id	sid	tid	token	lemma	pos
cana-1800	1	1	communications	communication	NOUN
cana-1800	1	2	on	on	ADP
cana-1800	1	3	applied	apply	VERB
cana-1800	1	4	nonlinear	nonlinear	ADJ
cana-1800	1	5	analysis	analysis	NOUN
cana-1800	1	6	issn	issn	NOUN
cana-1800	1	7	:	:	PUNCT
cana-1800	1	8	1074	1074	NUM
cana-1800	1	9	-	-	PUNCT
cana-1800	1	10	133x	133x	NUM
cana-1800	1	11	vol	vol	NOUN
cana-1800	1	12	32	32	NUM
cana-1800	1	13	no	no	NOUN
cana-1800	1	14	.	.	NOUN
cana-1800	1	15	2	2	NUM
cana-1800	1	16	(	(	PUNCT
cana-1800	1	17	2025	2025	NUM
cana-1800	1	18	)	)	PUNCT
cana-1800	1	19	507	507	NUM
cana-1800	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1800	1	21	characterization	characterization	NOUN
cana-1800	1	22	of	of	ADP
cana-1800	1	23	bipolar	bipolar	ADJ
cana-1800	1	24	valued	value	VERB
cana-1800	1	25	vague	vague	ADJ
cana-1800	1	26	ideals	ideal	NOUN
cana-1800	1	27	of	of	ADP
cana-1800	1	28	a	a	DET
cana-1800	1	29	semiring	semiring	NOUN
cana-1800	1	30	k.	k.	PROPN
cana-1800	1	31	anitha1	anitha1	PROPN
cana-1800	1	32	,	,	PUNCT
cana-1800	1	33	m.	m.	NOUN
cana-1800	1	34	muthusamy2	muthusamy2	PROPN
cana-1800	1	35	,	,	PUNCT
cana-1800	1	36	k.	k.	PROPN
cana-1800	1	37	arjunan3	arjunan3	PROPN
cana-1800	1	38	1research	1research	NUM
cana-1800	1	39	scholar	scholar	NOUN
cana-1800	1	40	,	,	PUNCT
cana-1800	1	41	department	department	NOUN
cana-1800	1	42	of	of	ADP
cana-1800	1	43	mathematics	mathematics	PROPN
cana-1800	1	44	,	,	PUNCT
cana-1800	1	45	dr	dr	PROPN
cana-1800	1	46	.	.	PROPN
cana-1800	1	47	zakir	zakir	PROPN
cana-1800	1	48	husain	husain	PROPN
cana-1800	1	49	college	college	PROPN
cana-1800	1	50	,	,	PUNCT
cana-1800	1	51	ilayangudi-630702	ilayangudi-630702	ADJ
cana-1800	1	52	,	,	PUNCT
cana-1800	1	53	tamilnadu	tamilnadu	ADJ
cana-1800	1	54	,	,	PUNCT
cana-1800	1	55	india	india	PROPN
cana-1800	1	56	.	.	PUNCT
cana-1800	2	1	email:anitha.kesav13@gmail.com	email:anitha.kesav13@gmail.com	X
cana-1800	3	1	2department	2department	NUM
cana-1800	3	2	of	of	ADP
cana-1800	3	3	mathematics	mathematics	PROPN
cana-1800	3	4	,	,	PUNCT
cana-1800	3	5	dr	dr	PROPN
cana-1800	3	6	.	.	PROPN
cana-1800	3	7	zakir	zakir	PROPN
cana-1800	3	8	husain	husain	PROPN
cana-1800	3	9	college	college	PROPN
cana-1800	3	10	,	,	PUNCT
cana-1800	3	11	ilayangudi-630702	ilayangudi-630702	ADJ
cana-1800	3	12	,	,	PUNCT
cana-1800	3	13	tamilnadu	tamilnadu	ADJ
cana-1800	3	14	,	,	PUNCT
cana-1800	3	15	india	india	PROPN
cana-1800	3	16	.	.	PUNCT
cana-1800	3	17	email	email	NOUN
cana-1800	3	18	:	:	PUNCT
cana-1800	3	19	msamy0207@yahoo.com	msamy0207@yahoo.com	X
cana-1800	4	1	3department	3department	NUM
cana-1800	4	2	of	of	ADP
cana-1800	4	3	mathematics	mathematic	NOUN
cana-1800	4	4	,	,	PUNCT
cana-1800	4	5	alagappa	alagappa	VERB
cana-1800	4	6	government	government	NOUN
cana-1800	4	7	arts	arts	PROPN
cana-1800	4	8	college	college	PROPN
cana-1800	4	9	,	,	PUNCT
cana-1800	4	10	karaikudi	karaikudi	PROPN
cana-1800	4	11	–	–	PUNCT
cana-1800	4	12	630003	630003	NUM
cana-1800	4	13	,	,	PUNCT
cana-1800	4	14	tamilnadu	tamilnadu	NOUN
cana-1800	4	15	,	,	PUNCT
cana-1800	5	1	india	india	PROPN
cana-1800	5	2	.	.	PUNCT
cana-1800	5	3	email	email	NOUN
cana-1800	5	4	:	:	PUNCT
cana-1800	5	5	arjunan.karmegam@gmail.com	arjunan.karmegam@gmail.com	X
cana-1800	5	6	article	article	NOUN
cana-1800	5	7	history	history	NOUN
cana-1800	5	8	:	:	PUNCT
cana-1800	5	9	received	receive	VERB
cana-1800	5	10	:	:	PUNCT
cana-1800	5	11	05	05	NUM
cana-1800	5	12	-	-	SYM
cana-1800	5	13	08	08	NUM
cana-1800	5	14	-	-	PUNCT
cana-1800	5	15	2024	2024	NUM
cana-1800	5	16	revised	revise	VERB
cana-1800	5	17	:	:	PUNCT
cana-1800	5	18	13	13	NUM
cana-1800	5	19	-	-	SYM
cana-1800	5	20	09	09	NUM
cana-1800	5	21	-	-	PUNCT
cana-1800	5	22	2024	2024	NUM
cana-1800	5	23	accepted	accept	VERB
cana-1800	5	24	:	:	PUNCT
cana-1800	5	25	21	21	NUM
cana-1800	5	26	-	-	SYM
cana-1800	5	27	09	09	NUM
cana-1800	5	28	-	-	PUNCT
cana-1800	5	29	2024	2024	NUM
cana-1800	5	30	abstract	abstract	NOUN
cana-1800	5	31	:	:	PUNCT
cana-1800	5	32	bipolar	bipolar	ADJ
cana-1800	5	33	valued	value	VERB
cana-1800	5	34	vague	vague	ADJ
cana-1800	5	35	ideal	ideal	NOUN
cana-1800	5	36	of	of	ADP
cana-1800	5	37	a	a	DET
cana-1800	5	38	semiring	semiring	NOUN
cana-1800	5	39	(	(	PUNCT
cana-1800	5	40	bvvi	bvvi	NOUN
cana-1800	5	41	)	)	PUNCT
cana-1800	5	42	is	be	AUX
cana-1800	5	43	described	describe	VERB
cana-1800	5	44	and	and	CCONJ
cana-1800	5	45	examined	examine	VERB
cana-1800	5	46	in	in	ADP
cana-1800	5	47	the	the	DET
cana-1800	5	48	present	present	ADJ
cana-1800	5	49	paper	paper	NOUN
cana-1800	5	50	.	.	PUNCT
cana-1800	6	1	some	some	DET
cana-1800	6	2	characterization	characterization	NOUN
cana-1800	6	3	theorems	theorem	NOUN
cana-1800	6	4	are	be	AUX
cana-1800	6	5	introduced	introduce	VERB
cana-1800	6	6	in	in	ADP
cana-1800	6	7	this	this	DET
cana-1800	6	8	paper	paper	NOUN
cana-1800	6	9	and	and	CCONJ
cana-1800	6	10	intersection	intersection	NOUN
cana-1800	6	11	,	,	PUNCT
cana-1800	6	12	product	product	NOUN
cana-1800	6	13	and	and	CCONJ
cana-1800	6	14	strongest	strong	ADJ
cana-1800	6	15	bipolar	bipolar	ADJ
cana-1800	6	16	valued	value	VERB
cana-1800	6	17	vague	vague	ADJ
cana-1800	6	18	relation	relation	NOUN
cana-1800	6	19	of	of	ADP
cana-1800	6	20	bipolar	bipolar	ADJ
cana-1800	6	21	valued	value	VERB
cana-1800	6	22	vague	vague	ADJ
cana-1800	6	23	ideals	ideal	NOUN
cana-1800	6	24	(	(	PUNCT
cana-1800	6	25	bvvi	bvvi	NOUN
cana-1800	6	26	)	)	PUNCT
cana-1800	6	27	of	of	ADP
cana-1800	6	28	a	a	DET
cana-1800	6	29	semirings	semiring	NOUN
cana-1800	6	30	are	be	AUX
cana-1800	6	31	introduced	introduce	VERB
cana-1800	6	32	.	.	PUNCT
cana-1800	7	1	keywords	keyword	NOUN
cana-1800	7	2	:	:	PUNCT
cana-1800	7	3	fuzzy	fuzzy	ADJ
cana-1800	7	4	subset	subset	NOUN
cana-1800	7	5	,	,	PUNCT
cana-1800	7	6	vague	vague	ADJ
cana-1800	7	7	subset	subset	NOUN
cana-1800	7	8	,	,	PUNCT
cana-1800	7	9	bipolar	bipolar	ADJ
cana-1800	7	10	valued	value	VERB
cana-1800	7	11	fuzzy	fuzzy	ADJ
cana-1800	7	12	subset	subset	NOUN
cana-1800	7	13	,	,	PUNCT
cana-1800	7	14	bipolar	bipolar	ADJ
cana-1800	7	15	valued	value	VERB
cana-1800	7	16	vague	vague	ADJ
cana-1800	7	17	subset	subset	NOUN
cana-1800	7	18	,	,	PUNCT
cana-1800	7	19	bipolar	bipolar	ADJ
cana-1800	7	20	valued	value	VERB
cana-1800	7	21	vague	vague	ADJ
cana-1800	7	22	ideals	ideal	NOUN
cana-1800	7	23	.	.	PUNCT
cana-1800	8	1	1	1	NUM
cana-1800	8	2	introduction	introduction	NOUN
cana-1800	8	3	zadeh	zadeh	PROPN
cana-1800	8	4	's	's	PART
cana-1800	8	5	[	[	X
cana-1800	8	6	17	17	NUM
cana-1800	8	7	]	]	PUNCT
cana-1800	8	8	research	research	NOUN
cana-1800	8	9	from	from	ADP
cana-1800	8	10	1965	1965	NUM
cana-1800	8	11	was	be	AUX
cana-1800	8	12	the	the	DET
cana-1800	8	13	first	first	ADJ
cana-1800	8	14	to	to	PART
cana-1800	8	15	create	create	VERB
cana-1800	8	16	the	the	DET
cana-1800	8	17	idea	idea	NOUN
cana-1800	8	18	of	of	ADP
cana-1800	8	19	a	a	DET
cana-1800	8	20	fuzzy	fuzzy	ADJ
cana-1800	8	21	subset	subset	NOUN
cana-1800	8	22	of	of	ADP
cana-1800	8	23	a	a	DET
cana-1800	8	24	set	set	NOUN
cana-1800	8	25	,	,	PUNCT
cana-1800	8	26	and	and	CCONJ
cana-1800	8	27	the	the	DET
cana-1800	8	28	mathematical	mathematical	ADJ
cana-1800	8	29	construct	construct	NOUN
cana-1800	8	30	known	know	VERB
cana-1800	8	31	as	as	ADP
cana-1800	8	32	a	a	DET
cana-1800	8	33	fuzzy	fuzzy	ADJ
cana-1800	8	34	set	set	NOUN
cana-1800	8	35	is	be	AUX
cana-1800	8	36	useful	useful	ADJ
cana-1800	8	37	for	for	ADP
cana-1800	8	38	expressing	express	VERB
cana-1800	8	39	a	a	DET
cana-1800	8	40	group	group	NOUN
cana-1800	8	41	of	of	ADP
cana-1800	8	42	things	thing	NOUN
cana-1800	8	43	with	with	ADP
cana-1800	8	44	ambiguous	ambiguous	ADJ
cana-1800	8	45	borders	border	NOUN
cana-1800	8	46	.	.	PUNCT
cana-1800	9	1	there	there	PRON
cana-1800	9	2	have	have	AUX
cana-1800	9	3	been	be	AUX
cana-1800	9	4	many	many	ADJ
cana-1800	9	5	generalisations	generalisation	NOUN
cana-1800	9	6	of	of	ADP
cana-1800	9	7	this	this	DET
cana-1800	9	8	core	core	NOUN
cana-1800	9	9	idea	idea	NOUN
cana-1800	9	10	since	since	SCONJ
cana-1800	9	11	it	it	PRON
cana-1800	9	12	has	have	AUX
cana-1800	9	13	developed	develop	VERB
cana-1800	9	14	into	into	ADP
cana-1800	9	15	a	a	DET
cana-1800	9	16	lively	lively	ADJ
cana-1800	9	17	area	area	NOUN
cana-1800	9	18	of	of	ADP
cana-1800	9	19	research	research	NOUN
cana-1800	9	20	in	in	ADP
cana-1800	9	21	other	other	ADJ
cana-1800	9	22	fields	field	NOUN
cana-1800	9	23	,	,	PUNCT
cana-1800	9	24	including	include	VERB
cana-1800	9	25	intuitionistic	intuitionistic	ADJ
cana-1800	9	26	fuzzy	fuzzy	ADJ
cana-1800	9	27	sets	set	NOUN
cana-1800	9	28	,	,	PUNCT
cana-1800	9	29	interval	interval	NOUN
cana-1800	9	30	valued	value	VERB
cana-1800	9	31	fuzzy	fuzzy	ADJ
cana-1800	9	32	sets	set	NOUN
cana-1800	9	33	,	,	PUNCT
cana-1800	9	34	vague	vague	ADJ
cana-1800	9	35	sets	set	NOUN
cana-1800	9	36	,	,	PUNCT
cana-1800	9	37	soft	soft	ADJ
cana-1800	9	38	sets	set	NOUN
cana-1800	9	39	,	,	PUNCT
cana-1800	9	40	etc	etc	X
cana-1800	9	41	.	.	X
cana-1800	9	42	grattan	grattan	PROPN
cana-1800	9	43	-	-	PUNCT
cana-1800	9	44	guiness	guiness	PROPN
cana-1800	9	45	fuzzy	fuzzy	ADJ
cana-1800	9	46	membership	membership	NOUN
cana-1800	9	47	mapped	map	VERB
cana-1800	9	48	onto	onto	ADP
cana-1800	9	49	interval	interval	NOUN
cana-1800	9	50	and	and	CCONJ
cana-1800	9	51	multiple	multiple	ADJ
cana-1800	9	52	valued	value	VERB
cana-1800	9	53	quantities	quantity	NOUN
cana-1800	9	54	was	be	AUX
cana-1800	9	55	discussed	discuss	VERB
cana-1800	9	56	in	in	ADP
cana-1800	9	57	[	[	X
cana-1800	9	58	9	9	NUM
cana-1800	9	59	]	]	PUNCT
cana-1800	9	60	.	.	PUNCT
cana-1800	10	1	a	a	DET
cana-1800	10	2	fuzzy	fuzzy	ADJ
cana-1800	10	3	set	set	NOUN
cana-1800	10	4	extension	extension	NOUN
cana-1800	10	5	known	know	VERB
cana-1800	10	6	as	as	ADP
cana-1800	10	7	a	a	DET
cana-1800	10	8	vague	vague	ADJ
cana-1800	10	9	set	set	NOUN
cana-1800	10	10	is	be	AUX
cana-1800	10	11	a	a	DET
cana-1800	10	12	special	special	ADJ
cana-1800	10	13	instance	instance	NOUN
cana-1800	10	14	of	of	ADP
cana-1800	10	15	a	a	DET
cana-1800	10	16	fuzzy	fuzzy	ADJ
cana-1800	10	17	set	set	NOUN
cana-1800	10	18	that	that	PRON
cana-1800	10	19	depends	depend	VERB
cana-1800	10	20	on	on	ADP
cana-1800	10	21	the	the	DET
cana-1800	10	22	context	context	NOUN
cana-1800	10	23	.	.	PUNCT
cana-1800	11	1	d.j.buehrer	d.j.buehrer	NOUN
cana-1800	11	2	and	and	CCONJ
cana-1800	11	3	w.l	w.l	PROPN
cana-1800	11	4	.	.	PROPN
cana-1800	11	5	gau	gau	PROPN
cana-1800	12	1	[	[	X
cana-1800	12	2	8	8	NUM
cana-1800	12	3	]	]	PUNCT
cana-1800	12	4	introduced	introduce	VERB
cana-1800	12	5	the	the	DET
cana-1800	12	6	vague	vague	ADJ
cana-1800	12	7	set	set	NOUN
cana-1800	12	8	.	.	PUNCT
cana-1800	13	1	lee	lee	PROPN
cana-1800	13	2	presented	present	VERB
cana-1800	13	3	the	the	DET
cana-1800	13	4	idea	idea	NOUN
cana-1800	13	5	of	of	ADP
cana-1800	13	6	bipolar	bipolar	ADJ
cana-1800	13	7	valued	value	VERB
cana-1800	13	8	fuzzy	fuzzy	ADJ
cana-1800	13	9	sets	set	NOUN
cana-1800	13	10	in	in	ADP
cana-1800	13	11	his	his	PRON
cana-1800	13	12	article	article	NOUN
cana-1800	13	13	from	from	ADP
cana-1800	13	14	[	[	X
cana-1800	13	15	8	8	NUM
cana-1800	13	16	]	]	PUNCT
cana-1800	13	17	.	.	PUNCT
cana-1800	14	1	fuzzy	fuzzy	ADJ
cana-1800	14	2	sets	set	NOUN
cana-1800	14	3	with	with	ADP
cana-1800	14	4	membership	membership	NOUN
cana-1800	14	5	degree	degree	NOUN
cana-1800	14	6	ranges	range	NOUN
cana-1800	14	7	that	that	PRON
cana-1800	14	8	vary	vary	VERB
cana-1800	14	9	from	from	ADP
cana-1800	14	10	[	[	X
cana-1800	14	11	0	0	NUM
cana-1800	14	12	,	,	PUNCT
cana-1800	14	13	1	1	NUM
cana-1800	14	14	]	]	PUNCT
cana-1800	14	15	to	to	ADP
cana-1800	14	16	[	[	X
cana-1800	14	17	-1	-1	INTJ
cana-1800	14	18	,	,	PUNCT
cana-1800	14	19	1	1	NUM
cana-1800	14	20	]	]	PUNCT
cana-1800	14	21	are	be	AUX
cana-1800	14	22	considered	consider	VERB
cana-1800	14	23	extensions	extension	NOUN
cana-1800	14	24	of	of	ADP
cana-1800	14	25	fuzzy	fuzzy	ADJ
cana-1800	14	26	sets	set	NOUN
cana-1800	14	27	.	.	PUNCT
cana-1800	15	1	in	in	ADP
cana-1800	15	2	a	a	DET
cana-1800	15	3	bipolar	bipolar	ADJ
cana-1800	15	4	valued	value	VERB
cana-1800	15	5	fuzzy	fuzzy	ADJ
cana-1800	15	6	set	set	NOUN
cana-1800	15	7	,	,	PUNCT
cana-1800	15	8	elements	element	NOUN
cana-1800	15	9	with	with	ADP
cana-1800	15	10	a	a	DET
cana-1800	15	11	membership	membership	NOUN
cana-1800	15	12	degree	degree	NOUN
cana-1800	15	13	of	of	ADP
cana-1800	15	14	0	0	NUM
cana-1800	15	15	are	be	AUX
cana-1800	15	16	unrelated	unrelated	ADJ
cana-1800	15	17	to	to	ADP
cana-1800	15	18	the	the	DET
cana-1800	15	19	associated	associated	ADJ
cana-1800	15	20	property	property	NOUN
cana-1800	15	21	,	,	PUNCT
cana-1800	15	22	those	those	PRON
cana-1800	15	23	with	with	ADP
cana-1800	15	24	a	a	DET
cana-1800	15	25	membership	membership	NOUN
cana-1800	15	26	degree	degree	NOUN
cana-1800	15	27	of	of	ADP
cana-1800	15	28	(	(	PUNCT
cana-1800	15	29	0	0	NUM
cana-1800	15	30	,	,	PUNCT
cana-1800	15	31	1	1	NUM
cana-1800	15	32	]	]	PUNCT
cana-1800	15	33	are	be	AUX
cana-1800	15	34	somewhat	somewhat	ADV
cana-1800	15	35	in	in	ADP
cana-1800	15	36	agreement	agreement	NOUN
cana-1800	15	37	with	with	ADP
cana-1800	15	38	the	the	DET
cana-1800	15	39	property	property	NOUN
cana-1800	15	40	,	,	PUNCT
cana-1800	15	41	and	and	CCONJ
cana-1800	15	42	those	those	PRON
cana-1800	15	43	with	with	ADP
cana-1800	15	44	a	a	DET
cana-1800	15	45	membership	membership	NOUN
cana-1800	15	46	degree	degree	NOUN
cana-1800	15	47	of	of	ADP
cana-1800	15	48	[	[	X
cana-1800	15	49	-1	-1	X
cana-1800	15	50	,	,	PUNCT
cana-1800	15	51	0	0	NUM
cana-1800	15	52	)	)	PUNCT
cana-1800	15	53	are	be	AUX
cana-1800	15	54	somewhat	somewhat	ADV
cana-1800	15	55	in	in	ADP
cana-1800	15	56	agreement	agreement	NOUN
cana-1800	15	57	with	with	ADP
cana-1800	15	58	the	the	DET
cana-1800	15	59	implicit	implicit	ADJ
cana-1800	15	60	counter	counter	NOUN
cana-1800	15	61	property	property	NOUN
cana-1800	15	62	.	.	PUNCT
cana-1800	16	1	both	both	PRON
cana-1800	16	2	intuitionistic	intuitionistic	ADJ
cana-1800	16	3	and	and	CCONJ
cana-1800	16	4	bipolar	bipolar	ADJ
cana-1800	16	5	valued	value	VERB
cana-1800	16	6	fuzzy	fuzzy	ADJ
cana-1800	16	7	sets	set	NOUN
cana-1800	16	8	have	have	VERB
cana-1800	16	9	a	a	DET
cana-1800	16	10	similar	similar	ADJ
cana-1800	16	11	appearance	appearance	NOUN
cana-1800	16	12	.	.	PUNCT
cana-1800	17	1	they	they	PRON
cana-1800	17	2	differ	differ	VERB
cana-1800	17	3	from	from	ADP
cana-1800	17	4	one	one	NUM
cana-1800	17	5	another	another	DET
cana-1800	17	6	,	,	PUNCT
cana-1800	17	7	but	but	CCONJ
cana-1800	17	8	,	,	PUNCT
cana-1800	17	9	[	[	X
cana-1800	17	10	10,11	10,11	NOUN
cana-1800	17	11	]	]	PUNCT
cana-1800	17	12	.	.	PUNCT
cana-1800	18	1	azriel	azriel	PROPN
cana-1800	18	2	rosenfeld[5	rosenfeld[5	PROPN
cana-1800	18	3	]	]	PUNCT
cana-1800	18	4	introduced	introduce	VERB
cana-1800	18	5	the	the	DET
cana-1800	18	6	fuzzy	fuzzy	ADJ
cana-1800	18	7	subgroup	subgroup	NOUN
cana-1800	18	8	.	.	PUNCT
cana-1800	19	1	[	[	X
cana-1800	19	2	5	5	NUM
cana-1800	19	3	]	]	PUNCT
cana-1800	19	4	.	.	PUNCT
cana-1800	20	1	the	the	DET
cana-1800	20	2	vague	vague	ADJ
cana-1800	20	3	groups	group	NOUN
cana-1800	20	4	were	be	AUX
cana-1800	20	5	introduced	introduce	VERB
cana-1800	20	6	by	by	ADP
cana-1800	20	7	ranjit	ranjit	PROPN
cana-1800	20	8	biswas	biswas	PROPN
cana-1800	20	9	(	(	PUNCT
cana-1800	20	10	[	[	X
cana-1800	20	11	13	13	NUM
cana-1800	20	12	]	]	NUM
cana-1800	20	13	)	)	PUNCT
cana-1800	20	14	.	.	PUNCT
cana-1800	21	1	a	a	DET
cana-1800	21	2	new	new	ADJ
cana-1800	21	3	class	class	NOUN
cana-1800	21	4	of	of	ADP
cana-1800	21	5	generalised	generalised	ADJ
cana-1800	21	6	bipolar	bipolar	ADJ
cana-1800	21	7	ambiguous	ambiguous	ADJ
cana-1800	21	8	sets	set	NOUN
cana-1800	21	9	has	have	AUX
cana-1800	21	10	been	be	AUX
cana-1800	21	11	presented	present	VERB
cana-1800	21	12	by	by	ADP
cana-1800	21	13	s.	s.	PROPN
cana-1800	21	14	cicily	cicily	ADV
cana-1800	21	15	flora	flora	PROPN
cana-1800	21	16	and	and	CCONJ
cana-1800	21	17	i.	i.	NOUN
cana-1800	21	18	arockiarani	arockiarani	PROPN
cana-1800	21	19	in	in	ADP
cana-1800	21	20	[	[	X
cana-1800	21	21	7	7	NUM
cana-1800	21	22	]	]	PUNCT
cana-1800	21	23	.	.	PUNCT
cana-1800	22	1	described	describe	VERB
cana-1800	22	2	as	as	ADP
cana-1800	22	3	bipolar	bipolar	ADJ
cana-1800	22	4	valued	value	VERB
cana-1800	22	5	fuzzy	fuzzy	ADJ
cana-1800	22	6	subgroups	subgroup	NOUN
cana-1800	22	7	of	of	ADP
cana-1800	22	8	a	a	DET
cana-1800	22	9	group	group	NOUN
cana-1800	22	10	by	by	ADP
cana-1800	22	11	anitha.m.s	anitha.m.	NOUN
cana-1800	22	12	.	.	PUNCT
cana-1800	22	13	,	,	PUNCT
cana-1800	22	14	et	et	PROPN
cana-1800	22	15	.	.	PUNCT
cana-1800	23	1	al.[1	al.[1	PROPN
cana-1800	23	2	]	]	PUNCT
cana-1800	23	3	and	and	CCONJ
cana-1800	23	4	the	the	DET
cana-1800	23	5	bipolar	bipolar	ADJ
cana-1800	23	6	interval	interval	NOUN
cana-1800	23	7	valued	value	VERB
cana-1800	23	8	fuzzy	fuzzy	ADJ
cana-1800	23	9	subgroups	subgroup	NOUN
cana-1800	23	10	of	of	ADP
cana-1800	23	11	a	a	DET
cana-1800	23	12	group	group	NOUN
cana-1800	23	13	were	be	AUX
cana-1800	23	14	defined	define	VERB
cana-1800	23	15	by	by	ADP
cana-1800	23	16	a.	a.	NOUN
cana-1800	23	17	balasubramanian	balasubramanian	PROPN
cana-1800	23	18	.	.	PUNCT
cana-1800	24	1	[	[	X
cana-1800	24	2	6	6	NUM
cana-1800	24	3	]	]	PUNCT
cana-1800	24	4	.	.	PUNCT
cana-1800	25	1	k.	k.	PROPN
cana-1800	25	2	murugalingam	murugalingam	PROPN
cana-1800	25	3	and	and	CCONJ
cana-1800	25	4	k.	k.	PROPN
cana-1800	25	5	arjunan	arjunan	PROPN
cana-1800	25	6	talked	talk	VERB
cana-1800	25	7	about	about	ADP
cana-1800	25	8	the	the	DET
cana-1800	25	9	intervalvalued	intervalvalue	VERB
cana-1800	25	10	fuzzy	fuzzy	ADJ
cana-1800	25	11	subsemiring	subsemiring	NOUN
cana-1800	25	12	of	of	ADP
cana-1800	25	13	a	a	DET
cana-1800	25	14	semiring	semiring	NOUN
cana-1800	25	15	in	in	ADP
cana-1800	25	16	their	their	PRON
cana-1800	25	17	discussion	discussion	NOUN
cana-1800	25	18	in	in	ADP
cana-1800	25	19	[	[	X
cana-1800	25	20	12	12	NUM
cana-1800	25	21	]	]	PUNCT
cana-1800	25	22	.	.	PUNCT
cana-1800	26	1	and	and	CCONJ
cana-1800	26	2	yasodara.b	yasodara.b	PROPN
cana-1800	26	3	and	and	CCONJ
cana-1800	26	4	ke.sathappan	ke.sathappan	X
cana-1800	27	1	[	[	X
cana-1800	27	2	14	14	NUM
cana-1800	27	3	]	]	PUNCT
cana-1800	27	4	developed	develop	VERB
cana-1800	27	5	bipolar	bipolar	ADJ
cana-1800	27	6	valued	value	VERB
cana-1800	27	7	multi	multi	NOUN
cana-1800	27	8	fuzzy	fuzzy	ADJ
cana-1800	27	9	semiring	semiring	NOUN
cana-1800	27	10	subsemirings	subsemiring	NOUN
cana-1800	27	11	.	.	PUNCT
cana-1800	28	1	bipolar	bipolar	ADJ
cana-1800	28	2	valued	value	VERB
cana-1800	28	3	vague	vague	ADJ
cana-1800	28	4	subsemirings	subsemiring	NOUN
cana-1800	28	5	of	of	ADP
cana-1800	28	6	a	a	DET
cana-1800	28	7	semiring	semiring	NOUN
cana-1800	28	8	were	be	AUX
cana-1800	28	9	defined	define	VERB
cana-1800	28	10	by	by	ADP
cana-1800	28	11	anitha.k	anitha.k	PROPN
cana-1800	28	12	.	.	PUNCT
cana-1800	28	13	,	,	PUNCT
cana-1800	28	14	et	et	PROPN
cana-1800	28	15	al	al	PROPN
cana-1800	29	1	[	[	X
cana-1800	29	2	2,3,4].this	2,3,4].this	NUM
cana-1800	29	3	article	article	NOUN
cana-1800	29	4	makes	make	VERB
cana-1800	29	5	use	use	NOUN
cana-1800	29	6	of	of	ADP
cana-1800	29	7	the	the	DET
cana-1800	29	8	idea	idea	NOUN
cana-1800	29	9	of	of	ADP
cana-1800	29	10	bipolar	bipolar	ADJ
cana-1800	29	11	valued	value	VERB
cana-1800	29	12	vague	vague	ADJ
cana-1800	29	13	ideals	ideal	NOUN
cana-1800	29	14	(	(	PUNCT
cana-1800	29	15	bvvi	bvvi	NOUN
cana-1800	29	16	)	)	PUNCT
cana-1800	29	17	of	of	ADP
cana-1800	29	18	a	a	DET
cana-1800	29	19	semiring	semiring	NOUN
cana-1800	29	20	.	.	PUNCT
cana-1800	30	1	communications	communication	NOUN
cana-1800	30	2	on	on	ADP
cana-1800	30	3	applied	apply	VERB
cana-1800	30	4	nonlinear	nonlinear	ADJ
cana-1800	30	5	analysis	analysis	NOUN
cana-1800	30	6	issn	issn	NOUN
cana-1800	30	7	:	:	PUNCT
cana-1800	30	8	1074	1074	NUM
cana-1800	30	9	-	-	PUNCT
cana-1800	30	10	133x	133x	NUM
cana-1800	30	11	vol	vol	NOUN
cana-1800	30	12	32	32	NUM
cana-1800	30	13	no	no	NOUN
cana-1800	30	14	.	.	NOUN
cana-1800	30	15	2	2	NUM
cana-1800	30	16	(	(	PUNCT
cana-1800	30	17	2025	2025	NUM
cana-1800	30	18	)	)	PUNCT
cana-1800	30	19	508	508	NUM
cana-1800	30	20	https://internationalpubls.com	https://internationalpubls.com	SYM
cana-1800	30	21	2	2	NUM
cana-1800	30	22	preliminaries	preliminary	NOUN
cana-1800	30	23	in	in	ADP
cana-1800	30	24	this	this	DET
cana-1800	30	25	step	step	NOUN
cana-1800	30	26	,	,	PUNCT
cana-1800	30	27	we	we	PRON
cana-1800	30	28	recollect	recollect	VERB
cana-1800	30	29	a	a	DET
cana-1800	30	30	few	few	ADJ
cana-1800	30	31	key	key	ADJ
cana-1800	30	32	standards	standard	NOUN
cana-1800	30	33	and	and	CCONJ
cana-1800	30	34	definitions	definition	NOUN
cana-1800	30	35	that	that	PRON
cana-1800	30	36	are	be	AUX
cana-1800	30	37	likely	likely	ADJ
cana-1800	30	38	to	to	PART
cana-1800	30	39	be	be	AUX
cana-1800	30	40	significant	significant	ADJ
cana-1800	30	41	for	for	ADP
cana-1800	30	42	this	this	DET
cana-1800	30	43	work	work	NOUN
cana-1800	30	44	.	.	PUNCT
cana-1800	31	1	definition	definition	NOUN
cana-1800	31	2	2.1	2.1	NUM
cana-1800	32	1	[	[	X
cana-1800	32	2	12	12	NUM
cana-1800	32	3	]	]	PUNCT
cana-1800	32	4	a	a	DET
cana-1800	32	5	mapping	mapping	NOUN
cana-1800	32	6	𝔜	𝔜	NOUN
cana-1800	32	7	:	:	PUNCT
cana-1800	32	8	𝔊	𝔊	PROPN
cana-1800	32	9	⟶	⟶	ADJ
cana-1800	32	10	[	[	X
cana-1800	32	11	0	0	NUM
cana-1800	32	12	,	,	PUNCT
cana-1800	32	13	1	1	NUM
cana-1800	32	14	]	]	PUNCT
cana-1800	32	15	𝑖𝑠	𝑖𝑠	CCONJ
cana-1800	32	16	𝑐𝑎𝑙𝑙𝑒𝑑	𝑐𝑎𝑙𝑙𝑒𝑑	PROPN
cana-1800	32	17	𝑓𝑢𝑧𝑧𝑦	𝑓𝑢𝑧𝑧𝑦	NOUN
cana-1800	32	18	𝑠𝑢𝑏𝑠𝑒𝑡	𝑠𝑢𝑏𝑠𝑒𝑡	NOUN
cana-1800	32	19	𝑜𝑓	𝑜𝑓	ADP
cana-1800	32	20	𝑡ℎ𝑒	𝑡ℎ𝑒	PROPN
cana-1800	32	21	𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑎𝑙	𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑎𝑙	PROPN
cana-1800	32	22	𝑠𝑒𝑡	𝑠𝑒𝑡	PROPN
cana-1800	32	23	𝔊.	𝔊.	NOUN
cana-1800	32	24	definition	definition	NOUN
cana-1800	32	25	2.2	2.2	NUM
cana-1800	32	26	[	[	SYM
cana-1800	32	27	5	5	NUM
cana-1800	32	28	]	]	X
cana-1800	32	29	𝑇ℎ𝑒	𝑇ℎ𝑒	PROPN
cana-1800	32	30	𝑜𝑟𝑑𝑒𝑟𝑒𝑑	𝑜𝑟𝑑𝑒𝑟𝑒𝑑	VERB
cana-1800	32	31	𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒	𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒	ADV
cana-1800	32	32	𝔑	𝔑	PROPN
cana-1800	32	33	=	=	SYM
cana-1800	32	34	{	{	PUNCT
cana-1800	32	35	(	(	PUNCT
cana-1800	32	36	𝔶	𝔶	PROPN
cana-1800	32	37	,	,	PUNCT
cana-1800	32	38	𝒱𝔑(𝔶	𝒱𝔑(𝔶	NUM
cana-1800	32	39	)	)	PUNCT
cana-1800	32	40	):	):	PUNCT
cana-1800	32	41	𝔶	𝔶	PROPN
cana-1800	32	42	∈	∈	PROPN
cana-1800	32	43	𝕋	𝕋	PROPN
cana-1800	32	44	}	}	PUNCT
cana-1800	32	45	𝑖𝑠	𝑖𝑠	NOUN
cana-1800	32	46	𝑐𝑎𝑙𝑙𝑒𝑑	𝑐𝑎𝑙𝑙𝑒𝑑	VERB
cana-1800	32	47	a	a	DET
cana-1800	32	48	vague	vague	ADJ
cana-1800	32	49	set	set	NOUN
cana-1800	32	50	𝑜𝑓	𝑜𝑓	ADP
cana-1800	32	51	𝑡ℎ𝑒	𝑡ℎ𝑒	CCONJ
cana-1800	32	52	𝑠𝑒𝑡	𝑠𝑒𝑡	PROPN
cana-1800	32	53	𝕋	𝕋	PROPN
cana-1800	32	54	,	,	PUNCT
cana-1800	32	55	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	NOUN
cana-1800	32	56	𝒱𝔑(𝔶	𝒱𝔑(𝔶	NUM
cana-1800	32	57	)	)	PUNCT
cana-1800	33	1	=	=	SYM
cana-1800	33	2	[	[	PUNCT
cana-1800	33	3	𝒯𝔑(𝔶	𝒯𝔑(𝔶	NOUN
cana-1800	33	4	)	)	PUNCT
cana-1800	33	5	,	,	PUNCT
cana-1800	33	6	1	1	NUM
cana-1800	33	7	−	−	PROPN
cana-1800	33	8	ℱ𝔑(𝔶	ℱ𝔑(𝔶	NUM
cana-1800	33	9	)	)	PUNCT
cana-1800	33	10	]	]	PUNCT
cana-1800	33	11	,	,	PUNCT
cana-1800	33	12	𝒯𝔑	𝒯𝔑	PROPN
cana-1800	33	13	:	:	PUNCT
cana-1800	33	14	𝕋	𝕋	NOUN
cana-1800	33	15	→	→	SYM
cana-1800	33	16	[	[	X
cana-1800	33	17	0,1	0,1	NUM
cana-1800	33	18	]	]	PUNCT
cana-1800	33	19	𝑖𝑠	𝑖𝑠	CCONJ
cana-1800	33	20	𝑎	𝑎	NOUN
cana-1800	33	21	𝑡𝑟𝑢𝑡ℎ	𝑡𝑟𝑢𝑡ℎ	NOUN
cana-1800	33	22	𝑚𝑒𝑚𝑏𝑒𝑟𝑠ℎ𝑖𝑝	𝑚𝑒𝑚𝑏𝑒𝑟𝑠ℎ𝑖𝑝	ADJ
cana-1800	33	23	map	map	NOUN
cana-1800	33	24	and	and	CCONJ
cana-1800	33	25	ℱ𝔑	ℱ𝔑	NOUN
cana-1800	33	26	:	:	PUNCT
cana-1800	33	27	𝕋	𝕋	X
cana-1800	33	28	→	→	SYM
cana-1800	33	29	[	[	X
cana-1800	33	30	0,1	0,1	NUM
cana-1800	33	31	]	]	PUNCT
cana-1800	33	32	is	be	AUX
cana-1800	33	33	a	a	DET
cana-1800	33	34	false	false	ADJ
cana-1800	33	35	membership	membership	NOUN
cana-1800	33	36	map	map	NOUN
cana-1800	33	37	.	.	PUNCT
cana-1800	34	1	example	example	NOUN
cana-1800	34	2	2.3	2.3	NUM
cana-1800	34	3	𝔑	𝔑	NOUN
cana-1800	34	4	=	=	PUNCT
cana-1800	34	5	{	{	PUNCT
cana-1800	34	6	(	(	PUNCT
cana-1800	34	7	𝔶	𝔶	INTJ
cana-1800	34	8	,	,	PUNCT
cana-1800	34	9	[	[	X
cana-1800	34	10	0.4	0.4	NUM
cana-1800	34	11	,	,	PUNCT
cana-1800	34	12	0.7	0.7	NUM
cana-1800	34	13	]	]	PUNCT
cana-1800	34	14	)	)	PUNCT
cana-1800	34	15	,	,	PUNCT
cana-1800	34	16	(	(	PUNCT
cana-1800	34	17	𝔳	𝔳	X
cana-1800	34	18	,	,	PUNCT
cana-1800	34	19	[	[	X
cana-1800	34	20	0.5	0.5	NUM
cana-1800	34	21	,	,	PUNCT
cana-1800	34	22	0.8	0.8	NUM
cana-1800	34	23	]	]	PUNCT
cana-1800	34	24	)	)	PUNCT
cana-1800	34	25	,	,	PUNCT
cana-1800	34	26	(	(	PUNCT
cana-1800	34	27	𝓀	𝓀	X
cana-1800	34	28	,	,	PUNCT
cana-1800	34	29	[	[	X
cana-1800	34	30	0.6	0.6	NUM
cana-1800	34	31	,	,	PUNCT
cana-1800	34	32	0.9	0.9	NUM
cana-1800	34	33	]	]	PUNCT
cana-1800	34	34	)	)	PUNCT
cana-1800	34	35	}	}	PUNCT
cana-1800	34	36	is	be	AUX
cana-1800	34	37	a	a	DET
cana-1800	34	38	vague	vague	ADJ
cana-1800	34	39	subset	subset	NOUN
cana-1800	34	40	of	of	ADP
cana-1800	34	41	the	the	DET
cana-1800	34	42	universal	universal	ADJ
cana-1800	34	43	set	set	NOUN
cana-1800	34	44	𝕋	𝕋	NOUN
cana-1800	34	45	=	=	PUNCT
cana-1800	34	46	{	{	PUNCT
cana-1800	34	47	𝔶	𝔶	PROPN
cana-1800	34	48	,	,	PUNCT
cana-1800	34	49	𝔳	𝔳	PROPN
cana-1800	34	50	,	,	PUNCT
cana-1800	34	51	𝓀	𝓀	NOUN
cana-1800	34	52	}	}	PUNCT
cana-1800	34	53	.	.	PUNCT
cana-1800	35	1	definition	definition	NOUN
cana-1800	35	2	2.4	2.4	NUM
cana-1800	35	3	[	[	SYM
cana-1800	35	4	9	9	X
cana-1800	35	5	]	]	X
cana-1800	35	6	𝑇ℎ𝑒	𝑇ℎ𝑒	PROPN
cana-1800	35	7	𝑜𝑟𝑑𝑒𝑟𝑒𝑑	𝑜𝑟𝑑𝑒𝑟𝑒𝑑	VERB
cana-1800	35	8	𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒	𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒	ADV
cana-1800	35	9	𝔗	𝔗	PROPN
cana-1800	35	10	=	=	X
cana-1800	35	11	{	{	PUNCT
cana-1800	35	12	(	(	PUNCT
cana-1800	35	13	𝔳	𝔳	PROPN
cana-1800	35	14	,	,	PUNCT
cana-1800	35	15	𝔗+(𝔳	𝔗+(𝔳	NOUN
cana-1800	35	16	)	)	PUNCT
cana-1800	35	17	,	,	PUNCT
cana-1800	35	18	𝔗−(𝔳	𝔗−(𝔳	NOUN
cana-1800	35	19	)	)	PUNCT
cana-1800	35	20	):	):	PUNCT
cana-1800	35	21	𝔳	𝔳	PROPN
cana-1800	35	22	∈	∈	PROPN
cana-1800	35	23	𝕋	𝕋	PROPN
cana-1800	35	24	}	}	PUNCT
cana-1800	35	25	𝑖𝑠	𝑖𝑠	NOUN
cana-1800	35	26	𝑐𝑎𝑙𝑙𝑒𝑑	𝑐𝑎𝑙𝑙𝑒𝑑	NOUN
cana-1800	35	27	a	a	DET
cana-1800	35	28	bipolar	bipolar	ADJ
cana-1800	35	29	𝑣𝑎𝑙𝑢𝑒𝑑	𝑣𝑎𝑙𝑢𝑒𝑑	VERB
cana-1800	35	30	𝑓𝑢𝑧𝑧𝑦	𝑓𝑢𝑧𝑧𝑦	NOUN
cana-1800	35	31	𝑠𝑢𝑏𝑠𝑒𝑡	𝑠𝑢𝑏𝑠𝑒𝑡	NOUN
cana-1800	35	32	𝑜𝑓	𝑜𝑓	ADP
cana-1800	35	33	𝕋	𝕋	PROPN
cana-1800	35	34	,	,	PUNCT
cana-1800	35	35	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	NOUN
cana-1800	35	36	𝔗+	𝔗+	NOUN
cana-1800	35	37	:	:	PUNCT
cana-1800	35	38	𝕋	𝕋	PROPN
cana-1800	35	39	→	→	SYM
cana-1800	35	40	[	[	X
cana-1800	35	41	0,1	0,1	NUM
cana-1800	35	42	]	]	PUNCT
cana-1800	35	43	𝑖𝑠	𝑖𝑠	CCONJ
cana-1800	35	44	𝑎	𝑎	DET
cana-1800	35	45	𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒	𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒	NOUN
cana-1800	35	46	𝑚𝑒𝑚𝑏𝑒𝑟𝑠ℎ𝑖𝑝	𝑚𝑒𝑚𝑏𝑒𝑟𝑠ℎ𝑖𝑝	ADJ
cana-1800	35	47	map	map	NOUN
cana-1800	35	48	and	and	CCONJ
cana-1800	35	49	𝔗−	𝔗−	NUM
cana-1800	35	50	:	:	PUNCT
cana-1800	35	51	𝕋	𝕋	NOUN
cana-1800	35	52	→	→	SYM
cana-1800	35	53	[	[	X
cana-1800	35	54	−1,0	−1,0	X
cana-1800	35	55	]	]	X
cana-1800	35	56	is	be	AUX
cana-1800	35	57	a	a	DET
cana-1800	35	58	negative	negative	ADJ
cana-1800	35	59	membership	membership	NOUN
cana-1800	35	60	map	map	NOUN
cana-1800	35	61	.	.	PUNCT
cana-1800	36	1	example	example	NOUN
cana-1800	36	2	2.5	2.5	NUM
cana-1800	36	3	𝔑	𝔑	NOUN
cana-1800	36	4	=	=	PUNCT
cana-1800	36	5	{	{	PUNCT
cana-1800	36	6	(	(	PUNCT
cana-1800	36	7	𝔶	𝔶	NOUN
cana-1800	36	8	,	,	PUNCT
cana-1800	36	9	0.05	0.05	NUM
cana-1800	36	10	,	,	PUNCT
cana-1800	36	11	−0.003	−0.003	ADJ
cana-1800	36	12	)	)	PUNCT
cana-1800	36	13	,	,	PUNCT
cana-1800	36	14	(	(	PUNCT
cana-1800	36	15	𝔳	𝔳	X
cana-1800	36	16	,	,	PUNCT
cana-1800	36	17	0.04	0.04	NUM
cana-1800	36	18	,	,	PUNCT
cana-1800	36	19	−0.6	−0.6	PROPN
cana-1800	36	20	)	)	PUNCT
cana-1800	36	21	,	,	PUNCT
cana-1800	36	22	(	(	PUNCT
cana-1800	36	23	𝓀	𝓀	X
cana-1800	36	24	,	,	PUNCT
cana-1800	36	25	0.004	0.004	NUM
cana-1800	36	26	,	,	PUNCT
cana-1800	36	27	−0.07	−0.07	NOUN
cana-1800	36	28	)	)	PUNCT
cana-1800	36	29	}	}	PUNCT
cana-1800	36	30	is	be	AUX
cana-1800	36	31	a	a	DET
cana-1800	36	32	bipolar	bipolar	ADJ
cana-1800	36	33	valued	value	VERB
cana-1800	36	34	fuzzy	fuzzy	ADJ
cana-1800	36	35	subset	subset	NOUN
cana-1800	36	36	of	of	ADP
cana-1800	36	37	the	the	DET
cana-1800	36	38	set	set	NOUN
cana-1800	36	39	𝕋	𝕋	NOUN
cana-1800	36	40	=	=	PUNCT
cana-1800	36	41	{	{	PUNCT
cana-1800	36	42	𝔶	𝔶	PROPN
cana-1800	36	43	,	,	PUNCT
cana-1800	36	44	𝔳	𝔳	X
cana-1800	36	45	,	,	PUNCT
cana-1800	36	46	𝓀	𝓀	PROPN
cana-1800	36	47	}	}	PUNCT
cana-1800	36	48	.	.	PUNCT
cana-1800	37	1	definition	definition	NOUN
cana-1800	37	2	2.6	2.6	NUM
cana-1800	38	1	[	[	X
cana-1800	38	2	7	7	X
cana-1800	38	3	]	]	X
cana-1800	38	4	𝑇ℎ𝑒	𝑇ℎ𝑒	PROPN
cana-1800	38	5	𝑜𝑟𝑑𝑒𝑟𝑒𝑑	𝑜𝑟𝑑𝑒𝑟𝑒𝑑	VERB
cana-1800	38	6	𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒	𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒	ADJ
cana-1800	38	7	𝔈	𝔈	NOUN
cana-1800	38	8	=	=	SYM
cana-1800	38	9	{	{	PUNCT
cana-1800	38	10	(	(	PUNCT
cana-1800	38	11	𝔳	𝔳	PROPN
cana-1800	38	12	,	,	PUNCT
cana-1800	38	13	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	38	14	+	+	PROPN
cana-1800	38	15	(	(	PUNCT
cana-1800	38	16	𝔳	𝔳	NOUN
cana-1800	38	17	)	)	PUNCT
cana-1800	38	18	,	,	PUNCT
cana-1800	38	19	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	38	20	−(𝔳	−(𝔳	NOUN
cana-1800	38	21	)	)	PUNCT
cana-1800	38	22	):	):	PUNCT
cana-1800	38	23	𝔳	𝔳	PROPN
cana-1800	38	24	∈	∈	PROPN
cana-1800	38	25	𝕋	𝕋	PROPN
cana-1800	38	26	}	}	PUNCT
cana-1800	38	27	𝑖𝑠	𝑖𝑠	NOUN
cana-1800	38	28	𝑐𝑎𝑙𝑙𝑒𝑑	𝑐𝑎𝑙𝑙𝑒𝑑	NOUN
cana-1800	38	29	a	a	DET
cana-1800	38	30	bipolar	bipolar	ADJ
cana-1800	38	31	𝑣𝑎𝑙𝑢𝑒𝑑	𝑣𝑎𝑙𝑢𝑒𝑑	VERB
cana-1800	39	1	𝑣𝑎𝑔𝑢𝑒	𝑣𝑎𝑔𝑢𝑒	PROPN
cana-1800	39	2	𝑠𝑢𝑏𝑠𝑒𝑡	𝑠𝑢𝑏𝑠𝑒𝑡	NOUN
cana-1800	39	3	(	(	PUNCT
cana-1800	39	4	𝔹𝕍𝕍𝕊𝕊	𝔹𝕍𝕍𝕊𝕊	PROPN
cana-1800	39	5	)	)	PUNCT
cana-1800	40	1	𝑜𝑓	𝑜𝑓	ADP
cana-1800	40	2	𝕋	𝕋	PROPN
cana-1800	40	3	,	,	PUNCT
cana-1800	40	4	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	ADJ
cana-1800	40	5	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	40	6	+	+	PROPN
cana-1800	40	7	(	(	PUNCT
cana-1800	40	8	𝔳	𝔳	PROPN
cana-1800	40	9	)	)	PUNCT
cana-1800	40	10	=	=	NOUN
cana-1800	41	1	[	[	PUNCT
cana-1800	41	2	𝒯𝔈	𝒯𝔈	PROPN
cana-1800	41	3	+	+	NOUN
cana-1800	41	4	(	(	PUNCT
cana-1800	41	5	𝔳	𝔳	NOUN
cana-1800	41	6	)	)	PUNCT
cana-1800	41	7	,	,	PUNCT
cana-1800	41	8	1	1	NUM
cana-1800	41	9	−	−	NOUN
cana-1800	41	10	ℱ𝔈	ℱ𝔈	PROPN
cana-1800	41	11	+	+	NOUN
cana-1800	41	12	(	(	PUNCT
cana-1800	41	13	𝔳	𝔳	NOUN
cana-1800	41	14	)	)	PUNCT
cana-1800	41	15	]	]	PUNCT
cana-1800	41	16	and	and	CCONJ
cana-1800	41	17	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	41	18	−(𝔳	−(𝔳	NOUN
cana-1800	41	19	)	)	PUNCT
cana-1800	41	20	=	=	PUNCT
cana-1800	42	1	[	[	PUNCT
cana-1800	42	2	−1	−1	NOUN
cana-1800	42	3	−	−	PROPN
cana-1800	42	4	ℱ𝔈	ℱ𝔈	PROPN
cana-1800	42	5	−(𝔳	−(𝔳	VERB
cana-1800	42	6	)	)	PUNCT
cana-1800	42	7	,	,	PUNCT
cana-1800	42	8	𝒯𝔈	𝒯𝔈	PROPN
cana-1800	42	9	−(𝔳	−(𝔳	VERB
cana-1800	42	10	)	)	PUNCT
cana-1800	42	11	]	]	PUNCT
cana-1800	42	12	,	,	PUNCT
cana-1800	42	13	𝒯𝔈	𝒯𝔈	PROPN
cana-1800	43	1	+	+	NOUN
cana-1800	43	2	:	:	PUNCT
cana-1800	43	3	𝕋	𝕋	NOUN
cana-1800	43	4	→	→	SYM
cana-1800	43	5	[	[	X
cana-1800	43	6	0	0	NUM
cana-1800	43	7	,	,	PUNCT
cana-1800	43	8	1	1	NUM
cana-1800	43	9	]	]	PUNCT
cana-1800	43	10	,	,	PUNCT
cana-1800	43	11	ℱ𝔈	ℱ𝔈	PROPN
cana-1800	43	12	+	+	NOUN
cana-1800	43	13	:	:	PUNCT
cana-1800	43	14	𝕋	𝕋	NOUN
cana-1800	43	15	→	→	SYM
cana-1800	43	16	[	[	X
cana-1800	43	17	0	0	NUM
cana-1800	43	18	,	,	PUNCT
cana-1800	43	19	1	1	NUM
cana-1800	43	20	]	]	PUNCT
cana-1800	43	21	,	,	PUNCT
cana-1800	43	22	𝒯𝔈	𝒯𝔈	PROPN
cana-1800	43	23	−	−	NOUN
cana-1800	43	24	:	:	PUNCT
cana-1800	43	25	𝕋	𝕋	X
cana-1800	43	26	→	→	SYM
cana-1800	43	27	[	[	X
cana-1800	43	28	−1	−1	NOUN
cana-1800	43	29	,	,	PUNCT
cana-1800	43	30	0	0	NUM
cana-1800	43	31	]	]	PUNCT
cana-1800	43	32	and	and	CCONJ
cana-1800	43	33	ℱ𝔈	ℱ𝔈	PROPN
cana-1800	43	34	−	−	NOUN
cana-1800	43	35	:	:	PUNCT
cana-1800	43	36	𝕋	𝕋	X
cana-1800	43	37	→	→	SYM
cana-1800	43	38	[	[	X
cana-1800	43	39	−1	−1	NOUN
cana-1800	43	40	,	,	PUNCT
cana-1800	43	41	0	0	NUM
cana-1800	43	42	]	]	X
cana-1800	43	43	𝑠𝑢𝑐ℎ	𝑠𝑢𝑐ℎ	NOUN
cana-1800	43	44	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	PROPN
cana-1800	43	45	𝒯𝔈	𝒯𝔈	PROPN
cana-1800	44	1	+	+	PROPN
cana-1800	44	2	(	(	PUNCT
cana-1800	44	3	𝔳	𝔳	NOUN
cana-1800	44	4	)	)	PUNCT
cana-1800	44	5	+	+	CCONJ
cana-1800	44	6	ℱ𝔈	ℱ𝔈	PROPN
cana-1800	45	1	+	+	ADJ
cana-1800	45	2	(	(	PUNCT
cana-1800	45	3	𝔳	𝔳	NOUN
cana-1800	45	4	)	)	PUNCT
cana-1800	45	5	≤	≤	NUM
cana-1800	45	6	1	1	NUM
cana-1800	45	7	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-1800	45	8	−	−	PROPN
cana-1800	45	9	1	1	NUM
cana-1800	45	10	≤	≤	PUNCT
cana-1800	45	11	ℱ𝔈	ℱ𝔈	PROPN
cana-1800	45	12	−(𝔳	−(𝔳	NOUN
cana-1800	45	13	)	)	PUNCT
cana-1800	46	1	+	+	CCONJ
cana-1800	46	2	𝒯𝔈	𝒯𝔈	PROPN
cana-1800	46	3	−(𝔳	−(𝔳	NOUN
cana-1800	46	4	)	)	PUNCT
cana-1800	46	5	.	.	PUNCT
cana-1800	47	1	example	example	NOUN
cana-1800	48	1	2.7	2.7	NUM
cana-1800	48	2	𝔈	𝔈	NOUN
cana-1800	48	3	=	=	PRON
cana-1800	48	4	{	{	PUNCT
cana-1800	48	5	(	(	PUNCT
cana-1800	48	6	𝔶	𝔶	NOUN
cana-1800	48	7	,	,	PUNCT
cana-1800	48	8	[	[	X
cana-1800	48	9	0.03	0.03	NUM
cana-1800	48	10	,	,	PUNCT
cana-1800	48	11	0,6	0,6	NUM
cana-1800	48	12	]	]	PUNCT
cana-1800	48	13	,	,	PUNCT
cana-1800	48	14	[	[	X
cana-1800	48	15	−0.6	−0.6	X
cana-1800	48	16	,	,	PUNCT
cana-1800	48	17	−0.02	−0.02	NOUN
cana-1800	48	18	]	]	X
cana-1800	48	19	)	)	PUNCT
cana-1800	48	20	,	,	PUNCT
cana-1800	48	21	(	(	PUNCT
cana-1800	48	22	𝔳	𝔳	X
cana-1800	48	23	,	,	PUNCT
cana-1800	48	24	[	[	X
cana-1800	48	25	0.002	0.002	NUM
cana-1800	48	26	,	,	PUNCT
cana-1800	48	27	0.04	0.04	NUM
cana-1800	48	28	]	]	PUNCT
cana-1800	48	29	,	,	PUNCT
cana-1800	48	30	[	[	X
cana-1800	48	31	−0.05	−0.05	ADV
cana-1800	48	32	,	,	PUNCT
cana-1800	48	33	−0.004	−0.004	NOUN
cana-1800	48	34	]	]	PUNCT
cana-1800	48	35	)	)	PUNCT
cana-1800	48	36	,	,	PUNCT
cana-1800	48	37	(	(	PUNCT
cana-1800	48	38	𝓀	𝓀	X
cana-1800	48	39	,	,	PUNCT
cana-1800	48	40	[	[	X
cana-1800	48	41	0.02	0.02	NUM
cana-1800	48	42	,	,	PUNCT
cana-1800	48	43	0.7	0.7	NUM
cana-1800	48	44	]	]	PUNCT
cana-1800	48	45	,	,	PUNCT
cana-1800	48	46	[	[	PUNCT
cana-1800	48	47	−0.05	−0.05	ADV
cana-1800	48	48	,	,	PUNCT
cana-1800	48	49	−0.005	−0.005	PROPN
cana-1800	48	50	]	]	PUNCT
cana-1800	48	51	)	)	PUNCT
cana-1800	48	52	}	}	PUNCT
cana-1800	48	53	is	be	AUX
cana-1800	48	54	a	a	DET
cana-1800	48	55	𝔹𝕍𝕍𝕊𝕊	𝔹𝕍𝕍𝕊𝕊	NOUN
cana-1800	48	56	of	of	ADP
cana-1800	48	57	𝕋	𝕋	NOUN
cana-1800	48	58	=	=	PUNCT
cana-1800	48	59	{	{	PUNCT
cana-1800	48	60	𝔶	𝔶	PROPN
cana-1800	48	61	,	,	PUNCT
cana-1800	48	62	𝔳	𝔳	PROPN
cana-1800	48	63	,	,	PUNCT
cana-1800	48	64	𝓀	𝓀	NOUN
cana-1800	48	65	}	}	PUNCT
cana-1800	48	66	.	.	PUNCT
cana-1800	49	1	definition	definition	NOUN
cana-1800	49	2	2.8	2.8	NUM
cana-1800	49	3	.	.	PUNCT
cana-1800	50	1	[	[	X
cana-1800	50	2	4	4	X
cana-1800	50	3	]	]	PUNCT
cana-1800	50	4	let	let	VERB
cana-1800	50	5	a	a	DET
cana-1800	50	6	=	=	X
cana-1800	50	7			X
cana-1800	51	1	+	+	CCONJ
cana-1800	51	2	av	av	PROPN
cana-1800	51	3	,	,	PUNCT
cana-1800	51	4	−	−	PROPN
cana-1800	51	5	av	av	PROPN
cana-1800	51	6			PROPN
cana-1800	51	7	and	and	CCONJ
cana-1800	51	8	b	b	X
cana-1800	51	9	=	=	X
cana-1800	51	10			PUNCT
cana-1800	51	11	+	+	CCONJ
cana-1800	51	12	bv	bv	PROPN
cana-1800	51	13	,	,	PUNCT
cana-1800	51	14	−	−	PROPN
cana-1800	51	15	bv	bv	PROPN
cana-1800	51	16			PROPN
cana-1800	51	17	be	be	AUX
cana-1800	51	18	two	two	NUM
cana-1800	51	19	𝔹𝕍𝕍𝕊𝕊s	𝔹𝕍𝕍𝕊𝕊	NOUN
cana-1800	51	20	of	of	ADP
cana-1800	51	21	a	a	DET
cana-1800	51	22	set	set	NOUN
cana-1800	51	23	x.	x.	NOUN
cana-1800	51	24	we	we	PRON
cana-1800	51	25	define	define	VERB
cana-1800	51	26	the	the	DET
cana-1800	51	27	following	follow	VERB
cana-1800	51	28	relations	relation	NOUN
cana-1800	51	29	and	and	CCONJ
cana-1800	51	30	operations	operation	NOUN
cana-1800	51	31	:	:	PUNCT
cana-1800	51	32	(	(	PUNCT
cana-1800	51	33	i	i	NOUN
cana-1800	51	34	)	)	PUNCT
cana-1800	52	1	[	[	X
cana-1800	52	2	a	a	X
cana-1800	52	3	]	]	X
cana-1800	52	4			PROPN
cana-1800	53	1	[	[	X
cana-1800	53	2	b	b	X
cana-1800	53	3	]	]	X
cana-1800	53	4	if	if	SCONJ
cana-1800	53	5	and	and	CCONJ
cana-1800	53	6	only	only	ADV
cana-1800	53	7	if	if	SCONJ
cana-1800	53	8	+	+	NUM
cana-1800	53	9	av	av	PROPN
cana-1800	53	10	(	(	PUNCT
cana-1800	53	11	u	u	NOUN
cana-1800	53	12	)	)	PUNCT
cana-1800	53	13	≤	≤	NOUN
cana-1800	54	1	+	+	CCONJ
cana-1800	54	2	bv	bv	PROPN
cana-1800	54	3	(	(	PUNCT
cana-1800	54	4	u	u	NOUN
cana-1800	54	5	)	)	PUNCT
cana-1800	54	6	and	and	CCONJ
cana-1800	54	7	−	−	PROPN
cana-1800	54	8	av	av	PROPN
cana-1800	54	9	(	(	PUNCT
cana-1800	54	10	u	u	NOUN
cana-1800	54	11	)	)	PUNCT
cana-1800	54	12	≥	≥	NOUN
cana-1800	54	13	−	−	PROPN
cana-1800	54	14	bv	bv	PROPN
cana-1800	54	15	(	(	PUNCT
cana-1800	54	16	u	u	NOUN
cana-1800	54	17	)	)	PUNCT
cana-1800	54	18	,	,	PUNCT
cana-1800	54	19			VERB
cana-1800	54	20	ux	ux	PROPN
cana-1800	54	21	.	.	PUNCT
cana-1800	55	1	(	(	PUNCT
cana-1800	55	2	ii	ii	NOUN
cana-1800	55	3	)	)	PUNCT
cana-1800	56	1	[	[	X
cana-1800	56	2	a	a	X
cana-1800	56	3	]	]	X
cana-1800	56	4	=	=	PUNCT
cana-1800	57	1	[	[	X
cana-1800	57	2	b	b	X
cana-1800	57	3	]	]	X
cana-1800	57	4	if	if	SCONJ
cana-1800	57	5	and	and	CCONJ
cana-1800	57	6	only	only	ADV
cana-1800	57	7	if	if	SCONJ
cana-1800	57	8	+	+	NUM
cana-1800	57	9	av	av	PROPN
cana-1800	57	10	(	(	PUNCT
cana-1800	57	11	u	u	NOUN
cana-1800	57	12	)	)	PUNCT
cana-1800	57	13	=	=	PUNCT
cana-1800	58	1	+	+	NUM
cana-1800	58	2	bv	bv	PROPN
cana-1800	58	3	(	(	PUNCT
cana-1800	58	4	u	u	NOUN
cana-1800	58	5	)	)	PUNCT
cana-1800	58	6	and	and	CCONJ
cana-1800	58	7	−	−	PROPN
cana-1800	59	1	av	av	PROPN
cana-1800	59	2	(	(	PUNCT
cana-1800	59	3	u	u	NOUN
cana-1800	59	4	)	)	PUNCT
cana-1800	59	5	=	=	SYM
cana-1800	59	6	−	−	PROPN
cana-1800	59	7	bv	bv	PROPN
cana-1800	59	8	(	(	PUNCT
cana-1800	59	9	u	u	NOUN
cana-1800	59	10	)	)	PUNCT
cana-1800	59	11	,	,	PUNCT
cana-1800	59	12			VERB
cana-1800	59	13	ux	ux	PROPN
cana-1800	59	14	.	.	PUNCT
cana-1800	60	1	(	(	PUNCT
cana-1800	60	2	iii	iii	X
cana-1800	60	3	)	)	PUNCT
cana-1800	61	1	[	[	X
cana-1800	61	2	a][b	a][b	X
cana-1800	61	3	]	]	X
cana-1800	61	4	=	=	X
cana-1800	61	5	{	{	PUNCT
cana-1800	61	6			X
cana-1800	61	7	u	u	NOUN
cana-1800	61	8	,	,	PUNCT
cana-1800	61	9	rmin	rmin	NOUN
cana-1800	61	10	(	(	PUNCT
cana-1800	61	11	+	+	CCONJ
cana-1800	61	12	av	av	PROPN
cana-1800	61	13	(	(	PUNCT
cana-1800	61	14	u	u	NOUN
cana-1800	61	15	)	)	PUNCT
cana-1800	61	16	,	,	PUNCT
cana-1800	61	17	+	+	CCONJ
cana-1800	61	18	bv	bv	PROPN
cana-1800	61	19	(	(	PUNCT
cana-1800	61	20	u	u	NOUN
cana-1800	61	21	)	)	PUNCT
cana-1800	61	22	)	)	PUNCT
cana-1800	61	23	,	,	PUNCT
cana-1800	61	24	rmax	rmax	X
cana-1800	61	25	(	(	PUNCT
cana-1800	61	26	−	−	PROPN
cana-1800	61	27	av	av	PROPN
cana-1800	61	28	(	(	PUNCT
cana-1800	61	29	u	u	NOUN
cana-1800	61	30	)	)	PUNCT
cana-1800	61	31	,	,	PUNCT
cana-1800	61	32	−	−	PROPN
cana-1800	61	33	bv	bv	PROPN
cana-1800	61	34	(	(	PUNCT
cana-1800	61	35	u	u	NOUN
cana-1800	61	36	)	)	PUNCT
cana-1800	61	37	)	)	PUNCT
cana-1800	62	1			PROPN
cana-1800	62	2	/	/	SYM
cana-1800	62	3	ux	ux	PUNCT
cana-1800	62	4	}	}	PUNCT
cana-1800	62	5	.	.	PUNCT
cana-1800	63	1	(	(	PUNCT
cana-1800	63	2	iv	iv	X
cana-1800	63	3	)	)	PUNCT
cana-1800	64	1	[	[	X
cana-1800	64	2	a][b	a][b	X
cana-1800	64	3	]	]	X
cana-1800	64	4	=	=	X
cana-1800	64	5	{	{	PUNCT
cana-1800	64	6			NOUN
cana-1800	64	7	u	u	NOUN
cana-1800	64	8	,	,	PUNCT
cana-1800	64	9	rmax	rmax	ADJ
cana-1800	64	10	(	(	PUNCT
cana-1800	64	11	+	+	X
cana-1800	64	12	av	av	PROPN
cana-1800	64	13	(	(	PUNCT
cana-1800	64	14	u	u	NOUN
cana-1800	64	15	)	)	PUNCT
cana-1800	64	16	,	,	PUNCT
cana-1800	64	17	+	+	CCONJ
cana-1800	64	18	bv	bv	PROPN
cana-1800	64	19	(	(	PUNCT
cana-1800	64	20	u	u	NOUN
cana-1800	64	21	)	)	PUNCT
cana-1800	64	22	)	)	PUNCT
cana-1800	64	23	,	,	PUNCT
cana-1800	64	24	rmin	rmin	NOUN
cana-1800	64	25	(	(	PUNCT
cana-1800	64	26	−	−	PROPN
cana-1800	64	27	av	av	PROPN
cana-1800	64	28	(	(	PUNCT
cana-1800	64	29	u	u	NOUN
cana-1800	64	30	)	)	PUNCT
cana-1800	64	31	,	,	PUNCT
cana-1800	64	32	−	−	PROPN
cana-1800	64	33	bv	bv	PROPN
cana-1800	64	34	(	(	PUNCT
cana-1800	64	35	u))	u))	PROPN
cana-1800	64	36	/	/	SYM
cana-1800	64	37	ux	ux	PRON
cana-1800	64	38	}	}	PUNCT
cana-1800	64	39	.	.	PUNCT
cana-1800	65	1	here	here	ADV
cana-1800	65	2	rmin	rmin	VERB
cana-1800	65	3	(	(	PUNCT
cana-1800	65	4	+	+	CCONJ
cana-1800	65	5	av	av	PROPN
cana-1800	65	6	(	(	PUNCT
cana-1800	65	7	u	u	NOUN
cana-1800	65	8	)	)	PUNCT
cana-1800	65	9	,	,	PUNCT
cana-1800	65	10	+	+	CCONJ
cana-1800	65	11	bv	bv	PROPN
cana-1800	65	12	(	(	PUNCT
cana-1800	65	13	u	u	NOUN
cana-1800	65	14	)	)	PUNCT
cana-1800	65	15	)	)	PUNCT
cana-1800	66	1	=	=	PUNCT
cana-1800	66	2	[	[	PUNCT
cana-1800	66	3	min	min	NOUN
cana-1800	66	4	{	{	PUNCT
cana-1800	66	5	)	)	PUNCT
cana-1800	66	6	,	,	PUNCT
cana-1800	66	7	(	(	PUNCT
cana-1800	66	8	xt	xt	ADP
cana-1800	66	9	a	a	DET
cana-1800	66	10	+	+	NOUN
cana-1800	66	11	)	)	PUNCT
cana-1800	66	12	(	(	PUNCT
cana-1800	66	13	xtb	xtb	X
cana-1800	67	1	+	+	PUNCT
cana-1800	67	2	}	}	PUNCT
cana-1800	67	3	,	,	PUNCT
cana-1800	67	4	min	min	X
cana-1800	67	5	{	{	PUNCT
cana-1800	67	6	1−	1−	NUM
cana-1800	67	7	)	)	PUNCT
cana-1800	67	8	(	(	PUNCT
cana-1800	67	9	xf	xf	PROPN
cana-1800	67	10	a	a	PROPN
cana-1800	67	11	+	+	X
cana-1800	67	12	,	,	PUNCT
cana-1800	67	13	1−	1−	NUM
cana-1800	67	14	)	)	PUNCT
cana-1800	67	15	(	(	PUNCT
cana-1800	67	16	xf	xf	PROPN
cana-1800	67	17	b	b	PROPN
cana-1800	67	18	+	+	CCONJ
cana-1800	67	19	}	}	PUNCT
cana-1800	67	20	]	]	PUNCT
cana-1800	67	21	,	,	PUNCT
cana-1800	67	22	rmax	rmax	X
cana-1800	67	23	(	(	PUNCT
cana-1800	67	24	+	+	CCONJ
cana-1800	67	25	av	av	PROPN
cana-1800	67	26	(	(	PUNCT
cana-1800	67	27	u	u	NOUN
cana-1800	67	28	)	)	PUNCT
cana-1800	67	29	,	,	PUNCT
cana-1800	67	30	+	+	CCONJ
cana-1800	67	31	bv	bv	PROPN
cana-1800	67	32	(	(	PUNCT
cana-1800	67	33	u	u	NOUN
cana-1800	67	34	)	)	PUNCT
cana-1800	67	35	)	)	PUNCT
cana-1800	68	1	=	=	PUNCT
cana-1800	69	1	[	[	X
cana-1800	69	2	max	max	X
cana-1800	69	3	{	{	PUNCT
cana-1800	69	4	)	)	PUNCT
cana-1800	69	5	,	,	PUNCT
cana-1800	69	6	(	(	PUNCT
cana-1800	69	7	xt	xt	ADP
cana-1800	69	8	a	a	DET
cana-1800	69	9	+	+	NOUN
cana-1800	69	10	)	)	PUNCT
cana-1800	69	11	(	(	PUNCT
cana-1800	69	12	xtb	xtb	X
cana-1800	69	13	+	+	PUNCT
cana-1800	69	14	}	}	PUNCT
cana-1800	69	15	,	,	PUNCT
cana-1800	69	16	max	max	PROPN
cana-1800	69	17	{	{	PUNCT
cana-1800	69	18	1−	1−	NUM
cana-1800	69	19	)	)	PUNCT
cana-1800	69	20	(	(	PUNCT
cana-1800	69	21	xf	xf	PROPN
cana-1800	69	22	a	a	PROPN
cana-1800	69	23	+	+	X
cana-1800	69	24	,	,	PUNCT
cana-1800	69	25	1−	1−	NUM
cana-1800	69	26	)	)	PUNCT
cana-1800	69	27	(	(	PUNCT
cana-1800	69	28	xf	xf	PROPN
cana-1800	69	29	b	b	PROPN
cana-1800	69	30	+	+	CCONJ
cana-1800	69	31	}	}	PUNCT
cana-1800	69	32	]	]	PUNCT
cana-1800	69	33	,	,	PUNCT
cana-1800	69	34	rmin	rmin	NOUN
cana-1800	69	35	(	(	PUNCT
cana-1800	69	36	−	−	PROPN
cana-1800	69	37	av	av	PROPN
cana-1800	69	38	(	(	PUNCT
cana-1800	69	39	u	u	NOUN
cana-1800	69	40	)	)	PUNCT
cana-1800	69	41	,	,	PUNCT
cana-1800	69	42	−	−	PROPN
cana-1800	69	43	bv	bv	PROPN
cana-1800	69	44	(	(	PUNCT
cana-1800	69	45	u	u	NOUN
cana-1800	69	46	)	)	PUNCT
cana-1800	69	47	)	)	PUNCT
cana-1800	70	1	=	=	PUNCT
cana-1800	71	1	[	[	X
cana-1800	71	2	min	min	X
cana-1800	71	3	{	{	PUNCT
cana-1800	71	4	−1−	−1−	NOUN
cana-1800	71	5	)	)	PUNCT
cana-1800	71	6	(	(	PUNCT
cana-1800	71	7	xf	xf	PROPN
cana-1800	71	8	a	a	DET
cana-1800	71	9	−	−	PROPN
cana-1800	71	10	,	,	PUNCT
cana-1800	71	11	−1−	−1−	PROPN
cana-1800	71	12	)	)	PUNCT
cana-1800	71	13	(	(	PUNCT
cana-1800	71	14	xf	xf	PROPN
cana-1800	71	15	b	b	PROPN
cana-1800	71	16	−	−	PROPN
cana-1800	71	17	}	}	PUNCT
cana-1800	71	18	,	,	PUNCT
cana-1800	71	19	min	min	NOUN
cana-1800	71	20	{	{	PUNCT
cana-1800	71	21	)	)	PUNCT
cana-1800	71	22	,	,	PUNCT
cana-1800	71	23	(	(	PUNCT
cana-1800	71	24	xt	xt	ADP
cana-1800	71	25	a	a	DET
cana-1800	71	26	−	−	NOUN
cana-1800	71	27	)	)	PUNCT
cana-1800	71	28	(	(	PUNCT
cana-1800	71	29	xtb	xtb	NUM
cana-1800	71	30	−	−	NOUN
cana-1800	71	31	}	}	PUNCT
cana-1800	71	32	]	]	PUNCT
cana-1800	71	33	,	,	PUNCT
cana-1800	71	34	rmax	rmax	X
cana-1800	71	35	(	(	PUNCT
cana-1800	71	36	−	−	PROPN
cana-1800	71	37	av	av	PROPN
cana-1800	71	38	(	(	PUNCT
cana-1800	71	39	u	u	NOUN
cana-1800	71	40	)	)	PUNCT
cana-1800	71	41	,	,	PUNCT
cana-1800	71	42	−	−	PROPN
cana-1800	71	43	bv	bv	PROPN
cana-1800	71	44	(	(	PUNCT
cana-1800	71	45	u	u	NOUN
cana-1800	71	46	)	)	PUNCT
cana-1800	71	47	)	)	PUNCT
cana-1800	72	1	=	=	PUNCT
cana-1800	73	1	[	[	X
cana-1800	73	2	max	max	X
cana-1800	73	3	{	{	PUNCT
cana-1800	73	4	−1−	−1−	PROPN
cana-1800	73	5	)	)	PUNCT
cana-1800	73	6	(	(	PUNCT
cana-1800	73	7	xf	xf	PROPN
cana-1800	73	8	a	a	DET
cana-1800	73	9	−	−	PROPN
cana-1800	73	10	,	,	PUNCT
cana-1800	73	11	−1−	−1−	PROPN
cana-1800	73	12	)	)	PUNCT
cana-1800	73	13	(	(	PUNCT
cana-1800	73	14	xf	xf	PROPN
cana-1800	73	15	b	b	PROPN
cana-1800	73	16	−	−	PROPN
cana-1800	73	17	}	}	PUNCT
cana-1800	73	18	,	,	PUNCT
cana-1800	73	19	max	max	PROPN
cana-1800	73	20	{	{	PUNCT
cana-1800	73	21	)	)	PUNCT
cana-1800	73	22	,	,	PUNCT
cana-1800	73	23	(	(	PUNCT
cana-1800	73	24	xt	xt	ADP
cana-1800	73	25	a	a	DET
cana-1800	73	26	−	−	NOUN
cana-1800	73	27	)	)	PUNCT
cana-1800	73	28	(	(	PUNCT
cana-1800	73	29	xtb	xtb	NUM
cana-1800	73	30	−	−	NOUN
cana-1800	73	31	}	}	PUNCT
cana-1800	73	32	]	]	PUNCT
cana-1800	73	33	.	.	PUNCT
cana-1800	74	1	definition	definition	NOUN
cana-1800	74	2	2.9	2.9	NUM
cana-1800	74	3	.	.	PUNCT
cana-1800	75	1	a	a	DET
cana-1800	75	2	𝔹𝕍𝕍𝕊𝕊	𝔹𝕍𝕍𝕊𝕊	PROPN
cana-1800	75	3	𝔈	𝔈	NOUN
cana-1800	75	4	=	=	PUNCT
cana-1800	75	5	〈	〈	PROPN
cana-1800	75	6	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	75	7	+	+	PROPN
cana-1800	75	8	,	,	PUNCT
cana-1800	75	9	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	75	10	−	−	PROPN
cana-1800	75	11	〉	〉	NOUN
cana-1800	75	12	𝑜𝑓	𝑜𝑓	ADP
cana-1800	75	13	𝑎	𝑎	NOUN
cana-1800	75	14	𝑠𝑒𝑚𝑖𝑟𝑖𝑛𝑔	𝑠𝑒𝑚𝑖𝑟𝑖𝑛𝑔	NOUN
cana-1800	75	15	ℝ	ℝ	PROPN
cana-1800	75	16	is	be	AUX
cana-1800	75	17	said	say	VERB
cana-1800	75	18	to	to	PART
cana-1800	75	19	be	be	AUX
cana-1800	75	20	a	a	DET
cana-1800	75	21	bipolar	bipolar	ADJ
cana-1800	75	22	valued	value	VERB
cana-1800	75	23	vague	vague	ADJ
cana-1800	75	24	subsemiring	subsemire	VERB
cana-1800	75	25	(	(	PUNCT
cana-1800	75	26	𝔹𝕍𝕍𝕊𝕊ℝ	𝔹𝕍𝕍𝕊𝕊ℝ	PROPN
cana-1800	75	27	)	)	PUNCT
cana-1800	76	1	𝑜𝑓	𝑜𝑓	ADP
cana-1800	76	2	ℝ	ℝ	PROPN
cana-1800	76	3	if	if	SCONJ
cana-1800	76	4	(	(	PUNCT
cana-1800	76	5	i	i	NOUN
cana-1800	76	6	)	)	PUNCT
cana-1800	76	7	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	76	8	+	+	PROPN
cana-1800	76	9	(	(	PUNCT
cana-1800	76	10	𝔳	𝔳	PROPN
cana-1800	76	11	+	+	NUM
cana-1800	76	12	𝔥	𝔥	NOUN
cana-1800	76	13	)	)	PUNCT
cana-1800	76	14			NUM
cana-1800	76	15	rmin	rmin	NOUN
cana-1800	76	16	{	{	PUNCT
cana-1800	76	17	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	76	18	+	+	PROPN
cana-1800	76	19	(	(	PUNCT
cana-1800	76	20	𝔳	𝔳	NOUN
cana-1800	76	21	)	)	PUNCT
cana-1800	76	22	,	,	PUNCT
cana-1800	76	23	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	76	24	+	+	PROPN
cana-1800	76	25	(	(	PUNCT
cana-1800	76	26	𝔥	𝔥	NOUN
cana-1800	76	27	)	)	PUNCT
cana-1800	76	28	}	}	PUNCT
cana-1800	76	29	,	,	PUNCT
cana-1800	76	30	communications	communication	NOUN
cana-1800	76	31	on	on	ADP
cana-1800	76	32	applied	apply	VERB
cana-1800	76	33	nonlinear	nonlinear	ADJ
cana-1800	76	34	analysis	analysis	NOUN
cana-1800	76	35	issn	issn	NOUN
cana-1800	76	36	:	:	PUNCT
cana-1800	76	37	1074	1074	NUM
cana-1800	76	38	-	-	PUNCT
cana-1800	76	39	133x	133x	NUM
cana-1800	76	40	vol	vol	NOUN
cana-1800	76	41	32	32	NUM
cana-1800	76	42	no	no	NOUN
cana-1800	76	43	.	.	NOUN
cana-1800	76	44	2	2	NUM
cana-1800	76	45	(	(	PUNCT
cana-1800	76	46	2025	2025	NUM
cana-1800	76	47	)	)	PUNCT
cana-1800	76	48	509	509	NUM
cana-1800	76	49	https://internationalpubls.com	https://internationalpubls.com	X
cana-1800	76	50	(	(	PUNCT
cana-1800	76	51	ii	ii	NOUN
cana-1800	76	52	)	)	PUNCT
cana-1800	77	1	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	77	2	+	+	PROPN
cana-1800	77	3	(	(	PUNCT
cana-1800	77	4	𝔳𝔥	𝔳𝔥	NOUN
cana-1800	77	5	)	)	PUNCT
cana-1800	77	6			PROPN
cana-1800	77	7	rmin	rmin	NOUN
cana-1800	77	8	{	{	PUNCT
cana-1800	77	9	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	77	10	+	+	PROPN
cana-1800	77	11	(	(	PUNCT
cana-1800	77	12	𝔳	𝔳	NOUN
cana-1800	77	13	)	)	PUNCT
cana-1800	77	14	,	,	PUNCT
cana-1800	77	15	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	77	16	+	+	PROPN
cana-1800	77	17	(	(	PUNCT
cana-1800	77	18	𝔥	𝔥	NOUN
cana-1800	77	19	)	)	PUNCT
cana-1800	77	20	}	}	PUNCT
cana-1800	77	21	,	,	PUNCT
cana-1800	77	22	(	(	PUNCT
cana-1800	77	23	iii	iii	X
cana-1800	77	24	)	)	PUNCT
cana-1800	77	25	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	77	26	−(𝔳	−(𝔳	NOUN
cana-1800	77	27	+	+	CCONJ
cana-1800	77	28	𝔥	𝔥	NOUN
cana-1800	77	29	)	)	PUNCT
cana-1800	77	30			NOUN
cana-1800	77	31	rmax	rmax	ADJ
cana-1800	77	32	{	{	PUNCT
cana-1800	77	33	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	77	34	−(𝔳	−(𝔳	NOUN
cana-1800	77	35	)	)	PUNCT
cana-1800	77	36	,	,	PUNCT
cana-1800	77	37	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	77	38	−(𝔥	−(𝔥	NOUN
cana-1800	77	39	)	)	PUNCT
cana-1800	77	40	}	}	PUNCT
cana-1800	77	41	,	,	PUNCT
cana-1800	77	42	(	(	PUNCT
cana-1800	77	43	iv	iv	X
cana-1800	77	44	)	)	PUNCT
cana-1800	77	45	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	77	46	−(𝔳𝔥)	−(𝔳𝔥)	PROPN
cana-1800	77	47	rmax	rmax	NOUN
cana-1800	77	48	{	{	PUNCT
cana-1800	77	49	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	77	50	−(𝔳	−(𝔳	NOUN
cana-1800	77	51	)	)	PUNCT
cana-1800	77	52	,	,	PUNCT
cana-1800	77	53	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	77	54	−(𝔥	−(𝔥	NOUN
cana-1800	77	55	)	)	PUNCT
cana-1800	77	56	}	}	PUNCT
cana-1800	77	57	,	,	PUNCT
cana-1800	77	58	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1800	77	59	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-1800	77	60	𝔳	𝔳	PROPN
cana-1800	77	61	,	,	PUNCT
cana-1800	77	62	𝔥	𝔥	PROPN
cana-1800	77	63	∈	∈	PROPN
cana-1800	77	64	ℝ.	ℝ.	PROPN
cana-1800	77	65	example	example	NOUN
cana-1800	77	66	2.10	2.10	NUM
cana-1800	77	67	.	.	PUNCT
cana-1800	78	1	let	let	VERB
cana-1800	78	2	r	r	NOUN
cana-1800	78	3	=	=	SYM
cana-1800	78	4	z3	z3	PROPN
cana-1800	78	5	=	=	PUNCT
cana-1800	78	6	{	{	PUNCT
cana-1800	78	7	0	0	NUM
cana-1800	78	8	,	,	PUNCT
cana-1800	78	9	1	1	NUM
cana-1800	78	10	,	,	PUNCT
cana-1800	78	11	2	2	NUM
cana-1800	78	12	}	}	PUNCT
cana-1800	78	13	be	be	AUX
cana-1800	78	14	a	a	DET
cana-1800	78	15	semiring	semiring	NOUN
cana-1800	78	16	in	in	ADP
cana-1800	78	17	terms	term	NOUN
cana-1800	78	18	of	of	ADP
cana-1800	78	19	standard	standard	ADJ
cana-1800	78	20	addition	addition	NOUN
cana-1800	78	21	and	and	CCONJ
cana-1800	78	22	multiplication	multiplication	NOUN
cana-1800	78	23	.	.	PUNCT
cana-1800	79	1	then	then	ADV
cana-1800	79	2	a	a	PRON
cana-1800	79	3	=	=	X
cana-1800	79	4	{	{	PUNCT
cana-1800	79	5	<	<	NOUN
cana-1800	79	6	0	0	NUM
cana-1800	79	7	,	,	PUNCT
cana-1800	79	8	[	[	X
cana-1800	79	9	0.5	0.5	NUM
cana-1800	79	10	,	,	PUNCT
cana-1800	79	11	0.7	0.7	NUM
cana-1800	79	12	]	]	PUNCT
cana-1800	79	13	,	,	PUNCT
cana-1800	79	14	[	[	X
cana-1800	79	15	−	−	NOUN
cana-1800	79	16	0.8	0.8	NUM
cana-1800	79	17	,	,	PUNCT
cana-1800	79	18	−	−	PROPN
cana-1800	79	19	0.5	0.5	NUM
cana-1800	79	20	]	]	PUNCT
cana-1800	79	21	>	>	PUNCT
cana-1800	79	22	,	,	PUNCT
cana-1800	79	23	<	<	X
cana-1800	79	24	1	1	NUM
cana-1800	79	25	,	,	PUNCT
cana-1800	79	26	[	[	X
cana-1800	79	27	0.4	0.4	NUM
cana-1800	79	28	,	,	PUNCT
cana-1800	79	29	0.6	0.6	NUM
cana-1800	79	30	]	]	PUNCT
cana-1800	79	31	,	,	PUNCT
cana-1800	79	32	[	[	X
cana-1800	79	33	−0.7	−0.7	PROPN
cana-1800	79	34	,	,	PUNCT
cana-1800	79	35	−0.4	−0.4	NUM
cana-1800	79	36	]	]	X
cana-1800	79	37	>	>	X
cana-1800	79	38	,	,	PUNCT
cana-1800	79	39	<	<	X
cana-1800	79	40	2	2	NUM
cana-1800	79	41	,	,	PUNCT
cana-1800	79	42	[	[	X
cana-1800	79	43	0.4	0.4	NUM
cana-1800	79	44	,	,	PUNCT
cana-1800	79	45	0.6	0.6	NUM
cana-1800	79	46	]	]	PUNCT
cana-1800	79	47	,	,	PUNCT
cana-1800	79	48	[	[	X
cana-1800	79	49	−0.7	−0.7	PROPN
cana-1800	79	50	,	,	PUNCT
cana-1800	79	51	−0.4	−0.4	NUM
cana-1800	79	52	]	]	PUNCT
cana-1800	79	53	>	>	X
cana-1800	79	54	}	}	PUNCT
cana-1800	79	55	is	be	AUX
cana-1800	79	56	a	a	DET
cana-1800	79	57	bvvssr	bvvssr	NOUN
cana-1800	79	58	of	of	ADP
cana-1800	79	59	r.	r.	PROPN
cana-1800	79	60	3	3	PROPN
cana-1800	79	61	bipolar	bipolar	ADJ
cana-1800	79	62	valued	value	VERB
cana-1800	79	63	vague	vague	ADJ
cana-1800	79	64	ideals	ideal	NOUN
cana-1800	79	65	:	:	PUNCT
cana-1800	79	66	this	this	DET
cana-1800	79	67	section	section	NOUN
cana-1800	79	68	introduced	introduce	VERB
cana-1800	79	69	bipolar	bipolar	ADJ
cana-1800	79	70	valued	value	VERB
cana-1800	79	71	vague	vague	ADJ
cana-1800	79	72	ideals	ideal	NOUN
cana-1800	79	73	(	(	PUNCT
cana-1800	79	74	bvvis	bvvis	NOUN
cana-1800	79	75	)	)	PUNCT
cana-1800	79	76	and	and	CCONJ
cana-1800	79	77	looked	look	VERB
cana-1800	79	78	at	at	ADP
cana-1800	79	79	their	their	PRON
cana-1800	79	80	characteristics	characteristic	NOUN
cana-1800	79	81	.	.	PUNCT
cana-1800	80	1	definition	definition	NOUN
cana-1800	80	2	3.1	3.1	NUM
cana-1800	80	3	.	.	PUNCT
cana-1800	81	1	a	a	DET
cana-1800	81	2	𝔹𝕍𝕍𝕊𝕊	𝔹𝕍𝕍𝕊𝕊	PROPN
cana-1800	81	3	𝔈	𝔈	NOUN
cana-1800	81	4	=	=	PUNCT
cana-1800	81	5	〈	〈	PROPN
cana-1800	81	6	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	81	7	+	+	PROPN
cana-1800	81	8	,	,	PUNCT
cana-1800	81	9	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	81	10	−	−	PROPN
cana-1800	81	11	〉	〉	NOUN
cana-1800	81	12	𝑜𝑓	𝑜𝑓	ADP
cana-1800	81	13	𝑎	𝑎	NOUN
cana-1800	81	14	𝑠𝑒𝑚𝑖𝑟𝑖𝑛𝑔	𝑠𝑒𝑚𝑖𝑟𝑖𝑛𝑔	NOUN
cana-1800	81	15	ℝ	ℝ	PROPN
cana-1800	81	16	is	be	AUX
cana-1800	81	17	said	say	VERB
cana-1800	81	18	to	to	PART
cana-1800	81	19	be	be	AUX
cana-1800	81	20	a	a	DET
cana-1800	81	21	bipolar	bipolar	ADJ
cana-1800	81	22	valued	value	VERB
cana-1800	81	23	vague	vague	ADJ
cana-1800	81	24	ideal	ideal	NOUN
cana-1800	81	25	(	(	PUNCT
cana-1800	81	26	𝔹𝕍𝕍𝕀	𝔹𝕍𝕍𝕀	NOUN
cana-1800	81	27	)	)	PUNCT
cana-1800	81	28	𝑜𝑓	𝑜𝑓	ADP
cana-1800	81	29	ℝ	ℝ	PROPN
cana-1800	82	1	if	if	SCONJ
cana-1800	82	2	(	(	PUNCT
cana-1800	82	3	i	i	NOUN
cana-1800	82	4	)	)	PUNCT
cana-1800	82	5	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	82	6	+	+	PROPN
cana-1800	82	7	(	(	PUNCT
cana-1800	82	8	𝔳	𝔳	PROPN
cana-1800	82	9	+	+	NUM
cana-1800	82	10	𝔥	𝔥	NOUN
cana-1800	82	11	)	)	PUNCT
cana-1800	82	12			NUM
cana-1800	82	13	rmin	rmin	NOUN
cana-1800	82	14	{	{	PUNCT
cana-1800	82	15	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	82	16	+	+	PROPN
cana-1800	82	17	(	(	PUNCT
cana-1800	82	18	𝔳	𝔳	NOUN
cana-1800	82	19	)	)	PUNCT
cana-1800	82	20	,	,	PUNCT
cana-1800	82	21	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	82	22	+	+	PROPN
cana-1800	82	23	(	(	PUNCT
cana-1800	82	24	𝔥	𝔥	NOUN
cana-1800	82	25	)	)	PUNCT
cana-1800	82	26	}	}	PUNCT
cana-1800	82	27	,	,	PUNCT
cana-1800	82	28	(	(	PUNCT
cana-1800	82	29	ii	ii	NOUN
cana-1800	82	30	)	)	PUNCT
cana-1800	82	31	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	82	32	+	+	PROPN
cana-1800	82	33	(	(	PUNCT
cana-1800	82	34	𝔳𝔥	𝔳𝔥	NOUN
cana-1800	82	35	)	)	PUNCT
cana-1800	82	36			NUM
cana-1800	82	37	rmax	rmax	NOUN
cana-1800	82	38	{	{	PUNCT
cana-1800	82	39	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	82	40	+	+	PROPN
cana-1800	82	41	(	(	PUNCT
cana-1800	82	42	𝔳	𝔳	NOUN
cana-1800	82	43	)	)	PUNCT
cana-1800	82	44	,	,	PUNCT
cana-1800	82	45	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	82	46	+	+	PROPN
cana-1800	82	47	(	(	PUNCT
cana-1800	82	48	𝔥	𝔥	NOUN
cana-1800	82	49	)	)	PUNCT
cana-1800	82	50	}	}	PUNCT
cana-1800	82	51	,	,	PUNCT
cana-1800	82	52	(	(	PUNCT
cana-1800	82	53	iii	iii	X
cana-1800	82	54	)	)	PUNCT
cana-1800	82	55	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	82	56	−(𝔳	−(𝔳	NOUN
cana-1800	82	57	+	+	CCONJ
cana-1800	82	58	𝔥	𝔥	NOUN
cana-1800	82	59	)	)	PUNCT
cana-1800	82	60			NOUN
cana-1800	82	61	rmax	rmax	ADJ
cana-1800	82	62	{	{	PUNCT
cana-1800	82	63	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	82	64	−(𝔳	−(𝔳	NOUN
cana-1800	82	65	)	)	PUNCT
cana-1800	82	66	,	,	PUNCT
cana-1800	82	67	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	82	68	−(𝔥	−(𝔥	NOUN
cana-1800	82	69	)	)	PUNCT
cana-1800	82	70	}	}	PUNCT
cana-1800	82	71	,	,	PUNCT
cana-1800	82	72	(	(	PUNCT
cana-1800	82	73	iv	iv	X
cana-1800	82	74	)	)	PUNCT
cana-1800	82	75	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	82	76	−(𝔳𝔥)	−(𝔳𝔥)	PROPN
cana-1800	82	77	rmin	rmin	NOUN
cana-1800	82	78	{	{	PUNCT
cana-1800	82	79	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	82	80	−(𝔳	−(𝔳	NOUN
cana-1800	82	81	)	)	PUNCT
cana-1800	82	82	,	,	PUNCT
cana-1800	82	83	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	82	84	−(𝔥	−(𝔥	NOUN
cana-1800	82	85	)	)	PUNCT
cana-1800	82	86	}	}	PUNCT
cana-1800	82	87	,	,	PUNCT
cana-1800	82	88	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1800	82	89	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-1800	82	90	𝔳	𝔳	PROPN
cana-1800	82	91	,	,	PUNCT
cana-1800	82	92	𝔥	𝔥	PROPN
cana-1800	82	93	∈	∈	PROPN
cana-1800	82	94	ℝ.	ℝ.	PROPN
cana-1800	82	95	theorem	theorem	VERB
cana-1800	82	96	3.2	3.2	NUM
cana-1800	82	97	.	.	PUNCT
cana-1800	83	1	let	let	VERB
cana-1800	83	2	𝔈	𝔈	PROPN
cana-1800	83	3	=	=	PUNCT
cana-1800	83	4	〈	〈	PROPN
cana-1800	83	5	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	83	6	+	+	PROPN
cana-1800	83	7	,	,	PUNCT
cana-1800	83	8	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	83	9	−	−	PROPN
cana-1800	83	10	〉	〉	NOUN
cana-1800	83	11	be	be	VERB
cana-1800	83	12	a	a	DET
cana-1800	83	13	𝔹𝕍𝕍𝕀	𝔹𝕍𝕍𝕀	NOUN
cana-1800	83	14	of	of	ADP
cana-1800	83	15	a	a	DET
cana-1800	83	16	semiring	semire	VERB
cana-1800	83	17	ℝ.	ℝ.	PROPN
cana-1800	83	18	(	(	PUNCT
cana-1800	83	19	i	i	NOUN
cana-1800	83	20	)	)	PUNCT
cana-1800	83	21	if	if	SCONJ
cana-1800	83	22	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	83	23	+	+	PROPN
cana-1800	83	24	(	(	PUNCT
cana-1800	83	25	𝔳	𝔳	PROPN
cana-1800	83	26	+	+	NUM
cana-1800	83	27	𝔥	𝔥	NOUN
cana-1800	83	28	)	)	PUNCT
cana-1800	83	29	=	=	PUNCT
cana-1800	84	1	[	[	X
cana-1800	84	2	0	0	X
cana-1800	84	3	]	]	PUNCT
cana-1800	84	4	then	then	ADV
cana-1800	84	5	either	either	CCONJ
cana-1800	84	6	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	84	7	+	+	PROPN
cana-1800	84	8	(	(	PUNCT
cana-1800	84	9	𝔳)=	𝔳)=	NOUN
cana-1800	85	1	[	[	X
cana-1800	85	2	0	0	X
cana-1800	85	3	]	]	PUNCT
cana-1800	85	4	or	or	CCONJ
cana-1800	85	5	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	85	6	+	+	PROPN
cana-1800	85	7	(	(	PUNCT
cana-1800	85	8	𝔥	𝔥	NOUN
cana-1800	85	9	)	)	PUNCT
cana-1800	85	10	=	=	PUNCT
cana-1800	86	1	[	[	X
cana-1800	86	2	0	0	X
cana-1800	86	3	]	]	X
cana-1800	86	4	for	for	ADP
cana-1800	86	5	𝔳	𝔳	PROPN
cana-1800	86	6	,	,	PUNCT
cana-1800	86	7	𝔥	𝔥	PROPN
cana-1800	86	8	∈	∈	PROPN
cana-1800	86	9	ℝ	ℝ	PROPN
cana-1800	86	10	(	(	PUNCT
cana-1800	86	11	ii	ii	NOUN
cana-1800	86	12	)	)	PUNCT
cana-1800	86	13	if	if	SCONJ
cana-1800	86	14	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	86	15	+	+	PROPN
cana-1800	86	16	(	(	PUNCT
cana-1800	86	17	𝔳𝔥	𝔳𝔥	NOUN
cana-1800	86	18	)	)	PUNCT
cana-1800	86	19	=	=	PUNCT
cana-1800	87	1	[	[	X
cana-1800	87	2	0	0	X
cana-1800	87	3	]	]	PUNCT
cana-1800	87	4	then	then	ADV
cana-1800	87	5	either	either	CCONJ
cana-1800	87	6	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	87	7	+	+	PROPN
cana-1800	87	8	(	(	PUNCT
cana-1800	87	9	𝔳	𝔳	PROPN
cana-1800	87	10	)	)	PUNCT
cana-1800	87	11	=	=	PUNCT
cana-1800	88	1	[	[	X
cana-1800	88	2	0	0	X
cana-1800	88	3	]	]	PUNCT
cana-1800	88	4	or	or	CCONJ
cana-1800	88	5	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	88	6	+	+	PROPN
cana-1800	88	7	(	(	PUNCT
cana-1800	88	8	𝔥)=	𝔥)=	ADJ
cana-1800	88	9	[	[	X
cana-1800	88	10	0	0	X
cana-1800	88	11	]	]	PUNCT
cana-1800	88	12	for	for	ADP
cana-1800	88	13	𝔳	𝔳	PROPN
cana-1800	88	14	,	,	PUNCT
cana-1800	88	15	𝔥	𝔥	PROPN
cana-1800	88	16	∈	∈	PROPN
cana-1800	88	17	ℝ	ℝ	PROPN
cana-1800	88	18	(	(	PUNCT
cana-1800	88	19	iii	iii	NOUN
cana-1800	88	20	)	)	PUNCT
cana-1800	88	21	if	if	SCONJ
cana-1800	88	22	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	88	23	−(𝔳	−(𝔳	NOUN
cana-1800	88	24	+	+	CCONJ
cana-1800	88	25	𝔥)=	𝔥)=	ADJ
cana-1800	88	26	[	[	X
cana-1800	88	27	0	0	X
cana-1800	88	28	]	]	PUNCT
cana-1800	88	29	then	then	ADV
cana-1800	88	30	either	either	CCONJ
cana-1800	88	31	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	88	32	−(𝔳	−(𝔳	NOUN
cana-1800	88	33	)	)	PUNCT
cana-1800	88	34	=	=	PUNCT
cana-1800	89	1	[	[	X
cana-1800	89	2	0	0	X
cana-1800	89	3	]	]	PUNCT
cana-1800	89	4	or	or	CCONJ
cana-1800	89	5	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	89	6	−(𝔥	−(𝔥	NOUN
cana-1800	89	7	)	)	PUNCT
cana-1800	89	8	=	=	PUNCT
cana-1800	90	1	[	[	X
cana-1800	90	2	0	0	X
cana-1800	90	3	]	]	X
cana-1800	90	4	for	for	ADP
cana-1800	90	5	𝔳	𝔳	PROPN
cana-1800	90	6	,	,	PUNCT
cana-1800	90	7	𝔥	𝔥	PROPN
cana-1800	90	8	∈	∈	PROPN
cana-1800	90	9	ℝ	ℝ	PROPN
cana-1800	90	10	(	(	PUNCT
cana-1800	90	11	iv	iv	X
cana-1800	90	12	)	)	PUNCT
cana-1800	90	13	if	if	SCONJ
cana-1800	90	14	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	90	15	−(𝔳𝔥	−(𝔳𝔥	PROPN
cana-1800	90	16	)	)	PUNCT
cana-1800	90	17	=	=	PUNCT
cana-1800	91	1	[	[	X
cana-1800	91	2	0	0	X
cana-1800	91	3	]	]	PUNCT
cana-1800	91	4	then	then	ADV
cana-1800	91	5	either	either	CCONJ
cana-1800	91	6	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	91	7	−(𝔳	−(𝔳	NOUN
cana-1800	91	8	)	)	PUNCT
cana-1800	91	9	=	=	PUNCT
cana-1800	92	1	[	[	X
cana-1800	92	2	0	0	X
cana-1800	92	3	]	]	PUNCT
cana-1800	92	4	or	or	CCONJ
cana-1800	92	5	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	92	6	−(𝔥	−(𝔥	NOUN
cana-1800	92	7	)	)	PUNCT
cana-1800	92	8	=	=	PUNCT
cana-1800	93	1	[	[	X
cana-1800	93	2	0	0	X
cana-1800	93	3	]	]	X
cana-1800	93	4	for	for	ADP
cana-1800	93	5	𝔳	𝔳	PROPN
cana-1800	93	6	,	,	PUNCT
cana-1800	93	7	𝔥	𝔥	PROPN
cana-1800	93	8	∈	∈	PROPN
cana-1800	93	9	ℝ	ℝ	PROPN
cana-1800	93	10	proof	proof	NOUN
cana-1800	93	11	.	.	PUNCT
cana-1800	94	1	let	let	VERB
cana-1800	94	2	𝔳	𝔳	NOUN
cana-1800	94	3	,	,	PUNCT
cana-1800	94	4	𝔥	𝔥	PROPN
cana-1800	94	5	∈	∈	PROPN
cana-1800	94	6	ℝ.	ℝ.	PROPN
cana-1800	94	7	(	(	PUNCT
cana-1800	94	8	i	i	NOUN
cana-1800	94	9	)	)	PUNCT
cana-1800	94	10	by	by	ADP
cana-1800	94	11	the	the	DET
cana-1800	94	12	definition	definition	NOUN
cana-1800	94	13	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	94	14	+	+	PROPN
cana-1800	94	15	(	(	PUNCT
cana-1800	94	16	𝔳	𝔳	PROPN
cana-1800	94	17	+	+	NUM
cana-1800	94	18	𝔥	𝔥	NOUN
cana-1800	94	19	)	)	PUNCT
cana-1800	94	20			X
cana-1800	94	21	rmin	rmin	VERB
cana-1800	94	22	{	{	PUNCT
cana-1800	94	23	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	94	24	+	+	PROPN
cana-1800	94	25	(	(	PUNCT
cana-1800	94	26	𝔳	𝔳	NOUN
cana-1800	94	27	)	)	PUNCT
cana-1800	94	28	,	,	PUNCT
cana-1800	94	29	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	94	30	+	+	PROPN
cana-1800	94	31	(	(	PUNCT
cana-1800	94	32	𝔥	𝔥	NOUN
cana-1800	94	33	)	)	PUNCT
cana-1800	94	34	}	}	PUNCT
cana-1800	94	35	which	which	PRON
cana-1800	94	36	implies	imply	VERB
cana-1800	94	37	that	that	SCONJ
cana-1800	94	38	[	[	X
cana-1800	94	39	0	0	NUM
cana-1800	94	40	]	]	X
cana-1800	94	41			X
cana-1800	94	42	rmin	rmin	PROPN
cana-1800	94	43	{	{	PUNCT
cana-1800	94	44	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	94	45	+	+	PROPN
cana-1800	94	46	(	(	PUNCT
cana-1800	94	47	𝔳	𝔳	NOUN
cana-1800	94	48	)	)	PUNCT
cana-1800	94	49	,	,	PUNCT
cana-1800	94	50	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	94	51	+	+	PROPN
cana-1800	94	52	(	(	PUNCT
cana-1800	94	53	𝔥	𝔥	NOUN
cana-1800	94	54	)	)	PUNCT
cana-1800	94	55	}	}	PUNCT
cana-1800	94	56	.	.	PUNCT
cana-1800	95	1	therefore	therefore	ADV
cana-1800	95	2	either	either	CCONJ
cana-1800	95	3	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	95	4	+	+	PROPN
cana-1800	95	5	(	(	PUNCT
cana-1800	95	6	𝔳	𝔳	PROPN
cana-1800	95	7	)	)	PUNCT
cana-1800	95	8	=	=	PUNCT
cana-1800	96	1	[	[	X
cana-1800	96	2	0	0	X
cana-1800	96	3	]	]	PUNCT
cana-1800	96	4	or	or	CCONJ
cana-1800	96	5	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	96	6	+	+	PROPN
cana-1800	96	7	(	(	PUNCT
cana-1800	96	8	𝔥	𝔥	NOUN
cana-1800	96	9	)	)	PUNCT
cana-1800	96	10	=	=	PUNCT
cana-1800	97	1	[	[	X
cana-1800	97	2	0	0	NUM
cana-1800	97	3	]	]	PUNCT
cana-1800	97	4	.	.	PUNCT
cana-1800	98	1	(	(	PUNCT
cana-1800	98	2	ii	ii	NOUN
cana-1800	98	3	)	)	PUNCT
cana-1800	98	4	by	by	ADP
cana-1800	98	5	the	the	DET
cana-1800	98	6	definition	definition	NOUN
cana-1800	98	7	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	98	8	+	+	PROPN
cana-1800	98	9	(	(	PUNCT
cana-1800	98	10	𝔳𝔥	𝔳𝔥	NOUN
cana-1800	98	11	)	)	PUNCT
cana-1800	98	12			NUM
cana-1800	98	13	rmax	rmax	NOUN
cana-1800	98	14	{	{	PUNCT
cana-1800	98	15	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	98	16	+	+	PROPN
cana-1800	98	17	(	(	PUNCT
cana-1800	98	18	𝔳	𝔳	NOUN
cana-1800	98	19	)	)	PUNCT
cana-1800	98	20	,	,	PUNCT
cana-1800	98	21	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	98	22	+	+	PROPN
cana-1800	98	23	(	(	PUNCT
cana-1800	98	24	𝔥	𝔥	NOUN
cana-1800	98	25	)	)	PUNCT
cana-1800	98	26	}	}	PUNCT
cana-1800	98	27	which	which	PRON
cana-1800	98	28	implies	imply	VERB
cana-1800	98	29	that	that	SCONJ
cana-1800	98	30	[	[	X
cana-1800	98	31	0	0	NUM
cana-1800	98	32	]	]	PUNCT
cana-1800	98	33			NUM
cana-1800	98	34	rmax	rmax	NOUN
cana-1800	98	35	{	{	PUNCT
cana-1800	98	36	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	98	37	+	+	PROPN
cana-1800	98	38	(	(	PUNCT
cana-1800	98	39	𝔳	𝔳	NOUN
cana-1800	98	40	)	)	PUNCT
cana-1800	98	41	,	,	PUNCT
cana-1800	98	42	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	98	43	+	+	PROPN
cana-1800	98	44	(	(	PUNCT
cana-1800	98	45	𝔥	𝔥	NOUN
cana-1800	98	46	)	)	PUNCT
cana-1800	98	47	}	}	PUNCT
cana-1800	98	48	.	.	PUNCT
cana-1800	99	1	therefore	therefore	ADV
cana-1800	99	2	either	either	CCONJ
cana-1800	99	3	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	99	4	+	+	PROPN
cana-1800	99	5	(	(	PUNCT
cana-1800	99	6	𝔳	𝔳	PROPN
cana-1800	99	7	)	)	PUNCT
cana-1800	99	8	=	=	PUNCT
cana-1800	100	1	[	[	X
cana-1800	100	2	0	0	X
cana-1800	100	3	]	]	PUNCT
cana-1800	100	4	or	or	CCONJ
cana-1800	100	5	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	100	6	+	+	PROPN
cana-1800	100	7	(	(	PUNCT
cana-1800	100	8	𝔥)=	𝔥)=	ADJ
cana-1800	100	9	[	[	X
cana-1800	100	10	0	0	NUM
cana-1800	100	11	]	]	PUNCT
cana-1800	100	12	.	.	PUNCT
cana-1800	101	1	(	(	PUNCT
cana-1800	101	2	iii	iii	NOUN
cana-1800	101	3	)	)	PUNCT
cana-1800	101	4	by	by	ADP
cana-1800	101	5	the	the	DET
cana-1800	101	6	definition	definition	NOUN
cana-1800	101	7	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	101	8	−(𝔳	−(𝔳	VERB
cana-1800	101	9	+	+	CCONJ
cana-1800	101	10	𝔥)≤	𝔥)≤	PROPN
cana-1800	101	11	rmax	rmax	NOUN
cana-1800	101	12	{	{	PUNCT
cana-1800	101	13	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	101	14	−(𝔳	−(𝔳	NOUN
cana-1800	101	15	)	)	PUNCT
cana-1800	101	16	,	,	PUNCT
cana-1800	101	17	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	101	18	−(𝔥	−(𝔥	NOUN
cana-1800	101	19	)	)	PUNCT
cana-1800	101	20	}	}	PUNCT
cana-1800	101	21	which	which	PRON
cana-1800	101	22	implies	imply	VERB
cana-1800	101	23	that	that	SCONJ
cana-1800	101	24	[	[	X
cana-1800	101	25	0	0	X
cana-1800	101	26	]	]	PUNCT
cana-1800	101	27	≤	≤	NUM
cana-1800	101	28	rmax	rmax	NOUN
cana-1800	101	29	{	{	PUNCT
cana-1800	101	30	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	101	31	−(𝔳	−(𝔳	NOUN
cana-1800	101	32	)	)	PUNCT
cana-1800	101	33	,	,	PUNCT
cana-1800	101	34	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	101	35	−(𝔥	−(𝔥	NOUN
cana-1800	101	36	)	)	PUNCT
cana-1800	101	37	}	}	PUNCT
cana-1800	101	38	.	.	PUNCT
cana-1800	102	1	therefore	therefore	ADV
cana-1800	102	2	either	either	CCONJ
cana-1800	102	3	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	102	4	−(𝔳	−(𝔳	NOUN
cana-1800	102	5	)	)	PUNCT
cana-1800	102	6	=	=	PUNCT
cana-1800	103	1	[	[	X
cana-1800	103	2	0	0	X
cana-1800	103	3	]	]	PUNCT
cana-1800	103	4	or	or	CCONJ
cana-1800	103	5	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	103	6	−(𝔥	−(𝔥	NOUN
cana-1800	103	7	)	)	PUNCT
cana-1800	103	8	=	=	PUNCT
cana-1800	104	1	[	[	X
cana-1800	104	2	0	0	X
cana-1800	104	3	]	]	PUNCT
cana-1800	104	4	.	.	PUNCT
cana-1800	105	1	(	(	PUNCT
cana-1800	105	2	iv	iv	X
cana-1800	105	3	)	)	PUNCT
cana-1800	105	4	by	by	ADP
cana-1800	105	5	the	the	DET
cana-1800	105	6	definition	definition	NOUN
cana-1800	105	7	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	105	8	−(𝔳𝔥	−(𝔳𝔥	PROPN
cana-1800	105	9	)	)	PUNCT
cana-1800	105	10	≤	≤	NUM
cana-1800	106	1	rmin	rmin	NOUN
cana-1800	106	2	{	{	PUNCT
cana-1800	106	3	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	106	4	−(𝔳	−(𝔳	NOUN
cana-1800	106	5	)	)	PUNCT
cana-1800	106	6	,	,	PUNCT
cana-1800	106	7	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	106	8	−(𝔥	−(𝔥	NOUN
cana-1800	106	9	)	)	PUNCT
cana-1800	106	10	}	}	PUNCT
cana-1800	106	11	which	which	PRON
cana-1800	106	12	implies	imply	VERB
cana-1800	106	13	that	that	SCONJ
cana-1800	107	1	[	[	X
cana-1800	107	2	0	0	X
cana-1800	107	3	]	]	X
cana-1800	107	4	≤	≤	NUM
cana-1800	107	5	rmin	rmin	NOUN
cana-1800	107	6	{	{	PUNCT
cana-1800	107	7	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	107	8	−(𝔳	−(𝔳	NOUN
cana-1800	107	9	)	)	PUNCT
cana-1800	107	10	,	,	PUNCT
cana-1800	107	11	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	107	12	−(𝔥	−(𝔥	NOUN
cana-1800	107	13	)	)	PUNCT
cana-1800	107	14	}	}	PUNCT
cana-1800	107	15	.	.	PUNCT
cana-1800	108	1	therefore	therefore	ADV
cana-1800	108	2	either	either	CCONJ
cana-1800	108	3	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	108	4	−(𝔳	−(𝔳	NOUN
cana-1800	108	5	)	)	PUNCT
cana-1800	108	6	=	=	PUNCT
cana-1800	109	1	[	[	X
cana-1800	109	2	0	0	X
cana-1800	109	3	]	]	PUNCT
cana-1800	109	4	or	or	CCONJ
cana-1800	109	5	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	109	6	−(𝔥	−(𝔥	NOUN
cana-1800	109	7	)	)	PUNCT
cana-1800	109	8	=	=	PUNCT
cana-1800	110	1	[	[	X
cana-1800	110	2	0	0	NUM
cana-1800	110	3	]	]	PUNCT
cana-1800	110	4	.	.	PUNCT
cana-1800	111	1	theorem	theorem	VERB
cana-1800	111	2	3.3	3.3	NUM
cana-1800	111	3	.	.	PUNCT
cana-1800	112	1	𝐼𝑓	𝐼𝑓	VERB
cana-1800	112	2	𝔈	𝔈	NOUN
cana-1800	112	3	=	=	SYM
cana-1800	112	4	〈	〈	PROPN
cana-1800	112	5	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	112	6	+	+	PROPN
cana-1800	112	7	,	,	PUNCT
cana-1800	112	8	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	112	9	−	−	PROPN
cana-1800	112	10	〉	〉	NOUN
cana-1800	112	11	𝑖𝑠	𝑖𝑠	ADP
cana-1800	112	12	𝑎	𝑎	DET
cana-1800	112	13	𝔹𝕍𝕍𝕀	𝔹𝕍𝕍𝕀	NOUN
cana-1800	112	14	𝑜𝑓	𝑜𝑓	ADP
cana-1800	112	15	𝑎	𝑎	NOUN
cana-1800	112	16	𝑠𝑒𝑚𝑖𝑟𝑖𝑛𝑔	𝑠𝑒𝑚𝑖𝑟𝑖𝑛𝑔	NOUN
cana-1800	112	17	ℜ	ℜ	PROPN
cana-1800	112	18	,	,	PUNCT
cana-1800	112	19	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	VERB
cana-1800	112	20	ℋ	ℋ	NOUN
cana-1800	112	21	=	=	SYM
cana-1800	112	22	{	{	PUNCT
cana-1800	112	23	𝔬ℜ	𝔬ℜ	NOUN
cana-1800	112	24	/	/	SYM
cana-1800	112	25	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	112	26	+	+	PROPN
cana-1800	112	27	(	(	PUNCT
cana-1800	112	28	𝔬	𝔬	NOUN
cana-1800	112	29	)	)	PUNCT
cana-1800	112	30	=	=	PUNCT
cana-1800	113	1	[	[	X
cana-1800	113	2	1	1	NUM
cana-1800	113	3	]	]	PUNCT
cana-1800	113	4	,	,	PUNCT
cana-1800	113	5	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	113	6	−(𝔬	−(𝔬	VERB
cana-1800	113	7	)	)	PUNCT
cana-1800	113	8	=	=	PUNCT
cana-1800	114	1	[	[	X
cana-1800	114	2	−1	−1	X
cana-1800	114	3	]	]	PUNCT
cana-1800	114	4	}	}	PUNCT
cana-1800	114	5	is	be	AUX
cana-1800	114	6	either	either	CCONJ
cana-1800	114	7	a	a	DET
cana-1800	114	8	subideal	subideal	NOUN
cana-1800	114	9	or	or	CCONJ
cana-1800	114	10	empty	empty	ADJ
cana-1800	114	11	of	of	ADP
cana-1800	114	12	ℜ.	ℜ.	PROPN
cana-1800	114	13	proof	proof	NOUN
cana-1800	114	14	.	.	PUNCT
cana-1800	115	1	there	there	PRON
cana-1800	115	2	is	be	VERB
cana-1800	115	3	no	no	DET
cana-1800	115	4	𝔬ℜ	𝔬ℜ	ADJ
cana-1800	115	5	𝑠𝑢𝑐ℎ	𝑠𝑢𝑐ℎ	NOUN
cana-1800	116	1	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	ADJ
cana-1800	116	2	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	116	3	+	+	PROPN
cana-1800	116	4	(	(	PUNCT
cana-1800	116	5	𝔬	𝔬	NOUN
cana-1800	116	6	)	)	PUNCT
cana-1800	116	7	=	=	PUNCT
cana-1800	117	1	[	[	X
cana-1800	117	2	1	1	X
cana-1800	117	3	]	]	PUNCT
cana-1800	117	4	and	and	CCONJ
cana-1800	117	5	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	117	6	−(𝔬	−(𝔬	VERB
cana-1800	117	7	)	)	PUNCT
cana-1800	117	8	=	=	PUNCT
cana-1800	118	1	[	[	X
cana-1800	118	2	−1	−1	NOUN
cana-1800	118	3	]	]	PUNCT
cana-1800	118	4	,	,	PUNCT
cana-1800	118	5	then	then	ADV
cana-1800	118	6	ℋ	ℋ	PROPN
cana-1800	118	7	is	be	AUX
cana-1800	118	8	empty	empty	ADJ
cana-1800	118	9	.	.	PUNCT
cana-1800	119	1	if	if	SCONJ
cana-1800	119	2	𝔬	𝔬	PROPN
cana-1800	119	3	and	and	CCONJ
cana-1800	119	4	𝔰	𝔰	PROPN
cana-1800	119	5	in	in	ADP
cana-1800	119	6	ℋ	ℋ	PROPN
cana-1800	119	7	,	,	PUNCT
cana-1800	119	8	then	then	ADV
cana-1800	119	9	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	119	10	+	+	PROPN
cana-1800	119	11	(	(	PUNCT
cana-1800	119	12	𝔬	𝔬	NOUN
cana-1800	119	13	+	+	NUM
cana-1800	119	14	𝔰	𝔰	NOUN
cana-1800	119	15	)	)	PUNCT
cana-1800	119	16			NUM
cana-1800	119	17	rmin	rmin	NOUN
cana-1800	119	18	{	{	PUNCT
cana-1800	119	19	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	119	20	+	+	PROPN
cana-1800	119	21	(	(	PUNCT
cana-1800	119	22	𝔬	𝔬	NOUN
cana-1800	119	23	)	)	PUNCT
cana-1800	119	24	,	,	PUNCT
cana-1800	119	25	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	119	26	+	+	PROPN
cana-1800	119	27	(	(	PUNCT
cana-1800	119	28	𝔰	𝔰	NOUN
cana-1800	119	29	)	)	PUNCT
cana-1800	119	30	}	}	PUNCT
cana-1800	119	31	=	=	PUNCT
cana-1800	119	32	rmin{[1	rmin{[1	PROPN
cana-1800	119	33	]	]	PUNCT
cana-1800	119	34	,	,	PUNCT
cana-1800	119	35	[	[	X
cana-1800	119	36	1	1	NUM
cana-1800	119	37	]	]	PUNCT
cana-1800	119	38	}	}	PUNCT
cana-1800	119	39	=	=	PUNCT
cana-1800	120	1	[	[	X
cana-1800	120	2	1	1	NUM
cana-1800	120	3	]	]	PUNCT
cana-1800	120	4	.	.	PUNCT
cana-1800	121	1	thus	thus	ADV
cana-1800	121	2	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	121	3	+	+	PROPN
cana-1800	121	4	(	(	PUNCT
cana-1800	121	5	𝔬+	𝔬+	NOUN
cana-1800	121	6	𝔰	𝔰	X
cana-1800	121	7	)	)	PUNCT
cana-1800	121	8	=[	=[	NOUN
cana-1800	121	9	1	1	NUM
cana-1800	121	10	]	]	PUNCT
cana-1800	121	11	.	.	PUNCT
cana-1800	122	1	and	and	CCONJ
cana-1800	122	2	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	122	3	+	+	PROPN
cana-1800	122	4	(	(	PUNCT
cana-1800	122	5	𝔬𝔰	𝔬𝔰	INTJ
cana-1800	122	6	)	)	PUNCT
cana-1800	122	7			NUM
cana-1800	122	8	rmax	rmax	NOUN
cana-1800	122	9	{	{	PUNCT
cana-1800	122	10	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	122	11	+	+	PROPN
cana-1800	122	12	(	(	PUNCT
cana-1800	122	13	𝔬	𝔬	NOUN
cana-1800	122	14	)	)	PUNCT
cana-1800	122	15	,	,	PUNCT
cana-1800	122	16	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	122	17	+	+	PROPN
cana-1800	122	18	(	(	PUNCT
cana-1800	122	19	𝔰	𝔰	NOUN
cana-1800	122	20	)	)	PUNCT
cana-1800	122	21	}	}	PUNCT
cana-1800	122	22	=	=	SYM
cana-1800	122	23	rmax{[1	rmax{[1	PROPN
cana-1800	122	24	]	]	PUNCT
cana-1800	122	25	,	,	PUNCT
cana-1800	122	26	[	[	X
cana-1800	122	27	1]}=	1]}=	NUM
cana-1800	122	28	[	[	X
cana-1800	122	29	1	1	NUM
cana-1800	122	30	]	]	PUNCT
cana-1800	122	31	.	.	PUNCT
cana-1800	123	1	so	so	ADV
cana-1800	123	2	,	,	PUNCT
cana-1800	123	3	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	123	4	+	+	PROPN
cana-1800	123	5	(	(	PUNCT
cana-1800	123	6	𝔬𝔰	𝔬𝔰	INTJ
cana-1800	123	7	)	)	PUNCT
cana-1800	123	8	=[	=[	NOUN
cana-1800	123	9	1	1	NUM
cana-1800	123	10	]	]	PUNCT
cana-1800	123	11	.	.	PUNCT
cana-1800	124	1	also	also	ADV
cana-1800	124	2	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	124	3	−(𝔬	−(𝔬	VERB
cana-1800	124	4	+	+	CCONJ
cana-1800	124	5	𝔰	𝔰	X
cana-1800	124	6	)	)	PUNCT
cana-1800	124	7			NOUN
cana-1800	124	8	rmax	rmax	ADJ
cana-1800	124	9	{	{	PUNCT
cana-1800	124	10	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	124	11	−(𝔬	−(𝔬	PROPN
cana-1800	124	12	)	)	PUNCT
cana-1800	124	13	,	,	PUNCT
cana-1800	124	14	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	124	15	−(𝔰)}=	−(𝔰)}=	PROPN
cana-1800	124	16	rmax{[−1],[−1	rmax{[−1],[−1	PROPN
cana-1800	124	17	]	]	PUNCT
cana-1800	124	18	}	}	PUNCT
cana-1800	124	19	=	=	PUNCT
cana-1800	125	1	[	[	X
cana-1800	125	2	−1	−1	NOUN
cana-1800	125	3	]	]	X
cana-1800	125	4	.	.	PUNCT
cana-1800	126	1	that	that	PRON
cana-1800	126	2	is	be	AUX
cana-1800	126	3	,	,	PUNCT
cana-1800	126	4	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	126	5	−(𝔬+	−(𝔬+	PROPN
cana-1800	126	6	𝔰	𝔰	NOUN
cana-1800	126	7	)	)	PUNCT
cana-1800	126	8	=	=	PUNCT
cana-1800	127	1	[	[	X
cana-1800	127	2	−1	−1	NOUN
cana-1800	127	3	]	]	PUNCT
cana-1800	127	4	.	.	PUNCT
cana-1800	128	1	and	and	CCONJ
cana-1800	128	2	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	128	3	−(𝔬𝔰	−(𝔬𝔰	PROPN
cana-1800	128	4	)	)	PUNCT
cana-1800	128	5			NOUN
cana-1800	128	6	rmin	rmin	VERB
cana-1800	128	7	{	{	PUNCT
cana-1800	128	8	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	128	9	−(𝔬	−(𝔬	PROPN
cana-1800	128	10	)	)	PUNCT
cana-1800	128	11	,	,	PUNCT
cana-1800	128	12	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	128	13	−(𝔰	−(𝔰	NOUN
cana-1800	128	14	)	)	PUNCT
cana-1800	128	15	}	}	PUNCT
cana-1800	129	1	=	=	SYM
cana-1800	129	2	rmin{[−1	rmin{[−1	X
cana-1800	129	3	]	]	X
cana-1800	129	4	,	,	PUNCT
cana-1800	130	1	[	[	X
cana-1800	130	2	−1	−1	NOUN
cana-1800	130	3	]	]	X
cana-1800	130	4	}	}	PUNCT
cana-1800	130	5	=	=	PUNCT
cana-1800	131	1	[	[	X
cana-1800	131	2	−1	−1	X
cana-1800	131	3	]	]	X
cana-1800	131	4	.	.	PUNCT
cana-1800	132	1	so	so	ADV
cana-1800	132	2	,	,	PUNCT
cana-1800	132	3	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	132	4	−(𝔬𝔰	−(𝔬𝔰	ADV
cana-1800	132	5	)	)	PUNCT
cana-1800	133	1	=	=	PUNCT
cana-1800	134	1	[	[	X
cana-1800	134	2	−1	−1	NOUN
cana-1800	134	3	]	]	X
cana-1800	134	4	.	.	PUNCT
cana-1800	135	1	that	that	PRON
cana-1800	135	2	is	be	AUX
cana-1800	135	3	𝔬+	𝔬+	ADJ
cana-1800	135	4	𝔰	𝔰	NOUN
cana-1800	135	5	,	,	PUNCT
cana-1800	135	6	𝔬𝔰	𝔬𝔰	ADJ
cana-1800	135	7	ℋ.	ℋ.	PROPN
cana-1800	135	8	hence	hence	ADV
cana-1800	135	9	ℋ	ℋ	PROPN
cana-1800	135	10	𝑖𝑠	𝑖𝑠	NOUN
cana-1800	135	11	𝑎	𝑎	DET
cana-1800	135	12	𝑠𝑢𝑖𝑑𝑒𝑎𝑙	𝑠𝑢𝑖𝑑𝑒𝑎𝑙	NOUN
cana-1800	135	13	𝑜𝑓	𝑜𝑓	INTJ
cana-1800	135	14	ℜ.	ℜ.	PROPN
cana-1800	135	15	communications	communication	NOUN
cana-1800	135	16	on	on	ADP
cana-1800	135	17	applied	apply	VERB
cana-1800	135	18	nonlinear	nonlinear	ADJ
cana-1800	135	19	analysis	analysis	NOUN
cana-1800	135	20	issn	issn	NOUN
cana-1800	135	21	:	:	PUNCT
cana-1800	135	22	1074	1074	NUM
cana-1800	135	23	-	-	PUNCT
cana-1800	135	24	133x	133x	NUM
cana-1800	135	25	vol	vol	NOUN
cana-1800	135	26	32	32	NUM
cana-1800	135	27	no	no	NOUN
cana-1800	135	28	.	.	NOUN
cana-1800	135	29	2	2	NUM
cana-1800	135	30	(	(	PUNCT
cana-1800	135	31	2025	2025	NUM
cana-1800	135	32	)	)	PUNCT
cana-1800	135	33	510	510	NUM
cana-1800	135	34	https://internationalpubls.com	https://internationalpubls.com	X
cana-1800	135	35	theorem	theorem	VERB
cana-1800	135	36	3.4	3.4	NUM
cana-1800	135	37	.	.	PUNCT
cana-1800	136	1	if	if	SCONJ
cana-1800	136	2	𝕰	𝕰	PROPN
cana-1800	136	3	=	=	SYM
cana-1800	136	4	〈	〈	PROPN
cana-1800	136	5	𝓥𝕰	𝓥𝕰	PROPN
cana-1800	136	6	+	+	ADP
cana-1800	136	7	,	,	PUNCT
cana-1800	136	8	𝓥𝕰	𝓥𝕰	PROPN
cana-1800	136	9	−	−	NUM
cana-1800	136	10	〉	〉	NOUN
cana-1800	136	11	and	and	CCONJ
cana-1800	136	12	𝕳	𝕳	NOUN
cana-1800	136	13	=	=	SYM
cana-1800	136	14	〈	〈	NOUN
cana-1800	136	15	𝓥𝕳	𝓥𝕳	NOUN
cana-1800	136	16	+	+	ADV
cana-1800	136	17	,	,	PUNCT
cana-1800	136	18	𝓥𝕳	𝓥𝕳	NOUN
cana-1800	136	19	−	−	NOUN
cana-1800	136	20	〉	〉	NOUN
cana-1800	136	21	are	be	AUX
cana-1800	136	22	two	two	NUM
cana-1800	136	23	𝔹𝕍𝕍𝕀s	𝔹𝕍𝕍𝕀	NOUN
cana-1800	136	24	of	of	ADP
cana-1800	136	25	a	a	DET
cana-1800	136	26	ring	ring	NOUN
cana-1800	136	27	r	r	NOUN
cana-1800	136	28	,	,	PUNCT
cana-1800	136	29	then	then	ADV
cana-1800	136	30	their	their	PRON
cana-1800	136	31	intersection	intersection	NOUN
cana-1800	136	32	𝕰	𝕰	NOUN
cana-1800	136	33			PUNCT
cana-1800	136	34	𝕳	𝕳	NOUN
cana-1800	136	35	is	be	AUX
cana-1800	136	36	a	a	DET
cana-1800	136	37	𝔹𝕍𝕍𝕀	𝔹𝕍𝕍𝕀	NOUN
cana-1800	136	38	of	of	ADP
cana-1800	136	39	ℝ	ℝ	PROPN
cana-1800	136	40	.	.	PUNCT
cana-1800	137	1	proof	proof	NOUN
cana-1800	137	2	.	.	PUNCT
cana-1800	138	1	let	let	VERB
cana-1800	138	2	ҫ	ҫ	PRON
cana-1800	138	3	=	=	SYM
cana-1800	138	4	𝔈	𝔈	PROPN
cana-1800	138	5			X
cana-1800	138	6	ℌ	ℌ	PROPN
cana-1800	138	7	and	and	CCONJ
cana-1800	138	8	let	let	VERB
cana-1800	138	9	𝔳	𝔳	ADP
cana-1800	138	10	,	,	PUNCT
cana-1800	138	11	𝔥	𝔥	PROPN
cana-1800	138	12	∈	∈	PROPN
cana-1800	138	13	ℝ.	ℝ.	PROPN
cana-1800	138	14	now	now	ADV
cana-1800	138	15	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	138	16	+	+	PROPN
cana-1800	138	17	(	(	PUNCT
cana-1800	138	18	𝔳	𝔳	PROPN
cana-1800	138	19	+	+	NUM
cana-1800	138	20	𝔥	𝔥	NOUN
cana-1800	138	21	)	)	PUNCT
cana-1800	138	22	=	=	VERB
cana-1800	138	23	rmin	rmin	NOUN
cana-1800	138	24	{	{	PUNCT
cana-1800	138	25	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	138	26	+	+	PROPN
cana-1800	138	27	(	(	PUNCT
cana-1800	138	28	𝔳	𝔳	PROPN
cana-1800	138	29	+	+	NUM
cana-1800	138	30	𝔥	𝔥	NOUN
cana-1800	138	31	)	)	PUNCT
cana-1800	138	32	,	,	PUNCT
cana-1800	138	33	𝒱ℌ	𝒱ℌ	PROPN
cana-1800	139	1	+	+	PROPN
cana-1800	139	2	(	(	PUNCT
cana-1800	139	3	𝔳	𝔳	PROPN
cana-1800	139	4	+	+	NUM
cana-1800	139	5	𝔥	𝔥	NOUN
cana-1800	139	6	)	)	PUNCT
cana-1800	139	7	}	}	PUNCT
cana-1800	139	8			X
cana-1800	139	9	rmin{rmin	rmin{rmin	PROPN
cana-1800	139	10	{	{	PUNCT
cana-1800	139	11	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	139	12	+	+	PROPN
cana-1800	139	13	(	(	PUNCT
cana-1800	139	14	𝔳	𝔳	NOUN
cana-1800	139	15	)	)	PUNCT
cana-1800	139	16	,	,	PUNCT
cana-1800	139	17	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	139	18	+	+	PROPN
cana-1800	139	19	(	(	PUNCT
cana-1800	139	20	𝔥	𝔥	NOUN
cana-1800	139	21	)	)	PUNCT
cana-1800	139	22	}	}	PUNCT
cana-1800	139	23	,	,	PUNCT
cana-1800	139	24	rmin	rmin	NOUN
cana-1800	139	25	{	{	PUNCT
cana-1800	139	26	𝒱ℌ	𝒱ℌ	PROPN
cana-1800	139	27	+	+	PROPN
cana-1800	139	28	(	(	PUNCT
cana-1800	139	29	𝔳	𝔳	NOUN
cana-1800	139	30	)	)	PUNCT
cana-1800	139	31	,	,	PUNCT
cana-1800	139	32	𝒱ℌ	𝒱ℌ	PROPN
cana-1800	139	33	+	+	PROPN
cana-1800	139	34	(	(	PUNCT
cana-1800	139	35	𝔥	𝔥	NOUN
cana-1800	139	36	)	)	PUNCT
cana-1800	139	37	}	}	PUNCT
cana-1800	139	38	}	}	PUNCT
cana-1800	139	39			X
cana-1800	139	40	rmin{rmin	rmin{rmin	PROPN
cana-1800	139	41	{	{	PUNCT
cana-1800	139	42	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	139	43	+	+	PROPN
cana-1800	139	44	(	(	PUNCT
cana-1800	139	45	𝔳	𝔳	NOUN
cana-1800	139	46	)	)	PUNCT
cana-1800	139	47	,	,	PUNCT
cana-1800	139	48	𝒱ℌ	𝒱ℌ	PROPN
cana-1800	139	49	+	+	ADJ
cana-1800	139	50	(	(	PUNCT
cana-1800	139	51	𝔳	𝔳	NOUN
cana-1800	139	52	)	)	PUNCT
cana-1800	139	53	}	}	PUNCT
cana-1800	139	54	,	,	PUNCT
cana-1800	139	55	rmin	rmin	NOUN
cana-1800	139	56	{	{	PUNCT
cana-1800	139	57	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	139	58	+	+	PROPN
cana-1800	139	59	(	(	PUNCT
cana-1800	139	60	𝔥	𝔥	NOUN
cana-1800	139	61	)	)	PUNCT
cana-1800	139	62	,	,	PUNCT
cana-1800	139	63	𝒱ℌ	𝒱ℌ	PROPN
cana-1800	139	64	+	+	PROPN
cana-1800	139	65	(	(	PUNCT
cana-1800	139	66	𝔥	𝔥	NOUN
cana-1800	139	67	)	)	PUNCT
cana-1800	139	68	}	}	PUNCT
cana-1800	139	69	}	}	PUNCT
cana-1800	139	70	=	=	SYM
cana-1800	139	71	rmin	rmin	NOUN
cana-1800	139	72	{	{	PUNCT
cana-1800	139	73	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	139	74	+	+	PROPN
cana-1800	139	75	(	(	PUNCT
cana-1800	139	76	𝔳	𝔳	NOUN
cana-1800	139	77	)	)	PUNCT
cana-1800	139	78	,	,	PUNCT
cana-1800	139	79	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	139	80	+	+	PROPN
cana-1800	139	81	(	(	PUNCT
cana-1800	139	82	𝔥	𝔥	NOUN
cana-1800	139	83	)	)	PUNCT
cana-1800	139	84	}	}	PUNCT
cana-1800	139	85	.	.	PUNCT
cana-1800	140	1	therefore	therefore	ADV
cana-1800	140	2	𝒱ҫ	𝒱ҫ	PRON
cana-1800	140	3	+	+	PROPN
cana-1800	140	4	(	(	PUNCT
cana-1800	140	5	𝔳	𝔳	PROPN
cana-1800	140	6	+	+	NUM
cana-1800	140	7	𝔥	𝔥	NOUN
cana-1800	140	8	)	)	PUNCT
cana-1800	140	9			NUM
cana-1800	140	10	rmin	rmin	NOUN
cana-1800	140	11	{	{	PUNCT
cana-1800	140	12	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	140	13	+	+	PROPN
cana-1800	140	14	(	(	PUNCT
cana-1800	140	15	𝔳	𝔳	NOUN
cana-1800	140	16	)	)	PUNCT
cana-1800	140	17	,	,	PUNCT
cana-1800	140	18	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	140	19	+	+	PROPN
cana-1800	140	20	(	(	PUNCT
cana-1800	140	21	𝔥	𝔥	NOUN
cana-1800	140	22	)	)	PUNCT
cana-1800	140	23	,	,	PUNCT
cana-1800	140	24	for	for	ADP
cana-1800	140	25	all	all	DET
cana-1800	140	26	𝔳	𝔳	NOUN
cana-1800	140	27	,	,	PUNCT
cana-1800	140	28	𝔥	𝔥	PROPN
cana-1800	140	29	∈	∈	PROPN
cana-1800	140	30	ℝ.	ℝ.	PROPN
cana-1800	140	31	and	and	CCONJ
cana-1800	140	32	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	140	33	+	+	PROPN
cana-1800	140	34	(	(	PUNCT
cana-1800	140	35	𝔳𝔥)=	𝔳𝔥)=	ADJ
cana-1800	140	36	rmin	rmin	NOUN
cana-1800	140	37	{	{	PUNCT
cana-1800	140	38	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	140	39	+	+	PROPN
cana-1800	140	40	(	(	PUNCT
cana-1800	140	41	𝔳𝔥	𝔳𝔥	NOUN
cana-1800	140	42	)	)	PUNCT
cana-1800	140	43	,	,	PUNCT
cana-1800	140	44	𝒱ℌ	𝒱ℌ	PROPN
cana-1800	140	45	+	+	NOUN
cana-1800	140	46	(	(	PUNCT
cana-1800	140	47	𝔳𝔥	𝔳𝔥	NOUN
cana-1800	140	48	)	)	PUNCT
cana-1800	140	49	}	}	PUNCT
cana-1800	140	50			NUM
cana-1800	140	51	rmin{rmax	rmin{rmax	NOUN
cana-1800	140	52	{	{	PUNCT
cana-1800	140	53	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	140	54	+	+	PROPN
cana-1800	140	55	(	(	PUNCT
cana-1800	140	56	𝔳	𝔳	NOUN
cana-1800	140	57	)	)	PUNCT
cana-1800	140	58	,	,	PUNCT
cana-1800	140	59	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	140	60	+	+	PROPN
cana-1800	140	61	(	(	PUNCT
cana-1800	140	62	𝔥	𝔥	NOUN
cana-1800	140	63	)	)	PUNCT
cana-1800	140	64	}	}	PUNCT
cana-1800	140	65	,	,	PUNCT
cana-1800	140	66	rmax	rmax	ADJ
cana-1800	140	67	{	{	PUNCT
cana-1800	140	68	𝒱ℌ	𝒱ℌ	PROPN
cana-1800	140	69	+	+	PROPN
cana-1800	140	70	(	(	PUNCT
cana-1800	140	71	𝔳	𝔳	NOUN
cana-1800	140	72	)	)	PUNCT
cana-1800	140	73	,	,	PUNCT
cana-1800	140	74	𝒱ℌ	𝒱ℌ	PROPN
cana-1800	140	75	+	+	PROPN
cana-1800	140	76	(	(	PUNCT
cana-1800	140	77	𝔥	𝔥	NOUN
cana-1800	140	78	)	)	PUNCT
cana-1800	140	79	}	}	PUNCT
cana-1800	140	80	}	}	PUNCT
cana-1800	140	81			NUM
cana-1800	140	82	rmax{rmin	rmax{rmin	NOUN
cana-1800	140	83	{	{	PUNCT
cana-1800	140	84	𝒱ℌ	𝒱ℌ	PROPN
cana-1800	140	85	+	+	PROPN
cana-1800	140	86	(	(	PUNCT
cana-1800	140	87	𝔳	𝔳	NOUN
cana-1800	140	88	)	)	PUNCT
cana-1800	140	89	,	,	PUNCT
cana-1800	140	90	𝒱ℌ	𝒱ℌ	PROPN
cana-1800	141	1	+	+	PROPN
cana-1800	141	2	(	(	PUNCT
cana-1800	141	3	𝔥	𝔥	NOUN
cana-1800	141	4	)	)	PUNCT
cana-1800	141	5	}	}	PUNCT
cana-1800	141	6	,	,	PUNCT
cana-1800	141	7	rmin	rmin	NOUN
cana-1800	141	8	{	{	PUNCT
cana-1800	141	9	𝒱ℌ	𝒱ℌ	PROPN
cana-1800	141	10	+	+	PROPN
cana-1800	141	11	(	(	PUNCT
cana-1800	141	12	𝔳	𝔳	NOUN
cana-1800	141	13	)	)	PUNCT
cana-1800	141	14	,	,	PUNCT
cana-1800	141	15	𝒱ℌ	𝒱ℌ	PROPN
cana-1800	141	16	+	+	PROPN
cana-1800	141	17	(	(	PUNCT
cana-1800	141	18	𝔥	𝔥	NOUN
cana-1800	141	19	)	)	PUNCT
cana-1800	141	20	}	}	PUNCT
cana-1800	141	21	}	}	PUNCT
cana-1800	141	22	=	=	SYM
cana-1800	141	23	rmax	rmax	ADJ
cana-1800	141	24	{	{	PUNCT
cana-1800	141	25	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	141	26	+	+	PROPN
cana-1800	141	27	(	(	PUNCT
cana-1800	141	28	𝔳	𝔳	NOUN
cana-1800	141	29	)	)	PUNCT
cana-1800	141	30	,	,	PUNCT
cana-1800	141	31	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	141	32	+	+	PROPN
cana-1800	141	33	(	(	PUNCT
cana-1800	141	34	𝔥	𝔥	NOUN
cana-1800	141	35	)	)	PUNCT
cana-1800	141	36	}	}	PUNCT
cana-1800	141	37	.	.	PUNCT
cana-1800	142	1	therefore	therefore	ADV
cana-1800	142	2	𝒱ҫ	𝒱ҫ	PRON
cana-1800	142	3	+	+	PROPN
cana-1800	142	4	(	(	PUNCT
cana-1800	142	5	𝔳𝔥	𝔳𝔥	NOUN
cana-1800	142	6	)	)	PUNCT
cana-1800	142	7			NUM
cana-1800	142	8	rmax	rmax	NOUN
cana-1800	142	9	{	{	PUNCT
cana-1800	142	10	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	142	11	+	+	PROPN
cana-1800	142	12	(	(	PUNCT
cana-1800	142	13	𝔳	𝔳	NOUN
cana-1800	142	14	)	)	PUNCT
cana-1800	142	15	,	,	PUNCT
cana-1800	142	16	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	142	17	+	+	PROPN
cana-1800	142	18	(	(	PUNCT
cana-1800	142	19	𝔥	𝔥	NOUN
cana-1800	142	20	)	)	PUNCT
cana-1800	142	21	}	}	PUNCT
cana-1800	142	22	,	,	PUNCT
cana-1800	142	23	for	for	ADP
cana-1800	142	24	all	all	DET
cana-1800	142	25	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	142	26	+	+	PROPN
cana-1800	142	27	(	(	PUNCT
cana-1800	142	28	𝔳	𝔳	PROPN
cana-1800	142	29	+	+	NUM
cana-1800	142	30	𝔥	𝔥	NOUN
cana-1800	142	31	)	)	PUNCT
cana-1800	142	32	.	.	PUNCT
cana-1800	143	1	also	also	ADV
cana-1800	143	2	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	143	3	−(𝔳	−(𝔳	VERB
cana-1800	143	4	+	+	CCONJ
cana-1800	143	5	𝔥	𝔥	NOUN
cana-1800	143	6	)	)	PUNCT
cana-1800	143	7	=	=	SYM
cana-1800	143	8	rmax	rmax	ADJ
cana-1800	143	9	{	{	PUNCT
cana-1800	143	10	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	143	11	−(𝔳	−(𝔳	NOUN
cana-1800	143	12	+	+	CCONJ
cana-1800	143	13	𝔥	𝔥	NOUN
cana-1800	143	14	)	)	PUNCT
cana-1800	143	15	,	,	PUNCT
cana-1800	143	16	𝒱ℌ	𝒱ℌ	PROPN
cana-1800	143	17	−(𝔳	−(𝔳	VERB
cana-1800	143	18	+	+	CCONJ
cana-1800	143	19	𝔥	𝔥	NOUN
cana-1800	143	20	)	)	PUNCT
cana-1800	143	21	}	}	PUNCT
cana-1800	143	22			NUM
cana-1800	143	23	rmax{rmax	rmax{rmax	NOUN
cana-1800	143	24	{	{	PUNCT
cana-1800	143	25	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	143	26	−(𝔳	−(𝔳	NOUN
cana-1800	143	27	)	)	PUNCT
cana-1800	143	28	,	,	PUNCT
cana-1800	143	29	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	143	30	−(𝔥	−(𝔥	NOUN
cana-1800	143	31	)	)	PUNCT
cana-1800	143	32	}	}	PUNCT
cana-1800	143	33	,	,	PUNCT
cana-1800	143	34	rmax	rmax	ADJ
cana-1800	143	35	{	{	PUNCT
cana-1800	143	36	𝒱ℌ	𝒱ℌ	PROPN
cana-1800	143	37	−(𝔳	−(𝔳	NOUN
cana-1800	143	38	)	)	PUNCT
cana-1800	143	39	,	,	PUNCT
cana-1800	143	40	𝒱ℌ	𝒱ℌ	PROPN
cana-1800	143	41	−(𝔥)}}	−(𝔥)}}	PRON
cana-1800	143	42	rmax{rmax	rmax{rmax	X
cana-1800	143	43	{	{	PUNCT
cana-1800	143	44	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	143	45	−(𝔳	−(𝔳	NOUN
cana-1800	143	46	)	)	PUNCT
cana-1800	143	47	,	,	PUNCT
cana-1800	143	48	𝒱ℌ	𝒱ℌ	PROPN
cana-1800	143	49	−(𝔳	−(𝔳	VERB
cana-1800	143	50	)	)	PUNCT
cana-1800	143	51	,	,	PUNCT
cana-1800	143	52	rmax	rmax	ADJ
cana-1800	143	53	{	{	PUNCT
cana-1800	143	54	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	143	55	−(𝔥	−(𝔥	PROPN
cana-1800	143	56	)	)	PUNCT
cana-1800	143	57	,	,	PUNCT
cana-1800	143	58	𝒱ℌ	𝒱ℌ	PROPN
cana-1800	143	59	−(𝔥	−(𝔥	NOUN
cana-1800	143	60	)	)	PUNCT
cana-1800	143	61	}	}	PUNCT
cana-1800	143	62	}	}	PUNCT
cana-1800	144	1	=	=	SYM
cana-1800	144	2	rmax	rmax	ADJ
cana-1800	144	3	{	{	PUNCT
cana-1800	144	4	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	144	5	−(𝔳	−(𝔳	NOUN
cana-1800	144	6	)	)	PUNCT
cana-1800	144	7	,	,	PUNCT
cana-1800	144	8	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	144	9	−(𝔥	−(𝔥	NOUN
cana-1800	144	10	)	)	PUNCT
cana-1800	144	11	}	}	PUNCT
cana-1800	144	12	.	.	PUNCT
cana-1800	145	1	therefore	therefore	ADV
cana-1800	145	2	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	145	3	−(𝔳	−(𝔳	VERB
cana-1800	145	4	+	+	CCONJ
cana-1800	145	5	𝔥	𝔥	NOUN
cana-1800	145	6	)	)	PUNCT
cana-1800	145	7			NOUN
cana-1800	145	8	rmax	rmax	ADJ
cana-1800	145	9	{	{	PUNCT
cana-1800	145	10	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	145	11	−(𝔳	−(𝔳	NOUN
cana-1800	145	12	)	)	PUNCT
cana-1800	145	13	,	,	PUNCT
cana-1800	145	14	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	145	15	−(𝔥	−(𝔥	NOUN
cana-1800	145	16	)	)	PUNCT
cana-1800	145	17	}	}	PUNCT
cana-1800	145	18	,	,	PUNCT
cana-1800	145	19	for	for	ADP
cana-1800	145	20	all𝔳	all𝔳	ADV
cana-1800	145	21	,	,	PUNCT
cana-1800	145	22	𝔥	𝔥	PROPN
cana-1800	145	23	∈	∈	PROPN
cana-1800	145	24	ℝ.	ℝ.	PROPN
cana-1800	145	25	and	and	CCONJ
cana-1800	145	26	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	145	27	−(𝔳𝔥	−(𝔳𝔥	PROPN
cana-1800	145	28	)	)	PUNCT
cana-1800	146	1	=	=	SYM
cana-1800	146	2	rmax	rmax	ADJ
cana-1800	146	3	{	{	PUNCT
cana-1800	146	4	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	146	5	−(𝔳𝔥	−(𝔳𝔥	PROPN
cana-1800	146	6	)	)	PUNCT
cana-1800	146	7	,	,	PUNCT
cana-1800	146	8	𝒱ℌ	𝒱ℌ	PROPN
cana-1800	146	9	−(𝔳𝔥	−(𝔳𝔥	PROPN
cana-1800	146	10	)	)	PUNCT
cana-1800	146	11	}	}	PUNCT
cana-1800	146	12			NOUN
cana-1800	146	13	rmax{rmin	rmax{rmin	NOUN
cana-1800	146	14	{	{	PUNCT
cana-1800	146	15	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	146	16	−(𝔳	−(𝔳	NOUN
cana-1800	146	17	)	)	PUNCT
cana-1800	146	18	,	,	PUNCT
cana-1800	146	19	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	146	20	−(𝔥	−(𝔥	NOUN
cana-1800	146	21	)	)	PUNCT
cana-1800	146	22	}	}	PUNCT
cana-1800	146	23	,	,	PUNCT
cana-1800	146	24	rmin	rmin	NOUN
cana-1800	146	25	{	{	PUNCT
cana-1800	146	26	𝒱ℌ	𝒱ℌ	PROPN
cana-1800	146	27	−(𝔳	−(𝔳	VERB
cana-1800	146	28	)	)	PUNCT
cana-1800	146	29	,	,	PUNCT
cana-1800	146	30	𝒱ℌ	𝒱ℌ	PROPN
cana-1800	146	31	−(𝔥	−(𝔥	NOUN
cana-1800	146	32	)	)	PUNCT
cana-1800	146	33	}	}	PUNCT
cana-1800	146	34	}	}	PUNCT
cana-1800	146	35			NUM
cana-1800	146	36	rmin{rmax	rmin{rmax	NOUN
cana-1800	146	37	{	{	PUNCT
cana-1800	146	38	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	146	39	−(𝔳	−(𝔳	NOUN
cana-1800	146	40	)	)	PUNCT
cana-1800	146	41	,	,	PUNCT
cana-1800	146	42	𝒱ℌ	𝒱ℌ	PROPN
cana-1800	146	43	−(𝔳	−(𝔳	VERB
cana-1800	146	44	)	)	PUNCT
cana-1800	146	45	}	}	PUNCT
cana-1800	146	46	,	,	PUNCT
cana-1800	146	47	rmax	rmax	ADJ
cana-1800	146	48	{	{	PUNCT
cana-1800	146	49	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	146	50	−(𝔥	−(𝔥	PROPN
cana-1800	146	51	)	)	PUNCT
cana-1800	146	52	,	,	PUNCT
cana-1800	146	53	𝒱ℌ	𝒱ℌ	PROPN
cana-1800	146	54	−(𝔥)}}=	−(𝔥)}}=	VERB
cana-1800	146	55	rmin	rmin	NOUN
cana-1800	146	56	{	{	PUNCT
cana-1800	146	57	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	146	58	−(𝔳	−(𝔳	NOUN
cana-1800	146	59	)	)	PUNCT
cana-1800	146	60	,	,	PUNCT
cana-1800	146	61	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	146	62	−(𝔥	−(𝔥	NOUN
cana-1800	146	63	)	)	PUNCT
cana-1800	146	64	}	}	PUNCT
cana-1800	146	65	.	.	PUNCT
cana-1800	147	1	therefore	therefore	ADV
cana-1800	147	2	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	147	3	−(𝔳𝔥	−(𝔳𝔥	PROPN
cana-1800	147	4	)	)	PUNCT
cana-1800	147	5			NOUN
cana-1800	147	6	rmin	rmin	VERB
cana-1800	147	7	{	{	PUNCT
cana-1800	147	8	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	147	9	−(𝔳	−(𝔳	NOUN
cana-1800	147	10	)	)	PUNCT
cana-1800	147	11	,	,	PUNCT
cana-1800	147	12	𝒱ҫ	𝒱ҫ	PROPN
cana-1800	147	13	−(𝔥	−(𝔥	NOUN
cana-1800	147	14	)	)	PUNCT
cana-1800	147	15	}	}	PUNCT
cana-1800	147	16	,	,	PUNCT
cana-1800	147	17	for	for	ADP
cana-1800	147	18	all	all	DET
cana-1800	147	19	𝔳	𝔳	NOUN
cana-1800	147	20	,	,	PUNCT
cana-1800	147	21	𝔥	𝔥	PROPN
cana-1800	147	22	∈	∈	PROPN
cana-1800	147	23	ℝ.	ℝ.	PROPN
cana-1800	147	24	hence	hence	ADV
cana-1800	147	25	𝔈	𝔈	NOUN
cana-1800	147	26			PUNCT
cana-1800	147	27	ℌis	ℌis	PROPN
cana-1800	147	28	a	a	DET
cana-1800	147	29	𝔹𝕍𝕍𝕀	𝔹𝕍𝕍𝕀	NOUN
cana-1800	147	30	of	of	ADP
cana-1800	147	31	ҫ	ҫ	NOUN
cana-1800	147	32	=	=	SYM
cana-1800	147	33	𝔈	𝔈	PROPN
cana-1800	147	34			PUNCT
cana-1800	147	35	ℌ.	ℌ.	PROPN
cana-1800	147	36	theorem	theorem	VERB
cana-1800	147	37	3.5	3.5	NUM
cana-1800	147	38	.	.	PUNCT
cana-1800	148	1	the	the	DET
cana-1800	148	2	intersection	intersection	NOUN
cana-1800	148	3	of	of	ADP
cana-1800	148	4	a	a	DET
cana-1800	148	5	family	family	NOUN
cana-1800	148	6	of	of	ADP
cana-1800	148	7	𝔹𝕍𝕍𝕀s	𝔹𝕍𝕍𝕀s	PROPN
cana-1800	148	8	of	of	ADP
cana-1800	148	9	a	a	DET
cana-1800	148	10	semiring	semire	VERB
cana-1800	148	11	ℝ	ℝ	NOUN
cana-1800	148	12	is	be	AUX
cana-1800	148	13	a	a	DET
cana-1800	148	14	𝔹𝕍𝕍𝕀	𝔹𝕍𝕍𝕀	NOUN
cana-1800	148	15	of	of	ADP
cana-1800	148	16	ℝ.	ℝ.	PROPN
cana-1800	148	17	proof	proof	NOUN
cana-1800	148	18	.	.	PUNCT
cana-1800	149	1	the	the	DET
cana-1800	149	2	proof	proof	NOUN
cana-1800	149	3	follows	follow	VERB
cana-1800	149	4	from	from	ADP
cana-1800	149	5	the	the	DET
cana-1800	149	6	theorem	theorem	ADJ
cana-1800	149	7	3.4	3.4	NUM
cana-1800	149	8	.	.	PUNCT
cana-1800	150	1	theorem	theorem	VERB
cana-1800	150	2	3.6	3.6	NUM
cana-1800	150	3	.	.	PUNCT
cana-1800	151	1	if	if	SCONJ
cana-1800	151	2	𝕰	𝕰	PROPN
cana-1800	151	3	=	=	SYM
cana-1800	151	4	〈	〈	PROPN
cana-1800	151	5	𝓥𝕰	𝓥𝕰	PROPN
cana-1800	151	6	+	+	ADP
cana-1800	151	7	,	,	PUNCT
cana-1800	151	8	𝓥𝕰	𝓥𝕰	PROPN
cana-1800	151	9	−	−	NUM
cana-1800	151	10	〉	〉	NOUN
cana-1800	151	11	and	and	CCONJ
cana-1800	151	12	𝕳	𝕳	NOUN
cana-1800	151	13	=	=	SYM
cana-1800	151	14	〈	〈	NOUN
cana-1800	151	15	𝓥𝕳	𝓥𝕳	NOUN
cana-1800	151	16	+	+	ADV
cana-1800	151	17	,	,	PUNCT
cana-1800	151	18	𝓥𝕳	𝓥𝕳	NOUN
cana-1800	151	19	−	−	NOUN
cana-1800	151	20	〉	〉	NOUN
cana-1800	151	21	are	be	AUX
cana-1800	151	22	two	two	NUM
cana-1800	151	23	𝔹𝕍𝕍𝕀s	𝔹𝕍𝕍𝕀	NOUN
cana-1800	151	24	of	of	ADP
cana-1800	151	25	a	a	DET
cana-1800	151	26	semiring	semire	VERB
cana-1800	151	27	ℝ	ℝ	PROPN
cana-1800	151	28	,	,	PUNCT
cana-1800	151	29	then	then	ADV
cana-1800	151	30	their	their	PRON
cana-1800	151	31	union	union	NOUN
cana-1800	151	32	𝕰	𝕰	PROPN
cana-1800	151	33			NOUN
cana-1800	151	34	𝕳	𝕳	PROPN
cana-1800	151	35	need	need	AUX
cana-1800	151	36	not	not	PART
cana-1800	151	37	be	be	AUX
cana-1800	151	38	a	a	DET
cana-1800	151	39	𝔹𝕍𝕍𝕀	𝔹𝕍𝕍𝕀	NOUN
cana-1800	151	40	of	of	ADP
cana-1800	151	41	ℝ.	ℝ.	PROPN
cana-1800	151	42	proof	proof	NOUN
cana-1800	151	43	.	.	PUNCT
cana-1800	152	1	it	it	PRON
cana-1800	152	2	can	can	AUX
cana-1800	152	3	be	be	AUX
cana-1800	152	4	easily	easily	ADV
cana-1800	152	5	proved	prove	VERB
cana-1800	152	6	.	.	PUNCT
cana-1800	153	1	remark	remark	VERB
cana-1800	153	2	3.7	3.7	NUM
cana-1800	153	3	.	.	PUNCT
cana-1800	154	1	if	if	SCONJ
cana-1800	154	2	one	one	PRON
cana-1800	154	3	is	be	AUX
cana-1800	154	4	contained	contain	VERB
cana-1800	154	5	other	other	ADJ
cana-1800	154	6	,	,	PUNCT
cana-1800	154	7	then	then	ADV
cana-1800	154	8	the	the	DET
cana-1800	154	9	union	union	NOUN
cana-1800	154	10	is	be	AUX
cana-1800	154	11	a	a	DET
cana-1800	154	12	𝔹𝕍𝕍𝕀s	𝔹𝕍𝕍𝕀s	NOUN
cana-1800	154	13	of	of	ADP
cana-1800	154	14	a	a	DET
cana-1800	154	15	semiring	semire	VERB
cana-1800	154	16	ℝ.	ℝ.	PROPN
cana-1800	154	17	definition	definition	NOUN
cana-1800	154	18	3.8	3.8	NUM
cana-1800	154	19	.	.	PUNCT
cana-1800	155	1	let	let	VERB
cana-1800	155	2	𝔈	𝔈	PROPN
cana-1800	155	3	=	=	PUNCT
cana-1800	155	4	〈	〈	PROPN
cana-1800	155	5	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	155	6	+	+	PROPN
cana-1800	155	7	,	,	PUNCT
cana-1800	155	8	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	155	9	−	−	NUM
cana-1800	155	10	〉	〉	NOUN
cana-1800	155	11	and	and	CCONJ
cana-1800	155	12	𝔓	𝔓	NOUN
cana-1800	155	13	=	=	SYM
cana-1800	155	14	〈	〈	PROPN
cana-1800	155	15	𝒱𝔓	𝒱𝔓	PROPN
cana-1800	155	16	+	+	PROPN
cana-1800	155	17	,	,	PUNCT
cana-1800	155	18	𝒱𝔓	𝒱𝔓	PROPN
cana-1800	155	19	−	−	NOUN
cana-1800	155	20	〉	〉	NOUN
cana-1800	155	21	be	be	VERB
cana-1800	155	22	any	any	DET
cana-1800	155	23	two	two	NUM
cana-1800	155	24	𝔹𝕍𝕍𝕊𝕊𝑠	𝔹𝕍𝕍𝕊𝕊𝑠	NOUN
cana-1800	155	25	of	of	ADP
cana-1800	155	26	sets	set	NOUN
cana-1800	155	27	ℜ1	ℜ1	PRON
cana-1800	155	28	and	and	CCONJ
cana-1800	155	29	ℜ2	ℜ2	NOUN
cana-1800	155	30	.	.	PUNCT
cana-1800	156	1	the	the	DET
cana-1800	156	2	product	product	NOUN
cana-1800	156	3	of	of	ADP
cana-1800	156	4	𝔈	𝔈	PROPN
cana-1800	156	5	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-1800	156	6	𝔓	𝔓	PROPN
cana-1800	156	7	,	,	PUNCT
cana-1800	156	8	denoted	denote	VERB
cana-1800	156	9	by	by	ADP
cana-1800	156	10	𝔈	𝔈	PROPN
cana-1800	156	11	×	×	PROPN
cana-1800	156	12	𝔓	𝔓	PROPN
cana-1800	156	13	,	,	PUNCT
cana-1800	156	14	is	be	AUX
cana-1800	156	15	defined	define	VERB
cana-1800	156	16	as	as	ADP
cana-1800	156	17	𝔈	𝔈	PROPN
cana-1800	156	18	×	×	NOUN
cana-1800	156	19	𝔓	𝔓	NOUN
cana-1800	156	20	=	=	PUNCT
cana-1800	156	21	{	{	PUNCT
cana-1800	156	22	(𝔳	(𝔳	PROPN
cana-1800	156	23	,	,	PUNCT
cana-1800	156	24	𝔥	𝔥	NOUN
cana-1800	156	25	)	)	PUNCT
cana-1800	156	26	,	,	PUNCT
cana-1800	156	27	𝒱𝔈×𝔓	𝒱𝔈×𝔓	PROPN
cana-1800	157	1	+	+	CCONJ
cana-1800	157	2	(	(	PUNCT
cana-1800	157	3	𝔳	𝔳	PROPN
cana-1800	157	4	,	,	PUNCT
cana-1800	157	5	𝔥	𝔥	NOUN
cana-1800	157	6	)	)	PUNCT
cana-1800	157	7	,	,	PUNCT
cana-1800	157	8	𝒱𝔈×𝔓	𝒱𝔈×𝔓	PROPN
cana-1800	157	9	−	−	PROPN
cana-1800	157	10	(	(	PUNCT
cana-1800	157	11	𝔳	𝔳	PROPN
cana-1800	157	12	,	,	PUNCT
cana-1800	157	13	𝔥)	𝔥)	PROPN
cana-1800	157	14	/	/	PUNCT
cana-1800	157	15	for	for	ADP
cana-1800	157	16	all	all	DET
cana-1800	157	17	𝔳	𝔳	DET
cana-1800	157	18	∈	∈	ADJ
cana-1800	157	19	ℜ1	ℜ1	NOUN
cana-1800	157	20	and	and	CCONJ
cana-1800	157	21	𝔥	𝔥	PROPN
cana-1800	157	22	∈	∈	PROPN
cana-1800	157	23	ℜ2	ℜ2	ADV
cana-1800	157	24	}	}	PUNCT
cana-1800	157	25	,	,	PUNCT
cana-1800	157	26	where	where	SCONJ
cana-1800	157	27	𝒱𝔈×𝔓	𝒱𝔈×𝔓	PUNCT
cana-1800	157	28	+	+	CCONJ
cana-1800	157	29	(	(	PUNCT
cana-1800	157	30	𝔳	𝔳	PROPN
cana-1800	157	31	,	,	PUNCT
cana-1800	157	32	𝔥	𝔥	NOUN
cana-1800	157	33	)	)	PUNCT
cana-1800	157	34	=	=	SYM
cana-1800	157	35	rmin	rmin	NOUN
cana-1800	157	36	{	{	PUNCT
cana-1800	157	37	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	157	38	+	+	PROPN
cana-1800	157	39	(	(	PUNCT
cana-1800	157	40	𝔳	𝔳	NOUN
cana-1800	157	41	)	)	PUNCT
cana-1800	157	42	,	,	PUNCT
cana-1800	157	43	𝒱𝔓	𝒱𝔓	PROPN
cana-1800	157	44	+	+	PROPN
cana-1800	157	45	(	(	PUNCT
cana-1800	157	46	𝔥	𝔥	NOUN
cana-1800	157	47	)	)	PUNCT
cana-1800	157	48	}	}	PUNCT
cana-1800	157	49	and	and	CCONJ
cana-1800	157	50	𝒱𝔈×𝔓	𝒱𝔈×𝔓	PROPN
cana-1800	157	51	−	−	PROPN
cana-1800	157	52	(	(	PUNCT
cana-1800	157	53	𝔳	𝔳	PROPN
cana-1800	157	54	,	,	PUNCT
cana-1800	157	55	𝔥	𝔥	NOUN
cana-1800	157	56	)	)	PUNCT
cana-1800	157	57	=	=	SYM
cana-1800	157	58	rmax	rmax	ADJ
cana-1800	157	59	{	{	PUNCT
cana-1800	157	60	𝒱𝔈	𝒱𝔈	PROPN
cana-1800	157	61	−(𝔳	−(𝔳	NOUN
cana-1800	157	62	)	)	PUNCT
cana-1800	157	63	,	,	PUNCT
cana-1800	157	64	𝒱𝔓	𝒱𝔓	PROPN
cana-1800	157	65	−(𝔥	−(𝔥	NOUN
cana-1800	157	66	)	)	PUNCT
cana-1800	157	67	}	}	PUNCT
cana-1800	157	68	for	for	ADP
cana-1800	157	69	all	all	DET
cana-1800	157	70	𝔳	𝔳	DET
cana-1800	157	71	∈	∈	ADJ
cana-1800	157	72	ℜ1	ℜ1	NOUN
cana-1800	157	73	and	and	CCONJ
cana-1800	157	74	𝔥	𝔥	PROPN
cana-1800	157	75	∈	∈	PROPN
cana-1800	157	76	ℜ2	ℜ2	PROPN
cana-1800	157	77	.	.	PUNCT
cana-1800	158	1	theorem	theorem	VERB
cana-1800	158	2	3.9	3.9	NUM
cana-1800	158	3	.	.	PUNCT
cana-1800	159	1	if	if	SCONJ
cana-1800	159	2	a	a	DET
cana-1800	159	3	=	=	X
cana-1800	159	4			X
cana-1800	159	5	+	+	CCONJ
cana-1800	159	6	av	av	PROPN
cana-1800	159	7	,	,	PUNCT
cana-1800	159	8	−	−	PROPN
cana-1800	159	9	av	av	PROPN
cana-1800	159	10			PROPN
cana-1800	159	11	and	and	CCONJ
cana-1800	159	12	b	b	X
cana-1800	159	13	=	=	X
cana-1800	159	14			PUNCT
cana-1800	159	15	+	+	CCONJ
cana-1800	159	16	bv	bv	PROPN
cana-1800	159	17	,	,	PUNCT
cana-1800	159	18	−	−	PROPN
cana-1800	159	19	bv	bv	PROPN
cana-1800	159	20			PROPN
cana-1800	159	21	are	be	AUX
cana-1800	159	22	any	any	DET
cana-1800	159	23	two	two	NUM
cana-1800	159	24	𝔹𝕍𝕍𝕀s	𝔹𝕍𝕍𝕀	NOUN
cana-1800	159	25	of	of	ADP
cana-1800	159	26	the	the	DET
cana-1800	159	27	semirings	semiring	NOUN
cana-1800	159	28	r1	r1	PROPN
cana-1800	159	29	and	and	CCONJ
cana-1800	159	30	r2	r2	PROPN
cana-1800	159	31	respectively	respectively	ADV
cana-1800	159	32	,	,	PUNCT
cana-1800	159	33	then	then	ADV
cana-1800	159	34	a×b	a×b	PUNCT
cana-1800	159	35	=	=	SYM
cana-1800	159	36			PUNCT
cana-1800	159	37	+	+	PUNCT
cana-1800	159	38	bav	bav	NOUN
cana-1800	159	39	,	,	PUNCT
cana-1800	159	40	−	−	PROPN
cana-1800	159	41	bav	bav	NOUN
cana-1800	159	42			PROPN
cana-1800	159	43	is	be	AUX
cana-1800	159	44	a	a	DET
cana-1800	159	45	𝔹𝕍𝕍𝕀	𝔹𝕍𝕍𝕀	NOUN
cana-1800	159	46	of	of	ADP
cana-1800	159	47	r1×r2	r1×r2	PROPN
cana-1800	159	48	.	.	PUNCT
cana-1800	160	1	proof	proof	NOUN
cana-1800	160	2	.	.	PUNCT
cana-1800	161	1	let	let	VERB
cana-1800	161	2	x1	x1	NUM
cana-1800	161	3	,	,	PUNCT
cana-1800	161	4	x2	x2	PRON
cana-1800	161	5	be	be	VERB
cana-1800	161	6	in	in	ADP
cana-1800	161	7	r1	r1	NOUN
cana-1800	161	8	,	,	PUNCT
cana-1800	161	9	y1	y1	NOUN
cana-1800	161	10	and	and	CCONJ
cana-1800	161	11	y2	y2	NOUN
cana-1800	161	12	be	be	AUX
cana-1800	161	13	in	in	ADP
cana-1800	161	14	r2	r2	PROPN
cana-1800	161	15	.	.	PUNCT
cana-1800	162	1	then	then	ADV
cana-1800	162	2	(	(	PUNCT
cana-1800	162	3	x1	x1	PROPN
cana-1800	162	4	,	,	PUNCT
cana-1800	162	5	y1	y1	INTJ
cana-1800	162	6	)	)	PUNCT
cana-1800	162	7	and	and	CCONJ
cana-1800	162	8	(	(	PUNCT
cana-1800	162	9	x2	x2	PROPN
cana-1800	162	10	,	,	PUNCT
cana-1800	162	11	y2	y2	PROPN
cana-1800	162	12	)	)	PUNCT
cana-1800	162	13	are	be	AUX
cana-1800	162	14	in	in	ADP
cana-1800	162	15	r1×r2	r1×r2	PROPN
cana-1800	162	16	.	.	PUNCT
cana-1800	163	1	now	now	ADV
cana-1800	163	2	,	,	PUNCT
cana-1800	163	3	+	+	PUNCT
cana-1800	163	4	bav	bav	X
cana-1800	164	1	[	[	X
cana-1800	164	2	(	(	PUNCT
cana-1800	164	3	x1	x1	ADJ
cana-1800	164	4	,	,	PUNCT
cana-1800	164	5	y1)+(x2	y1)+(x2	NOUN
cana-1800	164	6	,	,	PUNCT
cana-1800	164	7	y2	y2	PROPN
cana-1800	164	8	)	)	PUNCT
cana-1800	164	9	]	]	PUNCT
cana-1800	165	1	=	=	PUNCT
cana-1800	166	1	+	+	NUM
cana-1800	166	2	bav	bav	NOUN
cana-1800	166	3	(	(	PUNCT
cana-1800	166	4	x1+x2	x1+x2	PROPN
cana-1800	166	5	,	,	PUNCT
cana-1800	166	6	y1+y2	y1+y2	NOUN
cana-1800	166	7	)	)	PUNCT
cana-1800	166	8	=	=	SYM
cana-1800	166	9	rmin	rmin	NOUN
cana-1800	166	10	{	{	PUNCT
cana-1800	166	11	+	+	NUM
cana-1800	166	12	av	av	PROPN
cana-1800	166	13	(	(	PUNCT
cana-1800	166	14	x1+x2	x1+x2	PROPN
cana-1800	166	15	)	)	PUNCT
cana-1800	166	16	,	,	PUNCT
cana-1800	166	17	+	+	CCONJ
cana-1800	166	18	bv	bv	PROPN
cana-1800	166	19	(	(	PUNCT
cana-1800	166	20	y1+y2	y1+y2	NOUN
cana-1800	166	21	)	)	PUNCT
cana-1800	166	22	}	}	PUNCT
cana-1800	166	23			X
cana-1800	166	24	rmin{rmin	rmin{rmin	NOUN
cana-1800	166	25	{	{	PUNCT
cana-1800	166	26	+	+	CCONJ
cana-1800	166	27	av	av	PROPN
cana-1800	166	28	(	(	PUNCT
cana-1800	166	29	x1	x1	PROPN
cana-1800	166	30	)	)	PUNCT
cana-1800	166	31	,	,	PUNCT
cana-1800	166	32	+	+	CCONJ
cana-1800	166	33	av	av	PROPN
cana-1800	166	34	(	(	PUNCT
cana-1800	166	35	x2	x2	PROPN
cana-1800	166	36	)	)	PUNCT
cana-1800	166	37	}	}	PUNCT
cana-1800	166	38	,	,	PUNCT
cana-1800	166	39	rmin	rmin	NOUN
cana-1800	166	40	{	{	PUNCT
cana-1800	166	41	+	+	NUM
cana-1800	166	42	bv	bv	PROPN
cana-1800	166	43	(	(	PUNCT
cana-1800	166	44	y1	y1	PROPN
cana-1800	166	45	)	)	PUNCT
cana-1800	166	46	,	,	PUNCT
cana-1800	166	47	+	+	CCONJ
cana-1800	166	48	bv	bv	PROPN
cana-1800	166	49	(	(	PUNCT
cana-1800	166	50	y2	y2	PROPN
cana-1800	166	51	)	)	PUNCT
cana-1800	166	52	}	}	PUNCT
cana-1800	166	53	}	}	PUNCT
cana-1800	166	54	=	=	SYM
cana-1800	166	55	rmin{rmin	rmin{rmin	NOUN
cana-1800	166	56	{	{	PUNCT
cana-1800	166	57	+	+	NUM
cana-1800	166	58	av	av	PROPN
cana-1800	166	59	(	(	PUNCT
cana-1800	166	60	x1	x1	PROPN
cana-1800	166	61	)	)	PUNCT
cana-1800	166	62	,	,	PUNCT
cana-1800	166	63	+	+	CCONJ
cana-1800	166	64	bv	bv	PROPN
cana-1800	166	65	(	(	PUNCT
cana-1800	166	66	y1	y1	PROPN
cana-1800	166	67	)	)	PUNCT
cana-1800	166	68	}	}	PUNCT
cana-1800	166	69	,	,	PUNCT
cana-1800	166	70	rmin	rmin	NOUN
cana-1800	166	71	{	{	PUNCT
cana-1800	166	72	+	+	NUM
cana-1800	166	73	av	av	PROPN
cana-1800	166	74	(	(	PUNCT
cana-1800	166	75	x2	x2	PROPN
cana-1800	166	76	)	)	PUNCT
cana-1800	166	77	,	,	PUNCT
cana-1800	166	78	+	+	CCONJ
cana-1800	166	79	bv	bv	PROPN
cana-1800	166	80	(	(	PUNCT
cana-1800	166	81	y2	y2	PROPN
cana-1800	166	82	)	)	PUNCT
cana-1800	166	83	}	}	PUNCT
cana-1800	166	84	}	}	PUNCT
cana-1800	166	85	=	=	SYM
cana-1800	166	86	rmin	rmin	NOUN
cana-1800	166	87	{	{	PUNCT
cana-1800	166	88	+	+	ADJ
cana-1800	166	89	bav	bav	NOUN
cana-1800	166	90	(	(	PUNCT
cana-1800	166	91	x1	x1	PROPN
cana-1800	166	92	,	,	PUNCT
cana-1800	166	93	y1	y1	PROPN
cana-1800	166	94	)	)	PUNCT
cana-1800	166	95	,	,	PUNCT
cana-1800	166	96	+	+	NUM
cana-1800	166	97	bav	bav	NOUN
cana-1800	166	98	(	(	PUNCT
cana-1800	166	99	x2	x2	PROPN
cana-1800	166	100	,	,	PUNCT
cana-1800	166	101	y2	y2	PROPN
cana-1800	166	102	)	)	PUNCT
cana-1800	166	103	}	}	PUNCT
cana-1800	166	104	.	.	PUNCT
cana-1800	167	1	therefore	therefore	ADV
cana-1800	167	2	+	+	X
cana-1800	167	3	bav	bav	NOUN
cana-1800	167	4	[	[	PUNCT
cana-1800	167	5	(	(	PUNCT
cana-1800	167	6	x1	x1	ADJ
cana-1800	167	7	,	,	PUNCT
cana-1800	167	8	y1)+(x2	y1)+(x2	NOUN
cana-1800	167	9	,	,	PUNCT
cana-1800	167	10	y2	y2	PROPN
cana-1800	167	11	)	)	PUNCT
cana-1800	167	12	]	]	PUNCT
cana-1800	167	13			NUM
cana-1800	167	14	rmin	rmin	VERB
cana-1800	167	15	{	{	PUNCT
cana-1800	167	16	+	+	NUM
cana-1800	167	17	bav	bav	NOUN
cana-1800	167	18	(	(	PUNCT
cana-1800	167	19	x1	x1	PROPN
cana-1800	167	20	,	,	PUNCT
cana-1800	167	21	y1	y1	PROPN
cana-1800	167	22	)	)	PUNCT
cana-1800	167	23	,	,	PUNCT
cana-1800	167	24	+	+	NUM
cana-1800	167	25	bav	bav	NOUN
cana-1800	167	26	(	(	PUNCT
cana-1800	167	27	x2	x2	PROPN
cana-1800	167	28	,	,	PUNCT
cana-1800	167	29	y2	y2	PROPN
cana-1800	167	30	)	)	PUNCT
cana-1800	167	31	}	}	PUNCT
cana-1800	167	32	.	.	PUNCT
cana-1800	168	1	and	and	CCONJ
cana-1800	169	1	+	+	PUNCT
cana-1800	169	2	bav	bav	X
cana-1800	170	1	[	[	X
cana-1800	170	2	(	(	PUNCT
cana-1800	170	3	x1	x1	ADJ
cana-1800	170	4	,	,	PUNCT
cana-1800	170	5	y1)(x2	y1)(x2	NOUN
cana-1800	170	6	,	,	PUNCT
cana-1800	170	7	y2	y2	PROPN
cana-1800	170	8	)	)	PUNCT
cana-1800	170	9	]	]	PUNCT
cana-1800	171	1	=	=	PUNCT
cana-1800	172	1	+	+	NUM
cana-1800	172	2	bav	bav	NOUN
cana-1800	172	3	(	(	PUNCT
cana-1800	172	4	x1x2	x1x2	NOUN
cana-1800	172	5	,	,	PUNCT
cana-1800	172	6	y1y2	y1y2	NOUN
cana-1800	172	7	)	)	PUNCT
cana-1800	172	8	=	=	SYM
cana-1800	172	9	rmin	rmin	NOUN
cana-1800	172	10	{	{	PUNCT
cana-1800	172	11	+	+	NOUN
cana-1800	172	12	av	av	PROPN
cana-1800	172	13	(	(	PUNCT
cana-1800	172	14	x1x2	x1x2	NOUN
cana-1800	172	15	)	)	PUNCT
cana-1800	172	16	,	,	PUNCT
cana-1800	172	17	+	+	CCONJ
cana-1800	172	18	bv	bv	PROPN
cana-1800	172	19	(	(	PUNCT
cana-1800	172	20	y1y2	y1y2	PROPN
cana-1800	172	21	)	)	PUNCT
cana-1800	172	22	}	}	PUNCT
cana-1800	172	23			NUM
cana-1800	172	24	rmin{rmax	rmin{rmax	NOUN
cana-1800	172	25	{	{	PUNCT
cana-1800	172	26	+	+	NOUN
cana-1800	172	27	av	av	PROPN
cana-1800	172	28	(	(	PUNCT
cana-1800	172	29	x1	x1	PROPN
cana-1800	172	30	)	)	PUNCT
cana-1800	172	31	,	,	PUNCT
cana-1800	172	32	+	+	CCONJ
cana-1800	172	33	av	av	PROPN
cana-1800	172	34	(	(	PUNCT
cana-1800	172	35	x2	x2	PROPN
cana-1800	172	36	)	)	PUNCT
cana-1800	172	37	}	}	PUNCT
cana-1800	172	38	,	,	PUNCT
cana-1800	172	39	rmax	rmax	ADJ
cana-1800	172	40	{	{	PUNCT
cana-1800	172	41	+	+	NUM
cana-1800	172	42	bv	bv	PROPN
cana-1800	172	43	(	(	PUNCT
cana-1800	172	44	y1	y1	PROPN
cana-1800	172	45	)	)	PUNCT
cana-1800	172	46	,	,	PUNCT
cana-1800	172	47	+	+	CCONJ
cana-1800	172	48	bv	bv	PROPN
cana-1800	172	49	(	(	PUNCT
cana-1800	172	50	y2	y2	PROPN
cana-1800	172	51	)	)	PUNCT
cana-1800	172	52	}	}	PUNCT
cana-1800	172	53	}	}	PUNCT
cana-1800	172	54	=	=	SYM
cana-1800	172	55	rmax{rmin	rmax{rmin	NOUN
cana-1800	172	56	{	{	PUNCT
cana-1800	172	57	+	+	X
cana-1800	172	58	av	av	PROPN
cana-1800	172	59	(	(	PUNCT
cana-1800	172	60	x1	x1	PROPN
cana-1800	172	61	)	)	PUNCT
cana-1800	172	62	,	,	PUNCT
cana-1800	172	63	+	+	CCONJ
cana-1800	172	64	bv	bv	PROPN
cana-1800	172	65	(	(	PUNCT
cana-1800	172	66	y1	y1	PROPN
cana-1800	172	67	)	)	PUNCT
cana-1800	172	68	}	}	PUNCT
cana-1800	172	69	,	,	PUNCT
cana-1800	172	70	rmin	rmin	NOUN
cana-1800	172	71	{	{	PUNCT
cana-1800	172	72	+	+	NUM
cana-1800	172	73	av	av	PROPN
cana-1800	172	74	(	(	PUNCT
cana-1800	172	75	x2	x2	PROPN
cana-1800	172	76	)	)	PUNCT
cana-1800	172	77	,	,	PUNCT
cana-1800	172	78	+	+	CCONJ
cana-1800	172	79	bv	bv	PROPN
cana-1800	172	80	(	(	PUNCT
cana-1800	172	81	y2	y2	PROPN
cana-1800	172	82	)	)	PUNCT
cana-1800	172	83	}	}	PUNCT
cana-1800	172	84	}	}	PUNCT
cana-1800	172	85	=	=	SYM
cana-1800	172	86	rmax	rmax	ADJ
cana-1800	172	87	{	{	PUNCT
cana-1800	172	88	+	+	NUM
cana-1800	172	89	bav	bav	NOUN
cana-1800	172	90	(	(	PUNCT
cana-1800	172	91	x1	x1	PROPN
cana-1800	172	92	,	,	PUNCT
cana-1800	172	93	y1	y1	PROPN
cana-1800	172	94	)	)	PUNCT
cana-1800	172	95	,	,	PUNCT
cana-1800	172	96	+	+	NUM
cana-1800	172	97	bav	bav	NOUN
cana-1800	172	98	(	(	PUNCT
cana-1800	172	99	x2	x2	PROPN
cana-1800	172	100	,	,	PUNCT
cana-1800	172	101	y2)}.therefore	y2)}.therefore	PUNCT
cana-1800	172	102	+	+	PUNCT
cana-1800	172	103	bav	bav	X
cana-1800	173	1	[	[	X
cana-1800	173	2	(	(	PUNCT
cana-1800	173	3	x1	x1	PROPN
cana-1800	173	4	,	,	PUNCT
cana-1800	173	5	y1	y1	PROPN
cana-1800	173	6	)	)	PUNCT
cana-1800	173	7	(	(	PUNCT
cana-1800	173	8	x2	x2	PROPN
cana-1800	173	9	,	,	PUNCT
cana-1800	173	10	y2	y2	PROPN
cana-1800	173	11	)	)	PUNCT
cana-1800	173	12	]	]	PUNCT
cana-1800	174	1			NUM
cana-1800	174	2	rmax	rmax	NOUN
cana-1800	174	3	{	{	PUNCT
cana-1800	174	4	+	+	NUM
cana-1800	174	5	bav	bav	NOUN
cana-1800	174	6	(	(	PUNCT
cana-1800	174	7	x1	x1	PROPN
cana-1800	174	8	,	,	PUNCT
cana-1800	174	9	y1	y1	PROPN
cana-1800	174	10	)	)	PUNCT
cana-1800	174	11	,	,	PUNCT
cana-1800	174	12	+	+	NUM
cana-1800	174	13	bav	bav	NOUN
cana-1800	174	14	(	(	PUNCT
cana-1800	174	15	x2	x2	PROPN
cana-1800	174	16	,	,	PUNCT
cana-1800	174	17	y2	y2	PROPN
cana-1800	174	18	)	)	PUNCT
cana-1800	174	19	}	}	PUNCT
cana-1800	174	20	.	.	PUNCT
cana-1800	175	1	also	also	ADV
cana-1800	175	2	−	−	PART
cana-1800	175	3	bav	bav	NOUN
cana-1800	176	1	[	[	X
cana-1800	176	2	(	(	PUNCT
cana-1800	176	3	x1	x1	ADJ
cana-1800	176	4	,	,	PUNCT
cana-1800	176	5	y1)+(x2	y1)+(x2	NOUN
cana-1800	176	6	,	,	PUNCT
cana-1800	176	7	y2	y2	PROPN
cana-1800	176	8	)	)	PUNCT
cana-1800	176	9	]	]	PUNCT
cana-1800	177	1	=	=	PUNCT
cana-1800	178	1	−	−	PROPN
cana-1800	178	2	bav	bav	NOUN
cana-1800	178	3	(	(	PUNCT
cana-1800	178	4	x1+x2	x1+x2	PROPN
cana-1800	178	5	,	,	PUNCT
cana-1800	178	6	y1+y2	y1+y2	PROPN
cana-1800	178	7	)	)	PUNCT
cana-1800	178	8	=	=	SYM
cana-1800	178	9	rmax	rmax	ADJ
cana-1800	178	10	{	{	PUNCT
cana-1800	178	11	−	−	PROPN
cana-1800	178	12	av	av	PROPN
cana-1800	178	13	(	(	PUNCT
cana-1800	178	14	x1+x2	x1+x2	PROPN
cana-1800	178	15	)	)	PUNCT
cana-1800	178	16	,	,	PUNCT
cana-1800	178	17	−	−	PROPN
cana-1800	178	18	bv	bv	PROPN
cana-1800	178	19	(	(	PUNCT
cana-1800	178	20	y1+y2	y1+y2	PROPN
cana-1800	178	21	)	)	PUNCT
cana-1800	178	22	}	}	PUNCT
cana-1800	178	23	≤	≤	NUM
cana-1800	178	24	rmax{rmax	rmax{rmax	X
cana-1800	178	25	{	{	PUNCT
cana-1800	178	26	communications	communication	NOUN
cana-1800	178	27	on	on	ADP
cana-1800	178	28	applied	apply	VERB
cana-1800	178	29	nonlinear	nonlinear	ADJ
cana-1800	178	30	analysis	analysis	NOUN
cana-1800	178	31	issn	issn	NOUN
cana-1800	178	32	:	:	PUNCT
cana-1800	178	33	1074	1074	NUM
cana-1800	178	34	-	-	PUNCT
cana-1800	178	35	133x	133x	NUM
cana-1800	178	36	vol	vol	NOUN
cana-1800	178	37	32	32	NUM
cana-1800	178	38	no	no	NOUN
cana-1800	178	39	.	.	NOUN
cana-1800	178	40	2	2	NUM
cana-1800	178	41	(	(	PUNCT
cana-1800	178	42	2025	2025	NUM
cana-1800	178	43	)	)	PUNCT
cana-1800	178	44	511	511	NUM
cana-1800	178	45	https://internationalpubls.com	https://internationalpubls.com	X
cana-1800	178	46	−	−	PROPN
cana-1800	179	1	av	av	INTJ
cana-1800	179	2	(	(	PUNCT
cana-1800	179	3	x1	x1	PROPN
cana-1800	179	4	)	)	PUNCT
cana-1800	179	5	,	,	PUNCT
cana-1800	179	6	−	−	PROPN
cana-1800	179	7	av	av	PROPN
cana-1800	179	8	(	(	PUNCT
cana-1800	179	9	x2	x2	PROPN
cana-1800	179	10	)	)	PUNCT
cana-1800	179	11	}	}	PUNCT
cana-1800	179	12	,	,	PUNCT
cana-1800	179	13	rmax	rmax	ADJ
cana-1800	179	14	{	{	PUNCT
cana-1800	179	15	−	−	PROPN
cana-1800	179	16	bv	bv	PROPN
cana-1800	179	17	(	(	PUNCT
cana-1800	179	18	y1	y1	PROPN
cana-1800	179	19	)	)	PUNCT
cana-1800	179	20	,	,	PUNCT
cana-1800	179	21	(	(	PUNCT
cana-1800	179	22	y2	y2	NOUN
cana-1800	179	23	)	)	PUNCT
cana-1800	179	24	}	}	PUNCT
cana-1800	179	25	}	}	PUNCT
cana-1800	179	26	=	=	SYM
cana-1800	179	27	rmax{rmax	rmax{rmax	X
cana-1800	179	28	{	{	PUNCT
cana-1800	179	29	−	−	PROPN
cana-1800	179	30	av	av	PROPN
cana-1800	179	31	(	(	PUNCT
cana-1800	179	32	x1	x1	PROPN
cana-1800	179	33	)	)	PUNCT
cana-1800	179	34	,	,	PUNCT
cana-1800	179	35	−	−	PROPN
cana-1800	179	36	bv	bv	PROPN
cana-1800	179	37	(	(	PUNCT
cana-1800	179	38	y1	y1	PROPN
cana-1800	179	39	)	)	PUNCT
cana-1800	179	40	}	}	PUNCT
cana-1800	179	41	,	,	PUNCT
cana-1800	179	42	rmax	rmax	ADJ
cana-1800	179	43	{	{	PUNCT
cana-1800	179	44	−	−	PROPN
cana-1800	179	45	av	av	INTJ
cana-1800	179	46	(	(	PUNCT
cana-1800	179	47	x2	x2	PROPN
cana-1800	179	48	)	)	PUNCT
cana-1800	179	49	,	,	PUNCT
cana-1800	179	50	−	−	PROPN
cana-1800	179	51	bv	bv	PROPN
cana-1800	179	52	(	(	PUNCT
cana-1800	179	53	y2	y2	PROPN
cana-1800	179	54	)	)	PUNCT
cana-1800	179	55	}	}	PUNCT
cana-1800	179	56	}	}	PUNCT
cana-1800	179	57	=	=	SYM
cana-1800	179	58	rmax	rmax	ADJ
cana-1800	179	59	{	{	PUNCT
cana-1800	179	60	−	−	PROPN
cana-1800	179	61	bav	bav	NOUN
cana-1800	179	62	(	(	PUNCT
cana-1800	179	63	x1	x1	PROPN
cana-1800	179	64	,	,	PUNCT
cana-1800	179	65	y1	y1	PROPN
cana-1800	179	66	)	)	PUNCT
cana-1800	179	67	,	,	PUNCT
cana-1800	179	68	−	−	PROPN
cana-1800	179	69	bav	bav	NOUN
cana-1800	179	70	(	(	PUNCT
cana-1800	179	71	x2	x2	PROPN
cana-1800	179	72	,	,	PUNCT
cana-1800	179	73	y2	y2	PROPN
cana-1800	179	74	)	)	PUNCT
cana-1800	179	75	}	}	PUNCT
cana-1800	179	76	.	.	PUNCT
cana-1800	180	1	therefore	therefore	ADV
cana-1800	180	2	−	−	X
cana-1800	180	3	bav	bav	NOUN
cana-1800	181	1	[	[	X
cana-1800	181	2	(	(	PUNCT
cana-1800	181	3	x1	x1	ADJ
cana-1800	181	4	,	,	PUNCT
cana-1800	181	5	y1)+(x2	y1)+(x2	NOUN
cana-1800	181	6	,	,	PUNCT
cana-1800	181	7	y2	y2	PROPN
cana-1800	181	8	)	)	PUNCT
cana-1800	181	9	]	]	PUNCT
cana-1800	182	1	≤	≤	NUM
cana-1800	182	2	rmax	rmax	NOUN
cana-1800	182	3	{	{	PUNCT
cana-1800	182	4	−	−	PROPN
cana-1800	182	5	bav	bav	NOUN
cana-1800	182	6	(	(	PUNCT
cana-1800	182	7	x1	x1	PROPN
cana-1800	182	8	,	,	PUNCT
cana-1800	182	9	y1	y1	PROPN
cana-1800	182	10	)	)	PUNCT
cana-1800	182	11	,	,	PUNCT
cana-1800	182	12	−	−	PROPN
cana-1800	182	13	bav	bav	NOUN
cana-1800	182	14	(	(	PUNCT
cana-1800	182	15	x2	x2	PROPN
cana-1800	182	16	,	,	PUNCT
cana-1800	182	17	y2	y2	PROPN
cana-1800	182	18	)	)	PUNCT
cana-1800	182	19	}	}	PUNCT
cana-1800	182	20	.	.	PUNCT
cana-1800	183	1	and	and	CCONJ
cana-1800	183	2	−	−	PROPN
cana-1800	183	3	bav	bav	NOUN
cana-1800	184	1	[	[	X
cana-1800	184	2	(	(	PUNCT
cana-1800	184	3	x1	x1	ADJ
cana-1800	184	4	,	,	PUNCT
cana-1800	184	5	y1)(x2	y1)(x2	NOUN
cana-1800	184	6	,	,	PUNCT
cana-1800	184	7	y2	y2	PROPN
cana-1800	184	8	)	)	PUNCT
cana-1800	184	9	]	]	PUNCT
cana-1800	185	1	=	=	PUNCT
cana-1800	186	1	−	−	PROPN
cana-1800	186	2	bav	bav	NOUN
cana-1800	186	3	(	(	PUNCT
cana-1800	186	4	x1x2	x1x2	NOUN
cana-1800	186	5	,	,	PUNCT
cana-1800	186	6	y1y2	y1y2	NOUN
cana-1800	186	7	)	)	PUNCT
cana-1800	186	8	=	=	SYM
cana-1800	186	9	rmax	rmax	ADJ
cana-1800	186	10	{	{	PUNCT
cana-1800	186	11	−	−	PROPN
cana-1800	186	12	av	av	PROPN
cana-1800	186	13	(	(	PUNCT
cana-1800	186	14	x1x2	x1x2	NOUN
cana-1800	186	15	)	)	PUNCT
cana-1800	186	16	,	,	PUNCT
cana-1800	186	17	−	−	PROPN
cana-1800	186	18	bv	bv	PROPN
cana-1800	186	19	(	(	PUNCT
cana-1800	186	20	y1y2	y1y2	PROPN
cana-1800	186	21	)	)	PUNCT
cana-1800	186	22	}	}	PUNCT
cana-1800	186	23	≤	≤	NUM
cana-1800	186	24	rmax{rmin	rmax{rmin	NOUN
cana-1800	186	25	{	{	PUNCT
cana-1800	186	26	−	−	PROPN
cana-1800	186	27	av	av	PROPN
cana-1800	186	28	(	(	PUNCT
cana-1800	186	29	x1	x1	PROPN
cana-1800	186	30	)	)	PUNCT
cana-1800	186	31	,	,	PUNCT
cana-1800	186	32	−	−	PROPN
cana-1800	186	33	av	av	PROPN
cana-1800	186	34	(	(	PUNCT
cana-1800	186	35	x2	x2	PROPN
cana-1800	186	36	)	)	PUNCT
cana-1800	186	37	}	}	PUNCT
cana-1800	186	38	,	,	PUNCT
cana-1800	186	39	rmin	rmin	NOUN
cana-1800	186	40	{	{	PUNCT
cana-1800	186	41	−	−	PROPN
cana-1800	186	42	bv	bv	PROPN
cana-1800	186	43	(	(	PUNCT
cana-1800	186	44	y1	y1	PROPN
cana-1800	186	45	)	)	PUNCT
cana-1800	186	46	,	,	PUNCT
cana-1800	186	47	−	−	PROPN
cana-1800	186	48	bv	bv	PROPN
cana-1800	186	49	(	(	PUNCT
cana-1800	186	50	y2	y2	PROPN
cana-1800	186	51	)	)	PUNCT
cana-1800	186	52	}	}	PUNCT
cana-1800	186	53	}	}	PUNCT
cana-1800	186	54	=	=	PUNCT
cana-1800	186	55	rmin{rmax	rmin{rmax	NOUN
cana-1800	186	56	{	{	PUNCT
cana-1800	186	57	−	−	PROPN
cana-1800	186	58	av	av	PROPN
cana-1800	186	59	(	(	PUNCT
cana-1800	186	60	x1	x1	PROPN
cana-1800	186	61	)	)	PUNCT
cana-1800	186	62	,	,	PUNCT
cana-1800	186	63	−	−	PROPN
cana-1800	186	64	bv	bv	PROPN
cana-1800	186	65	(	(	PUNCT
cana-1800	186	66	y1	y1	PROPN
cana-1800	186	67	)	)	PUNCT
cana-1800	186	68	}	}	PUNCT
cana-1800	186	69	,	,	PUNCT
cana-1800	186	70	rmax	rmax	ADJ
cana-1800	186	71	{	{	PUNCT
cana-1800	186	72	−	−	PROPN
cana-1800	186	73	av	av	INTJ
cana-1800	186	74	(	(	PUNCT
cana-1800	186	75	x2	x2	PROPN
cana-1800	186	76	)	)	PUNCT
cana-1800	186	77	,	,	PUNCT
cana-1800	186	78	−	−	PROPN
cana-1800	186	79	bv	bv	PROPN
cana-1800	186	80	(	(	PUNCT
cana-1800	186	81	y2	y2	PROPN
cana-1800	186	82	)	)	PUNCT
cana-1800	186	83	}	}	PUNCT
cana-1800	186	84	}	}	PUNCT
cana-1800	186	85	=	=	SYM
cana-1800	186	86	rmin	rmin	NOUN
cana-1800	186	87	{	{	PUNCT
cana-1800	186	88	−	−	PROPN
cana-1800	186	89	bav	bav	NOUN
cana-1800	186	90	(	(	PUNCT
cana-1800	186	91	x1	x1	PROPN
cana-1800	186	92	,	,	PUNCT
cana-1800	186	93	y1	y1	PROPN
cana-1800	186	94	)	)	PUNCT
cana-1800	186	95	,	,	PUNCT
cana-1800	186	96	−	−	PROPN
cana-1800	186	97	bav	bav	NOUN
cana-1800	186	98	(	(	PUNCT
cana-1800	186	99	x2	x2	PROPN
cana-1800	186	100	,	,	PUNCT
cana-1800	186	101	y2	y2	PROPN
cana-1800	186	102	)	)	PUNCT
cana-1800	186	103	}	}	PUNCT
cana-1800	186	104	.	.	PUNCT
cana-1800	187	1	therefore	therefore	ADV
cana-1800	187	2	−	−	X
cana-1800	187	3	bav	bav	NOUN
cana-1800	188	1	[	[	X
cana-1800	188	2	(	(	PUNCT
cana-1800	188	3	x1	x1	ADJ
cana-1800	188	4	,	,	PUNCT
cana-1800	188	5	y1)(x2	y1)(x2	NOUN
cana-1800	188	6	,	,	PUNCT
cana-1800	188	7	y2	y2	PROPN
cana-1800	188	8	)	)	PUNCT
cana-1800	188	9	]	]	PUNCT
cana-1800	189	1	≤	≤	NUM
cana-1800	189	2	rmin	rmin	NOUN
cana-1800	189	3	{	{	PUNCT
cana-1800	189	4	−	−	PROPN
cana-1800	189	5	bav	bav	NOUN
cana-1800	189	6	(	(	PUNCT
cana-1800	189	7	x1	x1	PROPN
cana-1800	189	8	,	,	PUNCT
cana-1800	189	9	y1	y1	PROPN
cana-1800	189	10	)	)	PUNCT
cana-1800	189	11	,	,	PUNCT
cana-1800	189	12	−	−	PROPN
cana-1800	189	13	bav	bav	NOUN
cana-1800	189	14	(	(	PUNCT
cana-1800	189	15	x2	x2	PROPN
cana-1800	189	16	,	,	PUNCT
cana-1800	189	17	y2	y2	PROPN
cana-1800	189	18	)	)	PUNCT
cana-1800	189	19	}	}	PUNCT
cana-1800	189	20	.	.	PUNCT
cana-1800	190	1	hence	hence	ADV
cana-1800	190	2	a×b	a×b	PROPN
cana-1800	190	3	is	be	AUX
cana-1800	190	4	a	a	DET
cana-1800	190	5	𝔹𝕍𝕍𝕀	𝔹𝕍𝕍𝕀	NOUN
cana-1800	190	6	of	of	ADP
cana-1800	190	7	r1×r2	r1×r2	PROPN
cana-1800	190	8	.	.	PUNCT
cana-1800	191	1	theorem	theorem	VERB
cana-1800	191	2	3.10	3.10	NUM
cana-1800	191	3	.	.	PUNCT
cana-1800	192	1	a	a	DET
cana-1800	192	2	product	product	NOUN
cana-1800	192	3	of	of	ADP
cana-1800	192	4	𝔹𝕍𝕍𝕀s	𝔹𝕍𝕍𝕀	NOUN
cana-1800	192	5	of	of	ADP
cana-1800	192	6	the	the	DET
cana-1800	192	7	semirings	semiring	NOUN
cana-1800	192	8	is	be	AUX
cana-1800	192	9	also	also	ADV
cana-1800	192	10	a	a	DET
cana-1800	192	11	𝔹𝕍𝕍𝕀	𝔹𝕍𝕍𝕀	NOUN
cana-1800	192	12	in	in	ADV
cana-1800	192	13	.	.	PUNCT
cana-1800	193	1	proof	proof	NOUN
cana-1800	193	2	.	.	PUNCT
cana-1800	194	1	from	from	ADP
cana-1800	194	2	the	the	DET
cana-1800	194	3	theorem	theorem	NOUN
cana-1800	194	4	3.9	3.9	NUM
cana-1800	194	5	,	,	PUNCT
cana-1800	194	6	the	the	DET
cana-1800	194	7	proof	proof	NOUN
cana-1800	194	8	follows	follow	VERB
cana-1800	194	9	.	.	PUNCT
cana-1800	195	1	definition	definition	NOUN
cana-1800	195	2	3.11	3.11	NUM
cana-1800	195	3	.	.	PUNCT
cana-1800	196	1	let	let	VERB
cana-1800	196	2	a	a	PRON
cana-1800	196	3	=	=	X
cana-1800	196	4			X
cana-1800	196	5	+	+	CCONJ
cana-1800	196	6	av	av	PROPN
cana-1800	196	7	,	,	PUNCT
cana-1800	196	8	−	−	PROPN
cana-1800	196	9	av	av	PROPN
cana-1800	196	10			PROPN
cana-1800	196	11	be	be	AUX
cana-1800	196	12	a	a	DET
cana-1800	196	13	𝔹𝕍𝕍𝕊𝕊	𝔹𝕍𝕍𝕊𝕊	NOUN
cana-1800	196	14	in	in	ADP
cana-1800	196	15	a	a	DET
cana-1800	196	16	set	set	NOUN
cana-1800	196	17	s	s	PROPN
cana-1800	196	18	,	,	PUNCT
cana-1800	196	19	the	the	DET
cana-1800	196	20	strongest	strong	ADJ
cana-1800	196	21	𝔹𝕍𝕍	𝔹𝕍𝕍	PROPN
cana-1800	196	22	relation	relation	NOUN
cana-1800	196	23	on	on	ADP
cana-1800	196	24	s	s	NOUN
cana-1800	196	25	,	,	PUNCT
cana-1800	196	26	that	that	PRON
cana-1800	196	27	is	be	AUX
cana-1800	196	28	a	a	DET
cana-1800	196	29	𝔹𝕍𝕍	𝔹𝕍𝕍	PROPN
cana-1800	196	30	relation	relation	NOUN
cana-1800	196	31	on	on	ADP
cana-1800	196	32	a	a	PRON
cana-1800	196	33	is	be	AUX
cana-1800	196	34	v	v	NOUN
cana-1800	196	35	=	=	SYM
cana-1800	196	36	{	{	PUNCT
cana-1800	196	37	(x	(x	PROPN
cana-1800	196	38	,	,	PUNCT
cana-1800	196	39	y	y	NOUN
cana-1800	196	40	)	)	PUNCT
cana-1800	196	41	,	,	PUNCT
cana-1800	197	1	+	+	CCONJ
cana-1800	197	2	vv	vv	ADJ
cana-1800	197	3	(	(	PUNCT
cana-1800	197	4	x	x	NOUN
cana-1800	197	5	,	,	PUNCT
cana-1800	197	6	y	y	PROPN
cana-1800	197	7	)	)	PUNCT
cana-1800	197	8	,	,	PUNCT
cana-1800	197	9	−	−	PROPN
cana-1800	198	1	vv	vv	INTJ
cana-1800	198	2	(	(	PUNCT
cana-1800	198	3	x	x	X
cana-1800	198	4	,	,	PUNCT
cana-1800	198	5	y)	y)	PROPN
cana-1800	198	6	/	/	SYM
cana-1800	198	7	x	x	NOUN
cana-1800	198	8	,	,	PUNCT
cana-1800	198	9	ys	ys	PROPN
cana-1800	198	10	}	}	PUNCT
cana-1800	198	11	given	give	VERB
cana-1800	198	12	by	by	ADP
cana-1800	198	13	+	+	X
cana-1800	198	14	vv	vv	ADJ
cana-1800	198	15	(	(	PUNCT
cana-1800	198	16	x	x	NOUN
cana-1800	198	17	,	,	PUNCT
cana-1800	198	18	y	y	NOUN
cana-1800	198	19	)	)	PUNCT
cana-1800	198	20	=	=	SYM
cana-1800	198	21	rmin	rmin	NOUN
cana-1800	198	22	{	{	PUNCT
cana-1800	198	23	+	+	NOUN
cana-1800	198	24	av	av	PROPN
cana-1800	198	25	(	(	PUNCT
cana-1800	198	26	x	x	NOUN
cana-1800	198	27	)	)	PUNCT
cana-1800	198	28	,	,	PUNCT
cana-1800	198	29	+	+	CCONJ
cana-1800	198	30	av	av	PROPN
cana-1800	198	31	(	(	PUNCT
cana-1800	198	32	y	y	NOUN
cana-1800	198	33	)	)	PUNCT
cana-1800	198	34	}	}	PUNCT
cana-1800	198	35	and	and	CCONJ
cana-1800	198	36	−	−	X
cana-1800	199	1	vv	vv	INTJ
cana-1800	199	2	(	(	PUNCT
cana-1800	199	3	x	x	NOUN
cana-1800	199	4	,	,	PUNCT
cana-1800	199	5	y	y	NOUN
cana-1800	199	6	)	)	PUNCT
cana-1800	199	7	=	=	SYM
cana-1800	200	1	rmax	rmax	ADJ
cana-1800	200	2	{	{	PUNCT
cana-1800	200	3	−	−	PROPN
cana-1800	200	4	av	av	INTJ
cana-1800	200	5	(	(	PUNCT
cana-1800	200	6	x	x	NOUN
cana-1800	200	7	)	)	PUNCT
cana-1800	200	8	,	,	PUNCT
cana-1800	200	9	−	−	PROPN
cana-1800	200	10	av	av	PROPN
cana-1800	200	11	(	(	PUNCT
cana-1800	200	12	y	y	NOUN
cana-1800	200	13	)	)	PUNCT
cana-1800	200	14	}	}	PUNCT
cana-1800	200	15	,	,	PUNCT
cana-1800	200	16			NOUN
cana-1800	200	17	x	x	SYM
cana-1800	200	18	,	,	PUNCT
cana-1800	200	19	ys	ys	PROPN
cana-1800	200	20	.	.	PUNCT
cana-1800	200	21	theorem	theorem	VERB
cana-1800	200	22	3.12	3.12	NUM
cana-1800	200	23	.	.	PUNCT
cana-1800	201	1	let	let	VERB
cana-1800	201	2	a	a	PRON
cana-1800	201	3	=	=	X
cana-1800	201	4			X
cana-1800	201	5	+	+	CCONJ
cana-1800	201	6	av	av	PROPN
cana-1800	201	7	,	,	PUNCT
cana-1800	201	8	−	−	PROPN
cana-1800	201	9	av	av	PROPN
cana-1800	201	10			PROPN
cana-1800	201	11	be	be	AUX
cana-1800	201	12	a	a	DET
cana-1800	201	13	𝔹𝕍𝕍𝕊𝕊	𝔹𝕍𝕍𝕊𝕊	NOUN
cana-1800	201	14	of	of	ADP
cana-1800	201	15	a	a	DET
cana-1800	201	16	semiring	semire	VERB
cana-1800	201	17	r	r	NOUN
cana-1800	201	18	and	and	CCONJ
cana-1800	201	19	v	v	NOUN
cana-1800	201	20	=	=	PUNCT
cana-1800	201	21			PUNCT
cana-1800	202	1	+	+	CCONJ
cana-1800	202	2	vv	vv	NOUN
cana-1800	202	3	,	,	PUNCT
cana-1800	202	4	−	−	PROPN
cana-1800	202	5	vv	vv	ADP
cana-1800	202	6			PROPN
cana-1800	202	7	be	be	AUX
cana-1800	202	8	the	the	DET
cana-1800	202	9	strongest	strong	ADJ
cana-1800	202	10	𝔹𝕍𝕍	𝔹𝕍𝕍	PROPN
cana-1800	202	11	relation	relation	NOUN
cana-1800	202	12	of	of	ADP
cana-1800	202	13	r.	r.	PROPN
cana-1800	202	14	then	then	ADV
cana-1800	202	15	a	a	PRON
cana-1800	202	16	is	be	AUX
cana-1800	202	17	a	a	DET
cana-1800	202	18	𝔹𝕍𝕍𝕀	𝔹𝕍𝕍𝕀	NOUN
cana-1800	202	19	of	of	ADP
cana-1800	202	20	r	r	NOUN
cana-1800	202	21			PROPN
cana-1800	202	22	v	v	NOUN
cana-1800	202	23	is	be	AUX
cana-1800	202	24	a	a	DET
cana-1800	202	25	𝔹𝕍𝕍𝕀	𝔹𝕍𝕍𝕀	NOUN
cana-1800	202	26	of	of	ADP
cana-1800	202	27	r×r	r×r	PROPN
cana-1800	202	28	.	.	PROPN
cana-1800	202	29	proof	proof	NOUN
cana-1800	202	30	.	.	PUNCT
cana-1800	203	1	suppose	suppose	VERB
cana-1800	203	2	that	that	SCONJ
cana-1800	203	3	a	a	PRON
cana-1800	203	4	is	be	AUX
cana-1800	203	5	a	a	DET
cana-1800	203	6	𝔹𝕍𝕍𝕀	𝔹𝕍𝕍𝕀	NOUN
cana-1800	203	7	of	of	ADP
cana-1800	203	8	r.	r.	PROPN
cana-1800	203	9	then	then	ADV
cana-1800	203	10	for	for	ADP
cana-1800	203	11	any	any	DET
cana-1800	203	12	x	x	SYM
cana-1800	203	13	=	=	SYM
cana-1800	203	14	(	(	PUNCT
cana-1800	203	15	x1	x1	PROPN
cana-1800	203	16	,	,	PUNCT
cana-1800	203	17	x2	x2	PROPN
cana-1800	203	18	)	)	PUNCT
cana-1800	203	19	,	,	PUNCT
cana-1800	203	20	y	y	PROPN
cana-1800	203	21	=	=	SYM
cana-1800	203	22	(	(	PUNCT
cana-1800	203	23	y1	y1	INTJ
cana-1800	203	24	,	,	PUNCT
cana-1800	203	25	y2	y2	PROPN
cana-1800	203	26	)	)	PUNCT
cana-1800	203	27	are	be	AUX
cana-1800	203	28	in	in	ADP
cana-1800	203	29	r×r	r×r	PROPN
cana-1800	203	30	.	.	PUNCT
cana-1800	204	1	now	now	ADV
cana-1800	204	2	+	+	NUM
cana-1800	204	3	vv	vv	X
cana-1800	204	4	(	(	PUNCT
cana-1800	204	5	x+y	x+y	NUM
cana-1800	204	6	)	)	PUNCT
cana-1800	204	7	=	=	PUNCT
cana-1800	205	1	+	+	CCONJ
cana-1800	205	2	vv	vv	X
cana-1800	205	3	[	[	X
cana-1800	205	4	(	(	PUNCT
cana-1800	205	5	x1	x1	ADJ
cana-1800	205	6	,	,	PUNCT
cana-1800	205	7	x2)+(y1	x2)+(y1	NUM
cana-1800	205	8	,	,	PUNCT
cana-1800	205	9	y2	y2	NOUN
cana-1800	205	10	)	)	PUNCT
cana-1800	205	11	]	]	PUNCT
cana-1800	206	1	=	=	PUNCT
cana-1800	207	1	+	+	CCONJ
cana-1800	207	2	vv	vv	X
cana-1800	207	3	(	(	PUNCT
cana-1800	207	4	x1+y1	x1+y1	PROPN
cana-1800	207	5	,	,	PUNCT
cana-1800	207	6	x2+y2	x2+y2	NOUN
cana-1800	207	7	)	)	PUNCT
cana-1800	207	8	=	=	SYM
cana-1800	207	9	rmin	rmin	NOUN
cana-1800	207	10	{	{	PUNCT
cana-1800	207	11	+	+	NUM
cana-1800	207	12	av	av	PROPN
cana-1800	207	13	(	(	PUNCT
cana-1800	207	14	x1+y1	x1+y1	PROPN
cana-1800	207	15	)	)	PUNCT
cana-1800	207	16	,	,	PUNCT
cana-1800	207	17	+	+	CCONJ
cana-1800	207	18	av	av	PROPN
cana-1800	207	19	(	(	PUNCT
cana-1800	207	20	x2+y2	x2+y2	NOUN
cana-1800	207	21	)	)	PUNCT
cana-1800	207	22	}	}	PUNCT
cana-1800	207	23			X
cana-1800	207	24	rmin{rmin	rmin{rmin	NOUN
cana-1800	207	25	{	{	PUNCT
cana-1800	207	26	+	+	NUM
cana-1800	207	27	av	av	PROPN
cana-1800	207	28	(	(	PUNCT
cana-1800	207	29	x1	x1	PROPN
cana-1800	207	30	)	)	PUNCT
cana-1800	207	31	,	,	PUNCT
cana-1800	207	32	+	+	CCONJ
cana-1800	207	33	av	av	PROPN
cana-1800	207	34	(	(	PUNCT
cana-1800	207	35	y1	y1	NOUN
cana-1800	207	36	)	)	PUNCT
cana-1800	207	37	}	}	PUNCT
cana-1800	207	38	,	,	PUNCT
cana-1800	207	39	rmin	rmin	NOUN
cana-1800	207	40	{	{	PUNCT
cana-1800	207	41	+	+	NUM
cana-1800	207	42	av	av	PROPN
cana-1800	207	43	(	(	PUNCT
cana-1800	207	44	x2	x2	PROPN
cana-1800	207	45	)	)	PUNCT
cana-1800	207	46	,	,	PUNCT
cana-1800	207	47	+	+	CCONJ
cana-1800	207	48	av	av	PROPN
cana-1800	207	49	(	(	PUNCT
cana-1800	207	50	y2	y2	PROPN
cana-1800	207	51	)	)	PUNCT
cana-1800	207	52	}	}	PUNCT
cana-1800	207	53	}	}	PUNCT
cana-1800	207	54	=	=	SYM
cana-1800	207	55	rmin{rmin	rmin{rmin	NOUN
cana-1800	207	56	{	{	PUNCT
cana-1800	207	57	+	+	NUM
cana-1800	207	58	av	av	PROPN
cana-1800	207	59	(	(	PUNCT
cana-1800	207	60	x1	x1	PROPN
cana-1800	207	61	)	)	PUNCT
cana-1800	207	62	,	,	PUNCT
cana-1800	207	63	+	+	CCONJ
cana-1800	207	64	av	av	PROPN
cana-1800	207	65	(	(	PUNCT
cana-1800	207	66	x2	x2	PROPN
cana-1800	207	67	)	)	PUNCT
cana-1800	207	68	}	}	PUNCT
cana-1800	207	69	,	,	PUNCT
cana-1800	207	70	rmin	rmin	VERB
cana-1800	207	71	{	{	PUNCT
cana-1800	207	72	+	+	CCONJ
cana-1800	207	73	av	av	PROPN
cana-1800	207	74	(	(	PUNCT
cana-1800	207	75	y1	y1	PROPN
cana-1800	207	76	)	)	PUNCT
cana-1800	207	77	,	,	PUNCT
cana-1800	207	78	+	+	CCONJ
cana-1800	207	79	av	av	PROPN
cana-1800	207	80	(	(	PUNCT
cana-1800	207	81	y2	y2	PROPN
cana-1800	207	82	)	)	PUNCT
cana-1800	207	83	}	}	PUNCT
cana-1800	207	84	}	}	PUNCT
cana-1800	207	85	=	=	SYM
cana-1800	207	86	rmin	rmin	NOUN
cana-1800	207	87	{	{	PUNCT
cana-1800	207	88	+	+	X
cana-1800	207	89	vv	vv	CCONJ
cana-1800	207	90	(	(	PUNCT
cana-1800	207	91	x1	x1	PROPN
cana-1800	207	92	,	,	PUNCT
cana-1800	207	93	x2	x2	PROPN
cana-1800	207	94	)	)	PUNCT
cana-1800	207	95	,	,	PUNCT
cana-1800	207	96	+	+	CCONJ
cana-1800	207	97	vv	vv	X
cana-1800	207	98	(	(	PUNCT
cana-1800	207	99	y1	y1	INTJ
cana-1800	207	100	,	,	PUNCT
cana-1800	207	101	y2	y2	PROPN
cana-1800	207	102	)	)	PUNCT
cana-1800	207	103	}	}	PUNCT
cana-1800	207	104	=	=	SYM
cana-1800	207	105	rmin	rmin	NOUN
cana-1800	207	106	{	{	PUNCT
cana-1800	207	107	+	+	NUM
cana-1800	207	108	vv	vv	NOUN
cana-1800	207	109	(	(	PUNCT
cana-1800	207	110	x	x	NOUN
cana-1800	207	111	)	)	PUNCT
cana-1800	207	112	,	,	PUNCT
cana-1800	207	113	+	+	CCONJ
cana-1800	207	114	vv	vv	CCONJ
cana-1800	207	115	(	(	PUNCT
cana-1800	207	116	y	y	NOUN
cana-1800	207	117	)	)	PUNCT
cana-1800	207	118	}	}	PUNCT
cana-1800	207	119	.	.	PUNCT
cana-1800	208	1	therefore	therefore	ADV
cana-1800	208	2	+	+	CCONJ
cana-1800	208	3	vv	vv	ADJ
cana-1800	208	4	(	(	PUNCT
cana-1800	208	5	x+y	x+y	NUM
cana-1800	208	6	)	)	PUNCT
cana-1800	208	7			NUM
cana-1800	208	8	rmin	rmin	NOUN
cana-1800	208	9	{	{	PUNCT
cana-1800	208	10	+	+	NUM
cana-1800	208	11	vv	vv	NOUN
cana-1800	208	12	(	(	PUNCT
cana-1800	208	13	x	x	NOUN
cana-1800	208	14	)	)	PUNCT
cana-1800	208	15	,	,	PUNCT
cana-1800	208	16	+	+	CCONJ
cana-1800	208	17	vv	vv	CCONJ
cana-1800	208	18	(	(	PUNCT
cana-1800	208	19	y	y	NOUN
cana-1800	208	20	)	)	PUNCT
cana-1800	208	21	}	}	PUNCT
cana-1800	208	22	,	,	PUNCT
cana-1800	208	23			NOUN
cana-1800	208	24	x	x	PROPN
cana-1800	208	25	,	,	PUNCT
cana-1800	208	26	yr×r	yr×r	PROPN
cana-1800	208	27	.	.	PROPN
cana-1800	208	28	and	and	CCONJ
cana-1800	209	1	+	+	NUM
cana-1800	209	2	vv	vv	CCONJ
cana-1800	209	3	(	(	PUNCT
cana-1800	209	4	xy	xy	NOUN
cana-1800	209	5	)	)	PUNCT
cana-1800	209	6	=	=	PUNCT
cana-1800	210	1	+	+	CCONJ
cana-1800	210	2	vv	vv	X
cana-1800	210	3	[	[	X
cana-1800	210	4	(	(	PUNCT
cana-1800	210	5	x1	x1	PROPN
cana-1800	210	6	,	,	PUNCT
cana-1800	210	7	x2)(y1	x2)(y1	NUM
cana-1800	210	8	,	,	PUNCT
cana-1800	210	9	y2	y2	PROPN
cana-1800	210	10	)	)	PUNCT
cana-1800	210	11	]	]	PUNCT
cana-1800	211	1	=	=	PUNCT
cana-1800	212	1	+	+	CCONJ
cana-1800	212	2	vv	vv	ADJ
cana-1800	212	3	(	(	PUNCT
cana-1800	212	4	x1y1	x1y1	X
cana-1800	212	5	,	,	PUNCT
cana-1800	212	6	x2y2	x2y2	X
cana-1800	212	7	)	)	PUNCT
cana-1800	212	8	=	=	SYM
cana-1800	212	9	rmin	rmin	NOUN
cana-1800	212	10	{	{	PUNCT
cana-1800	212	11	+	+	NOUN
cana-1800	212	12	av	av	PROPN
cana-1800	212	13	(	(	PUNCT
cana-1800	212	14	x1y1	x1y1	PROPN
cana-1800	212	15	)	)	PUNCT
cana-1800	212	16	,	,	PUNCT
cana-1800	212	17	+	+	CCONJ
cana-1800	212	18	av	av	PROPN
cana-1800	212	19	(	(	PUNCT
cana-1800	212	20	x2y2	x2y2	NOUN
cana-1800	212	21	)	)	PUNCT
cana-1800	212	22	}	}	PUNCT
cana-1800	212	23			NUM
cana-1800	212	24	rmin{rmax	rmin{rmax	NOUN
cana-1800	212	25	{	{	PUNCT
cana-1800	212	26	+	+	NOUN
cana-1800	212	27	av	av	PROPN
cana-1800	212	28	(	(	PUNCT
cana-1800	212	29	x1	x1	PROPN
cana-1800	212	30	)	)	PUNCT
cana-1800	212	31	,	,	PUNCT
cana-1800	212	32	+	+	CCONJ
cana-1800	212	33	av	av	PROPN
cana-1800	212	34	(	(	PUNCT
cana-1800	212	35	y1	y1	NOUN
cana-1800	212	36	)	)	PUNCT
cana-1800	212	37	}	}	PUNCT
cana-1800	212	38	,	,	PUNCT
cana-1800	212	39	rmax	rmax	ADJ
cana-1800	212	40	{	{	PUNCT
cana-1800	212	41	+	+	NUM
cana-1800	212	42	av	av	PROPN
cana-1800	212	43	(	(	PUNCT
cana-1800	212	44	x2	x2	PROPN
cana-1800	212	45	)	)	PUNCT
cana-1800	212	46	,	,	PUNCT
cana-1800	212	47	+	+	CCONJ
cana-1800	212	48	av	av	PROPN
cana-1800	212	49	(	(	PUNCT
cana-1800	212	50	y2	y2	PROPN
cana-1800	212	51	)	)	PUNCT
cana-1800	212	52	}	}	PUNCT
cana-1800	212	53	}	}	PUNCT
cana-1800	212	54			NUM
cana-1800	212	55	rmax{rmin	rmax{rmin	NOUN
cana-1800	212	56	{	{	PUNCT
cana-1800	212	57	+	+	X
cana-1800	212	58	av	av	PROPN
cana-1800	212	59	(	(	PUNCT
cana-1800	212	60	x1	x1	PROPN
cana-1800	212	61	)	)	PUNCT
cana-1800	212	62	,	,	PUNCT
cana-1800	212	63	+	+	CCONJ
cana-1800	212	64	av	av	PROPN
cana-1800	212	65	(	(	PUNCT
cana-1800	212	66	x2	x2	PROPN
cana-1800	212	67	)	)	PUNCT
cana-1800	212	68	}	}	PUNCT
cana-1800	212	69	,	,	PUNCT
cana-1800	212	70	rmin	rmin	NOUN
cana-1800	212	71	{	{	PUNCT
cana-1800	212	72	+	+	NUM
cana-1800	212	73	av	av	PROPN
cana-1800	212	74	(	(	PUNCT
cana-1800	212	75	y1	y1	PROPN
cana-1800	212	76	)	)	PUNCT
cana-1800	212	77	,	,	PUNCT
cana-1800	212	78	+	+	CCONJ
cana-1800	212	79	av	av	PROPN
cana-1800	212	80	(	(	PUNCT
cana-1800	212	81	y2	y2	PROPN
cana-1800	212	82	)	)	PUNCT
cana-1800	212	83	}	}	PUNCT
cana-1800	212	84	}	}	PUNCT
cana-1800	212	85	=	=	SYM
cana-1800	212	86	rmax	rmax	ADJ
cana-1800	212	87	{	{	PUNCT
cana-1800	212	88	+	+	X
cana-1800	212	89	vv	vv	ADP
cana-1800	212	90	(	(	PUNCT
cana-1800	212	91	x1	x1	PROPN
cana-1800	212	92	,	,	PUNCT
cana-1800	212	93	x2	x2	PROPN
cana-1800	212	94	)	)	PUNCT
cana-1800	212	95	,	,	PUNCT
cana-1800	212	96	+	+	CCONJ
cana-1800	213	1	vv	vv	X
cana-1800	213	2	(	(	PUNCT
cana-1800	213	3	y1	y1	INTJ
cana-1800	213	4	,	,	PUNCT
cana-1800	213	5	y2)}=	y2)}=	SYM
cana-1800	213	6	rmax	rmax	NOUN
cana-1800	213	7	{	{	PUNCT
cana-1800	213	8	+	+	NUM
cana-1800	213	9	vv	vv	ADP
cana-1800	213	10	(	(	PUNCT
cana-1800	213	11	x	x	NOUN
cana-1800	213	12	)	)	PUNCT
cana-1800	213	13	,	,	PUNCT
cana-1800	213	14	+	+	CCONJ
cana-1800	213	15	vv	vv	CCONJ
cana-1800	213	16	(	(	PUNCT
cana-1800	213	17	y	y	NOUN
cana-1800	213	18	)	)	PUNCT
cana-1800	213	19	}	}	PUNCT
cana-1800	213	20	.	.	PUNCT
cana-1800	214	1	therefore	therefore	ADV
cana-1800	214	2	+	+	CCONJ
cana-1800	214	3	vv	vv	X
cana-1800	214	4	(	(	PUNCT
cana-1800	214	5	xy	xy	NOUN
cana-1800	214	6	)	)	PUNCT
cana-1800	214	7			NUM
cana-1800	214	8	rmax	rmax	NOUN
cana-1800	214	9	{	{	PUNCT
cana-1800	214	10	+	+	NUM
cana-1800	214	11	vv	vv	ADP
cana-1800	214	12	(	(	PUNCT
cana-1800	214	13	x	x	NOUN
cana-1800	214	14	)	)	PUNCT
cana-1800	214	15	,	,	PUNCT
cana-1800	214	16	+	+	CCONJ
cana-1800	215	1	vv	vv	CCONJ
cana-1800	215	2	(	(	PUNCT
cana-1800	215	3	y	y	NOUN
cana-1800	215	4	)	)	PUNCT
cana-1800	215	5	}	}	PUNCT
cana-1800	215	6	,	,	PUNCT
cana-1800	215	7			NOUN
cana-1800	215	8	x	x	PROPN
cana-1800	215	9	,	,	PUNCT
cana-1800	215	10	yr×r	yr×r	PROPN
cana-1800	215	11	.	.	PROPN
cana-1800	216	1	also	also	ADV
cana-1800	216	2	we	we	PRON
cana-1800	216	3	have	have	VERB
cana-1800	216	4	−	−	NUM
cana-1800	216	5	vv	vv	NOUN
cana-1800	216	6	(	(	PUNCT
cana-1800	216	7	x+y	x+y	NUM
cana-1800	216	8	)	)	PUNCT
cana-1800	217	1	=	=	SYM
cana-1800	217	2	−	−	NOUN
cana-1800	218	1	vv	vv	ADP
cana-1800	219	1	[	[	X
cana-1800	219	2	(	(	PUNCT
cana-1800	219	3	x1	x1	ADJ
cana-1800	219	4	,	,	PUNCT
cana-1800	219	5	x2)+(y1	x2)+(y1	NUM
cana-1800	219	6	,	,	PUNCT
cana-1800	219	7	y2	y2	NOUN
cana-1800	219	8	)	)	PUNCT
cana-1800	219	9	]	]	PUNCT
cana-1800	220	1	=	=	PUNCT
cana-1800	220	2	−	−	PROPN
cana-1800	221	1	vv	vv	INTJ
cana-1800	221	2	(	(	PUNCT
cana-1800	221	3	x1+y1	x1+y1	PROPN
cana-1800	221	4	,	,	PUNCT
cana-1800	221	5	x2+y2	x2+y2	NOUN
cana-1800	221	6	)	)	PUNCT
cana-1800	221	7	=	=	SYM
cana-1800	222	1	rmax	rmax	ADJ
cana-1800	222	2	{	{	PUNCT
cana-1800	222	3	−	−	PROPN
cana-1800	222	4	av	av	X
cana-1800	222	5	(	(	PUNCT
cana-1800	222	6	x1+y1	x1+y1	PROPN
cana-1800	222	7	)	)	PUNCT
cana-1800	222	8	,	,	PUNCT
cana-1800	222	9	−	−	PROPN
cana-1800	222	10	av	av	PROPN
cana-1800	222	11	(	(	PUNCT
cana-1800	222	12	x2+y2	x2+y2	NOUN
cana-1800	222	13	)	)	PUNCT
cana-1800	222	14	}	}	PUNCT
cana-1800	222	15			NUM
cana-1800	222	16	rmax{rmax	rmax{rmax	NOUN
cana-1800	222	17	{	{	PUNCT
cana-1800	222	18	−	−	PROPN
cana-1800	222	19	av	av	PROPN
cana-1800	222	20	(	(	PUNCT
cana-1800	222	21	x1	x1	PROPN
cana-1800	222	22	)	)	PUNCT
cana-1800	222	23	,	,	PUNCT
cana-1800	222	24	−	−	PROPN
cana-1800	222	25	av	av	PROPN
cana-1800	222	26	(	(	PUNCT
cana-1800	222	27	y1	y1	PROPN
cana-1800	222	28	)	)	PUNCT
cana-1800	222	29	}	}	PUNCT
cana-1800	222	30	,	,	PUNCT
cana-1800	222	31	rmax	rmax	ADJ
cana-1800	222	32	{	{	PUNCT
cana-1800	222	33	−	−	PROPN
cana-1800	222	34	av	av	INTJ
cana-1800	222	35	(	(	PUNCT
cana-1800	222	36	x2	x2	PROPN
cana-1800	222	37	)	)	PUNCT
cana-1800	222	38	,	,	PUNCT
cana-1800	222	39	−	−	PROPN
cana-1800	222	40	av	av	PROPN
cana-1800	222	41	(	(	PUNCT
cana-1800	222	42	y2	y2	PROPN
cana-1800	222	43	)	)	PUNCT
cana-1800	222	44	}	}	PUNCT
cana-1800	222	45	}	}	PUNCT
cana-1800	222	46	=	=	SYM
cana-1800	222	47	rmax{rmax	rmax{rmax	NOUN
cana-1800	222	48	{	{	PUNCT
cana-1800	222	49	−	−	PROPN
cana-1800	222	50	av	av	PROPN
cana-1800	222	51	(	(	PUNCT
cana-1800	222	52	x1	x1	PROPN
cana-1800	222	53	)	)	PUNCT
cana-1800	222	54	,	,	PUNCT
cana-1800	222	55	−	−	PROPN
cana-1800	222	56	av	av	PROPN
cana-1800	222	57	(	(	PUNCT
cana-1800	222	58	x2	x2	PROPN
cana-1800	222	59	)	)	PUNCT
cana-1800	222	60	}	}	PUNCT
cana-1800	222	61	,	,	PUNCT
cana-1800	222	62	rmax	rmax	ADJ
cana-1800	222	63	{	{	PUNCT
cana-1800	222	64	−	−	PROPN
cana-1800	222	65	av	av	PROPN
cana-1800	222	66	(	(	PUNCT
cana-1800	222	67	y1	y1	PROPN
cana-1800	222	68	)	)	PUNCT
cana-1800	222	69	,	,	PUNCT
cana-1800	222	70	−	−	PROPN
cana-1800	222	71	av	av	PROPN
cana-1800	222	72	(	(	PUNCT
cana-1800	222	73	y2)}}=	y2)}}=	PROPN
cana-1800	222	74	rmax	rmax	NOUN
cana-1800	222	75	{	{	PUNCT
cana-1800	222	76	−	−	PROPN
cana-1800	222	77	vv	vv	X
cana-1800	222	78	(	(	PUNCT
cana-1800	222	79	x1	x1	PROPN
cana-1800	222	80	,	,	PUNCT
cana-1800	222	81	x2	x2	PROPN
cana-1800	222	82	)	)	PUNCT
cana-1800	222	83	,	,	PUNCT
cana-1800	222	84	−	−	PROPN
cana-1800	223	1	vv	vv	PROPN
cana-1800	223	2	(	(	PUNCT
cana-1800	223	3	y1	y1	INTJ
cana-1800	223	4	,	,	PUNCT
cana-1800	223	5	y2	y2	PROPN
cana-1800	223	6	)	)	PUNCT
cana-1800	223	7	}	}	PUNCT
cana-1800	224	1	=	=	SYM
cana-1800	224	2	rmax	rmax	ADJ
cana-1800	224	3	{	{	PUNCT
cana-1800	224	4	−	−	PROPN
cana-1800	224	5	vv	vv	INTJ
cana-1800	224	6	(	(	PUNCT
cana-1800	224	7	x	x	NOUN
cana-1800	224	8	)	)	PUNCT
cana-1800	224	9	,	,	PUNCT
cana-1800	224	10	−	−	PROPN
cana-1800	225	1	vv	vv	INTJ
cana-1800	225	2	(	(	PUNCT
cana-1800	225	3	y	y	NOUN
cana-1800	225	4	)	)	PUNCT
cana-1800	225	5	}	}	PUNCT
cana-1800	225	6	.	.	PUNCT
cana-1800	226	1	therefore	therefore	ADV
cana-1800	226	2	−	−	PROPN
cana-1800	226	3	vv	vv	PROPN
cana-1800	226	4	(	(	PUNCT
cana-1800	226	5	x+y	x+y	NUM
cana-1800	226	6	)	)	PUNCT
cana-1800	226	7			NUM
cana-1800	226	8	rmax	rmax	NOUN
cana-1800	226	9	{	{	PUNCT
cana-1800	226	10	−	−	PROPN
cana-1800	226	11	vv	vv	INTJ
cana-1800	226	12	(	(	PUNCT
cana-1800	226	13	x	x	NOUN
cana-1800	226	14	)	)	PUNCT
cana-1800	226	15	,	,	PUNCT
cana-1800	226	16	−	−	PROPN
cana-1800	227	1	vv	vv	INTJ
cana-1800	227	2	(	(	PUNCT
cana-1800	227	3	y	y	NOUN
cana-1800	227	4	)	)	PUNCT
cana-1800	227	5	}	}	PUNCT
cana-1800	227	6	,	,	PUNCT
cana-1800	227	7			NOUN
cana-1800	227	8	x	x	PROPN
cana-1800	227	9	,	,	PUNCT
cana-1800	227	10	yr×r	yr×r	PROPN
cana-1800	227	11	.	.	PROPN
cana-1800	227	12	and	and	CCONJ
cana-1800	227	13	−	−	PROPN
cana-1800	228	1	vv	vv	INTJ
cana-1800	228	2	(	(	PUNCT
cana-1800	228	3	xy	xy	NOUN
cana-1800	228	4	)	)	PUNCT
cana-1800	228	5	=	=	SYM
cana-1800	229	1	−	−	NOUN
cana-1800	229	2	vv	vv	ADP
cana-1800	229	3	[	[	X
cana-1800	229	4	(	(	PUNCT
cana-1800	229	5	x1	x1	PROPN
cana-1800	229	6	,	,	PUNCT
cana-1800	229	7	x2)(y1	x2)(y1	NUM
cana-1800	229	8	,	,	PUNCT
cana-1800	229	9	y2	y2	PROPN
cana-1800	229	10	)	)	PUNCT
cana-1800	229	11	]	]	PUNCT
cana-1800	230	1	=	=	PUNCT
cana-1800	230	2	−	−	PROPN
cana-1800	231	1	vv	vv	INTJ
cana-1800	231	2	(	(	PUNCT
cana-1800	231	3	x1y1	x1y1	X
cana-1800	231	4	,	,	PUNCT
cana-1800	231	5	x2y2	x2y2	X
cana-1800	231	6	)	)	PUNCT
cana-1800	231	7	=	=	SYM
cana-1800	232	1	rmax	rmax	ADJ
cana-1800	232	2	{	{	PUNCT
cana-1800	232	3	−	−	PROPN
cana-1800	232	4	av	av	PROPN
cana-1800	232	5	(	(	PUNCT
cana-1800	232	6	x1y1	x1y1	PROPN
cana-1800	232	7	)	)	PUNCT
cana-1800	232	8	,	,	PUNCT
cana-1800	232	9	−	−	PROPN
cana-1800	232	10	av	av	PROPN
cana-1800	232	11	(	(	PUNCT
cana-1800	232	12	x2y2	x2y2	NOUN
cana-1800	232	13	)	)	PUNCT
cana-1800	232	14	}	}	PUNCT
cana-1800	232	15			NOUN
cana-1800	232	16	rmax{rmin	rmax{rmin	NOUN
cana-1800	232	17	{	{	PUNCT
cana-1800	232	18	−	−	PROPN
cana-1800	232	19	av	av	PROPN
cana-1800	232	20	(	(	PUNCT
cana-1800	232	21	x1	x1	PROPN
cana-1800	232	22	)	)	PUNCT
cana-1800	232	23	,	,	PUNCT
cana-1800	232	24	−	−	PROPN
cana-1800	232	25	av	av	PROPN
cana-1800	232	26	(	(	PUNCT
cana-1800	232	27	y1	y1	PROPN
cana-1800	232	28	)	)	PUNCT
cana-1800	232	29	}	}	PUNCT
cana-1800	232	30	,	,	PUNCT
cana-1800	232	31	rmin	rmin	NOUN
cana-1800	232	32	{	{	PUNCT
cana-1800	232	33	−	−	PROPN
cana-1800	232	34	av	av	INTJ
cana-1800	232	35	(	(	PUNCT
cana-1800	232	36	x2	x2	PROPN
cana-1800	232	37	)	)	PUNCT
cana-1800	232	38	,	,	PUNCT
cana-1800	232	39	−	−	PROPN
cana-1800	232	40	av	av	PROPN
cana-1800	232	41	(	(	PUNCT
cana-1800	232	42	y2	y2	PROPN
cana-1800	232	43	)	)	PUNCT
cana-1800	232	44	}	}	PUNCT
cana-1800	232	45	}	}	PUNCT
cana-1800	232	46	=	=	PUNCT
cana-1800	232	47	rmin{rmax	rmin{rmax	NOUN
cana-1800	232	48	{	{	PUNCT
cana-1800	232	49	−	−	PROPN
cana-1800	232	50	av	av	PROPN
cana-1800	232	51	(	(	PUNCT
cana-1800	232	52	x1	x1	PROPN
cana-1800	232	53	)	)	PUNCT
cana-1800	232	54	,	,	PUNCT
cana-1800	232	55	−	−	PROPN
cana-1800	232	56	av	av	PROPN
cana-1800	232	57	(	(	PUNCT
cana-1800	232	58	x2	x2	PROPN
cana-1800	232	59	)	)	PUNCT
cana-1800	232	60	}	}	PUNCT
cana-1800	232	61	,	,	PUNCT
cana-1800	232	62	rmax	rmax	ADJ
cana-1800	232	63	{	{	PUNCT
cana-1800	232	64	−	−	PROPN
cana-1800	232	65	av	av	PROPN
cana-1800	232	66	(	(	PUNCT
cana-1800	232	67	y1	y1	PROPN
cana-1800	232	68	)	)	PUNCT
cana-1800	232	69	,	,	PUNCT
cana-1800	232	70	−	−	PROPN
cana-1800	233	1	av	av	PROPN
cana-1800	233	2	(	(	PUNCT
cana-1800	233	3	y2)}}=	y2)}}=	NOUN
cana-1800	233	4	rmin	rmin	NOUN
cana-1800	233	5	{	{	PUNCT
cana-1800	233	6	−	−	PROPN
cana-1800	233	7	vv	vv	X
cana-1800	233	8	(	(	PUNCT
cana-1800	233	9	x1	x1	PROPN
cana-1800	233	10	,	,	PUNCT
cana-1800	233	11	x2	x2	PROPN
cana-1800	233	12	)	)	PUNCT
cana-1800	233	13	,	,	PUNCT
cana-1800	233	14	−	−	PROPN
cana-1800	233	15	vv	vv	PROPN
cana-1800	233	16	(	(	PUNCT
cana-1800	233	17	y1	y1	INTJ
cana-1800	233	18	,	,	PUNCT
cana-1800	233	19	y2	y2	PROPN
cana-1800	233	20	)	)	PUNCT
cana-1800	233	21	}	}	PUNCT
cana-1800	233	22	=	=	SYM
cana-1800	233	23	rmin	rmin	NOUN
cana-1800	233	24	{	{	PUNCT
cana-1800	233	25	−	−	PROPN
cana-1800	233	26	vv	vv	INTJ
cana-1800	233	27	(	(	PUNCT
cana-1800	233	28	x	x	NOUN
cana-1800	233	29	)	)	PUNCT
cana-1800	233	30	,	,	PUNCT
cana-1800	233	31	−	−	PROPN
cana-1800	234	1	vv	vv	INTJ
cana-1800	234	2	(	(	PUNCT
cana-1800	234	3	y	y	NOUN
cana-1800	234	4	)	)	PUNCT
cana-1800	234	5	}	}	PUNCT
cana-1800	234	6	.	.	PUNCT
cana-1800	235	1	therefore	therefore	ADV
cana-1800	235	2	−	−	AUX
cana-1800	235	3	vv	vv	INTJ
cana-1800	235	4	(	(	PUNCT
cana-1800	235	5	xy	xy	NOUN
cana-1800	235	6	)	)	PUNCT
cana-1800	235	7			NOUN
cana-1800	235	8	rmin	rmin	VERB
cana-1800	235	9	{	{	PUNCT
cana-1800	235	10	−	−	PROPN
cana-1800	235	11	vv	vv	INTJ
cana-1800	235	12	(	(	PUNCT
cana-1800	235	13	x	x	NOUN
cana-1800	235	14	)	)	PUNCT
cana-1800	235	15	,	,	PUNCT
cana-1800	235	16	−	−	PROPN
cana-1800	236	1	vv	vv	INTJ
cana-1800	236	2	(	(	PUNCT
cana-1800	236	3	y	y	NOUN
cana-1800	236	4	)	)	PUNCT
cana-1800	236	5	}	}	PUNCT
cana-1800	236	6	,	,	PUNCT
cana-1800	236	7			NOUN
cana-1800	236	8	x	x	PROPN
cana-1800	236	9	,	,	PUNCT
cana-1800	236	10	yr×r	yr×r	PROPN
cana-1800	236	11	.	.	PUNCT
cana-1800	237	1	this	this	PRON
cana-1800	237	2	proves	prove	VERB
cana-1800	237	3	that	that	SCONJ
cana-1800	237	4	v	v	NOUN
cana-1800	237	5	is	be	AUX
cana-1800	237	6	a	a	DET
cana-1800	237	7	𝔹𝕍𝕍𝕀	𝔹𝕍𝕍𝕀	NOUN
cana-1800	237	8	of	of	ADP
cana-1800	237	9	r×r	r×r	PROPN
cana-1800	237	10	.	.	PUNCT
cana-1800	237	11	conversely	conversely	ADV
cana-1800	237	12	assume	assume	VERB
cana-1800	237	13	that	that	SCONJ
cana-1800	237	14	v	v	NOUN
cana-1800	237	15	is	be	AUX
cana-1800	237	16	a	a	DET
cana-1800	237	17	𝔹𝕍𝕍𝕀	𝔹𝕍𝕍𝕀	NOUN
cana-1800	237	18	of	of	ADP
cana-1800	237	19	r×r	r×r	PROPN
cana-1800	237	20	,	,	PUNCT
cana-1800	237	21	then	then	ADV
cana-1800	237	22	for	for	ADP
cana-1800	237	23	any	any	PRON
cana-1800	237	24	x	x	SYM
cana-1800	237	25	=	=	SYM
cana-1800	237	26	(	(	PUNCT
cana-1800	237	27	x1	x1	PROPN
cana-1800	237	28	,	,	PUNCT
cana-1800	237	29	x2	x2	PROPN
cana-1800	237	30	)	)	PUNCT
cana-1800	237	31	and	and	CCONJ
cana-1800	237	32	y	y	PROPN
cana-1800	237	33	=	=	SYM
cana-1800	237	34	(	(	PUNCT
cana-1800	237	35	y1	y1	INTJ
cana-1800	237	36	,	,	PUNCT
cana-1800	237	37	y2	y2	PROPN
cana-1800	237	38	)	)	PUNCT
cana-1800	237	39	are	be	AUX
cana-1800	237	40	in	in	ADP
cana-1800	237	41	r×r	r×r	PROPN
cana-1800	237	42	,	,	PUNCT
cana-1800	237	43	we	we	PRON
cana-1800	237	44	have	have	AUX
cana-1800	237	45	rmin	rmin	VERB
cana-1800	237	46	{	{	PUNCT
cana-1800	237	47	+	+	NUM
cana-1800	237	48	av	av	PROPN
cana-1800	237	49	(	(	PUNCT
cana-1800	237	50	x1+y1	x1+y1	PROPN
cana-1800	237	51	)	)	PUNCT
cana-1800	237	52	,	,	PUNCT
cana-1800	238	1	+	+	CCONJ
cana-1800	238	2	av	av	PROPN
cana-1800	238	3	(	(	PUNCT
cana-1800	238	4	x2+y2	x2+y2	NOUN
cana-1800	238	5	)	)	PUNCT
cana-1800	238	6	}	}	PUNCT
cana-1800	239	1	=	=	PUNCT
cana-1800	240	1	+	+	NUM
cana-1800	240	2	vv	vv	X
cana-1800	240	3	(	(	PUNCT
cana-1800	240	4	x1+y1	x1+y1	PROPN
cana-1800	240	5	,	,	PUNCT
cana-1800	240	6	x2+y2	x2+y2	NOUN
cana-1800	240	7	)	)	PUNCT
cana-1800	241	1	=	=	PUNCT
cana-1800	242	1	+	+	CCONJ
cana-1800	242	2	vv	vv	PART
cana-1800	242	3	[	[	X
cana-1800	242	4	(	(	PUNCT
cana-1800	242	5	x1	x1	ADJ
cana-1800	242	6	,	,	PUNCT
cana-1800	242	7	x2)+(y1	x2)+(y1	NUM
cana-1800	242	8	,	,	PUNCT
cana-1800	242	9	y2	y2	NOUN
cana-1800	242	10	)	)	PUNCT
cana-1800	242	11	]	]	PUNCT
cana-1800	243	1	=	=	PUNCT
cana-1800	244	1	+	+	CCONJ
cana-1800	244	2	vv	vv	PROPN
cana-1800	244	3	(	(	PUNCT
cana-1800	244	4	x+y	x+y	NUM
cana-1800	244	5	)	)	PUNCT
cana-1800	244	6			NUM
cana-1800	244	7	rmin	rmin	NOUN
cana-1800	244	8	{	{	PUNCT
cana-1800	244	9	+	+	NUM
cana-1800	244	10	vv	vv	NOUN
cana-1800	244	11	(	(	PUNCT
cana-1800	244	12	x	x	NOUN
cana-1800	244	13	)	)	PUNCT
cana-1800	244	14	,	,	PUNCT
cana-1800	244	15	+	+	CCONJ
cana-1800	244	16	vv	vv	CCONJ
cana-1800	244	17	(	(	PUNCT
cana-1800	244	18	y	y	NOUN
cana-1800	244	19	)	)	PUNCT
cana-1800	244	20	}	}	PUNCT
cana-1800	244	21	=	=	SYM
cana-1800	244	22	rmin	rmin	NOUN
cana-1800	244	23	{	{	PUNCT
cana-1800	245	1	+	+	NUM
cana-1800	245	2	vv	vv	X
cana-1800	245	3	(	(	PUNCT
cana-1800	245	4	x1	x1	PROPN
cana-1800	245	5	,	,	PUNCT
cana-1800	245	6	x2	x2	PROPN
cana-1800	245	7	)	)	PUNCT
cana-1800	245	8	,	,	PUNCT
cana-1800	246	1	+	+	CCONJ
cana-1800	246	2	vv	vv	X
cana-1800	246	3	(	(	PUNCT
cana-1800	246	4	y1	y1	INTJ
cana-1800	246	5	,	,	PUNCT
cana-1800	246	6	y2	y2	PROPN
cana-1800	246	7	)	)	PUNCT
cana-1800	246	8	}	}	PUNCT
cana-1800	246	9	=	=	SYM
cana-1800	246	10	rmin{rmin	rmin{rmin	NOUN
cana-1800	246	11	{	{	PUNCT
cana-1800	246	12	+	+	NUM
cana-1800	246	13	av	av	PROPN
cana-1800	246	14	(	(	PUNCT
cana-1800	246	15	x1	x1	PROPN
cana-1800	246	16	)	)	PUNCT
cana-1800	246	17	,	,	PUNCT
cana-1800	246	18	+	+	CCONJ
cana-1800	246	19	av	av	PROPN
cana-1800	246	20	(	(	PUNCT
cana-1800	246	21	x2	x2	PROPN
cana-1800	246	22	)	)	PUNCT
cana-1800	246	23	}	}	PUNCT
cana-1800	246	24	,	,	PUNCT
cana-1800	246	25	rmin	rmin	NOUN
cana-1800	246	26	{	{	PUNCT
cana-1800	246	27	+	+	NUM
cana-1800	246	28	av	av	PROPN
cana-1800	246	29	(	(	PUNCT
cana-1800	246	30	y1	y1	PROPN
cana-1800	246	31	)	)	PUNCT
cana-1800	246	32	,	,	PUNCT
cana-1800	246	33	+	+	CCONJ
cana-1800	246	34	av	av	PROPN
cana-1800	246	35	(	(	PUNCT
cana-1800	246	36	y2	y2	PROPN
cana-1800	246	37	)	)	PUNCT
cana-1800	246	38	}	}	PUNCT
cana-1800	246	39	}	}	PUNCT
cana-1800	246	40	.	.	PUNCT
cana-1800	247	1	if	if	SCONJ
cana-1800	247	2	+	+	NUM
cana-1800	247	3	av	av	PROPN
cana-1800	247	4	communications	communication	NOUN
cana-1800	247	5	on	on	ADP
cana-1800	247	6	applied	apply	VERB
cana-1800	247	7	nonlinear	nonlinear	ADJ
cana-1800	247	8	analysis	analysis	NOUN
cana-1800	247	9	issn	issn	NOUN
cana-1800	247	10	:	:	PUNCT
cana-1800	247	11	1074	1074	NUM
cana-1800	247	12	-	-	PUNCT
cana-1800	247	13	133x	133x	NUM
cana-1800	247	14	vol	vol	NOUN
cana-1800	247	15	32	32	NUM
cana-1800	247	16	no	no	NOUN
cana-1800	247	17	.	.	NOUN
cana-1800	247	18	2	2	NUM
cana-1800	247	19	(	(	PUNCT
cana-1800	247	20	2025	2025	NUM
cana-1800	247	21	)	)	PUNCT
cana-1800	247	22	512	512	NUM
cana-1800	247	23	https://internationalpubls.com	https://internationalpubls.com	X
cana-1800	247	24	(	(	PUNCT
cana-1800	247	25	x1+y1	x1+y1	PROPN
cana-1800	247	26	)	)	PUNCT
cana-1800	247	27			NOUN
cana-1800	247	28	+	+	CCONJ
cana-1800	247	29	av	av	PROPN
cana-1800	247	30	(	(	PUNCT
cana-1800	247	31	x2+y2	x2+y2	PROPN
cana-1800	247	32	)	)	PUNCT
cana-1800	247	33	,	,	PUNCT
cana-1800	247	34	we	we	PRON
cana-1800	247	35	get	get	VERB
cana-1800	247	36	,	,	PUNCT
cana-1800	247	37	+	+	CCONJ
cana-1800	247	38	av	av	PROPN
cana-1800	247	39	(	(	PUNCT
cana-1800	247	40	x1+y1	x1+y1	PROPN
cana-1800	247	41	)	)	PUNCT
cana-1800	247	42			NUM
cana-1800	247	43	rmin	rmin	NOUN
cana-1800	247	44	{	{	PUNCT
cana-1800	247	45	+	+	NUM
cana-1800	247	46	av	av	PROPN
cana-1800	247	47	(	(	PUNCT
cana-1800	247	48	x1	x1	PROPN
cana-1800	247	49	)	)	PUNCT
cana-1800	247	50	,	,	PUNCT
cana-1800	248	1	+	+	CCONJ
cana-1800	248	2	av	av	PROPN
cana-1800	248	3	(	(	PUNCT
cana-1800	248	4	y1	y1	NOUN
cana-1800	248	5	)	)	PUNCT
cana-1800	248	6	}	}	PUNCT
cana-1800	248	7	,	,	PUNCT
cana-1800	248	8			NOUN
cana-1800	248	9	x1	x1	PROPN
cana-1800	248	10	,	,	PUNCT
cana-1800	248	11	y1r	y1r	PROPN
cana-1800	248	12	.	.	PUNCT
cana-1800	249	1	and	and	CCONJ
cana-1800	249	2	rmin	rmin	VERB
cana-1800	249	3	{	{	PUNCT
cana-1800	250	1	+	+	CCONJ
cana-1800	250	2	av	av	PROPN
cana-1800	250	3	(	(	PUNCT
cana-1800	250	4	x1y1	x1y1	PROPN
cana-1800	250	5	)	)	PUNCT
cana-1800	250	6	,	,	PUNCT
cana-1800	250	7	+	+	CCONJ
cana-1800	250	8	av	av	PROPN
cana-1800	250	9	(	(	PUNCT
cana-1800	250	10	x2y2	x2y2	NOUN
cana-1800	250	11	)	)	PUNCT
cana-1800	250	12	}	}	PUNCT
cana-1800	250	13	=	=	PUNCT
cana-1800	251	1	+	+	NUM
cana-1800	251	2	vv	vv	ADJ
cana-1800	251	3	(	(	PUNCT
cana-1800	251	4	x1y1	x1y1	X
cana-1800	251	5	,	,	PUNCT
cana-1800	251	6	x2y2	x2y2	X
cana-1800	251	7	)	)	PUNCT
cana-1800	252	1	=	=	PUNCT
cana-1800	253	1	+	+	CCONJ
cana-1800	253	2	vv	vv	X
cana-1800	253	3	[	[	X
cana-1800	253	4	(	(	PUNCT
cana-1800	253	5	x1	x1	PROPN
cana-1800	253	6	,	,	PUNCT
cana-1800	253	7	x2)(y1	x2)(y1	NUM
cana-1800	253	8	,	,	PUNCT
cana-1800	253	9	y2	y2	PROPN
cana-1800	253	10	)	)	PUNCT
cana-1800	253	11	]	]	PUNCT
cana-1800	254	1	=	=	PUNCT
cana-1800	255	1	+	+	CCONJ
cana-1800	255	2	vv	vv	CCONJ
cana-1800	255	3	(	(	PUNCT
cana-1800	255	4	xy	xy	NOUN
cana-1800	255	5	)	)	PUNCT
cana-1800	255	6			NUM
cana-1800	255	7	rmax	rmax	NOUN
cana-1800	255	8	{	{	PUNCT
cana-1800	255	9	+	+	NUM
cana-1800	255	10	vv	vv	ADP
cana-1800	255	11	(	(	PUNCT
cana-1800	255	12	x	x	NOUN
cana-1800	255	13	)	)	PUNCT
cana-1800	255	14	,	,	PUNCT
cana-1800	255	15	+	+	CCONJ
cana-1800	255	16	vv	vv	CCONJ
cana-1800	255	17	(	(	PUNCT
cana-1800	255	18	y	y	NOUN
cana-1800	255	19	)	)	PUNCT
cana-1800	255	20	}	}	PUNCT
cana-1800	255	21	=	=	SYM
cana-1800	255	22	rmax	rmax	ADJ
cana-1800	255	23	{	{	PUNCT
cana-1800	255	24	+	+	X
cana-1800	255	25	vv	vv	ADP
cana-1800	255	26	(	(	PUNCT
cana-1800	255	27	x1	x1	PROPN
cana-1800	255	28	,	,	PUNCT
cana-1800	255	29	x2	x2	PROPN
cana-1800	255	30	)	)	PUNCT
cana-1800	255	31	,	,	PUNCT
cana-1800	255	32	+	+	CCONJ
cana-1800	255	33	vv	vv	X
cana-1800	255	34	(	(	PUNCT
cana-1800	255	35	y1	y1	INTJ
cana-1800	255	36	,	,	PUNCT
cana-1800	255	37	y2	y2	PROPN
cana-1800	255	38	)	)	PUNCT
cana-1800	255	39	}	}	PUNCT
cana-1800	255	40	=	=	PUNCT
cana-1800	255	41	rmax{rmin	rmax{rmin	NOUN
cana-1800	255	42	{	{	PUNCT
cana-1800	255	43	+	+	X
cana-1800	255	44	av	av	PROPN
cana-1800	255	45	(	(	PUNCT
cana-1800	255	46	x1	x1	PROPN
cana-1800	255	47	)	)	PUNCT
cana-1800	255	48	,	,	PUNCT
cana-1800	255	49	+	+	CCONJ
cana-1800	255	50	av	av	PROPN
cana-1800	255	51	(	(	PUNCT
cana-1800	255	52	x2	x2	PROPN
cana-1800	255	53	)	)	PUNCT
cana-1800	255	54	}	}	PUNCT
cana-1800	255	55	,	,	PUNCT
cana-1800	255	56	rmin	rmin	NOUN
cana-1800	255	57	{	{	PUNCT
cana-1800	255	58	+	+	NUM
cana-1800	255	59	av	av	PROPN
cana-1800	255	60	(	(	PUNCT
cana-1800	255	61	y1	y1	PROPN
cana-1800	255	62	)	)	PUNCT
cana-1800	255	63	,	,	PUNCT
cana-1800	255	64	+	+	CCONJ
cana-1800	255	65	av	av	PROPN
cana-1800	255	66	(	(	PUNCT
cana-1800	255	67	y2	y2	PROPN
cana-1800	255	68	)	)	PUNCT
cana-1800	255	69	}	}	PUNCT
cana-1800	255	70	}	}	PUNCT
cana-1800	255	71	.	.	PUNCT
cana-1800	256	1	if	if	SCONJ
cana-1800	256	2	+	+	CCONJ
cana-1800	256	3	av	av	PROPN
cana-1800	256	4	(	(	PUNCT
cana-1800	256	5	x1y1	x1y1	PROPN
cana-1800	256	6	)	)	PUNCT
cana-1800	256	7			NOUN
cana-1800	256	8	+	+	CCONJ
cana-1800	256	9	av	av	PROPN
cana-1800	256	10	(	(	PUNCT
cana-1800	256	11	x2y2	x2y2	NOUN
cana-1800	256	12	)	)	PUNCT
cana-1800	256	13	,	,	PUNCT
cana-1800	256	14	we	we	PRON
cana-1800	256	15	get	get	VERB
cana-1800	256	16	+	+	CCONJ
cana-1800	256	17	av	av	PROPN
cana-1800	256	18	(	(	PUNCT
cana-1800	256	19	x1y1	x1y1	NOUN
cana-1800	256	20	)	)	PUNCT
cana-1800	256	21			NUM
cana-1800	256	22	rmax	rmax	NOUN
cana-1800	256	23	{	{	PUNCT
cana-1800	257	1	+	+	NOUN
cana-1800	257	2	av	av	PROPN
cana-1800	257	3	(	(	PUNCT
cana-1800	257	4	x1	x1	PROPN
cana-1800	257	5	)	)	PUNCT
cana-1800	257	6	,	,	PUNCT
cana-1800	257	7	+	+	CCONJ
cana-1800	257	8	av	av	PROPN
cana-1800	257	9	(	(	PUNCT
cana-1800	257	10	y1	y1	NOUN
cana-1800	257	11	)	)	PUNCT
cana-1800	257	12	}	}	PUNCT
cana-1800	257	13	,	,	PUNCT
cana-1800	257	14			NOUN
cana-1800	257	15	x1	x1	PROPN
cana-1800	257	16	,	,	PUNCT
cana-1800	257	17	y1r	y1r	PROPN
cana-1800	257	18	.	.	PUNCT
cana-1800	258	1	also	also	ADV
cana-1800	258	2	we	we	PRON
cana-1800	258	3	have	have	AUX
cana-1800	258	4	rmax	rmax	VERB
cana-1800	258	5	{	{	PUNCT
cana-1800	258	6	−	−	PROPN
cana-1800	258	7	av	av	X
cana-1800	258	8	(	(	PUNCT
cana-1800	258	9	x1+y1	x1+y1	PROPN
cana-1800	258	10	)	)	PUNCT
cana-1800	258	11	,	,	PUNCT
cana-1800	258	12	−	−	PROPN
cana-1800	258	13	av	av	PROPN
cana-1800	258	14	(	(	PUNCT
cana-1800	258	15	x2+y2	x2+y2	NOUN
cana-1800	258	16	)	)	PUNCT
cana-1800	258	17	}	}	PUNCT
cana-1800	259	1	=	=	PUNCT
cana-1800	259	2	−	−	PROPN
cana-1800	260	1	vv	vv	CCONJ
cana-1800	260	2	(	(	PUNCT
cana-1800	260	3	x1+y1	x1+y1	PROPN
cana-1800	260	4	,	,	PUNCT
cana-1800	260	5	x2+y2	x2+y2	NOUN
cana-1800	260	6	)	)	PUNCT
cana-1800	260	7	=	=	SYM
cana-1800	261	1	−	−	NOUN
cana-1800	261	2	vv	vv	ADP
cana-1800	261	3	[	[	X
cana-1800	261	4	(	(	PUNCT
cana-1800	261	5	x1	x1	ADJ
cana-1800	261	6	,	,	PUNCT
cana-1800	261	7	x2)+(y1	x2)+(y1	NUM
cana-1800	261	8	,	,	PUNCT
cana-1800	261	9	y2	y2	NOUN
cana-1800	261	10	)	)	PUNCT
cana-1800	261	11	]	]	PUNCT
cana-1800	262	1	=	=	PUNCT
cana-1800	262	2	−	−	PROPN
cana-1800	263	1	vv	vv	CCONJ
cana-1800	263	2	(	(	PUNCT
cana-1800	263	3	x+y	x+y	NUM
cana-1800	263	4	)	)	PUNCT
cana-1800	263	5			NUM
cana-1800	263	6	rmax	rmax	NOUN
cana-1800	263	7	{	{	PUNCT
cana-1800	263	8	−	−	PROPN
cana-1800	263	9	vv	vv	INTJ
cana-1800	263	10	(	(	PUNCT
cana-1800	263	11	x	x	NOUN
cana-1800	263	12	)	)	PUNCT
cana-1800	263	13	,	,	PUNCT
cana-1800	263	14	−	−	PROPN
cana-1800	264	1	vv	vv	INTJ
cana-1800	264	2	(	(	PUNCT
cana-1800	264	3	y	y	NOUN
cana-1800	264	4	)	)	PUNCT
cana-1800	264	5	}	}	PUNCT
cana-1800	264	6	=	=	SYM
cana-1800	264	7	rmax	rmax	ADJ
cana-1800	264	8	{	{	PUNCT
cana-1800	264	9	−	−	PROPN
cana-1800	264	10	vv	vv	X
cana-1800	264	11	(	(	PUNCT
cana-1800	264	12	x1	x1	PROPN
cana-1800	264	13	,	,	PUNCT
cana-1800	264	14	x2	x2	PROPN
cana-1800	264	15	)	)	PUNCT
cana-1800	264	16	,	,	PUNCT
cana-1800	264	17	−	−	PROPN
cana-1800	265	1	vv	vv	INTJ
cana-1800	265	2	(	(	PUNCT
cana-1800	265	3	y1	y1	INTJ
cana-1800	265	4	,	,	PUNCT
cana-1800	265	5	y2)}=	y2)}=	NUM
cana-1800	265	6	rmax{rmax	rmax{rmax	NOUN
cana-1800	265	7	{	{	PUNCT
cana-1800	265	8	−	−	PROPN
cana-1800	265	9	av	av	PROPN
cana-1800	265	10	(	(	PUNCT
cana-1800	265	11	x1	x1	PROPN
cana-1800	265	12	)	)	PUNCT
cana-1800	265	13	,	,	PUNCT
cana-1800	265	14	−	−	PROPN
cana-1800	265	15	av	av	PROPN
cana-1800	265	16	(	(	PUNCT
cana-1800	265	17	x2	x2	PROPN
cana-1800	265	18	)	)	PUNCT
cana-1800	265	19	}	}	PUNCT
cana-1800	265	20	,	,	PUNCT
cana-1800	265	21	rmax	rmax	ADJ
cana-1800	265	22	{	{	PUNCT
cana-1800	265	23	−	−	PROPN
cana-1800	265	24	av	av	PROPN
cana-1800	265	25	(	(	PUNCT
cana-1800	265	26	y1	y1	PROPN
cana-1800	265	27	)	)	PUNCT
cana-1800	265	28	,	,	PUNCT
cana-1800	265	29	−	−	PROPN
cana-1800	265	30	av	av	PROPN
cana-1800	265	31	(	(	PUNCT
cana-1800	265	32	y2	y2	PROPN
cana-1800	265	33	)	)	PUNCT
cana-1800	265	34	}	}	PUNCT
cana-1800	265	35	}	}	PUNCT
cana-1800	265	36	.	.	PUNCT
cana-1800	266	1	if	if	SCONJ
cana-1800	266	2	+	+	CCONJ
cana-1800	266	3	av	av	PROPN
cana-1800	266	4	(	(	PUNCT
cana-1800	266	5	x1+y1	x1+y1	PROPN
cana-1800	266	6	)	)	PUNCT
cana-1800	266	7			PUNCT
cana-1800	266	8	+	+	CCONJ
cana-1800	266	9	av	av	PROPN
cana-1800	266	10	(	(	PUNCT
cana-1800	266	11	x2+y2	x2+y2	PROPN
cana-1800	266	12	)	)	PUNCT
cana-1800	266	13	,	,	PUNCT
cana-1800	266	14	we	we	PRON
cana-1800	266	15	get	get	VERB
cana-1800	266	16	−	−	PROPN
cana-1800	266	17	av	av	INTJ
cana-1800	266	18	(	(	PUNCT
cana-1800	266	19	x1+y1	x1+y1	PROPN
cana-1800	266	20	)	)	PUNCT
cana-1800	266	21			NOUN
cana-1800	266	22	rmax	rmax	VERB
cana-1800	266	23	{	{	PUNCT
cana-1800	266	24	−	−	PROPN
cana-1800	266	25	av	av	INTJ
cana-1800	266	26	(	(	PUNCT
cana-1800	266	27	x1	x1	PROPN
cana-1800	266	28	)	)	PUNCT
cana-1800	266	29	,	,	PUNCT
cana-1800	266	30	−	−	PROPN
cana-1800	266	31	av	av	PROPN
cana-1800	266	32	(	(	PUNCT
cana-1800	266	33	y1	y1	PROPN
cana-1800	266	34	)	)	PUNCT
cana-1800	266	35	}	}	PUNCT
cana-1800	266	36	,	,	PUNCT
cana-1800	266	37			NOUN
cana-1800	266	38	x1	x1	PROPN
cana-1800	266	39	,	,	PUNCT
cana-1800	266	40	y1r	y1r	PROPN
cana-1800	266	41	.	.	PUNCT
cana-1800	267	1	and	and	CCONJ
cana-1800	267	2	rmax	rmax	ADJ
cana-1800	267	3	{	{	PUNCT
cana-1800	267	4	−	−	PROPN
cana-1800	267	5	av	av	PROPN
cana-1800	267	6	(	(	PUNCT
cana-1800	267	7	x1y1	x1y1	PROPN
cana-1800	267	8	)	)	PUNCT
cana-1800	267	9	,	,	PUNCT
cana-1800	267	10	−	−	PROPN
cana-1800	268	1	av	av	PROPN
cana-1800	268	2	(	(	PUNCT
cana-1800	268	3	x2y2	x2y2	NOUN
cana-1800	268	4	)	)	PUNCT
cana-1800	268	5	}	}	PUNCT
cana-1800	268	6	=	=	PUNCT
cana-1800	268	7	−	−	PROPN
cana-1800	269	1	vv	vv	CCONJ
cana-1800	269	2	(	(	PUNCT
cana-1800	269	3	x1y1	x1y1	X
cana-1800	269	4	,	,	PUNCT
cana-1800	269	5	x2y2	x2y2	X
cana-1800	269	6	)	)	PUNCT
cana-1800	269	7	=	=	SYM
cana-1800	270	1	−	−	NOUN
cana-1800	270	2	vv	vv	ADP
cana-1800	270	3	[	[	X
cana-1800	270	4	(	(	PUNCT
cana-1800	270	5	x1	x1	PROPN
cana-1800	270	6	,	,	PUNCT
cana-1800	270	7	x2)(y1	x2)(y1	NUM
cana-1800	270	8	,	,	PUNCT
cana-1800	270	9	y2	y2	PROPN
cana-1800	270	10	)	)	PUNCT
cana-1800	270	11	]	]	PUNCT
cana-1800	271	1	=	=	PUNCT
cana-1800	271	2	−	−	PROPN
cana-1800	272	1	vv	vv	CCONJ
cana-1800	272	2	(	(	PUNCT
cana-1800	272	3	xy	xy	NOUN
cana-1800	272	4	)	)	PUNCT
cana-1800	272	5			NOUN
cana-1800	272	6	rmin	rmin	VERB
cana-1800	272	7	{	{	PUNCT
cana-1800	272	8	−	−	PROPN
cana-1800	272	9	vv	vv	INTJ
cana-1800	272	10	(	(	PUNCT
cana-1800	272	11	x	x	NOUN
cana-1800	272	12	)	)	PUNCT
cana-1800	272	13	,	,	PUNCT
cana-1800	272	14	−	−	PROPN
cana-1800	273	1	vv	vv	INTJ
cana-1800	273	2	(	(	PUNCT
cana-1800	273	3	y	y	NOUN
cana-1800	273	4	)	)	PUNCT
cana-1800	273	5	}	}	PUNCT
cana-1800	273	6	=	=	SYM
cana-1800	273	7	rmin	rmin	NOUN
cana-1800	273	8	{	{	PUNCT
cana-1800	273	9	−	−	PROPN
cana-1800	273	10	vv	vv	X
cana-1800	273	11	(	(	PUNCT
cana-1800	273	12	x1	x1	PROPN
cana-1800	273	13	,	,	PUNCT
cana-1800	273	14	x2	x2	PROPN
cana-1800	273	15	)	)	PUNCT
cana-1800	273	16	,	,	PUNCT
cana-1800	273	17	−	−	PROPN
cana-1800	274	1	vv	vv	PROPN
cana-1800	274	2	(	(	PUNCT
cana-1800	274	3	y1	y1	INTJ
cana-1800	274	4	,	,	PUNCT
cana-1800	274	5	y2	y2	PROPN
cana-1800	274	6	)	)	PUNCT
cana-1800	274	7	}	}	PUNCT
cana-1800	274	8	=	=	SYM
cana-1800	274	9	rmin{rmax	rmin{rmax	NOUN
cana-1800	274	10	{	{	PUNCT
cana-1800	274	11	−	−	PROPN
cana-1800	274	12	av	av	PROPN
cana-1800	274	13	(	(	PUNCT
cana-1800	274	14	x1	x1	PROPN
cana-1800	274	15	)	)	PUNCT
cana-1800	274	16	,	,	PUNCT
cana-1800	274	17	−	−	PROPN
cana-1800	274	18	av	av	PROPN
cana-1800	274	19	(	(	PUNCT
cana-1800	274	20	x2	x2	PROPN
cana-1800	274	21	)	)	PUNCT
cana-1800	274	22	}	}	PUNCT
cana-1800	274	23	,	,	PUNCT
cana-1800	274	24	rmax	rmax	ADJ
cana-1800	274	25	{	{	PUNCT
cana-1800	274	26	−	−	PROPN
cana-1800	274	27	av	av	PROPN
cana-1800	274	28	(	(	PUNCT
cana-1800	274	29	y1	y1	PROPN
cana-1800	274	30	)	)	PUNCT
cana-1800	274	31	,	,	PUNCT
cana-1800	274	32	−	−	PROPN
cana-1800	274	33	av	av	PROPN
cana-1800	274	34	(	(	PUNCT
cana-1800	274	35	y2	y2	PROPN
cana-1800	274	36	)	)	PUNCT
cana-1800	274	37	}	}	PUNCT
cana-1800	274	38	}	}	PUNCT
cana-1800	274	39	.	.	PUNCT
cana-1800	275	1	if	if	SCONJ
cana-1800	275	2	+	+	CCONJ
cana-1800	275	3	av	av	PROPN
cana-1800	275	4	(	(	PUNCT
cana-1800	275	5	x1y1	x1y1	X
cana-1800	275	6	)	)	PUNCT
cana-1800	275	7			X
cana-1800	275	8	+	+	CCONJ
cana-1800	275	9	av	av	PROPN
cana-1800	275	10	(	(	PUNCT
cana-1800	275	11	x2y2	x2y2	NOUN
cana-1800	275	12	)	)	PUNCT
cana-1800	275	13	,	,	PUNCT
cana-1800	275	14	we	we	PRON
cana-1800	275	15	get	get	VERB
cana-1800	275	16	−	−	PROPN
cana-1800	275	17	av	av	PROPN
cana-1800	275	18	(	(	PUNCT
cana-1800	275	19	x1y1	x1y1	PROPN
cana-1800	275	20	)	)	PUNCT
cana-1800	275	21			NOUN
cana-1800	275	22	rmin	rmin	VERB
cana-1800	275	23	{	{	PUNCT
cana-1800	275	24	−	−	PROPN
cana-1800	275	25	av	av	PROPN
cana-1800	275	26	(	(	PUNCT
cana-1800	275	27	x1	x1	PROPN
cana-1800	275	28	)	)	PUNCT
cana-1800	275	29	,	,	PUNCT
cana-1800	275	30	−	−	PROPN
cana-1800	275	31	av	av	INTJ
cana-1800	275	32	(	(	PUNCT
cana-1800	275	33	y1)},	y1)},	PROPN
cana-1800	275	34	x1	x1	PROPN
cana-1800	275	35	,	,	PUNCT
cana-1800	275	36	y1r	y1r	PROPN
cana-1800	275	37	.	.	PUNCT
cana-1800	276	1	hence	hence	ADV
cana-1800	276	2	a	a	PRON
cana-1800	276	3	is	be	AUX
cana-1800	276	4	a	a	DET
cana-1800	276	5	𝔹𝕍𝕍𝕀	𝔹𝕍𝕍𝕀	NOUN
cana-1800	276	6	of	of	ADP
cana-1800	276	7	r.	r.	NOUN
cana-1800	276	8	4	4	NUM
cana-1800	276	9	conclusion	conclusion	NOUN
cana-1800	276	10	the	the	DET
cana-1800	276	11	concept	concept	NOUN
cana-1800	276	12	of	of	ADP
cana-1800	276	13	characterization	characterization	NOUN
cana-1800	276	14	of	of	ADP
cana-1800	276	15	bipolar	bipolar	ADJ
cana-1800	276	16	valued	value	VERB
cana-1800	276	17	vague	vague	ADJ
cana-1800	276	18	ideal	ideal	NOUN
cana-1800	276	19	a	a	DET
cana-1800	276	20	semiring	semiring	NOUN
cana-1800	276	21	is	be	AUX
cana-1800	276	22	discussed	discuss	VERB
cana-1800	276	23	in	in	ADP
cana-1800	276	24	this	this	DET
cana-1800	276	25	section	section	NOUN
cana-1800	276	26	and	and	CCONJ
cana-1800	276	27	bipolar	bipolar	ADJ
cana-1800	276	28	valued	value	VERB
cana-1800	276	29	vague	vague	ADJ
cana-1800	276	30	ideal	ideal	NOUN
cana-1800	276	31	a	a	DET
cana-1800	276	32	semiring	semire	VERB
cana-1800	276	33	properties	property	NOUN
cana-1800	276	34	have	have	AUX
cana-1800	276	35	been	be	AUX
cana-1800	276	36	introduced	introduce	VERB
cana-1800	276	37	.	.	PUNCT
cana-1800	277	1	these	these	DET
cana-1800	277	2	ideas	idea	NOUN
cana-1800	277	3	are	be	AUX
cana-1800	277	4	applied	apply	VERB
cana-1800	277	5	to	to	ADP
cana-1800	277	6	further	further	ADJ
cana-1800	277	7	research	research	NOUN
cana-1800	277	8	in	in	ADP
cana-1800	277	9	the	the	DET
cana-1800	277	10	creation	creation	NOUN
cana-1800	277	11	of	of	ADP
cana-1800	277	12	bipolar	bipolar	ADJ
cana-1800	277	13	valued	value	VERB
cana-1800	277	14	vague	vague	ADJ
cana-1800	277	15	semiring	semire	VERB
cana-1800	277	16	subsemirings	subsemiring	NOUN
cana-1800	277	17	.	.	PUNCT
cana-1800	278	1	as	as	ADP
cana-1800	278	2	a	a	DET
cana-1800	278	3	result	result	NOUN
cana-1800	278	4	,	,	PUNCT
cana-1800	278	5	our	our	PRON
cana-1800	278	6	upcoming	upcoming	ADJ
cana-1800	278	7	research	research	NOUN
cana-1800	278	8	will	will	AUX
cana-1800	278	9	examine	examine	VERB
cana-1800	278	10	some	some	PRON
cana-1800	278	11	of	of	ADP
cana-1800	278	12	the	the	DET
cana-1800	278	13	qualities	quality	NOUN
cana-1800	278	14	based	base	VERB
cana-1800	278	15	on	on	ADP
cana-1800	278	16	the	the	DET
cana-1800	278	17	idea	idea	NOUN
cana-1800	278	18	of	of	ADP
cana-1800	278	19	bipolar	bipolar	ADJ
cana-1800	278	20	valued	value	VERB
cana-1800	278	21	vague	vague	ADJ
cana-1800	278	22	ideals	ideal	NOUN
cana-1800	278	23	with	with	ADP
cana-1800	278	24	translations	translation	NOUN
cana-1800	278	25	.	.	PUNCT
cana-1800	279	1	references	reference	NOUN
cana-1800	279	2	:	:	PUNCT
cana-1800	280	1	[	[	X
cana-1800	280	2	1	1	NUM
cana-1800	280	3	]	]	PUNCT
cana-1800	280	4	anitha.m.s	anitha.m.s	ADV
cana-1800	280	5	.	.	PUNCT
cana-1800	280	6	,	,	PUNCT
cana-1800	280	7	muruganantha	muruganantha	PROPN
cana-1800	280	8	prasad	prasad	PROPN
cana-1800	280	9	&	&	CCONJ
cana-1800	280	10	k.arjunan	k.arjunan	PROPN
cana-1800	280	11	,	,	PUNCT
cana-1800	280	12	notes	note	NOUN
cana-1800	280	13	on	on	ADP
cana-1800	280	14	bipolar	bipolar	ADJ
cana-1800	280	15	valued	value	VERB
cana-1800	280	16	fuzzy	fuzzy	ADJ
cana-1800	280	17	subgroups	subgroup	NOUN
cana-1800	280	18	of	of	ADP
cana-1800	280	19	a	a	DET
cana-1800	280	20	group	group	NOUN
cana-1800	280	21	,	,	PUNCT
cana-1800	280	22	bulletin	bulletin	NOUN
cana-1800	280	23	of	of	ADP
cana-1800	280	24	society	society	NOUN
cana-1800	280	25	for	for	ADP
cana-1800	280	26	mathematical	mathematical	ADJ
cana-1800	280	27	services	service	NOUN
cana-1800	280	28	and	and	CCONJ
cana-1800	280	29	standards	standard	NOUN
cana-1800	280	30	,	,	PUNCT
cana-1800	280	31	vol	vol	NOUN
cana-1800	280	32	.	.	NOUN
cana-1800	280	33	2	2	NUM
cana-1800	280	34	no	no	NOUN
cana-1800	280	35	.	.	NOUN
cana-1800	280	36	3	3	NUM
cana-1800	280	37	(	(	PUNCT
cana-1800	280	38	2013	2013	NUM
cana-1800	280	39	)	)	PUNCT
cana-1800	280	40	,	,	PUNCT
cana-1800	280	41	pp	pp	ADP
cana-1800	280	42	.	.	PUNCT
cana-1800	281	1	52	52	NUM
cana-1800	281	2	-	-	SYM
cana-1800	281	3	59	59	NUM
cana-1800	281	4	.	.	PUNCT
cana-1800	282	1	[	[	X
cana-1800	282	2	2	2	NUM
cana-1800	282	3	]	]	PUNCT
cana-1800	282	4	anitha.k	anitha.k	ADJ
cana-1800	282	5	,	,	PUNCT
cana-1800	282	6	m.muthusamy	m.muthusamy	NOUN
cana-1800	282	7	&	&	CCONJ
cana-1800	282	8	k.arjunan	k.arjunan	PROPN
cana-1800	282	9	,	,	PUNCT
cana-1800	282	10	“	"	PUNCT
cana-1800	282	11	bipolar	bipolar	ADJ
cana-1800	282	12	valued	value	VERB
cana-1800	282	13	vague	vague	ADJ
cana-1800	282	14	subsemiring	subsemiring	NOUN
cana-1800	282	15	of	of	ADP
cana-1800	282	16	a	a	DET
cana-1800	282	17	semiring	semiring	NOUN
cana-1800	282	18	”	"	PUNCT
cana-1800	282	19	,	,	PUNCT
cana-1800	282	20	journal	journal	NOUN
cana-1800	282	21	of	of	ADP
cana-1800	282	22	shanghai	shanghai	PROPN
cana-1800	282	23	jiaotong	jiaotong	PROPN
cana-1800	282	24	university	university	PROPN
cana-1800	282	25	,	,	PUNCT
cana-1800	282	26	vol	vol	NOUN
cana-1800	282	27	.	.	PROPN
cana-1800	282	28	16	16	NUM
cana-1800	282	29	,	,	PUNCT
cana-1800	282	30	issue	issue	NOUN
cana-1800	282	31	8	8	NUM
cana-1800	282	32	(	(	PUNCT
cana-1800	282	33	2020	2020	NUM
cana-1800	282	34	)	)	PUNCT
cana-1800	282	35	,	,	PUNCT
cana-1800	282	36	129	129	NUM
cana-1800	282	37	–	–	SYM
cana-1800	282	38	135	135	NUM
cana-1800	283	1	[	[	X
cana-1800	283	2	3	3	NUM
cana-1800	283	3	]	]	PUNCT
cana-1800	283	4	anitha.k	anitha.k	ADJ
cana-1800	283	5	,	,	PUNCT
cana-1800	283	6	m.muthusamy	m.muthusamy	NOUN
cana-1800	283	7	and	and	CCONJ
cana-1800	283	8	k.arjunan	k.arjunan	NOUN
cana-1800	283	9	,	,	PUNCT
cana-1800	283	10	“	"	PUNCT
cana-1800	283	11	homomorphism	homomorphism	NOUN
cana-1800	283	12	and	and	CCONJ
cana-1800	283	13	antihomomorphism	antihomomorphism	NOUN
cana-1800	283	14	functions	function	NOUN
cana-1800	283	15	in	in	ADP
cana-1800	283	16	bipolar	bipolar	ADJ
cana-1800	283	17	valued	value	VERB
cana-1800	283	18	vague	vague	ADJ
cana-1800	283	19	subsemiring	subsemiring	NOUN
cana-1800	283	20	of	of	ADP
cana-1800	283	21	a	a	DET
cana-1800	283	22	semiring	semiring	NOUN
cana-1800	283	23	”	"	PUNCT
cana-1800	283	24	,	,	PUNCT
cana-1800	283	25	international	international	ADJ
cana-1800	283	26	journal	journal	NOUN
cana-1800	283	27	of	of	ADP
cana-1800	283	28	mathematical	mathematical	ADJ
cana-1800	283	29	archive	archive	NOUN
cana-1800	283	30	,	,	PUNCT
cana-1800	283	31	12(3	12(3	NUM
cana-1800	283	32	)	)	PUNCT
cana-1800	283	33	,	,	PUNCT
cana-1800	283	34	(	(	PUNCT
cana-1800	283	35	2021	2021	NUM
cana-1800	283	36	)	)	PUNCT
cana-1800	283	37	,	,	PUNCT
cana-1800	283	38	23	23	NUM
cana-1800	283	39	–	–	SYM
cana-1800	283	40	28	28	NUM
cana-1800	283	41	.	.	PUNCT
cana-1800	284	1	[	[	X
cana-1800	284	2	4	4	NUM
cana-1800	284	3	]	]	PUNCT
cana-1800	284	4	anitha.k	anitha.k	ADJ
cana-1800	284	5	,	,	PUNCT
cana-1800	284	6	m.muthusamy	m.muthusamy	NOUN
cana-1800	284	7	&	&	CCONJ
cana-1800	284	8	k.arjunan	k.arjunan	PROPN
cana-1800	284	9	,	,	PUNCT
cana-1800	284	10	“	"	PUNCT
cana-1800	284	11	bipolar	bipolar	ADJ
cana-1800	284	12	valued	value	VERB
cana-1800	284	13	vague	vague	ADJ
cana-1800	284	14	normal	normal	ADJ
cana-1800	284	15	subsemirings	subsemiring	NOUN
cana-1800	284	16	of	of	ADP
cana-1800	284	17	a	a	DET
cana-1800	284	18	semiring	semiring	NOUN
cana-1800	284	19	with	with	ADP
cana-1800	284	20	translation	translation	NOUN
cana-1800	284	21	”	"	PUNCT
cana-1800	284	22	,	,	PUNCT
cana-1800	284	23	journal	journal	NOUN
cana-1800	284	24	for	for	ADP
cana-1800	284	25	basic	basic	ADJ
cana-1800	284	26	sciences	science	NOUN
cana-1800	284	27	,	,	PUNCT
cana-1800	284	28	vol.23	vol.23	NOUN
cana-1800	284	29	,	,	PUNCT
cana-1800	284	30	issue	issue	NOUN
cana-1800	284	31	.1(2023	.1(2023	NUM
cana-1800	284	32	)	)	PUNCT
cana-1800	284	33	,	,	PUNCT
cana-1800	284	34	185	185	NUM
cana-1800	284	35	-	-	SYM
cana-1800	284	36	192	192	NUM
cana-1800	284	37	.	.	PUNCT
cana-1800	285	1	[	[	X
cana-1800	285	2	5	5	X
cana-1800	285	3	]	]	PUNCT
cana-1800	285	4	azriel	azriel	PROPN
cana-1800	285	5	rosenfeld	rosenfeld	PROPN
cana-1800	285	6	,	,	PUNCT
cana-1800	285	7	fuzzy	fuzzy	ADJ
cana-1800	285	8	groups	group	NOUN
cana-1800	285	9	,	,	PUNCT
cana-1800	285	10	journal	journal	NOUN
cana-1800	285	11	of	of	ADP
cana-1800	285	12	mathematical	mathematical	ADJ
cana-1800	285	13	analysis	analysis	NOUN
cana-1800	285	14	and	and	CCONJ
cana-1800	285	15	applications	application	NOUN
cana-1800	285	16	35(1971	35(1971	NUM
cana-1800	285	17	)	)	PUNCT
cana-1800	285	18	,	,	PUNCT
cana-1800	285	19	512	512	NUM
cana-1800	285	20	-	-	SYM
cana-1800	285	21	517	517	NUM
cana-1800	285	22	.	.	PUNCT
cana-1800	286	1	[	[	X
cana-1800	286	2	6	6	NUM
cana-1800	286	3	]	]	SYM
cana-1800	286	4	balasubramanian.a	balasubramanian.a	PROPN
cana-1800	286	5	,	,	PUNCT
cana-1800	286	6	k.l.muruganantha	k.l.muruganantha	PROPN
cana-1800	286	7	prasad	prasad	PROPN
cana-1800	286	8	&	&	CCONJ
cana-1800	286	9	k.arjunan	k.arjunan	PROPN
cana-1800	286	10	,	,	PUNCT
cana-1800	286	11	“	"	PUNCT
cana-1800	286	12	properties	property	NOUN
cana-1800	286	13	of	of	ADP
cana-1800	286	14	bipolar	bipolar	ADJ
cana-1800	286	15	interval	interval	NOUN
cana-1800	286	16	valued	value	VERB
cana-1800	286	17	fuzzy	fuzzy	ADJ
cana-1800	286	18	subgroups	subgroup	NOUN
cana-1800	286	19	of	of	ADP
cana-1800	286	20	a	a	DET
cana-1800	286	21	group	group	NOUN
cana-1800	286	22	”	"	PUNCT
cana-1800	286	23	,	,	PUNCT
cana-1800	286	24	international	international	ADJ
cana-1800	286	25	journal	journal	NOUN
cana-1800	286	26	of	of	ADP
cana-1800	286	27	scientific	scientific	ADJ
cana-1800	286	28	research	research	NOUN
cana-1800	286	29	,	,	PUNCT
cana-1800	286	30	vol	vol	NOUN
cana-1800	286	31	.	.	PROPN
cana-1800	286	32	4	4	NUM
cana-1800	286	33	,	,	PUNCT
cana-1800	286	34	iss	iss	PROPN
cana-1800	286	35	.	.	PROPN
cana-1800	286	36	4	4	NUM
cana-1800	286	37	(	(	PUNCT
cana-1800	286	38	2015	2015	NUM
cana-1800	286	39	)	)	PUNCT
cana-1800	286	40	,	,	PUNCT
cana-1800	286	41	262	262	NUM
cana-1800	286	42	268	268	NUM
cana-1800	286	43	.	.	PUNCT
cana-1800	287	1	[	[	X
cana-1800	287	2	7	7	X
cana-1800	287	3	]	]	X
cana-1800	287	4	cicily	cicily	ADV
cana-1800	287	5	flora	flora	NOUN
cana-1800	287	6	.	.	PUNCT
cana-1800	288	1	s	s	PART
cana-1800	288	2	and	and	CCONJ
cana-1800	288	3	arockiarani.i	arockiarani.i	PROPN
cana-1800	288	4	,	,	PUNCT
cana-1800	288	5	a	a	DET
cana-1800	288	6	new	new	ADJ
cana-1800	288	7	class	class	NOUN
cana-1800	288	8	of	of	ADP
cana-1800	288	9	generalized	generalized	ADJ
cana-1800	288	10	bipolar	bipolar	ADJ
cana-1800	288	11	vague	vague	ADJ
cana-1800	288	12	sets	set	NOUN
cana-1800	288	13	,	,	PUNCT
cana-1800	288	14	international	international	ADJ
cana-1800	288	15	journal	journal	NOUN
cana-1800	288	16	of	of	ADP
cana-1800	288	17	information	information	NOUN
cana-1800	288	18	research	research	NOUN
cana-1800	288	19	and	and	CCONJ
cana-1800	288	20	review	review	NOUN
cana-1800	288	21	,	,	PUNCT
cana-1800	288	22	3(11	3(11	NUM
cana-1800	288	23	)	)	PUNCT
cana-1800	288	24	,	,	PUNCT
cana-1800	288	25	(	(	PUNCT
cana-1800	288	26	2016	2016	NUM
cana-1800	288	27	)	)	PUNCT
cana-1800	288	28	,	,	PUNCT
cana-1800	288	29	3058−	3058−	NUM
cana-1800	288	30	3065	3065	NUM
cana-1800	288	31	.	.	PUNCT
cana-1800	289	1	[	[	X
cana-1800	289	2	8	8	NUM
cana-1800	289	3	]	]	X
cana-1800	289	4	gau	gau	NOUN
cana-1800	289	5	w.l	w.l	PROPN
cana-1800	289	6	and	and	CCONJ
cana-1800	289	7	buehrer	buehrer	PROPN
cana-1800	289	8	d.j	d.j	PROPN
cana-1800	289	9	,	,	PUNCT
cana-1800	289	10	vague	vague	ADJ
cana-1800	289	11	sets	set	NOUN
cana-1800	289	12	,	,	PUNCT
cana-1800	289	13	ieee	ieee	NOUN
cana-1800	289	14	transactions	transaction	NOUN
cana-1800	289	15	on	on	ADP
cana-1800	289	16	systems	system	NOUN
cana-1800	289	17	,	,	PUNCT
cana-1800	289	18	man	man	NOUN
cana-1800	289	19	and	and	CCONJ
cana-1800	289	20	cybernetics	cybernetic	NOUN
cana-1800	289	21	,	,	PUNCT
cana-1800	289	22	23(1993	23(1993	NUM
cana-1800	289	23	)	)	PUNCT
cana-1800	289	24	,	,	PUNCT
cana-1800	289	25	610	610	NUM
cana-1800	289	26	−	−	NOUN
cana-1800	289	27	614	614	NUM
cana-1800	289	28	.	.	PUNCT
cana-1800	290	1	[	[	X
cana-1800	290	2	9	9	NUM
cana-1800	290	3	]	]	X
cana-1800	290	4	grattan	grattan	PROPN
cana-1800	290	5	-	-	PUNCT
cana-1800	290	6	guiness	guiness	PROPN
cana-1800	290	7	,	,	PUNCT
cana-1800	290	8	“	"	PUNCT
cana-1800	290	9	fuzzy	fuzzy	ADJ
cana-1800	290	10	membership	membership	NOUN
cana-1800	290	11	mapped	map	VERB
cana-1800	290	12	onto	onto	ADP
cana-1800	290	13	interval	interval	NOUN
cana-1800	290	14	and	and	CCONJ
cana-1800	290	15	many	many	ADJ
cana-1800	290	16	valued	value	VERB
cana-1800	290	17	quantities	quantity	NOUN
cana-1800	290	18	”	"	PUNCT
cana-1800	290	19	,	,	PUNCT
cana-1800	290	20	z.math.logik	z.math.logik	PROPN
cana-1800	290	21	.	.	PUNCT
cana-1800	290	22	grundladen	grundladen	PROPN
cana-1800	290	23	math	math	NOUN
cana-1800	290	24	.	.	PUNCT
cana-1800	291	1	22	22	NUM
cana-1800	291	2	(	(	PUNCT
cana-1800	291	3	1975	1975	NUM
cana-1800	291	4	)	)	PUNCT
cana-1800	291	5	,	,	PUNCT
cana-1800	291	6	149	149	NUM
cana-1800	291	7	−	−	NUM
cana-1800	291	8	160	160	NUM
cana-1800	291	9	.	.	PUNCT
cana-1800	292	1	[	[	X
cana-1800	292	2	10	10	NUM
cana-1800	292	3	]	]	X
cana-1800	292	4	k.m.lee	k.m.lee	PROPN
cana-1800	292	5	,	,	PUNCT
cana-1800	292	6	bipolar	bipolar	ADJ
cana-1800	292	7	valued	value	VERB
cana-1800	292	8	fuzzy	fuzzy	ADJ
cana-1800	292	9	sets	set	NOUN
cana-1800	292	10	and	and	CCONJ
cana-1800	292	11	their	their	PRON
cana-1800	292	12	operations	operation	NOUN
cana-1800	292	13	.	.	PUNCT
cana-1800	293	1	proc	proc	NOUN
cana-1800	293	2	.	.	PUNCT
cana-1800	294	1	int	int	NOUN
cana-1800	294	2	.	.	PUNCT
cana-1800	294	3	conf	conf	PROPN
cana-1800	294	4	.	.	PUNCT
cana-1800	295	1	on	on	ADP
cana-1800	295	2	intelligent	intelligent	ADJ
cana-1800	295	3	technologies	technology	NOUN
cana-1800	295	4	,	,	PUNCT
cana-1800	295	5	bangkok	bangkok	PROPN
cana-1800	295	6	,	,	PUNCT
cana-1800	295	7	thailand	thailand	PROPN
cana-1800	295	8	(	(	PUNCT
cana-1800	295	9	2000	2000	NUM
cana-1800	295	10	)	)	PUNCT
cana-1800	295	11	,	,	PUNCT
cana-1800	295	12	307	307	NUM
cana-1800	295	13	-	-	SYM
cana-1800	295	14	312	312	NUM
cana-1800	295	15	.	.	PUNCT
cana-1800	295	16	communications	communication	NOUN
cana-1800	295	17	on	on	ADP
cana-1800	295	18	applied	apply	VERB
cana-1800	295	19	nonlinear	nonlinear	ADJ
cana-1800	295	20	analysis	analysis	NOUN
cana-1800	295	21	issn	issn	NOUN
cana-1800	295	22	:	:	PUNCT
cana-1800	295	23	1074	1074	NUM
cana-1800	295	24	-	-	PUNCT
cana-1800	295	25	133x	133x	NUM
cana-1800	295	26	vol	vol	NOUN
cana-1800	295	27	32	32	NUM
cana-1800	295	28	no	no	NOUN
cana-1800	295	29	.	.	NOUN
cana-1800	295	30	2	2	NUM
cana-1800	295	31	(	(	PUNCT
cana-1800	295	32	2025	2025	NUM
cana-1800	295	33	)	)	PUNCT
cana-1800	295	34	513	513	NUM
cana-1800	295	35	https://internationalpubls.com	https://internationalpubls.com	X
cana-1800	296	1	[	[	X
cana-1800	296	2	11	11	NUM
cana-1800	296	3	]	]	X
cana-1800	296	4	k.m.lee	k.m.lee	PROPN
cana-1800	296	5	,	,	PUNCT
cana-1800	296	6	comparison	comparison	NOUN
cana-1800	296	7	of	of	ADP
cana-1800	296	8	interval	interval	NOUN
cana-1800	296	9	valued	value	VERB
cana-1800	296	10	fuzzy	fuzzy	ADJ
cana-1800	296	11	sets	set	NOUN
cana-1800	296	12	,	,	PUNCT
cana-1800	296	13	intuitionistic	intuitionistic	ADJ
cana-1800	296	14	fuzzy	fuzzy	ADJ
cana-1800	296	15	sets	set	NOUN
cana-1800	296	16	and	and	CCONJ
cana-1800	296	17	bipolar	bipolar	ADJ
cana-1800	296	18	valued	value	VERB
cana-1800	296	19	fuzzy	fuzzy	ADJ
cana-1800	296	20	sets	set	NOUN
cana-1800	296	21	.	.	PUNCT
cana-1800	297	1	j.	j.	PROPN
cana-1800	297	2	fuzzy	fuzzy	ADJ
cana-1800	297	3	logic	logic	NOUN
cana-1800	297	4	intelligent	intelligent	ADJ
cana-1800	297	5	systems	system	NOUN
cana-1800	297	6	,	,	PUNCT
cana-1800	297	7	14	14	NUM
cana-1800	297	8	(	(	PUNCT
cana-1800	297	9	2	2	NUM
cana-1800	297	10	)	)	PUNCT
cana-1800	297	11	(	(	PUNCT
cana-1800	297	12	2004	2004	NUM
cana-1800	297	13	)	)	PUNCT
cana-1800	297	14	,	,	PUNCT
cana-1800	297	15	125	125	NUM
cana-1800	297	16	-	-	SYM
cana-1800	297	17	129	129	NUM
cana-1800	297	18	.	.	PUNCT
cana-1800	298	1	[	[	X
cana-1800	298	2	12	12	NUM
cana-1800	298	3	]	]	X
cana-1800	298	4	k.murugalingam	k.murugalingam	PROPN
cana-1800	298	5	&	&	CCONJ
cana-1800	298	6	k.arjunan	k.arjunan	PROPN
cana-1800	298	7	,	,	PUNCT
cana-1800	298	8	a	a	DET
cana-1800	298	9	study	study	NOUN
cana-1800	298	10	on	on	ADP
cana-1800	298	11	interval	interval	NOUN
cana-1800	298	12	valued	value	VERB
cana-1800	298	13	fuzzy	fuzzy	ADJ
cana-1800	298	14	subsemiring	subsemiring	NOUN
cana-1800	298	15	of	of	ADP
cana-1800	298	16	a	a	DET
cana-1800	298	17	semiring	semiring	NOUN
cana-1800	298	18	,	,	PUNCT
cana-1800	298	19	international	international	ADJ
cana-1800	298	20	journal	journal	NOUN
cana-1800	298	21	of	of	ADP
cana-1800	298	22	applied	apply	VERB
cana-1800	298	23	mathematics	mathematic	NOUN
cana-1800	298	24	modeling	modeling	NOUN
cana-1800	298	25	,	,	PUNCT
cana-1800	298	26	vol.1	vol.1	PROPN
cana-1800	298	27	,	,	PUNCT
cana-1800	298	28	no.5	no.5	PROPN
cana-1800	298	29	,	,	PUNCT
cana-1800	298	30	1	1	NUM
cana-1800	298	31	-	-	SYM
cana-1800	298	32	6	6	NUM
cana-1800	298	33	,	,	PUNCT
cana-1800	298	34	(	(	PUNCT
cana-1800	298	35	2013	2013	NUM
cana-1800	298	36	)	)	PUNCT
cana-1800	298	37	.	.	PUNCT
cana-1800	299	1	[	[	X
cana-1800	299	2	13	13	NUM
cana-1800	299	3	]	]	SYM
cana-1800	299	4	ranjitbiswas	ranjitbiswas	ADJ
cana-1800	299	5	,	,	PUNCT
cana-1800	299	6	vague	vague	ADJ
cana-1800	299	7	groups	group	NOUN
cana-1800	299	8	,	,	PUNCT
cana-1800	299	9	international	international	ADJ
cana-1800	299	10	journal	journal	NOUN
cana-1800	299	11	of	of	ADP
cana-1800	299	12	computational	computational	ADJ
cana-1800	299	13	coginition	coginition	NOUN
cana-1800	299	14	,	,	PUNCT
cana-1800	299	15	4(2	4(2	NUM
cana-1800	299	16	)	)	PUNCT
cana-1800	299	17	,	,	PUNCT
cana-1800	299	18	(	(	PUNCT
cana-1800	299	19	2006	2006	NUM
cana-1800	299	20	)	)	PUNCT
cana-1800	299	21	,	,	PUNCT
cana-1800	299	22	20	20	NUM
cana-1800	299	23	−	−	NOUN
cana-1800	299	24	23	23	NUM
cana-1800	299	25	.	.	PUNCT
cana-1800	300	1	[	[	X
cana-1800	300	2	14	14	NUM
cana-1800	300	3	]	]	SYM
cana-1800	300	4	yasodara.b	yasodara.b	PUNCT
cana-1800	300	5	and	and	CCONJ
cana-1800	300	6	ke.sathappan	ke.sathappan	NOUN
cana-1800	300	7	,	,	PUNCT
cana-1800	300	8	“	"	PUNCT
cana-1800	300	9	bipolar	bipolar	ADJ
cana-1800	300	10	-	-	PUNCT
cana-1800	300	11	valued	value	VERB
cana-1800	300	12	multi	multi	ADJ
cana-1800	300	13	fuzzy	fuzzy	ADJ
cana-1800	300	14	subsemirings	subsemiring	NOUN
cana-1800	300	15	of	of	ADP
cana-1800	300	16	a	a	DET
cana-1800	300	17	semiring	semiring	NOUN
cana-1800	300	18	”	"	PUNCT
cana-1800	300	19	,	,	PUNCT
cana-1800	300	20	international	international	ADJ
cana-1800	300	21	journal	journal	NOUN
cana-1800	300	22	of	of	ADP
cana-1800	300	23	mathematical	mathematical	ADJ
cana-1800	300	24	archive	archive	NOUN
cana-1800	300	25	,	,	PUNCT
cana-1800	300	26	6(9	6(9	NUM
cana-1800	300	27	)	)	PUNCT
cana-1800	300	28	(	(	PUNCT
cana-1800	300	29	2015	2015	NUM
cana-1800	300	30	)	)	PUNCT
cana-1800	300	31	,	,	PUNCT
cana-1800	300	32	75	75	NUM
cana-1800	300	33	−	−	NOUN
cana-1800	300	34	80	80	NUM
cana-1800	300	35	.	.	PUNCT
cana-1800	301	1	[	[	X
cana-1800	301	2	15	15	NUM
cana-1800	301	3	]	]	PUNCT
cana-1800	301	4	b.yasodara	b.yasodara	ADP
cana-1800	301	5	and	and	CCONJ
cana-1800	301	6	ke.sathappan	ke.sathappan	NOUN
cana-1800	301	7	,	,	PUNCT
cana-1800	301	8	homomorphism	homomorphism	PROPN
cana-1800	301	9	and	and	CCONJ
cana-1800	301	10	anti	anti	NOUN
cana-1800	301	11	-	-	NOUN
cana-1800	301	12	homomorphism	homomorphism	NOUN
cana-1800	301	13	of	of	ADP
cana-1800	301	14	bipolar	bipolar	ADV
cana-1800	301	15	-	-	PUNCT
cana-1800	301	16	valued	value	VERB
cana-1800	301	17	vague	vague	ADJ
cana-1800	301	18	subsemirings	subsemiring	NOUN
cana-1800	301	19	of	of	ADP
cana-1800	301	20	a	a	DET
cana-1800	301	21	semiring	semiring	NOUN
cana-1800	301	22	,	,	PUNCT
cana-1800	301	23	bulletin	bulletin	NOUN
cana-1800	301	24	of	of	ADP
cana-1800	301	25	mathematics	mathematic	NOUN
cana-1800	301	26	and	and	CCONJ
cana-1800	301	27	statistic	statistic	ADJ
cana-1800	301	28	research	research	NOUN
cana-1800	301	29	,	,	PUNCT
cana-1800	301	30	vol.3	vol.3	PROPN
cana-1800	301	31	,	,	PUNCT
cana-1800	301	32	iss	iss	PROPN
cana-1800	301	33	.	.	PUNCT
cana-1800	302	1	3(2015	3(2015	NUM
cana-1800	302	2	)	)	PUNCT
cana-1800	303	1	,	,	PUNCT
cana-1800	303	2	229	229	NUM
cana-1800	303	3	-	-	SYM
cana-1800	303	4	233	233	NUM
cana-1800	303	5	.	.	PUNCT
cana-1800	304	1	[	[	X
cana-1800	304	2	16	16	NUM
cana-1800	304	3	]	]	PUNCT
cana-1800	304	4	m.	m.	NOUN
cana-1800	304	5	s.	s.	PROPN
cana-1800	304	6	anitha	anitha	PROPN
cana-1800	304	7	and	and	CCONJ
cana-1800	304	8	b.	b.	PROPN
cana-1800	304	9	yasodara	yasodara	PROPN
cana-1800	304	10	,	,	PUNCT
cana-1800	304	11	properties	property	NOUN
cana-1800	304	12	of	of	ADP
cana-1800	304	13	bipolar	bipolar	ADV
cana-1800	304	14	-	-	PUNCT
cana-1800	304	15	valued	value	VERB
cana-1800	304	16	fuzzy	fuzzy	ADJ
cana-1800	304	17	subsemigroups	subsemigroup	NOUN
cana-1800	304	18	of	of	ADP
cana-1800	304	19	a	a	DET
cana-1800	304	20	semigroup	semigroup	NOUN
cana-1800	304	21	,	,	PUNCT
cana-1800	304	22	journal	journal	NOUN
cana-1800	304	23	of	of	ADP
cana-1800	304	24	discrete	discrete	ADJ
cana-1800	304	25	mathematical	mathematical	ADJ
cana-1800	304	26	sciences	science	NOUN
cana-1800	304	27	and	and	CCONJ
cana-1800	304	28	cryptography,(2019	cryptography,(2019	NOUN
cana-1800	304	29	)	)	PUNCT
cana-1800	304	30	,	,	PUNCT
cana-1800	304	31	711	711	NOUN
cana-1800	304	32	-	-	SYM
cana-1800	304	33	717	717	NUM
cana-1800	304	34	.	.	PUNCT
cana-1800	305	1	[	[	X
cana-1800	305	2	17	17	NUM
cana-1800	305	3	]	]	PUNCT
cana-1800	305	4	l.a.zadeh	l.a.zadeh	NOUN
cana-1800	305	5	,	,	PUNCT
cana-1800	305	6	fuzzy	fuzzy	ADJ
cana-1800	305	7	sets	set	NOUN
cana-1800	305	8	,	,	PUNCT
cana-1800	305	9	inform	inform	NOUN
cana-1800	305	10	.	.	PUNCT
cana-1800	306	1	and	and	CCONJ
cana-1800	306	2	control	control	NOUN
cana-1800	306	3	,	,	PUNCT
cana-1800	306	4	8(1965	8(1965	NUM
cana-1800	306	5	)	)	PUNCT
cana-1800	306	6	,	,	PUNCT
cana-1800	306	7	338	338	NUM
cana-1800	306	8	-	-	SYM
cana-1800	306	9	353	353	NUM
cana-1800	306	10	.	.	PUNCT