150	19	.	.	PUNCT
cana-5363	151	1	then	then	ADV
cana-5363	151	2	1	1	X
cana-5363	151	3	.	.	PUNCT
cana-5363	151	4	{	{	PUNCT
cana-5363	151	5	⟨	⟨	ADP
cana-5363	151	6	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	151	7	0.4,0.1	0.4,0.1	PROPN
cana-5363	151	8	⟩	⟩	NOUN
cana-5363	151	9	,	,	PUNCT
cana-5363	151	10	⟨	⟨	VERB
cana-5363	151	11	𝑠2	𝑠2	NOUN
cana-5363	151	12	0.1,0.5	0.1,0.5	PROPN
cana-5363	151	13	⟩	⟩	NOUN
cana-5363	151	14	,	,	PUNCT
cana-5363	151	15	⟨	⟨	VERB
cana-5363	151	16	𝑠3	𝑠3	PROPN
cana-5363	151	17	0.2,0.45	0.2,0.45	PUNCT
cana-5363	151	18	⟩	⟩	NOUN
cana-5363	151	19	}	}	PUNCT
cana-5363	151	20	is	be	AUX
cana-5363	151	21	a	a	DET
cana-5363	151	22	𝒫ℱ𝔑𝑜	𝒫ℱ𝔑𝑜	PROPN
cana-5363	151	23	(	(	PUNCT
cana-5363	151	24	resp	resp	NOUN
cana-5363	151	25	.	.	PUNCT
cana-5363	152	1	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	152	2	)	)	PUNCT
cana-5363	152	3	set	set	VERB
cana-5363	152	4	but	but	CCONJ
cana-5363	152	5	not	not	PART
cana-5363	152	6	𝒫ℱ𝔑𝛿𝑜	𝒫ℱ𝔑𝛿𝑜	PROPN
cana-5363	152	7	(	(	PUNCT
cana-5363	152	8	resp	resp	NOUN
cana-5363	152	9	.	.	PUNCT
cana-5363	153	1	𝒫ℱ𝔑𝛿𝒮𝑜	𝒫ℱ𝔑𝛿𝒮𝑜	NOUN
cana-5363	153	2	)	)	PUNCT
cana-5363	153	3	set	set	NOUN
cana-5363	153	4	.	.	PUNCT
cana-5363	154	1	2	2	X
cana-5363	154	2	.	.	X
cana-5363	154	3	{	{	PUNCT
cana-5363	154	4	⟨	⟨	ADP
cana-5363	154	5	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	154	6	0.25,0.3	0.25,0.3	NOUN
cana-5363	154	7	⟩	⟩	NOUN
cana-5363	154	8	,	,	PUNCT
cana-5363	154	9	⟨	⟨	VERB
cana-5363	154	10	𝑠2	𝑠2	NOUN
cana-5363	154	11	0.5,0.1	0.5,0.1	PROPN
cana-5363	154	12	⟩	⟩	NOUN
cana-5363	154	13	,	,	PUNCT
cana-5363	154	14	⟨	⟨	VERB
cana-5363	154	15	𝑠3	𝑠3	PROPN
cana-5363	154	16	0.45,0.2	0.45,0.2	NOUN
cana-5363	155	1	⟩	⟩	NOUN
cana-5363	155	2	}	}	PUNCT
cana-5363	155	3	is	be	AUX
cana-5363	155	4	a	a	DET
cana-5363	155	5	𝒫ℱ𝔑𝛿𝒮𝑜	𝒫ℱ𝔑𝛿𝒮𝑜	PROPN
cana-5363	155	6	(	(	PUNCT
cana-5363	155	7	resp	resp	NOUN
cana-5363	155	8	.	.	PUNCT
cana-5363	156	1	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	156	2	)	)	PUNCT
cana-5363	156	3	set	set	VERB
cana-5363	156	4	but	but	CCONJ
cana-5363	156	5	not	not	PART
cana-5363	156	6	𝒫ℱ𝔑𝛿𝑜	𝒫ℱ𝔑𝛿𝑜	PROPN
cana-5363	156	7	(	(	PUNCT
cana-5363	156	8	resp	resp	NOUN
cana-5363	156	9	.	.	PUNCT
cana-5363	157	1	𝒫ℱ𝔑𝒫𝑜	𝒫ℱ𝔑𝒫𝑜	NOUN
cana-5363	157	2	)	)	PUNCT
cana-5363	157	3	set	set	NOUN
cana-5363	157	4	.	.	PUNCT
cana-5363	158	1	3	3	X
cana-5363	158	2	.	.	PUNCT
cana-5363	158	3	{	{	PUNCT
cana-5363	158	4	⟨	⟨	ADP
cana-5363	158	5	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	158	6	0.45,0.35	0.45,0.35	NOUN
cana-5363	158	7	⟩	⟩	NOUN
cana-5363	158	8	,	,	PUNCT
cana-5363	158	9	⟨	⟨	VERB
cana-5363	158	10	𝑠2	𝑠2	PROPN
cana-5363	158	11	0.25,0.45	0.25,0.45	ADJ
cana-5363	158	12	⟩	⟩	NOUN
cana-5363	158	13	,	,	PUNCT
cana-5363	158	14	⟨	⟨	VERB
cana-5363	158	15	𝑠3	𝑠3	PROPN
cana-5363	159	1	0.3,0.2	0.3,0.2	PROPN
cana-5363	159	2	⟩	⟩	NOUN
cana-5363	159	3	}	}	PUNCT
cana-5363	159	4	is	be	AUX
cana-5363	159	5	a	a	DET
cana-5363	159	6	𝒫ℱ𝔑𝒫𝑜	𝒫ℱ𝔑𝒫𝑜	NOUN
cana-5363	159	7	set	set	NOUN
cana-5363	159	8	but	but	CCONJ
cana-5363	159	9	not	not	PART
cana-5363	159	10	𝒫ℱ𝔑𝑜	𝒫ℱ𝔑𝑜	PROPN
cana-5363	159	11	set	set	NOUN
cana-5363	159	12	.	.	PUNCT
cana-5363	160	1	remark	remark	PROPN
cana-5363	160	2	3.2	3.2	NUM
cana-5363	160	3	according	accord	VERB
cana-5363	160	4	to	to	ADP
cana-5363	160	5	definition	definition	NOUN
cana-5363	160	6	3.4	3.4	NUM
cana-5363	160	7	and	and	CCONJ
cana-5363	160	8	theorem	theorem	VERB
cana-5363	160	9	3.1	3.1	NUM
cana-5363	160	10	,	,	PUNCT
cana-5363	160	11	the	the	DET
cana-5363	160	12	following	follow	VERB
cana-5363	160	13	diagram	diagram	NOUN
cana-5363	160	14	holds	hold	VERB
cana-5363	160	15	for	for	ADP
cana-5363	160	16	any	any	DET
cana-5363	160	17	set	set	NOUN
cana-5363	160	18	in	in	ADP
cana-5363	160	19	𝒫ℱ𝒩𝑡𝑠.	𝒫ℱ𝒩𝑡𝑠.	PROPN
cana-5363	160	20	lemma	lemma	PROPN
cana-5363	160	21	3.1	3.1	NUM
cana-5363	160	22	let	let	NOUN
cana-5363	160	23	(	(	PUNCT
cana-5363	160	24	𝑈	𝑈	PROPN
cana-5363	160	25	,	,	PUNCT
cana-5363	160	26	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	160	27	)	)	PUNCT
cana-5363	160	28	)	)	PUNCT
cana-5363	160	29	be	be	AUX
cana-5363	160	30	a	a	DET
cana-5363	160	31	𝒫ℱ𝔑𝑡𝑠.	𝒫ℱ𝔑𝑡𝑠.	NOUN
cana-5363	160	32	then	then	ADV
cana-5363	160	33	the	the	DET
cana-5363	160	34	following	follow	VERB
cana-5363	160	35	statements	statement	NOUN
cana-5363	160	36	are	be	AUX
cana-5363	160	37	hold	hold	ADJ
cana-5363	160	38	.	.	PUNCT
cana-5363	161	1	(	(	PUNCT
cana-5363	161	2	i	i	NOUN
cana-5363	161	3	)	)	PUNCT
cana-5363	161	4	the	the	DET
cana-5363	161	5	union	union	NOUN
cana-5363	161	6	of	of	ADP
cana-5363	161	7	arbitrary	arbitrary	ADJ
cana-5363	161	8	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	161	9	sets	set	NOUN
cana-5363	161	10	is	be	AUX
cana-5363	161	11	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	PROPN
cana-5363	161	12	,	,	PUNCT
cana-5363	161	13	(	(	PUNCT
cana-5363	161	14	ii	ii	NOUN
cana-5363	161	15	)	)	PUNCT
cana-5363	161	16	the	the	DET
cana-5363	161	17	intersection	intersection	NOUN
cana-5363	161	18	of	of	ADP
cana-5363	161	19	arbitrary	arbitrary	ADJ
cana-5363	161	20	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	ADJ
cana-5363	161	21	sets	set	NOUN
cana-5363	161	22	is	be	AUX
cana-5363	161	23	𝒫ℱ𝔑𝑍𝑐.	𝒫ℱ𝔑𝑍𝑐.	PROPN
cana-5363	161	24	proof	proof	NOUN
cana-5363	161	25	.	.	PUNCT
cana-5363	162	1	(	(	PUNCT
cana-5363	162	2	i	i	NOUN
cana-5363	162	3	)	)	PUNCT
cana-5363	162	4	let	let	VERB
cana-5363	162	5	{	{	PUNCT
cana-5363	162	6	𝐾𝑖	𝐾𝑖	INTJ
cana-5363	162	7	,	,	PUNCT
cana-5363	162	8	𝑖	𝑖	SYM
cana-5363	162	9	∈	∈	PROPN
cana-5363	162	10	𝐼	𝐼	PROPN
cana-5363	162	11	}	}	PUNCT
cana-5363	162	12	be	be	VERB
cana-5363	162	13	a	a	DET
cana-5363	162	14	family	family	NOUN
cana-5363	162	15	of	of	ADP
cana-5363	162	16	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	162	17	sets	set	NOUN
cana-5363	162	18	.	.	PUNCT
cana-5363	163	1	then	then	ADV
cana-5363	163	2	𝐾𝑖	𝐾𝑖	PROPN
cana-5363	163	3	⊆	⊆	NUM
cana-5363	163	4	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾𝑖	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾𝑖	NOUN
cana-5363	163	5	)	)	PUNCT
cana-5363	163	6	)	)	PUNCT
cana-5363	163	7	∪	∪	ADP
cana-5363	163	8	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾𝑖	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾𝑖	NOUN
cana-5363	163	9	)	)	PUNCT
cana-5363	163	10	)	)	PUNCT
cana-5363	163	11	and	and	CCONJ
cana-5363	163	12	hence	hence	ADV
cana-5363	163	13	∪𝑖	∪𝑖	NUM
cana-5363	164	1	𝐾𝑖	𝐾𝑖	PROPN
cana-5363	164	2	⊆	⊆	NUM
cana-5363	164	3	∪𝑖	∪𝑖	NUM
cana-5363	164	4	(	(	PUNCT
cana-5363	164	5	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾𝑖	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾𝑖	NOUN
cana-5363	164	6	)	)	PUNCT
cana-5363	164	7	)	)	PUNCT
cana-5363	164	8	∩	∩	ADJ
cana-5363	164	9	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾𝑖	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾𝑖	NOUN
cana-5363	164	10	)	)	PUNCT
cana-5363	164	11	)	)	PUNCT
cana-5363	164	12	)	)	PUNCT
cana-5363	165	1	⊆	⊆	X
cana-5363	165	2	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(∪𝑖	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(∪𝑖	X
cana-5363	165	3	𝐾𝑖	𝐾𝑖	PROPN
cana-5363	165	4	)	)	PUNCT
cana-5363	165	5	)	)	PUNCT
cana-5363	165	6	∩	∩	NOUN
cana-5363	165	7	communications	communication	NOUN
cana-5363	165	8	on	on	ADP
cana-5363	165	9	applied	apply	VERB
cana-5363	165	10	nonlinear	nonlinear	ADJ
cana-5363	165	11	analysis	analysis	NOUN
cana-5363	165	12	issn	issn	NOUN
cana-5363	165	13	:	:	PUNCT
cana-5363	165	14	1074	1074	NUM
cana-5363	165	15	-	-	PUNCT
cana-5363	165	16	133x	133x	NUM
cana-5363	165	17	vol	vol	VERB
cana-5363	165	18	32	32	NUM
cana-5363	165	19	no	no	NOUN
cana-5363	165	20	.	.	PUNCT
cana-5363	166	1	10s	10	NOUN
cana-5363	166	2	(	(	PUNCT
cana-5363	166	3	2025	2025	NUM
cana-5363	166	4	)	)	PUNCT
cana-5363	166	5	2011	2011	NUM
cana-5363	166	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5363	166	7	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(∪𝑖	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(∪𝑖	PROPN
cana-5363	166	8	𝐾𝑖	𝐾𝑖	PROPN
cana-5363	166	9	)	)	PUNCT
cana-5363	166	10	)	)	PUNCT
cana-5363	166	11	,	,	PUNCT
cana-5363	166	12	for	for	ADP
cana-5363	166	13	all	all	DET
cana-5363	166	14	𝑖	𝑖	SYM
cana-5363	166	15	∈	∈	NOUN
cana-5363	166	16	𝐼.	𝐼.	NOUN
cana-5363	166	17	thus	thus	ADV
cana-5363	166	18	∪𝑖	∪𝑖	NUM
cana-5363	167	1	𝐾𝑖	𝐾𝑖	PROPN
cana-5363	167	2	is	be	AUX
cana-5363	167	3	𝒫ℱ𝔑𝑍𝑜.	𝒫ℱ𝔑𝑍𝑜.	PROPN
cana-5363	167	4	(	(	PUNCT
cana-5363	167	5	ii	ii	NOUN
cana-5363	167	6	)	)	PUNCT
cana-5363	167	7	it	it	PRON
cana-5363	167	8	follows	follow	VERB
cana-5363	167	9	from	from	ADP
cana-5363	167	10	(	(	PUNCT
cana-5363	167	11	i	i	NOUN
cana-5363	167	12	)	)	PUNCT
cana-5363	167	13	.	.	PUNCT
cana-5363	168	1	remark	remark	VERB
cana-5363	168	2	3.3	3.3	NUM
cana-5363	168	3	by	by	ADP
cana-5363	168	4	the	the	DET
cana-5363	168	5	following	following	NOUN
cana-5363	168	6	we	we	PRON
cana-5363	168	7	show	show	VERB
cana-5363	168	8	that	that	SCONJ
cana-5363	168	9	the	the	DET
cana-5363	168	10	intersection	intersection	NOUN
cana-5363	168	11	of	of	ADP
cana-5363	168	12	any	any	DET
cana-5363	168	13	two	two	NUM
cana-5363	168	14	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	168	15	sets	set	NOUN
cana-5363	168	16	is	be	AUX
cana-5363	168	17	not	not	PART
cana-5363	168	18	𝒫ℱ𝔑𝑍𝑜.	𝒫ℱ𝔑𝑍𝑜.	PROPN
cana-5363	168	19	example	example	NOUN
cana-5363	168	20	3.2	3.2	NUM
cana-5363	168	21	in	in	ADP
cana-5363	168	22	example	example	NOUN
cana-5363	168	23	3.1	3.1	NUM
cana-5363	168	24	,	,	PUNCT
cana-5363	168	25	let	let	VERB
cana-5363	168	26	𝐴	𝐴	PROPN
cana-5363	168	27	=	=	PRON
cana-5363	168	28	{	{	PUNCT
cana-5363	168	29	⟨	⟨	ADP
cana-5363	168	30	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	168	31	0.3	0.3	NUM
cana-5363	168	32	⟩	⟩	NOUN
cana-5363	168	33	,	,	PUNCT
cana-5363	168	34	⟨	⟨	VERB
cana-5363	168	35	𝑠2	𝑠2	PROPN
cana-5363	168	36	0.5	0.5	NUM
cana-5363	168	37	⟩	⟩	NOUN
cana-5363	168	38	,	,	PUNCT
cana-5363	168	39	⟨	⟨	VERB
cana-5363	168	40	𝑠3	𝑠3	PROPN
cana-5363	168	41	0.5	0.5	NUM
cana-5363	168	42	⟩	⟩	NOUN
cana-5363	168	43	}	}	PUNCT
cana-5363	168	44	and	and	CCONJ
cana-5363	168	45	𝐵	𝐵	NOUN
cana-5363	168	46	=	=	PUNCT
cana-5363	168	47	{	{	PUNCT
cana-5363	168	48	⟨	⟨	ADP
cana-5363	168	49	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	168	50	0.1	0.1	NUM
cana-5363	168	51	⟩	⟩	NOUN
cana-5363	168	52	,	,	PUNCT
cana-5363	168	53	⟨	⟨	VERB
cana-5363	168	54	𝑠2	𝑠2	PROPN
cana-5363	168	55	0.2	0.2	NUM
cana-5363	168	56	⟩	⟩	NOUN
cana-5363	168	57	,	,	PUNCT
cana-5363	169	1	⟨	⟨	VERB
cana-5363	169	2	𝑠3	𝑠3	PROPN
cana-5363	169	3	0.7	0.7	NUM
cana-5363	169	4	⟩	⟩	NOUN
cana-5363	169	5	}	}	PUNCT
cana-5363	169	6	are	be	AUX
cana-5363	169	7	𝒫ℱ𝒩𝑍𝑜	𝒫ℱ𝒩𝑍𝑜	NOUN
cana-5363	169	8	sets	set	NOUN
cana-5363	169	9	but	but	CCONJ
cana-5363	169	10	𝐴	𝐴	PROPN
cana-5363	169	11	∩	∩	ADJ
cana-5363	169	12	𝐵	𝐵	NOUN
cana-5363	169	13	=	=	PUNCT
cana-5363	169	14	{	{	PUNCT
cana-5363	169	15	⟨	⟨	ADP
cana-5363	169	16	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	169	17	0.1	0.1	NUM
cana-5363	169	18	⟩	⟩	NOUN
cana-5363	169	19	,	,	PUNCT
cana-5363	169	20	⟨	⟨	VERB
cana-5363	169	21	𝑠2	𝑠2	PROPN
cana-5363	169	22	0.2	0.2	NUM
cana-5363	169	23	⟩	⟩	NOUN
cana-5363	169	24	,	,	PUNCT
cana-5363	169	25	⟨	⟨	VERB
cana-5363	169	26	𝑠3	𝑠3	PROPN
cana-5363	169	27	0.5	0.5	NUM
cana-5363	169	28	⟩	⟩	NOUN
cana-5363	169	29	}	}	PUNCT
cana-5363	169	30	is	be	AUX
cana-5363	169	31	not	not	PART
cana-5363	169	32	𝒫ℱ𝒩𝑍𝑜	𝒫ℱ𝒩𝑍𝑜	NOUN
cana-5363	169	33	set	set	VERB
cana-5363	169	34	.	.	PUNCT
cana-5363	170	1	theorem	theorem	VERB
cana-5363	170	2	3.2	3.2	NUM
cana-5363	170	3	let	let	VERB
cana-5363	170	4	𝐾	𝐾	PRON
cana-5363	170	5	be	be	AUX
cana-5363	170	6	a	a	DET
cana-5363	170	7	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	NOUN
cana-5363	170	8	set	set	NOUN
cana-5363	170	9	in	in	ADP
cana-5363	170	10	(	(	PUNCT
cana-5363	170	11	𝑈	𝑈	PROPN
cana-5363	170	12	,	,	PUNCT
cana-5363	170	13	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	170	14	)	)	PUNCT
cana-5363	170	15	)	)	PUNCT
cana-5363	171	1	then	then	ADV
cana-5363	171	2	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	171	3	)	)	PUNCT
cana-5363	171	4	−	−	PROPN
cana-5363	172	1	𝐾	𝐾	PROPN
cana-5363	172	2	does	do	AUX
cana-5363	172	3	not	not	PART
cana-5363	172	4	contain	contain	VERB
cana-5363	172	5	any	any	DET
cana-5363	172	6	non	non	ADJ
cana-5363	172	7	-	-	ADJ
cana-5363	172	8	empty	empty	ADJ
cana-5363	172	9	𝒫ℱ𝔑𝑐	𝒫ℱ𝔑𝑐	NOUN
cana-5363	172	10	set	set	VERB
cana-5363	172	11	in	in	ADP
cana-5363	172	12	(	(	PUNCT
cana-5363	172	13	𝑈	𝑈	PROPN
cana-5363	172	14	,	,	PUNCT
cana-5363	172	15	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	172	16	)	)	PUNCT
cana-5363	172	17	)	)	PUNCT
cana-5363	172	18	.	.	PUNCT
cana-5363	173	1	proof	proof	NOUN
cana-5363	173	2	.	.	PUNCT
cana-5363	174	1	let	let	VERB
cana-5363	174	2	𝐾	𝐾	PRON
cana-5363	174	3	be	be	AUX
cana-5363	174	4	a	a	DET
cana-5363	174	5	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	NOUN
cana-5363	174	6	set	set	NOUN
cana-5363	174	7	in	in	ADP
cana-5363	174	8	(	(	PUNCT
cana-5363	174	9	𝑈	𝑈	PROPN
cana-5363	174	10	,	,	PUNCT
cana-5363	174	11	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	174	12	)	)	PUNCT
cana-5363	174	13	)	)	PUNCT
cana-5363	175	1	and	and	CCONJ
cana-5363	175	2	𝑆	𝑆	PROPN
cana-5363	175	3	be	be	VERB
cana-5363	175	4	a	a	DET
cana-5363	175	5	𝒫ℱ𝔑𝑐	𝒫ℱ𝔑𝑐	NOUN
cana-5363	175	6	subset	subset	NOUN
cana-5363	175	7	of	of	ADP
cana-5363	175	8	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	175	9	)	)	PUNCT
cana-5363	175	10	−	−	PROPN
cana-5363	175	11	𝐾	𝐾	PROPN
cana-5363	175	12	.	.	PUNCT
cana-5363	176	1	that	that	PRON
cana-5363	176	2	is	be	AUX
cana-5363	176	3	,	,	PUNCT
cana-5363	176	4	𝑆	𝑆	PROPN
cana-5363	176	5	⊆	⊆	NUM
cana-5363	176	6	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	176	7	)	)	PUNCT
cana-5363	176	8	−	−	PROPN
cana-5363	177	1	𝐾	𝐾	PROPN
cana-5363	177	2	implies	imply	VERB
cana-5363	177	3	𝑆	𝑆	PROPN
cana-5363	177	4	⊆	⊆	NUM
cana-5363	177	5	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	177	6	)	)	PUNCT
cana-5363	177	7	∩	∩	NOUN
cana-5363	177	8	(	(	PUNCT
cana-5363	177	9	1𝒫	1𝒫	NOUN
cana-5363	177	10	−	−	PROPN
cana-5363	177	11	𝐾	𝐾	PROPN
cana-5363	177	12	)	)	PUNCT
cana-5363	177	13	.	.	PUNCT
cana-5363	178	1	that	that	PRON
cana-5363	178	2	is	be	AUX
cana-5363	178	3	𝑆	𝑆	PROPN
cana-5363	178	4	⊆	⊆	NUM
cana-5363	178	5	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	178	6	)	)	PUNCT
cana-5363	178	7	and	and	CCONJ
cana-5363	178	8	𝑆	𝑆	PROPN
cana-5363	178	9	⊆	⊆	NUM
cana-5363	178	10	(	(	PUNCT
cana-5363	178	11	1𝒫	1𝒫	NOUN
cana-5363	178	12	−	−	PROPN
cana-5363	178	13	𝐾	𝐾	PROPN
cana-5363	178	14	)	)	PUNCT
cana-5363	178	15	which	which	PRON
cana-5363	178	16	implies	imply	VERB
cana-5363	178	17	𝐾	𝐾	PROPN
cana-5363	178	18	⊆	⊆	NUM
cana-5363	178	19	(	(	PUNCT
cana-5363	178	20	1𝒫	1𝒫	NOUN
cana-5363	178	21	−	−	PROPN
cana-5363	178	22	𝑆	𝑆	PROPN
cana-5363	178	23	)	)	PUNCT
cana-5363	178	24	where	where	SCONJ
cana-5363	178	25	1𝒫	1𝒫	PROPN
cana-5363	178	26	−	−	PROPN
cana-5363	178	27	𝑆	𝑆	PROPN
cana-5363	178	28	is	be	AUX
cana-5363	178	29	a	a	DET
cana-5363	178	30	𝒫ℱ𝔑𝑜	𝒫ℱ𝔑𝑜	PROPN
cana-5363	178	31	set	set	NOUN
cana-5363	178	32	.	.	PUNCT
cana-5363	179	1	since	since	SCONJ
cana-5363	179	2	𝐾	𝐾	PROPN
cana-5363	179	3	is	be	AUX
cana-5363	179	4	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	PROPN
cana-5363	179	5	,	,	PUNCT
cana-5363	179	6	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	179	7	)	)	PUNCT
cana-5363	179	8	⊆	⊆	NUM
cana-5363	179	9	(	(	PUNCT
cana-5363	179	10	1𝒫	1𝒫	NOUN
cana-5363	179	11	−	−	PROPN
cana-5363	179	12	𝑆	𝑆	PROPN
cana-5363	179	13	)	)	PUNCT
cana-5363	179	14	.	.	PUNCT
cana-5363	180	1	that	that	PRON
cana-5363	180	2	is	be	AUX
cana-5363	180	3	𝑆	𝑆	PROPN
cana-5363	180	4	⊆	⊆	NUM
cana-5363	180	5	(	(	PUNCT
cana-5363	180	6	1𝒫	1𝒫	INTJ
cana-5363	180	7	−	−	PROPN
cana-5363	180	8	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	180	9	)	)	PUNCT
cana-5363	180	10	)	)	PUNCT
cana-5363	180	11	.	.	PUNCT
cana-5363	181	1	thus	thus	ADV
cana-5363	181	2	𝑆	𝑆	PROPN
cana-5363	181	3	⊆	⊆	NUM
cana-5363	181	4	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	181	5	)	)	PUNCT
cana-5363	181	6	∩	∩	NOUN
cana-5363	181	7	(	(	PUNCT
cana-5363	181	8	1𝒫	1𝒫	INTJ
cana-5363	181	9	−	−	PROPN
cana-5363	181	10	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	181	11	)	)	PUNCT
cana-5363	181	12	)	)	PUNCT
cana-5363	182	1	=	=	PUNCT
cana-5363	183	1	0𝑃.	0𝑃.	NOUN
cana-5363	184	1	hence	hence	ADV
cana-5363	184	2	𝑆	𝑆	PROPN
cana-5363	184	3	=	=	SYM
cana-5363	184	4	0𝑃.	0𝑃.	NUM
cana-5363	184	5	therefore	therefore	ADV
cana-5363	184	6	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	184	7	)	)	PUNCT
cana-5363	184	8	−	−	PROPN
cana-5363	185	1	𝐾	𝐾	PROPN
cana-5363	185	2	does	do	AUX
cana-5363	185	3	not	not	PART
cana-5363	185	4	contain	contain	VERB
cana-5363	185	5	any	any	DET
cana-5363	185	6	non	non	ADJ
cana-5363	185	7	-	-	ADJ
cana-5363	185	8	empty	empty	ADJ
cana-5363	185	9	𝒫ℱ𝔑𝑐	𝒫ℱ𝔑𝑐	NOUN
cana-5363	185	10	set	set	VERB
cana-5363	185	11	in	in	ADP
cana-5363	185	12	(	(	PUNCT
cana-5363	185	13	𝑈	𝑈	PROPN
cana-5363	185	14	,	,	PUNCT
cana-5363	185	15	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	185	16	)	)	PUNCT
cana-5363	185	17	)	)	PUNCT
cana-5363	185	18	.	.	PUNCT
cana-5363	186	1	remark	remark	VERB
cana-5363	186	2	3.4	3.4	NUM
cana-5363	186	3	the	the	DET
cana-5363	186	4	converse	converse	NOUN
cana-5363	186	5	of	of	ADP
cana-5363	186	6	the	the	DET
cana-5363	186	7	theorem	theorem	ADJ
cana-5363	186	8	3.2	3.2	NUM
cana-5363	186	9	need	need	AUX
cana-5363	186	10	not	not	PART
cana-5363	186	11	be	be	AUX
cana-5363	186	12	true	true	ADJ
cana-5363	186	13	as	as	SCONJ
cana-5363	186	14	seen	see	VERB
cana-5363	186	15	from	from	ADP
cana-5363	186	16	the	the	DET
cana-5363	186	17	following	follow	VERB
cana-5363	186	18	example	example	NOUN
cana-5363	186	19	.	.	PUNCT
cana-5363	187	1	example	example	NOUN
cana-5363	187	2	3.3	3.3	NUM
cana-5363	187	3	in	in	ADP
cana-5363	187	4	example	example	NOUN
cana-5363	187	5	3.1	3.1	NUM
cana-5363	187	6	,	,	PUNCT
cana-5363	187	7	let	let	VERB
cana-5363	187	8	𝐾	𝐾	PROPN
cana-5363	187	9	=	=	PRON
cana-5363	187	10	{	{	PUNCT
cana-5363	187	11	𝑙2	𝑙2	PROPN
cana-5363	187	12	,	,	PUNCT
cana-5363	187	13	𝑙3	𝑙3	ADJ
cana-5363	187	14	,	,	PUNCT
cana-5363	187	15	𝑙4	𝑙4	PROPN
cana-5363	187	16	}	}	PUNCT
cana-5363	187	17	be	be	VERB
cana-5363	187	18	any	any	DET
cana-5363	187	19	subset	subset	NOUN
cana-5363	187	20	of	of	ADP
cana-5363	187	21	𝑈	𝑈	PROPN
cana-5363	187	22	then	then	ADV
cana-5363	187	23	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	187	24	)	)	PUNCT
cana-5363	187	25	−	−	PROPN
cana-5363	187	26	𝐾	𝐾	PROPN
cana-5363	187	27	=	=	PROPN
cana-5363	187	28	𝑈	𝑈	PROPN
cana-5363	187	29	−	−	PROPN
cana-5363	187	30	{	{	PUNCT
cana-5363	187	31	𝑙2	𝑙2	PROPN
cana-5363	187	32	,	,	PUNCT
cana-5363	187	33	𝑙3	𝑙3	ADJ
cana-5363	187	34	,	,	PUNCT
cana-5363	187	35	𝑙4	𝑙4	PROPN
cana-5363	187	36	}	}	PUNCT
cana-5363	187	37	=	=	SYM
cana-5363	187	38	{	{	PUNCT
cana-5363	187	39	𝑙1	𝑙1	PROPN
cana-5363	187	40	}	}	PUNCT
cana-5363	187	41	which	which	PRON
cana-5363	187	42	contain	contain	VERB
cana-5363	187	43	any	any	DET
cana-5363	187	44	non	non	ADJ
cana-5363	187	45	-	-	ADJ
cana-5363	187	46	empty	empty	ADJ
cana-5363	187	47	𝒫ℱ𝔑𝑐	𝒫ℱ𝔑𝑐	NOUN
cana-5363	187	48	in	in	ADP
cana-5363	187	49	𝑈	𝑈	PROPN
cana-5363	187	50	,	,	PUNCT
cana-5363	187	51	but	but	CCONJ
cana-5363	187	52	𝐾	𝐾	PROPN
cana-5363	187	53	is	be	AUX
cana-5363	187	54	not	not	PART
cana-5363	187	55	a	a	DET
cana-5363	187	56	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	NOUN
cana-5363	187	57	in	in	ADP
cana-5363	187	58	𝑈.	𝑈.	PROPN
cana-5363	187	59	theorem	theorem	VERB
cana-5363	187	60	3.3	3.3	NUM
cana-5363	187	61	let	let	VERB
cana-5363	187	62	𝐾	𝐾	PRON
cana-5363	187	63	be	be	AUX
cana-5363	187	64	a	a	DET
cana-5363	187	65	𝒫ℱ𝔑	𝒫ℱ𝔑	NOUN
cana-5363	187	66	set	set	VERB
cana-5363	187	67	in	in	ADP
cana-5363	187	68	(	(	PUNCT
cana-5363	187	69	𝑈	𝑈	PROPN
cana-5363	187	70	,	,	PUNCT
cana-5363	187	71	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	187	72	)	)	PUNCT
cana-5363	187	73	)	)	PUNCT
cana-5363	188	1	then	then	ADV
cana-5363	188	2	𝐾	𝐾	PROPN
cana-5363	188	3	is	be	AUX
cana-5363	188	4	𝒫ℱ𝔑𝑍𝑐𝑠	𝒫ℱ𝔑𝑍𝑐𝑠	PROPN
cana-5363	188	5	if	if	SCONJ
cana-5363	188	6	and	and	CCONJ
cana-5363	188	7	only	only	ADV
cana-5363	188	8	if	if	SCONJ
cana-5363	188	9	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	188	10	)	)	PUNCT
cana-5363	188	11	−	−	PROPN
cana-5363	189	1	𝐾	𝐾	PROPN
cana-5363	189	2	is	be	AUX
cana-5363	189	3	a	a	DET
cana-5363	189	4	𝒫ℱ𝔑𝑐	𝒫ℱ𝔑𝑐	ADJ
cana-5363	189	5	set	set	VERB
cana-5363	189	6	in	in	ADP
cana-5363	189	7	(	(	PUNCT
cana-5363	189	8	𝑈	𝑈	PROPN
cana-5363	189	9	,	,	PUNCT
cana-5363	189	10	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	189	11	)	)	PUNCT
cana-5363	189	12	)	)	PUNCT
cana-5363	189	13	.	.	PUNCT
cana-5363	190	1	proof	proof	NOUN
cana-5363	190	2	.	.	PUNCT
cana-5363	191	1	let	let	VERB
cana-5363	191	2	𝐾	𝐾	PRON
cana-5363	191	3	be	be	AUX
cana-5363	191	4	a	a	DET
cana-5363	191	5	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	ADJ
cana-5363	191	6	set	set	NOUN
cana-5363	191	7	.	.	PUNCT
cana-5363	192	1	assume	assume	VERB
cana-5363	192	2	that	that	SCONJ
cana-5363	192	3	𝐾	𝐾	PROPN
cana-5363	192	4	is	be	AUX
cana-5363	192	5	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	PROPN
cana-5363	192	6	then	then	ADV
cana-5363	192	7	we	we	PRON
cana-5363	192	8	have	have	VERB
cana-5363	192	9	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	192	10	)	)	PUNCT
cana-5363	193	1	=	=	SYM
cana-5363	193	2	𝐾	𝐾	PROPN
cana-5363	193	3	,	,	PUNCT
cana-5363	193	4	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	193	5	)	)	PUNCT
cana-5363	193	6	−	−	PROPN
cana-5363	194	1	𝐾	𝐾	NOUN
cana-5363	194	2	=	=	SYM
cana-5363	194	3	0𝑃	0𝑃	NOUN
cana-5363	194	4	which	which	PRON
cana-5363	194	5	is	be	AUX
cana-5363	194	6	𝒫ℱ𝔑𝑐𝑠.	𝒫ℱ𝔑𝑐𝑠.	PROPN
cana-5363	194	7	conversely	conversely	ADV
cana-5363	194	8	,	,	PUNCT
cana-5363	194	9	assume	assume	VERB
cana-5363	194	10	that	that	SCONJ
cana-5363	194	11	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	194	12	)	)	PUNCT
cana-5363	194	13	−	−	PROPN
cana-5363	195	1	𝐾	𝐾	PROPN
cana-5363	195	2	be	be	VERB
cana-5363	195	3	𝒫ℱ𝔑𝑐𝑠	𝒫ℱ𝔑𝑐𝑠	PROPN
cana-5363	195	4	and	and	CCONJ
cana-5363	195	5	𝐾	𝐾	PROPN
cana-5363	195	6	is	be	AUX
cana-5363	195	7	a	a	DET
cana-5363	195	8	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	NOUN
cana-5363	195	9	set	set	NOUN
cana-5363	195	10	in	in	ADP
cana-5363	195	11	𝑈	𝑈	PROPN
cana-5363	195	12	.	.	PUNCT
cana-5363	196	1	now	now	ADV
cana-5363	196	2	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	196	3	)	)	PUNCT
cana-5363	197	1	−	−	PROPN
cana-5363	198	1	𝐾	𝐾	PROPN
cana-5363	198	2	is	be	AUX
cana-5363	198	3	a	a	DET
cana-5363	198	4	𝒫ℱ𝔑𝑐	𝒫ℱ𝔑𝑐	NOUN
cana-5363	198	5	subset	subset	NOUN
cana-5363	198	6	of	of	ADP
cana-5363	198	7	itself	itself	PRON
cana-5363	198	8	.	.	PUNCT
cana-5363	199	1	therefore	therefore	ADV
cana-5363	199	2	by	by	ADP
cana-5363	199	3	theorem	theorem	NOUN
cana-5363	199	4	3.2	3.2	NUM
cana-5363	199	5	,	,	PUNCT
cana-5363	199	6	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	199	7	)	)	PUNCT
cana-5363	199	8	−	−	PROPN
cana-5363	199	9	𝐾	𝐾	NOUN
cana-5363	199	10	=	=	NOUN
cana-5363	199	11	0𝑃	0𝑃	PROPN
cana-5363	199	12	.	.	PUNCT
cana-5363	200	1	that	that	PRON
cana-5363	200	2	is	be	AUX
cana-5363	200	3	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	200	4	)	)	PUNCT
cana-5363	200	5	=	=	SYM
cana-5363	200	6	𝐾	𝐾	PROPN
cana-5363	200	7	,	,	PUNCT
cana-5363	200	8	implies	imply	VERB
cana-5363	200	9	𝐾	𝐾	PROPN
cana-5363	200	10	is	be	AUX
cana-5363	200	11	𝒫ℱ𝔑𝑍𝑐𝑠.	𝒫ℱ𝔑𝑍𝑐𝑠.	X
cana-5363	200	12	theorem	theorem	VERB
cana-5363	200	13	3.4	3.4	NUM
cana-5363	200	14	if	if	SCONJ
cana-5363	200	15	𝐾	𝐾	PROPN
cana-5363	200	16	is	be	AUX
cana-5363	200	17	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	NOUN
cana-5363	200	18	set	set	VERB
cana-5363	200	19	in	in	ADP
cana-5363	200	20	(	(	PUNCT
cana-5363	200	21	𝑈	𝑈	PROPN
cana-5363	200	22	,	,	PUNCT
cana-5363	200	23	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	200	24	)	)	PUNCT
cana-5363	200	25	)	)	PUNCT
cana-5363	201	1	and	and	CCONJ
cana-5363	201	2	𝐾	𝐾	PROPN
cana-5363	201	3	⊆	⊆	NUM
cana-5363	201	4	𝐿	𝐿	PROPN
cana-5363	201	5	⊆	⊆	NUM
cana-5363	201	6	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	201	7	)	)	PUNCT
cana-5363	201	8	,	,	PUNCT
cana-5363	201	9	then	then	ADV
cana-5363	201	10	𝐿	𝐿	PROPN
cana-5363	201	11	is	be	AUX
cana-5363	201	12	also	also	ADV
cana-5363	201	13	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	NOUN
cana-5363	201	14	in	in	ADP
cana-5363	201	15	(	(	PUNCT
cana-5363	201	16	𝑈	𝑈	PROPN
cana-5363	201	17	,	,	PUNCT
cana-5363	201	18	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	201	19	)	)	PUNCT
cana-5363	201	20	)	)	PUNCT
cana-5363	201	21	.	.	PUNCT
cana-5363	202	1	proof	proof	NOUN
cana-5363	202	2	.	.	PUNCT
cana-5363	203	1	let	let	VERB
cana-5363	203	2	𝐾	𝐾	PRON
cana-5363	203	3	be	be	AUX
cana-5363	203	4	a	a	DET
cana-5363	203	5	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	NOUN
cana-5363	203	6	set	set	NOUN
cana-5363	203	7	in	in	ADP
cana-5363	203	8	𝑈	𝑈	PROPN
cana-5363	203	9	and	and	CCONJ
cana-5363	203	10	𝐾	𝐾	PROPN
cana-5363	203	11	⊆	⊆	NUM
cana-5363	203	12	𝐿	𝐿	PROPN
cana-5363	203	13	⊆	⊆	NUM
cana-5363	203	14	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	NOUN
cana-5363	203	15	)	)	PUNCT
cana-5363	203	16	.	.	PUNCT
cana-5363	204	1	let	let	VERB
cana-5363	204	2	𝐿	𝐿	PROPN
cana-5363	204	3	⊆	⊆	NUM
cana-5363	204	4	𝑂	𝑂	PROPN
cana-5363	204	5	where	where	SCONJ
cana-5363	204	6	𝑂	𝑂	PROPN
cana-5363	204	7	be	be	AUX
cana-5363	204	8	𝒫ℱ𝔑𝑐	𝒫ℱ𝔑𝑐	NOUN
cana-5363	204	9	set	set	VERB
cana-5363	204	10	in	in	ADP
cana-5363	204	11	𝑈.	𝑈.	PROPN
cana-5363	204	12	since	since	SCONJ
cana-5363	204	13	𝐾	𝐾	PROPN
cana-5363	204	14	⊆	⊆	NUM
cana-5363	204	15	𝐿	𝐿	PROPN
cana-5363	204	16	,	,	PUNCT
cana-5363	204	17	implies	imply	VERB
cana-5363	204	18	𝐾	𝐾	PROPN
cana-5363	204	19	⊂	⊂	PROPN
cana-5363	204	20	𝑂	𝑂	PROPN
cana-5363	204	21	and	and	CCONJ
cana-5363	204	22	𝐾	𝐾	PROPN
cana-5363	204	23	is	be	AUX
cana-5363	204	24	𝒫ℱ𝔑𝑍𝑐𝑠	𝒫ℱ𝔑𝑍𝑐𝑠	PROPN
cana-5363	204	25	,	,	PUNCT
cana-5363	204	26	implies	imply	VERB
cana-5363	204	27	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	NOUN
cana-5363	204	28	)	)	PUNCT
cana-5363	204	29	⊆	⊆	NUM
cana-5363	204	30	𝑂	𝑂	NOUN
cana-5363	204	31	.	.	PUNCT
cana-5363	205	1	by	by	ADP
cana-5363	205	2	hypothesis	hypothesis	NOUN
cana-5363	205	3	𝐿	𝐿	PROPN
cana-5363	205	4	⊆	⊆	NUM
cana-5363	205	5	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	205	6	)	)	PUNCT
cana-5363	205	7	,	,	PUNCT
cana-5363	205	8	implies	imply	VERB
cana-5363	205	9	𝒫ℱ𝔑𝑍𝑐𝑙(𝐿	𝒫ℱ𝔑𝑍𝑐𝑙(𝐿	PROPN
cana-5363	205	10	)	)	PUNCT
cana-5363	205	11	⊆	⊆	NUM
cana-5363	205	12	𝒫ℱ𝔑𝑍𝑐𝑙(𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	NOUN
cana-5363	205	13	)	)	PUNCT
cana-5363	205	14	)	)	PUNCT
cana-5363	206	1	=	=	SYM
cana-5363	206	2	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	206	3	)	)	PUNCT
cana-5363	206	4	⊆	⊆	NUM
cana-5363	206	5	𝑂	𝑂	PROPN
cana-5363	206	6	,	,	PUNCT
cana-5363	206	7	which	which	PRON
cana-5363	206	8	implies	imply	VERB
cana-5363	206	9	𝒫ℱ𝔑𝑍𝑐𝑙(𝐿	𝒫ℱ𝔑𝑍𝑐𝑙(𝐿	PROPN
cana-5363	206	10	)	)	PUNCT
cana-5363	206	11	⊆	⊆	NUM
cana-5363	206	12	𝑂	𝑂	NOUN
cana-5363	206	13	.	.	PUNCT
cana-5363	207	1	therefore	therefore	ADV
cana-5363	207	2	𝐿	𝐿	PROPN
cana-5363	207	3	is	be	AUX
cana-5363	207	4	𝒫ℱ𝔑𝑍𝑐𝑠	𝒫ℱ𝔑𝑍𝑐𝑠	PROPN
cana-5363	207	5	in	in	ADP
cana-5363	207	6	𝑈.	𝑈.	PROPN
cana-5363	207	7	theorem	theorem	VERB
cana-5363	207	8	3.5	3.5	NUM
cana-5363	207	9	if	if	SCONJ
cana-5363	207	10	a	a	DET
cana-5363	207	11	subset	subset	NOUN
cana-5363	207	12	𝐾	𝐾	PROPN
cana-5363	207	13	of	of	ADP
cana-5363	207	14	𝑈	𝑈	PROPN
cana-5363	207	15	is	be	AUX
cana-5363	207	16	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	ADJ
cana-5363	207	17	set	set	NOUN
cana-5363	207	18	,	,	PUNCT
cana-5363	207	19	then	then	ADV
cana-5363	207	20	𝒫ℱ𝔑𝑐𝑙({𝑥𝑟	𝒫ℱ𝔑𝑐𝑙({𝑥𝑟	ADP
cana-5363	207	21	}	}	PUNCT
cana-5363	207	22	)	)	PUNCT
cana-5363	207	23	∩	∩	NOUN
cana-5363	207	24	𝐾	𝐾	PROPN
cana-5363	207	25	≠	≠	PROPN
cana-5363	207	26	0𝑃	0𝑃	NOUN
cana-5363	207	27	for	for	ADP
cana-5363	207	28	each	each	DET
cana-5363	207	29	𝑥𝑟	𝑥𝑟	ADP
cana-5363	207	30	∈	∈	PROPN
cana-5363	207	31	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	207	32	)	)	PUNCT
cana-5363	207	33	.	.	PUNCT
cana-5363	208	1	proof	proof	NOUN
cana-5363	208	2	.	.	PUNCT
cana-5363	209	1	suppose	suppose	VERB
cana-5363	209	2	𝐾	𝐾	PROPN
cana-5363	209	3	is	be	AUX
cana-5363	209	4	a	a	DET
cana-5363	209	5	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	NOUN
cana-5363	209	6	set	set	NOUN
cana-5363	209	7	and	and	CCONJ
cana-5363	209	8	𝑥𝑟	𝑥𝑟	ADP
cana-5363	209	9	∈	∈	PROPN
cana-5363	209	10	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	209	11	)	)	PUNCT
cana-5363	209	12	.	.	PUNCT
cana-5363	210	1	if	if	SCONJ
cana-5363	210	2	possible	possible	ADJ
cana-5363	210	3	𝒫ℱ𝔑𝑐𝑙({𝑥𝑟	𝒫ℱ𝔑𝑐𝑙({𝑥𝑟	ADP
cana-5363	210	4	}	}	PUNCT
cana-5363	210	5	)	)	PUNCT
cana-5363	210	6	∩	∩	NOUN
cana-5363	210	7	𝐾	𝐾	PROPN
cana-5363	210	8	=	=	ADJ
cana-5363	210	9	0𝑃.	0𝑃.	NOUN
cana-5363	211	1	then	then	ADV
cana-5363	211	2	𝐾	𝐾	PROPN
cana-5363	211	3	⊆	⊆	NUM
cana-5363	211	4	1𝒫	1𝒫	NOUN
cana-5363	211	5	−	−	NOUN
cana-5363	211	6	𝒫ℱ𝔑𝑐𝑙({𝑥𝑟	𝒫ℱ𝔑𝑐𝑙({𝑥𝑟	PROPN
cana-5363	211	7	}	}	PUNCT
cana-5363	211	8	)	)	PUNCT
cana-5363	211	9	and	and	CCONJ
cana-5363	211	10	1𝒫	1𝒫	INTJ
cana-5363	212	1	−	−	NOUN
cana-5363	212	2	𝒫ℱ𝔑𝑐𝑙({𝑥𝑟	𝒫ℱ𝔑𝑐𝑙({𝑥𝑟	PROPN
cana-5363	212	3	}	}	PUNCT
cana-5363	212	4	)	)	PUNCT
cana-5363	212	5	is	be	AUX
cana-5363	212	6	a	a	DET
cana-5363	212	7	𝒫ℱ𝔑𝑜	𝒫ℱ𝔑𝑜	NOUN
cana-5363	212	8	set	set	VERB
cana-5363	212	9	containing	contain	VERB
cana-5363	212	10	𝐾	𝐾	PROPN
cana-5363	212	11	.	.	PUNCT
cana-5363	213	1	since	since	SCONJ
cana-5363	213	2	𝐾	𝐾	PROPN
cana-5363	213	3	is	be	AUX
cana-5363	213	4	communications	communication	NOUN
cana-5363	213	5	on	on	ADP
cana-5363	213	6	applied	apply	VERB
cana-5363	213	7	nonlinear	nonlinear	ADJ
cana-5363	213	8	analysis	analysis	NOUN
cana-5363	213	9	issn	issn	NOUN
cana-5363	213	10	:	:	PUNCT
cana-5363	213	11	1074	1074	NUM
cana-5363	213	12	-	-	PUNCT
cana-5363	213	13	133x	133x	NUM
cana-5363	213	14	vol	vol	VERB
cana-5363	213	15	32	32	NUM
cana-5363	213	16	no	no	NOUN
cana-5363	213	17	.	.	PUNCT
cana-5363	214	1	10s	10	NOUN
cana-5363	214	2	(	(	PUNCT
cana-5363	214	3	2025	2025	NUM
cana-5363	214	4	)	)	PUNCT
cana-5363	214	5	2012	2012	NUM
cana-5363	214	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5363	214	7	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	PROPN
cana-5363	214	8	set	set	NOUN
cana-5363	214	9	,	,	PUNCT
cana-5363	214	10	implies	imply	VERB
cana-5363	214	11	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	NOUN
cana-5363	214	12	)	)	PUNCT
cana-5363	214	13	⊆	⊆	NUM
cana-5363	214	14	1𝒫	1𝒫	NOUN
cana-5363	214	15	−	−	NOUN
cana-5363	214	16	𝒫ℱ𝔑𝑐𝑙({𝑥𝑟	𝒫ℱ𝔑𝑐𝑙({𝑥𝑟	PROPN
cana-5363	214	17	}	}	PUNCT
cana-5363	214	18	)	)	PUNCT
cana-5363	214	19	which	which	PRON
cana-5363	214	20	is	be	AUX
cana-5363	214	21	a	a	DET
cana-5363	214	22	contradiction	contradiction	NOUN
cana-5363	214	23	to	to	ADP
cana-5363	214	24	𝑥𝑟	𝑥𝑟	VERB
cana-5363	214	25	∈	∈	PROPN
cana-5363	214	26	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	214	27	)	)	PUNCT
cana-5363	214	28	.	.	PUNCT
cana-5363	215	1	therefore	therefore	ADV
cana-5363	215	2	,	,	PUNCT
cana-5363	215	3	𝒫ℱ𝔑𝑐𝑙({𝑥𝑟	𝒫ℱ𝔑𝑐𝑙({𝑥𝑟	ADP
cana-5363	215	4	}	}	PUNCT
cana-5363	215	5	)	)	PUNCT
cana-5363	215	6	∩	∩	NOUN
cana-5363	215	7	𝐾	𝐾	PROPN
cana-5363	215	8	≠	≠	PROPN
cana-5363	215	9	0𝑃.	0𝑃.	PRON
cana-5363	215	10	theorem	theorem	VERB
cana-5363	215	11	3.6	3.6	NUM
cana-5363	215	12	if	if	SCONJ
cana-5363	215	13	𝒫ℱ𝔑𝑍𝑂(𝑈	𝒫ℱ𝔑𝑍𝑂(𝑈	NUM
cana-5363	215	14	,	,	PUNCT
cana-5363	215	15	𝐴	𝐴	PROPN
cana-5363	215	16	)	)	PUNCT
cana-5363	215	17	=	=	PUNCT
cana-5363	216	1	𝒫ℱ𝔑𝑍𝐶(𝑈	𝒫ℱ𝔑𝑍𝐶(𝑈	PROPN
cana-5363	216	2	,	,	PUNCT
cana-5363	216	3	𝐴	𝐴	PROPN
cana-5363	216	4	)	)	PUNCT
cana-5363	216	5	,	,	PUNCT
cana-5363	216	6	then	then	ADV
cana-5363	216	7	𝒫ℱ𝔑𝑍𝐶(𝑈	𝒫ℱ𝔑𝑍𝐶(𝑈	NUM
cana-5363	216	8	,	,	PUNCT
cana-5363	216	9	𝐴	𝐴	PROPN
cana-5363	216	10	)	)	PUNCT
cana-5363	216	11	=	=	SYM
cana-5363	216	12	𝑃(𝑈	𝑃(𝑈	NOUN
cana-5363	216	13	)	)	PUNCT
cana-5363	216	14	is	be	AUX
cana-5363	216	15	the	the	DET
cana-5363	216	16	power	power	NOUN
cana-5363	216	17	set	set	NOUN
cana-5363	216	18	of	of	ADP
cana-5363	216	19	𝑈.	𝑈.	PROPN
cana-5363	216	20	proof	proof	NOUN
cana-5363	216	21	.	.	PUNCT
cana-5363	217	1	suppose	suppose	VERB
cana-5363	217	2	𝐾	𝐾	PROPN
cana-5363	217	3	⊆	⊆	NUM
cana-5363	217	4	𝑂	𝑂	PROPN
cana-5363	217	5	,	,	PUNCT
cana-5363	217	6	where	where	SCONJ
cana-5363	217	7	𝑂	𝑂	PROPN
cana-5363	217	8	is	be	AUX
cana-5363	217	9	𝒫ℱ𝔑𝑜	𝒫ℱ𝔑𝑜	NOUN
cana-5363	217	10	in	in	ADP
cana-5363	217	11	𝑈	𝑈	PROPN
cana-5363	217	12	.	.	PUNCT
cana-5363	218	1	since	since	SCONJ
cana-5363	218	2	every	every	DET
cana-5363	218	3	𝒫ℱ𝔑𝑜	𝒫ℱ𝔑𝑜	NOUN
cana-5363	218	4	set	set	VERB
cana-5363	218	5	is	be	AUX
cana-5363	218	6	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	218	7	,	,	PUNCT
cana-5363	218	8	𝑂	𝑂	PROPN
cana-5363	218	9	is	be	AUX
cana-5363	218	10	𝒫ℱ𝔑𝑍𝑜.	𝒫ℱ𝔑𝑍𝑜.	PROPN
cana-5363	218	11	by	by	ADP
cana-5363	218	12	hypothesis	hypothesis	NOUN
cana-5363	218	13	,	,	PUNCT
cana-5363	218	14	𝑂	𝑂	PROPN
cana-5363	218	15	is	be	AUX
cana-5363	218	16	𝒫ℱ𝔑𝑍𝑐.	𝒫ℱ𝔑𝑍𝑐.	PROPN
cana-5363	218	17	hence	hence	ADV
cana-5363	218	18	,	,	PUNCT
cana-5363	218	19	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	218	20	)	)	PUNCT
cana-5363	218	21	⊆	⊆	NUM
cana-5363	218	22	𝑂.	𝑂.	PROPN
cana-5363	218	23	therefore	therefore	ADV
cana-5363	218	24	,	,	PUNCT
cana-5363	218	25	𝐾	𝐾	PROPN
cana-5363	218	26	is	be	AUX
cana-5363	218	27	a	a	DET
cana-5363	218	28	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	ADJ
cana-5363	218	29	set	set	NOUN
cana-5363	218	30	.	.	PUNCT
cana-5363	219	1	since	since	ADV
cana-5363	219	2	,	,	PUNCT
cana-5363	219	3	𝐾	𝐾	PROPN
cana-5363	219	4	is	be	AUX
cana-5363	219	5	arbitrary	arbitrary	ADJ
cana-5363	219	6	,	,	PUNCT
cana-5363	219	7	by	by	ADP
cana-5363	219	8	theorem	theorem	NOUN
cana-5363	219	9	3.5	3.5	NUM
cana-5363	219	10	,	,	PUNCT
cana-5363	219	11	every	every	DET
cana-5363	219	12	subset	subset	NOUN
cana-5363	219	13	of	of	ADP
cana-5363	219	14	𝑈	𝑈	PROPN
cana-5363	219	15	is	be	AUX
cana-5363	219	16	𝒫ℱ𝔑𝑍𝑐.	𝒫ℱ𝔑𝑍𝑐.	NOUN
cana-5363	219	17	thus	thus	ADV
cana-5363	219	18	𝒫ℱ𝔑𝑍𝐶(𝑈	𝒫ℱ𝔑𝑍𝐶(𝑈	NUM
cana-5363	219	19	,	,	PUNCT
cana-5363	219	20	𝐴	𝐴	PROPN
cana-5363	219	21	)	)	PUNCT
cana-5363	220	1	=	=	SYM
cana-5363	220	2	𝑃(𝑈	𝑃(𝑈	NOUN
cana-5363	220	3	)	)	PUNCT
cana-5363	220	4	.	.	PUNCT
cana-5363	221	1	definition	definition	NOUN
cana-5363	221	2	3.6	3.6	NUM
cana-5363	221	3	the	the	DET
cana-5363	221	4	intersection	intersection	NOUN
cana-5363	221	5	of	of	ADP
cana-5363	221	6	all	all	DET
cana-5363	221	7	𝒫ℱ𝔑𝑜	𝒫ℱ𝔑𝑜	NOUN
cana-5363	221	8	subset	subset	NOUN
cana-5363	221	9	of	of	ADP
cana-5363	221	10	𝑈	𝑈	PROPN
cana-5363	221	11	containing	contain	VERB
cana-5363	221	12	𝐾	𝐾	PROPN
cana-5363	221	13	is	be	AUX
cana-5363	221	14	called	call	VERB
cana-5363	221	15	the	the	DET
cana-5363	221	16	pythagorean	pythagorean	ADJ
cana-5363	221	17	fuzzy	fuzzy	ADJ
cana-5363	221	18	nano	nano	NOUN
cana-5363	221	19	kernel	kernel	NOUN
cana-5363	221	20	of	of	ADP
cana-5363	221	21	𝐾	𝐾	PROPN
cana-5363	221	22	(	(	PUNCT
cana-5363	221	23	briefly	briefly	ADV
cana-5363	221	24	,	,	PUNCT
cana-5363	221	25	𝒫ℱ𝔑𝑘𝑒𝑟(𝐾	𝒫ℱ𝔑𝑘𝑒𝑟(𝐾	ADJ
cana-5363	221	26	)	)	PUNCT
cana-5363	221	27	)	)	PUNCT
cana-5363	221	28	,	,	PUNCT
cana-5363	221	29	this	this	PRON
cana-5363	221	30	means	mean	VERB
cana-5363	221	31	𝒫ℱ𝔑𝑘𝑒𝑟(𝐾	𝒫ℱ𝔑𝑘𝑒𝑟(𝐾	NOUN
cana-5363	221	32	)	)	PUNCT
cana-5363	221	33	=	=	NOUN
cana-5363	221	34	∩	∩	X
cana-5363	221	35	{	{	PUNCT
cana-5363	221	36	𝐺	𝐺	PROPN
cana-5363	221	37	∈	∈	PROPN
cana-5363	221	38	𝒫ℱ𝔑𝑂(𝑈	𝒫ℱ𝔑𝑂(𝑈	PROPN
cana-5363	221	39	,	,	PUNCT
cana-5363	221	40	𝐴	𝐴	PROPN
cana-5363	221	41	):	):	PUNCT
cana-5363	221	42	𝐾	𝐾	PROPN
cana-5363	221	43	⊆	⊆	NUM
cana-5363	221	44	𝐺	𝐺	PROPN
cana-5363	221	45	}	}	PUNCT
cana-5363	221	46	.	.	PUNCT
cana-5363	222	1	theorem	theorem	VERB
cana-5363	222	2	3.7	3.7	NUM
cana-5363	222	3	a	a	DET
cana-5363	222	4	subset	subset	NOUN
cana-5363	222	5	𝐾	𝐾	NOUN
cana-5363	222	6	is	be	AUX
cana-5363	222	7	a	a	DET
cana-5363	222	8	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	NOUN
cana-5363	222	9	set	set	NOUN
cana-5363	222	10	iff	iff	PROPN
cana-5363	222	11	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	222	12	)	)	PUNCT
cana-5363	222	13	⊆	⊆	NUM
cana-5363	222	14	𝒫ℱ𝔑𝑘𝑒𝑟(𝐾	𝒫ℱ𝔑𝑘𝑒𝑟(𝐾	NOUN
cana-5363	222	15	)	)	PUNCT
cana-5363	222	16	.	.	PUNCT
cana-5363	223	1	proof	proof	NOUN
cana-5363	223	2	.	.	PUNCT
cana-5363	224	1	suppose	suppose	VERB
cana-5363	224	2	𝐾	𝐾	PROPN
cana-5363	224	3	is	be	AUX
cana-5363	224	4	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	ADJ
cana-5363	224	5	set	set	NOUN
cana-5363	224	6	,	,	PUNCT
cana-5363	224	7	then	then	ADV
cana-5363	224	8	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	224	9	)	)	PUNCT
cana-5363	224	10	⊆	⊆	NUM
cana-5363	224	11	𝑂	𝑂	NOUN
cana-5363	224	12	whenever	whenever	SCONJ
cana-5363	224	13	𝐾	𝐾	PROPN
cana-5363	224	14	⊆	⊆	NUM
cana-5363	224	15	𝑂	𝑂	NOUN
cana-5363	224	16	and	and	CCONJ
cana-5363	224	17	𝑂	𝑂	PROPN
cana-5363	224	18	is	be	AUX
cana-5363	224	19	𝒫ℱ𝔑𝑜.	𝒫ℱ𝔑𝑜.	NOUN
cana-5363	224	20	let	let	VERB
cana-5363	224	21	𝑥𝑟	𝑥𝑟	PRON
cana-5363	224	22	∈	∈	PROPN
cana-5363	224	23	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	224	24	)	)	PUNCT
cana-5363	224	25	.	.	PUNCT
cana-5363	225	1	if	if	SCONJ
cana-5363	225	2	𝑥𝑟	𝑥𝑟	PRON
cana-5363	225	3	∉	∉	PROPN
cana-5363	225	4	𝒫ℱ𝔑𝑘𝑒𝑟(𝐾	𝒫ℱ𝔑𝑘𝑒𝑟(𝐾	VERB
cana-5363	225	5	)	)	PUNCT
cana-5363	225	6	,	,	PUNCT
cana-5363	225	7	then	then	ADV
cana-5363	225	8	there	there	PRON
cana-5363	225	9	exist	exist	VERB
cana-5363	225	10	a	a	DET
cana-5363	225	11	𝒫ℱ𝔑𝑜	𝒫ℱ𝔑𝑜	NOUN
cana-5363	225	12	set	set	VERB
cana-5363	225	13	𝑂	𝑂	NOUN
cana-5363	225	14	containing	contain	VERB
cana-5363	225	15	𝐾	𝐾	PROPN
cana-5363	225	16	such	such	ADJ
cana-5363	225	17	that	that	SCONJ
cana-5363	225	18	𝑥𝑟	𝑥𝑟	PROPN
cana-5363	225	19	∉	∉	PROPN
cana-5363	225	20	𝑂.	𝑂.	PROPN
cana-5363	225	21	since	since	SCONJ
cana-5363	225	22	𝑂	𝑂	PROPN
cana-5363	225	23	is	be	AUX
cana-5363	225	24	a	a	DET
cana-5363	225	25	𝒫ℱ𝔑𝑜	𝒫ℱ𝔑𝑜	NOUN
cana-5363	225	26	set	set	VERB
cana-5363	225	27	containing	contain	VERB
cana-5363	225	28	𝐾	𝐾	PROPN
cana-5363	225	29	,	,	PUNCT
cana-5363	225	30	implies	imply	VERB
cana-5363	225	31	𝑥𝑟	𝑥𝑟	PROPN
cana-5363	225	32	∉	∉	PROPN
cana-5363	225	33	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	225	34	)	)	PUNCT
cana-5363	225	35	,	,	PUNCT
cana-5363	225	36	which	which	PRON
cana-5363	225	37	is	be	AUX
cana-5363	225	38	a	a	DET
cana-5363	225	39	contradiction	contradiction	NOUN
cana-5363	225	40	.	.	PUNCT
cana-5363	226	1	therefore	therefore	ADV
cana-5363	226	2	𝑥𝑟	𝑥𝑟	VERB
cana-5363	226	3	∈	∈	PROPN
cana-5363	226	4	𝒫ℱ𝔑𝑘𝑒𝑟(𝐾	𝒫ℱ𝔑𝑘𝑒𝑟(𝐾	NOUN
cana-5363	226	5	)	)	PUNCT
cana-5363	226	6	.	.	PUNCT
cana-5363	227	1	conversely	conversely	ADV
cana-5363	227	2	,	,	PUNCT
cana-5363	227	3	let	let	VERB
cana-5363	227	4	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	NOUN
cana-5363	227	5	)	)	PUNCT
cana-5363	227	6	⊆	⊆	NUM
cana-5363	227	7	𝒫ℱ𝔑𝑘𝑒𝑟(𝐾	𝒫ℱ𝔑𝑘𝑒𝑟(𝐾	NOUN
cana-5363	227	8	)	)	PUNCT
cana-5363	227	9	.	.	PUNCT
cana-5363	228	1	if	if	SCONJ
cana-5363	228	2	𝑂	𝑂	PROPN
cana-5363	228	3	is	be	AUX
cana-5363	228	4	a	a	DET
cana-5363	228	5	𝒫ℱ𝔑𝑜	𝒫ℱ𝔑𝑜	NOUN
cana-5363	228	6	set	set	VERB
cana-5363	228	7	containing	contain	VERB
cana-5363	228	8	𝐾	𝐾	PROPN
cana-5363	228	9	,	,	PUNCT
cana-5363	228	10	then	then	ADV
cana-5363	228	11	𝒫ℱ𝔑𝑘𝑒𝑟(𝐾	𝒫ℱ𝔑𝑘𝑒𝑟(𝐾	CCONJ
cana-5363	228	12	)	)	PUNCT
cana-5363	228	13	⊆	⊆	NUM
cana-5363	228	14	𝑂	𝑂	NOUN
cana-5363	228	15	,	,	PUNCT
cana-5363	228	16	which	which	PRON
cana-5363	228	17	implies	imply	VERB
cana-5363	228	18	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	228	19	)	)	PUNCT
cana-5363	228	20	⊆	⊆	NUM
cana-5363	228	21	𝑂.	𝑂.	PROPN
cana-5363	228	22	therefore	therefore	ADV
cana-5363	228	23	,	,	PUNCT
cana-5363	228	24	𝐾	𝐾	PROPN
cana-5363	228	25	is	be	AUX
cana-5363	228	26	a	a	DET
cana-5363	228	27	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	ADJ
cana-5363	228	28	set	set	NOUN
cana-5363	228	29	.	.	PUNCT
cana-5363	229	1	theorem	theorem	VERB
cana-5363	229	2	3.8	3.8	NUM
cana-5363	229	3	if	if	SCONJ
cana-5363	229	4	𝐾	𝐾	PROPN
cana-5363	229	5	⊆	⊆	NUM
cana-5363	229	6	𝑉	𝑉	PROPN
cana-5363	229	7	⊆	⊆	NUM
cana-5363	229	8	1𝑃	1𝑃	NOUN
cana-5363	229	9	and	and	CCONJ
cana-5363	229	10	suppose	suppose	VERB
cana-5363	229	11	that	that	SCONJ
cana-5363	229	12	𝐾	𝐾	PROPN
cana-5363	229	13	is	be	AUX
cana-5363	229	14	a	a	DET
cana-5363	229	15	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	NOUN
cana-5363	229	16	set	set	NOUN
cana-5363	229	17	in	in	ADP
cana-5363	229	18	1𝑃	1𝑃	NOUN
cana-5363	229	19	,	,	PUNCT
cana-5363	229	20	then	then	ADV
cana-5363	229	21	𝐾	𝐾	PROPN
cana-5363	229	22	is	be	AUX
cana-5363	229	23	𝒫ℱ𝔑𝑍𝑐𝑠	𝒫ℱ𝔑𝑍𝑐𝑠	PROPN
cana-5363	229	24	relative	relative	ADJ
cana-5363	229	25	to	to	ADP
cana-5363	229	26	𝑉.	𝑉.	NOUN
cana-5363	229	27	proof	proof	NOUN
cana-5363	229	28	.	.	PUNCT
cana-5363	230	1	given	give	VERB
cana-5363	230	2	that	that	SCONJ
cana-5363	230	3	𝐾	𝐾	PROPN
cana-5363	230	4	⊆	⊆	NUM
cana-5363	230	5	𝑉	𝑉	PROPN
cana-5363	230	6	⊆	⊆	NUM
cana-5363	230	7	1𝑃	1𝑃	NOUN
cana-5363	230	8	and	and	CCONJ
cana-5363	230	9	let	let	VERB
cana-5363	230	10	𝐾	𝐾	PROPN
cana-5363	230	11	⊆	⊆	NUM
cana-5363	230	12	𝑉	𝑉	PROPN
cana-5363	230	13	∩	∩	NOUN
cana-5363	230	14	𝑂	𝑂	NOUN
cana-5363	230	15	where	where	SCONJ
cana-5363	230	16	𝑂	𝑂	PROPN
cana-5363	230	17	is	be	AUX
cana-5363	230	18	𝒫ℱ𝔑𝑜𝑠	𝒫ℱ𝔑𝑜𝑠	NUM
cana-5363	230	19	in	in	ADP
cana-5363	230	20	𝑈	𝑈	PROPN
cana-5363	230	21	.	.	PUNCT
cana-5363	231	1	since	since	SCONJ
cana-5363	231	2	𝐾	𝐾	PROPN
cana-5363	231	3	is	be	AUX
cana-5363	231	4	a	a	DET
cana-5363	231	5	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	NOUN
cana-5363	231	6	set	set	NOUN
cana-5363	231	7	in	in	ADP
cana-5363	231	8	𝑈	𝑈	PROPN
cana-5363	231	9	,	,	PUNCT
cana-5363	231	10	𝐾	𝐾	PROPN
cana-5363	231	11	⊆	⊆	NUM
cana-5363	231	12	𝑂	𝑂	PROPN
cana-5363	231	13	which	which	PRON
cana-5363	231	14	implies	imply	VERB
cana-5363	231	15	that	that	PRON
cana-5363	231	16	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	231	17	)	)	PUNCT
cana-5363	232	1	⊆	⊆	NUM
cana-5363	232	2	𝑂.	𝑂.	PROPN
cana-5363	232	3	that	that	PRON
cana-5363	232	4	is	be	AUX
cana-5363	232	5	,	,	PUNCT
cana-5363	232	6	𝑉	𝑉	PROPN
cana-5363	232	7	∩	∩	ADJ
cana-5363	232	8	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	232	9	)	)	PUNCT
cana-5363	232	10	⊆	⊆	NUM
cana-5363	232	11	𝑉	𝑉	PROPN
cana-5363	232	12	∩	∩	ADJ
cana-5363	232	13	𝑂	𝑂	PROPN
cana-5363	232	14	,	,	PUNCT
cana-5363	232	15	where	where	SCONJ
cana-5363	232	16	𝑉	𝑉	PROPN
cana-5363	232	17	∩	∩	NOUN
cana-5363	232	18	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	232	19	)	)	PUNCT
cana-5363	232	20	is	be	AUX
cana-5363	232	21	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	232	22	)	)	PUNCT
cana-5363	232	23	in	in	ADP
cana-5363	232	24	𝑉.	𝑉.	NOUN
cana-5363	232	25	thus	thus	ADV
cana-5363	232	26	𝐾	𝐾	PROPN
cana-5363	232	27	is	be	AUX
cana-5363	232	28	𝒫ℱ𝔑𝑍𝑐𝑠	𝒫ℱ𝔑𝑍𝑐𝑠	PROPN
cana-5363	232	29	relative	relative	ADJ
cana-5363	232	30	to	to	ADP
cana-5363	232	31	𝑉.	𝑉.	PROPN
cana-5363	232	32	lemma	lemma	PROPN
cana-5363	232	33	3.2	3.2	NUM
cana-5363	232	34	let	let	VERB
cana-5363	232	35	𝐾	𝐾	PROPN
cana-5363	232	36	and	and	CCONJ
cana-5363	232	37	𝐿	𝐿	PROPN
cana-5363	232	38	be	be	VERB
cana-5363	232	39	two	two	NUM
cana-5363	232	40	𝑝𝑓𝑠	𝑝𝑓𝑠	NOUN
cana-5363	232	41	’s	’s	NOUN
cana-5363	232	42	of	of	ADP
cana-5363	232	43	(	(	PUNCT
cana-5363	232	44	𝑈	𝑈	PROPN
cana-5363	232	45	,	,	PUNCT
cana-5363	232	46	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	232	47	)	)	PUNCT
cana-5363	232	48	)	)	PUNCT
cana-5363	232	49	.	.	PUNCT
cana-5363	233	1	then	then	ADV
cana-5363	233	2	:	:	PUNCT
cana-5363	233	3	(	(	PUNCT
cana-5363	233	4	i	i	NOUN
cana-5363	233	5	)	)	PUNCT
cana-5363	233	6	1𝒫	1𝒫	NOUN
cana-5363	233	7	−	−	NOUN
cana-5363	233	8	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	PUNCT
cana-5363	233	9	)	)	PUNCT
cana-5363	234	1	=	=	SYM
cana-5363	234	2	𝒫ℱ𝔑𝛿𝑐𝑙(1𝒫	𝒫ℱ𝔑𝛿𝑐𝑙(1𝒫	NOUN
cana-5363	234	3	−	−	PROPN
cana-5363	234	4	𝐾	𝐾	PROPN
cana-5363	234	5	)	)	PUNCT
cana-5363	234	6	and	and	CCONJ
cana-5363	234	7	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(1𝒫	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(1𝒫	PROPN
cana-5363	234	8	−	−	PROPN
cana-5363	234	9	𝐾	𝐾	PROPN
cana-5363	234	10	)	)	PUNCT
cana-5363	234	11	=	=	SYM
cana-5363	234	12	1𝒫	1𝒫	NUM
cana-5363	234	13	−	−	PROPN
cana-5363	234	14	𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	NOUN
cana-5363	234	15	)	)	PUNCT
cana-5363	234	16	,	,	PUNCT
cana-5363	234	17	(	(	PUNCT
cana-5363	234	18	ii	ii	NOUN
cana-5363	234	19	)	)	PUNCT
cana-5363	234	20	𝒫ℱ𝔑𝑐𝑙(𝐾	𝒫ℱ𝔑𝑐𝑙(𝐾	NOUN
cana-5363	234	21	)	)	PUNCT
cana-5363	234	22	⊆	⊆	NUM
cana-5363	234	23	𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	NOUN
cana-5363	234	24	)	)	PUNCT
cana-5363	234	25	(	(	PUNCT
cana-5363	234	26	resp	resp	NOUN
cana-5363	234	27	.	.	PUNCT
cana-5363	234	28	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	NOUN
cana-5363	234	29	)	)	PUNCT
cana-5363	234	30	⊆	⊆	NUM
cana-5363	234	31	𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	NOUN
cana-5363	234	32	)	)	PUNCT
cana-5363	234	33	)	)	PUNCT
cana-5363	234	34	,	,	PUNCT
cana-5363	234	35	for	for	ADP
cana-5363	234	36	any	any	DET
cana-5363	234	37	subset	subset	NOUN
cana-5363	234	38	𝐾	𝐾	PROPN
cana-5363	234	39	of	of	ADP
cana-5363	234	40	𝑈	𝑈	PROPN
cana-5363	234	41	,	,	PUNCT
cana-5363	234	42	(	(	PUNCT
cana-5363	234	43	iii	iii	X
cana-5363	234	44	)	)	PUNCT
cana-5363	234	45	𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	NOUN
cana-5363	234	46	∪	∪	PROPN
cana-5363	234	47	𝐿	𝐿	PROPN
cana-5363	234	48	)	)	PUNCT
cana-5363	234	49	=	=	SYM
cana-5363	234	50	𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	NOUN
cana-5363	234	51	)	)	PUNCT
cana-5363	234	52	∪	∪	ADP
cana-5363	234	53	𝒫ℱ𝔑𝛿𝑐𝑙(𝐿	𝒫ℱ𝔑𝛿𝑐𝑙(𝐿	PROPN
cana-5363	234	54	)	)	PUNCT
cana-5363	234	55	,	,	PUNCT
cana-5363	234	56	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	PUNCT
cana-5363	234	57	∩	∩	ADJ
cana-5363	234	58	𝐿	𝐿	PROPN
cana-5363	234	59	)	)	PUNCT
cana-5363	234	60	=	=	SYM
cana-5363	234	61	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	NOUN
cana-5363	234	62	)	)	PUNCT
cana-5363	234	63	∩	∩	NOUN
cana-5363	234	64	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	ADJ
cana-5363	234	65	)	)	PUNCT
cana-5363	234	66	.	.	PUNCT
cana-5363	235	1	proposition	proposition	NOUN
cana-5363	235	2	3.1	3.1	NUM
cana-5363	235	3	let	let	VERB
cana-5363	235	4	𝐾	𝐾	PRON
cana-5363	235	5	be	be	AUX
cana-5363	235	6	a	a	DET
cana-5363	235	7	𝑝𝑓𝑠	𝑝𝑓𝑠	NOUN
cana-5363	235	8	in	in	ADP
cana-5363	235	9	a	a	DET
cana-5363	235	10	𝒫ℱ𝒩𝑡𝑠	𝒫ℱ𝒩𝑡𝑠	PROPN
cana-5363	235	11	(	(	PUNCT
cana-5363	235	12	𝑈	𝑈	PROPN
cana-5363	235	13	,	,	PUNCT
cana-5363	235	14	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	235	15	)	)	PUNCT
cana-5363	235	16	)	)	PUNCT
cana-5363	235	17	.	.	PUNCT
cana-5363	236	1	then	then	ADV
cana-5363	236	2	:	:	PUNCT
cana-5363	236	3	(	(	PUNCT
cana-5363	236	4	i	i	NOUN
cana-5363	236	5	)	)	PUNCT
cana-5363	236	6	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	PROPN
cana-5363	236	7	)	)	PUNCT
cana-5363	236	8	=	=	SYM
cana-5363	236	9	𝐾	𝐾	PROPN
cana-5363	236	10	∪	∪	VERB
cana-5363	236	11	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	PROPN
cana-5363	236	12	)	)	PUNCT
cana-5363	236	13	)	)	PUNCT
cana-5363	236	14	,	,	PUNCT
cana-5363	236	15	𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	PROPN
cana-5363	236	16	)	)	PUNCT
cana-5363	236	17	=	=	SYM
cana-5363	236	18	𝐾	𝐾	PROPN
cana-5363	236	19	∩	∩	NOUN
cana-5363	236	20	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	NOUN
cana-5363	236	21	)	)	PUNCT
cana-5363	236	22	)	)	PUNCT
cana-5363	236	23	,	,	PUNCT
cana-5363	236	24	(	(	PUNCT
cana-5363	236	25	ii	ii	NOUN
cana-5363	236	26	)	)	PUNCT
cana-5363	236	27	𝒫ℱ𝔑𝛿𝒮𝑐𝑙(1𝒫	𝒫ℱ𝔑𝛿𝒮𝑐𝑙(1𝒫	NOUN
cana-5363	236	28	−	−	PROPN
cana-5363	236	29	𝐾	𝐾	PROPN
cana-5363	236	30	)	)	PUNCT
cana-5363	236	31	=	=	SYM
cana-5363	236	32	1𝒫	1𝒫	NUM
cana-5363	236	33	−	−	PROPN
cana-5363	236	34	𝒫ℱ𝔑𝛿𝒮𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝒮𝑖𝑛𝑡(𝐾	PROPN
cana-5363	236	35	)	)	PUNCT
cana-5363	236	36	,	,	PUNCT
cana-5363	236	37	𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾	𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾	PROPN
cana-5363	236	38	∩	∩	ADJ
cana-5363	236	39	𝐿	𝐿	NOUN
cana-5363	236	40	)	)	PUNCT
cana-5363	236	41	⊆	⊆	NUM
cana-5363	236	42	𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾	𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾	PROPN
cana-5363	236	43	)	)	PUNCT
cana-5363	236	44	∪	∪	ADP
cana-5363	236	45	𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐿	𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐿	PROPN
cana-5363	236	46	)	)	PUNCT
cana-5363	236	47	,	,	PUNCT
cana-5363	236	48	lemma	lemma	PROPN
cana-5363	236	49	3.3	3.3	NUM
cana-5363	236	50	the	the	DET
cana-5363	236	51	following	follow	VERB
cana-5363	236	52	hold	hold	NOUN
cana-5363	236	53	for	for	ADP
cana-5363	236	54	a	a	DET
cana-5363	236	55	𝑝𝑓𝑠	𝑝𝑓𝑠	NOUN
cana-5363	236	56	𝐻	𝐻	PROPN
cana-5363	236	57	in	in	ADP
cana-5363	236	58	a	a	DET
cana-5363	236	59	𝒫ℱ𝒩𝑡𝑠	𝒫ℱ𝒩𝑡𝑠	PROPN
cana-5363	236	60	(	(	PUNCT
cana-5363	236	61	𝑈	𝑈	PROPN
cana-5363	236	62	,	,	PUNCT
cana-5363	236	63	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	236	64	)	)	PUNCT
cana-5363	236	65	)	)	PUNCT
cana-5363	236	66	.	.	PUNCT
cana-5363	237	1	1	1	X
cana-5363	237	2	.	.	X
cana-5363	237	3	𝒫ℱ𝔑𝒫𝑐𝑙(𝐻	𝒫ℱ𝔑𝒫𝑐𝑙(𝐻	NOUN
cana-5363	237	4	)	)	PUNCT
cana-5363	237	5	=	=	SYM
cana-5363	237	6	𝐻	𝐻	PROPN
cana-5363	237	7	∩	∩	NOUN
cana-5363	237	8	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝐻	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝐻	NUM
cana-5363	237	9	)	)	PUNCT
cana-5363	237	10	)	)	PUNCT
cana-5363	237	11	and	and	CCONJ
cana-5363	237	12	𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐻	𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐻	VERB
cana-5363	237	13	)	)	PUNCT
cana-5363	237	14	=	=	SYM
cana-5363	237	15	𝐻	𝐻	PROPN
cana-5363	237	16	∩	∩	NOUN
cana-5363	237	17	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐻	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐻	NOUN
cana-5363	237	18	)	)	PUNCT
cana-5363	237	19	)	)	PUNCT
cana-5363	237	20	,	,	PUNCT
cana-5363	237	21	communications	communication	NOUN
cana-5363	237	22	on	on	ADP
cana-5363	237	23	applied	apply	VERB
cana-5363	237	24	nonlinear	nonlinear	ADJ
cana-5363	237	25	analysis	analysis	NOUN
cana-5363	237	26	issn	issn	NOUN
cana-5363	237	27	:	:	PUNCT
cana-5363	237	28	1074	1074	NUM
cana-5363	237	29	-	-	PUNCT
cana-5363	237	30	133x	133x	NUM
cana-5363	237	31	vol	vol	VERB
cana-5363	237	32	32	32	NUM
cana-5363	237	33	no	no	NOUN
cana-5363	237	34	.	.	PUNCT
cana-5363	238	1	10s	10	NOUN
cana-5363	238	2	(	(	PUNCT
cana-5363	238	3	2025	2025	NUM
cana-5363	238	4	)	)	PUNCT
cana-5363	238	5	2013	2013	NUM
cana-5363	238	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5363	238	7	2	2	X
cana-5363	238	8	.	.	X
cana-5363	239	1	𝒫ℱ𝔑𝛿𝒮𝑖𝑛𝑡(𝐻	𝒫ℱ𝔑𝛿𝒮𝑖𝑛𝑡(𝐻	ADJ
cana-5363	239	2	)	)	PUNCT
cana-5363	240	1	=	=	SYM
cana-5363	240	2	𝐻	𝐻	PROPN
cana-5363	240	3	∩	∩	NOUN
cana-5363	240	4	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐻	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐻	NOUN
cana-5363	240	5	)	)	PUNCT
cana-5363	240	6	)	)	PUNCT
cana-5363	240	7	and	and	CCONJ
cana-5363	240	8	𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐻	𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐻	NOUN
cana-5363	240	9	)	)	PUNCT
cana-5363	240	10	=	=	SYM
cana-5363	241	1	𝐻	𝐻	PROPN
cana-5363	241	2	∪	∪	VERB
cana-5363	241	3	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐻	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐻	NOUN
cana-5363	241	4	)	)	PUNCT
cana-5363	241	5	)	)	PUNCT
cana-5363	241	6	.	.	PUNCT
cana-5363	242	1	lemma	lemma	PROPN
cana-5363	242	2	3.4	3.4	NUM
cana-5363	242	3	the	the	DET
cana-5363	242	4	following	follow	VERB
cana-5363	242	5	hold	hold	NOUN
cana-5363	242	6	for	for	ADP
cana-5363	242	7	a	a	DET
cana-5363	242	8	𝑝𝑓𝑠	𝑝𝑓𝑠	NOUN
cana-5363	242	9	𝐻	𝐻	PROPN
cana-5363	242	10	in	in	ADP
cana-5363	242	11	a	a	DET
cana-5363	242	12	𝒫ℱ𝒩𝑡𝑠	𝒫ℱ𝒩𝑡𝑠	PROPN
cana-5363	242	13	(	(	PUNCT
cana-5363	242	14	𝑈	𝑈	PROPN
cana-5363	242	15	,	,	PUNCT
cana-5363	242	16	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	242	17	)	)	PUNCT
cana-5363	242	18	)	)	PUNCT
cana-5363	242	19	.	.	PUNCT
cana-5363	243	1	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐻	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐻	NOUN
cana-5363	243	2	)	)	PUNCT
cana-5363	243	3	)	)	PUNCT
cana-5363	244	1	=	=	SYM
cana-5363	244	2	𝒫ℱ𝔑𝛿𝑐𝑙(𝒫ℱ𝔑	𝒫ℱ𝔑𝛿𝑐𝑙(𝒫ℱ𝔑	NUM
cana-5363	244	3	𝛿𝑖𝑛𝑡(𝐻	𝛿𝑖𝑛𝑡(𝐻	NOUN
cana-5363	244	4	)	)	PUNCT
cana-5363	244	5	)	)	PUNCT
cana-5363	244	6	and	and	CCONJ
cana-5363	244	7	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐻	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐻	NOUN
cana-5363	244	8	)	)	PUNCT
cana-5363	244	9	)	)	PUNCT
cana-5363	245	1	=	=	PUNCT
cana-5363	245	2	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐻	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐻	NOUN
cana-5363	245	3	)	)	PUNCT
cana-5363	245	4	)	)	PUNCT
cana-5363	245	5	.	.	PUNCT
cana-5363	246	1	theorem	theorem	VERB
cana-5363	246	2	3.9	3.9	NUM
cana-5363	246	3	let	let	VERB
cana-5363	246	4	(	(	PUNCT
cana-5363	246	5	𝑈	𝑈	NOUN
cana-5363	246	6	,	,	PUNCT
cana-5363	246	7	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	246	8	)	)	PUNCT
cana-5363	246	9	)	)	PUNCT
cana-5363	247	1	be	be	AUX
cana-5363	247	2	a	a	DET
cana-5363	247	3	𝒫ℱ𝔑𝑡𝑠.	𝒫ℱ𝔑𝑡𝑠.	PROPN
cana-5363	247	4	then	then	ADV
cana-5363	247	5	:	:	PUNCT
cana-5363	247	6	(	(	PUNCT
cana-5363	247	7	i	i	NOUN
cana-5363	247	8	)	)	PUNCT
cana-5363	247	9	if	if	SCONJ
cana-5363	247	10	𝐾	𝐾	PROPN
cana-5363	247	11	∈	∈	PROPN
cana-5363	247	12	𝒫ℱ𝔑𝛿𝑂(𝑈	𝒫ℱ𝔑𝛿𝑂(𝑈	PROPN
cana-5363	247	13	,	,	PUNCT
cana-5363	247	14	𝐴	𝐴	PROPN
cana-5363	247	15	)	)	PUNCT
cana-5363	247	16	and	and	CCONJ
cana-5363	247	17	𝐿	𝐿	PROPN
cana-5363	247	18	∈	∈	PROPN
cana-5363	247	19	𝒫ℱ𝔑𝑍𝑂(𝑈	𝒫ℱ𝔑𝑍𝑂(𝑈	NUM
cana-5363	247	20	,	,	PUNCT
cana-5363	247	21	𝐴	𝐴	PROPN
cana-5363	247	22	)	)	PUNCT
cana-5363	247	23	,	,	PUNCT
cana-5363	247	24	then	then	ADV
cana-5363	247	25	𝐾	𝐾	PROPN
cana-5363	247	26	∩	∩	ADJ
cana-5363	247	27	𝐿	𝐿	PROPN
cana-5363	247	28	is	be	AUX
cana-5363	247	29	𝒫ℱ𝔑𝑍𝑜𝑠	𝒫ℱ𝔑𝑍𝑜𝑠	NUM
cana-5363	247	30	,	,	PUNCT
cana-5363	247	31	(	(	PUNCT
cana-5363	247	32	ii	ii	NOUN
cana-5363	247	33	)	)	PUNCT
cana-5363	247	34	if	if	SCONJ
cana-5363	247	35	𝐾	𝐾	PROPN
cana-5363	247	36	∈	∈	PROPN
cana-5363	247	37	𝒫ℱ𝔑𝑎𝑂(𝑈	𝒫ℱ𝔑𝑎𝑂(𝑈	NOUN
cana-5363	247	38	,	,	PUNCT
cana-5363	247	39	𝐴	𝐴	PROPN
cana-5363	247	40	)	)	PUNCT
cana-5363	247	41	and	and	CCONJ
cana-5363	247	42	𝐿	𝐿	PROPN
cana-5363	247	43	∈	∈	PROPN
cana-5363	247	44	𝒫ℱ𝔑𝑍𝑂(𝑈	𝒫ℱ𝔑𝑍𝑂(𝑈	NUM
cana-5363	247	45	,	,	PUNCT
cana-5363	247	46	𝐴	𝐴	PROPN
cana-5363	247	47	)	)	PUNCT
cana-5363	247	48	,	,	PUNCT
cana-5363	247	49	then	then	ADV
cana-5363	247	50	𝐾	𝐾	PROPN
cana-5363	247	51	∩	∩	ADJ
cana-5363	247	52	𝐿	𝐿	PROPN
cana-5363	247	53	∈	∈	PROPN
cana-5363	247	54	𝒫ℱ𝔑𝑍𝑜𝑠.	𝒫ℱ𝔑𝑍𝑜𝑠.	NOUN
cana-5363	247	55	proof	proof	NOUN
cana-5363	247	56	.	.	PUNCT
cana-5363	248	1	(	(	PUNCT
cana-5363	248	2	i	i	NOUN
cana-5363	248	3	)	)	PUNCT
cana-5363	248	4	suppose	suppose	VERB
cana-5363	248	5	that	that	SCONJ
cana-5363	248	6	𝐾	𝐾	PROPN
cana-5363	248	7	∈	∈	PROPN
cana-5363	248	8	𝒫ℱ𝔑𝛿𝑂(𝑈	𝒫ℱ𝔑𝛿𝑂(𝑈	PROPN
cana-5363	248	9	,	,	PUNCT
cana-5363	248	10	𝐴	𝐴	PROPN
cana-5363	248	11	)	)	PUNCT
cana-5363	248	12	.	.	PUNCT
cana-5363	249	1	then	then	ADV
cana-5363	249	2	𝐾	𝐾	PROPN
cana-5363	249	3	=	=	PUNCT
cana-5363	249	4	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	X
cana-5363	249	5	)	)	PUNCT
cana-5363	249	6	.	.	PUNCT
cana-5363	250	1	since	since	SCONJ
cana-5363	250	2	𝐿	𝐿	PROPN
cana-5363	250	3	∈	∈	PROPN
cana-5363	250	4	𝒫ℱ𝔑𝑍𝑂(𝑈	𝒫ℱ𝔑𝑍𝑂(𝑈	NUM
cana-5363	250	5	,	,	PUNCT
cana-5363	250	6	𝐴	𝐴	PROPN
cana-5363	250	7	)	)	PUNCT
cana-5363	250	8	,	,	PUNCT
cana-5363	250	9	then	then	ADV
cana-5363	250	10	𝐿	𝐿	PROPN
cana-5363	250	11	⊆	⊆	NUM
cana-5363	250	12	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑	PROPN
cana-5363	250	13	𝛿𝑖𝑛𝑡(𝐿	𝛿𝑖𝑛𝑡(𝐿	NOUN
cana-5363	250	14	)	)	PUNCT
cana-5363	250	15	)	)	PUNCT
cana-5363	250	16	∪	∪	ADP
cana-5363	250	17	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿	NUM
cana-5363	250	18	)	)	PUNCT
cana-5363	250	19	)	)	PUNCT
cana-5363	250	20	and	and	CCONJ
cana-5363	250	21	hence	hence	ADV
cana-5363	250	22	𝐾	𝐾	PROPN
cana-5363	250	23	∩	∩	ADJ
cana-5363	250	24	𝐿	𝐿	NOUN
cana-5363	250	25	⊆	⊆	NUM
cana-5363	250	26	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	NUM
cana-5363	250	27	)	)	PUNCT
cana-5363	250	28	∩	∩	NOUN
cana-5363	250	29	(	(	PUNCT
cana-5363	250	30	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	NOUN
cana-5363	250	31	)	)	PUNCT
cana-5363	250	32	)	)	PUNCT
cana-5363	250	33	∪	∪	ADP
cana-5363	250	34	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿	NUM
cana-5363	250	35	)	)	PUNCT
cana-5363	250	36	)	)	PUNCT
cana-5363	250	37	)	)	PUNCT
cana-5363	251	1	=	=	SYM
cana-5363	251	2	(	(	PUNCT
cana-5363	251	3	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	NOUN
cana-5363	251	4	)	)	PUNCT
cana-5363	251	5	∩	∩	ADJ
cana-5363	251	6	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	PROPN
cana-5363	251	7	)	)	PUNCT
cana-5363	251	8	)	)	PUNCT
cana-5363	251	9	)	)	PUNCT
cana-5363	252	1	∪	∪	ADP
cana-5363	252	2	(	(	PUNCT
cana-5363	252	3	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	NOUN
cana-5363	252	4	)	)	PUNCT
cana-5363	252	5	∩	∩	NOUN
cana-5363	252	6	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿	NUM
cana-5363	252	7	)	)	PUNCT
cana-5363	252	8	)	)	PUNCT
cana-5363	253	1	⊆	⊆	X
cana-5363	253	2	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	PROPN
cana-5363	253	3	)	)	PUNCT
cana-5363	253	4	∩	∩	NOUN
cana-5363	253	5	(	(	PUNCT
cana-5363	253	6	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	ADJ
cana-5363	253	7	)	)	PUNCT
cana-5363	253	8	)	)	PUNCT
cana-5363	253	9	)	)	PUNCT
cana-5363	253	10	∪	∪	ADP
cana-5363	253	11	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	PROPN
cana-5363	253	12	)	)	PUNCT
cana-5363	253	13	∩	∩	NOUN
cana-5363	253	14	𝒫ℱ𝔑𝑐𝑙(𝐿	𝒫ℱ𝔑𝑐𝑙(𝐿	NOUN
cana-5363	253	15	)	)	PUNCT
cana-5363	253	16	)	)	PUNCT
cana-5363	254	1	⊆	⊆	NUM
cana-5363	254	2	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	PROPN
cana-5363	254	3	∩	∩	ADJ
cana-5363	254	4	𝐿	𝐿	PROPN
cana-5363	254	5	)	)	PUNCT
cana-5363	254	6	)	)	PUNCT
cana-5363	254	7	∪	∪	ADP
cana-5363	254	8	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	X
cana-5363	254	9	∩	∩	PROPN
cana-5363	254	10	𝐿	𝐿	PROPN
cana-5363	254	11	)	)	PUNCT
cana-5363	254	12	)	)	PUNCT
cana-5363	254	13	.	.	PUNCT
cana-5363	255	1	thus	thus	ADV
cana-5363	255	2	𝐾	𝐾	PROPN
cana-5363	255	3	∩	∩	ADJ
cana-5363	255	4	𝐿	𝐿	PROPN
cana-5363	255	5	⊆	⊆	NUM
cana-5363	255	6	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	PROPN
cana-5363	255	7	∩	∩	ADJ
cana-5363	255	8	𝐿	𝐿	PROPN
cana-5363	255	9	)	)	PUNCT
cana-5363	255	10	)	)	PUNCT
cana-5363	255	11	∪	∪	ADP
cana-5363	255	12	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	X
cana-5363	255	13	∩	∩	PROPN
cana-5363	255	14	𝐿	𝐿	PROPN
cana-5363	255	15	)	)	PUNCT
cana-5363	255	16	)	)	PUNCT
cana-5363	255	17	.	.	PUNCT
cana-5363	256	1	therefore	therefore	ADV
cana-5363	256	2	,	,	PUNCT
cana-5363	256	3	𝐾	𝐾	PROPN
cana-5363	256	4	∩	∩	ADJ
cana-5363	256	5	𝐿	𝐿	PROPN
cana-5363	256	6	is	be	AUX
cana-5363	256	7	𝒫ℱ𝔑𝑍𝑜𝑠.	𝒫ℱ𝔑𝑍𝑜𝑠.	NOUN
cana-5363	256	8	(	(	PUNCT
cana-5363	256	9	ii	ii	NOUN
cana-5363	256	10	)	)	PUNCT
cana-5363	256	11	suppose	suppose	VERB
cana-5363	256	12	that	that	SCONJ
cana-5363	256	13	𝐾	𝐾	PROPN
cana-5363	256	14	∈	∈	PROPN
cana-5363	256	15	𝒫ℱ𝔑𝑎𝑂(𝑈	𝒫ℱ𝔑𝑎𝑂(𝑈	NOUN
cana-5363	256	16	,	,	PUNCT
cana-5363	256	17	𝐴	𝐴	PROPN
cana-5363	256	18	)	)	PUNCT
cana-5363	256	19	and	and	CCONJ
cana-5363	256	20	𝐿	𝐿	PROPN
cana-5363	256	21	∈	∈	PROPN
cana-5363	256	22	𝒫ℱ𝔑𝑍𝑂(𝑈	𝒫ℱ𝔑𝑍𝑂(𝑈	NUM
cana-5363	256	23	,	,	PUNCT
cana-5363	256	24	𝐴	𝐴	PROPN
cana-5363	256	25	)	)	PUNCT
cana-5363	256	26	,	,	PUNCT
cana-5363	256	27	the	the	DET
cana-5363	256	28	𝐾	𝐾	PROPN
cana-5363	256	29	∩	∩	NOUN
cana-5363	256	30	𝐿	𝐿	PROPN
cana-5363	256	31	⊆	⊆	NUM
cana-5363	256	32	𝒫ℱ𝔑𝑖𝑛𝑡	𝒫ℱ𝔑𝑖𝑛𝑡	PROPN
cana-5363	256	33	(	(	PUNCT
cana-5363	256	34	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	PROPN
cana-5363	256	35	)	)	PUNCT
cana-5363	256	36	)	)	PUNCT
cana-5363	256	37	)	)	PUNCT
cana-5363	256	38	∩	∩	NOUN
cana-5363	256	39	(	(	PUNCT
cana-5363	256	40	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	NOUN
cana-5363	256	41	)	)	PUNCT
cana-5363	256	42	)	)	PUNCT
cana-5363	256	43	∪	∪	ADP
cana-5363	256	44	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿	NUM
cana-5363	256	45	)	)	PUNCT
cana-5363	256	46	)	)	PUNCT
cana-5363	256	47	)	)	PUNCT
cana-5363	257	1	=	=	PRON
cana-5363	257	2	(	(	PUNCT
cana-5363	257	3	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	PROPN
cana-5363	257	4	)	)	PUNCT
cana-5363	257	5	)	)	PUNCT
cana-5363	257	6	)	)	PUNCT
cana-5363	258	1	∩	∩	ADJ
cana-5363	258	2	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	PROPN
cana-5363	258	3	)	)	PUNCT
cana-5363	258	4	)	)	PUNCT
cana-5363	258	5	)	)	PUNCT
cana-5363	259	1	∪	∪	ADV
cana-5363	259	2	(	(	PUNCT
cana-5363	259	3	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙	PROPN
cana-5363	259	4	(	(	PUNCT
cana-5363	259	5	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	NOUN
cana-5363	259	6	)	)	PUNCT
cana-5363	259	7	)	)	PUNCT
cana-5363	259	8	)	)	PUNCT
cana-5363	259	9	∩	∩	X
cana-5363	259	10	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿	NOUN
cana-5363	259	11	)	)	PUNCT
cana-5363	259	12	)	)	PUNCT
cana-5363	259	13	)	)	PUNCT
cana-5363	260	1	⊆	⊆	NUM
cana-5363	260	2	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	NOUN
cana-5363	260	3	)	)	PUNCT
cana-5363	260	4	)	)	PUNCT
cana-5363	260	5	∩	∩	NOUN
cana-5363	260	6	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	ADJ
cana-5363	260	7	)	)	PUNCT
cana-5363	260	8	)	)	PUNCT
cana-5363	260	9	∪	∪	ADP
cana-5363	260	10	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	NOUN
cana-5363	260	11	)	)	PUNCT
cana-5363	260	12	)	)	PUNCT
cana-5363	260	13	∩	∩	X
cana-5363	260	14	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿	NOUN
cana-5363	260	15	)	)	PUNCT
cana-5363	260	16	)	)	PUNCT
cana-5363	260	17	)	)	PUNCT
cana-5363	261	1	⊆	⊆	NUM
cana-5363	261	2	𝒫ℱ𝔑𝑐𝑙	𝒫ℱ𝔑𝑐𝑙	PROPN
cana-5363	261	3	(	(	PUNCT
cana-5363	261	4	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	PROPN
cana-5363	261	5	)	)	PUNCT
cana-5363	261	6	∩	∩	NOUN
cana-5363	261	7	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	ADJ
cana-5363	261	8	)	)	PUNCT
cana-5363	261	9	)	)	PUNCT
cana-5363	261	10	)	)	PUNCT
cana-5363	261	11	∪	∪	ADP
cana-5363	261	12	𝒫ℱ𝔑𝑖𝑛𝑡	𝒫ℱ𝔑𝑖𝑛𝑡	PROPN
cana-5363	261	13	(	(	PUNCT
cana-5363	261	14	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	PROPN
cana-5363	261	15	)	)	PUNCT
cana-5363	261	16	)	)	PUNCT
cana-5363	261	17	∩	∩	X
cana-5363	261	18	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿	NOUN
cana-5363	261	19	)	)	PUNCT
cana-5363	261	20	)	)	PUNCT
cana-5363	261	21	)	)	PUNCT
cana-5363	261	22	and	and	CCONJ
cana-5363	261	23	hence	hence	ADV
cana-5363	261	24	𝐾	𝐾	PROPN
cana-5363	261	25	∩	∩	ADJ
cana-5363	261	26	𝐿	𝐿	PROPN
cana-5363	261	27	⊆	⊆	NUM
cana-5363	261	28	(	(	PUNCT
cana-5363	261	29	𝐾	𝐾	PROPN
cana-5363	261	30	∩	∩	ADJ
cana-5363	261	31	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	PROPN
cana-5363	261	32	)	)	PUNCT
cana-5363	261	33	∩	∩	NOUN
cana-5363	261	34	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	ADJ
cana-5363	261	35	)	)	PUNCT
cana-5363	261	36	)	)	PUNCT
cana-5363	261	37	)	)	PUNCT
cana-5363	261	38	∪	∪	ADV
cana-5363	261	39	(	(	PUNCT
cana-5363	261	40	𝐾	𝐾	PROPN
cana-5363	261	41	∩	∩	NOUN
cana-5363	261	42	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	NOUN
cana-5363	261	43	)	)	PUNCT
cana-5363	261	44	)	)	PUNCT
cana-5363	261	45	∩	∩	X
cana-5363	261	46	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿	NOUN
cana-5363	261	47	)	)	PUNCT
cana-5363	261	48	)	)	PUNCT
cana-5363	261	49	)	)	PUNCT
cana-5363	261	50	)	)	PUNCT
cana-5363	262	1	⊆	⊆	X
cana-5363	262	2	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	PROPN
cana-5363	262	3	)	)	PUNCT
cana-5363	262	4	∩	∩	NOUN
cana-5363	262	5	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	ADJ
cana-5363	262	6	)	)	PUNCT
cana-5363	262	7	)	)	PUNCT
cana-5363	262	8	∪	∪	ADP
cana-5363	262	9	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	PROPN
cana-5363	262	10	)	)	PUNCT
cana-5363	262	11	∩	∩	PROPN
cana-5363	262	12	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐿	NOUN
cana-5363	262	13	)	)	PUNCT
cana-5363	262	14	)	)	PUNCT
cana-5363	262	15	)	)	PUNCT
cana-5363	262	16	)	)	PUNCT
cana-5363	263	1	⊆	⊆	X
cana-5363	263	2	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	PROPN
cana-5363	263	3	)	)	PUNCT
cana-5363	263	4	∩	∩	NOUN
cana-5363	263	5	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	ADJ
cana-5363	263	6	)	)	PUNCT
cana-5363	263	7	)	)	PUNCT
cana-5363	263	8	∪	∪	ADP
cana-5363	263	9	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	PROPN
cana-5363	263	10	)	)	PUNCT
cana-5363	263	11	∩	∩	NOUN
cana-5363	263	12	𝒫ℱ𝔑𝑐𝑙(𝐿	𝒫ℱ𝔑𝑐𝑙(𝐿	NUM
cana-5363	263	13	)	)	PUNCT
cana-5363	263	14	)	)	PUNCT
cana-5363	263	15	)	)	PUNCT
cana-5363	264	1	⊆	⊆	NUM
cana-5363	264	2	𝒫ℱ𝔑𝑐𝑙(𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝔑𝛿𝑖𝑛𝑡(𝐾	NOUN
cana-5363	264	3	)	)	PUNCT
cana-5363	264	4	∩	∩	NOUN
cana-5363	264	5	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	ADJ
cana-5363	264	6	)	)	PUNCT
cana-5363	264	7	)	)	PUNCT
cana-5363	265	1	∩	∩	PROPN
cana-5363	265	2	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	NOUN
cana-5363	265	3	)	)	PUNCT
cana-5363	265	4	∩	∩	PROPN
cana-5363	265	5	𝐿	𝐿	PROPN
cana-5363	265	6	)	)	PUNCT
cana-5363	265	7	)	)	PUNCT
cana-5363	265	8	.	.	PUNCT
cana-5363	266	1	since	since	SCONJ
cana-5363	266	2	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	NOUN
cana-5363	266	3	)	)	PUNCT
cana-5363	266	4	∩	∩	NOUN
cana-5363	266	5	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	X
cana-5363	266	6	)	)	PUNCT
cana-5363	266	7	⊆	⊆	NUM
cana-5363	266	8	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	NOUN
cana-5363	266	9	)	)	PUNCT
cana-5363	266	10	⊆	⊆	NUM
cana-5363	266	11	𝐾	𝐾	PROPN
cana-5363	266	12	which	which	PRON
cana-5363	266	13	is	be	AUX
cana-5363	266	14	𝒫ℱ𝔑𝛿𝑜	𝒫ℱ𝔑𝛿𝑜	PROPN
cana-5363	266	15	in	in	ADP
cana-5363	266	16	𝐾	𝐾	PROPN
cana-5363	266	17	,	,	PUNCT
cana-5363	266	18	communications	communication	NOUN
cana-5363	266	19	on	on	ADP
cana-5363	266	20	applied	apply	VERB
cana-5363	266	21	nonlinear	nonlinear	ADJ
cana-5363	266	22	analysis	analysis	NOUN
cana-5363	266	23	issn	issn	NOUN
cana-5363	266	24	:	:	PUNCT
cana-5363	266	25	1074	1074	NUM
cana-5363	266	26	-	-	PUNCT
cana-5363	266	27	133x	133x	NUM
cana-5363	266	28	vol	vol	VERB
cana-5363	266	29	32	32	NUM
cana-5363	266	30	no	no	NOUN
cana-5363	266	31	.	.	PUNCT
cana-5363	266	32	10s	10	NOUN
cana-5363	266	33	(	(	PUNCT
cana-5363	266	34	2025	2025	NUM
cana-5363	266	35	)	)	PUNCT
cana-5363	266	36	2014	2014	NUM
cana-5363	267	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5363	267	2	then	then	ADV
cana-5363	267	3	𝐾	𝐾	PROPN
cana-5363	267	4	∩	∩	ADJ
cana-5363	267	5	𝐿	𝐿	PROPN
cana-5363	267	6	⊆	⊆	NUM
cana-5363	267	7	𝒫ℱ𝔑𝑐𝑙𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	PROPN
cana-5363	267	8	)	)	PUNCT
cana-5363	267	9	∩	∩	NOUN
cana-5363	267	10	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐿	ADJ
cana-5363	267	11	)	)	PUNCT
cana-5363	267	12	)	)	PUNCT
cana-5363	267	13	∪	∪	ADP
cana-5363	267	14	𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	NOUN
cana-5363	267	15	∪	∪	ADP
cana-5363	267	16	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	PROPN
cana-5363	267	17	)	)	PUNCT
cana-5363	267	18	∩	∩	ADJ
cana-5363	267	19	𝐿	𝐿	PROPN
cana-5363	267	20	)	)	PUNCT
cana-5363	267	21	)	)	PUNCT
cana-5363	268	1	⊆	⊆	NUM
cana-5363	268	2	𝒫ℱ𝔑𝑐𝑙𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	NOUN
cana-5363	268	3	∩	∩	ADJ
cana-5363	268	4	𝐿	𝐿	PROPN
cana-5363	268	5	)	)	PUNCT
cana-5363	268	6	∪	∪	ADP
cana-5363	268	7	𝒫ℱ𝔑𝑖𝑛𝑡𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	NOUN
cana-5363	268	8	)	)	PUNCT
cana-5363	268	9	∩	∩	ADJ
cana-5363	268	10	𝐿	𝐿	PROPN
cana-5363	268	11	)	)	PUNCT
cana-5363	268	12	⊆	⊆	NUM
cana-5363	268	13	𝒫ℱ𝔑𝑐𝑙𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	NOUN
cana-5363	268	14	∩	∩	ADJ
cana-5363	268	15	𝐿	𝐿	NOUN
cana-5363	268	16	)	)	PUNCT
cana-5363	268	17	∪	∪	ADJ
cana-5363	268	18	𝒫ℱ𝔑𝑖𝑛𝑡𝒫ℱ𝔑𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡𝒫ℱ𝔑𝑐𝑙(𝐾	NUM
cana-5363	268	19	∩	∩	ADJ
cana-5363	268	20	𝐿	𝐿	PROPN
cana-5363	268	21	)	)	PUNCT
cana-5363	268	22	.	.	PUNCT
cana-5363	269	1	therefore	therefore	ADV
cana-5363	269	2	𝐾	𝐾	PROPN
cana-5363	269	3	∩	∩	NOUN
cana-5363	269	4	𝐿	𝐿	PROPN
cana-5363	269	5	∈	∈	PROPN
cana-5363	269	6	𝒫ℱ𝔑𝑍𝑂(𝑈	𝒫ℱ𝔑𝑍𝑂(𝑈	NUM
cana-5363	269	7	,	,	PUNCT
cana-5363	269	8	𝐴	𝐴	PROPN
cana-5363	269	9	)	)	PUNCT
cana-5363	269	10	.	.	PUNCT
cana-5363	270	1	proposition	proposition	NOUN
cana-5363	270	2	3.2	3.2	NUM
cana-5363	270	3	let	let	VERB
cana-5363	270	4	(	(	PUNCT
cana-5363	270	5	𝑈	𝑈	PROPN
cana-5363	270	6	,	,	PUNCT
cana-5363	270	7	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	270	8	)	)	PUNCT
cana-5363	270	9	)	)	PUNCT
cana-5363	270	10	be	be	AUX
cana-5363	270	11	a	a	DET
cana-5363	270	12	𝒫ℱ𝔑𝑡𝑠	𝒫ℱ𝔑𝑡𝑠	PROPN
cana-5363	270	13	.	.	PUNCT
cana-5363	271	1	then	then	ADV
cana-5363	271	2	the	the	DET
cana-5363	271	3	closure	closure	NOUN
cana-5363	271	4	of	of	ADP
cana-5363	271	5	a	a	DET
cana-5363	271	6	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	271	7	set	set	NOUN
cana-5363	271	8	of	of	ADP
cana-5363	271	9	𝐴	𝐴	PROPN
cana-5363	271	10	is	be	AUX
cana-5363	271	11	𝒫ℱ𝔑𝒮𝑜𝑠.	𝒫ℱ𝔑𝒮𝑜𝑠.	ADJ
cana-5363	271	12	proof	proof	NOUN
cana-5363	271	13	.	.	PUNCT
cana-5363	272	1	let	let	VERB
cana-5363	272	2	𝐾	𝐾	PROPN
cana-5363	272	3	∈	∈	PROPN
cana-5363	272	4	𝒫ℱ𝔑𝑍𝑂(𝑈	𝒫ℱ𝔑𝑍𝑂(𝑈	NUM
cana-5363	272	5	,	,	PUNCT
cana-5363	272	6	𝐴	𝐴	PROPN
cana-5363	272	7	)	)	PUNCT
cana-5363	272	8	.	.	PUNCT
cana-5363	273	1	then	then	ADV
cana-5363	273	2	𝒫ℱ𝔑𝑐𝑙(𝐾	𝒫ℱ𝔑𝑐𝑙(𝐾	NOUN
cana-5363	273	3	)	)	PUNCT
cana-5363	273	4	⊆	⊆	NUM
cana-5363	273	5	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	NOUN
cana-5363	273	6	)	)	PUNCT
cana-5363	273	7	)	)	PUNCT
cana-5363	274	1	⊆	⊆	NUM
cana-5363	274	2	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	NOUN
cana-5363	274	3	)	)	PUNCT
cana-5363	274	4	)	)	PUNCT
cana-5363	274	5	)	)	PUNCT
cana-5363	275	1	⊆	⊆	X
cana-5363	275	2	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	NOUN
cana-5363	275	3	)	)	PUNCT
cana-5363	275	4	)	)	PUNCT
cana-5363	275	5	∪	∪	ADP
cana-5363	275	6	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	PROPN
cana-5363	275	7	)	)	PUNCT
cana-5363	275	8	)	)	PUNCT
cana-5363	275	9	)	)	PUNCT
cana-5363	276	1	=	=	SYM
cana-5363	276	2	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	PROPN
cana-5363	276	3	)	)	PUNCT
cana-5363	276	4	)	)	PUNCT
cana-5363	276	5	)	)	PUNCT
cana-5363	276	6	.	.	PUNCT
cana-5363	277	1	therefore	therefore	ADV
cana-5363	277	2	,	,	PUNCT
cana-5363	277	3	𝒫ℱ𝔑𝑐𝑙(𝐾	𝒫ℱ𝔑𝑐𝑙(𝐾	NOUN
cana-5363	277	4	)	)	PUNCT
cana-5363	277	5	is	be	AUX
cana-5363	277	6	𝒫ℱ𝔑𝒮𝑜𝑠.	𝒫ℱ𝔑𝒮𝑜𝑠.	NOUN
cana-5363	277	7	proposition	proposition	NOUN
cana-5363	277	8	3.3	3.3	NUM
cana-5363	277	9	let	let	VERB
cana-5363	277	10	𝐾	𝐾	PRON
cana-5363	277	11	be	be	AUX
cana-5363	277	12	a	a	DET
cana-5363	277	13	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	277	14	set	set	NOUN
cana-5363	277	15	of	of	ADP
cana-5363	277	16	a	a	DET
cana-5363	277	17	𝒫ℱ𝔑𝑡𝑠	𝒫ℱ𝔑𝑡𝑠	PROPN
cana-5363	277	18	(	(	PUNCT
cana-5363	277	19	𝑈	𝑈	PROPN
cana-5363	277	20	,	,	PUNCT
cana-5363	277	21	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	277	22	)	)	PUNCT
cana-5363	277	23	)	)	PUNCT
cana-5363	277	24	and	and	CCONJ
cana-5363	277	25	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	NOUN
cana-5363	277	26	)	)	PUNCT
cana-5363	278	1	=	=	PRON
cana-5363	278	2	0𝑃.	0𝑃.	NOUN
cana-5363	279	1	then	then	ADV
cana-5363	279	2	𝐾	𝐾	PROPN
cana-5363	279	3	is	be	AUX
cana-5363	279	4	𝒫ℱ𝔑𝒫𝑜𝑠.	𝒫ℱ𝔑𝒫𝑜𝑠.	ADJ
cana-5363	279	5	proof	proof	NOUN
cana-5363	279	6	.	.	PUNCT
cana-5363	280	1	obvious	obvious	ADJ
cana-5363	280	2	.	.	PUNCT
cana-5363	281	1	theorem	theorem	VERB
cana-5363	281	2	3.10	3.10	NUM
cana-5363	281	3	let	let	VERB
cana-5363	281	4	𝐾	𝐾	PROPN
cana-5363	281	5	and	and	CCONJ
cana-5363	281	6	𝐿	𝐿	PROPN
cana-5363	281	7	be	be	VERB
cana-5363	281	8	two	two	NUM
cana-5363	281	9	subsets	subset	NOUN
cana-5363	281	10	of	of	ADP
cana-5363	281	11	a	a	DET
cana-5363	281	12	𝒫ℱ𝔑𝑡𝑠	𝒫ℱ𝔑𝑡𝑠	PROPN
cana-5363	281	13	(	(	PUNCT
cana-5363	281	14	𝑈	𝑈	PROPN
cana-5363	281	15	,	,	PUNCT
cana-5363	281	16	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	281	17	)	)	PUNCT
cana-5363	281	18	)	)	PUNCT
cana-5363	281	19	.	.	PUNCT
cana-5363	282	1	then	then	ADV
cana-5363	282	2	the	the	DET
cana-5363	282	3	following	follow	VERB
cana-5363	282	4	are	be	AUX
cana-5363	282	5	hold	hold	ADJ
cana-5363	282	6	:	:	PUNCT
cana-5363	282	7	(	(	PUNCT
cana-5363	282	8	i	i	NOUN
cana-5363	282	9	)	)	PUNCT
cana-5363	282	10	𝒫ℱ𝔑𝑍𝑐𝑙(1𝒫	𝒫ℱ𝔑𝑍𝑐𝑙(1𝒫	PROPN
cana-5363	282	11	−	−	PROPN
cana-5363	282	12	𝐾	𝐾	PROPN
cana-5363	282	13	)	)	PUNCT
cana-5363	282	14	=	=	SYM
cana-5363	282	15	1𝒫	1𝒫	NUM
cana-5363	282	16	−	−	PROPN
cana-5363	283	1	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	PROPN
cana-5363	283	2	)	)	PUNCT
cana-5363	283	3	,	,	PUNCT
cana-5363	283	4	(	(	PUNCT
cana-5363	283	5	ii	ii	NOUN
cana-5363	283	6	)	)	PUNCT
cana-5363	283	7	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(1𝒫	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(1𝒫	PROPN
cana-5363	283	8	−	−	PROPN
cana-5363	283	9	𝐾	𝐾	PROPN
cana-5363	283	10	)	)	PUNCT
cana-5363	283	11	=	=	SYM
cana-5363	283	12	1𝒫	1𝒫	NUM
cana-5363	283	13	−	−	PROPN
cana-5363	283	14	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	283	15	)	)	PUNCT
cana-5363	283	16	,	,	PUNCT
cana-5363	283	17	(	(	PUNCT
cana-5363	283	18	iii	iii	X
cana-5363	283	19	)	)	PUNCT
cana-5363	283	20	if	if	SCONJ
cana-5363	283	21	𝐾	𝐾	PROPN
cana-5363	283	22	⊆	⊆	NUM
cana-5363	283	23	𝐿	𝐿	PROPN
cana-5363	283	24	,	,	PUNCT
cana-5363	283	25	then	then	ADV
cana-5363	283	26	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	283	27	)	)	PUNCT
cana-5363	283	28	⊆	⊆	NUM
cana-5363	283	29	𝒫ℱ𝔑𝑍𝑐𝑙(𝐿	𝒫ℱ𝔑𝑍𝑐𝑙(𝐿	PROPN
cana-5363	283	30	)	)	PUNCT
cana-5363	283	31	and	and	CCONJ
cana-5363	283	32	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	PROPN
cana-5363	283	33	)	)	PUNCT
cana-5363	283	34	⊆	⊆	NUM
cana-5363	283	35	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐿	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐿	NOUN
cana-5363	283	36	)	)	PUNCT
cana-5363	283	37	,	,	PUNCT
cana-5363	283	38	(	(	PUNCT
cana-5363	283	39	iv	iv	X
cana-5363	283	40	)	)	PUNCT
cana-5363	283	41	𝑙	𝑙	PRON
cana-5363	283	42	∈	∈	PROPN
cana-5363	283	43	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	283	44	)	)	PUNCT
cana-5363	283	45	iff	iff	NOUN
cana-5363	283	46	for	for	ADP
cana-5363	283	47	each	each	DET
cana-5363	283	48	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	283	49	set	set	VERB
cana-5363	283	50	𝐴	𝐴	PROPN
cana-5363	283	51	contains	contain	VERB
cana-5363	283	52	𝑙	𝑙	X
cana-5363	283	53	,	,	PUNCT
cana-5363	283	54	𝐴	𝐴	PROPN
cana-5363	283	55	∩	∩	NOUN
cana-5363	283	56	𝐾	𝐾	PROPN
cana-5363	283	57	≠	≠	PROPN
cana-5363	283	58	0𝑃	0𝑃	NOUN
cana-5363	283	59	,	,	PUNCT
cana-5363	283	60	(	(	PUNCT
cana-5363	283	61	v	v	NOUN
cana-5363	283	62	)	)	PUNCT
cana-5363	283	63	𝑙	𝑙	PRON
cana-5363	283	64	∈	∈	PROPN
cana-5363	283	65	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	PROPN
cana-5363	283	66	)	)	PUNCT
cana-5363	283	67	iff	iff	PROPN
cana-5363	283	68	there	there	PRON
cana-5363	283	69	exist	exist	VERB
cana-5363	283	70	a	a	DET
cana-5363	283	71	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	283	72	set	set	VERB
cana-5363	283	73	𝑊	𝑊	PRON
cana-5363	283	74	such	such	DET
cana-5363	283	75	that	that	SCONJ
cana-5363	283	76	𝑙	𝑙	PRON
cana-5363	283	77	∈	∈	PROPN
cana-5363	283	78	𝑊	𝑊	PROPN
cana-5363	283	79	⊆	⊆	NUM
cana-5363	283	80	𝐾	𝐾	PROPN
cana-5363	283	81	,	,	PUNCT
cana-5363	283	82	(	(	PUNCT
cana-5363	283	83	vi	vi	NOUN
cana-5363	283	84	)	)	PUNCT
cana-5363	283	85	𝒫ℱ𝔑𝑍𝑐𝑙(𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	NOUN
cana-5363	283	86	)	)	PUNCT
cana-5363	283	87	)	)	PUNCT
cana-5363	283	88	=	=	SYM
cana-5363	283	89	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	283	90	)	)	PUNCT
cana-5363	283	91	and	and	CCONJ
cana-5363	283	92	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	NOUN
cana-5363	283	93	)	)	PUNCT
cana-5363	283	94	)	)	PUNCT
cana-5363	284	1	=	=	SYM
cana-5363	284	2	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	NOUN
cana-5363	284	3	)	)	PUNCT
cana-5363	284	4	,	,	PUNCT
cana-5363	284	5	(	(	PUNCT
cana-5363	284	6	vii	vii	PROPN
cana-5363	284	7	)	)	PUNCT
cana-5363	284	8	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	284	9	)	)	PUNCT
cana-5363	284	10	∪	∪	PROPN
cana-5363	284	11	𝒫ℱ𝔑𝑍𝑐𝑙(𝐿	𝒫ℱ𝔑𝑍𝑐𝑙(𝐿	PROPN
cana-5363	284	12	)	)	PUNCT
cana-5363	284	13	⊆	⊆	NUM
cana-5363	284	14	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	284	15	∪	∪	X
cana-5363	284	16	𝐿	𝐿	PROPN
cana-5363	284	17	)	)	PUNCT
cana-5363	284	18	and	and	CCONJ
cana-5363	284	19	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	NOUN
cana-5363	284	20	)	)	PUNCT
cana-5363	284	21	∪	∪	ADP
cana-5363	284	22	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐿	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐿	NOUN
cana-5363	284	23	)	)	PUNCT
cana-5363	284	24	⊆	⊆	NUM
cana-5363	284	25	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	PROPN
cana-5363	284	26	∪	∪	PROPN
cana-5363	284	27	𝐿	𝐿	PROPN
cana-5363	284	28	)	)	PUNCT
cana-5363	284	29	,	,	PUNCT
cana-5363	284	30	(	(	PUNCT
cana-5363	284	31	viii	viii	NOUN
cana-5363	284	32	)	)	PUNCT
cana-5363	284	33	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	PROPN
cana-5363	284	34	∩	∩	PROPN
cana-5363	284	35	𝐿	𝐿	PROPN
cana-5363	284	36	)	)	PUNCT
cana-5363	284	37	⊆	⊆	NUM
cana-5363	284	38	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	NOUN
cana-5363	284	39	)	)	PUNCT
cana-5363	284	40	∩	∩	ADJ
cana-5363	284	41	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐿	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐿	NOUN
cana-5363	284	42	)	)	PUNCT
cana-5363	284	43	and	and	CCONJ
cana-5363	284	44	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	284	45	∩	∩	ADJ
cana-5363	284	46	𝐿	𝐿	PROPN
cana-5363	284	47	)	)	PUNCT
cana-5363	284	48	⊆	⊆	NUM
cana-5363	284	49	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	284	50	)	)	PUNCT
cana-5363	284	51	∩	∩	PROPN
cana-5363	284	52	𝒫ℱ𝔑𝑍𝑐𝑙(𝐿	𝒫ℱ𝔑𝑍𝑐𝑙(𝐿	PROPN
cana-5363	284	53	)	)	PUNCT
cana-5363	284	54	.	.	PUNCT
cana-5363	285	1	proof	proof	NOUN
cana-5363	285	2	.	.	PUNCT
cana-5363	286	1	(	(	PUNCT
cana-5363	286	2	i	i	NOUN
cana-5363	286	3	)	)	PUNCT
cana-5363	286	4	it	it	PRON
cana-5363	286	5	follows	follow	VERB
cana-5363	286	6	from	from	ADP
cana-5363	286	7	definition	definition	NOUN
cana-5363	286	8	3.5	3.5	NUM
cana-5363	286	9	.	.	PUNCT
cana-5363	287	1	remark	remark	NOUN
cana-5363	287	2	3.5	3.5	NUM
cana-5363	287	3	by	by	ADP
cana-5363	287	4	the	the	DET
cana-5363	287	5	following	following	ADJ
cana-5363	287	6	example	example	NOUN
cana-5363	287	7	we	we	PRON
cana-5363	287	8	show	show	VERB
cana-5363	287	9	that	that	SCONJ
cana-5363	287	10	the	the	DET
cana-5363	287	11	inclusion	inclusion	NOUN
cana-5363	287	12	relation	relation	NOUN
cana-5363	287	13	in	in	ADP
cana-5363	287	14	parts	part	NOUN
cana-5363	287	15	(	(	PUNCT
cana-5363	287	16	vii	vii	PROPN
cana-5363	287	17	)	)	PUNCT
cana-5363	287	18	and	and	CCONJ
cana-5363	287	19	(	(	PUNCT
cana-5363	287	20	viii	viii	NOUN
cana-5363	287	21	)	)	PUNCT
cana-5363	287	22	of	of	ADP
cana-5363	287	23	the	the	DET
cana-5363	287	24	above	above	ADJ
cana-5363	287	25	theorem	theorem	NOUN
cana-5363	287	26	can	can	AUX
cana-5363	287	27	not	not	PART
cana-5363	287	28	be	be	AUX
cana-5363	287	29	replaced	replace	VERB
cana-5363	287	30	by	by	ADP
cana-5363	287	31	equality	equality	NOUN
cana-5363	287	32	.	.	PUNCT
cana-5363	288	1	example	example	NOUN
cana-5363	288	2	3.4	3.4	NUM
cana-5363	288	3	in	in	ADP
cana-5363	288	4	example	example	NOUN
cana-5363	288	5	3.1	3.1	NUM
cana-5363	288	6	,	,	PUNCT
cana-5363	288	7	the	the	DET
cana-5363	288	8	sets	set	NOUN
cana-5363	288	9	1	1	NUM
cana-5363	288	10	.	.	PUNCT
cana-5363	289	1	𝐴	𝐴	PROPN
cana-5363	289	2	=	=	PRON
cana-5363	289	3	{	{	PUNCT
cana-5363	289	4	⟨	⟨	ADP
cana-5363	289	5	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	289	6	0.1	0.1	NUM
cana-5363	289	7	⟩	⟩	NOUN
cana-5363	289	8	,	,	PUNCT
cana-5363	289	9	⟨	⟨	VERB
cana-5363	289	10	𝑠2	𝑠2	PROPN
cana-5363	289	11	0.5	0.5	NUM
cana-5363	289	12	⟩	⟩	NOUN
cana-5363	289	13	,	,	PUNCT
cana-5363	289	14	⟨	⟨	VERB
cana-5363	289	15	𝑠3	𝑠3	PROPN
cana-5363	289	16	0.4	0.4	NUM
cana-5363	289	17	⟩	⟩	NOUN
cana-5363	289	18	}	}	PUNCT
cana-5363	289	19	and	and	CCONJ
cana-5363	289	20	𝐵	𝐵	NOUN
cana-5363	289	21	=	=	PUNCT
cana-5363	289	22	{	{	PUNCT
cana-5363	289	23	⟨	⟨	ADP
cana-5363	289	24	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	289	25	0.2	0.2	NUM
cana-5363	289	26	⟩	⟩	NOUN
cana-5363	289	27	,	,	PUNCT
cana-5363	289	28	⟨	⟨	VERB
cana-5363	289	29	𝑠2	𝑠2	PROPN
cana-5363	289	30	0.4	0.4	NUM
cana-5363	289	31	⟩	⟩	NOUN
cana-5363	289	32	,	,	PUNCT
cana-5363	289	33	⟨	⟨	VERB
cana-5363	289	34	𝑠3	𝑠3	PROPN
cana-5363	289	35	0.6	0.6	NUM
cana-5363	289	36	⟩	⟩	NOUN
cana-5363	289	37	}	}	PUNCT
cana-5363	289	38	,	,	PUNCT
cana-5363	289	39	then	then	ADV
cana-5363	289	40	𝐴	𝐴	PROPN
cana-5363	289	41	∨	∨	NOUN
cana-5363	289	42	𝐵	𝐵	NOUN
cana-5363	289	43	=	=	PUNCT
cana-5363	289	44	{	{	PUNCT
cana-5363	289	45	⟨	⟨	ADP
cana-5363	289	46	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	289	47	0.2	0.2	NUM
cana-5363	289	48	⟩	⟩	NOUN
cana-5363	289	49	,	,	PUNCT
cana-5363	289	50	⟨	⟨	VERB
cana-5363	289	51	𝑠2	𝑠2	PROPN
cana-5363	289	52	0.5	0.5	NUM
cana-5363	289	53	⟩	⟩	NOUN
cana-5363	289	54	,	,	PUNCT
cana-5363	289	55	⟨	⟨	VERB
cana-5363	289	56	𝑠3	𝑠3	PROPN
cana-5363	289	57	0.6	0.6	NUM
cana-5363	289	58	⟩	⟩	NOUN
cana-5363	289	59	}	}	PUNCT
cana-5363	289	60	.	.	PUNCT
cana-5363	290	1	communications	communication	NOUN
cana-5363	290	2	on	on	ADP
cana-5363	290	3	applied	apply	VERB
cana-5363	290	4	nonlinear	nonlinear	ADJ
cana-5363	290	5	analysis	analysis	NOUN
cana-5363	290	6	issn	issn	NOUN
cana-5363	290	7	:	:	PUNCT
cana-5363	290	8	1074	1074	NUM
cana-5363	290	9	-	-	PUNCT
cana-5363	290	10	133x	133x	NUM
cana-5363	290	11	vol	vol	VERB
cana-5363	290	12	32	32	NUM
cana-5363	290	13	no	no	NOUN
cana-5363	290	14	.	.	PUNCT
cana-5363	291	1	10s	10	NOUN
cana-5363	291	2	(	(	PUNCT
cana-5363	291	3	2025	2025	NUM
cana-5363	291	4	)	)	PUNCT
cana-5363	291	5	2015	2015	NUM
cana-5363	291	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5363	291	7	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐴	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐴	NUM
cana-5363	291	8	)	)	PUNCT
cana-5363	291	9	=	=	NOUN
cana-5363	291	10	{	{	PUNCT
cana-5363	291	11	⟨	⟨	ADP
cana-5363	291	12	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	291	13	0.1	0.1	NUM
cana-5363	291	14	⟩	⟩	NOUN
cana-5363	291	15	,	,	PUNCT
cana-5363	291	16	⟨	⟨	VERB
cana-5363	291	17	𝑠2	𝑠2	PROPN
cana-5363	291	18	0.3	0.3	NUM
cana-5363	291	19	⟩	⟩	NOUN
cana-5363	291	20	,	,	PUNCT
cana-5363	291	21	⟨	⟨	VERB
cana-5363	291	22	𝑠3	𝑠3	PROPN
cana-5363	291	23	0.4	0.4	NUM
cana-5363	291	24	⟩	⟩	NOUN
cana-5363	291	25	}	}	PUNCT
cana-5363	291	26	,	,	PUNCT
cana-5363	291	27	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐵	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐵	NOUN
cana-5363	291	28	)	)	PUNCT
cana-5363	291	29	=	=	PRON
cana-5363	291	30	{	{	PUNCT
cana-5363	291	31	⟨	⟨	ADP
cana-5363	291	32	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	291	33	0.2	0.2	NUM
cana-5363	291	34	⟩	⟩	NOUN
cana-5363	291	35	,	,	PUNCT
cana-5363	291	36	⟨	⟨	VERB
cana-5363	291	37	𝑠2	𝑠2	PROPN
cana-5363	291	38	0.4	0.4	NUM
cana-5363	291	39	⟩	⟩	NOUN
cana-5363	291	40	,	,	PUNCT
cana-5363	291	41	⟨	⟨	VERB
cana-5363	291	42	𝑠3	𝑠3	PROPN
cana-5363	291	43	0.6	0.6	NUM
cana-5363	291	44	⟩	⟩	NOUN
cana-5363	291	45	}	}	PUNCT
cana-5363	291	46	and	and	CCONJ
cana-5363	291	47	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐴	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐴	ADV
cana-5363	291	48	∨	∨	NUM
cana-5363	291	49	𝐵	𝐵	NOUN
cana-5363	291	50	)	)	PUNCT
cana-5363	291	51	=	=	NOUN
cana-5363	291	52	{	{	PUNCT
cana-5363	291	53	⟨	⟨	ADP
cana-5363	291	54	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	291	55	0.2	0.2	NUM
cana-5363	291	56	⟩	⟩	NOUN
cana-5363	291	57	,	,	PUNCT
cana-5363	291	58	⟨	⟨	VERB
cana-5363	291	59	𝑠2	𝑠2	PROPN
cana-5363	291	60	0.5	0.5	NUM
cana-5363	291	61	⟩	⟩	NOUN
cana-5363	291	62	,	,	PUNCT
cana-5363	291	63	⟨	⟨	VERB
cana-5363	291	64	𝑠3	𝑠3	PROPN
cana-5363	291	65	0.6	0.6	NUM
cana-5363	291	66	⟩	⟩	NOUN
cana-5363	291	67	}	}	PUNCT
cana-5363	291	68	.	.	PUNCT
cana-5363	292	1	thus	thus	ADV
cana-5363	292	2	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐴	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐴	ADJ
cana-5363	292	3	∨	∨	NUM
cana-5363	292	4	𝐵	𝐵	NOUN
cana-5363	292	5	)	)	PUNCT
cana-5363	292	6	<	<	X
cana-5363	292	7	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐴	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐴	X
cana-5363	292	8	)	)	PUNCT
cana-5363	292	9	∨	∨	NUM
cana-5363	292	10	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐵	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐵	NOUN
cana-5363	292	11	)	)	PUNCT
cana-5363	292	12	.	.	PUNCT
cana-5363	293	1	2	2	X
cana-5363	293	2	.	.	X
cana-5363	293	3	𝐶	𝐶	PROPN
cana-5363	293	4	=	=	PRON
cana-5363	293	5	{	{	PUNCT
cana-5363	293	6	⟨	⟨	ADP
cana-5363	293	7	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	293	8	0.1	0.1	NUM
cana-5363	293	9	⟩	⟩	NOUN
cana-5363	293	10	,	,	PUNCT
cana-5363	293	11	⟨	⟨	VERB
cana-5363	293	12	𝑠2	𝑠2	PROPN
cana-5363	293	13	0.5	0.5	NUM
cana-5363	293	14	⟩	⟩	NOUN
cana-5363	293	15	,	,	PUNCT
cana-5363	293	16	⟨	⟨	VERB
cana-5363	293	17	𝑠3	𝑠3	PROPN
cana-5363	293	18	0.7	0.7	NUM
cana-5363	293	19	⟩	⟩	NOUN
cana-5363	293	20	}	}	PUNCT
cana-5363	293	21	and	and	CCONJ
cana-5363	293	22	𝐷	𝐷	NOUN
cana-5363	293	23	=	=	PUNCT
cana-5363	293	24	{	{	PUNCT
cana-5363	293	25	⟨	⟨	ADP
cana-5363	293	26	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	293	27	0.2	0.2	NUM
cana-5363	293	28	⟩	⟩	NOUN
cana-5363	293	29	,	,	PUNCT
cana-5363	293	30	⟨	⟨	VERB
cana-5363	293	31	𝑠2	𝑠2	PROPN
cana-5363	293	32	0.4	0.4	NUM
cana-5363	293	33	⟩	⟩	NOUN
cana-5363	293	34	,	,	PUNCT
cana-5363	293	35	⟨	⟨	VERB
cana-5363	293	36	𝑠3	𝑠3	PROPN
cana-5363	293	37	0.6	0.6	NUM
cana-5363	293	38	⟩	⟩	NOUN
cana-5363	293	39	}	}	PUNCT
cana-5363	293	40	,	,	PUNCT
cana-5363	293	41	then	then	ADV
cana-5363	293	42	𝐶	𝐶	PROPN
cana-5363	293	43	∧	∧	PROPN
cana-5363	293	44	𝐷	𝐷	PROPN
cana-5363	293	45	=	=	PUNCT
cana-5363	293	46	{	{	PUNCT
cana-5363	293	47	⟨	⟨	ADP
cana-5363	293	48	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	293	49	0.1	0.1	NUM
cana-5363	293	50	⟩	⟩	NOUN
cana-5363	293	51	,	,	PUNCT
cana-5363	293	52	⟨	⟨	VERB
cana-5363	293	53	𝑠2	𝑠2	PROPN
cana-5363	293	54	0.4	0.4	NUM
cana-5363	293	55	⟩	⟩	NOUN
cana-5363	293	56	,	,	PUNCT
cana-5363	293	57	⟨	⟨	VERB
cana-5363	293	58	𝑠3	𝑠3	PROPN
cana-5363	293	59	0.6	0.6	NUM
cana-5363	293	60	⟩	⟩	NOUN
cana-5363	293	61	}	}	PUNCT
cana-5363	293	62	.	.	PUNCT
cana-5363	294	1	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐶	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐶	VERB
cana-5363	294	2	)	)	PUNCT
cana-5363	294	3	=	=	NOUN
cana-5363	294	4	{	{	PUNCT
cana-5363	294	5	⟨	⟨	ADP
cana-5363	294	6	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	294	7	0.1	0.1	NUM
cana-5363	294	8	⟩	⟩	NOUN
cana-5363	294	9	,	,	PUNCT
cana-5363	294	10	⟨	⟨	VERB
cana-5363	294	11	𝑠2	𝑠2	PROPN
cana-5363	294	12	0.5	0.5	NUM
cana-5363	294	13	⟩	⟩	NOUN
cana-5363	294	14	,	,	PUNCT
cana-5363	294	15	⟨	⟨	VERB
cana-5363	294	16	𝑠3	𝑠3	PROPN
cana-5363	294	17	0.7	0.7	NUM
cana-5363	294	18	⟩	⟩	NOUN
cana-5363	294	19	}	}	PUNCT
cana-5363	294	20	,	,	PUNCT
cana-5363	294	21	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐷	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐷	NOUN
cana-5363	294	22	)	)	PUNCT
cana-5363	294	23	=	=	NOUN
cana-5363	294	24	{	{	PUNCT
cana-5363	294	25	⟨	⟨	ADP
cana-5363	294	26	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	294	27	0.2	0.2	NUM
cana-5363	294	28	⟩	⟩	NOUN
cana-5363	294	29	,	,	PUNCT
cana-5363	294	30	⟨	⟨	VERB
cana-5363	294	31	𝑠2	𝑠2	PROPN
cana-5363	294	32	0.4	0.4	NUM
cana-5363	294	33	⟩	⟩	NOUN
cana-5363	294	34	,	,	PUNCT
cana-5363	294	35	⟨	⟨	VERB
cana-5363	294	36	𝑠3	𝑠3	PROPN
cana-5363	294	37	0.6	0.6	NUM
cana-5363	294	38	⟩	⟩	NOUN
cana-5363	294	39	}	}	PUNCT
cana-5363	294	40	and	and	CCONJ
cana-5363	294	41	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐶	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐶	VERB
cana-5363	294	42	∧	∧	PROPN
cana-5363	294	43	𝐷	𝐷	PROPN
cana-5363	294	44	)	)	PUNCT
cana-5363	294	45	=	=	PRON
cana-5363	294	46	{	{	PUNCT
cana-5363	294	47	⟨	⟨	ADP
cana-5363	294	48	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	294	49	0.1	0.1	NUM
cana-5363	294	50	⟩	⟩	NOUN
cana-5363	294	51	,	,	PUNCT
cana-5363	294	52	⟨	⟨	VERB
cana-5363	294	53	𝑠2	𝑠2	PROPN
cana-5363	294	54	0.3	0.3	NUM
cana-5363	294	55	⟩	⟩	NOUN
cana-5363	294	56	,	,	PUNCT
cana-5363	294	57	⟨	⟨	VERB
cana-5363	294	58	𝑠3	𝑠3	PROPN
cana-5363	294	59	0.4	0.4	NUM
cana-5363	294	60	⟩	⟩	NOUN
cana-5363	294	61	}	}	PUNCT
cana-5363	294	62	.	.	PUNCT
cana-5363	295	1	thus	thus	ADV
cana-5363	295	2	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐶	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐶	VERB
cana-5363	295	3	)	)	PUNCT
cana-5363	295	4	∧	∧	NOUN
cana-5363	295	5	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐷	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐷	NOUN
cana-5363	295	6	)	)	PUNCT
cana-5363	295	7	<	<	X
cana-5363	295	8	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐶	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐶	VERB
cana-5363	295	9	∧	∧	PROPN
cana-5363	295	10	𝐷	𝐷	PROPN
cana-5363	295	11	)	)	PUNCT
cana-5363	295	12	.	.	PUNCT
cana-5363	296	1	theorem	theorem	VERB
cana-5363	296	2	3.11	3.11	NUM
cana-5363	296	3	let	let	NOUN
cana-5363	296	4	(	(	PUNCT
cana-5363	296	5	𝑈	𝑈	PROPN
cana-5363	296	6	,	,	PUNCT
cana-5363	296	7	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	296	8	)	)	PUNCT
cana-5363	296	9	)	)	PUNCT
cana-5363	296	10	be	be	AUX
cana-5363	296	11	a	a	DET
cana-5363	296	12	𝒫ℱ𝔑𝑡𝑠	𝒫ℱ𝔑𝑡𝑠	PROPN
cana-5363	296	13	and	and	CCONJ
cana-5363	296	14	𝐾	𝐾	PROPN
cana-5363	296	15	⊆	⊆	NUM
cana-5363	296	16	𝑆.	𝑆.	PROPN
cana-5363	296	17	then	then	ADV
cana-5363	296	18	𝐾	𝐾	PROPN
cana-5363	296	19	is	be	AUX
cana-5363	296	20	a	a	DET
cana-5363	296	21	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	296	22	set	set	VERB
cana-5363	296	23	iff	iff	PROPN
cana-5363	296	24	𝐾	𝐾	PROPN
cana-5363	296	25	=	=	PUNCT
cana-5363	296	26	𝒫ℱ𝔑𝛿𝒮𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝒮𝑖𝑛𝑡(𝐾	PROPN
cana-5363	296	27	)	)	PUNCT
cana-5363	296	28	∪	∪	PROPN
cana-5363	296	29	𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	PROPN
cana-5363	296	30	)	)	PUNCT
cana-5363	296	31	.	.	PUNCT
cana-5363	297	1	proof	proof	NOUN
cana-5363	297	2	.	.	PUNCT
cana-5363	298	1	let	let	VERB
cana-5363	298	2	𝐾	𝐾	PRON
cana-5363	298	3	be	be	AUX
cana-5363	298	4	a	a	DET
cana-5363	298	5	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	298	6	set	set	NOUN
cana-5363	298	7	.	.	PUNCT
cana-5363	299	1	then	then	ADV
cana-5363	299	2	𝐾	𝐾	PROPN
cana-5363	299	3	⊆	⊆	NUM
cana-5363	299	4	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	PROPN
cana-5363	299	5	)	)	PUNCT
cana-5363	299	6	)	)	PUNCT
cana-5363	299	7	∪	∪	ADP
cana-5363	299	8	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	NOUN
cana-5363	299	9	)	)	PUNCT
cana-5363	299	10	)	)	PUNCT
cana-5363	299	11	and	and	CCONJ
cana-5363	299	12	hence	hence	ADV
cana-5363	299	13	by	by	ADP
cana-5363	299	14	proposition	proposition	NOUN
cana-5363	299	15	3.1	3.1	NUM
cana-5363	299	16	and	and	CCONJ
cana-5363	299	17	lemma	lemma	PROPN
cana-5363	299	18	3.3	3.3	NUM
cana-5363	299	19	,	,	PUNCT
cana-5363	299	20	𝒫ℱ𝔑𝛿𝒮𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝒮𝑖𝑛𝑡(𝐾	PROPN
cana-5363	299	21	)	)	PUNCT
cana-5363	299	22	∪	∪	PROPN
cana-5363	299	23	𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	PROPN
cana-5363	299	24	)	)	PUNCT
cana-5363	299	25	=	=	PUNCT
cana-5363	300	1	(	(	PUNCT
cana-5363	300	2	𝑆	𝑆	PROPN
cana-5363	300	3	∩	∩	NOUN
cana-5363	300	4	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	PROPN
cana-5363	300	5	)	)	PUNCT
cana-5363	300	6	)	)	PUNCT
cana-5363	300	7	)	)	PUNCT
cana-5363	300	8	∪	∪	ADV
cana-5363	300	9	(	(	PUNCT
cana-5363	300	10	𝑆	𝑆	PROPN
cana-5363	300	11	∩	∩	NOUN
cana-5363	300	12	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	NOUN
cana-5363	300	13	)	)	PUNCT
cana-5363	300	14	)	)	PUNCT
cana-5363	300	15	)	)	PUNCT
cana-5363	301	1	=	=	SYM
cana-5363	301	2	𝐾	𝐾	PROPN
cana-5363	301	3	∩	∩	NOUN
cana-5363	301	4	(	(	PUNCT
cana-5363	301	5	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿	NOUN
cana-5363	301	6	𝑖𝑛𝑡(𝐾	𝑖𝑛𝑡(𝐾	NUM
cana-5363	301	7	)	)	PUNCT
cana-5363	301	8	)	)	PUNCT
cana-5363	301	9	∪	∪	ADP
cana-5363	301	10	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑐𝑙(𝐾	NOUN
cana-5363	301	11	)	)	PUNCT
cana-5363	301	12	)	)	PUNCT
cana-5363	301	13	)	)	PUNCT
cana-5363	302	1	=	=	PUNCT
cana-5363	302	2	𝐾.	𝐾.	PROPN
cana-5363	302	3	the	the	DET
cana-5363	302	4	converse	converse	NOUN
cana-5363	302	5	,	,	PUNCT
cana-5363	302	6	it	it	PRON
cana-5363	302	7	follows	follow	VERB
cana-5363	302	8	from	from	ADP
cana-5363	302	9	proposition	proposition	NOUN
cana-5363	302	10	3.1	3.1	NUM
cana-5363	302	11	and	and	CCONJ
cana-5363	302	12	lemma	lemma	PROPN
cana-5363	302	13	3.3	3.3	NUM
cana-5363	302	14	.	.	PUNCT
cana-5363	303	1	proposition	proposition	NOUN
cana-5363	303	2	3.4	3.4	NUM
cana-5363	303	3	let	let	VERB
cana-5363	303	4	(	(	PUNCT
cana-5363	303	5	𝑈	𝑈	NOUN
cana-5363	303	6	,	,	PUNCT
cana-5363	303	7	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	303	8	)	)	PUNCT
cana-5363	303	9	)	)	PUNCT
cana-5363	303	10	be	be	AUX
cana-5363	303	11	a	a	DET
cana-5363	303	12	𝒫ℱ𝔑𝑡𝑠	𝒫ℱ𝔑𝑡𝑠	PROPN
cana-5363	303	13	and	and	CCONJ
cana-5363	303	14	𝐾	𝐾	PROPN
cana-5363	303	15	⊆	⊆	NUM
cana-5363	303	16	𝑆.	𝑆.	PROPN
cana-5363	303	17	then	then	ADV
cana-5363	303	18	𝐾	𝐾	PROPN
cana-5363	303	19	is	be	AUX
cana-5363	303	20	a	a	DET
cana-5363	303	21	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	ADJ
cana-5363	303	22	set	set	NOUN
cana-5363	303	23	iff	iff	PROPN
cana-5363	303	24	𝐾	𝐾	PROPN
cana-5363	303	25	=	=	PUNCT
cana-5363	303	26	𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾	𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾	PROPN
cana-5363	303	27	)	)	PUNCT
cana-5363	303	28	∩	∩	ADJ
cana-5363	303	29	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	PROPN
cana-5363	303	30	)	)	PUNCT
cana-5363	303	31	.	.	PUNCT
cana-5363	304	1	proof	proof	NOUN
cana-5363	304	2	.	.	PUNCT
cana-5363	305	1	it	it	PRON
cana-5363	305	2	follows	follow	VERB
cana-5363	305	3	from	from	ADP
cana-5363	305	4	theorem	theorem	ADJ
cana-5363	305	5	3.11	3.11	NUM
cana-5363	305	6	.	.	PUNCT
cana-5363	306	1	theorem	theorem	NOUN
cana-5363	306	2	3.12	3.12	NUM
cana-5363	306	3	let	let	VERB
cana-5363	306	4	𝐾	𝐾	PRON
cana-5363	306	5	be	be	AUX
cana-5363	306	6	a	a	DET
cana-5363	306	7	subset	subset	NOUN
cana-5363	306	8	of	of	ADP
cana-5363	306	9	a	a	DET
cana-5363	306	10	space	space	NOUN
cana-5363	306	11	(	(	PUNCT
cana-5363	306	12	𝑈	𝑈	PROPN
cana-5363	306	13	,	,	PUNCT
cana-5363	306	14	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	306	15	)	)	PUNCT
cana-5363	306	16	)	)	PUNCT
cana-5363	306	17	.	.	PUNCT
cana-5363	307	1	then	then	ADV
cana-5363	307	2	:	:	PUNCT
cana-5363	307	3	(	(	PUNCT
cana-5363	307	4	i	i	NOUN
cana-5363	307	5	)	)	PUNCT
cana-5363	307	6	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	307	7	)	)	PUNCT
cana-5363	307	8	=	=	PUNCT
cana-5363	307	9	𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾	𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾	PROPN
cana-5363	307	10	)	)	PUNCT
cana-5363	307	11	∩	∩	ADJ
cana-5363	307	12	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	PROPN
cana-5363	307	13	)	)	PUNCT
cana-5363	307	14	,	,	PUNCT
cana-5363	307	15	(	(	PUNCT
cana-5363	307	16	ii	ii	NOUN
cana-5363	307	17	)	)	PUNCT
cana-5363	307	18	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	NOUN
cana-5363	307	19	)	)	PUNCT
cana-5363	308	1	=	=	SYM
cana-5363	308	2	𝒫ℱ𝔑𝛿𝒮𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝒮𝑖𝑛𝑡(𝐾	PROPN
cana-5363	308	3	)	)	PUNCT
cana-5363	308	4	∪	∪	PROPN
cana-5363	308	5	𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	PROPN
cana-5363	308	6	)	)	PUNCT
cana-5363	308	7	.	.	PUNCT
cana-5363	309	1	proof	proof	NOUN
cana-5363	309	2	.	.	PUNCT
cana-5363	310	1	(	(	PUNCT
cana-5363	310	2	i	i	NOUN
cana-5363	310	3	)	)	PUNCT
cana-5363	310	4	it	it	PRON
cana-5363	310	5	is	be	AUX
cana-5363	310	6	easy	easy	ADJ
cana-5363	310	7	to	to	PART
cana-5363	310	8	see	see	VERB
cana-5363	310	9	that	that	DET
cana-5363	310	10	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	310	11	)	)	PUNCT
cana-5363	310	12	⊆	⊆	NUM
cana-5363	310	13	𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾	𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾	NOUN
cana-5363	310	14	)	)	PUNCT
cana-5363	310	15	∩	∩	ADJ
cana-5363	310	16	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	PROPN
cana-5363	310	17	)	)	PUNCT
cana-5363	310	18	.	.	PUNCT
cana-5363	311	1	also	also	ADV
cana-5363	311	2	,	,	PUNCT
cana-5363	311	3	𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾	𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾	PROPN
cana-5363	311	4	)	)	PUNCT
cana-5363	311	5	∩	∩	ADJ
cana-5363	311	6	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	PROPN
cana-5363	311	7	)	)	PUNCT
cana-5363	311	8	=	=	PUNCT
cana-5363	312	1	(	(	PUNCT
cana-5363	312	2	𝐾	𝐾	PROPN
cana-5363	312	3	∪	∪	ADJ
cana-5363	312	4	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	PROPN
cana-5363	312	5	)	)	PUNCT
cana-5363	312	6	)	)	PUNCT
cana-5363	312	7	∩	∩	NOUN
cana-5363	312	8	(	(	PUNCT
cana-5363	312	9	𝐾	𝐾	PROPN
cana-5363	312	10	∪	∪	ADP
cana-5363	312	11	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	PROPN
cana-5363	312	12	)	)	PUNCT
cana-5363	312	13	)	)	PUNCT
cana-5363	313	1	=	=	PUNCT
cana-5363	313	2	𝐾	𝐾	NOUN
cana-5363	313	3	∪	∪	X
cana-5363	313	4	(	(	PUNCT
cana-5363	313	5	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	PROPN
cana-5363	313	6	)	)	PUNCT
cana-5363	313	7	)	)	PUNCT
cana-5363	313	8	∩	∩	PROPN
cana-5363	313	9	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	PROPN
cana-5363	313	10	)	)	PUNCT
cana-5363	313	11	)	)	PUNCT
cana-5363	313	12	)	)	PUNCT
cana-5363	313	13	.	.	PUNCT
cana-5363	314	1	since	since	SCONJ
cana-5363	314	2	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	314	3	)	)	PUNCT
cana-5363	314	4	is	be	AUX
cana-5363	314	5	𝒫ℱ𝔑𝑍𝑐𝑠	𝒫ℱ𝔑𝑍𝑐𝑠	PROPN
cana-5363	314	6	,	,	PUNCT
cana-5363	314	7	then	then	ADV
cana-5363	314	8	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	314	9	)	)	PUNCT
cana-5363	314	10	⊆	⊆	NUM
cana-5363	314	11	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	NOUN
cana-5363	314	12	)	)	PUNCT
cana-5363	314	13	)	)	PUNCT
cana-5363	314	14	)	)	PUNCT
cana-5363	315	1	∩	∩	NOUN
cana-5363	315	2	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	NOUN
cana-5363	315	3	)	)	PUNCT
cana-5363	315	4	)	)	PUNCT
cana-5363	315	5	)	)	PUNCT
cana-5363	316	1	⊇	⊇	PROPN
cana-5363	316	2	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	PROPN
cana-5363	316	3	)	)	PUNCT
cana-5363	316	4	)	)	PUNCT
cana-5363	316	5	∩	∩	PROPN
cana-5363	316	6	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	PROPN
cana-5363	316	7	)	)	PUNCT
cana-5363	316	8	)	)	PUNCT
cana-5363	316	9	.	.	PUNCT
cana-5363	317	1	thus	thus	ADV
cana-5363	317	2	𝐾	𝐾	PRON
cana-5363	317	3	∪	∪	NOUN
cana-5363	317	4	(	(	PUNCT
cana-5363	317	5	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	PROPN
cana-5363	317	6	)	)	PUNCT
cana-5363	317	7	)	)	PUNCT
cana-5363	317	8	∩	∩	PROPN
cana-5363	317	9	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	PROPN
cana-5363	317	10	)	)	PUNCT
cana-5363	317	11	)	)	PUNCT
cana-5363	317	12	)	)	PUNCT
cana-5363	318	1	⊆	⊆	NUM
cana-5363	318	2	𝐾	𝐾	PROPN
cana-5363	318	3	∪	∪	ADJ
cana-5363	318	4	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	318	5	)	)	PUNCT
cana-5363	318	6	=	=	SYM
cana-5363	318	7	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	318	8	)	)	PUNCT
cana-5363	318	9	and	and	CCONJ
cana-5363	318	10	hence	hence	ADV
cana-5363	318	11	,	,	PUNCT
cana-5363	318	12	𝒫ℱ𝔑𝛿𝒮	𝒫ℱ𝔑𝛿𝒮	PROPN
cana-5363	318	13	𝑐𝑙(𝐾	𝑐𝑙(𝐾	PROPN
cana-5363	318	14	)	)	PUNCT
cana-5363	318	15	∩	∩	ADJ
cana-5363	318	16	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	PROPN
cana-5363	318	17	)	)	PUNCT
cana-5363	318	18	⊆	⊆	NUM
cana-5363	318	19	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	NOUN
cana-5363	318	20	)	)	PUNCT
cana-5363	318	21	.	.	PUNCT
cana-5363	319	1	so	so	ADV
cana-5363	319	2	,	,	PUNCT
cana-5363	319	3	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	319	4	)	)	PUNCT
cana-5363	319	5	=	=	SYM
cana-5363	319	6	𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾	𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾	PROPN
cana-5363	319	7	)	)	PUNCT
cana-5363	319	8	∩	∩	ADJ
cana-5363	319	9	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	PROPN
cana-5363	319	10	)	)	PUNCT
cana-5363	319	11	.	.	PUNCT
cana-5363	320	1	(	(	PUNCT
cana-5363	320	2	ii	ii	X
cana-5363	320	3	)	)	PUNCT
cana-5363	320	4	it	it	PRON
cana-5363	320	5	follows	follow	VERB
cana-5363	320	6	from	from	ADP
cana-5363	320	7	(	(	PUNCT
cana-5363	320	8	i	i	NOUN
cana-5363	320	9	)	)	PUNCT
cana-5363	320	10	.	.	PUNCT
cana-5363	321	1	theorem	theorem	ADJ
cana-5363	321	2	3.13	3.13	NUM
cana-5363	321	3	let	let	VERB
cana-5363	321	4	𝐾	𝐾	PRON
cana-5363	321	5	be	be	AUX
cana-5363	321	6	a	a	DET
cana-5363	321	7	pythagorean	pythagorean	ADJ
cana-5363	321	8	fuzzy	fuzzy	ADJ
cana-5363	321	9	subset	subset	NOUN
cana-5363	321	10	of	of	ADP
cana-5363	321	11	a	a	DET
cana-5363	321	12	space	space	NOUN
cana-5363	321	13	(	(	PUNCT
cana-5363	321	14	𝑈	𝑈	PROPN
cana-5363	321	15	,	,	PUNCT
cana-5363	321	16	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	321	17	)	)	PUNCT
cana-5363	321	18	)	)	PUNCT
cana-5363	322	1	then	then	ADV
cana-5363	322	2	(	(	PUNCT
cana-5363	322	3	i	i	NOUN
cana-5363	322	4	)	)	PUNCT
cana-5363	322	5	𝐾	𝐾	PROPN
cana-5363	322	6	is	be	AUX
cana-5363	322	7	a	a	DET
cana-5363	322	8	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	322	9	set	set	VERB
cana-5363	322	10	iff	iff	PROPN
cana-5363	322	11	𝐾	𝐾	PROPN
cana-5363	322	12	=	=	PUNCT
cana-5363	322	13	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	PROPN
cana-5363	322	14	)	)	PUNCT
cana-5363	322	15	,	,	PUNCT
cana-5363	322	16	(	(	PUNCT
cana-5363	322	17	ii	ii	NOUN
cana-5363	322	18	)	)	PUNCT
cana-5363	322	19	𝐾	𝐾	NOUN
cana-5363	322	20	is	be	AUX
cana-5363	322	21	a	a	DET
cana-5363	322	22	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	ADJ
cana-5363	322	23	set	set	NOUN
cana-5363	322	24	iff	iff	PROPN
cana-5363	322	25	𝐾	𝐾	PROPN
cana-5363	322	26	=	=	PROPN
cana-5363	322	27	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	322	28	)	)	PUNCT
cana-5363	322	29	.	.	PUNCT
cana-5363	323	1	communications	communication	NOUN
cana-5363	323	2	on	on	ADP
cana-5363	323	3	applied	apply	VERB
cana-5363	323	4	nonlinear	nonlinear	ADJ
cana-5363	323	5	analysis	analysis	NOUN
cana-5363	323	6	issn	issn	NOUN
cana-5363	323	7	:	:	PUNCT
cana-5363	323	8	1074	1074	NUM
cana-5363	323	9	-	-	PUNCT
cana-5363	323	10	133x	133x	NUM
cana-5363	323	11	vol	vol	VERB
cana-5363	323	12	32	32	NUM
cana-5363	323	13	no	no	NOUN
cana-5363	323	14	.	.	PUNCT
cana-5363	324	1	10s	10	NOUN
cana-5363	324	2	(	(	PUNCT
cana-5363	324	3	2025	2025	NUM
cana-5363	324	4	)	)	PUNCT
cana-5363	324	5	2016	2016	NUM
cana-5363	324	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5363	324	7	proof	proof	NOUN
cana-5363	324	8	.	.	PUNCT
cana-5363	325	1	(	(	PUNCT
cana-5363	325	2	i	i	NOUN
cana-5363	325	3	)	)	PUNCT
cana-5363	325	4	it	it	PRON
cana-5363	325	5	follows	follow	VERB
cana-5363	325	6	from	from	ADP
cana-5363	325	7	theorems	theorems	PROPN
cana-5363	325	8	3.11	3.11	NUM
cana-5363	325	9	&	&	CCONJ
cana-5363	325	10	3.12	3.12	NUM
cana-5363	325	11	.	.	PUNCT
cana-5363	326	1	lemma	lemma	PROPN
cana-5363	326	2	3.5	3.5	NUM
cana-5363	326	3	let	let	VERB
cana-5363	326	4	𝐾	𝐾	PRON
cana-5363	326	5	be	be	AUX
cana-5363	326	6	a	a	DET
cana-5363	326	7	𝑝𝑓𝑠	𝑝𝑓𝑠	NOUN
cana-5363	326	8	of	of	ADP
cana-5363	326	9	a	a	DET
cana-5363	326	10	𝒫ℱ𝔑𝑡𝑠	𝒫ℱ𝔑𝑡𝑠	PROPN
cana-5363	326	11	(	(	PUNCT
cana-5363	326	12	𝑈	𝑈	PROPN
cana-5363	326	13	,	,	PUNCT
cana-5363	326	14	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	326	15	)	)	PUNCT
cana-5363	326	16	)	)	PUNCT
cana-5363	326	17	.	.	PUNCT
cana-5363	327	1	then	then	ADV
cana-5363	327	2	the	the	DET
cana-5363	327	3	following	follow	VERB
cana-5363	327	4	statement	statement	NOUN
cana-5363	327	5	are	be	AUX
cana-5363	327	6	hold	hold	ADJ
cana-5363	327	7	:	:	PUNCT
cana-5363	327	8	(	(	PUNCT
cana-5363	327	9	i	i	NOUN
cana-5363	327	10	)	)	PUNCT
cana-5363	327	11	𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	PROPN
cana-5363	327	12	)	)	PUNCT
cana-5363	327	13	)	)	PUNCT
cana-5363	328	1	=	=	SYM
cana-5363	328	2	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	PROPN
cana-5363	328	3	)	)	PUNCT
cana-5363	328	4	∩	∩	ADJ
cana-5363	328	5	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	PROPN
cana-5363	328	6	)	)	PUNCT
cana-5363	328	7	)	)	PUNCT
cana-5363	328	8	,	,	PUNCT
cana-5363	328	9	(	(	PUNCT
cana-5363	328	10	ii	ii	NOUN
cana-5363	328	11	)	)	PUNCT
cana-5363	328	12	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	PROPN
cana-5363	328	13	)	)	PUNCT
cana-5363	328	14	)	)	PUNCT
cana-5363	329	1	=	=	SYM
cana-5363	330	1	𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	PROPN
cana-5363	330	2	)	)	PUNCT
cana-5363	330	3	∪	∪	ADP
cana-5363	330	4	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑐𝑙(𝒫ℱ𝔑𝛿𝑖𝑛𝑡(𝐾	PROPN
cana-5363	330	5	)	)	PUNCT
cana-5363	330	6	)	)	PUNCT
cana-5363	330	7	.	.	PUNCT
cana-5363	331	1	proof	proof	NOUN
cana-5363	331	2	.	.	PUNCT
cana-5363	332	1	(	(	PUNCT
cana-5363	332	2	i	i	NOUN
cana-5363	332	3	)	)	PUNCT
cana-5363	332	4	by	by	ADP
cana-5363	332	5	lemma	lemma	PROPN
cana-5363	332	6	3.4	3.4	NUM
cana-5363	332	7	,	,	PUNCT
cana-5363	332	8	𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	PROPN
cana-5363	332	9	)	)	PUNCT
cana-5363	332	10	)	)	PUNCT
cana-5363	333	1	=	=	SYM
cana-5363	333	2	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	PROPN
cana-5363	333	3	)	)	PUNCT
cana-5363	333	4	∩	∩	NOUN
cana-5363	333	5	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	PROPN
cana-5363	333	6	)	)	PUNCT
cana-5363	333	7	)	)	PUNCT
cana-5363	333	8	)	)	PUNCT
cana-5363	334	1	=	=	PUNCT
cana-5363	335	1	𝒫ℱ𝔑𝒫	𝒫ℱ𝔑𝒫	PROPN
cana-5363	335	2	𝑐𝑙(𝐾	𝑐𝑙(𝐾	NUM
cana-5363	335	3	)	)	PUNCT
cana-5363	335	4	∩	∩	NOUN
cana-5363	335	5	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	PROPN
cana-5363	335	6	∪	∪	ADP
cana-5363	335	7	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑	PROPN
cana-5363	335	8	𝑐𝑙(𝐾	𝑐𝑙(𝐾	PROPN
cana-5363	335	9	)	)	PUNCT
cana-5363	335	10	)	)	PUNCT
cana-5363	335	11	)	)	PUNCT
cana-5363	335	12	)	)	PUNCT
cana-5363	336	1	=	=	PUNCT
cana-5363	336	2	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	PROPN
cana-5363	336	3	)	)	PUNCT
cana-5363	336	4	∩	∩	ADJ
cana-5363	336	5	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	PROPN
cana-5363	336	6	)	)	PUNCT
cana-5363	336	7	)	)	PUNCT
cana-5363	336	8	.	.	PUNCT
cana-5363	337	1	(	(	PUNCT
cana-5363	337	2	ii	ii	X
cana-5363	337	3	)	)	PUNCT
cana-5363	337	4	it	it	PRON
cana-5363	337	5	follows	follow	VERB
cana-5363	337	6	from	from	ADP
cana-5363	337	7	(	(	PUNCT
cana-5363	337	8	i	i	NOUN
cana-5363	337	9	)	)	PUNCT
cana-5363	337	10	.	.	PUNCT
cana-5363	338	1	proposition	proposition	NOUN
cana-5363	338	2	3.5	3.5	NUM
cana-5363	338	3	let	let	VERB
cana-5363	338	4	𝐾	𝐾	PRON
cana-5363	338	5	be	be	AUX
cana-5363	338	6	a	a	DET
cana-5363	338	7	𝑝𝑓𝑠	𝑝𝑓𝑠	NOUN
cana-5363	338	8	of	of	ADP
cana-5363	338	9	a	a	DET
cana-5363	338	10	𝒫ℱ𝔑𝑡𝑠	𝒫ℱ𝔑𝑡𝑠	PROPN
cana-5363	338	11	(	(	PUNCT
cana-5363	338	12	𝑈	𝑈	PROPN
cana-5363	338	13	,	,	PUNCT
cana-5363	338	14	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	338	15	)	)	PUNCT
cana-5363	338	16	)	)	PUNCT
cana-5363	338	17	.	.	PUNCT
cana-5363	339	1	then	then	ADV
cana-5363	339	2	:	:	PUNCT
cana-5363	339	3	(	(	PUNCT
cana-5363	339	4	i	i	NOUN
cana-5363	339	5	)	)	PUNCT
cana-5363	339	6	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	339	7	)	)	PUNCT
cana-5363	339	8	=	=	SYM
cana-5363	339	9	𝐾	𝐾	PROPN
cana-5363	339	10	∪	∪	VERB
cana-5363	339	11	𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	PROPN
cana-5363	339	12	)	)	PUNCT
cana-5363	339	13	)	)	PUNCT
cana-5363	339	14	,	,	PUNCT
cana-5363	339	15	(	(	PUNCT
cana-5363	339	16	ii	ii	NOUN
cana-5363	339	17	)	)	PUNCT
cana-5363	339	18	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	NOUN
cana-5363	339	19	)	)	PUNCT
cana-5363	339	20	=	=	SYM
cana-5363	339	21	𝐾	𝐾	PROPN
cana-5363	339	22	∩	∩	NOUN
cana-5363	339	23	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	NOUN
cana-5363	339	24	)	)	PUNCT
cana-5363	339	25	)	)	PUNCT
cana-5363	339	26	.	.	PUNCT
cana-5363	340	1	proof	proof	NOUN
cana-5363	340	2	.	.	PUNCT
cana-5363	341	1	(	(	PUNCT
cana-5363	341	2	i	i	NOUN
cana-5363	341	3	)	)	PUNCT
cana-5363	341	4	by	by	ADP
cana-5363	341	5	lemma	lemma	PROPN
cana-5363	341	6	3.5	3.5	NUM
cana-5363	341	7	,	,	PUNCT
cana-5363	341	8	𝐾	𝐾	PROPN
cana-5363	341	9	∪	∪	ADJ
cana-5363	341	10	𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	PROPN
cana-5363	341	11	)	)	PUNCT
cana-5363	341	12	)	)	PUNCT
cana-5363	342	1	=	=	PUNCT
cana-5363	342	2	𝐾	𝐾	NOUN
cana-5363	342	3	∪	∪	NOUN
cana-5363	342	4	(	(	PUNCT
cana-5363	342	5	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	PROPN
cana-5363	342	6	)	)	PUNCT
cana-5363	342	7	∩	∩	ADJ
cana-5363	342	8	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	PROPN
cana-5363	342	9	)	)	PUNCT
cana-5363	342	10	)	)	PUNCT
cana-5363	342	11	)	)	PUNCT
cana-5363	343	1	=	=	PUNCT
cana-5363	343	2	(	(	PUNCT
cana-5363	343	3	𝐾	𝐾	PROPN
cana-5363	343	4	∪	∪	ADJ
cana-5363	343	5	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	PROPN
cana-5363	343	6	)	)	PUNCT
cana-5363	343	7	)	)	PUNCT
cana-5363	343	8	∩	∩	NOUN
cana-5363	343	9	(	(	PUNCT
cana-5363	343	10	𝐾	𝐾	PROPN
cana-5363	343	11	∪	∪	ADJ
cana-5363	343	12	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝒫ℱ𝔑𝛿𝑐𝑙(𝐾	PROPN
cana-5363	343	13	)	)	PUNCT
cana-5363	343	14	)	)	PUNCT
cana-5363	343	15	)	)	PUNCT
cana-5363	344	1	=	=	SYM
cana-5363	344	2	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	PROPN
cana-5363	344	3	)	)	PUNCT
cana-5363	344	4	∩	∩	NOUN
cana-5363	344	5	𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾	𝒫ℱ𝔑𝛿𝒮𝑐𝑙(𝐾	PROPN
cana-5363	344	6	)	)	PUNCT
cana-5363	344	7	=	=	SYM
cana-5363	344	8	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	344	9	)	)	PUNCT
cana-5363	344	10	.	.	PUNCT
cana-5363	345	1	(	(	PUNCT
cana-5363	345	2	ii	ii	X
cana-5363	345	3	)	)	PUNCT
cana-5363	345	4	it	it	PRON
cana-5363	345	5	follows	follow	VERB
cana-5363	345	6	from	from	ADP
cana-5363	345	7	(	(	PUNCT
cana-5363	345	8	i	i	NOUN
cana-5363	345	9	)	)	PUNCT
cana-5363	345	10	.	.	PUNCT
cana-5363	346	1	theorem	theorem	VERB
cana-5363	346	2	3.14	3.14	NUM
cana-5363	346	3	let	let	VERB
cana-5363	346	4	𝐾	𝐾	PRON
cana-5363	346	5	be	be	AUX
cana-5363	346	6	a	a	DET
cana-5363	346	7	fuzzy	fuzzy	ADJ
cana-5363	346	8	subset	subset	NOUN
cana-5363	346	9	of	of	ADP
cana-5363	346	10	a	a	DET
cana-5363	346	11	𝒫ℱ𝔑𝑡𝑠	𝒫ℱ𝔑𝑡𝑠	PROPN
cana-5363	346	12	(	(	PUNCT
cana-5363	346	13	𝑈	𝑈	PROPN
cana-5363	346	14	,	,	PUNCT
cana-5363	346	15	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	346	16	)	)	PUNCT
cana-5363	346	17	)	)	PUNCT
cana-5363	346	18	.	.	PUNCT
cana-5363	347	1	then	then	ADV
cana-5363	347	2	the	the	DET
cana-5363	347	3	following	follow	VERB
cana-5363	347	4	are	be	AUX
cana-5363	347	5	equivalent	equivalent	ADJ
cana-5363	347	6	:	:	PUNCT
cana-5363	347	7	(	(	PUNCT
cana-5363	347	8	i	i	NOUN
cana-5363	347	9	)	)	PUNCT
cana-5363	347	10	𝐾	𝐾	PROPN
cana-5363	347	11	is	be	AUX
cana-5363	347	12	a	a	DET
cana-5363	347	13	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	347	14	set	set	NOUN
cana-5363	347	15	,	,	PUNCT
cana-5363	347	16	(	(	PUNCT
cana-5363	347	17	ii	ii	NOUN
cana-5363	347	18	)	)	PUNCT
cana-5363	347	19	𝐾	𝐾	PROPN
cana-5363	347	20	⊆	⊆	NUM
cana-5363	347	21	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	NUM
cana-5363	347	22	)	)	PUNCT
cana-5363	347	23	)	)	PUNCT
cana-5363	347	24	,	,	PUNCT
cana-5363	347	25	(	(	PUNCT
cana-5363	347	26	iii	iii	X
cana-5363	347	27	)	)	PUNCT
cana-5363	347	28	there	there	PRON
cana-5363	347	29	exists	exist	VERB
cana-5363	347	30	𝑂	𝑂	PROPN
cana-5363	347	31	∈	∈	PROPN
cana-5363	347	32	𝒫ℱ𝔑𝒫𝑂(𝐴	𝒫ℱ𝔑𝒫𝑂(𝐴	NOUN
cana-5363	347	33	)	)	PUNCT
cana-5363	347	34	such	such	ADJ
cana-5363	347	35	that	that	SCONJ
cana-5363	347	36	𝑂	𝑂	PROPN
cana-5363	347	37	⊆	⊆	NUM
cana-5363	347	38	𝐾	𝐾	PROPN
cana-5363	347	39	⊆	⊆	NUM
cana-5363	347	40	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝑂	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝑂	NOUN
cana-5363	347	41	)	)	PUNCT
cana-5363	347	42	,	,	PUNCT
cana-5363	347	43	(	(	PUNCT
cana-5363	347	44	iv	iv	X
cana-5363	347	45	)	)	PUNCT
cana-5363	347	46	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝐾	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝐾	PROPN
cana-5363	347	47	)	)	PUNCT
cana-5363	347	48	=	=	SYM
cana-5363	347	49	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	NOUN
cana-5363	347	50	)	)	PUNCT
cana-5363	347	51	)	)	PUNCT
cana-5363	347	52	.	.	PUNCT
cana-5363	348	1	proof	proof	NOUN
cana-5363	348	2	.	.	PUNCT
cana-5363	349	1	(	(	PUNCT
cana-5363	349	2	i	i	NOUN
cana-5363	349	3	)	)	PUNCT
cana-5363	349	4	⇒	⇒	PROPN
cana-5363	349	5	(	(	PUNCT
cana-5363	349	6	ii	ii	PROPN
cana-5363	349	7	):	):	PUNCT
cana-5363	349	8	let	let	VERB
cana-5363	349	9	𝐾	𝐾	PRON
cana-5363	349	10	be	be	AUX
cana-5363	349	11	a	a	DET
cana-5363	349	12	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	349	13	set	set	NOUN
cana-5363	349	14	.	.	PUNCT
cana-5363	350	1	then	then	ADV
cana-5363	350	2	by	by	ADP
cana-5363	350	3	theorem	theorem	NOUN
cana-5363	350	4	3.13	3.13	NUM
cana-5363	350	5	,	,	PUNCT
cana-5363	350	6	𝐾	𝐾	PROPN
cana-5363	350	7	=	=	PUNCT
cana-5363	350	8	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	PROPN
cana-5363	350	9	)	)	PUNCT
cana-5363	350	10	and	and	CCONJ
cana-5363	350	11	by	by	ADP
cana-5363	350	12	proposition	proposition	NOUN
cana-5363	350	13	3.5	3.5	NUM
cana-5363	350	14	,	,	PUNCT
cana-5363	350	15	𝐾	𝐾	PROPN
cana-5363	350	16	=	=	SYM
cana-5363	350	17	𝐾	𝐾	PROPN
cana-5363	350	18	∩	∩	NOUN
cana-5363	350	19	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	NOUN
cana-5363	350	20	)	)	PUNCT
cana-5363	350	21	)	)	PUNCT
cana-5363	350	22	and	and	CCONJ
cana-5363	350	23	hence	hence	ADV
cana-5363	350	24	,	,	PUNCT
cana-5363	350	25	𝐾	𝐾	PROPN
cana-5363	350	26	⊆	⊆	NUM
cana-5363	350	27	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	NUM
cana-5363	350	28	)	)	PUNCT
cana-5363	350	29	)	)	PUNCT
cana-5363	350	30	.	.	PUNCT
cana-5363	351	1	(	(	PUNCT
cana-5363	351	2	ii	ii	NOUN
cana-5363	351	3	)	)	PUNCT
cana-5363	351	4	⇒	⇒	NOUN
cana-5363	351	5	(	(	PUNCT
cana-5363	351	6	i	i	NOUN
cana-5363	351	7	):	):	PUNCT
cana-5363	351	8	let	let	VERB
cana-5363	351	9	𝐾	𝐾	PROPN
cana-5363	351	10	⊆	⊆	NUM
cana-5363	351	11	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	NUM
cana-5363	351	12	)	)	PUNCT
cana-5363	351	13	)	)	PUNCT
cana-5363	351	14	.	.	PUNCT
cana-5363	352	1	then	then	ADV
cana-5363	352	2	by	by	ADP
cana-5363	352	3	proposition	proposition	NOUN
cana-5363	352	4	3.5	3.5	NUM
cana-5363	352	5	,	,	PUNCT
cana-5363	352	6	𝐾	𝐾	PROPN
cana-5363	352	7	⊆	⊆	NUM
cana-5363	352	8	𝐾	𝐾	PROPN
cana-5363	352	9	∩	∩	NOUN
cana-5363	352	10	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	NOUN
cana-5363	352	11	)	)	PUNCT
cana-5363	352	12	)	)	PUNCT
cana-5363	353	1	=	=	SYM
cana-5363	353	2	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	NOUN
cana-5363	353	3	)	)	PUNCT
cana-5363	353	4	,	,	PUNCT
cana-5363	353	5	and	and	CCONJ
cana-5363	353	6	hence	hence	ADV
cana-5363	353	7	𝐾	𝐾	PROPN
cana-5363	353	8	=	=	PUNCT
cana-5363	353	9	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	NOUN
cana-5363	353	10	)	)	PUNCT
cana-5363	353	11	.	.	PUNCT
cana-5363	354	1	thus	thus	ADV
cana-5363	354	2	𝐾	𝐾	PROPN
cana-5363	354	3	is	be	AUX
cana-5363	354	4	𝒫ℱ𝔑𝑍𝑜𝑠.	𝒫ℱ𝔑𝑍𝑜𝑠.	NOUN
cana-5363	354	5	(	(	PUNCT
cana-5363	354	6	ii	ii	NOUN
cana-5363	354	7	)	)	PUNCT
cana-5363	354	8	⇒	⇒	NOUN
cana-5363	354	9	(	(	PUNCT
cana-5363	354	10	iii	iii	X
cana-5363	354	11	):	):	PUNCT
cana-5363	354	12	it	it	PRON
cana-5363	354	13	follows	follow	VERB
cana-5363	354	14	from	from	ADP
cana-5363	354	15	putting	put	VERB
cana-5363	354	16	𝑂	𝑂	PROPN
cana-5363	354	17	=	=	SYM
cana-5363	354	18	𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	PROPN
cana-5363	354	19	)	)	PUNCT
cana-5363	354	20	,	,	PUNCT
cana-5363	354	21	(	(	PUNCT
cana-5363	354	22	iii	iii	X
cana-5363	354	23	)	)	PUNCT
cana-5363	354	24	⇒	⇒	NOUN
cana-5363	354	25	(	(	PUNCT
cana-5363	354	26	ii	ii	NOUN
cana-5363	354	27	)	)	PUNCT
cana-5363	354	28	.	.	PUNCT
cana-5363	355	1	let	let	VERB
cana-5363	355	2	there	there	PRON
cana-5363	355	3	exists	exist	VERB
cana-5363	355	4	𝑂	𝑂	PROPN
cana-5363	355	5	∈	∈	PROPN
cana-5363	355	6	𝒫ℱ𝔑𝒫𝑂(𝐴	𝒫ℱ𝔑𝒫𝑂(𝐴	NOUN
cana-5363	355	7	)	)	PUNCT
cana-5363	355	8	such	such	ADJ
cana-5363	355	9	that	that	SCONJ
cana-5363	355	10	𝑂	𝑂	PROPN
cana-5363	355	11	⊆	⊆	NUM
cana-5363	355	12	𝐾	𝐾	PROPN
cana-5363	355	13	⊆	⊆	NUM
cana-5363	355	14	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝑂	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝑂	NOUN
cana-5363	355	15	)	)	PUNCT
cana-5363	355	16	.	.	PUNCT
cana-5363	356	1	since	since	SCONJ
cana-5363	356	2	𝑂	𝑂	PROPN
cana-5363	356	3	⊆	⊆	NUM
cana-5363	356	4	𝐾	𝐾	PROPN
cana-5363	356	5	,	,	PUNCT
cana-5363	356	6	then	then	ADV
cana-5363	356	7	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝑂	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝑂	VERB
cana-5363	356	8	)	)	PUNCT
cana-5363	356	9	⊆	⊆	NUM
cana-5363	356	10	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	NOUN
cana-5363	356	11	)	)	PUNCT
cana-5363	356	12	)	)	PUNCT
cana-5363	356	13	,	,	PUNCT
cana-5363	356	14	therefore	therefore	ADV
cana-5363	356	15	𝐾	𝐾	PROPN
cana-5363	356	16	⊆	⊆	NUM
cana-5363	356	17	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝑂	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝑂	NOUN
cana-5363	356	18	)	)	PUNCT
cana-5363	356	19	⊆	⊆	NUM
cana-5363	356	20	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝒫ℱ𝔑𝒫𝑖𝑛𝑡(𝐾	NOUN
cana-5363	356	21	)	)	PUNCT
cana-5363	356	22	)	)	PUNCT
cana-5363	356	23	.	.	PUNCT
cana-5363	357	1	(	(	PUNCT
cana-5363	357	2	ii	ii	NOUN
cana-5363	357	3	)	)	PUNCT
cana-5363	357	4	⇔	⇔	X
cana-5363	357	5	(	(	PUNCT
cana-5363	357	6	iv	iv	NUM
cana-5363	357	7	):	):	PUNCT
cana-5363	357	8	it	it	PRON
cana-5363	357	9	is	be	AUX
cana-5363	357	10	clear	clear	ADJ
cana-5363	357	11	.	.	PUNCT
cana-5363	358	1	theorem	theorem	VERB
cana-5363	358	2	3.15	3.15	NUM
cana-5363	358	3	let	let	VERB
cana-5363	358	4	𝐾	𝐾	PRON
cana-5363	358	5	be	be	AUX
cana-5363	358	6	a	a	DET
cana-5363	358	7	𝑝𝑓𝑠	𝑝𝑓𝑠	NOUN
cana-5363	358	8	of	of	ADP
cana-5363	358	9	a	a	DET
cana-5363	358	10	𝒫ℱ𝔑𝑡𝑠	𝒫ℱ𝔑𝑡𝑠	PROPN
cana-5363	358	11	(	(	PUNCT
cana-5363	358	12	𝑈	𝑈	PROPN
cana-5363	358	13	,	,	PUNCT
cana-5363	358	14	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	358	15	)	)	PUNCT
cana-5363	358	16	)	)	PUNCT
cana-5363	358	17	.	.	PUNCT
cana-5363	359	1	then	then	ADV
cana-5363	359	2	the	the	DET
cana-5363	359	3	following	follow	VERB
cana-5363	359	4	are	be	AUX
cana-5363	359	5	equivalent	equivalent	ADJ
cana-5363	359	6	:	:	PUNCT
cana-5363	359	7	(	(	PUNCT
cana-5363	359	8	i	i	NOUN
cana-5363	359	9	)	)	PUNCT
cana-5363	359	10	𝐾	𝐾	PROPN
cana-5363	359	11	is	be	AUX
cana-5363	359	12	a	a	DET
cana-5363	359	13	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	ADJ
cana-5363	359	14	set	set	NOUN
cana-5363	359	15	,	,	PUNCT
cana-5363	359	16	communications	communication	NOUN
cana-5363	359	17	on	on	ADP
cana-5363	359	18	applied	apply	VERB
cana-5363	359	19	nonlinear	nonlinear	ADJ
cana-5363	359	20	analysis	analysis	NOUN
cana-5363	359	21	issn	issn	NOUN
cana-5363	359	22	:	:	PUNCT
cana-5363	359	23	1074	1074	NUM
cana-5363	359	24	-	-	PUNCT
cana-5363	359	25	133x	133x	NUM
cana-5363	359	26	vol	vol	VERB
cana-5363	359	27	32	32	NUM
cana-5363	359	28	no	no	NOUN
cana-5363	359	29	.	.	PUNCT
cana-5363	360	1	10s	10	NOUN
cana-5363	360	2	(	(	PUNCT
cana-5363	360	3	2025	2025	NUM
cana-5363	360	4	)	)	PUNCT
cana-5363	360	5	2017	2017	NUM
cana-5363	360	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5363	360	7	(	(	PUNCT
cana-5363	360	8	ii	ii	NOUN
cana-5363	360	9	)	)	PUNCT
cana-5363	360	10	𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	PROPN
cana-5363	360	11	)	)	PUNCT
cana-5363	360	12	)	)	PUNCT
cana-5363	361	1	⊆	⊆	X
cana-5363	361	2	𝐾	𝐾	PROPN
cana-5363	361	3	,	,	PUNCT
cana-5363	361	4	(	(	PUNCT
cana-5363	361	5	iii	iii	X
cana-5363	361	6	)	)	PUNCT
cana-5363	361	7	there	there	PRON
cana-5363	361	8	exists	exist	VERB
cana-5363	361	9	𝑂	𝑂	PROPN
cana-5363	361	10	∈	∈	PROPN
cana-5363	361	11	𝒫ℱ𝔑𝑃𝐶(𝐴	𝒫ℱ𝔑𝑃𝐶(𝐴	NOUN
cana-5363	361	12	)	)	PUNCT
cana-5363	361	13	such	such	ADJ
cana-5363	361	14	that	that	SCONJ
cana-5363	361	15	𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝑂	𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝑂	PROPN
cana-5363	361	16	)	)	PUNCT
cana-5363	362	1	⊆	⊆	NUM
cana-5363	362	2	𝐾	𝐾	PROPN
cana-5363	362	3	⊆	⊆	NUM
cana-5363	362	4	𝑂	𝑂	PROPN
cana-5363	362	5	,	,	PUNCT
cana-5363	362	6	(	(	PUNCT
cana-5363	362	7	iv	iv	X
cana-5363	362	8	)	)	PUNCT
cana-5363	362	9	𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝐾	PROPN
cana-5363	362	10	)	)	PUNCT
cana-5363	362	11	=	=	PUNCT
cana-5363	362	12	𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	𝒫ℱ𝔑𝛿𝒫𝑖𝑛𝑡(𝒫ℱ𝔑𝒫𝑐𝑙(𝐾	PROPN
cana-5363	362	13	)	)	PUNCT
cana-5363	362	14	)	)	PUNCT
cana-5363	362	15	.	.	PUNCT
cana-5363	363	1	proof	proof	NOUN
cana-5363	363	2	.	.	PUNCT
cana-5363	364	1	it	it	PRON
cana-5363	364	2	follows	follow	VERB
cana-5363	364	3	from	from	ADP
cana-5363	364	4	theorem	theorem	ADJ
cana-5363	364	5	3.14	3.14	NUM
cana-5363	364	6	.	.	PUNCT
cana-5363	365	1	proposition	proposition	NOUN
cana-5363	365	2	3.6	3.6	NUM
cana-5363	365	3	if	if	SCONJ
cana-5363	365	4	𝐾	𝐾	PROPN
cana-5363	365	5	is	be	AUX
cana-5363	365	6	a	a	DET
cana-5363	365	7	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	365	8	set	set	NOUN
cana-5363	365	9	of	of	ADP
cana-5363	365	10	a	a	DET
cana-5363	365	11	𝒫ℱ𝔑𝑡𝑠	𝒫ℱ𝔑𝑡𝑠	PROPN
cana-5363	365	12	(	(	PUNCT
cana-5363	365	13	𝑈	𝑈	PROPN
cana-5363	365	14	,	,	PUNCT
cana-5363	365	15	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	365	16	)	)	PUNCT
cana-5363	365	17	)	)	PUNCT
cana-5363	366	1	such	such	ADJ
cana-5363	366	2	that	that	SCONJ
cana-5363	366	3	𝐾	𝐾	PROPN
cana-5363	366	4	⊆	⊆	NUM
cana-5363	366	5	𝐿	𝐿	PROPN
cana-5363	366	6	⊆	⊆	NUM
cana-5363	366	7	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝐾	𝒫ℱ𝔑𝛿𝒫𝑐𝑙(𝐾	NUM
cana-5363	366	8	)	)	PUNCT
cana-5363	366	9	,	,	PUNCT
cana-5363	366	10	then	then	ADV
cana-5363	366	11	𝐿	𝐿	PROPN
cana-5363	366	12	is	be	AUX
cana-5363	366	13	𝒫ℱ𝔑𝑍𝑜.	𝒫ℱ𝔑𝑍𝑜.	PROPN
cana-5363	366	14	proof	proof	NOUN
cana-5363	366	15	.	.	PUNCT
cana-5363	367	1	it	it	PRON
cana-5363	367	2	is	be	AUX
cana-5363	367	3	clear	clear	ADJ
cana-5363	367	4	.	.	PUNCT
cana-5363	368	1	definition	definition	NOUN
cana-5363	368	2	3.7	3.7	NUM
cana-5363	368	3	a	a	DET
cana-5363	368	4	set	set	ADJ
cana-5363	368	5	𝐾	𝐾	PROPN
cana-5363	368	6	of	of	ADP
cana-5363	368	7	a	a	DET
cana-5363	368	8	𝒫ℱ𝔑𝑡𝑠	𝒫ℱ𝔑𝑡𝑠	PROPN
cana-5363	368	9	(	(	PUNCT
cana-5363	368	10	𝑈	𝑈	PROPN
cana-5363	368	11	,	,	PUNCT
cana-5363	368	12	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	368	13	)	)	PUNCT
cana-5363	368	14	)	)	PUNCT
cana-5363	368	15	is	be	AUX
cana-5363	368	16	said	say	VERB
cana-5363	368	17	to	to	PART
cana-5363	368	18	be	be	AUX
cana-5363	368	19	locally	locally	ADV
cana-5363	368	20	𝒫ℱ𝔑𝑍𝑐𝑠	𝒫ℱ𝔑𝑍𝑐𝑠	PROPN
cana-5363	368	21	if	if	SCONJ
cana-5363	368	22	𝐾	𝐾	PROPN
cana-5363	368	23	=	=	SYM
cana-5363	368	24	𝑂	𝑂	PROPN
cana-5363	368	25	∩	∩	ADJ
cana-5363	368	26	𝑆	𝑆	PROPN
cana-5363	368	27	,	,	PUNCT
cana-5363	368	28	where	where	SCONJ
cana-5363	368	29	𝑂	𝑂	PROPN
cana-5363	368	30	∈	∈	NOUN
cana-5363	368	31	𝑂(𝐴	𝑂(𝐴	PROPN
cana-5363	368	32	)	)	PUNCT
cana-5363	368	33	and	and	CCONJ
cana-5363	368	34	𝑆	𝑆	PROPN
cana-5363	368	35	∈	∈	PROPN
cana-5363	368	36	𝒫ℱ𝔑𝑍𝐶(𝐴	𝒫ℱ𝔑𝑍𝐶(𝐴	NUM
cana-5363	368	37	)	)	PUNCT
cana-5363	368	38	.	.	PUNCT
cana-5363	369	1	theorem	theorem	NOUN
cana-5363	369	2	3.16	3.16	NUM
cana-5363	369	3	let	let	VERB
cana-5363	369	4	𝐻	𝐻	PRON
cana-5363	369	5	be	be	AUX
cana-5363	369	6	a	a	DET
cana-5363	369	7	𝑝𝑓𝑠	𝑝𝑓𝑠	NOUN
cana-5363	369	8	of	of	ADP
cana-5363	369	9	a	a	DET
cana-5363	369	10	𝒫ℱ𝔑𝑡𝑠	𝒫ℱ𝔑𝑡𝑠	PROPN
cana-5363	369	11	(	(	PUNCT
cana-5363	369	12	𝑈	𝑈	PROPN
cana-5363	369	13	,	,	PUNCT
cana-5363	369	14	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	369	15	)	)	PUNCT
cana-5363	369	16	)	)	PUNCT
cana-5363	369	17	.	.	PUNCT
cana-5363	370	1	then	then	ADV
cana-5363	370	2	𝐻	𝐻	PROPN
cana-5363	370	3	is	be	AUX
cana-5363	370	4	locally	locally	ADV
cana-5363	370	5	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	ADJ
cana-5363	370	6	iff	iff	PROPN
cana-5363	370	7	𝐻	𝐻	NOUN
cana-5363	370	8	=	=	NOUN
cana-5363	370	9	𝑂	𝑂	NOUN
cana-5363	370	10	∩	∩	ADJ
cana-5363	370	11	𝒫ℱ𝔑𝑍𝑐𝑙(𝐻	𝒫ℱ𝔑𝑍𝑐𝑙(𝐻	NOUN
cana-5363	370	12	)	)	PUNCT
cana-5363	370	13	.	.	PUNCT
cana-5363	371	1	proof	proof	NOUN
cana-5363	371	2	.	.	PUNCT
cana-5363	372	1	since	since	SCONJ
cana-5363	372	2	𝐻	𝐻	PROPN
cana-5363	372	3	is	be	AUX
cana-5363	372	4	a	a	DET
cana-5363	372	5	locally	locally	ADV
cana-5363	372	6	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	ADJ
cana-5363	372	7	set	set	NOUN
cana-5363	372	8	,	,	PUNCT
cana-5363	372	9	then	then	ADV
cana-5363	372	10	𝐻	𝐻	PROPN
cana-5363	372	11	=	=	NOUN
cana-5363	372	12	𝑂	𝑂	PROPN
cana-5363	372	13	∩	∩	ADJ
cana-5363	372	14	𝑆	𝑆	PROPN
cana-5363	372	15	,	,	PUNCT
cana-5363	372	16	where	where	SCONJ
cana-5363	372	17	𝑂	𝑂	PROPN
cana-5363	372	18	∈	∈	NOUN
cana-5363	372	19	𝑂(𝐴	𝑂(𝐴	PROPN
cana-5363	372	20	)	)	PUNCT
cana-5363	372	21	and	and	CCONJ
cana-5363	372	22	𝑆	𝑆	PROPN
cana-5363	372	23	∈	∈	PROPN
cana-5363	372	24	𝒫ℱ𝔑𝑍𝐶(𝐴	𝒫ℱ𝔑𝑍𝐶(𝐴	NUM
cana-5363	372	25	)	)	PUNCT
cana-5363	372	26	and	and	CCONJ
cana-5363	372	27	hence	hence	ADV
cana-5363	372	28	𝐻	𝐻	NOUN
cana-5363	372	29	⊆	⊆	NUM
cana-5363	372	30	𝒫ℱ𝔑𝑍𝑐𝑙(𝐻	𝒫ℱ𝔑𝑍𝑐𝑙(𝐻	NOUN
cana-5363	372	31	)	)	PUNCT
cana-5363	372	32	⊆	⊆	NUM
cana-5363	372	33	𝒫ℱ𝔑𝑍𝑐𝑙(𝑆	𝒫ℱ𝔑𝑍𝑐𝑙(𝑆	PROPN
cana-5363	372	34	)	)	PUNCT
cana-5363	372	35	=	=	SYM
cana-5363	373	1	𝑆.	𝑆.	VERB
cana-5363	373	2	thus	thus	ADV
cana-5363	373	3	𝐻	𝐻	PROPN
cana-5363	373	4	⊆	⊆	NUM
cana-5363	373	5	𝑂	𝑂	NOUN
cana-5363	373	6	∩	∩	ADJ
cana-5363	373	7	𝒫ℱ𝔑𝑍𝑐𝑙(𝐻	𝒫ℱ𝔑𝑍𝑐𝑙(𝐻	NOUN
cana-5363	373	8	)	)	PUNCT
cana-5363	373	9	⊆	⊆	NUM
cana-5363	373	10	𝑂	𝑂	PROPN
cana-5363	373	11	∩	∩	ADJ
cana-5363	373	12	𝒫ℱ𝔑𝑍𝑐𝑙(𝑆	𝒫ℱ𝔑𝑍𝑐𝑙(𝑆	PROPN
cana-5363	373	13	)	)	PUNCT
cana-5363	373	14	=	=	SYM
cana-5363	373	15	𝐻.	𝐻.	PROPN
cana-5363	373	16	therefore	therefore	ADV
cana-5363	373	17	𝐻	𝐻	PROPN
cana-5363	373	18	=	=	NOUN
cana-5363	373	19	𝑂	𝑂	NOUN
cana-5363	373	20	∩	∩	ADJ
cana-5363	373	21	𝒫ℱ𝔑𝑍𝑐𝑙(𝐻	𝒫ℱ𝔑𝑍𝑐𝑙(𝐻	NOUN
cana-5363	373	22	)	)	PUNCT
cana-5363	373	23	.	.	PUNCT
cana-5363	374	1	the	the	DET
cana-5363	374	2	converse	converse	NOUN
cana-5363	374	3	is	be	AUX
cana-5363	374	4	clear	clear	ADJ
cana-5363	374	5	.	.	PUNCT
cana-5363	375	1	theorem	theorem	VERB
cana-5363	375	2	3.17	3.17	NUM
cana-5363	375	3	let	let	VERB
cana-5363	375	4	𝐾	𝐾	PRON
cana-5363	375	5	be	be	AUX
cana-5363	375	6	a	a	DET
cana-5363	375	7	locally	locally	ADV
cana-5363	375	8	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	ADJ
cana-5363	375	9	set	set	NOUN
cana-5363	375	10	of	of	ADP
cana-5363	375	11	a	a	DET
cana-5363	375	12	space	space	NOUN
cana-5363	375	13	(	(	PUNCT
cana-5363	375	14	𝑈	𝑈	PROPN
cana-5363	375	15	,	,	PUNCT
cana-5363	375	16	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	375	17	)	)	PUNCT
cana-5363	375	18	)	)	PUNCT
cana-5363	375	19	.	.	PUNCT
cana-5363	376	1	then	then	ADV
cana-5363	376	2	the	the	DET
cana-5363	376	3	following	following	ADJ
cana-5363	376	4	statements	statement	NOUN
cana-5363	376	5	are	be	AUX
cana-5363	376	6	hold	hold	ADJ
cana-5363	376	7	:	:	PUNCT
cana-5363	376	8	(	(	PUNCT
cana-5363	376	9	i	i	NOUN
cana-5363	376	10	)	)	PUNCT
cana-5363	376	11	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	376	12	)	)	PUNCT
cana-5363	376	13	−	−	PROPN
cana-5363	377	1	𝐾	𝐾	PROPN
cana-5363	377	2	is	be	AUX
cana-5363	377	3	a	a	DET
cana-5363	377	4	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	ADJ
cana-5363	377	5	set	set	NOUN
cana-5363	377	6	,	,	PUNCT
cana-5363	377	7	(	(	PUNCT
cana-5363	377	8	ii	ii	NOUN
cana-5363	377	9	)	)	PUNCT
cana-5363	377	10	(	(	PUNCT
cana-5363	377	11	𝐾	𝐾	PROPN
cana-5363	377	12	∪	∪	NOUN
cana-5363	377	13	(	(	PUNCT
cana-5363	377	14	1𝒫	1𝒫	INTJ
cana-5363	377	15	−	−	PROPN
cana-5363	377	16	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	377	17	)	)	PUNCT
cana-5363	377	18	)	)	PUNCT
cana-5363	377	19	)	)	PUNCT
cana-5363	377	20	is	be	AUX
cana-5363	377	21	a	a	DET
cana-5363	377	22	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	PROPN
cana-5363	377	23	,	,	PUNCT
cana-5363	377	24	(	(	PUNCT
cana-5363	377	25	iii	iii	X
cana-5363	377	26	)	)	PUNCT
cana-5363	377	27	𝐾	𝐾	PROPN
cana-5363	377	28	⊆	⊆	NUM
cana-5363	377	29	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	PROPN
cana-5363	377	30	∪	∪	X
cana-5363	377	31	(	(	PUNCT
cana-5363	377	32	1𝒫	1𝒫	INTJ
cana-5363	377	33	−	−	PROPN
cana-5363	377	34	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	377	35	)	)	PUNCT
cana-5363	377	36	)	)	PUNCT
cana-5363	377	37	)	)	PUNCT
cana-5363	377	38	.	.	PUNCT
cana-5363	378	1	proof	proof	NOUN
cana-5363	378	2	.	.	PUNCT
cana-5363	379	1	(	(	PUNCT
cana-5363	379	2	i	i	NOUN
cana-5363	379	3	)	)	PUNCT
cana-5363	379	4	if	if	SCONJ
cana-5363	379	5	𝐾	𝐾	PROPN
cana-5363	379	6	is	be	AUX
cana-5363	379	7	a	a	DET
cana-5363	379	8	locally	locally	ADV
cana-5363	379	9	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	ADJ
cana-5363	379	10	set	set	NOUN
cana-5363	379	11	,	,	PUNCT
cana-5363	379	12	then	then	ADV
cana-5363	379	13	there	there	PRON
cana-5363	379	14	exists	exist	VERB
cana-5363	379	15	an	an	DET
cana-5363	379	16	𝒫ℱ𝔑𝑜	𝒫ℱ𝔑𝑜	NOUN
cana-5363	379	17	set	set	VERB
cana-5363	379	18	𝑂	𝑂	PROPN
cana-5363	379	19	such	such	DET
cana-5363	379	20	that	that	DET
cana-5363	379	21	𝐾	𝐾	NOUN
cana-5363	379	22	=	=	SYM
cana-5363	379	23	𝑂	𝑂	PROPN
cana-5363	379	24	∩	∩	ADJ
cana-5363	379	25	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	379	26	)	)	PUNCT
cana-5363	379	27	.	.	PUNCT
cana-5363	380	1	hence	hence	ADV
cana-5363	380	2	,	,	PUNCT
cana-5363	380	3	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	380	4	)	)	PUNCT
cana-5363	380	5	−	−	PROPN
cana-5363	380	6	𝐾	𝐾	PROPN
cana-5363	380	7	=	=	PROPN
cana-5363	380	8	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	380	9	)	)	PUNCT
cana-5363	380	10	−	−	PROPN
cana-5363	381	1	(	(	PUNCT
cana-5363	381	2	𝑂	𝑂	PROPN
cana-5363	381	3	∩	∩	ADJ
cana-5363	381	4	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	381	5	)	)	PUNCT
cana-5363	381	6	)	)	PUNCT
cana-5363	382	1	=	=	SYM
cana-5363	382	2	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	382	3	)	)	PUNCT
cana-5363	382	4	∩	∩	NOUN
cana-5363	382	5	(	(	PUNCT
cana-5363	382	6	1𝒫	1𝒫	INTJ
cana-5363	382	7	−	−	PROPN
cana-5363	382	8	(	(	PUNCT
cana-5363	382	9	𝑂	𝑂	PROPN
cana-5363	382	10	∩	∩	ADJ
cana-5363	382	11	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	382	12	)	)	PUNCT
cana-5363	382	13	)	)	PUNCT
cana-5363	382	14	)	)	PUNCT
cana-5363	383	1	=	=	SYM
cana-5363	383	2	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	383	3	)	)	PUNCT
cana-5363	383	4	∩	∩	NOUN
cana-5363	383	5	(	(	PUNCT
cana-5363	383	6	1𝒫	1𝒫	INTJ
cana-5363	383	7	−	−	NOUN
cana-5363	383	8	𝑂	𝑂	PROPN
cana-5363	383	9	)	)	PUNCT
cana-5363	383	10	∪	∪	NOUN
cana-5363	383	11	(	(	PUNCT
cana-5363	383	12	1𝒫	1𝒫	INTJ
cana-5363	383	13	−	−	PROPN
cana-5363	383	14	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	383	15	)	)	PUNCT
cana-5363	383	16	)	)	PUNCT
cana-5363	383	17	)	)	PUNCT
cana-5363	384	1	=	=	SYM
cana-5363	384	2	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	384	3	)	)	PUNCT
cana-5363	384	4	∩	∩	NOUN
cana-5363	384	5	(	(	PUNCT
cana-5363	384	6	1𝒫	1𝒫	INTJ
cana-5363	384	7	−	−	PROPN
cana-5363	384	8	𝑂	𝑂	PROPN
cana-5363	384	9	)	)	PUNCT
cana-5363	384	10	which	which	PRON
cana-5363	384	11	is	be	AUX
cana-5363	384	12	𝒫ℱ𝔑𝑍𝑐.	𝒫ℱ𝔑𝑍𝑐.	PROPN
cana-5363	384	13	(	(	PUNCT
cana-5363	384	14	ii	ii	NOUN
cana-5363	384	15	)	)	PUNCT
cana-5363	384	16	since	since	SCONJ
cana-5363	384	17	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	384	18	)	)	PUNCT
cana-5363	384	19	−	−	PROPN
cana-5363	385	1	𝐾	𝐾	PROPN
cana-5363	385	2	is	be	AUX
cana-5363	385	3	𝒫ℱ𝔑𝑍𝑐	𝒫ℱ𝔑𝑍𝑐	PROPN
cana-5363	385	4	,	,	PUNCT
cana-5363	385	5	then	then	ADV
cana-5363	385	6	1𝒫	1𝒫	INTJ
cana-5363	385	7	−	−	PROPN
cana-5363	386	1	(	(	PUNCT
cana-5363	386	2	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	386	3	)	)	PUNCT
cana-5363	386	4	−	−	PROPN
cana-5363	386	5	𝐾	𝐾	PROPN
cana-5363	386	6	)	)	PUNCT
cana-5363	386	7	is	be	AUX
cana-5363	386	8	a	a	DET
cana-5363	386	9	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	386	10	set	set	NOUN
cana-5363	386	11	.	.	PUNCT
cana-5363	387	1	since	since	SCONJ
cana-5363	387	2	1𝒫	1𝒫	NUM
cana-5363	387	3	−	−	PROPN
cana-5363	387	4	(	(	PUNCT
cana-5363	387	5	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	387	6	)	)	PUNCT
cana-5363	387	7	−	−	PROPN
cana-5363	387	8	𝐾	𝐾	PROPN
cana-5363	387	9	)	)	PUNCT
cana-5363	387	10	=	=	SYM
cana-5363	387	11	(	(	PUNCT
cana-5363	387	12	(	(	PUNCT
cana-5363	387	13	1𝒫	1𝒫	INTJ
cana-5363	387	14	−	−	PROPN
cana-5363	387	15	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	387	16	)	)	PUNCT
cana-5363	387	17	)	)	PUNCT
cana-5363	387	18	∪	∪	ADP
cana-5363	387	19	(	(	PUNCT
cana-5363	387	20	1𝒫	1𝒫	NOUN
cana-5363	387	21	∩	∩	ADJ
cana-5363	387	22	𝐾	𝐾	NOUN
cana-5363	387	23	)	)	PUNCT
cana-5363	387	24	)	)	PUNCT
cana-5363	388	1	=	=	PUNCT
cana-5363	388	2	(	(	PUNCT
cana-5363	388	3	𝐾	𝐾	NOUN
cana-5363	388	4	∪	∪	X
cana-5363	388	5	(	(	PUNCT
cana-5363	388	6	1𝒫	1𝒫	INTJ
cana-5363	388	7	−	−	PROPN
cana-5363	388	8	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	388	9	)	)	PUNCT
cana-5363	388	10	)	)	PUNCT
cana-5363	388	11	)	)	PUNCT
cana-5363	388	12	,	,	PUNCT
cana-5363	388	13	then	then	ADV
cana-5363	388	14	𝐾	𝐾	PROPN
cana-5363	388	15	∪	∪	NOUN
cana-5363	388	16	(	(	PUNCT
cana-5363	388	17	1𝒫	1𝒫	INTJ
cana-5363	388	18	−	−	PROPN
cana-5363	388	19	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	𝒫ℱ𝔑𝑍𝑐𝑙(𝐾	PROPN
cana-5363	388	20	)	)	PUNCT
cana-5363	388	21	)	)	PUNCT
cana-5363	388	22	is	be	AUX
cana-5363	388	23	𝒫ℱ𝔑𝑍𝑜𝑠.	𝒫ℱ𝔑𝑍𝑜𝑠.	NOUN
cana-5363	388	24	(	(	PUNCT
cana-5363	388	25	iii	iii	X
cana-5363	388	26	)	)	PUNCT
cana-5363	388	27	it	it	PRON
cana-5363	388	28	follows	follow	VERB
cana-5363	388	29	from	from	ADP
cana-5363	388	30	(	(	PUNCT
cana-5363	388	31	ii	ii	NOUN
cana-5363	388	32	)	)	PUNCT
cana-5363	388	33	.	.	PUNCT
cana-5363	389	1	definition	definition	NOUN
cana-5363	389	2	3.8	3.8	NUM
cana-5363	389	3	a	a	DET
cana-5363	389	4	pythagorean	pythagorean	ADJ
cana-5363	389	5	fuzzy	fuzzy	NOUN
cana-5363	389	6	set	set	VERB
cana-5363	389	7	𝐾	𝐾	PROPN
cana-5363	389	8	of	of	ADP
cana-5363	389	9	a	a	DET
cana-5363	389	10	space	space	NOUN
cana-5363	389	11	(	(	PUNCT
cana-5363	389	12	𝑈	𝑈	PROPN
cana-5363	389	13	,	,	PUNCT
cana-5363	389	14	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	389	15	)	)	PUNCT
cana-5363	389	16	)	)	PUNCT
cana-5363	389	17	is	be	AUX
cana-5363	389	18	said	say	VERB
cana-5363	389	19	to	to	PART
cana-5363	389	20	be	be	AUX
cana-5363	389	21	pythagorean	pythagorean	PROPN
cana-5363	389	22	fuzzy	fuzzy	ADJ
cana-5363	389	23	nano	nano	NOUN
cana-5363	389	24	𝐷(𝑐	𝐷(𝑐	PROPN
cana-5363	389	25	,	,	PUNCT
cana-5363	389	26	𝑧	𝑧	NOUN
cana-5363	389	27	)	)	PUNCT
cana-5363	389	28	(	(	PUNCT
cana-5363	389	29	briefly	briefly	ADV
cana-5363	389	30	,	,	PUNCT
cana-5363	389	31	𝒫ℱ𝔑𝐷(𝑐	𝒫ℱ𝔑𝐷(𝑐	NOUN
cana-5363	389	32	,	,	PUNCT
cana-5363	389	33	𝑧	𝑧	NOUN
cana-5363	389	34	)	)	PUNCT
cana-5363	389	35	)	)	PUNCT
cana-5363	390	1	iff	iff	PROPN
cana-5363	390	2	𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑖𝑛𝑡(𝐾	NOUN
cana-5363	390	3	)	)	PUNCT
cana-5363	390	4	=	=	SYM
cana-5363	390	5	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	𝒫ℱ𝔑𝑍𝑖𝑛𝑡(𝐾	NOUN
cana-5363	390	6	)	)	PUNCT
cana-5363	390	7	.	.	PUNCT
cana-5363	391	1	remark	remark	VERB
cana-5363	391	2	3.6	3.6	NUM
cana-5363	391	3	one	one	NOUN
cana-5363	391	4	may	may	AUX
cana-5363	391	5	notice	notice	VERB
cana-5363	391	6	that	that	SCONJ
cana-5363	391	7	the	the	DET
cana-5363	391	8	concepts	concept	NOUN
cana-5363	391	9	of	of	ADP
cana-5363	391	10	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	391	11	and	and	CCONJ
cana-5363	391	12	𝒫ℱ𝔑𝐷(𝑐	𝒫ℱ𝔑𝐷(𝑐	NOUN
cana-5363	391	13	,	,	PUNCT
cana-5363	391	14	𝑧	𝑧	PART
cana-5363	391	15	)	)	PUNCT
cana-5363	391	16	are	be	AUX
cana-5363	391	17	independent	independent	ADJ
cana-5363	391	18	and	and	CCONJ
cana-5363	391	19	by	by	ADP
cana-5363	391	20	we	we	PRON
cana-5363	391	21	show	show	VERB
cana-5363	391	22	this	this	PRON
cana-5363	391	23	the	the	DET
cana-5363	391	24	following	follow	VERB
cana-5363	391	25	example	example	NOUN
cana-5363	391	26	.	.	PUNCT
cana-5363	392	1	example	example	NOUN
cana-5363	392	2	3.5	3.5	NUM
cana-5363	392	3	in	in	ADP
cana-5363	392	4	example	example	NOUN
cana-5363	392	5	3.1	3.1	NUM
cana-5363	392	6	,	,	PUNCT
cana-5363	392	7	the	the	DET
cana-5363	392	8	nano	nano	NOUN
cana-5363	392	9	sets	set	VERB
cana-5363	392	10	communications	communication	NOUN
cana-5363	392	11	on	on	ADP
cana-5363	392	12	applied	apply	VERB
cana-5363	392	13	nonlinear	nonlinear	ADJ
cana-5363	392	14	analysis	analysis	NOUN
cana-5363	392	15	issn	issn	NOUN
cana-5363	392	16	:	:	PUNCT
cana-5363	392	17	1074	1074	NUM
cana-5363	392	18	-	-	PUNCT
cana-5363	392	19	133x	133x	NUM
cana-5363	392	20	vol	vol	VERB
cana-5363	392	21	32	32	NUM
cana-5363	392	22	no	no	NOUN
cana-5363	392	23	.	.	PUNCT
cana-5363	393	1	10s	10	NOUN
cana-5363	393	2	(	(	PUNCT
cana-5363	393	3	2025	2025	NUM
cana-5363	393	4	)	)	PUNCT
cana-5363	393	5	2018	2018	NUM
cana-5363	393	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5363	393	7	1	1	X
cana-5363	393	8	.	.	PUNCT
cana-5363	393	9	{	{	PUNCT
cana-5363	394	1	⟨	⟨	ADP
cana-5363	394	2	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	394	3	0.3	0.3	NUM
cana-5363	394	4	⟩	⟩	NOUN
cana-5363	394	5	,	,	PUNCT
cana-5363	394	6	⟨	⟨	VERB
cana-5363	394	7	𝑠2	𝑠2	PROPN
cana-5363	394	8	0.5	0.5	NUM
cana-5363	394	9	⟩	⟩	NOUN
cana-5363	394	10	,	,	PUNCT
cana-5363	394	11	⟨	⟨	VERB
cana-5363	394	12	𝑠3	𝑠3	PROPN
cana-5363	394	13	0.7	0.7	NUM
cana-5363	394	14	⟩	⟩	NOUN
cana-5363	394	15	}	}	PUNCT
cana-5363	394	16	is	be	AUX
cana-5363	394	17	a	a	DET
cana-5363	394	18	𝒫ℱ𝔑𝑍𝑜𝑠	𝒫ℱ𝔑𝑍𝑜𝑠	PROPN
cana-5363	394	19	but	but	CCONJ
cana-5363	394	20	not	not	PART
cana-5363	394	21	𝒫ℱ𝔑𝐷(𝑐	𝒫ℱ𝔑𝐷(𝑐	NOUN
cana-5363	394	22	,	,	PUNCT
cana-5363	394	23	𝑧	𝑧	NOUN
cana-5363	394	24	)	)	PUNCT
cana-5363	394	25	.	.	PUNCT
cana-5363	395	1	2	2	X
cana-5363	395	2	.	.	PUNCT
cana-5363	395	3	{	{	PUNCT
cana-5363	395	4	⟨	⟨	ADP
cana-5363	395	5	𝑠1,𝑠4	𝑠1,𝑠4	PROPN
cana-5363	395	6	0.1	0.1	NUM
cana-5363	395	7	⟩	⟩	NOUN
cana-5363	395	8	,	,	PUNCT
cana-5363	395	9	⟨	⟨	VERB
cana-5363	395	10	𝑠2	𝑠2	PROPN
cana-5363	395	11	0.3	0.3	NUM
cana-5363	395	12	⟩	⟩	NOUN
cana-5363	395	13	,	,	PUNCT
cana-5363	395	14	⟨	⟨	VERB
cana-5363	395	15	𝑠3	𝑠3	PROPN
cana-5363	395	16	0.5	0.5	NUM
cana-5363	395	17	⟩	⟩	NOUN
cana-5363	395	18	}	}	PUNCT
cana-5363	395	19	is	be	AUX
cana-5363	395	20	a	a	DET
cana-5363	395	21	𝒫ℱ𝔑𝐷(𝑐	𝒫ℱ𝔑𝐷(𝑐	NOUN
cana-5363	395	22	,	,	PUNCT
cana-5363	395	23	𝑧	𝑧	NOUN
cana-5363	395	24	)	)	PUNCT
cana-5363	395	25	but	but	CCONJ
cana-5363	395	26	not	not	PART
cana-5363	395	27	𝒫ℱ𝔑𝑍𝑜𝑠.	𝒫ℱ𝔑𝑍𝑜𝑠.	NOUN
cana-5363	395	28	theorem	theorem	VERB
cana-5363	395	29	3.18	3.18	NUM
cana-5363	395	30	let	let	VERB
cana-5363	395	31	𝐾	𝐾	PRON
cana-5363	395	32	be	be	AUX
cana-5363	395	33	a	a	DET
cana-5363	395	34	𝑝𝑓𝑠	𝑝𝑓𝑠	NOUN
cana-5363	395	35	of	of	ADP
cana-5363	395	36	𝒫ℱ𝔑𝑡𝑠	𝒫ℱ𝔑𝑡𝑠	PROPN
cana-5363	395	37	(	(	PUNCT
cana-5363	395	38	𝑈	𝑈	PROPN
cana-5363	395	39	,	,	PUNCT
cana-5363	395	40	𝜏ℛ(𝐴	𝜏ℛ(𝐴	NOUN
cana-5363	395	41	)	)	PUNCT
cana-5363	395	42	)	)	PUNCT
cana-5363	395	43	.	.	PUNCT
cana-5363	396	1	then	then	ADV
cana-5363	396	2	the	the	DET
cana-5363	396	3	following	follow	VERB
cana-5363	396	4	are	be	AUX
cana-5363	396	5	equivalent	equivalent	ADJ
cana-5363	396	6	:	:	PUNCT
cana-5363	396	7	1	1	X
cana-5363	396	8	.	.	X
cana-5363	397	1	𝐾	𝐾	NOUN
cana-5363	397	2	is	be	AUX
cana-5363	397	3	an	an	DET
cana-5363	397	4	𝒫ℱ𝔑𝑜	𝒫ℱ𝔑𝑜	PROPN
cana-5363	397	5	set	set	NOUN
cana-5363	397	6	,	,	PUNCT
cana-5363	397	7	2	2	NUM
cana-5363	397	8	.	.	X
cana-5363	398	1	𝐾	𝐾	NOUN
cana-5363	398	2	is	be	AUX
cana-5363	398	3	𝒫ℱ𝔑𝑍𝑜	𝒫ℱ𝔑𝑍𝑜	NOUN
cana-5363	398	4	and	and	CCONJ
cana-5363	398	5	𝒫ℱ𝔑𝐷(𝑐	𝒫ℱ𝔑𝐷(𝑐	NOUN
cana-5363	398	6	,	,	PUNCT
cana-5363	398	7	𝑧	𝑧	NOUN
cana-5363	398	8	)	)	PUNCT
cana-5363	398	9	.	.	PUNCT
cana-5363	399	1	proof	proof	NOUN
cana-5363	399	2	.	.	PUNCT
cana-5363	400	1	obvious	obvious	ADJ
cana-5363	400	2	.	.	PUNCT
cana-5363	401	1	4	4	NUM
cana-5363	401	2	application	application	NOUN
cana-5363	401	3	entropy	entropy	NOUN
cana-5363	401	4	as	as	ADP
cana-5363	401	5	a	a	DET
cana-5363	401	6	measure	measure	NOUN
cana-5363	401	7	of	of	ADP
cana-5363	401	8	fuzziness	fuzziness	NOUN
cana-5363	401	9	was	be	AUX
cana-5363	401	10	first	first	ADV
cana-5363	401	11	proposed	propose	VERB
cana-5363	401	12	by	by	ADP
cana-5363	401	13	zadeh	zadeh	PROPN
cana-5363	401	14	[	[	X
cana-5363	401	15	21	21	NUM
cana-5363	401	16	]	]	PUNCT
cana-5363	401	17	.	.	PUNCT
cana-5363	402	1	later	later	ADV
cana-5363	402	2	many	many	ADJ
cana-5363	402	3	mathematicians	mathematician	NOUN
cana-5363	402	4	defined	define	VERB
cana-5363	402	5	several	several	ADJ
cana-5363	402	6	entropy	entropy	NOUN
cana-5363	402	7	measures	measure	NOUN
cana-5363	402	8	.	.	PUNCT
cana-5363	403	1	in	in	ADP
cana-5363	403	2	this	this	DET
cana-5363	403	3	section	section	NOUN
cana-5363	403	4	,	,	PUNCT
cana-5363	403	5	we	we	PRON
cana-5363	403	6	focus	focus	VERB
cana-5363	403	7	on	on	ADP
cana-5363	403	8	defining	define	VERB
cana-5363	403	9	an	an	DET
cana-5363	403	10	entropy	entropy	NOUN
cana-5363	403	11	measure	measure	NOUN
cana-5363	403	12	for	for	ADP
cana-5363	403	13	𝑝𝑓𝑠	𝑝𝑓𝑠	PROPN
cana-5363	403	14	that	that	PRON
cana-5363	403	15	connects	connect	VERB
cana-5363	403	16	the	the	DET
cana-5363	403	17	degree	degree	NOUN
cana-5363	403	18	of	of	ADP
cana-5363	403	19	membership	membership	NOUN
cana-5363	403	20	and	and	CCONJ
cana-5363	403	21	non	non	ADJ
cana-5363	403	22	-	-	NOUN
cana-5363	403	23	membership	membership	NOUN
cana-5363	403	24	.	.	PUNCT
cana-5363	404	1	as	as	ADP
cana-5363	404	2	an	an	DET
cana-5363	404	3	example	example	NOUN
cana-5363	404	4	,	,	PUNCT
cana-5363	404	5	we	we	PRON
cana-5363	404	6	have	have	AUX
cana-5363	404	7	applied	apply	VERB
cana-5363	404	8	the	the	DET
cana-5363	404	9	proposed	propose	VERB
cana-5363	404	10	entropy	entropy	NOUN
cana-5363	404	11	measure	measure	NOUN
cana-5363	404	12	in	in	ADP
cana-5363	404	13	decision	decision	NOUN
cana-5363	404	14	making	making	NOUN
cana-5363	404	15	.	.	PUNCT
cana-5363	405	1	definition	definition	NOUN
cana-5363	405	2	4.1	4.1	NUM
cana-5363	405	3	let	let	VERB
cana-5363	405	4	𝐴	𝐴	PROPN
cana-5363	405	5	=	=	PUNCT
cana-5363	405	6	{	{	PUNCT
cana-5363	405	7	<	<	X
cana-5363	405	8	𝑥	𝑥	X
cana-5363	405	9	,	,	PUNCT
cana-5363	405	10	𝜇𝐴(𝑥	𝜇𝐴(𝑥	NUM
cana-5363	405	11	)	)	PUNCT
cana-5363	405	12	,	,	PUNCT
cana-5363	405	13	𝜆𝐴(𝑥)|𝑥	𝜆𝐴(𝑥)|𝑥	NOUN
cana-5363	405	14	∈	∈	PROPN
cana-5363	405	15	𝑋	𝑋	PROPN
cana-5363	405	16	}	}	PUNCT
cana-5363	405	17	be	be	AUX
cana-5363	405	18	a	a	DET
cana-5363	405	19	𝑝𝑓𝑠	𝑝𝑓𝑠	NOUN
cana-5363	405	20	in	in	ADP
cana-5363	405	21	𝑋.	𝑋.	PROPN
cana-5363	405	22	the	the	DET
cana-5363	405	23	new	new	ADJ
cana-5363	405	24	entropy	entropy	NOUN
cana-5363	405	25	measure	measure	NOUN
cana-5363	405	26	for	for	ADP
cana-5363	405	27	𝐴	𝐴	PROPN
cana-5363	405	28	denoted	denote	VERB
cana-5363	405	29	by	by	ADP
cana-5363	405	30	휀𝑝𝑓𝑠(𝐴	휀𝑝𝑓𝑠(𝐴	NOUN
cana-5363	405	31	)	)	PUNCT
cana-5363	405	32	,	,	PUNCT
cana-5363	405	33	is	be	AUX
cana-5363	405	34	a	a	DET
cana-5363	405	35	function	function	NOUN
cana-5363	405	36	,	,	PUNCT
cana-5363	405	37	휀𝑝𝑓𝑠	휀𝑝𝑓𝑠	NOUN
cana-5363	405	38	:	:	PUNCT
cana-5363	405	39	𝜏𝑝𝑓𝑠(𝑋	𝜏𝑝𝑓𝑠(𝑋	PROPN
cana-5363	405	40	)	)	PUNCT
cana-5363	405	41	→	→	PUNCT
cana-5363	406	1	[	[	X
cana-5363	406	2	0,1	0,1	NUM
cana-5363	406	3	]	]	PUNCT
cana-5363	406	4	and	and	CCONJ
cana-5363	406	5	is	be	AUX
cana-5363	406	6	defined	define	VERB
cana-5363	406	7	as	as	ADP
cana-5363	406	8	휀𝑝𝑓𝑠(𝐴	휀𝑝𝑓𝑠(𝐴	X
cana-5363	406	9	)	)	PUNCT
cana-5363	407	1	=	=	SYM
cana-5363	408	1	1	1	NUM
cana-5363	408	2	−	−	NUM
cana-5363	408	3	1	1	NUM
cana-5363	408	4	𝑛	𝑛	PRON
cana-5363	408	5	∑𝑛	∑𝑛	PROPN
cana-5363	408	6	𝑖=1	𝑖=1	PROPN
cana-5363	408	7	(	(	PUNCT
cana-5363	408	8	𝜇𝐴	𝜇𝐴	ADP
cana-5363	408	9	−	−	PROPN
cana-5363	408	10	𝜆𝐴)2	𝜆𝐴)2	NUM
cana-5363	408	11	;	;	PUNCT
cana-5363	408	12	𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦`𝑥𝑖	𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦`𝑥𝑖	NUM
cana-5363	408	13	∈	∈	PROPN
cana-5363	408	14	𝐴	𝐴	PROPN
cana-5363	408	15	,	,	PUNCT
cana-5363	408	16	where	where	SCONJ
cana-5363	408	17	𝜏𝑝𝑓𝑠(𝑋	𝜏𝑝𝑓𝑠(𝑋	NOUN
cana-5363	408	18	)	)	PUNCT
cana-5363	408	19	denote	denote	VERB
cana-5363	408	20	the	the	DET
cana-5363	408	21	family	family	NOUN
cana-5363	408	22	of	of	ADP
cana-5363	408	23	all	all	PRON
cana-5363	408	24	𝑝𝑓𝑠	𝑝𝑓𝑠	PROPN
cana-5363	408	25	’s	’s	NOUN
cana-5363	408	26	on	on	ADP
cana-5363	408	27	𝑋.	𝑋.	PROPN
cana-5363	408	28	example	example	NOUN
cana-5363	408	29	4.1	4.1	NUM
cana-5363	408	30	the	the	DET
cana-5363	408	31	association	association	NOUN
cana-5363	408	32	of	of	ADP
cana-5363	408	33	the	the	DET
cana-5363	408	34	tourism	tourism	NOUN
cana-5363	408	35	wants	want	VERB
cana-5363	408	36	to	to	PART
cana-5363	408	37	announce	announce	VERB
cana-5363	408	38	that	that	SCONJ
cana-5363	408	39	best	good	ADJ
cana-5363	408	40	“	"	PUNCT
cana-5363	408	41	hotel	hotel	NOUN
cana-5363	408	42	of	of	ADP
cana-5363	408	43	the	the	DET
cana-5363	408	44	year	year	NOUN
cana-5363	408	45	"	"	PUNCT
cana-5363	408	46	,	,	PUNCT
cana-5363	408	47	for	for	ADP
cana-5363	408	48	each	each	DET
cana-5363	408	49	year	year	NOUN
cana-5363	408	50	.	.	PUNCT
cana-5363	409	1	the	the	DET
cana-5363	409	2	actual	actual	ADJ
cana-5363	409	3	problem	problem	NOUN
cana-5363	409	4	is	be	AUX
cana-5363	409	5	,	,	PUNCT
cana-5363	409	6	they	they	PRON
cana-5363	409	7	want	want	VERB
cana-5363	409	8	to	to	PART
cana-5363	409	9	select	select	VERB
cana-5363	409	10	the	the	DET
cana-5363	409	11	best	good	ADJ
cana-5363	409	12	hotel	hotel	NOUN
cana-5363	409	13	based	base	VERB
cana-5363	409	14	on	on	ADP
cana-5363	409	15	reviews	review	NOUN
cana-5363	409	16	and	and	CCONJ
cana-5363	409	17	ratings	rating	NOUN
cana-5363	409	18	.	.	PUNCT
cana-5363	410	1	there	there	PRON
cana-5363	410	2	are	be	VERB
cana-5363	410	3	four	four	NUM
cana-5363	410	4	nominees	nominee	NOUN
cana-5363	410	5	namely	namely	ADV
cana-5363	410	6	hotel	hotel	NOUN
cana-5363	410	7	1	1	NUM
cana-5363	410	8	,	,	PUNCT
cana-5363	410	9	hotel	hotel	NOUN
cana-5363	410	10	2	2	NUM
cana-5363	410	11	,	,	PUNCT
cana-5363	410	12	hotel	hotel	NOUN
cana-5363	410	13	3	3	NUM
cana-5363	410	14	and	and	CCONJ
cana-5363	410	15	hotel	hotel	NOUN
cana-5363	410	16	4	4	NUM
cana-5363	410	17	for	for	ADP
cana-5363	410	18	this	this	DET
cana-5363	410	19	award	award	NOUN
cana-5363	410	20	and	and	CCONJ
cana-5363	410	21	they	they	PRON
cana-5363	410	22	have	have	VERB
cana-5363	410	23	to	to	PART
cana-5363	410	24	reviewed	review	VERB
cana-5363	410	25	based	base	VERB
cana-5363	410	26	on	on	ADP
cana-5363	410	27	the	the	DET
cana-5363	410	28	four	four	NUM
cana-5363	410	29	criteria	criterion	NOUN
cana-5363	410	30	namely	namely	ADV
cana-5363	410	31	ambiance	ambiance	NOUN
cana-5363	410	32	,	,	PUNCT
cana-5363	410	33	good	good	ADJ
cana-5363	410	34	food	food	NOUN
cana-5363	410	35	,	,	PUNCT
cana-5363	410	36	clean	clean	ADJ
cana-5363	410	37	and	and	CCONJ
cana-5363	410	38	tidy	tidy	ADJ
cana-5363	410	39	,	,	PUNCT
cana-5363	410	40	cyber	cyber	ADJ
cana-5363	410	41	security	security	NOUN
cana-5363	410	42	facility	facility	NOUN
cana-5363	410	43	.	.	PUNCT
cana-5363	411	1	here	here	ADV
cana-5363	411	2	we	we	PRON
cana-5363	411	3	use	use	VERB
cana-5363	411	4	entropy	entropy	NOUN
cana-5363	411	5	measure	measure	NOUN
cana-5363	411	6	to	to	PART
cana-5363	411	7	find	find	VERB
cana-5363	411	8	the	the	DET
cana-5363	411	9	best	good	ADJ
cana-5363	411	10	hotel	hotel	NOUN
cana-5363	411	11	by	by	ADP
cana-5363	411	12	the	the	DET
cana-5363	411	13	overall	overall	ADJ
cana-5363	411	14	entropy	entropy	NOUN
cana-5363	411	15	measure	measure	NOUN
cana-5363	411	16	with	with	ADP
cana-5363	411	17	the	the	DET
cana-5363	411	18	pythagorean	pythagorean	ADJ
cana-5363	411	19	fuzzy	fuzzy	ADJ
cana-5363	411	20	sets	set	NOUN
cana-5363	411	21	.	.	PUNCT
cana-5363	412	1	table	table	NOUN
cana-5363	412	2	1	1	NUM
cana-5363	412	3	.	.	PUNCT
cana-5363	413	1	reviews	review	NOUN
cana-5363	413	2	of	of	ADP
cana-5363	413	3	the	the	DET
cana-5363	413	4	hotels	hotel	NOUN
cana-5363	413	5	based	base	VERB
cana-5363	413	6	on	on	ADP
cana-5363	413	7	the	the	DET
cana-5363	413	8	criteria	criterion	NOUN
cana-5363	413	9	criteria	criterion	NOUN
cana-5363	413	10	1	1	NUM
cana-5363	413	11	(	(	PUNCT
cana-5363	413	12	𝐶1	𝐶1	NOUN
cana-5363	413	13	)	)	PUNCT
cana-5363	413	14	criteria	criterion	NOUN
cana-5363	413	15	2	2	NUM
cana-5363	413	16	(	(	PUNCT
cana-5363	413	17	𝐶2	𝐶2	ADJ
cana-5363	413	18	)	)	PUNCT
cana-5363	413	19	criteria	criterion	NOUN
cana-5363	413	20	3	3	NUM
cana-5363	413	21	(	(	PUNCT
cana-5363	413	22	𝐶3	𝐶3	NOUN
cana-5363	413	23	)	)	PUNCT
cana-5363	413	24	criteria	criterion	NOUN
cana-5363	413	25	4	4	NUM
cana-5363	413	26	(	(	PUNCT
cana-5363	413	27	𝐶4	𝐶4	NOUN
cana-5363	413	28	)	)	PUNCT
cana-5363	413	29	hotel	hotel	NOUN
cana-5363	413	30	1	1	NUM
cana-5363	413	31	(	(	PUNCT
cana-5363	413	32	𝐻1	𝐻1	PROPN
cana-5363	413	33	)	)	PUNCT
cana-5363	413	34	<	<	X
cana-5363	413	35	𝐻1	𝐻1	PROPN
cana-5363	413	36	,	,	PUNCT
cana-5363	413	37	𝐶1	𝐶1	PRON
cana-5363	413	38	;	;	PUNCT
cana-5363	414	1	0.9,0.3	0.9,0.3	PROPN
cana-5363	414	2	>	>	X
cana-5363	414	3	<	<	X
cana-5363	414	4	𝐻1	𝐻1	PROPN
cana-5363	414	5	,	,	PUNCT
cana-5363	414	6	𝐶2	𝐶2	ADJ
cana-5363	414	7	;	;	PUNCT
cana-5363	414	8	0.7,0.6	0.7,0.6	X
cana-5363	414	9	>	>	X
cana-5363	414	10	<	<	X
cana-5363	414	11	𝐻1	𝐻1	PROPN
cana-5363	414	12	,	,	PUNCT
cana-5363	414	13	𝐶3	𝐶3	NOUN
cana-5363	414	14	;	;	PUNCT
cana-5363	414	15	0.5,0.8	0.5,0.8	X
cana-5363	414	16	>	>	X
cana-5363	414	17	<	<	X
cana-5363	414	18	𝐻1	𝐻1	PROPN
cana-5363	414	19	,	,	PUNCT
cana-5363	414	20	𝐶4	𝐶4	NOUN
cana-5363	414	21	;	;	PUNCT
cana-5363	414	22	0.6,0.4	0.6,0.4	NUM
cana-5363	414	23	>	>	X
cana-5363	414	24	hotel	hotel	NOUN
cana-5363	414	25	2	2	NUM
cana-5363	414	26	(	(	PUNCT
cana-5363	414	27	𝐻2	𝐻2	PROPN
cana-5363	414	28	)	)	PUNCT
cana-5363	414	29	<	<	X
cana-5363	414	30	𝐻2	𝐻2	PROPN
cana-5363	414	31	,	,	PUNCT
cana-5363	414	32	𝐶1	𝐶1	PRON
cana-5363	414	33	;	;	PUNCT
cana-5363	414	34	0.7,0.1	0.7,0.1	PROPN
cana-5363	414	35	>	>	X
cana-5363	414	36	<	<	X
cana-5363	414	37	𝐻2	𝐻2	PROPN
cana-5363	414	38	,	,	PUNCT
cana-5363	414	39	𝐶2	𝐶2	ADJ
cana-5363	414	40	;	;	PUNCT
cana-5363	414	41	0.9,0.2	0.9,0.2	X
cana-5363	414	42	>	>	X
cana-5363	414	43	<	<	X
cana-5363	414	44	𝐻2	𝐻2	PROPN
cana-5363	414	45	,	,	PUNCT
cana-5363	414	46	𝐶3	𝐶3	ADJ
cana-5363	414	47	;	;	PUNCT
cana-5363	414	48	0.8,0.1	0.8,0.1	SYM
cana-5363	414	49	>	>	X
cana-5363	414	50	<	<	X
cana-5363	414	51	𝐻2	𝐻2	PROPN
cana-5363	414	52	,	,	PUNCT
cana-5363	414	53	𝐶4	𝐶4	NOUN
cana-5363	414	54	;	;	PUNCT
cana-5363	414	55	0.6,0.3	0.6,0.3	PROPN
cana-5363	414	56	>	>	X
cana-5363	414	57	hotel	hotel	PROPN
cana-5363	414	58	3	3	NUM
cana-5363	414	59	(	(	PUNCT
cana-5363	414	60	𝐻3	𝐻3	PROPN
cana-5363	414	61	)	)	PUNCT
cana-5363	414	62	<	<	X
cana-5363	414	63	𝐻3	𝐻3	PROPN
cana-5363	414	64	,	,	PUNCT
cana-5363	414	65	𝐶1	𝐶1	PRON
cana-5363	414	66	;	;	PUNCT
cana-5363	415	1	0.8,0.4	0.8,0.4	NUM
cana-5363	415	2	>	>	X
cana-5363	415	3	<	<	X
cana-5363	415	4	𝐻3	𝐻3	PROPN
cana-5363	415	5	,	,	PUNCT
cana-5363	415	6	𝐶2	𝐶2	ADJ
cana-5363	415	7	;	;	PUNCT
cana-5363	415	8	0.7,0.5	0.7,0.5	NUM
cana-5363	415	9	>	>	X
cana-5363	415	10	<	<	X
cana-5363	415	11	𝐻3	𝐻3	PROPN
cana-5363	415	12	,	,	PUNCT
cana-5363	415	13	𝐶3	𝐶3	NOUN
cana-5363	415	14	;	;	PUNCT
cana-5363	415	15	0.6,0.2	0.6,0.2	PROPN
cana-5363	415	16	>	>	X
cana-5363	415	17	<	<	X
cana-5363	415	18	𝐻3	𝐻3	PROPN
cana-5363	415	19	,	,	PUNCT
cana-5363	415	20	𝐶4	𝐶4	NOUN
cana-5363	415	21	;	;	PUNCT
cana-5363	415	22	0.7,0.5	0.7,0.5	NUM
cana-5363	415	23	>	>	SYM
cana-5363	415	24	hotel	hotel	NOUN
cana-5363	415	25	4	4	NUM
cana-5363	415	26	(	(	PUNCT
cana-5363	415	27	𝐻4	𝐻4	PROPN
cana-5363	415	28	)	)	PUNCT
cana-5363	415	29	<	<	X
cana-5363	415	30	𝐻4	𝐻4	PROPN
cana-5363	415	31	,	,	PUNCT
cana-5363	415	32	𝐶1	𝐶1	PRON
cana-5363	415	33	;	;	PUNCT
cana-5363	416	1	0.7,0.2	0.7,0.2	PROPN
cana-5363	416	2	>	>	X
cana-5363	416	3	<	<	X
cana-5363	416	4	𝐻4	𝐻4	PROPN
cana-5363	416	5	,	,	PUNCT
cana-5363	416	6	𝐶2	𝐶2	PROPN
cana-5363	416	7	;	;	PUNCT
cana-5363	416	8	0.8,0.2	0.8,0.2	PROPN
cana-5363	416	9	>	>	X
cana-5363	416	10	<	<	X
cana-5363	416	11	𝐻4	𝐻4	PROPN
cana-5363	416	12	,	,	PUNCT
cana-5363	416	13	𝐶3	𝐶3	NOUN
cana-5363	416	14	;	;	PUNCT
cana-5363	416	15	0.8,0.4	0.8,0.4	NUM
cana-5363	416	16	>	>	X
cana-5363	416	17	<	<	X
cana-5363	416	18	𝐻4	𝐻4	PROPN
cana-5363	416	19	,	,	PUNCT
cana-5363	416	20	𝐶4	𝐶4	NOUN
cana-5363	416	21	;	;	PUNCT
cana-5363	416	22	0.6,0.6	0.6,0.6	PROPN
cana-5363	416	23	>	>	PUNCT
cana-5363	416	24	clearly	clearly	ADV
cana-5363	416	25	,	,	PUNCT
cana-5363	416	26	all	all	DET
cana-5363	416	27	values	value	NOUN
cana-5363	416	28	in	in	ADP
cana-5363	416	29	the	the	DET
cana-5363	416	30	table	table	NOUN
cana-5363	416	31	1	1	NUM
cana-5363	416	32	are	be	AUX
cana-5363	416	33	𝑝𝑓𝑠	𝑝𝑓𝑠	PROPN
cana-5363	416	34	’s	’s	NOUN
cana-5363	416	35	.	.	PUNCT
cana-5363	417	1	now	now	ADV
cana-5363	417	2	we	we	PRON
cana-5363	417	3	calculate	calculate	VERB
cana-5363	417	4	the	the	DET
cana-5363	417	5	휀𝑝𝑓𝑠	휀𝑝𝑓𝑠	NOUN
cana-5363	417	6	of	of	ADP
cana-5363	417	7	each	each	DET
cana-5363	417	8	hotel	hotel	NOUN
cana-5363	417	9	.	.	PUNCT
cana-5363	418	1	table	table	NOUN
cana-5363	418	2	2	2	NUM
cana-5363	418	3	.	.	PUNCT
cana-5363	418	4	entropy	entropy	PROPN
cana-5363	418	5	measure	measure	NOUN
cana-5363	418	6	of	of	ADP
cana-5363	418	7	each	each	DET
cana-5363	418	8	hotel	hotel	NOUN
cana-5363	418	9	.	.	PUNCT
cana-5363	419	1	휀𝑝𝑓𝑠(𝐻𝑖	휀𝑝𝑓𝑠(𝐻𝑖	PROPN
cana-5363	419	2	)	)	PUNCT
cana-5363	419	3	𝐻1	𝐻1	VERB
cana-5363	419	4	0.87	0.87	NUM
cana-5363	419	5	communications	communication	NOUN
cana-5363	419	6	on	on	ADP
cana-5363	419	7	applied	apply	VERB
cana-5363	419	8	nonlinear	nonlinear	ADJ
cana-5363	419	9	analysis	analysis	NOUN
cana-5363	419	10	issn	issn	NOUN
cana-5363	419	11	:	:	PUNCT
cana-5363	419	12	1074	1074	NUM
cana-5363	419	13	-	-	PUNCT
cana-5363	419	14	133x	133x	NUM
cana-5363	419	15	vol	vol	VERB
cana-5363	419	16	32	32	NUM
cana-5363	419	17	no	no	NOUN
cana-5363	419	18	.	.	PUNCT
cana-5363	420	1	10s	10	NOUN
cana-5363	420	2	(	(	PUNCT
cana-5363	420	3	2025	2025	NUM
cana-5363	420	4	)	)	PUNCT
cana-5363	420	5	2019	2019	NUM
cana-5363	420	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5363	420	7	𝐻2	𝐻2	VERB
cana-5363	420	8	0.64	0.64	NUM
cana-5363	420	9	𝐻3	𝐻3	NOUN
cana-5363	420	10	0.9	0.9	NUM
cana-5363	420	11	𝐻4	𝐻4	NOUN
cana-5363	420	12	0.81	0.81	NUM
cana-5363	420	13	from	from	ADP
cana-5363	420	14	table	table	NOUN
cana-5363	420	15	2	2	NUM
cana-5363	420	16	,	,	PUNCT
cana-5363	420	17	clearly	clearly	ADV
cana-5363	420	18	that	that	PRON
cana-5363	420	19	휀𝑝𝑓𝑠(𝐻2	휀𝑝𝑓𝑠(𝐻2	NUM
cana-5363	420	20	)	)	PUNCT
cana-5363	420	21	<	<	X
cana-5363	420	22	휀𝑝𝑓𝑠(𝐻4	휀𝑝𝑓𝑠(𝐻4	NOUN
cana-5363	420	23	)	)	PUNCT
cana-5363	420	24	<	<	X
cana-5363	420	25	휀𝑝𝑓𝑠(𝐻1	휀𝑝𝑓𝑠(𝐻1	X
cana-5363	420	26	)	)	PUNCT
cana-5363	420	27	<	<	X
cana-5363	420	28	휀𝑝𝑓𝑠(𝐻3	휀𝑝𝑓𝑠(𝐻3	NOUN
cana-5363	420	29	)	)	PUNCT
cana-5363	420	30	.	.	PUNCT
cana-5363	421	1	hence	hence	ADV
cana-5363	421	2	we	we	PRON
cana-5363	421	3	conclude	conclude	VERB
cana-5363	421	4	that	that	DET
cana-5363	421	5	𝐻2	𝐻2	PROPN
cana-5363	421	6	is	be	AUX
cana-5363	421	7	the	the	DET
cana-5363	421	8	best	good	ADJ
cana-5363	421	9	hotel	hotel	NOUN
cana-5363	421	10	of	of	ADP
cana-5363	421	11	the	the	DET
cana-5363	421	12	year	year	NOUN
cana-5363	421	13	with	with	ADP
cana-5363	421	14	less	less	ADJ
cana-5363	421	15	fuzziness	fuzziness	NOUN
cana-5363	421	16	.	.	PUNCT
cana-5363	422	1	5	5	NUM
cana-5363	422	2	conclusion	conclusion	NOUN
cana-5363	422	3	in	in	ADP
cana-5363	422	4	this	this	DET
cana-5363	422	5	paper	paper	NOUN
cana-5363	422	6	,	,	PUNCT
cana-5363	422	7	we	we	PRON
cana-5363	422	8	have	have	AUX
cana-5363	422	9	studied	study	VERB
cana-5363	422	10	a	a	DET
cana-5363	422	11	new	new	ADJ
cana-5363	422	12	class	class	NOUN
cana-5363	422	13	of	of	ADP
cana-5363	422	14	sets	set	NOUN
cana-5363	422	15	called	call	VERB
cana-5363	422	16	pythagorean	pythagorean	PROPN
cana-5363	422	17	fuzzy	fuzzy	ADJ
cana-5363	422	18	nano	nano	NOUN
cana-5363	422	19	𝑍-open	𝑍-open	ADJ
cana-5363	422	20	sets	set	NOUN
cana-5363	422	21	in	in	ADP
cana-5363	422	22	pythagorean	pythagorean	PROPN
cana-5363	422	23	fuzzy	fuzzy	ADJ
cana-5363	422	24	nano	nano	PROPN
cana-5363	422	25	topological	topological	ADJ
cana-5363	422	26	spaces	space	NOUN
cana-5363	422	27	and	and	CCONJ
cana-5363	422	28	their	their	PRON
cana-5363	422	29	properties	property	NOUN
cana-5363	422	30	,	,	PUNCT
cana-5363	422	31	and	and	CCONJ
cana-5363	422	32	also	also	ADV
cana-5363	422	33	discussed	discuss	VERB
cana-5363	422	34	about	about	ADP
cana-5363	422	35	pythagorean	pythagorean	PROPN
cana-5363	422	36	fuzzy	fuzzy	ADJ
cana-5363	422	37	nano	nano	PROPN
cana-5363	422	38	𝑍-closure	𝑍-closure	PROPN
cana-5363	422	39	,	,	PUNCT
cana-5363	422	40	pythagorean	pythagorean	ADJ
cana-5363	422	41	fuzzy	fuzzy	ADJ
cana-5363	422	42	nano	nano	NOUN
cana-5363	422	43	𝑍-interior	𝑍-interior	PROPN
cana-5363	422	44	and	and	CCONJ
cana-5363	422	45	their	their	PRON
cana-5363	422	46	relations	relation	NOUN
cana-5363	422	47	with	with	ADP
cana-5363	422	48	already	already	ADV
cana-5363	422	49	existing	exist	VERB
cana-5363	422	50	well	well	ADV
cana-5363	422	51	known	know	VERB
cana-5363	422	52	fuzzy	fuzzy	ADJ
cana-5363	422	53	sets	set	NOUN
cana-5363	422	54	.	.	PUNCT
cana-5363	423	1	in	in	ADP
cana-5363	423	2	future	future	NOUN
cana-5363	423	3	,	,	PUNCT
cana-5363	423	4	this	this	PRON
cana-5363	423	5	can	can	AUX
cana-5363	423	6	be	be	AUX
cana-5363	423	7	extended	extend	VERB
cana-5363	423	8	to	to	PART
cana-5363	423	9	pythagorean	pythagorean	VERB
cana-5363	423	10	fuzzy	fuzzy	ADJ
cana-5363	423	11	nano	nano	NOUN
cana-5363	423	12	𝑍	𝑍	PROPN
cana-5363	423	13	continuous	continuous	ADJ
cana-5363	423	14	function	function	NOUN
cana-5363	423	15	,	,	PUNCT
cana-5363	423	16	pythagorean	pythagorean	PROPN
cana-5363	423	17	fuzzy	fuzzy	ADJ
cana-5363	423	18	nano	nano	NOUN
cana-5363	423	19	𝑍	𝑍	VERB
cana-5363	423	20	open	open	ADJ
cana-5363	423	21	mapping	mapping	NOUN
cana-5363	423	22	,	,	PUNCT
cana-5363	423	23	pythagorean	pythagorean	PROPN
cana-5363	423	24	fuzzy	fuzzy	ADJ
cana-5363	423	25	nano	nano	NOUN
cana-5363	423	26	𝑍	𝑍	NOUN
cana-5363	423	27	closed	closed	ADJ
cana-5363	423	28	mapping	mapping	NOUN
cana-5363	423	29	and	and	CCONJ
cana-5363	423	30	pythagorean	pythagorean	PROPN
cana-5363	423	31	fuzzy	fuzzy	ADJ
cana-5363	423	32	nano	nano	NOUN
cana-5363	423	33	𝑍	𝑍	NOUN
cana-5363	423	34	homeomorphic	homeomorphic	ADJ
cana-5363	423	35	functions	function	NOUN
cana-5363	423	36	.	.	PUNCT
cana-5363	424	1	references	reference	NOUN
cana-5363	424	2	[	[	X
cana-5363	424	3	1	1	NUM
cana-5363	424	4	]	]	PUNCT
cana-5363	424	5	d.	d.	PROPN
cana-5363	424	6	ajay	ajay	PROPN
cana-5363	424	7	and	and	CCONJ
cana-5363	424	8	j.	j.	PROPN
cana-5363	424	9	joseline	joseline	PROPN
cana-5363	424	10	charisma	charisma	PROPN
cana-5363	424	11	,	,	PUNCT
cana-5363	424	12	pythagorean	pythagorean	PROPN
cana-5363	424	13	nano	nano	PROPN
cana-5363	424	14	topological	topological	ADJ
cana-5363	424	15	space	space	NOUN
cana-5363	424	16	,	,	PUNCT
cana-5363	424	17	international	international	ADJ
cana-5363	424	18	journal	journal	NOUN
cana-5363	424	19	of	of	ADP
cana-5363	424	20	recent	recent	ADJ
cana-5363	424	21	technology	technology	NOUN
cana-5363	424	22	and	and	CCONJ
cana-5363	424	23	engineering	engineering	NOUN
cana-5363	424	24	,	,	PUNCT
cana-5363	424	25	8	8	NUM
cana-5363	424	26	(	(	PUNCT
cana-5363	424	27	2020	2020	NUM
cana-5363	424	28	)	)	PUNCT
cana-5363	424	29	,	,	PUNCT
cana-5363	424	30	3415	3415	NUM
cana-5363	424	31	-	-	SYM
cana-5363	424	32	3419	3419	NUM
cana-5363	424	33	.	.	PUNCT
cana-5363	425	1	[	[	X
cana-5363	425	2	2	2	NUM
cana-5363	425	3	]	]	X
cana-5363	425	4	d.	d.	PROPN
cana-5363	425	5	ajay	ajay	PROPN
cana-5363	425	6	and	and	CCONJ
cana-5363	425	7	j.	j.	PROPN
cana-5363	425	8	joseline	joseline	PROPN
cana-5363	425	9	charisma	charisma	PROPN
cana-5363	425	10	,	,	PUNCT
cana-5363	425	11	on	on	ADP
cana-5363	425	12	weak	weak	ADJ
cana-5363	425	13	forms	form	NOUN
cana-5363	425	14	of	of	ADP
cana-5363	425	15	pythagorean	pythagorean	PROPN
cana-5363	425	16	nano	nano	NOUN
cana-5363	425	17	open	open	ADJ
cana-5363	425	18	sets	set	NOUN
cana-5363	425	19	,	,	PUNCT
cana-5363	425	20	advances	advance	NOUN
cana-5363	425	21	in	in	ADP
cana-5363	425	22	mathematics	mathematic	NOUN
cana-5363	425	23	:	:	PUNCT
cana-5363	425	24	scientific	scientific	ADJ
cana-5363	425	25	journal	journal	NOUN
cana-5363	425	26	,	,	PUNCT
cana-5363	425	27	9	9	NUM
cana-5363	425	28	(	(	PUNCT
cana-5363	425	29	2020	2020	NUM
cana-5363	425	30	)	)	PUNCT
cana-5363	425	31	,	,	PUNCT
cana-5363	425	32	5953	5953	NUM
cana-5363	425	33	-	-	SYM
cana-5363	425	34	5963	5963	NUM
cana-5363	425	35	.	.	PUNCT
cana-5363	426	1	[	[	X
cana-5363	426	2	3	3	NUM
cana-5363	426	3	]	]	PUNCT
cana-5363	426	4	x.	x.	NOUN
cana-5363	426	5	arul	arul	PROPN
cana-5363	426	6	selvaraj	selvaraj	PROPN
cana-5363	426	7	and	and	CCONJ
cana-5363	426	8	u.	u.	NOUN
cana-5363	426	9	balakrishna	balakrishna	NOUN
cana-5363	426	10	,	,	PUNCT
cana-5363	426	11	𝑍-open	𝑍-open	ADJ
cana-5363	426	12	sets	set	NOUN
cana-5363	426	13	in	in	ADP
cana-5363	426	14	nano	nano	ADJ
cana-5363	426	15	topological	topological	ADJ
cana-5363	426	16	spaces	space	NOUN
cana-5363	426	17	,	,	PUNCT
cana-5363	426	18	aip	aip	PROPN
cana-5363	426	19	conference	conference	NOUN
cana-5363	426	20	proceedings	proceeding	NOUN
cana-5363	426	21	,	,	PUNCT
cana-5363	426	22	2364	2364	NUM
cana-5363	426	23	(	(	PUNCT
cana-5363	426	24	2021	2021	NUM
cana-5363	426	25	)	)	PUNCT
cana-5363	426	26	,	,	PUNCT
cana-5363	426	27	020037	020037	NUM
cana-5363	426	28	.	.	PUNCT
cana-5363	427	1	[	[	X
cana-5363	427	2	4	4	X
cana-5363	427	3	]	]	PUNCT
cana-5363	427	4	k.	k.	PROPN
cana-5363	427	5	t.	t.	PROPN
cana-5363	427	6	atanassov	atanassov	PROPN
cana-5363	427	7	,	,	PUNCT
cana-5363	427	8	intuitionistic	intuitionistic	ADJ
cana-5363	427	9	fuzzy	fuzzy	ADJ
cana-5363	427	10	sets	set	NOUN
cana-5363	427	11	,	,	PUNCT
cana-5363	427	12	vii	vii	PROPN
cana-5363	427	13	itkr	itkr	PROPN
cana-5363	427	14	’s	’s	PART
cana-5363	427	15	session	session	NOUN
cana-5363	427	16	,	,	PUNCT
cana-5363	427	17	sofia	sofia	PROPN
cana-5363	427	18	(	(	PUNCT
cana-5363	427	19	deposed	depose	VERB
cana-5363	427	20	in	in	ADP
cana-5363	427	21	central	central	ADJ
cana-5363	427	22	sci.-technical	sci.-technical	ADJ
cana-5363	427	23	library	library	NOUN
cana-5363	427	24	of	of	ADP
cana-5363	427	25	bulg	bulg	PROPN
cana-5363	427	26	.	.	PUNCT
cana-5363	428	1	acad	acad	PROPN
cana-5363	428	2	.	.	PROPN
cana-5363	428	3	of	of	ADP
cana-5363	428	4	sci	sci	PROPN
cana-5363	428	5	.	.	PROPN
cana-5363	428	6	,	,	PUNCT
cana-5363	428	7	1697/84	1697/84	NUM
cana-5363	428	8	)	)	PUNCT
cana-5363	428	9	(	(	PUNCT
cana-5363	428	10	in	in	ADP
cana-5363	428	11	bulgarian	bulgarian	NOUN
cana-5363	428	12	)	)	PUNCT
cana-5363	428	13	,	,	PUNCT
cana-5363	428	14	(	(	PUNCT
cana-5363	428	15	1983	1983	NUM
cana-5363	428	16	)	)	PUNCT
cana-5363	428	17	.	.	PUNCT
cana-5363	429	1	[	[	X
cana-5363	429	2	5	5	X
cana-5363	429	3	]	]	PUNCT
cana-5363	429	4	c.	c.	PROPN
cana-5363	429	5	l.	l.	PROPN
cana-5363	429	6	chang	chang	PROPN
cana-5363	429	7	,	,	PUNCT
cana-5363	429	8	fuzzy	fuzzy	ADJ
cana-5363	429	9	topological	topological	ADJ
cana-5363	429	10	spaces	space	NOUN
cana-5363	429	11	,	,	PUNCT
cana-5363	429	12	j.	j.	PROPN
cana-5363	429	13	math	math	PROPN
cana-5363	429	14	.	.	PUNCT
cana-5363	430	1	anal	anal	PROPN
cana-5363	430	2	.	.	PUNCT
cana-5363	431	1	appl	appl	PROPN
cana-5363	431	2	.	.	PROPN
cana-5363	431	3	,	,	PUNCT
cana-5363	431	4	24	24	NUM
cana-5363	431	5	(	(	PUNCT
cana-5363	431	6	1968	1968	NUM
cana-5363	431	7	)	)	PUNCT
cana-5363	431	8	,	,	PUNCT
cana-5363	431	9	182	182	NUM
cana-5363	431	10	-	-	SYM
cana-5363	431	11	190	190	NUM
cana-5363	431	12	.	.	PUNCT
cana-5363	432	1	[	[	X
cana-5363	432	2	6	6	NUM
cana-5363	432	3	]	]	X
cana-5363	432	4	d.	d.	PROPN
cana-5363	432	5	coker	coker	PROPN
cana-5363	432	6	,	,	PUNCT
cana-5363	432	7	an	an	DET
cana-5363	432	8	introduction	introduction	NOUN
cana-5363	432	9	to	to	ADP
cana-5363	432	10	intuitionistic	intuitionistic	ADJ
cana-5363	432	11	fuzzy	fuzzy	ADJ
cana-5363	432	12	topological	topological	ADJ
cana-5363	432	13	spaces	space	NOUN
cana-5363	432	14	,	,	PUNCT
cana-5363	432	15	fuzzy	fuzzy	ADJ
cana-5363	432	16	sets	set	NOUN
cana-5363	432	17	and	and	CCONJ
cana-5363	432	18	systems	system	NOUN
cana-5363	432	19	,	,	PUNCT
cana-5363	432	20	88	88	NUM
cana-5363	432	21	(	(	PUNCT
cana-5363	432	22	1997	1997	NUM
cana-5363	432	23	)	)	PUNCT
cana-5363	432	24	,	,	PUNCT
cana-5363	432	25	81	81	NUM
cana-5363	432	26	-	-	SYM
cana-5363	432	27	89	89	NUM
cana-5363	432	28	.	.	PUNCT
cana-5363	433	1	[	[	X
cana-5363	433	2	7	7	X
cana-5363	433	3	]	]	X
cana-5363	433	4	r.	r.	PROPN
cana-5363	433	5	lowen	lowen	PROPN
cana-5363	433	6	,	,	PUNCT
cana-5363	433	7	fuzzy	fuzzy	ADJ
cana-5363	433	8	topological	topological	ADJ
cana-5363	433	9	spaces	space	NOUN
cana-5363	433	10	and	and	CCONJ
cana-5363	433	11	fuzzy	fuzzy	ADJ
cana-5363	433	12	compactness	compactness	NOUN
cana-5363	433	13	,	,	PUNCT
cana-5363	433	14	journal	journal	NOUN
cana-5363	433	15	of	of	ADP
cana-5363	433	16	mathematical	mathematical	ADJ
cana-5363	433	17	analysis	analysis	NOUN
cana-5363	433	18	and	and	CCONJ
cana-5363	433	19	applications	application	NOUN
cana-5363	433	20	,	,	PUNCT
cana-5363	433	21	56	56	NUM
cana-5363	433	22	(	(	PUNCT
cana-5363	433	23	3	3	NUM
cana-5363	433	24	)	)	PUNCT
cana-5363	433	25	(	(	PUNCT
cana-5363	433	26	1976	1976	NUM
cana-5363	433	27	)	)	PUNCT
cana-5363	433	28	,	,	PUNCT
cana-5363	433	29	621–633	621–633	NOUN
cana-5363	433	30	.	.	PUNCT
cana-5363	434	1	[	[	X
cana-5363	434	2	8	8	NUM
cana-5363	434	3	]	]	PUNCT
cana-5363	434	4	a.	a.	NOUN
cana-5363	434	5	i.	i.	PROPN
cana-5363	434	6	el	el	PROPN
cana-5363	434	7	-	-	PUNCT
cana-5363	434	8	magharabi	magharabi	PROPN
cana-5363	434	9	and	and	CCONJ
cana-5363	434	10	a.	a.	NOUN
cana-5363	434	11	m.	m.	PROPN
cana-5363	434	12	mubarki	mubarki	NOUN
cana-5363	434	13	,	,	PUNCT
cana-5363	434	14	𝑍-open	𝑍-open	ADJ
cana-5363	434	15	sets	set	NOUN
cana-5363	434	16	and	and	CCONJ
cana-5363	434	17	𝑍-continuity	𝑍-continuity	NOUN
cana-5363	434	18	in	in	ADP
cana-5363	434	19	topological	topological	ADJ
cana-5363	434	20	spaces	space	NOUN
cana-5363	434	21	,	,	PUNCT
cana-5363	434	22	international	international	ADJ
cana-5363	434	23	journal	journal	NOUN
cana-5363	434	24	of	of	ADP
cana-5363	434	25	mathematical	mathematical	ADJ
cana-5363	434	26	archive	archive	NOUN
cana-5363	434	27	,	,	PUNCT
cana-5363	434	28	2	2	NUM
cana-5363	434	29	(	(	PUNCT
cana-5363	434	30	10	10	NUM
cana-5363	434	31	)	)	PUNCT
cana-5363	434	32	(	(	PUNCT
cana-5363	434	33	2011	2011	NUM
cana-5363	434	34	)	)	PUNCT
cana-5363	434	35	,	,	PUNCT
cana-5363	434	36	1819	1819	NUM
cana-5363	434	37	-	-	SYM
cana-5363	434	38	1827	1827	NUM
cana-5363	434	39	.	.	PUNCT
cana-5363	435	1	[	[	X
cana-5363	435	2	9	9	NUM
cana-5363	435	3	]	]	PUNCT
cana-5363	435	4	m.	m.	NOUN
cana-5363	435	5	lellis	lellis	PROPN
cana-5363	435	6	thivagar	thivagar	PROPN
cana-5363	435	7	and	and	CCONJ
cana-5363	435	8	c.	c.	PROPN
cana-5363	435	9	richard	richard	PROPN
cana-5363	435	10	,	,	PUNCT
cana-5363	435	11	on	on	ADP
cana-5363	435	12	nano	nano	NOUN
cana-5363	435	13	forms	form	NOUN
cana-5363	435	14	of	of	ADP
cana-5363	435	15	weekly	weekly	ADJ
cana-5363	435	16	open	open	ADJ
cana-5363	435	17	sets	set	NOUN
cana-5363	435	18	,	,	PUNCT
cana-5363	435	19	international	international	ADJ
cana-5363	435	20	journal	journal	NOUN
cana-5363	435	21	of	of	ADP
cana-5363	435	22	mathematics	mathematics	PROPN
cana-5363	435	23	and	and	CCONJ
cana-5363	435	24	statistics	statistic	NOUN
cana-5363	435	25	invention	invention	NOUN
cana-5363	435	26	,	,	PUNCT
cana-5363	435	27	1	1	NUM
cana-5363	435	28	(	(	PUNCT
cana-5363	435	29	1	1	NUM
cana-5363	435	30	)	)	PUNCT
cana-5363	435	31	(	(	PUNCT
cana-5363	435	32	2013	2013	NUM
cana-5363	435	33	)	)	PUNCT
cana-5363	435	34	,	,	PUNCT
cana-5363	435	35	31	31	NUM
cana-5363	435	36	-	-	SYM
cana-5363	435	37	37	37	NUM
cana-5363	435	38	.	.	PUNCT
cana-5363	436	1	[	[	X
cana-5363	436	2	10	10	NUM
cana-5363	436	3	]	]	PUNCT
cana-5363	436	4	m.	m.	NOUN
cana-5363	436	5	lellis	lellis	PROPN
cana-5363	436	6	thivagar	thivagar	PROPN
cana-5363	436	7	,	,	PUNCT
cana-5363	436	8	s.	s.	PROPN
cana-5363	436	9	jafari	jafari	PROPN
cana-5363	436	10	,	,	PUNCT
cana-5363	436	11	v.	v.	ADP
cana-5363	436	12	sutha	sutha	PROPN
cana-5363	436	13	devi	devi	PROPN
cana-5363	436	14	and	and	CCONJ
cana-5363	436	15	v.	v.	ADP
cana-5363	436	16	antonysamy	antonysamy	NOUN
cana-5363	436	17	,	,	PUNCT
cana-5363	436	18	a	a	DET
cana-5363	436	19	novel	novel	ADJ
cana-5363	436	20	approach	approach	NOUN
cana-5363	436	21	to	to	ADP
cana-5363	436	22	nano	nano	NOUN
cana-5363	436	23	topology	topology	NOUN
cana-5363	436	24	via	via	ADP
cana-5363	436	25	neutrosophic	neutrosophic	ADJ
cana-5363	436	26	sets	set	NOUN
cana-5363	436	27	,	,	PUNCT
cana-5363	436	28	neutrosophic	neutrosophic	ADJ
cana-5363	436	29	sets	set	NOUN
cana-5363	436	30	and	and	CCONJ
cana-5363	436	31	systems	system	NOUN
cana-5363	436	32	,	,	PUNCT
cana-5363	436	33	20	20	NUM
cana-5363	436	34	(	(	PUNCT
cana-5363	436	35	2018	2018	NUM
cana-5363	436	36	)	)	PUNCT
cana-5363	436	37	,	,	PUNCT
cana-5363	436	38	86	86	NUM
cana-5363	436	39	-	-	SYM
cana-5363	436	40	94	94	NUM
cana-5363	436	41	.	.	PUNCT
cana-5363	437	1	[	[	X
cana-5363	437	2	11	11	NUM
cana-5363	437	3	]	]	X
cana-5363	437	4	murat	murat	PROPN
cana-5363	437	5	olgun	olgun	PROPN
cana-5363	437	6	,	,	PUNCT
cana-5363	437	7	mehmet	mehmet	PROPN
cana-5363	437	8	unver	unver	PROPN
cana-5363	437	9	and	and	CCONJ
cana-5363	437	10	seyhmus	seyhmus	VERB
cana-5363	437	11	yardimci	yardimci	PROPN
cana-5363	437	12	(	(	PUNCT
cana-5363	437	13	2019	2019	NUM
cana-5363	437	14	)	)	PUNCT
cana-5363	437	15	,	,	PUNCT
cana-5363	437	16	pythagorean	pythagorean	VERB
cana-5363	437	17	fuzzy	fuzzy	ADJ
cana-5363	437	18	topological	topological	ADJ
cana-5363	437	19	communications	communication	NOUN
cana-5363	437	20	on	on	ADP
cana-5363	437	21	applied	apply	VERB
cana-5363	437	22	nonlinear	nonlinear	ADJ
cana-5363	437	23	analysis	analysis	NOUN
cana-5363	437	24	issn	issn	NOUN
cana-5363	437	25	:	:	PUNCT
cana-5363	437	26	1074	1074	NUM
cana-5363	437	27	-	-	PUNCT
cana-5363	437	28	133x	133x	NUM
cana-5363	437	29	vol	vol	VERB
cana-5363	437	30	32	32	NUM
cana-5363	437	31	no	no	NOUN
cana-5363	437	32	.	.	PUNCT
cana-5363	438	1	10s	10	NOUN
cana-5363	438	2	(	(	PUNCT
cana-5363	438	3	2025	2025	NUM
cana-5363	438	4	)	)	PUNCT
cana-5363	438	5	2020	2020	NUM
cana-5363	439	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5363	439	2	spaces	space	NOUN
cana-5363	439	3	,	,	PUNCT
cana-5363	439	4	complex	complex	ADJ
cana-5363	439	5	&	&	CCONJ
cana-5363	439	6	intelligent	intelligent	ADJ
cana-5363	439	7	systems	system	NOUN
cana-5363	439	8	,	,	PUNCT
cana-5363	439	9	177	177	NUM
cana-5363	439	10	-	-	SYM
cana-5363	439	11	183	183	NUM
cana-5363	439	12	.	.	PUNCT
cana-5363	440	1	https://doi.org/10.1007/s40747-019-0095-2	https://doi.org/10.1007/s40747-019-0095-2	NUM
cana-5363	440	2	.	.	PUNCT
cana-5363	441	1	[	[	X
cana-5363	441	2	12	12	NUM
cana-5363	441	3	]	]	PUNCT
cana-5363	441	4	a.	a.	NOUN
cana-5363	441	5	padma	padma	NOUN
cana-5363	441	6	,	,	PUNCT
cana-5363	441	7	m.	m.	NOUN
cana-5363	441	8	saraswathi	saraswathi	PROPN
cana-5363	441	9	,	,	PUNCT
cana-5363	441	10	a.	a.	NOUN
cana-5363	441	11	vadivel	vadivel	NOUN
cana-5363	441	12	and	and	CCONJ
cana-5363	441	13	g.	g.	PROPN
cana-5363	441	14	saravanakumar	saravanakumar	PROPN
cana-5363	441	15	,	,	PUNCT
cana-5363	441	16	new	new	ADJ
cana-5363	441	17	notions	notion	NOUN
cana-5363	441	18	of	of	ADP
cana-5363	441	19	nano	nano	NOUN
cana-5363	441	20	𝑀-open	𝑀-open	PROPN
cana-5363	441	21	sets	set	NOUN
cana-5363	441	22	,	,	PUNCT
cana-5363	441	23	malaya	malaya	PROPN
cana-5363	441	24	journal	journal	PROPN
cana-5363	441	25	of	of	ADP
cana-5363	441	26	matematik	matematik	PROPN
cana-5363	441	27	,	,	PUNCT
cana-5363	441	28	s	s	X
cana-5363	441	29	(	(	PUNCT
cana-5363	441	30	1	1	NUM
cana-5363	441	31	)	)	PUNCT
cana-5363	441	32	(	(	PUNCT
cana-5363	441	33	2019	2019	NUM
cana-5363	441	34	)	)	PUNCT
cana-5363	441	35	,	,	PUNCT
cana-5363	441	36	656	656	NUM
cana-5363	441	37	-	-	SYM
cana-5363	441	38	660	660	NUM
cana-5363	441	39	.	.	PUNCT
cana-5363	442	1	[	[	X
cana-5363	442	2	13	13	NUM
cana-5363	442	3	]	]	PUNCT
cana-5363	442	4	v.	v.	ADP
cana-5363	442	5	pankajam	pankajam	PROPN
cana-5363	442	6	and	and	CCONJ
cana-5363	442	7	k.	k.	PROPN
cana-5363	442	8	kavitha	kavitha	PROPN
cana-5363	442	9	,	,	PUNCT
cana-5363	442	10	𝛿	𝛿	DET
cana-5363	442	11	-open	-open	ADJ
cana-5363	442	12	sets	set	NOUN
cana-5363	442	13	and	and	CCONJ
cana-5363	442	14	𝛿	𝛿	DET
cana-5363	442	15	-nano	-nano	ADJ
cana-5363	442	16	continuity	continuity	NOUN
cana-5363	442	17	in	in	ADP
cana-5363	442	18	𝛿	𝛿	DET
cana-5363	442	19	-nano	-nano	ADJ
cana-5363	442	20	topological	topological	ADJ
cana-5363	442	21	spaces	space	NOUN
cana-5363	442	22	,	,	PUNCT
cana-5363	442	23	international	international	ADJ
cana-5363	442	24	journal	journal	NOUN
cana-5363	442	25	of	of	ADP
cana-5363	442	26	innovative	innovative	ADJ
cana-5363	442	27	science	science	NOUN
cana-5363	442	28	and	and	CCONJ
cana-5363	442	29	research	research	NOUN
cana-5363	442	30	technology	technology	NOUN
cana-5363	442	31	,	,	PUNCT
cana-5363	442	32	2	2	NUM
cana-5363	442	33	(	(	PUNCT
cana-5363	442	34	12	12	NUM
cana-5363	442	35	)	)	PUNCT
cana-5363	442	36	(	(	PUNCT
cana-5363	442	37	2017	2017	NUM
cana-5363	442	38	)	)	PUNCT
cana-5363	442	39	,	,	PUNCT
cana-5363	442	40	110	110	NUM
cana-5363	442	41	-	-	SYM
cana-5363	442	42	118	118	NUM
cana-5363	442	43	.	.	PUNCT
cana-5363	443	1	[	[	X
cana-5363	443	2	14	14	NUM
cana-5363	443	3	]	]	X
cana-5363	443	4	shiventhiradevi	shiventhiradevi	ADJ
cana-5363	443	5	sathaananthan	sathaananthan	NOUN
cana-5363	443	6	,	,	PUNCT
cana-5363	443	7	s.	s.	PROPN
cana-5363	443	8	tamilselvan	tamilselvan	PROPN
cana-5363	443	9	,	,	PUNCT
cana-5363	443	10	a.	a.	NOUN
cana-5363	443	11	vadivel	vadivel	NOUN
cana-5363	443	12	and	and	CCONJ
cana-5363	443	13	g.	g.	PROPN
cana-5363	443	14	saravanakumar	saravanakumar	PROPN
cana-5363	443	15	,	,	PUNCT
cana-5363	443	16	fuzzy	fuzzy	ADJ
cana-5363	443	17	𝒵	𝒵	PROPN
cana-5363	443	18	closed	close	VERB
cana-5363	443	19	sets	set	NOUN
cana-5363	443	20	in	in	ADP
cana-5363	443	21	double	double	ADJ
cana-5363	443	22	fuzzy	fuzzy	ADJ
cana-5363	443	23	topological	topological	ADJ
cana-5363	443	24	spaces	space	NOUN
cana-5363	443	25	,	,	PUNCT
cana-5363	443	26	aip	aip	PROPN
cana-5363	443	27	conf	conf	PROPN
cana-5363	443	28	proc	proc	PROPN
cana-5363	443	29	.	.	PROPN
cana-5363	443	30	,	,	PUNCT
cana-5363	443	31	2277	2277	NUM
cana-5363	443	32	(	(	PUNCT
cana-5363	443	33	2020	2020	NUM
cana-5363	443	34	)	)	PUNCT
cana-5363	443	35	,	,	PUNCT
cana-5363	443	36	090001	090001	NUM
cana-5363	443	37	.	.	PUNCT
cana-5363	444	1	[	[	X
cana-5363	444	2	15	15	NUM
cana-5363	444	3	]	]	X
cana-5363	444	4	shiventhiradevi	shiventhiradevi	ADJ
cana-5363	444	5	sathaananthan	sathaananthan	NOUN
cana-5363	444	6	,	,	PUNCT
cana-5363	444	7	a.	a.	NOUN
cana-5363	444	8	vadivel	vadivel	NOUN
cana-5363	444	9	,	,	PUNCT
cana-5363	444	10	s.	s.	PROPN
cana-5363	444	11	tamilselvan	tamilselvan	PROPN
cana-5363	444	12	and	and	CCONJ
cana-5363	444	13	g.	g.	PROPN
cana-5363	444	14	saravanakumar	saravanakumar	PROPN
cana-5363	444	15	,	,	PUNCT
cana-5363	444	16	generalized	generalize	VERB
cana-5363	444	17	fuzzy	fuzzy	ADJ
cana-5363	444	18	𝒵	𝒵	PROPN
cana-5363	444	19	closed	close	VERB
cana-5363	444	20	sets	set	NOUN
cana-5363	444	21	in	in	ADP
cana-5363	444	22	double	double	ADJ
cana-5363	444	23	fuzzy	fuzzy	ADJ
cana-5363	444	24	topological	topological	ADJ
cana-5363	444	25	spaces	space	NOUN
cana-5363	444	26	,	,	PUNCT
cana-5363	444	27	adv	adv	PROPN
cana-5363	444	28	.	.	PUNCT
cana-5363	444	29	math	math	PROPN
cana-5363	444	30	:	:	PUNCT
cana-5363	445	1	sci	sci	PROPN
cana-5363	445	2	.	.	PUNCT
cana-5363	445	3	j.	j.	PROPN
cana-5363	445	4	,	,	PUNCT
cana-5363	445	5	9	9	NUM
cana-5363	445	6	(	(	PUNCT
cana-5363	445	7	4	4	NUM
cana-5363	445	8	)	)	PUNCT
cana-5363	445	9	(	(	PUNCT
cana-5363	445	10	2020	2020	NUM
cana-5363	445	11	)	)	PUNCT
cana-5363	445	12	,	,	PUNCT
cana-5363	445	13	2107	2107	NUM
cana-5363	445	14	-	-	SYM
cana-5363	445	15	2112	2112	NUM
cana-5363	445	16	.	.	PUNCT
cana-5363	446	1	[	[	X
cana-5363	446	2	16	16	NUM
cana-5363	446	3	]	]	X
cana-5363	446	4	r.	r.	PROPN
cana-5363	446	5	thangammal	thangammal	PROPN
cana-5363	446	6	,	,	PUNCT
cana-5363	446	7	m.	m.	NOUN
cana-5363	446	8	saraswathi	saraswathi	PROPN
cana-5363	446	9	,	,	PUNCT
cana-5363	446	10	a.	a.	NOUN
cana-5363	446	11	vadivel	vadivel	NOUN
cana-5363	446	12	and	and	CCONJ
cana-5363	446	13	c.	c.	PROPN
cana-5363	446	14	john	john	PROPN
cana-5363	446	15	sundar	sundar	PROPN
cana-5363	446	16	,	,	PUNCT
cana-5363	446	17	fuzzy	fuzzy	ADJ
cana-5363	446	18	nano	nano	NOUN
cana-5363	446	19	𝑍-open	𝑍-open	ADJ
cana-5363	446	20	sets	set	NOUN
cana-5363	446	21	in	in	ADP
cana-5363	446	22	fuzzy	fuzzy	ADJ
cana-5363	446	23	nano	nano	NOUN
cana-5363	446	24	topological	topological	ADJ
cana-5363	446	25	spaces	space	NOUN
cana-5363	446	26	,	,	PUNCT
cana-5363	446	27	journal	journal	NOUN
cana-5363	446	28	of	of	ADP
cana-5363	446	29	linear	linear	PROPN
cana-5363	446	30	and	and	CCONJ
cana-5363	446	31	topological	topological	ADJ
cana-5363	446	32	algebra	algebra	NOUN
cana-5363	446	33	vol	vol	NOUN
cana-5363	446	34	.	.	PROPN
cana-5363	447	1	11	11	NUM
cana-5363	447	2	,	,	PUNCT
cana-5363	447	3	no	no	INTJ
cana-5363	447	4	.	.	NOUN
cana-5363	447	5	01	01	NUM
cana-5363	447	6	,	,	PUNCT
cana-5363	447	7	2022	2022	NUM
cana-5363	447	8	,	,	PUNCT
cana-5363	447	9	2738	2738	NUM
cana-5363	447	10	.	.	PUNCT
cana-5363	448	1	[	[	X
cana-5363	448	2	17	17	NUM
cana-5363	448	3	]	]	PUNCT
cana-5363	448	4	a.	a.	NOUN
cana-5363	448	5	vadivel	vadivel	NOUN
cana-5363	448	6	,	,	PUNCT
cana-5363	448	7	c.	c.	PROPN
cana-5363	448	8	john	john	PROPN
cana-5363	448	9	sundar	sundar	PROPN
cana-5363	448	10	,	,	PUNCT
cana-5363	448	11	k.	k.	PROPN
cana-5363	448	12	kirubadevi	kirubadevi	PROPN
cana-5363	448	13	and	and	CCONJ
cana-5363	448	14	s.	s.	PROPN
cana-5363	448	15	tamilselvan	tamilselvan	PROPN
cana-5363	448	16	,	,	PUNCT
cana-5363	448	17	more	more	ADJ
cana-5363	448	18	on	on	ADP
cana-5363	448	19	neutrosophic	neutrosophic	ADJ
cana-5363	448	20	nano	nano	NOUN
cana-5363	448	21	open	open	ADJ
cana-5363	448	22	sets	set	NOUN
cana-5363	448	23	,	,	PUNCT
cana-5363	448	24	international	international	ADJ
cana-5363	448	25	journal	journal	NOUN
cana-5363	448	26	of	of	ADP
cana-5363	448	27	neutrosophic	neutrosophic	ADJ
cana-5363	448	28	science	science	NOUN
cana-5363	448	29	(	(	PUNCT
cana-5363	448	30	ijns	ijns	PROPN
cana-5363	448	31	)	)	PUNCT
cana-5363	448	32	,	,	PUNCT
cana-5363	448	33	18	18	NUM
cana-5363	448	34	(	(	PUNCT
cana-5363	448	35	4	4	NUM
cana-5363	448	36	)	)	PUNCT
cana-5363	448	37	(	(	PUNCT
cana-5363	448	38	2022	2022	NUM
cana-5363	448	39	)	)	PUNCT
cana-5363	448	40	,	,	PUNCT
cana-5363	448	41	204	204	NUM
cana-5363	448	42	-	-	SYM
cana-5363	448	43	222	222	NUM
cana-5363	448	44	.	.	PUNCT
cana-5363	449	1	[	[	X
cana-5363	449	2	18	18	NUM
cana-5363	449	3	]	]	X
cana-5363	449	4	r.	r.	PROPN
cana-5363	449	5	r.	r.	PROPN
cana-5363	449	6	yager	yager	PROPN
cana-5363	449	7	(	(	PUNCT
cana-5363	449	8	2013	2013	NUM
cana-5363	449	9	)	)	PUNCT
cana-5363	449	10	,	,	PUNCT
cana-5363	449	11	pythagorean	pythagorean	PROPN
cana-5363	449	12	membership	membership	NOUN
cana-5363	449	13	grades	grade	NOUN
cana-5363	449	14	in	in	ADP
cana-5363	449	15	multicriteria	multicriteria	PROPN
cana-5363	449	16	decision	decision	NOUN
cana-5363	449	17	making	making	NOUN
cana-5363	449	18	,	,	PUNCT
cana-5363	449	19	in	in	ADP
cana-5363	449	20	:	:	PUNCT
cana-5363	449	21	technical	technical	ADJ
cana-5363	449	22	report	report	NOUN
cana-5363	449	23	𝑀𝐼𝐼-3301	𝑀𝐼𝐼-3301	PROPN
cana-5363	449	24	.	.	PUNCT
cana-5363	450	1	machine	machine	NOUN
cana-5363	450	2	intelligence	intelligence	PROPN
cana-5363	450	3	institute	institute	PROPN
cana-5363	450	4	,	,	PUNCT
cana-5363	450	5	iona	iona	PROPN
cana-5363	450	6	college	college	PROPN
cana-5363	450	7	,	,	PUNCT
cana-5363	450	8	new	new	ADJ
cana-5363	450	9	rochelle	rochelle	NOUN
cana-5363	450	10	.	.	PUNCT
cana-5363	451	1	[	[	X
cana-5363	451	2	19	19	NUM
cana-5363	451	3	]	]	X
cana-5363	451	4	r.	r.	PROPN
cana-5363	451	5	r.	r.	PROPN
cana-5363	451	6	yager	yager	PROPN
cana-5363	451	7	(	(	PUNCT
cana-5363	451	8	2013	2013	NUM
cana-5363	451	9	)	)	PUNCT
cana-5363	451	10	,	,	PUNCT
cana-5363	451	11	pythagorean	pythagorean	PROPN
cana-5363	451	12	fuzzy	fuzzy	ADJ
cana-5363	451	13	subsets	subset	NOUN
cana-5363	451	14	,	,	PUNCT
cana-5363	451	15	in	in	ADP
cana-5363	451	16	:	:	PUNCT
cana-5363	451	17	proceedings	proceeding	NOUN
cana-5363	451	18	of	of	ADP
cana-5363	451	19	the	the	DET
cana-5363	451	20	joint	joint	ADJ
cana-5363	451	21	𝐼𝐹𝑆𝐴	𝐼𝐹𝑆𝐴	PROPN
cana-5363	451	22	world	world	PROPN
cana-5363	451	23	congress	congress	PROPN
cana-5363	451	24	𝑁𝐴𝐹𝐼𝑃𝑆	𝑁𝐴𝐹𝐼𝑃𝑆	PROPN
cana-5363	451	25	annual	annual	ADJ
cana-5363	451	26	meeting	meeting	NOUN
cana-5363	451	27	,	,	PUNCT
cana-5363	451	28	57	57	NUM
cana-5363	451	29	-	-	SYM
cana-5363	451	30	61	61	NUM
cana-5363	451	31	.	.	PUNCT
cana-5363	452	1	[	[	X
cana-5363	452	2	20	20	NUM
cana-5363	452	3	]	]	PUNCT
cana-5363	452	4	r.	r.	PROPN
cana-5363	452	5	r.	r.	PROPN
cana-5363	452	6	yager	yager	PROPN
cana-5363	452	7	(	(	PUNCT
cana-5363	452	8	2014	2014	NUM
cana-5363	452	9	)	)	PUNCT
cana-5363	452	10	,	,	PUNCT
cana-5363	452	11	pythagorean	pythagorean	PROPN
cana-5363	452	12	membership	membership	NOUN
cana-5363	452	13	grades	grade	NOUN
cana-5363	452	14	in	in	ADP
cana-5363	452	15	multicriteria	multicriteria	PROPN
cana-5363	452	16	decision	decision	NOUN
cana-5363	452	17	making	making	NOUN
cana-5363	452	18	,	,	PUNCT
cana-5363	452	19	𝐼𝐸𝐸𝐸	𝐼𝐸𝐸𝐸	PROPN
cana-5363	452	20	trans	trans	PROPN
cana-5363	452	21	fuzzy	fuzzy	PROPN
cana-5363	452	22	syst	syst	PROPN
cana-5363	452	23	.	.	PUNCT
cana-5363	453	1	22	22	NUM
cana-5363	453	2	(	(	PUNCT
cana-5363	453	3	4	4	NUM
cana-5363	453	4	)	)	PUNCT
cana-5363	453	5	,	,	PUNCT
cana-5363	453	6	958	958	NUM
cana-5363	453	7	-	-	SYM
cana-5363	453	8	965	965	NUM
cana-5363	453	9	.	.	PUNCT
cana-5363	454	1	[	[	X
cana-5363	454	2	21	21	NUM
cana-5363	454	3	]	]	X
cana-5363	454	4	l.	l.	PROPN
cana-5363	454	5	a.	a.	PROPN
cana-5363	454	6	zadeh	zadeh	PROPN
cana-5363	454	7	,	,	PUNCT
cana-5363	454	8	fuzzy	fuzzy	ADJ
cana-5363	454	9	sets	set	NOUN
cana-5363	454	10	,	,	PUNCT
cana-5363	454	11	information	information	NOUN
cana-5363	454	12	and	and	CCONJ
cana-5363	454	13	control	control	NOUN
cana-5363	454	14	,	,	PUNCT
cana-5363	454	15	8	8	NUM
cana-5363	454	16	(	(	PUNCT
cana-5363	454	17	1965	1965	NUM
cana-5363	454	18	)	)	PUNCT
cana-5363	454	19	,	,	PUNCT
cana-5363	454	20	338	338	NUM
cana-5363	454	21	-	-	SYM
cana-5363	454	22	353	353	NUM
cana-5363	454	23	.	.	PUNCT