id	sid	tid	token	lemma	pos
cana-4565	1	1	communications	communication	NOUN
cana-4565	1	2	on	on	ADP
cana-4565	1	3	applied	apply	VERB
cana-4565	1	4	nonlinear	nonlinear	ADJ
cana-4565	1	5	analysis	analysis	NOUN
cana-4565	1	6	issn	issn	NOUN
cana-4565	1	7	:	:	PUNCT
cana-4565	1	8	1074	1074	NUM
cana-4565	1	9	-	-	PUNCT
cana-4565	1	10	133x	133x	NUM
cana-4565	1	11	vol	vol	NOUN
cana-4565	1	12	32	32	NUM
cana-4565	1	13	no	no	NOUN
cana-4565	1	14	.	.	PUNCT
cana-4565	2	1	9s	9s	NUM
cana-4565	2	2	(	(	PUNCT
cana-4565	2	3	2025	2025	NUM
cana-4565	2	4	)	)	PUNCT
cana-4565	2	5	2615	2615	NUM
cana-4565	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4565	2	7	various	various	ADJ
cana-4565	2	8	types	type	NOUN
cana-4565	2	9	of	of	ADP
cana-4565	2	10	translation	translation	NOUN
cana-4565	2	11	in	in	ADP
cana-4565	2	12	bipolar	bipolar	ADJ
cana-4565	2	13	valued	value	VERB
cana-4565	2	14	multi	multi	NOUN
cana-4565	2	15	i	i	PRON
cana-4565	2	16	-	-	PUNCT
cana-4565	2	17	fuzzy	fuzzy	ADJ
cana-4565	2	18	normal	normal	ADJ
cana-4565	2	19	subrings	subring	NOUN
cana-4565	2	20	of	of	ADP
cana-4565	2	21	a	a	DET
cana-4565	2	22	ring	ring	NOUN
cana-4565	2	23	1k.vairamuthu	1k.vairamuthu	PROPN
cana-4565	2	24	&	&	CCONJ
cana-4565	2	25	2s	2s	PROPN
cana-4565	2	26	.	.	PUNCT
cana-4565	3	1	loganathan	loganathan	PROPN
cana-4565	3	2	1	1	NUM
cana-4565	3	3	department	department	NOUN
cana-4565	3	4	of	of	ADP
cana-4565	3	5	mathematics	mathematic	NOUN
cana-4565	3	6	,	,	PUNCT
cana-4565	3	7	sethupathy	sethupathy	ADJ
cana-4565	3	8	government	government	NOUN
cana-4565	3	9	arts	art	NOUN
cana-4565	3	10	college(affiliated	college(affiliate	VERB
cana-4565	3	11	to	to	PART
cana-4565	3	12	alagappa	alagappa	VERB
cana-4565	3	13	university	university	PROPN
cana-4565	3	14	,	,	PUNCT
cana-4565	3	15	karaikudi	karaikudi	PROPN
cana-4565	3	16	)	)	PUNCT
cana-4565	3	17	,	,	PUNCT
cana-4565	3	18	ramanathapuram	ramanathapuram	NOUN
cana-4565	3	19	-623	-623	PROPN
cana-4565	3	20	502	502	NUM
cana-4565	3	21	,	,	PUNCT
cana-4565	3	22	tamilnadu	tamilnadu	ADJ
cana-4565	3	23	,	,	PUNCT
cana-4565	3	24	india	india	PROPN
cana-4565	3	25	.	.	PUNCT
cana-4565	3	26	email	email	NOUN
cana-4565	3	27	:	:	PUNCT
cana-4565	3	28	vairammathi83@gmail.com	vairammathi83@gmail.com	X
cana-4565	3	29	2department	2department	NUM
cana-4565	3	30	of	of	ADP
cana-4565	3	31	mathematics	mathematic	NOUN
cana-4565	3	32	,	,	PUNCT
cana-4565	3	33	sethupathy	sethupathy	ADJ
cana-4565	3	34	government	government	NOUN
cana-4565	3	35	arts	art	NOUN
cana-4565	3	36	college(affiliated	college(affiliate	VERB
cana-4565	3	37	to	to	PART
cana-4565	3	38	alagappa	alagappa	VERB
cana-4565	3	39	university	university	PROPN
cana-4565	3	40	,	,	PUNCT
cana-4565	3	41	karaikudi	karaikudi	PROPN
cana-4565	3	42	)	)	PUNCT
cana-4565	3	43	,	,	PUNCT
cana-4565	3	44	ramanathapuram	ramanathapuram	NOUN
cana-4565	3	45	-623	-623	PROPN
cana-4565	3	46	502	502	NUM
cana-4565	3	47	,	,	PUNCT
cana-4565	3	48	tamilnadu	tamilnadu	ADJ
cana-4565	3	49	,	,	PUNCT
cana-4565	3	50	india	india	PROPN
cana-4565	3	51	.	.	PUNCT
cana-4565	3	52	email	email	NOUN
cana-4565	3	53	:	:	PUNCT
cana-4565	3	54	logaamaths2010@gmail.com	logaamaths2010@gmail.com	X
cana-4565	3	55	article	article	NOUN
cana-4565	3	56	history	history	NOUN
cana-4565	3	57	:	:	PUNCT
cana-4565	3	58	received	receive	VERB
cana-4565	3	59	:	:	PUNCT
cana-4565	3	60	12	12	NUM
cana-4565	3	61	-	-	SYM
cana-4565	3	62	01	01	NUM
cana-4565	3	63	-	-	PUNCT
cana-4565	3	64	2025	2025	NUM
cana-4565	3	65	revised	revise	VERB
cana-4565	3	66	:	:	PUNCT
cana-4565	3	67	15	15	NUM
cana-4565	3	68	-	-	NUM
cana-4565	3	69	02	02	NUM
cana-4565	3	70	-	-	PUNCT
cana-4565	3	71	2025	2025	NUM
cana-4565	3	72	accepted	accept	VERB
cana-4565	3	73	:	:	PUNCT
cana-4565	3	74	01	01	NUM
cana-4565	3	75	-	-	SYM
cana-4565	3	76	03	03	NUM
cana-4565	3	77	-	-	PUNCT
cana-4565	3	78	2025	2025	NUM
cana-4565	3	79	abstract	abstract	NOUN
cana-4565	3	80	:	:	PUNCT
cana-4565	3	81	in	in	ADP
cana-4565	3	82	this	this	DET
cana-4565	3	83	paper	paper	NOUN
cana-4565	3	84	,	,	PUNCT
cana-4565	3	85	various	various	ADJ
cana-4565	3	86	types	type	NOUN
cana-4565	3	87	of	of	ADP
cana-4565	3	88	translation	translation	NOUN
cana-4565	3	89	in	in	ADP
cana-4565	3	90	b_v	b_v	NOUN
cana-4565	3	91	mifnsr	mifnsr	NOUN
cana-4565	3	92	of	of	ADP
cana-4565	3	93	a	a	DET
cana-4565	3	94	ring	ring	NOUN
cana-4565	3	95	are	be	AUX
cana-4565	3	96	studied	study	VERB
cana-4565	3	97	and	and	CCONJ
cana-4565	3	98	dealt	deal	VERB
cana-4565	3	99	.	.	PUNCT
cana-4565	4	1	some	some	DET
cana-4565	4	2	theorems	theorem	NOUN
cana-4565	4	3	are	be	AUX
cana-4565	4	4	given	give	VERB
cana-4565	4	5	and	and	CCONJ
cana-4565	4	6	,	,	PUNCT
cana-4565	4	7	they	they	PRON
cana-4565	4	8	are	be	AUX
cana-4565	4	9	proved	prove	VERB
cana-4565	4	10	.	.	PUNCT
cana-4565	5	1	keywords	keyword	NOUN
cana-4565	5	2	:	:	PUNCT
cana-4565	5	3	interval	interval	NOUN
cana-4565	5	4	valued	value	VERB
cana-4565	5	5	fuzzy	fuzzy	ADJ
cana-4565	5	6	subset	subset	NOUN
cana-4565	5	7	,	,	PUNCT
cana-4565	5	8	bipolar	bipolar	ADJ
cana-4565	5	9	valued	value	VERB
cana-4565	5	10	fuzzy	fuzzy	ADJ
cana-4565	5	11	subset	subset	NOUN
cana-4565	5	12	,	,	PUNCT
cana-4565	5	13	〖	〖	PROPN
cana-4565	5	14	b〗_v	b〗_v	NUM
cana-4565	5	15	mifs	mif	NOUN
cana-4565	5	16	,	,	PUNCT
cana-4565	5	17	〖	〖	PROPN
cana-4565	5	18	b〗_v	b〗_v	PROPN
cana-4565	5	19	mifsr	mifsr	PROPN
cana-4565	5	20	,	,	PUNCT
cana-4565	5	21	〖	〖	PROPN
cana-4565	5	22	b〗_v	b〗_v	NUM
cana-4565	5	23	mifnsr	mifnsr	NOUN
cana-4565	5	24	and	and	CCONJ
cana-4565	5	25	translations	translation	NOUN
cana-4565	5	26	.	.	PUNCT
cana-4565	6	1	introduction	introduction	NOUN
cana-4565	6	2	.	.	PUNCT
cana-4565	7	1	zadeh	zadeh	PROPN
cana-4565	8	1	[	[	X
cana-4565	8	2	14]had	14]had	PROPN
cana-4565	8	3	introduced	introduce	VERB
cana-4565	8	4	the	the	DET
cana-4565	8	5	fuzzy	fuzzy	ADJ
cana-4565	8	6	subset	subset	NOUN
cana-4565	8	7	in	in	ADP
cana-4565	8	8	1965	1965	NUM
cana-4565	8	9	.	.	PUNCT
cana-4565	9	1	it	it	PRON
cana-4565	9	2	is	be	AUX
cana-4565	9	3	one	one	NUM
cana-4565	9	4	of	of	ADP
cana-4565	9	5	the	the	DET
cana-4565	9	6	generalizations	generalization	NOUN
cana-4565	9	7	of	of	ADP
cana-4565	9	8	crisp	crisp	ADJ
cana-4565	9	9	set	set	NOUN
cana-4565	9	10	.	.	PUNCT
cana-4565	10	1	𝐺𝑟𝑜𝑢𝑝	𝐺𝑟𝑜𝑢𝑝	PROPN
cana-4565	10	2	𝑤𝑎𝑠	𝑤𝑎𝑠	PROPN
cana-4565	10	3	𝑔𝑒𝑛𝑒𝑟𝑎𝑙𝑖𝑧𝑒𝑑	𝑔𝑒𝑛𝑒𝑟𝑎𝑙𝑖𝑧𝑒𝑑	VERB
cana-4565	10	4	𝑎𝑠	𝑎𝑠	ADP
cana-4565	10	5	𝑓𝑢𝑧𝑧𝑦	𝑓𝑢𝑧𝑧𝑦	PROPN
cana-4565	10	6	𝑔𝑟𝑜𝑢𝑝	𝑔𝑟𝑜𝑢𝑝	PROPN
cana-4565	10	7	𝑏𝑦	𝑏𝑦	PROPN
cana-4565	10	8	azriel	azriel	PROPN
cana-4565	10	9	rosenfeld	rosenfeld	PROPN
cana-4565	11	1	[	[	X
cana-4565	11	2	3	3	NUM
cana-4565	11	3	]	]	PUNCT
cana-4565	11	4	.	.	PUNCT
cana-4565	12	1	𝐴𝑓𝑡𝑒𝑟	𝐴𝑓𝑡𝑒𝑟	PROPN
cana-4565	12	2	,	,	PUNCT
cana-4565	12	3	𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡	𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡	PROPN
cana-4565	12	4	𝑡𝑦𝑝𝑒𝑠	𝑡𝑦𝑝𝑒𝑠	NOUN
cana-4565	12	5	𝑜𝑓	𝑜𝑓	ADP
cana-4565	12	6	𝑓𝑢𝑧𝑧𝑦	𝑓𝑢𝑧𝑧𝑦	NOUN
cana-4565	12	7	𝑤𝑒𝑟𝑒	𝑤𝑒𝑟𝑒	NOUN
cana-4565	12	8	𝑖𝑛𝑡𝑟𝑜𝑑𝑢𝑐𝑒𝑑	𝑖𝑛𝑡𝑟𝑜𝑑𝑢𝑐𝑒𝑑	NOUN
cana-4565	12	9	𝑏𝑦	𝑏𝑦	VERB
cana-4565	12	10	𝑣𝑎𝑟𝑖𝑜𝑢𝑠	𝑣𝑎𝑟𝑖𝑜𝑢𝑠	ADJ
cana-4565	12	11	𝑎𝑢𝑡ℎ𝑜𝑟𝑠.	𝑎𝑢𝑡ℎ𝑜𝑟𝑠.	NOUN
cana-4565	12	12	in	in	ADP
cana-4565	12	13	1994	1994	NUM
cana-4565	12	14	,	,	PUNCT
cana-4565	12	15	bipolar	bipolar	ADJ
cana-4565	12	16	valued	value	VERB
cana-4565	12	17	fuzzy	fuzzy	ADJ
cana-4565	12	18	subset	subset	NOUN
cana-4565	12	19	was	be	AUX
cana-4565	12	20	introduced	introduce	VERB
cana-4565	12	21	by	by	ADP
cana-4565	12	22	w.r.zhang[15	w.r.zhang[15	PROPN
cana-4565	12	23	]	]	PUNCT
cana-4565	12	24	.	.	PUNCT
cana-4565	13	1	bipolar	bipolar	PROPN
cana-4565	13	2	valued	value	VERB
cana-4565	13	3	multi	multi	NOUN
cana-4565	13	4	i	i	PRON
cana-4565	13	5	-	-	PUNCT
cana-4565	13	6	fuzzy	fuzzy	ADJ
cana-4565	13	7	subring	subring	NOUN
cana-4565	13	8	has	have	AUX
cana-4565	13	9	been	be	AUX
cana-4565	13	10	introduced	introduce	VERB
cana-4565	13	11	by	by	ADP
cana-4565	13	12	k.vairamuthu	k.vairamuthu	PROPN
cana-4565	13	13	,	,	PUNCT
cana-4565	13	14	s.loganathan[11	s.loganathan[11	ADV
cana-4565	13	15	]	]	PUNCT
cana-4565	13	16	.	.	PUNCT
cana-4565	14	1	the	the	DET
cana-4565	14	2	following	follow	VERB
cana-4565	14	3	papers	paper	NOUN
cana-4565	14	4	[	[	X
cana-4565	14	5	1	1	NUM
cana-4565	14	6	]	]	PUNCT
cana-4565	14	7	,	,	PUNCT
cana-4565	14	8	[	[	X
cana-4565	14	9	2	2	NUM
cana-4565	14	10	]	]	PUNCT
cana-4565	14	11	,	,	PUNCT
cana-4565	14	12	[	[	X
cana-4565	14	13	4	4	NUM
cana-4565	14	14	]	]	PUNCT
cana-4565	14	15	,	,	PUNCT
cana-4565	14	16	[	[	X
cana-4565	14	17	5	5	NUM
cana-4565	14	18	]	]	PUNCT
cana-4565	14	19	,	,	PUNCT
cana-4565	14	20	[	[	X
cana-4565	14	21	6	6	NUM
cana-4565	14	22	]	]	PUNCT
cana-4565	14	23	,	,	PUNCT
cana-4565	14	24	[	[	X
cana-4565	14	25	7	7	NUM
cana-4565	14	26	]	]	PUNCT
cana-4565	14	27	,	,	PUNCT
cana-4565	14	28	[	[	X
cana-4565	14	29	8	8	NUM
cana-4565	14	30	]	]	PUNCT
cana-4565	14	31	,	,	PUNCT
cana-4565	14	32	[	[	X
cana-4565	14	33	9	9	NUM
cana-4565	14	34	]	]	PUNCT
cana-4565	14	35	,	,	PUNCT
cana-4565	14	36	[	[	X
cana-4565	14	37	10	10	NUM
cana-4565	14	38	]	]	PUNCT
cana-4565	14	39	,	,	PUNCT
cana-4565	14	40	[	[	X
cana-4565	14	41	12	12	NUM
cana-4565	14	42	]	]	PUNCT
cana-4565	14	43	and	and	CCONJ
cana-4565	14	44	[	[	X
cana-4565	14	45	13	13	NUM
cana-4565	14	46	]	]	PUNCT
cana-4565	14	47	were	be	AUX
cana-4565	14	48	useful	useful	ADJ
cana-4565	14	49	to	to	PART
cana-4565	14	50	write	write	VERB
cana-4565	14	51	the	the	DET
cana-4565	14	52	this	this	DET
cana-4565	14	53	paper	paper	NOUN
cana-4565	14	54	.	.	PUNCT
cana-4565	15	1	in	in	ADP
cana-4565	15	2	this	this	DET
cana-4565	15	3	paper	paper	NOUN
cana-4565	15	4	,	,	PUNCT
cana-4565	15	5	𝑣𝑎𝑟𝑖𝑜𝑢𝑠	𝑣𝑎𝑟𝑖𝑜𝑢𝑠	ADJ
cana-4565	15	6	𝑡𝑦𝑝𝑒𝑠	𝑡𝑦𝑝𝑒𝑠	NOUN
cana-4565	15	7	𝑜𝑓	𝑜𝑓	ADP
cana-4565	15	8	𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛	𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛	NOUN
cana-4565	15	9	𝑖𝑛	𝑖𝑛	X
cana-4565	15	10	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	15	11	𝑜𝑓	𝑜𝑓	ADP
cana-4565	15	12	𝑎	𝑎	PROPN
cana-4565	15	13	𝑟𝑖𝑛𝑔	𝑟𝑖𝑛𝑔	ADJ
cana-4565	15	14	𝑎𝑟𝑒	𝑎𝑟𝑒	NOUN
cana-4565	15	15	𝑖𝑛𝑡𝑟𝑜𝑑𝑢𝑐𝑒𝑑	𝑖𝑛𝑡𝑟𝑜𝑑𝑢𝑐𝑒𝑑	NOUN
cana-4565	15	16	and	and	CCONJ
cana-4565	15	17	established	establish	VERB
cana-4565	15	18	some	some	DET
cana-4565	15	19	results	result	NOUN
cana-4565	15	20	.	.	PUNCT
cana-4565	16	1	1.preliminaries	1.preliminaries	X
cana-4565	16	2	.	.	PUNCT
cana-4565	17	1	definition	definition	NOUN
cana-4565	17	2	1.1	1.1	NUM
cana-4565	17	3	[	[	X
cana-4565	17	4	14	14	NUM
cana-4565	17	5	]	]	PUNCT
cana-4565	17	6	𝐴	𝐴	PROPN
cana-4565	17	7	𝑚𝑎𝑝	𝑚𝑎𝑝	NOUN
cana-4565	17	8	ℜ	ℜ	PROPN
cana-4565	17	9	:	:	PUNCT
cana-4565	17	10	𝕄	𝕄	PROPN
cana-4565	17	11	→	→	SYM
cana-4565	17	12	𝔻[0,1	𝔻[0,1	PROPN
cana-4565	17	13	]	]	PUNCT
cana-4565	17	14	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	17	15	𝑠𝑎𝑖𝑑	𝑠𝑎𝑖𝑑	PROPN
cana-4565	17	16	𝑡𝑜	𝑡𝑜	PROPN
cana-4565	17	17	𝑏𝑒	𝑏𝑒	PROPN
cana-4565	17	18	𝑎𝑛	𝑎𝑛	PROPN
cana-4565	17	19	𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙	𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙	PROPN
cana-4565	17	20	𝑣𝑎𝑙𝑢𝑒𝑑	𝑣𝑎𝑙𝑢𝑒𝑑	VERB
cana-4565	17	21	𝑓𝑢𝑧𝑧𝑦	𝑓𝑢𝑧𝑧𝑦	PROPN
cana-4565	17	22	𝑠𝑢𝑏𝑠𝑒𝑡	𝑠𝑢𝑏𝑠𝑒𝑡	NOUN
cana-4565	17	23	𝑜𝑓	𝑜𝑓	ADP
cana-4565	17	24	𝕄	𝕄	PROPN
cana-4565	17	25	,	,	PUNCT
cana-4565	17	26	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	NOUN
cana-4565	17	27	𝔻[0,1	𝔻[0,1	PROPN
cana-4565	17	28	]	]	PUNCT
cana-4565	17	29	𝑚𝑒𝑎𝑛𝑠	𝑚𝑒𝑎𝑛𝑠	NOUN
cana-4565	17	30	𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑖𝑜𝑛	𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑖𝑜𝑛	PROPN
cana-4565	17	31	𝑜𝑓	𝑜𝑓	ADP
cana-4565	17	32	𝑎𝑙𝑙	𝑎𝑙𝑙	PROPN
cana-4565	17	33	𝑐𝑙𝑜𝑠𝑒𝑑	𝑐𝑙𝑜𝑠𝑒𝑑	VERB
cana-4565	17	34	𝑠𝑢𝑏𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙	𝑠𝑢𝑏𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙	PROPN
cana-4565	17	35	𝑜𝑓	𝑜𝑓	ADP
cana-4565	17	36	[	[	X
cana-4565	17	37	0	0	NUM
cana-4565	17	38	,	,	PUNCT
cana-4565	17	39	1	1	NUM
cana-4565	17	40	]	]	PUNCT
cana-4565	17	41	.	.	PUNCT
cana-4565	18	1	definition	definition	NOUN
cana-4565	18	2	1.2	1.2	NUM
cana-4565	18	3	[	[	X
cana-4565	18	4	15	15	X
cana-4565	18	5	]	]	X
cana-4565	18	6	𝑇ℎ𝑒	𝑇ℎ𝑒	PROPN
cana-4565	18	7	𝑜𝑟𝑑𝑒𝑟𝑒𝑑	𝑜𝑟𝑑𝑒𝑟𝑒𝑑	VERB
cana-4565	18	8	𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒	𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒	ADV
cana-4565	18	9	𝔗	𝔗	PROPN
cana-4565	18	10	=	=	X
cana-4565	18	11	{	{	PUNCT
cana-4565	18	12	(	(	PUNCT
cana-4565	18	13	𝔷	𝔷	PROPN
cana-4565	18	14	,	,	PUNCT
cana-4565	18	15	𝔗+(𝔷	𝔗+(𝔷	NOUN
cana-4565	18	16	)	)	PUNCT
cana-4565	18	17	,	,	PUNCT
cana-4565	18	18	𝔗−(𝔷	𝔗−(𝔷	ADJ
cana-4565	18	19	)	)	PUNCT
cana-4565	18	20	):	):	PUNCT
cana-4565	18	21	𝔷	𝔷	PROPN
cana-4565	18	22	∈	∈	PROPN
cana-4565	18	23	𝕎	𝕎	PROPN
cana-4565	18	24	}	}	PUNCT
cana-4565	18	25	𝑖𝑠	𝑖𝑠	NOUN
cana-4565	18	26	𝑐𝑎𝑙𝑙𝑒𝑑	𝑐𝑎𝑙𝑙𝑒𝑑	NOUN
cana-4565	18	27	a	a	DET
cana-4565	18	28	bipolar	bipolar	ADJ
cana-4565	18	29	𝑣𝑎𝑙𝑢𝑒𝑑	𝑣𝑎𝑙𝑢𝑒𝑑	VERB
cana-4565	18	30	𝑓𝑢𝑧𝑧𝑦	𝑓𝑢𝑧𝑧𝑦	NOUN
cana-4565	18	31	𝑠𝑢𝑏𝑠𝑒𝑡	𝑠𝑢𝑏𝑠𝑒𝑡	NOUN
cana-4565	18	32	𝑜𝑓	𝑜𝑓	ADP
cana-4565	18	33	𝕨	𝕨	PROPN
cana-4565	18	34	,	,	PUNCT
cana-4565	18	35	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	ADJ
cana-4565	18	36	𝔗+	𝔗+	NOUN
cana-4565	18	37	:	:	PUNCT
cana-4565	18	38	𝕨	𝕨	PROPN
cana-4565	18	39	→	→	X
cana-4565	18	40	[	[	X
cana-4565	18	41	0,1	0,1	NUM
cana-4565	18	42	]	]	PUNCT
cana-4565	18	43	𝑖𝑠	𝑖𝑠	CCONJ
cana-4565	18	44	𝑎	𝑎	DET
cana-4565	18	45	𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒	𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒	NOUN
cana-4565	18	46	𝑚𝑒𝑚𝑏𝑒𝑟𝑠ℎ𝑖𝑝	𝑚𝑒𝑚𝑏𝑒𝑟𝑠ℎ𝑖𝑝	ADJ
cana-4565	18	47	map	map	NOUN
cana-4565	18	48	and	and	CCONJ
cana-4565	18	49	𝔗−	𝔗−	NUM
cana-4565	18	50	:	:	PUNCT
cana-4565	18	51	𝕨	𝕨	PROPN
cana-4565	18	52	→	→	SYM
cana-4565	18	53	[	[	X
cana-4565	18	54	−1,0	−1,0	X
cana-4565	18	55	]	]	X
cana-4565	18	56	is	be	AUX
cana-4565	18	57	a	a	DET
cana-4565	18	58	negative	negative	ADJ
cana-4565	18	59	membership	membership	NOUN
cana-4565	18	60	map	map	NOUN
cana-4565	18	61	.	.	PUNCT
cana-4565	19	1	definition	definition	NOUN
cana-4565	19	2	1.3	1.3	NUM
cana-4565	20	1	[	[	X
cana-4565	20	2	11	11	NUM
cana-4565	20	3	]	]	X
cana-4565	20	4	𝑇ℎ𝑒	𝑇ℎ𝑒	PROPN
cana-4565	20	5	𝑜𝑟𝑑𝑒𝑟𝑒𝑑	𝑜𝑟𝑑𝑒𝑟𝑒𝑑	VERB
cana-4565	20	6	𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒	𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒	ADV
cana-4565	20	7	𝔗	𝔗	PROPN
cana-4565	20	8	=	=	X
cana-4565	20	9	{	{	PUNCT
cana-4565	20	10	(	(	PUNCT
cana-4565	20	11	𝔷	𝔷	PROPN
cana-4565	20	12	,	,	PUNCT
cana-4565	20	13	𝔗1	𝔗1	PROPN
cana-4565	20	14	+	+	PROPN
cana-4565	20	15	(	(	PUNCT
cana-4565	20	16	𝔷	𝔷	NOUN
cana-4565	20	17	)	)	PUNCT
cana-4565	20	18	,	,	PUNCT
cana-4565	20	19	𝔗2	𝔗2	PROPN
cana-4565	20	20	+	+	PROPN
cana-4565	20	21	(	(	PUNCT
cana-4565	20	22	𝔷	𝔷	NOUN
cana-4565	20	23	)	)	PUNCT
cana-4565	20	24	,	,	PUNCT
cana-4565	20	25	…	…	PUNCT
cana-4565	20	26	,	,	PUNCT
cana-4565	20	27	𝔗𝑛	𝔗𝑛	PROPN
cana-4565	20	28	+	+	ADJ
cana-4565	20	29	(	(	PUNCT
cana-4565	20	30	𝔷	𝔷	NOUN
cana-4565	20	31	)	)	PUNCT
cana-4565	20	32	,	,	PUNCT
cana-4565	20	33	𝔗1	𝔗1	PROPN
cana-4565	20	34	−(𝔷	−(𝔷	PROPN
cana-4565	20	35	)	)	PUNCT
cana-4565	20	36	,	,	PUNCT
cana-4565	20	37	𝔗2	𝔗2	NOUN
cana-4565	20	38	−(𝔷	−(𝔷	NOUN
cana-4565	20	39	)	)	PUNCT
cana-4565	20	40	,	,	PUNCT
cana-4565	20	41	…	…	PUNCT
cana-4565	20	42	,	,	PUNCT
cana-4565	20	43	𝔗𝑛	𝔗𝑛	PROPN
cana-4565	20	44	−(𝔷	−(𝔷	NOUN
cana-4565	20	45	)	)	PUNCT
cana-4565	20	46	):	):	PUNCT
cana-4565	20	47	𝔷	𝔷	PROPN
cana-4565	20	48	∈	∈	PROPN
cana-4565	20	49	𝕎	𝕎	PROPN
cana-4565	20	50	}	}	PUNCT
cana-4565	20	51	𝑖𝑠	𝑖𝑠	NOUN
cana-4565	20	52	𝑐𝑎𝑙𝑙𝑒𝑑	𝑐𝑎𝑙𝑙𝑒𝑑	NOUN
cana-4565	20	53	𝑎	𝑎	PRON
cana-4565	20	54	𝑏𝑖𝑝𝑜𝑙𝑎𝑟	𝑏𝑖𝑝𝑜𝑙𝑎𝑟	NOUN
cana-4565	20	55	𝑣𝑎𝑙𝑢𝑒𝑑	𝑣𝑎𝑙𝑢𝑒𝑑	VERB
cana-4565	20	56	𝑚𝑢𝑙𝑡𝑖	𝑚𝑢𝑙𝑡𝑖	NOUN
cana-4565	20	57	𝐼	𝐼	ADP
cana-4565	20	58	−	−	PROPN
cana-4565	20	59	𝑓𝑢𝑧𝑧𝑦	𝑓𝑢𝑧𝑧𝑦	NOUN
cana-4565	20	60	𝑠𝑢𝑏𝑠𝑒𝑡	𝑠𝑢𝑏𝑠𝑒𝑡	NOUN
cana-4565	20	61	(	(	PUNCT
cana-4565	20	62	𝐵𝑉𝑀𝐼𝐹𝑆	𝐵𝑉𝑀𝐼𝐹𝑆	PROPN
cana-4565	20	63	)	)	PUNCT
cana-4565	20	64	of	of	ADP
cana-4565	20	65	𝕨	𝕨	PROPN
cana-4565	20	66	,	,	PUNCT
cana-4565	20	67	where	where	SCONJ
cana-4565	20	68	𝔗𝑖	𝔗𝑖	ADP
cana-4565	20	69	+	+	ADJ
cana-4565	20	70	:	:	PUNCT
cana-4565	20	71	𝕨	𝕨	PROPN
cana-4565	20	72	→	→	SYM
cana-4565	20	73	𝔻[0,1	𝔻[0,1	PROPN
cana-4565	20	74	]	]	PUNCT
cana-4565	20	75	𝑖𝑠	𝑖𝑠	CCONJ
cana-4565	20	76	𝑎	𝑎	DET
cana-4565	20	77	𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒	𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒	NOUN
cana-4565	20	78	𝑚𝑒𝑚𝑏𝑒𝑟𝑠ℎ𝑖𝑝	𝑚𝑒𝑚𝑏𝑒𝑟𝑠ℎ𝑖𝑝	ADJ
cana-4565	20	79	map	map	NOUN
cana-4565	20	80	and	and	CCONJ
cana-4565	20	81	𝔗𝑖	𝔗𝑖	ADP
cana-4565	20	82	−	−	NOUN
cana-4565	20	83	:	:	PUNCT
cana-4565	20	84	𝕨	𝕨	PROPN
cana-4565	20	85	→	→	SYM
cana-4565	20	86	𝔻[−1,0	𝔻[−1,0	PROPN
cana-4565	20	87	]	]	PUNCT
cana-4565	20	88	is	be	AUX
cana-4565	20	89	a	a	DET
cana-4565	20	90	negative	negative	ADJ
cana-4565	20	91	membership	membership	NOUN
cana-4565	20	92	map	map	NOUN
cana-4565	20	93	.	.	PUNCT
cana-4565	21	1	communications	communication	NOUN
cana-4565	21	2	on	on	ADP
cana-4565	21	3	applied	apply	VERB
cana-4565	21	4	nonlinear	nonlinear	ADJ
cana-4565	21	5	analysis	analysis	NOUN
cana-4565	21	6	issn	issn	NOUN
cana-4565	21	7	:	:	PUNCT
cana-4565	21	8	1074	1074	NUM
cana-4565	21	9	-	-	PUNCT
cana-4565	21	10	133x	133x	NUM
cana-4565	21	11	vol	vol	NOUN
cana-4565	21	12	32	32	NUM
cana-4565	21	13	no	no	NOUN
cana-4565	21	14	.	.	PUNCT
cana-4565	22	1	9s	9s	NUM
cana-4565	22	2	(	(	PUNCT
cana-4565	22	3	2025	2025	NUM
cana-4565	22	4	)	)	PUNCT
cana-4565	22	5	2616	2616	NUM
cana-4565	22	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4565	22	7	definition	definition	NOUN
cana-4565	22	8	1.4	1.4	NUM
cana-4565	23	1	[	[	X
cana-4565	23	2	11	11	NUM
cana-4565	23	3	]	]	PUNCT
cana-4565	23	4	𝐴	𝐴	PROPN
cana-4565	23	5	𝐵𝑉𝑀𝐼𝐹𝑆	𝐵𝑉𝑀𝐼𝐹𝑆	PROPN
cana-4565	23	6	ħ	ħ	PROPN
cana-4565	23	7	=	=	X
cana-4565	23	8			NOUN
cana-4565	23	9	ħ1	ħ1	VERB
cana-4565	23	10	+	+	PROPN
cana-4565	23	11	,	,	PUNCT
cana-4565	23	12	ħ2	ħ2	ADJ
cana-4565	23	13	+	+	PROPN
cana-4565	23	14	,	,	PUNCT
cana-4565	23	15	…	…	PUNCT
cana-4565	23	16	,	,	PUNCT
cana-4565	23	17	ħ𝑛	ħ𝑛	X
cana-4565	24	1	+	+	ADJ
cana-4565	24	2	,	,	PUNCT
cana-4565	24	3	ħ1	ħ1	VERB
cana-4565	24	4	−	−	PROPN
cana-4565	24	5	,	,	PUNCT
cana-4565	24	6	ħ2	ħ2	ADJ
cana-4565	24	7	−,	−,	NOUN
cana-4565	24	8	…	…	PUNCT
cana-4565	24	9	,ħ𝑛	,ħ𝑛	PUNCT
cana-4565	24	10	−	−	ADJ
cana-4565	24	11	𝑜𝑓	𝑜𝑓	ADP
cana-4565	24	12	𝑎	𝑎	NOUN
cana-4565	24	13	𝑟𝑖𝑛𝑔	𝑟𝑖𝑛𝑔	NOUN
cana-4565	24	14	ṏ	ṏ	X
cana-4565	24	15	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	24	16	𝑠𝑎𝑖𝑑	𝑠𝑎𝑖𝑑	VERB
cana-4565	24	17	to	to	ADP
cana-4565	24	18	𝑏𝑒	𝑏𝑒	NUM
cana-4565	24	19	𝑎	𝑎	DET
cana-4565	24	20	bipolar	bipolar	ADJ
cana-4565	24	21	valued	value	VERB
cana-4565	24	22	multi	multi	NOUN
cana-4565	25	1	i	i	PRON
cana-4565	25	2	−	−	VERB
cana-4565	25	3	fuzzy	fuzzy	ADJ
cana-4565	25	4	subring	subring	NOUN
cana-4565	25	5	of	of	ADP
cana-4565	25	6	ṏ	ṏ	PROPN
cana-4565	25	7	(	(	PUNCT
cana-4565	25	8	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	PROPN
cana-4565	25	9	)	)	PUNCT
cana-4565	25	10	𝑖𝑓	𝑖𝑓	ADP
cana-4565	25	11	ħ	ħ	PROPN
cana-4565	25	12	ℎ𝑎𝑠	ℎ𝑎𝑠	NOUN
cana-4565	25	13	,	,	PUNCT
cana-4565	25	14	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-4565	25	15	𝑎𝑙𝑙	𝑎𝑙𝑙	X
cana-4565	25	16	𝑖	𝑖	SYM
cana-4565	25	17	,	,	PUNCT
cana-4565	25	18	(	(	PUNCT
cana-4565	25	19	i	i	NOUN
cana-4565	25	20	)	)	PUNCT
cana-4565	25	21	ħ𝑖	ħ𝑖	ADP
cana-4565	26	1	+	+	ADJ
cana-4565	26	2	(	(	PUNCT
cana-4565	26	3	ჲ	ჲ	PROPN
cana-4565	26	4	−	−	NUM
cana-4565	26	5	ծ	ծ	SYM
cana-4565	26	6	)	)	PUNCT
cana-4565	26	7	≥	≥	NOUN
cana-4565	26	8	𝑟𝑚𝑖𝑛{ħ𝑖	𝑟𝑚𝑖𝑛{ħ𝑖	NOUN
cana-4565	26	9	+	+	ADJ
cana-4565	26	10	(	(	PUNCT
cana-4565	26	11	ჲ	ჲ	NOUN
cana-4565	26	12	)	)	PUNCT
cana-4565	26	13	,	,	PUNCT
cana-4565	26	14	ħ𝑖	ħ𝑖	ADP
cana-4565	26	15	+	+	ADJ
cana-4565	26	16	(	(	PUNCT
cana-4565	26	17	ծ	ծ	NOUN
cana-4565	26	18	)	)	PUNCT
cana-4565	26	19	}	}	PUNCT
cana-4565	26	20	,	,	PUNCT
cana-4565	26	21	(	(	PUNCT
cana-4565	26	22	ii	ii	NOUN
cana-4565	26	23	)	)	PUNCT
cana-4565	26	24	ħ𝑖	ħ𝑖	ADP
cana-4565	26	25	+	+	ADJ
cana-4565	26	26	(	(	PUNCT
cana-4565	26	27	ჲծ	ჲծ	NOUN
cana-4565	26	28	)	)	PUNCT
cana-4565	26	29	≥	≥	NOUN
cana-4565	26	30	𝑟𝑚𝑖𝑛{ħ𝑖	𝑟𝑚𝑖𝑛{ħ𝑖	NOUN
cana-4565	26	31	+	+	ADJ
cana-4565	26	32	(	(	PUNCT
cana-4565	26	33	ჲ	ჲ	NOUN
cana-4565	26	34	)	)	PUNCT
cana-4565	26	35	,	,	PUNCT
cana-4565	26	36	ħ𝑖	ħ𝑖	ADP
cana-4565	26	37	+	+	ADJ
cana-4565	26	38	(	(	PUNCT
cana-4565	26	39	ծ	ծ	NOUN
cana-4565	26	40	)	)	PUNCT
cana-4565	26	41	}	}	PUNCT
cana-4565	26	42	,	,	PUNCT
cana-4565	26	43	(	(	PUNCT
cana-4565	26	44	iii	iii	X
cana-4565	26	45	)	)	PUNCT
cana-4565	26	46	ħ𝑖	ħ𝑖	ADP
cana-4565	26	47	−(ჲ	−(ჲ	PROPN
cana-4565	26	48	−	−	PROPN
cana-4565	26	49	ծ	ծ	SYM
cana-4565	26	50	)	)	PUNCT
cana-4565	26	51	≤	≤	NOUN
cana-4565	26	52	𝑟𝑚𝑎𝑥{ħ𝑖	𝑟𝑚𝑎𝑥{ħ𝑖	PROPN
cana-4565	26	53	−(ჲ	−(ჲ	NOUN
cana-4565	26	54	)	)	PUNCT
cana-4565	26	55	,	,	PUNCT
cana-4565	26	56	ħ𝑖	ħ𝑖	ADP
cana-4565	26	57	−(ծ	−(ծ	NOUN
cana-4565	26	58	)	)	PUNCT
cana-4565	26	59	}	}	PUNCT
cana-4565	26	60	,	,	PUNCT
cana-4565	26	61	(	(	PUNCT
cana-4565	26	62	iv	iv	X
cana-4565	26	63	)	)	PUNCT
cana-4565	26	64	ħ𝑖	ħ𝑖	ADP
cana-4565	26	65	−(ჲծ	−(ჲծ	NOUN
cana-4565	26	66	)	)	PUNCT
cana-4565	26	67	≤	≤	NOUN
cana-4565	26	68	𝑟𝑚𝑎𝑥{ħ𝑖	𝑟𝑚𝑎𝑥{ħ𝑖	PROPN
cana-4565	26	69	−(ჲ	−(ჲ	NOUN
cana-4565	26	70	)	)	PUNCT
cana-4565	26	71	,	,	PUNCT
cana-4565	26	72	ħ𝑖	ħ𝑖	ADP
cana-4565	26	73	−(ծ	−(ծ	NOUN
cana-4565	26	74	)	)	PUNCT
cana-4565	26	75	}	}	PUNCT
cana-4565	26	76	,	,	PUNCT
cana-4565	26	77	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-4565	26	78	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-4565	26	79	ჲ	ჲ	X
cana-4565	26	80	,	,	PUNCT
cana-4565	26	81	ծ	ծ	PROPN
cana-4565	26	82	∈	∈	PROPN
cana-4565	26	83	ṏ	ṏ	NOUN
cana-4565	26	84	,	,	PUNCT
cana-4565	26	85	where	where	SCONJ
cana-4565	26	86	𝑟𝑚𝑖𝑛{[𝔯	𝑟𝑚𝑖𝑛{[𝔯	NOUN
cana-4565	26	87	,	,	PUNCT
cana-4565	26	88	ą	ą	PROPN
cana-4565	26	89	]	]	X
cana-4565	26	90	,	,	PUNCT
cana-4565	26	91	[	[	X
cana-4565	26	92	ъ	ъ	X
cana-4565	26	93	,	,	PUNCT
cana-4565	26	94	ծ	ծ	X
cana-4565	26	95	]	]	X
cana-4565	26	96	}	}	PUNCT
cana-4565	26	97	=	=	PUNCT
cana-4565	27	1	[	[	X
cana-4565	27	2	min{𝔯	min{𝔯	NOUN
cana-4565	27	3	,	,	PUNCT
cana-4565	27	4	ъ	ъ	NOUN
cana-4565	27	5	}	}	PUNCT
cana-4565	27	6	,	,	PUNCT
cana-4565	27	7	min{ą	min{ą	NOUN
cana-4565	27	8	,	,	PUNCT
cana-4565	27	9	ծ	ծ	X
cana-4565	27	10	}	}	PUNCT
cana-4565	27	11	]	]	PUNCT
cana-4565	27	12	and	and	CCONJ
cana-4565	27	13	𝑟𝑚𝑎𝑥{[𝔯	𝑟𝑚𝑎𝑥{[𝔯	NOUN
cana-4565	27	14	,	,	PUNCT
cana-4565	27	15	ą	ą	PROPN
cana-4565	27	16	]	]	X
cana-4565	27	17	,	,	PUNCT
cana-4565	27	18	[	[	X
cana-4565	27	19	ъ	ъ	X
cana-4565	27	20	,	,	PUNCT
cana-4565	27	21	ծ	ծ	X
cana-4565	27	22	]	]	X
cana-4565	27	23	}	}	PUNCT
cana-4565	27	24	=	=	PUNCT
cana-4565	28	1	[	[	X
cana-4565	28	2	max{𝔯	max{𝔯	X
cana-4565	28	3	,	,	PUNCT
cana-4565	28	4	ъ	ъ	NOUN
cana-4565	28	5	}	}	PUNCT
cana-4565	28	6	,	,	PUNCT
cana-4565	28	7	max{ą	max{ą	PROPN
cana-4565	28	8	,	,	PUNCT
cana-4565	28	9	ծ	ծ	X
cana-4565	28	10	}	}	PUNCT
cana-4565	28	11	]	]	PUNCT
cana-4565	28	12	.	.	PUNCT
cana-4565	28	13	example	example	NOUN
cana-4565	28	14	1.5	1.5	NUM
cana-4565	28	15	let	let	VERB
cana-4565	28	16	𝑅	𝑅	PROPN
cana-4565	28	17	=	=	PROPN
cana-4565	28	18	𝕫3	𝕫3	PROPN
cana-4565	28	19	=	=	SYM
cana-4565	28	20	{	{	PUNCT
cana-4565	28	21	0	0	NUM
cana-4565	28	22	,	,	PUNCT
cana-4565	28	23	1	1	NUM
cana-4565	28	24	,	,	PUNCT
cana-4565	28	25	2	2	NUM
cana-4565	28	26	}	}	PUNCT
cana-4565	28	27	𝑏𝑒	𝑏𝑒	NOUN
cana-4565	28	28	𝑎	𝑎	DET
cana-4565	28	29	𝑟𝑖𝑛𝑔	𝑟𝑖𝑛𝑔	ADJ
cana-4565	28	30	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	NOUN
cana-4565	28	31	⨁3	⨁3	NOUN
cana-4565	28	32	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-4565	28	33	⨂3	⨂3	PROPN
cana-4565	28	34	.	.	PUNCT
cana-4565	29	1	𝑇ℎ𝑒𝑛	𝑇ℎ𝑒𝑛	NOUN
cana-4565	29	2	ℭ	ℭ	NOUN
cana-4565	29	3	=	=	SYM
cana-4565	29	4	{	{	PUNCT
cana-4565	29	5	(	(	PUNCT
cana-4565	29	6	0	0	NUM
cana-4565	29	7	,	,	PUNCT
cana-4565	29	8	[	[	X
cana-4565	29	9	0.72	0.72	NUM
cana-4565	29	10	,	,	PUNCT
cana-4565	29	11	0.81	0.81	NUM
cana-4565	29	12	]	]	PUNCT
cana-4565	29	13	,	,	PUNCT
cana-4565	29	14	[	[	X
cana-4565	29	15	0.91	0.91	NUM
cana-4565	29	16	,	,	PUNCT
cana-4565	29	17	1	1	NUM
cana-4565	29	18	]	]	PUNCT
cana-4565	29	19	,	,	PUNCT
cana-4565	29	20	[	[	X
cana-4565	29	21	0.52	0.52	NUM
cana-4565	29	22	,	,	PUNCT
cana-4565	29	23	0.63	0.63	NUM
cana-4565	29	24	]	]	PUNCT
cana-4565	29	25	,	,	PUNCT
cana-4565	29	26	[	[	X
cana-4565	29	27	−	−	PROPN
cana-4565	29	28	0.91,−0.81	0.91,−0.81	NOUN
cana-4565	29	29	]	]	PUNCT
cana-4565	29	30	,	,	PUNCT
cana-4565	30	1	[	[	X
cana-4565	30	2	−	−	X
cana-4565	30	3	1,−0.91	1,−0.91	NUM
cana-4565	30	4	]	]	PUNCT
cana-4565	30	5	,	,	PUNCT
cana-4565	30	6	[	[	X
cana-4565	30	7	−	−	NOUN
cana-4565	30	8	0.81	0.81	NUM
cana-4565	30	9	,	,	PUNCT
cana-4565	30	10	−0.72	−0.72	NOUN
cana-4565	30	11	]	]	X
cana-4565	30	12	)	)	PUNCT
cana-4565	30	13	,	,	PUNCT
cana-4565	30	14	(	(	PUNCT
cana-4565	30	15	1	1	NUM
cana-4565	30	16	,	,	PUNCT
cana-4565	30	17	[	[	X
cana-4565	30	18	0.51	0.51	NUM
cana-4565	30	19	,	,	PUNCT
cana-4565	30	20	0.61	0.61	NUM
cana-4565	30	21	]	]	PUNCT
cana-4565	30	22	,	,	PUNCT
cana-4565	30	23	[	[	X
cana-4565	30	24	0.71	0.71	NUM
cana-4565	30	25	,	,	PUNCT
cana-4565	30	26	0.81	0.81	NUM
cana-4565	30	27	]	]	PUNCT
cana-4565	30	28	,	,	PUNCT
cana-4565	30	29	[	[	X
cana-4565	30	30	0.31	0.31	NUM
cana-4565	30	31	,	,	PUNCT
cana-4565	30	32	0.41],[−	0.41],[−	NUM
cana-4565	31	1	0.71,−0.61	0.71,−0.61	PROPN
cana-4565	31	2	]	]	PUNCT
cana-4565	31	3	,	,	PUNCT
cana-4565	31	4	[	[	X
cana-4565	31	5	−	−	X
cana-4565	31	6	0.61,−0.51	0.61,−0.51	NOUN
cana-4565	31	7	]	]	X
cana-4565	31	8	,	,	PUNCT
cana-4565	31	9	[	[	X
cana-4565	31	10	−	−	NOUN
cana-4565	31	11	0.51	0.51	NUM
cana-4565	31	12	,	,	PUNCT
cana-4565	31	13	−0.41	−0.41	NOUN
cana-4565	31	14	]	]	PUNCT
cana-4565	31	15	)	)	PUNCT
cana-4565	31	16	,	,	PUNCT
cana-4565	31	17	(	(	PUNCT
cana-4565	31	18	2	2	NUM
cana-4565	31	19	,	,	PUNCT
cana-4565	31	20	[	[	X
cana-4565	31	21	0.51	0.51	NUM
cana-4565	31	22	,	,	PUNCT
cana-4565	31	23	0.61	0.61	NUM
cana-4565	31	24	]	]	PUNCT
cana-4565	31	25	,	,	PUNCT
cana-4565	31	26	[	[	X
cana-4565	31	27	0.71	0.71	NUM
cana-4565	31	28	,	,	PUNCT
cana-4565	31	29	0.81	0.81	NUM
cana-4565	31	30	]	]	PUNCT
cana-4565	31	31	,	,	PUNCT
cana-4565	32	1	[	[	X
cana-4565	32	2	0.31	0.31	NUM
cana-4565	32	3	,	,	PUNCT
cana-4565	32	4	0.41	0.41	NUM
cana-4565	32	5	]	]	PUNCT
cana-4565	32	6	,	,	PUNCT
cana-4565	33	1	[	[	X
cana-4565	33	2	−0.71	−0.71	ADP
cana-4565	33	3	,	,	PUNCT
cana-4565	33	4	−0.61	−0.61	NOUN
cana-4565	33	5	]	]	PUNCT
cana-4565	33	6	,	,	PUNCT
cana-4565	34	1	[	[	X
cana-4565	34	2	−0.61,−0.51	−0.61,−0.51	X
cana-4565	34	3	]	]	X
cana-4565	34	4	,	,	PUNCT
cana-4565	34	5	[	[	X
cana-4565	34	6	−0.51	−0.51	X
cana-4565	34	7	,	,	PUNCT
cana-4565	34	8	−0.41	−0.41	NOUN
cana-4565	34	9	]	]	PUNCT
cana-4565	34	10	)	)	PUNCT
cana-4565	34	11	}	}	PUNCT
cana-4565	34	12	is	be	AUX
cana-4565	34	13	a	a	DET
cana-4565	34	14	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	NOUN
cana-4565	34	15	of	of	ADP
cana-4565	34	16	r.	r.	PROPN
cana-4565	34	17	definition	definition	NOUN
cana-4565	34	18	1.6	1.6	NUM
cana-4565	34	19	𝐴	𝐴	PROPN
cana-4565	34	20	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	NOUN
cana-4565	34	21	ɮ	ɮ	NOUN
cana-4565	34	22	=	=	SYM
cana-4565	34	23			X
cana-4565	34	24	ɮ1	ɮ1	PROPN
cana-4565	34	25	+	+	NOUN
cana-4565	34	26	,	,	PUNCT
cana-4565	34	27	ɮ2	ɮ2	NOUN
cana-4565	34	28	+	+	PROPN
cana-4565	34	29	,	,	PUNCT
cana-4565	34	30	…	…	PUNCT
cana-4565	34	31	,	,	PUNCT
cana-4565	34	32	ɮ𝑛	ɮ𝑛	ADP
cana-4565	34	33	+	+	PROPN
cana-4565	34	34	,	,	PUNCT
cana-4565	34	35	ɮ1	ɮ1	PROPN
cana-4565	34	36	−	−	PROPN
cana-4565	34	37	,	,	PUNCT
cana-4565	34	38	ɮ2	ɮ2	ADJ
cana-4565	34	39	−,	−,	NOUN
cana-4565	34	40	…	…	PUNCT
cana-4565	34	41	,ɮ𝑛	,ɮ𝑛	PUNCT
cana-4565	34	42	−	−	ADJ
cana-4565	34	43	𝑜𝑓	𝑜𝑓	ADP
cana-4565	34	44	𝑎	𝑎	NOUN
cana-4565	34	45	𝑟𝑖𝑛𝑔	𝑟𝑖𝑛𝑔	NOUN
cana-4565	34	46	₢	₢	ADP
cana-4565	34	47	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	34	48	𝑠𝑎𝑖𝑑	𝑠𝑎𝑖𝑑	VERB
cana-4565	34	49	to	to	ADP
cana-4565	34	50	𝑏𝑒	𝑏𝑒	NUM
cana-4565	34	51	𝑎	𝑎	DET
cana-4565	34	52	bipolar	bipolar	ADJ
cana-4565	34	53	valued	value	VERB
cana-4565	34	54	multi	multi	NOUN
cana-4565	35	1	i	i	PRON
cana-4565	35	2	−	−	VERB
cana-4565	35	3	fuzzy	fuzzy	ADJ
cana-4565	35	4	normal	normal	ADJ
cana-4565	35	5	subring	subring	NOUN
cana-4565	35	6	of	of	ADP
cana-4565	35	7	r	r	NOUN
cana-4565	35	8	(	(	PUNCT
cana-4565	35	9	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	35	10	)	)	PUNCT
cana-4565	35	11	𝑖𝑓	𝑖𝑓	ADP
cana-4565	35	12	ɮ	ɮ	NOUN
cana-4565	35	13	ℎ𝑎𝑠	ℎ𝑎𝑠	NOUN
cana-4565	35	14	,	,	PUNCT
cana-4565	35	15	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-4565	35	16	𝑎𝑙𝑙	𝑎𝑙𝑙	X
cana-4565	35	17	𝑖	𝑖	SYM
cana-4565	35	18	,	,	PUNCT
cana-4565	35	19	(	(	PUNCT
cana-4565	35	20	i	i	NOUN
cana-4565	35	21	)	)	PUNCT
cana-4565	35	22	ɮ𝑖	ɮ𝑖	NOUN
cana-4565	35	23	+	+	PROPN
cana-4565	35	24	(	(	PUNCT
cana-4565	35	25	𝔶𝔴	𝔶𝔴	NOUN
cana-4565	35	26	)	)	PUNCT
cana-4565	35	27	=	=	NOUN
cana-4565	35	28	ɮ𝑖	ɮ𝑖	NOUN
cana-4565	35	29	+	+	NOUN
cana-4565	35	30	(	(	PUNCT
cana-4565	35	31	𝔴𝔶	𝔴𝔶	NOUN
cana-4565	35	32	)	)	PUNCT
cana-4565	35	33	,	,	PUNCT
cana-4565	35	34	(	(	PUNCT
cana-4565	35	35	ii	ii	NOUN
cana-4565	35	36	)	)	PUNCT
cana-4565	35	37	ɮ𝑖	ɮ𝑖	NOUN
cana-4565	35	38	−(𝔶𝔴	−(𝔶𝔴	PROPN
cana-4565	35	39	)	)	PUNCT
cana-4565	35	40	=	=	PRON
cana-4565	35	41	ɮ𝑖	ɮ𝑖	NOUN
cana-4565	35	42	−(𝔴𝔶	−(𝔴𝔶	PROPN
cana-4565	35	43	)	)	PUNCT
cana-4565	35	44	,	,	PUNCT
cana-4565	35	45	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-4565	35	46	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-4565	35	47	𝔶	𝔶	PROPN
cana-4565	35	48	,	,	PUNCT
cana-4565	35	49	𝔴	𝔴	PROPN
cana-4565	35	50	∈	∈	PROPN
cana-4565	35	51	₢	₢	ADP
cana-4565	35	52	.	.	PUNCT
cana-4565	36	1	definition	definition	NOUN
cana-4565	36	2	1.7	1.7	NUM
cana-4565	37	1	[	[	X
cana-4565	37	2	11	11	NUM
cana-4565	37	3	]	]	PUNCT
cana-4565	37	4	let	let	VERB
cana-4565	37	5	œ	œ	PROPN
cana-4565	37	6	=	=	PUNCT
cana-4565	37	7			SYM
cana-4565	37	8	œ1	œ1	PROPN
cana-4565	37	9	+	+	PROPN
cana-4565	37	10	,	,	PUNCT
cana-4565	37	11	œ2	œ2	NOUN
cana-4565	37	12	+	+	PROPN
cana-4565	37	13	,	,	PUNCT
cana-4565	37	14	…	…	PUNCT
cana-4565	37	15	,	,	PUNCT
cana-4565	37	16	œ𝑛	œ𝑛	ADP
cana-4565	37	17	+	+	ADJ
cana-4565	37	18	,	,	PUNCT
cana-4565	37	19	œ1	œ1	ADJ
cana-4565	37	20	−	−	PROPN
cana-4565	37	21	,	,	PUNCT
cana-4565	37	22	œ2	œ2	ADJ
cana-4565	37	23	−,	−,	NOUN
cana-4565	37	24	…	…	PUNCT
cana-4565	37	25	,œ𝑛	,œ𝑛	PUNCT
cana-4565	37	26	−	−	NUM
cana-4565	37	27	and	and	CCONJ
cana-4565	37	28	ϥ	ϥ	NOUN
cana-4565	37	29	=	=	SYM
cana-4565	37	30			SYM
cana-4565	37	31	ϥ1	ϥ1	NOUN
cana-4565	37	32	+	+	PROPN
cana-4565	37	33	,	,	PUNCT
cana-4565	37	34	ϥ2	ϥ2	PROPN
cana-4565	37	35	+	+	PROPN
cana-4565	37	36	,	,	PUNCT
cana-4565	37	37	…	…	PUNCT
cana-4565	37	38	,	,	PUNCT
cana-4565	37	39	ϥ𝑛	ϥ𝑛	VERB
cana-4565	37	40	+	+	ADJ
cana-4565	37	41	,	,	PUNCT
cana-4565	37	42	ϥ1	ϥ1	VERB
cana-4565	37	43	−	−	PROPN
cana-4565	37	44	,	,	PUNCT
cana-4565	37	45	ϥ2	ϥ2	PROPN
cana-4565	37	46	−,	−,	PROPN
cana-4565	37	47	…	…	PUNCT
cana-4565	37	48	,ϥ𝑛	,ϥ𝑛	PUNCT
cana-4565	37	49	−	−	PROPN
cana-4565	37	50	be	be	AUX
cana-4565	37	51	𝑡𝑤𝑜	𝑡𝑤𝑜	PROPN
cana-4565	37	52	𝐵𝑉𝑀𝐼𝐹𝑆𝑠	𝐵𝑉𝑀𝐼𝐹𝑆𝑠	NOUN
cana-4565	37	53	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	NOUN
cana-4565	37	54	𝑑𝑒𝑔𝑟𝑒𝑒	𝑑𝑒𝑔𝑟𝑒𝑒	VERB
cana-4565	37	55	𝑛	𝑛	PRON
cana-4565	37	56	𝑜𝑓	𝑜𝑓	ADP
cana-4565	37	57	𝑎	𝑎	X
cana-4565	37	58	𝑠𝑒𝑡	𝑠𝑒𝑡	NOUN
cana-4565	37	59	𝒲.	𝒲.	PROPN
cana-4565	37	60	𝑇ℎ𝑒𝑛	𝑇ℎ𝑒𝑛	PROPN
cana-4565	37	61	(	(	PUNCT
cana-4565	37	62	i	i	NOUN
cana-4565	37	63	)	)	PUNCT
cana-4565	37	64	œ	œ	PROPN
cana-4565	38	1	⊂	⊂	X
cana-4565	38	2	ϥ	ϥ	PROPN
cana-4565	38	3	𝑖𝑓	𝑖𝑓	X
cana-4565	38	4	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-4565	38	5	𝑜𝑛𝑙𝑦	𝑜𝑛𝑙𝑦	ADV
cana-4565	38	6	𝑖𝑓	𝑖𝑓	PUNCT
cana-4565	38	7			PROPN
cana-4565	38	8	𝑖	𝑖	PROPN
cana-4565	38	9	,	,	PUNCT
cana-4565	38	10	œ𝑖	œ𝑖	VERB
cana-4565	38	11	+	+	PROPN
cana-4565	38	12	(	(	PUNCT
cana-4565	38	13	ҁ	ҁ	NOUN
cana-4565	38	14	)	)	PUNCT
cana-4565	38	15	≤	≤	NOUN
cana-4565	38	16	ϥ𝑖	ϥ𝑖	X
cana-4565	39	1	+	+	ADJ
cana-4565	39	2	(	(	PUNCT
cana-4565	39	3	ҁ	ҁ	NOUN
cana-4565	39	4	)	)	PUNCT
cana-4565	39	5	and	and	CCONJ
cana-4565	39	6	œ𝑖	œ𝑖	VERB
cana-4565	39	7	−(ҁ	−(ҁ	ADV
cana-4565	39	8	)	)	PUNCT
cana-4565	39	9	≥	≥	NOUN
cana-4565	39	10	ϥ𝑖	ϥ𝑖	NOUN
cana-4565	39	11	−(ҁ),	−(ҁ),	NOUN
cana-4565	39	12	ҁ	ҁ	PRON
cana-4565	39	13	∈	∈	PROPN
cana-4565	39	14	𝒲.	𝒲.	PROPN
cana-4565	39	15	(	(	PUNCT
cana-4565	39	16	ii	ii	NOUN
cana-4565	39	17	)	)	PUNCT
cana-4565	39	18	œ	œ	PROPN
cana-4565	39	19	∩	∩	NOUN
cana-4565	39	20	ϥ	ϥ	NOUN
cana-4565	39	21	=	=	SYM
cana-4565	39	22	{	{	PUNCT
cana-4565	39	23	ҁ	ҁ	NOUN
cana-4565	39	24	,	,	PUNCT
cana-4565	39	25	rmin(œ1	rmin(œ1	NOUN
cana-4565	39	26	+	+	NOUN
cana-4565	39	27	(	(	PUNCT
cana-4565	39	28	ҁ	ҁ	NOUN
cana-4565	39	29	)	)	PUNCT
cana-4565	39	30	,	,	PUNCT
cana-4565	39	31	ϥ1	ϥ1	NOUN
cana-4565	39	32	+	+	PROPN
cana-4565	39	33	(	(	PUNCT
cana-4565	39	34	ҁ	ҁ	NOUN
cana-4565	39	35	)	)	PUNCT
cana-4565	39	36	)	)	PUNCT
cana-4565	40	1	,	,	PUNCT
cana-4565	40	2	rmin(œ2	rmin(œ2	NOUN
cana-4565	40	3	+	+	NOUN
cana-4565	40	4	(	(	PUNCT
cana-4565	40	5	ҁ	ҁ	NOUN
cana-4565	40	6	)	)	PUNCT
cana-4565	40	7	,	,	PUNCT
cana-4565	40	8	ϥ2	ϥ2	PROPN
cana-4565	40	9	+	+	PROPN
cana-4565	40	10	(	(	PUNCT
cana-4565	40	11	ҁ	ҁ	NOUN
cana-4565	40	12	)	)	PUNCT
cana-4565	40	13	)	)	PUNCT
cana-4565	40	14	,	,	PUNCT
cana-4565	40	15	…	…	PUNCT
cana-4565	40	16	,	,	PUNCT
cana-4565	40	17	rmin(œ𝑛	rmin(œ𝑛	PROPN
cana-4565	40	18	+	+	PROPN
cana-4565	40	19	(	(	PUNCT
cana-4565	40	20	ҁ	ҁ	NOUN
cana-4565	40	21	)	)	PUNCT
cana-4565	40	22	,	,	PUNCT
cana-4565	40	23	ϥ𝑛	ϥ𝑛	VERB
cana-4565	40	24	+	+	NOUN
cana-4565	40	25	(	(	PUNCT
cana-4565	40	26	ҁ	ҁ	NOUN
cana-4565	40	27	)	)	PUNCT
cana-4565	40	28	)	)	PUNCT
cana-4565	40	29	,	,	PUNCT
cana-4565	40	30	rmax(œ1	rmax(œ1	NOUN
cana-4565	40	31	−(ҁ	−(ҁ	VERB
cana-4565	40	32	)	)	PUNCT
cana-4565	40	33	,	,	PUNCT
cana-4565	40	34	ϥ1	ϥ1	NOUN
cana-4565	40	35	−(ҁ	−(ҁ	NOUN
cana-4565	40	36	)	)	PUNCT
cana-4565	40	37	)	)	PUNCT
cana-4565	40	38	,	,	PUNCT
cana-4565	40	39	rmax(œ2	rmax(œ2	NOUN
cana-4565	40	40	−(ҁ	−(ҁ	VERB
cana-4565	40	41	)	)	PUNCT
cana-4565	40	42	,	,	PUNCT
cana-4565	40	43	ϥ2	ϥ2	PROPN
cana-4565	40	44	−(ҁ	−(ҁ	PROPN
cana-4565	40	45	)	)	PUNCT
cana-4565	40	46	)	)	PUNCT
cana-4565	40	47	,	,	PUNCT
cana-4565	40	48	…	…	PUNCT
cana-4565	40	49	,	,	PUNCT
cana-4565	40	50	rmax(œ𝑛	rmax(œ𝑛	ADP
cana-4565	40	51	−(ҁ	−(ҁ	NOUN
cana-4565	40	52	)	)	PUNCT
cana-4565	40	53	,	,	PUNCT
cana-4565	40	54	ϥ𝑛	ϥ𝑛	NOUN
cana-4565	40	55	−(ҁ))	−(ҁ))	ADJ
cana-4565	40	56	/	/	SYM
cana-4565	40	57	ҁ	ҁ	PRON
cana-4565	40	58	∈	∈	NOUN
cana-4565	40	59	𝒲	𝒲	NOUN
cana-4565	40	60	}	}	PUNCT
cana-4565	40	61	.	.	PUNCT
cana-4565	41	1	definition	definition	NOUN
cana-4565	41	2	1.8	1.8	NUM
cana-4565	41	3	.	.	PUNCT
cana-4565	42	1	𝐿𝑒𝑡	𝐿𝑒𝑡	NOUN
cana-4565	42	2	ж	ж	NOUN
cana-4565	42	3	=	=	PUNCT
cana-4565	42	4			X
cana-4565	42	5	ж1	ж1	X
cana-4565	42	6	+	+	PROPN
cana-4565	42	7	,	,	PUNCT
cana-4565	42	8	ж2	ж2	PROPN
cana-4565	42	9	+	+	PROPN
cana-4565	42	10	,	,	PUNCT
cana-4565	42	11	…	…	PUNCT
cana-4565	42	12	,	,	PUNCT
cana-4565	42	13	ж𝑛	ж𝑛	X
cana-4565	43	1	+	+	ADJ
cana-4565	43	2	,	,	PUNCT
cana-4565	43	3	ж1	ж1	PROPN
cana-4565	43	4	−	−	PROPN
cana-4565	43	5	,	,	PUNCT
cana-4565	43	6	ж2	ж2	PROPN
cana-4565	43	7	−,	−,	PROPN
cana-4565	43	8	…	…	PUNCT
cana-4565	43	9	,ж𝑛	,ж𝑛	PUNCT
cana-4565	43	10	−	−	PROPN
cana-4565	43	11	be	be	AUX
cana-4565	43	12	𝐵𝑉𝑀𝐼𝐹𝑆	𝐵𝑉𝑀𝐼𝐹𝑆	PROPN
cana-4565	43	13	of	of	ADP
cana-4565	43	14	the	the	DET
cana-4565	43	15	set	set	NOUN
cana-4565	43	16	𝒲.	𝒲.	PROPN
cana-4565	43	17	the	the	DET
cana-4565	43	18	transformations	transformation	NOUN
cana-4565	43	19	are	be	AUX
cana-4565	43	20	defined	define	VERB
cana-4565	43	21	as	as	ADP
cana-4565	43	22	,	,	PUNCT
cana-4565	43	23			NOUN
cana-4565	43	24	i	i	NOUN
cana-4565	43	25	=	=	NOUN
cana-4565	43	26	1	1	NUM
cana-4565	43	27	,	,	PUNCT
cana-4565	43	28	2	2	NUM
cana-4565	43	29	,	,	PUNCT
cana-4565	43	30	…	…	PUNCT
cana-4565	43	31	,	,	PUNCT
cana-4565	43	32	n	n	CCONJ
cana-4565	43	33	,	,	PUNCT
cana-4565	43	34	(	(	PUNCT
cana-4565	43	35	i	i	NOUN
cana-4565	43	36	)	)	PUNCT
cana-4565	43	37	≬	≬	PROPN
cana-4565	43	38	(	(	PUNCT
cana-4565	43	39	ж	ж	X
cana-4565	43	40	)	)	PUNCT
cana-4565	43	41	=	=	NOUN
cana-4565	43	42			X
cana-4565	43	43	≬(ж1	≬(ж1	X
cana-4565	43	44	+	+	NOUN
cana-4565	43	45	)	)	PUNCT
cana-4565	43	46	,	,	PUNCT
cana-4565	43	47	≬(ж2	≬(ж2	PROPN
cana-4565	43	48	+	+	ADJ
cana-4565	43	49	)	)	PUNCT
cana-4565	43	50	,	,	PUNCT
cana-4565	43	51	…	…	PUNCT
cana-4565	43	52	,	,	PUNCT
cana-4565	43	53	≬(ж𝑛	≬(ж𝑛	PROPN
cana-4565	43	54	+	+	NUM
cana-4565	43	55	)	)	PUNCT
cana-4565	43	56	,	,	PUNCT
cana-4565	43	57	≬(ж1	≬(ж1	PROPN
cana-4565	43	58	−	−	NOUN
cana-4565	43	59	)	)	PUNCT
cana-4565	43	60	,	,	PUNCT
cana-4565	43	61	≬(ж2	≬(ж2	PROPN
cana-4565	43	62	−	−	NUM
cana-4565	43	63	)	)	PUNCT
cana-4565	43	64	,	,	PUNCT
cana-4565	43	65	…	…	PUNCT
cana-4565	43	66	,	,	PUNCT
cana-4565	43	67	≬(ж𝑛	≬(ж𝑛	PROPN
cana-4565	43	68	−	−	NUM
cana-4565	43	69	)	)	PUNCT
cana-4565	43	70			PROPN
cana-4565	43	71	,	,	PUNCT
cana-4565	43	72	where	where	SCONJ
cana-4565	43	73	≬(ж𝑖	≬(ж𝑖	NOUN
cana-4565	43	74	+	+	NOUN
cana-4565	43	75	)	)	PUNCT
cana-4565	43	76	(	(	PUNCT
cana-4565	43	77	𝜚	𝜚	NOUN
cana-4565	43	78	)	)	PUNCT
cana-4565	44	1	=	=	SYM
cana-4565	44	2	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
cana-4565	44	3	{	{	PUNCT
cana-4565	45	1	[	[	X
cana-4565	45	2	½	½	NOUN
cana-4565	45	3	,	,	PUNCT
cana-4565	45	4	½	½	NOUN
cana-4565	45	5	]	]	PUNCT
cana-4565	45	6	,	,	PUNCT
cana-4565	45	7	ж𝑖	ж𝑖	ADP
cana-4565	45	8	+	+	PROPN
cana-4565	45	9	(	(	PUNCT
cana-4565	45	10	𝜚	𝜚	NOUN
cana-4565	45	11	)	)	PUNCT
cana-4565	45	12	}	}	PUNCT
cana-4565	45	13	and	and	CCONJ
cana-4565	45	14	≬(ж𝑖	≬(ж𝑖	NOUN
cana-4565	45	15	−)(𝜚	−)(𝜚	NOUN
cana-4565	45	16	)	)	PUNCT
cana-4565	45	17	=	=	VERB
cana-4565	46	1	𝑟𝑚𝑎𝑥	𝑟𝑚𝑎𝑥	NOUN
cana-4565	46	2	{	{	PUNCT
cana-4565	46	3	[	[	X
cana-4565	46	4	−½	−½	X
cana-4565	46	5	,	,	PUNCT
cana-4565	46	6	−½	−½	PROPN
cana-4565	46	7	]	]	PUNCT
cana-4565	46	8	,	,	PUNCT
cana-4565	46	9	ж𝑖	ж𝑖	ADP
cana-4565	46	10	−(𝜚	−(𝜚	NOUN
cana-4565	46	11	)	)	PUNCT
cana-4565	46	12	}	}	PUNCT
cana-4565	46	13	,	,	PUNCT
cana-4565	46	14			NOUN
cana-4565	46	15	𝜚	𝜚	NOUN
cana-4565	46	16	∈	∈	NOUN
cana-4565	46	17	𝒲.	𝒲.	PROPN
cana-4565	46	18	(	(	PUNCT
cana-4565	46	19	ii	ii	NOUN
cana-4565	46	20	)	)	PUNCT
cana-4565	46	21	⋈	⋈	PROPN
cana-4565	46	22	(	(	PUNCT
cana-4565	46	23	ж	ж	X
cana-4565	46	24	)	)	PUNCT
cana-4565	46	25	=	=	SYM
cana-4565	46	26			NOUN
cana-4565	46	27	⋈(ж1	⋈(ж1	PROPN
cana-4565	46	28	+	+	ADJ
cana-4565	46	29	)	)	PUNCT
cana-4565	46	30	,	,	PUNCT
cana-4565	46	31	⋈(ж2	⋈(ж2	PROPN
cana-4565	46	32	+	+	ADJ
cana-4565	46	33	)	)	PUNCT
cana-4565	46	34	,	,	PUNCT
cana-4565	46	35	…	…	PUNCT
cana-4565	46	36	,	,	PUNCT
cana-4565	46	37	⋈(ж𝑛	⋈(ж𝑛	PROPN
cana-4565	46	38	+	+	PROPN
cana-4565	46	39	)	)	PUNCT
cana-4565	46	40	,	,	PUNCT
cana-4565	46	41	⋈(ж1	⋈(ж1	PROPN
cana-4565	46	42	−	−	NOUN
cana-4565	46	43	)	)	PUNCT
cana-4565	46	44	,	,	PUNCT
cana-4565	46	45	⋈(ж2	⋈(ж2	PROPN
cana-4565	46	46	−	−	NUM
cana-4565	46	47	)	)	PUNCT
cana-4565	46	48	,	,	PUNCT
cana-4565	46	49	…	…	PUNCT
cana-4565	46	50	,	,	PUNCT
cana-4565	46	51	⋈(ж𝑛	⋈(ж𝑛	PROPN
cana-4565	46	52	−	−	NUM
cana-4565	46	53	)	)	PUNCT
cana-4565	46	54			PROPN
cana-4565	46	55	,	,	PUNCT
cana-4565	46	56	where	where	SCONJ
cana-4565	46	57	⋈(ж𝑖	⋈(ж𝑖	PROPN
cana-4565	46	58	+	+	NOUN
cana-4565	46	59	)	)	PUNCT
cana-4565	46	60	(	(	PUNCT
cana-4565	46	61	𝜚	𝜚	NOUN
cana-4565	46	62	)	)	PUNCT
cana-4565	46	63	=	=	VERB
cana-4565	46	64	𝑟𝑚𝑎𝑥	𝑟𝑚𝑎𝑥	NOUN
cana-4565	46	65	{	{	PUNCT
cana-4565	46	66	[	[	X
cana-4565	46	67	½	½	NOUN
cana-4565	46	68	,	,	PUNCT
cana-4565	46	69	½	½	NOUN
cana-4565	46	70	]	]	PUNCT
cana-4565	46	71	,	,	PUNCT
cana-4565	46	72	ж𝑖	ж𝑖	ADP
cana-4565	46	73	+	+	PROPN
cana-4565	46	74	(	(	PUNCT
cana-4565	46	75	𝜚	𝜚	NOUN
cana-4565	46	76	)	)	PUNCT
cana-4565	46	77	}	}	PUNCT
cana-4565	46	78	and	and	CCONJ
cana-4565	46	79	⋈(ж𝑖	⋈(ж𝑖	PROPN
cana-4565	46	80	−)(𝜚	−)(𝜚	NUM
cana-4565	46	81	)	)	PUNCT
cana-4565	46	82	=	=	SYM
cana-4565	46	83	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
cana-4565	46	84	{	{	PUNCT
cana-4565	47	1	[	[	X
cana-4565	47	2	−½	−½	X
cana-4565	47	3	,	,	PUNCT
cana-4565	47	4	−½	−½	PROPN
cana-4565	47	5	]	]	PUNCT
cana-4565	47	6	,	,	PUNCT
cana-4565	47	7	ж𝑖	ж𝑖	ADP
cana-4565	47	8	−(𝜚	−(𝜚	NOUN
cana-4565	47	9	)	)	PUNCT
cana-4565	47	10	}	}	PUNCT
cana-4565	47	11	,	,	PUNCT
cana-4565	47	12			NOUN
cana-4565	47	13	𝜚	𝜚	VERB
cana-4565	47	14	∈	∈	NOUN
cana-4565	47	15	𝒲.	𝒲.	PROPN
cana-4565	47	16	(	(	PUNCT
cana-4565	47	17	iii	iii	NOUN
cana-4565	47	18	)	)	PUNCT
cana-4565	47	19	𝔔(𝜛,𝜍)(ж	𝔔(𝜛,𝜍)(ж	PROPN
cana-4565	47	20	)	)	PUNCT
cana-4565	47	21	=	=	SYM
cana-4565	47	22			X
cana-4565	47	23	𝔔(𝜛,𝜍)(ж1	𝔔(𝜛,𝜍)(ж1	NOUN
cana-4565	47	24	+	+	NOUN
cana-4565	47	25	)	)	PUNCT
cana-4565	47	26	,	,	PUNCT
cana-4565	47	27	𝔔(𝜛,𝜍)(ж2	𝔔(𝜛,𝜍)(ж2	PROPN
cana-4565	47	28	+	+	NOUN
cana-4565	47	29	)	)	PUNCT
cana-4565	47	30	,	,	PUNCT
cana-4565	47	31	…	…	PUNCT
cana-4565	47	32	,	,	PUNCT
cana-4565	47	33	𝔔(𝜛,𝜍)(ж𝑛	𝔔(𝜛,𝜍)(ж𝑛	X
cana-4565	47	34	+	+	NOUN
cana-4565	47	35	)	)	PUNCT
cana-4565	47	36	,	,	PUNCT
cana-4565	47	37	𝔔(𝜛,𝜍)(ж1	𝔔(𝜛,𝜍)(ж1	PROPN
cana-4565	47	38	−	−	PROPN
cana-4565	47	39	)	)	PUNCT
cana-4565	47	40	,	,	PUNCT
cana-4565	47	41	𝔔(𝜛,𝜍)(ж2	𝔔(𝜛,𝜍)(ж2	ADJ
cana-4565	47	42	−	−	NOUN
cana-4565	47	43	)	)	PUNCT
cana-4565	47	44	,	,	PUNCT
cana-4565	47	45	…	…	PUNCT
cana-4565	47	46	,	,	PUNCT
cana-4565	47	47	𝔔(𝜛,𝜍)(ж𝑛	𝔔(𝜛,𝜍)(ж𝑛	NUM
cana-4565	47	48	−	−	NUM
cana-4565	47	49	)	)	PUNCT
cana-4565	47	50			PROPN
cana-4565	47	51	,	,	PUNCT
cana-4565	47	52	where	where	SCONJ
cana-4565	47	53	𝔔(𝜛,𝜍)(ж𝑖	𝔔(𝜛,𝜍)(ж𝑖	NOUN
cana-4565	47	54	+	+	NOUN
cana-4565	47	55	)	)	PUNCT
cana-4565	47	56	(	(	PUNCT
cana-4565	47	57	𝜚	𝜚	NOUN
cana-4565	47	58	)	)	PUNCT
cana-4565	47	59	=	=	SYM
cana-4565	47	60	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
cana-4565	47	61	{	{	PUNCT
cana-4565	47	62	𝜛𝑖	𝜛𝑖	NOUN
cana-4565	47	63	,	,	PUNCT
cana-4565	47	64	ж𝑖	ж𝑖	ADP
cana-4565	47	65	+	+	PROPN
cana-4565	47	66	(	(	PUNCT
cana-4565	47	67	𝜚	𝜚	NOUN
cana-4565	47	68	)	)	PUNCT
cana-4565	47	69	}	}	PUNCT
cana-4565	47	70	and	and	CCONJ
cana-4565	47	71	𝔔(𝜛,𝜍)(ж𝑖	𝔔(𝜛,𝜍)(ж𝑖	NOUN
cana-4565	47	72	−)(𝜚	−)(𝜚	NOUN
cana-4565	47	73	)	)	PUNCT
cana-4565	47	74	=	=	VERB
cana-4565	48	1	𝑟𝑚𝑎𝑥	𝑟𝑚𝑎𝑥	NOUN
cana-4565	48	2	{	{	PUNCT
cana-4565	48	3	𝜍𝑖	𝜍𝑖	NOUN
cana-4565	48	4	,	,	PUNCT
cana-4565	48	5	ж𝑖	ж𝑖	ADP
cana-4565	48	6	−(𝜚	−(𝜚	NOUN
cana-4565	48	7	)	)	PUNCT
cana-4565	48	8	}	}	PUNCT
cana-4565	48	9	,	,	PUNCT
cana-4565	48	10			NOUN
cana-4565	48	11	𝜚	𝜚	NOUN
cana-4565	48	12	∈	∈	PROPN
cana-4565	48	13	𝒲	𝒲	PROPN
cana-4565	48	14	,	,	PUNCT
cana-4565	48	15	𝜛	𝜛	X
cana-4565	48	16	=	=	SYM
cana-4565	48	17	(	(	PUNCT
cana-4565	48	18	𝜛1	𝜛1	NOUN
cana-4565	48	19	,	,	PUNCT
cana-4565	48	20	𝜛2	𝜛2	NOUN
cana-4565	48	21	,	,	PUNCT
cana-4565	48	22	…	…	PUNCT
cana-4565	48	23	,	,	PUNCT
cana-4565	48	24	𝜛𝑛	𝜛𝑛	NOUN
cana-4565	48	25	)	)	PUNCT
cana-4565	48	26	and	and	CCONJ
cana-4565	48	27	𝜍	𝜍	X
cana-4565	48	28	=	=	SYM
cana-4565	48	29	(	(	PUNCT
cana-4565	48	30	𝜍1	𝜍1	PROPN
cana-4565	48	31	,	,	PUNCT
cana-4565	48	32	𝜍2	𝜍2	NOUN
cana-4565	48	33	,	,	PUNCT
cana-4565	48	34	…	…	PUNCT
cana-4565	48	35	,	,	PUNCT
cana-4565	48	36	𝜍𝑛	𝜍𝑛	NOUN
cana-4565	48	37	)	)	PUNCT
cana-4565	48	38	,	,	PUNCT
cana-4565	48	39	𝜛𝑖	𝜛𝑖	NOUN
cana-4565	48	40	∈	∈	PROPN
cana-4565	48	41	𝐷[0	𝐷[0	PROPN
cana-4565	48	42	,	,	PUNCT
cana-4565	48	43	1	1	NUM
cana-4565	48	44	]	]	X
cana-4565	48	45	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-4565	48	46	𝜍𝑖	𝜍𝑖	PROPN
cana-4565	48	47	∈	∈	PROPN
cana-4565	48	48	𝐷[−1	𝐷[−1	NOUN
cana-4565	48	49	,	,	PUNCT
cana-4565	48	50	0	0	NUM
cana-4565	48	51	]	]	PUNCT
cana-4565	48	52	.	.	PUNCT
cana-4565	49	1	(	(	PUNCT
cana-4565	49	2	iv	iv	X
cana-4565	49	3	)	)	PUNCT
cana-4565	49	4	ℜ(𝜛,𝜍)(ж	ℜ(𝜛,𝜍)(ж	PROPN
cana-4565	49	5	)	)	PUNCT
cana-4565	50	1	=	=	PUNCT
cana-4565	50	2			PUNCT
cana-4565	51	1	ℜ(𝜛,𝜍)(ж1	ℜ(𝜛,𝜍)(ж1	NOUN
cana-4565	51	2	+	+	NOUN
cana-4565	51	3	)	)	PUNCT
cana-4565	51	4	,	,	PUNCT
cana-4565	51	5	ℜ(𝜛,𝜍)(ж2	ℜ(𝜛,𝜍)(ж2	PROPN
cana-4565	51	6	+	+	PROPN
cana-4565	51	7	)	)	PUNCT
cana-4565	51	8	,	,	PUNCT
cana-4565	51	9	…	…	PUNCT
cana-4565	51	10	,	,	PUNCT
cana-4565	51	11	ℜ(𝜛,𝜍)(ж𝑛	ℜ(𝜛,𝜍)(ж𝑛	X
cana-4565	51	12	+	+	NOUN
cana-4565	51	13	)	)	PUNCT
cana-4565	51	14	,	,	PUNCT
cana-4565	51	15	ℜ(𝜛,𝜍)(ж1	ℜ(𝜛,𝜍)(ж1	PROPN
cana-4565	51	16	−	−	NOUN
cana-4565	51	17	)	)	PUNCT
cana-4565	51	18	,	,	PUNCT
cana-4565	51	19	ℜ(𝜛,𝜍)(ж2	ℜ(𝜛,𝜍)(ж2	PROPN
cana-4565	51	20	−	−	PROPN
cana-4565	51	21	)	)	PUNCT
cana-4565	51	22	,	,	PUNCT
cana-4565	51	23	…	…	PUNCT
cana-4565	51	24	,	,	PUNCT
cana-4565	51	25	ℜ(𝜛,𝜍)(ж𝑛	ℜ(𝜛,𝜍)(ж𝑛	ADJ
cana-4565	51	26	−	−	NOUN
cana-4565	51	27	)	)	PUNCT
cana-4565	51	28			PROPN
cana-4565	51	29	,	,	PUNCT
cana-4565	51	30	where	where	SCONJ
cana-4565	51	31	ℜ(𝜛,𝜍)(ж𝑖	ℜ(𝜛,𝜍)(ж𝑖	PROPN
cana-4565	51	32	+	+	NOUN
cana-4565	51	33	)	)	PUNCT
cana-4565	51	34	(	(	PUNCT
cana-4565	51	35	𝜚	𝜚	NOUN
cana-4565	51	36	)	)	PUNCT
cana-4565	51	37	=	=	PRON
cana-4565	52	1	𝑟𝑚𝑎𝑥	𝑟𝑚𝑎𝑥	NOUN
cana-4565	52	2	{	{	PUNCT
cana-4565	52	3	𝜛𝑖	𝜛𝑖	NOUN
cana-4565	52	4	,	,	PUNCT
cana-4565	52	5	ж𝑖	ж𝑖	ADP
cana-4565	52	6	+	+	PROPN
cana-4565	52	7	(	(	PUNCT
cana-4565	52	8	𝜚	𝜚	NOUN
cana-4565	52	9	)	)	PUNCT
cana-4565	52	10	}	}	PUNCT
cana-4565	52	11	and	and	CCONJ
cana-4565	52	12	ℜ(𝜛,𝜍)(ж𝑖	ℜ(𝜛,𝜍)(ж𝑖	VERB
cana-4565	52	13	−)(𝜚	−)(𝜚	NOUN
cana-4565	52	14	)	)	PUNCT
cana-4565	52	15	=	=	SYM
cana-4565	52	16	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
cana-4565	52	17	{	{	PUNCT
cana-4565	52	18	𝜍𝑖	𝜍𝑖	NOUN
cana-4565	52	19	,	,	PUNCT
cana-4565	52	20	ж𝑖	ж𝑖	ADP
cana-4565	52	21	−(𝜚	−(𝜚	NOUN
cana-4565	52	22	)	)	PUNCT
cana-4565	52	23	}	}	PUNCT
cana-4565	52	24	,	,	PUNCT
cana-4565	52	25	communications	communication	NOUN
cana-4565	52	26	on	on	ADP
cana-4565	52	27	applied	apply	VERB
cana-4565	52	28	nonlinear	nonlinear	ADJ
cana-4565	52	29	analysis	analysis	NOUN
cana-4565	52	30	issn	issn	NOUN
cana-4565	52	31	:	:	PUNCT
cana-4565	52	32	1074	1074	NUM
cana-4565	52	33	-	-	PUNCT
cana-4565	52	34	133x	133x	NUM
cana-4565	52	35	vol	vol	NOUN
cana-4565	52	36	32	32	NUM
cana-4565	53	1	no	no	NOUN
cana-4565	53	2	.	.	PUNCT
cana-4565	54	1	9s	9s	NUM
cana-4565	54	2	(	(	PUNCT
cana-4565	54	3	2025	2025	NUM
cana-4565	54	4	)	)	PUNCT
cana-4565	54	5	2617	2617	NUM
cana-4565	54	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-4565	54	7			VERB
cana-4565	54	8	𝜚	𝜚	NOUN
cana-4565	54	9	∈	∈	PROPN
cana-4565	54	10	𝒲	𝒲	PROPN
cana-4565	54	11	,	,	PUNCT
cana-4565	54	12	𝜛	𝜛	X
cana-4565	54	13	=	=	SYM
cana-4565	54	14	(	(	PUNCT
cana-4565	54	15	𝜛1	𝜛1	NOUN
cana-4565	54	16	,	,	PUNCT
cana-4565	54	17	𝜛2	𝜛2	NOUN
cana-4565	54	18	,	,	PUNCT
cana-4565	54	19	…	…	PUNCT
cana-4565	54	20	,	,	PUNCT
cana-4565	54	21	𝜛𝑛	𝜛𝑛	NOUN
cana-4565	54	22	)	)	PUNCT
cana-4565	54	23	and	and	CCONJ
cana-4565	54	24	𝜍	𝜍	X
cana-4565	54	25	=	=	SYM
cana-4565	54	26	(	(	PUNCT
cana-4565	54	27	𝜍1	𝜍1	PROPN
cana-4565	54	28	,	,	PUNCT
cana-4565	54	29	𝜍2	𝜍2	NOUN
cana-4565	54	30	,	,	PUNCT
cana-4565	54	31	…	…	PUNCT
cana-4565	54	32	,	,	PUNCT
cana-4565	54	33	𝜍𝑛	𝜍𝑛	NOUN
cana-4565	54	34	)	)	PUNCT
cana-4565	54	35	,	,	PUNCT
cana-4565	54	36	𝜛𝑖	𝜛𝑖	NOUN
cana-4565	54	37	∈	∈	PROPN
cana-4565	54	38	𝐷[0	𝐷[0	PROPN
cana-4565	54	39	,	,	PUNCT
cana-4565	54	40	1	1	NUM
cana-4565	54	41	]	]	X
cana-4565	54	42	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-4565	54	43	𝜍𝑖	𝜍𝑖	PROPN
cana-4565	54	44	∈	∈	PROPN
cana-4565	54	45	𝐷[−1	𝐷[−1	NOUN
cana-4565	54	46	,	,	PUNCT
cana-4565	54	47	0	0	NUM
cana-4565	54	48	]	]	PUNCT
cana-4565	54	49	.	.	PUNCT
cana-4565	55	1	(	(	PUNCT
cana-4565	55	2	v	v	NOUN
cana-4565	55	3	)	)	PUNCT
cana-4565	55	4	𝔖(𝜛,𝜍)(ж	𝔖(𝜛,𝜍)(ж	NUM
cana-4565	55	5	)	)	PUNCT
cana-4565	56	1	=	=	SYM
cana-4565	56	2			NOUN
cana-4565	56	3	𝔖(𝜛,𝜍)(ж1	𝔖(𝜛,𝜍)(ж1	NUM
cana-4565	56	4	+	+	NOUN
cana-4565	56	5	)	)	PUNCT
cana-4565	56	6	,	,	PUNCT
cana-4565	56	7	𝔖(𝜛,𝜍)(ж2	𝔖(𝜛,𝜍)(ж2	PROPN
cana-4565	56	8	+	+	NOUN
cana-4565	56	9	)	)	PUNCT
cana-4565	56	10	,	,	PUNCT
cana-4565	56	11	…	…	PUNCT
cana-4565	56	12	,	,	PUNCT
cana-4565	56	13	𝔖(𝜛,𝜍)(ж𝑛	𝔖(𝜛,𝜍)(ж𝑛	NUM
cana-4565	56	14	+	+	X
cana-4565	56	15	)	)	PUNCT
cana-4565	56	16	,	,	PUNCT
cana-4565	56	17	𝔖(𝜛,𝜍)(ж1	𝔖(𝜛,𝜍)(ж1	NUM
cana-4565	56	18	−	−	NUM
cana-4565	56	19	)	)	PUNCT
cana-4565	56	20	,	,	PUNCT
cana-4565	56	21	𝔖(𝜛,𝜍)(ж2	𝔖(𝜛,𝜍)(ж2	PROPN
cana-4565	56	22	−	−	NOUN
cana-4565	56	23	)	)	PUNCT
cana-4565	56	24	,	,	PUNCT
cana-4565	56	25	…	…	PUNCT
cana-4565	56	26	,	,	PUNCT
cana-4565	56	27	𝔖(𝜛,𝜍)(ж𝑛	𝔖(𝜛,𝜍)(ж𝑛	NUM
cana-4565	56	28	−	−	NOUN
cana-4565	56	29	)	)	PUNCT
cana-4565	56	30			PROPN
cana-4565	56	31	,	,	PUNCT
cana-4565	56	32	where	where	SCONJ
cana-4565	56	33	𝔖(𝜛,𝜍)(ж𝑖	𝔖(𝜛,𝜍)(ж𝑖	NOUN
cana-4565	56	34	+	+	NOUN
cana-4565	56	35	)	)	PUNCT
cana-4565	56	36	(	(	PUNCT
cana-4565	56	37	𝜚	𝜚	NOUN
cana-4565	56	38	)	)	PUNCT
cana-4565	56	39	=	=	VERB
cana-4565	57	1	𝜛𝑖ж𝑖	𝜛𝑖ж𝑖	NOUN
cana-4565	57	2	+	+	NOUN
cana-4565	57	3	(	(	PUNCT
cana-4565	57	4	𝜚	𝜚	NOUN
cana-4565	57	5	)	)	PUNCT
cana-4565	57	6	and	and	CCONJ
cana-4565	57	7	𝔖(𝜛,𝜍)(ж𝑖	𝔖(𝜛,𝜍)(ж𝑖	VERB
cana-4565	57	8	−)(𝜚	−)(𝜚	NOUN
cana-4565	57	9	)	)	PUNCT
cana-4565	57	10	=	=	NOUN
cana-4565	57	11	−𝜍𝑖ж𝑖	−𝜍𝑖ж𝑖	NOUN
cana-4565	57	12	−(𝜚	−(𝜚	NOUN
cana-4565	57	13	)	)	PUNCT
cana-4565	57	14	,	,	PUNCT
cana-4565	57	15			NOUN
cana-4565	57	16	𝜚	𝜚	NOUN
cana-4565	57	17	∈	∈	PROPN
cana-4565	57	18	𝒲	𝒲	PROPN
cana-4565	57	19	,	,	PUNCT
cana-4565	57	20	𝜛	𝜛	X
cana-4565	57	21	=	=	SYM
cana-4565	57	22	(	(	PUNCT
cana-4565	57	23	𝜛1	𝜛1	NOUN
cana-4565	57	24	,	,	PUNCT
cana-4565	57	25	𝜛2	𝜛2	NOUN
cana-4565	57	26	,	,	PUNCT
cana-4565	57	27	…	…	PUNCT
cana-4565	57	28	,	,	PUNCT
cana-4565	57	29	𝜛𝑛	𝜛𝑛	NOUN
cana-4565	57	30	)	)	PUNCT
cana-4565	57	31	and	and	CCONJ
cana-4565	57	32	𝜍	𝜍	X
cana-4565	57	33	=	=	SYM
cana-4565	57	34	(	(	PUNCT
cana-4565	57	35	𝜍1	𝜍1	PROPN
cana-4565	57	36	,	,	PUNCT
cana-4565	57	37	𝜍2	𝜍2	NOUN
cana-4565	57	38	,	,	PUNCT
cana-4565	57	39	…	…	PUNCT
cana-4565	57	40	,	,	PUNCT
cana-4565	57	41	𝜍𝑛	𝜍𝑛	NOUN
cana-4565	57	42	)	)	PUNCT
cana-4565	57	43	,	,	PUNCT
cana-4565	57	44	𝜛𝑖	𝜛𝑖	NOUN
cana-4565	57	45	∈	∈	PROPN
cana-4565	58	1	[	[	X
cana-4565	58	2	0	0	NUM
cana-4565	58	3	,	,	PUNCT
cana-4565	58	4	1	1	NUM
cana-4565	58	5	]	]	X
cana-4565	58	6	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-4565	58	7	𝜍𝑖	𝜍𝑖	PROPN
cana-4565	58	8	∈	∈	PROPN
cana-4565	59	1	[	[	X
cana-4565	59	2	−1	−1	NOUN
cana-4565	59	3	,	,	PUNCT
cana-4565	59	4	0	0	NUM
cana-4565	59	5	]	]	PUNCT
cana-4565	59	6	.	.	X
cana-4565	59	7	2	2	NUM
cana-4565	59	8	–	–	PUNCT
cana-4565	59	9	theorems	theorem	NOUN
cana-4565	59	10	.	.	PUNCT
cana-4565	59	11	theorem	theorem	VERB
cana-4565	59	12	2.1	2.1	NUM
cana-4565	59	13	.	.	PUNCT
cana-4565	60	1	𝐼𝑓	𝐼𝑓	VERB
cana-4565	60	2	њ	њ	NOUN
cana-4565	60	3	=	=	NOUN
cana-4565	60	4			NOUN
cana-4565	60	5	њ1	њ1	NOUN
cana-4565	60	6	+	+	NOUN
cana-4565	60	7	,	,	PUNCT
cana-4565	60	8	њ2	њ2	PROPN
cana-4565	60	9	+	+	PROPN
cana-4565	60	10	,	,	PUNCT
cana-4565	60	11	…	…	PUNCT
cana-4565	60	12	,	,	PUNCT
cana-4565	60	13	њ𝑛	њ𝑛	ADP
cana-4565	60	14	+	+	ADJ
cana-4565	60	15	,	,	PUNCT
cana-4565	60	16	њ1	њ1	NOUN
cana-4565	60	17	−	−	NOUN
cana-4565	60	18	,	,	PUNCT
cana-4565	60	19	њ2	њ2	NOUN
cana-4565	60	20	−	−	PROPN
cana-4565	60	21	,	,	PUNCT
cana-4565	60	22	…	…	PUNCT
cana-4565	60	23	,	,	PUNCT
cana-4565	60	24	њ𝑛	њ𝑛	VERB
cana-4565	60	25	−	−	NUM
cana-4565	60	26	and	and	CCONJ
cana-4565	60	27	𝔉	𝔉	PROPN
cana-4565	60	28	=	=	SYM
cana-4565	60	29			X
cana-4565	60	30	𝔉1	𝔉1	VERB
cana-4565	60	31	+	+	ADP
cana-4565	60	32	,	,	PUNCT
cana-4565	60	33	𝔉2	𝔉2	ADJ
cana-4565	60	34	+	+	ADJ
cana-4565	60	35	,	,	PUNCT
cana-4565	60	36	…	…	PUNCT
cana-4565	60	37	,	,	PUNCT
cana-4565	60	38	𝔉𝑛	𝔉𝑛	PROPN
cana-4565	60	39	+	+	ADV
cana-4565	60	40	,	,	PUNCT
cana-4565	60	41	𝔉1	𝔉1	VERB
cana-4565	60	42	−	−	PROPN
cana-4565	60	43	,	,	PUNCT
cana-4565	60	44	𝔉2	𝔉2	PROPN
cana-4565	60	45	−	−	PROPN
cana-4565	60	46	,	,	PUNCT
cana-4565	60	47	…	…	PUNCT
cana-4565	60	48	,	,	PUNCT
cana-4565	61	1	𝔉𝑛	𝔉𝑛	PROPN
cana-4565	61	2	−	−	NUM
cana-4565	61	3	𝑎𝑟𝑒	𝑎𝑟𝑒	NOUN
cana-4565	61	4	𝑡𝑤𝑜	𝑡𝑤𝑜	ADJ
cana-4565	61	5	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	NOUN
cana-4565	61	6	of	of	ADP
cana-4565	61	7	a	a	DET
cana-4565	61	8	ring	ring	NOUN
cana-4565	61	9	ĕ1	ĕ1	NOUN
cana-4565	61	10	,	,	PUNCT
cana-4565	61	11	then	then	ADV
cana-4565	61	12	their	their	PRON
cana-4565	61	13	intersection	intersection	NOUN
cana-4565	61	14	њ	њ	X
cana-4565	61	15	∩	∩	ADJ
cana-4565	61	16	𝔉	𝔉	PROPN
cana-4565	61	17	𝑖𝑠	𝑖𝑠	NOUN
cana-4565	61	18	𝑎𝑙𝑠𝑜	𝑎𝑙𝑠𝑜	NOUN
cana-4565	61	19	𝑎	𝑎	DET
cana-4565	61	20	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	61	21	of	of	ADP
cana-4565	61	22	ĕ1	ĕ1	PROPN
cana-4565	61	23	.	.	PUNCT
cana-4565	62	1	theorem	theorem	VERB
cana-4565	62	2	2.2	2.2	NUM
cana-4565	62	3	.	.	PUNCT
cana-4565	63	1	𝐼𝑓	𝐼𝑓	VERB
cana-4565	63	2	ѝ	ѝ	NOUN
cana-4565	63	3	=	=	NOUN
cana-4565	63	4			ADP
cana-4565	63	5	ѝ1	ѝ1	ADJ
cana-4565	63	6	+	+	PROPN
cana-4565	63	7	,	,	PUNCT
cana-4565	63	8	ѝ2	ѝ2	VERB
cana-4565	63	9	+	+	PROPN
cana-4565	63	10	,	,	PUNCT
cana-4565	63	11	…	…	PUNCT
cana-4565	63	12	,	,	PUNCT
cana-4565	63	13	ѝ𝑛	ѝ𝑛	NOUN
cana-4565	63	14	+	+	NOUN
cana-4565	63	15	,	,	PUNCT
cana-4565	63	16	ѝ1	ѝ1	ADJ
cana-4565	63	17	−	−	NOUN
cana-4565	63	18	,	,	PUNCT
cana-4565	63	19	ѝ2	ѝ2	VERB
cana-4565	63	20	−	−	PROPN
cana-4565	63	21	,	,	PUNCT
cana-4565	63	22	…	…	PUNCT
cana-4565	63	23	,	,	PUNCT
cana-4565	63	24	ѝ𝑛	ѝ𝑛	ADP
cana-4565	63	25	−	−	PROPN
cana-4565	63	26	is	be	AUX
cana-4565	63	27	𝑎	𝑎	DET
cana-4565	63	28	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	NOUN
cana-4565	63	29	of	of	ADP
cana-4565	63	30	a	a	DET
cana-4565	63	31	ring	ring	NOUN
cana-4565	63	32	ß1	ß1	NOUN
cana-4565	63	33	,	,	PUNCT
cana-4565	63	34	then	then	ADV
cana-4565	63	35	≬	≬	PROPN
cana-4565	63	36	(	(	PUNCT
cana-4565	63	37	ѝ)𝑖𝑠	ѝ)𝑖𝑠	PROPN
cana-4565	63	38	𝑎𝑙𝑠𝑜	𝑎𝑙𝑠𝑜	NOUN
cana-4565	63	39	𝑎𝐵𝑉𝑀𝐼𝐹𝑆𝑅	𝑎𝐵𝑉𝑀𝐼𝐹𝑆𝑅	PROPN
cana-4565	63	40	of	of	ADP
cana-4565	63	41	ß1	ß1	PROPN
cana-4565	63	42	.	.	PUNCT
cana-4565	64	1	theorem	theorem	VERB
cana-4565	64	2	2.3	2.3	NUM
cana-4565	64	3	.	.	PUNCT
cana-4565	65	1	𝐼𝑓	𝐼𝑓	ADJ
cana-4565	65	2	ℭ	ℭ	NOUN
cana-4565	65	3	=	=	PUNCT
cana-4565	65	4			NOUN
cana-4565	65	5	ℭ1	ℭ1	INTJ
cana-4565	65	6	+	+	ADV
cana-4565	65	7	,	,	PUNCT
cana-4565	65	8	ℭ2	ℭ2	PROPN
cana-4565	65	9	+	+	PROPN
cana-4565	65	10	,	,	PUNCT
cana-4565	65	11	…	…	PUNCT
cana-4565	65	12	,	,	PUNCT
cana-4565	66	1	ℭ𝑛	ℭ𝑛	ADP
cana-4565	66	2	+	+	ADJ
cana-4565	66	3	,	,	PUNCT
cana-4565	66	4	ℭ1	ℭ1	ADP
cana-4565	66	5	−	−	PROPN
cana-4565	66	6	,	,	PUNCT
cana-4565	66	7	ℭ2	ℭ2	PROPN
cana-4565	66	8	−	−	PROPN
cana-4565	66	9	,	,	PUNCT
cana-4565	66	10	…	…	PUNCT
cana-4565	66	11	,	,	PUNCT
cana-4565	66	12	ℭ𝑛	ℭ𝑛	SCONJ
cana-4565	66	13	−	−	PROPN
cana-4565	66	14	is	be	AUX
cana-4565	66	15	𝑎	𝑎	DET
cana-4565	66	16	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	66	17	of	of	ADP
cana-4565	66	18	a	a	DET
cana-4565	66	19	ring	ring	NOUN
cana-4565	66	20	℧	℧	PROPN
cana-4565	66	21	1	1	NUM
cana-4565	66	22	,	,	PUNCT
cana-4565	66	23	then	then	ADV
cana-4565	66	24	≬	≬	PROPN
cana-4565	66	25	(	(	PUNCT
cana-4565	66	26	ℭ)𝑖𝑠	ℭ)𝑖𝑠	NOUN
cana-4565	66	27	𝑎𝑙𝑠𝑜	𝑎𝑙𝑠𝑜	NOUN
cana-4565	66	28	𝑎	𝑎	DET
cana-4565	66	29	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	66	30	of	of	ADP
cana-4565	66	31	the	the	DET
cana-4565	66	32	ring	ring	NOUN
cana-4565	66	33	℧	℧	NOUN
cana-4565	66	34	1	1	NUM
cana-4565	66	35	.	.	PUNCT
cana-4565	67	1	proof	proof	NOUN
cana-4565	67	2	.	.	PUNCT
cana-4565	68	1	let	let	VERB
cana-4565	68	2	𝜁	𝜁	NOUN
cana-4565	68	3	,	,	PUNCT
cana-4565	68	4	𝜐	𝜐	VERB
cana-4565	68	5	be	be	AUX
cana-4565	68	6	in	in	ADP
cana-4565	68	7	℧	℧	NOUN
cana-4565	68	8	1	1	NUM
cana-4565	68	9	.	.	PUNCT
cana-4565	69	1	for	for	ADP
cana-4565	69	2	all	all	DET
cana-4565	69	3	i	i	PRON
cana-4565	69	4	=	=	NOUN
cana-4565	69	5	1	1	NUM
cana-4565	69	6	,	,	PUNCT
cana-4565	69	7	2	2	NUM
cana-4565	69	8	,	,	PUNCT
cana-4565	69	9	…	…	PUNCT
cana-4565	69	10	,	,	PUNCT
cana-4565	69	11	n	n	CCONJ
cana-4565	69	12	,	,	PUNCT
cana-4565	69	13	by	by	ADP
cana-4565	69	14	theorem	theorem	NOUN
cana-4565	69	15	2.2	2.2	NUM
cana-4565	69	16	,	,	PUNCT
cana-4565	69	17	≬(ℭ	≬(ℭ	PROPN
cana-4565	69	18	)	)	PUNCT
cana-4565	69	19	is	be	AUX
cana-4565	69	20	a	a	DET
cana-4565	69	21	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	NOUN
cana-4565	69	22	of	of	ADP
cana-4565	69	23	℧	℧	NOUN
cana-4565	69	24	1	1	NUM
cana-4565	69	25	,	,	PUNCT
cana-4565	69	26	≬(ℭ𝑖	≬(ℭ𝑖	NOUN
cana-4565	69	27	+	+	NOUN
cana-4565	69	28	)	)	PUNCT
cana-4565	69	29	(	(	PUNCT
cana-4565	69	30	𝜁𝜐	𝜁𝜐	NOUN
cana-4565	69	31	)	)	PUNCT
cana-4565	69	32	=	=	SYM
cana-4565	69	33	rmin{[½,½	rmin{[½,½	NOUN
cana-4565	69	34	]	]	PUNCT
cana-4565	69	35	,	,	PUNCT
cana-4565	70	1	ℭ𝑖	ℭ𝑖	PROPN
cana-4565	70	2	+	+	PROPN
cana-4565	70	3	(	(	PUNCT
cana-4565	70	4	𝜁𝜐)}=	𝜁𝜐)}=	PROPN
cana-4565	70	5	rmin{[½,½	rmin{[½,½	NOUN
cana-4565	70	6	]	]	PUNCT
cana-4565	70	7	,	,	PUNCT
cana-4565	70	8	ℭ𝑖	ℭ𝑖	PROPN
cana-4565	70	9	+	+	PROPN
cana-4565	70	10	(	(	PUNCT
cana-4565	70	11	𝜐𝜁)}=	𝜐𝜁)}=	PROPN
cana-4565	70	12	≬(ℭ𝑖	≬(ℭ𝑖	PROPN
cana-4565	70	13	+	+	PROPN
cana-4565	70	14	)	)	PUNCT
cana-4565	70	15	(	(	PUNCT
cana-4565	70	16	𝜐𝜁	𝜐𝜁	NOUN
cana-4565	70	17	)	)	PUNCT
cana-4565	70	18	,	,	PUNCT
cana-4565	70	19	for	for	ADP
cana-4565	70	20	all	all	DET
cana-4565	70	21	𝜁	𝜁	NOUN
cana-4565	70	22	,	,	PUNCT
cana-4565	70	23	𝜐	𝜐	X
cana-4565	70	24	in	in	ADP
cana-4565	70	25	℧	℧	NOUN
cana-4565	70	26	1	1	NUM
cana-4565	70	27	.	.	PUNCT
cana-4565	70	28	also	also	ADV
cana-4565	70	29	≬(ℭ𝑖	≬(ℭ𝑖	PROPN
cana-4565	70	30	−)(𝜁𝜐	−)(𝜁𝜐	NOUN
cana-4565	70	31	)	)	PUNCT
cana-4565	70	32	=	=	PUNCT
cana-4565	70	33	rmax{[−½,−½	rmax{[−½,−½	NOUN
cana-4565	70	34	]	]	X
cana-4565	70	35	,	,	PUNCT
cana-4565	70	36	ℭ𝑖	ℭ𝑖	PROPN
cana-4565	70	37	−(𝜁𝜐	−(𝜁𝜐	PROPN
cana-4565	70	38	)	)	PUNCT
cana-4565	70	39	}	}	PUNCT
cana-4565	70	40	=	=	PUNCT
cana-4565	70	41	rmax{[−½,−½	rmax{[−½,−½	NOUN
cana-4565	70	42	]	]	X
cana-4565	70	43	,	,	PUNCT
cana-4565	70	44	ℭ𝑖	ℭ𝑖	PROPN
cana-4565	70	45	−(𝜐𝜁)}=	−(𝜐𝜁)}=	PROPN
cana-4565	70	46	≬(ℭ𝑖	≬(ℭ𝑖	PROPN
cana-4565	70	47	−)(𝜐𝜁	−)(𝜐𝜁	PROPN
cana-4565	70	48	)	)	PUNCT
cana-4565	70	49	,	,	PUNCT
cana-4565	70	50	for	for	ADP
cana-4565	70	51	all	all	DET
cana-4565	70	52	𝜁	𝜁	NOUN
cana-4565	70	53	,	,	PUNCT
cana-4565	70	54	𝜐	𝜐	X
cana-4565	70	55	in	in	ADP
cana-4565	70	56	℧	℧	NOUN
cana-4565	70	57	1	1	NUM
cana-4565	70	58	.	.	PUNCT
cana-4565	70	59	hence	hence	ADV
cana-4565	70	60	≬(ℭ	≬(ℭ	PROPN
cana-4565	70	61	)	)	PUNCT
cana-4565	70	62	is	be	AUX
cana-4565	70	63	a	a	DET
cana-4565	70	64	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	70	65	of	of	ADP
cana-4565	70	66	℧	℧	NOUN
cana-4565	70	67	1	1	NUM
cana-4565	70	68	.	.	PUNCT
cana-4565	70	69	corollary	corollary	ADJ
cana-4565	70	70	2.4	2.4	NUM
cana-4565	70	71	.	.	PUNCT
cana-4565	71	1	if	if	SCONJ
cana-4565	71	2	𝔓	𝔓	PROPN
cana-4565	71	3	and	and	CCONJ
cana-4565	71	4	𝔚	𝔚	PROPN
cana-4565	71	5	are	be	AUX
cana-4565	71	6	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	NOUN
cana-4565	71	7	𝑜𝑓	𝑜𝑓	ADP
cana-4565	71	8	𝑡ℎ𝑒	𝑡ℎ𝑒	NUM
cana-4565	71	9	𝑟𝑖𝑛𝑔	𝑟𝑖𝑛𝑔	NOUN
cana-4565	71	10	℧	℧	PROPN
cana-4565	71	11	1	1	NUM
cana-4565	71	12	,	,	PUNCT
cana-4565	71	13	then	then	ADV
cana-4565	71	14	≬(𝔓	≬(𝔓	NOUN
cana-4565	71	15	∩	∩	ADJ
cana-4565	71	16	𝔚	𝔚	NOUN
cana-4565	71	17	)	)	PUNCT
cana-4565	71	18	is	be	AUX
cana-4565	71	19	a	a	DET
cana-4565	71	20	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	71	21	of	of	ADP
cana-4565	71	22	℧	℧	NOUN
cana-4565	71	23	1	1	NUM
cana-4565	71	24	.	.	PUNCT
cana-4565	72	1	proof	proof	NOUN
cana-4565	72	2	.	.	PUNCT
cana-4565	73	1	from	from	ADP
cana-4565	73	2	the	the	DET
cana-4565	73	3	above	above	ADJ
cana-4565	73	4	theorems	theorem	NOUN
cana-4565	73	5	,	,	PUNCT
cana-4565	73	6	it	it	PRON
cana-4565	73	7	is	be	AUX
cana-4565	73	8	trivial	trivial	ADJ
cana-4565	73	9	.	.	PUNCT
cana-4565	74	1	corollary	corollary	ADJ
cana-4565	74	2	2.5	2.5	NUM
cana-4565	74	3	.	.	PUNCT
cana-4565	75	1	if	if	SCONJ
cana-4565	75	2	𝔓	𝔓	PROPN
cana-4565	75	3	and	and	CCONJ
cana-4565	75	4	𝔚	𝔚	PROPN
cana-4565	75	5	are	be	AUX
cana-4565	75	6	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	NOUN
cana-4565	75	7	𝑜𝑓	𝑜𝑓	ADP
cana-4565	75	8	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	75	9	𝑟𝑖𝑛𝑔𝑠	𝑟𝑖𝑛𝑔𝑠	NOUN
cana-4565	75	10	℧	℧	PROPN
cana-4565	75	11	1and	1and	NUM
cana-4565	75	12	℧	℧	PROPN
cana-4565	75	13	2	2	NUM
cana-4565	75	14	,	,	PUNCT
cana-4565	75	15	then	then	ADV
cana-4565	75	16	≬	≬	PROPN
cana-4565	75	17	𝔓	𝔓	PROPN
cana-4565	75	18	∩≬	∩≬	PROPN
cana-4565	75	19	𝔚	𝔚	PROPN
cana-4565	75	20	is	be	AUX
cana-4565	75	21	a	a	DET
cana-4565	75	22	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	75	23	of	of	ADP
cana-4565	75	24	℧	℧	NOUN
cana-4565	75	25	1	1	NUM
cana-4565	75	26	∩	∩	NOUN
cana-4565	75	27	℧	℧	PROPN
cana-4565	75	28	2	2	NUM
cana-4565	75	29	.	.	PUNCT
cana-4565	76	1	proof	proof	NOUN
cana-4565	76	2	.	.	PUNCT
cana-4565	77	1	from	from	ADP
cana-4565	77	2	the	the	DET
cana-4565	77	3	above	above	ADJ
cana-4565	77	4	theorems	theorem	NOUN
cana-4565	77	5	,	,	PUNCT
cana-4565	77	6	it	it	PRON
cana-4565	77	7	is	be	AUX
cana-4565	77	8	trivial	trivial	ADJ
cana-4565	77	9	.	.	PUNCT
cana-4565	78	1	corollary	corollary	ADJ
cana-4565	78	2	2.6	2.6	NUM
cana-4565	78	3	.	.	PUNCT
cana-4565	79	1	if	if	SCONJ
cana-4565	79	2	𝔓	𝔓	PROPN
cana-4565	79	3	and	and	CCONJ
cana-4565	79	4	𝔚	𝔚	PROPN
cana-4565	79	5	are	be	AUX
cana-4565	79	6	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	NOUN
cana-4565	79	7	𝑜𝑓	𝑜𝑓	ADP
cana-4565	79	8	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	79	9	𝑟𝑖𝑛𝑔𝑠	𝑟𝑖𝑛𝑔𝑠	NOUN
cana-4565	79	10	℧	℧	PROPN
cana-4565	79	11	1	1	NUM
cana-4565	79	12	,	,	PUNCT
cana-4565	79	13	then	then	ADV
cana-4565	79	14	≬	≬	PROPN
cana-4565	79	15	𝔓	𝔓	PROPN
cana-4565	79	16	∩≬	∩≬	PROPN
cana-4565	79	17	𝔚	𝔚	PROPN
cana-4565	79	18	is	be	AUX
cana-4565	79	19	a	a	DET
cana-4565	79	20	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	79	21	of	of	ADP
cana-4565	79	22	℧	℧	NOUN
cana-4565	79	23	1	1	NUM
cana-4565	79	24	.	.	PUNCT
cana-4565	80	1	proof	proof	NOUN
cana-4565	80	2	.	.	PUNCT
cana-4565	81	1	from	from	ADP
cana-4565	81	2	the	the	DET
cana-4565	81	3	above	above	ADJ
cana-4565	81	4	theorems	theorem	NOUN
cana-4565	81	5	,	,	PUNCT
cana-4565	81	6	it	it	PRON
cana-4565	81	7	is	be	AUX
cana-4565	81	8	trivial	trivial	ADJ
cana-4565	81	9	.	.	PUNCT
cana-4565	82	1	theorem	theorem	VERB
cana-4565	82	2	2.7	2.7	NUM
cana-4565	82	3	.	.	PUNCT
cana-4565	83	1	𝐼𝑓	𝐼𝑓	PROPN
cana-4565	83	2	𝔓1	𝔓1	PROPN
cana-4565	83	3	,	,	PUNCT
cana-4565	83	4	𝔓2	𝔓2	NOUN
cana-4565	83	5	,	,	PUNCT
cana-4565	83	6	…	…	PUNCT
cana-4565	83	7	,	,	PUNCT
cana-4565	83	8	𝔓𝑚𝑎𝑟𝑒	𝔓𝑚𝑎𝑟𝑒	PROPN
cana-4565	83	9	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	83	10	of	of	ADP
cana-4565	83	11	the	the	DET
cana-4565	83	12	rings	ring	NOUN
cana-4565	83	13	℧	℧	PROPN
cana-4565	83	14	1	1	NUM
cana-4565	83	15	,	,	PUNCT
cana-4565	83	16	℧	℧	NOUN
cana-4565	83	17	2	2	NUM
cana-4565	83	18	,	,	PUNCT
cana-4565	83	19	…	…	PUNCT
cana-4565	83	20	,	,	PUNCT
cana-4565	83	21	℧	℧	NOUN
cana-4565	83	22	m	m	VERB
cana-4565	83	23	respectively	respectively	ADV
cana-4565	83	24	,	,	PUNCT
cana-4565	83	25	then	then	ADV
cana-4565	83	26	≬	≬	PROPN
cana-4565	83	27	(	(	PUNCT
cana-4565	83	28	𝔓1	𝔓1	PROPN
cana-4565	83	29	∩	∩	PROPN
cana-4565	83	30	𝔓2	𝔓2	NOUN
cana-4565	83	31	∩	∩	NOUN
cana-4565	83	32	…	…	PUNCT
cana-4565	83	33	∩	∩	ADJ
cana-4565	83	34	𝔓𝑚	𝔓𝑚	NOUN
cana-4565	83	35	)	)	PUNCT
cana-4565	83	36	is	be	AUX
cana-4565	83	37	a	a	DET
cana-4565	83	38	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	83	39	of	of	ADP
cana-4565	83	40	the	the	DET
cana-4565	83	41	ring	ring	NOUN
cana-4565	83	42	℧	℧	NOUN
cana-4565	83	43	1	1	NUM
cana-4565	83	44	∩	∩	X
cana-4565	83	45	℧	℧	NOUN
cana-4565	83	46	2	2	NUM
cana-4565	83	47	∩	∩	NOUN
cana-4565	83	48	…	…	PUNCT
cana-4565	83	49	∩	∩	ADJ
cana-4565	83	50	℧	℧	NOUN
cana-4565	83	51	m.	m.	NOUN
cana-4565	83	52	proof	proof	NOUN
cana-4565	83	53	.	.	PUNCT
cana-4565	84	1	from	from	ADP
cana-4565	84	2	the	the	DET
cana-4565	84	3	above	above	ADJ
cana-4565	84	4	theorems	theorem	NOUN
cana-4565	84	5	,	,	PUNCT
cana-4565	84	6	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	84	7	𝑝𝑟𝑜𝑜𝑓	𝑝𝑟𝑜𝑜𝑓	NOUN
cana-4565	84	8	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	84	9	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	NOUN
cana-4565	84	10	corollary	corollary	NOUN
cana-4565	84	11	2.8	2.8	NUM
cana-4565	84	12	.	.	PUNCT
cana-4565	85	1	𝐼𝑓	𝐼𝑓	PROPN
cana-4565	85	2	𝔓1	𝔓1	PROPN
cana-4565	85	3	,	,	PUNCT
cana-4565	85	4	𝔓2	𝔓2	NOUN
cana-4565	85	5	,	,	PUNCT
cana-4565	85	6	…	…	PUNCT
cana-4565	85	7	,	,	PUNCT
cana-4565	85	8	𝔓𝑚	𝔓𝑚	PROPN
cana-4565	85	9	𝑎𝑟𝑒	𝑎𝑟𝑒	VERB
cana-4565	85	10	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	85	11	of	of	ADP
cana-4565	85	12	the	the	DET
cana-4565	85	13	rings	ring	NOUN
cana-4565	85	14	℧	℧	PROPN
cana-4565	85	15	1	1	NUM
cana-4565	85	16	,	,	PUNCT
cana-4565	85	17	℧	℧	NOUN
cana-4565	85	18	2	2	NUM
cana-4565	85	19	,	,	PUNCT
cana-4565	85	20	…	…	PUNCT
cana-4565	85	21	,	,	PUNCT
cana-4565	85	22	℧	℧	NOUN
cana-4565	85	23	m	m	VERB
cana-4565	85	24	respectively	respectively	ADV
cana-4565	85	25	,	,	PUNCT
cana-4565	85	26	then	then	ADV
cana-4565	85	27	≬	≬	PROPN
cana-4565	85	28	𝔓1	𝔓1	PROPN
cana-4565	85	29	∩≬	∩≬	PROPN
cana-4565	85	30	𝔓2	𝔓2	PROPN
cana-4565	85	31	∩	∩	PROPN
cana-4565	85	32	…	…	PUNCT
cana-4565	86	1	∩≬	∩≬	PROPN
cana-4565	86	2	𝔓𝑚	𝔓𝑚	PROPN
cana-4565	86	3	is	be	AUX
cana-4565	86	4	a	a	DET
cana-4565	86	5	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	86	6	of	of	ADP
cana-4565	86	7	the	the	DET
cana-4565	86	8	ring	ring	NOUN
cana-4565	86	9	℧	℧	NOUN
cana-4565	86	10	1	1	NUM
cana-4565	86	11	∩	∩	X
cana-4565	86	12	℧	℧	NOUN
cana-4565	86	13	2	2	NUM
cana-4565	86	14	∩	∩	NOUN
cana-4565	86	15	…	…	PUNCT
cana-4565	86	16	∩	∩	ADJ
cana-4565	86	17	℧	℧	NOUN
cana-4565	86	18	m.	m.	NOUN
cana-4565	86	19	proof	proof	NOUN
cana-4565	86	20	.	.	PUNCT
cana-4565	87	1	from	from	ADP
cana-4565	87	2	the	the	DET
cana-4565	87	3	above	above	ADJ
cana-4565	87	4	theorems	theorem	NOUN
cana-4565	87	5	,	,	PUNCT
cana-4565	87	6	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	87	7	𝑝𝑟𝑜𝑜𝑓	𝑝𝑟𝑜𝑜𝑓	NOUN
cana-4565	87	8	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	87	9	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	PROPN
cana-4565	87	10	corollary	corollary	ADJ
cana-4565	87	11	2.9	2.9	NUM
cana-4565	87	12	.	.	PUNCT
cana-4565	88	1	𝐼𝑓	𝐼𝑓	PROPN
cana-4565	88	2	𝔓1	𝔓1	PROPN
cana-4565	88	3	,	,	PUNCT
cana-4565	88	4	𝔓2	𝔓2	NOUN
cana-4565	88	5	,	,	PUNCT
cana-4565	88	6	…	…	PUNCT
cana-4565	88	7	,	,	PUNCT
cana-4565	88	8	𝔓𝑚	𝔓𝑚	PROPN
cana-4565	88	9	are	be	AUX
cana-4565	88	10	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	88	11	of	of	ADP
cana-4565	88	12	the	the	DET
cana-4565	88	13	ring	ring	NOUN
cana-4565	88	14	℧	℧	PROPN
cana-4565	88	15	1	1	NUM
cana-4565	88	16	,	,	PUNCT
cana-4565	88	17	then	then	ADV
cana-4565	88	18	≬	≬	PROPN
cana-4565	88	19	(	(	PUNCT
cana-4565	88	20	𝔓1	𝔓1	PROPN
cana-4565	88	21	∩	∩	PROPN
cana-4565	88	22	𝔓2	𝔓2	NOUN
cana-4565	88	23	∩	∩	NOUN
cana-4565	88	24	…	…	PUNCT
cana-4565	88	25	∩	∩	ADJ
cana-4565	88	26	𝔓𝑚	𝔓𝑚	NOUN
cana-4565	88	27	)	)	PUNCT
cana-4565	88	28	is	be	AUX
cana-4565	88	29	a	a	DET
cana-4565	88	30	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	88	31	𝑜𝑓	𝑜𝑓	ADP
cana-4565	88	32	the	the	DET
cana-4565	88	33	ring	ring	NOUN
cana-4565	88	34	℧	℧	PROPN
cana-4565	88	35	1	1	NUM
cana-4565	88	36	.	.	PUNCT
cana-4565	88	37	communications	communication	NOUN
cana-4565	88	38	on	on	ADP
cana-4565	88	39	applied	apply	VERB
cana-4565	88	40	nonlinear	nonlinear	ADJ
cana-4565	88	41	analysis	analysis	NOUN
cana-4565	88	42	issn	issn	NOUN
cana-4565	88	43	:	:	PUNCT
cana-4565	88	44	1074	1074	NUM
cana-4565	88	45	-	-	PUNCT
cana-4565	88	46	133x	133x	NUM
cana-4565	88	47	vol	vol	NOUN
cana-4565	88	48	32	32	NUM
cana-4565	88	49	no	no	NOUN
cana-4565	88	50	.	.	PUNCT
cana-4565	89	1	9s	9s	NUM
cana-4565	89	2	(	(	PUNCT
cana-4565	89	3	2025	2025	NUM
cana-4565	89	4	)	)	PUNCT
cana-4565	89	5	2618	2618	NUM
cana-4565	90	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-4565	90	2	proof	proof	NOUN
cana-4565	90	3	.	.	PUNCT
cana-4565	91	1	from	from	ADP
cana-4565	91	2	the	the	DET
cana-4565	91	3	above	above	ADJ
cana-4565	91	4	theorems	theorem	NOUN
cana-4565	91	5	,	,	PUNCT
cana-4565	91	6	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	91	7	𝑝𝑟𝑜𝑜𝑓	𝑝𝑟𝑜𝑜𝑓	NOUN
cana-4565	91	8	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	91	9	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	NOUN
cana-4565	91	10	corollary	corollary	ADJ
cana-4565	91	11	2.10	2.10	NUM
cana-4565	91	12	.	.	PUNCT
cana-4565	92	1	𝐼𝑓	𝐼𝑓	PROPN
cana-4565	92	2	𝔓1	𝔓1	PROPN
cana-4565	92	3	,	,	PUNCT
cana-4565	92	4	𝔓2	𝔓2	NOUN
cana-4565	92	5	,	,	PUNCT
cana-4565	92	6	…	…	PUNCT
cana-4565	92	7	,	,	PUNCT
cana-4565	92	8	𝔓𝑚	𝔓𝑚	PROPN
cana-4565	92	9	are	be	AUX
cana-4565	92	10	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	92	11	of	of	ADP
cana-4565	92	12	the	the	DET
cana-4565	92	13	ring	ring	NOUN
cana-4565	92	14	℧	℧	PROPN
cana-4565	92	15	1	1	NUM
cana-4565	92	16	,	,	PUNCT
cana-4565	92	17	then	then	ADV
cana-4565	92	18	≬	≬	PROPN
cana-4565	92	19	𝔓1	𝔓1	PROPN
cana-4565	92	20	∩≬	∩≬	PROPN
cana-4565	92	21	𝔓2	𝔓2	PROPN
cana-4565	92	22	∩	∩	PROPN
cana-4565	92	23	…	…	PUNCT
cana-4565	92	24	∩≬	∩≬	PROPN
cana-4565	92	25	𝔓𝑚	𝔓𝑚	PROPN
cana-4565	92	26	is	be	AUX
cana-4565	92	27	a	a	DET
cana-4565	92	28	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	92	29	of	of	ADP
cana-4565	92	30	the	the	DET
cana-4565	92	31	ring	ring	NOUN
cana-4565	92	32	℧	℧	NOUN
cana-4565	92	33	1	1	NUM
cana-4565	92	34	.	.	PUNCT
cana-4565	93	1	proof	proof	NOUN
cana-4565	93	2	.	.	PUNCT
cana-4565	94	1	from	from	ADP
cana-4565	94	2	the	the	DET
cana-4565	94	3	above	above	ADJ
cana-4565	94	4	theorems	theorem	NOUN
cana-4565	94	5	,	,	PUNCT
cana-4565	94	6	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	94	7	𝑝𝑟𝑜𝑜𝑓	𝑝𝑟𝑜𝑜𝑓	NOUN
cana-4565	94	8	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	94	9	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	NOUN
cana-4565	94	10	theorem	theorem	VERB
cana-4565	94	11	2.11	2.11	NUM
cana-4565	94	12	.	.	PUNCT
cana-4565	95	1	𝐼𝑓	𝐼𝑓	VERB
cana-4565	95	2	њ	њ	NOUN
cana-4565	95	3	=	=	NOUN
cana-4565	95	4			NOUN
cana-4565	95	5	њ1	њ1	NOUN
cana-4565	95	6	+	+	NOUN
cana-4565	95	7	,	,	PUNCT
cana-4565	95	8	њ2	њ2	PROPN
cana-4565	95	9	+	+	PROPN
cana-4565	95	10	,	,	PUNCT
cana-4565	95	11	…	…	PUNCT
cana-4565	95	12	,	,	PUNCT
cana-4565	95	13	њ𝑛	њ𝑛	ADP
cana-4565	95	14	+	+	ADJ
cana-4565	95	15	,	,	PUNCT
cana-4565	95	16	њ1	њ1	NOUN
cana-4565	95	17	−	−	NOUN
cana-4565	95	18	,	,	PUNCT
cana-4565	95	19	њ2	њ2	NOUN
cana-4565	95	20	−	−	PROPN
cana-4565	95	21	,	,	PUNCT
cana-4565	95	22	…	…	PUNCT
cana-4565	95	23	,	,	PUNCT
cana-4565	95	24	њ𝑛	њ𝑛	ADP
cana-4565	95	25	−	−	PROPN
cana-4565	95	26	is	be	AUX
cana-4565	95	27	𝑎	𝑎	DET
cana-4565	95	28	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	NOUN
cana-4565	95	29	of	of	ADP
cana-4565	95	30	a	a	DET
cana-4565	95	31	ring	ring	NOUN
cana-4565	95	32	ѝ1	ѝ1	ADJ
cana-4565	95	33	,	,	PUNCT
cana-4565	95	34	then	then	ADV
cana-4565	95	35	⋈	⋈	PROPN
cana-4565	95	36	(	(	PUNCT
cana-4565	95	37	њ)𝑖𝑠	њ)𝑖𝑠	PROPN
cana-4565	95	38	𝑎𝑙𝑠𝑜	𝑎𝑙𝑠𝑜	VERB
cana-4565	95	39	𝑎𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝑎𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	95	40	of	of	ADP
cana-4565	95	41	ѝ1	ѝ1	ADJ
cana-4565	95	42	.	.	PUNCT
cana-4565	96	1	theorem	theorem	VERB
cana-4565	96	2	2.12	2.12	NUM
cana-4565	96	3	.	.	PUNCT
cana-4565	97	1	𝐼𝑓	𝐼𝑓	ADJ
cana-4565	97	2	ℭ	ℭ	NOUN
cana-4565	97	3	=	=	PUNCT
cana-4565	97	4			NOUN
cana-4565	97	5	ℭ1	ℭ1	INTJ
cana-4565	97	6	+	+	ADV
cana-4565	97	7	,	,	PUNCT
cana-4565	97	8	ℭ2	ℭ2	PROPN
cana-4565	97	9	+	+	PROPN
cana-4565	97	10	,	,	PUNCT
cana-4565	97	11	…	…	PUNCT
cana-4565	97	12	,	,	PUNCT
cana-4565	98	1	ℭ𝑛	ℭ𝑛	ADP
cana-4565	98	2	+	+	ADJ
cana-4565	98	3	,	,	PUNCT
cana-4565	98	4	ℭ1	ℭ1	ADP
cana-4565	98	5	−	−	PROPN
cana-4565	98	6	,	,	PUNCT
cana-4565	98	7	ℭ2	ℭ2	PROPN
cana-4565	98	8	−	−	PROPN
cana-4565	98	9	,	,	PUNCT
cana-4565	98	10	…	…	PUNCT
cana-4565	98	11	,	,	PUNCT
cana-4565	98	12	ℭ𝑛	ℭ𝑛	SCONJ
cana-4565	98	13	−	−	PROPN
cana-4565	98	14	is	be	AUX
cana-4565	98	15	𝑎	𝑎	DET
cana-4565	98	16	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	98	17	of	of	ADP
cana-4565	98	18	a	a	DET
cana-4565	98	19	ring	ring	NOUN
cana-4565	98	20	℧	℧	PROPN
cana-4565	98	21	1	1	NUM
cana-4565	98	22	,	,	PUNCT
cana-4565	98	23	then	then	ADV
cana-4565	98	24	⋈	⋈	PROPN
cana-4565	98	25	(	(	PUNCT
cana-4565	98	26	ℭ)𝑖𝑠	ℭ)𝑖𝑠	VERB
cana-4565	98	27	𝑎𝑙𝑠𝑜	𝑎𝑙𝑠𝑜	NOUN
cana-4565	98	28	𝑎𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝑎𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	98	29	of	of	ADP
cana-4565	98	30	the	the	DET
cana-4565	98	31	ring	ring	NOUN
cana-4565	98	32	℧	℧	NOUN
cana-4565	98	33	1	1	NUM
cana-4565	98	34	.	.	PUNCT
cana-4565	99	1	proof	proof	NOUN
cana-4565	99	2	.	.	PUNCT
cana-4565	100	1	let	let	VERB
cana-4565	100	2	𝜁	𝜁	NOUN
cana-4565	100	3	,	,	PUNCT
cana-4565	100	4	𝜐	𝜐	VERB
cana-4565	100	5	be	be	AUX
cana-4565	100	6	in	in	ADP
cana-4565	100	7	℧	℧	NOUN
cana-4565	100	8	1	1	NUM
cana-4565	100	9	.	.	PUNCT
cana-4565	101	1	for	for	ADP
cana-4565	101	2	all	all	DET
cana-4565	101	3	i	i	PRON
cana-4565	101	4	=	=	NOUN
cana-4565	101	5	1	1	NUM
cana-4565	101	6	,	,	PUNCT
cana-4565	101	7	2	2	NUM
cana-4565	101	8	,	,	PUNCT
cana-4565	101	9	…	…	PUNCT
cana-4565	101	10	,	,	PUNCT
cana-4565	101	11	n	n	CCONJ
cana-4565	101	12	,	,	PUNCT
cana-4565	101	13	by	by	ADP
cana-4565	101	14	theorem	theorem	NOUN
cana-4565	101	15	2.11	2.11	NUM
cana-4565	101	16	,	,	PUNCT
cana-4565	101	17	⋈(ℭ	⋈(ℭ	NUM
cana-4565	101	18	)	)	PUNCT
cana-4565	101	19	is	be	AUX
cana-4565	101	20	a	a	DET
cana-4565	101	21	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	NOUN
cana-4565	101	22	of	of	ADP
cana-4565	101	23	℧	℧	NOUN
cana-4565	101	24	1	1	NUM
cana-4565	101	25	,	,	PUNCT
cana-4565	101	26	⋈(ℭ𝑖	⋈(ℭ𝑖	NOUN
cana-4565	101	27	+	+	NOUN
cana-4565	101	28	)	)	PUNCT
cana-4565	101	29	(	(	PUNCT
cana-4565	101	30	𝜁𝜐	𝜁𝜐	NOUN
cana-4565	101	31	)	)	PUNCT
cana-4565	101	32	=	=	PUNCT
cana-4565	102	1	rmax{[½,½	rmax{[½,½	NOUN
cana-4565	102	2	]	]	PUNCT
cana-4565	102	3	,	,	PUNCT
cana-4565	102	4	ℭ𝑖	ℭ𝑖	PROPN
cana-4565	102	5	+	+	PROPN
cana-4565	102	6	(	(	PUNCT
cana-4565	102	7	𝜁𝜐)}=	𝜁𝜐)}=	PROPN
cana-4565	102	8	rmax{[½,½	rmax{[½,½	NOUN
cana-4565	102	9	]	]	PUNCT
cana-4565	102	10	,	,	PUNCT
cana-4565	102	11	ℭ𝑖	ℭ𝑖	PROPN
cana-4565	102	12	+	+	PROPN
cana-4565	102	13	(	(	PUNCT
cana-4565	102	14	𝜐𝜁)}=	𝜐𝜁)}=	ADJ
cana-4565	102	15	⋈(ℭ𝑖	⋈(ℭ𝑖	NOUN
cana-4565	102	16	+	+	NOUN
cana-4565	102	17	)	)	PUNCT
cana-4565	102	18	(	(	PUNCT
cana-4565	102	19	𝜐𝜁	𝜐𝜁	NOUN
cana-4565	102	20	)	)	PUNCT
cana-4565	102	21	,	,	PUNCT
cana-4565	102	22	for	for	ADP
cana-4565	102	23	all	all	DET
cana-4565	102	24	𝜁	𝜁	NOUN
cana-4565	102	25	,	,	PUNCT
cana-4565	102	26	𝜐	𝜐	X
cana-4565	102	27	in	in	ADP
cana-4565	102	28	℧	℧	NOUN
cana-4565	102	29	1	1	NUM
cana-4565	102	30	.	.	PUNCT
cana-4565	102	31	also	also	ADV
cana-4565	102	32	⋈(ℭ𝑖	⋈(ℭ𝑖	NOUN
cana-4565	102	33	−)(𝜁𝜐	−)(𝜁𝜐	NOUN
cana-4565	102	34	)	)	PUNCT
cana-4565	102	35	=	=	SYM
cana-4565	102	36	rmin{[−½,−½	rmin{[−½,−½	PROPN
cana-4565	102	37	]	]	PUNCT
cana-4565	102	38	,	,	PUNCT
cana-4565	102	39	ℭ𝑖	ℭ𝑖	PROPN
cana-4565	102	40	−(𝜁𝜐	−(𝜁𝜐	PROPN
cana-4565	102	41	)	)	PUNCT
cana-4565	102	42	}	}	PUNCT
cana-4565	102	43	=	=	SYM
cana-4565	102	44	rmin{[−½,−½	rmin{[−½,−½	NOUN
cana-4565	102	45	]	]	PUNCT
cana-4565	102	46	,	,	PUNCT
cana-4565	102	47	ℭ𝑖	ℭ𝑖	PROPN
cana-4565	102	48	−(𝜐𝜁)}=	−(𝜐𝜁)}=	PRON
cana-4565	102	49	⋈(ℭ𝑖	⋈(ℭ𝑖	NOUN
cana-4565	102	50	−)(𝜐𝜁	−)(𝜐𝜁	PROPN
cana-4565	102	51	)	)	PUNCT
cana-4565	102	52	,	,	PUNCT
cana-4565	102	53	for	for	ADP
cana-4565	102	54	all	all	DET
cana-4565	102	55	𝜁	𝜁	NOUN
cana-4565	102	56	,	,	PUNCT
cana-4565	102	57	𝜐	𝜐	X
cana-4565	102	58	in	in	ADP
cana-4565	102	59	℧	℧	NOUN
cana-4565	102	60	1	1	NUM
cana-4565	102	61	.	.	PUNCT
cana-4565	102	62	hence	hence	ADV
cana-4565	102	63	⋈(ℭ	⋈(ℭ	NUM
cana-4565	102	64	)	)	PUNCT
cana-4565	102	65	is	be	AUX
cana-4565	102	66	a	a	DET
cana-4565	102	67	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	102	68	of	of	ADP
cana-4565	102	69	℧	℧	NOUN
cana-4565	102	70	1	1	NUM
cana-4565	102	71	.	.	PUNCT
cana-4565	102	72	corollary	corollary	ADJ
cana-4565	102	73	2.13	2.13	NUM
cana-4565	102	74	.	.	PUNCT
cana-4565	103	1	if	if	SCONJ
cana-4565	103	2	𝔓	𝔓	PROPN
cana-4565	103	3	and	and	CCONJ
cana-4565	103	4	𝔚	𝔚	PROPN
cana-4565	103	5	are	be	AUX
cana-4565	103	6	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	NOUN
cana-4565	103	7	𝑜𝑓	𝑜𝑓	ADP
cana-4565	103	8	𝑡ℎ𝑒	𝑡ℎ𝑒	NUM
cana-4565	103	9	𝑟𝑖𝑛𝑔	𝑟𝑖𝑛𝑔	NOUN
cana-4565	103	10	℧	℧	PROPN
cana-4565	103	11	1	1	NUM
cana-4565	103	12	,	,	PUNCT
cana-4565	103	13	then	then	ADV
cana-4565	103	14	⋈(𝔓	⋈(𝔓	PRON
cana-4565	103	15	∩	∩	ADJ
cana-4565	103	16	𝔚	𝔚	NOUN
cana-4565	103	17	)	)	PUNCT
cana-4565	103	18	is	be	AUX
cana-4565	103	19	a	a	DET
cana-4565	103	20	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	103	21	of	of	ADP
cana-4565	103	22	℧	℧	NOUN
cana-4565	103	23	1	1	NUM
cana-4565	103	24	.	.	PUNCT
cana-4565	104	1	proof	proof	NOUN
cana-4565	104	2	.	.	PUNCT
cana-4565	105	1	from	from	ADP
cana-4565	105	2	the	the	DET
cana-4565	105	3	above	above	ADJ
cana-4565	105	4	theorems	theorem	NOUN
cana-4565	105	5	,	,	PUNCT
cana-4565	105	6	it	it	PRON
cana-4565	105	7	is	be	AUX
cana-4565	105	8	trivial	trivial	ADJ
cana-4565	105	9	.	.	PUNCT
cana-4565	106	1	corollary	corollary	ADJ
cana-4565	106	2	2.14	2.14	NUM
cana-4565	106	3	.	.	PUNCT
cana-4565	107	1	if	if	SCONJ
cana-4565	107	2	𝔓	𝔓	PROPN
cana-4565	107	3	and	and	CCONJ
cana-4565	107	4	𝔚	𝔚	PROPN
cana-4565	107	5	are	be	AUX
cana-4565	107	6	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	NOUN
cana-4565	107	7	𝑜𝑓	𝑜𝑓	ADP
cana-4565	107	8	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	107	9	𝑟𝑖𝑛𝑔𝑠	𝑟𝑖𝑛𝑔𝑠	NOUN
cana-4565	107	10	℧	℧	PROPN
cana-4565	107	11	1and	1and	NUM
cana-4565	107	12	℧	℧	PROPN
cana-4565	107	13	2	2	NUM
cana-4565	107	14	,	,	PUNCT
cana-4565	107	15	then	then	ADV
cana-4565	107	16	⋈	⋈	PROPN
cana-4565	107	17	𝔓	𝔓	PROPN
cana-4565	107	18	∩⋈	∩⋈	NOUN
cana-4565	107	19	𝔚	𝔚	PROPN
cana-4565	107	20	is	be	AUX
cana-4565	107	21	a	a	DET
cana-4565	107	22	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	107	23	of	of	ADP
cana-4565	107	24	℧	℧	NOUN
cana-4565	107	25	1	1	NUM
cana-4565	107	26	∩	∩	NOUN
cana-4565	107	27	℧	℧	PROPN
cana-4565	107	28	2	2	NUM
cana-4565	107	29	.	.	PUNCT
cana-4565	108	1	proof	proof	NOUN
cana-4565	108	2	.	.	PUNCT
cana-4565	109	1	from	from	ADP
cana-4565	109	2	the	the	DET
cana-4565	109	3	above	above	ADJ
cana-4565	109	4	theorems	theorem	NOUN
cana-4565	109	5	,	,	PUNCT
cana-4565	109	6	it	it	PRON
cana-4565	109	7	is	be	AUX
cana-4565	109	8	trivial	trivial	ADJ
cana-4565	109	9	.	.	PUNCT
cana-4565	110	1	corollary	corollary	ADJ
cana-4565	110	2	2.15	2.15	NUM
cana-4565	110	3	.	.	PUNCT
cana-4565	111	1	if	if	SCONJ
cana-4565	111	2	𝔓	𝔓	PROPN
cana-4565	111	3	and	and	CCONJ
cana-4565	111	4	𝔚	𝔚	PROPN
cana-4565	111	5	are	be	AUX
cana-4565	111	6	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	NOUN
cana-4565	111	7	𝑜𝑓	𝑜𝑓	ADP
cana-4565	111	8	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	111	9	𝑟𝑖𝑛𝑔𝑠	𝑟𝑖𝑛𝑔𝑠	NOUN
cana-4565	111	10	℧	℧	PROPN
cana-4565	111	11	1	1	NUM
cana-4565	111	12	,	,	PUNCT
cana-4565	111	13	then	then	ADV
cana-4565	111	14	⋈	⋈	PROPN
cana-4565	111	15	𝔓	𝔓	PROPN
cana-4565	111	16	∩⋈	∩⋈	NOUN
cana-4565	111	17	𝔚	𝔚	PROPN
cana-4565	111	18	is	be	AUX
cana-4565	111	19	a	a	DET
cana-4565	111	20	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	111	21	of	of	ADP
cana-4565	111	22	℧	℧	NOUN
cana-4565	111	23	1	1	NUM
cana-4565	111	24	.	.	PUNCT
cana-4565	112	1	proof	proof	NOUN
cana-4565	112	2	.	.	PUNCT
cana-4565	113	1	from	from	ADP
cana-4565	113	2	the	the	DET
cana-4565	113	3	above	above	ADJ
cana-4565	113	4	theorems	theorem	NOUN
cana-4565	113	5	,	,	PUNCT
cana-4565	113	6	it	it	PRON
cana-4565	113	7	is	be	AUX
cana-4565	113	8	trivial	trivial	ADJ
cana-4565	113	9	.	.	PUNCT
cana-4565	114	1	theorem	theorem	VERB
cana-4565	114	2	2.16	2.16	NUM
cana-4565	114	3	.	.	PUNCT
cana-4565	115	1	𝐼𝑓	𝐼𝑓	PROPN
cana-4565	115	2	𝔓1	𝔓1	PROPN
cana-4565	115	3	,	,	PUNCT
cana-4565	115	4	𝔓2	𝔓2	NOUN
cana-4565	115	5	,	,	PUNCT
cana-4565	115	6	…	…	PUNCT
cana-4565	115	7	,	,	PUNCT
cana-4565	115	8	𝔓𝑚𝑎𝑟𝑒	𝔓𝑚𝑎𝑟𝑒	PROPN
cana-4565	115	9	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	115	10	of	of	ADP
cana-4565	115	11	the	the	DET
cana-4565	115	12	rings	ring	NOUN
cana-4565	115	13	℧	℧	PROPN
cana-4565	115	14	1	1	NUM
cana-4565	115	15	,	,	PUNCT
cana-4565	115	16	℧	℧	NOUN
cana-4565	115	17	2	2	NUM
cana-4565	115	18	,	,	PUNCT
cana-4565	115	19	…	…	PUNCT
cana-4565	115	20	,	,	PUNCT
cana-4565	115	21	℧	℧	NOUN
cana-4565	115	22	m	m	VERB
cana-4565	115	23	respectively	respectively	ADV
cana-4565	115	24	,	,	PUNCT
cana-4565	115	25	then	then	ADV
cana-4565	115	26	⋈	⋈	PROPN
cana-4565	115	27	(	(	PUNCT
cana-4565	115	28	𝔓1	𝔓1	PROPN
cana-4565	115	29	∩	∩	PROPN
cana-4565	115	30	𝔓2	𝔓2	NOUN
cana-4565	115	31	∩	∩	NOUN
cana-4565	115	32	…	…	PUNCT
cana-4565	115	33	∩	∩	ADJ
cana-4565	115	34	𝔓𝑚	𝔓𝑚	NOUN
cana-4565	115	35	)	)	PUNCT
cana-4565	115	36	is	be	AUX
cana-4565	115	37	a	a	DET
cana-4565	115	38	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	115	39	of	of	ADP
cana-4565	115	40	the	the	DET
cana-4565	115	41	ring	ring	NOUN
cana-4565	115	42	℧	℧	NOUN
cana-4565	115	43	1	1	NUM
cana-4565	115	44	∩	∩	X
cana-4565	115	45	℧	℧	NOUN
cana-4565	115	46	2	2	NUM
cana-4565	115	47	∩	∩	NOUN
cana-4565	115	48	…	…	PUNCT
cana-4565	115	49	∩	∩	ADJ
cana-4565	115	50	℧	℧	NOUN
cana-4565	115	51	m.	m.	NOUN
cana-4565	115	52	proof	proof	NOUN
cana-4565	115	53	.	.	PUNCT
cana-4565	116	1	from	from	ADP
cana-4565	116	2	the	the	DET
cana-4565	116	3	above	above	ADJ
cana-4565	116	4	theorems	theorem	NOUN
cana-4565	116	5	,	,	PUNCT
cana-4565	116	6	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	116	7	𝑝𝑟𝑜𝑜𝑓	𝑝𝑟𝑜𝑜𝑓	NOUN
cana-4565	116	8	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	116	9	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	PROPN
cana-4565	116	10	corollary	corollary	NOUN
cana-4565	116	11	2.17	2.17	NUM
cana-4565	116	12	.	.	PUNCT
cana-4565	117	1	𝐼𝑓	𝐼𝑓	PROPN
cana-4565	117	2	𝔓1	𝔓1	PROPN
cana-4565	117	3	,	,	PUNCT
cana-4565	117	4	𝔓2	𝔓2	NOUN
cana-4565	117	5	,	,	PUNCT
cana-4565	117	6	…	…	PUNCT
cana-4565	117	7	,	,	PUNCT
cana-4565	117	8	𝔓𝑚	𝔓𝑚	PROPN
cana-4565	117	9	𝑎𝑟𝑒	𝑎𝑟𝑒	VERB
cana-4565	117	10	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	117	11	of	of	ADP
cana-4565	117	12	the	the	DET
cana-4565	117	13	rings	ring	NOUN
cana-4565	117	14	℧	℧	PROPN
cana-4565	117	15	1	1	NUM
cana-4565	117	16	,	,	PUNCT
cana-4565	117	17	℧	℧	NOUN
cana-4565	117	18	2	2	NUM
cana-4565	117	19	,	,	PUNCT
cana-4565	117	20	…	…	PUNCT
cana-4565	117	21	,	,	PUNCT
cana-4565	117	22	℧	℧	NOUN
cana-4565	117	23	m	m	VERB
cana-4565	117	24	respectively	respectively	ADV
cana-4565	117	25	,	,	PUNCT
cana-4565	117	26	then	then	ADV
cana-4565	117	27	⋈	⋈	PROPN
cana-4565	117	28	𝔓1	𝔓1	PROPN
cana-4565	117	29	∩⋈	∩⋈	PROPN
cana-4565	117	30	𝔓2	𝔓2	PROPN
cana-4565	117	31	∩	∩	NOUN
cana-4565	117	32	…	…	PUNCT
cana-4565	117	33	∩⋈	∩⋈	NOUN
cana-4565	118	1	𝔓𝑚	𝔓𝑚	NOUN
cana-4565	118	2	is	be	AUX
cana-4565	118	3	a	a	DET
cana-4565	118	4	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	118	5	of	of	ADP
cana-4565	118	6	the	the	DET
cana-4565	118	7	ring	ring	NOUN
cana-4565	118	8	℧	℧	NOUN
cana-4565	118	9	1	1	NUM
cana-4565	118	10	∩	∩	X
cana-4565	118	11	℧	℧	NOUN
cana-4565	118	12	2	2	NUM
cana-4565	118	13	∩	∩	NOUN
cana-4565	118	14	…	…	PUNCT
cana-4565	118	15	∩	∩	ADJ
cana-4565	118	16	℧	℧	NOUN
cana-4565	118	17	m.	m.	NOUN
cana-4565	118	18	proof	proof	NOUN
cana-4565	118	19	.	.	PUNCT
cana-4565	119	1	from	from	ADP
cana-4565	119	2	the	the	DET
cana-4565	119	3	above	above	ADJ
cana-4565	119	4	theorems	theorem	NOUN
cana-4565	119	5	,	,	PUNCT
cana-4565	119	6	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	119	7	𝑝𝑟𝑜𝑜𝑓	𝑝𝑟𝑜𝑜𝑓	NOUN
cana-4565	119	8	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	119	9	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	PROPN
cana-4565	119	10	corollary	corollary	NOUN
cana-4565	119	11	2.18	2.18	NUM
cana-4565	119	12	.	.	PUNCT
cana-4565	120	1	𝐼𝑓	𝐼𝑓	PROPN
cana-4565	120	2	𝔓1	𝔓1	PROPN
cana-4565	120	3	,	,	PUNCT
cana-4565	120	4	𝔓2	𝔓2	NOUN
cana-4565	120	5	,	,	PUNCT
cana-4565	120	6	…	…	PUNCT
cana-4565	120	7	,	,	PUNCT
cana-4565	120	8	𝔓𝑚	𝔓𝑚	PROPN
cana-4565	120	9	are	be	AUX
cana-4565	120	10	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	120	11	of	of	ADP
cana-4565	120	12	the	the	DET
cana-4565	120	13	ring	ring	NOUN
cana-4565	120	14	℧	℧	PROPN
cana-4565	120	15	1	1	NUM
cana-4565	120	16	,	,	PUNCT
cana-4565	120	17	then	then	ADV
cana-4565	120	18	⋈	⋈	PROPN
cana-4565	120	19	(	(	PUNCT
cana-4565	120	20	𝔓1	𝔓1	PROPN
cana-4565	120	21	∩	∩	PROPN
cana-4565	120	22	𝔓2	𝔓2	NOUN
cana-4565	120	23	∩	∩	NOUN
cana-4565	120	24	…	…	PUNCT
cana-4565	120	25	∩	∩	ADJ
cana-4565	120	26	𝔓𝑚	𝔓𝑚	NOUN
cana-4565	120	27	)	)	PUNCT
cana-4565	120	28	is	be	AUX
cana-4565	120	29	a	a	DET
cana-4565	120	30	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	120	31	𝑜𝑓	𝑜𝑓	ADP
cana-4565	120	32	the	the	DET
cana-4565	120	33	ring	ring	NOUN
cana-4565	120	34	℧	℧	NOUN
cana-4565	120	35	1	1	NUM
cana-4565	120	36	.	.	PUNCT
cana-4565	121	1	proof	proof	NOUN
cana-4565	121	2	.	.	PUNCT
cana-4565	122	1	from	from	ADP
cana-4565	122	2	the	the	DET
cana-4565	122	3	above	above	ADJ
cana-4565	122	4	theorems	theorem	NOUN
cana-4565	122	5	,	,	PUNCT
cana-4565	122	6	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	122	7	𝑝𝑟𝑜𝑜𝑓	𝑝𝑟𝑜𝑜𝑓	NOUN
cana-4565	122	8	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	122	9	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	NOUN
cana-4565	122	10	corollary	corollary	NOUN
cana-4565	122	11	2.19	2.19	NUM
cana-4565	122	12	.	.	PUNCT
cana-4565	123	1	𝐼𝑓	𝐼𝑓	PROPN
cana-4565	123	2	𝔓1	𝔓1	PROPN
cana-4565	123	3	,	,	PUNCT
cana-4565	123	4	𝔓2	𝔓2	NOUN
cana-4565	123	5	,	,	PUNCT
cana-4565	123	6	…	…	PUNCT
cana-4565	123	7	,	,	PUNCT
cana-4565	123	8	𝔓𝑚	𝔓𝑚	PROPN
cana-4565	123	9	are	be	AUX
cana-4565	123	10	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	123	11	of	of	ADP
cana-4565	123	12	the	the	DET
cana-4565	123	13	ring	ring	NOUN
cana-4565	123	14	℧	℧	PROPN
cana-4565	123	15	1	1	NUM
cana-4565	123	16	,	,	PUNCT
cana-4565	123	17	then	then	ADV
cana-4565	123	18	⋈	⋈	PROPN
cana-4565	123	19	𝔓1	𝔓1	PROPN
cana-4565	123	20	∩⋈	∩⋈	PROPN
cana-4565	123	21	𝔓2	𝔓2	PROPN
cana-4565	123	22	∩	∩	NOUN
cana-4565	123	23	…	…	PUNCT
cana-4565	123	24	∩	∩	NOUN
cana-4565	124	1	⋈	⋈	PROPN
cana-4565	125	1	𝔓𝑚	𝔓𝑚	PROPN
cana-4565	125	2	is	be	AUX
cana-4565	125	3	a	a	DET
cana-4565	125	4	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	125	5	of	of	ADP
cana-4565	125	6	℧	℧	PROPN
cana-4565	125	7	1	1	NUM
cana-4565	125	8	.	.	PUNCT
cana-4565	125	9	communications	communication	NOUN
cana-4565	125	10	on	on	ADP
cana-4565	125	11	applied	apply	VERB
cana-4565	125	12	nonlinear	nonlinear	ADJ
cana-4565	125	13	analysis	analysis	NOUN
cana-4565	125	14	issn	issn	NOUN
cana-4565	125	15	:	:	PUNCT
cana-4565	125	16	1074	1074	NUM
cana-4565	125	17	-	-	PUNCT
cana-4565	125	18	133x	133x	NUM
cana-4565	125	19	vol	vol	NOUN
cana-4565	125	20	32	32	NUM
cana-4565	125	21	no	no	NOUN
cana-4565	125	22	.	.	PUNCT
cana-4565	126	1	9s	9s	NUM
cana-4565	126	2	(	(	PUNCT
cana-4565	126	3	2025	2025	NUM
cana-4565	126	4	)	)	PUNCT
cana-4565	126	5	2619	2619	NUM
cana-4565	127	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-4565	127	2	proof	proof	NOUN
cana-4565	127	3	.	.	PUNCT
cana-4565	128	1	from	from	ADP
cana-4565	128	2	the	the	DET
cana-4565	128	3	above	above	ADJ
cana-4565	128	4	theorems	theorem	NOUN
cana-4565	128	5	,	,	PUNCT
cana-4565	128	6	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	128	7	𝑝𝑟𝑜𝑜𝑓	𝑝𝑟𝑜𝑜𝑓	NOUN
cana-4565	128	8	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	128	9	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	NOUN
cana-4565	128	10	theorem	theorem	VERB
cana-4565	128	11	2.20	2.20	NUM
cana-4565	128	12	.	.	PUNCT
cana-4565	129	1	𝐼𝑓	𝐼𝑓	VERB
cana-4565	129	2	ⅎ	ⅎ	PROPN
cana-4565	129	3	=	=	NOUN
cana-4565	129	4			NOUN
cana-4565	129	5	ⅎ1	ⅎ1	PROPN
cana-4565	129	6	+	+	PROPN
cana-4565	129	7	,	,	PUNCT
cana-4565	129	8	ⅎ2	ⅎ2	PROPN
cana-4565	129	9	+	+	PROPN
cana-4565	129	10	,	,	PUNCT
cana-4565	129	11	…	…	PUNCT
cana-4565	129	12	,	,	PUNCT
cana-4565	129	13	ⅎ𝑛	ⅎ𝑛	ADP
cana-4565	129	14	+	+	PROPN
cana-4565	129	15	,	,	PUNCT
cana-4565	129	16	ⅎ1	ⅎ1	ADJ
cana-4565	129	17	−	−	PROPN
cana-4565	129	18	,	,	PUNCT
cana-4565	129	19	ⅎ2	ⅎ2	PROPN
cana-4565	129	20	−	−	PROPN
cana-4565	129	21	,	,	PUNCT
cana-4565	129	22	…	…	PUNCT
cana-4565	129	23	,	,	PUNCT
cana-4565	129	24	ⅎ𝑛	ⅎ𝑛	ADP
cana-4565	129	25	−	−	PROPN
cana-4565	129	26	is	be	AUX
cana-4565	129	27	𝑎	𝑎	DET
cana-4565	129	28	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	NOUN
cana-4565	129	29	of	of	ADP
cana-4565	129	30	a	a	DET
cana-4565	129	31	ring	ring	NOUN
cana-4565	129	32	ℵ1	ℵ1	NOUN
cana-4565	129	33	,	,	PUNCT
cana-4565	129	34	then	then	ADV
cana-4565	129	35	𝔔(𝜛,𝜍)(ⅎ	𝔔(𝜛,𝜍)(ⅎ	PROPN
cana-4565	129	36	)	)	PUNCT
cana-4565	129	37	is	be	AUX
cana-4565	129	38	a	a	DET
cana-4565	129	39	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	NOUN
cana-4565	129	40	of	of	ADP
cana-4565	129	41	ℵ1	ℵ1	PROPN
cana-4565	129	42	,	,	PUNCT
cana-4565	129	43	where	where	SCONJ
cana-4565	129	44	𝜛	𝜛	X
cana-4565	129	45	=	=	PRON
cana-4565	129	46	(	(	PUNCT
cana-4565	129	47	𝜛1	𝜛1	NOUN
cana-4565	129	48	,	,	PUNCT
cana-4565	129	49	𝜛2	𝜛2	NOUN
cana-4565	129	50	,	,	PUNCT
cana-4565	129	51	…	…	PUNCT
cana-4565	129	52	,	,	PUNCT
cana-4565	129	53	𝜛𝑛	𝜛𝑛	NOUN
cana-4565	129	54	)	)	PUNCT
cana-4565	129	55	and	and	CCONJ
cana-4565	129	56	𝜍	𝜍	X
cana-4565	129	57	=	=	SYM
cana-4565	129	58	(	(	PUNCT
cana-4565	129	59	𝜍1	𝜍1	PROPN
cana-4565	129	60	,	,	PUNCT
cana-4565	129	61	𝜍2	𝜍2	NOUN
cana-4565	129	62	,	,	PUNCT
cana-4565	129	63	…	…	PUNCT
cana-4565	129	64	,	,	PUNCT
cana-4565	129	65	𝜍𝑛	𝜍𝑛	NOUN
cana-4565	129	66	)	)	PUNCT
cana-4565	129	67	,	,	PUNCT
cana-4565	129	68	𝜛𝑖	𝜛𝑖	NOUN
cana-4565	129	69	∈	∈	PROPN
cana-4565	129	70	𝐷[0	𝐷[0	PROPN
cana-4565	129	71	,	,	PUNCT
cana-4565	129	72	1	1	NUM
cana-4565	129	73	]	]	X
cana-4565	129	74	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-4565	129	75	𝜍𝑖	𝜍𝑖	PROPN
cana-4565	129	76	∈	∈	PROPN
cana-4565	129	77	𝐷[−1	𝐷[−1	NOUN
cana-4565	129	78	,	,	PUNCT
cana-4565	129	79	0	0	NUM
cana-4565	129	80	]	]	PUNCT
cana-4565	129	81	.	.	PUNCT
cana-4565	130	1	theorem	theorem	VERB
cana-4565	130	2	2.21	2.21	NUM
cana-4565	130	3	.	.	PUNCT
cana-4565	131	1	𝐼𝑓	𝐼𝑓	VERB
cana-4565	131	2	𝔓	𝔓	NOUN
cana-4565	131	3	=	=	PUNCT
cana-4565	131	4			X
cana-4565	131	5	𝔓1	𝔓1	NOUN
cana-4565	131	6	+	+	ADJ
cana-4565	131	7	,	,	PUNCT
cana-4565	131	8	𝔓2	𝔓2	VERB
cana-4565	131	9	+	+	ADJ
cana-4565	131	10	,	,	PUNCT
cana-4565	131	11	…	…	PUNCT
cana-4565	131	12	,	,	PUNCT
cana-4565	131	13	𝔓𝑛	𝔓𝑛	PROPN
cana-4565	131	14	+	+	PROPN
cana-4565	131	15	,	,	PUNCT
cana-4565	131	16	𝔓1	𝔓1	ADP
cana-4565	131	17	−	−	PROPN
cana-4565	131	18	,	,	PUNCT
cana-4565	131	19	𝔓2	𝔓2	NOUN
cana-4565	131	20	−	−	NUM
cana-4565	131	21	,	,	PUNCT
cana-4565	131	22	…	…	PUNCT
cana-4565	131	23	,	,	PUNCT
cana-4565	131	24	𝔓𝑛	𝔓𝑛	PROPN
cana-4565	131	25	−	−	PROPN
cana-4565	131	26	is	be	AUX
cana-4565	131	27	𝑎	𝑎	DET
cana-4565	131	28	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	131	29	of	of	ADP
cana-4565	131	30	a	a	DET
cana-4565	131	31	ring	ring	NOUN
cana-4565	131	32	ℜ1	ℜ1	NOUN
cana-4565	131	33	,	,	PUNCT
cana-4565	131	34	then	then	ADV
cana-4565	131	35	𝔔(𝜛,𝜍)(𝔓	𝔔(𝜛,𝜍)(𝔓	ADV
cana-4565	131	36	)	)	PUNCT
cana-4565	131	37	is	be	AUX
cana-4565	131	38	a	a	DET
cana-4565	131	39	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	131	40	of	of	ADP
cana-4565	131	41	ℜ1	ℜ1	PROPN
cana-4565	131	42	,	,	PUNCT
cana-4565	131	43	where	where	SCONJ
cana-4565	131	44	𝜛	𝜛	X
cana-4565	131	45	=	=	PRON
cana-4565	131	46	(	(	PUNCT
cana-4565	131	47	𝜛1	𝜛1	NOUN
cana-4565	131	48	,	,	PUNCT
cana-4565	131	49	𝜛2	𝜛2	NOUN
cana-4565	131	50	,	,	PUNCT
cana-4565	131	51	…	…	PUNCT
cana-4565	131	52	,	,	PUNCT
cana-4565	131	53	𝜛𝑛	𝜛𝑛	NOUN
cana-4565	131	54	)	)	PUNCT
cana-4565	131	55	and	and	CCONJ
cana-4565	131	56	𝜍	𝜍	X
cana-4565	131	57	=	=	SYM
cana-4565	131	58	(	(	PUNCT
cana-4565	131	59	𝜍1	𝜍1	PROPN
cana-4565	131	60	,	,	PUNCT
cana-4565	131	61	𝜍2	𝜍2	NOUN
cana-4565	131	62	,	,	PUNCT
cana-4565	131	63	…	…	PUNCT
cana-4565	131	64	,	,	PUNCT
cana-4565	131	65	𝜍𝑛	𝜍𝑛	NOUN
cana-4565	131	66	)	)	PUNCT
cana-4565	131	67	,	,	PUNCT
cana-4565	131	68	𝜛𝑖	𝜛𝑖	NOUN
cana-4565	131	69	∈	∈	PROPN
cana-4565	131	70	𝐷[0	𝐷[0	PROPN
cana-4565	131	71	,	,	PUNCT
cana-4565	131	72	1	1	NUM
cana-4565	131	73	]	]	X
cana-4565	131	74	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-4565	131	75	𝜍𝑖	𝜍𝑖	PROPN
cana-4565	131	76	∈	∈	PROPN
cana-4565	131	77	𝐷[−1	𝐷[−1	NOUN
cana-4565	131	78	,	,	PUNCT
cana-4565	131	79	0	0	NUM
cana-4565	131	80	]	]	PUNCT
cana-4565	131	81	.	.	PUNCT
cana-4565	132	1	proof	proof	NOUN
cana-4565	132	2	.	.	PUNCT
cana-4565	133	1	let	let	VERB
cana-4565	133	2	𝜉	𝜉	X
cana-4565	133	3	,	,	PUNCT
cana-4565	133	4	ℏ	ℏ	PROPN
cana-4565	133	5	be	be	VERB
cana-4565	133	6	in	in	ADP
cana-4565	133	7	ℜ1	ℜ1	PROPN
cana-4565	133	8	,	,	PUNCT
cana-4565	133	9	𝜛𝑖	𝜛𝑖	ADP
cana-4565	133	10	∈	∈	PROPN
cana-4565	133	11	𝐷[0	𝐷[0	PROPN
cana-4565	133	12	,	,	PUNCT
cana-4565	133	13	1]𝑎𝑛𝑑	1]𝑎𝑛𝑑	NUM
cana-4565	133	14	𝜍𝑖	𝜍𝑖	ADP
cana-4565	133	15	∈	∈	PROPN
cana-4565	133	16	𝐷[−1	𝐷[−1	X
cana-4565	133	17	,	,	PUNCT
cana-4565	133	18	0	0	NUM
cana-4565	133	19	]	]	PUNCT
cana-4565	133	20	.	.	PUNCT
cana-4565	134	1	for	for	ADP
cana-4565	134	2	all	all	DET
cana-4565	134	3	i	i	PRON
cana-4565	134	4	=	=	NOUN
cana-4565	134	5	1	1	NUM
cana-4565	134	6	,	,	PUNCT
cana-4565	134	7	2	2	NUM
cana-4565	134	8	,	,	PUNCT
cana-4565	134	9	…	…	PUNCT
cana-4565	134	10	,	,	PUNCT
cana-4565	134	11	n	n	CCONJ
cana-4565	134	12	,	,	PUNCT
cana-4565	134	13	by	by	ADP
cana-4565	134	14	theorem	theorem	NOUN
cana-4565	134	15	2.20	2.20	NUM
cana-4565	134	16	,	,	PUNCT
cana-4565	134	17	𝔔(𝜛,𝜍)(𝔓	𝔔(𝜛,𝜍)(𝔓	ADV
cana-4565	134	18	)	)	PUNCT
cana-4565	134	19	is	be	AUX
cana-4565	134	20	a	a	DET
cana-4565	134	21	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	NOUN
cana-4565	134	22	of	of	ADP
cana-4565	134	23	ℜ1	ℜ1	PROPN
cana-4565	134	24	,	,	PUNCT
cana-4565	134	25	𝔔(𝜛,𝜍)(𝔓𝑖	𝔔(𝜛,𝜍)(𝔓𝑖	VERB
cana-4565	135	1	+	+	ADJ
cana-4565	136	1	)	)	PUNCT
cana-4565	136	2	(	(	PUNCT
cana-4565	136	3	𝜉ℏ	𝜉ℏ	NOUN
cana-4565	136	4	)	)	PUNCT
cana-4565	136	5	=	=	SYM
cana-4565	136	6	rmin{𝜛𝑖	rmin{𝜛𝑖	NOUN
cana-4565	136	7	,	,	PUNCT
cana-4565	136	8	𝔓𝑖	𝔓𝑖	PROPN
cana-4565	136	9	+	+	PROPN
cana-4565	136	10	(	(	PUNCT
cana-4565	136	11	𝜉ℏ	𝜉ℏ	NOUN
cana-4565	136	12	)	)	PUNCT
cana-4565	136	13	}	}	PUNCT
cana-4565	136	14	=	=	SYM
cana-4565	136	15	rmin{𝜛𝑖	rmin{𝜛𝑖	NOUN
cana-4565	136	16	,	,	PUNCT
cana-4565	136	17	𝔓𝑖	𝔓𝑖	PROPN
cana-4565	136	18	+	+	PROPN
cana-4565	136	19	(	(	PUNCT
cana-4565	136	20	ℏ𝜉)}=	ℏ𝜉)}=	NOUN
cana-4565	136	21	𝔔(𝜛,𝜍)(𝔓𝑖	𝔔(𝜛,𝜍)(𝔓𝑖	NOUN
cana-4565	136	22	+	+	NOUN
cana-4565	136	23	)	)	PUNCT
cana-4565	136	24	(	(	PUNCT
cana-4565	136	25	ℏ𝜉	ℏ𝜉	NOUN
cana-4565	136	26	)	)	PUNCT
cana-4565	136	27	,	,	PUNCT
cana-4565	136	28	for	for	ADP
cana-4565	136	29	all	all	DET
cana-4565	136	30	𝜉	𝜉	NOUN
cana-4565	136	31	,	,	PUNCT
cana-4565	136	32	ℏ	ℏ	PROPN
cana-4565	136	33	in	in	ADP
cana-4565	136	34	ℜ1	ℜ1	PROPN
cana-4565	136	35	.	.	PUNCT
cana-4565	137	1	and	and	CCONJ
cana-4565	137	2	𝔔(𝜛,𝜍)(𝔓𝑖	𝔔(𝜛,𝜍)(𝔓𝑖	VERB
cana-4565	137	3	−)(𝜉ℏ	−)(𝜉ℏ	NOUN
cana-4565	137	4	)	)	PUNCT
cana-4565	138	1	=	=	SYM
cana-4565	138	2	rmax{𝜍𝑖	rmax{𝜍𝑖	PROPN
cana-4565	138	3	,	,	PUNCT
cana-4565	138	4	𝔓𝑖	𝔓𝑖	PROPN
cana-4565	138	5	−(𝜉ℏ	−(𝜉ℏ	PROPN
cana-4565	138	6	)	)	PUNCT
cana-4565	138	7	}	}	PUNCT
cana-4565	138	8	=	=	SYM
cana-4565	138	9	rmax{𝜍𝑖	rmax{𝜍𝑖	PROPN
cana-4565	138	10	,	,	PUNCT
cana-4565	138	11	𝔓𝑖	𝔓𝑖	PROPN
cana-4565	138	12	−(ℏ𝜉)}=	−(ℏ𝜉)}=	PROPN
cana-4565	138	13	𝔔(𝜛,𝜍)(𝔓𝑖	𝔔(𝜛,𝜍)(𝔓𝑖	VERB
cana-4565	138	14	−)(ℏ𝜉	−)(ℏ𝜉	NOUN
cana-4565	138	15	)	)	PUNCT
cana-4565	138	16	,	,	PUNCT
cana-4565	138	17	for	for	ADP
cana-4565	138	18	all	all	DET
cana-4565	138	19	𝜉	𝜉	NOUN
cana-4565	138	20	,	,	PUNCT
cana-4565	138	21	ℏ	ℏ	PROPN
cana-4565	138	22	in	in	ADP
cana-4565	138	23	ℜ1	ℜ1	PROPN
cana-4565	138	24	.	.	PUNCT
cana-4565	139	1	hence	hence	ADV
cana-4565	139	2	𝔔(𝜛,𝜍)(𝔓	𝔔(𝜛,𝜍)(𝔓	ADV
cana-4565	139	3	)	)	PUNCT
cana-4565	139	4	is	be	AUX
cana-4565	139	5	a	a	DET
cana-4565	139	6	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	139	7	of	of	ADP
cana-4565	139	8	ℜ1	ℜ1	PROPN
cana-4565	139	9	.	.	PUNCT
cana-4565	140	1	corollary	corollary	ADJ
cana-4565	140	2	2.22	2.22	NUM
cana-4565	140	3	.	.	PUNCT
cana-4565	141	1	if	if	SCONJ
cana-4565	141	2	𝔓	𝔓	PROPN
cana-4565	141	3	and	and	CCONJ
cana-4565	141	4	𝔚	𝔚	PROPN
cana-4565	141	5	are	be	AUX
cana-4565	141	6	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	NOUN
cana-4565	141	7	𝑜𝑓	𝑜𝑓	ADP
cana-4565	141	8	𝑡ℎ𝑒	𝑡ℎ𝑒	NUM
cana-4565	141	9	𝑟𝑖𝑛𝑔	𝑟𝑖𝑛𝑔	NOUN
cana-4565	141	10	℧	℧	PROPN
cana-4565	141	11	1	1	NUM
cana-4565	141	12	,	,	PUNCT
cana-4565	141	13	then	then	ADV
cana-4565	141	14	𝔔(𝜛,𝜍)(𝔓	𝔔(𝜛,𝜍)(𝔓	ADV
cana-4565	141	15	∩	∩	ADJ
cana-4565	141	16	𝔚	𝔚	NOUN
cana-4565	141	17	)	)	PUNCT
cana-4565	141	18	is	be	AUX
cana-4565	141	19	a	a	DET
cana-4565	141	20	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	141	21	of	of	ADP
cana-4565	141	22	℧	℧	NOUN
cana-4565	141	23	1	1	NUM
cana-4565	141	24	.	.	PUNCT
cana-4565	142	1	proof	proof	NOUN
cana-4565	142	2	.	.	PUNCT
cana-4565	143	1	from	from	ADP
cana-4565	143	2	the	the	DET
cana-4565	143	3	above	above	ADJ
cana-4565	143	4	theorems	theorem	NOUN
cana-4565	143	5	,	,	PUNCT
cana-4565	143	6	it	it	PRON
cana-4565	143	7	is	be	AUX
cana-4565	143	8	trivial	trivial	ADJ
cana-4565	143	9	.	.	PUNCT
cana-4565	144	1	corollary	corollary	ADJ
cana-4565	144	2	2.23	2.23	NUM
cana-4565	144	3	.	.	PUNCT
cana-4565	145	1	if	if	SCONJ
cana-4565	145	2	𝔓	𝔓	PROPN
cana-4565	145	3	and	and	CCONJ
cana-4565	145	4	𝔚	𝔚	PROPN
cana-4565	145	5	are	be	AUX
cana-4565	145	6	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	NOUN
cana-4565	145	7	𝑜𝑓	𝑜𝑓	ADP
cana-4565	145	8	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	145	9	𝑟𝑖𝑛𝑔𝑠	𝑟𝑖𝑛𝑔𝑠	NOUN
cana-4565	145	10	℧	℧	PROPN
cana-4565	145	11	1and	1and	NUM
cana-4565	145	12	℧	℧	PROPN
cana-4565	145	13	2	2	NUM
cana-4565	145	14	,	,	PUNCT
cana-4565	145	15	then	then	ADV
cana-4565	145	16	𝔔(𝜛,𝜍)𝔓	𝔔(𝜛,𝜍)𝔓	PROPN
cana-4565	145	17	∩	∩	PROPN
cana-4565	145	18	𝔔(𝜛,𝜍)𝔚	𝔔(𝜛,𝜍)𝔚	PROPN
cana-4565	145	19	is	be	AUX
cana-4565	145	20	a	a	DET
cana-4565	145	21	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	145	22	of	of	ADP
cana-4565	145	23	℧	℧	NOUN
cana-4565	145	24	1	1	NUM
cana-4565	145	25	∩	∩	NOUN
cana-4565	145	26	℧	℧	PROPN
cana-4565	145	27	2	2	NUM
cana-4565	145	28	.	.	PUNCT
cana-4565	146	1	proof	proof	NOUN
cana-4565	146	2	.	.	PUNCT
cana-4565	147	1	from	from	ADP
cana-4565	147	2	the	the	DET
cana-4565	147	3	above	above	ADJ
cana-4565	147	4	theorems	theorem	NOUN
cana-4565	147	5	,	,	PUNCT
cana-4565	147	6	it	it	PRON
cana-4565	147	7	is	be	AUX
cana-4565	147	8	trivial	trivial	ADJ
cana-4565	147	9	.	.	PUNCT
cana-4565	148	1	corollary	corollary	ADJ
cana-4565	148	2	2.24	2.24	NUM
cana-4565	148	3	.	.	PUNCT
cana-4565	149	1	if	if	SCONJ
cana-4565	149	2	𝔓	𝔓	PROPN
cana-4565	149	3	and	and	CCONJ
cana-4565	149	4	𝔚	𝔚	PROPN
cana-4565	149	5	are	be	AUX
cana-4565	149	6	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	NOUN
cana-4565	149	7	𝑜𝑓	𝑜𝑓	ADP
cana-4565	149	8	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	149	9	𝑟𝑖𝑛𝑔𝑠	𝑟𝑖𝑛𝑔𝑠	NOUN
cana-4565	149	10	℧	℧	NOUN
cana-4565	149	11	1	1	NUM
cana-4565	149	12	,	,	PUNCT
cana-4565	149	13	then	then	ADV
cana-4565	149	14	𝔔(𝜛,𝜍)𝔓	𝔔(𝜛,𝜍)𝔓	PROPN
cana-4565	149	15	∩	∩	PROPN
cana-4565	149	16	𝔔(𝜛,𝜍)𝔚	𝔔(𝜛,𝜍)𝔚	PROPN
cana-4565	149	17	is	be	AUX
cana-4565	149	18	a	a	DET
cana-4565	149	19	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	149	20	of	of	ADP
cana-4565	149	21	℧	℧	NOUN
cana-4565	149	22	1	1	NUM
cana-4565	149	23	.	.	PUNCT
cana-4565	149	24	proof	proof	NOUN
cana-4565	149	25	.	.	PUNCT
cana-4565	150	1	from	from	ADP
cana-4565	150	2	the	the	DET
cana-4565	150	3	above	above	ADJ
cana-4565	150	4	theorems	theorem	NOUN
cana-4565	150	5	,	,	PUNCT
cana-4565	150	6	it	it	PRON
cana-4565	150	7	is	be	AUX
cana-4565	150	8	trivial	trivial	ADJ
cana-4565	150	9	.	.	PUNCT
cana-4565	151	1	theorem	theorem	VERB
cana-4565	151	2	2.25	2.25	NUM
cana-4565	151	3	.	.	PUNCT
cana-4565	152	1	𝐼𝑓	𝐼𝑓	PROPN
cana-4565	152	2	𝔓1	𝔓1	PROPN
cana-4565	152	3	,	,	PUNCT
cana-4565	152	4	𝔓2	𝔓2	NOUN
cana-4565	152	5	,	,	PUNCT
cana-4565	152	6	…	…	PUNCT
cana-4565	152	7	,	,	PUNCT
cana-4565	152	8	𝔓𝑚𝑎𝑟𝑒	𝔓𝑚𝑎𝑟𝑒	PROPN
cana-4565	152	9	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	152	10	of	of	ADP
cana-4565	152	11	the	the	DET
cana-4565	152	12	rings	ring	NOUN
cana-4565	152	13	℧	℧	PROPN
cana-4565	152	14	1	1	NUM
cana-4565	152	15	,	,	PUNCT
cana-4565	152	16	℧	℧	NOUN
cana-4565	152	17	2	2	NUM
cana-4565	152	18	,	,	PUNCT
cana-4565	152	19	…	…	PUNCT
cana-4565	152	20	,	,	PUNCT
cana-4565	152	21	℧	℧	NOUN
cana-4565	152	22	m	m	VERB
cana-4565	152	23	respectively	respectively	ADV
cana-4565	152	24	,	,	PUNCT
cana-4565	152	25	then	then	ADV
cana-4565	152	26	𝔔(𝜛,𝜍)(𝔓1	𝔔(𝜛,𝜍)(𝔓1	PROPN
cana-4565	152	27	∩	∩	PROPN
cana-4565	152	28	𝔓2	𝔓2	NOUN
cana-4565	152	29	∩	∩	NOUN
cana-4565	152	30	…	…	PUNCT
cana-4565	152	31	∩	∩	ADJ
cana-4565	152	32	𝔓𝑚	𝔓𝑚	NOUN
cana-4565	152	33	)	)	PUNCT
cana-4565	152	34	is	be	AUX
cana-4565	152	35	a	a	DET
cana-4565	152	36	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	152	37	of	of	ADP
cana-4565	152	38	the	the	DET
cana-4565	152	39	ring	ring	NOUN
cana-4565	152	40	℧	℧	NOUN
cana-4565	152	41	1	1	NUM
cana-4565	152	42	∩	∩	X
cana-4565	152	43	℧	℧	NOUN
cana-4565	152	44	2	2	NUM
cana-4565	152	45	∩	∩	NOUN
cana-4565	152	46	…	…	PUNCT
cana-4565	152	47	∩	∩	ADJ
cana-4565	152	48	℧	℧	NOUN
cana-4565	152	49	m.	m.	NOUN
cana-4565	152	50	proof	proof	NOUN
cana-4565	152	51	.	.	PUNCT
cana-4565	153	1	from	from	ADP
cana-4565	153	2	the	the	DET
cana-4565	153	3	above	above	ADJ
cana-4565	153	4	theorems	theorem	NOUN
cana-4565	153	5	,	,	PUNCT
cana-4565	153	6	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	153	7	𝑝𝑟𝑜𝑜𝑓	𝑝𝑟𝑜𝑜𝑓	NOUN
cana-4565	153	8	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	153	9	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	NOUN
cana-4565	153	10	corollary	corollary	NOUN
cana-4565	153	11	2.26	2.26	NUM
cana-4565	153	12	.	.	PUNCT
cana-4565	154	1	𝐼𝑓	𝐼𝑓	PROPN
cana-4565	154	2	𝔓1	𝔓1	PROPN
cana-4565	154	3	,	,	PUNCT
cana-4565	154	4	𝔓2	𝔓2	NOUN
cana-4565	154	5	,	,	PUNCT
cana-4565	154	6	…	…	PUNCT
cana-4565	154	7	,	,	PUNCT
cana-4565	154	8	𝔓𝑚	𝔓𝑚	PROPN
cana-4565	154	9	𝑎𝑟𝑒	𝑎𝑟𝑒	VERB
cana-4565	154	10	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	154	11	of	of	ADP
cana-4565	154	12	the	the	DET
cana-4565	154	13	rings	ring	NOUN
cana-4565	154	14	℧	℧	PROPN
cana-4565	154	15	1	1	NUM
cana-4565	154	16	,	,	PUNCT
cana-4565	154	17	℧	℧	NOUN
cana-4565	154	18	2	2	NUM
cana-4565	154	19	,	,	PUNCT
cana-4565	154	20	…	…	PUNCT
cana-4565	154	21	,	,	PUNCT
cana-4565	154	22	℧	℧	NOUN
cana-4565	154	23	m	m	VERB
cana-4565	154	24	respectively	respectively	ADV
cana-4565	154	25	,	,	PUNCT
cana-4565	154	26	then	then	ADV
cana-4565	154	27	𝔔(𝜛,𝜍)𝔓1	𝔔(𝜛,𝜍)𝔓1	PROPN
cana-4565	154	28	∩	∩	PROPN
cana-4565	154	29	𝔔(𝜛,𝜍)𝔓2	𝔔(𝜛,𝜍)𝔓2	NOUN
cana-4565	154	30	∩	∩	NOUN
cana-4565	154	31	…	…	PUNCT
cana-4565	154	32	∩	∩	NOUN
cana-4565	154	33	𝔔(𝜛,𝜍)𝔓𝑚	𝔔(𝜛,𝜍)𝔓𝑚	NOUN
cana-4565	154	34	is	be	AUX
cana-4565	154	35	a	a	DET
cana-4565	154	36	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	154	37	of	of	ADP
cana-4565	154	38	the	the	DET
cana-4565	154	39	ring	ring	NOUN
cana-4565	154	40	℧	℧	NOUN
cana-4565	154	41	1	1	NUM
cana-4565	154	42	∩	∩	X
cana-4565	154	43	℧	℧	NOUN
cana-4565	154	44	2	2	NUM
cana-4565	154	45	∩	∩	NOUN
cana-4565	154	46	…	…	PUNCT
cana-4565	154	47	∩	∩	ADJ
cana-4565	154	48	℧	℧	NOUN
cana-4565	154	49	m.	m.	NOUN
cana-4565	154	50	proof	proof	NOUN
cana-4565	154	51	.	.	PUNCT
cana-4565	155	1	from	from	ADP
cana-4565	155	2	the	the	DET
cana-4565	155	3	above	above	ADJ
cana-4565	155	4	theorems	theorem	NOUN
cana-4565	155	5	,	,	PUNCT
cana-4565	155	6	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	155	7	𝑝𝑟𝑜𝑜𝑓	𝑝𝑟𝑜𝑜𝑓	NOUN
cana-4565	155	8	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	155	9	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	PROPN
cana-4565	155	10	corollary	corollary	NOUN
cana-4565	155	11	2.27	2.27	NUM
cana-4565	155	12	.	.	PUNCT
cana-4565	156	1	𝐼𝑓	𝐼𝑓	PROPN
cana-4565	156	2	𝔓1	𝔓1	PROPN
cana-4565	156	3	,	,	PUNCT
cana-4565	156	4	𝔓2	𝔓2	NOUN
cana-4565	156	5	,	,	PUNCT
cana-4565	156	6	…	…	PUNCT
cana-4565	156	7	,	,	PUNCT
cana-4565	156	8	𝔓𝑚	𝔓𝑚	PROPN
cana-4565	156	9	are	be	AUX
cana-4565	156	10	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	156	11	of	of	ADP
cana-4565	156	12	the	the	DET
cana-4565	156	13	ring	ring	NOUN
cana-4565	156	14	℧	℧	PROPN
cana-4565	156	15	1	1	NUM
cana-4565	156	16	,	,	PUNCT
cana-4565	156	17	then	then	ADV
cana-4565	156	18	𝔔(𝜛,𝜍)(𝔓1	𝔔(𝜛,𝜍)(𝔓1	PROPN
cana-4565	156	19	∩	∩	PROPN
cana-4565	156	20	𝔓2	𝔓2	NOUN
cana-4565	156	21	∩	∩	NOUN
cana-4565	156	22	…	…	PUNCT
cana-4565	156	23	∩	∩	ADJ
cana-4565	156	24	𝔓𝑚	𝔓𝑚	NOUN
cana-4565	156	25	)	)	PUNCT
cana-4565	156	26	is	be	AUX
cana-4565	156	27	a	a	DET
cana-4565	156	28	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	156	29	𝑜𝑓	𝑜𝑓	ADP
cana-4565	156	30	℧	℧	PROPN
cana-4565	156	31	1	1	NUM
cana-4565	156	32	.	.	PUNCT
cana-4565	156	33	proof	proof	NOUN
cana-4565	156	34	.	.	PUNCT
cana-4565	157	1	from	from	ADP
cana-4565	157	2	the	the	DET
cana-4565	157	3	above	above	ADJ
cana-4565	157	4	theorems	theorem	NOUN
cana-4565	157	5	,	,	PUNCT
cana-4565	157	6	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	157	7	𝑝𝑟𝑜𝑜𝑓	𝑝𝑟𝑜𝑜𝑓	NOUN
cana-4565	157	8	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	157	9	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	PROPN
cana-4565	157	10	corollary	corollary	NOUN
cana-4565	157	11	2.28	2.28	NUM
cana-4565	157	12	.	.	PUNCT
cana-4565	158	1	𝐼𝑓	𝐼𝑓	PROPN
cana-4565	158	2	𝔓1	𝔓1	PROPN
cana-4565	158	3	,	,	PUNCT
cana-4565	158	4	𝔓2	𝔓2	NOUN
cana-4565	158	5	,	,	PUNCT
cana-4565	158	6	…	…	PUNCT
cana-4565	158	7	,	,	PUNCT
cana-4565	158	8	𝔓𝑚	𝔓𝑚	PROPN
cana-4565	158	9	are	be	AUX
cana-4565	158	10	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	158	11	of	of	ADP
cana-4565	158	12	the	the	DET
cana-4565	158	13	ring	ring	NOUN
cana-4565	158	14	℧	℧	PROPN
cana-4565	158	15	1	1	NUM
cana-4565	158	16	,	,	PUNCT
cana-4565	158	17	then	then	ADV
cana-4565	158	18	𝔔(𝜛,𝜍)𝔓1	𝔔(𝜛,𝜍)𝔓1	PROPN
cana-4565	158	19	∩	∩	PROPN
cana-4565	158	20	𝔔(𝜛,𝜍)𝔓2	𝔔(𝜛,𝜍)𝔓2	NOUN
cana-4565	158	21	∩	∩	NOUN
cana-4565	158	22	…	…	PUNCT
cana-4565	158	23	∩	∩	NOUN
cana-4565	158	24	𝔔(𝜛,𝜍)𝔓𝑚	𝔔(𝜛,𝜍)𝔓𝑚	NOUN
cana-4565	158	25	is	be	AUX
cana-4565	158	26	a	a	DET
cana-4565	158	27	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	158	28	of	of	ADP
cana-4565	158	29	℧	℧	PROPN
cana-4565	158	30	1	1	NUM
cana-4565	158	31	.	.	PUNCT
cana-4565	158	32	communications	communication	NOUN
cana-4565	158	33	on	on	ADP
cana-4565	158	34	applied	apply	VERB
cana-4565	158	35	nonlinear	nonlinear	ADJ
cana-4565	158	36	analysis	analysis	NOUN
cana-4565	158	37	issn	issn	NOUN
cana-4565	158	38	:	:	PUNCT
cana-4565	158	39	1074	1074	NUM
cana-4565	158	40	-	-	PUNCT
cana-4565	158	41	133x	133x	NUM
cana-4565	158	42	vol	vol	NOUN
cana-4565	158	43	32	32	NUM
cana-4565	158	44	no	no	NOUN
cana-4565	158	45	.	.	PUNCT
cana-4565	159	1	9s	9s	NUM
cana-4565	159	2	(	(	PUNCT
cana-4565	159	3	2025	2025	NUM
cana-4565	159	4	)	)	PUNCT
cana-4565	159	5	2620	2620	NUM
cana-4565	160	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-4565	160	2	proof	proof	NOUN
cana-4565	160	3	.	.	PUNCT
cana-4565	161	1	from	from	ADP
cana-4565	161	2	the	the	DET
cana-4565	161	3	above	above	ADJ
cana-4565	161	4	theorems	theorem	NOUN
cana-4565	161	5	,	,	PUNCT
cana-4565	161	6	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	161	7	𝑝𝑟𝑜𝑜𝑓	𝑝𝑟𝑜𝑜𝑓	NOUN
cana-4565	161	8	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	161	9	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	NOUN
cana-4565	161	10	theorem	theorem	VERB
cana-4565	161	11	2.29	2.29	NUM
cana-4565	161	12	.	.	PUNCT
cana-4565	162	1	𝐼𝑓	𝐼𝑓	ADJ
cana-4565	162	2	℔	℔	NOUN
cana-4565	162	3	=	=	NOUN
cana-4565	162	4			PUNCT
cana-4565	162	5	℔	℔	X
cana-4565	162	6	1	1	NUM
cana-4565	162	7	+	+	ADJ
cana-4565	162	8	,	,	PUNCT
cana-4565	162	9	℔	℔	X
cana-4565	162	10	2	2	NUM
cana-4565	162	11	+	+	ADJ
cana-4565	162	12	,	,	PUNCT
cana-4565	162	13	…	…	PUNCT
cana-4565	162	14	,	,	PUNCT
cana-4565	162	15	℔	℔	NOUN
cana-4565	162	16	𝑛	𝑛	PRON
cana-4565	162	17	+	+	ADJ
cana-4565	162	18	,	,	PUNCT
cana-4565	162	19	℔	℔	NOUN
cana-4565	162	20	1	1	NUM
cana-4565	162	21	−	−	NOUN
cana-4565	162	22	,	,	PUNCT
cana-4565	162	23	℔	℔	NOUN
cana-4565	162	24	2	2	NUM
cana-4565	162	25	−	−	NOUN
cana-4565	162	26	,	,	PUNCT
cana-4565	162	27	…	…	PUNCT
cana-4565	162	28	,	,	PUNCT
cana-4565	162	29	℔	℔	ADP
cana-4565	162	30	𝑛	𝑛	DET
cana-4565	162	31	−	−	PROPN
cana-4565	162	32	is	be	AUX
cana-4565	162	33	𝑎	𝑎	DET
cana-4565	162	34	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	NOUN
cana-4565	162	35	of	of	ADP
cana-4565	162	36	a	a	DET
cana-4565	162	37	ring	ring	NOUN
cana-4565	162	38	ṏ1	ṏ1	PROPN
cana-4565	162	39	,	,	PUNCT
cana-4565	162	40	then	then	ADV
cana-4565	162	41	ℜ(𝜛,𝜍	ℜ(𝜛,𝜍	NOUN
cana-4565	162	42	)	)	PUNCT
cana-4565	162	43	(	(	PUNCT
cana-4565	162	44	℔	℔	NOUN
cana-4565	162	45	)	)	PUNCT
cana-4565	162	46	is	be	AUX
cana-4565	162	47	a	a	DET
cana-4565	162	48	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	NOUN
cana-4565	162	49	of	of	ADP
cana-4565	162	50	ṏ1	ṏ1	NOUN
cana-4565	162	51	,	,	PUNCT
cana-4565	162	52	where	where	SCONJ
cana-4565	162	53	𝜛	𝜛	X
cana-4565	162	54	=	=	PRON
cana-4565	162	55	(	(	PUNCT
cana-4565	162	56	𝜛1	𝜛1	NOUN
cana-4565	162	57	,	,	PUNCT
cana-4565	162	58	𝜛2	𝜛2	NOUN
cana-4565	162	59	,	,	PUNCT
cana-4565	162	60	…	…	PUNCT
cana-4565	162	61	,	,	PUNCT
cana-4565	162	62	𝜛𝑛	𝜛𝑛	NOUN
cana-4565	162	63	)	)	PUNCT
cana-4565	162	64	and	and	CCONJ
cana-4565	162	65	𝜍	𝜍	X
cana-4565	162	66	=	=	SYM
cana-4565	162	67	(	(	PUNCT
cana-4565	162	68	𝜍1	𝜍1	PROPN
cana-4565	162	69	,	,	PUNCT
cana-4565	162	70	𝜍2	𝜍2	NOUN
cana-4565	162	71	,	,	PUNCT
cana-4565	162	72	…	…	PUNCT
cana-4565	162	73	,	,	PUNCT
cana-4565	162	74	𝜍𝑛	𝜍𝑛	NOUN
cana-4565	162	75	)	)	PUNCT
cana-4565	162	76	,	,	PUNCT
cana-4565	162	77	𝜛𝑖	𝜛𝑖	NOUN
cana-4565	162	78	∈	∈	PROPN
cana-4565	162	79	𝐷[0	𝐷[0	PROPN
cana-4565	162	80	,	,	PUNCT
cana-4565	162	81	1	1	NUM
cana-4565	162	82	]	]	X
cana-4565	162	83	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-4565	162	84	𝜍𝑖	𝜍𝑖	PROPN
cana-4565	162	85	∈	∈	PROPN
cana-4565	162	86	𝐷[−1	𝐷[−1	NOUN
cana-4565	162	87	,	,	PUNCT
cana-4565	162	88	0	0	NUM
cana-4565	162	89	]	]	PUNCT
cana-4565	162	90	.	.	PUNCT
cana-4565	163	1	theorem	theorem	VERB
cana-4565	163	2	2.30	2.30	NUM
cana-4565	163	3	.	.	PUNCT
cana-4565	164	1	𝐼𝑓	𝐼𝑓	VERB
cana-4565	164	2	ℭ	ℭ	NOUN
cana-4565	164	3	=	=	PUNCT
cana-4565	164	4			NOUN
cana-4565	164	5	ℭ1	ℭ1	INTJ
cana-4565	164	6	+	+	ADV
cana-4565	164	7	,	,	PUNCT
cana-4565	164	8	ℭ2	ℭ2	PROPN
cana-4565	164	9	+	+	PROPN
cana-4565	164	10	,	,	PUNCT
cana-4565	164	11	…	…	PUNCT
cana-4565	164	12	,	,	PUNCT
cana-4565	165	1	ℭ𝑛	ℭ𝑛	ADP
cana-4565	165	2	+	+	ADJ
cana-4565	165	3	,	,	PUNCT
cana-4565	165	4	ℭ1	ℭ1	ADP
cana-4565	165	5	−	−	PROPN
cana-4565	165	6	,	,	PUNCT
cana-4565	165	7	ℭ2	ℭ2	PROPN
cana-4565	165	8	−	−	PROPN
cana-4565	165	9	,	,	PUNCT
cana-4565	165	10	…	…	PUNCT
cana-4565	165	11	,	,	PUNCT
cana-4565	165	12	ℭ𝑛	ℭ𝑛	SCONJ
cana-4565	165	13	−	−	PROPN
cana-4565	165	14	is	be	AUX
cana-4565	165	15	𝑎	𝑎	DET
cana-4565	165	16	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	165	17	of	of	ADP
cana-4565	165	18	a	a	DET
cana-4565	165	19	ring	ring	NOUN
cana-4565	165	20	ℝ1	ℝ1	PROPN
cana-4565	165	21	,	,	PUNCT
cana-4565	165	22	then	then	ADV
cana-4565	165	23	ℜ(𝜛,𝜍)(ℭ	ℜ(𝜛,𝜍)(ℭ	PROPN
cana-4565	165	24	)	)	PUNCT
cana-4565	165	25	is	be	AUX
cana-4565	165	26	a	a	DET
cana-4565	165	27	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	165	28	of	of	ADP
cana-4565	165	29	ℝ1	ℝ1	PROPN
cana-4565	165	30	,	,	PUNCT
cana-4565	165	31	where	where	SCONJ
cana-4565	165	32	𝜛	𝜛	X
cana-4565	165	33	=	=	PRON
cana-4565	165	34	(	(	PUNCT
cana-4565	165	35	𝜛1	𝜛1	NOUN
cana-4565	165	36	,	,	PUNCT
cana-4565	165	37	𝜛2	𝜛2	NOUN
cana-4565	165	38	,	,	PUNCT
cana-4565	165	39	…	…	PUNCT
cana-4565	165	40	,	,	PUNCT
cana-4565	165	41	𝜛𝑛	𝜛𝑛	NOUN
cana-4565	165	42	)	)	PUNCT
cana-4565	165	43	and	and	CCONJ
cana-4565	165	44	𝜍	𝜍	X
cana-4565	165	45	=	=	SYM
cana-4565	165	46	(	(	PUNCT
cana-4565	165	47	𝜍1	𝜍1	PROPN
cana-4565	165	48	,	,	PUNCT
cana-4565	165	49	𝜍2	𝜍2	NOUN
cana-4565	165	50	,	,	PUNCT
cana-4565	165	51	…	…	PUNCT
cana-4565	165	52	,	,	PUNCT
cana-4565	165	53	𝜍𝑛	𝜍𝑛	NOUN
cana-4565	165	54	)	)	PUNCT
cana-4565	165	55	,	,	PUNCT
cana-4565	165	56	𝜛𝑖	𝜛𝑖	NOUN
cana-4565	165	57	∈	∈	PROPN
cana-4565	165	58	𝐷[0	𝐷[0	PROPN
cana-4565	165	59	,	,	PUNCT
cana-4565	165	60	1	1	NUM
cana-4565	165	61	]	]	X
cana-4565	165	62	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-4565	165	63	𝜍𝑖	𝜍𝑖	PROPN
cana-4565	165	64	∈	∈	PROPN
cana-4565	165	65	𝐷[−1	𝐷[−1	NOUN
cana-4565	165	66	,	,	PUNCT
cana-4565	165	67	0	0	NUM
cana-4565	165	68	]	]	PUNCT
cana-4565	165	69	.	.	PUNCT
cana-4565	166	1	proof	proof	NOUN
cana-4565	166	2	.	.	PUNCT
cana-4565	167	1	let	let	VERB
cana-4565	167	2	𝜉	𝜉	X
cana-4565	167	3	,	,	PUNCT
cana-4565	167	4	ℏ	ℏ	PROPN
cana-4565	167	5	be	be	VERB
cana-4565	167	6	in	in	ADP
cana-4565	167	7	ℝ1	ℝ1	PROPN
cana-4565	167	8	,	,	PUNCT
cana-4565	167	9	𝜛𝑖	𝜛𝑖	ADP
cana-4565	167	10	∈	∈	PROPN
cana-4565	167	11	𝐷[0	𝐷[0	PROPN
cana-4565	167	12	,	,	PUNCT
cana-4565	167	13	1]𝑎𝑛𝑑	1]𝑎𝑛𝑑	NUM
cana-4565	167	14	𝜍𝑖	𝜍𝑖	ADP
cana-4565	167	15	∈	∈	PROPN
cana-4565	167	16	𝐷[−1	𝐷[−1	X
cana-4565	167	17	,	,	PUNCT
cana-4565	167	18	0	0	NUM
cana-4565	167	19	]	]	PUNCT
cana-4565	167	20	.	.	PUNCT
cana-4565	168	1	for	for	ADP
cana-4565	168	2	all	all	DET
cana-4565	168	3	i	i	PRON
cana-4565	168	4	=	=	NOUN
cana-4565	168	5	1	1	NUM
cana-4565	168	6	,	,	PUNCT
cana-4565	168	7	2	2	NUM
cana-4565	168	8	,	,	PUNCT
cana-4565	168	9	…	…	PUNCT
cana-4565	168	10	,	,	PUNCT
cana-4565	168	11	n	n	CCONJ
cana-4565	168	12	,	,	PUNCT
cana-4565	168	13	by	by	ADP
cana-4565	168	14	theorem	theorem	NOUN
cana-4565	168	15	2.29	2.29	NUM
cana-4565	168	16	,	,	PUNCT
cana-4565	168	17	ℜ(𝜛,𝜍)(ℭ	ℜ(𝜛,𝜍)(ℭ	PROPN
cana-4565	168	18	)	)	PUNCT
cana-4565	168	19	is	be	AUX
cana-4565	168	20	a	a	DET
cana-4565	168	21	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	NOUN
cana-4565	168	22	of	of	ADP
cana-4565	168	23	ℝ1	ℝ1	PROPN
cana-4565	168	24	,	,	PUNCT
cana-4565	168	25	ℜ(𝜛,𝜍)(ℭ𝑖	ℜ(𝜛,𝜍)(ℭ𝑖	X
cana-4565	168	26	+	+	PROPN
cana-4565	168	27	)	)	PUNCT
cana-4565	168	28	(	(	PUNCT
cana-4565	168	29	𝜉ℏ	𝜉ℏ	NOUN
cana-4565	168	30	)	)	PUNCT
cana-4565	168	31	=	=	PUNCT
cana-4565	168	32	rmax{𝜛𝑖	rmax{𝜛𝑖	NOUN
cana-4565	168	33	,	,	PUNCT
cana-4565	168	34	ℭ𝑖	ℭ𝑖	PROPN
cana-4565	168	35	+	+	PROPN
cana-4565	168	36	(	(	PUNCT
cana-4565	168	37	𝜉ℏ	𝜉ℏ	NOUN
cana-4565	168	38	)	)	PUNCT
cana-4565	168	39	}	}	PUNCT
cana-4565	168	40	=	=	PUNCT
cana-4565	168	41	rmax{𝜛𝑖	rmax{𝜛𝑖	NOUN
cana-4565	168	42	,	,	PUNCT
cana-4565	168	43	ℭ𝑖	ℭ𝑖	PROPN
cana-4565	168	44	+	+	PROPN
cana-4565	168	45	(	(	PUNCT
cana-4565	168	46	ℏ𝜉)}=	ℏ𝜉)}=	PROPN
cana-4565	168	47	ℜ(𝜛,𝜍)(ℭ𝑖	ℜ(𝜛,𝜍)(ℭ𝑖	PROPN
cana-4565	168	48	+	+	PROPN
cana-4565	168	49	)	)	PUNCT
cana-4565	168	50	(	(	PUNCT
cana-4565	168	51	ℏ𝜉	ℏ𝜉	NOUN
cana-4565	168	52	)	)	PUNCT
cana-4565	168	53	,	,	PUNCT
cana-4565	168	54	for	for	ADP
cana-4565	168	55	all	all	DET
cana-4565	168	56	𝜉	𝜉	NOUN
cana-4565	168	57	,	,	PUNCT
cana-4565	168	58	ℏ	ℏ	PROPN
cana-4565	168	59	in	in	ADP
cana-4565	168	60	ℝ1	ℝ1	PROPN
cana-4565	168	61	.	.	PUNCT
cana-4565	169	1	and	and	CCONJ
cana-4565	169	2	ℜ(𝜛,𝜍)(ℭ𝑖	ℜ(𝜛,𝜍)(ℭ𝑖	PROPN
cana-4565	169	3	−)(𝜉ℏ	−)(𝜉ℏ	PROPN
cana-4565	169	4	)	)	PUNCT
cana-4565	170	1	=	=	SYM
cana-4565	170	2	rmin{𝜍𝑖	rmin{𝜍𝑖	NUM
cana-4565	170	3	,	,	PUNCT
cana-4565	170	4	ℭ𝑖	ℭ𝑖	PROPN
cana-4565	170	5	−(𝜉ℏ	−(𝜉ℏ	PROPN
cana-4565	170	6	)	)	PUNCT
cana-4565	170	7	}	}	PUNCT
cana-4565	170	8	=	=	SYM
cana-4565	170	9	rmin{𝜍𝑖	rmin{𝜍𝑖	NUM
cana-4565	170	10	,	,	PUNCT
cana-4565	170	11	ℭ𝑖	ℭ𝑖	PROPN
cana-4565	170	12	−(ℏ𝜉)}=	−(ℏ𝜉)}=	PROPN
cana-4565	170	13	ℜ(𝜛,𝜍)(ℭ𝑖	ℜ(𝜛,𝜍)(ℭ𝑖	NOUN
cana-4565	170	14	−)(ℏ𝜉	−)(ℏ𝜉	NOUN
cana-4565	170	15	)	)	PUNCT
cana-4565	170	16	,	,	PUNCT
cana-4565	170	17	for	for	ADP
cana-4565	170	18	all	all	DET
cana-4565	170	19	𝜉	𝜉	NOUN
cana-4565	170	20	,	,	PUNCT
cana-4565	170	21	ℏ	ℏ	PROPN
cana-4565	170	22	in	in	ADP
cana-4565	170	23	ℝ1	ℝ1	PROPN
cana-4565	170	24	.	.	PUNCT
cana-4565	171	1	hence	hence	ADV
cana-4565	171	2	ℜ(𝜛,𝜍)(ℭ	ℜ(𝜛,𝜍)(ℭ	PROPN
cana-4565	171	3	)	)	PUNCT
cana-4565	171	4	is	be	AUX
cana-4565	171	5	a	a	DET
cana-4565	171	6	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	171	7	of	of	ADP
cana-4565	171	8	ℝ1	ℝ1	PROPN
cana-4565	171	9	.	.	PUNCT
cana-4565	172	1	corollary	corollary	ADJ
cana-4565	172	2	2.31	2.31	NUM
cana-4565	172	3	.	.	PUNCT
cana-4565	173	1	if	if	SCONJ
cana-4565	173	2	𝔓	𝔓	PROPN
cana-4565	173	3	and	and	CCONJ
cana-4565	173	4	𝔚	𝔚	PROPN
cana-4565	173	5	are	be	AUX
cana-4565	173	6	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	NOUN
cana-4565	173	7	𝑜𝑓	𝑜𝑓	ADP
cana-4565	173	8	𝑡ℎ𝑒	𝑡ℎ𝑒	NUM
cana-4565	173	9	𝑟𝑖𝑛𝑔	𝑟𝑖𝑛𝑔	NOUN
cana-4565	173	10	℧	℧	PROPN
cana-4565	173	11	1	1	NUM
cana-4565	173	12	,	,	PUNCT
cana-4565	173	13	then	then	ADV
cana-4565	173	14	ℜ(𝜛,𝜍)(𝔓	ℜ(𝜛,𝜍)(𝔓	ADV
cana-4565	173	15	∩	∩	ADJ
cana-4565	173	16	𝔚	𝔚	NOUN
cana-4565	173	17	)	)	PUNCT
cana-4565	173	18	is	be	AUX
cana-4565	173	19	a	a	DET
cana-4565	173	20	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	173	21	of	of	ADP
cana-4565	173	22	℧	℧	NOUN
cana-4565	173	23	1	1	NUM
cana-4565	173	24	.	.	PUNCT
cana-4565	174	1	proof	proof	NOUN
cana-4565	174	2	.	.	PUNCT
cana-4565	175	1	from	from	ADP
cana-4565	175	2	the	the	DET
cana-4565	175	3	above	above	ADJ
cana-4565	175	4	theorems	theorem	NOUN
cana-4565	175	5	,	,	PUNCT
cana-4565	175	6	it	it	PRON
cana-4565	175	7	is	be	AUX
cana-4565	175	8	trivial	trivial	ADJ
cana-4565	175	9	.	.	PUNCT
cana-4565	176	1	corollary	corollary	ADJ
cana-4565	176	2	2.32	2.32	NUM
cana-4565	176	3	.	.	PUNCT
cana-4565	177	1	if	if	SCONJ
cana-4565	177	2	𝔓	𝔓	PROPN
cana-4565	177	3	and	and	CCONJ
cana-4565	177	4	𝔚	𝔚	PROPN
cana-4565	177	5	are	be	AUX
cana-4565	177	6	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	NOUN
cana-4565	177	7	𝑜𝑓	𝑜𝑓	ADP
cana-4565	177	8	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	177	9	𝑟𝑖𝑛𝑔𝑠	𝑟𝑖𝑛𝑔𝑠	NOUN
cana-4565	177	10	℧	℧	PROPN
cana-4565	177	11	1and	1and	NUM
cana-4565	177	12	℧	℧	PROPN
cana-4565	177	13	2	2	NUM
cana-4565	177	14	,	,	PUNCT
cana-4565	177	15	then	then	ADV
cana-4565	177	16	ℜ(𝜛,𝜍)𝔓	ℜ(𝜛,𝜍)𝔓	NOUN
cana-4565	177	17	∩	∩	NOUN
cana-4565	177	18	ℜ(𝜛,𝜍)𝔚	ℜ(𝜛,𝜍)𝔚	PROPN
cana-4565	177	19	is	be	AUX
cana-4565	177	20	a	a	DET
cana-4565	177	21	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	177	22	of	of	ADP
cana-4565	177	23	℧	℧	NOUN
cana-4565	177	24	1	1	NUM
cana-4565	177	25	∩	∩	NOUN
cana-4565	177	26	℧	℧	PROPN
cana-4565	177	27	2	2	NUM
cana-4565	177	28	.	.	PUNCT
cana-4565	178	1	proof	proof	NOUN
cana-4565	178	2	.	.	PUNCT
cana-4565	179	1	from	from	ADP
cana-4565	179	2	the	the	DET
cana-4565	179	3	above	above	ADJ
cana-4565	179	4	theorems	theorem	NOUN
cana-4565	179	5	,	,	PUNCT
cana-4565	179	6	it	it	PRON
cana-4565	179	7	is	be	AUX
cana-4565	179	8	trivial	trivial	ADJ
cana-4565	179	9	.	.	PUNCT
cana-4565	180	1	corollary	corollary	ADJ
cana-4565	180	2	2.33	2.33	NUM
cana-4565	180	3	.	.	PUNCT
cana-4565	181	1	if	if	SCONJ
cana-4565	181	2	𝔓	𝔓	PROPN
cana-4565	181	3	and	and	CCONJ
cana-4565	181	4	𝔚	𝔚	PROPN
cana-4565	181	5	are	be	AUX
cana-4565	181	6	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	NOUN
cana-4565	181	7	𝑜𝑓	𝑜𝑓	ADP
cana-4565	181	8	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	181	9	𝑟𝑖𝑛𝑔𝑠	𝑟𝑖𝑛𝑔𝑠	NOUN
cana-4565	181	10	℧	℧	NOUN
cana-4565	181	11	1	1	NUM
cana-4565	181	12	,	,	PUNCT
cana-4565	181	13	then	then	ADV
cana-4565	181	14	ℜ(𝜛,𝜍)𝔓	ℜ(𝜛,𝜍)𝔓	NOUN
cana-4565	181	15	∩	∩	NOUN
cana-4565	181	16	ℜ(𝜛,𝜍)𝔚	ℜ(𝜛,𝜍)𝔚	PROPN
cana-4565	181	17	is	be	AUX
cana-4565	181	18	a	a	DET
cana-4565	181	19	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	181	20	of	of	ADP
cana-4565	181	21	℧	℧	NOUN
cana-4565	181	22	1	1	NUM
cana-4565	181	23	.	.	PUNCT
cana-4565	181	24	proof	proof	NOUN
cana-4565	181	25	.	.	PUNCT
cana-4565	182	1	from	from	ADP
cana-4565	182	2	the	the	DET
cana-4565	182	3	above	above	ADJ
cana-4565	182	4	theorems	theorem	NOUN
cana-4565	182	5	,	,	PUNCT
cana-4565	182	6	it	it	PRON
cana-4565	182	7	is	be	AUX
cana-4565	182	8	trivial	trivial	ADJ
cana-4565	182	9	.	.	PUNCT
cana-4565	183	1	theorem	theorem	VERB
cana-4565	183	2	2.34	2.34	NUM
cana-4565	183	3	.	.	PUNCT
cana-4565	184	1	𝐼𝑓	𝐼𝑓	PROPN
cana-4565	184	2	𝔓1	𝔓1	PROPN
cana-4565	184	3	,	,	PUNCT
cana-4565	184	4	𝔓2	𝔓2	NOUN
cana-4565	184	5	,	,	PUNCT
cana-4565	184	6	…	…	PUNCT
cana-4565	184	7	,	,	PUNCT
cana-4565	184	8	𝔓𝑚𝑎𝑟𝑒	𝔓𝑚𝑎𝑟𝑒	PROPN
cana-4565	184	9	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	184	10	of	of	ADP
cana-4565	184	11	the	the	DET
cana-4565	184	12	rings	ring	NOUN
cana-4565	184	13	℧	℧	PROPN
cana-4565	184	14	1	1	NUM
cana-4565	184	15	,	,	PUNCT
cana-4565	184	16	℧	℧	NOUN
cana-4565	184	17	2	2	NUM
cana-4565	184	18	,	,	PUNCT
cana-4565	184	19	…	…	PUNCT
cana-4565	184	20	,	,	PUNCT
cana-4565	184	21	℧	℧	NOUN
cana-4565	184	22	m	m	VERB
cana-4565	184	23	respectively	respectively	ADV
cana-4565	184	24	,	,	PUNCT
cana-4565	184	25	then	then	ADV
cana-4565	184	26	ℜ(𝜛,𝜍)(𝔓1	ℜ(𝜛,𝜍)(𝔓1	PROPN
cana-4565	184	27	∩	∩	PROPN
cana-4565	184	28	𝔓2	𝔓2	NOUN
cana-4565	184	29	∩	∩	NOUN
cana-4565	184	30	…	…	PUNCT
cana-4565	184	31	∩	∩	ADJ
cana-4565	184	32	𝔓𝑚	𝔓𝑚	NOUN
cana-4565	184	33	)	)	PUNCT
cana-4565	184	34	is	be	AUX
cana-4565	184	35	a	a	DET
cana-4565	184	36	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	184	37	of	of	ADP
cana-4565	184	38	the	the	DET
cana-4565	184	39	ring	ring	NOUN
cana-4565	184	40	℧	℧	NOUN
cana-4565	184	41	1	1	NUM
cana-4565	184	42	∩	∩	X
cana-4565	184	43	℧	℧	NOUN
cana-4565	184	44	2	2	NUM
cana-4565	184	45	∩	∩	NOUN
cana-4565	184	46	…	…	PUNCT
cana-4565	184	47	∩	∩	ADJ
cana-4565	184	48	℧	℧	NOUN
cana-4565	184	49	m.	m.	NOUN
cana-4565	184	50	proof	proof	NOUN
cana-4565	184	51	.	.	PUNCT
cana-4565	185	1	from	from	ADP
cana-4565	185	2	the	the	DET
cana-4565	185	3	above	above	ADJ
cana-4565	185	4	theorems	theorem	NOUN
cana-4565	185	5	,	,	PUNCT
cana-4565	185	6	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	185	7	𝑝𝑟𝑜𝑜𝑓	𝑝𝑟𝑜𝑜𝑓	NOUN
cana-4565	185	8	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	185	9	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	PROPN
cana-4565	185	10	corollary	corollary	NOUN
cana-4565	185	11	2.35	2.35	NUM
cana-4565	185	12	.	.	PUNCT
cana-4565	186	1	𝐼𝑓	𝐼𝑓	PROPN
cana-4565	186	2	𝔓1	𝔓1	PROPN
cana-4565	186	3	,	,	PUNCT
cana-4565	186	4	𝔓2	𝔓2	NOUN
cana-4565	186	5	,	,	PUNCT
cana-4565	186	6	…	…	PUNCT
cana-4565	186	7	,	,	PUNCT
cana-4565	186	8	𝔓𝑚	𝔓𝑚	PROPN
cana-4565	186	9	𝑎𝑟𝑒	𝑎𝑟𝑒	VERB
cana-4565	186	10	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	186	11	of	of	ADP
cana-4565	186	12	the	the	DET
cana-4565	186	13	rings	ring	NOUN
cana-4565	186	14	℧	℧	PROPN
cana-4565	186	15	1	1	NUM
cana-4565	186	16	,	,	PUNCT
cana-4565	186	17	℧	℧	NOUN
cana-4565	186	18	2	2	NUM
cana-4565	186	19	,	,	PUNCT
cana-4565	186	20	…	…	PUNCT
cana-4565	186	21	,	,	PUNCT
cana-4565	186	22	℧	℧	NOUN
cana-4565	186	23	m	m	VERB
cana-4565	186	24	respectively	respectively	ADV
cana-4565	186	25	,	,	PUNCT
cana-4565	186	26	then	then	ADV
cana-4565	186	27	ℜ(𝜛,𝜍)𝔓1	ℜ(𝜛,𝜍)𝔓1	PROPN
cana-4565	186	28	∩	∩	ADJ
cana-4565	186	29	ℜ(𝜛,𝜍)𝔓2	ℜ(𝜛,𝜍)𝔓2	NOUN
cana-4565	186	30	∩	∩	NOUN
cana-4565	186	31	…	…	PUNCT
cana-4565	186	32	∩	∩	NOUN
cana-4565	186	33	ℜ(𝜛,𝜍)𝔓𝑚	ℜ(𝜛,𝜍)𝔓𝑚	NOUN
cana-4565	186	34	is	be	AUX
cana-4565	186	35	a	a	DET
cana-4565	186	36	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	186	37	of	of	ADP
cana-4565	186	38	the	the	DET
cana-4565	186	39	ring	ring	NOUN
cana-4565	186	40	℧	℧	NOUN
cana-4565	186	41	1	1	NUM
cana-4565	186	42	∩	∩	X
cana-4565	186	43	℧	℧	NOUN
cana-4565	186	44	2	2	NUM
cana-4565	186	45	∩	∩	NOUN
cana-4565	186	46	…	…	PUNCT
cana-4565	186	47	∩	∩	ADJ
cana-4565	186	48	℧	℧	NOUN
cana-4565	186	49	m.	m.	NOUN
cana-4565	186	50	proof	proof	NOUN
cana-4565	186	51	.	.	PUNCT
cana-4565	187	1	from	from	ADP
cana-4565	187	2	the	the	DET
cana-4565	187	3	above	above	ADJ
cana-4565	187	4	theorems	theorem	NOUN
cana-4565	187	5	,	,	PUNCT
cana-4565	187	6	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	187	7	𝑝𝑟𝑜𝑜𝑓	𝑝𝑟𝑜𝑜𝑓	NOUN
cana-4565	187	8	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	187	9	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	PROPN
cana-4565	187	10	corollary	corollary	NOUN
cana-4565	187	11	2.36	2.36	NUM
cana-4565	187	12	.	.	PUNCT
cana-4565	188	1	𝐼𝑓	𝐼𝑓	PROPN
cana-4565	188	2	𝔓1	𝔓1	PROPN
cana-4565	188	3	,	,	PUNCT
cana-4565	188	4	𝔓2	𝔓2	NOUN
cana-4565	188	5	,	,	PUNCT
cana-4565	188	6	…	…	PUNCT
cana-4565	188	7	,	,	PUNCT
cana-4565	188	8	𝔓𝑚	𝔓𝑚	PROPN
cana-4565	188	9	are	be	AUX
cana-4565	188	10	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	188	11	of	of	ADP
cana-4565	188	12	the	the	DET
cana-4565	188	13	ring	ring	NOUN
cana-4565	188	14	℧	℧	PROPN
cana-4565	188	15	1	1	NUM
cana-4565	188	16	,	,	PUNCT
cana-4565	188	17	then	then	ADV
cana-4565	188	18	ℜ(𝜛,𝜍)(𝔓1	ℜ(𝜛,𝜍)(𝔓1	PROPN
cana-4565	188	19	∩	∩	PROPN
cana-4565	188	20	𝔓2	𝔓2	NOUN
cana-4565	188	21	∩	∩	NOUN
cana-4565	188	22	…	…	PUNCT
cana-4565	188	23	∩	∩	ADJ
cana-4565	188	24	𝔓𝑚	𝔓𝑚	NOUN
cana-4565	188	25	)	)	PUNCT
cana-4565	188	26	is	be	AUX
cana-4565	188	27	a	a	DET
cana-4565	188	28	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	188	29	𝑜𝑓	𝑜𝑓	ADP
cana-4565	188	30	℧	℧	PROPN
cana-4565	188	31	1	1	NUM
cana-4565	188	32	.	.	PUNCT
cana-4565	189	1	proof	proof	NOUN
cana-4565	189	2	.	.	PUNCT
cana-4565	190	1	from	from	ADP
cana-4565	190	2	the	the	DET
cana-4565	190	3	above	above	ADJ
cana-4565	190	4	theorems	theorem	NOUN
cana-4565	190	5	,	,	PUNCT
cana-4565	190	6	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	190	7	𝑝𝑟𝑜𝑜𝑓	𝑝𝑟𝑜𝑜𝑓	NOUN
cana-4565	190	8	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	190	9	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	PROPN
cana-4565	190	10	corollary	corollary	NOUN
cana-4565	190	11	2.37	2.37	NUM
cana-4565	190	12	.	.	PUNCT
cana-4565	191	1	𝐼𝑓	𝐼𝑓	PROPN
cana-4565	191	2	𝔓1	𝔓1	PROPN
cana-4565	191	3	,	,	PUNCT
cana-4565	191	4	𝔓2	𝔓2	NOUN
cana-4565	191	5	,	,	PUNCT
cana-4565	191	6	…	…	PUNCT
cana-4565	191	7	,	,	PUNCT
cana-4565	191	8	𝔓𝑚	𝔓𝑚	PROPN
cana-4565	191	9	are	be	AUX
cana-4565	191	10	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	191	11	of	of	ADP
cana-4565	191	12	the	the	DET
cana-4565	191	13	ring	ring	NOUN
cana-4565	191	14	℧	℧	PROPN
cana-4565	191	15	1	1	NUM
cana-4565	191	16	,	,	PUNCT
cana-4565	191	17	then	then	ADV
cana-4565	191	18	ℜ(𝜛,𝜍)𝔓1	ℜ(𝜛,𝜍)𝔓1	PROPN
cana-4565	191	19	∩	∩	ADJ
cana-4565	191	20	ℜ(𝜛,𝜍)𝔓2	ℜ(𝜛,𝜍)𝔓2	NOUN
cana-4565	191	21	∩	∩	NOUN
cana-4565	191	22	…	…	PUNCT
cana-4565	191	23	∩	∩	NOUN
cana-4565	191	24	ℜ(𝜛,𝜍)𝔓𝑚	ℜ(𝜛,𝜍)𝔓𝑚	NOUN
cana-4565	191	25	is	be	AUX
cana-4565	191	26	a	a	DET
cana-4565	191	27	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	191	28	of	of	ADP
cana-4565	191	29	℧	℧	PROPN
cana-4565	191	30	1	1	NUM
cana-4565	191	31	.	.	PUNCT
cana-4565	191	32	communications	communication	NOUN
cana-4565	191	33	on	on	ADP
cana-4565	191	34	applied	apply	VERB
cana-4565	191	35	nonlinear	nonlinear	ADJ
cana-4565	191	36	analysis	analysis	NOUN
cana-4565	191	37	issn	issn	NOUN
cana-4565	191	38	:	:	PUNCT
cana-4565	191	39	1074	1074	NUM
cana-4565	191	40	-	-	PUNCT
cana-4565	191	41	133x	133x	NUM
cana-4565	191	42	vol	vol	NOUN
cana-4565	191	43	32	32	NUM
cana-4565	191	44	no	no	NOUN
cana-4565	191	45	.	.	PUNCT
cana-4565	192	1	9s	9s	NUM
cana-4565	192	2	(	(	PUNCT
cana-4565	192	3	2025	2025	NUM
cana-4565	192	4	)	)	PUNCT
cana-4565	192	5	2621	2621	NUM
cana-4565	193	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-4565	193	2	proof	proof	NOUN
cana-4565	193	3	.	.	PUNCT
cana-4565	194	1	from	from	ADP
cana-4565	194	2	the	the	DET
cana-4565	194	3	above	above	ADJ
cana-4565	194	4	theorems	theorem	NOUN
cana-4565	194	5	,	,	PUNCT
cana-4565	194	6	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	194	7	𝑝𝑟𝑜𝑜𝑓	𝑝𝑟𝑜𝑜𝑓	NOUN
cana-4565	194	8	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	194	9	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	NOUN
cana-4565	194	10	theorem	theorem	VERB
cana-4565	194	11	2.38	2.38	NUM
cana-4565	194	12	.	.	PUNCT
cana-4565	195	1	𝐼𝑓	𝐼𝑓	VERB
cana-4565	195	2	ж	ж	X
cana-4565	195	3	=	=	SYM
cana-4565	195	4			X
cana-4565	195	5	ж1	ж1	X
cana-4565	195	6	+	+	PROPN
cana-4565	195	7	,	,	PUNCT
cana-4565	195	8	ж2	ж2	PROPN
cana-4565	195	9	+	+	PROPN
cana-4565	195	10	,	,	PUNCT
cana-4565	195	11	…	…	PUNCT
cana-4565	195	12	,	,	PUNCT
cana-4565	195	13	ж𝑛	ж𝑛	X
cana-4565	195	14	+	+	ADJ
cana-4565	195	15	,	,	PUNCT
cana-4565	195	16	ж1	ж1	PROPN
cana-4565	195	17	−	−	PROPN
cana-4565	195	18	,	,	PUNCT
cana-4565	195	19	ж2	ж2	VERB
cana-4565	195	20	−	−	PROPN
cana-4565	195	21	,	,	PUNCT
cana-4565	195	22	…	…	PUNCT
cana-4565	195	23	,	,	PUNCT
cana-4565	195	24	ж𝑛	ж𝑛	PROPN
cana-4565	195	25	−	−	PROPN
cana-4565	195	26	is	be	AUX
cana-4565	195	27	𝑎	𝑎	DET
cana-4565	195	28	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	NOUN
cana-4565	195	29	of	of	ADP
cana-4565	195	30	a	a	DET
cana-4565	195	31	ring	ring	NOUN
cana-4565	195	32	ɮ1	ɮ1	PROPN
cana-4565	195	33	,	,	PUNCT
cana-4565	195	34	then	then	ADV
cana-4565	195	35	𝔖(𝜛,𝜍)(ж	𝔖(𝜛,𝜍)(ж	NUM
cana-4565	195	36	)	)	PUNCT
cana-4565	195	37	is	be	AUX
cana-4565	195	38	a	a	DET
cana-4565	195	39	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	NOUN
cana-4565	195	40	of	of	ADP
cana-4565	195	41	ɮ1	ɮ1	PROPN
cana-4565	195	42	,	,	PUNCT
cana-4565	195	43	where	where	SCONJ
cana-4565	195	44	𝜛	𝜛	NOUN
cana-4565	195	45	=	=	PRON
cana-4565	195	46	(	(	PUNCT
cana-4565	195	47	𝜛1	𝜛1	NOUN
cana-4565	195	48	,	,	PUNCT
cana-4565	195	49	𝜛2	𝜛2	NOUN
cana-4565	195	50	,	,	PUNCT
cana-4565	195	51	…	…	PUNCT
cana-4565	195	52	,	,	PUNCT
cana-4565	195	53	𝜛𝑛	𝜛𝑛	NOUN
cana-4565	195	54	)	)	PUNCT
cana-4565	195	55	and	and	CCONJ
cana-4565	195	56	𝜍	𝜍	X
cana-4565	195	57	=	=	SYM
cana-4565	195	58	(	(	PUNCT
cana-4565	195	59	𝜍1	𝜍1	PROPN
cana-4565	195	60	,	,	PUNCT
cana-4565	195	61	𝜍2	𝜍2	NOUN
cana-4565	195	62	,	,	PUNCT
cana-4565	195	63	…	…	PUNCT
cana-4565	195	64	,	,	PUNCT
cana-4565	195	65	𝜍𝑛	𝜍𝑛	NOUN
cana-4565	195	66	)	)	PUNCT
cana-4565	195	67	,	,	PUNCT
cana-4565	195	68	𝜛𝑖	𝜛𝑖	NOUN
cana-4565	195	69	∈	∈	PROPN
cana-4565	196	1	[	[	X
cana-4565	196	2	0	0	NUM
cana-4565	196	3	,	,	PUNCT
cana-4565	196	4	1	1	NUM
cana-4565	196	5	]	]	X
cana-4565	196	6	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-4565	196	7	𝜍𝑖	𝜍𝑖	PROPN
cana-4565	196	8	∈	∈	PROPN
cana-4565	197	1	[	[	X
cana-4565	197	2	−1	−1	NOUN
cana-4565	197	3	,	,	PUNCT
cana-4565	197	4	0	0	NUM
cana-4565	197	5	]	]	PUNCT
cana-4565	197	6	.	.	PUNCT
cana-4565	198	1	theorem	theorem	VERB
cana-4565	198	2	2.39	2.39	NUM
cana-4565	198	3	.	.	PUNCT
cana-4565	199	1	𝐼𝑓	𝐼𝑓	VERB
cana-4565	199	2	ℬ	ℬ	NOUN
cana-4565	199	3	=	=	NOUN
cana-4565	199	4			PRON
cana-4565	199	5	ℬ1	ℬ1	NOUN
cana-4565	199	6	+	+	NOUN
cana-4565	199	7	,	,	PUNCT
cana-4565	199	8	ℬ2	ℬ2	NOUN
cana-4565	199	9	+	+	PROPN
cana-4565	199	10	,	,	PUNCT
cana-4565	199	11	…	…	PUNCT
cana-4565	199	12	,	,	PUNCT
cana-4565	199	13	ℬ𝑛	ℬ𝑛	PROPN
cana-4565	199	14	+	+	PROPN
cana-4565	199	15	,	,	PUNCT
cana-4565	199	16	ℬ1	ℬ1	NOUN
cana-4565	199	17	−	−	NOUN
cana-4565	199	18	,	,	PUNCT
cana-4565	199	19	ℬ2	ℬ2	NOUN
cana-4565	199	20	−	−	PROPN
cana-4565	199	21	,	,	PUNCT
cana-4565	199	22	…	…	PUNCT
cana-4565	199	23	,	,	PUNCT
cana-4565	199	24	ℬ𝑛	ℬ𝑛	ADP
cana-4565	199	25	−	−	PROPN
cana-4565	199	26	is	be	AUX
cana-4565	199	27	𝑎	𝑎	DET
cana-4565	199	28	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	199	29	of	of	ADP
cana-4565	199	30	a	a	DET
cana-4565	199	31	ring	ring	NOUN
cana-4565	199	32	℧	℧	PROPN
cana-4565	199	33	1	1	NUM
cana-4565	199	34	,	,	PUNCT
cana-4565	199	35	then	then	ADV
cana-4565	199	36	𝔖(𝜛,𝜍)(ℬ	𝔖(𝜛,𝜍)(ℬ	ADV
cana-4565	199	37	)	)	PUNCT
cana-4565	199	38	is	be	AUX
cana-4565	199	39	a	a	DET
cana-4565	199	40	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	199	41	of	of	ADP
cana-4565	199	42	℧	℧	PROPN
cana-4565	199	43	1	1	NUM
cana-4565	199	44	,	,	PUNCT
cana-4565	199	45	where	where	SCONJ
cana-4565	199	46	𝜛	𝜛	NOUN
cana-4565	199	47	=	=	PRON
cana-4565	199	48	(	(	PUNCT
cana-4565	199	49	𝜛1	𝜛1	NOUN
cana-4565	199	50	,	,	PUNCT
cana-4565	199	51	𝜛2	𝜛2	NOUN
cana-4565	199	52	,	,	PUNCT
cana-4565	199	53	…	…	PUNCT
cana-4565	199	54	,	,	PUNCT
cana-4565	199	55	𝜛𝑛	𝜛𝑛	NOUN
cana-4565	199	56	)	)	PUNCT
cana-4565	199	57	and	and	CCONJ
cana-4565	199	58	𝜍	𝜍	X
cana-4565	199	59	=	=	SYM
cana-4565	199	60	(	(	PUNCT
cana-4565	199	61	𝜍1	𝜍1	PROPN
cana-4565	199	62	,	,	PUNCT
cana-4565	199	63	𝜍2	𝜍2	NOUN
cana-4565	199	64	,	,	PUNCT
cana-4565	199	65	…	…	PUNCT
cana-4565	199	66	,	,	PUNCT
cana-4565	199	67	𝜍𝑛	𝜍𝑛	NOUN
cana-4565	199	68	)	)	PUNCT
cana-4565	199	69	,	,	PUNCT
cana-4565	199	70	𝜛𝑖	𝜛𝑖	NOUN
cana-4565	199	71	∈	∈	PROPN
cana-4565	200	1	[	[	X
cana-4565	200	2	0	0	NUM
cana-4565	200	3	,	,	PUNCT
cana-4565	200	4	1	1	NUM
cana-4565	200	5	]	]	X
cana-4565	200	6	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-4565	200	7	𝜍𝑖	𝜍𝑖	PROPN
cana-4565	200	8	∈	∈	PROPN
cana-4565	201	1	[	[	X
cana-4565	201	2	−1	−1	NOUN
cana-4565	201	3	,	,	PUNCT
cana-4565	201	4	0	0	NUM
cana-4565	201	5	]	]	PUNCT
cana-4565	201	6	.	.	PUNCT
cana-4565	202	1	proof	proof	NOUN
cana-4565	202	2	.	.	PUNCT
cana-4565	203	1	let	let	VERB
cana-4565	203	2	ℴ	ℴ	NOUN
cana-4565	203	3	,	,	PUNCT
cana-4565	203	4	𝜐	𝜐	PRON
cana-4565	203	5	be	be	AUX
cana-4565	203	6	in	in	ADP
cana-4565	203	7	℧	℧	PROPN
cana-4565	203	8	1	1	NUM
cana-4565	203	9	,	,	PUNCT
cana-4565	203	10	𝜛𝑖	𝜛𝑖	ADP
cana-4565	203	11	∈	∈	PROPN
cana-4565	204	1	[	[	X
cana-4565	204	2	0	0	NUM
cana-4565	204	3	,	,	PUNCT
cana-4565	204	4	1]𝑎𝑛𝑑	1]𝑎𝑛𝑑	NUM
cana-4565	204	5	𝜍𝑖	𝜍𝑖	ADP
cana-4565	204	6	∈	∈	PROPN
cana-4565	205	1	[	[	X
cana-4565	205	2	−1	−1	NOUN
cana-4565	205	3	,	,	PUNCT
cana-4565	205	4	0	0	NUM
cana-4565	205	5	]	]	PUNCT
cana-4565	205	6	.	.	PUNCT
cana-4565	206	1	for	for	ADP
cana-4565	206	2	all	all	DET
cana-4565	206	3	i	i	PRON
cana-4565	206	4	=	=	NOUN
cana-4565	206	5	1	1	NUM
cana-4565	206	6	,	,	PUNCT
cana-4565	206	7	2	2	NUM
cana-4565	206	8	,	,	PUNCT
cana-4565	206	9	…	…	PUNCT
cana-4565	206	10	,	,	PUNCT
cana-4565	206	11	n	n	CCONJ
cana-4565	206	12	,	,	PUNCT
cana-4565	206	13	by	by	ADP
cana-4565	206	14	theorem	theorem	NOUN
cana-4565	206	15	2.38	2.38	NUM
cana-4565	206	16	,	,	PUNCT
cana-4565	206	17	𝔖(𝜛,𝜍)(ℬ	𝔖(𝜛,𝜍)(ℬ	ADV
cana-4565	206	18	)	)	PUNCT
cana-4565	206	19	is	be	AUX
cana-4565	206	20	a	a	DET
cana-4565	206	21	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑆𝑅	NOUN
cana-4565	206	22	of	of	ADP
cana-4565	206	23	℧	℧	NOUN
cana-4565	206	24	1	1	NUM
cana-4565	206	25	,	,	PUNCT
cana-4565	206	26	𝔖(𝜛,𝜍)(ℬ𝑖	𝔖(𝜛,𝜍)(ℬ𝑖	ADJ
cana-4565	206	27	+	+	X
cana-4565	206	28	)	)	PUNCT
cana-4565	206	29	(	(	PUNCT
cana-4565	206	30	ℴ𝜐	ℴ𝜐	X
cana-4565	206	31	)	)	PUNCT
cana-4565	206	32	=	=	PUNCT
cana-4565	206	33	𝜛𝑖	𝜛𝑖	NOUN
cana-4565	206	34	ℬ𝑖	ℬ𝑖	PROPN
cana-4565	206	35	+	+	PROPN
cana-4565	206	36	(	(	PUNCT
cana-4565	206	37	ℴ𝜐	ℴ𝜐	INTJ
cana-4565	206	38	)	)	PUNCT
cana-4565	206	39	=	=	PUNCT
cana-4565	206	40	𝜛𝑖	𝜛𝑖	NOUN
cana-4565	206	41	ℬ𝑖	ℬ𝑖	PROPN
cana-4565	206	42	+	+	PROPN
cana-4565	206	43	(	(	PUNCT
cana-4565	206	44	𝜐ℴ	𝜐ℴ	X
cana-4565	206	45	)	)	PUNCT
cana-4565	206	46	=	=	PUNCT
cana-4565	207	1	𝔖(𝜛,𝜍)(ℬ𝑖	𝔖(𝜛,𝜍)(ℬ𝑖	PUNCT
cana-4565	207	2	+	+	X
cana-4565	207	3	)	)	PUNCT
cana-4565	207	4	(	(	PUNCT
cana-4565	207	5	𝜐ℴ	𝜐ℴ	X
cana-4565	207	6	)	)	PUNCT
cana-4565	207	7	,	,	PUNCT
cana-4565	207	8	for	for	ADP
cana-4565	207	9	all	all	DET
cana-4565	207	10	ℴ	ℴ	NOUN
cana-4565	207	11	,	,	PUNCT
cana-4565	207	12	𝜐	𝜐	X
cana-4565	207	13	in	in	ADP
cana-4565	207	14	℧	℧	PROPN
cana-4565	207	15	1	1	NUM
cana-4565	207	16	.	.	PUNCT
cana-4565	208	1	and	and	CCONJ
cana-4565	208	2	𝔖(𝜛,𝜍)(ℬ𝑖	𝔖(𝜛,𝜍)(ℬ𝑖	NOUN
cana-4565	208	3	−)(ℴ𝜐	−)(ℴ𝜐	ADJ
cana-4565	208	4	)	)	PUNCT
cana-4565	209	1	=	=	PRON
cana-4565	209	2	(	(	PUNCT
cana-4565	209	3	−𝜍𝑖	−𝜍𝑖	NOUN
cana-4565	209	4	)	)	PUNCT
cana-4565	210	1	ℬ𝑖	ℬ𝑖	PROPN
cana-4565	210	2	−(ℴ𝜐	−(ℴ𝜐	PROPN
cana-4565	210	3	)	)	PUNCT
cana-4565	210	4	=	=	SYM
cana-4565	210	5	(	(	PUNCT
cana-4565	210	6	−𝜍𝑖	−𝜍𝑖	NOUN
cana-4565	210	7	)	)	PUNCT
cana-4565	210	8	ℬ𝑖	ℬ𝑖	PROPN
cana-4565	210	9	−(𝜐ℴ	−(𝜐ℴ	PROPN
cana-4565	210	10	)	)	PUNCT
cana-4565	210	11	=	=	PUNCT
cana-4565	211	1	𝔖(𝜛,𝜍)(ℬ𝑖	𝔖(𝜛,𝜍)(ℬ𝑖	NUM
cana-4565	211	2	−)(𝜐ℴ	−)(𝜐ℴ	NOUN
cana-4565	211	3	)	)	PUNCT
cana-4565	211	4	,	,	PUNCT
cana-4565	211	5	for	for	ADP
cana-4565	211	6	all	all	DET
cana-4565	211	7	ℴ	ℴ	NOUN
cana-4565	211	8	,	,	PUNCT
cana-4565	211	9	𝜐	𝜐	X
cana-4565	211	10	in	in	ADP
cana-4565	211	11	℧	℧	NOUN
cana-4565	211	12	1	1	NUM
cana-4565	211	13	.	.	PUNCT
cana-4565	211	14	hence	hence	ADV
cana-4565	211	15	𝔖(𝜛,𝜍)(ℬ	𝔖(𝜛,𝜍)(ℬ	ADV
cana-4565	211	16	)	)	PUNCT
cana-4565	211	17	is	be	AUX
cana-4565	211	18	a	a	DET
cana-4565	211	19	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	211	20	of	of	ADP
cana-4565	211	21	℧	℧	NOUN
cana-4565	211	22	1	1	NUM
cana-4565	211	23	.	.	PUNCT
cana-4565	211	24	corollary	corollary	ADJ
cana-4565	211	25	2.40	2.40	NUM
cana-4565	211	26	.	.	PUNCT
cana-4565	212	1	if	if	SCONJ
cana-4565	212	2	𝔓	𝔓	PROPN
cana-4565	212	3	and	and	CCONJ
cana-4565	212	4	𝔚	𝔚	PROPN
cana-4565	212	5	are	be	AUX
cana-4565	212	6	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	NOUN
cana-4565	212	7	𝑜𝑓	𝑜𝑓	ADP
cana-4565	212	8	𝑡ℎ𝑒	𝑡ℎ𝑒	NUM
cana-4565	212	9	𝑟𝑖𝑛𝑔	𝑟𝑖𝑛𝑔	NOUN
cana-4565	212	10	℧	℧	PROPN
cana-4565	212	11	1	1	NUM
cana-4565	212	12	,	,	PUNCT
cana-4565	212	13	then	then	ADV
cana-4565	212	14	𝔖(𝜛,𝜍)(𝔓	𝔖(𝜛,𝜍)(𝔓	ADV
cana-4565	212	15	∩	∩	ADJ
cana-4565	212	16	𝔚	𝔚	PROPN
cana-4565	212	17	)	)	PUNCT
cana-4565	212	18	is	be	AUX
cana-4565	212	19	a	a	DET
cana-4565	212	20	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	212	21	of	of	ADP
cana-4565	212	22	℧	℧	NOUN
cana-4565	212	23	1	1	NUM
cana-4565	212	24	.	.	PUNCT
cana-4565	213	1	proof	proof	NOUN
cana-4565	213	2	.	.	PUNCT
cana-4565	214	1	from	from	ADP
cana-4565	214	2	the	the	DET
cana-4565	214	3	above	above	ADJ
cana-4565	214	4	theorems	theorem	NOUN
cana-4565	214	5	,	,	PUNCT
cana-4565	214	6	it	it	PRON
cana-4565	214	7	is	be	AUX
cana-4565	214	8	trivial	trivial	ADJ
cana-4565	214	9	.	.	PUNCT
cana-4565	215	1	corollary	corollary	ADJ
cana-4565	215	2	2.41	2.41	NUM
cana-4565	215	3	.	.	PUNCT
cana-4565	216	1	if	if	SCONJ
cana-4565	216	2	𝔓	𝔓	PROPN
cana-4565	216	3	and	and	CCONJ
cana-4565	216	4	𝔚	𝔚	PROPN
cana-4565	216	5	are	be	AUX
cana-4565	216	6	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	NOUN
cana-4565	216	7	𝑜𝑓	𝑜𝑓	ADP
cana-4565	216	8	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	216	9	𝑟𝑖𝑛𝑔𝑠	𝑟𝑖𝑛𝑔𝑠	NOUN
cana-4565	216	10	℧	℧	PROPN
cana-4565	216	11	1and	1and	NUM
cana-4565	216	12	℧	℧	PROPN
cana-4565	216	13	2	2	NUM
cana-4565	216	14	,	,	PUNCT
cana-4565	216	15	then	then	ADV
cana-4565	216	16	𝔖(𝜛,𝜍)𝔓	𝔖(𝜛,𝜍)𝔓	NOUN
cana-4565	216	17	∩	∩	NOUN
cana-4565	216	18	𝔖(𝜛,𝜍)𝔚	𝔖(𝜛,𝜍)𝔚	PROPN
cana-4565	216	19	is	be	AUX
cana-4565	216	20	a	a	DET
cana-4565	216	21	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	216	22	of	of	ADP
cana-4565	216	23	℧	℧	NOUN
cana-4565	216	24	1	1	NUM
cana-4565	216	25	∩	∩	NOUN
cana-4565	216	26	℧	℧	PROPN
cana-4565	216	27	2	2	NUM
cana-4565	216	28	.	.	PUNCT
cana-4565	217	1	proof	proof	NOUN
cana-4565	217	2	.	.	PUNCT
cana-4565	218	1	from	from	ADP
cana-4565	218	2	the	the	DET
cana-4565	218	3	above	above	ADJ
cana-4565	218	4	theorems	theorem	NOUN
cana-4565	218	5	,	,	PUNCT
cana-4565	218	6	it	it	PRON
cana-4565	218	7	is	be	AUX
cana-4565	218	8	trivial	trivial	ADJ
cana-4565	218	9	.	.	PUNCT
cana-4565	219	1	corollary	corollary	ADJ
cana-4565	219	2	2.42	2.42	NUM
cana-4565	219	3	.	.	PUNCT
cana-4565	220	1	if	if	SCONJ
cana-4565	220	2	𝔓	𝔓	PROPN
cana-4565	220	3	and	and	CCONJ
cana-4565	220	4	𝔚	𝔚	PROPN
cana-4565	220	5	are	be	AUX
cana-4565	220	6	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅𝑠	NOUN
cana-4565	220	7	𝑜𝑓	𝑜𝑓	ADP
cana-4565	220	8	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	220	9	𝑟𝑖𝑛𝑔𝑠	𝑟𝑖𝑛𝑔𝑠	NOUN
cana-4565	220	10	℧	℧	NOUN
cana-4565	220	11	1	1	NUM
cana-4565	220	12	,	,	PUNCT
cana-4565	220	13	then	then	ADV
cana-4565	220	14	𝔖(𝜛,𝜍)𝔓	𝔖(𝜛,𝜍)𝔓	NOUN
cana-4565	220	15	∩	∩	NOUN
cana-4565	220	16	𝔖(𝜛,𝜍)𝔚	𝔖(𝜛,𝜍)𝔚	PROPN
cana-4565	220	17	is	be	AUX
cana-4565	220	18	a	a	DET
cana-4565	220	19	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	220	20	of	of	ADP
cana-4565	220	21	℧	℧	NOUN
cana-4565	220	22	1	1	NUM
cana-4565	220	23	.	.	PUNCT
cana-4565	221	1	proof	proof	NOUN
cana-4565	221	2	.	.	PUNCT
cana-4565	222	1	from	from	ADP
cana-4565	222	2	the	the	DET
cana-4565	222	3	above	above	ADJ
cana-4565	222	4	theorems	theorem	NOUN
cana-4565	222	5	,	,	PUNCT
cana-4565	222	6	it	it	PRON
cana-4565	222	7	is	be	AUX
cana-4565	222	8	trivial	trivial	ADJ
cana-4565	222	9	.	.	PUNCT
cana-4565	223	1	theorem	theorem	VERB
cana-4565	223	2	2.43	2.43	NUM
cana-4565	223	3	.	.	PUNCT
cana-4565	224	1	𝐼𝑓	𝐼𝑓	PROPN
cana-4565	224	2	𝔓1	𝔓1	PROPN
cana-4565	224	3	,	,	PUNCT
cana-4565	224	4	𝔓2	𝔓2	NOUN
cana-4565	224	5	,	,	PUNCT
cana-4565	224	6	…	…	PUNCT
cana-4565	224	7	,	,	PUNCT
cana-4565	224	8	𝔓𝑚𝑎𝑟𝑒	𝔓𝑚𝑎𝑟𝑒	PROPN
cana-4565	224	9	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	224	10	of	of	ADP
cana-4565	224	11	the	the	DET
cana-4565	224	12	rings	ring	NOUN
cana-4565	224	13	℧	℧	PROPN
cana-4565	224	14	1	1	NUM
cana-4565	224	15	,	,	PUNCT
cana-4565	224	16	℧	℧	NOUN
cana-4565	224	17	2	2	NUM
cana-4565	224	18	,	,	PUNCT
cana-4565	224	19	…	…	PUNCT
cana-4565	224	20	,	,	PUNCT
cana-4565	224	21	℧	℧	NOUN
cana-4565	224	22	m	m	VERB
cana-4565	224	23	respectively	respectively	ADV
cana-4565	224	24	,	,	PUNCT
cana-4565	224	25	then	then	ADV
cana-4565	224	26	𝔖(𝜛,𝜍)(𝔓1	𝔖(𝜛,𝜍)(𝔓1	NUM
cana-4565	224	27	∩	∩	PROPN
cana-4565	224	28	𝔓2	𝔓2	NOUN
cana-4565	224	29	∩	∩	NOUN
cana-4565	224	30	…	…	PUNCT
cana-4565	224	31	∩	∩	ADJ
cana-4565	224	32	𝔓𝑚	𝔓𝑚	NOUN
cana-4565	224	33	)	)	PUNCT
cana-4565	224	34	is	be	AUX
cana-4565	224	35	a	a	DET
cana-4565	224	36	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	224	37	of	of	ADP
cana-4565	224	38	the	the	DET
cana-4565	224	39	ring	ring	NOUN
cana-4565	224	40	℧	℧	NOUN
cana-4565	224	41	1	1	NUM
cana-4565	224	42	∩	∩	X
cana-4565	224	43	℧	℧	NOUN
cana-4565	224	44	2	2	NUM
cana-4565	224	45	∩	∩	NOUN
cana-4565	224	46	…	…	PUNCT
cana-4565	224	47	∩	∩	ADJ
cana-4565	224	48	℧	℧	NOUN
cana-4565	224	49	m.	m.	NOUN
cana-4565	224	50	proof	proof	NOUN
cana-4565	224	51	.	.	PUNCT
cana-4565	225	1	from	from	ADP
cana-4565	225	2	the	the	DET
cana-4565	225	3	above	above	ADJ
cana-4565	225	4	theorems	theorem	NOUN
cana-4565	225	5	,	,	PUNCT
cana-4565	225	6	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	225	7	𝑝𝑟𝑜𝑜𝑓	𝑝𝑟𝑜𝑜𝑓	NOUN
cana-4565	225	8	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	225	9	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	NOUN
cana-4565	225	10	corollary	corollary	NOUN
cana-4565	225	11	2.44	2.44	NUM
cana-4565	225	12	.	.	PUNCT
cana-4565	226	1	𝐼𝑓	𝐼𝑓	PROPN
cana-4565	226	2	𝔓1	𝔓1	PROPN
cana-4565	226	3	,	,	PUNCT
cana-4565	226	4	𝔓2	𝔓2	NOUN
cana-4565	226	5	,	,	PUNCT
cana-4565	226	6	…	…	PUNCT
cana-4565	226	7	,	,	PUNCT
cana-4565	226	8	𝔓𝑚	𝔓𝑚	PROPN
cana-4565	226	9	𝑎𝑟𝑒	𝑎𝑟𝑒	VERB
cana-4565	226	10	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	226	11	of	of	ADP
cana-4565	226	12	the	the	DET
cana-4565	226	13	rings	ring	NOUN
cana-4565	226	14	℧	℧	PROPN
cana-4565	226	15	1	1	NUM
cana-4565	226	16	,	,	PUNCT
cana-4565	226	17	℧	℧	NOUN
cana-4565	226	18	2	2	NUM
cana-4565	226	19	,	,	PUNCT
cana-4565	226	20	…	…	PUNCT
cana-4565	226	21	,	,	PUNCT
cana-4565	226	22	℧	℧	NOUN
cana-4565	226	23	m	m	VERB
cana-4565	226	24	respectively	respectively	ADV
cana-4565	226	25	,	,	PUNCT
cana-4565	226	26	then	then	ADV
cana-4565	226	27	𝔖(𝜛,𝜍)𝔓1	𝔖(𝜛,𝜍)𝔓1	NUM
cana-4565	226	28	∩	∩	ADJ
cana-4565	226	29	𝔖(𝜛,𝜍)𝔓2	𝔖(𝜛,𝜍)𝔓2	NOUN
cana-4565	226	30	∩	∩	NOUN
cana-4565	226	31	…	…	PUNCT
cana-4565	226	32	∩	∩	NOUN
cana-4565	226	33	𝔖(𝜛,𝜍)𝔓𝑚	𝔖(𝜛,𝜍)𝔓𝑚	NOUN
cana-4565	226	34	is	be	AUX
cana-4565	226	35	a	a	DET
cana-4565	226	36	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	226	37	of	of	ADP
cana-4565	226	38	the	the	DET
cana-4565	226	39	ring	ring	NOUN
cana-4565	226	40	℧	℧	NOUN
cana-4565	226	41	1	1	NUM
cana-4565	226	42	∩	∩	X
cana-4565	226	43	℧	℧	NOUN
cana-4565	226	44	2	2	NUM
cana-4565	226	45	∩	∩	NOUN
cana-4565	226	46	…	…	PUNCT
cana-4565	226	47	∩	∩	ADJ
cana-4565	226	48	℧	℧	NOUN
cana-4565	226	49	m.	m.	NOUN
cana-4565	226	50	proof	proof	NOUN
cana-4565	226	51	.	.	PUNCT
cana-4565	227	1	from	from	ADP
cana-4565	227	2	the	the	DET
cana-4565	227	3	above	above	ADJ
cana-4565	227	4	theorems	theorem	NOUN
cana-4565	227	5	,	,	PUNCT
cana-4565	227	6	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	227	7	𝑝𝑟𝑜𝑜𝑓	𝑝𝑟𝑜𝑜𝑓	NOUN
cana-4565	227	8	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	227	9	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	NOUN
cana-4565	227	10	corollary	corollary	NOUN
cana-4565	227	11	2.45	2.45	NUM
cana-4565	227	12	.	.	PUNCT
cana-4565	228	1	𝐼𝑓	𝐼𝑓	PROPN
cana-4565	228	2	𝔓1	𝔓1	PROPN
cana-4565	228	3	,	,	PUNCT
cana-4565	228	4	𝔓2	𝔓2	NOUN
cana-4565	228	5	,	,	PUNCT
cana-4565	228	6	…	…	PUNCT
cana-4565	228	7	,	,	PUNCT
cana-4565	228	8	𝔓𝑚	𝔓𝑚	PROPN
cana-4565	228	9	are	be	AUX
cana-4565	228	10	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	228	11	of	of	ADP
cana-4565	228	12	the	the	DET
cana-4565	228	13	ring	ring	NOUN
cana-4565	228	14	℧	℧	PROPN
cana-4565	228	15	1	1	NUM
cana-4565	228	16	,	,	PUNCT
cana-4565	228	17	then	then	ADV
cana-4565	228	18	𝔖(𝜛,𝜍)(𝔓1	𝔖(𝜛,𝜍)(𝔓1	NUM
cana-4565	228	19	∩	∩	NOUN
cana-4565	228	20	𝔓2	𝔓2	NOUN
cana-4565	228	21	∩	∩	NOUN
cana-4565	228	22	…	…	PUNCT
cana-4565	228	23	∩	∩	ADJ
cana-4565	228	24	𝔓𝑚	𝔓𝑚	NOUN
cana-4565	228	25	)	)	PUNCT
cana-4565	228	26	is	be	AUX
cana-4565	228	27	a	a	DET
cana-4565	228	28	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	228	29	𝑜𝑓	𝑜𝑓	ADP
cana-4565	228	30	the	the	DET
cana-4565	228	31	ring	ring	NOUN
cana-4565	228	32	℧	℧	NOUN
cana-4565	228	33	1	1	NUM
cana-4565	228	34	.	.	PUNCT
cana-4565	229	1	proof	proof	NOUN
cana-4565	229	2	.	.	PUNCT
cana-4565	230	1	from	from	ADP
cana-4565	230	2	the	the	DET
cana-4565	230	3	above	above	ADJ
cana-4565	230	4	theorems	theorem	NOUN
cana-4565	230	5	,	,	PUNCT
cana-4565	230	6	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	230	7	𝑝𝑟𝑜𝑜𝑓	𝑝𝑟𝑜𝑜𝑓	NOUN
cana-4565	230	8	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	230	9	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	PROPN
cana-4565	230	10	corollary	corollary	NOUN
cana-4565	230	11	2.46	2.46	NUM
cana-4565	230	12	.	.	PUNCT
cana-4565	231	1	𝐼𝑓	𝐼𝑓	PROPN
cana-4565	231	2	𝔓1	𝔓1	PROPN
cana-4565	231	3	,	,	PUNCT
cana-4565	231	4	𝔓2	𝔓2	NOUN
cana-4565	231	5	,	,	PUNCT
cana-4565	231	6	…	…	PUNCT
cana-4565	231	7	,	,	PUNCT
cana-4565	231	8	𝔓𝑚	𝔓𝑚	PROPN
cana-4565	231	9	are	be	AUX
cana-4565	231	10	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅s	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	NOUN
cana-4565	231	11	of	of	ADP
cana-4565	231	12	the	the	DET
cana-4565	231	13	ring	ring	NOUN
cana-4565	231	14	℧	℧	PROPN
cana-4565	231	15	1	1	NUM
cana-4565	231	16	,	,	PUNCT
cana-4565	231	17	then	then	ADV
cana-4565	231	18	𝔖(𝜛,𝜍)𝔓1	𝔖(𝜛,𝜍)𝔓1	NUM
cana-4565	231	19	∩	∩	ADJ
cana-4565	231	20	𝔖(𝜛,𝜍)𝔓2	𝔖(𝜛,𝜍)𝔓2	NOUN
cana-4565	231	21	∩	∩	NOUN
cana-4565	231	22	…	…	PUNCT
cana-4565	231	23	∩	∩	NOUN
cana-4565	231	24	𝔖(𝜛,𝜍)𝔓𝑚	𝔖(𝜛,𝜍)𝔓𝑚	NOUN
cana-4565	231	25	is	be	AUX
cana-4565	231	26	a	a	DET
cana-4565	231	27	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	𝐵𝑉𝑀𝐼𝐹𝑁𝑆𝑅	PROPN
cana-4565	231	28	of	of	ADP
cana-4565	231	29	the	the	DET
cana-4565	231	30	ring	ring	NOUN
cana-4565	231	31	℧	℧	NOUN
cana-4565	231	32	1	1	NUM
cana-4565	231	33	.	.	PUNCT
cana-4565	231	34	communications	communication	NOUN
cana-4565	231	35	on	on	ADP
cana-4565	231	36	applied	apply	VERB
cana-4565	231	37	nonlinear	nonlinear	ADJ
cana-4565	231	38	analysis	analysis	NOUN
cana-4565	231	39	issn	issn	NOUN
cana-4565	231	40	:	:	PUNCT
cana-4565	231	41	1074	1074	NUM
cana-4565	231	42	-	-	PUNCT
cana-4565	231	43	133x	133x	NUM
cana-4565	231	44	vol	vol	NOUN
cana-4565	231	45	32	32	NUM
cana-4565	231	46	no	no	NOUN
cana-4565	231	47	.	.	PUNCT
cana-4565	232	1	9s	9s	NUM
cana-4565	232	2	(	(	PUNCT
cana-4565	232	3	2025	2025	NUM
cana-4565	232	4	)	)	PUNCT
cana-4565	232	5	2622	2622	NUM
cana-4565	233	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-4565	233	2	proof	proof	NOUN
cana-4565	233	3	.	.	PUNCT
cana-4565	234	1	from	from	ADP
cana-4565	234	2	the	the	DET
cana-4565	234	3	above	above	ADJ
cana-4565	234	4	theorems	theorem	NOUN
cana-4565	234	5	,	,	PUNCT
cana-4565	234	6	𝑡ℎ𝑒	𝑡ℎ𝑒	ADJ
cana-4565	234	7	𝑝𝑟𝑜𝑜𝑓	𝑝𝑟𝑜𝑜𝑓	NOUN
cana-4565	234	8	𝑖𝑠	𝑖𝑠	PROPN
cana-4565	234	9	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	𝑡𝑟𝑖𝑣𝑖𝑎𝑙.	ADJ
cana-4565	234	10	conclusion	conclusion	NOUN
cana-4565	234	11	in	in	ADP
cana-4565	234	12	this	this	DET
cana-4565	234	13	paper	paper	NOUN
cana-4565	234	14	,	,	PUNCT
cana-4565	234	15	𝑣𝑎𝑟𝑖𝑜𝑢𝑠	𝑣𝑎𝑟𝑖𝑜𝑢𝑠	ADJ
cana-4565	234	16	𝑡𝑦𝑝𝑒𝑠	𝑡𝑦𝑝𝑒𝑠	NOUN
cana-4565	234	17	𝑜𝑓	𝑜𝑓	ADP
cana-4565	234	18	𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛	𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛	NOUN
cana-4565	234	19	𝑜𝑓	𝑜𝑓	ADP
cana-4565	234	20	bipolar	bipolar	PROPN
cana-4565	234	21	valued	value	VERB
cana-4565	234	22	multi	multi	NOUN
cana-4565	234	23	ifuzzy	ifuzzy	VERB
cana-4565	234	24	normal	normal	ADJ
cana-4565	234	25	subring	subring	NOUN
cana-4565	234	26	of	of	ADP
cana-4565	234	27	a	a	DET
cana-4565	234	28	ring	ring	NOUN
cana-4565	234	29	have	have	AUX
cana-4565	234	30	been	be	AUX
cana-4565	234	31	introduced	introduce	VERB
cana-4565	234	32	.	.	PUNCT
cana-4565	235	1	some	some	DET
cana-4565	235	2	useful	useful	ADJ
cana-4565	235	3	theorems	theorem	NOUN
cana-4565	235	4	have	have	AUX
cana-4565	235	5	been	be	AUX
cana-4565	235	6	found	find	VERB
cana-4565	235	7	and	and	CCONJ
cana-4565	235	8	using	use	VERB
cana-4565	235	9	these	these	DET
cana-4565	235	10	theorem	theorem	NOUN
cana-4565	235	11	we	we	PRON
cana-4565	235	12	can	can	AUX
cana-4565	235	13	find	find	VERB
cana-4565	235	14	more	more	ADJ
cana-4565	235	15	results	result	NOUN
cana-4565	235	16	.	.	PUNCT
cana-4565	236	1	it	it	PRON
cana-4565	236	2	can	can	AUX
cana-4565	236	3	be	be	AUX
cana-4565	236	4	extended	extend	VERB
cana-4565	236	5	into	into	ADP
cana-4565	236	6	different	different	ADJ
cana-4565	236	7	types	type	NOUN
cana-4565	236	8	of	of	ADP
cana-4565	236	9	algebra	algebra	NOUN
cana-4565	236	10	.	.	PUNCT
cana-4565	237	1	references	reference	NOUN
cana-4565	237	2	[	[	X
cana-4565	237	3	1	1	NUM
cana-4565	237	4	]	]	PUNCT
cana-4565	237	5	.	.	PUNCT
cana-4565	238	1	anitha.m.s	anitha.m.s	ADP
cana-4565	238	2	.	.	PUNCT
cana-4565	238	3	,	,	PUNCT
cana-4565	238	4	muruganantha	muruganantha	PROPN
cana-4565	238	5	prasad	prasad	PROPN
cana-4565	238	6	&	&	CCONJ
cana-4565	238	7	k.arjunan	k.arjunan	PROPN
cana-4565	238	8	,	,	PUNCT
cana-4565	238	9	“	"	PUNCT
cana-4565	238	10	notes	note	NOUN
cana-4565	238	11	on	on	ADP
cana-4565	238	12	bipolar	bipolar	ADV
cana-4565	238	13	-	-	PUNCT
cana-4565	238	14	valued	value	VERB
cana-4565	238	15	fuzzy	fuzzy	ADJ
cana-4565	238	16	subgroups	subgroup	NOUN
cana-4565	238	17	of	of	ADP
cana-4565	238	18	a	a	DET
cana-4565	238	19	group	group	NOUN
cana-4565	238	20	”	"	PUNCT
cana-4565	238	21	,	,	PUNCT
cana-4565	238	22	bulletin	bulletin	NOUN
cana-4565	238	23	of	of	ADP
cana-4565	238	24	society	society	NOUN
cana-4565	238	25	for	for	ADP
cana-4565	238	26	mathematical	mathematical	ADJ
cana-4565	238	27	services	service	NOUN
cana-4565	238	28	and	and	CCONJ
cana-4565	238	29	standards	standard	NOUN
cana-4565	238	30	,	,	PUNCT
cana-4565	238	31	vol	vol	NOUN
cana-4565	238	32	.	.	NOUN
cana-4565	238	33	2	2	NUM
cana-4565	239	1	no	no	NOUN
cana-4565	239	2	.	.	NOUN
cana-4565	239	3	3	3	NUM
cana-4565	239	4	(	(	PUNCT
cana-4565	239	5	2013	2013	NUM
cana-4565	239	6	)	)	PUNCT
cana-4565	239	7	,	,	PUNCT
cana-4565	239	8	pp	pp	ADP
cana-4565	239	9	.	.	PUNCT
cana-4565	240	1	52	52	NUM
cana-4565	240	2	−	−	NOUN
cana-4565	240	3	59	59	NUM
cana-4565	240	4	.	.	PUNCT
cana-4565	241	1	[	[	X
cana-4565	241	2	2	2	NUM
cana-4565	241	3	]	]	PUNCT
cana-4565	241	4	.	.	PUNCT
cana-4565	242	1	arsham	arsham	PROPN
cana-4565	242	2	borumand	borumand	PROPN
cana-4565	242	3	saeid	saeid	PROPN
cana-4565	242	4	,	,	PUNCT
cana-4565	242	5	“	"	PUNCT
cana-4565	242	6	bipolar	bipolar	ADJ
cana-4565	242	7	-	-	PUNCT
cana-4565	242	8	valued	value	VERB
cana-4565	242	9	fuzzy	fuzzy	ADJ
cana-4565	242	10	bck	bck	PROPN
cana-4565	242	11	/	/	SYM
cana-4565	242	12	bci	bci	NOUN
cana-4565	242	13	-	-	PUNCT
cana-4565	242	14	algebras	algebra	NOUN
cana-4565	242	15	”	"	PUNCT
cana-4565	242	16	,	,	PUNCT
cana-4565	242	17	world	world	NOUN
cana-4565	242	18	applied	apply	VERB
cana-4565	242	19	sciences	science	NOUN
cana-4565	242	20	journal	journal	NOUN
cana-4565	242	21	,	,	PUNCT
cana-4565	242	22	7	7	NUM
cana-4565	242	23	(	(	PUNCT
cana-4565	242	24	11	11	NUM
cana-4565	242	25	)	)	PUNCT
cana-4565	242	26	(	(	PUNCT
cana-4565	242	27	2009	2009	NUM
cana-4565	242	28	)	)	PUNCT
cana-4565	242	29	,	,	PUNCT
cana-4565	242	30	1404	1404	NUM
cana-4565	242	31	−	−	NOUN
cana-4565	242	32	1411	1411	NUM
cana-4565	242	33	.	.	PUNCT
cana-4565	243	1	[	[	X
cana-4565	243	2	3	3	NUM
cana-4565	243	3	]	]	PUNCT
cana-4565	243	4	.	.	PUNCT
cana-4565	244	1	azriel	azriel	PROPN
cana-4565	244	2	rosenfeld	rosenfeld	PROPN
cana-4565	244	3	,	,	PUNCT
cana-4565	244	4	“	"	PUNCT
cana-4565	244	5	fuzzy	fuzzy	ADJ
cana-4565	244	6	groups	group	NOUN
cana-4565	244	7	”	"	PUNCT
cana-4565	244	8	,	,	PUNCT
cana-4565	244	9	journal	journal	NOUN
cana-4565	244	10	of	of	ADP
cana-4565	244	11	mathematical	mathematical	ADJ
cana-4565	244	12	analysis	analysis	NOUN
cana-4565	244	13	and	and	CCONJ
cana-4565	244	14	applications	application	NOUN
cana-4565	244	15	,	,	PUNCT
cana-4565	244	16	35(1971	35(1971	NUM
cana-4565	244	17	)	)	PUNCT
cana-4565	244	18	,	,	PUNCT
cana-4565	244	19	512	512	NUM
cana-4565	244	20	−	−	NUM
cana-4565	244	21	517	517	NUM
cana-4565	244	22	.	.	PUNCT
cana-4565	245	1	[	[	X
cana-4565	245	2	4	4	NUM
cana-4565	245	3	]	]	PUNCT
cana-4565	245	4	.	.	PUNCT
cana-4565	246	1	balasubramanian.a	balasubramanian.a	PROPN
cana-4565	246	2	,	,	PUNCT
cana-4565	246	3	k.l.muruganantha	k.l.muruganantha	PROPN
cana-4565	246	4	prasad	prasad	PROPN
cana-4565	246	5	&	&	CCONJ
cana-4565	246	6	k.arjunan	k.arjunan	PROPN
cana-4565	246	7	,	,	PUNCT
cana-4565	246	8	“	"	PUNCT
cana-4565	246	9	properties	property	NOUN
cana-4565	246	10	of	of	ADP
cana-4565	246	11	bipolar	bipolar	ADJ
cana-4565	246	12	interval	interval	NOUN
cana-4565	246	13	valued	value	VERB
cana-4565	246	14	fuzzy	fuzzy	ADJ
cana-4565	246	15	subgroups	subgroup	NOUN
cana-4565	246	16	of	of	ADP
cana-4565	246	17	a	a	DET
cana-4565	246	18	group	group	NOUN
cana-4565	246	19	”	"	PUNCT
cana-4565	246	20	,	,	PUNCT
cana-4565	246	21	international	international	ADJ
cana-4565	246	22	journal	journal	NOUN
cana-4565	246	23	of	of	ADP
cana-4565	246	24	scientific	scientific	ADJ
cana-4565	246	25	research	research	NOUN
cana-4565	246	26	,	,	PUNCT
cana-4565	246	27	vol	vol	NOUN
cana-4565	246	28	.	.	PROPN
cana-4565	246	29	4	4	NUM
cana-4565	246	30	,	,	PUNCT
cana-4565	246	31	iss	iss	PROPN
cana-4565	246	32	.	.	PROPN
cana-4565	246	33	4	4	NUM
cana-4565	246	34	(	(	PUNCT
cana-4565	246	35	2015	2015	NUM
cana-4565	246	36	)	)	PUNCT
cana-4565	246	37	,	,	PUNCT
cana-4565	246	38	262	262	NUM
cana-4565	246	39	268	268	NUM
cana-4565	246	40	.	.	PUNCT
cana-4565	247	1	[	[	X
cana-4565	247	2	5	5	NUM
cana-4565	247	3	]	]	PUNCT
cana-4565	247	4	.	.	PUNCT
cana-4565	248	1	grattan	grattan	PROPN
cana-4565	248	2	-	-	PUNCT
cana-4565	248	3	guiness	guiness	PROPN
cana-4565	248	4	,	,	PUNCT
cana-4565	248	5	“	"	PUNCT
cana-4565	248	6	fuzzy	fuzzy	ADJ
cana-4565	248	7	membership	membership	NOUN
cana-4565	248	8	mapped	map	VERB
cana-4565	248	9	onto	onto	ADP
cana-4565	248	10	interval	interval	NOUN
cana-4565	248	11	and	and	CCONJ
cana-4565	248	12	many	many	ADJ
cana-4565	248	13	valued	value	VERB
cana-4565	248	14	quantities	quantity	NOUN
cana-4565	248	15	”	"	PUNCT
cana-4565	248	16	,	,	PUNCT
cana-4565	248	17	z.math.logik	z.math.logik	PROPN
cana-4565	248	18	.	.	PUNCT
cana-4565	248	19	grundladen	grundladen	PROPN
cana-4565	248	20	math	math	NOUN
cana-4565	248	21	.	.	PUNCT
cana-4565	249	1	22	22	NUM
cana-4565	249	2	(	(	PUNCT
cana-4565	249	3	1975	1975	NUM
cana-4565	249	4	)	)	PUNCT
cana-4565	249	5	,	,	PUNCT
cana-4565	249	6	149	149	NUM
cana-4565	249	7	−	−	NUM
cana-4565	249	8	160	160	NUM
cana-4565	249	9	.	.	PUNCT
cana-4565	250	1	[	[	X
cana-4565	250	2	6	6	NUM
cana-4565	250	3	]	]	PUNCT
cana-4565	250	4	.	.	PUNCT
cana-4565	251	1	kyoung	kyoung	PROPN
cana-4565	251	2	ja	ja	PROPN
cana-4565	251	3	lee	lee	PROPN
cana-4565	251	4	,	,	PUNCT
cana-4565	251	5	“	"	PUNCT
cana-4565	251	6	bipolar	bipolar	ADJ
cana-4565	251	7	fuzzy	fuzzy	ADJ
cana-4565	251	8	subalgebras	subalgebra	NOUN
cana-4565	251	9	and	and	CCONJ
cana-4565	251	10	bipolar	bipolar	ADJ
cana-4565	251	11	fuzzy	fuzzy	ADJ
cana-4565	251	12	ideals	ideal	NOUN
cana-4565	251	13	of	of	ADP
cana-4565	251	14	bck	bck	PROPN
cana-4565	251	15	/	/	SYM
cana-4565	251	16	bci	bci	PROPN
cana-4565	251	17	algebras	algebra	NOUN
cana-4565	251	18	”	"	PUNCT
cana-4565	251	19	,	,	PUNCT
cana-4565	251	20	bull	bull	NOUN
cana-4565	251	21	.	.	PUNCT
cana-4565	252	1	malays.math	malays.math	PROPN
cana-4565	252	2	.	.	PUNCT
cana-4565	253	1	sci	sci	PROPN
cana-4565	253	2	.	.	PUNCT
cana-4565	253	3	soc	soc	PROPN
cana-4565	253	4	.	.	PUNCT
cana-4565	253	5	,	,	PUNCT
cana-4565	253	6	(	(	PUNCT
cana-4565	253	7	2	2	X
cana-4565	253	8	)	)	PUNCT
cana-4565	253	9	32(3	32(3	NUM
cana-4565	253	10	)	)	PUNCT
cana-4565	253	11	(	(	PUNCT
cana-4565	253	12	2009	2009	NUM
cana-4565	253	13	)	)	PUNCT
cana-4565	253	14	,	,	PUNCT
cana-4565	253	15	361	361	NUM
cana-4565	253	16	–	–	PUNCT
cana-4565	253	17	373	373	NUM
cana-4565	253	18	.	.	PUNCT
cana-4565	254	1	[	[	X
cana-4565	254	2	7	7	NUM
cana-4565	254	3	]	]	PUNCT
cana-4565	254	4	.	.	PUNCT
cana-4565	255	1	k.m.lee	k.m.lee	PROPN
cana-4565	255	2	,	,	PUNCT
cana-4565	255	3	“	"	PUNCT
cana-4565	255	4	bipolar	bipolar	ADJ
cana-4565	255	5	-	-	PUNCT
cana-4565	255	6	valued	value	VERB
cana-4565	255	7	fuzzy	fuzzy	ADJ
cana-4565	255	8	sets	set	NOUN
cana-4565	255	9	and	and	CCONJ
cana-4565	255	10	their	their	PRON
cana-4565	255	11	operations	operation	NOUN
cana-4565	255	12	”	"	PUNCT
cana-4565	255	13	.	.	PUNCT
cana-4565	256	1	proc	proc	NOUN
cana-4565	256	2	.	.	PUNCT
cana-4565	257	1	int	int	NOUN
cana-4565	257	2	.	.	PUNCT
cana-4565	257	3	conf	conf	PROPN
cana-4565	257	4	.	.	PUNCT
cana-4565	258	1	on	on	ADP
cana-4565	258	2	intelligent	intelligent	ADJ
cana-4565	258	3	technologies	technology	NOUN
cana-4565	258	4	,	,	PUNCT
cana-4565	258	5	bangkok	bangkok	PROPN
cana-4565	258	6	,	,	PUNCT
cana-4565	258	7	thailand	thailand	PROPN
cana-4565	258	8	,	,	PUNCT
cana-4565	258	9	(	(	PUNCT
cana-4565	258	10	2000	2000	NUM
cana-4565	258	11	)	)	PUNCT
cana-4565	258	12	,	,	PUNCT
cana-4565	258	13	307	307	NUM
cana-4565	258	14	−	−	NUM
cana-4565	258	15	312	312	NUM
cana-4565	258	16	.	.	PUNCT
cana-4565	259	1	[	[	X
cana-4565	259	2	8	8	NUM
cana-4565	259	3	]	]	PUNCT
cana-4565	259	4	.	.	PUNCT
cana-4565	260	1	k.m.lee	k.m.lee	PROPN
cana-4565	260	2	,	,	PUNCT
cana-4565	260	3	“	"	PUNCT
cana-4565	260	4	comparison	comparison	NOUN
cana-4565	260	5	of	of	ADP
cana-4565	260	6	interval	interval	NOUN
cana-4565	260	7	-	-	PUNCT
cana-4565	260	8	valued	value	VERB
cana-4565	260	9	fuzzy	fuzzy	ADJ
cana-4565	260	10	sets	set	NOUN
cana-4565	260	11	,	,	PUNCT
cana-4565	260	12	intuitionistic	intuitionistic	ADJ
cana-4565	260	13	fuzzy	fuzzy	ADJ
cana-4565	260	14	sets	set	NOUN
cana-4565	260	15	and	and	CCONJ
cana-4565	260	16	bipolarvalued	bipolarvalue	VERB
cana-4565	260	17	fuzzy	fuzzy	ADJ
cana-4565	260	18	sets	set	NOUN
cana-4565	260	19	”	"	PUNCT
cana-4565	260	20	.	.	PUNCT
cana-4565	261	1	j.	j.	PROPN
cana-4565	261	2	fuzzy	fuzzy	ADJ
cana-4565	261	3	logic	logic	NOUN
cana-4565	261	4	intelligent	intelligent	ADJ
cana-4565	261	5	systems	system	NOUN
cana-4565	261	6	,	,	PUNCT
cana-4565	261	7	14	14	NUM
cana-4565	261	8	(	(	PUNCT
cana-4565	261	9	2	2	NUM
cana-4565	261	10	)	)	PUNCT
cana-4565	261	11	(	(	PUNCT
cana-4565	261	12	2004	2004	NUM
cana-4565	261	13	)	)	PUNCT
cana-4565	261	14	,	,	PUNCT
cana-4565	261	15	125	125	NUM
cana-4565	261	16	−129	−129	NOUN
cana-4565	261	17	.	.	PUNCT
cana-4565	262	1	[	[	X
cana-4565	262	2	9	9	NUM
cana-4565	262	3	]	]	PUNCT
cana-4565	262	4	.	.	PUNCT
cana-4565	263	1	murugalingam.k	murugalingam.k	PROPN
cana-4565	263	2	and	and	CCONJ
cana-4565	263	3	k.arjunan	k.arjunan	NOUN
cana-4565	263	4	,	,	PUNCT
cana-4565	263	5	“	"	PUNCT
cana-4565	263	6	a	a	DET
cana-4565	263	7	study	study	NOUN
cana-4565	263	8	on	on	ADP
cana-4565	263	9	interval	interval	NOUN
cana-4565	263	10	valued	value	VERB
cana-4565	263	11	fuzzy	fuzzy	ADJ
cana-4565	263	12	subsemirings	subsemiring	NOUN
cana-4565	263	13	of	of	ADP
cana-4565	263	14	a	a	DET
cana-4565	263	15	semiring	semiring	NOUN
cana-4565	263	16	”	"	PUNCT
cana-4565	263	17	,	,	PUNCT
cana-4565	263	18	international	international	ADJ
cana-4565	263	19	journal	journal	NOUN
cana-4565	263	20	of	of	ADP
cana-4565	263	21	applied	apply	VERB
cana-4565	263	22	mathematics	mathematic	NOUN
cana-4565	263	23	and	and	CCONJ
cana-4565	263	24	modeling	modeling	NOUN
cana-4565	263	25	,	,	PUNCT
cana-4565	263	26	vol	vol	NOUN
cana-4565	263	27	.	.	PROPN
cana-4565	263	28	1	1	NUM
cana-4565	263	29	,	,	PUNCT
cana-4565	263	30	no	no	INTJ
cana-4565	263	31	.	.	NOUN
cana-4565	263	32	5	5	NUM
cana-4565	263	33	(	(	PUNCT
cana-4565	263	34	2013	2013	NUM
cana-4565	263	35	)	)	PUNCT
cana-4565	263	36	,	,	PUNCT
cana-4565	263	37	1	1	NUM
cana-4565	263	38	−	−	PROPN
cana-4565	263	39	6	6	NUM
cana-4565	263	40	.	.	PUNCT
cana-4565	264	1	[	[	X
cana-4565	264	2	10	10	NUM
cana-4565	264	3	]	]	PUNCT
cana-4565	264	4	.	.	PUNCT
cana-4565	265	1	sabu	sabu	PROPN
cana-4565	265	2	sebastian	sebastian	PROPN
cana-4565	265	3	,	,	PUNCT
cana-4565	265	4	t.v.ramakrishnan	t.v.ramakrishnan	NOUN
cana-4565	265	5	,	,	PUNCT
cana-4565	265	6	“	"	PUNCT
cana-4565	265	7	multi	multi	X
cana-4565	265	8	fuzzy	fuzzy	ADJ
cana-4565	265	9	sets	set	NOUN
cana-4565	265	10	”	"	PUNCT
cana-4565	265	11	,	,	PUNCT
cana-4565	265	12	international	international	PROPN
cana-4565	265	13	mathematical	mathematical	ADJ
cana-4565	265	14	forum	forum	PROPN
cana-4565	265	15	,	,	PUNCT
cana-4565	265	16	5	5	NUM
cana-4565	265	17	,	,	PUNCT
cana-4565	265	18	no.50	no.50	PROPN
cana-4565	265	19	(	(	PUNCT
cana-4565	265	20	2010	2010	NUM
cana-4565	265	21	)	)	PUNCT
cana-4565	265	22	,	,	PUNCT
cana-4565	265	23	2471	2471	NUM
cana-4565	265	24	−2476	−2476	NOUN
cana-4565	265	25	.	.	PUNCT
cana-4565	266	1	[	[	X
cana-4565	266	2	11	11	NUM
cana-4565	266	3	]	]	PUNCT
cana-4565	266	4	.	.	PUNCT
cana-4565	266	5	vairamuthu.k	vairamuthu.k	PROPN
cana-4565	266	6	,	,	PUNCT
cana-4565	266	7	s.loganathan	s.loganathan	PROPN
cana-4565	266	8	,	,	PUNCT
cana-4565	266	9	“	"	PUNCT
cana-4565	266	10	product	product	NOUN
cana-4565	266	11	in	in	ADP
cana-4565	266	12	bipolar	bipolar	ADJ
cana-4565	266	13	valued	value	VERB
cana-4565	266	14	multi	multi	NOUN
cana-4565	266	15	i	i	PRON
cana-4565	266	16	-	-	PUNCT
cana-4565	266	17	fuzzy	fuzzy	ADJ
cana-4565	266	18	subrings	subring	NOUN
cana-4565	266	19	of	of	ADP
cana-4565	266	20	a	a	DET
cana-4565	266	21	ring	ring	NOUN
cana-4565	266	22	”	"	PUNCT
cana-4565	266	23	,	,	PUNCT
cana-4565	266	24	gradiva	gradiva	PROPN
cana-4565	266	25	review	review	PROPN
cana-4565	266	26	journal	journal	PROPN
cana-4565	266	27	,	,	PUNCT
cana-4565	266	28	vol.8	vol.8	PROPN
cana-4565	266	29	,	,	PUNCT
cana-4565	266	30	issue	issue	NOUN
cana-4565	266	31	11(2022	11(2022	NOUN
cana-4565	266	32	)	)	PUNCT
cana-4565	266	33	.	.	PUNCT
cana-4565	267	1	[	[	X
cana-4565	267	2	12	12	NUM
cana-4565	267	3	]	]	PUNCT
cana-4565	267	4	.	.	PUNCT
cana-4565	268	1	vairamuthu.k	vairamuthu.k	PROPN
cana-4565	268	2	,	,	PUNCT
cana-4565	268	3	s.loganathan	s.loganathan	PROPN
cana-4565	268	4	,	,	PUNCT
cana-4565	268	5	“	"	PUNCT
cana-4565	268	6	bipolar	bipolar	ADJ
cana-4565	268	7	valued	value	VERB
cana-4565	268	8	multi	multi	NOUN
cana-4565	268	9	i	i	PRON
cana-4565	268	10	-	-	PUNCT
cana-4565	268	11	fuzzy	fuzzy	ADJ
cana-4565	268	12	subrings	subring	NOUN
cana-4565	268	13	of	of	ADP
cana-4565	268	14	a	a	DET
cana-4565	268	15	ring	ring	NOUN
cana-4565	268	16	”	"	PUNCT
cana-4565	268	17	,	,	PUNCT
cana-4565	268	18	journal	journal	NOUN
cana-4565	268	19	for	for	ADP
cana-4565	268	20	basic	basic	ADJ
cana-4565	268	21	sciences	science	NOUN
cana-4565	268	22	,	,	PUNCT
cana-4565	268	23	vol.23	vol.23	NOUN
cana-4565	268	24	,	,	PUNCT
cana-4565	268	25	issue	issue	NOUN
cana-4565	268	26	1(2023	1(2023	NUM
cana-4565	268	27	)	)	PUNCT
cana-4565	268	28	.	.	PUNCT
cana-4565	269	1	[	[	X
cana-4565	269	2	13	13	NUM
cana-4565	269	3	]	]	PUNCT
cana-4565	269	4	.	.	PUNCT
cana-4565	270	1	yasodara.s	yasodara.s	PROPN
cana-4565	270	2	,	,	PUNCT
cana-4565	270	3	ke	ke	NOUN
cana-4565	270	4	.	.	PUNCT
cana-4565	270	5	sathappan	sathappan	ADJ
cana-4565	270	6	,	,	PUNCT
cana-4565	270	7	“	"	PUNCT
cana-4565	270	8	bipolar	bipolar	ADJ
cana-4565	270	9	-	-	PUNCT
cana-4565	270	10	valued	value	VERB
cana-4565	270	11	multi	multi	ADJ
cana-4565	270	12	fuzzy	fuzzy	ADJ
cana-4565	270	13	subsemirings	subsemiring	NOUN
cana-4565	270	14	of	of	ADP
cana-4565	270	15	a	a	DET
cana-4565	270	16	semiring	semiring	NOUN
cana-4565	270	17	”	"	PUNCT
cana-4565	270	18	,	,	PUNCT
cana-4565	270	19	international	international	ADJ
cana-4565	270	20	journal	journal	NOUN
cana-4565	270	21	of	of	ADP
cana-4565	270	22	mathematical	mathematical	ADJ
cana-4565	270	23	archive	archive	NOUN
cana-4565	270	24	,	,	PUNCT
cana-4565	270	25	6(9	6(9	NUM
cana-4565	270	26	)	)	PUNCT
cana-4565	270	27	(	(	PUNCT
cana-4565	270	28	2015	2015	NUM
cana-4565	270	29	)	)	PUNCT
cana-4565	270	30	,	,	PUNCT
cana-4565	270	31	75	75	NUM
cana-4565	270	32	−80	−80	NOUN
cana-4565	270	33	.	.	PUNCT
cana-4565	271	1	[	[	X
cana-4565	271	2	14	14	NUM
cana-4565	271	3	]	]	PUNCT
cana-4565	271	4	.	.	PUNCT
cana-4565	272	1	l.a.zadeh	l.a.zadeh	NOUN
cana-4565	272	2	,	,	PUNCT
cana-4565	272	3	fuzzy	fuzzy	ADJ
cana-4565	272	4	sets	set	NOUN
cana-4565	272	5	,	,	PUNCT
cana-4565	272	6	inform	inform	NOUN
cana-4565	272	7	.	.	PUNCT
cana-4565	273	1	and	and	CCONJ
cana-4565	273	2	control	control	NOUN
cana-4565	273	3	,	,	PUNCT
cana-4565	273	4	8(1965	8(1965	NUM
cana-4565	273	5	)	)	PUNCT
cana-4565	273	6	,	,	PUNCT
cana-4565	273	7	338	338	NUM
cana-4565	273	8	−353	−353	NOUN
cana-4565	273	9	.	.	PUNCT
cana-4565	274	1	[	[	X
cana-4565	274	2	15	15	NUM
cana-4565	274	3	]	]	PUNCT
cana-4565	274	4	.	.	PUNCT
cana-4565	275	1	w.r.zhang	w.r.zhang	PROPN
cana-4565	275	2	,	,	PUNCT
cana-4565	275	3	bipolar	bipolar	ADJ
cana-4565	275	4	fuzzy	fuzzy	ADJ
cana-4565	275	5	sets	set	NOUN
cana-4565	275	6	and	and	CCONJ
cana-4565	275	7	relations	relation	NOUN
cana-4565	275	8	,	,	PUNCT
cana-4565	275	9	a	a	DET
cana-4565	275	10	computational	computational	ADJ
cana-4565	275	11	frame	frame	NOUN
cana-4565	275	12	work	work	NOUN
cana-4565	275	13	for	for	ADP
cana-4565	275	14	cognitive	cognitive	ADJ
cana-4565	275	15	modeling	modeling	NOUN
cana-4565	275	16	and	and	CCONJ
cana-4565	275	17	multiple	multiple	ADJ
cana-4565	275	18	decision	decision	NOUN
cana-4565	275	19	analysis	analysis	NOUN
cana-4565	275	20	,	,	PUNCT
cana-4565	275	21	proceedings	proceeding	NOUN
cana-4565	275	22	of	of	ADP
cana-4565	275	23	fuzzy	fuzzy	ADJ
cana-4565	275	24	ieee	ieee	NOUN
cana-4565	275	25	conferences	conference	NOUN
cana-4565	275	26	,	,	PUNCT
cana-4565	275	27	(	(	PUNCT
cana-4565	275	28	1994	1994	NUM
cana-4565	275	29	)	)	PUNCT
cana-4565	275	30	,	,	PUNCT
cana-4565	275	31	305−	305−	NUM
cana-4565	275	32	309	309	NUM
cana-4565	275	33	.	.	PUNCT