id	sid	tid	token	lemma	pos
cana-1298	1	1	communications	communication	NOUN
cana-1298	1	2	on	on	ADP
cana-1298	1	3	applied	apply	VERB
cana-1298	1	4	nonlinear	nonlinear	ADJ
cana-1298	1	5	analysis	analysis	NOUN
cana-1298	1	6	issn	issn	NOUN
cana-1298	1	7	:	:	PUNCT
cana-1298	1	8	1074	1074	NUM
cana-1298	1	9	-	-	PUNCT
cana-1298	1	10	133x	133x	NUM
cana-1298	1	11	vol	vol	NOUN
cana-1298	1	12	31	31	NUM
cana-1298	1	13	no	no	NOUN
cana-1298	1	14	.	.	PUNCT
cana-1298	2	1	7s	7	NOUN
cana-1298	2	2	(	(	PUNCT
cana-1298	2	3	2024	2024	NUM
cana-1298	2	4	)	)	PUNCT
cana-1298	2	5	231	231	NUM
cana-1298	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1298	2	7	some	some	DET
cana-1298	2	8	product	product	NOUN
cana-1298	2	9	in	in	ADP
cana-1298	2	10	bipolar	bipolar	ADJ
cana-1298	2	11	valued	value	VERB
cana-1298	2	12	multi	multi	NOUN
cana-1298	3	1	i	i	PRON
cana-1298	3	2	-	-	PUNCT
cana-1298	3	3	fuzzy	fuzzy	ADJ
cana-1298	3	4	subrings	subring	NOUN
cana-1298	3	5	of	of	ADP
cana-1298	3	6	a	a	DET
cana-1298	3	7	ring	ring	NOUN
cana-1298	3	8	1k.vairamuthu	1k.vairamuthu	NUM
cana-1298	3	9	,	,	PUNCT
cana-1298	3	10	2	2	NUM
cana-1298	3	11	s.	s.	NOUN
cana-1298	3	12	loganathan	loganathan	PROPN
cana-1298	3	13	1	1	NUM
cana-1298	3	14	department	department	NOUN
cana-1298	3	15	of	of	ADP
cana-1298	3	16	mathematics	mathematic	NOUN
cana-1298	3	17	,	,	PUNCT
cana-1298	3	18	sethupathy	sethupathy	ADJ
cana-1298	3	19	government	government	NOUN
cana-1298	3	20	arts	arts	PROPN
cana-1298	3	21	college	college	PROPN
cana-1298	3	22	,	,	PUNCT
cana-1298	3	23	ramanathapuram	ramanathapuram	NOUN
cana-1298	3	24	-623	-623	PROPN
cana-1298	3	25	502	502	NUM
cana-1298	3	26	,	,	PUNCT
cana-1298	3	27	affiliated	affiliate	VERB
cana-1298	3	28	to	to	PART
cana-1298	3	29	alagappa	alagappa	VERB
cana-1298	3	30	university	university	NOUN
cana-1298	3	31	,	,	PUNCT
cana-1298	3	32	tamilnadu	tamilnadu	NOUN
cana-1298	3	33	,	,	PUNCT
cana-1298	3	34	india	india	PROPN
cana-1298	3	35	.	.	PUNCT
cana-1298	4	1	email	email	NOUN
cana-1298	4	2	:	:	PUNCT
cana-1298	4	3	vairammathi83@gmail.com	vairammathi83@gmail.com	X
cana-1298	4	4	2department	2department	NUM
cana-1298	4	5	of	of	ADP
cana-1298	4	6	mathematics	mathematic	NOUN
cana-1298	4	7	,	,	PUNCT
cana-1298	4	8	sethupathy	sethupathy	ADJ
cana-1298	4	9	government	government	NOUN
cana-1298	4	10	arts	arts	PROPN
cana-1298	4	11	college	college	PROPN
cana-1298	4	12	,	,	PUNCT
cana-1298	4	13	ramanathapuram	ramanathapuram	NOUN
cana-1298	4	14	-623	-623	PROPN
cana-1298	4	15	502	502	NUM
cana-1298	4	16	,	,	PUNCT
cana-1298	4	17	affiliated	affiliate	VERB
cana-1298	4	18	to	to	PART
cana-1298	4	19	alagappa	alagappa	VERB
cana-1298	4	20	university	university	NOUN
cana-1298	4	21	,	,	PUNCT
cana-1298	4	22	tamilnadu	tamilnadu	NOUN
cana-1298	4	23	,	,	PUNCT
cana-1298	4	24	india	india	PROPN
cana-1298	4	25	.	.	PUNCT
cana-1298	4	26	email	email	NOUN
cana-1298	4	27	:	:	PUNCT
cana-1298	4	28	logaamaths2010@gmail.com	logaamaths2010@gmail.com	X
cana-1298	4	29	article	article	NOUN
cana-1298	4	30	history	history	NOUN
cana-1298	4	31	:	:	PUNCT
cana-1298	4	32	received	receive	VERB
cana-1298	4	33	:	:	PUNCT
cana-1298	4	34	01	01	NUM
cana-1298	4	35	-	-	PUNCT
cana-1298	4	36	06	06	NUM
cana-1298	4	37	-	-	PUNCT
cana-1298	4	38	2024	2024	NUM
cana-1298	4	39	revised	revise	VERB
cana-1298	4	40	:	:	PUNCT
cana-1298	4	41	03	03	NUM
cana-1298	4	42	-	-	PUNCT
cana-1298	4	43	07	07	NUM
cana-1298	4	44	-	-	PUNCT
cana-1298	4	45	2024	2024	NUM
cana-1298	4	46	accepted	accept	VERB
cana-1298	4	47	:	:	PUNCT
cana-1298	4	48	29	29	NUM
cana-1298	4	49	-	-	SYM
cana-1298	4	50	07	07	NUM
cana-1298	4	51	-	-	PUNCT
cana-1298	4	52	2024	2024	NUM
cana-1298	4	53	abstract	abstract	NOUN
cana-1298	4	54	:	:	PUNCT
cana-1298	4	55	the	the	DET
cana-1298	4	56	cited	cite	VERB
cana-1298	4	57	sources	source	NOUN
cana-1298	4	58	help	help	VERB
cana-1298	4	59	to	to	PART
cana-1298	4	60	construct	construct	VERB
cana-1298	4	61	this	this	DET
cana-1298	4	62	paper	paper	NOUN
cana-1298	4	63	.	.	PUNCT
cana-1298	5	1	here	here	ADV
cana-1298	5	2	,	,	PUNCT
cana-1298	5	3	theorems	theorem	NOUN
cana-1298	5	4	on	on	ADP
cana-1298	5	5	products	product	NOUN
cana-1298	5	6	in	in	ADP
cana-1298	5	7	bipolar	bipolar	ADJ
cana-1298	5	8	valued	value	VERB
cana-1298	5	9	multi	multi	ADJ
cana-1298	5	10	-	-	ADJ
cana-1298	5	11	i	i	NOUN
cana-1298	5	12	-	-	PUNCT
cana-1298	5	13	fuzzy	fuzzy	ADJ
cana-1298	5	14	subrings	subring	NOUN
cana-1298	5	15	of	of	ADP
cana-1298	5	16	rings	ring	NOUN
cana-1298	5	17	are	be	AUX
cana-1298	5	18	presented	present	VERB
cana-1298	5	19	together	together	ADV
cana-1298	5	20	with	with	ADP
cana-1298	5	21	their	their	PRON
cana-1298	5	22	attributes	attribute	NOUN
cana-1298	5	23	,	,	PUNCT
cana-1298	5	24	which	which	PRON
cana-1298	5	25	are	be	AUX
cana-1298	5	26	stated	state	VERB
cana-1298	5	27	and	and	CCONJ
cana-1298	5	28	demonstrated	demonstrate	VERB
cana-1298	5	29	.	.	PUNCT
cana-1298	6	1	keywords	keyword	NOUN
cana-1298	6	2	:	:	PUNCT
cana-1298	6	3	interval	interval	NOUN
cana-1298	6	4	-	-	PUNCT
cana-1298	6	5	valued	value	VERB
cana-1298	6	6	fuzzy	fuzzy	ADJ
cana-1298	6	7	subset	subset	NOUN
cana-1298	6	8	,	,	PUNCT
cana-1298	6	9	bipolar	bipolar	ADJ
cana-1298	6	10	valued	value	VERB
cana-1298	6	11	fuzzy	fuzzy	ADJ
cana-1298	6	12	subset	subset	NOUN
cana-1298	6	13	,	,	PUNCT
cana-1298	6	14	bipolar	bipolar	ADJ
cana-1298	6	15	valued	value	VERB
cana-1298	6	16	multi	multi	NOUN
cana-1298	6	17	fuzzy	fuzzy	ADJ
cana-1298	6	18	subset	subset	NOUN
cana-1298	6	19	,	,	PUNCT
cana-1298	6	20	bipolar	bipolar	PROPN
cana-1298	6	21	valued	value	VERB
cana-1298	6	22	multi	multi	NOUN
cana-1298	6	23	i	i	PRON
cana-1298	6	24	-	-	PUNCT
cana-1298	6	25	fuzzy	fuzzy	ADJ
cana-1298	6	26	subset	subset	NOUN
cana-1298	6	27	,	,	PUNCT
cana-1298	6	28	bipolar	bipolar	ADJ
cana-1298	6	29	valued	value	VERB
cana-1298	6	30	multi	multi	ADJ
cana-1298	6	31	fuzzy	fuzzy	ADJ
cana-1298	6	32	subring	subring	NOUN
cana-1298	6	33	,	,	PUNCT
cana-1298	6	34	bipolar	bipolar	ADJ
cana-1298	6	35	valued	value	VERB
cana-1298	6	36	multi	multi	NOUN
cana-1298	7	1	i	i	PRON
cana-1298	7	2	-	-	PUNCT
cana-1298	7	3	fuzzy	fuzzy	ADJ
cana-1298	7	4	subring	subring	NOUN
cana-1298	7	5	,	,	PUNCT
cana-1298	7	6	product	product	NOUN
cana-1298	7	7	,	,	PUNCT
cana-1298	7	8	and	and	CCONJ
cana-1298	7	9	strongest	strong	ADJ
cana-1298	7	10	relation	relation	NOUN
cana-1298	7	11	.	.	PUNCT
cana-1298	8	1	introduction	introduction	NOUN
cana-1298	8	2	the	the	DET
cana-1298	8	3	concept	concept	NOUN
cana-1298	8	4	of	of	ADP
cana-1298	8	5	a	a	DET
cana-1298	8	6	fuzzy	fuzzy	ADJ
cana-1298	8	7	subset	subset	NOUN
cana-1298	8	8	of	of	ADP
cana-1298	8	9	a	a	DET
cana-1298	8	10	set	set	NOUN
cana-1298	8	11	was	be	AUX
cana-1298	8	12	first	first	ADV
cana-1298	8	13	suggested	suggest	VERB
cana-1298	8	14	by	by	ADP
cana-1298	8	15	zadeh	zadeh	PROPN
cana-1298	9	1	[	[	X
cana-1298	9	2	17	17	NUM
cana-1298	9	3	]	]	PUNCT
cana-1298	9	4	in	in	ADP
cana-1298	9	5	1965	1965	NUM
cana-1298	9	6	.	.	PUNCT
cana-1298	10	1	fuzzy	fuzzy	ADJ
cana-1298	10	2	sets	set	NOUN
cana-1298	10	3	are	be	AUX
cana-1298	10	4	a	a	DET
cana-1298	10	5	helpful	helpful	ADJ
cana-1298	10	6	mathematical	mathematical	ADJ
cana-1298	10	7	structure	structure	NOUN
cana-1298	10	8	that	that	PRON
cana-1298	10	9	can	can	AUX
cana-1298	10	10	be	be	AUX
cana-1298	10	11	used	use	VERB
cana-1298	10	12	to	to	PART
cana-1298	10	13	describe	describe	VERB
cana-1298	10	14	a	a	DET
cana-1298	10	15	group	group	NOUN
cana-1298	10	16	of	of	ADP
cana-1298	10	17	objects	object	NOUN
cana-1298	10	18	whose	whose	DET
cana-1298	10	19	boundaries	boundary	NOUN
cana-1298	10	20	are	be	AUX
cana-1298	10	21	not	not	PART
cana-1298	10	22	clearly	clearly	ADV
cana-1298	10	23	defined	define	VERB
cana-1298	10	24	.	.	PUNCT
cana-1298	11	1	since	since	SCONJ
cana-1298	11	2	then	then	ADV
cana-1298	11	3	,	,	PUNCT
cana-1298	11	4	there	there	PRON
cana-1298	11	5	have	have	AUX
cana-1298	11	6	been	be	AUX
cana-1298	11	7	many	many	ADJ
cana-1298	11	8	generalizations	generalization	NOUN
cana-1298	11	9	of	of	ADP
cana-1298	11	10	this	this	DET
cana-1298	11	11	basic	basic	ADJ
cana-1298	11	12	idea	idea	NOUN
cana-1298	11	13	,	,	PUNCT
cana-1298	11	14	including	include	VERB
cana-1298	11	15	intuitionistic	intuitionistic	ADJ
cana-1298	11	16	fuzzy	fuzzy	ADJ
cana-1298	11	17	sets	set	NOUN
cana-1298	11	18	,	,	PUNCT
cana-1298	11	19	interval	interval	NOUN
cana-1298	11	20	-	-	PUNCT
cana-1298	11	21	valued	value	VERB
cana-1298	11	22	fuzzy	fuzzy	ADJ
cana-1298	11	23	sets	set	NOUN
cana-1298	11	24	,	,	PUNCT
cana-1298	11	25	vague	vague	ADJ
cana-1298	11	26	sets	set	NOUN
cana-1298	11	27	,	,	PUNCT
cana-1298	11	28	soft	soft	ADJ
cana-1298	11	29	sets	set	NOUN
cana-1298	11	30	,	,	PUNCT
cana-1298	11	31	etc	etc	X
cana-1298	11	32	.	.	X
cana-1298	12	1	it	it	PRON
cana-1298	12	2	has	have	AUX
cana-1298	12	3	also	also	ADV
cana-1298	12	4	become	become	VERB
cana-1298	12	5	a	a	DET
cana-1298	12	6	burgeoning	burgeon	VERB
cana-1298	12	7	field	field	NOUN
cana-1298	12	8	of	of	ADP
cana-1298	12	9	study	study	NOUN
cana-1298	12	10	in	in	ADP
cana-1298	12	11	other	other	ADJ
cana-1298	12	12	disciplines	discipline	NOUN
cana-1298	12	13	.	.	PUNCT
cana-1298	13	1	1	1	NUM
cana-1298	13	2	,	,	PUNCT
cana-1298	13	3	1	1	NUM
cana-1298	13	4	]	]	PUNCT
cana-1298	13	5	are	be	AUX
cana-1298	13	6	called	call	VERB
cana-1298	13	7	bipolar	bipolar	ADV
cana-1298	13	8	-	-	PUNCT
cana-1298	13	9	valued	value	VERB
cana-1298	13	10	fuzzy	fuzzy	ADJ
cana-1298	13	11	sets	set	NOUN
cana-1298	13	12	.	.	PUNCT
cana-1298	14	1	intuitionistic	intuitionistic	ADJ
cana-1298	14	2	fuzzy	fuzzy	ADJ
cana-1298	14	3	sets	set	NOUN
cana-1298	14	4	and	and	CCONJ
cana-1298	14	5	bipolar	bipolar	ADV
cana-1298	14	6	-	-	PUNCT
cana-1298	14	7	valued	value	VERB
cana-1298	14	8	fuzzy	fuzzy	ADJ
cana-1298	14	9	sets	set	NOUN
cana-1298	14	10	have	have	VERB
cana-1298	14	11	a	a	DET
cana-1298	14	12	similar	similar	ADJ
cana-1298	14	13	appearance	appearance	NOUN
cana-1298	14	14	.	.	PUNCT
cana-1298	15	1	they	they	PRON
cana-1298	15	2	differ	differ	VERB
cana-1298	15	3	from	from	ADP
cana-1298	15	4	one	one	NUM
cana-1298	15	5	another	another	DET
cana-1298	15	6	,	,	PUNCT
cana-1298	15	7	nevertheless	nevertheless	ADV
cana-1298	15	8	[	[	X
cana-1298	15	9	9	9	NUM
cana-1298	15	10	,	,	PUNCT
cana-1298	15	11	10	10	NUM
cana-1298	15	12	]	]	PUNCT
cana-1298	15	13	.	.	PUNCT
cana-1298	16	1	azriel	azriel	PROPN
cana-1298	16	2	rosenfeld	rosenfeld	PROPN
cana-1298	16	3	introduced	introduce	VERB
cana-1298	16	4	the	the	DET
cana-1298	16	5	fuzzy	fuzzy	ADJ
cana-1298	16	6	group	group	NOUN
cana-1298	16	7	[	[	X
cana-1298	16	8	4	4	NUM
cana-1298	16	9	]	]	PUNCT
cana-1298	16	10	.	.	PUNCT
cana-1298	17	1	following	follow	VERB
cana-1298	17	2	that	that	PRON
cana-1298	17	3	,	,	PUNCT
cana-1298	17	4	anthony	anthony	PROPN
cana-1298	17	5	j.	j.	PROPN
cana-1298	17	6	m.	m.	PROPN
cana-1298	17	7	and	and	CCONJ
cana-1298	17	8	h.	h.	PROPN
cana-1298	17	9	sherwood	sherwood	PROPN
cana-1298	17	10	proposed	propose	VERB
cana-1298	17	11	fuzzy	fuzzy	ADJ
cana-1298	17	12	groups	group	NOUN
cana-1298	17	13	redefined[2	redefined[2	PROPN
cana-1298	17	14	]	]	PUNCT
cana-1298	17	15	,	,	PUNCT
cana-1298	17	16	and	and	CCONJ
cana-1298	17	17	chitra	chitra	PROPN
cana-1298	17	18	v.	v.	PROPN
cana-1298	17	19	and	and	CCONJ
cana-1298	17	20	k.	k.	PROPN
cana-1298	17	21	arjunan	arjunan	PROPN
cana-1298	17	22	extended	extend	VERB
cana-1298	17	23	q	q	ADJ
cana-1298	17	24	-	-	PUNCT
cana-1298	17	25	fuzzy	fuzzy	ADJ
cana-1298	17	26	principles	principle	NOUN
cana-1298	17	27	to	to	ADP
cana-1298	17	28	nearring[6	nearring[6	ADV
cana-1298	17	29	]	]	PUNCT
cana-1298	17	30	.	.	PUNCT
cana-1298	18	1	t.v	t.v	PROPN
cana-1298	18	2	.	.	PROPN
cana-1298	18	3	ramakrishnan	ramakrishnan	PROPN
cana-1298	18	4	and	and	CCONJ
cana-1298	18	5	sabu	sabu	PROPN
cana-1298	18	6	sebastian	sebastian	PROPN
cana-1298	18	7	introduced	introduce	VERB
cana-1298	18	8	multi	multi	NOUN
cana-1298	18	9	fuzzy	fuzzy	ADJ
cana-1298	18	10	sets[12	sets[12	PROPN
cana-1298	18	11	]	]	PUNCT
cana-1298	18	12	.	.	PUNCT
cana-1298	19	1	the	the	DET
cana-1298	19	2	concept	concept	NOUN
cana-1298	19	3	of	of	ADP
cana-1298	19	4	bipolar	bipolar	ADV
cana-1298	19	5	-	-	PUNCT
cana-1298	19	6	valued	value	VERB
cana-1298	19	7	fuzzy	fuzzy	ADJ
cana-1298	19	8	sets	set	NOUN
cana-1298	19	9	was	be	AUX
cana-1298	19	10	suggested	suggest	VERB
cana-1298	19	11	by	by	ADP
cana-1298	19	12	lee	lee	PROPN
cana-1298	20	1	[	[	X
cana-1298	20	2	9	9	NUM
cana-1298	20	3	]	]	PUNCT
cana-1298	20	4	.	.	PUNCT
cana-1298	21	1	fuzzy	fuzzy	ADJ
cana-1298	21	2	sets	set	NOUN
cana-1298	21	3	that	that	PRON
cana-1298	21	4	have	have	VERB
cana-1298	21	5	their	their	PRON
cana-1298	21	6	membership	membership	NOUN
cana-1298	21	7	degree	degree	NOUN
cana-1298	21	8	range	range	NOUN
cana-1298	21	9	expanded	expand	VERB
cana-1298	21	10	from	from	ADP
cana-1298	21	11	[	[	X
cana-1298	21	12	0	0	NUM
cana-1298	21	13	,	,	PUNCT
cana-1298	21	14	1	1	NUM
cana-1298	21	15	]	]	PUNCT
cana-1298	21	16	to	to	ADP
cana-1298	21	17	[	[	X
cana-1298	21	18	−1	−1	NOUN
cana-1298	21	19	,	,	PUNCT
cana-1298	21	20	1	1	NUM
cana-1298	21	21	]	]	PUNCT
cana-1298	21	22	.	.	PUNCT
cana-1298	22	1	following	follow	VERB
cana-1298	22	2	that	that	PRON
cana-1298	22	3	,	,	PUNCT
cana-1298	22	4	anitha	anitha	PROPN
cana-1298	22	5	m.s	m.s	PROPN
cana-1298	22	6	et	et	PROPN
cana-1298	22	7	al	al	PROPN
cana-1298	22	8	.	.	PUNCT
cana-1298	23	1	[	[	X
cana-1298	23	2	1	1	X
cana-1298	23	3	]	]	PUNCT
cana-1298	23	4	introduced	introduce	VERB
cana-1298	23	5	bipolarvalued	bipolarvalue	VERB
cana-1298	23	6	fuzzy	fuzzy	ADJ
cana-1298	23	7	subgroups	subgroup	NOUN
cana-1298	23	8	of	of	ADP
cana-1298	23	9	a	a	DET
cana-1298	23	10	group	group	NOUN
cana-1298	23	11	,	,	PUNCT
cana-1298	23	12	while	while	SCONJ
cana-1298	23	13	arsham	arsham	PROPN
cana-1298	23	14	borum	borum	PROPN
cana-1298	23	15	and	and	CCONJ
cana-1298	23	16	saeid	saeid	PROPN
cana-1298	24	1	[	[	X
cana-1298	24	2	3	3	NUM
cana-1298	24	3	]	]	PUNCT
cana-1298	24	4	introduced	introduce	VERB
cana-1298	24	5	bipolar	bipolar	ADV
cana-1298	24	6	-	-	PUNCT
cana-1298	24	7	valued	value	VERB
cana-1298	24	8	fuzzy	fuzzy	ADJ
cana-1298	24	9	bck	bck	PROPN
cana-1298	24	10	/	/	SYM
cana-1298	24	11	bci	bci	NOUN
cana-1298	24	12	-	-	PUNCT
cana-1298	24	13	algebras	algebra	NOUN
cana-1298	24	14	.	.	PUNCT
cana-1298	25	1	balasubramanian	balasubramanian	PROPN
cana-1298	25	2	introduced	introduce	VERB
cana-1298	25	3	properties	property	NOUN
cana-1298	25	4	of	of	ADP
cana-1298	25	5	bipolar	bipolar	ADJ
cana-1298	25	6	interval	interval	NOUN
cana-1298	25	7	-	-	PUNCT
cana-1298	25	8	valued	value	VERB
cana-1298	25	9	fuzzy	fuzzy	ADJ
cana-1298	25	10	subgroups	subgroup	NOUN
cana-1298	25	11	of	of	ADP
cana-1298	25	12	a	a	DET
cana-1298	25	13	group	group	NOUN
cana-1298	25	14	and	and	CCONJ
cana-1298	25	15	associates	associate	NOUN
cana-1298	25	16	[	[	X
cana-1298	25	17	5	5	NUM
cana-1298	25	18	]	]	PUNCT
cana-1298	25	19	.	.	PUNCT
cana-1298	26	1	kyoung	kyoung	PROPN
cana-1298	26	2	ja	ja	PROPN
cana-1298	26	3	lee	lee	PROPN
cana-1298	26	4	introduced	introduce	VERB
cana-1298	26	5	bipolar	bipolar	ADJ
cana-1298	26	6	fuzzy	fuzzy	ADJ
cana-1298	26	7	subalgebras	subalgebra	NOUN
cana-1298	26	8	and	and	CCONJ
cana-1298	26	9	bipolar	bipolar	ADJ
cana-1298	26	10	fuzzy	fuzzy	ADJ
cana-1298	26	11	ideals	ideal	NOUN
cana-1298	26	12	of	of	ADP
cana-1298	26	13	bck	bck	PROPN
cana-1298	26	14	/	/	SYM
cana-1298	26	15	bci	bci	PROPN
cana-1298	26	16	-	-	PUNCT
cana-1298	26	17	algebras[8	algebras[8	PROPN
cana-1298	26	18	]	]	PUNCT
cana-1298	26	19	.	.	PUNCT
cana-1298	27	1	murugalingam.k	murugalingam.k	PROPN
cana-1298	27	2	and	and	CCONJ
cana-1298	27	3	k.	k.	PROPN
cana-1298	27	4	arjunan[11	arjunan[11	PROPN
cana-1298	27	5	]	]	PUNCT
cana-1298	27	6	presented	present	VERB
cana-1298	27	7	a	a	DET
cana-1298	27	8	study	study	NOUN
cana-1298	27	9	on	on	ADP
cana-1298	27	10	interval	interval	NOUN
cana-1298	27	11	-	-	PUNCT
cana-1298	27	12	valued	value	VERB
cana-1298	27	13	fuzzy	fuzzy	ADJ
cana-1298	27	14	subsemirings	subsemiring	NOUN
cana-1298	27	15	of	of	ADP
cana-1298	27	16	a	a	DET
cana-1298	27	17	semiring	semiring	NOUN
cana-1298	27	18	,	,	PUNCT
cana-1298	27	19	while	while	SCONJ
cana-1298	27	20	shanmugapriya.m.m	shanmugapriya.m.m	PROPN
cana-1298	27	21	&	&	CCONJ
cana-1298	27	22	k.	k.	PROPN
cana-1298	27	23	arjunan[13	arjunan[13	PROPN
cana-1298	27	24	]	]	PUNCT
cana-1298	27	25	presented	present	VERB
cana-1298	27	26	the	the	DET
cana-1298	27	27	(	(	PUNCT
cana-1298	27	28	q	q	NOUN
cana-1298	27	29	,	,	PUNCT
cana-1298	27	30	l)-fuzzy	l)-fuzzy	ADJ
cana-1298	27	31	subnearrings	subnearring	NOUN
cana-1298	27	32	of	of	ADP
cana-1298	27	33	a	a	DET
cana-1298	27	34	nearing	nearing	NOUN
cana-1298	27	35	.	.	PUNCT
cana-1298	28	1	somasundra	somasundra	PROPN
cana-1298	28	2	moorthy	moorthy	PROPN
cana-1298	28	3	's	's	PART
cana-1298	28	4	work	work	NOUN
cana-1298	28	5	,	,	PUNCT
cana-1298	28	6	"	"	PUNCT
cana-1298	28	7	a	a	DET
cana-1298	28	8	study	study	NOUN
cana-1298	28	9	on	on	ADP
cana-1298	28	10	interval	interval	NOUN
cana-1298	28	11	valued	value	VERB
cana-1298	28	12	fuzzy	fuzzy	ADJ
cana-1298	28	13	,	,	PUNCT
cana-1298	28	14	anti	anti	ADJ
cana-1298	28	15	-	-	ADJ
cana-1298	28	16	fuzzy	fuzzy	ADJ
cana-1298	28	17	,	,	PUNCT
cana-1298	28	18	intuitionistic	intuitionistic	ADJ
cana-1298	28	19	fuzzy	fuzzy	ADJ
cana-1298	28	20	subrings	subring	NOUN
cana-1298	28	21	of	of	ADP
cana-1298	28	22	a	a	DET
cana-1298	28	23	ring	ring	NOUN
cana-1298	28	24	,	,	PUNCT
cana-1298	28	25	[	[	X
cana-1298	28	26	14	14	NUM
cana-1298	28	27	]	]	PUNCT
cana-1298	28	28	,	,	PUNCT
cana-1298	28	29	writing	write	VERB
cana-1298	28	30	this	this	DET
cana-1298	28	31	work	work	NOUN
cana-1298	28	32	benefited	benefit	VERB
cana-1298	28	33	from	from	ADP
cana-1298	28	34	the	the	DET
cana-1298	28	35	thesis	thesis	NOUN
cana-1298	28	36	.	.	PUNCT
cana-1298	29	1	the	the	DET
cana-1298	29	2	idea	idea	NOUN
cana-1298	29	3	of	of	ADP
cana-1298	29	4	product	product	NOUN
cana-1298	29	5	in	in	ADP
cana-1298	29	6	the	the	DET
cana-1298	29	7	bipolar	bipolar	ADJ
cana-1298	29	8	valued	value	VERB
cana-1298	29	9	multi	multi	NOUN
cana-1298	29	10	i	i	PRON
cana-1298	29	11	-	-	PUNCT
cana-1298	29	12	fuzzy	fuzzy	ADJ
cana-1298	29	13	subring	subring	NOUN
cana-1298	29	14	of	of	ADP
cana-1298	29	15	an	an	PRON
cana-1298	29	16	is	be	AUX
cana-1298	29	17	explored	explore	VERB
cana-1298	29	18	in	in	ADP
cana-1298	29	19	this	this	DET
cana-1298	29	20	article	article	NOUN
cana-1298	29	21	.	.	PUNCT
cana-1298	30	1	1	1	X
cana-1298	30	2	.	.	X
cana-1298	30	3	prelirrminaries	prelirrminarie	NOUN
cana-1298	30	4	.	.	PUNCT
cana-1298	31	1	definition	definition	NOUN
cana-1298	31	2	1.1	1.1	NUM
cana-1298	31	3	.	.	PUNCT
cana-1298	32	1	[	[	X
cana-1298	32	2	17	17	NUM
cana-1298	32	3	]	]	PUNCT
cana-1298	32	4	an	an	DET
cana-1298	32	5	interval	interval	NOUN
cana-1298	32	6	-	-	PUNCT
cana-1298	32	7	valued	value	VERB
cana-1298	32	8	fuzzy	fuzzy	NOUN
cana-1298	32	9	subset	subset	VERB
cana-1298	32	10	ƒ	ƒ	PRON
cana-1298	32	11	of	of	ADP
cana-1298	32	12	the	the	DET
cana-1298	32	13	set	set	VERB
cana-1298	32	14			PROPN
cana-1298	32	15	is	be	AUX
cana-1298	32	16	a	a	DET
cana-1298	32	17	function	function	NOUN
cana-1298	32	18	ƒ	ƒ	NOUN
cana-1298	32	19	:	:	PUNCT
cana-1298	32	20			PROPN
cana-1298	32	21	→d[0	→d[0	PUNCT
cana-1298	32	22	,	,	PUNCT
cana-1298	32	23	1	1	NUM
cana-1298	32	24	]	]	PUNCT
cana-1298	32	25	.	.	PUNCT
cana-1298	33	1	here	here	ADV
cana-1298	33	2	d[0	d[0	ADJ
cana-1298	33	3	,	,	PUNCT
cana-1298	33	4	1	1	NUM
cana-1298	33	5	]	]	PUNCT
cana-1298	33	6	denotes	denote	VERB
cana-1298	33	7	the	the	DET
cana-1298	33	8	family	family	NOUN
cana-1298	33	9	of	of	ADP
cana-1298	33	10	all	all	DET
cana-1298	33	11	closed	closed	ADJ
cana-1298	33	12	subintervals	subinterval	NOUN
cana-1298	33	13	of	of	ADP
cana-1298	33	14	[	[	X
cana-1298	33	15	0	0	NUM
cana-1298	33	16	,	,	PUNCT
cana-1298	33	17	1	1	NUM
cana-1298	33	18	]	]	PUNCT
cana-1298	33	19	.	.	PUNCT
cana-1298	34	1	communications	communication	NOUN
cana-1298	34	2	on	on	ADP
cana-1298	34	3	applied	apply	VERB
cana-1298	34	4	nonlinear	nonlinear	ADJ
cana-1298	34	5	analysis	analysis	NOUN
cana-1298	34	6	issn	issn	NOUN
cana-1298	34	7	:	:	PUNCT
cana-1298	34	8	1074	1074	NUM
cana-1298	34	9	-	-	PUNCT
cana-1298	34	10	133x	133x	NUM
cana-1298	34	11	vol	vol	NOUN
cana-1298	34	12	31	31	NUM
cana-1298	34	13	no	no	NOUN
cana-1298	34	14	.	.	PUNCT
cana-1298	35	1	7s	7	NOUN
cana-1298	35	2	(	(	PUNCT
cana-1298	35	3	2024	2024	NUM
cana-1298	35	4	)	)	PUNCT
cana-1298	35	5	232	232	NUM
cana-1298	36	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1298	36	2	definition	definition	NOUN
cana-1298	36	3	1.2	1.2	NUM
cana-1298	36	4	.	.	PUNCT
cana-1298	37	1	[	[	X
cana-1298	37	2	9	9	X
cana-1298	37	3	]	]	X
cana-1298	37	4	𝑇ℎ𝑒	𝑇ℎ𝑒	PROPN
cana-1298	37	5	𝑜𝑟𝑑𝑒𝑟𝑒𝑑	𝑜𝑟𝑑𝑒𝑟𝑒𝑑	VERB
cana-1298	37	6	𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒	𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒	ADV
cana-1298	37	7	𝔗	𝔗	PROPN
cana-1298	37	8	=	=	X
cana-1298	37	9	{	{	PUNCT
cana-1298	37	10	(	(	PUNCT
cana-1298	37	11	𝔷	𝔷	PROPN
cana-1298	37	12	,	,	PUNCT
cana-1298	37	13	𝔗+(𝔷	𝔗+(𝔷	NOUN
cana-1298	37	14	)	)	PUNCT
cana-1298	37	15	,	,	PUNCT
cana-1298	37	16	𝔗−(𝔷	𝔗−(𝔷	ADJ
cana-1298	37	17	)	)	PUNCT
cana-1298	37	18	):	):	PUNCT
cana-1298	37	19	𝔷	𝔷	PROPN
cana-1298	37	20	∈	∈	PROPN
cana-1298	37	21	𝕎	𝕎	PROPN
cana-1298	37	22	}	}	PUNCT
cana-1298	37	23	𝑖𝑠	𝑖𝑠	NOUN
cana-1298	37	24	𝑐𝑎𝑙𝑙𝑒𝑑	𝑐𝑎𝑙𝑙𝑒𝑑	NOUN
cana-1298	37	25	a	a	DET
cana-1298	37	26	bipolar	bipolar	ADJ
cana-1298	37	27	valued	value	VERB
cana-1298	37	28	𝑓𝑢𝑧𝑧𝑦	𝑓𝑢𝑧𝑧𝑦	NOUN
cana-1298	37	29	𝑠𝑢𝑏𝑠𝑒𝑡(𝔹𝕍𝔽𝕊	𝑠𝑢𝑏𝑠𝑒𝑡(𝔹𝕍𝔽𝕊	NOUN
cana-1298	37	30	)	)	PUNCT
cana-1298	37	31	𝑜𝑓	𝑜𝑓	ADP
cana-1298	37	32	𝕨	𝕨	PROPN
cana-1298	37	33	,	,	PUNCT
cana-1298	37	34	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	ADJ
cana-1298	37	35	𝔗+	𝔗+	NOUN
cana-1298	37	36	:	:	PUNCT
cana-1298	37	37	𝕨	𝕨	PROPN
cana-1298	37	38	→	→	X
cana-1298	37	39	[	[	X
cana-1298	37	40	0,1	0,1	NUM
cana-1298	37	41	]	]	PUNCT
cana-1298	37	42	𝑖𝑠	𝑖𝑠	CCONJ
cana-1298	37	43	𝑎	𝑎	DET
cana-1298	37	44	𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒	𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒	NOUN
cana-1298	37	45	membership	membership	NOUN
cana-1298	37	46	map	map	NOUN
cana-1298	37	47	and	and	CCONJ
cana-1298	37	48	𝔗−	𝔗−	NUM
cana-1298	37	49	:	:	PUNCT
cana-1298	37	50	𝕨	𝕨	PROPN
cana-1298	37	51	→	→	SYM
cana-1298	37	52	[	[	X
cana-1298	37	53	−1,0	−1,0	X
cana-1298	37	54	]	]	X
cana-1298	37	55	is	be	AUX
cana-1298	37	56	a	a	DET
cana-1298	37	57	negative	negative	ADJ
cana-1298	37	58	membership	membership	NOUN
cana-1298	37	59	map	map	NOUN
cana-1298	37	60	.	.	PUNCT
cana-1298	38	1	example	example	NOUN
cana-1298	38	2	1.3	1.3	NUM
cana-1298	38	3	.	.	PUNCT
cana-1298	39	1	let	let	VERB
cana-1298	39	2			PROPN
cana-1298	40	1	=	=	PRON
cana-1298	40	2	{	{	PUNCT
cana-1298	40	3			PROPN
cana-1298	40	4	,	,	PUNCT
cana-1298	40	5			X
cana-1298	40	6	,	,	PUNCT
cana-1298	40	7			PROPN
cana-1298	40	8	}	}	PUNCT
cana-1298	40	9	be	be	AUX
cana-1298	40	10	a	a	DET
cana-1298	40	11	set	set	NOUN
cana-1298	40	12	.	.	PUNCT
cana-1298	41	1	then	then	ADV
cana-1298	41	2	𝜑	𝜑	X
cana-1298	41	3	=	=	PUNCT
cana-1298	41	4	{	{	PUNCT
cana-1298	41	5			PROPN
cana-1298	41	6	,	,	PUNCT
cana-1298	41	7	0.4	0.4	NUM
cana-1298	41	8	,	,	PUNCT
cana-1298	41	9	−0.7	−0.7	PROPN
cana-1298	41	10	,	,	PUNCT
cana-1298	41	11			ADP
cana-1298	41	12	,	,	PUNCT
cana-1298	41	13	0.9	0.9	NUM
cana-1298	41	14	,	,	PUNCT
cana-1298	41	15	−0.3	−0.3	PROPN
cana-1298	41	16	,	,	PUNCT
cana-1298	41	17			NUM
cana-1298	41	18	,	,	PUNCT
cana-1298	41	19	0.8	0.8	NUM
cana-1298	41	20	,	,	PUNCT
cana-1298	41	21	−0.03	−0.03	CCONJ
cana-1298	41	22	}	}	PUNCT
cana-1298	41	23	is	be	AUX
cana-1298	41	24	a	a	DET
cana-1298	41	25	bipolar	bipolar	ADJ
cana-1298	41	26	valued	value	VERB
cana-1298	41	27	fuzzy	fuzzy	ADJ
cana-1298	41	28	subset	subset	NOUN
cana-1298	41	29	of	of	ADP
cana-1298	41	30	.	.	PUNCT
cana-1298	41	31	definition	definition	NOUN
cana-1298	41	32	1.4	1.4	NUM
cana-1298	41	33	.	.	PUNCT
cana-1298	42	1	[	[	X
cana-1298	42	2	16	16	X
cana-1298	42	3	]	]	X
cana-1298	42	4	𝑇ℎ𝑒	𝑇ℎ𝑒	PROPN
cana-1298	42	5	𝑜𝑟𝑑𝑒𝑟𝑒𝑑	𝑜𝑟𝑑𝑒𝑟𝑒𝑑	VERB
cana-1298	42	6	𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒	𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒	ADV
cana-1298	42	7	℘	℘	NOUN
cana-1298	42	8	=	=	SYM
cana-1298	42	9	{	{	PUNCT
cana-1298	42	10	(	(	PUNCT
cana-1298	42	11	𝔷	𝔷	PROPN
cana-1298	42	12	,	,	PUNCT
cana-1298	42	13	℘1	℘1	VERB
cana-1298	42	14	+	+	ADJ
cana-1298	42	15	(	(	PUNCT
cana-1298	42	16	𝔷	𝔷	NOUN
cana-1298	42	17	)	)	PUNCT
cana-1298	42	18	,	,	PUNCT
cana-1298	42	19	℘2	℘2	PROPN
cana-1298	42	20	+	+	PROPN
cana-1298	42	21	(	(	PUNCT
cana-1298	42	22	𝔷	𝔷	NOUN
cana-1298	42	23	)	)	PUNCT
cana-1298	42	24	,	,	PUNCT
cana-1298	42	25	…	…	PUNCT
cana-1298	42	26	,	,	PUNCT
cana-1298	42	27	℘𝑛	℘𝑛	NOUN
cana-1298	42	28	+	+	NOUN
cana-1298	42	29	(	(	PUNCT
cana-1298	42	30	𝔷	𝔷	NOUN
cana-1298	42	31	)	)	PUNCT
cana-1298	42	32	,	,	PUNCT
cana-1298	42	33	℘1	℘1	VERB
cana-1298	42	34	−(𝔷	−(𝔷	NOUN
cana-1298	42	35	)	)	PUNCT
cana-1298	42	36	,	,	PUNCT
cana-1298	42	37	℘2	℘2	NOUN
cana-1298	42	38	−(𝔷	−(𝔷	NOUN
cana-1298	42	39	)	)	PUNCT
cana-1298	42	40	,	,	PUNCT
cana-1298	42	41	…	…	PUNCT
cana-1298	42	42	,	,	PUNCT
cana-1298	42	43	℘𝑛	℘𝑛	NOUN
cana-1298	42	44	−(𝔷	−(𝔷	NOUN
cana-1298	42	45	)	)	PUNCT
cana-1298	42	46	)	)	PUNCT
cana-1298	43	1	∶	∶	NOUN
cana-1298	43	2	𝔷	𝔷	X
cana-1298	43	3	∈	∈	PROPN
cana-1298	43	4	ℳ	ℳ	PROPN
cana-1298	43	5	}	}	PUNCT
cana-1298	43	6	𝑖𝑠	𝑖𝑠	NOUN
cana-1298	43	7	𝑐𝑎𝑙𝑙𝑒𝑑	𝑐𝑎𝑙𝑙𝑒𝑑	VERB
cana-1298	43	8	a	a	DET
cana-1298	43	9	bipolar	bipolar	ADV
cana-1298	43	10	-	-	PUNCT
cana-1298	43	11	valued	value	VERB
cana-1298	43	12	multi	multi	ADJ
cana-1298	43	13	-	-	ADJ
cana-1298	43	14	fuzzy	fuzzy	ADJ
cana-1298	43	15	subset	subset	NOUN
cana-1298	43	16	(	(	PUNCT
cana-1298	43	17	𝔹𝕍𝕄𝔽𝕊)𝑜𝑓	𝔹𝕍𝕄𝔽𝕊)𝑜𝑓	PROPN
cana-1298	43	18	ℳ𝑤𝑖𝑡ℎ	ℳ𝑤𝑖𝑡ℎ	PROPN
cana-1298	43	19	𝑜𝑟𝑑𝑒𝑟	𝑜𝑟𝑑𝑒𝑟	NOUN
cana-1298	43	20	𝑛	𝑛	PROPN
cana-1298	43	21	,	,	PUNCT
cana-1298	43	22	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	ADJ
cana-1298	43	23	℘𝑖	℘𝑖	NOUN
cana-1298	43	24	+	+	PROPN
cana-1298	43	25	:	:	PUNCT
cana-1298	43	26	ℳ	ℳ	NOUN
cana-1298	43	27	→	→	SYM
cana-1298	43	28	[	[	X
cana-1298	43	29	0,1	0,1	NUM
cana-1298	43	30	]	]	X
cana-1298	43	31	𝑎𝑟𝑒	𝑎𝑟𝑒	NOUN
cana-1298	43	32	𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒	𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒	NOUN
cana-1298	43	33	membership	membership	NOUN
cana-1298	43	34	maps	map	NOUN
cana-1298	43	35	and	and	CCONJ
cana-1298	43	36	℘𝑖	℘𝑖	NOUN
cana-1298	43	37	−	−	PROPN
cana-1298	43	38	:	:	PUNCT
cana-1298	43	39	ℳ	ℳ	NOUN
cana-1298	43	40	→	→	SYM
cana-1298	43	41	[	[	X
cana-1298	43	42	−1,0	−1,0	X
cana-1298	43	43	]	]	X
cana-1298	43	44	are	be	AUX
cana-1298	43	45	negative	negative	ADJ
cana-1298	43	46	membership	membership	NOUN
cana-1298	43	47	maps	map	NOUN
cana-1298	43	48	,	,	PUNCT
cana-1298	43	49	where	where	SCONJ
cana-1298	43	50	i	i	PRON
cana-1298	43	51	=	=	NOUN
cana-1298	43	52	1	1	NUM
cana-1298	43	53	,	,	PUNCT
cana-1298	43	54	2	2	NUM
cana-1298	43	55	,	,	PUNCT
cana-1298	43	56	…	…	PUNCT
cana-1298	43	57	,	,	PUNCT
cana-1298	43	58	n.	n.	PROPN
cana-1298	43	59	example	example	NOUN
cana-1298	43	60	1.5	1.5	NUM
cana-1298	43	61	.	.	PUNCT
cana-1298	44	1	let	let	VERB
cana-1298	44	2			PROPN
cana-1298	45	1	=	=	PRON
cana-1298	45	2	{	{	PUNCT
cana-1298	45	3			PROPN
cana-1298	45	4	,	,	PUNCT
cana-1298	45	5			X
cana-1298	45	6	,	,	PUNCT
cana-1298	45	7			PROPN
cana-1298	45	8	}	}	PUNCT
cana-1298	45	9	be	be	AUX
cana-1298	45	10	a	a	DET
cana-1298	45	11	set	set	NOUN
cana-1298	45	12	.	.	PUNCT
cana-1298	46	1	then	then	ADV
cana-1298	46	2	𝜑	𝜑	X
cana-1298	46	3	=	=	PUNCT
cana-1298	46	4	{	{	PUNCT
cana-1298	46	5			PROPN
cana-1298	46	6	,	,	PUNCT
cana-1298	46	7	0.4	0.4	NUM
cana-1298	46	8	,	,	PUNCT
cana-1298	46	9	0.5	0.5	NUM
cana-1298	46	10	,	,	PUNCT
cana-1298	46	11	0.2	0.2	NUM
cana-1298	46	12	,	,	PUNCT
cana-1298	46	13	−0.7	−0.7	PROPN
cana-1298	46	14	,	,	PUNCT
cana-1298	46	15	−0.4	−0.4	NUM
cana-1298	46	16	,	,	PUNCT
cana-1298	46	17	−0.1	−0.1	PROPN
cana-1298	46	18	,	,	PUNCT
cana-1298	46	19			ADP
cana-1298	46	20	,	,	PUNCT
cana-1298	46	21	0.9	0.9	NUM
cana-1298	46	22	,	,	PUNCT
cana-1298	46	23	0.5	0.5	NUM
cana-1298	46	24	,	,	PUNCT
cana-1298	46	25	0.8	0.8	NUM
cana-1298	46	26	,	,	PUNCT
cana-1298	46	27	−0.3	−0.3	PROPN
cana-1298	46	28	,	,	PUNCT
cana-1298	46	29	−0.2	−0.2	PROPN
cana-1298	46	30	,	,	PUNCT
cana-1298	46	31	−0.8	−0.8	PROPN
cana-1298	46	32	,	,	PUNCT
cana-1298	46	33			NUM
cana-1298	46	34	,	,	PUNCT
cana-1298	46	35	0.8	0.8	NUM
cana-1298	46	36	,	,	PUNCT
cana-1298	46	37	0.1	0.1	NUM
cana-1298	46	38	,	,	PUNCT
cana-1298	46	39	0.4	0.4	NUM
cana-1298	46	40	,	,	PUNCT
cana-1298	46	41	−0.4	−0.4	NUM
cana-1298	46	42	,	,	PUNCT
cana-1298	46	43	−0.3	−0.3	NUM
cana-1298	46	44	,	,	PUNCT
cana-1298	46	45	−0.6	−0.6	ADP
cana-1298	46	46	}	}	PUNCT
cana-1298	46	47	is	be	AUX
cana-1298	46	48	a	a	DET
cana-1298	46	49	bipolar	bipolar	ADJ
cana-1298	46	50	valued	value	VERB
cana-1298	46	51	multi	multi	ADJ
cana-1298	46	52	fuzzy	fuzzy	ADJ
cana-1298	46	53	subset	subset	NOUN
cana-1298	46	54	of	of	ADP
cana-1298	46	55			PROPN
cana-1298	46	56	with	with	ADP
cana-1298	46	57	order	order	NOUN
cana-1298	46	58	3	3	NUM
cana-1298	46	59	.	.	PUNCT
cana-1298	46	60	definition	definition	NOUN
cana-1298	46	61	1.6	1.6	NUM
cana-1298	46	62	.	.	PUNCT
cana-1298	47	1	[	[	X
cana-1298	47	2	15	15	X
cana-1298	47	3	]	]	X
cana-1298	47	4	𝑇ℎ𝑒	𝑇ℎ𝑒	PROPN
cana-1298	47	5	𝑜𝑟𝑑𝑒𝑟𝑒𝑑	𝑜𝑟𝑑𝑒𝑟𝑒𝑑	VERB
cana-1298	47	6	𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒	𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒	ADV
cana-1298	47	7	℘	℘	NOUN
cana-1298	47	8	=	=	SYM
cana-1298	47	9	{	{	PUNCT
cana-1298	47	10	(	(	PUNCT
cana-1298	47	11	𝔷	𝔷	PROPN
cana-1298	47	12	,	,	PUNCT
cana-1298	47	13	℘1	℘1	VERB
cana-1298	47	14	+	+	ADJ
cana-1298	47	15	(	(	PUNCT
cana-1298	47	16	𝔷	𝔷	NOUN
cana-1298	47	17	)	)	PUNCT
cana-1298	47	18	,	,	PUNCT
cana-1298	47	19	℘2	℘2	PROPN
cana-1298	47	20	+	+	PROPN
cana-1298	47	21	(	(	PUNCT
cana-1298	47	22	𝔷	𝔷	NOUN
cana-1298	47	23	)	)	PUNCT
cana-1298	47	24	,	,	PUNCT
cana-1298	47	25	…	…	PUNCT
cana-1298	47	26	,	,	PUNCT
cana-1298	47	27	℘𝑛	℘𝑛	NOUN
cana-1298	47	28	+	+	NOUN
cana-1298	47	29	(	(	PUNCT
cana-1298	47	30	𝔷	𝔷	NOUN
cana-1298	47	31	)	)	PUNCT
cana-1298	47	32	,	,	PUNCT
cana-1298	47	33	℘1	℘1	VERB
cana-1298	47	34	−(𝔷	−(𝔷	NOUN
cana-1298	47	35	)	)	PUNCT
cana-1298	47	36	,	,	PUNCT
cana-1298	47	37	℘2	℘2	NOUN
cana-1298	47	38	−(𝔷	−(𝔷	NOUN
cana-1298	47	39	)	)	PUNCT
cana-1298	47	40	,	,	PUNCT
cana-1298	47	41	…	…	PUNCT
cana-1298	47	42	,	,	PUNCT
cana-1298	47	43	℘𝑛	℘𝑛	NOUN
cana-1298	47	44	−(𝔷	−(𝔷	NOUN
cana-1298	47	45	)	)	PUNCT
cana-1298	47	46	)	)	PUNCT
cana-1298	48	1	∶	∶	NOUN
cana-1298	48	2	𝔷	𝔷	X
cana-1298	48	3	∈	∈	PROPN
cana-1298	48	4	ℳ	ℳ	PROPN
cana-1298	48	5	}	}	PUNCT
cana-1298	48	6	𝑖𝑠	𝑖𝑠	NOUN
cana-1298	48	7	𝑐𝑎𝑙𝑙𝑒𝑑	𝑐𝑎𝑙𝑙𝑒𝑑	VERB
cana-1298	48	8	a	a	DET
cana-1298	48	9	bipolar	bipolar	ADV
cana-1298	48	10	-	-	PUNCT
cana-1298	48	11	valued	value	VERB
cana-1298	48	12	multi	multi	ADJ
cana-1298	48	13	-	-	ADJ
cana-1298	48	14	i	i	NOUN
cana-1298	48	15	-	-	PUNCT
cana-1298	48	16	fuzzy	fuzzy	ADJ
cana-1298	48	17	subset	subset	NOUN
cana-1298	48	18	(	(	PUNCT
cana-1298	48	19	𝔹𝕍𝕄𝕀𝔽𝕊)𝑜𝑓	𝔹𝕍𝕄𝕀𝔽𝕊)𝑜𝑓	PROPN
cana-1298	48	20	ℳ𝑤𝑖𝑡ℎ	ℳ𝑤𝑖𝑡ℎ	PROPN
cana-1298	48	21	𝑜𝑟𝑑𝑒𝑟	𝑜𝑟𝑑𝑒𝑟	NOUN
cana-1298	48	22	𝑛	𝑛	PROPN
cana-1298	48	23	,	,	PUNCT
cana-1298	48	24	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	ADJ
cana-1298	48	25	℘𝑖	℘𝑖	NOUN
cana-1298	48	26	+	+	PROPN
cana-1298	48	27	:	:	PUNCT
cana-1298	48	28	ℳ	ℳ	PROPN
cana-1298	48	29	→	→	SYM
cana-1298	48	30	𝐷[0,1	𝐷[0,1	NOUN
cana-1298	48	31	]	]	X
cana-1298	48	32	𝑎𝑟𝑒	𝑎𝑟𝑒	NOUN
cana-1298	48	33	𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒	𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒	NOUN
cana-1298	48	34	membership	membership	NOUN
cana-1298	48	35	maps	map	NOUN
cana-1298	48	36	and	and	CCONJ
cana-1298	48	37	℘𝑖	℘𝑖	NOUN
cana-1298	48	38	−	−	PROPN
cana-1298	48	39	:	:	PUNCT
cana-1298	48	40	ℳ	ℳ	NOUN
cana-1298	48	41	→	→	SYM
cana-1298	48	42	𝐷[−1,0	𝐷[−1,0	NOUN
cana-1298	48	43	]	]	PUNCT
cana-1298	48	44	are	be	AUX
cana-1298	48	45	negative	negative	ADJ
cana-1298	48	46	membership	membership	NOUN
cana-1298	48	47	maps	map	NOUN
cana-1298	48	48	,	,	PUNCT
cana-1298	48	49	where	where	SCONJ
cana-1298	48	50	i	i	PRON
cana-1298	48	51	=	=	NOUN
cana-1298	48	52	1	1	NUM
cana-1298	48	53	,	,	PUNCT
cana-1298	48	54	2	2	NUM
cana-1298	48	55	,	,	PUNCT
cana-1298	48	56	…	…	PUNCT
cana-1298	48	57	,	,	PUNCT
cana-1298	48	58	n.	n.	NOUN
cana-1298	48	59	here	here	ADV
cana-1298	48	60	d[0	d[0	PROPN
cana-1298	48	61	,	,	PUNCT
cana-1298	48	62	1	1	NUM
cana-1298	48	63	]	]	PUNCT
cana-1298	48	64	denotes	denote	VERB
cana-1298	48	65	the	the	DET
cana-1298	48	66	family	family	NOUN
cana-1298	48	67	of	of	ADP
cana-1298	48	68	all	all	DET
cana-1298	48	69	closed	closed	ADJ
cana-1298	48	70	subintervals	subinterval	NOUN
cana-1298	48	71	of	of	ADP
cana-1298	48	72	[	[	X
cana-1298	48	73	0	0	NUM
cana-1298	48	74	,	,	PUNCT
cana-1298	48	75	1	1	NUM
cana-1298	48	76	]	]	PUNCT
cana-1298	48	77	and	and	CCONJ
cana-1298	48	78	𝐷[−1,0	𝐷[−1,0	NOUN
cana-1298	48	79	]	]	X
cana-1298	48	80	denotes	denote	VERB
cana-1298	48	81	the	the	DET
cana-1298	48	82	family	family	NOUN
cana-1298	48	83	of	of	ADP
cana-1298	48	84	all	all	DET
cana-1298	48	85	closed	closed	ADJ
cana-1298	48	86	subintervals	subinterval	NOUN
cana-1298	48	87	of	of	ADP
cana-1298	48	88	[	[	X
cana-1298	48	89	−1	−1	NOUN
cana-1298	48	90	,	,	PUNCT
cana-1298	48	91	0	0	NUM
cana-1298	48	92	]	]	PUNCT
cana-1298	48	93	.	.	PUNCT
cana-1298	49	1	note	note	VERB
cana-1298	49	2	that	that	SCONJ
cana-1298	50	1	[	[	X
cana-1298	50	2	0	0	X
cana-1298	50	3	]	]	X
cana-1298	50	4	=	=	PUNCT
cana-1298	51	1	[	[	X
cana-1298	51	2	0	0	NUM
cana-1298	51	3	,	,	PUNCT
cana-1298	51	4	0	0	NUM
cana-1298	51	5	]	]	PUNCT
cana-1298	51	6	,	,	PUNCT
cana-1298	51	7	[	[	X
cana-1298	51	8	1	1	X
cana-1298	51	9	]	]	PUNCT
cana-1298	51	10	=	=	PUNCT
cana-1298	52	1	[	[	X
cana-1298	52	2	1	1	NUM
cana-1298	52	3	,	,	PUNCT
cana-1298	52	4	1	1	NUM
cana-1298	52	5	]	]	PUNCT
cana-1298	52	6	and	and	CCONJ
cana-1298	52	7	[	[	X
cana-1298	52	8	−1	−1	X
cana-1298	52	9	]	]	X
cana-1298	52	10	=	=	PUNCT
cana-1298	53	1	[	[	X
cana-1298	53	2	−1	−1	NOUN
cana-1298	53	3	,	,	PUNCT
cana-1298	53	4	−1	−1	NOUN
cana-1298	53	5	]	]	PUNCT
cana-1298	53	6	.	.	PUNCT
cana-1298	54	1	example	example	NOUN
cana-1298	54	2	1.7	1.7	NUM
cana-1298	54	3	.	.	PUNCT
cana-1298	55	1	let	let	VERB
cana-1298	55	2			PROPN
cana-1298	56	1	=	=	PRON
cana-1298	56	2	{	{	PUNCT
cana-1298	56	3			PROPN
cana-1298	56	4	,	,	PUNCT
cana-1298	56	5			X
cana-1298	56	6	,	,	PUNCT
cana-1298	56	7			PROPN
cana-1298	56	8	}	}	PUNCT
cana-1298	56	9	be	be	AUX
cana-1298	56	10	a	a	DET
cana-1298	56	11	set	set	NOUN
cana-1298	56	12	.	.	PUNCT
cana-1298	57	1	then	then	ADV
cana-1298	57	2	𝜑	𝜑	X
cana-1298	57	3	=	=	PUNCT
cana-1298	57	4	{	{	PUNCT
cana-1298	57	5			NOUN
cana-1298	57	6	,	,	PUNCT
cana-1298	57	7	[	[	X
cana-1298	57	8	0.4	0.4	NUM
cana-1298	57	9	,	,	PUNCT
cana-1298	57	10	0.6	0.6	NUM
cana-1298	57	11	]	]	PUNCT
cana-1298	57	12	,	,	PUNCT
cana-1298	57	13	[	[	X
cana-1298	57	14	0.5	0.5	NUM
cana-1298	57	15	,	,	PUNCT
cana-1298	57	16	0.7	0.7	NUM
cana-1298	57	17	]	]	PUNCT
cana-1298	57	18	,	,	PUNCT
cana-1298	57	19	[	[	X
cana-1298	57	20	0.2	0.2	NUM
cana-1298	57	21	,	,	PUNCT
cana-1298	57	22	0.6	0.6	NUM
cana-1298	57	23	]	]	PUNCT
cana-1298	57	24	,	,	PUNCT
cana-1298	58	1	[	[	X
cana-1298	58	2	−0.7	−0.7	PROPN
cana-1298	58	3	,	,	PUNCT
cana-1298	58	4	−0.4	−0.4	NUM
cana-1298	58	5	]	]	PUNCT
cana-1298	58	6	,	,	PUNCT
cana-1298	58	7	[	[	X
cana-1298	58	8	−0.4	−0.4	X
cana-1298	58	9	,	,	PUNCT
cana-1298	58	10	−0.1	−0.1	PROPN
cana-1298	58	11	]	]	X
cana-1298	58	12	,	,	PUNCT
cana-1298	59	1	[	[	X
cana-1298	59	2	−0.3	−0.3	NUM
cana-1298	59	3	,	,	PUNCT
cana-1298	59	4	−0.1]	−0.1]	NUM
cana-1298	59	5	,	,	PUNCT
cana-1298	59	6			ADP
cana-1298	59	7	,	,	PUNCT
cana-1298	59	8	[	[	X
cana-1298	59	9	0.5	0.5	NUM
cana-1298	59	10	,	,	PUNCT
cana-1298	59	11	0.9	0.9	NUM
cana-1298	59	12	]	]	PUNCT
cana-1298	59	13	,	,	PUNCT
cana-1298	59	14	[	[	X
cana-1298	59	15	0.5	0.5	NUM
cana-1298	59	16	,	,	PUNCT
cana-1298	59	17	0.7	0.7	NUM
cana-1298	59	18	]	]	PUNCT
cana-1298	59	19	,	,	PUNCT
cana-1298	59	20	[	[	X
cana-1298	59	21	0.8	0.8	NUM
cana-1298	59	22	,	,	PUNCT
cana-1298	59	23	0.9	0.9	NUM
cana-1298	59	24	]	]	PUNCT
cana-1298	59	25	,	,	PUNCT
cana-1298	60	1	[	[	X
cana-1298	60	2	−0.3	−0.3	PROPN
cana-1298	60	3	,	,	PUNCT
cana-1298	60	4	−0.2	−0.2	PROPN
cana-1298	60	5	]	]	PUNCT
cana-1298	60	6	,	,	PUNCT
cana-1298	61	1	[	[	X
cana-1298	61	2	−0.2	−0.2	PROPN
cana-1298	61	3	,	,	PUNCT
cana-1298	61	4	−0.1	−0.1	PROPN
cana-1298	61	5	]	]	X
cana-1298	61	6	,	,	PUNCT
cana-1298	61	7	[	[	X
cana-1298	61	8	−0.8	−0.8	ADJ
cana-1298	61	9	,	,	PUNCT
cana-1298	61	10	−0.5]	−0.5]	NOUN
cana-1298	61	11	,	,	PUNCT
cana-1298	61	12			NUM
cana-1298	61	13	,	,	PUNCT
cana-1298	61	14	[	[	X
cana-1298	61	15	0.8	0.8	NUM
cana-1298	61	16	,	,	PUNCT
cana-1298	61	17	0.9	0.9	NUM
cana-1298	61	18	]	]	PUNCT
cana-1298	61	19	,	,	PUNCT
cana-1298	61	20	[	[	X
cana-1298	61	21	0.1	0.1	NUM
cana-1298	61	22	,	,	PUNCT
cana-1298	61	23	0.6	0.6	NUM
cana-1298	61	24	]	]	PUNCT
cana-1298	61	25	,	,	PUNCT
cana-1298	61	26	[	[	X
cana-1298	61	27	0.4	0.4	NUM
cana-1298	61	28	,	,	PUNCT
cana-1298	61	29	0.7	0.7	NUM
cana-1298	61	30	]	]	PUNCT
cana-1298	61	31	,	,	PUNCT
cana-1298	61	32	[	[	X
cana-1298	61	33	−0.4	−0.4	X
cana-1298	61	34	,	,	PUNCT
cana-1298	61	35	−0.2	−0.2	PROPN
cana-1298	61	36	]	]	PUNCT
cana-1298	61	37	,	,	PUNCT
cana-1298	62	1	[	[	X
cana-1298	62	2	−0.3	−0.3	PROPN
cana-1298	62	3	,	,	PUNCT
cana-1298	62	4	−0.1	−0.1	PROPN
cana-1298	62	5	]	]	X
cana-1298	62	6	,	,	PUNCT
cana-1298	63	1	[	[	X
cana-1298	63	2	−0.6	−0.6	PROPN
cana-1298	63	3	,	,	PUNCT
cana-1298	63	4	−0.2]	−0.2]	NUM
cana-1298	63	5	}	}	PUNCT
cana-1298	63	6	is	be	AUX
cana-1298	63	7	a	a	DET
cana-1298	63	8	bipolar	bipolar	ADJ
cana-1298	63	9	valued	value	VERB
cana-1298	63	10	multi	multi	PROPN
cana-1298	63	11	ifuzzy	ifuzzy	PROPN
cana-1298	63	12	subset	subset	NOUN
cana-1298	63	13	of	of	ADP
cana-1298	63	14			PROPN
cana-1298	63	15	with	with	ADP
cana-1298	63	16	order	order	NOUN
cana-1298	63	17	3	3	NUM
cana-1298	63	18	.	.	PUNCT
cana-1298	63	19	definition	definition	NOUN
cana-1298	63	20	1.8	1.8	NUM
cana-1298	63	21	.	.	PUNCT
cana-1298	64	1	[	[	X
cana-1298	64	2	15	15	NUM
cana-1298	64	3	]	]	X
cana-1298	64	4	𝐴	𝐴	NOUN
cana-1298	64	5	𝔹𝕍𝕄𝕀𝔽𝕊	𝔹𝕍𝕄𝕀𝔽𝕊	ADJ
cana-1298	64	6	℘	℘	PROPN
cana-1298	64	7	=	=	SYM
cana-1298	64	8			NOUN
cana-1298	64	9	℘1	℘1	VERB
cana-1298	64	10	+	+	PROPN
cana-1298	64	11	,	,	PUNCT
cana-1298	64	12	℘2	℘2	NOUN
cana-1298	64	13	+	+	PROPN
cana-1298	64	14	,	,	PUNCT
cana-1298	64	15	…	…	PUNCT
cana-1298	64	16	,	,	PUNCT
cana-1298	64	17	℘𝑛	℘𝑛	NOUN
cana-1298	64	18	+	+	ADV
cana-1298	64	19	,	,	PUNCT
cana-1298	64	20	℘1	℘1	VERB
cana-1298	64	21	−	−	PROPN
cana-1298	64	22	,	,	PUNCT
cana-1298	64	23	℘2	℘2	NOUN
cana-1298	64	24	−	−	PROPN
cana-1298	64	25	,	,	PUNCT
cana-1298	64	26	…	…	PUNCT
cana-1298	64	27	,	,	PUNCT
cana-1298	64	28	℘𝑛	℘𝑛	NOUN
cana-1298	64	29	−	−	PRON
cana-1298	64	30			NOUN
cana-1298	64	31	of	of	ADP
cana-1298	64	32	a	a	DET
cana-1298	64	33	ring	ring	NOUN
cana-1298	64	34	𝔜	𝔜	NOUN
cana-1298	64	35	𝑖𝑠	𝑖𝑠	PROPN
cana-1298	64	36	𝑠𝑎𝑖𝑑	𝑠𝑎𝑖𝑑	VERB
cana-1298	64	37	to	to	PART
cana-1298	64	38	be	be	AUX
cana-1298	64	39	a	a	DET
cana-1298	64	40	bipolar	bipolar	ADJ
cana-1298	64	41	valued	value	VERB
cana-1298	64	42	multi	multi	NOUN
cana-1298	65	1	i	i	PRON
cana-1298	65	2	−	−	VERB
cana-1298	65	3	fuzzy	fuzzy	ADJ
cana-1298	65	4	subring	subring	NOUN
cana-1298	65	5	of	of	ADP
cana-1298	65	6	𝔜	𝔜	PROPN
cana-1298	65	7	(	(	PUNCT
cana-1298	65	8	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	65	9	)	)	PUNCT
cana-1298	65	10	𝑖𝑓	𝑖𝑓	ADP
cana-1298	65	11	℘	℘	PROPN
cana-1298	65	12	ℎ𝑎𝑠	ℎ𝑎𝑠	NOUN
cana-1298	65	13	𝑡ℎ𝑒	𝑡ℎ𝑒	ADP
cana-1298	65	14	𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔	𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔	ADJ
cana-1298	65	15	𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛	𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛	NOUN
cana-1298	65	16	,	,	PUNCT
cana-1298	65	17	(	(	PUNCT
cana-1298	65	18	i	i	NOUN
cana-1298	65	19	)	)	PUNCT
cana-1298	65	20	℘𝑖	℘𝑖	VERB
cana-1298	66	1	+	+	PROPN
cana-1298	66	2	(	(	PUNCT
cana-1298	66	3	𝔶	𝔶	ADP
cana-1298	66	4	−	−	PROPN
cana-1298	66	5	𝔴	𝔴	NUM
cana-1298	66	6	)	)	PUNCT
cana-1298	66	7	≥	≥	NOUN
cana-1298	66	8	𝑟𝑚𝑖𝑛{℘𝑖	𝑟𝑚𝑖𝑛{℘𝑖	NOUN
cana-1298	67	1	+	+	NOUN
cana-1298	67	2	(	(	PUNCT
cana-1298	67	3	𝔶	𝔶	ADJ
cana-1298	67	4	)	)	PUNCT
cana-1298	67	5	,	,	PUNCT
cana-1298	67	6	℘𝑖	℘𝑖	NOUN
cana-1298	67	7	+	+	PROPN
cana-1298	67	8	(	(	PUNCT
cana-1298	67	9	𝔴	𝔴	NOUN
cana-1298	67	10	)	)	PUNCT
cana-1298	67	11	}	}	PUNCT
cana-1298	67	12	,	,	PUNCT
cana-1298	67	13	(	(	PUNCT
cana-1298	67	14	ii	ii	NOUN
cana-1298	67	15	)	)	PUNCT
cana-1298	68	1	℘𝑖	℘𝑖	NOUN
cana-1298	68	2	+	+	PROPN
cana-1298	68	3	(	(	PUNCT
cana-1298	68	4	𝔶𝔴	𝔶𝔴	NOUN
cana-1298	68	5	)	)	PUNCT
cana-1298	68	6	≥	≥	NOUN
cana-1298	68	7	𝑟𝑚𝑖𝑛{℘𝑖	𝑟𝑚𝑖𝑛{℘𝑖	NOUN
cana-1298	69	1	+	+	NOUN
cana-1298	69	2	(	(	PUNCT
cana-1298	69	3	𝔶	𝔶	ADJ
cana-1298	69	4	)	)	PUNCT
cana-1298	70	1	,	,	PUNCT
cana-1298	70	2	℘𝑖	℘𝑖	NOUN
cana-1298	70	3	+	+	PROPN
cana-1298	70	4	(	(	PUNCT
cana-1298	70	5	𝔴	𝔴	NOUN
cana-1298	70	6	)	)	PUNCT
cana-1298	70	7	}	}	PUNCT
cana-1298	70	8	,	,	PUNCT
cana-1298	70	9	(	(	PUNCT
cana-1298	70	10	iii	iii	NOUN
cana-1298	70	11	)	)	PUNCT
cana-1298	70	12	℘𝑖	℘𝑖	NOUN
cana-1298	70	13	−(𝔶	−(𝔶	NOUN
cana-1298	70	14	−	−	PROPN
cana-1298	70	15	𝔴	𝔴	NOUN
cana-1298	70	16	)	)	PUNCT
cana-1298	70	17	≤	≤	NUM
cana-1298	70	18	𝑟𝑚𝑎𝑥{℘𝑖	𝑟𝑚𝑎𝑥{℘𝑖	NOUN
cana-1298	70	19	−(𝔶	−(𝔶	NOUN
cana-1298	70	20	)	)	PUNCT
cana-1298	70	21	,	,	PUNCT
cana-1298	70	22	℘𝑖	℘𝑖	PROPN
cana-1298	70	23	−(𝔴	−(𝔴	PROPN
cana-1298	70	24	)	)	PUNCT
cana-1298	70	25	}	}	PUNCT
cana-1298	70	26	,	,	PUNCT
cana-1298	70	27	(	(	PUNCT
cana-1298	70	28	iv	iv	X
cana-1298	70	29	)	)	PUNCT
cana-1298	70	30	℘𝑖	℘𝑖	NOUN
cana-1298	70	31	−(𝔶𝔴	−(𝔶𝔴	PROPN
cana-1298	70	32	)	)	PUNCT
cana-1298	70	33	≤	≤	NUM
cana-1298	70	34	𝑟𝑚𝑎𝑥{℘𝑖	𝑟𝑚𝑎𝑥{℘𝑖	NOUN
cana-1298	70	35	−(𝔶	−(𝔶	NOUN
cana-1298	70	36	)	)	PUNCT
cana-1298	70	37	,	,	PUNCT
cana-1298	70	38	℘𝑖	℘𝑖	PROPN
cana-1298	70	39	−(𝔴	−(𝔴	PROPN
cana-1298	70	40	)	)	PUNCT
cana-1298	70	41	}	}	PUNCT
cana-1298	70	42	,	,	PUNCT
cana-1298	70	43	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1298	70	44	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-1298	70	45	𝔶	𝔶	PROPN
cana-1298	70	46	,	,	PUNCT
cana-1298	70	47	𝔴	𝔴	PROPN
cana-1298	70	48	∈	∈	PROPN
cana-1298	70	49	𝔜.	𝔜.	PROPN
cana-1298	70	50	example	example	NOUN
cana-1298	70	51	1.9	1.9	NUM
cana-1298	70	52	.	.	PUNCT
cana-1298	71	1	let	let	VERB
cana-1298	71	2	𝕫3	𝕫3	PROPN
cana-1298	71	3	=	=	SYM
cana-1298	71	4	{	{	PUNCT
cana-1298	71	5	0	0	NUM
cana-1298	71	6	,	,	PUNCT
cana-1298	71	7	1	1	NUM
cana-1298	71	8	,	,	PUNCT
cana-1298	71	9	2	2	NUM
cana-1298	71	10	}	}	PUNCT
cana-1298	71	11	𝑏𝑒	𝑏𝑒	NOUN
cana-1298	71	12	𝑎	𝑎	DET
cana-1298	71	13	𝑟𝑖𝑛𝑔	𝑟𝑖𝑛𝑔	ADJ
cana-1298	71	14	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	NOUN
cana-1298	71	15	⊕3	⊕3	PROPN
cana-1298	71	16	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-1298	71	17	⊗3	⊗3	PROPN
cana-1298	71	18	.	.	PUNCT
cana-1298	72	1	then	then	ADV
cana-1298	72	2	℘	℘	PROPN
cana-1298	72	3	is	be	AUX
cana-1298	72	4	defined	define	VERB
cana-1298	72	5	as	as	ADP
cana-1298	72	6	℘	℘	PROPN
cana-1298	72	7	=	=	SYM
cana-1298	72	8	{	{	PUNCT
cana-1298	72	9	(	(	PUNCT
cana-1298	72	10	0	0	NUM
cana-1298	72	11	,	,	PUNCT
cana-1298	72	12	[	[	X
cana-1298	72	13	0.7	0.7	NUM
cana-1298	72	14	,	,	PUNCT
cana-1298	72	15	0.8	0.8	NUM
cana-1298	72	16	]	]	PUNCT
cana-1298	72	17	,	,	PUNCT
cana-1298	72	18	[	[	X
cana-1298	72	19	0.8	0.8	NUM
cana-1298	72	20	,	,	PUNCT
cana-1298	72	21	0.9	0.9	NUM
cana-1298	72	22	]	]	PUNCT
cana-1298	72	23	,	,	PUNCT
cana-1298	72	24	[	[	X
cana-1298	72	25	0.9	0.9	NUM
cana-1298	72	26	,	,	PUNCT
cana-1298	72	27	1.0	1.0	NUM
cana-1298	72	28	]	]	PUNCT
cana-1298	72	29	,	,	PUNCT
cana-1298	73	1	[	[	X
cana-1298	73	2	−	−	NOUN
cana-1298	73	3	0.9	0.9	NUM
cana-1298	73	4	,	,	PUNCT
cana-1298	73	5	−0.8	−0.8	PROPN
cana-1298	73	6	]	]	PUNCT
cana-1298	73	7	,	,	PUNCT
cana-1298	74	1	[	[	X
cana-1298	74	2	−	−	PROPN
cana-1298	74	3	0.8	0.8	NUM
cana-1298	74	4	,	,	PUNCT
cana-1298	74	5	−0.7	−0.7	PROPN
cana-1298	74	6	]	]	PUNCT
cana-1298	74	7	,	,	PUNCT
cana-1298	74	8	[	[	X
cana-1298	74	9	−	−	NOUN
cana-1298	74	10	0.7	0.7	NUM
cana-1298	74	11	,	,	PUNCT
cana-1298	74	12	−0.6	−0.6	PROPN
cana-1298	74	13	]	]	X
cana-1298	74	14	)	)	PUNCT
cana-1298	74	15	,	,	PUNCT
cana-1298	74	16	(	(	PUNCT
cana-1298	74	17	1	1	X
cana-1298	74	18	,	,	PUNCT
cana-1298	74	19	[	[	X
cana-1298	74	20	0.5	0.5	NUM
cana-1298	74	21	,	,	PUNCT
cana-1298	74	22	0.6	0.6	NUM
cana-1298	74	23	]	]	PUNCT
cana-1298	74	24	,	,	PUNCT
cana-1298	74	25	[	[	X
cana-1298	74	26	0.6	0.6	NUM
cana-1298	74	27	,	,	PUNCT
cana-1298	74	28	0.7	0.7	NUM
cana-1298	74	29	]	]	PUNCT
cana-1298	74	30	,	,	PUNCT
cana-1298	74	31	[	[	X
cana-1298	74	32	0.7	0.7	NUM
cana-1298	74	33	,	,	PUNCT
cana-1298	74	34	0.8	0.8	NUM
cana-1298	74	35	]	]	PUNCT
cana-1298	74	36	,	,	PUNCT
cana-1298	75	1	[	[	X
cana-1298	75	2	−	−	X
cana-1298	75	3	0.6	0.6	NUM
cana-1298	75	4	,	,	PUNCT
cana-1298	75	5	−0.5	−0.5	PROPN
cana-1298	75	6	]	]	PUNCT
cana-1298	75	7	,	,	PUNCT
cana-1298	75	8	[	[	X
cana-1298	75	9	−	−	NOUN
cana-1298	75	10	0.5	0.5	NUM
cana-1298	75	11	,	,	PUNCT
cana-1298	75	12	−0.4	−0.4	NUM
cana-1298	75	13	]	]	PUNCT
cana-1298	75	14	,	,	PUNCT
cana-1298	76	1	[	[	X
cana-1298	76	2	−	−	PROPN
cana-1298	76	3	0.4	0.4	NUM
cana-1298	76	4	,	,	PUNCT
cana-1298	76	5	−0.3	−0.3	PROPN
cana-1298	76	6	]	]	PUNCT
cana-1298	76	7	)	)	PUNCT
cana-1298	76	8	,	,	PUNCT
cana-1298	76	9	(	(	PUNCT
cana-1298	76	10	2	2	X
cana-1298	76	11	,	,	PUNCT
cana-1298	76	12	[	[	X
cana-1298	76	13	0.5	0.5	NUM
cana-1298	76	14	,	,	PUNCT
cana-1298	76	15	0.6	0.6	NUM
cana-1298	76	16	]	]	PUNCT
cana-1298	76	17	,	,	PUNCT
cana-1298	76	18	[	[	X
cana-1298	76	19	0.6	0.6	NUM
cana-1298	76	20	,	,	PUNCT
cana-1298	76	21	0.7	0.7	NUM
cana-1298	76	22	]	]	PUNCT
cana-1298	76	23	,	,	PUNCT
cana-1298	76	24	[	[	X
cana-1298	76	25	0.7	0.7	NUM
cana-1298	76	26	,	,	PUNCT
cana-1298	76	27	0.8	0.8	NUM
cana-1298	76	28	]	]	PUNCT
cana-1298	76	29	,	,	PUNCT
cana-1298	77	1	[	[	X
cana-1298	77	2	−0.6	−0.6	PROPN
cana-1298	77	3	,	,	PUNCT
cana-1298	77	4	−0.5	−0.5	PROPN
cana-1298	77	5	]	]	PUNCT
cana-1298	77	6	,	,	PUNCT
cana-1298	77	7	[	[	X
cana-1298	77	8	−	−	NOUN
cana-1298	77	9	0.5	0.5	NUM
cana-1298	77	10	,	,	PUNCT
cana-1298	77	11	−0.4	−0.4	NUM
cana-1298	77	12	]	]	PUNCT
cana-1298	77	13	,	,	PUNCT
cana-1298	77	14	[	[	X
cana-1298	77	15	−	−	PROPN
cana-1298	77	16	0.4	0.4	NUM
cana-1298	77	17	,	,	PUNCT
cana-1298	77	18	−0.3	−0.3	PROPN
cana-1298	77	19	]	]	PUNCT
cana-1298	77	20	)	)	PUNCT
cana-1298	77	21	}	}	PUNCT
cana-1298	77	22	,	,	PUNCT
cana-1298	77	23	is	be	AUX
cana-1298	77	24	a	a	DET
cana-1298	77	25	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	77	26	of	of	ADP
cana-1298	77	27	𝕫3	𝕫3	PROPN
cana-1298	77	28	.	.	PUNCT
cana-1298	78	1	definition	definition	NOUN
cana-1298	78	2	1.10	1.10	NUM
cana-1298	78	3	.	.	PUNCT
cana-1298	79	1	𝐿𝑒𝑡	𝐿𝑒𝑡	PROPN
cana-1298	79	2	𝔎	𝔎	PROPN
cana-1298	79	3	=	=	PUNCT
cana-1298	79	4			PRON
cana-1298	79	5	𝔎1	𝔎1	VERB
cana-1298	79	6	+	+	NOUN
cana-1298	79	7	,	,	PUNCT
cana-1298	79	8	𝔎2	𝔎2	VERB
cana-1298	79	9	+	+	ADV
cana-1298	79	10	,	,	PUNCT
cana-1298	79	11	…	…	PUNCT
cana-1298	79	12	,	,	PUNCT
cana-1298	79	13	𝔎𝑛	𝔎𝑛	PROPN
cana-1298	79	14	+	+	PROPN
cana-1298	79	15	,	,	PUNCT
cana-1298	79	16	𝔎1	𝔎1	PROPN
cana-1298	79	17	−	−	PROPN
cana-1298	79	18	,	,	PUNCT
cana-1298	79	19	𝔎2	𝔎2	VERB
cana-1298	79	20	−	−	PROPN
cana-1298	79	21	,	,	PUNCT
cana-1298	79	22	…	…	PUNCT
cana-1298	79	23	,	,	PUNCT
cana-1298	80	1	𝔎𝑛	𝔎𝑛	PROPN
cana-1298	80	2	−	−	ADJ
cana-1298	80	3	𝑏𝑒	𝑏𝑒	NOUN
cana-1298	80	4	𝔹𝕍𝕄𝕀𝔽𝕊	𝔹𝕍𝕄𝕀𝔽𝕊	ADV
cana-1298	80	5	of	of	ADP
cana-1298	80	6	the	the	DET
cana-1298	80	7	set	set	NOUN
cana-1298	80	8	𝔏1	𝔏1	PROPN
cana-1298	80	9	,	,	PUNCT
cana-1298	80	10	the	the	DET
cana-1298	80	11	strongest	strong	ADJ
cana-1298	80	12	𝔹𝕍𝕄𝕀𝔽	𝔹𝕍𝕄𝕀𝔽	PROPN
cana-1298	80	13	𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛	𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛	NOUN
cana-1298	80	14	𝑜𝑛	𝑜𝑛	PROPN
cana-1298	80	15	𝔏1	𝔏1	PROPN
cana-1298	80	16	,	,	PUNCT
cana-1298	80	17	that	that	PRON
cana-1298	80	18	is	be	AUX
cana-1298	80	19	a	a	DET
cana-1298	80	20	𝔹𝕍𝕄𝕀𝔽	𝔹𝕍𝕄𝕀𝔽	ADJ
cana-1298	80	21	𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛	𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛	NOUN
cana-1298	80	22	on	on	ADP
cana-1298	80	23	𝔎	𝔎	PROPN
cana-1298	80	24	𝑖𝑠	𝑖𝑠	NOUN
cana-1298	80	25	℘	℘	PROPN
cana-1298	80	26	=	=	SYM
cana-1298	80	27	{	{	PUNCT
cana-1298	80	28	(𝜚	(𝜚	PROPN
cana-1298	80	29	,	,	PUNCT
cana-1298	80	30	𝜁	𝜁	PROPN
cana-1298	80	31	)	)	PUNCT
cana-1298	80	32	,	,	PUNCT
cana-1298	80	33	℘1	℘1	VERB
cana-1298	80	34	+	+	ADJ
cana-1298	80	35	(	(	PUNCT
cana-1298	80	36	𝜚	𝜚	NOUN
cana-1298	80	37	,	,	PUNCT
cana-1298	80	38	𝜁	𝜁	NOUN
cana-1298	80	39	)	)	PUNCT
cana-1298	80	40	,	,	PUNCT
cana-1298	80	41	℘2	℘2	PROPN
cana-1298	80	42	+	+	PROPN
cana-1298	80	43	(	(	PUNCT
cana-1298	80	44	𝜚	𝜚	NOUN
cana-1298	80	45	,	,	PUNCT
cana-1298	80	46	𝜁	𝜁	NOUN
cana-1298	80	47	)	)	PUNCT
cana-1298	80	48	,	,	PUNCT
cana-1298	80	49	…	…	PUNCT
cana-1298	80	50	,	,	PUNCT
cana-1298	80	51	℘𝑛	℘𝑛	NOUN
cana-1298	80	52	+	+	NOUN
cana-1298	80	53	(	(	PUNCT
cana-1298	80	54	𝜚	𝜚	NOUN
cana-1298	80	55	,	,	PUNCT
cana-1298	80	56	𝜁	𝜁	NOUN
cana-1298	80	57	)	)	PUNCT
cana-1298	80	58	,	,	PUNCT
cana-1298	80	59	℘1	℘1	ADJ
cana-1298	80	60	−(𝜚	−(𝜚	NOUN
cana-1298	80	61	,	,	PUNCT
cana-1298	80	62	𝜁	𝜁	NOUN
cana-1298	80	63	)	)	PUNCT
cana-1298	80	64	,	,	PUNCT
cana-1298	80	65	℘2	℘2	NOUN
cana-1298	80	66	−(𝜚	−(𝜚	NOUN
cana-1298	80	67	,	,	PUNCT
cana-1298	80	68	𝜁	𝜁	NOUN
cana-1298	80	69	)	)	PUNCT
cana-1298	80	70	,	,	PUNCT
cana-1298	80	71	…	…	PUNCT
cana-1298	80	72	,	,	PUNCT
cana-1298	80	73	℘𝑛	℘𝑛	NOUN
cana-1298	80	74	−(𝜚	−(𝜚	NOUN
cana-1298	80	75	,	,	PUNCT
cana-1298	80	76	𝜁)	𝜁)	NOUN
cana-1298	80	77	/	/	PUNCT
cana-1298	80	78	for	for	ADP
cana-1298	80	79	all	all	DET
cana-1298	80	80	𝜚	𝜚	NOUN
cana-1298	80	81	,	,	PUNCT
cana-1298	80	82	𝜁𝔏1	𝜁𝔏1	PROPN
cana-1298	80	83	}	}	PUNCT
cana-1298	80	84	,	,	PUNCT
cana-1298	80	85	where	where	SCONJ
cana-1298	80	86	℘𝑖	℘𝑖	PROPN
cana-1298	80	87	+	+	PROPN
cana-1298	80	88	(	(	PUNCT
cana-1298	80	89	𝜚	𝜚	NOUN
cana-1298	80	90	,	,	PUNCT
cana-1298	80	91	𝜁	𝜁	NOUN
cana-1298	80	92	)	)	PUNCT
cana-1298	80	93	=	=	PUNCT
cana-1298	80	94	rmin{𝔎i	rmin{𝔎i	NOUN
cana-1298	80	95	+	+	NOUN
cana-1298	80	96	(	(	PUNCT
cana-1298	80	97	𝜚	𝜚	NOUN
cana-1298	80	98	)	)	PUNCT
cana-1298	80	99	,	,	PUNCT
cana-1298	80	100	𝔎i	𝔎i	PROPN
cana-1298	80	101	+	+	PROPN
cana-1298	80	102	(	(	PUNCT
cana-1298	80	103	𝜁	𝜁	NOUN
cana-1298	80	104	)	)	PUNCT
cana-1298	80	105	}	}	PUNCT
cana-1298	80	106	and	and	CCONJ
cana-1298	80	107	℘𝑖	℘𝑖	NOUN
cana-1298	80	108	−(𝜚	−(𝜚	NOUN
cana-1298	80	109	,	,	PUNCT
cana-1298	80	110	𝜁	𝜁	NOUN
cana-1298	80	111	)	)	PUNCT
cana-1298	80	112	=	=	NOUN
cana-1298	80	113	rmax{𝔎i	rmax{𝔎i	NOUN
cana-1298	80	114	−(𝜚	−(𝜚	NOUN
cana-1298	80	115	)	)	PUNCT
cana-1298	80	116	,	,	PUNCT
cana-1298	80	117	𝔎i	𝔎i	PROPN
cana-1298	80	118	−(𝜁	−(𝜁	NOUN
cana-1298	80	119	)	)	PUNCT
cana-1298	80	120	}	}	PUNCT
cana-1298	80	121	,	,	PUNCT
cana-1298	80	122	for	for	ADP
cana-1298	80	123	all	all	DET
cana-1298	80	124	𝜚	𝜚	NOUN
cana-1298	80	125	,	,	PUNCT
cana-1298	80	126	𝜁𝔏1	𝜁𝔏1	PROPN
cana-1298	80	127	,	,	PUNCT
cana-1298	80	128	i	i	PRON
cana-1298	80	129	=	=	NOUN
cana-1298	80	130	1	1	NUM
cana-1298	80	131	,	,	PUNCT
cana-1298	80	132	2	2	NUM
cana-1298	80	133	,	,	PUNCT
cana-1298	80	134	…	…	PUNCT
cana-1298	80	135	,	,	PUNCT
cana-1298	80	136	n.	n.	ADJ
cana-1298	80	137	definition	definition	NOUN
cana-1298	80	138	1.11	1.11	NUM
cana-1298	80	139	.	.	PUNCT
cana-1298	81	1	𝐿𝑒𝑡	𝐿𝑒𝑡	PROPN
cana-1298	81	2	𝔎	𝔎	PROPN
cana-1298	81	3	=	=	PUNCT
cana-1298	81	4			PRON
cana-1298	81	5	𝔎1	𝔎1	VERB
cana-1298	81	6	+	+	NOUN
cana-1298	81	7	,	,	PUNCT
cana-1298	81	8	𝔎2	𝔎2	VERB
cana-1298	81	9	+	+	ADV
cana-1298	81	10	,	,	PUNCT
cana-1298	81	11	…	…	PUNCT
cana-1298	81	12	,	,	PUNCT
cana-1298	81	13	𝔎𝑛	𝔎𝑛	PROPN
cana-1298	81	14	+	+	PROPN
cana-1298	81	15	,	,	PUNCT
cana-1298	81	16	𝔎1	𝔎1	PROPN
cana-1298	81	17	−	−	PROPN
cana-1298	81	18	,	,	PUNCT
cana-1298	81	19	𝔎2	𝔎2	VERB
cana-1298	81	20	−	−	PROPN
cana-1298	81	21	,	,	PUNCT
cana-1298	81	22	…	…	PUNCT
cana-1298	81	23	,	,	PUNCT
cana-1298	81	24	𝔎𝑛	𝔎𝑛	PROPN
cana-1298	81	25	−	−	ADJ
cana-1298	81	26	and	and	CCONJ
cana-1298	81	27	℘	℘	PROPN
cana-1298	81	28	=	=	SYM
cana-1298	81	29			NOUN
cana-1298	81	30	℘1	℘1	VERB
cana-1298	81	31	+	+	PROPN
cana-1298	81	32	,	,	PUNCT
cana-1298	81	33	℘2	℘2	NOUN
cana-1298	81	34	+	+	PROPN
cana-1298	81	35	,	,	PUNCT
cana-1298	81	36	…	…	PUNCT
cana-1298	81	37	,	,	PUNCT
cana-1298	81	38	℘𝑛	℘𝑛	NOUN
cana-1298	82	1	+	+	ADV
cana-1298	82	2	,	,	PUNCT
cana-1298	82	3	℘1	℘1	VERB
cana-1298	82	4	−	−	PROPN
cana-1298	82	5	,	,	PUNCT
cana-1298	82	6	℘2	℘2	NOUN
cana-1298	82	7	−	−	PROPN
cana-1298	82	8	,	,	PUNCT
cana-1298	82	9	…	…	PUNCT
cana-1298	82	10	,	,	PUNCT
cana-1298	82	11	℘𝑛	℘𝑛	NOUN
cana-1298	82	12	−	−	ADJ
cana-1298	82	13	𝑏𝑒	𝑏𝑒	ADP
cana-1298	82	14	𝔹𝕍𝕄𝕀𝔽𝕊s	𝔹𝕍𝕄𝕀𝔽𝕊	NOUN
cana-1298	82	15	of	of	ADP
cana-1298	82	16	the	the	DET
cana-1298	82	17	sets	set	NOUN
cana-1298	82	18	𝔏1	𝔏1	NOUN
cana-1298	82	19	and	and	CCONJ
cana-1298	82	20	𝔏2	𝔏2	NOUN
cana-1298	82	21	respectively	respectively	ADV
cana-1298	82	22	.	.	PUNCT
cana-1298	83	1	the	the	DET
cana-1298	83	2	product	product	NOUN
cana-1298	83	3	of	of	ADP
cana-1298	83	4	𝔎	𝔎	PROPN
cana-1298	83	5	and	and	CCONJ
cana-1298	83	6	℘	℘	NOUN
cana-1298	83	7	,	,	PUNCT
cana-1298	83	8	denoted	denote	VERB
cana-1298	83	9	by	by	ADP
cana-1298	83	10	𝔎	𝔎	PROPN
cana-1298	83	11	×	×	NOUN
cana-1298	83	12	℘	℘	PROPN
cana-1298	83	13	,	,	PUNCT
cana-1298	83	14	is	be	AUX
cana-1298	83	15	defined	define	VERB
cana-1298	83	16	as	as	ADP
cana-1298	83	17	𝔎	𝔎	PROPN
cana-1298	83	18	×	×	NOUN
cana-1298	83	19	℘	℘	NOUN
cana-1298	83	20	=	=	SYM
cana-1298	83	21	{	{	PUNCT
cana-1298	83	22	(𝜚	(𝜚	PROPN
cana-1298	83	23	,	,	PUNCT
cana-1298	83	24	𝜁	𝜁	PROPN
cana-1298	83	25	)	)	PUNCT
cana-1298	83	26	,	,	PUNCT
cana-1298	83	27	(	(	PUNCT
cana-1298	83	28	𝔎1×℘1	𝔎1×℘1	X
cana-1298	83	29	)	)	PUNCT
cana-1298	83	30	+	+	PROPN
cana-1298	83	31	(	(	PUNCT
cana-1298	83	32	𝜚	𝜚	NOUN
cana-1298	83	33	,	,	PUNCT
cana-1298	83	34	𝜁	𝜁	NOUN
cana-1298	83	35	)	)	PUNCT
cana-1298	83	36	,	,	PUNCT
cana-1298	83	37	(	(	PUNCT
cana-1298	83	38	𝔎2×℘2	𝔎2×℘2	ADV
cana-1298	83	39	)	)	PUNCT
cana-1298	84	1	+	+	PROPN
cana-1298	84	2	(	(	PUNCT
cana-1298	84	3	𝜚	𝜚	NOUN
cana-1298	84	4	,	,	PUNCT
cana-1298	84	5	𝜁	𝜁	NOUN
cana-1298	84	6	)	)	PUNCT
cana-1298	84	7	,	,	PUNCT
cana-1298	84	8	…	…	PUNCT
cana-1298	84	9	,	,	PUNCT
cana-1298	84	10	(	(	PUNCT
cana-1298	84	11	𝔎n×℘n	𝔎n×℘n	NOUN
cana-1298	84	12	)	)	PUNCT
cana-1298	85	1	+	+	PROPN
cana-1298	85	2	(	(	PUNCT
cana-1298	85	3	𝜚	𝜚	NOUN
cana-1298	85	4	,	,	PUNCT
cana-1298	85	5	𝜁	𝜁	NOUN
cana-1298	85	6	)	)	PUNCT
cana-1298	85	7	,	,	PUNCT
cana-1298	85	8	(	(	PUNCT
cana-1298	85	9	𝔎1×℘1)−(𝜚	𝔎1×℘1)−(𝜚	NOUN
cana-1298	85	10	,	,	PUNCT
cana-1298	85	11	𝜁	𝜁	NOUN
cana-1298	85	12	)	)	PUNCT
cana-1298	85	13	,	,	PUNCT
cana-1298	85	14	communications	communication	NOUN
cana-1298	85	15	on	on	ADP
cana-1298	85	16	applied	apply	VERB
cana-1298	85	17	nonlinear	nonlinear	ADJ
cana-1298	85	18	analysis	analysis	NOUN
cana-1298	85	19	issn	issn	NOUN
cana-1298	85	20	:	:	PUNCT
cana-1298	85	21	1074	1074	NUM
cana-1298	85	22	-	-	PUNCT
cana-1298	85	23	133x	133x	NUM
cana-1298	85	24	vol	vol	NOUN
cana-1298	85	25	31	31	NUM
cana-1298	85	26	no	no	NOUN
cana-1298	85	27	.	.	PUNCT
cana-1298	86	1	7s	7	NOUN
cana-1298	86	2	(	(	PUNCT
cana-1298	86	3	2024	2024	NUM
cana-1298	86	4	)	)	PUNCT
cana-1298	86	5	233	233	NUM
cana-1298	86	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1298	86	7	(	(	PUNCT
cana-1298	86	8	𝔎2×℘2)−(𝜚	𝔎2×℘2)−(𝜚	NOUN
cana-1298	86	9	,	,	PUNCT
cana-1298	86	10	𝜁	𝜁	NOUN
cana-1298	86	11	)	)	PUNCT
cana-1298	86	12	,	,	PUNCT
cana-1298	86	13	…	…	PUNCT
cana-1298	86	14	,	,	PUNCT
cana-1298	86	15	(	(	PUNCT
cana-1298	86	16	𝔎n×℘n)−(𝜚	𝔎n×℘n)−(𝜚	NOUN
cana-1298	86	17	,	,	PUNCT
cana-1298	86	18	𝜁)	𝜁)	NOUN
cana-1298	86	19	/	/	PUNCT
cana-1298	86	20	for	for	ADP
cana-1298	86	21	all	all	DET
cana-1298	86	22	(	(	PUNCT
cana-1298	86	23	𝜚	𝜚	NOUN
cana-1298	86	24	,	,	PUNCT
cana-1298	86	25	𝜁)𝔏1	𝜁)𝔏1	ADV
cana-1298	86	26	×	×	NOUN
cana-1298	86	27	𝔏2	𝔏2	NOUN
cana-1298	86	28	}	}	PUNCT
cana-1298	86	29	,	,	PUNCT
cana-1298	86	30	where	where	SCONJ
cana-1298	86	31	(	(	PUNCT
cana-1298	86	32	𝔎i×℘i	𝔎i×℘i	X
cana-1298	86	33	)	)	PUNCT
cana-1298	86	34	+	+	PROPN
cana-1298	86	35	(	(	PUNCT
cana-1298	86	36	𝜚	𝜚	NOUN
cana-1298	86	37	,	,	PUNCT
cana-1298	86	38	𝜁	𝜁	NOUN
cana-1298	86	39	)	)	PUNCT
cana-1298	86	40	=	=	PUNCT
cana-1298	86	41	rmin{𝔎i	rmin{𝔎i	NOUN
cana-1298	86	42	+	+	NOUN
cana-1298	86	43	(	(	PUNCT
cana-1298	86	44	𝜚	𝜚	NOUN
cana-1298	86	45	)	)	PUNCT
cana-1298	86	46	,	,	PUNCT
cana-1298	86	47	℘i	℘i	ADJ
cana-1298	86	48	+	+	PROPN
cana-1298	86	49	(	(	PUNCT
cana-1298	86	50	𝜁	𝜁	NOUN
cana-1298	86	51	)	)	PUNCT
cana-1298	86	52	}	}	PUNCT
cana-1298	86	53	and	and	CCONJ
cana-1298	86	54	(	(	PUNCT
cana-1298	86	55	𝔎i×℘i)−(𝜚	𝔎i×℘i)−(𝜚	NOUN
cana-1298	86	56	,	,	PUNCT
cana-1298	86	57	𝜁	𝜁	NOUN
cana-1298	86	58	)	)	PUNCT
cana-1298	86	59	=	=	NOUN
cana-1298	86	60	rmax{𝔎i	rmax{𝔎i	NOUN
cana-1298	86	61	−(𝜚	−(𝜚	NOUN
cana-1298	86	62	)	)	PUNCT
cana-1298	86	63	,	,	PUNCT
cana-1298	86	64	℘i	℘i	ADJ
cana-1298	86	65	−(𝜁	−(𝜁	NOUN
cana-1298	86	66	)	)	PUNCT
cana-1298	86	67	}	}	PUNCT
cana-1298	86	68	,	,	PUNCT
cana-1298	86	69	i	i	PRON
cana-1298	86	70	=	=	NOUN
cana-1298	86	71	1	1	NUM
cana-1298	86	72	,	,	PUNCT
cana-1298	86	73	2	2	NUM
cana-1298	86	74	,	,	PUNCT
cana-1298	86	75	…	…	PUNCT
cana-1298	86	76	,	,	PUNCT
cana-1298	86	77	n.	n.	NOUN
cana-1298	86	78	2	2	NUM
cana-1298	86	79	.	.	PUNCT
cana-1298	87	1	some	some	DET
cana-1298	87	2	theorems	theorem	NOUN
cana-1298	87	3	.	.	PUNCT
cana-1298	87	4	theorem	theorem	VERB
cana-1298	87	5	2.1	2.1	NUM
cana-1298	87	6	.	.	PUNCT
cana-1298	88	1	𝐼𝑓	𝐼𝑓	VERB
cana-1298	88	2	𝔎	𝔎	NOUN
cana-1298	88	3	=	=	PUNCT
cana-1298	88	4			PRON
cana-1298	88	5	𝔎1	𝔎1	VERB
cana-1298	88	6	+	+	NOUN
cana-1298	88	7	,	,	PUNCT
cana-1298	88	8	𝔎2	𝔎2	VERB
cana-1298	88	9	+	+	PROPN
cana-1298	88	10	,	,	PUNCT
cana-1298	88	11	…	…	PUNCT
cana-1298	88	12	,	,	PUNCT
cana-1298	88	13	𝔎𝑛	𝔎𝑛	PROPN
cana-1298	88	14	+	+	PROPN
cana-1298	88	15	,	,	PUNCT
cana-1298	88	16	𝔎1	𝔎1	PROPN
cana-1298	88	17	−	−	PROPN
cana-1298	88	18	,	,	PUNCT
cana-1298	88	19	𝔎2	𝔎2	VERB
cana-1298	88	20	−	−	PROPN
cana-1298	88	21	,	,	PUNCT
cana-1298	88	22	…	…	PUNCT
cana-1298	88	23	,	,	PUNCT
cana-1298	88	24	𝔎𝑛	𝔎𝑛	PROPN
cana-1298	88	25	−	−	ADJ
cana-1298	88	26	and	and	CCONJ
cana-1298	88	27	℘	℘	PROPN
cana-1298	88	28	=	=	SYM
cana-1298	88	29			NOUN
cana-1298	88	30	℘1	℘1	VERB
cana-1298	88	31	+	+	PROPN
cana-1298	88	32	,	,	PUNCT
cana-1298	88	33	℘2	℘2	NOUN
cana-1298	88	34	+	+	PROPN
cana-1298	88	35	,	,	PUNCT
cana-1298	88	36	…	…	PUNCT
cana-1298	88	37	,	,	PUNCT
cana-1298	88	38	℘𝑛	℘𝑛	NOUN
cana-1298	89	1	+	+	ADV
cana-1298	89	2	,	,	PUNCT
cana-1298	89	3	℘1	℘1	VERB
cana-1298	89	4	−	−	PROPN
cana-1298	89	5	,	,	PUNCT
cana-1298	89	6	℘2	℘2	NOUN
cana-1298	89	7	−	−	PROPN
cana-1298	89	8	,	,	PUNCT
cana-1298	89	9	…	…	PUNCT
cana-1298	89	10	,	,	PUNCT
cana-1298	89	11	℘𝑛	℘𝑛	VERB
cana-1298	89	12	−	−	PROPN
cana-1298	89	13	𝑎𝑟𝑒	𝑎𝑟𝑒	NOUN
cana-1298	89	14	𝔹𝕍𝕄𝕀𝔽𝕊ℝs	𝔹𝕍𝕄𝕀𝔽𝕊ℝs	PROPN
cana-1298	89	15	of	of	ADP
cana-1298	89	16	the	the	DET
cana-1298	89	17	rings	ring	NOUN
cana-1298	89	18	𝔏1	𝔏1	PROPN
cana-1298	89	19	and	and	CCONJ
cana-1298	89	20	𝔏2	𝔏2	NOUN
cana-1298	89	21	respectively	respectively	ADV
cana-1298	89	22	,	,	PUNCT
cana-1298	89	23	then	then	ADV
cana-1298	89	24	𝔎	𝔎	PROPN
cana-1298	89	25	×	×	NOUN
cana-1298	89	26	℘	℘	PROPN
cana-1298	89	27	is	be	AUX
cana-1298	89	28	a	a	DET
cana-1298	89	29	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	89	30	of	of	ADP
cana-1298	89	31	the	the	DET
cana-1298	89	32	ring	ring	NOUN
cana-1298	89	33	𝔏1	𝔏1	PROPN
cana-1298	89	34	×	×	PROPN
cana-1298	89	35	𝔏2	𝔏2	NOUN
cana-1298	89	36	.	.	PUNCT
cana-1298	90	1	proof	proof	NOUN
cana-1298	90	2	.	.	PUNCT
cana-1298	91	1	let	let	VERB
cana-1298	91	2	𝜚	𝜚	X
cana-1298	91	3	,	,	PUNCT
cana-1298	91	4	𝜐	𝜐	VERB
cana-1298	91	5	be	be	AUX
cana-1298	91	6	in	in	ADP
cana-1298	91	7	𝔏1	𝔏1	PROPN
cana-1298	91	8	and	and	CCONJ
cana-1298	91	9	𝜁	𝜁	PROPN
cana-1298	91	10	,	,	PUNCT
cana-1298	91	11	𝜉	𝜉	AUX
cana-1298	91	12	be	be	AUX
cana-1298	91	13	in	in	ADP
cana-1298	91	14	𝔏2	𝔏2	NOUN
cana-1298	91	15	.	.	PUNCT
cana-1298	92	1	then	then	ADV
cana-1298	92	2	(	(	PUNCT
cana-1298	92	3	𝜚	𝜚	NOUN
cana-1298	92	4	,	,	PUNCT
cana-1298	92	5	𝜁	𝜁	NOUN
cana-1298	92	6	)	)	PUNCT
cana-1298	92	7	and	and	CCONJ
cana-1298	92	8	(	(	PUNCT
cana-1298	92	9	𝜐	𝜐	NOUN
cana-1298	92	10	,	,	PUNCT
cana-1298	92	11	𝜉	𝜉	X
cana-1298	92	12	)	)	PUNCT
cana-1298	92	13	are	be	AUX
cana-1298	92	14	in	in	ADP
cana-1298	92	15	𝔏1×𝔏2	𝔏1×𝔏2	PROPN
cana-1298	92	16	.	.	PUNCT
cana-1298	93	1	for	for	ADP
cana-1298	93	2	all	all	DET
cana-1298	93	3	i	i	PRON
cana-1298	93	4	,	,	PUNCT
cana-1298	93	5	i	i	NOUN
cana-1298	93	6	=	=	NOUN
cana-1298	93	7	1	1	NUM
cana-1298	93	8	,	,	PUNCT
cana-1298	93	9	2	2	NUM
cana-1298	93	10	,	,	PUNCT
cana-1298	93	11	…	…	PUNCT
cana-1298	93	12	,	,	PUNCT
cana-1298	93	13	n	n	CCONJ
cana-1298	93	14	,	,	PUNCT
cana-1298	93	15	(	(	PUNCT
cana-1298	93	16	𝔎i×℘i	𝔎i×℘i	X
cana-1298	93	17	)	)	PUNCT
cana-1298	93	18	+	+	PUNCT
cana-1298	93	19	[	[	X
cana-1298	93	20	(	(	PUNCT
cana-1298	93	21	𝜚	𝜚	NOUN
cana-1298	93	22	,	,	PUNCT
cana-1298	93	23	𝜁)−(𝜐	𝜁)−(𝜐	NOUN
cana-1298	93	24	,	,	PUNCT
cana-1298	93	25	𝜉	𝜉	NOUN
cana-1298	93	26	)	)	PUNCT
cana-1298	93	27	]	]	PUNCT
cana-1298	93	28	=	=	SYM
cana-1298	93	29	(	(	PUNCT
cana-1298	93	30	𝔎i×℘i	𝔎i×℘i	X
cana-1298	93	31	)	)	PUNCT
cana-1298	94	1	+	+	ADJ
cana-1298	94	2	(	(	PUNCT
cana-1298	94	3	𝜚−𝜐	𝜚−𝜐	NOUN
cana-1298	94	4	,	,	PUNCT
cana-1298	94	5	𝜁−	𝜁−	ADJ
cana-1298	94	6	𝜉	𝜉	NOUN
cana-1298	94	7	)	)	PUNCT
cana-1298	94	8	=	=	PUNCT
cana-1298	94	9	rmin{𝔎i	rmin{𝔎i	NOUN
cana-1298	94	10	+	+	NOUN
cana-1298	94	11	(	(	PUNCT
cana-1298	94	12	𝜚−𝜐	𝜚−𝜐	NOUN
cana-1298	94	13	)	)	PUNCT
cana-1298	94	14	,	,	PUNCT
cana-1298	94	15	℘i	℘i	ADJ
cana-1298	94	16	+	+	PROPN
cana-1298	94	17	(	(	PUNCT
cana-1298	94	18	𝜁−	𝜁−	ADJ
cana-1298	94	19	𝜉	𝜉	NOUN
cana-1298	94	20	)	)	PUNCT
cana-1298	94	21	}	}	PUNCT
cana-1298	94	22			X
cana-1298	94	23	rmin	rmin	VERB
cana-1298	94	24	{	{	PUNCT
cana-1298	94	25	rmin{𝔎i	rmin{𝔎i	NOUN
cana-1298	94	26	+	+	PROPN
cana-1298	94	27	(	(	PUNCT
cana-1298	94	28	𝜚	𝜚	NOUN
cana-1298	94	29	)	)	PUNCT
cana-1298	94	30	,	,	PUNCT
cana-1298	95	1	𝔎i	𝔎i	PROPN
cana-1298	95	2	+	+	PROPN
cana-1298	95	3	(	(	PUNCT
cana-1298	95	4	𝜐	𝜐	NOUN
cana-1298	95	5	)	)	PUNCT
cana-1298	95	6	}	}	PUNCT
cana-1298	95	7	,	,	PUNCT
cana-1298	95	8	rmin{℘i	rmin{℘i	NOUN
cana-1298	95	9	+	+	PROPN
cana-1298	95	10	(	(	PUNCT
cana-1298	95	11	𝜁	𝜁	ADJ
cana-1298	95	12	)	)	PUNCT
cana-1298	95	13	,	,	PUNCT
cana-1298	95	14	℘i	℘i	ADJ
cana-1298	95	15	+	+	PROPN
cana-1298	95	16	(	(	PUNCT
cana-1298	95	17	𝜉	𝜉	NOUN
cana-1298	95	18	)	)	PUNCT
cana-1298	95	19	}	}	PUNCT
cana-1298	95	20	}	}	PUNCT
cana-1298	95	21	=	=	SYM
cana-1298	95	22	rmin{rmin{𝔎i	rmin{rmin{𝔎i	NOUN
cana-1298	95	23	+	+	PROPN
cana-1298	95	24	(	(	PUNCT
cana-1298	95	25	𝜚	𝜚	NOUN
cana-1298	95	26	)	)	PUNCT
cana-1298	95	27	,	,	PUNCT
cana-1298	95	28	℘i	℘i	ADJ
cana-1298	95	29	+	+	PROPN
cana-1298	95	30	(	(	PUNCT
cana-1298	95	31	𝜁	𝜁	NOUN
cana-1298	95	32	)	)	PUNCT
cana-1298	95	33	}	}	PUNCT
cana-1298	95	34	,	,	PUNCT
cana-1298	95	35	rmin{𝔎i	rmin{𝔎i	NOUN
cana-1298	95	36	+	+	PROPN
cana-1298	95	37	(	(	PUNCT
cana-1298	95	38	𝜐	𝜐	NOUN
cana-1298	95	39	)	)	PUNCT
cana-1298	95	40	,	,	PUNCT
cana-1298	95	41	℘i	℘i	ADJ
cana-1298	95	42	+	+	PROPN
cana-1298	95	43	(	(	PUNCT
cana-1298	95	44	𝜉	𝜉	NOUN
cana-1298	95	45	)	)	PUNCT
cana-1298	95	46	}	}	PUNCT
cana-1298	95	47	}	}	PUNCT
cana-1298	95	48	=	=	SYM
cana-1298	95	49	rmin{(𝔎i×℘i	rmin{(𝔎i×℘i	X
cana-1298	95	50	)	)	PUNCT
cana-1298	95	51	+	+	ADJ
cana-1298	95	52	(	(	PUNCT
cana-1298	95	53	𝜚	𝜚	NOUN
cana-1298	95	54	,	,	PUNCT
cana-1298	95	55	𝜁	𝜁	NOUN
cana-1298	95	56	)	)	PUNCT
cana-1298	95	57	,	,	PUNCT
cana-1298	95	58	(	(	PUNCT
cana-1298	95	59	𝔎i×℘i	𝔎i×℘i	X
cana-1298	95	60	)	)	PUNCT
cana-1298	95	61	+	+	PROPN
cana-1298	95	62	(	(	PUNCT
cana-1298	95	63	𝜐	𝜐	NOUN
cana-1298	95	64	,	,	PUNCT
cana-1298	95	65	𝜉	𝜉	NOUN
cana-1298	95	66	)	)	PUNCT
cana-1298	95	67	}	}	PUNCT
cana-1298	95	68	,	,	PUNCT
cana-1298	95	69	for	for	ADP
cana-1298	95	70	all	all	PRON
cana-1298	95	71	(	(	PUNCT
cana-1298	95	72	𝜚	𝜚	NOUN
cana-1298	95	73	,	,	PUNCT
cana-1298	95	74	𝜁	𝜁	NOUN
cana-1298	95	75	)	)	PUNCT
cana-1298	95	76	,	,	PUNCT
cana-1298	95	77	(	(	PUNCT
cana-1298	95	78	𝜐	𝜐	NOUN
cana-1298	95	79	,	,	PUNCT
cana-1298	95	80	𝜉	𝜉	NOUN
cana-1298	95	81	)	)	PUNCT
cana-1298	95	82	in	in	ADP
cana-1298	95	83	𝔏1×𝔏2	𝔏1×𝔏2	PROPN
cana-1298	95	84	.	.	PUNCT
cana-1298	95	85	and	and	CCONJ
cana-1298	95	86	(	(	PUNCT
cana-1298	95	87	𝔎i×℘i	𝔎i×℘i	X
cana-1298	95	88	)	)	PUNCT
cana-1298	96	1	+	+	PUNCT
cana-1298	96	2	[	[	X
cana-1298	96	3	(	(	PUNCT
cana-1298	96	4	𝜚	𝜚	NOUN
cana-1298	96	5	,	,	PUNCT
cana-1298	96	6	𝜁)(𝜐	𝜁)(𝜐	NOUN
cana-1298	96	7	,	,	PUNCT
cana-1298	96	8	𝜉	𝜉	NOUN
cana-1298	96	9	)	)	PUNCT
cana-1298	96	10	]	]	PUNCT
cana-1298	97	1	=	=	SYM
cana-1298	97	2	(	(	PUNCT
cana-1298	97	3	𝔎i×℘i	𝔎i×℘i	X
cana-1298	97	4	)	)	PUNCT
cana-1298	97	5	+	+	ADJ
cana-1298	97	6	(	(	PUNCT
cana-1298	97	7	𝜚𝜐	𝜚𝜐	ADP
cana-1298	97	8	,	,	PUNCT
cana-1298	97	9	𝜁	𝜁	DET
cana-1298	97	10	𝜉	𝜉	X
cana-1298	97	11	)	)	PUNCT
cana-1298	97	12	=	=	PUNCT
cana-1298	97	13	rmin{𝔎i	rmin{𝔎i	NOUN
cana-1298	97	14	+	+	NOUN
cana-1298	97	15	(	(	PUNCT
cana-1298	97	16	𝜚𝜐	𝜚𝜐	NOUN
cana-1298	97	17	)	)	PUNCT
cana-1298	97	18	,	,	PUNCT
cana-1298	97	19	℘i	℘i	ADJ
cana-1298	97	20	+	+	PROPN
cana-1298	97	21	(	(	PUNCT
cana-1298	97	22	𝜁	𝜁	PRON
cana-1298	97	23	𝜉	𝜉	NOUN
cana-1298	97	24	)	)	PUNCT
cana-1298	97	25	}	}	PUNCT
cana-1298	97	26			X
cana-1298	97	27	rmin	rmin	VERB
cana-1298	97	28	{	{	PUNCT
cana-1298	97	29	rmin{𝔎i	rmin{𝔎i	NOUN
cana-1298	97	30	+	+	PROPN
cana-1298	97	31	(	(	PUNCT
cana-1298	97	32	𝜚	𝜚	NOUN
cana-1298	97	33	)	)	PUNCT
cana-1298	97	34	,	,	PUNCT
cana-1298	97	35	𝔎i	𝔎i	PROPN
cana-1298	97	36	+	+	PROPN
cana-1298	97	37	(	(	PUNCT
cana-1298	97	38	𝜐	𝜐	NOUN
cana-1298	97	39	)	)	PUNCT
cana-1298	97	40	}	}	PUNCT
cana-1298	97	41	,	,	PUNCT
cana-1298	97	42	rmin{℘i	rmin{℘i	NOUN
cana-1298	97	43	+	+	PROPN
cana-1298	97	44	(	(	PUNCT
cana-1298	97	45	𝜁	𝜁	ADJ
cana-1298	97	46	)	)	PUNCT
cana-1298	97	47	,	,	PUNCT
cana-1298	97	48	℘i	℘i	ADJ
cana-1298	97	49	+	+	PROPN
cana-1298	97	50	(	(	PUNCT
cana-1298	97	51	𝜉	𝜉	NOUN
cana-1298	97	52	)	)	PUNCT
cana-1298	97	53	}	}	PUNCT
cana-1298	97	54	}	}	PUNCT
cana-1298	97	55	=	=	SYM
cana-1298	97	56	rmin{rmin{𝔎i	rmin{rmin{𝔎i	NOUN
cana-1298	97	57	+	+	PROPN
cana-1298	97	58	(	(	PUNCT
cana-1298	97	59	𝜚	𝜚	NOUN
cana-1298	97	60	)	)	PUNCT
cana-1298	97	61	,	,	PUNCT
cana-1298	97	62	℘i	℘i	ADJ
cana-1298	97	63	+	+	PROPN
cana-1298	97	64	(	(	PUNCT
cana-1298	97	65	𝜁	𝜁	NOUN
cana-1298	97	66	)	)	PUNCT
cana-1298	97	67	}	}	PUNCT
cana-1298	97	68	,	,	PUNCT
cana-1298	97	69	rmin{𝔎i	rmin{𝔎i	NOUN
cana-1298	97	70	+	+	PROPN
cana-1298	97	71	(	(	PUNCT
cana-1298	97	72	𝜐	𝜐	NOUN
cana-1298	97	73	)	)	PUNCT
cana-1298	97	74	,	,	PUNCT
cana-1298	97	75	℘i	℘i	ADJ
cana-1298	97	76	+	+	PROPN
cana-1298	97	77	(	(	PUNCT
cana-1298	97	78	𝜉	𝜉	NOUN
cana-1298	97	79	)	)	PUNCT
cana-1298	97	80	}	}	PUNCT
cana-1298	97	81	}	}	PUNCT
cana-1298	97	82	=	=	SYM
cana-1298	97	83	rmin	rmin	NOUN
cana-1298	97	84	{	{	PUNCT
cana-1298	97	85	(	(	PUNCT
cana-1298	97	86	𝔎i×℘i	𝔎i×℘i	X
cana-1298	97	87	)	)	PUNCT
cana-1298	97	88	+	+	PROPN
cana-1298	97	89	(	(	PUNCT
cana-1298	97	90	𝜚	𝜚	NOUN
cana-1298	97	91	,	,	PUNCT
cana-1298	97	92	𝜁	𝜁	NOUN
cana-1298	97	93	)	)	PUNCT
cana-1298	97	94	,	,	PUNCT
cana-1298	97	95	(	(	PUNCT
cana-1298	97	96	𝔎i×℘i	𝔎i×℘i	X
cana-1298	97	97	)	)	PUNCT
cana-1298	97	98	+	+	PROPN
cana-1298	97	99	(	(	PUNCT
cana-1298	97	100	𝜐	𝜐	NOUN
cana-1298	97	101	,	,	PUNCT
cana-1298	97	102	𝜉	𝜉	NOUN
cana-1298	97	103	)	)	PUNCT
cana-1298	97	104	}	}	PUNCT
cana-1298	97	105	,	,	PUNCT
cana-1298	97	106	for	for	ADP
cana-1298	97	107	all	all	PRON
cana-1298	97	108	(	(	PUNCT
cana-1298	97	109	𝜚	𝜚	NOUN
cana-1298	97	110	,	,	PUNCT
cana-1298	97	111	𝜁	𝜁	NOUN
cana-1298	97	112	)	)	PUNCT
cana-1298	97	113	,	,	PUNCT
cana-1298	97	114	(	(	PUNCT
cana-1298	97	115	𝜐	𝜐	NOUN
cana-1298	97	116	,	,	PUNCT
cana-1298	97	117	𝜉	𝜉	NOUN
cana-1298	97	118	)	)	PUNCT
cana-1298	97	119	in	in	ADP
cana-1298	97	120	𝔏1×𝔏2	𝔏1×𝔏2	PROPN
cana-1298	97	121	.	.	PUNCT
cana-1298	98	1	also	also	ADV
cana-1298	98	2	(	(	PUNCT
cana-1298	98	3	𝔎i×℘i)−[(𝜚	𝔎i×℘i)−[(𝜚	ADJ
cana-1298	98	4	,	,	PUNCT
cana-1298	98	5	𝜁)−(𝜐	𝜁)−(𝜐	NOUN
cana-1298	98	6	,	,	PUNCT
cana-1298	98	7	𝜉	𝜉	NOUN
cana-1298	98	8	)	)	PUNCT
cana-1298	98	9	]	]	PUNCT
cana-1298	99	1	=	=	SYM
cana-1298	99	2	(	(	PUNCT
cana-1298	99	3	𝔎i×℘i)−(𝜚−𝜐	𝔎i×℘i)−(𝜚−𝜐	NOUN
cana-1298	99	4	,	,	PUNCT
cana-1298	99	5	𝜁−	𝜁−	ADJ
cana-1298	99	6	𝜉	𝜉	NOUN
cana-1298	99	7	)	)	PUNCT
cana-1298	99	8	=	=	SYM
cana-1298	99	9	rmax{𝔎i	rmax{𝔎i	NOUN
cana-1298	99	10	−(𝜚−𝜐	−(𝜚−𝜐	NUM
cana-1298	99	11	)	)	PUNCT
cana-1298	99	12	,	,	PUNCT
cana-1298	99	13	℘i	℘i	ADJ
cana-1298	99	14	−(𝜁−	−(𝜁−	NOUN
cana-1298	99	15	𝜉	𝜉	NOUN
cana-1298	99	16	)	)	PUNCT
cana-1298	99	17	}	}	PUNCT
cana-1298	99	18			NUM
cana-1298	99	19	rmax	rmax	ADJ
cana-1298	99	20	{	{	PUNCT
cana-1298	99	21	rmax{𝔎i	rmax{𝔎i	NOUN
cana-1298	99	22	−(𝜚	−(𝜚	NOUN
cana-1298	99	23	)	)	PUNCT
cana-1298	99	24	,	,	PUNCT
cana-1298	99	25	𝔎i	𝔎i	PROPN
cana-1298	99	26	−(𝜐	−(𝜐	NOUN
cana-1298	99	27	)	)	PUNCT
cana-1298	99	28	}	}	PUNCT
cana-1298	99	29	,	,	PUNCT
cana-1298	99	30	rmax{℘i	rmax{℘i	NOUN
cana-1298	99	31	−(𝜁	−(𝜁	NOUN
cana-1298	99	32	)	)	PUNCT
cana-1298	99	33	,	,	PUNCT
cana-1298	99	34	℘i	℘i	ADJ
cana-1298	99	35	−	−	PROPN
cana-1298	99	36	(	(	PUNCT
cana-1298	99	37	𝜉	𝜉	NOUN
cana-1298	99	38	)	)	PUNCT
cana-1298	99	39	}	}	PUNCT
cana-1298	99	40	}	}	PUNCT
cana-1298	99	41	=	=	SYM
cana-1298	99	42	rmax{rmax{𝔎i	rmax{rmax{𝔎i	NOUN
cana-1298	99	43	−(𝜚	−(𝜚	NOUN
cana-1298	99	44	)	)	PUNCT
cana-1298	99	45	,	,	PUNCT
cana-1298	99	46	℘i	℘i	ADJ
cana-1298	99	47	−(𝜁	−(𝜁	NOUN
cana-1298	99	48	)	)	PUNCT
cana-1298	99	49	}	}	PUNCT
cana-1298	99	50	,	,	PUNCT
cana-1298	99	51	rmax{𝔎i	rmax{𝔎i	NOUN
cana-1298	99	52	−(𝜐	−(𝜐	PROPN
cana-1298	99	53	)	)	PUNCT
cana-1298	99	54	,	,	PUNCT
cana-1298	99	55	℘i	℘i	ADJ
cana-1298	99	56	−	−	PROPN
cana-1298	99	57	(	(	PUNCT
cana-1298	99	58	𝜉	𝜉	NOUN
cana-1298	99	59	)	)	PUNCT
cana-1298	99	60	}	}	PUNCT
cana-1298	99	61	}	}	PUNCT
cana-1298	99	62	=	=	SYM
cana-1298	99	63	rmax	rmax	ADJ
cana-1298	99	64	{	{	PUNCT
cana-1298	99	65	(	(	PUNCT
cana-1298	99	66	𝔎i×℘i)−(𝜚	𝔎i×℘i)−(𝜚	NOUN
cana-1298	99	67	,	,	PUNCT
cana-1298	99	68	𝜁	𝜁	NOUN
cana-1298	99	69	)	)	PUNCT
cana-1298	99	70	,	,	PUNCT
cana-1298	99	71	(	(	PUNCT
cana-1298	99	72	𝔎i×℘i)−(𝜐	𝔎i×℘i)−(𝜐	PROPN
cana-1298	99	73	,	,	PUNCT
cana-1298	99	74	𝜉	𝜉	NOUN
cana-1298	99	75	)	)	PUNCT
cana-1298	99	76	}	}	PUNCT
cana-1298	99	77	,	,	PUNCT
cana-1298	99	78	for	for	ADP
cana-1298	99	79	all	all	PRON
cana-1298	99	80	(	(	PUNCT
cana-1298	99	81	𝜚	𝜚	NOUN
cana-1298	99	82	,	,	PUNCT
cana-1298	99	83	𝜁	𝜁	NOUN
cana-1298	99	84	)	)	PUNCT
cana-1298	99	85	,	,	PUNCT
cana-1298	99	86	(	(	PUNCT
cana-1298	99	87	𝜐	𝜐	NOUN
cana-1298	99	88	,	,	PUNCT
cana-1298	99	89	𝜉	𝜉	NOUN
cana-1298	99	90	)	)	PUNCT
cana-1298	99	91	in	in	ADP
cana-1298	99	92	𝔏1×𝔏2	𝔏1×𝔏2	PROPN
cana-1298	99	93	.	.	PUNCT
cana-1298	100	1	and	and	CCONJ
cana-1298	100	2	(	(	PUNCT
cana-1298	100	3	𝔎i×℘i)−[(𝜚	𝔎i×℘i)−[(𝜚	ADJ
cana-1298	100	4	,	,	PUNCT
cana-1298	100	5	𝜁)(𝜐	𝜁)(𝜐	NOUN
cana-1298	100	6	,	,	PUNCT
cana-1298	100	7	𝜉	𝜉	NOUN
cana-1298	100	8	)	)	PUNCT
cana-1298	100	9	]	]	PUNCT
cana-1298	101	1	=	=	SYM
cana-1298	101	2	(	(	PUNCT
cana-1298	101	3	𝔎i×℘i)−(𝜚𝜐	𝔎i×℘i)−(𝜚𝜐	PROPN
cana-1298	101	4	,	,	PUNCT
cana-1298	101	5	𝜁𝜉	𝜁𝜉	NOUN
cana-1298	101	6	)	)	PUNCT
cana-1298	101	7	=	=	NOUN
cana-1298	101	8	rmax{𝔎i	rmax{𝔎i	NOUN
cana-1298	101	9	−(𝜚𝜐	−(𝜚𝜐	NOUN
cana-1298	101	10	)	)	PUNCT
cana-1298	101	11	,	,	PUNCT
cana-1298	101	12	℘i	℘i	ADJ
cana-1298	101	13	−(𝜁𝜉	−(𝜁𝜉	NOUN
cana-1298	101	14	)	)	PUNCT
cana-1298	101	15	}	}	PUNCT
cana-1298	101	16			NUM
cana-1298	101	17	rmax	rmax	ADJ
cana-1298	101	18	{	{	PUNCT
cana-1298	101	19	rmax{𝔎i	rmax{𝔎i	NOUN
cana-1298	101	20	−(𝜚	−(𝜚	NOUN
cana-1298	101	21	)	)	PUNCT
cana-1298	101	22	,	,	PUNCT
cana-1298	101	23	𝔎i	𝔎i	PROPN
cana-1298	101	24	−(𝜐	−(𝜐	NOUN
cana-1298	101	25	)	)	PUNCT
cana-1298	101	26	}	}	PUNCT
cana-1298	101	27	,	,	PUNCT
cana-1298	101	28	rmax{℘i	rmax{℘i	NOUN
cana-1298	101	29	−(𝜁	−(𝜁	NOUN
cana-1298	101	30	)	)	PUNCT
cana-1298	101	31	,	,	PUNCT
cana-1298	101	32	℘i	℘i	ADJ
cana-1298	101	33	−	−	PROPN
cana-1298	101	34	(	(	PUNCT
cana-1298	101	35	𝜉	𝜉	NOUN
cana-1298	101	36	)	)	PUNCT
cana-1298	101	37	}	}	PUNCT
cana-1298	101	38	}	}	PUNCT
cana-1298	101	39	=	=	SYM
cana-1298	101	40	rmax{max{𝔎i	rmax{max{𝔎i	NOUN
cana-1298	101	41	−(𝜚	−(𝜚	NOUN
cana-1298	101	42	)	)	PUNCT
cana-1298	101	43	,	,	PUNCT
cana-1298	101	44	℘i	℘i	ADJ
cana-1298	101	45	−(𝜁	−(𝜁	NOUN
cana-1298	101	46	)	)	PUNCT
cana-1298	101	47	}	}	PUNCT
cana-1298	101	48	,	,	PUNCT
cana-1298	101	49	rmax{𝔎i	rmax{𝔎i	NOUN
cana-1298	101	50	−(𝜐	−(𝜐	PROPN
cana-1298	101	51	)	)	PUNCT
cana-1298	101	52	,	,	PUNCT
cana-1298	101	53	℘i	℘i	ADJ
cana-1298	101	54	−	−	PROPN
cana-1298	101	55	(	(	PUNCT
cana-1298	101	56	𝜉	𝜉	NOUN
cana-1298	101	57	)	)	PUNCT
cana-1298	101	58	}	}	PUNCT
cana-1298	101	59	}	}	PUNCT
cana-1298	101	60	=	=	SYM
cana-1298	101	61	rmax{(𝔎i×℘i)−(𝜚	rmax{(𝔎i×℘i)−(𝜚	NOUN
cana-1298	101	62	,	,	PUNCT
cana-1298	101	63	𝜁	𝜁	NOUN
cana-1298	101	64	)	)	PUNCT
cana-1298	101	65	,	,	PUNCT
cana-1298	101	66	(	(	PUNCT
cana-1298	101	67	𝔎i×℘i)−(𝜐	𝔎i×℘i)−(𝜐	PROPN
cana-1298	101	68	,	,	PUNCT
cana-1298	101	69	𝜉	𝜉	NOUN
cana-1298	101	70	)	)	PUNCT
cana-1298	101	71	}	}	PUNCT
cana-1298	101	72	,	,	PUNCT
cana-1298	101	73	for	for	ADP
cana-1298	101	74	all	all	PRON
cana-1298	101	75	(	(	PUNCT
cana-1298	101	76	𝜚	𝜚	NOUN
cana-1298	101	77	,	,	PUNCT
cana-1298	101	78	𝜁	𝜁	NOUN
cana-1298	101	79	)	)	PUNCT
cana-1298	101	80	,	,	PUNCT
cana-1298	101	81	(	(	PUNCT
cana-1298	101	82	𝜐	𝜐	NOUN
cana-1298	101	83	,	,	PUNCT
cana-1298	101	84	𝜉	𝜉	NOUN
cana-1298	101	85	)	)	PUNCT
cana-1298	101	86	in	in	ADP
cana-1298	101	87	𝔏1×𝔏2	𝔏1×𝔏2	PROPN
cana-1298	101	88	.	.	PUNCT
cana-1298	102	1	hence	hence	ADV
cana-1298	102	2	𝔎×℘	𝔎×℘	PROPN
cana-1298	102	3	is	be	AUX
cana-1298	102	4	a	a	DET
cana-1298	102	5	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	102	6	of	of	ADP
cana-1298	102	7	𝔏1×𝔏2	𝔏1×𝔏2	PROPN
cana-1298	102	8	.	.	PUNCT
cana-1298	103	1	theorem	theorem	VERB
cana-1298	103	2	2.2	2.2	NUM
cana-1298	103	3	.	.	PUNCT
cana-1298	104	1	𝐼𝑓	𝐼𝑓	NOUN
cana-1298	104	2	℘1	℘1	NOUN
cana-1298	104	3	,	,	PUNCT
cana-1298	104	4	℘2	℘2	NOUN
cana-1298	104	5	,	,	PUNCT
cana-1298	104	6	…	…	PUNCT
cana-1298	104	7	,	,	PUNCT
cana-1298	104	8	℘𝑚	℘𝑚	NOUN
cana-1298	104	9	𝑎𝑟𝑒	𝑎𝑟𝑒	PROPN
cana-1298	104	10	𝔹𝕍𝕄𝕀𝔽𝕊ℝs	𝔹𝕍𝕄𝕀𝔽𝕊ℝs	PROPN
cana-1298	104	11	of	of	ADP
cana-1298	104	12	the	the	DET
cana-1298	104	13	rings	ring	NOUN
cana-1298	104	14	𝔏1	𝔏1	PROPN
cana-1298	104	15	,	,	PUNCT
cana-1298	104	16	𝔏2	𝔏2	NOUN
cana-1298	104	17	,	,	PUNCT
cana-1298	104	18	…	…	PUNCT
cana-1298	104	19	,	,	PUNCT
cana-1298	104	20	𝔏m	𝔏m	AUX
cana-1298	104	21	respectively	respectively	ADV
cana-1298	104	22	,	,	PUNCT
cana-1298	104	23	then	then	ADV
cana-1298	104	24	℘1	℘1	VERB
cana-1298	104	25	×	×	PROPN
cana-1298	104	26	℘2	℘2	NOUN
cana-1298	104	27	×	×	NOUN
cana-1298	104	28	…	…	PUNCT
cana-1298	104	29	×	×	NOUN
cana-1298	104	30	℘𝑚	℘𝑚	NOUN
cana-1298	104	31	is	be	AUX
cana-1298	104	32	a	a	DET
cana-1298	104	33	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	104	34	of	of	ADP
cana-1298	104	35	the	the	DET
cana-1298	104	36	ring	ring	NOUN
cana-1298	104	37	𝔏1	𝔏1	PROPN
cana-1298	104	38	×	×	PROPN
cana-1298	104	39	𝔏2	𝔏2	NOUN
cana-1298	104	40	×	×	PROPN
cana-1298	104	41	…	…	PUNCT
cana-1298	104	42	×	×	PROPN
cana-1298	104	43	𝔏m	𝔏m	PROPN
cana-1298	104	44	.	.	PUNCT
cana-1298	105	1	proof	proof	NOUN
cana-1298	105	2	.	.	PUNCT
cana-1298	106	1	𝐹𝑟𝑜𝑚	𝐹𝑟𝑜𝑚	PROPN
cana-1298	106	2	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-1298	106	3	𝑡ℎ𝑒𝑜𝑟𝑒𝑚	𝑡ℎ𝑒𝑜𝑟𝑒𝑚	NOUN
cana-1298	106	4	2.1	2.1	NUM
cana-1298	106	5	,	,	PUNCT
cana-1298	106	6	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-1298	106	7	𝑝𝑟𝑜𝑜𝑓	𝑝𝑟𝑜𝑜𝑓	NOUN
cana-1298	106	8	𝑖𝑠	𝑖𝑠	PROPN
cana-1298	106	9	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	NOUN
cana-1298	106	10	theorem	theorem	VERB
cana-1298	106	11	2.3	2.3	NUM
cana-1298	106	12	.	.	PUNCT
cana-1298	107	1	𝐼𝑓	𝐼𝑓	NOUN
cana-1298	107	2	𝔎×℘	𝔎×℘	NUM
cana-1298	107	3	is	be	AUX
cana-1298	107	4	a	a	DET
cana-1298	107	5	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	107	6	with	with	ADP
cana-1298	107	7	degree	degree	NOUN
cana-1298	107	8	n	n	PROPN
cana-1298	107	9	of	of	ADP
cana-1298	107	10	a	a	DET
cana-1298	107	11	ring	ring	NOUN
cana-1298	107	12	𝔏1×𝔏2	𝔏1×𝔏2	PROPN
cana-1298	107	13	,	,	PUNCT
cana-1298	107	14	then	then	ADV
cana-1298	107	15	for	for	ADP
cana-1298	107	16	all	all	DET
cana-1298	107	17	i	i	PRON
cana-1298	107	18	,	,	PUNCT
cana-1298	107	19	i	i	NOUN
cana-1298	107	20	=	=	NOUN
cana-1298	107	21	1	1	NUM
cana-1298	107	22	,	,	PUNCT
cana-1298	107	23	2	2	NUM
cana-1298	107	24	,	,	PUNCT
cana-1298	107	25	…	…	PUNCT
cana-1298	107	26	,	,	PUNCT
cana-1298	107	27	n	n	CCONJ
cana-1298	107	28	,	,	PUNCT
cana-1298	107	29	(	(	PUNCT
cana-1298	107	30	𝔎i×℘i	𝔎i×℘i	X
cana-1298	107	31	)	)	PUNCT
cana-1298	107	32	+	+	ADJ
cana-1298	107	33	(	(	PUNCT
cana-1298	107	34	−𝜐	−𝜐	ADJ
cana-1298	107	35	,	,	PUNCT
cana-1298	107	36	𝜉−1	𝜉−1	NOUN
cana-1298	107	37	)	)	PUNCT
cana-1298	107	38	=	=	SYM
cana-1298	107	39	(	(	PUNCT
cana-1298	107	40	𝔎i×℘i	𝔎i×℘i	X
cana-1298	107	41	)	)	PUNCT
cana-1298	107	42	+	+	PROPN
cana-1298	107	43	(	(	PUNCT
cana-1298	107	44	𝜐	𝜐	NOUN
cana-1298	107	45	,	,	PUNCT
cana-1298	107	46	𝜉	𝜉	NOUN
cana-1298	107	47	)	)	PUNCT
cana-1298	107	48	,	,	PUNCT
cana-1298	107	49	(	(	PUNCT
cana-1298	107	50	𝔎i×℘i)−(−𝜐	𝔎i×℘i)−(−𝜐	NOUN
cana-1298	107	51	,	,	PUNCT
cana-1298	107	52	𝜉−1	𝜉−1	NOUN
cana-1298	107	53	)	)	PUNCT
cana-1298	107	54	=	=	SYM
cana-1298	107	55	(	(	PUNCT
cana-1298	107	56	𝔎i×℘i)−(𝜐	𝔎i×℘i)−(𝜐	PROPN
cana-1298	107	57	,	,	PUNCT
cana-1298	107	58	𝜉	𝜉	NOUN
cana-1298	107	59	)	)	PUNCT
cana-1298	107	60	,	,	PUNCT
cana-1298	107	61	(	(	PUNCT
cana-1298	107	62	𝔎i×℘i	𝔎i×℘i	X
cana-1298	107	63	)	)	PUNCT
cana-1298	108	1	+	+	PROPN
cana-1298	108	2	(	(	PUNCT
cana-1298	108	3	𝜐	𝜐	NOUN
cana-1298	108	4	,	,	PUNCT
cana-1298	108	5	𝜉	𝜉	NOUN
cana-1298	108	6	)	)	PUNCT
cana-1298	108	7			PROPN
cana-1298	108	8	(	(	PUNCT
cana-1298	108	9	𝔎i×℘i	𝔎i×℘i	X
cana-1298	108	10	)	)	PUNCT
cana-1298	108	11	+	+	ADJ
cana-1298	108	12	(	(	PUNCT
cana-1298	108	13	0	0	NUM
cana-1298	108	14	,	,	PUNCT
cana-1298	108	15	1	1	NUM
cana-1298	108	16	)	)	PUNCT
cana-1298	108	17	and	and	CCONJ
cana-1298	108	18	(	(	PUNCT
cana-1298	108	19	𝔎i×℘i)−(𝜐	𝔎i×℘i)−(𝜐	PROPN
cana-1298	108	20	,	,	PUNCT
cana-1298	108	21	𝜉	𝜉	NOUN
cana-1298	108	22	)	)	PUNCT
cana-1298	108	23	≥	≥	X
cana-1298	108	24	(	(	PUNCT
cana-1298	108	25	𝔎i×℘i)−(0	𝔎i×℘i)−(0	NOUN
cana-1298	108	26	,	,	PUNCT
cana-1298	108	27	1	1	NUM
cana-1298	108	28	)	)	PUNCT
cana-1298	108	29	,	,	PUNCT
cana-1298	108	30	for	for	ADP
cana-1298	108	31	all	all	PRON
cana-1298	108	32	(	(	PUNCT
cana-1298	108	33	𝜐	𝜐	NOUN
cana-1298	108	34	,	,	PUNCT
cana-1298	108	35	𝜉	𝜉	NOUN
cana-1298	108	36	)	)	PUNCT
cana-1298	108	37	in	in	ADP
cana-1298	108	38	𝔏1×𝔏2	𝔏1×𝔏2	PROPN
cana-1298	108	39	,	,	PUNCT
cana-1298	108	40	where	where	SCONJ
cana-1298	108	41	(	(	PUNCT
cana-1298	108	42	0	0	NUM
cana-1298	108	43	,	,	PUNCT
cana-1298	108	44	1	1	NUM
cana-1298	108	45	)	)	PUNCT
cana-1298	108	46	is	be	AUX
cana-1298	108	47	the	the	DET
cana-1298	108	48	identity	identity	NOUN
cana-1298	108	49	element	element	NOUN
cana-1298	108	50	of	of	ADP
cana-1298	108	51	𝔏1×𝔏2	𝔏1×𝔏2	PROPN
cana-1298	108	52	.	.	PUNCT
cana-1298	109	1	proof	proof	NOUN
cana-1298	109	2	.	.	PUNCT
cana-1298	110	1	let	let	VERB
cana-1298	110	2	(	(	PUNCT
cana-1298	110	3	𝜐	𝜐	NOUN
cana-1298	110	4	,	,	PUNCT
cana-1298	110	5	𝜉	𝜉	AUX
cana-1298	110	6	)	)	PUNCT
cana-1298	110	7	be	be	VERB
cana-1298	110	8	in	in	ADP
cana-1298	110	9	𝔏1×𝔏2	𝔏1×𝔏2	PROPN
cana-1298	110	10	and	and	CCONJ
cana-1298	110	11	(	(	PUNCT
cana-1298	110	12	0	0	NUM
cana-1298	110	13	,	,	PUNCT
cana-1298	110	14	1	1	NUM
cana-1298	110	15	)	)	PUNCT
cana-1298	110	16	be	be	AUX
cana-1298	110	17	the	the	DET
cana-1298	110	18	identity	identity	NOUN
cana-1298	110	19	element	element	NOUN
cana-1298	110	20	of	of	ADP
cana-1298	110	21	𝔏1×𝔏2	𝔏1×𝔏2	PROPN
cana-1298	110	22	.	.	PUNCT
cana-1298	111	1	for	for	ADP
cana-1298	111	2	all	all	DET
cana-1298	111	3	i	i	PRON
cana-1298	111	4	,	,	PUNCT
cana-1298	111	5	i	i	NOUN
cana-1298	111	6	=	=	NOUN
cana-1298	111	7	1	1	NUM
cana-1298	111	8	,	,	PUNCT
cana-1298	111	9	2	2	NUM
cana-1298	111	10	,	,	PUNCT
cana-1298	111	11	…	…	PUNCT
cana-1298	111	12	,	,	PUNCT
cana-1298	111	13	n	n	CCONJ
cana-1298	111	14	,	,	PUNCT
cana-1298	111	15	(	(	PUNCT
cana-1298	111	16	𝔎i×℘i	𝔎i×℘i	X
cana-1298	111	17	)	)	PUNCT
cana-1298	112	1	+	+	PROPN
cana-1298	112	2	(	(	PUNCT
cana-1298	112	3	𝜐	𝜐	NOUN
cana-1298	112	4	,	,	PUNCT
cana-1298	112	5	𝜉	𝜉	NOUN
cana-1298	112	6	)	)	PUNCT
cana-1298	112	7	=	=	SYM
cana-1298	112	8	(	(	PUNCT
cana-1298	112	9	𝔎i×℘i	𝔎i×℘i	X
cana-1298	112	10	)	)	PUNCT
cana-1298	112	11	+	+	ADJ
cana-1298	112	12	(	(	PUNCT
cana-1298	112	13	−(−𝜐	−(−𝜐	NOUN
cana-1298	112	14	)	)	PUNCT
cana-1298	112	15	,	,	PUNCT
cana-1298	112	16	(	(	PUNCT
cana-1298	112	17	𝜉−1)−1	𝜉−1)−1	NOUN
cana-1298	112	18	)	)	PUNCT
cana-1298	112	19	≥	≥	NOUN
cana-1298	112	20	(	(	PUNCT
cana-1298	112	21	𝔎i×℘i	𝔎i×℘i	X
cana-1298	112	22	)	)	PUNCT
cana-1298	112	23	+	+	ADJ
cana-1298	112	24	(	(	PUNCT
cana-1298	112	25	−𝜐	−𝜐	ADJ
cana-1298	112	26	,	,	PUNCT
cana-1298	112	27	𝜉−1	𝜉−1	NOUN
cana-1298	112	28	)	)	PUNCT
cana-1298	112	29	≥	≥	NOUN
cana-1298	112	30	(	(	PUNCT
cana-1298	112	31	𝔎i×℘i	𝔎i×℘i	X
cana-1298	112	32	)	)	PUNCT
cana-1298	112	33	+	+	PROPN
cana-1298	112	34	(	(	PUNCT
cana-1298	112	35	𝜐	𝜐	NOUN
cana-1298	112	36	,	,	PUNCT
cana-1298	112	37	𝜉	𝜉	X
cana-1298	112	38	)	)	PUNCT
cana-1298	112	39	.	.	PUNCT
cana-1298	113	1	thus	thus	ADV
cana-1298	113	2	(	(	PUNCT
cana-1298	113	3	𝔎i×℘i	𝔎i×℘i	X
cana-1298	113	4	)	)	PUNCT
cana-1298	113	5	+	+	ADJ
cana-1298	113	6	(	(	PUNCT
cana-1298	113	7	−𝜐	−𝜐	ADJ
cana-1298	113	8	,	,	PUNCT
cana-1298	113	9	𝜉−1	𝜉−1	NOUN
cana-1298	113	10	)	)	PUNCT
cana-1298	113	11	=	=	SYM
cana-1298	113	12	(	(	PUNCT
cana-1298	113	13	𝔎i×℘i	𝔎i×℘i	X
cana-1298	113	14	)	)	PUNCT
cana-1298	114	1	+	+	PROPN
cana-1298	114	2	(	(	PUNCT
cana-1298	114	3	𝜐	𝜐	NOUN
cana-1298	114	4	,	,	PUNCT
cana-1298	114	5	𝜉	𝜉	NOUN
cana-1298	114	6	)	)	PUNCT
cana-1298	114	7	,	,	PUNCT
cana-1298	114	8	for	for	ADP
cana-1298	114	9	all	all	PRON
cana-1298	114	10	(	(	PUNCT
cana-1298	114	11	𝜐	𝜐	NOUN
cana-1298	114	12	,	,	PUNCT
cana-1298	114	13	𝜉	𝜉	NOUN
cana-1298	114	14	)	)	PUNCT
cana-1298	114	15	in	in	ADP
cana-1298	114	16	𝔏1×𝔏2	𝔏1×𝔏2	PROPN
cana-1298	114	17	.	.	PUNCT
cana-1298	115	1	and	and	CCONJ
cana-1298	115	2	(	(	PUNCT
cana-1298	115	3	𝔎i×℘i)−(𝜐	𝔎i×℘i)−(𝜐	PROPN
cana-1298	115	4	,	,	PUNCT
cana-1298	115	5	𝜉	𝜉	NOUN
cana-1298	115	6	)	)	PUNCT
cana-1298	115	7	=	=	SYM
cana-1298	115	8	(	(	PUNCT
cana-1298	115	9	𝔎i×℘i)−(−(−𝜐	𝔎i×℘i)−(−(−𝜐	NOUN
cana-1298	115	10	)	)	PUNCT
cana-1298	115	11	,	,	PUNCT
cana-1298	115	12	(	(	PUNCT
cana-1298	115	13	𝜉−1)−1	𝜉−1)−1	NOUN
cana-1298	115	14	)	)	PUNCT
cana-1298	115	15			NOUN
cana-1298	115	16	(	(	PUNCT
cana-1298	115	17	𝔎i×℘i)−(−𝜐	𝔎i×℘i)−(−𝜐	NOUN
cana-1298	115	18	,	,	PUNCT
cana-1298	115	19	𝜉−1	𝜉−1	NOUN
cana-1298	115	20	)	)	PUNCT
cana-1298	115	21			NOUN
cana-1298	115	22	(	(	PUNCT
cana-1298	115	23	𝔎i×℘i)−(𝜐	𝔎i×℘i)−(𝜐	PROPN
cana-1298	115	24	,	,	PUNCT
cana-1298	115	25	𝜉	𝜉	NOUN
cana-1298	115	26	)	)	PUNCT
cana-1298	115	27	.	.	PUNCT
cana-1298	116	1	thus	thus	ADV
cana-1298	116	2	(	(	PUNCT
cana-1298	116	3	𝔎i×℘i)−(−𝜐	𝔎i×℘i)−(−𝜐	NOUN
cana-1298	116	4	,	,	PUNCT
cana-1298	116	5	𝜉−1	𝜉−1	NOUN
cana-1298	116	6	)	)	PUNCT
cana-1298	116	7	=	=	SYM
cana-1298	116	8	(	(	PUNCT
cana-1298	116	9	𝔎i×℘i)−(𝜐	𝔎i×℘i)−(𝜐	PROPN
cana-1298	116	10	,	,	PUNCT
cana-1298	116	11	𝜉	𝜉	NOUN
cana-1298	116	12	)	)	PUNCT
cana-1298	116	13	,	,	PUNCT
cana-1298	116	14	for	for	ADP
cana-1298	116	15	all	all	PRON
cana-1298	116	16	(	(	PUNCT
cana-1298	116	17	𝜐	𝜐	NOUN
cana-1298	116	18	,	,	PUNCT
cana-1298	116	19	𝜉	𝜉	NOUN
cana-1298	116	20	)	)	PUNCT
cana-1298	116	21	in	in	ADP
cana-1298	116	22	𝔏1×𝔏2	𝔏1×𝔏2	PROPN
cana-1298	116	23	.	.	PUNCT
cana-1298	117	1	also	also	ADV
cana-1298	117	2	(	(	PUNCT
cana-1298	117	3	𝔎i×℘i	𝔎i×℘i	X
cana-1298	117	4	)	)	PUNCT
cana-1298	117	5	+	+	ADJ
cana-1298	117	6	(	(	PUNCT
cana-1298	117	7	0	0	NUM
cana-1298	117	8	,	,	PUNCT
cana-1298	117	9	1	1	NUM
cana-1298	117	10	)	)	PUNCT
cana-1298	117	11	=	=	SYM
cana-1298	117	12	(	(	PUNCT
cana-1298	117	13	𝔎i×℘i	𝔎i×℘i	X
cana-1298	117	14	)	)	PUNCT
cana-1298	118	1	+	+	PROPN
cana-1298	118	2	(	(	PUNCT
cana-1298	118	3	𝜐	𝜐	PROPN
cana-1298	118	4	−	−	PROPN
cana-1298	118	5	𝜐	𝜐	NOUN
cana-1298	118	6	)	)	PUNCT
cana-1298	118	7	,	,	PUNCT
cana-1298	118	8	𝜉𝜉−1	𝜉𝜉−1	PROPN
cana-1298	118	9	)	)	PUNCT
cana-1298	118	10	=	=	VERB
cana-1298	118	11	rmin	rmin	NOUN
cana-1298	118	12	{	{	PUNCT
cana-1298	118	13	𝔎i	𝔎i	PROPN
cana-1298	118	14	+	+	PROPN
cana-1298	118	15	(	(	PUNCT
cana-1298	118	16	𝜐	𝜐	NOUN
cana-1298	118	17	−	−	PROPN
cana-1298	118	18	𝜐	𝜐	NOUN
cana-1298	118	19	)	)	PUNCT
cana-1298	118	20	,	,	PUNCT
cana-1298	118	21	℘i	℘i	ADJ
cana-1298	118	22	+	+	PROPN
cana-1298	118	23	(	(	PUNCT
cana-1298	118	24	𝜉𝜉−1	𝜉𝜉−1	NOUN
cana-1298	118	25	)	)	PUNCT
cana-1298	118	26	}	}	PUNCT
cana-1298	118	27			X
cana-1298	118	28	rmin	rmin	VERB
cana-1298	118	29	{	{	PUNCT
cana-1298	118	30	rmin	rmin	NOUN
cana-1298	118	31	{	{	PUNCT
cana-1298	118	32	𝔎i	𝔎i	PROPN
cana-1298	118	33	+	+	PROPN
cana-1298	118	34	(	(	PUNCT
cana-1298	118	35	𝜐	𝜐	NOUN
cana-1298	118	36	)	)	PUNCT
cana-1298	118	37	,	,	PUNCT
cana-1298	118	38	𝔎i	𝔎i	PROPN
cana-1298	118	39	+	+	PROPN
cana-1298	118	40	(	(	PUNCT
cana-1298	118	41	𝜐	𝜐	NOUN
cana-1298	118	42	)	)	PUNCT
cana-1298	118	43	}	}	PUNCT
cana-1298	118	44	,	,	PUNCT
cana-1298	118	45	rmin	rmin	VERB
cana-1298	118	46	{	{	PUNCT
cana-1298	118	47	℘i	℘i	ADJ
cana-1298	118	48	+	+	PROPN
cana-1298	118	49	(	(	PUNCT
cana-1298	118	50	𝜉	𝜉	NOUN
cana-1298	118	51	)	)	PUNCT
cana-1298	118	52	,	,	PUNCT
cana-1298	118	53	℘i	℘i	ADJ
cana-1298	118	54	+	+	PROPN
cana-1298	118	55	(	(	PUNCT
cana-1298	118	56	𝜉	𝜉	NOUN
cana-1298	118	57	)	)	PUNCT
cana-1298	118	58	}	}	PUNCT
cana-1298	118	59	}	}	PUNCT
cana-1298	118	60	=	=	SYM
cana-1298	118	61	rmin	rmin	NOUN
cana-1298	118	62	{	{	PUNCT
cana-1298	118	63	𝔎i	𝔎i	PROPN
cana-1298	118	64	+	+	PROPN
cana-1298	118	65	(	(	PUNCT
cana-1298	118	66	𝜐	𝜐	NOUN
cana-1298	118	67	)	)	PUNCT
cana-1298	118	68	,	,	PUNCT
cana-1298	118	69	℘i	℘i	ADJ
cana-1298	118	70	+	+	PROPN
cana-1298	118	71	(	(	PUNCT
cana-1298	118	72	𝜉	𝜉	NOUN
cana-1298	118	73	)	)	PUNCT
cana-1298	118	74	}	}	PUNCT
cana-1298	118	75	=	=	SYM
cana-1298	118	76	(	(	PUNCT
cana-1298	118	77	𝔎i×℘i	𝔎i×℘i	X
cana-1298	118	78	)	)	PUNCT
cana-1298	118	79	+	+	PROPN
cana-1298	118	80	(	(	PUNCT
cana-1298	118	81	𝜐	𝜐	NOUN
cana-1298	118	82	,	,	PUNCT
cana-1298	118	83	𝜉	𝜉	NOUN
cana-1298	118	84	)	)	PUNCT
cana-1298	118	85	.	.	PUNCT
cana-1298	119	1	thus	thus	ADV
cana-1298	119	2	(	(	PUNCT
cana-1298	119	3	𝔎i×℘i	𝔎i×℘i	X
cana-1298	119	4	)	)	PUNCT
cana-1298	119	5	+	+	PROPN
cana-1298	119	6	(	(	PUNCT
cana-1298	119	7	𝜐	𝜐	NOUN
cana-1298	119	8	,	,	PUNCT
cana-1298	119	9	𝜉	𝜉	NOUN
cana-1298	119	10	)	)	PUNCT
cana-1298	119	11			PROPN
cana-1298	119	12	(	(	PUNCT
cana-1298	119	13	𝔎i×℘i	𝔎i×℘i	X
cana-1298	119	14	)	)	PUNCT
cana-1298	119	15	+	+	ADJ
cana-1298	119	16	(	(	PUNCT
cana-1298	119	17	0	0	NUM
cana-1298	119	18	,	,	PUNCT
cana-1298	119	19	1	1	NUM
cana-1298	119	20	)	)	PUNCT
cana-1298	119	21	,	,	PUNCT
cana-1298	119	22	for	for	ADP
cana-1298	119	23	all	all	DET
cana-1298	119	24	(	(	PUNCT
cana-1298	119	25	𝜐	𝜐	NOUN
cana-1298	119	26	,	,	PUNCT
cana-1298	119	27	𝜉	𝜉	NOUN
cana-1298	119	28	)	)	PUNCT
cana-1298	119	29	in	in	ADP
cana-1298	119	30	𝔏1×𝔏2	𝔏1×𝔏2	PROPN
cana-1298	119	31	.	.	PUNCT
cana-1298	120	1	and	and	CCONJ
cana-1298	120	2	(	(	PUNCT
cana-1298	120	3	𝔎i×℘i)−(0	𝔎i×℘i)−(0	NOUN
cana-1298	120	4	,	,	PUNCT
cana-1298	120	5	1	1	NUM
cana-1298	120	6	)	)	PUNCT
cana-1298	120	7	=	=	SYM
cana-1298	121	1	(	(	PUNCT
cana-1298	121	2	𝔎i×℘i)−(𝜐	𝔎i×℘i)−(𝜐	ADJ
cana-1298	121	3	−	−	PROPN
cana-1298	121	4	𝜐	𝜐	NOUN
cana-1298	121	5	)	)	PUNCT
cana-1298	121	6	,	,	PUNCT
cana-1298	121	7	𝜉𝜉−1	𝜉𝜉−1	PROPN
cana-1298	121	8	)	)	PUNCT
cana-1298	121	9	=	=	SYM
cana-1298	121	10	rmax{𝔎i	rmax{𝔎i	NOUN
cana-1298	121	11	−(𝜐	−(𝜐	PROPN
cana-1298	121	12	−	−	PROPN
cana-1298	121	13	𝜐	𝜐	NOUN
cana-1298	121	14	)	)	PUNCT
cana-1298	121	15	,	,	PUNCT
cana-1298	121	16	℘i	℘i	PROPN
cana-1298	121	17	−(𝜉𝜉−1	−(𝜉𝜉−1	PROPN
cana-1298	121	18	)	)	PUNCT
cana-1298	121	19	}	}	PUNCT
cana-1298	121	20			NUM
cana-1298	121	21	rmax	rmax	ADJ
cana-1298	121	22	{	{	PUNCT
cana-1298	121	23	rmax	rmax	ADJ
cana-1298	121	24	{	{	PUNCT
cana-1298	121	25	𝔎i	𝔎i	ADJ
cana-1298	121	26	−(𝜐	−(𝜐	NOUN
cana-1298	121	27	)	)	PUNCT
cana-1298	121	28	,	,	PUNCT
cana-1298	121	29	𝔎i	𝔎i	ADJ
cana-1298	121	30	−(𝜐	−(𝜐	NOUN
cana-1298	121	31	)	)	PUNCT
cana-1298	121	32	}	}	PUNCT
cana-1298	121	33	,	,	PUNCT
cana-1298	121	34	rmax{℘i	rmax{℘i	NOUN
cana-1298	121	35	−(𝜉	−(𝜉	NOUN
cana-1298	121	36	)	)	PUNCT
cana-1298	121	37	,	,	PUNCT
cana-1298	121	38	℘i	℘i	ADJ
cana-1298	121	39	−(𝜉	−(𝜉	NOUN
cana-1298	121	40	)	)	PUNCT
cana-1298	121	41	}	}	PUNCT
cana-1298	121	42	}	}	PUNCT
cana-1298	121	43	=	=	SYM
cana-1298	121	44	rmax{𝔎i	rmax{𝔎i	NOUN
cana-1298	121	45	−(𝜐	−(𝜐	NOUN
cana-1298	121	46	)	)	PUNCT
cana-1298	121	47	,	,	PUNCT
cana-1298	121	48	℘i	℘i	ADJ
cana-1298	121	49	−(𝜉	−(𝜉	NOUN
cana-1298	121	50	)	)	PUNCT
cana-1298	121	51	}	}	PUNCT
cana-1298	121	52	=	=	SYM
cana-1298	121	53	(	(	PUNCT
cana-1298	121	54	𝔎i×℘i)−(𝜐	𝔎i×℘i)−(𝜐	PROPN
cana-1298	121	55	,	,	PUNCT
cana-1298	121	56	𝜉	𝜉	NOUN
cana-1298	121	57	)	)	PUNCT
cana-1298	121	58	.	.	PUNCT
cana-1298	122	1	thus	thus	ADV
cana-1298	122	2	(	(	PUNCT
cana-1298	122	3	𝔎i×℘i)−(𝜐	𝔎i×℘i)−(𝜐	ADJ
cana-1298	122	4	,	,	PUNCT
cana-1298	122	5	𝜉	𝜉	NOUN
cana-1298	122	6	)	)	PUNCT
cana-1298	122	7	≥	≥	X
cana-1298	122	8	(	(	PUNCT
cana-1298	122	9	𝔎i×℘i)−(0	𝔎i×℘i)−(0	NOUN
cana-1298	122	10	,	,	PUNCT
cana-1298	122	11	1	1	NUM
cana-1298	122	12	)	)	PUNCT
cana-1298	122	13	,	,	PUNCT
cana-1298	122	14	for	for	ADP
cana-1298	122	15	all	all	PRON
cana-1298	122	16	(	(	PUNCT
cana-1298	122	17	𝜐	𝜐	NOUN
cana-1298	122	18	,	,	PUNCT
cana-1298	122	19	𝜉	𝜉	NOUN
cana-1298	122	20	)	)	PUNCT
cana-1298	122	21	in	in	ADP
cana-1298	122	22	𝔏1×𝔏2	𝔏1×𝔏2	PROPN
cana-1298	122	23	.	.	PUNCT
cana-1298	122	24	theorem	theorem	VERB
cana-1298	122	25	2.4	2.4	NUM
cana-1298	122	26	.	.	PUNCT
cana-1298	123	1	𝐿𝑒𝑡	𝐿𝑒𝑡	PROPN
cana-1298	123	2	𝔜	𝔜	PROPN
cana-1298	123	3	and	and	CCONJ
cana-1298	123	4	𝔚	𝔚	NOUN
cana-1298	123	5	𝑏𝑒	𝑏𝑒	NOUN
cana-1298	123	6	𝑎𝑛𝑦	𝑎𝑛𝑦	INTJ
cana-1298	123	7	𝑡𝑤𝑜	𝑡𝑤𝑜	VERB
cana-1298	123	8	𝔹𝕍𝕄𝕀𝔽𝕊s	𝔹𝕍𝕄𝕀𝔽𝕊	NOUN
cana-1298	123	9	of	of	ADP
cana-1298	123	10	the	the	DET
cana-1298	123	11	rings	ring	NOUN
cana-1298	123	12	ℌ1	ℌ1	NOUN
cana-1298	123	13	and	and	CCONJ
cana-1298	123	14	ℌ2	ℌ2	VERB
cana-1298	123	15	communications	communication	NOUN
cana-1298	123	16	on	on	ADP
cana-1298	123	17	applied	apply	VERB
cana-1298	123	18	nonlinear	nonlinear	ADJ
cana-1298	123	19	analysis	analysis	NOUN
cana-1298	123	20	issn	issn	NOUN
cana-1298	123	21	:	:	PUNCT
cana-1298	123	22	1074	1074	NUM
cana-1298	123	23	-	-	PUNCT
cana-1298	123	24	133x	133x	NUM
cana-1298	123	25	vol	vol	NOUN
cana-1298	123	26	31	31	NUM
cana-1298	123	27	no	no	NOUN
cana-1298	123	28	.	.	PUNCT
cana-1298	124	1	7s	7	NOUN
cana-1298	124	2	(	(	PUNCT
cana-1298	124	3	2024	2024	NUM
cana-1298	124	4	)	)	PUNCT
cana-1298	124	5	234	234	NUM
cana-1298	124	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1298	124	7	respectively	respectively	ADV
cana-1298	124	8	.	.	PUNCT
cana-1298	125	1	if	if	SCONJ
cana-1298	125	2	𝔜	𝔜	PROPN
cana-1298	125	3	×	×	NOUN
cana-1298	125	4	𝔚	𝔚	NOUN
cana-1298	125	5	is	be	AUX
cana-1298	125	6	a	a	DET
cana-1298	125	7	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	125	8	of	of	ADP
cana-1298	125	9	the	the	DET
cana-1298	125	10	ring	ring	NOUN
cana-1298	125	11	ℌ1	ℌ1	PROPN
cana-1298	125	12	×	×	PROPN
cana-1298	125	13	ℌ2	ℌ2	VERB
cana-1298	125	14	,	,	PUNCT
cana-1298	125	15	then	then	ADV
cana-1298	125	16	at	at	ADP
cana-1298	125	17	least	least	ADJ
cana-1298	125	18	one	one	NUM
cana-1298	125	19	of	of	ADP
cana-1298	125	20	the	the	DET
cana-1298	125	21	following	follow	VERB
cana-1298	125	22	two	two	NUM
cana-1298	125	23	statements	statement	NOUN
cana-1298	125	24	must	must	AUX
cana-1298	125	25	hold	hold	VERB
cana-1298	125	26	;	;	PUNCT
cana-1298	125	27	(	(	PUNCT
cana-1298	125	28	i	i	NOUN
cana-1298	125	29	)	)	PUNCT
cana-1298	125	30	for	for	ADP
cana-1298	125	31	all	all	DET
cana-1298	125	32	i	i	PRON
cana-1298	125	33	=	=	NOUN
cana-1298	125	34	1	1	NUM
cana-1298	125	35	,	,	PUNCT
cana-1298	125	36	2	2	NUM
cana-1298	125	37	,	,	PUNCT
cana-1298	125	38	…	…	PUNCT
cana-1298	125	39	,	,	PUNCT
cana-1298	125	40	n	n	CCONJ
cana-1298	125	41	,	,	PUNCT
cana-1298	125	42	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	125	43	+	+	PROPN
cana-1298	125	44	(	(	PUNCT
cana-1298	125	45	𝔬	𝔬	NOUN
cana-1298	125	46	)	)	PUNCT
cana-1298	125	47	≥	≥	NOUN
cana-1298	126	1	𝔜𝑖	𝔜𝑖	NOUN
cana-1298	126	2	+	+	PROPN
cana-1298	126	3	(	(	PUNCT
cana-1298	126	4	𝜚	𝜚	NOUN
cana-1298	126	5	)	)	PUNCT
cana-1298	126	6	,	,	PUNCT
cana-1298	126	7	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	126	8	−(𝔬	−(𝔬	VERB
cana-1298	126	9	)	)	PUNCT
cana-1298	126	10	≤	≤	PUNCT
cana-1298	127	1	𝔜𝑖	𝔜𝑖	NOUN
cana-1298	127	2	−(𝜚	−(𝜚	NOUN
cana-1298	127	3	)	)	PUNCT
cana-1298	127	4	,	,	PUNCT
cana-1298	127	5	for	for	ADP
cana-1298	127	6	all	all	DET
cana-1298	127	7	𝜚ℌ1	𝜚ℌ1	NOUN
cana-1298	127	8	,	,	PUNCT
cana-1298	127	9	(	(	PUNCT
cana-1298	127	10	ii	ii	NOUN
cana-1298	127	11	)	)	PUNCT
cana-1298	127	12	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	127	13	+	+	PROPN
cana-1298	127	14	(	(	PUNCT
cana-1298	127	15	𝜁	𝜁	NOUN
cana-1298	127	16	)	)	PUNCT
cana-1298	127	17	≤	≤	PUNCT
cana-1298	128	1	𝔜𝑖	𝔜𝑖	NOUN
cana-1298	128	2	+	+	PROPN
cana-1298	128	3	(	(	PUNCT
cana-1298	128	4	𝔢	𝔢	NOUN
cana-1298	128	5	)	)	PUNCT
cana-1298	128	6	,	,	PUNCT
cana-1298	128	7	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	128	8	−(𝜁	−(𝜁	PROPN
cana-1298	128	9	)	)	PUNCT
cana-1298	128	10	≥	≥	NOUN
cana-1298	128	11	𝔜𝑖	𝔜𝑖	NOUN
cana-1298	128	12	−(𝔢	−(𝔢	NOUN
cana-1298	128	13	)	)	PUNCT
cana-1298	128	14	,	,	PUNCT
cana-1298	128	15	for	for	ADP
cana-1298	128	16	all	all	DET
cana-1298	128	17	𝜁ℌ2	𝜁ℌ2	NOUN
cana-1298	128	18	,	,	PUNCT
cana-1298	128	19	where	where	SCONJ
cana-1298	128	20	𝔢	𝔢	NOUN
cana-1298	128	21	,	,	PUNCT
cana-1298	128	22	𝔬	𝔬	PROPN
cana-1298	128	23	𝑎𝑟𝑒	𝑎𝑟𝑒	PROPN
cana-1298	128	24	𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦	𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦	PROPN
cana-1298	128	25	𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠	𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠	NOUN
cana-1298	128	26	𝑜𝑓	𝑜𝑓	AUX
cana-1298	128	27	ℌ1and	ℌ1and	PROPN
cana-1298	128	28	ℌ2	ℌ2	VERB
cana-1298	128	29	.	.	PUNCT
cana-1298	129	1	proof	proof	NOUN
cana-1298	129	2	.	.	PUNCT
cana-1298	130	1	by	by	ADP
cana-1298	130	2	contraposition	contraposition	NOUN
cana-1298	130	3	,	,	PUNCT
cana-1298	130	4	suppose	suppose	VERB
cana-1298	130	5	that	that	SCONJ
cana-1298	130	6	none	none	NOUN
cana-1298	130	7	of	of	ADP
cana-1298	130	8	the	the	DET
cana-1298	130	9	statements	statement	NOUN
cana-1298	130	10	(	(	PUNCT
cana-1298	130	11	i	i	NOUN
cana-1298	130	12	)	)	PUNCT
cana-1298	130	13	and	and	CCONJ
cana-1298	130	14	(	(	PUNCT
cana-1298	130	15	ii	ii	NOUN
cana-1298	130	16	)	)	PUNCT
cana-1298	130	17	holds	hold	VERB
cana-1298	130	18	.	.	PUNCT
cana-1298	131	1	for	for	ADP
cana-1298	131	2	𝜚	𝜚	PROPN
cana-1298	131	3	∈	∈	PROPN
cana-1298	131	4	ℌ1	ℌ1	NOUN
cana-1298	131	5	and	and	CCONJ
cana-1298	131	6	𝜁	𝜁	PROPN
cana-1298	131	7	∈	∈	PROPN
cana-1298	131	8	ℌ2	ℌ2	VERB
cana-1298	131	9	such	such	ADJ
cana-1298	131	10	that	that	SCONJ
cana-1298	131	11	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	131	12	+	+	PROPN
cana-1298	131	13	(	(	PUNCT
cana-1298	131	14	𝔬	𝔬	NOUN
cana-1298	131	15	)	)	PUNCT
cana-1298	131	16	<	<	X
cana-1298	132	1	𝔜𝑖	𝔜𝑖	NOUN
cana-1298	132	2	+	+	PROPN
cana-1298	132	3	(	(	PUNCT
cana-1298	132	4	𝜚	𝜚	NOUN
cana-1298	132	5	)	)	PUNCT
cana-1298	132	6	,	,	PUNCT
cana-1298	132	7	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	132	8	−(𝔬	−(𝔬	VERB
cana-1298	132	9	)	)	PUNCT
cana-1298	132	10	>	>	PUNCT
cana-1298	133	1	𝔜𝑖	𝔜𝑖	NOUN
cana-1298	133	2	−(𝜚	−(𝜚	NOUN
cana-1298	133	3	)	)	PUNCT
cana-1298	133	4	and	and	CCONJ
cana-1298	133	5	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	133	6	+	+	PROPN
cana-1298	133	7	(	(	PUNCT
cana-1298	133	8	𝜁	𝜁	NOUN
cana-1298	133	9	)	)	PUNCT
cana-1298	133	10	>	>	PUNCT
cana-1298	134	1	𝔜𝑖	𝔜𝑖	NOUN
cana-1298	134	2	+	+	PROPN
cana-1298	134	3	(	(	PUNCT
cana-1298	134	4	𝔢	𝔢	NOUN
cana-1298	134	5	)	)	PUNCT
cana-1298	134	6	,	,	PUNCT
cana-1298	134	7	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	134	8	−(𝜁	−(𝜁	PROPN
cana-1298	134	9	)	)	PUNCT
cana-1298	134	10	<	<	X
cana-1298	134	11	𝔜𝑖	𝔜𝑖	NOUN
cana-1298	134	12	−(𝔢	−(𝔢	NOUN
cana-1298	134	13	)	)	PUNCT
cana-1298	134	14	.	.	PUNCT
cana-1298	135	1	for	for	ADP
cana-1298	135	2	all	all	DET
cana-1298	135	3	i	i	PRON
cana-1298	135	4	=	=	NOUN
cana-1298	135	5	1	1	NUM
cana-1298	135	6	,	,	PUNCT
cana-1298	135	7	2	2	NUM
cana-1298	135	8	,	,	PUNCT
cana-1298	135	9	…	…	PUNCT
cana-1298	135	10	,	,	PUNCT
cana-1298	135	11	n	n	CCONJ
cana-1298	135	12	,	,	PUNCT
cana-1298	135	13	(	(	PUNCT
cana-1298	135	14	𝔜i×𝔚i	𝔜i×𝔚i	NOUN
cana-1298	135	15	)	)	PUNCT
cana-1298	135	16	+	+	PROPN
cana-1298	135	17	(	(	PUNCT
cana-1298	135	18	𝜚	𝜚	NOUN
cana-1298	135	19	,	,	PUNCT
cana-1298	135	20	𝜁	𝜁	NOUN
cana-1298	135	21	)	)	PUNCT
cana-1298	135	22	=	=	VERB
cana-1298	135	23	rmin{𝔜i	rmin{𝔜i	VERB
cana-1298	135	24	+	+	ADJ
cana-1298	135	25	(	(	PUNCT
cana-1298	135	26	𝜚	𝜚	NOUN
cana-1298	135	27	)	)	PUNCT
cana-1298	135	28	,	,	PUNCT
cana-1298	136	1	𝔚i	𝔚i	PROPN
cana-1298	136	2	+	+	ADJ
cana-1298	136	3	(	(	PUNCT
cana-1298	136	4	𝜁	𝜁	NOUN
cana-1298	136	5	)	)	PUNCT
cana-1298	136	6	}	}	PUNCT
cana-1298	136	7	>	>	X
cana-1298	136	8	rmin	rmin	PROPN
cana-1298	136	9	{	{	PUNCT
cana-1298	136	10	𝔜i	𝔜i	PROPN
cana-1298	136	11	+	+	PROPN
cana-1298	136	12	(	(	PUNCT
cana-1298	136	13	𝔢	𝔢	NOUN
cana-1298	136	14	)	)	PUNCT
cana-1298	136	15	,	,	PUNCT
cana-1298	137	1	𝔚i	𝔚i	PROPN
cana-1298	137	2	+	+	ADJ
cana-1298	137	3	(	(	PUNCT
cana-1298	137	4	𝔬	𝔬	NOUN
cana-1298	137	5	)	)	PUNCT
cana-1298	137	6	}	}	PUNCT
cana-1298	137	7	=	=	SYM
cana-1298	137	8	(	(	PUNCT
cana-1298	137	9	𝔜i×𝔚i	𝔜i×𝔚i	NOUN
cana-1298	137	10	)	)	PUNCT
cana-1298	137	11	+	+	PROPN
cana-1298	137	12	(	(	PUNCT
cana-1298	137	13	𝔢	𝔢	PROPN
cana-1298	137	14	,	,	PUNCT
cana-1298	137	15	𝔬	𝔬	NOUN
cana-1298	137	16	)	)	PUNCT
cana-1298	137	17	.	.	PUNCT
cana-1298	138	1	also	also	ADV
cana-1298	138	2	(	(	PUNCT
cana-1298	138	3	𝔜i×𝔚i	𝔜i×𝔚i	X
cana-1298	138	4	)	)	PUNCT
cana-1298	138	5	+	+	PROPN
cana-1298	138	6	(	(	PUNCT
cana-1298	138	7	𝜚	𝜚	NOUN
cana-1298	138	8	,	,	PUNCT
cana-1298	138	9	𝜁	𝜁	NOUN
cana-1298	138	10	)	)	PUNCT
cana-1298	138	11	=	=	VERB
cana-1298	138	12	rmax{𝔜i	rmax{𝔜i	VERB
cana-1298	139	1	+	+	ADJ
cana-1298	139	2	(	(	PUNCT
cana-1298	139	3	𝜚	𝜚	NOUN
cana-1298	139	4	)	)	PUNCT
cana-1298	139	5	,	,	PUNCT
cana-1298	140	1	𝔚i	𝔚i	PROPN
cana-1298	140	2	+	+	ADJ
cana-1298	140	3	(	(	PUNCT
cana-1298	140	4	𝜁	𝜁	NOUN
cana-1298	140	5	)	)	PUNCT
cana-1298	140	6	}	}	PUNCT
cana-1298	140	7	<	<	X
cana-1298	140	8	rmax	rmax	ADJ
cana-1298	140	9	{	{	PUNCT
cana-1298	140	10	𝔜i	𝔜i	PROPN
cana-1298	140	11	+	+	PROPN
cana-1298	140	12	(	(	PUNCT
cana-1298	140	13	𝔢	𝔢	NOUN
cana-1298	140	14	)	)	PUNCT
cana-1298	140	15	,	,	PUNCT
cana-1298	141	1	𝔚i	𝔚i	PROPN
cana-1298	141	2	+	+	NOUN
cana-1298	141	3	(	(	PUNCT
cana-1298	141	4	𝔬)}=	𝔬)}=	X
cana-1298	141	5	(	(	PUNCT
cana-1298	141	6	𝔜i×𝔚i	𝔜i×𝔚i	NOUN
cana-1298	141	7	)	)	PUNCT
cana-1298	141	8	+	+	PROPN
cana-1298	141	9	(	(	PUNCT
cana-1298	141	10	𝔢	𝔢	PROPN
cana-1298	141	11	,	,	PUNCT
cana-1298	141	12	𝔬	𝔬	NOUN
cana-1298	141	13	)	)	PUNCT
cana-1298	141	14	.	.	PUNCT
cana-1298	142	1	thus	thus	ADV
cana-1298	142	2	𝔜	𝔜	PROPN
cana-1298	142	3	×	×	NOUN
cana-1298	142	4	𝔚	𝔚	NOUN
cana-1298	142	5	is	be	AUX
cana-1298	142	6	not	not	PART
cana-1298	142	7	a	a	DET
cana-1298	142	8	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	142	9	of	of	ADP
cana-1298	142	10	the	the	DET
cana-1298	142	11	ring	ring	NOUN
cana-1298	142	12	ℌ1	ℌ1	PROPN
cana-1298	142	13	×	×	PROPN
cana-1298	142	14	ℌ2	ℌ2	ADJ
cana-1298	142	15	.	.	PUNCT
cana-1298	143	1	hence	hence	ADV
cana-1298	143	2	either	either	CCONJ
cana-1298	143	3	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	143	4	+	+	PROPN
cana-1298	143	5	(	(	PUNCT
cana-1298	143	6	𝔬	𝔬	NOUN
cana-1298	143	7	)	)	PUNCT
cana-1298	143	8	≥	≥	NOUN
cana-1298	144	1	𝔜𝑖	𝔜𝑖	NOUN
cana-1298	144	2	+	+	PROPN
cana-1298	144	3	(	(	PUNCT
cana-1298	144	4	𝜚	𝜚	NOUN
cana-1298	144	5	)	)	PUNCT
cana-1298	144	6	,	,	PUNCT
cana-1298	144	7	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	144	8	−(𝔬	−(𝔬	VERB
cana-1298	144	9	)	)	PUNCT
cana-1298	144	10	≤	≤	PUNCT
cana-1298	145	1	𝔜𝑖	𝔜𝑖	NOUN
cana-1298	145	2	−(𝜚	−(𝜚	NOUN
cana-1298	145	3	)	)	PUNCT
cana-1298	145	4	,	,	PUNCT
cana-1298	145	5	for	for	ADP
cana-1298	145	6	all	all	DET
cana-1298	145	7	𝜚ℌ1	𝜚ℌ1	NOUN
cana-1298	145	8	or	or	CCONJ
cana-1298	145	9	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	145	10	+	+	PROPN
cana-1298	145	11	(	(	PUNCT
cana-1298	145	12	𝜁	𝜁	NOUN
cana-1298	145	13	)	)	PUNCT
cana-1298	145	14	≤	≤	PUNCT
cana-1298	146	1	𝔜𝑖	𝔜𝑖	NOUN
cana-1298	146	2	+	+	PROPN
cana-1298	146	3	(	(	PUNCT
cana-1298	146	4	𝔢	𝔢	NOUN
cana-1298	146	5	)	)	PUNCT
cana-1298	146	6	,	,	PUNCT
cana-1298	146	7	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	146	8	−(𝜁	−(𝜁	PROPN
cana-1298	146	9	)	)	PUNCT
cana-1298	146	10	≥	≥	NOUN
cana-1298	146	11	𝔜𝑖	𝔜𝑖	NOUN
cana-1298	146	12	−(𝔢	−(𝔢	NOUN
cana-1298	146	13	)	)	PUNCT
cana-1298	146	14	,	,	PUNCT
cana-1298	146	15	for	for	ADP
cana-1298	146	16	all	all	DET
cana-1298	146	17	𝜁ℌ2	𝜁ℌ2	NOUN
cana-1298	146	18	.	.	PUNCT
cana-1298	146	19	theorem	theorem	VERB
cana-1298	146	20	2.5	2.5	NUM
cana-1298	146	21	.	.	PUNCT
cana-1298	147	1	𝐿𝑒𝑡	𝐿𝑒𝑡	PROPN
cana-1298	147	2	𝔓	𝔓	PROPN
cana-1298	147	3	and	and	CCONJ
cana-1298	147	4	𝔚	𝔚	NOUN
cana-1298	147	5	𝑏𝑒	𝑏𝑒	NOUN
cana-1298	147	6	𝑎𝑛𝑦	𝑎𝑛𝑦	INTJ
cana-1298	147	7	𝑡𝑤𝑜	𝑡𝑤𝑜	VERB
cana-1298	147	8	𝔹𝕍𝕄𝕀𝔽𝕊s	𝔹𝕍𝕄𝕀𝔽𝕊	NOUN
cana-1298	147	9	of	of	ADP
cana-1298	147	10	the	the	DET
cana-1298	147	11	rings	ring	NOUN
cana-1298	147	12	𝔒1	𝔒1	NOUN
cana-1298	147	13	and	and	CCONJ
cana-1298	147	14	𝔒2	𝔒2	NOUN
cana-1298	147	15	respectively	respectively	ADV
cana-1298	147	16	and	and	CCONJ
cana-1298	147	17	𝔓	𝔓	NOUN
cana-1298	147	18	×	×	NOUN
cana-1298	147	19	𝔚	𝔚	NOUN
cana-1298	147	20	be	be	AUX
cana-1298	147	21	a	a	DET
cana-1298	147	22	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	147	23	of	of	ADP
cana-1298	147	24	the	the	DET
cana-1298	147	25	ring	ring	NOUN
cana-1298	147	26	𝔒1	𝔒1	PROPN
cana-1298	147	27	×	×	PROPN
cana-1298	147	28	𝔒2	𝔒2	NOUN
cana-1298	147	29	.	.	PUNCT
cana-1298	148	1	then	then	ADV
cana-1298	148	2	the	the	DET
cana-1298	148	3	following	follow	VERB
cana-1298	148	4	are	be	AUX
cana-1298	148	5	true	true	ADJ
cana-1298	148	6	;	;	PUNCT
cana-1298	148	7	(	(	PUNCT
cana-1298	148	8	i	i	NOUN
cana-1298	148	9	)	)	PUNCT
cana-1298	148	10	for	for	ADP
cana-1298	148	11	all	all	DET
cana-1298	148	12	i	i	PRON
cana-1298	148	13	=	=	NOUN
cana-1298	148	14	1	1	NUM
cana-1298	148	15	,	,	PUNCT
cana-1298	148	16	2	2	NUM
cana-1298	148	17	,	,	PUNCT
cana-1298	148	18	…	…	PUNCT
cana-1298	148	19	,	,	PUNCT
cana-1298	148	20	n	n	CCONJ
cana-1298	148	21	,	,	PUNCT
cana-1298	148	22	if	if	SCONJ
cana-1298	148	23	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	148	24	+	+	PROPN
cana-1298	148	25	(	(	PUNCT
cana-1298	148	26	𝔬	𝔬	NOUN
cana-1298	148	27	)	)	PUNCT
cana-1298	148	28	≥	≥	NOUN
cana-1298	148	29	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	148	30	+	+	PROPN
cana-1298	148	31	(	(	PUNCT
cana-1298	148	32	𝜚	𝜚	NOUN
cana-1298	148	33	)	)	PUNCT
cana-1298	148	34	,	,	PUNCT
cana-1298	148	35	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	148	36	−(𝔬	−(𝔬	VERB
cana-1298	148	37	)	)	PUNCT
cana-1298	148	38	≤	≤	NOUN
cana-1298	149	1	𝔓𝑖	𝔓𝑖	PROPN
cana-1298	149	2	−(𝜚	−(𝜚	NOUN
cana-1298	149	3	)	)	PUNCT
cana-1298	149	4	,	,	PUNCT
cana-1298	149	5	for	for	ADP
cana-1298	149	6	all	all	DET
cana-1298	149	7	𝜚𝔒1	𝜚𝔒1	NOUN
cana-1298	149	8	,	,	PUNCT
cana-1298	149	9	then	then	ADV
cana-1298	149	10	𝔓	𝔓	PROPN
cana-1298	149	11	is	be	AUX
cana-1298	149	12	a	a	DET
cana-1298	149	13	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	149	14	of	of	ADP
cana-1298	149	15	𝔒1	𝔒1	NUM
cana-1298	149	16	;	;	PUNCT
cana-1298	149	17	(	(	PUNCT
cana-1298	149	18	ii	ii	NOUN
cana-1298	149	19	)	)	PUNCT
cana-1298	149	20	if	if	SCONJ
cana-1298	149	21	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	149	22	+	+	PROPN
cana-1298	149	23	(	(	PUNCT
cana-1298	149	24	𝜁	𝜁	ADJ
cana-1298	149	25	)	)	PUNCT
cana-1298	149	26	≤	≤	NOUN
cana-1298	150	1	𝔓𝑖	𝔓𝑖	PROPN
cana-1298	150	2	+	+	PROPN
cana-1298	150	3	(	(	PUNCT
cana-1298	150	4	𝔢	𝔢	NOUN
cana-1298	150	5	)	)	PUNCT
cana-1298	150	6	,	,	PUNCT
cana-1298	150	7	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	150	8	−(𝜁	−(𝜁	PROPN
cana-1298	150	9	)	)	PUNCT
cana-1298	150	10	≥	≥	NOUN
cana-1298	150	11	𝔓𝑖	𝔓𝑖	PROPN
cana-1298	150	12	−(𝔢	−(𝔢	NOUN
cana-1298	150	13	)	)	PUNCT
cana-1298	150	14	,	,	PUNCT
cana-1298	150	15	for	for	ADP
cana-1298	150	16	all	all	DET
cana-1298	150	17	𝜁𝔒2	𝜁𝔒2	NOUN
cana-1298	150	18	,	,	PUNCT
cana-1298	150	19	then	then	ADV
cana-1298	150	20	𝔚	𝔚	PROPN
cana-1298	150	21	is	be	AUX
cana-1298	150	22	a	a	DET
cana-1298	150	23	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	150	24	of	of	ADP
cana-1298	150	25	𝔒2	𝔒2	NOUN
cana-1298	150	26	;	;	PUNCT
cana-1298	150	27	where	where	SCONJ
cana-1298	150	28	𝔢	𝔢	X
cana-1298	150	29	,	,	PUNCT
cana-1298	150	30	𝔬	𝔬	PROPN
cana-1298	150	31	𝑎𝑟𝑒	𝑎𝑟𝑒	PROPN
cana-1298	150	32	𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦	𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦	PROPN
cana-1298	150	33	𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠	𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠	NOUN
cana-1298	150	34	𝑜𝑓	𝑜𝑓	ADP
cana-1298	150	35	𝔒1and	𝔒1and	NOUN
cana-1298	150	36	𝔒2	𝔒2	NOUN
cana-1298	150	37	.	.	PUNCT
cana-1298	151	1	proof	proof	NOUN
cana-1298	151	2	.	.	PUNCT
cana-1298	152	1	let	let	VERB
cana-1298	152	2	𝜚	𝜚	X
cana-1298	152	3	,	,	PUNCT
cana-1298	152	4	𝜐	𝜐	VERB
cana-1298	152	5	be	be	VERB
cana-1298	152	6	in	in	ADP
cana-1298	152	7	𝔒1	𝔒1	ADJ
cana-1298	152	8	.	.	PUNCT
cana-1298	153	1	then	then	ADV
cana-1298	153	2	(	(	PUNCT
cana-1298	153	3	𝜚	𝜚	NOUN
cana-1298	153	4	,	,	PUNCT
cana-1298	153	5	𝔬	𝔬	NOUN
cana-1298	153	6	)	)	PUNCT
cana-1298	153	7	and	and	CCONJ
cana-1298	153	8	(	(	PUNCT
cana-1298	153	9	𝜐	𝜐	NOUN
cana-1298	153	10	,	,	PUNCT
cana-1298	153	11	𝔬	𝔬	NOUN
cana-1298	153	12	)	)	PUNCT
cana-1298	153	13	are	be	AUX
cana-1298	153	14	in	in	ADP
cana-1298	153	15	𝔒1×𝔒2	𝔒1×𝔒2	PROPN
cana-1298	153	16	.	.	PUNCT
cana-1298	154	1	for	for	ADP
cana-1298	154	2	all	all	DET
cana-1298	154	3	i	i	PRON
cana-1298	154	4	,	,	PUNCT
cana-1298	154	5	i	i	NOUN
cana-1298	154	6	=	=	NOUN
cana-1298	154	7	1	1	NUM
cana-1298	154	8	,	,	PUNCT
cana-1298	154	9	2	2	NUM
cana-1298	154	10	,	,	PUNCT
cana-1298	154	11	…	…	PUNCT
cana-1298	154	12	,	,	PUNCT
cana-1298	154	13	n	n	CCONJ
cana-1298	154	14	,	,	PUNCT
cana-1298	154	15	(	(	PUNCT
cana-1298	154	16	i	i	NOUN
cana-1298	154	17	)	)	PUNCT
cana-1298	154	18	𝔓i	𝔓i	PROPN
cana-1298	154	19	+	+	ADJ
cana-1298	154	20	(	(	PUNCT
cana-1298	154	21	𝜚−𝜐	𝜚−𝜐	NOUN
cana-1298	154	22	)	)	PUNCT
cana-1298	154	23	=	=	PUNCT
cana-1298	154	24	rmin{𝔓i	rmin{𝔓i	NOUN
cana-1298	154	25	+	+	NOUN
cana-1298	154	26	(	(	PUNCT
cana-1298	154	27	𝜚−𝜐	𝜚−𝜐	NOUN
cana-1298	154	28	)	)	PUNCT
cana-1298	154	29	,	,	PUNCT
cana-1298	154	30	𝔚i	𝔚i	PROPN
cana-1298	154	31	+	+	NOUN
cana-1298	154	32	(	(	PUNCT
cana-1298	154	33	𝔬−	𝔬−	NOUN
cana-1298	154	34	𝔬	𝔬	NOUN
cana-1298	154	35	)	)	PUNCT
cana-1298	154	36	}	}	PUNCT
cana-1298	154	37	=	=	SYM
cana-1298	154	38	(	(	PUNCT
cana-1298	154	39	𝔓i×𝔚i	𝔓i×𝔚i	NOUN
cana-1298	154	40	)	)	PUNCT
cana-1298	154	41	+	+	NOUN
cana-1298	154	42	(	(	PUNCT
cana-1298	154	43	𝜚−𝜐	𝜚−𝜐	NOUN
cana-1298	154	44	,	,	PUNCT
cana-1298	154	45	𝔬−	𝔬−	NOUN
cana-1298	154	46	𝔬	𝔬	NOUN
cana-1298	154	47	)	)	PUNCT
cana-1298	154	48	=	=	SYM
cana-1298	154	49	(	(	PUNCT
cana-1298	154	50	𝔓i×𝔚i	𝔓i×𝔚i	NOUN
cana-1298	154	51	)	)	PUNCT
cana-1298	154	52	+	+	PROPN
cana-1298	154	53	[	[	X
cana-1298	154	54	(	(	PUNCT
cana-1298	154	55	𝜚	𝜚	NOUN
cana-1298	154	56	,	,	PUNCT
cana-1298	154	57	𝔬)−(𝜐	𝔬)−(𝜐	PROPN
cana-1298	154	58	,	,	PUNCT
cana-1298	154	59	𝔬	𝔬	NOUN
cana-1298	154	60	)	)	PUNCT
cana-1298	154	61	]	]	PUNCT
cana-1298	154	62			NUM
cana-1298	154	63	rmin{(𝔓i×𝔚i	rmin{(𝔓i×𝔚i	NOUN
cana-1298	154	64	)	)	PUNCT
cana-1298	155	1	+	+	PROPN
cana-1298	155	2	(	(	PUNCT
cana-1298	155	3	𝜚	𝜚	NOUN
cana-1298	155	4	,	,	PUNCT
cana-1298	155	5	𝔬	𝔬	NOUN
cana-1298	155	6	)	)	PUNCT
cana-1298	155	7	,	,	PUNCT
cana-1298	155	8	(	(	PUNCT
cana-1298	155	9	𝔓i×𝔚i	𝔓i×𝔚i	X
cana-1298	155	10	)	)	PUNCT
cana-1298	155	11	+	+	PROPN
cana-1298	155	12	(	(	PUNCT
cana-1298	155	13	𝜐	𝜐	PROPN
cana-1298	155	14	,	,	PUNCT
cana-1298	155	15	𝔬	𝔬	NOUN
cana-1298	155	16	)	)	PUNCT
cana-1298	155	17	}	}	PUNCT
cana-1298	155	18	=	=	PUNCT
cana-1298	156	1	rmin{rmin{𝔓i	rmin{rmin{𝔓i	NOUN
cana-1298	157	1	+	+	ADJ
cana-1298	157	2	(	(	PUNCT
cana-1298	157	3	𝜚	𝜚	NOUN
cana-1298	157	4	)	)	PUNCT
cana-1298	157	5	,	,	PUNCT
cana-1298	158	1	𝔚i	𝔚i	PROPN
cana-1298	158	2	+	+	ADJ
cana-1298	158	3	(	(	PUNCT
cana-1298	158	4	𝔬	𝔬	NOUN
cana-1298	158	5	)	)	PUNCT
cana-1298	158	6	}	}	PUNCT
cana-1298	158	7	,	,	PUNCT
cana-1298	158	8	rmin{𝔓i	rmin{𝔓i	NOUN
cana-1298	158	9	+	+	PROPN
cana-1298	158	10	(	(	PUNCT
cana-1298	158	11	𝜐	𝜐	NOUN
cana-1298	158	12	)	)	PUNCT
cana-1298	158	13	,	,	PUNCT
cana-1298	158	14	𝔚i	𝔚i	PROPN
cana-1298	158	15	+	+	ADJ
cana-1298	158	16	(	(	PUNCT
cana-1298	158	17	𝔬	𝔬	NOUN
cana-1298	158	18	)	)	PUNCT
cana-1298	158	19	}	}	PUNCT
cana-1298	158	20	}	}	PUNCT
cana-1298	158	21	=	=	SYM
cana-1298	158	22	rmin	rmin	NOUN
cana-1298	158	23	{	{	PUNCT
cana-1298	158	24	𝔓i	𝔓i	PROPN
cana-1298	158	25	+	+	PROPN
cana-1298	158	26	(	(	PUNCT
cana-1298	158	27	𝜚	𝜚	NOUN
cana-1298	158	28	)	)	PUNCT
cana-1298	158	29	,	,	PUNCT
cana-1298	158	30	𝔓i	𝔓i	PROPN
cana-1298	158	31	+	+	PROPN
cana-1298	158	32	(	(	PUNCT
cana-1298	158	33	𝜐	𝜐	NOUN
cana-1298	158	34	)	)	PUNCT
cana-1298	158	35	}	}	PUNCT
cana-1298	158	36	,	,	PUNCT
cana-1298	158	37	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1298	158	38	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-1298	158	39	𝜚	𝜚	X
cana-1298	158	40	,	,	PUNCT
cana-1298	158	41	𝜐	𝜐	PROPN
cana-1298	158	42	in	in	ADP
cana-1298	158	43	𝔒1	𝔒1	NUM
cana-1298	158	44	.	.	PUNCT
cana-1298	159	1	and	and	CCONJ
cana-1298	159	2	𝔓i	𝔓i	PROPN
cana-1298	159	3	+	+	PROPN
cana-1298	159	4	(	(	PUNCT
cana-1298	159	5	𝜚𝜐	𝜚𝜐	NOUN
cana-1298	159	6	)	)	PUNCT
cana-1298	159	7	=	=	PUNCT
cana-1298	160	1	rmin{𝔓i	rmin{𝔓i	NOUN
cana-1298	161	1	+	+	NOUN
cana-1298	161	2	(	(	PUNCT
cana-1298	161	3	𝜚𝜐	𝜚𝜐	NOUN
cana-1298	161	4	)	)	PUNCT
cana-1298	161	5	,	,	PUNCT
cana-1298	162	1	𝔚i	𝔚i	PROPN
cana-1298	162	2	+	+	NOUN
cana-1298	162	3	(	(	PUNCT
cana-1298	162	4	𝔬𝔬)}=	𝔬𝔬)}=	NOUN
cana-1298	162	5	(	(	PUNCT
cana-1298	162	6	𝔓i×𝔚i	𝔓i×𝔚i	NOUN
cana-1298	162	7	)	)	PUNCT
cana-1298	162	8	+	+	ADJ
cana-1298	162	9	(	(	PUNCT
cana-1298	162	10	𝜚𝜐	𝜚𝜐	PROPN
cana-1298	162	11	,	,	PUNCT
cana-1298	162	12	𝔬𝔬	𝔬𝔬	ADJ
cana-1298	162	13	)	)	PUNCT
cana-1298	162	14	=	=	SYM
cana-1298	162	15	(	(	PUNCT
cana-1298	162	16	𝔓i×𝔚i	𝔓i×𝔚i	NOUN
cana-1298	162	17	)	)	PUNCT
cana-1298	162	18	+	+	PROPN
cana-1298	162	19	[	[	X
cana-1298	162	20	(	(	PUNCT
cana-1298	162	21	𝜚	𝜚	NOUN
cana-1298	162	22	,	,	PUNCT
cana-1298	162	23	𝔬)(𝜐	𝔬)(𝜐	NUM
cana-1298	162	24	,	,	PUNCT
cana-1298	162	25	𝔬	𝔬	NOUN
cana-1298	162	26	)	)	PUNCT
cana-1298	162	27	]	]	PUNCT
cana-1298	162	28			NUM
cana-1298	162	29	rmin{(𝔓i×𝔚i	rmin{(𝔓i×𝔚i	NOUN
cana-1298	162	30	)	)	PUNCT
cana-1298	163	1	+	+	PROPN
cana-1298	163	2	(	(	PUNCT
cana-1298	163	3	𝜚	𝜚	NOUN
cana-1298	163	4	,	,	PUNCT
cana-1298	163	5	𝔬	𝔬	NOUN
cana-1298	163	6	)	)	PUNCT
cana-1298	163	7	,	,	PUNCT
cana-1298	163	8	(	(	PUNCT
cana-1298	163	9	𝔓i×𝔚i	𝔓i×𝔚i	X
cana-1298	163	10	)	)	PUNCT
cana-1298	163	11	+	+	PROPN
cana-1298	163	12	(	(	PUNCT
cana-1298	163	13	𝜐	𝜐	PROPN
cana-1298	163	14	,	,	PUNCT
cana-1298	163	15	𝔬	𝔬	NOUN
cana-1298	163	16	)	)	PUNCT
cana-1298	163	17	}	}	PUNCT
cana-1298	163	18	=	=	PUNCT
cana-1298	164	1	rmin{rmin{𝔓i	rmin{rmin{𝔓i	NOUN
cana-1298	165	1	+	+	ADJ
cana-1298	165	2	(	(	PUNCT
cana-1298	165	3	𝜚	𝜚	NOUN
cana-1298	165	4	)	)	PUNCT
cana-1298	165	5	,	,	PUNCT
cana-1298	166	1	𝔚i	𝔚i	PROPN
cana-1298	166	2	+	+	ADJ
cana-1298	166	3	(	(	PUNCT
cana-1298	166	4	𝔬	𝔬	NOUN
cana-1298	166	5	)	)	PUNCT
cana-1298	166	6	}	}	PUNCT
cana-1298	166	7	,	,	PUNCT
cana-1298	166	8	rmin{𝔓i	rmin{𝔓i	NOUN
cana-1298	166	9	+	+	PROPN
cana-1298	166	10	(	(	PUNCT
cana-1298	166	11	𝜐	𝜐	NOUN
cana-1298	166	12	)	)	PUNCT
cana-1298	166	13	,	,	PUNCT
cana-1298	166	14	𝔚i	𝔚i	PROPN
cana-1298	166	15	+	+	ADJ
cana-1298	166	16	(	(	PUNCT
cana-1298	166	17	𝔬	𝔬	NOUN
cana-1298	166	18	)	)	PUNCT
cana-1298	166	19	}	}	PUNCT
cana-1298	166	20	}	}	PUNCT
cana-1298	166	21	=	=	SYM
cana-1298	166	22	rmin	rmin	NOUN
cana-1298	166	23	{	{	PUNCT
cana-1298	166	24	𝔓i	𝔓i	PROPN
cana-1298	166	25	+	+	PROPN
cana-1298	166	26	(	(	PUNCT
cana-1298	166	27	𝜚	𝜚	NOUN
cana-1298	166	28	)	)	PUNCT
cana-1298	166	29	,	,	PUNCT
cana-1298	166	30	𝔓i	𝔓i	PROPN
cana-1298	166	31	+	+	PROPN
cana-1298	166	32	(	(	PUNCT
cana-1298	166	33	𝜐	𝜐	NOUN
cana-1298	166	34	)	)	PUNCT
cana-1298	166	35	}	}	PUNCT
cana-1298	166	36	,	,	PUNCT
cana-1298	166	37	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1298	166	38	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-1298	166	39	𝜚	𝜚	X
cana-1298	166	40	,	,	PUNCT
cana-1298	166	41	𝜐	𝜐	PROPN
cana-1298	166	42	in	in	ADP
cana-1298	166	43	𝔒1	𝔒1	NUM
cana-1298	166	44	.	.	PUNCT
cana-1298	167	1	also	also	ADV
cana-1298	167	2	𝔓𝑖	𝔓𝑖	PROPN
cana-1298	167	3	−(𝜚−𝜐	−(𝜚−𝜐	NUM
cana-1298	167	4	)	)	PUNCT
cana-1298	167	5	=	=	SYM
cana-1298	167	6	rmax{𝔓𝑖	rmax{𝔓𝑖	NOUN
cana-1298	167	7	−(𝜚−𝜐	−(𝜚−𝜐	NUM
cana-1298	167	8	)	)	PUNCT
cana-1298	167	9	,	,	PUNCT
cana-1298	167	10	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	167	11	−(𝔬−	−(𝔬−	NOUN
cana-1298	167	12	𝔬)}=	𝔬)}=	PUNCT
cana-1298	167	13	(	(	PUNCT
cana-1298	167	14	𝔓i×𝔚i)−(𝜚−𝜐	𝔓i×𝔚i)−(𝜚−𝜐	NUM
cana-1298	167	15	,	,	PUNCT
cana-1298	167	16	𝔬−	𝔬−	NOUN
cana-1298	167	17	𝔬	𝔬	NOUN
cana-1298	167	18	)	)	PUNCT
cana-1298	167	19	=	=	SYM
cana-1298	167	20	(	(	PUNCT
cana-1298	167	21	𝔓i×𝔚i)−[(𝜚	𝔓i×𝔚i)−[(𝜚	PROPN
cana-1298	167	22	,	,	PUNCT
cana-1298	167	23	𝔬)−(𝜐	𝔬)−(𝜐	PROPN
cana-1298	167	24	,	,	PUNCT
cana-1298	167	25	𝔬	𝔬	NOUN
cana-1298	167	26	)	)	PUNCT
cana-1298	167	27	]	]	PUNCT
cana-1298	167	28			NUM
cana-1298	167	29	rmax{(𝔓i×𝔚i)−(𝜚	rmax{(𝔓i×𝔚i)−(𝜚	NOUN
cana-1298	167	30	,	,	PUNCT
cana-1298	167	31	𝔬	𝔬	NOUN
cana-1298	167	32	)	)	PUNCT
cana-1298	167	33	,	,	PUNCT
cana-1298	167	34	(	(	PUNCT
cana-1298	167	35	𝔓i×𝔚i)−(𝜐	𝔓i×𝔚i)−(𝜐	ADJ
cana-1298	167	36	,	,	PUNCT
cana-1298	167	37	𝔬	𝔬	NOUN
cana-1298	167	38	)	)	PUNCT
cana-1298	167	39	}	}	PUNCT
cana-1298	167	40	=	=	SYM
cana-1298	167	41	rmax{rmax{𝔓𝑖	rmax{rmax{𝔓𝑖	NOUN
cana-1298	167	42	−(𝜚	−(𝜚	NOUN
cana-1298	167	43	)	)	PUNCT
cana-1298	167	44	,	,	PUNCT
cana-1298	167	45	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	167	46	−(𝔬	−(𝔬	VERB
cana-1298	167	47	)	)	PUNCT
cana-1298	167	48	}	}	PUNCT
cana-1298	167	49	,	,	PUNCT
cana-1298	167	50	rmax{𝔓𝑖	rmax{𝔓𝑖	X
cana-1298	167	51	−(𝜐	−(𝜐	PROPN
cana-1298	167	52	)	)	PUNCT
cana-1298	167	53	,	,	PUNCT
cana-1298	167	54	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	167	55	−(𝔬	−(𝔬	VERB
cana-1298	167	56	)	)	PUNCT
cana-1298	167	57	}	}	PUNCT
cana-1298	167	58	}	}	PUNCT
cana-1298	167	59	=	=	SYM
cana-1298	167	60	rmax	rmax	ADJ
cana-1298	167	61	{	{	PUNCT
cana-1298	167	62	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	167	63	−(𝜚	−(𝜚	NOUN
cana-1298	167	64	)	)	PUNCT
cana-1298	167	65	,	,	PUNCT
cana-1298	167	66	𝔓𝑖	𝔓𝑖	PROPN
cana-1298	167	67	−(𝜐	−(𝜐	NOUN
cana-1298	167	68	)	)	PUNCT
cana-1298	167	69	}	}	PUNCT
cana-1298	167	70	,	,	PUNCT
cana-1298	167	71	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1298	167	72	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-1298	167	73	𝜚	𝜚	X
cana-1298	167	74	,	,	PUNCT
cana-1298	167	75	𝜐	𝜐	PROPN
cana-1298	167	76	in	in	ADP
cana-1298	167	77	𝔒1	𝔒1	NUM
cana-1298	167	78	.	.	PUNCT
cana-1298	168	1	and	and	CCONJ
cana-1298	168	2	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	168	3	−(𝜚𝜐	−(𝜚𝜐	NOUN
cana-1298	168	4	)	)	PUNCT
cana-1298	168	5	=	=	PUNCT
cana-1298	169	1	rmax{𝔓𝑖	rmax{𝔓𝑖	NOUN
cana-1298	169	2	−(𝜚𝜐	−(𝜚𝜐	NOUN
cana-1298	169	3	)	)	PUNCT
cana-1298	169	4	,	,	PUNCT
cana-1298	169	5	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	169	6	−(𝔬𝔬	−(𝔬𝔬	PROPN
cana-1298	169	7	)	)	PUNCT
cana-1298	169	8	}	}	PUNCT
cana-1298	169	9	=	=	SYM
cana-1298	169	10	(	(	PUNCT
cana-1298	169	11	𝔓i×𝔚i)−(𝜚𝜐	𝔓i×𝔚i)−(𝜚𝜐	NOUN
cana-1298	169	12	,	,	PUNCT
cana-1298	169	13	𝔬𝔬	𝔬𝔬	ADJ
cana-1298	169	14	)	)	PUNCT
cana-1298	169	15	=	=	SYM
cana-1298	169	16	(	(	PUNCT
cana-1298	169	17	𝔓i×𝔚i)−[(𝜚	𝔓i×𝔚i)−[(𝜚	NOUN
cana-1298	169	18	,	,	PUNCT
cana-1298	169	19	𝔬)(𝜐	𝔬)(𝜐	NUM
cana-1298	169	20	,	,	PUNCT
cana-1298	169	21	𝔬	𝔬	NOUN
cana-1298	169	22	)	)	PUNCT
cana-1298	169	23	]	]	PUNCT
cana-1298	169	24			NUM
cana-1298	169	25	rmax{(𝔓i×𝔚i)−(𝜚	rmax{(𝔓i×𝔚i)−(𝜚	NOUN
cana-1298	169	26	,	,	PUNCT
cana-1298	169	27	𝔬	𝔬	NOUN
cana-1298	169	28	)	)	PUNCT
cana-1298	169	29	,	,	PUNCT
cana-1298	169	30	(	(	PUNCT
cana-1298	169	31	𝔓i×𝔚i)−(𝜐	𝔓i×𝔚i)−(𝜐	ADJ
cana-1298	169	32	,	,	PUNCT
cana-1298	169	33	𝔬	𝔬	NOUN
cana-1298	169	34	)	)	PUNCT
cana-1298	169	35	}	}	PUNCT
cana-1298	169	36	=	=	SYM
cana-1298	169	37	rmax{rmax{𝔓𝑖	rmax{rmax{𝔓𝑖	NOUN
cana-1298	169	38	−(𝜚	−(𝜚	NOUN
cana-1298	169	39	)	)	PUNCT
cana-1298	169	40	,	,	PUNCT
cana-1298	169	41	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	169	42	−(𝔬	−(𝔬	VERB
cana-1298	169	43	)	)	PUNCT
cana-1298	169	44	}	}	PUNCT
cana-1298	169	45	,	,	PUNCT
cana-1298	169	46	rmax{𝔓𝑖	rmax{𝔓𝑖	X
cana-1298	169	47	−(𝜐	−(𝜐	PROPN
cana-1298	169	48	)	)	PUNCT
cana-1298	169	49	,	,	PUNCT
cana-1298	169	50	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	169	51	−(𝔬	−(𝔬	VERB
cana-1298	169	52	)	)	PUNCT
cana-1298	169	53	}	}	PUNCT
cana-1298	169	54	}	}	PUNCT
cana-1298	169	55	=	=	SYM
cana-1298	169	56	rmax	rmax	ADJ
cana-1298	169	57	{	{	PUNCT
cana-1298	169	58	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	169	59	−(𝜚	−(𝜚	NOUN
cana-1298	169	60	)	)	PUNCT
cana-1298	169	61	,	,	PUNCT
cana-1298	169	62	𝔓𝑖	𝔓𝑖	PROPN
cana-1298	169	63	−(𝜐	−(𝜐	NOUN
cana-1298	169	64	)	)	PUNCT
cana-1298	169	65	}	}	PUNCT
cana-1298	169	66	,	,	PUNCT
cana-1298	169	67	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1298	169	68	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-1298	169	69	𝜚	𝜚	X
cana-1298	169	70	,	,	PUNCT
cana-1298	169	71	𝜐	𝜐	PROPN
cana-1298	169	72	in	in	ADP
cana-1298	169	73	𝔒1	𝔒1	NUM
cana-1298	169	74	.	.	PUNCT
cana-1298	170	1	hence	hence	ADV
cana-1298	170	2	𝔓	𝔓	PROPN
cana-1298	170	3	is	be	AUX
cana-1298	170	4	a	a	DET
cana-1298	170	5	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	170	6	of	of	ADP
cana-1298	170	7	𝔒1	𝔒1	X
cana-1298	170	8	.	.	PUNCT
cana-1298	171	1	(	(	PUNCT
cana-1298	171	2	ii	ii	NOUN
cana-1298	171	3	)	)	PUNCT
cana-1298	171	4	let	let	VERB
cana-1298	171	5	𝜚	𝜚	NOUN
cana-1298	171	6	,	,	PUNCT
cana-1298	171	7	𝜐	𝜐	VERB
cana-1298	171	8	be	be	VERB
cana-1298	171	9	in	in	ADP
cana-1298	171	10	𝔒2	𝔒2	NOUN
cana-1298	171	11	.	.	PUNCT
cana-1298	172	1	then	then	ADV
cana-1298	172	2	(	(	PUNCT
cana-1298	172	3	𝔢	𝔢	NOUN
cana-1298	172	4	,	,	PUNCT
cana-1298	172	5	𝜚	𝜚	NOUN
cana-1298	172	6	)	)	PUNCT
cana-1298	172	7	and	and	CCONJ
cana-1298	172	8	(	(	PUNCT
cana-1298	172	9	𝔢	𝔢	PROPN
cana-1298	172	10	,	,	PUNCT
cana-1298	172	11	𝜐	𝜐	NOUN
cana-1298	172	12	)	)	PUNCT
cana-1298	172	13	are	be	AUX
cana-1298	172	14	in	in	ADP
cana-1298	172	15	𝔒1×𝔒2	𝔒1×𝔒2	PROPN
cana-1298	172	16	.	.	PUNCT
cana-1298	173	1	for	for	ADP
cana-1298	173	2	all	all	DET
cana-1298	173	3	i	i	PRON
cana-1298	173	4	,	,	PUNCT
cana-1298	173	5	i	i	NOUN
cana-1298	173	6	=	=	NOUN
cana-1298	173	7	1	1	NUM
cana-1298	173	8	,	,	PUNCT
cana-1298	173	9	2	2	NUM
cana-1298	173	10	,	,	PUNCT
cana-1298	173	11	…	…	PUNCT
cana-1298	173	12	,	,	PUNCT
cana-1298	173	13	n	n	CCONJ
cana-1298	173	14	,	,	PUNCT
cana-1298	173	15	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	173	16	+	+	PROPN
cana-1298	173	17	(	(	PUNCT
cana-1298	173	18	𝜚−𝜐	𝜚−𝜐	NOUN
cana-1298	173	19	)	)	PUNCT
cana-1298	173	20	=	=	PUNCT
cana-1298	173	21	rmin{𝔓i	rmin{𝔓i	NOUN
cana-1298	173	22	+	+	NOUN
cana-1298	173	23	(	(	PUNCT
cana-1298	173	24	𝔢	𝔢	PROPN
cana-1298	173	25	−	−	PROPN
cana-1298	173	26	𝔢	𝔢	PROPN
cana-1298	173	27	)	)	PUNCT
cana-1298	173	28	,	,	PUNCT
cana-1298	173	29	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	173	30	+	+	PROPN
cana-1298	173	31	(	(	PUNCT
cana-1298	173	32	𝜚−𝜐)}=	𝜚−𝜐)}=	PRON
cana-1298	173	33	(	(	PUNCT
cana-1298	173	34	𝔓i×𝔚i	𝔓i×𝔚i	NOUN
cana-1298	173	35	)	)	PUNCT
cana-1298	173	36	+	+	PROPN
cana-1298	173	37	(	(	PUNCT
cana-1298	173	38	𝔢	𝔢	PROPN
cana-1298	173	39	−	−	PROPN
cana-1298	173	40	𝔢	𝔢	PROPN
cana-1298	173	41	,	,	PUNCT
cana-1298	173	42	𝜚−𝜐	𝜚−𝜐	NOUN
cana-1298	173	43	)	)	PUNCT
cana-1298	173	44	=	=	SYM
cana-1298	173	45	(	(	PUNCT
cana-1298	173	46	𝔓i×𝔚i	𝔓i×𝔚i	NOUN
cana-1298	173	47	)	)	PUNCT
cana-1298	173	48	+	+	PROPN
cana-1298	173	49	[	[	X
cana-1298	173	50	(	(	PUNCT
cana-1298	173	51	𝔢	𝔢	NOUN
cana-1298	173	52	,	,	PUNCT
cana-1298	173	53	𝜚)−	𝜚)−	PROPN
cana-1298	173	54	(	(	PUNCT
cana-1298	173	55	𝔢	𝔢	PROPN
cana-1298	173	56	,	,	PUNCT
cana-1298	173	57	𝜐	𝜐	NOUN
cana-1298	173	58	)	)	PUNCT
cana-1298	173	59	]	]	PUNCT
cana-1298	173	60			NUM
cana-1298	173	61	rmin{(𝔓i×𝔚i	rmin{(𝔓i×𝔚i	NOUN
cana-1298	173	62	)	)	PUNCT
cana-1298	174	1	+	+	PROPN
cana-1298	174	2	(	(	PUNCT
cana-1298	174	3	𝔢	𝔢	PROPN
cana-1298	174	4	,	,	PUNCT
cana-1298	174	5	𝜚	𝜚	NOUN
cana-1298	174	6	)	)	PUNCT
cana-1298	174	7	,	,	PUNCT
cana-1298	174	8	(	(	PUNCT
cana-1298	174	9	𝔓i×𝔚i	𝔓i×𝔚i	X
cana-1298	174	10	)	)	PUNCT
cana-1298	174	11	+	+	PROPN
cana-1298	174	12	(	(	PUNCT
cana-1298	174	13	𝔢	𝔢	PROPN
cana-1298	174	14	,	,	PUNCT
cana-1298	174	15	𝜐	𝜐	NOUN
cana-1298	174	16	)	)	PUNCT
cana-1298	174	17	}	}	PUNCT
cana-1298	174	18	=	=	PUNCT
cana-1298	175	1	rmin{rmin{𝔓i	rmin{rmin{𝔓i	X
cana-1298	176	1	+	+	ADJ
cana-1298	176	2	(	(	PUNCT
cana-1298	176	3	𝔢	𝔢	NOUN
cana-1298	176	4	)	)	PUNCT
cana-1298	176	5	,	,	PUNCT
cana-1298	176	6	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	176	7	+	+	PROPN
cana-1298	176	8	(	(	PUNCT
cana-1298	176	9	𝜚	𝜚	NOUN
cana-1298	176	10	)	)	PUNCT
cana-1298	176	11	}	}	PUNCT
cana-1298	176	12	,	,	PUNCT
cana-1298	176	13	rmin{𝔓i	rmin{𝔓i	NOUN
cana-1298	176	14	+	+	NOUN
cana-1298	176	15	(	(	PUNCT
cana-1298	176	16	𝔢	𝔢	NOUN
cana-1298	176	17	)	)	PUNCT
cana-1298	176	18	,	,	PUNCT
cana-1298	176	19	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	176	20	+	+	PROPN
cana-1298	176	21	(	(	PUNCT
cana-1298	176	22	𝜐	𝜐	NOUN
cana-1298	176	23	)	)	PUNCT
cana-1298	176	24	}	}	PUNCT
cana-1298	176	25	}	}	PUNCT
cana-1298	176	26	=	=	VERB
cana-1298	176	27	rmin{𝔚𝑖	rmin{𝔚𝑖	ADJ
cana-1298	177	1	+	+	ADJ
cana-1298	177	2	(	(	PUNCT
cana-1298	177	3	𝜚	𝜚	NOUN
cana-1298	177	4	)	)	PUNCT
cana-1298	177	5	,	,	PUNCT
cana-1298	177	6	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	177	7	+	+	PROPN
cana-1298	177	8	(	(	PUNCT
cana-1298	177	9	𝜐	𝜐	NOUN
cana-1298	177	10	)	)	PUNCT
cana-1298	177	11	}	}	PUNCT
cana-1298	177	12	,	,	PUNCT
cana-1298	177	13	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1298	177	14	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-1298	177	15	𝜚	𝜚	NOUN
cana-1298	177	16	,	,	PUNCT
cana-1298	177	17	𝜐	𝜐	PROPN
cana-1298	177	18	in	in	ADP
cana-1298	177	19	𝔒2	𝔒2	NOUN
cana-1298	177	20	.	.	PUNCT
cana-1298	178	1	and	and	CCONJ
cana-1298	178	2	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	178	3	+	+	PROPN
cana-1298	178	4	(	(	PUNCT
cana-1298	178	5	𝜚𝜐	𝜚𝜐	ADP
cana-1298	178	6	)	)	PUNCT
cana-1298	178	7	=	=	PUNCT
cana-1298	178	8	rmin{𝔓i	rmin{𝔓i	NOUN
cana-1298	179	1	+	+	NOUN
cana-1298	179	2	(	(	PUNCT
cana-1298	179	3	𝔢𝔢	𝔢𝔢	NOUN
cana-1298	179	4	)	)	PUNCT
cana-1298	179	5	,	,	PUNCT
cana-1298	179	6	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	179	7	+	+	PROPN
cana-1298	179	8	(	(	PUNCT
cana-1298	179	9	𝜚𝜐)}=	𝜚𝜐)}=	PROPN
cana-1298	179	10	(	(	PUNCT
cana-1298	179	11	𝔓i×𝔚i	𝔓i×𝔚i	NOUN
cana-1298	179	12	)	)	PUNCT
cana-1298	179	13	+	+	NOUN
cana-1298	179	14	(	(	PUNCT
cana-1298	179	15	𝔢𝔢	𝔢𝔢	NOUN
cana-1298	179	16	,	,	PUNCT
cana-1298	179	17	𝜚𝜐	𝜚𝜐	ADP
cana-1298	179	18	)	)	PUNCT
cana-1298	179	19	=	=	SYM
cana-1298	179	20	(	(	PUNCT
cana-1298	179	21	𝔓i×𝔚i	𝔓i×𝔚i	NOUN
cana-1298	179	22	)	)	PUNCT
cana-1298	179	23	+	+	PROPN
cana-1298	179	24	[	[	X
cana-1298	179	25	(	(	PUNCT
cana-1298	179	26	𝔢	𝔢	NOUN
cana-1298	179	27	,	,	PUNCT
cana-1298	179	28	𝜚)(𝔢	𝜚)(𝔢	PRON
cana-1298	179	29	,	,	PUNCT
cana-1298	179	30	𝜐	𝜐	NOUN
cana-1298	179	31	)	)	PUNCT
cana-1298	179	32	]	]	PUNCT
cana-1298	179	33			NUM
cana-1298	179	34	rmin{(𝔓i×𝔚i	rmin{(𝔓i×𝔚i	NOUN
cana-1298	179	35	)	)	PUNCT
cana-1298	180	1	+	+	PROPN
cana-1298	180	2	(	(	PUNCT
cana-1298	180	3	𝔢	𝔢	PROPN
cana-1298	180	4	,	,	PUNCT
cana-1298	180	5	𝜚	𝜚	NOUN
cana-1298	180	6	)	)	PUNCT
cana-1298	180	7	,	,	PUNCT
cana-1298	180	8	(	(	PUNCT
cana-1298	180	9	𝔓i×𝔚i	𝔓i×𝔚i	X
cana-1298	180	10	)	)	PUNCT
cana-1298	180	11	+	+	PROPN
cana-1298	180	12	(	(	PUNCT
cana-1298	180	13	𝔢	𝔢	PROPN
cana-1298	180	14	,	,	PUNCT
cana-1298	180	15	𝜐	𝜐	NOUN
cana-1298	180	16	)	)	PUNCT
cana-1298	180	17	}	}	PUNCT
cana-1298	180	18	=	=	PUNCT
cana-1298	181	1	rmin{rmin{𝔓i	rmin{rmin{𝔓i	X
cana-1298	182	1	+	+	ADJ
cana-1298	182	2	(	(	PUNCT
cana-1298	182	3	𝔢	𝔢	NOUN
cana-1298	182	4	)	)	PUNCT
cana-1298	182	5	,	,	PUNCT
cana-1298	182	6	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	182	7	+	+	PROPN
cana-1298	182	8	(	(	PUNCT
cana-1298	182	9	𝜚	𝜚	NOUN
cana-1298	182	10	)	)	PUNCT
cana-1298	182	11	}	}	PUNCT
cana-1298	182	12	,	,	PUNCT
cana-1298	182	13	rmin{𝔓i	rmin{𝔓i	NOUN
cana-1298	182	14	+	+	NOUN
cana-1298	182	15	(	(	PUNCT
cana-1298	182	16	𝔢	𝔢	NOUN
cana-1298	182	17	)	)	PUNCT
cana-1298	182	18	,	,	PUNCT
cana-1298	182	19	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	182	20	+	+	PROPN
cana-1298	182	21	(	(	PUNCT
cana-1298	182	22	𝜐	𝜐	NOUN
cana-1298	182	23	)	)	PUNCT
cana-1298	182	24	}	}	PUNCT
cana-1298	182	25	}	}	PUNCT
cana-1298	182	26	=	=	VERB
cana-1298	182	27	rmin{𝔚𝑖	rmin{𝔚𝑖	ADJ
cana-1298	183	1	+	+	ADJ
cana-1298	183	2	(	(	PUNCT
cana-1298	183	3	𝜚	𝜚	NOUN
cana-1298	183	4	)	)	PUNCT
cana-1298	183	5	,	,	PUNCT
cana-1298	183	6	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	183	7	+	+	PROPN
cana-1298	183	8	(	(	PUNCT
cana-1298	183	9	𝜐	𝜐	NOUN
cana-1298	183	10	)	)	PUNCT
cana-1298	183	11	}	}	PUNCT
cana-1298	183	12	,	,	PUNCT
cana-1298	183	13	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1298	183	14	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-1298	183	15	𝜚	𝜚	NOUN
cana-1298	183	16	,	,	PUNCT
cana-1298	183	17	𝜐	𝜐	PROPN
cana-1298	183	18	in	in	ADP
cana-1298	183	19	𝔒2	𝔒2	NOUN
cana-1298	183	20	.	.	PUNCT
cana-1298	184	1	also	also	ADV
cana-1298	184	2	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	184	3	−(𝜚−𝜐	−(𝜚−𝜐	NUM
cana-1298	184	4	)	)	PUNCT
cana-1298	184	5	=	=	SYM
cana-1298	184	6	rmax{𝔓𝑖	rmax{𝔓𝑖	NOUN
cana-1298	184	7	−(𝔢	−(𝔢	PROPN
cana-1298	184	8	−	−	PROPN
cana-1298	184	9	𝔢	𝔢	PROPN
cana-1298	184	10	)	)	PUNCT
cana-1298	184	11	,	,	PUNCT
cana-1298	184	12	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	184	13	−(𝜚−𝜐)}=	−(𝜚−𝜐)}=	NOUN
cana-1298	184	14	(	(	PUNCT
cana-1298	184	15	𝔓i×𝔚i)−(𝔢	𝔓i×𝔚i)−(𝔢	VERB
cana-1298	184	16	−	−	PROPN
cana-1298	184	17	𝔢	𝔢	NOUN
cana-1298	184	18	,	,	PUNCT
cana-1298	184	19	𝜚−𝜐	𝜚−𝜐	NOUN
cana-1298	184	20	)	)	PUNCT
cana-1298	184	21	=	=	SYM
cana-1298	184	22	(	(	PUNCT
cana-1298	184	23	𝔓i×𝔚i)−[(𝔢	𝔓i×𝔚i)−[(𝔢	NOUN
cana-1298	184	24	,	,	PUNCT
cana-1298	184	25	𝜚)−	𝜚)−	PROPN
cana-1298	184	26	(	(	PUNCT
cana-1298	184	27	𝔢	𝔢	PROPN
cana-1298	184	28	,	,	PUNCT
cana-1298	184	29	𝜐	𝜐	NOUN
cana-1298	184	30	)	)	PUNCT
cana-1298	184	31	]	]	PUNCT
cana-1298	185	1			NUM
cana-1298	185	2	rmax{(𝔓i×𝔚i)−(𝔢	rmax{(𝔓i×𝔚i)−(𝔢	NOUN
cana-1298	185	3	,	,	PUNCT
cana-1298	185	4	𝜚	𝜚	NOUN
cana-1298	185	5	)	)	PUNCT
cana-1298	185	6	,	,	PUNCT
cana-1298	185	7	(	(	PUNCT
cana-1298	185	8	𝔓i×𝔚i)−(𝔢	𝔓i×𝔚i)−(𝔢	NOUN
cana-1298	185	9	,	,	PUNCT
cana-1298	185	10	𝜐	𝜐	NOUN
cana-1298	185	11	)	)	PUNCT
cana-1298	185	12	}	}	PUNCT
cana-1298	185	13	=	=	SYM
cana-1298	185	14	rmax{rmax{𝔓𝑖	rmax{rmax{𝔓𝑖	PROPN
cana-1298	185	15	−(𝔢	−(𝔢	NOUN
cana-1298	185	16	)	)	PUNCT
cana-1298	185	17	,	,	PUNCT
cana-1298	185	18	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	185	19	−(𝜚	−(𝜚	NOUN
cana-1298	185	20	)	)	PUNCT
cana-1298	185	21	}	}	PUNCT
cana-1298	185	22	,	,	PUNCT
cana-1298	185	23	rmax{𝔓𝑖	rmax{𝔓𝑖	PROPN
cana-1298	185	24	−(𝔢	−(𝔢	PROPN
cana-1298	185	25	)	)	PUNCT
cana-1298	185	26	,	,	PUNCT
cana-1298	185	27	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	185	28	−(𝜐	−(𝜐	NOUN
cana-1298	185	29	)	)	PUNCT
cana-1298	185	30	}	}	PUNCT
cana-1298	185	31	}	}	PUNCT
cana-1298	185	32	=	=	PUNCT
cana-1298	185	33	rmax{𝔚𝑖	rmax{𝔚𝑖	ADJ
cana-1298	185	34	−(𝜚	−(𝜚	NOUN
cana-1298	185	35	)	)	PUNCT
cana-1298	185	36	,	,	PUNCT
cana-1298	185	37	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	185	38	−(𝜐	−(𝜐	NOUN
cana-1298	185	39	)	)	PUNCT
cana-1298	185	40	}	}	PUNCT
cana-1298	185	41	,	,	PUNCT
cana-1298	185	42	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1298	185	43	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-1298	185	44	𝜚	𝜚	NOUN
cana-1298	185	45	,	,	PUNCT
cana-1298	185	46	𝜐	𝜐	PROPN
cana-1298	185	47	in	in	ADP
cana-1298	185	48	𝔒2	𝔒2	NOUN
cana-1298	185	49	.	.	PUNCT
cana-1298	186	1	communications	communication	NOUN
cana-1298	186	2	on	on	ADP
cana-1298	186	3	applied	apply	VERB
cana-1298	186	4	nonlinear	nonlinear	ADJ
cana-1298	186	5	analysis	analysis	NOUN
cana-1298	186	6	issn	issn	NOUN
cana-1298	186	7	:	:	PUNCT
cana-1298	186	8	1074	1074	NUM
cana-1298	186	9	-	-	PUNCT
cana-1298	186	10	133x	133x	NUM
cana-1298	186	11	vol	vol	NOUN
cana-1298	186	12	31	31	NUM
cana-1298	186	13	no	no	NOUN
cana-1298	186	14	.	.	PUNCT
cana-1298	187	1	7s	7	NOUN
cana-1298	187	2	(	(	PUNCT
cana-1298	187	3	2024	2024	NUM
cana-1298	187	4	)	)	PUNCT
cana-1298	187	5	235	235	NUM
cana-1298	187	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1298	187	7	and	and	CCONJ
cana-1298	187	8	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	187	9	−(𝜚𝜐	−(𝜚𝜐	NOUN
cana-1298	187	10	)	)	PUNCT
cana-1298	187	11	=	=	PUNCT
cana-1298	188	1	rmax{𝔓𝑖	rmax{𝔓𝑖	PROPN
cana-1298	188	2	−(𝔢𝔢	−(𝔢𝔢	PROPN
cana-1298	188	3	)	)	PUNCT
cana-1298	188	4	,	,	PUNCT
cana-1298	189	1	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	189	2	−(𝜚𝜐)}=	−(𝜚𝜐)}=	PRON
cana-1298	189	3	(	(	PUNCT
cana-1298	189	4	𝔓i×𝔚i)−(𝔢𝔢	𝔓i×𝔚i)−(𝔢𝔢	PROPN
cana-1298	189	5	,	,	PUNCT
cana-1298	189	6	𝜚𝜐	𝜚𝜐	ADP
cana-1298	189	7	)	)	PUNCT
cana-1298	189	8	=	=	SYM
cana-1298	189	9	(	(	PUNCT
cana-1298	189	10	𝔓i×𝔚i)−[(𝔢	𝔓i×𝔚i)−[(𝔢	NOUN
cana-1298	189	11	,	,	PUNCT
cana-1298	189	12	𝜚)(𝔢	𝜚)(𝔢	PRON
cana-1298	189	13	,	,	PUNCT
cana-1298	189	14	𝜐	𝜐	NOUN
cana-1298	189	15	)	)	PUNCT
cana-1298	189	16	]	]	PUNCT
cana-1298	190	1			NUM
cana-1298	190	2	rmax{(𝔓i×𝔚i)−(𝔢	rmax{(𝔓i×𝔚i)−(𝔢	NOUN
cana-1298	190	3	,	,	PUNCT
cana-1298	190	4	𝜚	𝜚	NOUN
cana-1298	190	5	)	)	PUNCT
cana-1298	190	6	,	,	PUNCT
cana-1298	190	7	(	(	PUNCT
cana-1298	190	8	𝔓i×𝔚i)−(𝔢	𝔓i×𝔚i)−(𝔢	NOUN
cana-1298	190	9	,	,	PUNCT
cana-1298	190	10	𝜐	𝜐	NOUN
cana-1298	190	11	)	)	PUNCT
cana-1298	190	12	}	}	PUNCT
cana-1298	190	13	=	=	SYM
cana-1298	190	14	rmax{rmax{𝔓𝑖	rmax{rmax{𝔓𝑖	PROPN
cana-1298	190	15	−(𝔢	−(𝔢	NOUN
cana-1298	190	16	)	)	PUNCT
cana-1298	190	17	,	,	PUNCT
cana-1298	190	18	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	190	19	−(𝜚	−(𝜚	NOUN
cana-1298	190	20	)	)	PUNCT
cana-1298	190	21	}	}	PUNCT
cana-1298	190	22	,	,	PUNCT
cana-1298	190	23	rmax{𝔓𝑖	rmax{𝔓𝑖	PROPN
cana-1298	190	24	−(𝔢	−(𝔢	PROPN
cana-1298	190	25	)	)	PUNCT
cana-1298	190	26	,	,	PUNCT
cana-1298	190	27	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	190	28	−(𝜐	−(𝜐	NOUN
cana-1298	190	29	)	)	PUNCT
cana-1298	190	30	}	}	PUNCT
cana-1298	190	31	}	}	PUNCT
cana-1298	190	32	=	=	PUNCT
cana-1298	190	33	rmax{𝔚𝑖	rmax{𝔚𝑖	ADJ
cana-1298	190	34	−(𝜚	−(𝜚	NOUN
cana-1298	190	35	)	)	PUNCT
cana-1298	190	36	,	,	PUNCT
cana-1298	190	37	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	190	38	−(𝜐	−(𝜐	NOUN
cana-1298	190	39	)	)	PUNCT
cana-1298	190	40	}	}	PUNCT
cana-1298	190	41	,	,	PUNCT
cana-1298	190	42	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1298	190	43	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-1298	190	44	𝜚	𝜚	NOUN
cana-1298	190	45	,	,	PUNCT
cana-1298	190	46	𝜐	𝜐	PROPN
cana-1298	190	47	in	in	ADP
cana-1298	190	48	𝔒2	𝔒2	NOUN
cana-1298	190	49	.	.	PUNCT
cana-1298	191	1	hence	hence	ADV
cana-1298	191	2	𝔚	𝔚	PROPN
cana-1298	191	3	is	be	AUX
cana-1298	191	4	a	a	DET
cana-1298	191	5	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	191	6	of	of	ADP
cana-1298	191	7	𝔒2	𝔒2	NOUN
cana-1298	191	8	.	.	PUNCT
cana-1298	192	1	theorem	theorem	VERB
cana-1298	192	2	2.6	2.6	NUM
cana-1298	192	3	.	.	PUNCT
cana-1298	193	1	𝐿𝑒𝑡	𝐿𝑒𝑡	PROPN
cana-1298	193	2	𝔓1	𝔓1	PROPN
cana-1298	193	3	,	,	PUNCT
cana-1298	193	4	𝔓2	𝔓2	NOUN
cana-1298	193	5	,	,	PUNCT
cana-1298	193	6	…	…	PUNCT
cana-1298	193	7	,	,	PUNCT
cana-1298	193	8	𝔓n	𝔓n	PROPN
cana-1298	193	9	𝑏𝑒	𝑏𝑒	NOUN
cana-1298	193	10	𝑡ℎ𝑒	𝑡ℎ𝑒	NUM
cana-1298	193	11	𝔹𝕍𝕄𝕀𝔽𝕊s	𝔹𝕍𝕄𝕀𝔽𝕊	NOUN
cana-1298	193	12	of	of	ADP
cana-1298	193	13	the	the	DET
cana-1298	193	14	rings	ring	NOUN
cana-1298	193	15	𝔒1	𝔒1	NOUN
cana-1298	193	16	,	,	PUNCT
cana-1298	193	17	𝔒2	𝔒2	NOUN
cana-1298	193	18	,	,	PUNCT
cana-1298	193	19	…	…	PUNCT
cana-1298	193	20	,	,	PUNCT
cana-1298	193	21	𝔒n	𝔒n	PROPN
cana-1298	193	22	respectively	respectively	ADV
cana-1298	193	23	and	and	CCONJ
cana-1298	193	24	𝔓1	𝔓1	ADP
cana-1298	193	25	×	×	PROPN
cana-1298	193	26	𝔓2	𝔓2	NOUN
cana-1298	193	27	×	×	NOUN
cana-1298	193	28	…	…	PUNCT
cana-1298	193	29	×	×	NOUN
cana-1298	193	30	𝔓n	𝔓n	PROPN
cana-1298	193	31	be	be	VERB
cana-1298	193	32	a	a	DET
cana-1298	193	33	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	193	34	of	of	ADP
cana-1298	193	35	the	the	DET
cana-1298	193	36	ring	ring	NOUN
cana-1298	193	37	𝔒1	𝔒1	X
cana-1298	193	38	×	×	PROPN
cana-1298	193	39	𝔒2	𝔒2	NOUN
cana-1298	193	40	×	×	NOUN
cana-1298	193	41	…	…	PUNCT
cana-1298	193	42	×	×	PROPN
cana-1298	193	43	𝔒n	𝔒n	PROPN
cana-1298	193	44	.	.	PUNCT
cana-1298	194	1	then	then	ADV
cana-1298	194	2	the	the	DET
cana-1298	194	3	following	follow	VERB
cana-1298	194	4	are	be	AUX
cana-1298	194	5	true	true	ADJ
cana-1298	194	6	;	;	PUNCT
cana-1298	194	7	for	for	ADP
cana-1298	194	8	all	all	DET
cana-1298	194	9	i	i	PROPN
cana-1298	194	10	,	,	PUNCT
cana-1298	194	11	j	j	PROPN
cana-1298	194	12	,	,	PUNCT
cana-1298	194	13	k	k	PROPN
cana-1298	194	14	=	=	SYM
cana-1298	194	15	1	1	NUM
cana-1298	194	16	,	,	PUNCT
cana-1298	194	17	2	2	NUM
cana-1298	194	18	,	,	PUNCT
cana-1298	194	19	…	…	PUNCT
cana-1298	194	20	,	,	PUNCT
cana-1298	194	21	n	n	CCONJ
cana-1298	194	22	,	,	PUNCT
cana-1298	194	23	if	if	SCONJ
cana-1298	194	24	𝔓𝑘𝑗	𝔓𝑘𝑗	PROPN
cana-1298	194	25	+	+	CCONJ
cana-1298	194	26	(	(	PUNCT
cana-1298	194	27	𝔬	𝔬	NOUN
cana-1298	194	28	)	)	PUNCT
cana-1298	194	29	≥	≥	NOUN
cana-1298	195	1	𝔓𝑖𝑗	𝔓𝑖𝑗	PROPN
cana-1298	195	2	+	+	CCONJ
cana-1298	195	3	(	(	PUNCT
cana-1298	195	4	𝜚	𝜚	NOUN
cana-1298	195	5	)	)	PUNCT
cana-1298	195	6	,	,	PUNCT
cana-1298	195	7	𝔓𝑘𝑗	𝔓𝑘𝑗	PROPN
cana-1298	195	8	−	−	PROPN
cana-1298	195	9	(	(	PUNCT
cana-1298	195	10	𝔬	𝔬	NOUN
cana-1298	195	11	)	)	PUNCT
cana-1298	195	12	≤	≤	NOUN
cana-1298	196	1	𝔓𝑖𝑗	𝔓𝑖𝑗	ADV
cana-1298	196	2	−	−	PROPN
cana-1298	196	3	(	(	PUNCT
cana-1298	196	4	𝜚	𝜚	NOUN
cana-1298	196	5	)	)	PUNCT
cana-1298	196	6	,	,	PUNCT
cana-1298	196	7	for	for	ADP
cana-1298	196	8	all	all	DET
cana-1298	196	9	𝜚𝔒i	𝜚𝔒i	NOUN
cana-1298	196	10	,	,	PUNCT
cana-1298	196	11	then	then	ADV
cana-1298	196	12	𝔓i	𝔓i	PROPN
cana-1298	196	13	is	be	AUX
cana-1298	196	14	a	a	DET
cana-1298	196	15	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	196	16	of	of	ADP
cana-1298	196	17	𝔒i	𝔒i	PROPN
cana-1298	196	18	.	.	PROPN
cana-1298	196	19	where	where	SCONJ
cana-1298	196	20	𝔢	𝔢	NOUN
cana-1298	196	21	,	,	PUNCT
cana-1298	196	22	𝔬	𝔬	PROPN
cana-1298	196	23	𝑎𝑟𝑒	𝑎𝑟𝑒	PROPN
cana-1298	196	24	𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦	𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦	PROPN
cana-1298	196	25	𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠	𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠	NOUN
cana-1298	196	26	𝑜𝑓	𝑜𝑓	ADP
cana-1298	196	27	𝔒iand	𝔒iand	PROPN
cana-1298	196	28	𝔒k	𝔒k	PROPN
cana-1298	196	29	.	.	PUNCT
cana-1298	197	1	proof	proof	NOUN
cana-1298	197	2	.	.	PUNCT
cana-1298	198	1	the	the	DET
cana-1298	198	2	proof	proof	NOUN
cana-1298	198	3	follows	follow	VERB
cana-1298	198	4	from	from	ADP
cana-1298	198	5	the	the	DET
cana-1298	198	6	theorem	theorem	ADJ
cana-1298	198	7	2.5	2.5	NUM
cana-1298	198	8	.	.	PUNCT
cana-1298	199	1	theorem	theorem	VERB
cana-1298	199	2	2.7	2.7	NUM
cana-1298	199	3	.	.	PUNCT
cana-1298	200	1	𝐼𝑓	𝐼𝑓	VERB
cana-1298	200	2	𝔓	𝔓	PROPN
cana-1298	200	3	×	×	NOUN
cana-1298	200	4	𝔚	𝔚	NOUN
cana-1298	200	5	𝑖𝑠	𝑖𝑠	CCONJ
cana-1298	200	6	𝑎	𝑎	DET
cana-1298	200	7	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	200	8	of	of	ADP
cana-1298	200	9	the	the	DET
cana-1298	200	10	ring	ring	NOUN
cana-1298	200	11	𝔎1	𝔎1	PROPN
cana-1298	200	12	×	×	PROPN
cana-1298	200	13	𝔎2	𝔎2	NOUN
cana-1298	200	14	,	,	PUNCT
cana-1298	200	15	then	then	ADV
cana-1298	200	16	𝔉	𝔉	PROPN
cana-1298	200	17	=	=	SYM
cana-1298	200	18	{	{	PUNCT
cana-1298	200	19	(	(	PUNCT
cana-1298	200	20	𝔥	𝔥	NOUN
cana-1298	200	21	,	,	PUNCT
cana-1298	200	22	𝔷	𝔷	NOUN
cana-1298	200	23	)	)	PUNCT
cana-1298	200	24	∈	∈	PROPN
cana-1298	200	25	𝔎1	𝔎1	PROPN
cana-1298	200	26	×	×	PROPN
cana-1298	200	27	𝔎2	𝔎2	NOUN
cana-1298	200	28	:	:	PUNCT
cana-1298	200	29	(	(	PUNCT
cana-1298	200	30	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	200	31	+	+	NOUN
cana-1298	200	32	×	×	PROPN
cana-1298	200	33	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	200	34	+	+	PROPN
cana-1298	200	35	)	)	PUNCT
cana-1298	200	36	(	(	PUNCT
cana-1298	200	37	𝔥	𝔥	NOUN
cana-1298	200	38	,	,	PUNCT
cana-1298	200	39	𝔷	𝔷	NOUN
cana-1298	200	40	)	)	PUNCT
cana-1298	200	41	=	=	PUNCT
cana-1298	201	1	(	(	PUNCT
cana-1298	201	2	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	201	3	+	+	NOUN
cana-1298	201	4	×	×	PROPN
cana-1298	201	5	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	201	6	+	+	PROPN
cana-1298	201	7	)	)	PUNCT
cana-1298	201	8	(	(	PUNCT
cana-1298	201	9	𝔬	𝔬	NOUN
cana-1298	201	10	,	,	PUNCT
cana-1298	201	11	𝔡	𝔡	NOUN
cana-1298	201	12	)	)	PUNCT
cana-1298	201	13	𝑎𝑛𝑑	𝑎𝑛𝑑	NOUN
cana-1298	201	14	(	(	PUNCT
cana-1298	201	15	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	201	16	−	−	PROPN
cana-1298	201	17	×	×	NOUN
cana-1298	201	18	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	201	19	−)(𝔥	−)(𝔥	NOUN
cana-1298	201	20	,	,	PUNCT
cana-1298	201	21	𝔷	𝔷	NOUN
cana-1298	201	22	)	)	PUNCT
cana-1298	201	23	=	=	PUNCT
cana-1298	202	1	(	(	PUNCT
cana-1298	202	2	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	202	3	−	−	PROPN
cana-1298	203	1	×	×	NOUN
cana-1298	204	1	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	204	2	−)(𝔬	−)(𝔬	PROPN
cana-1298	204	3	,	,	PUNCT
cana-1298	204	4	𝔡	𝔡	X
cana-1298	204	5	)	)	PUNCT
cana-1298	204	6	,	,	PUNCT
cana-1298	204	7	for	for	ADP
cana-1298	204	8	all	all	DET
cana-1298	204	9	i	i	PRON
cana-1298	204	10	=	=	NOUN
cana-1298	204	11	1	1	NUM
cana-1298	204	12	,	,	PUNCT
cana-1298	204	13	2	2	NUM
cana-1298	204	14	,	,	PUNCT
cana-1298	204	15	…	…	PUNCT
cana-1298	204	16	,	,	PUNCT
cana-1298	204	17	n	n	CCONJ
cana-1298	204	18	}	}	PUNCT
cana-1298	204	19	is	be	AUX
cana-1298	204	20	either	either	CCONJ
cana-1298	204	21	empty	empty	ADJ
cana-1298	204	22	or	or	CCONJ
cana-1298	204	23	𝑎	𝑎	DET
cana-1298	204	24	subring	subre	VERB
cana-1298	204	25	𝔎1	𝔎1	NOUN
cana-1298	204	26	×	×	PROPN
cana-1298	204	27	𝔎2	𝔎2	NOUN
cana-1298	204	28	,	,	PUNCT
cana-1298	204	29	where	where	SCONJ
cana-1298	204	30	𝔬	𝔬	NOUN
cana-1298	204	31	,	,	PUNCT
cana-1298	204	32	𝔡	𝔡	VERB
cana-1298	204	33	are	be	AUX
cana-1298	204	34	first	first	ADJ
cana-1298	204	35	operation	operation	NOUN
cana-1298	204	36	identity	identity	NOUN
cana-1298	204	37	elements	element	NOUN
cana-1298	204	38	of	of	ADP
cana-1298	204	39	𝔎1	𝔎1	PROPN
cana-1298	204	40	and	and	CCONJ
cana-1298	204	41	𝔎2	𝔎2	NOUN
cana-1298	204	42	.	.	PUNCT
cana-1298	205	1	proof	proof	NOUN
cana-1298	205	2	.	.	PUNCT
cana-1298	206	1	𝐼𝑓	𝐼𝑓	NOUN
cana-1298	206	2	𝑎𝑛𝑦	𝑎𝑛𝑦	VERB
cana-1298	206	3	𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠	𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠	PROPN
cana-1298	206	4	𝑛𝑜𝑡	𝑛𝑜𝑡	PROPN
cana-1298	206	5	𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑠	𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑠	PROPN
cana-1298	206	6	𝑡ℎ𝑒	𝑡ℎ𝑒	PROPN
cana-1298	206	7	𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛	𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛	NOUN
cana-1298	206	8	,	,	PUNCT
cana-1298	206	9	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	VERB
cana-1298	206	10	𝔉	𝔉	PROPN
cana-1298	206	11	𝑖𝑠	𝑖𝑠	PROPN
cana-1298	206	12	𝑒𝑚𝑝𝑡𝑦.	𝑒𝑚𝑝𝑡𝑦.	PROPN
cana-1298	206	13	let	let	VERB
cana-1298	206	14	(	(	PUNCT
cana-1298	206	15	𝔥1	𝔥1	NOUN
cana-1298	206	16	,	,	PUNCT
cana-1298	206	17	𝔷1	𝔷1	NOUN
cana-1298	206	18	)	)	PUNCT
cana-1298	206	19	,	,	PUNCT
cana-1298	206	20	(	(	PUNCT
cana-1298	206	21	𝔥2	𝔥2	PROPN
cana-1298	206	22	,	,	PUNCT
cana-1298	206	23	𝔷2	𝔷2	PROPN
cana-1298	206	24	)	)	PUNCT
cana-1298	207	1	∈	∈	PROPN
cana-1298	207	2	𝔉.	𝔉.	PROPN
cana-1298	207	3	𝐹or	𝐹or	PROPN
cana-1298	207	4	all	all	PRON
cana-1298	207	5	i	i	PRON
cana-1298	207	6	=	=	NOUN
cana-1298	207	7	1	1	NUM
cana-1298	207	8	,	,	PUNCT
cana-1298	207	9	2	2	NUM
cana-1298	207	10	,	,	PUNCT
cana-1298	207	11	…	…	PUNCT
cana-1298	207	12	,	,	PUNCT
cana-1298	207	13	n	n	CCONJ
cana-1298	207	14	,	,	PUNCT
cana-1298	207	15	then	then	ADV
cana-1298	207	16	(	(	PUNCT
cana-1298	207	17	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	207	18	+	+	NOUN
cana-1298	207	19	×	×	PROPN
cana-1298	207	20	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	207	21	+	+	PROPN
cana-1298	207	22	)	)	PUNCT
cana-1298	208	1	[	[	X
cana-1298	208	2	(	(	PUNCT
cana-1298	208	3	𝔥1	𝔥1	NOUN
cana-1298	208	4	,	,	PUNCT
cana-1298	208	5	𝔷1	𝔷1	NOUN
cana-1298	208	6	)	)	PUNCT
cana-1298	208	7	−	−	PROPN
cana-1298	208	8	(	(	PUNCT
cana-1298	208	9	𝔥2	𝔥2	PROPN
cana-1298	208	10	,	,	PUNCT
cana-1298	208	11	𝔷2	𝔷2	PROPN
cana-1298	208	12	)	)	PUNCT
cana-1298	208	13	]	]	PUNCT
cana-1298	209	1			NUM
cana-1298	209	2	rmin{(𝔓𝑖	rmin{(𝔓𝑖	PUNCT
cana-1298	209	3	+	+	CCONJ
cana-1298	209	4	×	×	PROPN
cana-1298	209	5	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	209	6	+	+	PROPN
cana-1298	209	7	)	)	PUNCT
cana-1298	209	8	(	(	PUNCT
cana-1298	209	9	𝔥1	𝔥1	NOUN
cana-1298	209	10	,	,	PUNCT
cana-1298	209	11	𝔷1	𝔷1	NOUN
cana-1298	209	12	)	)	PUNCT
cana-1298	209	13	,	,	PUNCT
cana-1298	209	14	(	(	PUNCT
cana-1298	209	15	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	209	16	+	+	NOUN
cana-1298	209	17	×	×	PROPN
cana-1298	209	18	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	209	19	+	+	PROPN
cana-1298	209	20	)	)	PUNCT
cana-1298	209	21	(	(	PUNCT
cana-1298	209	22	𝔥2	𝔥2	PROPN
cana-1298	209	23	,	,	PUNCT
cana-1298	209	24	𝔷2	𝔷2	PROPN
cana-1298	209	25	)	)	PUNCT
cana-1298	209	26	}	}	PUNCT
cana-1298	209	27	=	=	SYM
cana-1298	209	28	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
cana-1298	209	29	{	{	PUNCT
cana-1298	209	30	(	(	PUNCT
cana-1298	209	31	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	209	32	+	+	NOUN
cana-1298	209	33	×	×	PROPN
cana-1298	209	34	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	209	35	+	+	PROPN
cana-1298	209	36	)	)	PUNCT
cana-1298	209	37	(	(	PUNCT
cana-1298	209	38	𝔬	𝔬	NOUN
cana-1298	209	39	,	,	PUNCT
cana-1298	209	40	𝔡	𝔡	NOUN
cana-1298	209	41	)	)	PUNCT
cana-1298	209	42	,	,	PUNCT
cana-1298	209	43	(	(	PUNCT
cana-1298	209	44	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	209	45	+	+	NOUN
cana-1298	209	46	×	×	PROPN
cana-1298	209	47	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	209	48	+	+	PROPN
cana-1298	209	49	)	)	PUNCT
cana-1298	209	50	(	(	PUNCT
cana-1298	209	51	𝔬	𝔬	NOUN
cana-1298	209	52	,	,	PUNCT
cana-1298	209	53	𝔡	𝔡	NOUN
cana-1298	209	54	)	)	PUNCT
cana-1298	209	55	}	}	PUNCT
cana-1298	209	56	=	=	SYM
cana-1298	210	1	(	(	PUNCT
cana-1298	210	2	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	210	3	+	+	NOUN
cana-1298	210	4	×	×	PROPN
cana-1298	210	5	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	210	6	+	+	PROPN
cana-1298	210	7	)	)	PUNCT
cana-1298	210	8	(	(	PUNCT
cana-1298	210	9	𝔬	𝔬	NOUN
cana-1298	210	10	,	,	PUNCT
cana-1298	210	11	𝔡	𝔡	NOUN
cana-1298	210	12	)	)	PUNCT
cana-1298	210	13	.	.	PUNCT
cana-1298	211	1	thus	thus	ADV
cana-1298	211	2	(	(	PUNCT
cana-1298	211	3	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	211	4	+	+	NOUN
cana-1298	211	5	×	×	PROPN
cana-1298	211	6	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	211	7	+	+	PROPN
cana-1298	211	8	)	)	PUNCT
cana-1298	211	9	[	[	X
cana-1298	211	10	(	(	PUNCT
cana-1298	211	11	𝔥1	𝔥1	NOUN
cana-1298	211	12	,	,	PUNCT
cana-1298	211	13	𝔷1	𝔷1	NOUN
cana-1298	211	14	)	)	PUNCT
cana-1298	211	15	−	−	PROPN
cana-1298	212	1	(	(	PUNCT
cana-1298	213	1	𝔥2	𝔥2	PROPN
cana-1298	213	2	,	,	PUNCT
cana-1298	213	3	𝔷2	𝔷2	PROPN
cana-1298	213	4	)	)	PUNCT
cana-1298	213	5	]	]	PUNCT
cana-1298	214	1	=	=	PUNCT
cana-1298	214	2	(	(	PUNCT
cana-1298	214	3	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	214	4	+	+	NOUN
cana-1298	214	5	×	×	PROPN
cana-1298	214	6	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	214	7	+	+	PROPN
cana-1298	214	8	)	)	PUNCT
cana-1298	214	9	(	(	PUNCT
cana-1298	214	10	𝔬	𝔬	NOUN
cana-1298	214	11	,	,	PUNCT
cana-1298	214	12	𝔡	𝔡	NOUN
cana-1298	214	13	)	)	PUNCT
cana-1298	214	14	,	,	PUNCT
cana-1298	214	15	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1298	214	16	𝑎𝑙𝑙	𝑎𝑙𝑙	X
cana-1298	214	17	(	(	PUNCT
cana-1298	214	18	𝔥1	𝔥1	NOUN
cana-1298	214	19	,	,	PUNCT
cana-1298	214	20	𝔷1	𝔷1	NOUN
cana-1298	214	21	)	)	PUNCT
cana-1298	214	22	,	,	PUNCT
cana-1298	214	23	(	(	PUNCT
cana-1298	214	24	𝔥2	𝔥2	PROPN
cana-1298	214	25	,	,	PUNCT
cana-1298	214	26	𝔷2	𝔷2	PROPN
cana-1298	214	27	)	)	PUNCT
cana-1298	214	28	∈	∈	PROPN
cana-1298	214	29	𝔉.	𝔉.	PROPN
cana-1298	214	30	and	and	CCONJ
cana-1298	214	31	(	(	PUNCT
cana-1298	214	32	𝔓𝑖	𝔓𝑖	PROPN
cana-1298	214	33	−	−	PROPN
cana-1298	214	34	×	×	NOUN
cana-1298	214	35	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	214	36	−)[(𝔥1	−)[(𝔥1	NOUN
cana-1298	214	37	,	,	PUNCT
cana-1298	214	38	𝔷1	𝔷1	NOUN
cana-1298	214	39	)	)	PUNCT
cana-1298	214	40	−	−	PROPN
cana-1298	215	1	(	(	PUNCT
cana-1298	215	2	𝔥2	𝔥2	PROPN
cana-1298	215	3	,	,	PUNCT
cana-1298	215	4	𝔷2	𝔷2	PROPN
cana-1298	215	5	)	)	PUNCT
cana-1298	215	6	]	]	PUNCT
cana-1298	216	1			NOUN
cana-1298	216	2	rmax{(𝔓𝑖	rmax{(𝔓𝑖	NOUN
cana-1298	216	3	−	−	PROPN
cana-1298	216	4	×	×	NOUN
cana-1298	216	5	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	216	6	−)(𝔥1	−)(𝔥1	PROPN
cana-1298	216	7	,	,	PUNCT
cana-1298	216	8	𝔷1	𝔷1	NOUN
cana-1298	216	9	)	)	PUNCT
cana-1298	216	10	,	,	PUNCT
cana-1298	216	11	(	(	PUNCT
cana-1298	216	12	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	216	13	−	−	PROPN
cana-1298	217	1	×	×	NOUN
cana-1298	218	1	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	218	2	−)(𝔥2	−)(𝔥2	PROPN
cana-1298	218	3	,	,	PUNCT
cana-1298	218	4	𝔷2	𝔷2	ADV
cana-1298	218	5	)	)	PUNCT
cana-1298	218	6	}	}	PUNCT
cana-1298	219	1	=	=	PUNCT
cana-1298	219	2	𝑟𝑚𝑎𝑥	𝑟𝑚𝑎𝑥	NOUN
cana-1298	219	3	{	{	PUNCT
cana-1298	219	4	(	(	PUNCT
cana-1298	219	5	𝔓𝑖	𝔓𝑖	PROPN
cana-1298	219	6	−	−	PROPN
cana-1298	219	7	×	×	NOUN
cana-1298	219	8	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	219	9	−)(𝔬	−)(𝔬	PROPN
cana-1298	219	10	,	,	PUNCT
cana-1298	219	11	𝔡	𝔡	X
cana-1298	219	12	)	)	PUNCT
cana-1298	219	13	,	,	PUNCT
cana-1298	219	14	(	(	PUNCT
cana-1298	219	15	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	219	16	−	−	PROPN
cana-1298	219	17	×	×	NOUN
cana-1298	219	18	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	219	19	−)(𝔬	−)(𝔬	PROPN
cana-1298	219	20	,	,	PUNCT
cana-1298	219	21	𝔡	𝔡	X
cana-1298	219	22	)	)	PUNCT
cana-1298	219	23	}	}	PUNCT
cana-1298	219	24	=	=	SYM
cana-1298	219	25	(	(	PUNCT
cana-1298	219	26	𝔓𝑖	𝔓𝑖	PROPN
cana-1298	219	27	−	−	PROPN
cana-1298	219	28	×	×	NOUN
cana-1298	219	29	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	219	30	−)(𝔬	−)(𝔬	PROPN
cana-1298	219	31	,	,	PUNCT
cana-1298	219	32	𝔡	𝔡	X
cana-1298	219	33	)	)	PUNCT
cana-1298	219	34	.	.	PUNCT
cana-1298	220	1	thus	thus	ADV
cana-1298	220	2	(	(	PUNCT
cana-1298	220	3	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	220	4	−	−	PROPN
cana-1298	220	5	×	×	NOUN
cana-1298	220	6	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	220	7	−)[(𝔥1	−)[(𝔥1	NOUN
cana-1298	220	8	,	,	PUNCT
cana-1298	220	9	𝔷1	𝔷1	NOUN
cana-1298	220	10	)	)	PUNCT
cana-1298	220	11	−	−	PROPN
cana-1298	220	12	(	(	PUNCT
cana-1298	220	13	𝔥2	𝔥2	PROPN
cana-1298	220	14	,	,	PUNCT
cana-1298	220	15	𝔷2	𝔷2	PROPN
cana-1298	220	16	)	)	PUNCT
cana-1298	220	17	]	]	PUNCT
cana-1298	221	1	=	=	PUNCT
cana-1298	221	2	(	(	PUNCT
cana-1298	221	3	𝔓𝑖	𝔓𝑖	PROPN
cana-1298	221	4	−	−	PROPN
cana-1298	222	1	×	×	NOUN
cana-1298	223	1	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	223	2	−)(𝔬	−)(𝔬	PROPN
cana-1298	223	3	,	,	PUNCT
cana-1298	223	4	𝔡	𝔡	X
cana-1298	223	5	)	)	PUNCT
cana-1298	223	6	,	,	PUNCT
cana-1298	223	7	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1298	223	8	𝑎𝑙𝑙	𝑎𝑙𝑙	X
cana-1298	223	9	(	(	PUNCT
cana-1298	223	10	𝔥1	𝔥1	NOUN
cana-1298	223	11	,	,	PUNCT
cana-1298	223	12	𝔷1	𝔷1	NOUN
cana-1298	223	13	)	)	PUNCT
cana-1298	223	14	,	,	PUNCT
cana-1298	223	15	(	(	PUNCT
cana-1298	223	16	𝔥2	𝔥2	PROPN
cana-1298	223	17	,	,	PUNCT
cana-1298	223	18	𝔷2	𝔷2	PROPN
cana-1298	223	19	)	)	PUNCT
cana-1298	223	20	∈	∈	PROPN
cana-1298	223	21	𝔉.	𝔉.	PROPN
cana-1298	223	22	therefore	therefore	ADV
cana-1298	223	23	(	(	PUNCT
cana-1298	223	24	𝔥1	𝔥1	NOUN
cana-1298	223	25	,	,	PUNCT
cana-1298	223	26	𝔷1	𝔷1	NOUN
cana-1298	223	27	)	)	PUNCT
cana-1298	223	28	−	−	PROPN
cana-1298	224	1	(	(	PUNCT
cana-1298	224	2	𝔥2	𝔥2	PROPN
cana-1298	224	3	,	,	PUNCT
cana-1298	224	4	𝔷2	𝔷2	PROPN
cana-1298	224	5	)	)	PUNCT
cana-1298	224	6	∈	∈	PROPN
cana-1298	224	7	𝔉.	𝔉.	PROPN
cana-1298	224	8	𝐴𝑙𝑠𝑜	𝐴𝑙𝑠𝑜	PROPN
cana-1298	224	9	(	(	PUNCT
cana-1298	224	10	𝔓𝑖	𝔓𝑖	PROPN
cana-1298	224	11	+	+	NOUN
cana-1298	224	12	×	×	PROPN
cana-1298	224	13	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	224	14	+	+	PROPN
cana-1298	224	15	)	)	PUNCT
cana-1298	224	16	[	[	X
cana-1298	224	17	(	(	PUNCT
cana-1298	224	18	𝔥1	𝔥1	PROPN
cana-1298	224	19	,	,	PUNCT
cana-1298	224	20	𝔷1)(𝔥2	𝔷1)(𝔥2	NOUN
cana-1298	224	21	,	,	PUNCT
cana-1298	224	22	𝔷2	𝔷2	PROPN
cana-1298	224	23	)	)	PUNCT
cana-1298	224	24	]	]	PUNCT
cana-1298	224	25			NUM
cana-1298	224	26	rmin	rmin	VERB
cana-1298	224	27	{	{	PUNCT
cana-1298	224	28	(	(	PUNCT
cana-1298	224	29	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	224	30	+	+	NOUN
cana-1298	224	31	×	×	PROPN
cana-1298	224	32	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	224	33	+	+	PROPN
cana-1298	224	34	)	)	PUNCT
cana-1298	224	35	(	(	PUNCT
cana-1298	224	36	𝔥1	𝔥1	NOUN
cana-1298	224	37	,	,	PUNCT
cana-1298	224	38	𝔷1	𝔷1	NOUN
cana-1298	224	39	)	)	PUNCT
cana-1298	224	40	,	,	PUNCT
cana-1298	224	41	(	(	PUNCT
cana-1298	224	42	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	224	43	+	+	NOUN
cana-1298	224	44	×	×	PROPN
cana-1298	224	45	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	224	46	+	+	PROPN
cana-1298	224	47	)	)	PUNCT
cana-1298	224	48	(	(	PUNCT
cana-1298	224	49	𝔥2	𝔥2	PROPN
cana-1298	224	50	,	,	PUNCT
cana-1298	224	51	𝔷2	𝔷2	PROPN
cana-1298	224	52	)	)	PUNCT
cana-1298	224	53	}	}	PUNCT
cana-1298	224	54	=	=	SYM
cana-1298	224	55	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
cana-1298	224	56	{	{	PUNCT
cana-1298	224	57	(	(	PUNCT
cana-1298	224	58	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	224	59	+	+	NOUN
cana-1298	224	60	×	×	PROPN
cana-1298	224	61	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	224	62	+	+	PROPN
cana-1298	224	63	)	)	PUNCT
cana-1298	224	64	(	(	PUNCT
cana-1298	224	65	𝔬	𝔬	NOUN
cana-1298	224	66	,	,	PUNCT
cana-1298	224	67	𝔡	𝔡	NOUN
cana-1298	224	68	)	)	PUNCT
cana-1298	224	69	,	,	PUNCT
cana-1298	224	70	(	(	PUNCT
cana-1298	225	1	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	225	2	+	+	NOUN
cana-1298	225	3	×	×	PROPN
cana-1298	225	4	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	225	5	+	+	PROPN
cana-1298	225	6	)	)	PUNCT
cana-1298	225	7	(	(	PUNCT
cana-1298	225	8	𝔬	𝔬	NOUN
cana-1298	225	9	,	,	PUNCT
cana-1298	225	10	𝔡	𝔡	NOUN
cana-1298	225	11	)	)	PUNCT
cana-1298	225	12	}	}	PUNCT
cana-1298	225	13	=	=	SYM
cana-1298	225	14	(	(	PUNCT
cana-1298	225	15	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	225	16	+	+	NOUN
cana-1298	225	17	×	×	PROPN
cana-1298	225	18	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	225	19	+	+	PROPN
cana-1298	225	20	)	)	PUNCT
cana-1298	225	21	(	(	PUNCT
cana-1298	225	22	𝔬	𝔬	NOUN
cana-1298	225	23	,	,	PUNCT
cana-1298	225	24	𝔡	𝔡	NOUN
cana-1298	225	25	)	)	PUNCT
cana-1298	225	26	.	.	PUNCT
cana-1298	226	1	thus	thus	ADV
cana-1298	226	2	(	(	PUNCT
cana-1298	226	3	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	226	4	+	+	NOUN
cana-1298	226	5	×	×	PROPN
cana-1298	226	6	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	226	7	+	+	PROPN
cana-1298	226	8	)	)	PUNCT
cana-1298	226	9	[	[	X
cana-1298	226	10	(	(	PUNCT
cana-1298	226	11	𝔥1	𝔥1	PROPN
cana-1298	226	12	,	,	PUNCT
cana-1298	226	13	𝔷1)(𝔥2	𝔷1)(𝔥2	NOUN
cana-1298	226	14	,	,	PUNCT
cana-1298	226	15	𝔷2	𝔷2	PROPN
cana-1298	226	16	)	)	PUNCT
cana-1298	226	17	]	]	PUNCT
cana-1298	227	1	=	=	PUNCT
cana-1298	227	2	(	(	PUNCT
cana-1298	227	3	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	227	4	+	+	NOUN
cana-1298	227	5	×	×	PROPN
cana-1298	227	6	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	227	7	+	+	PROPN
cana-1298	227	8	)	)	PUNCT
cana-1298	227	9	(	(	PUNCT
cana-1298	227	10	𝔬	𝔬	NOUN
cana-1298	227	11	,	,	PUNCT
cana-1298	227	12	𝔡	𝔡	NOUN
cana-1298	227	13	)	)	PUNCT
cana-1298	227	14	,	,	PUNCT
cana-1298	227	15	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1298	227	16	𝑎𝑙𝑙	𝑎𝑙𝑙	X
cana-1298	227	17	(	(	PUNCT
cana-1298	227	18	𝔥1	𝔥1	NOUN
cana-1298	227	19	,	,	PUNCT
cana-1298	227	20	𝔷1	𝔷1	NOUN
cana-1298	227	21	)	)	PUNCT
cana-1298	227	22	,	,	PUNCT
cana-1298	227	23	(	(	PUNCT
cana-1298	227	24	𝔥2	𝔥2	PROPN
cana-1298	227	25	,	,	PUNCT
cana-1298	227	26	𝔷2	𝔷2	PROPN
cana-1298	227	27	)	)	PUNCT
cana-1298	227	28	∈	∈	PROPN
cana-1298	227	29	𝔉.	𝔉.	PROPN
cana-1298	227	30	and	and	CCONJ
cana-1298	227	31	(	(	PUNCT
cana-1298	227	32	𝔓𝑖	𝔓𝑖	PROPN
cana-1298	227	33	−	−	PROPN
cana-1298	227	34	×	×	NOUN
cana-1298	227	35	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	227	36	−)[(𝔥1	−)[(𝔥1	NOUN
cana-1298	227	37	,	,	PUNCT
cana-1298	227	38	𝔷1)(𝔥2	𝔷1)(𝔥2	NOUN
cana-1298	227	39	,	,	PUNCT
cana-1298	227	40	𝔷2	𝔷2	PROPN
cana-1298	227	41	)	)	PUNCT
cana-1298	227	42	]	]	PUNCT
cana-1298	227	43			NOUN
cana-1298	227	44	rmax{(𝔓𝑖	rmax{(𝔓𝑖	NOUN
cana-1298	227	45	−	−	PROPN
cana-1298	227	46	×	×	NOUN
cana-1298	227	47	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	227	48	−)(𝔥1	−)(𝔥1	PROPN
cana-1298	227	49	,	,	PUNCT
cana-1298	227	50	𝔷1	𝔷1	NOUN
cana-1298	227	51	)	)	PUNCT
cana-1298	227	52	,	,	PUNCT
cana-1298	227	53	(	(	PUNCT
cana-1298	227	54	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	227	55	−	−	PROPN
cana-1298	227	56	×	×	NOUN
cana-1298	227	57	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	227	58	−)(𝔥2	−)(𝔥2	PROPN
cana-1298	227	59	,	,	PUNCT
cana-1298	227	60	𝔷2	𝔷2	ADV
cana-1298	227	61	)	)	PUNCT
cana-1298	227	62	}	}	PUNCT
cana-1298	227	63	=	=	PUNCT
cana-1298	228	1	𝑟𝑚𝑎𝑥	𝑟𝑚𝑎𝑥	NOUN
cana-1298	228	2	{	{	PUNCT
cana-1298	228	3	(	(	PUNCT
cana-1298	228	4	𝔓𝑖	𝔓𝑖	PROPN
cana-1298	228	5	−	−	PROPN
cana-1298	228	6	×	×	NOUN
cana-1298	228	7	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	228	8	−)(𝔬	−)(𝔬	PROPN
cana-1298	228	9	,	,	PUNCT
cana-1298	228	10	𝔡	𝔡	X
cana-1298	228	11	)	)	PUNCT
cana-1298	228	12	,	,	PUNCT
cana-1298	228	13	(	(	PUNCT
cana-1298	228	14	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	228	15	−	−	PROPN
cana-1298	228	16	×	×	NOUN
cana-1298	228	17	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	228	18	−)(𝔬	−)(𝔬	PROPN
cana-1298	228	19	,	,	PUNCT
cana-1298	228	20	𝔡	𝔡	X
cana-1298	228	21	)	)	PUNCT
cana-1298	228	22	}	}	PUNCT
cana-1298	228	23	=	=	SYM
cana-1298	228	24	(	(	PUNCT
cana-1298	228	25	𝔓𝑖	𝔓𝑖	PROPN
cana-1298	228	26	−	−	PROPN
cana-1298	228	27	×	×	NOUN
cana-1298	228	28	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	228	29	−)(𝔬	−)(𝔬	PROPN
cana-1298	228	30	,	,	PUNCT
cana-1298	228	31	𝔡	𝔡	X
cana-1298	228	32	)	)	PUNCT
cana-1298	228	33	.	.	PUNCT
cana-1298	229	1	thus	thus	ADV
cana-1298	229	2	(	(	PUNCT
cana-1298	229	3	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	229	4	−	−	PROPN
cana-1298	229	5	×	×	NOUN
cana-1298	229	6	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	229	7	−	−	PROPN
cana-1298	229	8	)	)	PUNCT
cana-1298	230	1	[	[	X
cana-1298	230	2	(	(	PUNCT
cana-1298	230	3	𝔥1	𝔥1	NOUN
cana-1298	230	4	,	,	PUNCT
cana-1298	230	5	𝔷1)(𝔥2	𝔷1)(𝔥2	NOUN
cana-1298	230	6	,	,	PUNCT
cana-1298	230	7	𝔷2	𝔷2	PROPN
cana-1298	230	8	)	)	PUNCT
cana-1298	230	9	]	]	PUNCT
cana-1298	231	1	=	=	PUNCT
cana-1298	231	2	(	(	PUNCT
cana-1298	231	3	𝔓𝑖	𝔓𝑖	PROPN
cana-1298	231	4	−	−	PROPN
cana-1298	232	1	×	×	NOUN
cana-1298	233	1	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	233	2	−)(𝔬	−)(𝔬	PROPN
cana-1298	233	3	,	,	PUNCT
cana-1298	233	4	𝔡	𝔡	X
cana-1298	233	5	)	)	PUNCT
cana-1298	233	6	,	,	PUNCT
cana-1298	233	7	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1298	233	8	𝑎𝑙𝑙	𝑎𝑙𝑙	X
cana-1298	233	9	(	(	PUNCT
cana-1298	233	10	𝔥1	𝔥1	NOUN
cana-1298	233	11	,	,	PUNCT
cana-1298	233	12	𝔷1	𝔷1	NOUN
cana-1298	233	13	)	)	PUNCT
cana-1298	233	14	,	,	PUNCT
cana-1298	233	15	(	(	PUNCT
cana-1298	233	16	𝔥2	𝔥2	PROPN
cana-1298	233	17	,	,	PUNCT
cana-1298	233	18	𝔷2	𝔷2	PROPN
cana-1298	233	19	)	)	PUNCT
cana-1298	233	20	∈	∈	PROPN
cana-1298	233	21	𝔉.	𝔉.	PROPN
cana-1298	233	22	therefore	therefore	ADV
cana-1298	233	23	(	(	PUNCT
cana-1298	233	24	𝔥1	𝔥1	PROPN
cana-1298	233	25	,	,	PUNCT
cana-1298	233	26	𝔷1)(𝔥2	𝔷1)(𝔥2	NOUN
cana-1298	233	27	,	,	PUNCT
cana-1298	233	28	𝔷2	𝔷2	PROPN
cana-1298	233	29	)	)	PUNCT
cana-1298	233	30	∈	∈	PROPN
cana-1298	233	31	𝔉.	𝔉.	PROPN
cana-1298	233	32	𝐻𝑒𝑛𝑐𝑒	𝐻𝑒𝑛𝑐𝑒	PROPN
cana-1298	233	33	𝔉	𝔉	PROPN
cana-1298	233	34	𝑖𝑠	𝑖𝑠	CCONJ
cana-1298	233	35	𝑎	𝑎	DET
cana-1298	233	36	𝑠𝑢𝑏𝑟𝑖𝑛𝑔	𝑠𝑢𝑏𝑟𝑖𝑛𝑔	NOUN
cana-1298	233	37	𝑜𝑓	𝑜𝑓	ADP
cana-1298	233	38	𝔎1	𝔎1	PROPN
cana-1298	233	39	×	×	PROPN
cana-1298	233	40	𝔎2	𝔎2	NOUN
cana-1298	233	41	.	.	PUNCT
cana-1298	233	42	theorem	theorem	VERB
cana-1298	233	43	2.8	2.8	NUM
cana-1298	233	44	.	.	PUNCT
cana-1298	234	1	𝐼𝑓	𝐼𝑓	VERB
cana-1298	234	2	𝔓	𝔓	PROPN
cana-1298	234	3	×	×	NOUN
cana-1298	234	4	𝔚	𝔚	NOUN
cana-1298	234	5	𝑖𝑠	𝑖𝑠	CCONJ
cana-1298	234	6	𝑎	𝑎	DET
cana-1298	234	7	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	234	8	of	of	ADP
cana-1298	234	9	the	the	DET
cana-1298	234	10	ring	ring	NOUN
cana-1298	234	11	𝔊1	𝔊1	VERB
cana-1298	234	12	×	×	PROPN
cana-1298	234	13	𝔊2	𝔊2	NOUN
cana-1298	234	14	,	,	PUNCT
cana-1298	234	15	then	then	ADV
cana-1298	234	16	𝕐	𝕐	PROPN
cana-1298	234	17	=	=	SYM
cana-1298	234	18	{	{	PUNCT
cana-1298	234	19	(	(	PUNCT
cana-1298	234	20	𝔥	𝔥	NOUN
cana-1298	234	21	,	,	PUNCT
cana-1298	234	22	𝔷	𝔷	NOUN
cana-1298	234	23	)	)	PUNCT
cana-1298	234	24	∈	∈	PROPN
cana-1298	234	25	𝔊1	𝔊1	NOUN
cana-1298	234	26	×	×	PROPN
cana-1298	234	27	𝔊2	𝔊2	PROPN
cana-1298	234	28	:	:	PUNCT
cana-1298	234	29	(	(	PUNCT
cana-1298	234	30	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	234	31	+	+	NOUN
cana-1298	234	32	×	×	PROPN
cana-1298	234	33	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	234	34	+	+	PROPN
cana-1298	234	35	)	)	PUNCT
cana-1298	234	36	(	(	PUNCT
cana-1298	234	37	𝔥	𝔥	NOUN
cana-1298	234	38	,	,	PUNCT
cana-1298	234	39	𝔷	𝔷	NOUN
cana-1298	234	40	)	)	PUNCT
cana-1298	234	41	=	=	PUNCT
cana-1298	235	1	[	[	X
cana-1298	235	2	1	1	NUM
cana-1298	235	3	]	]	X
cana-1298	235	4	𝑎𝑛𝑑	𝑎𝑛𝑑	X
cana-1298	235	5	(	(	PUNCT
cana-1298	235	6	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	235	7	−	−	PROPN
cana-1298	236	1	×	×	NOUN
cana-1298	236	2	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	236	3	−)(𝔥	−)(𝔥	NOUN
cana-1298	236	4	,	,	PUNCT
cana-1298	236	5	𝔷	𝔷	NOUN
cana-1298	236	6	)	)	PUNCT
cana-1298	236	7	=	=	PUNCT
cana-1298	237	1	[	[	X
cana-1298	237	2	−1	−1	NOUN
cana-1298	237	3	]	]	X
cana-1298	237	4	,	,	PUNCT
cana-1298	237	5	for	for	ADP
cana-1298	237	6	all	all	DET
cana-1298	237	7	i	i	PRON
cana-1298	237	8	=	=	NOUN
cana-1298	237	9	1	1	NUM
cana-1298	237	10	,	,	PUNCT
cana-1298	237	11	2	2	NUM
cana-1298	237	12	,	,	PUNCT
cana-1298	237	13	…	…	PUNCT
cana-1298	237	14	,	,	PUNCT
cana-1298	237	15	n	n	CCONJ
cana-1298	237	16	}	}	PUNCT
cana-1298	237	17	is	be	AUX
cana-1298	237	18	either	either	CCONJ
cana-1298	237	19	empty	empty	ADJ
cana-1298	237	20	or	or	CCONJ
cana-1298	237	21	𝑎	𝑎	DET
cana-1298	237	22	subring	subring	NOUN
cana-1298	237	23	𝔊1	𝔊1	NOUN
cana-1298	237	24	×	×	PROPN
cana-1298	237	25	𝔊2	𝔊2	NOUN
cana-1298	237	26	.	.	PUNCT
cana-1298	238	1	proof	proof	NOUN
cana-1298	238	2	.	.	PUNCT
cana-1298	239	1	𝐼𝑓	𝐼𝑓	NOUN
cana-1298	239	2	𝑎𝑛𝑦	𝑎𝑛𝑦	VERB
cana-1298	239	3	𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠	𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠	PROPN
cana-1298	239	4	𝑛𝑜𝑡	𝑛𝑜𝑡	PROPN
cana-1298	239	5	𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑠	𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑠	PROPN
cana-1298	239	6	𝑡ℎ𝑒	𝑡ℎ𝑒	PROPN
cana-1298	239	7	𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛	𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛	NOUN
cana-1298	239	8	,	,	PUNCT
cana-1298	239	9	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	VERB
cana-1298	239	10	𝕐	𝕐	PROPN
cana-1298	239	11	𝑖𝑠	𝑖𝑠	ADV
cana-1298	239	12	𝑒𝑚𝑝𝑡𝑦.	𝑒𝑚𝑝𝑡𝑦.	ADJ
cana-1298	239	13	let	let	NOUN
cana-1298	239	14	(	(	PUNCT
cana-1298	239	15	𝔥1	𝔥1	NOUN
cana-1298	239	16	,	,	PUNCT
cana-1298	239	17	𝔷1	𝔷1	NOUN
cana-1298	239	18	)	)	PUNCT
cana-1298	239	19	,	,	PUNCT
cana-1298	239	20	(	(	PUNCT
cana-1298	239	21	𝔥2	𝔥2	PROPN
cana-1298	239	22	,	,	PUNCT
cana-1298	239	23	𝔷2	𝔷2	PROPN
cana-1298	239	24	)	)	PUNCT
cana-1298	240	1	∈	∈	PROPN
cana-1298	240	2	𝕐.	𝕐.	NOUN
cana-1298	240	3	𝐹or	𝐹or	PROPN
cana-1298	240	4	all	all	PRON
cana-1298	241	1	i	i	PRON
cana-1298	241	2	=	=	NOUN
cana-1298	241	3	1	1	NUM
cana-1298	241	4	,	,	PUNCT
cana-1298	241	5	2	2	NUM
cana-1298	241	6	,	,	PUNCT
cana-1298	241	7	…	…	PUNCT
cana-1298	241	8	,	,	PUNCT
cana-1298	241	9	n	n	CCONJ
cana-1298	241	10	,	,	PUNCT
cana-1298	241	11	then	then	ADV
cana-1298	241	12	(	(	PUNCT
cana-1298	241	13	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	241	14	+	+	NOUN
cana-1298	241	15	×	×	PROPN
cana-1298	241	16	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	241	17	+	+	PROPN
cana-1298	241	18	)	)	PUNCT
cana-1298	242	1	[	[	X
cana-1298	242	2	(	(	PUNCT
cana-1298	242	3	𝔥1	𝔥1	NOUN
cana-1298	242	4	,	,	PUNCT
cana-1298	242	5	𝔷1	𝔷1	NOUN
cana-1298	242	6	)	)	PUNCT
cana-1298	242	7	−	−	PROPN
cana-1298	242	8	(	(	PUNCT
cana-1298	242	9	𝔥2	𝔥2	PROPN
cana-1298	242	10	,	,	PUNCT
cana-1298	242	11	𝔷2	𝔷2	PROPN
cana-1298	242	12	)	)	PUNCT
cana-1298	242	13	]	]	PUNCT
cana-1298	243	1			NUM
cana-1298	243	2	rmin{(𝔓𝑖	rmin{(𝔓𝑖	PUNCT
cana-1298	243	3	+	+	CCONJ
cana-1298	243	4	×	×	PROPN
cana-1298	243	5	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	243	6	+	+	PROPN
cana-1298	243	7	)	)	PUNCT
cana-1298	243	8	(	(	PUNCT
cana-1298	243	9	𝔥1	𝔥1	NOUN
cana-1298	243	10	,	,	PUNCT
cana-1298	243	11	𝔷1	𝔷1	NOUN
cana-1298	243	12	)	)	PUNCT
cana-1298	243	13	,	,	PUNCT
cana-1298	243	14	(	(	PUNCT
cana-1298	243	15	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	243	16	+	+	NOUN
cana-1298	243	17	×	×	PROPN
cana-1298	243	18	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	243	19	+	+	PROPN
cana-1298	243	20	)	)	PUNCT
cana-1298	243	21	(	(	PUNCT
cana-1298	243	22	𝔥2	𝔥2	PROPN
cana-1298	243	23	,	,	PUNCT
cana-1298	243	24	𝔷2	𝔷2	PROPN
cana-1298	243	25	)	)	PUNCT
cana-1298	243	26	}	}	PUNCT
cana-1298	243	27	=	=	SYM
cana-1298	243	28	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
cana-1298	243	29	{	{	PUNCT
cana-1298	244	1	[	[	X
cana-1298	244	2	1	1	NUM
cana-1298	244	3	]	]	PUNCT
cana-1298	244	4	,	,	PUNCT
cana-1298	245	1	[	[	X
cana-1298	245	2	1	1	NUM
cana-1298	245	3	]	]	PUNCT
cana-1298	245	4	}	}	PUNCT
cana-1298	245	5	=	=	PUNCT
cana-1298	246	1	[	[	X
cana-1298	246	2	1	1	NUM
cana-1298	246	3	]	]	PUNCT
cana-1298	246	4	.	.	PUNCT
cana-1298	247	1	thus	thus	ADV
cana-1298	247	2	(	(	PUNCT
cana-1298	247	3	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	247	4	+	+	NOUN
cana-1298	247	5	×	×	PROPN
cana-1298	247	6	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	247	7	+	+	PROPN
cana-1298	247	8	)	)	PUNCT
cana-1298	247	9	[	[	X
cana-1298	247	10	(	(	PUNCT
cana-1298	247	11	𝔥1	𝔥1	NOUN
cana-1298	247	12	,	,	PUNCT
cana-1298	247	13	𝔷1	𝔷1	NOUN
cana-1298	247	14	)	)	PUNCT
cana-1298	247	15	−	−	PROPN
cana-1298	248	1	(	(	PUNCT
cana-1298	248	2	𝔥2	𝔥2	PROPN
cana-1298	248	3	,	,	PUNCT
cana-1298	248	4	𝔷2	𝔷2	PROPN
cana-1298	248	5	)	)	PUNCT
cana-1298	248	6	]	]	PUNCT
cana-1298	249	1	=	=	PUNCT
cana-1298	250	1	[	[	X
cana-1298	250	2	1	1	NUM
cana-1298	250	3	]	]	PUNCT
cana-1298	250	4	,	,	PUNCT
cana-1298	250	5	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1298	250	6	𝑎𝑙𝑙	𝑎𝑙𝑙	X
cana-1298	250	7	(	(	PUNCT
cana-1298	250	8	𝔥1	𝔥1	NOUN
cana-1298	250	9	,	,	PUNCT
cana-1298	250	10	𝔷1	𝔷1	NOUN
cana-1298	250	11	)	)	PUNCT
cana-1298	250	12	,	,	PUNCT
cana-1298	250	13	(	(	PUNCT
cana-1298	250	14	𝔥2	𝔥2	PROPN
cana-1298	250	15	,	,	PUNCT
cana-1298	250	16	𝔷2	𝔷2	PROPN
cana-1298	250	17	)	)	PUNCT
cana-1298	250	18	∈	∈	PROPN
cana-1298	250	19	𝕐.	𝕐.	NOUN
cana-1298	250	20	and	and	CCONJ
cana-1298	250	21	(	(	PUNCT
cana-1298	250	22	𝔓𝑖	𝔓𝑖	PROPN
cana-1298	250	23	−	−	PROPN
cana-1298	251	1	×	×	NOUN
cana-1298	251	2	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	251	3	−)[(𝔥1	−)[(𝔥1	NOUN
cana-1298	251	4	,	,	PUNCT
cana-1298	251	5	𝔷1	𝔷1	NOUN
cana-1298	251	6	)	)	PUNCT
cana-1298	251	7	−	−	PROPN
cana-1298	251	8	(	(	PUNCT
cana-1298	251	9	𝔥2	𝔥2	PROPN
cana-1298	251	10	,	,	PUNCT
cana-1298	251	11	𝔷2	𝔷2	PROPN
cana-1298	251	12	)	)	PUNCT
cana-1298	251	13	]	]	PUNCT
cana-1298	252	1			NOUN
cana-1298	252	2	rmax{(𝔓𝑖	rmax{(𝔓𝑖	NOUN
cana-1298	252	3	−	−	PROPN
cana-1298	252	4	×	×	NOUN
cana-1298	252	5	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	252	6	−)(𝔥1	−)(𝔥1	PROPN
cana-1298	252	7	,	,	PUNCT
cana-1298	252	8	𝔷1	𝔷1	NOUN
cana-1298	252	9	)	)	PUNCT
cana-1298	252	10	,	,	PUNCT
cana-1298	252	11	(	(	PUNCT
cana-1298	252	12	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	252	13	−	−	PROPN
cana-1298	253	1	×	×	NOUN
cana-1298	254	1	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	254	2	−)(𝔥2	−)(𝔥2	PROPN
cana-1298	254	3	,	,	PUNCT
cana-1298	254	4	𝔷2	𝔷2	ADV
cana-1298	254	5	)	)	PUNCT
cana-1298	254	6	}	}	PUNCT
cana-1298	255	1	=	=	SYM
cana-1298	255	2	𝑟𝑚𝑎𝑥	𝑟𝑚𝑎𝑥	NOUN
cana-1298	255	3	{	{	PUNCT
cana-1298	255	4	[	[	X
cana-1298	255	5	−1	−1	NOUN
cana-1298	255	6	]	]	X
cana-1298	255	7	,	,	PUNCT
cana-1298	255	8	[	[	X
cana-1298	255	9	−1	−1	NOUN
cana-1298	255	10	]	]	X
cana-1298	255	11	}	}	PUNCT
cana-1298	255	12	=	=	PUNCT
cana-1298	256	1	[	[	X
cana-1298	256	2	−1	−1	NOUN
cana-1298	256	3	]	]	PUNCT
cana-1298	256	4	.	.	PUNCT
cana-1298	257	1	thus	thus	ADV
cana-1298	257	2	(	(	PUNCT
cana-1298	257	3	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	257	4	−	−	PROPN
cana-1298	257	5	×	×	NOUN
cana-1298	257	6	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	257	7	−)[(𝔥1	−)[(𝔥1	NOUN
cana-1298	257	8	,	,	PUNCT
cana-1298	257	9	𝔷1	𝔷1	NOUN
cana-1298	257	10	)	)	PUNCT
cana-1298	257	11	−	−	PROPN
cana-1298	257	12	(	(	PUNCT
cana-1298	257	13	𝔥2	𝔥2	PROPN
cana-1298	257	14	,	,	PUNCT
cana-1298	257	15	𝔷2	𝔷2	PROPN
cana-1298	257	16	)	)	PUNCT
cana-1298	257	17	]	]	PUNCT
cana-1298	258	1	=	=	PUNCT
cana-1298	259	1	[	[	X
cana-1298	259	2	−1	−1	X
cana-1298	259	3	]	]	PUNCT
cana-1298	259	4	,	,	PUNCT
cana-1298	259	5	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1298	259	6	𝑎𝑙𝑙	𝑎𝑙𝑙	X
cana-1298	259	7	(	(	PUNCT
cana-1298	259	8	𝔥1	𝔥1	NOUN
cana-1298	259	9	,	,	PUNCT
cana-1298	259	10	𝔷1	𝔷1	NOUN
cana-1298	259	11	)	)	PUNCT
cana-1298	259	12	,	,	PUNCT
cana-1298	259	13	(	(	PUNCT
cana-1298	259	14	𝔥2	𝔥2	PROPN
cana-1298	259	15	,	,	PUNCT
cana-1298	259	16	𝔷2	𝔷2	PROPN
cana-1298	259	17	)	)	PUNCT
cana-1298	259	18	∈	∈	PROPN
cana-1298	259	19	𝕐.	𝕐.	NOUN
cana-1298	259	20	communications	communication	NOUN
cana-1298	259	21	on	on	ADP
cana-1298	259	22	applied	apply	VERB
cana-1298	259	23	nonlinear	nonlinear	ADJ
cana-1298	259	24	analysis	analysis	NOUN
cana-1298	259	25	issn	issn	NOUN
cana-1298	259	26	:	:	PUNCT
cana-1298	259	27	1074	1074	NUM
cana-1298	259	28	-	-	PUNCT
cana-1298	259	29	133x	133x	NUM
cana-1298	259	30	vol	vol	NOUN
cana-1298	259	31	31	31	NUM
cana-1298	259	32	no	no	NOUN
cana-1298	259	33	.	.	PUNCT
cana-1298	260	1	7s	7	NOUN
cana-1298	260	2	(	(	PUNCT
cana-1298	260	3	2024	2024	NUM
cana-1298	260	4	)	)	PUNCT
cana-1298	260	5	236	236	NUM
cana-1298	260	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1298	260	7	therefore	therefore	ADV
cana-1298	260	8	(	(	PUNCT
cana-1298	260	9	𝔥1	𝔥1	NOUN
cana-1298	260	10	,	,	PUNCT
cana-1298	260	11	𝔷1	𝔷1	NOUN
cana-1298	260	12	)	)	PUNCT
cana-1298	260	13	−	−	PROPN
cana-1298	260	14	(	(	PUNCT
cana-1298	260	15	𝔥2	𝔥2	PROPN
cana-1298	260	16	,	,	PUNCT
cana-1298	260	17	𝔷2	𝔷2	PROPN
cana-1298	260	18	)	)	PUNCT
cana-1298	260	19	∈	∈	PROPN
cana-1298	260	20	𝕐.	𝕐.	PROPN
cana-1298	260	21	𝐴𝑙𝑠𝑜	𝐴𝑙𝑠𝑜	PROPN
cana-1298	260	22	(	(	PUNCT
cana-1298	260	23	𝔓𝑖	𝔓𝑖	PROPN
cana-1298	260	24	+	+	NOUN
cana-1298	260	25	×	×	PROPN
cana-1298	260	26	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	260	27	+	+	PROPN
cana-1298	260	28	)	)	PUNCT
cana-1298	261	1	[	[	X
cana-1298	261	2	(	(	PUNCT
cana-1298	261	3	𝔥1	𝔥1	PROPN
cana-1298	261	4	,	,	PUNCT
cana-1298	261	5	𝔷1)(𝔥2	𝔷1)(𝔥2	NOUN
cana-1298	261	6	,	,	PUNCT
cana-1298	261	7	𝔷2	𝔷2	PROPN
cana-1298	261	8	)	)	PUNCT
cana-1298	261	9	]	]	PUNCT
cana-1298	261	10			NUM
cana-1298	261	11	rmin	rmin	VERB
cana-1298	261	12	{	{	PUNCT
cana-1298	261	13	(	(	PUNCT
cana-1298	261	14	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	261	15	+	+	NOUN
cana-1298	261	16	×	×	PROPN
cana-1298	261	17	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	261	18	+	+	PROPN
cana-1298	261	19	)	)	PUNCT
cana-1298	261	20	(	(	PUNCT
cana-1298	261	21	𝔥1	𝔥1	NOUN
cana-1298	261	22	,	,	PUNCT
cana-1298	261	23	𝔷1	𝔷1	NOUN
cana-1298	261	24	)	)	PUNCT
cana-1298	261	25	,	,	PUNCT
cana-1298	261	26	(	(	PUNCT
cana-1298	261	27	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	261	28	+	+	NOUN
cana-1298	261	29	×	×	PROPN
cana-1298	261	30	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	261	31	+	+	PROPN
cana-1298	261	32	)	)	PUNCT
cana-1298	261	33	(	(	PUNCT
cana-1298	261	34	𝔥2	𝔥2	PROPN
cana-1298	261	35	,	,	PUNCT
cana-1298	261	36	𝔷2	𝔷2	PROPN
cana-1298	261	37	)	)	PUNCT
cana-1298	261	38	}	}	PUNCT
cana-1298	262	1	=	=	SYM
cana-1298	262	2	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
cana-1298	262	3	{	{	PUNCT
cana-1298	262	4	[	[	X
cana-1298	262	5	1	1	NUM
cana-1298	262	6	]	]	PUNCT
cana-1298	262	7	,	,	PUNCT
cana-1298	262	8	[	[	X
cana-1298	262	9	1	1	NUM
cana-1298	262	10	]	]	PUNCT
cana-1298	262	11	}	}	PUNCT
cana-1298	262	12	=	=	PUNCT
cana-1298	263	1	[	[	X
cana-1298	263	2	1	1	NUM
cana-1298	263	3	]	]	PUNCT
cana-1298	263	4	.	.	PUNCT
cana-1298	264	1	thus	thus	ADV
cana-1298	264	2	(	(	PUNCT
cana-1298	264	3	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	264	4	+	+	NOUN
cana-1298	264	5	×	×	PROPN
cana-1298	264	6	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	264	7	+	+	PROPN
cana-1298	264	8	)	)	PUNCT
cana-1298	264	9	[	[	X
cana-1298	264	10	(	(	PUNCT
cana-1298	264	11	𝔥1	𝔥1	PROPN
cana-1298	264	12	,	,	PUNCT
cana-1298	264	13	𝔷1)(𝔥2	𝔷1)(𝔥2	NOUN
cana-1298	264	14	,	,	PUNCT
cana-1298	264	15	𝔷2	𝔷2	PROPN
cana-1298	264	16	)	)	PUNCT
cana-1298	264	17	]	]	PUNCT
cana-1298	265	1	=	=	PUNCT
cana-1298	266	1	[	[	X
cana-1298	266	2	1	1	NUM
cana-1298	266	3	]	]	PUNCT
cana-1298	266	4	,	,	PUNCT
cana-1298	266	5	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1298	266	6	𝑎𝑙𝑙	𝑎𝑙𝑙	X
cana-1298	266	7	(	(	PUNCT
cana-1298	266	8	𝔥1	𝔥1	NOUN
cana-1298	266	9	,	,	PUNCT
cana-1298	266	10	𝔷1	𝔷1	NOUN
cana-1298	266	11	)	)	PUNCT
cana-1298	266	12	,	,	PUNCT
cana-1298	266	13	(	(	PUNCT
cana-1298	266	14	𝔥2	𝔥2	PROPN
cana-1298	266	15	,	,	PUNCT
cana-1298	266	16	𝔷2	𝔷2	PROPN
cana-1298	266	17	)	)	PUNCT
cana-1298	266	18	∈	∈	PROPN
cana-1298	266	19	𝕐.	𝕐.	NOUN
cana-1298	266	20	and	and	CCONJ
cana-1298	266	21	(	(	PUNCT
cana-1298	266	22	𝔓𝑖	𝔓𝑖	PROPN
cana-1298	266	23	−	−	PROPN
cana-1298	267	1	×	×	NOUN
cana-1298	267	2	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	267	3	−)[(𝔥1	−)[(𝔥1	NOUN
cana-1298	267	4	,	,	PUNCT
cana-1298	267	5	𝔷1)(𝔥2	𝔷1)(𝔥2	NOUN
cana-1298	267	6	,	,	PUNCT
cana-1298	267	7	𝔷2	𝔷2	PROPN
cana-1298	267	8	)	)	PUNCT
cana-1298	267	9	]	]	PUNCT
cana-1298	267	10			NOUN
cana-1298	267	11	rmax{(𝔓𝑖	rmax{(𝔓𝑖	NOUN
cana-1298	267	12	−	−	PROPN
cana-1298	267	13	×	×	NOUN
cana-1298	267	14	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	267	15	−)(𝔥1	−)(𝔥1	PROPN
cana-1298	267	16	,	,	PUNCT
cana-1298	267	17	𝔷1	𝔷1	NOUN
cana-1298	267	18	)	)	PUNCT
cana-1298	267	19	,	,	PUNCT
cana-1298	268	1	(	(	PUNCT
cana-1298	268	2	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	268	3	−	−	PROPN
cana-1298	268	4	×	×	NOUN
cana-1298	268	5	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	268	6	−)(𝔥2	−)(𝔥2	PROPN
cana-1298	268	7	,	,	PUNCT
cana-1298	268	8	𝔷2	𝔷2	ADV
cana-1298	268	9	)	)	PUNCT
cana-1298	268	10	}	}	PUNCT
cana-1298	268	11	=	=	SYM
cana-1298	268	12	𝑟𝑚𝑎𝑥	𝑟𝑚𝑎𝑥	NOUN
cana-1298	268	13	{	{	PUNCT
cana-1298	268	14	[	[	X
cana-1298	268	15	−1	−1	NOUN
cana-1298	268	16	]	]	X
cana-1298	268	17	,	,	PUNCT
cana-1298	268	18	[	[	X
cana-1298	268	19	−1	−1	NOUN
cana-1298	268	20	]	]	X
cana-1298	268	21	}	}	PUNCT
cana-1298	268	22	=	=	PUNCT
cana-1298	269	1	[	[	X
cana-1298	269	2	−1	−1	NOUN
cana-1298	269	3	]	]	PUNCT
cana-1298	269	4	.	.	PUNCT
cana-1298	270	1	thus	thus	ADV
cana-1298	270	2	(	(	PUNCT
cana-1298	270	3	𝔓𝑖	𝔓𝑖	NOUN
cana-1298	270	4	−	−	PROPN
cana-1298	270	5	×	×	NOUN
cana-1298	270	6	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	270	7	−	−	PROPN
cana-1298	270	8	)	)	PUNCT
cana-1298	271	1	[	[	X
cana-1298	271	2	(	(	PUNCT
cana-1298	271	3	𝔥1	𝔥1	NOUN
cana-1298	271	4	,	,	PUNCT
cana-1298	271	5	𝔷1)(𝔥2	𝔷1)(𝔥2	NOUN
cana-1298	271	6	,	,	PUNCT
cana-1298	271	7	𝔷2	𝔷2	PROPN
cana-1298	271	8	)	)	PUNCT
cana-1298	271	9	]	]	PUNCT
cana-1298	272	1	=	=	PUNCT
cana-1298	273	1	[	[	X
cana-1298	273	2	−1	−1	X
cana-1298	273	3	]	]	PUNCT
cana-1298	273	4	,	,	PUNCT
cana-1298	273	5	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-1298	273	6	𝑎𝑙𝑙	𝑎𝑙𝑙	X
cana-1298	273	7	(	(	PUNCT
cana-1298	273	8	𝔥1	𝔥1	NOUN
cana-1298	273	9	,	,	PUNCT
cana-1298	273	10	𝔷1	𝔷1	NOUN
cana-1298	273	11	)	)	PUNCT
cana-1298	273	12	,	,	PUNCT
cana-1298	273	13	(	(	PUNCT
cana-1298	273	14	𝔥2	𝔥2	PROPN
cana-1298	273	15	,	,	PUNCT
cana-1298	273	16	𝔷2	𝔷2	PROPN
cana-1298	273	17	)	)	PUNCT
cana-1298	273	18	∈	∈	PROPN
cana-1298	273	19	𝕐.	𝕐.	NOUN
cana-1298	273	20	therefore	therefore	ADV
cana-1298	273	21	(	(	PUNCT
cana-1298	273	22	𝔥1	𝔥1	PROPN
cana-1298	273	23	,	,	PUNCT
cana-1298	273	24	𝔷1)(𝔥2	𝔷1)(𝔥2	NOUN
cana-1298	273	25	,	,	PUNCT
cana-1298	273	26	𝔷2	𝔷2	PROPN
cana-1298	273	27	)	)	PUNCT
cana-1298	273	28	∈	∈	PROPN
cana-1298	273	29	𝕐.	𝕐.	NOUN
cana-1298	273	30	𝐻𝑒𝑛𝑐𝑒	𝐻𝑒𝑛𝑐𝑒	PROPN
cana-1298	273	31	𝕐	𝕐	PROPN
cana-1298	273	32	𝑖𝑠	𝑖𝑠	ADP
cana-1298	273	33	𝑎	𝑎	DET
cana-1298	273	34	𝑠𝑢𝑏𝑟𝑖𝑛𝑔	𝑠𝑢𝑏𝑟𝑖𝑛𝑔	NOUN
cana-1298	273	35	𝑜𝑓	𝑜𝑓	ADP
cana-1298	273	36	𝔊1	𝔊1	PROPN
cana-1298	273	37	×	×	PROPN
cana-1298	273	38	𝔊2	𝔊2	PROPN
cana-1298	273	39	.	.	PUNCT
cana-1298	273	40	theorem	theorem	VERB
cana-1298	273	41	2.9	2.9	NUM
cana-1298	273	42	.	.	PUNCT
cana-1298	274	1	𝐿𝑒𝑡	𝐿𝑒𝑡	NOUN
cana-1298	274	2	𝔓	𝔓	PROPN
cana-1298	274	3	𝑏𝑒	𝑏𝑒	NOUN
cana-1298	274	4	𝑎	𝑎	DET
cana-1298	274	5	𝔹𝕍𝕄𝕀𝔽𝕊	𝔹𝕍𝕄𝕀𝔽𝕊	NOUN
cana-1298	274	6	of	of	ADP
cana-1298	274	7	a	a	DET
cana-1298	274	8	ring	ring	NOUN
cana-1298	274	9	ℨ	ℨ	NOUN
cana-1298	274	10	and	and	CCONJ
cana-1298	274	11	𝔐	𝔐	PRON
cana-1298	274	12	be	be	VERB
cana-1298	274	13	the	the	DET
cana-1298	274	14	stronget	stronget	NOUN
cana-1298	274	15	𝔹𝕍𝕄𝕀𝔽	𝔹𝕍𝕄𝕀𝔽	PROPN
cana-1298	274	16	relation	relation	NOUN
cana-1298	274	17	of	of	ADP
cana-1298	274	18	ℨ.	ℨ.	PROPN
cana-1298	274	19	then	then	ADV
cana-1298	274	20	𝔓	𝔓	PROPN
cana-1298	274	21	is	be	AUX
cana-1298	274	22	a	a	DET
cana-1298	274	23	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	274	24	of	of	ADP
cana-1298	274	25	ℨ	ℨ	PRON
cana-1298	274	26	if	if	SCONJ
cana-1298	275	1	and	and	CCONJ
cana-1298	275	2	only	only	ADV
cana-1298	275	3	if	if	SCONJ
cana-1298	275	4	𝔐	𝔐	PRON
cana-1298	275	5	is	be	AUX
cana-1298	275	6	a	a	DET
cana-1298	275	7	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	275	8	of	of	ADP
cana-1298	275	9	ℨ×ℨ.	ℨ×ℨ.	PROPN
cana-1298	275	10	proof	proof	NOUN
cana-1298	275	11	.	.	PUNCT
cana-1298	276	1	let	let	VERB
cana-1298	276	2	𝜚	𝜚	X
cana-1298	276	3	,	,	PUNCT
cana-1298	276	4	𝜐	𝜐	VERB
cana-1298	276	5	be	be	VERB
cana-1298	276	6	in	in	ADP
cana-1298	276	7	ℨ	ℨ	PROPN
cana-1298	276	8	and	and	CCONJ
cana-1298	276	9	𝜁	𝜁	NOUN
cana-1298	276	10	,	,	PUNCT
cana-1298	276	11	𝜉	𝜉	AUX
cana-1298	276	12	be	be	AUX
cana-1298	276	13	in	in	ADP
cana-1298	276	14	ℨ.	ℨ.	PROPN
cana-1298	276	15	then	then	ADV
cana-1298	276	16	(	(	PUNCT
cana-1298	276	17	𝜚	𝜚	NOUN
cana-1298	276	18	,	,	PUNCT
cana-1298	276	19	𝜁	𝜁	NOUN
cana-1298	276	20	)	)	PUNCT
cana-1298	276	21	and	and	CCONJ
cana-1298	276	22	(	(	PUNCT
cana-1298	276	23	𝜐	𝜐	NOUN
cana-1298	276	24	,	,	PUNCT
cana-1298	276	25	𝜉	𝜉	X
cana-1298	276	26	)	)	PUNCT
cana-1298	276	27	are	be	AUX
cana-1298	276	28	in	in	ADP
cana-1298	276	29	ℨ×ℨ.	ℨ×ℨ.	PROPN
cana-1298	276	30	for	for	ADP
cana-1298	276	31	all	all	DET
cana-1298	276	32	i	i	PRON
cana-1298	276	33	,	,	PUNCT
cana-1298	276	34	i	i	NOUN
cana-1298	276	35	=	=	NOUN
cana-1298	276	36	1	1	NUM
cana-1298	276	37	,	,	PUNCT
cana-1298	276	38	2	2	NUM
cana-1298	276	39	,	,	PUNCT
cana-1298	276	40	…	…	PUNCT
cana-1298	276	41	,	,	PUNCT
cana-1298	276	42	n	n	CCONJ
cana-1298	276	43	,	,	PUNCT
cana-1298	276	44	if	if	SCONJ
cana-1298	276	45	𝔓	𝔓	PROPN
cana-1298	276	46	is	be	AUX
cana-1298	276	47	a	a	DET
cana-1298	276	48	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	276	49	of	of	ADP
cana-1298	276	50	ℨ	ℨ	PROPN
cana-1298	276	51	,	,	PUNCT
cana-1298	276	52	then	then	ADV
cana-1298	277	1	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	277	2	+	+	PROPN
cana-1298	277	3	[	[	X
cana-1298	277	4	(	(	PUNCT
cana-1298	277	5	𝜚	𝜚	NOUN
cana-1298	277	6	,	,	PUNCT
cana-1298	277	7	𝜁)−(𝜐	𝜁)−(𝜐	NOUN
cana-1298	277	8	,	,	PUNCT
cana-1298	277	9	𝜉	𝜉	NOUN
cana-1298	277	10	)	)	PUNCT
cana-1298	277	11	]	]	PUNCT
cana-1298	278	1	=	=	PUNCT
cana-1298	278	2	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	278	3	+	+	PROPN
cana-1298	278	4	(	(	PUNCT
cana-1298	278	5	𝜚−𝜐	𝜚−𝜐	NOUN
cana-1298	278	6	,	,	PUNCT
cana-1298	278	7	𝜁−	𝜁−	ADJ
cana-1298	278	8	𝜉	𝜉	NOUN
cana-1298	278	9	)	)	PUNCT
cana-1298	278	10	=	=	PUNCT
cana-1298	278	11	rmin{𝔓i	rmin{𝔓i	NOUN
cana-1298	279	1	+	+	NOUN
cana-1298	279	2	(	(	PUNCT
cana-1298	279	3	𝜚−𝜐	𝜚−𝜐	NOUN
cana-1298	279	4	)	)	PUNCT
cana-1298	279	5	,	,	PUNCT
cana-1298	279	6	𝔓i	𝔓i	PROPN
cana-1298	279	7	+	+	PROPN
cana-1298	279	8	(	(	PUNCT
cana-1298	279	9	𝜁−	𝜁−	ADJ
cana-1298	279	10	𝜉	𝜉	NOUN
cana-1298	279	11	)	)	PUNCT
cana-1298	279	12	}	}	PUNCT
cana-1298	279	13			NUM
cana-1298	280	1	rmin{rmin{𝔓i	rmin{rmin{𝔓i	PROPN
cana-1298	280	2	+	+	ADJ
cana-1298	280	3	(	(	PUNCT
cana-1298	280	4	𝜚	𝜚	NOUN
cana-1298	280	5	)	)	PUNCT
cana-1298	280	6	,	,	PUNCT
cana-1298	281	1	𝔓i	𝔓i	PROPN
cana-1298	281	2	+	+	PROPN
cana-1298	281	3	(	(	PUNCT
cana-1298	281	4	𝜐	𝜐	NOUN
cana-1298	281	5	)	)	PUNCT
cana-1298	281	6	}	}	PUNCT
cana-1298	281	7	,	,	PUNCT
cana-1298	281	8	rmin{𝔓i	rmin{𝔓i	NOUN
cana-1298	281	9	+	+	NOUN
cana-1298	281	10	(	(	PUNCT
cana-1298	281	11	𝜁	𝜁	PROPN
cana-1298	281	12	)	)	PUNCT
cana-1298	281	13	,	,	PUNCT
cana-1298	281	14	𝔓i	𝔓i	PROPN
cana-1298	281	15	+	+	PROPN
cana-1298	281	16	(	(	PUNCT
cana-1298	281	17	𝜉	𝜉	NOUN
cana-1298	281	18	)	)	PUNCT
cana-1298	281	19	}	}	PUNCT
cana-1298	281	20	}	}	PUNCT
cana-1298	281	21	=	=	PUNCT
cana-1298	281	22	rmin{rmin{𝔓i	rmin{rmin{𝔓i	X
cana-1298	282	1	+	+	ADJ
cana-1298	282	2	(	(	PUNCT
cana-1298	282	3	𝜚	𝜚	NOUN
cana-1298	282	4	)	)	PUNCT
cana-1298	282	5	,	,	PUNCT
cana-1298	282	6	𝔓i	𝔓i	PROPN
cana-1298	282	7	+	+	PROPN
cana-1298	282	8	(	(	PUNCT
cana-1298	282	9	𝜁	𝜁	NOUN
cana-1298	282	10	)	)	PUNCT
cana-1298	282	11	}	}	PUNCT
cana-1298	282	12	,	,	PUNCT
cana-1298	282	13	rmin{𝔓i	rmin{𝔓i	NOUN
cana-1298	282	14	+	+	PROPN
cana-1298	282	15	(	(	PUNCT
cana-1298	282	16	𝜐	𝜐	NOUN
cana-1298	282	17	)	)	PUNCT
cana-1298	282	18	,	,	PUNCT
cana-1298	282	19	𝔓i	𝔓i	PROPN
cana-1298	282	20	+	+	PROPN
cana-1298	282	21	(	(	PUNCT
cana-1298	282	22	𝜉	𝜉	NOUN
cana-1298	282	23	)	)	PUNCT
cana-1298	282	24	}	}	PUNCT
cana-1298	282	25	}	}	PUNCT
cana-1298	282	26	=	=	VERB
cana-1298	282	27	rmin{𝔚𝑖	rmin{𝔚𝑖	ADJ
cana-1298	283	1	+	+	ADJ
cana-1298	283	2	(	(	PUNCT
cana-1298	283	3	𝜚	𝜚	NOUN
cana-1298	283	4	,	,	PUNCT
cana-1298	283	5	𝜁	𝜁	NOUN
cana-1298	283	6	)	)	PUNCT
cana-1298	283	7	,	,	PUNCT
cana-1298	283	8	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	283	9	+	+	PROPN
cana-1298	283	10	(	(	PUNCT
cana-1298	283	11	𝜐	𝜐	PROPN
cana-1298	283	12	,	,	PUNCT
cana-1298	283	13	𝜉	𝜉	NOUN
cana-1298	283	14	)	)	PUNCT
cana-1298	283	15	}	}	PUNCT
cana-1298	283	16	,	,	PUNCT
cana-1298	283	17	for	for	ADP
cana-1298	283	18	all	all	PRON
cana-1298	283	19	(	(	PUNCT
cana-1298	283	20	𝜚	𝜚	NOUN
cana-1298	283	21	,	,	PUNCT
cana-1298	283	22	𝜁	𝜁	NOUN
cana-1298	283	23	)	)	PUNCT
cana-1298	283	24	,	,	PUNCT
cana-1298	283	25	(	(	PUNCT
cana-1298	283	26	𝜐	𝜐	NOUN
cana-1298	283	27	,	,	PUNCT
cana-1298	283	28	𝜉	𝜉	NOUN
cana-1298	283	29	)	)	PUNCT
cana-1298	283	30	in	in	ADP
cana-1298	283	31	ℨ×ℨ.	ℨ×ℨ.	PROPN
cana-1298	283	32	and	and	CCONJ
cana-1298	283	33	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	283	34	+	+	PROPN
cana-1298	283	35	[	[	X
cana-1298	283	36	(	(	PUNCT
cana-1298	283	37	𝜚	𝜚	NOUN
cana-1298	283	38	,	,	PUNCT
cana-1298	283	39	𝜁)(𝜐	𝜁)(𝜐	NOUN
cana-1298	283	40	,	,	PUNCT
cana-1298	283	41	𝜉	𝜉	NOUN
cana-1298	283	42	)	)	PUNCT
cana-1298	283	43	]	]	PUNCT
cana-1298	284	1	=	=	PUNCT
cana-1298	284	2	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	284	3	+	+	PROPN
cana-1298	284	4	(	(	PUNCT
cana-1298	284	5	𝜚𝜐	𝜚𝜐	ADP
cana-1298	284	6	,	,	PUNCT
cana-1298	284	7	𝜁	𝜁	DET
cana-1298	284	8	𝜉	𝜉	X
cana-1298	284	9	)	)	PUNCT
cana-1298	284	10	=	=	PUNCT
cana-1298	285	1	rmin{𝔓i	rmin{𝔓i	NOUN
cana-1298	286	1	+	+	NOUN
cana-1298	286	2	(	(	PUNCT
cana-1298	286	3	𝜚𝜐	𝜚𝜐	NOUN
cana-1298	286	4	)	)	PUNCT
cana-1298	286	5	,	,	PUNCT
cana-1298	286	6	𝔓i	𝔓i	PROPN
cana-1298	286	7	+	+	PROPN
cana-1298	286	8	(	(	PUNCT
cana-1298	286	9	𝜁	𝜁	PROPN
cana-1298	286	10	𝜉)}	𝜉)}	PROPN
cana-1298	286	11	rmin{rmin{𝔓i	rmin{rmin{𝔓i	PROPN
cana-1298	287	1	+	+	PROPN
cana-1298	287	2	(	(	PUNCT
cana-1298	287	3	𝜚	𝜚	NOUN
cana-1298	287	4	)	)	PUNCT
cana-1298	287	5	,	,	PUNCT
cana-1298	287	6	𝔓i	𝔓i	PROPN
cana-1298	287	7	+	+	PROPN
cana-1298	287	8	(	(	PUNCT
cana-1298	287	9	𝜐	𝜐	NOUN
cana-1298	287	10	)	)	PUNCT
cana-1298	287	11	}	}	PUNCT
cana-1298	287	12	,	,	PUNCT
cana-1298	287	13	rmin{𝔓i	rmin{𝔓i	NOUN
cana-1298	287	14	+	+	NOUN
cana-1298	287	15	(	(	PUNCT
cana-1298	287	16	𝜁	𝜁	PROPN
cana-1298	287	17	)	)	PUNCT
cana-1298	287	18	,	,	PUNCT
cana-1298	287	19	𝔓i	𝔓i	PROPN
cana-1298	287	20	+	+	PROPN
cana-1298	287	21	(	(	PUNCT
cana-1298	287	22	𝜉	𝜉	NOUN
cana-1298	287	23	)	)	PUNCT
cana-1298	287	24	}	}	PUNCT
cana-1298	287	25	}	}	PUNCT
cana-1298	287	26	=	=	PUNCT
cana-1298	288	1	rmin{rmin{𝔓i	rmin{rmin{𝔓i	X
cana-1298	289	1	+	+	ADJ
cana-1298	289	2	(	(	PUNCT
cana-1298	289	3	𝜚	𝜚	NOUN
cana-1298	289	4	)	)	PUNCT
cana-1298	289	5	,	,	PUNCT
cana-1298	290	1	𝔓i	𝔓i	PROPN
cana-1298	290	2	+	+	PROPN
cana-1298	290	3	(	(	PUNCT
cana-1298	290	4	𝜁	𝜁	NOUN
cana-1298	290	5	)	)	PUNCT
cana-1298	290	6	}	}	PUNCT
cana-1298	290	7	,	,	PUNCT
cana-1298	290	8	rmin{𝔓i	rmin{𝔓i	NOUN
cana-1298	290	9	+	+	PROPN
cana-1298	290	10	(	(	PUNCT
cana-1298	290	11	𝜐	𝜐	NOUN
cana-1298	290	12	)	)	PUNCT
cana-1298	290	13	,	,	PUNCT
cana-1298	290	14	𝔓i	𝔓i	PROPN
cana-1298	290	15	+	+	PROPN
cana-1298	290	16	(	(	PUNCT
cana-1298	290	17	𝜉	𝜉	NOUN
cana-1298	290	18	)	)	PUNCT
cana-1298	290	19	}	}	PUNCT
cana-1298	290	20	}	}	PUNCT
cana-1298	290	21	=	=	SYM
cana-1298	290	22	rmin	rmin	NOUN
cana-1298	290	23	{	{	PUNCT
cana-1298	290	24	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	290	25	+	+	PROPN
cana-1298	290	26	(	(	PUNCT
cana-1298	290	27	𝜚	𝜚	NOUN
cana-1298	290	28	,	,	PUNCT
cana-1298	290	29	𝜁	𝜁	NOUN
cana-1298	290	30	)	)	PUNCT
cana-1298	290	31	,	,	PUNCT
cana-1298	290	32	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	290	33	+	+	PROPN
cana-1298	290	34	(	(	PUNCT
cana-1298	290	35	𝜐	𝜐	PROPN
cana-1298	290	36	,	,	PUNCT
cana-1298	290	37	𝜉	𝜉	NOUN
cana-1298	290	38	)	)	PUNCT
cana-1298	290	39	}	}	PUNCT
cana-1298	290	40	,	,	PUNCT
cana-1298	290	41	for	for	ADP
cana-1298	290	42	all	all	DET
cana-1298	290	43	(	(	PUNCT
cana-1298	290	44	𝜚	𝜚	NOUN
cana-1298	290	45	,	,	PUNCT
cana-1298	290	46	𝜁	𝜁	NOUN
cana-1298	290	47	)	)	PUNCT
cana-1298	290	48	,	,	PUNCT
cana-1298	290	49	(	(	PUNCT
cana-1298	290	50	𝜐	𝜐	NOUN
cana-1298	290	51	,	,	PUNCT
cana-1298	290	52	𝜉	𝜉	NOUN
cana-1298	290	53	)	)	PUNCT
cana-1298	290	54	in	in	ADP
cana-1298	290	55	ℨ×ℨ.	ℨ×ℨ.	PROPN
cana-1298	290	56	also	also	ADV
cana-1298	290	57	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	290	58	−[(𝜚	−[(𝜚	NUM
cana-1298	290	59	,	,	PUNCT
cana-1298	290	60	𝜁)−(𝜐	𝜁)−(𝜐	NOUN
cana-1298	290	61	,	,	PUNCT
cana-1298	290	62	𝜉	𝜉	NOUN
cana-1298	290	63	)	)	PUNCT
cana-1298	290	64	]	]	PUNCT
cana-1298	291	1	=	=	PUNCT
cana-1298	291	2	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	291	3	−(𝜚−𝜐	−(𝜚−𝜐	NUM
cana-1298	291	4	,	,	PUNCT
cana-1298	291	5	𝜁−	𝜁−	ADJ
cana-1298	291	6	𝜉	𝜉	NOUN
cana-1298	291	7	)	)	PUNCT
cana-1298	291	8	=	=	NOUN
cana-1298	291	9	rmax{𝔓i	rmax{𝔓i	NOUN
cana-1298	291	10	−(𝜚−𝜐	−(𝜚−𝜐	NOUN
cana-1298	291	11	)	)	PUNCT
cana-1298	291	12	,	,	PUNCT
cana-1298	291	13	𝔓i	𝔓i	PROPN
cana-1298	291	14	−(𝜁−	−(𝜁−	NOUN
cana-1298	291	15	𝜉	𝜉	NOUN
cana-1298	291	16	)	)	PUNCT
cana-1298	291	17	}	}	PUNCT
cana-1298	291	18			PROPN
cana-1298	291	19	rmax{rmax{𝔓i	rmax{rmax{𝔓i	NOUN
cana-1298	291	20	−(𝜚	−(𝜚	NOUN
cana-1298	291	21	)	)	PUNCT
cana-1298	291	22	,	,	PUNCT
cana-1298	291	23	𝔓i	𝔓i	PROPN
cana-1298	291	24	−(𝜐	−(𝜐	NOUN
cana-1298	291	25	)	)	PUNCT
cana-1298	291	26	}	}	PUNCT
cana-1298	291	27	,	,	PUNCT
cana-1298	291	28	rmax{𝔓i	rmax{𝔓i	NOUN
cana-1298	291	29	−(𝜁	−(𝜁	PROPN
cana-1298	291	30	)	)	PUNCT
cana-1298	291	31	,	,	PUNCT
cana-1298	291	32	𝔓i	𝔓i	PROPN
cana-1298	291	33	−	−	PROPN
cana-1298	291	34	(	(	PUNCT
cana-1298	291	35	𝜉	𝜉	NOUN
cana-1298	291	36	)	)	PUNCT
cana-1298	291	37	}	}	PUNCT
cana-1298	291	38	}	}	PUNCT
cana-1298	291	39	=	=	SYM
cana-1298	291	40	rmax{rmax{𝔓i	rmax{rmax{𝔓i	NOUN
cana-1298	291	41	−(𝜚	−(𝜚	NOUN
cana-1298	291	42	)	)	PUNCT
cana-1298	291	43	,	,	PUNCT
cana-1298	291	44	𝔓i	𝔓i	PROPN
cana-1298	291	45	−(𝜁	−(𝜁	NOUN
cana-1298	291	46	)	)	PUNCT
cana-1298	291	47	}	}	PUNCT
cana-1298	291	48	,	,	PUNCT
cana-1298	291	49	rmax{𝔓i	rmax{𝔓i	NOUN
cana-1298	291	50	−(𝜐	−(𝜐	PROPN
cana-1298	291	51	)	)	PUNCT
cana-1298	291	52	,	,	PUNCT
cana-1298	291	53	𝔓i	𝔓i	PROPN
cana-1298	291	54	−	−	PROPN
cana-1298	291	55	(	(	PUNCT
cana-1298	291	56	𝜉	𝜉	NOUN
cana-1298	291	57	)	)	PUNCT
cana-1298	291	58	}	}	PUNCT
cana-1298	291	59	}	}	PUNCT
cana-1298	291	60	=	=	SYM
cana-1298	291	61	rmax	rmax	ADJ
cana-1298	291	62	{	{	PUNCT
cana-1298	291	63	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	291	64	−(𝜚	−(𝜚	NOUN
cana-1298	291	65	,	,	PUNCT
cana-1298	291	66	𝜁	𝜁	NOUN
cana-1298	291	67	)	)	PUNCT
cana-1298	291	68	,	,	PUNCT
cana-1298	291	69	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	291	70	−(𝜐	−(𝜐	NOUN
cana-1298	291	71	,	,	PUNCT
cana-1298	291	72	𝜉	𝜉	NOUN
cana-1298	291	73	)	)	PUNCT
cana-1298	291	74	}	}	PUNCT
cana-1298	291	75	,	,	PUNCT
cana-1298	291	76	for	for	ADP
cana-1298	291	77	all	all	PRON
cana-1298	291	78	(	(	PUNCT
cana-1298	291	79	𝜚	𝜚	NOUN
cana-1298	291	80	,	,	PUNCT
cana-1298	291	81	𝜁	𝜁	NOUN
cana-1298	291	82	)	)	PUNCT
cana-1298	291	83	,	,	PUNCT
cana-1298	291	84	(	(	PUNCT
cana-1298	291	85	𝜐	𝜐	NOUN
cana-1298	291	86	,	,	PUNCT
cana-1298	291	87	𝜉	𝜉	NOUN
cana-1298	291	88	)	)	PUNCT
cana-1298	291	89	in	in	ADP
cana-1298	291	90	ℨ×ℨ.	ℨ×ℨ.	PROPN
cana-1298	291	91	and	and	CCONJ
cana-1298	291	92	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	291	93	−[(𝜚	−[(𝜚	NUM
cana-1298	291	94	,	,	PUNCT
cana-1298	291	95	𝜁)(𝜐	𝜁)(𝜐	VERB
cana-1298	291	96	,	,	PUNCT
cana-1298	291	97	𝜉	𝜉	NOUN
cana-1298	291	98	)	)	PUNCT
cana-1298	291	99	]	]	PUNCT
cana-1298	292	1	=	=	PUNCT
cana-1298	292	2	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	292	3	−(𝜚𝜐	−(𝜚𝜐	NOUN
cana-1298	292	4	,	,	PUNCT
cana-1298	292	5	𝜁𝜉	𝜁𝜉	NOUN
cana-1298	292	6	)	)	PUNCT
cana-1298	292	7	=	=	SYM
cana-1298	292	8	rmax{𝔓i	rmax{𝔓i	NOUN
cana-1298	292	9	−(𝜚𝜐	−(𝜚𝜐	NOUN
cana-1298	292	10	)	)	PUNCT
cana-1298	292	11	,	,	PUNCT
cana-1298	292	12	𝔓i	𝔓i	PROPN
cana-1298	292	13	−(𝜁𝜉	−(𝜁𝜉	NOUN
cana-1298	292	14	)	)	PUNCT
cana-1298	292	15	}	}	PUNCT
cana-1298	292	16			PROPN
cana-1298	292	17	rmax{rmax{𝔓i	rmax{rmax{𝔓i	NOUN
cana-1298	292	18	−(𝜚	−(𝜚	NOUN
cana-1298	292	19	)	)	PUNCT
cana-1298	292	20	,	,	PUNCT
cana-1298	292	21	𝔓i	𝔓i	PROPN
cana-1298	292	22	−(𝜐	−(𝜐	NOUN
cana-1298	292	23	)	)	PUNCT
cana-1298	292	24	}	}	PUNCT
cana-1298	292	25	,	,	PUNCT
cana-1298	292	26	rmax{𝔓i	rmax{𝔓i	NOUN
cana-1298	292	27	−(𝜁	−(𝜁	PROPN
cana-1298	292	28	)	)	PUNCT
cana-1298	292	29	,	,	PUNCT
cana-1298	292	30	𝔓i	𝔓i	PROPN
cana-1298	292	31	−(𝜉	−(𝜉	NOUN
cana-1298	292	32	)	)	PUNCT
cana-1298	292	33	}	}	PUNCT
cana-1298	292	34	}	}	PUNCT
cana-1298	292	35	=	=	SYM
cana-1298	292	36	rmax{rmax{𝔓i	rmax{rmax{𝔓i	NOUN
cana-1298	292	37	−(𝜚	−(𝜚	NOUN
cana-1298	292	38	)	)	PUNCT
cana-1298	292	39	,	,	PUNCT
cana-1298	292	40	𝔓i	𝔓i	PROPN
cana-1298	292	41	−(𝜁	−(𝜁	NOUN
cana-1298	292	42	)	)	PUNCT
cana-1298	292	43	}	}	PUNCT
cana-1298	292	44	,	,	PUNCT
cana-1298	292	45	rmax{𝔓i	rmax{𝔓i	NOUN
cana-1298	292	46	−(𝜐	−(𝜐	PROPN
cana-1298	292	47	)	)	PUNCT
cana-1298	292	48	,	,	PUNCT
cana-1298	292	49	𝔓i	𝔓i	PROPN
cana-1298	292	50	−(𝜉	−(𝜉	NOUN
cana-1298	292	51	)	)	PUNCT
cana-1298	292	52	}	}	PUNCT
cana-1298	292	53	}	}	PUNCT
cana-1298	292	54	=	=	SYM
cana-1298	292	55	max{𝔚𝑖	max{𝔚𝑖	NUM
cana-1298	292	56	−(𝜚	−(𝜚	NOUN
cana-1298	292	57	,	,	PUNCT
cana-1298	292	58	𝜁	𝜁	NOUN
cana-1298	292	59	)	)	PUNCT
cana-1298	292	60	,	,	PUNCT
cana-1298	292	61	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	292	62	−(𝜐	−(𝜐	NOUN
cana-1298	292	63	,	,	PUNCT
cana-1298	292	64	𝜉	𝜉	NOUN
cana-1298	292	65	)	)	PUNCT
cana-1298	292	66	}	}	PUNCT
cana-1298	292	67	,	,	PUNCT
cana-1298	292	68	for	for	ADP
cana-1298	292	69	all	all	PRON
cana-1298	292	70	(	(	PUNCT
cana-1298	292	71	𝜚	𝜚	NOUN
cana-1298	292	72	,	,	PUNCT
cana-1298	292	73	𝜁	𝜁	NOUN
cana-1298	292	74	)	)	PUNCT
cana-1298	292	75	,	,	PUNCT
cana-1298	292	76	(	(	PUNCT
cana-1298	292	77	𝜐	𝜐	NOUN
cana-1298	292	78	,	,	PUNCT
cana-1298	292	79	𝜉	𝜉	NOUN
cana-1298	292	80	)	)	PUNCT
cana-1298	292	81	in	in	ADP
cana-1298	292	82	ℨ×ℨ.	ℨ×ℨ.	PROPN
cana-1298	292	83	hence	hence	ADV
cana-1298	292	84	𝔚	𝔚	PROPN
cana-1298	292	85	is	be	AUX
cana-1298	292	86	a	a	DET
cana-1298	292	87	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	292	88	of	of	ADP
cana-1298	292	89	ℨ×ℨ.	ℨ×ℨ.	PUNCT
cana-1298	292	90	conversely	conversely	ADV
cana-1298	292	91	,	,	PUNCT
cana-1298	292	92	assume	assume	VERB
cana-1298	292	93	that	that	SCONJ
cana-1298	292	94	𝔚	𝔚	PROPN
cana-1298	292	95	is	be	AUX
cana-1298	292	96	a	a	DET
cana-1298	292	97	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	292	98	of	of	ADP
cana-1298	292	99	ℨ×ℨ.	ℨ×ℨ.	PROPN
cana-1298	292	100	for	for	ADP
cana-1298	292	101	all	all	DET
cana-1298	292	102	i	i	PRON
cana-1298	292	103	=	=	NOUN
cana-1298	292	104	1	1	NUM
cana-1298	292	105	,	,	PUNCT
cana-1298	292	106	2	2	NUM
cana-1298	292	107	,	,	PUNCT
cana-1298	292	108	…	…	PUNCT
cana-1298	292	109	,	,	PUNCT
cana-1298	292	110	n	n	CCONJ
cana-1298	292	111	,	,	PUNCT
cana-1298	292	112	rmin{𝔓i	rmin{𝔓i	NOUN
cana-1298	293	1	+	+	NOUN
cana-1298	293	2	(	(	PUNCT
cana-1298	293	3	𝜚−𝜐	𝜚−𝜐	NOUN
cana-1298	293	4	)	)	PUNCT
cana-1298	293	5	,	,	PUNCT
cana-1298	293	6	𝔓i	𝔓i	PROPN
cana-1298	293	7	+	+	PROPN
cana-1298	293	8	(	(	PUNCT
cana-1298	293	9	𝜁−	𝜁−	ADJ
cana-1298	293	10	𝜉	𝜉	NOUN
cana-1298	293	11	)	)	PUNCT
cana-1298	293	12	}	}	PUNCT
cana-1298	294	1	=	=	PUNCT
cana-1298	294	2	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	294	3	+	+	PROPN
cana-1298	294	4	(	(	PUNCT
cana-1298	294	5	𝜚−𝜐	𝜚−𝜐	NOUN
cana-1298	294	6	,	,	PUNCT
cana-1298	294	7	𝜁−	𝜁−	ADJ
cana-1298	294	8	𝜉	𝜉	NOUN
cana-1298	294	9	)	)	PUNCT
cana-1298	294	10	=	=	PUNCT
cana-1298	295	1	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	295	2	+	+	PROPN
cana-1298	295	3	[	[	X
cana-1298	295	4	(	(	PUNCT
cana-1298	295	5	𝜚	𝜚	NOUN
cana-1298	295	6	,	,	PUNCT
cana-1298	295	7	𝜁)−(𝜐	𝜁)−(𝜐	NOUN
cana-1298	295	8	,	,	PUNCT
cana-1298	295	9	𝜉	𝜉	NOUN
cana-1298	295	10	)	)	PUNCT
cana-1298	295	11	]	]	PUNCT
cana-1298	295	12			NUM
cana-1298	295	13	rmin{𝔚𝑖	rmin{𝔚𝑖	VERB
cana-1298	296	1	+	+	PROPN
cana-1298	296	2	(	(	PUNCT
cana-1298	296	3	𝜚	𝜚	NOUN
cana-1298	296	4	,	,	PUNCT
cana-1298	296	5	𝜁	𝜁	NOUN
cana-1298	296	6	)	)	PUNCT
cana-1298	296	7	,	,	PUNCT
cana-1298	296	8	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	296	9	+	+	PROPN
cana-1298	296	10	(	(	PUNCT
cana-1298	296	11	𝜐	𝜐	PROPN
cana-1298	296	12	,	,	PUNCT
cana-1298	296	13	𝜉	𝜉	NOUN
cana-1298	296	14	)	)	PUNCT
cana-1298	296	15	}	}	PUNCT
cana-1298	296	16	=	=	PUNCT
cana-1298	297	1	rmin{rmin{𝔓i	rmin{rmin{𝔓i	NOUN
cana-1298	298	1	+	+	ADJ
cana-1298	298	2	(	(	PUNCT
cana-1298	298	3	𝜚	𝜚	NOUN
cana-1298	298	4	)	)	PUNCT
cana-1298	298	5	,	,	PUNCT
cana-1298	298	6	𝔓i	𝔓i	PROPN
cana-1298	298	7	+	+	PROPN
cana-1298	298	8	(	(	PUNCT
cana-1298	298	9	𝜁	𝜁	NOUN
cana-1298	298	10	)	)	PUNCT
cana-1298	298	11	}	}	PUNCT
cana-1298	298	12	,	,	PUNCT
cana-1298	298	13	rmin{𝔓i	rmin{𝔓i	NOUN
cana-1298	298	14	+	+	PROPN
cana-1298	298	15	(	(	PUNCT
cana-1298	298	16	𝜐	𝜐	NOUN
cana-1298	298	17	)	)	PUNCT
cana-1298	298	18	,	,	PUNCT
cana-1298	298	19	𝔓i	𝔓i	PROPN
cana-1298	298	20	+	+	PROPN
cana-1298	298	21	(	(	PUNCT
cana-1298	298	22	𝜉	𝜉	NOUN
cana-1298	298	23	)	)	PUNCT
cana-1298	298	24	}	}	PUNCT
cana-1298	298	25	}	}	PUNCT
cana-1298	298	26	,	,	PUNCT
cana-1298	298	27	put	put	VERB
cana-1298	298	28	𝜁	𝜁	PROPN
cana-1298	298	29	=	=	SYM
cana-1298	298	30	𝔬	𝔬	PROPN
cana-1298	298	31	and	and	CCONJ
cana-1298	298	32	𝜉	𝜉	X
cana-1298	298	33	=	=	SYM
cana-1298	298	34	𝔬	𝔬	NOUN
cana-1298	298	35	,	,	PUNCT
cana-1298	298	36	where	where	SCONJ
cana-1298	298	37	𝔬	𝔬	NOUN
cana-1298	298	38	is	be	AUX
cana-1298	298	39	an	an	DET
cana-1298	298	40	first	first	ADJ
cana-1298	298	41	operation	operation	NOUN
cana-1298	298	42	identity	identity	NOUN
cana-1298	298	43	element	element	NOUN
cana-1298	298	44	of	of	ADP
cana-1298	298	45	ℨ	ℨ	NOUN
cana-1298	298	46	,	,	PUNCT
cana-1298	298	47	then	then	ADV
cana-1298	298	48	𝔓i	𝔓i	PROPN
cana-1298	298	49	+	+	PROPN
cana-1298	298	50	(	(	PUNCT
cana-1298	298	51	𝜚−𝜐	𝜚−𝜐	NOUN
cana-1298	298	52	)	)	PUNCT
cana-1298	298	53			NUM
cana-1298	298	54	rmin{𝔓i	rmin{𝔓i	NOUN
cana-1298	299	1	+	+	PROPN
cana-1298	299	2	(	(	PUNCT
cana-1298	299	3	𝜚	𝜚	NOUN
cana-1298	299	4	)	)	PUNCT
cana-1298	299	5	,	,	PUNCT
cana-1298	300	1	𝔓i	𝔓i	PROPN
cana-1298	300	2	+	+	PROPN
cana-1298	300	3	(	(	PUNCT
cana-1298	300	4	𝜐	𝜐	NOUN
cana-1298	300	5	)	)	PUNCT
cana-1298	300	6	}	}	PUNCT
cana-1298	300	7	,	,	PUNCT
cana-1298	300	8	for	for	ADP
cana-1298	300	9	all	all	DET
cana-1298	300	10	𝜚	𝜚	NOUN
cana-1298	300	11	,	,	PUNCT
cana-1298	300	12	𝜐	𝜐	PROPN
cana-1298	300	13	in	in	ADP
cana-1298	300	14	ℨ.	ℨ.	PROPN
cana-1298	300	15	and	and	CCONJ
cana-1298	300	16	rmin{𝔓i	rmin{𝔓i	NOUN
cana-1298	300	17	+	+	NOUN
cana-1298	300	18	(	(	PUNCT
cana-1298	300	19	𝜚𝜐	𝜚𝜐	NOUN
cana-1298	300	20	)	)	PUNCT
cana-1298	300	21	,	,	PUNCT
cana-1298	300	22	𝔓i	𝔓i	PROPN
cana-1298	300	23	+	+	PROPN
cana-1298	300	24	(	(	PUNCT
cana-1298	300	25	𝜁𝜉	𝜁𝜉	NOUN
cana-1298	300	26	)	)	PUNCT
cana-1298	300	27	}	}	PUNCT
cana-1298	300	28	=	=	PUNCT
cana-1298	301	1	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	301	2	+	+	PROPN
cana-1298	301	3	(	(	PUNCT
cana-1298	301	4	𝜚𝜐	𝜚𝜐	PROPN
cana-1298	301	5	,	,	PUNCT
cana-1298	301	6	𝜁𝜉	𝜁𝜉	NOUN
cana-1298	301	7	)	)	PUNCT
cana-1298	301	8	=	=	PUNCT
cana-1298	302	1	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	302	2	+	+	PROPN
cana-1298	302	3	[	[	X
cana-1298	302	4	(	(	PUNCT
cana-1298	302	5	𝜚	𝜚	NOUN
cana-1298	302	6	,	,	PUNCT
cana-1298	302	7	𝜁)(𝜐	𝜁)(𝜐	NOUN
cana-1298	302	8	,	,	PUNCT
cana-1298	302	9	𝜉	𝜉	NOUN
cana-1298	302	10	)	)	PUNCT
cana-1298	302	11	]	]	PUNCT
cana-1298	302	12			NUM
cana-1298	302	13	rmin{𝔚𝑖	rmin{𝔚𝑖	VERB
cana-1298	303	1	+	+	PROPN
cana-1298	303	2	(	(	PUNCT
cana-1298	303	3	𝜚	𝜚	NOUN
cana-1298	303	4	,	,	PUNCT
cana-1298	303	5	𝜁	𝜁	NOUN
cana-1298	303	6	)	)	PUNCT
cana-1298	303	7	,	,	PUNCT
cana-1298	303	8	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	303	9	+	+	PROPN
cana-1298	303	10	(	(	PUNCT
cana-1298	303	11	𝜐	𝜐	PROPN
cana-1298	303	12	,	,	PUNCT
cana-1298	303	13	𝜉	𝜉	NOUN
cana-1298	303	14	)	)	PUNCT
cana-1298	303	15	}	}	PUNCT
cana-1298	303	16	=	=	PUNCT
cana-1298	304	1	rmin{rmin{𝔓i	rmin{rmin{𝔓i	NOUN
cana-1298	305	1	+	+	ADJ
cana-1298	305	2	(	(	PUNCT
cana-1298	305	3	𝜚	𝜚	NOUN
cana-1298	305	4	)	)	PUNCT
cana-1298	305	5	,	,	PUNCT
cana-1298	305	6	𝔓i	𝔓i	PROPN
cana-1298	305	7	+	+	PROPN
cana-1298	305	8	(	(	PUNCT
cana-1298	305	9	𝜁	𝜁	NOUN
cana-1298	305	10	)	)	PUNCT
cana-1298	305	11	}	}	PUNCT
cana-1298	305	12	,	,	PUNCT
cana-1298	305	13	rmin{𝔓i	rmin{𝔓i	NOUN
cana-1298	305	14	+	+	PROPN
cana-1298	305	15	(	(	PUNCT
cana-1298	305	16	𝜐	𝜐	NOUN
cana-1298	305	17	)	)	PUNCT
cana-1298	305	18	,	,	PUNCT
cana-1298	305	19	𝔓i	𝔓i	PROPN
cana-1298	305	20	+	+	PROPN
cana-1298	305	21	(	(	PUNCT
cana-1298	305	22	𝜉	𝜉	NOUN
cana-1298	305	23	)	)	PUNCT
cana-1298	305	24	}	}	PUNCT
cana-1298	305	25	}	}	PUNCT
cana-1298	305	26	,	,	PUNCT
cana-1298	305	27	put	put	VERB
cana-1298	305	28	𝜁	𝜁	PROPN
cana-1298	305	29	=	=	SYM
cana-1298	305	30	𝔬	𝔬	PROPN
cana-1298	305	31	and	and	CCONJ
cana-1298	305	32	𝜉	𝜉	X
cana-1298	305	33	=	=	SYM
cana-1298	305	34	𝔬	𝔬	NOUN
cana-1298	305	35	,	,	PUNCT
cana-1298	305	36	where	where	SCONJ
cana-1298	305	37	𝔬	𝔬	NOUN
cana-1298	305	38	is	be	AUX
cana-1298	305	39	an	an	DET
cana-1298	305	40	first	first	ADJ
cana-1298	305	41	operation	operation	NOUN
cana-1298	305	42	identity	identity	NOUN
cana-1298	305	43	element	element	NOUN
cana-1298	305	44	of	of	ADP
cana-1298	305	45	ℨ	ℨ	NOUN
cana-1298	305	46	,	,	PUNCT
cana-1298	305	47	then	then	ADV
cana-1298	305	48	𝔓i	𝔓i	PROPN
cana-1298	305	49	+	+	PROPN
cana-1298	305	50	(	(	PUNCT
cana-1298	305	51	𝜚𝜐	𝜚𝜐	NOUN
cana-1298	305	52	)	)	PUNCT
cana-1298	305	53			NUM
cana-1298	305	54	rmin{𝔓i	rmin{𝔓i	NOUN
cana-1298	306	1	+	+	PROPN
cana-1298	306	2	(	(	PUNCT
cana-1298	306	3	𝜚	𝜚	NOUN
cana-1298	306	4	)	)	PUNCT
cana-1298	306	5	,	,	PUNCT
cana-1298	306	6	𝔓i	𝔓i	PROPN
cana-1298	306	7	+	+	PROPN
cana-1298	306	8	(	(	PUNCT
cana-1298	306	9	𝜐	𝜐	NOUN
cana-1298	306	10	)	)	PUNCT
cana-1298	306	11	}	}	PUNCT
cana-1298	306	12	,	,	PUNCT
cana-1298	306	13	for	for	ADP
cana-1298	306	14	all	all	DET
cana-1298	306	15	𝜚	𝜚	NOUN
cana-1298	306	16	,	,	PUNCT
cana-1298	306	17	𝜐	𝜐	PROPN
cana-1298	306	18	in	in	ADP
cana-1298	306	19	ℨ.	ℨ.	PROPN
cana-1298	306	20	also	also	ADV
cana-1298	306	21	rmax{𝔓i	rmax{𝔓i	NOUN
cana-1298	306	22	−(𝜚−𝜐	−(𝜚−𝜐	NOUN
cana-1298	306	23	)	)	PUNCT
cana-1298	306	24	,	,	PUNCT
cana-1298	306	25	𝔓i	𝔓i	PROPN
cana-1298	306	26	−(𝜁−	−(𝜁−	NOUN
cana-1298	306	27	𝜉	𝜉	NOUN
cana-1298	306	28	)	)	PUNCT
cana-1298	306	29	}	}	PUNCT
cana-1298	306	30	=	=	SYM
cana-1298	306	31	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	306	32	−(𝜚−𝜐	−(𝜚−𝜐	NUM
cana-1298	306	33	,	,	PUNCT
cana-1298	306	34	𝜁−	𝜁−	ADJ
cana-1298	306	35	𝜉	𝜉	NOUN
cana-1298	306	36	)	)	PUNCT
cana-1298	306	37	=	=	SYM
cana-1298	307	1	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	307	2	−[(𝜚	−[(𝜚	NUM
cana-1298	307	3	,	,	PUNCT
cana-1298	307	4	𝜁)−(𝜐	𝜁)−(𝜐	NOUN
cana-1298	307	5	,	,	PUNCT
cana-1298	307	6	𝜉	𝜉	NOUN
cana-1298	307	7	)	)	PUNCT
cana-1298	307	8	]	]	PUNCT
cana-1298	307	9			NUM
cana-1298	307	10	rmax{𝔚𝑖	rmax{𝔚𝑖	ADJ
cana-1298	307	11	−(𝜚	−(𝜚	NOUN
cana-1298	307	12	,	,	PUNCT
cana-1298	307	13	𝜁	𝜁	NOUN
cana-1298	307	14	)	)	PUNCT
cana-1298	307	15	,	,	PUNCT
cana-1298	307	16	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	307	17	−(𝜐	−(𝜐	NOUN
cana-1298	307	18	,	,	PUNCT
cana-1298	307	19	𝜉	𝜉	NOUN
cana-1298	307	20	)	)	PUNCT
cana-1298	307	21	}	}	PUNCT
cana-1298	307	22	=	=	SYM
cana-1298	307	23	rmax{rmax{𝔓i	rmax{rmax{𝔓i	NOUN
cana-1298	307	24	−(𝜚	−(𝜚	NOUN
cana-1298	307	25	)	)	PUNCT
cana-1298	307	26	,	,	PUNCT
cana-1298	307	27	𝔓i	𝔓i	PROPN
cana-1298	307	28	−(𝜁	−(𝜁	NOUN
cana-1298	307	29	)	)	PUNCT
cana-1298	307	30	}	}	PUNCT
cana-1298	307	31	,	,	PUNCT
cana-1298	307	32	rmax{𝔓i	rmax{𝔓i	NOUN
cana-1298	307	33	−(𝜐	−(𝜐	PROPN
cana-1298	307	34	)	)	PUNCT
cana-1298	307	35	,	,	PUNCT
cana-1298	307	36	𝔓i	𝔓i	PROPN
cana-1298	307	37	−(𝜉	−(𝜉	NOUN
cana-1298	307	38	)	)	PUNCT
cana-1298	307	39	}	}	PUNCT
cana-1298	307	40	}	}	PUNCT
cana-1298	307	41	,	,	PUNCT
cana-1298	307	42	put	put	VERB
cana-1298	307	43	𝜁	𝜁	PROPN
cana-1298	307	44	=	=	SYM
cana-1298	307	45	𝔬	𝔬	PROPN
cana-1298	307	46	and	and	CCONJ
cana-1298	307	47	𝜉	𝜉	X
cana-1298	307	48	=	=	SYM
cana-1298	307	49	𝔬	𝔬	NOUN
cana-1298	307	50	,	,	PUNCT
cana-1298	307	51	where	where	SCONJ
cana-1298	307	52	𝔬	𝔬	NOUN
cana-1298	307	53	is	be	AUX
cana-1298	307	54	an	an	DET
cana-1298	307	55	first	first	ADJ
cana-1298	307	56	operation	operation	NOUN
cana-1298	307	57	identity	identity	NOUN
cana-1298	307	58	element	element	NOUN
cana-1298	307	59	of	of	ADP
cana-1298	307	60	ℨ	ℨ	NOUN
cana-1298	307	61	,	,	PUNCT
cana-1298	307	62	then	then	ADV
cana-1298	307	63	𝔓i	𝔓i	PROPN
cana-1298	307	64	−(𝜚−𝜐	−(𝜚−𝜐	NOUN
cana-1298	307	65	)	)	PUNCT
cana-1298	307	66			NUM
cana-1298	307	67	rmax{𝔓i	rmax{𝔓i	NOUN
cana-1298	307	68	−(𝜚	−(𝜚	NOUN
cana-1298	307	69	)	)	PUNCT
cana-1298	307	70	,	,	PUNCT
cana-1298	307	71	𝔓i	𝔓i	PROPN
cana-1298	307	72	−(𝜐	−(𝜐	NOUN
cana-1298	307	73	)	)	PUNCT
cana-1298	307	74	}	}	PUNCT
cana-1298	307	75	,	,	PUNCT
cana-1298	307	76	for	for	ADP
cana-1298	307	77	all	all	DET
cana-1298	307	78	𝜚	𝜚	NOUN
cana-1298	307	79	,	,	PUNCT
cana-1298	307	80	𝜐	𝜐	PROPN
cana-1298	307	81	in	in	ADP
cana-1298	307	82	ℨ.	ℨ.	PROPN
cana-1298	307	83	communications	communication	NOUN
cana-1298	307	84	on	on	ADP
cana-1298	307	85	applied	apply	VERB
cana-1298	307	86	nonlinear	nonlinear	ADJ
cana-1298	307	87	analysis	analysis	NOUN
cana-1298	307	88	issn	issn	NOUN
cana-1298	307	89	:	:	PUNCT
cana-1298	307	90	1074	1074	NUM
cana-1298	307	91	-	-	PUNCT
cana-1298	307	92	133x	133x	NUM
cana-1298	307	93	vol	vol	NOUN
cana-1298	307	94	31	31	NUM
cana-1298	307	95	no	no	NOUN
cana-1298	307	96	.	.	PUNCT
cana-1298	308	1	7s	7	NOUN
cana-1298	308	2	(	(	PUNCT
cana-1298	308	3	2024	2024	NUM
cana-1298	308	4	)	)	PUNCT
cana-1298	308	5	237	237	NUM
cana-1298	308	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1298	308	7	and	and	CCONJ
cana-1298	308	8	rmax{𝔓i	rmax{𝔓i	NOUN
cana-1298	308	9	−(𝜚𝜐	−(𝜚𝜐	NOUN
cana-1298	308	10	)	)	PUNCT
cana-1298	308	11	,	,	PUNCT
cana-1298	308	12	𝔓i	𝔓i	PROPN
cana-1298	308	13	−(𝜁𝜉	−(𝜁𝜉	NOUN
cana-1298	308	14	)	)	PUNCT
cana-1298	308	15	}	}	PUNCT
cana-1298	309	1	=	=	SYM
cana-1298	309	2	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	309	3	−(𝜚𝜐	−(𝜚𝜐	NOUN
cana-1298	309	4	,	,	PUNCT
cana-1298	309	5	𝜁𝜉	𝜁𝜉	NOUN
cana-1298	309	6	)	)	PUNCT
cana-1298	309	7	=	=	SYM
cana-1298	310	1	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	310	2	−[(𝜚	−[(𝜚	NOUN
cana-1298	310	3	,	,	PUNCT
cana-1298	310	4	𝜁)(𝜐	𝜁)(𝜐	VERB
cana-1298	310	5	,	,	PUNCT
cana-1298	310	6	𝜉	𝜉	NOUN
cana-1298	310	7	)	)	PUNCT
cana-1298	310	8	]	]	PUNCT
cana-1298	310	9			NUM
cana-1298	310	10	rmax{𝔚𝑖	rmax{𝔚𝑖	ADJ
cana-1298	310	11	−(𝜚	−(𝜚	NOUN
cana-1298	310	12	,	,	PUNCT
cana-1298	310	13	𝜁	𝜁	NOUN
cana-1298	310	14	)	)	PUNCT
cana-1298	310	15	,	,	PUNCT
cana-1298	310	16	𝔚𝑖	𝔚𝑖	PROPN
cana-1298	310	17	−(𝜐	−(𝜐	NOUN
cana-1298	310	18	,	,	PUNCT
cana-1298	310	19	𝜉	𝜉	NOUN
cana-1298	310	20	)	)	PUNCT
cana-1298	310	21	}	}	PUNCT
cana-1298	310	22	=	=	SYM
cana-1298	310	23	rmax{rmax{𝔓i	rmax{rmax{𝔓i	NOUN
cana-1298	310	24	−(𝜚	−(𝜚	NOUN
cana-1298	310	25	)	)	PUNCT
cana-1298	310	26	,	,	PUNCT
cana-1298	310	27	𝔓i	𝔓i	PROPN
cana-1298	310	28	−(𝜁	−(𝜁	NOUN
cana-1298	310	29	)	)	PUNCT
cana-1298	310	30	}	}	PUNCT
cana-1298	310	31	,	,	PUNCT
cana-1298	310	32	rmax{𝔓i	rmax{𝔓i	NOUN
cana-1298	310	33	−(𝜐	−(𝜐	PROPN
cana-1298	310	34	)	)	PUNCT
cana-1298	310	35	,	,	PUNCT
cana-1298	310	36	𝔓i	𝔓i	PROPN
cana-1298	310	37	−(𝜉	−(𝜉	NOUN
cana-1298	310	38	)	)	PUNCT
cana-1298	310	39	}	}	PUNCT
cana-1298	310	40	}	}	PUNCT
cana-1298	310	41	,	,	PUNCT
cana-1298	310	42	put	put	VERB
cana-1298	310	43	𝜁	𝜁	PROPN
cana-1298	310	44	=	=	SYM
cana-1298	310	45	𝔬	𝔬	PROPN
cana-1298	310	46	and	and	CCONJ
cana-1298	310	47	𝜉	𝜉	X
cana-1298	310	48	=	=	SYM
cana-1298	310	49	𝔬	𝔬	NOUN
cana-1298	310	50	,	,	PUNCT
cana-1298	310	51	where	where	SCONJ
cana-1298	310	52	𝔬	𝔬	NOUN
cana-1298	310	53	is	be	AUX
cana-1298	310	54	an	an	DET
cana-1298	310	55	first	first	ADJ
cana-1298	310	56	operation	operation	NOUN
cana-1298	310	57	identity	identity	NOUN
cana-1298	310	58	element	element	NOUN
cana-1298	310	59	of	of	ADP
cana-1298	310	60	ℨ	ℨ	NOUN
cana-1298	310	61	,	,	PUNCT
cana-1298	310	62	then	then	ADV
cana-1298	310	63	𝔓i	𝔓i	PROPN
cana-1298	310	64	−(𝜚𝜐	−(𝜚𝜐	NOUN
cana-1298	310	65	)	)	PUNCT
cana-1298	310	66			NUM
cana-1298	310	67	rmax{𝔓i	rmax{𝔓i	NOUN
cana-1298	310	68	−(𝜚	−(𝜚	NOUN
cana-1298	310	69	)	)	PUNCT
cana-1298	310	70	,	,	PUNCT
cana-1298	310	71	𝔓i	𝔓i	PROPN
cana-1298	310	72	−(𝜐	−(𝜐	NOUN
cana-1298	310	73	)	)	PUNCT
cana-1298	310	74	}	}	PUNCT
cana-1298	310	75	,	,	PUNCT
cana-1298	310	76	for	for	ADP
cana-1298	310	77	all	all	DET
cana-1298	310	78	𝜚	𝜚	NOUN
cana-1298	310	79	,	,	PUNCT
cana-1298	310	80	𝜐	𝜐	PROPN
cana-1298	310	81	in	in	ADP
cana-1298	310	82	ℨ.	ℨ.	PROPN
cana-1298	310	83	hence	hence	ADV
cana-1298	310	84	𝔓	𝔓	PROPN
cana-1298	310	85	is	be	AUX
cana-1298	310	86	a	a	DET
cana-1298	310	87	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	310	88	of	of	ADP
cana-1298	310	89	ℨ.	ℨ.	PROPN
cana-1298	310	90	theorem	theorem	VERB
cana-1298	310	91	2.10	2.10	NUM
cana-1298	310	92	.	.	PUNCT
cana-1298	311	1	𝐿𝑒𝑡	𝐿𝑒𝑡	PROPN
cana-1298	311	2	𝔓1	𝔓1	PROPN
cana-1298	311	3	,	,	PUNCT
cana-1298	311	4	𝔓2	𝔓2	NOUN
cana-1298	311	5	,	,	PUNCT
cana-1298	311	6	…	…	PUNCT
cana-1298	311	7	,	,	PUNCT
cana-1298	311	8	𝔓𝑚	𝔓𝑚	NOUN
cana-1298	311	9	be	be	AUX
cana-1298	311	10	𝔹𝕍𝕄𝕀𝔽𝕊𝑠	𝔹𝕍𝕄𝕀𝔽𝕊𝑠	PROPN
cana-1298	311	11	of	of	ADP
cana-1298	311	12	a	a	DET
cana-1298	311	13	ring	ring	NOUN
cana-1298	311	14	ℨ	ℨ	NOUN
cana-1298	311	15	and	and	CCONJ
cana-1298	311	16	𝔐	𝔐	PRON
cana-1298	311	17	be	be	AUX
cana-1298	311	18	the	the	DET
cana-1298	311	19	strongest	strong	ADJ
cana-1298	311	20	𝔹𝕍𝕄𝕀𝔽	𝔹𝕍𝕄𝕀𝔽	PROPN
cana-1298	311	21	ndimensional	ndimensional	ADJ
cana-1298	311	22	relation	relation	NOUN
cana-1298	311	23	of	of	ADP
cana-1298	311	24	ℨ.	ℨ.	PROPN
cana-1298	311	25	then	then	ADV
cana-1298	311	26	𝔓1	𝔓1	PROPN
cana-1298	311	27	,	,	PUNCT
cana-1298	311	28	𝔓2	𝔓2	NOUN
cana-1298	311	29	,	,	PUNCT
cana-1298	311	30	…	…	PUNCT
cana-1298	311	31	,	,	PUNCT
cana-1298	311	32	𝔓𝑚	𝔓𝑚	PROPN
cana-1298	311	33	are	be	AUX
cana-1298	311	34	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	311	35	𝑜𝑓	𝑜𝑓	SCONJ
cana-1298	311	36	ℨ	ℨ	NOUN
cana-1298	311	37	𝑖𝑓	𝑖𝑓	VERB
cana-1298	311	38	𝑎𝑛𝑑	𝑎𝑛𝑑	NOUN
cana-1298	311	39	𝑜𝑛𝑙𝑦	𝑜𝑛𝑙𝑦	ADV
cana-1298	312	1	𝑖𝑓	𝑖𝑓	ADP
cana-1298	312	2	𝔐	𝔐	INTJ
cana-1298	312	3	𝑖𝑠	𝑖𝑠	NOUN
cana-1298	312	4	𝑎	𝑎	DET
cana-1298	312	5	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	𝔹𝕍𝕄𝕀𝔽𝕊ℝ	PROPN
cana-1298	312	6	𝑜𝑓	𝑜𝑓	ADP
cana-1298	312	7	ℨ×ℨ	ℨ×ℨ	NOUN
cana-1298	312	8	…	…	PUNCT
cana-1298	312	9	×ℨ	×ℨ	X
cana-1298	312	10	(	(	PUNCT
cana-1298	312	11	m	m	NOUN
cana-1298	312	12	times	time	NOUN
cana-1298	312	13	)	)	PUNCT
cana-1298	312	14	.	.	PUNCT
cana-1298	313	1	proof	proof	NOUN
cana-1298	313	2	.	.	PUNCT
cana-1298	314	1	𝐹𝑟𝑜𝑚	𝐹𝑟𝑜𝑚	PROPN
cana-1298	314	2	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-1298	314	3	𝑇ℎ𝑒𝑜𝑟𝑒𝑚	𝑇ℎ𝑒𝑜𝑟𝑒𝑚	PROPN
cana-1298	314	4	2.9	2.9	NUM
cana-1298	314	5	,	,	PUNCT
cana-1298	314	6	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-1298	314	7	𝑝𝑟𝑜𝑜𝑓	𝑝𝑟𝑜𝑜𝑓	NOUN
cana-1298	314	8	𝑖𝑠	𝑖𝑠	PROPN
cana-1298	314	9	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	NOUN
cana-1298	314	10	3	3	NUM
cana-1298	314	11	.	.	PUNCT
cana-1298	314	12	conclusion	conclusion	NOUN
cana-1298	314	13	properties	property	NOUN
cana-1298	314	14	of	of	ADP
cana-1298	314	15	transformations	transformation	NOUN
cana-1298	314	16	of	of	ADP
cana-1298	314	17	𝔹𝕍𝕄𝕀𝔽𝕊ℝof	𝔹𝕍𝕄𝕀𝔽𝕊ℝof	PROPN
cana-1298	314	18	a	a	DET
cana-1298	314	19	ring	ring	NOUN
cana-1298	314	20	have	have	AUX
cana-1298	314	21	been	be	AUX
cana-1298	314	22	discussed	discuss	VERB
cana-1298	314	23	.	.	PUNCT
cana-1298	315	1	the	the	DET
cana-1298	315	2	above	above	ADJ
cana-1298	315	3	concepts	concept	NOUN
cana-1298	315	4	can	can	AUX
cana-1298	315	5	be	be	AUX
cana-1298	315	6	extended	extend	VERB
cana-1298	315	7	into	into	ADP
cana-1298	315	8	bipolar	bipolar	ADJ
cana-1298	315	9	valued	value	VERB
cana-1298	315	10	multi	multi	NOUN
cana-1298	316	1	i	i	PRON
cana-1298	316	2	-	-	PUNCT
cana-1298	316	3	fuzzy	fuzzy	ADJ
cana-1298	316	4	subfield	subfield	NOUN
cana-1298	316	5	of	of	ADP
cana-1298	316	6	a	a	DET
cana-1298	316	7	field	field	NOUN
cana-1298	316	8	,	,	PUNCT
cana-1298	316	9	bipolar	bipolar	ADJ
cana-1298	316	10	interval	interval	NOUN
cana-1298	316	11	valued	value	VERB
cana-1298	316	12	multi	multi	NOUN
cana-1298	316	13	fuzzy	fuzzy	ADJ
cana-1298	316	14	subspace	subspace	NOUN
cana-1298	316	15	of	of	ADP
cana-1298	316	16	a	a	DET
cana-1298	316	17	linear	linear	ADJ
cana-1298	316	18	space	space	NOUN
cana-1298	316	19	and	and	CCONJ
cana-1298	316	20	any	any	DET
cana-1298	316	21	other	other	ADJ
cana-1298	316	22	algebraic	algebraic	ADJ
cana-1298	316	23	system	system	NOUN
cana-1298	316	24	.	.	PUNCT
cana-1298	317	1	references	reference	NOUN
cana-1298	317	2	[	[	X
cana-1298	317	3	1	1	NUM
cana-1298	317	4	]	]	PUNCT
cana-1298	317	5	anitha.m.s	anitha.m.s	ADV
cana-1298	317	6	.	.	PUNCT
cana-1298	317	7	,	,	PUNCT
cana-1298	317	8	muruganantha	muruganantha	PROPN
cana-1298	317	9	prasad	prasad	PROPN
cana-1298	317	10	&	&	CCONJ
cana-1298	317	11	k.arjunan	k.arjunan	PROPN
cana-1298	317	12	,	,	PUNCT
cana-1298	317	13	“	"	PUNCT
cana-1298	317	14	notes	note	NOUN
cana-1298	317	15	on	on	ADP
cana-1298	317	16	bipolar	bipolar	ADV
cana-1298	317	17	-	-	PUNCT
cana-1298	317	18	valued	value	VERB
cana-1298	317	19	fuzzy	fuzzy	ADJ
cana-1298	317	20	subgroups	subgroup	NOUN
cana-1298	317	21	of	of	ADP
cana-1298	317	22	a	a	DET
cana-1298	317	23	group	group	NOUN
cana-1298	317	24	”	"	PUNCT
cana-1298	317	25	,	,	PUNCT
cana-1298	317	26	bulletin	bulletin	NOUN
cana-1298	317	27	of	of	ADP
cana-1298	317	28	society	society	NOUN
cana-1298	317	29	for	for	ADP
cana-1298	317	30	mathematical	mathematical	ADJ
cana-1298	317	31	services	service	NOUN
cana-1298	317	32	and	and	CCONJ
cana-1298	317	33	standards	standard	NOUN
cana-1298	317	34	,	,	PUNCT
cana-1298	317	35	vol	vol	NOUN
cana-1298	317	36	.	.	NOUN
cana-1298	317	37	2	2	NUM
cana-1298	318	1	no	no	NOUN
cana-1298	318	2	.	.	NOUN
cana-1298	318	3	3	3	NUM
cana-1298	318	4	(	(	PUNCT
cana-1298	318	5	2013	2013	NUM
cana-1298	318	6	)	)	PUNCT
cana-1298	318	7	,	,	PUNCT
cana-1298	318	8	pp	pp	ADP
cana-1298	318	9	.	.	PUNCT
cana-1298	319	1	52	52	NUM
cana-1298	319	2	−	−	NOUN
cana-1298	319	3	59	59	NUM
cana-1298	319	4	.	.	PUNCT
cana-1298	320	1	[	[	X
cana-1298	320	2	2	2	NUM
cana-1298	320	3	]	]	PUNCT
cana-1298	320	4	anthony.j.m	anthony.j.m	NOUN
cana-1298	320	5	and	and	CCONJ
cana-1298	320	6	h.sherwood	h.sherwood	NOUN
cana-1298	320	7	,	,	PUNCT
cana-1298	320	8	“	"	PUNCT
cana-1298	320	9	fuzzy	fuzzy	ADJ
cana-1298	320	10	groups	group	NOUN
cana-1298	320	11	redefined	redefine	VERB
cana-1298	320	12	”	"	PUNCT
cana-1298	320	13	,	,	PUNCT
cana-1298	320	14	journal	journal	NOUN
cana-1298	320	15	of	of	ADP
cana-1298	320	16	mathematical	mathematical	ADJ
cana-1298	320	17	analysis	analysis	NOUN
cana-1298	320	18	and	and	CCONJ
cana-1298	320	19	applications	application	NOUN
cana-1298	320	20	,	,	PUNCT
cana-1298	320	21	69(1979	69(1979	NUM
cana-1298	320	22	)	)	PUNCT
cana-1298	320	23	,	,	PUNCT
cana-1298	320	24	124	124	NUM
cana-1298	320	25	−130	−130	ADJ
cana-1298	320	26	.	.	PUNCT
cana-1298	321	1	[	[	X
cana-1298	321	2	3	3	X
cana-1298	321	3	]	]	SYM
cana-1298	321	4	arsham	arsham	PROPN
cana-1298	321	5	borumand	borumand	PROPN
cana-1298	321	6	saeid	saeid	PROPN
cana-1298	321	7	,	,	PUNCT
cana-1298	321	8	“	"	PUNCT
cana-1298	321	9	bipolar	bipolar	ADJ
cana-1298	321	10	-	-	PUNCT
cana-1298	321	11	valued	value	VERB
cana-1298	321	12	fuzzy	fuzzy	ADJ
cana-1298	321	13	bck	bck	PROPN
cana-1298	321	14	/	/	SYM
cana-1298	321	15	bci	bci	NOUN
cana-1298	321	16	-	-	PUNCT
cana-1298	321	17	algebras	algebra	NOUN
cana-1298	321	18	”	"	PUNCT
cana-1298	321	19	,	,	PUNCT
cana-1298	321	20	world	world	NOUN
cana-1298	321	21	applied	apply	VERB
cana-1298	321	22	sciences	science	NOUN
cana-1298	321	23	journal	journal	NOUN
cana-1298	321	24	,	,	PUNCT
cana-1298	321	25	7	7	NUM
cana-1298	321	26	(	(	PUNCT
cana-1298	321	27	11	11	NUM
cana-1298	321	28	)	)	PUNCT
cana-1298	321	29	(	(	PUNCT
cana-1298	321	30	2009	2009	NUM
cana-1298	321	31	)	)	PUNCT
cana-1298	321	32	,	,	PUNCT
cana-1298	321	33	1404	1404	NUM
cana-1298	321	34	−	−	NOUN
cana-1298	321	35	1411	1411	NUM
cana-1298	321	36	.	.	PUNCT
cana-1298	322	1	[	[	X
cana-1298	322	2	4	4	X
cana-1298	322	3	]	]	PUNCT
cana-1298	322	4	azriel	azriel	PROPN
cana-1298	322	5	rosenfeld	rosenfeld	PROPN
cana-1298	322	6	,	,	PUNCT
cana-1298	322	7	“	"	PUNCT
cana-1298	322	8	fuzzy	fuzzy	ADJ
cana-1298	322	9	groups	group	NOUN
cana-1298	322	10	”	"	PUNCT
cana-1298	322	11	,	,	PUNCT
cana-1298	322	12	journal	journal	NOUN
cana-1298	322	13	of	of	ADP
cana-1298	322	14	mathematical	mathematical	ADJ
cana-1298	322	15	analysis	analysis	NOUN
cana-1298	322	16	and	and	CCONJ
cana-1298	322	17	applications	application	NOUN
cana-1298	322	18	,	,	PUNCT
cana-1298	322	19	35(1971	35(1971	NUM
cana-1298	322	20	)	)	PUNCT
cana-1298	322	21	,	,	PUNCT
cana-1298	322	22	512	512	NUM
cana-1298	322	23	−	−	NUM
cana-1298	322	24	517	517	NUM
cana-1298	322	25	.	.	PUNCT
cana-1298	323	1	[	[	X
cana-1298	323	2	5	5	NUM
cana-1298	323	3	]	]	PUNCT
cana-1298	323	4	balasubramanian.a	balasubramanian.a	PROPN
cana-1298	323	5	,	,	PUNCT
cana-1298	323	6	k.l.muruganantha	k.l.muruganantha	PROPN
cana-1298	323	7	prasad	prasad	PROPN
cana-1298	323	8	&	&	CCONJ
cana-1298	323	9	k.arjunan	k.arjunan	PROPN
cana-1298	323	10	,	,	PUNCT
cana-1298	323	11	“	"	PUNCT
cana-1298	323	12	properties	property	NOUN
cana-1298	323	13	of	of	ADP
cana-1298	323	14	bipolar	bipolar	ADJ
cana-1298	323	15	interval	interval	NOUN
cana-1298	323	16	valued	value	VERB
cana-1298	323	17	fuzzy	fuzzy	ADJ
cana-1298	323	18	subgroups	subgroup	NOUN
cana-1298	323	19	of	of	ADP
cana-1298	323	20	a	a	DET
cana-1298	323	21	group	group	NOUN
cana-1298	323	22	”	"	PUNCT
cana-1298	323	23	,	,	PUNCT
cana-1298	323	24	international	international	ADJ
cana-1298	323	25	journal	journal	NOUN
cana-1298	323	26	of	of	ADP
cana-1298	323	27	scientific	scientific	ADJ
cana-1298	323	28	research	research	NOUN
cana-1298	323	29	,	,	PUNCT
cana-1298	323	30	vol	vol	NOUN
cana-1298	323	31	.	.	PROPN
cana-1298	323	32	4	4	NUM
cana-1298	323	33	,	,	PUNCT
cana-1298	323	34	iss	iss	PROPN
cana-1298	323	35	.	.	PROPN
cana-1298	323	36	4	4	NUM
cana-1298	323	37	(	(	PUNCT
cana-1298	323	38	2015	2015	NUM
cana-1298	323	39	)	)	PUNCT
cana-1298	323	40	,	,	PUNCT
cana-1298	323	41	262	262	NUM
cana-1298	323	42	268	268	NUM
cana-1298	323	43	.	.	PUNCT
cana-1298	324	1	[	[	X
cana-1298	324	2	6	6	NUM
cana-1298	324	3	]	]	SYM
cana-1298	324	4	chitra.v	chitra.v	NOUN
cana-1298	324	5	&	&	CCONJ
cana-1298	324	6	k.arjunan	k.arjunan	PROPN
cana-1298	324	7	,	,	PUNCT
cana-1298	324	8	“	"	PUNCT
cana-1298	324	9	a	a	DET
cana-1298	324	10	study	study	NOUN
cana-1298	324	11	on	on	ADP
cana-1298	324	12	q	q	ADJ
cana-1298	324	13	-	-	PUNCT
cana-1298	324	14	fuzzy	fuzzy	ADJ
cana-1298	324	15	subnearrings	subnearring	NOUN
cana-1298	324	16	of	of	ADP
cana-1298	324	17	a	a	DET
cana-1298	324	18	nearing	nearing	NOUN
cana-1298	324	19	”	"	PUNCT
cana-1298	324	20	,	,	PUNCT
cana-1298	324	21	journal	journal	NOUN
cana-1298	324	22	of	of	ADP
cana-1298	324	23	advances	advance	NOUN
cana-1298	324	24	in	in	ADP
cana-1298	324	25	mathematics	mathematic	NOUN
cana-1298	324	26	,	,	PUNCT
cana-1298	324	27	vol	vol	NOUN
cana-1298	324	28	.	.	PROPN
cana-1298	324	29	4	4	NUM
cana-1298	324	30	,	,	PUNCT
cana-1298	324	31	no	no	INTJ
cana-1298	324	32	.	.	NOUN
cana-1298	324	33	1	1	NUM
cana-1298	324	34	(	(	PUNCT
cana-1298	324	35	2013	2013	NUM
cana-1298	324	36	)	)	PUNCT
cana-1298	324	37	,	,	PUNCT
cana-1298	324	38	320	320	NUM
cana-1298	324	39	−324	−324	ADJ
cana-1298	324	40	.	.	PUNCT
cana-1298	325	1	[	[	X
cana-1298	325	2	7	7	X
cana-1298	325	3	]	]	X
cana-1298	325	4	grattan	grattan	PROPN
cana-1298	325	5	-	-	PUNCT
cana-1298	325	6	guiness	guiness	PROPN
cana-1298	325	7	,	,	PUNCT
cana-1298	325	8	“	"	PUNCT
cana-1298	325	9	fuzzy	fuzzy	ADJ
cana-1298	325	10	membership	membership	NOUN
cana-1298	325	11	mapped	map	VERB
cana-1298	325	12	onto	onto	ADP
cana-1298	325	13	interval	interval	NOUN
cana-1298	325	14	and	and	CCONJ
cana-1298	325	15	many	many	ADJ
cana-1298	325	16	valued	value	VERB
cana-1298	325	17	quantities	quantity	NOUN
cana-1298	325	18	”	"	PUNCT
cana-1298	325	19	,	,	PUNCT
cana-1298	325	20	z.math.logik	z.math.logik	PROPN
cana-1298	325	21	.	.	PUNCT
cana-1298	325	22	grundladen	grundladen	PROPN
cana-1298	325	23	math	math	NOUN
cana-1298	325	24	.	.	PUNCT
cana-1298	326	1	22	22	NUM
cana-1298	326	2	(	(	PUNCT
cana-1298	326	3	1975	1975	NUM
cana-1298	326	4	)	)	PUNCT
cana-1298	326	5	,	,	PUNCT
cana-1298	326	6	149	149	NUM
cana-1298	326	7	−	−	NUM
cana-1298	326	8	160	160	NUM
cana-1298	326	9	.	.	PUNCT
cana-1298	327	1	[	[	X
cana-1298	327	2	8	8	NUM
cana-1298	327	3	]	]	X
cana-1298	327	4	kyoung	kyoung	PROPN
cana-1298	327	5	ja	ja	PROPN
cana-1298	327	6	lee	lee	PROPN
cana-1298	327	7	,	,	PUNCT
cana-1298	327	8	“	"	PUNCT
cana-1298	327	9	bipolar	bipolar	ADJ
cana-1298	327	10	fuzzy	fuzzy	ADJ
cana-1298	327	11	subalgebras	subalgebra	NOUN
cana-1298	327	12	and	and	CCONJ
cana-1298	327	13	bipolar	bipolar	ADJ
cana-1298	327	14	fuzzy	fuzzy	ADJ
cana-1298	327	15	ideals	ideal	NOUN
cana-1298	327	16	of	of	ADP
cana-1298	327	17	bck	bck	PROPN
cana-1298	327	18	/	/	SYM
cana-1298	327	19	bci	bci	PROPN
cana-1298	327	20	-	-	PUNCT
cana-1298	327	21	algebras	algebra	NOUN
cana-1298	327	22	”	"	PUNCT
cana-1298	327	23	,	,	PUNCT
cana-1298	327	24	bull	bull	NOUN
cana-1298	327	25	.	.	PUNCT
cana-1298	328	1	malays.math	malays.math	PROPN
cana-1298	328	2	.	.	PUNCT
cana-1298	329	1	sci	sci	PROPN
cana-1298	329	2	.	.	PUNCT
cana-1298	329	3	soc	soc	PROPN
cana-1298	329	4	.	.	PUNCT
cana-1298	329	5	,	,	PUNCT
cana-1298	329	6	(	(	PUNCT
cana-1298	329	7	2	2	X
cana-1298	329	8	)	)	PUNCT
cana-1298	329	9	32(3	32(3	NUM
cana-1298	329	10	)	)	PUNCT
cana-1298	329	11	(	(	PUNCT
cana-1298	329	12	2009	2009	NUM
cana-1298	329	13	)	)	PUNCT
cana-1298	329	14	,	,	PUNCT
cana-1298	329	15	361	361	NUM
cana-1298	329	16	–	–	PUNCT
cana-1298	329	17	373	373	NUM
cana-1298	329	18	.	.	PUNCT
cana-1298	330	1	[	[	X
cana-1298	330	2	9	9	NUM
cana-1298	330	3	]	]	X
cana-1298	330	4	k.m.lee	k.m.lee	PROPN
cana-1298	330	5	,	,	PUNCT
cana-1298	330	6	“	"	PUNCT
cana-1298	330	7	bipolar	bipolar	ADJ
cana-1298	330	8	-	-	PUNCT
cana-1298	330	9	valued	value	VERB
cana-1298	330	10	fuzzy	fuzzy	ADJ
cana-1298	330	11	sets	set	NOUN
cana-1298	330	12	and	and	CCONJ
cana-1298	330	13	their	their	PRON
cana-1298	330	14	operations	operation	NOUN
cana-1298	330	15	”	"	PUNCT
cana-1298	330	16	.	.	PUNCT
cana-1298	331	1	proc	proc	NOUN
cana-1298	331	2	.	.	PUNCT
cana-1298	332	1	int	int	NOUN
cana-1298	332	2	.	.	PUNCT
cana-1298	332	3	conf	conf	PROPN
cana-1298	332	4	.	.	PUNCT
cana-1298	333	1	on	on	ADP
cana-1298	333	2	intelligent	intelligent	ADJ
cana-1298	333	3	technologies	technology	NOUN
cana-1298	333	4	,	,	PUNCT
cana-1298	333	5	bangkok	bangkok	PROPN
cana-1298	333	6	,	,	PUNCT
cana-1298	333	7	thailand	thailand	PROPN
cana-1298	333	8	,	,	PUNCT
cana-1298	333	9	(	(	PUNCT
cana-1298	333	10	2000	2000	NUM
cana-1298	333	11	)	)	PUNCT
cana-1298	333	12	,	,	PUNCT
cana-1298	333	13	307	307	NUM
cana-1298	333	14	−	−	NUM
cana-1298	333	15	312	312	NUM
cana-1298	333	16	.	.	PUNCT
cana-1298	334	1	[	[	X
cana-1298	334	2	10	10	NUM
cana-1298	334	3	]	]	X
cana-1298	334	4	k.m.lee	k.m.lee	PROPN
cana-1298	334	5	,	,	PUNCT
cana-1298	334	6	“	"	PUNCT
cana-1298	334	7	comparison	comparison	NOUN
cana-1298	334	8	of	of	ADP
cana-1298	334	9	interval	interval	NOUN
cana-1298	334	10	-	-	PUNCT
cana-1298	334	11	valued	value	VERB
cana-1298	334	12	fuzzy	fuzzy	ADJ
cana-1298	334	13	sets	set	NOUN
cana-1298	334	14	,	,	PUNCT
cana-1298	334	15	intuitionistic	intuitionistic	ADJ
cana-1298	334	16	fuzzy	fuzzy	ADJ
cana-1298	334	17	sets	set	NOUN
cana-1298	334	18	and	and	CCONJ
cana-1298	334	19	bipolarvalued	bipolarvalue	VERB
cana-1298	334	20	fuzzy	fuzzy	ADJ
cana-1298	334	21	sets	set	NOUN
cana-1298	334	22	”	"	PUNCT
cana-1298	334	23	.	.	PUNCT
cana-1298	335	1	j.	j.	PROPN
cana-1298	335	2	fuzzy	fuzzy	ADJ
cana-1298	335	3	logic	logic	NOUN
cana-1298	335	4	intelligent	intelligent	ADJ
cana-1298	335	5	systems	system	NOUN
cana-1298	335	6	,	,	PUNCT
cana-1298	335	7	14	14	NUM
cana-1298	335	8	(	(	PUNCT
cana-1298	335	9	2	2	NUM
cana-1298	335	10	)	)	PUNCT
cana-1298	335	11	(	(	PUNCT
cana-1298	335	12	2004	2004	NUM
cana-1298	335	13	)	)	PUNCT
cana-1298	335	14	,	,	PUNCT
cana-1298	335	15	125	125	NUM
cana-1298	335	16	−129	−129	NOUN
cana-1298	335	17	.	.	PUNCT
cana-1298	336	1	[	[	X
cana-1298	336	2	11	11	NUM
cana-1298	336	3	]	]	SYM
cana-1298	336	4	murugalingam.k	murugalingam.k	NOUN
cana-1298	336	5	and	and	CCONJ
cana-1298	336	6	k.arjunan	k.arjunan	NOUN
cana-1298	336	7	,	,	PUNCT
cana-1298	336	8	“	"	PUNCT
cana-1298	336	9	a	a	DET
cana-1298	336	10	study	study	NOUN
cana-1298	336	11	on	on	ADP
cana-1298	336	12	interval	interval	NOUN
cana-1298	336	13	valued	value	VERB
cana-1298	336	14	fuzzy	fuzzy	ADJ
cana-1298	336	15	subsemirings	subsemiring	NOUN
cana-1298	336	16	of	of	ADP
cana-1298	336	17	a	a	DET
cana-1298	336	18	semiring	semiring	NOUN
cana-1298	336	19	”	"	PUNCT
cana-1298	336	20	,	,	PUNCT
cana-1298	336	21	international	international	ADJ
cana-1298	336	22	journal	journal	NOUN
cana-1298	336	23	of	of	ADP
cana-1298	336	24	applied	apply	VERB
cana-1298	336	25	mathematics	mathematic	NOUN
cana-1298	336	26	and	and	CCONJ
cana-1298	336	27	modeling	modeling	NOUN
cana-1298	336	28	,	,	PUNCT
cana-1298	336	29	vol	vol	NOUN
cana-1298	336	30	.	.	PROPN
cana-1298	336	31	1	1	NUM
cana-1298	336	32	,	,	PUNCT
cana-1298	336	33	no	no	INTJ
cana-1298	336	34	.	.	NOUN
cana-1298	336	35	5	5	NUM
cana-1298	336	36	(	(	PUNCT
cana-1298	336	37	2013	2013	NUM
cana-1298	336	38	)	)	PUNCT
cana-1298	336	39	,	,	PUNCT
cana-1298	336	40	1	1	NUM
cana-1298	336	41	−	−	PROPN
cana-1298	336	42	6	6	NUM
cana-1298	336	43	.	.	PUNCT
cana-1298	337	1	[	[	X
cana-1298	337	2	12	12	NUM
cana-1298	337	3	]	]	X
cana-1298	337	4	sabu	sabu	PROPN
cana-1298	337	5	sebastian	sebastian	PROPN
cana-1298	337	6	,	,	PUNCT
cana-1298	337	7	t.v.ramakrishnan	t.v.ramakrishnan	NOUN
cana-1298	337	8	,	,	PUNCT
cana-1298	337	9	“	"	PUNCT
cana-1298	337	10	multi	multi	X
cana-1298	337	11	fuzzy	fuzzy	ADJ
cana-1298	337	12	sets	set	NOUN
cana-1298	337	13	”	"	PUNCT
cana-1298	337	14	,	,	PUNCT
cana-1298	337	15	international	international	PROPN
cana-1298	337	16	mathematical	mathematical	ADJ
cana-1298	337	17	forum	forum	PROPN
cana-1298	337	18	,	,	PUNCT
cana-1298	337	19	5	5	NUM
cana-1298	337	20	,	,	PUNCT
cana-1298	337	21	no.50	no.50	PROPN
cana-1298	337	22	(	(	PUNCT
cana-1298	337	23	2010	2010	NUM
cana-1298	337	24	)	)	PUNCT
cana-1298	337	25	,	,	PUNCT
cana-1298	337	26	2471	2471	NUM
cana-1298	337	27	−2476	−2476	NOUN
cana-1298	337	28	.	.	PUNCT
cana-1298	338	1	[	[	X
cana-1298	338	2	13	13	NUM
cana-1298	338	3	]	]	PUNCT
cana-1298	338	4	shanmugapriya.m	shanmugapriya.m	PROPN
cana-1298	338	5	.	.	PUNCT
cana-1298	339	1	m	m	PROPN
cana-1298	339	2	&	&	CCONJ
cana-1298	339	3	k.arjunan	k.arjunan	PROPN
cana-1298	339	4	,	,	PUNCT
cana-1298	339	5	“	"	PUNCT
cana-1298	339	6	notes	note	NOUN
cana-1298	339	7	on	on	ADP
cana-1298	339	8	(	(	PUNCT
cana-1298	339	9	q	q	INTJ
cana-1298	339	10	,	,	PUNCT
cana-1298	339	11	l)-fuzzy	l)-fuzzy	ADJ
cana-1298	339	12	sub	sub	NOUN
cana-1298	339	13	-	-	NOUN
cana-1298	339	14	nearrings	nearring	NOUN
cana-1298	339	15	of	of	ADP
cana-1298	339	16	a	a	DET
cana-1298	339	17	nearing	nearing	NOUN
cana-1298	339	18	”	"	PUNCT
cana-1298	339	19	,	,	PUNCT
cana-1298	339	20	international	international	ADJ
cana-1298	339	21	journal	journal	NOUN
cana-1298	339	22	of	of	ADP
cana-1298	339	23	engg	engg	PROPN
cana-1298	339	24	.	.	PUNCT
cana-1298	340	1	research	research	NOUN
cana-1298	340	2	and	and	CCONJ
cana-1298	340	3	applications	application	NOUN
cana-1298	340	4	,	,	PUNCT
cana-1298	340	5	vol.2	vol.2	PROPN
cana-1298	340	6	,	,	PUNCT
cana-1298	340	7	issue	issue	NOUN
cana-1298	340	8	2	2	NUM
cana-1298	340	9	(	(	PUNCT
cana-1298	340	10	2012	2012	NUM
cana-1298	340	11	)	)	PUNCT
cana-1298	340	12	,	,	PUNCT
cana-1298	340	13	pp	pp	ADJ
cana-1298	340	14	.	.	PUNCT
cana-1298	341	1	1633	1633	NUM
cana-1298	342	1	−	−	NOUN
cana-1298	342	2	1637	1637	NUM
cana-1298	342	3	.	.	PUNCT
cana-1298	343	1	[	[	X
cana-1298	343	2	14	14	NUM
cana-1298	343	3	]	]	PUNCT
cana-1298	343	4	somasundra	somasundra	NOUN
cana-1298	343	5	moorthy.m.g	moorthy.m.g	NUM
cana-1298	343	6	.	.	PUNCT
cana-1298	343	7	,	,	PUNCT
cana-1298	343	8	“	"	PUNCT
cana-1298	343	9	a	a	DET
cana-1298	343	10	study	study	NOUN
cana-1298	343	11	on	on	ADP
cana-1298	343	12	interval	interval	NOUN
cana-1298	343	13	valued	value	VERB
cana-1298	343	14	fuzzy	fuzzy	ADJ
cana-1298	343	15	,	,	PUNCT
cana-1298	343	16	anti	anti	X
cana-1298	343	17	fuzzy	fuzzy	ADJ
cana-1298	343	18	,	,	PUNCT
cana-1298	343	19	intuitionistic	intuitionistic	ADJ
cana-1298	343	20	fuzzy	fuzzy	ADJ
cana-1298	343	21	subrings	subring	NOUN
cana-1298	343	22	of	of	ADP
cana-1298	343	23	a	a	DET
cana-1298	343	24	ring	ring	NOUN
cana-1298	343	25	”	"	PUNCT
cana-1298	343	26	,	,	PUNCT
cana-1298	343	27	ph.d	ph.d	PROPN
cana-1298	343	28	thesis	thesis	NOUN
cana-1298	343	29	,	,	PUNCT
cana-1298	343	30	bharathidasan	bharathidasan	ADJ
cana-1298	343	31	university	university	NOUN
cana-1298	343	32	,	,	PUNCT
cana-1298	343	33	trichy	trichy	NOUN
cana-1298	343	34	,	,	PUNCT
cana-1298	343	35	2014	2014	NUM
cana-1298	343	36	.	.	PUNCT
cana-1298	344	1	communications	communication	NOUN
cana-1298	344	2	on	on	ADP
cana-1298	344	3	applied	apply	VERB
cana-1298	344	4	nonlinear	nonlinear	ADJ
cana-1298	344	5	analysis	analysis	NOUN
cana-1298	344	6	issn	issn	NOUN
cana-1298	344	7	:	:	PUNCT
cana-1298	344	8	1074	1074	NUM
cana-1298	344	9	-	-	PUNCT
cana-1298	344	10	133x	133x	NUM
cana-1298	344	11	vol	vol	NOUN
cana-1298	344	12	31	31	NUM
cana-1298	344	13	no	no	NOUN
cana-1298	344	14	.	.	PUNCT
cana-1298	345	1	7s	7	NOUN
cana-1298	345	2	(	(	PUNCT
cana-1298	345	3	2024	2024	NUM
cana-1298	345	4	)	)	PUNCT
cana-1298	345	5	238	238	NUM
cana-1298	345	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1298	346	1	[	[	X
cana-1298	346	2	15	15	NUM
cana-1298	346	3	]	]	X
cana-1298	346	4	vairamuthu.k	vairamuthu.k	PROPN
cana-1298	346	5	and	and	CCONJ
cana-1298	346	6	loganathan.s	loganathan.s	PROPN
cana-1298	346	7	,	,	PUNCT
cana-1298	346	8	“	"	PUNCT
cana-1298	346	9	bipolar	bipolar	ADJ
cana-1298	346	10	valued	value	VERB
cana-1298	346	11	multi	multi	NOUN
cana-1298	346	12	i	i	PRON
cana-1298	346	13	-	-	PUNCT
cana-1298	346	14	fuzzy	fuzzy	ADJ
cana-1298	346	15	subrings	subring	NOUN
cana-1298	346	16	of	of	ADP
cana-1298	346	17	a	a	DET
cana-1298	346	18	ring	ring	NOUN
cana-1298	346	19	”	"	PUNCT
cana-1298	346	20	,	,	PUNCT
cana-1298	346	21	indian	indian	ADJ
cana-1298	346	22	journal	journal	NOUN
cana-1298	346	23	of	of	ADP
cana-1298	346	24	natural	natural	ADJ
cana-1298	346	25	sciences	science	NOUN
cana-1298	346	26	,	,	PUNCT
cana-1298	346	27	vol.13	vol.13	NOUN
cana-1298	346	28	,	,	PUNCT
cana-1298	346	29	issue	issue	NOUN
cana-1298	346	30	76	76	NUM
cana-1298	346	31	(	(	PUNCT
cana-1298	346	32	2023	2023	NUM
cana-1298	346	33	)	)	PUNCT
cana-1298	346	34	,	,	PUNCT
cana-1298	346	35	53008	53008	NUM
cana-1298	346	36	-	-	SYM
cana-1298	346	37	53013	53013	NUM
cana-1298	346	38	.	.	PUNCT
cana-1298	347	1	[	[	X
cana-1298	347	2	16	16	NUM
cana-1298	347	3	]	]	SYM
cana-1298	347	4	yasodara.s	yasodara.s	PROPN
cana-1298	347	5	,	,	PUNCT
cana-1298	347	6	ke	ke	PROPN
cana-1298	347	7	.	.	PUNCT
cana-1298	347	8	sathappan	sathappan	ADJ
cana-1298	347	9	,	,	PUNCT
cana-1298	347	10	“	"	PUNCT
cana-1298	347	11	bipolar	bipolar	ADJ
cana-1298	347	12	-	-	PUNCT
cana-1298	347	13	valued	value	VERB
cana-1298	347	14	multi	multi	ADJ
cana-1298	347	15	fuzzy	fuzzy	ADJ
cana-1298	347	16	subsemirings	subsemiring	NOUN
cana-1298	347	17	of	of	ADP
cana-1298	347	18	a	a	DET
cana-1298	347	19	semiring	semiring	NOUN
cana-1298	347	20	”	"	PUNCT
cana-1298	347	21	,	,	PUNCT
cana-1298	347	22	international	international	ADJ
cana-1298	347	23	journal	journal	NOUN
cana-1298	347	24	of	of	ADP
cana-1298	347	25	mathematical	mathematical	ADJ
cana-1298	347	26	archive	archive	NOUN
cana-1298	347	27	,	,	PUNCT
cana-1298	347	28	6(9	6(9	NUM
cana-1298	347	29	)	)	PUNCT
cana-1298	347	30	(	(	PUNCT
cana-1298	347	31	2015	2015	NUM
cana-1298	347	32	)	)	PUNCT
cana-1298	347	33	,	,	PUNCT
cana-1298	347	34	75	75	NUM
cana-1298	347	35	−80	−80	NOUN
cana-1298	347	36	.	.	PUNCT
cana-1298	348	1	[	[	X
cana-1298	348	2	17	17	NUM
cana-1298	348	3	]	]	PUNCT
cana-1298	348	4	l.a.zadeh	l.a.zadeh	NOUN
cana-1298	348	5	,	,	PUNCT
cana-1298	348	6	fuzzy	fuzzy	ADJ
cana-1298	348	7	sets	set	NOUN
cana-1298	348	8	,	,	PUNCT
cana-1298	348	9	inform	inform	NOUN
cana-1298	348	10	.	.	PUNCT
cana-1298	349	1	and	and	CCONJ
cana-1298	349	2	control	control	NOUN
cana-1298	349	3	,	,	PUNCT
cana-1298	349	4	8(1965	8(1965	NUM
cana-1298	349	5	)	)	PUNCT
cana-1298	349	6	,	,	PUNCT
cana-1298	349	7	338	338	NUM
cana-1298	349	8	−353	−353	NOUN
cana-1298	349	9	.	.	PUNCT
cana-1298	350	1	[	[	X
cana-1298	350	2	18	18	NUM
cana-1298	350	3	]	]	X
cana-1298	350	4	g.	g.	PROPN
cana-1298	350	5	srinivasa	srinivasa	PROPN
cana-1298	350	6	rao	rao	PROPN
cana-1298	350	7	,	,	PUNCT
cana-1298	350	8	d.	d.	PROPN
cana-1298	350	9	madhusudhanarao	madhusudhanarao	PROPN
cana-1298	350	10	and	and	CCONJ
cana-1298	350	11	p.	p.	PROPN
cana-1298	350	12	siva	siva	PROPN
cana-1298	350	13	prasad	prasad	PROPN
cana-1298	350	14	,	,	PUNCT
cana-1298	350	15	simple	simple	ADJ
cana-1298	350	16	ternary	ternary	ADJ
cana-1298	350	17	semi	semi	NOUN
cana-1298	350	18	-	-	NOUN
cana-1298	350	19	rings	ring	NOUN
cana-1298	350	20	,	,	PUNCT
cana-1298	350	21	the	the	DET
cana-1298	350	22	global	global	ADJ
cana-1298	350	23	journal	journal	NOUN
cana-1298	350	24	of	of	ADP
cana-1298	350	25	mathematics	mathematics	PROPN
cana-1298	350	26	&	&	CCONJ
cana-1298	350	27	mathematical	mathematical	PROPN
cana-1298	350	28	sciences	sciences	PROPN
cana-1298	350	29	,	,	PUNCT
cana-1298	350	30	9(2	9(2	NUM
cana-1298	350	31	)	)	PUNCT
cana-1298	350	32	(	(	PUNCT
cana-1298	350	33	2016	2016	NUM
cana-1298	350	34	)	)	PUNCT
cana-1298	350	35	,	,	PUNCT
cana-1298	350	36	185	185	NUM
cana-1298	350	37	-	-	SYM
cana-1298	350	38	196	196	NUM
cana-1298	350	39	.	.	PUNCT
cana-1298	351	1	[	[	X
cana-1298	351	2	19	19	NUM
cana-1298	351	3	]	]	PUNCT
cana-1298	351	4	g.	g.	PROPN
cana-1298	351	5	srinivasa	srinivasa	PROPN
cana-1298	351	6	rao	rao	PROPN
cana-1298	351	7	,	,	PUNCT
cana-1298	351	8	d.	d.	PROPN
cana-1298	351	9	madhusudhana	madhusudhana	PROPN
cana-1298	351	10	rao	rao	PROPN
cana-1298	351	11	,	,	PUNCT
cana-1298	351	12	a	a	DET
cana-1298	351	13	study	study	NOUN
cana-1298	351	14	on	on	ADP
cana-1298	351	15	ternary	ternary	ADJ
cana-1298	351	16	semi	semi	ADJ
cana-1298	351	17	rings	ring	NOUN
cana-1298	351	18	,	,	PUNCT
cana-1298	351	19	int	int	NOUN
cana-1298	351	20	.	.	PUNCT
cana-1298	352	1	j.	j.	PROPN
cana-1298	352	2	of	of	ADP
cana-1298	352	3	math	math	PROPN
cana-1298	352	4	.	.	PUNCT
cana-1298	353	1	archive	archive	NOUN
cana-1298	353	2	,	,	PUNCT
cana-1298	353	3	5(12	5(12	NUM
cana-1298	353	4	)	)	PUNCT
cana-1298	353	5	(	(	PUNCT
cana-1298	353	6	2014	2014	NUM
cana-1298	353	7	)	)	PUNCT
cana-1298	353	8	,	,	PUNCT
cana-1298	353	9	2430	2430	NUM
cana-1298	353	10	.	.	PUNCT