id	sid	tid	token	lemma	pos
ejpam-5914	1	1	european	european	PROPN
ejpam-5914	1	2	journal	journal	PROPN
ejpam-5914	1	3	of	of	ADP
ejpam-5914	1	4	pure	pure	ADJ
ejpam-5914	1	5	and	and	CCONJ
ejpam-5914	1	6	applied	applied	ADJ
ejpam-5914	1	7	mathematics	mathematic	NOUN
ejpam-5914	1	8	2025	2025	NUM
ejpam-5914	1	9	,	,	PUNCT
ejpam-5914	1	10	vol	vol	NOUN
ejpam-5914	1	11	.	.	PROPN
ejpam-5914	1	12	18	18	NUM
ejpam-5914	1	13	,	,	PUNCT
ejpam-5914	1	14	issue	issue	NOUN
ejpam-5914	1	15	2	2	NUM
ejpam-5914	1	16	,	,	PUNCT
ejpam-5914	1	17	article	article	NOUN
ejpam-5914	1	18	number	number	NOUN
ejpam-5914	1	19	5914	5914	NUM
ejpam-5914	1	20	issn	issn	VERB
ejpam-5914	1	21	1307	1307	NUM
ejpam-5914	1	22	-	-	SYM
ejpam-5914	1	23	5543	5543	NUM
ejpam-5914	1	24	–	–	PUNCT
ejpam-5914	1	25	ejpam.com	ejpam.com	X
ejpam-5914	1	26	published	publish	VERB
ejpam-5914	1	27	by	by	ADP
ejpam-5914	1	28	new	new	PROPN
ejpam-5914	1	29	york	york	PROPN
ejpam-5914	1	30	business	business	PROPN
ejpam-5914	1	31	global	global	ADJ
ejpam-5914	1	32	investigating	investigate	VERB
ejpam-5914	1	33	length	length	NOUN
ejpam-5914	1	34	and	and	CCONJ
ejpam-5914	1	35	mean	mean	ADJ
ejpam-5914	1	36	-	-	PUNCT
ejpam-5914	1	37	fuzzy	fuzzy	ADJ
ejpam-5914	1	38	subalgebras	subalgebra	NOUN
ejpam-5914	1	39	in	in	ADP
ejpam-5914	1	40	sheffer	sheffer	PROPN
ejpam-5914	1	41	stroke	stroke	PROPN
ejpam-5914	1	42	hilbert	hilbert	PROPN
ejpam-5914	1	43	algebras	algebras	PROPN
ejpam-5914	1	44	neelamegarajan	neelamegarajan	PROPN
ejpam-5914	1	45	rajesh1	rajesh1	PROPN
ejpam-5914	1	46	,	,	PUNCT
ejpam-5914	1	47	tahsin	tahsin	PROPN
ejpam-5914	1	48	oner2	oner2	VERB
ejpam-5914	1	49	,	,	PUNCT
ejpam-5914	1	50	aiyared	aiyare	VERB
ejpam-5914	1	51	iampan3,∗	iampan3,∗	NOUN
ejpam-5914	1	52	,	,	PUNCT
ejpam-5914	1	53	akbar	akbar	NOUN
ejpam-5914	1	54	rezaei4	rezaei4	NOUN
ejpam-5914	1	55	1	1	NUM
ejpam-5914	1	56	department	department	NOUN
ejpam-5914	1	57	of	of	ADP
ejpam-5914	1	58	mathematics	mathematic	NOUN
ejpam-5914	1	59	,	,	PUNCT
ejpam-5914	1	60	rajah	rajah	NOUN
ejpam-5914	1	61	serfoji	serfoji	ADJ
ejpam-5914	1	62	government	government	NOUN
ejpam-5914	1	63	college	college	NOUN
ejpam-5914	1	64	,	,	PUNCT
ejpam-5914	1	65	thanjavur-613005	thanjavur-613005	NOUN
ejpam-5914	1	66	,	,	PUNCT
ejpam-5914	1	67	tamil	tamil	PROPN
ejpam-5914	1	68	nadu	nadu	NOUN
ejpam-5914	1	69	,	,	PUNCT
ejpam-5914	1	70	india	india	PROPN
ejpam-5914	1	71	2	2	NUM
ejpam-5914	1	72	department	department	NOUN
ejpam-5914	1	73	of	of	ADP
ejpam-5914	1	74	mathematics	mathematic	NOUN
ejpam-5914	1	75	,	,	PUNCT
ejpam-5914	1	76	faculty	faculty	NOUN
ejpam-5914	1	77	of	of	ADP
ejpam-5914	1	78	science	science	NOUN
ejpam-5914	1	79	,	,	PUNCT
ejpam-5914	1	80	ege	ege	PROPN
ejpam-5914	1	81	university	university	NOUN
ejpam-5914	1	82	,	,	PUNCT
ejpam-5914	1	83	35100	35100	NUM
ejpam-5914	1	84	izmir	izmir	PROPN
ejpam-5914	1	85	,	,	PUNCT
ejpam-5914	1	86	turkey	turkey	PROPN
ejpam-5914	1	87	3	3	NUM
ejpam-5914	1	88	department	department	NOUN
ejpam-5914	1	89	of	of	ADP
ejpam-5914	1	90	mathematics	mathematic	NOUN
ejpam-5914	1	91	,	,	PUNCT
ejpam-5914	1	92	school	school	NOUN
ejpam-5914	1	93	of	of	ADP
ejpam-5914	1	94	science	science	NOUN
ejpam-5914	1	95	,	,	PUNCT
ejpam-5914	1	96	university	university	NOUN
ejpam-5914	1	97	of	of	ADP
ejpam-5914	1	98	phayao	phayao	NOUN
ejpam-5914	1	99	,	,	PUNCT
ejpam-5914	1	100	mae	mae	PROPN
ejpam-5914	1	101	ka	ka	PROPN
ejpam-5914	1	102	,	,	PUNCT
ejpam-5914	1	103	mueang	mueang	PROPN
ejpam-5914	1	104	,	,	PUNCT
ejpam-5914	1	105	phayao	phayao	NOUN
ejpam-5914	1	106	56000	56000	NUM
ejpam-5914	1	107	,	,	PUNCT
ejpam-5914	1	108	thailand	thailand	PROPN
ejpam-5914	1	109	4	4	NUM
ejpam-5914	1	110	department	department	NOUN
ejpam-5914	1	111	of	of	ADP
ejpam-5914	1	112	mathematics	mathematic	NOUN
ejpam-5914	1	113	,	,	PUNCT
ejpam-5914	1	114	faculty	faculty	NOUN
ejpam-5914	1	115	of	of	ADP
ejpam-5914	1	116	basic	basic	ADJ
ejpam-5914	1	117	science	science	NOUN
ejpam-5914	1	118	,	,	PUNCT
ejpam-5914	1	119	payame	payame	NOUN
ejpam-5914	1	120	noor	noor	PROPN
ejpam-5914	1	121	university	university	PROPN
ejpam-5914	1	122	,	,	PUNCT
ejpam-5914	1	123	p.o	p.o	PROPN
ejpam-5914	1	124	.	.	PROPN
ejpam-5914	1	125	box	box	PROPN
ejpam-5914	1	126	19395	19395	NUM
ejpam-5914	1	127	-	-	SYM
ejpam-5914	1	128	4697	4697	NUM
ejpam-5914	1	129	,	,	PUNCT
ejpam-5914	1	130	tehran	tehran	PROPN
ejpam-5914	1	131	,	,	PUNCT
ejpam-5914	1	132	iran	iran	PROPN
ejpam-5914	1	133	abstract	abstract	ADJ
ejpam-5914	1	134	.	.	PUNCT
ejpam-5914	2	1	the	the	DET
ejpam-5914	2	2	aim	aim	NOUN
ejpam-5914	2	3	of	of	ADP
ejpam-5914	2	4	this	this	DET
ejpam-5914	2	5	paper	paper	NOUN
ejpam-5914	2	6	is	be	AUX
ejpam-5914	2	7	to	to	PART
ejpam-5914	2	8	introduce	introduce	VERB
ejpam-5914	2	9	the	the	DET
ejpam-5914	2	10	notions	notion	NOUN
ejpam-5914	2	11	of	of	ADP
ejpam-5914	2	12	the	the	DET
ejpam-5914	2	13	length	length	NOUN
ejpam-5914	2	14	and	and	CCONJ
ejpam-5914	2	15	the	the	DET
ejpam-5914	2	16	mean	mean	NOUN
ejpam-5914	2	17	of	of	ADP
ejpam-5914	2	18	an	an	DET
ejpam-5914	2	19	interval	interval	NOUN
ejpam-5914	2	20	-	-	PUNCT
ejpam-5914	2	21	valued	value	VERB
ejpam-5914	2	22	fuzzy	fuzzy	ADJ
ejpam-5914	2	23	structure	structure	NOUN
ejpam-5914	2	24	in	in	ADP
ejpam-5914	2	25	sheffer	sheffer	PROPN
ejpam-5914	2	26	stroke	stroke	PROPN
ejpam-5914	2	27	hilbert	hilbert	PROPN
ejpam-5914	2	28	algebras	algebras	PROPN
ejpam-5914	2	29	.	.	PUNCT
ejpam-5914	3	1	the	the	DET
ejpam-5914	3	2	notions	notion	NOUN
ejpam-5914	3	3	of	of	ADP
ejpam-5914	3	4	length	length	NOUN
ejpam-5914	3	5	-	-	PUNCT
ejpam-5914	3	6	fuzzy	fuzzy	ADJ
ejpam-5914	3	7	subalgebras	subalgebra	NOUN
ejpam-5914	3	8	and	and	CCONJ
ejpam-5914	3	9	mean	mean	ADJ
ejpam-5914	3	10	-	-	PUNCT
ejpam-5914	3	11	fuzzy	fuzzy	ADJ
ejpam-5914	3	12	subalgebras	subalgebra	NOUN
ejpam-5914	3	13	of	of	ADP
ejpam-5914	3	14	sheffer	sheffer	PROPN
ejpam-5914	3	15	stroke	stroke	PROPN
ejpam-5914	3	16	hilbert	hilbert	PROPN
ejpam-5914	3	17	algebras	algebras	PROPN
ejpam-5914	3	18	are	be	AUX
ejpam-5914	3	19	introduced	introduce	VERB
ejpam-5914	3	20	,	,	PUNCT
ejpam-5914	3	21	and	and	CCONJ
ejpam-5914	3	22	related	related	ADJ
ejpam-5914	3	23	properties	property	NOUN
ejpam-5914	3	24	are	be	AUX
ejpam-5914	3	25	investigated	investigate	VERB
ejpam-5914	3	26	.	.	PUNCT
ejpam-5914	4	1	characterizations	characterization	NOUN
ejpam-5914	4	2	of	of	ADP
ejpam-5914	4	3	length	length	NOUN
ejpam-5914	4	4	-	-	PUNCT
ejpam-5914	4	5	fuzzy	fuzzy	ADJ
ejpam-5914	4	6	subalgebras	subalgebra	NOUN
ejpam-5914	4	7	and	and	CCONJ
ejpam-5914	4	8	mean	mean	ADJ
ejpam-5914	4	9	-	-	PUNCT
ejpam-5914	4	10	fuzzy	fuzzy	ADJ
ejpam-5914	4	11	subalgebras	subalgebra	NOUN
ejpam-5914	4	12	are	be	AUX
ejpam-5914	4	13	discussed	discuss	VERB
ejpam-5914	4	14	.	.	PUNCT
ejpam-5914	5	1	relations	relation	NOUN
ejpam-5914	5	2	between	between	ADP
ejpam-5914	5	3	length	length	NOUN
ejpam-5914	5	4	-	-	PUNCT
ejpam-5914	5	5	fuzzy	fuzzy	ADJ
ejpam-5914	5	6	subalgebras	subalgebra	NOUN
ejpam-5914	5	7	(	(	PUNCT
ejpam-5914	5	8	resp	resp	NOUN
ejpam-5914	5	9	.	.	PUNCT
ejpam-5914	5	10	,	,	PUNCT
ejpam-5914	5	11	mean	mean	ADJ
ejpam-5914	5	12	-	-	PUNCT
ejpam-5914	5	13	fuzzy	fuzzy	ADJ
ejpam-5914	5	14	subalgebras	subalgebra	NOUN
ejpam-5914	5	15	)	)	PUNCT
ejpam-5914	5	16	and	and	CCONJ
ejpam-5914	5	17	subalgebras	subalgebras	PROPN
ejpam-5914	5	18	are	be	AUX
ejpam-5914	5	19	established	establish	VERB
ejpam-5914	5	20	.	.	PUNCT
ejpam-5914	6	1	moreover	moreover	ADV
ejpam-5914	6	2	,	,	PUNCT
ejpam-5914	6	3	we	we	PRON
ejpam-5914	6	4	discuss	discuss	VERB
ejpam-5914	6	5	the	the	DET
ejpam-5914	6	6	relationships	relationship	NOUN
ejpam-5914	6	7	among	among	ADP
ejpam-5914	6	8	length	length	NOUN
ejpam-5914	6	9	-	-	PUNCT
ejpam-5914	6	10	fuzzy	fuzzy	ADJ
ejpam-5914	6	11	subalgebras	subalgebra	NOUN
ejpam-5914	6	12	(	(	PUNCT
ejpam-5914	6	13	resp	resp	NOUN
ejpam-5914	6	14	.	.	PUNCT
ejpam-5914	6	15	,	,	PUNCT
ejpam-5914	6	16	mean	mean	ADJ
ejpam-5914	6	17	-	-	PUNCT
ejpam-5914	6	18	fuzzy	fuzzy	ADJ
ejpam-5914	6	19	subalgebras	subalgebra	NOUN
ejpam-5914	6	20	)	)	PUNCT
ejpam-5914	6	21	and	and	CCONJ
ejpam-5914	6	22	upper	upper	ADJ
ejpam-5914	6	23	and	and	CCONJ
ejpam-5914	6	24	lower	low	ADJ
ejpam-5914	6	25	-	-	PUNCT
ejpam-5914	6	26	level	level	NOUN
ejpam-5914	6	27	subsets	subset	NOUN
ejpam-5914	6	28	of	of	ADP
ejpam-5914	6	29	the	the	DET
ejpam-5914	6	30	length	length	NOUN
ejpam-5914	6	31	(	(	PUNCT
ejpam-5914	6	32	resp	resp	NOUN
ejpam-5914	6	33	.	.	PUNCT
ejpam-5914	6	34	,	,	PUNCT
ejpam-5914	6	35	mean	mean	VERB
ejpam-5914	6	36	)	)	PUNCT
ejpam-5914	6	37	of	of	ADP
ejpam-5914	6	38	an	an	DET
ejpam-5914	6	39	interval	interval	NOUN
ejpam-5914	6	40	-	-	PUNCT
ejpam-5914	6	41	valued	value	VERB
ejpam-5914	6	42	fuzzy	fuzzy	ADJ
ejpam-5914	6	43	structure	structure	NOUN
ejpam-5914	6	44	in	in	ADP
ejpam-5914	6	45	sheffer	sheffer	PROPN
ejpam-5914	6	46	stroke	stroke	PROPN
ejpam-5914	6	47	hilbert	hilbert	PROPN
ejpam-5914	6	48	algebras	algebras	PROPN
ejpam-5914	6	49	.	.	PUNCT
ejpam-5914	7	1	2020	2020	NUM
ejpam-5914	7	2	mathematics	mathematics	PROPN
ejpam-5914	7	3	subject	subject	NOUN
ejpam-5914	7	4	classifications	classification	NOUN
ejpam-5914	7	5	:	:	PUNCT
ejpam-5914	7	6	20n05	20n05	NUM
ejpam-5914	7	7	,	,	PUNCT
ejpam-5914	7	8	94d05	94d05	NUM
ejpam-5914	7	9	,	,	PUNCT
ejpam-5914	7	10	03e72	03e72	X
ejpam-5914	7	11	key	key	ADJ
ejpam-5914	7	12	words	word	NOUN
ejpam-5914	7	13	and	and	CCONJ
ejpam-5914	7	14	phrases	phrase	NOUN
ejpam-5914	7	15	:	:	PUNCT
ejpam-5914	7	16	sheffer	sheffer	NOUN
ejpam-5914	7	17	stroke	stroke	PROPN
ejpam-5914	7	18	hilbert	hilbert	PROPN
ejpam-5914	7	19	algebra	algebra	PROPN
ejpam-5914	7	20	,	,	PUNCT
ejpam-5914	7	21	subalgebra	subalgebra	NOUN
ejpam-5914	7	22	,	,	PUNCT
ejpam-5914	7	23	length	length	NOUN
ejpam-5914	7	24	-	-	PUNCT
ejpam-5914	7	25	fuzzy	fuzzy	ADJ
ejpam-5914	7	26	subalgebra	subalgebra	NOUN
ejpam-5914	7	27	,	,	PUNCT
ejpam-5914	7	28	mean	mean	ADJ
ejpam-5914	7	29	-	-	PUNCT
ejpam-5914	7	30	fuzzy	fuzzy	ADJ
ejpam-5914	7	31	subalgebra	subalgebra	NOUN
ejpam-5914	7	32	1	1	NUM
ejpam-5914	7	33	.	.	X
ejpam-5914	7	34	introduction	introduction	NOUN
ejpam-5914	7	35	the	the	DET
ejpam-5914	7	36	sheffer	sheffer	NOUN
ejpam-5914	7	37	operation	operation	NOUN
ejpam-5914	7	38	,	,	PUNCT
ejpam-5914	7	39	also	also	ADV
ejpam-5914	7	40	known	know	VERB
ejpam-5914	7	41	as	as	ADP
ejpam-5914	7	42	the	the	DET
ejpam-5914	7	43	sheffer	sheffer	NOUN
ejpam-5914	7	44	stroke	stroke	NOUN
ejpam-5914	7	45	or	or	CCONJ
ejpam-5914	7	46	nand	nand	NOUN
ejpam-5914	7	47	operator	operator	NOUN
ejpam-5914	7	48	,	,	PUNCT
ejpam-5914	7	49	was	be	AUX
ejpam-5914	7	50	first	first	ADV
ejpam-5914	7	51	introduced	introduce	VERB
ejpam-5914	7	52	by	by	ADP
ejpam-5914	7	53	henry	henry	PROPN
ejpam-5914	7	54	maurice	maurice	PROPN
ejpam-5914	7	55	sheffer	sheffer	VERB
ejpam-5914	8	1	[	[	X
ejpam-5914	8	2	1	1	NUM
ejpam-5914	8	3	]	]	PUNCT
ejpam-5914	8	4	.	.	PUNCT
ejpam-5914	9	1	this	this	DET
ejpam-5914	9	2	operation	operation	NOUN
ejpam-5914	9	3	holds	hold	VERB
ejpam-5914	9	4	significance	significance	NOUN
ejpam-5914	9	5	because	because	SCONJ
ejpam-5914	9	6	it	it	PRON
ejpam-5914	9	7	can	can	AUX
ejpam-5914	9	8	be	be	AUX
ejpam-5914	9	9	used	use	VERB
ejpam-5914	9	10	independently	independently	ADV
ejpam-5914	9	11	,	,	PUNCT
ejpam-5914	9	12	without	without	ADP
ejpam-5914	9	13	any	any	DET
ejpam-5914	9	14	other	other	ADJ
ejpam-5914	9	15	logical	logical	ADJ
ejpam-5914	9	16	operators	operator	NOUN
ejpam-5914	9	17	,	,	PUNCT
ejpam-5914	9	18	to	to	PART
ejpam-5914	9	19	construct	construct	VERB
ejpam-5914	9	20	a	a	DET
ejpam-5914	9	21	logical	logical	ADJ
ejpam-5914	9	22	system	system	NOUN
ejpam-5914	9	23	.	.	PUNCT
ejpam-5914	10	1	this	this	PRON
ejpam-5914	10	2	means	mean	VERB
ejpam-5914	10	3	that	that	SCONJ
ejpam-5914	10	4	any	any	DET
ejpam-5914	10	5	axiom	axiom	NOUN
ejpam-5914	10	6	of	of	ADP
ejpam-5914	10	7	a	a	DET
ejpam-5914	10	8	logical	logical	ADJ
ejpam-5914	10	9	system	system	NOUN
ejpam-5914	10	10	can	can	AUX
ejpam-5914	10	11	be	be	AUX
ejpam-5914	10	12	restated	restate	VERB
ejpam-5914	10	13	using	use	VERB
ejpam-5914	10	14	only	only	ADV
ejpam-5914	10	15	the	the	DET
ejpam-5914	10	16	sheffer	sheffer	NOUN
ejpam-5914	10	17	operation	operation	NOUN
ejpam-5914	10	18	.	.	PUNCT
ejpam-5914	11	1	because	because	SCONJ
ejpam-5914	11	2	of	of	ADP
ejpam-5914	11	3	this	this	DET
ejpam-5914	11	4	property	property	NOUN
ejpam-5914	11	5	,	,	PUNCT
ejpam-5914	11	6	it	it	PRON
ejpam-5914	11	7	becomes	become	VERB
ejpam-5914	11	8	easier	easy	ADJ
ejpam-5914	11	9	to	to	PART
ejpam-5914	11	10	control	control	VERB
ejpam-5914	11	11	certain	certain	ADJ
ejpam-5914	11	12	properties	property	NOUN
ejpam-5914	11	13	of	of	ADP
ejpam-5914	11	14	the	the	DET
ejpam-5914	11	15	newly	newly	ADV
ejpam-5914	11	16	constructed	construct	VERB
ejpam-5914	11	17	logical	logical	ADJ
ejpam-5914	11	18	system	system	NOUN
ejpam-5914	11	19	.	.	PUNCT
ejpam-5914	12	1	additionally	additionally	ADV
ejpam-5914	12	2	,	,	PUNCT
ejpam-5914	12	3	it	it	PRON
ejpam-5914	12	4	’s	’	VERB
ejpam-5914	12	5	worth	worth	ADJ
ejpam-5914	12	6	noting	note	VERB
ejpam-5914	12	7	that	that	SCONJ
ejpam-5914	12	8	the	the	DET
ejpam-5914	12	9	axioms	axiom	NOUN
ejpam-5914	12	10	of	of	ADP
ejpam-5914	12	11	boolean	boolean	ADJ
ejpam-5914	12	12	algebra	algebra	NOUN
ejpam-5914	12	13	,	,	PUNCT
ejpam-5914	12	14	which	which	PRON
ejpam-5914	12	15	are	be	AUX
ejpam-5914	12	16	the	the	DET
ejpam-5914	12	17	algebraic	algebraic	ADJ
ejpam-5914	12	18	counterpart	counterpart	NOUN
ejpam-5914	12	19	of	of	ADP
ejpam-5914	12	20	classical	classical	ADJ
ejpam-5914	12	21	propositional	propositional	ADJ
ejpam-5914	12	22	calculus	calculus	NOUN
ejpam-5914	12	23	,	,	PUNCT
ejpam-5914	12	24	∗corresponding	∗corresponde	VERB
ejpam-5914	12	25	author	author	NOUN
ejpam-5914	12	26	.	.	PUNCT
ejpam-5914	13	1	doi	doi	NOUN
ejpam-5914	13	2	:	:	PUNCT
ejpam-5914	13	3	https://doi.org/10.29020/nybg.ejpam.v18i2.5914	https://doi.org/10.29020/nybg.ejpam.v18i2.5914	VERB
ejpam-5914	13	4	email	email	NOUN
ejpam-5914	13	5	addresses	address	NOUN
ejpam-5914	13	6	:	:	PUNCT
ejpam-5914	13	7	nrajesh	nrajesh	PROPN
ejpam-5914	13	8	topology@yahoo.co.in	topology@yahoo.co.in	PROPN
ejpam-5914	13	9	(	(	PUNCT
ejpam-5914	13	10	n.	n.	PROPN
ejpam-5914	13	11	rajesh	rajesh	PROPN
ejpam-5914	13	12	)	)	PUNCT
ejpam-5914	13	13	,	,	PUNCT
ejpam-5914	13	14	tahsin.oner@ege.edu.tr	tahsin.oner@ege.edu.tr	NOUN
ejpam-5914	13	15	(	(	PUNCT
ejpam-5914	13	16	t.	t.	NOUN
ejpam-5914	13	17	oner	oner	PROPN
ejpam-5914	13	18	)	)	PUNCT
ejpam-5914	13	19	,	,	PUNCT
ejpam-5914	13	20	aiyared.ia@up.ac.th	aiyared.ia@up.ac.th	NOUN
ejpam-5914	13	21	(	(	PUNCT
ejpam-5914	13	22	a.	a.	NOUN
ejpam-5914	13	23	iampan	iampan	PROPN
ejpam-5914	13	24	)	)	PUNCT
ejpam-5914	13	25	,	,	PUNCT
ejpam-5914	13	26	rezaei@pnu.ac.ir	rezaei@pnu.ac.ir	NOUN
ejpam-5914	13	27	(	(	PUNCT
ejpam-5914	13	28	a.	a.	NOUN
ejpam-5914	13	29	rezaei	rezaei	PROPN
ejpam-5914	13	30	)	)	PUNCT
ejpam-5914	13	31	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5914	14	1	1	1	NUM
ejpam-5914	14	2	copyright	copyright	NOUN
ejpam-5914	14	3	:	:	PUNCT
ejpam-5914	14	4	©	©	PROPN
ejpam-5914	14	5	2025	2025	NUM
ejpam-5914	14	6	the	the	DET
ejpam-5914	14	7	author(s	author(s	NOUN
ejpam-5914	14	8	)	)	PUNCT
ejpam-5914	14	9	.	.	PUNCT
ejpam-5914	15	1	(	(	PUNCT
ejpam-5914	15	2	cc	cc	NOUN
ejpam-5914	15	3	by	by	ADP
ejpam-5914	15	4	-	-	PUNCT
ejpam-5914	15	5	nc	nc	PROPN
ejpam-5914	15	6	4.0	4.0	NUM
ejpam-5914	15	7	)	)	PUNCT
ejpam-5914	15	8	n.	n.	PROPN
ejpam-5914	15	9	rajesh	rajesh	PROPN
ejpam-5914	15	10	et	et	PROPN
ejpam-5914	16	1	al	al	PROPN
ejpam-5914	16	2	.	.	PUNCT
ejpam-5914	16	3	/	/	SYM
ejpam-5914	16	4	eur	eur	PROPN
ejpam-5914	16	5	.	.	PUNCT
ejpam-5914	17	1	j.	j.	PROPN
ejpam-5914	17	2	pure	pure	PROPN
ejpam-5914	17	3	appl	appl	PROPN
ejpam-5914	17	4	.	.	PROPN
ejpam-5914	17	5	math	math	PROPN
ejpam-5914	17	6	,	,	PUNCT
ejpam-5914	17	7	18	18	NUM
ejpam-5914	17	8	(	(	PUNCT
ejpam-5914	17	9	2	2	NUM
ejpam-5914	17	10	)	)	PUNCT
ejpam-5914	17	11	(	(	PUNCT
ejpam-5914	17	12	2025	2025	NUM
ejpam-5914	17	13	)	)	PUNCT
ejpam-5914	17	14	,	,	PUNCT
ejpam-5914	17	15	5914	5914	NUM
ejpam-5914	17	16	2	2	NUM
ejpam-5914	17	17	of	of	ADP
ejpam-5914	17	18	21	21	NUM
ejpam-5914	17	19	can	can	AUX
ejpam-5914	17	20	be	be	AUX
ejpam-5914	17	21	expressed	express	VERB
ejpam-5914	17	22	solely	solely	ADV
ejpam-5914	17	23	using	use	VERB
ejpam-5914	17	24	the	the	DET
ejpam-5914	17	25	sheffer	sheffer	NOUN
ejpam-5914	17	26	operation	operation	NOUN
ejpam-5914	17	27	.	.	PUNCT
ejpam-5914	18	1	this	this	DET
ejpam-5914	18	2	highlights	highlight	VERB
ejpam-5914	18	3	the	the	DET
ejpam-5914	18	4	fundamental	fundamental	ADJ
ejpam-5914	18	5	nature	nature	NOUN
ejpam-5914	18	6	and	and	CCONJ
ejpam-5914	18	7	versatility	versatility	NOUN
ejpam-5914	18	8	of	of	ADP
ejpam-5914	18	9	the	the	DET
ejpam-5914	18	10	sheffer	sheffer	NOUN
ejpam-5914	18	11	operation	operation	NOUN
ejpam-5914	18	12	in	in	ADP
ejpam-5914	18	13	logical	logical	ADJ
ejpam-5914	18	14	and	and	CCONJ
ejpam-5914	18	15	algebraic	algebraic	ADJ
ejpam-5914	18	16	systems	system	NOUN
ejpam-5914	18	17	.	.	PUNCT
ejpam-5914	19	1	in	in	ADP
ejpam-5914	19	2	2002	2002	NUM
ejpam-5914	19	3	,	,	PUNCT
ejpam-5914	19	4	mccune	mccune	PROPN
ejpam-5914	19	5	et	et	PROPN
ejpam-5914	19	6	al	al	PROPN
ejpam-5914	19	7	.	.	PUNCT
ejpam-5914	20	1	[	[	X
ejpam-5914	20	2	2	2	NUM
ejpam-5914	20	3	]	]	PUNCT
ejpam-5914	20	4	applied	apply	VERB
ejpam-5914	20	5	the	the	DET
ejpam-5914	20	6	sheffer	sheffer	NOUN
ejpam-5914	20	7	stroke	stroke	NOUN
ejpam-5914	20	8	operation	operation	NOUN
ejpam-5914	20	9	for	for	ADP
ejpam-5914	20	10	boolean	boolean	ADJ
ejpam-5914	20	11	algebras	algebra	NOUN
ejpam-5914	20	12	,	,	PUNCT
ejpam-5914	20	13	and	and	CCONJ
ejpam-5914	20	14	it	it	PRON
ejpam-5914	20	15	is	be	AUX
ejpam-5914	20	16	shown	show	VERB
ejpam-5914	20	17	that	that	SCONJ
ejpam-5914	20	18	there	there	PRON
ejpam-5914	20	19	is	be	VERB
ejpam-5914	20	20	no	no	DET
ejpam-5914	20	21	shorter	short	ADJ
ejpam-5914	20	22	axiom	axiom	NOUN
ejpam-5914	20	23	in	in	ADP
ejpam-5914	20	24	terms	term	NOUN
ejpam-5914	20	25	of	of	ADP
ejpam-5914	20	26	the	the	DET
ejpam-5914	20	27	sheffer	sheffer	NOUN
ejpam-5914	20	28	stroke	stroke	NOUN
ejpam-5914	20	29	.	.	PUNCT
ejpam-5914	21	1	algebraic	algebraic	ADJ
ejpam-5914	21	2	structures	structure	NOUN
ejpam-5914	21	3	play	play	VERB
ejpam-5914	21	4	a	a	DET
ejpam-5914	21	5	prominent	prominent	ADJ
ejpam-5914	21	6	role	role	NOUN
ejpam-5914	21	7	in	in	ADP
ejpam-5914	21	8	mathematics	mathematic	NOUN
ejpam-5914	21	9	,	,	PUNCT
ejpam-5914	21	10	with	with	ADP
ejpam-5914	21	11	wide	wide	ADV
ejpam-5914	21	12	-	-	PUNCT
ejpam-5914	21	13	ranging	range	VERB
ejpam-5914	21	14	applications	application	NOUN
ejpam-5914	21	15	in	in	ADP
ejpam-5914	21	16	various	various	ADJ
ejpam-5914	21	17	disciplines	discipline	NOUN
ejpam-5914	21	18	,	,	PUNCT
ejpam-5914	21	19	including	include	VERB
ejpam-5914	21	20	theoretical	theoretical	ADJ
ejpam-5914	21	21	physics	physics	NOUN
ejpam-5914	21	22	,	,	PUNCT
ejpam-5914	21	23	computer	computer	NOUN
ejpam-5914	21	24	science	science	NOUN
ejpam-5914	21	25	,	,	PUNCT
ejpam-5914	21	26	control	control	NOUN
ejpam-5914	21	27	engineering	engineering	NOUN
ejpam-5914	21	28	,	,	PUNCT
ejpam-5914	21	29	information	information	NOUN
ejpam-5914	21	30	sciences	science	NOUN
ejpam-5914	21	31	,	,	PUNCT
ejpam-5914	21	32	coding	code	VERB
ejpam-5914	21	33	theory	theory	NOUN
ejpam-5914	21	34	,	,	PUNCT
ejpam-5914	21	35	and	and	CCONJ
ejpam-5914	21	36	topological	topological	ADJ
ejpam-5914	21	37	spaces	space	NOUN
ejpam-5914	21	38	,	,	PUNCT
ejpam-5914	21	39	among	among	ADP
ejpam-5914	21	40	others	other	NOUN
ejpam-5914	21	41	.	.	PUNCT
ejpam-5914	22	1	this	this	PRON
ejpam-5914	22	2	provides	provide	VERB
ejpam-5914	22	3	sufficient	sufficient	ADJ
ejpam-5914	22	4	motivation	motivation	NOUN
ejpam-5914	22	5	for	for	SCONJ
ejpam-5914	22	6	researchers	researcher	NOUN
ejpam-5914	22	7	to	to	PART
ejpam-5914	22	8	simplify	simplify	VERB
ejpam-5914	22	9	axioms	axiom	NOUN
ejpam-5914	22	10	for	for	ADP
ejpam-5914	22	11	various	various	ADJ
ejpam-5914	22	12	algebraic	algebraic	ADJ
ejpam-5914	22	13	structures	structure	NOUN
ejpam-5914	22	14	,	,	PUNCT
ejpam-5914	22	15	e.g.	e.g.	ADV
ejpam-5914	22	16	,	,	PUNCT
ejpam-5914	22	17	see	see	VERB
ejpam-5914	22	18	[	[	X
ejpam-5914	22	19	3–5	3–5	NOUN
ejpam-5914	22	20	]	]	PUNCT
ejpam-5914	22	21	.	.	PUNCT
ejpam-5914	23	1	in	in	ADP
ejpam-5914	23	2	1950	1950	NUM
ejpam-5914	23	3	,	,	PUNCT
ejpam-5914	23	4	henkin	henkin	X
ejpam-5914	23	5	[	[	X
ejpam-5914	23	6	6	6	NUM
ejpam-5914	23	7	]	]	PUNCT
ejpam-5914	23	8	introduced	introduce	VERB
ejpam-5914	23	9	the	the	DET
ejpam-5914	23	10	notion	notion	NOUN
ejpam-5914	23	11	of	of	ADP
ejpam-5914	23	12	“	"	PUNCT
ejpam-5914	23	13	implicative	implicative	ADJ
ejpam-5914	23	14	model	model	NOUN
ejpam-5914	23	15	”	"	PUNCT
ejpam-5914	23	16	as	as	ADP
ejpam-5914	23	17	a	a	DET
ejpam-5914	23	18	model	model	NOUN
ejpam-5914	23	19	of	of	ADP
ejpam-5914	23	20	positive	positive	ADJ
ejpam-5914	23	21	implicative	implicative	ADJ
ejpam-5914	23	22	propositional	propositional	ADJ
ejpam-5914	23	23	calculus	calculus	NOUN
ejpam-5914	23	24	.	.	PUNCT
ejpam-5914	24	1	in	in	ADP
ejpam-5914	24	2	1960	1960	NUM
ejpam-5914	24	3	,	,	PUNCT
ejpam-5914	24	4	monteiro	monteiro	PROPN
ejpam-5914	24	5	[	[	X
ejpam-5914	24	6	7	7	X
ejpam-5914	24	7	]	]	PUNCT
ejpam-5914	24	8	gave	give	VERB
ejpam-5914	24	9	the	the	DET
ejpam-5914	24	10	name	name	NOUN
ejpam-5914	24	11	“	"	PUNCT
ejpam-5914	24	12	hilbert	hilbert	PROPN
ejpam-5914	24	13	algebras	algebras	PROPN
ejpam-5914	24	14	”	"	PUNCT
ejpam-5914	24	15	to	to	ADP
ejpam-5914	24	16	the	the	DET
ejpam-5914	24	17	dual	dual	ADJ
ejpam-5914	24	18	algebras	algebra	NOUN
ejpam-5914	24	19	of	of	ADP
ejpam-5914	24	20	henkin	henkin	PROPN
ejpam-5914	24	21	’s	’s	PART
ejpam-5914	24	22	implicative	implicative	ADJ
ejpam-5914	24	23	models	model	NOUN
ejpam-5914	24	24	.	.	PUNCT
ejpam-5914	25	1	in	in	ADP
ejpam-5914	25	2	1966	1966	NUM
ejpam-5914	25	3	,	,	PUNCT
ejpam-5914	25	4	diego	diego	X
ejpam-5914	26	1	[	[	X
ejpam-5914	26	2	8	8	NUM
ejpam-5914	26	3	]	]	PUNCT
ejpam-5914	26	4	intensively	intensively	ADV
ejpam-5914	26	5	studied	study	VERB
ejpam-5914	26	6	and	and	CCONJ
ejpam-5914	26	7	developed	develop	VERB
ejpam-5914	26	8	some	some	DET
ejpam-5914	26	9	properties	property	NOUN
ejpam-5914	26	10	of	of	ADP
ejpam-5914	26	11	hilbert	hilbert	PROPN
ejpam-5914	26	12	algebras	algebras	PROPN
ejpam-5914	26	13	.	.	PUNCT
ejpam-5914	27	1	in	in	ADP
ejpam-5914	27	2	2021	2021	NUM
ejpam-5914	27	3	,	,	PUNCT
ejpam-5914	27	4	oner	oner	NOUN
ejpam-5914	27	5	et	et	PROPN
ejpam-5914	27	6	al	al	PROPN
ejpam-5914	27	7	.	.	PUNCT
ejpam-5914	28	1	[	[	X
ejpam-5914	28	2	9	9	NUM
ejpam-5914	28	3	]	]	PUNCT
ejpam-5914	28	4	investigated	investigate	VERB
ejpam-5914	28	5	the	the	DET
ejpam-5914	28	6	relation	relation	NOUN
ejpam-5914	28	7	between	between	ADP
ejpam-5914	28	8	sheffer	sheffer	PROPN
ejpam-5914	28	9	stroke	stroke	PROPN
ejpam-5914	28	10	and	and	CCONJ
ejpam-5914	28	11	hilbert	hilbert	PROPN
ejpam-5914	28	12	algebras	algebras	PROPN
ejpam-5914	28	13	.	.	PUNCT
ejpam-5914	29	1	also	also	ADV
ejpam-5914	29	2	,	,	PUNCT
ejpam-5914	29	3	see	see	VERB
ejpam-5914	29	4	[	[	X
ejpam-5914	29	5	10	10	NUM
ejpam-5914	29	6	]	]	PUNCT
ejpam-5914	29	7	.	.	PUNCT
ejpam-5914	30	1	in	in	ADP
ejpam-5914	30	2	1965	1965	NUM
ejpam-5914	30	3	,	,	PUNCT
ejpam-5914	30	4	zadeh	zadeh	PROPN
ejpam-5914	31	1	[	[	X
ejpam-5914	31	2	11	11	NUM
ejpam-5914	31	3	]	]	PUNCT
ejpam-5914	31	4	proposed	propose	VERB
ejpam-5914	31	5	a	a	DET
ejpam-5914	31	6	new	new	ADJ
ejpam-5914	31	7	theory	theory	NOUN
ejpam-5914	31	8	named	name	VERB
ejpam-5914	31	9	fuzzy	fuzzy	ADJ
ejpam-5914	31	10	set	set	NOUN
ejpam-5914	31	11	theory	theory	NOUN
ejpam-5914	31	12	.	.	PUNCT
ejpam-5914	32	1	then	then	ADV
ejpam-5914	32	2	several	several	ADJ
ejpam-5914	32	3	researchers	researcher	NOUN
ejpam-5914	32	4	studied	study	VERB
ejpam-5914	32	5	various	various	ADJ
ejpam-5914	32	6	extensions	extension	NOUN
ejpam-5914	32	7	and	and	CCONJ
ejpam-5914	32	8	generalizations	generalization	NOUN
ejpam-5914	32	9	of	of	ADP
ejpam-5914	32	10	this	this	DET
ejpam-5914	32	11	theory	theory	NOUN
ejpam-5914	32	12	,	,	PUNCT
ejpam-5914	32	13	e.g.	e.g.	ADV
ejpam-5914	32	14	,	,	PUNCT
ejpam-5914	32	15	intuitionistic	intuitionistic	ADJ
ejpam-5914	32	16	fuzzy	fuzzy	ADJ
ejpam-5914	32	17	sets	set	NOUN
ejpam-5914	32	18	[	[	X
ejpam-5914	32	19	12	12	NUM
ejpam-5914	32	20	]	]	PUNCT
ejpam-5914	32	21	,	,	PUNCT
ejpam-5914	32	22	l	l	ADJ
ejpam-5914	32	23	-	-	ADJ
ejpam-5914	32	24	fuzzy	fuzzy	ADJ
ejpam-5914	32	25	sets	set	NOUN
ejpam-5914	32	26	[	[	X
ejpam-5914	32	27	13	13	NUM
ejpam-5914	32	28	]	]	PUNCT
ejpam-5914	32	29	,	,	PUNCT
ejpam-5914	32	30	type-2	type-2	NUM
ejpam-5914	32	31	fuzzy	fuzzy	ADJ
ejpam-5914	32	32	sets	set	NOUN
ejpam-5914	32	33	[	[	X
ejpam-5914	32	34	14	14	NUM
ejpam-5914	32	35	]	]	PUNCT
ejpam-5914	32	36	,	,	PUNCT
ejpam-5914	32	37	interval	interval	NOUN
ejpam-5914	32	38	-	-	PUNCT
ejpam-5914	32	39	valued	value	VERB
ejpam-5914	32	40	fuzzy	fuzzy	ADJ
ejpam-5914	32	41	sets	set	NOUN
ejpam-5914	32	42	[	[	X
ejpam-5914	32	43	15	15	NUM
ejpam-5914	32	44	]	]	PUNCT
ejpam-5914	32	45	,	,	PUNCT
ejpam-5914	32	46	multi	multi	X
ejpam-5914	32	47	fuzzy	fuzzy	ADJ
ejpam-5914	32	48	sets	set	VERB
ejpam-5914	32	49	[	[	X
ejpam-5914	32	50	16	16	NUM
ejpam-5914	32	51	]	]	PUNCT
ejpam-5914	32	52	,	,	PUNCT
ejpam-5914	32	53	bipolarvalued	bipolarvalue	VERB
ejpam-5914	32	54	fuzzy	fuzzy	ADJ
ejpam-5914	32	55	sets	set	NOUN
ejpam-5914	32	56	[	[	X
ejpam-5914	32	57	17	17	NUM
ejpam-5914	32	58	]	]	PUNCT
ejpam-5914	32	59	,	,	PUNCT
ejpam-5914	32	60	m	m	ADJ
ejpam-5914	32	61	-	-	ADJ
ejpam-5914	32	62	polar	polar	ADJ
ejpam-5914	32	63	fuzzy	fuzzy	ADJ
ejpam-5914	32	64	sets	set	NOUN
ejpam-5914	32	65	[	[	X
ejpam-5914	32	66	18	18	NUM
ejpam-5914	32	67	]	]	PUNCT
ejpam-5914	32	68	,	,	PUNCT
ejpam-5914	32	69	and	and	CCONJ
ejpam-5914	32	70	neutrosophic	neutrosophic	ADJ
ejpam-5914	32	71	sets	set	NOUN
ejpam-5914	32	72	[	[	X
ejpam-5914	32	73	19	19	NUM
ejpam-5914	32	74	,	,	PUNCT
ejpam-5914	32	75	20	20	NUM
ejpam-5914	32	76	]	]	PUNCT
ejpam-5914	32	77	.	.	PUNCT
ejpam-5914	33	1	recently	recently	ADV
ejpam-5914	33	2	,	,	PUNCT
ejpam-5914	33	3	many	many	ADJ
ejpam-5914	33	4	researchers	researcher	NOUN
ejpam-5914	33	5	have	have	AUX
ejpam-5914	33	6	studied	study	VERB
ejpam-5914	33	7	and	and	CCONJ
ejpam-5914	33	8	applied	apply	VERB
ejpam-5914	33	9	concepts	concept	NOUN
ejpam-5914	33	10	of	of	ADP
ejpam-5914	33	11	fuzzy	fuzzy	ADJ
ejpam-5914	33	12	sets	set	NOUN
ejpam-5914	33	13	,	,	PUNCT
ejpam-5914	33	14	including	include	VERB
ejpam-5914	33	15	fuzzy	fuzzy	ADJ
ejpam-5914	33	16	(	(	PUNCT
ejpam-5914	33	17	weak	weak	ADJ
ejpam-5914	33	18	)	)	PUNCT
ejpam-5914	33	19	filters	filter	NOUN
ejpam-5914	33	20	and	and	CCONJ
ejpam-5914	33	21	deductive	deductive	ADJ
ejpam-5914	33	22	systems	system	NOUN
ejpam-5914	33	23	,	,	PUNCT
ejpam-5914	33	24	to	to	PART
ejpam-5914	33	25	sheffer	sheffer	VERB
ejpam-5914	33	26	stroke	stroke	PROPN
ejpam-5914	33	27	hilbert	hilbert	PROPN
ejpam-5914	33	28	algebras	algebras	PROPN
ejpam-5914	34	1	[	[	X
ejpam-5914	34	2	21–24	21–24	NUM
ejpam-5914	34	3	]	]	PUNCT
ejpam-5914	34	4	.	.	PUNCT
ejpam-5914	35	1	this	this	DET
ejpam-5914	35	2	paper	paper	NOUN
ejpam-5914	35	3	aims	aim	VERB
ejpam-5914	35	4	to	to	PART
ejpam-5914	35	5	introduce	introduce	VERB
ejpam-5914	35	6	and	and	CCONJ
ejpam-5914	35	7	explore	explore	VERB
ejpam-5914	35	8	the	the	DET
ejpam-5914	35	9	concepts	concept	NOUN
ejpam-5914	35	10	of	of	ADP
ejpam-5914	35	11	length	length	NOUN
ejpam-5914	35	12	and	and	CCONJ
ejpam-5914	35	13	mean	mean	VERB
ejpam-5914	35	14	within	within	ADP
ejpam-5914	35	15	interval	interval	NOUN
ejpam-5914	35	16	-	-	PUNCT
ejpam-5914	35	17	valued	value	VERB
ejpam-5914	35	18	fuzzy	fuzzy	ADJ
ejpam-5914	35	19	structures	structure	NOUN
ejpam-5914	35	20	in	in	ADP
ejpam-5914	35	21	sheffer	sheffer	PROPN
ejpam-5914	35	22	stroke	stroke	PROPN
ejpam-5914	35	23	hilbert	hilbert	PROPN
ejpam-5914	35	24	algebras	algebras	PROPN
ejpam-5914	35	25	.	.	PUNCT
ejpam-5914	36	1	specifically	specifically	ADV
ejpam-5914	36	2	,	,	PUNCT
ejpam-5914	36	3	we	we	PRON
ejpam-5914	36	4	define	define	VERB
ejpam-5914	36	5	and	and	CCONJ
ejpam-5914	36	6	analyze	analyze	VERB
ejpam-5914	36	7	the	the	DET
ejpam-5914	36	8	notions	notion	NOUN
ejpam-5914	36	9	of	of	ADP
ejpam-5914	36	10	length	length	NOUN
ejpam-5914	36	11	-	-	PUNCT
ejpam-5914	36	12	fuzzy	fuzzy	ADJ
ejpam-5914	36	13	subalgebras	subalgebra	NOUN
ejpam-5914	36	14	and	and	CCONJ
ejpam-5914	36	15	mean	mean	ADJ
ejpam-5914	36	16	-	-	PUNCT
ejpam-5914	36	17	fuzzy	fuzzy	ADJ
ejpam-5914	36	18	subalgebras	subalgebra	NOUN
ejpam-5914	36	19	,	,	PUNCT
ejpam-5914	36	20	investigating	investigate	VERB
ejpam-5914	36	21	their	their	PRON
ejpam-5914	36	22	key	key	ADJ
ejpam-5914	36	23	properties	property	NOUN
ejpam-5914	36	24	and	and	CCONJ
ejpam-5914	36	25	characterizations	characterization	NOUN
ejpam-5914	36	26	.	.	PUNCT
ejpam-5914	37	1	the	the	DET
ejpam-5914	37	2	study	study	NOUN
ejpam-5914	37	3	establishes	establish	VERB
ejpam-5914	37	4	relationships	relationship	NOUN
ejpam-5914	37	5	between	between	ADP
ejpam-5914	37	6	these	these	DET
ejpam-5914	37	7	fuzzy	fuzzy	ADJ
ejpam-5914	37	8	subalgebras	subalgebra	NOUN
ejpam-5914	37	9	and	and	CCONJ
ejpam-5914	37	10	traditional	traditional	ADJ
ejpam-5914	37	11	subalgebras	subalgebra	NOUN
ejpam-5914	37	12	,	,	PUNCT
ejpam-5914	37	13	providing	provide	VERB
ejpam-5914	37	14	a	a	DET
ejpam-5914	37	15	deeper	deep	ADJ
ejpam-5914	37	16	understanding	understanding	NOUN
ejpam-5914	37	17	of	of	ADP
ejpam-5914	37	18	their	their	PRON
ejpam-5914	37	19	structural	structural	ADJ
ejpam-5914	37	20	interaction	interaction	NOUN
ejpam-5914	37	21	.	.	PUNCT
ejpam-5914	38	1	additionally	additionally	ADV
ejpam-5914	38	2	,	,	PUNCT
ejpam-5914	38	3	we	we	PRON
ejpam-5914	38	4	examine	examine	VERB
ejpam-5914	38	5	the	the	DET
ejpam-5914	38	6	connections	connection	NOUN
ejpam-5914	38	7	between	between	ADP
ejpam-5914	38	8	length	length	NOUN
ejpam-5914	38	9	-	-	PUNCT
ejpam-5914	38	10	fuzzy	fuzzy	ADJ
ejpam-5914	38	11	(	(	PUNCT
ejpam-5914	38	12	resp	resp	NOUN
ejpam-5914	38	13	.	.	PUNCT
ejpam-5914	38	14	,	,	PUNCT
ejpam-5914	38	15	mean	mean	ADJ
ejpam-5914	38	16	-	-	PUNCT
ejpam-5914	38	17	fuzzy	fuzzy	ADJ
ejpam-5914	38	18	)	)	PUNCT
ejpam-5914	38	19	subalgebras	subalgebra	NOUN
ejpam-5914	38	20	and	and	CCONJ
ejpam-5914	38	21	their	their	PRON
ejpam-5914	38	22	corresponding	corresponding	ADJ
ejpam-5914	38	23	upper	upper	ADJ
ejpam-5914	38	24	and	and	CCONJ
ejpam-5914	38	25	lowerlevel	lowerlevel	NOUN
ejpam-5914	38	26	subsets	subset	NOUN
ejpam-5914	38	27	within	within	ADP
ejpam-5914	38	28	interval	interval	NOUN
ejpam-5914	38	29	-	-	PUNCT
ejpam-5914	38	30	valued	value	VERB
ejpam-5914	38	31	fuzzy	fuzzy	ADJ
ejpam-5914	38	32	structures	structure	NOUN
ejpam-5914	38	33	.	.	PUNCT
ejpam-5914	39	1	these	these	DET
ejpam-5914	39	2	findings	finding	NOUN
ejpam-5914	39	3	offer	offer	VERB
ejpam-5914	39	4	a	a	DET
ejpam-5914	39	5	comprehensive	comprehensive	ADJ
ejpam-5914	39	6	framework	framework	NOUN
ejpam-5914	39	7	for	for	ADP
ejpam-5914	39	8	studying	study	VERB
ejpam-5914	39	9	gradations	gradation	NOUN
ejpam-5914	39	10	of	of	ADP
ejpam-5914	39	11	membership	membership	NOUN
ejpam-5914	39	12	and	and	CCONJ
ejpam-5914	39	13	their	their	PRON
ejpam-5914	39	14	implications	implication	NOUN
ejpam-5914	39	15	in	in	ADP
ejpam-5914	39	16	sheffer	sheffer	PROPN
ejpam-5914	39	17	stroke	stroke	PROPN
ejpam-5914	39	18	hilbert	hilbert	PROPN
ejpam-5914	39	19	algebras	algebras	PROPN
ejpam-5914	39	20	,	,	PUNCT
ejpam-5914	39	21	laying	lay	VERB
ejpam-5914	39	22	the	the	DET
ejpam-5914	39	23	groundwork	groundwork	NOUN
ejpam-5914	39	24	for	for	ADP
ejpam-5914	39	25	further	further	ADJ
ejpam-5914	39	26	theoretical	theoretical	ADJ
ejpam-5914	39	27	development	development	NOUN
ejpam-5914	39	28	and	and	CCONJ
ejpam-5914	39	29	practical	practical	ADJ
ejpam-5914	39	30	applications	application	NOUN
ejpam-5914	39	31	in	in	ADP
ejpam-5914	39	32	fuzzy	fuzzy	ADJ
ejpam-5914	39	33	logic	logic	NOUN
ejpam-5914	39	34	and	and	CCONJ
ejpam-5914	39	35	algebraic	algebraic	ADJ
ejpam-5914	39	36	systems	system	NOUN
ejpam-5914	39	37	.	.	PUNCT
ejpam-5914	40	1	2	2	X
ejpam-5914	40	2	.	.	X
ejpam-5914	40	3	preliminaries	preliminary	NOUN
ejpam-5914	40	4	sheffer	sheffer	VERB
ejpam-5914	40	5	stroke	stroke	PROPN
ejpam-5914	40	6	hilbert	hilbert	PROPN
ejpam-5914	40	7	algebras	algebras	PROPN
ejpam-5914	40	8	constitute	constitute	VERB
ejpam-5914	40	9	a	a	DET
ejpam-5914	40	10	pivotal	pivotal	ADJ
ejpam-5914	40	11	framework	framework	NOUN
ejpam-5914	40	12	within	within	ADP
ejpam-5914	40	13	the	the	DET
ejpam-5914	40	14	realms	realm	NOUN
ejpam-5914	40	15	of	of	ADP
ejpam-5914	40	16	logic	logic	NOUN
ejpam-5914	40	17	and	and	CCONJ
ejpam-5914	40	18	lattice	lattice	PROPN
ejpam-5914	40	19	theory	theory	NOUN
ejpam-5914	40	20	,	,	PUNCT
ejpam-5914	40	21	distinguished	distinguish	VERB
ejpam-5914	40	22	by	by	ADP
ejpam-5914	40	23	the	the	DET
ejpam-5914	40	24	incorporation	incorporation	NOUN
ejpam-5914	40	25	of	of	ADP
ejpam-5914	40	26	the	the	DET
ejpam-5914	40	27	sheffer	sheffer	NOUN
ejpam-5914	40	28	stroke	stroke	NOUN
ejpam-5914	40	29	(	(	PUNCT
ejpam-5914	40	30	nand	nand	NOUN
ejpam-5914	40	31	)	)	PUNCT
ejpam-5914	40	32	operation	operation	NOUN
ejpam-5914	40	33	—	—	PUNCT
ejpam-5914	40	34	a	a	DET
ejpam-5914	40	35	cornerstone	cornerstone	NOUN
ejpam-5914	40	36	of	of	ADP
ejpam-5914	40	37	boolean	boolean	ADJ
ejpam-5914	40	38	algebra	algebra	NOUN
ejpam-5914	40	39	.	.	PUNCT
ejpam-5914	41	1	this	this	DET
ejpam-5914	41	2	integration	integration	NOUN
ejpam-5914	41	3	extends	extend	VERB
ejpam-5914	41	4	the	the	DET
ejpam-5914	41	5	classical	classical	ADJ
ejpam-5914	41	6	hilbert	hilbert	NOUN
ejpam-5914	41	7	algebra	algebra	PROPN
ejpam-5914	41	8	structure	structure	NOUN
ejpam-5914	41	9	,	,	PUNCT
ejpam-5914	41	10	enabling	enable	VERB
ejpam-5914	41	11	a	a	DET
ejpam-5914	41	12	more	more	ADV
ejpam-5914	41	13	versatile	versatile	ADJ
ejpam-5914	41	14	exploration	exploration	NOUN
ejpam-5914	41	15	of	of	ADP
ejpam-5914	41	16	logical	logical	ADJ
ejpam-5914	41	17	systems	system	NOUN
ejpam-5914	41	18	and	and	CCONJ
ejpam-5914	41	19	their	their	PRON
ejpam-5914	41	20	properties	property	NOUN
ejpam-5914	41	21	.	.	PUNCT
ejpam-5914	42	1	by	by	ADP
ejpam-5914	42	2	bridging	bridge	VERB
ejpam-5914	42	3	algebraic	algebraic	ADJ
ejpam-5914	42	4	theory	theory	NOUN
ejpam-5914	42	5	and	and	CCONJ
ejpam-5914	42	6	practical	practical	ADJ
ejpam-5914	42	7	applications	application	NOUN
ejpam-5914	42	8	,	,	PUNCT
ejpam-5914	42	9	sheffer	sheffer	NOUN
ejpam-5914	42	10	stroke	stroke	PROPN
ejpam-5914	42	11	hilbert	hilbert	PROPN
ejpam-5914	42	12	algebras	algebras	PROPN
ejpam-5914	42	13	provide	provide	VERB
ejpam-5914	42	14	a	a	DET
ejpam-5914	42	15	robust	robust	ADJ
ejpam-5914	42	16	toolset	toolset	NOUN
ejpam-5914	42	17	for	for	ADP
ejpam-5914	42	18	analyzing	analyze	VERB
ejpam-5914	42	19	and	and	CCONJ
ejpam-5914	42	20	modeling	model	VERB
ejpam-5914	42	21	complex	complex	ADJ
ejpam-5914	42	22	systems	system	NOUN
ejpam-5914	42	23	characterized	characterize	VERB
ejpam-5914	42	24	by	by	ADP
ejpam-5914	42	25	uncertainty	uncertainty	NOUN
ejpam-5914	42	26	,	,	PUNCT
ejpam-5914	42	27	fuzziness	fuzziness	NOUN
ejpam-5914	42	28	,	,	PUNCT
ejpam-5914	42	29	and	and	CCONJ
ejpam-5914	42	30	imprecision	imprecision	NOUN
ejpam-5914	42	31	.	.	PUNCT
ejpam-5914	43	1	these	these	DET
ejpam-5914	43	2	algebras	algebra	NOUN
ejpam-5914	43	3	are	be	AUX
ejpam-5914	43	4	particularly	particularly	ADV
ejpam-5914	43	5	relevant	relevant	ADJ
ejpam-5914	43	6	in	in	ADP
ejpam-5914	43	7	advancing	advance	VERB
ejpam-5914	43	8	fuzzy	fuzzy	ADJ
ejpam-5914	43	9	logic	logic	NOUN
ejpam-5914	43	10	,	,	PUNCT
ejpam-5914	43	11	decision	decision	NOUN
ejpam-5914	43	12	-	-	PUNCT
ejpam-5914	43	13	making	make	VERB
ejpam-5914	43	14	algorithms	algorithm	NOUN
ejpam-5914	43	15	,	,	PUNCT
ejpam-5914	43	16	and	and	CCONJ
ejpam-5914	43	17	computational	computational	ADJ
ejpam-5914	43	18	frameworks	framework	NOUN
ejpam-5914	43	19	,	,	PUNCT
ejpam-5914	43	20	offering	offer	VERB
ejpam-5914	43	21	insights	insight	NOUN
ejpam-5914	43	22	that	that	PRON
ejpam-5914	43	23	transcend	transcend	VERB
ejpam-5914	43	24	traditional	traditional	ADJ
ejpam-5914	43	25	logical	logical	ADJ
ejpam-5914	43	26	paradigms	paradigm	NOUN
ejpam-5914	43	27	.	.	PUNCT
ejpam-5914	44	1	moreover	moreover	ADV
ejpam-5914	44	2	,	,	PUNCT
ejpam-5914	44	3	their	their	PRON
ejpam-5914	44	4	study	study	NOUN
ejpam-5914	44	5	contributes	contribute	VERB
ejpam-5914	44	6	to	to	ADP
ejpam-5914	44	7	the	the	DET
ejpam-5914	44	8	broader	broad	ADJ
ejpam-5914	44	9	understanding	understanding	NOUN
ejpam-5914	44	10	of	of	ADP
ejpam-5914	44	11	algebraic	algebraic	ADJ
ejpam-5914	44	12	hierarchies	hierarchy	NOUN
ejpam-5914	44	13	,	,	PUNCT
ejpam-5914	44	14	enriching	enrich	VERB
ejpam-5914	44	15	both	both	CCONJ
ejpam-5914	44	16	foundational	foundational	ADJ
ejpam-5914	44	17	research	research	NOUN
ejpam-5914	44	18	n.	n.	PROPN
ejpam-5914	44	19	rajesh	rajesh	PROPN
ejpam-5914	44	20	et	et	PROPN
ejpam-5914	44	21	al	al	PROPN
ejpam-5914	44	22	.	.	PUNCT
ejpam-5914	44	23	/	/	SYM
ejpam-5914	44	24	eur	eur	PROPN
ejpam-5914	44	25	.	.	PUNCT
ejpam-5914	45	1	j.	j.	PROPN
ejpam-5914	45	2	pure	pure	PROPN
ejpam-5914	45	3	appl	appl	PROPN
ejpam-5914	45	4	.	.	PROPN
ejpam-5914	45	5	math	math	PROPN
ejpam-5914	45	6	,	,	PUNCT
ejpam-5914	45	7	18	18	NUM
ejpam-5914	45	8	(	(	PUNCT
ejpam-5914	45	9	2	2	NUM
ejpam-5914	45	10	)	)	PUNCT
ejpam-5914	45	11	(	(	PUNCT
ejpam-5914	45	12	2025	2025	NUM
ejpam-5914	45	13	)	)	PUNCT
ejpam-5914	45	14	,	,	PUNCT
ejpam-5914	45	15	5914	5914	NUM
ejpam-5914	45	16	3	3	NUM
ejpam-5914	45	17	of	of	ADP
ejpam-5914	45	18	21	21	NUM
ejpam-5914	45	19	and	and	CCONJ
ejpam-5914	45	20	real	real	ADJ
ejpam-5914	45	21	-	-	PUNCT
ejpam-5914	45	22	world	world	NOUN
ejpam-5914	45	23	problem	problem	NOUN
ejpam-5914	45	24	-	-	PUNCT
ejpam-5914	45	25	solving	solve	VERB
ejpam-5914	45	26	methodologies	methodology	NOUN
ejpam-5914	45	27	.	.	PUNCT
ejpam-5914	46	1	this	this	DET
ejpam-5914	46	2	unique	unique	ADJ
ejpam-5914	46	3	combination	combination	NOUN
ejpam-5914	46	4	of	of	ADP
ejpam-5914	46	5	theoretical	theoretical	ADJ
ejpam-5914	46	6	depth	depth	NOUN
ejpam-5914	46	7	and	and	CCONJ
ejpam-5914	46	8	practical	practical	ADJ
ejpam-5914	46	9	utility	utility	NOUN
ejpam-5914	46	10	underscores	underscore	VERB
ejpam-5914	46	11	their	their	PRON
ejpam-5914	46	12	significance	significance	NOUN
ejpam-5914	46	13	in	in	ADP
ejpam-5914	46	14	contemporary	contemporary	ADJ
ejpam-5914	46	15	mathematical	mathematical	ADJ
ejpam-5914	46	16	and	and	CCONJ
ejpam-5914	46	17	computational	computational	ADJ
ejpam-5914	46	18	research	research	NOUN
ejpam-5914	46	19	.	.	PUNCT
ejpam-5914	47	1	recall	recall	VERB
ejpam-5914	47	2	the	the	DET
ejpam-5914	47	3	definitions	definition	NOUN
ejpam-5914	47	4	and	and	CCONJ
ejpam-5914	47	5	results	result	NOUN
ejpam-5914	47	6	that	that	PRON
ejpam-5914	47	7	are	be	AUX
ejpam-5914	47	8	taken	take	VERB
ejpam-5914	47	9	from	from	ADP
ejpam-5914	47	10	[	[	X
ejpam-5914	47	11	1	1	NUM
ejpam-5914	47	12	,	,	PUNCT
ejpam-5914	47	13	9	9	NUM
ejpam-5914	47	14	,	,	PUNCT
ejpam-5914	47	15	15	15	NUM
ejpam-5914	47	16	,	,	PUNCT
ejpam-5914	47	17	25	25	NUM
ejpam-5914	47	18	]	]	PUNCT
ejpam-5914	47	19	for	for	ADP
ejpam-5914	47	20	the	the	DET
ejpam-5914	47	21	ready	ready	ADJ
ejpam-5914	47	22	reference	reference	NOUN
ejpam-5914	47	23	of	of	ADP
ejpam-5914	47	24	the	the	DET
ejpam-5914	47	25	reader	reader	NOUN
ejpam-5914	47	26	.	.	PUNCT
ejpam-5914	48	1	definition	definition	NOUN
ejpam-5914	48	2	1	1	NUM
ejpam-5914	48	3	.	.	PUNCT
ejpam-5914	49	1	[	[	X
ejpam-5914	49	2	1	1	X
ejpam-5914	49	3	]	]	X
ejpam-5914	49	4	let	let	VERB
ejpam-5914	49	5	⟨a	⟨a	NOUN
ejpam-5914	49	6	,	,	PUNCT
ejpam-5914	49	7	|⟩	|⟩	PROPN
ejpam-5914	49	8	be	be	VERB
ejpam-5914	49	9	a	a	DET
ejpam-5914	49	10	groupoid	groupoid	NOUN
ejpam-5914	49	11	.	.	PUNCT
ejpam-5914	50	1	the	the	DET
ejpam-5914	50	2	operation	operation	NOUN
ejpam-5914	50	3	|	|	ADV
ejpam-5914	50	4	is	be	AUX
ejpam-5914	50	5	said	say	VERB
ejpam-5914	50	6	to	to	PART
ejpam-5914	50	7	be	be	AUX
ejpam-5914	50	8	a	a	DET
ejpam-5914	50	9	sheffer	sheffer	NOUN
ejpam-5914	50	10	stroke	stroke	NOUN
ejpam-5914	50	11	operation	operation	NOUN
ejpam-5914	50	12	if	if	SCONJ
ejpam-5914	50	13	it	it	PRON
ejpam-5914	50	14	satisfies	satisfy	VERB
ejpam-5914	50	15	the	the	DET
ejpam-5914	50	16	following	follow	VERB
ejpam-5914	50	17	conditions	condition	NOUN
ejpam-5914	50	18	:	:	PUNCT
ejpam-5914	50	19	for	for	ADP
ejpam-5914	50	20	all	all	DET
ejpam-5914	50	21	x	x	NOUN
ejpam-5914	50	22	,	,	PUNCT
ejpam-5914	50	23	y	y	PROPN
ejpam-5914	50	24	,	,	PUNCT
ejpam-5914	50	25	z	z	PROPN
ejpam-5914	50	26	∈	∈	PROPN
ejpam-5914	50	27	a	a	DET
ejpam-5914	50	28	,	,	PUNCT
ejpam-5914	50	29	(	(	PUNCT
ejpam-5914	50	30	s1	s1	NOUN
ejpam-5914	50	31	)	)	PUNCT
ejpam-5914	50	32	(	(	PUNCT
ejpam-5914	50	33	x|(y|y))|(x|(y|y	x|(y|y))|(x|(y|y	PROPN
ejpam-5914	50	34	)	)	PUNCT
ejpam-5914	50	35	)	)	PUNCT
ejpam-5914	51	1	=	=	SYM
ejpam-5914	51	2	y|x	y|x	NOUN
ejpam-5914	51	3	,	,	PUNCT
ejpam-5914	51	4	(	(	PUNCT
ejpam-5914	51	5	s2	s2	PROPN
ejpam-5914	51	6	)	)	PUNCT
ejpam-5914	51	7	(	(	PUNCT
ejpam-5914	51	8	x|x)|((x|(y|y))|(x|(y|y	x|x)|((x|(y|y))|(x|(y|y	PROPN
ejpam-5914	51	9	)	)	PUNCT
ejpam-5914	51	10	)	)	PUNCT
ejpam-5914	51	11	)	)	PUNCT
ejpam-5914	52	1	=	=	SYM
ejpam-5914	52	2	x	x	X
ejpam-5914	52	3	,	,	PUNCT
ejpam-5914	52	4	(	(	PUNCT
ejpam-5914	52	5	s3	s3	PROPN
ejpam-5914	52	6	)	)	PUNCT
ejpam-5914	52	7	x|((y|z)|(y|z	x|((y|z)|(y|z	NUM
ejpam-5914	52	8	)	)	PUNCT
ejpam-5914	52	9	)	)	PUNCT
ejpam-5914	53	1	=	=	PUNCT
ejpam-5914	53	2	(	(	PUNCT
ejpam-5914	53	3	(	(	PUNCT
ejpam-5914	53	4	(	(	PUNCT
ejpam-5914	53	5	x|(y|y))|(x|(y|y))))|((x|(y|y))|(x|(y|y)))|z	x|(y|y))|(x|(y|y))))|((x|(y|y))|(x|(y|y)))|z	PROPN
ejpam-5914	53	6	,	,	PUNCT
ejpam-5914	53	7	(	(	PUNCT
ejpam-5914	53	8	s4	s4	PROPN
ejpam-5914	53	9	)	)	PUNCT
ejpam-5914	53	10	(	(	PUNCT
ejpam-5914	53	11	x|((x|x)|(y|y)))|(x|((x|x)|(y|y	x|((x|x)|(y|y)))|(x|((x|x)|(y|y	NUM
ejpam-5914	53	12	)	)	PUNCT
ejpam-5914	53	13	)	)	PUNCT
ejpam-5914	53	14	)	)	PUNCT
ejpam-5914	53	15	=	=	PUNCT
ejpam-5914	54	1	x.	x.	NOUN
ejpam-5914	54	2	definition	definition	NOUN
ejpam-5914	54	3	2	2	NUM
ejpam-5914	54	4	.	.	PUNCT
ejpam-5914	55	1	[	[	X
ejpam-5914	55	2	25	25	NUM
ejpam-5914	55	3	]	]	PUNCT
ejpam-5914	55	4	an	an	DET
ejpam-5914	55	5	algebra	algebra	NOUN
ejpam-5914	55	6	⟨a,→	⟨a,→	PROPN
ejpam-5914	55	7	,	,	PUNCT
ejpam-5914	55	8	0⟩	0⟩	PROPN
ejpam-5914	55	9	of	of	ADP
ejpam-5914	55	10	type	type	NOUN
ejpam-5914	55	11	(	(	PUNCT
ejpam-5914	55	12	2	2	NUM
ejpam-5914	55	13	,	,	PUNCT
ejpam-5914	55	14	0	0	NUM
ejpam-5914	55	15	)	)	PUNCT
ejpam-5914	55	16	is	be	AUX
ejpam-5914	55	17	called	call	VERB
ejpam-5914	55	18	a	a	DET
ejpam-5914	55	19	hilbert	hilbert	NOUN
ejpam-5914	55	20	algebra	algebra	NOUN
ejpam-5914	55	21	if	if	SCONJ
ejpam-5914	55	22	it	it	PRON
ejpam-5914	55	23	satisfies	satisfy	VERB
ejpam-5914	55	24	the	the	DET
ejpam-5914	55	25	following	follow	VERB
ejpam-5914	55	26	axioms	axiom	NOUN
ejpam-5914	55	27	:	:	PUNCT
ejpam-5914	55	28	for	for	ADP
ejpam-5914	55	29	all	all	DET
ejpam-5914	55	30	x	x	NOUN
ejpam-5914	55	31	,	,	PUNCT
ejpam-5914	55	32	y	y	PROPN
ejpam-5914	55	33	,	,	PUNCT
ejpam-5914	55	34	z	z	PROPN
ejpam-5914	55	35	∈	∈	PROPN
ejpam-5914	55	36	a	a	DET
ejpam-5914	55	37	,	,	PUNCT
ejpam-5914	55	38	(	(	PUNCT
ejpam-5914	55	39	h1	h1	PROPN
ejpam-5914	55	40	)	)	PUNCT
ejpam-5914	55	41	x	x	PUNCT
ejpam-5914	56	1	→	→	PUNCT
ejpam-5914	56	2	(	(	PUNCT
ejpam-5914	56	3	y	y	PROPN
ejpam-5914	56	4	→	→	SYM
ejpam-5914	56	5	x	x	X
ejpam-5914	56	6	)	)	PUNCT
ejpam-5914	56	7	=	=	SYM
ejpam-5914	56	8	0	0	NUM
ejpam-5914	56	9	,	,	PUNCT
ejpam-5914	56	10	(	(	PUNCT
ejpam-5914	56	11	h2	h2	NOUN
ejpam-5914	56	12	)	)	PUNCT
ejpam-5914	56	13	(	(	PUNCT
ejpam-5914	56	14	x	x	X
ejpam-5914	56	15	→	→	PUNCT
ejpam-5914	56	16	(	(	PUNCT
ejpam-5914	56	17	y	y	PROPN
ejpam-5914	56	18	→	→	SYM
ejpam-5914	56	19	z	z	NOUN
ejpam-5914	56	20	)	)	PUNCT
ejpam-5914	56	21	)	)	PUNCT
ejpam-5914	56	22	→	→	PUNCT
ejpam-5914	56	23	(	(	PUNCT
ejpam-5914	56	24	(	(	PUNCT
ejpam-5914	56	25	x	x	SYM
ejpam-5914	56	26	→	→	SYM
ejpam-5914	56	27	y	y	NOUN
ejpam-5914	56	28	)	)	PUNCT
ejpam-5914	56	29	→	→	SYM
ejpam-5914	56	30	(	(	PUNCT
ejpam-5914	56	31	x	x	X
ejpam-5914	56	32	→	→	SYM
ejpam-5914	56	33	z	z	NOUN
ejpam-5914	56	34	)	)	PUNCT
ejpam-5914	56	35	)	)	PUNCT
ejpam-5914	57	1	=	=	SYM
ejpam-5914	57	2	0	0	NUM
ejpam-5914	57	3	,	,	PUNCT
ejpam-5914	57	4	(	(	PUNCT
ejpam-5914	57	5	h3	h3	NOUN
ejpam-5914	57	6	)	)	PUNCT
ejpam-5914	57	7	x	x	PUNCT
ejpam-5914	58	1	→	→	PUNCT
ejpam-5914	58	2	y	y	PROPN
ejpam-5914	58	3	=	=	SYM
ejpam-5914	58	4	0	0	PROPN
ejpam-5914	58	5	and	and	CCONJ
ejpam-5914	58	6	y	y	PROPN
ejpam-5914	58	7	→	→	SYM
ejpam-5914	58	8	x	x	SYM
ejpam-5914	58	9	=	=	SYM
ejpam-5914	58	10	0	0	NUM
ejpam-5914	58	11	⇒	⇒	NOUN
ejpam-5914	58	12	x	x	PUNCT
ejpam-5914	59	1	=	=	PUNCT
ejpam-5914	59	2	y.	y.	NOUN
ejpam-5914	59	3	definition	definition	NOUN
ejpam-5914	59	4	3	3	NUM
ejpam-5914	59	5	.	.	PUNCT
ejpam-5914	60	1	[	[	X
ejpam-5914	60	2	9	9	NUM
ejpam-5914	60	3	]	]	X
ejpam-5914	60	4	a	a	DET
ejpam-5914	60	5	sheffer	sheffer	NOUN
ejpam-5914	60	6	stroke	stroke	NOUN
ejpam-5914	60	7	hilbert	hilbert	PROPN
ejpam-5914	60	8	algebra	algebra	PROPN
ejpam-5914	60	9	(	(	PUNCT
ejpam-5914	60	10	abbreviated	abbreviate	VERB
ejpam-5914	60	11	sha	sha	PROPN
ejpam-5914	60	12	)	)	PUNCT
ejpam-5914	60	13	is	be	AUX
ejpam-5914	60	14	a	a	DET
ejpam-5914	60	15	structure	structure	NOUN
ejpam-5914	60	16	⟨a	⟨a	PROPN
ejpam-5914	60	17	,	,	PUNCT
ejpam-5914	60	18	|	|	ADV
ejpam-5914	60	19	,	,	PUNCT
ejpam-5914	60	20	0⟩	0⟩	PROPN
ejpam-5914	60	21	of	of	ADP
ejpam-5914	60	22	type	type	NOUN
ejpam-5914	60	23	(	(	PUNCT
ejpam-5914	60	24	2	2	NUM
ejpam-5914	60	25	,	,	PUNCT
ejpam-5914	60	26	0	0	NUM
ejpam-5914	60	27	)	)	PUNCT
ejpam-5914	60	28	,	,	PUNCT
ejpam-5914	60	29	in	in	ADP
ejpam-5914	60	30	which	which	PRON
ejpam-5914	60	31	a	a	PRON
ejpam-5914	60	32	is	be	AUX
ejpam-5914	60	33	a	a	DET
ejpam-5914	60	34	nonempty	nonempty	ADJ
ejpam-5914	60	35	set	set	NOUN
ejpam-5914	60	36	,	,	PUNCT
ejpam-5914	60	37	|	|	ADV
ejpam-5914	60	38	is	be	AUX
ejpam-5914	60	39	a	a	DET
ejpam-5914	60	40	sheffer	sheffer	NOUN
ejpam-5914	60	41	stroke	stroke	NOUN
ejpam-5914	60	42	operation	operation	NOUN
ejpam-5914	60	43	on	on	ADP
ejpam-5914	60	44	a	a	PRON
ejpam-5914	60	45	,	,	PUNCT
ejpam-5914	60	46	and	and	CCONJ
ejpam-5914	60	47	0	0	NUM
ejpam-5914	60	48	is	be	AUX
ejpam-5914	60	49	the	the	DET
ejpam-5914	60	50	fixed	fix	VERB
ejpam-5914	60	51	element	element	NOUN
ejpam-5914	60	52	in	in	ADP
ejpam-5914	60	53	a	a	DET
ejpam-5914	60	54	such	such	ADJ
ejpam-5914	60	55	that	that	SCONJ
ejpam-5914	60	56	the	the	DET
ejpam-5914	60	57	following	follow	VERB
ejpam-5914	60	58	identities	identity	NOUN
ejpam-5914	60	59	are	be	AUX
ejpam-5914	60	60	satisfied	satisfied	ADJ
ejpam-5914	60	61	for	for	ADP
ejpam-5914	60	62	all	all	DET
ejpam-5914	60	63	x	x	NOUN
ejpam-5914	60	64	,	,	PUNCT
ejpam-5914	60	65	y	y	PROPN
ejpam-5914	60	66	,	,	PUNCT
ejpam-5914	60	67	z	z	PROPN
ejpam-5914	60	68	∈	∈	PROPN
ejpam-5914	60	69	a	a	DET
ejpam-5914	60	70	,	,	PUNCT
ejpam-5914	60	71	(	(	PUNCT
ejpam-5914	60	72	1	1	NUM
ejpam-5914	60	73	)	)	PUNCT
ejpam-5914	60	74	(	(	PUNCT
ejpam-5914	60	75	x|(p	x|(p	PROPN
ejpam-5914	60	76	|p	|p	PROPN
ejpam-5914	60	77	)	)	PUNCT
ejpam-5914	60	78	)	)	PUNCT
ejpam-5914	60	79	|(q|(r|r))|(q|(r|r	|(q|(r|r))|(q|(r|r	NUM
ejpam-5914	60	80	)	)	PUNCT
ejpam-5914	60	81	)	)	PUNCT
ejpam-5914	61	1	=	=	SYM
ejpam-5914	61	2	x|(x|x	x|(x|x	PROPN
ejpam-5914	61	3	)	)	PUNCT
ejpam-5914	61	4	,	,	PUNCT
ejpam-5914	61	5	where	where	SCONJ
ejpam-5914	61	6	p	p	X
ejpam-5914	61	7	:	:	PUNCT
ejpam-5914	61	8	=	=	SYM
ejpam-5914	61	9	y|(z|z	y|(z|z	PROPN
ejpam-5914	61	10	)	)	PUNCT
ejpam-5914	61	11	,	,	PUNCT
ejpam-5914	61	12	q	q	NOUN
ejpam-5914	61	13	:	:	PUNCT
ejpam-5914	61	14	=	=	SYM
ejpam-5914	61	15	x|(y|y	x|(y|y	PROPN
ejpam-5914	61	16	)	)	PUNCT
ejpam-5914	61	17	and	and	CCONJ
ejpam-5914	61	18	r	r	NOUN
ejpam-5914	61	19	:	:	PUNCT
ejpam-5914	61	20	=	=	PUNCT
ejpam-5914	61	21	x|(z|z	x|(z|z	PROPN
ejpam-5914	61	22	)	)	PUNCT
ejpam-5914	61	23	,	,	PUNCT
ejpam-5914	61	24	(	(	PUNCT
ejpam-5914	61	25	2	2	X
ejpam-5914	61	26	)	)	PUNCT
ejpam-5914	61	27	x|(y|y	x|(y|y	NUM
ejpam-5914	61	28	)	)	PUNCT
ejpam-5914	61	29	=	=	SYM
ejpam-5914	61	30	y|(x|x	y|(x|x	PROPN
ejpam-5914	61	31	)	)	PUNCT
ejpam-5914	61	32	=	=	SYM
ejpam-5914	61	33	x|(x|x	x|(x|x	PROPN
ejpam-5914	61	34	)	)	PUNCT
ejpam-5914	61	35	⇒	⇒	NOUN
ejpam-5914	61	36	x	x	PUNCT
ejpam-5914	62	1	=	=	PUNCT
ejpam-5914	62	2	y.	y.	NOUN
ejpam-5914	62	3	proposition	proposition	NOUN
ejpam-5914	62	4	1	1	NUM
ejpam-5914	62	5	.	.	PUNCT
ejpam-5914	63	1	[	[	X
ejpam-5914	63	2	9	9	NUM
ejpam-5914	63	3	]	]	X
ejpam-5914	63	4	let	let	VERB
ejpam-5914	63	5	⟨a	⟨a	NOUN
ejpam-5914	63	6	,	,	PUNCT
ejpam-5914	63	7	|	|	ADV
ejpam-5914	63	8	,	,	PUNCT
ejpam-5914	63	9	0⟩	0⟩	PROPN
ejpam-5914	63	10	be	be	VERB
ejpam-5914	63	11	a	a	DET
ejpam-5914	63	12	sheffer	sheffer	NOUN
ejpam-5914	63	13	stroke	stroke	NOUN
ejpam-5914	63	14	hilbert	hilbert	PROPN
ejpam-5914	63	15	algebra	algebra	PROPN
ejpam-5914	63	16	.	.	PUNCT
ejpam-5914	64	1	then	then	ADV
ejpam-5914	64	2	the	the	DET
ejpam-5914	64	3	binary	binary	PROPN
ejpam-5914	64	4	relation	relation	PROPN
ejpam-5914	64	5	x	x	SYM
ejpam-5914	64	6	≤	≤	ADJ
ejpam-5914	64	7	y	y	NOUN
ejpam-5914	64	8	if	if	SCONJ
ejpam-5914	65	1	and	and	CCONJ
ejpam-5914	65	2	only	only	ADV
ejpam-5914	65	3	if	if	SCONJ
ejpam-5914	65	4	x|(y|y	x|(y|y	PROPN
ejpam-5914	65	5	)	)	PUNCT
ejpam-5914	65	6	=	=	SYM
ejpam-5914	65	7	0	0	NUM
ejpam-5914	65	8	is	be	AUX
ejpam-5914	65	9	a	a	DET
ejpam-5914	65	10	partial	partial	ADJ
ejpam-5914	65	11	order	order	NOUN
ejpam-5914	65	12	on	on	ADP
ejpam-5914	65	13	a.	a.	NOUN
ejpam-5914	65	14	definition	definition	NOUN
ejpam-5914	65	15	4	4	NUM
ejpam-5914	65	16	.	.	PUNCT
ejpam-5914	66	1	[	[	X
ejpam-5914	66	2	9	9	NUM
ejpam-5914	66	3	]	]	PUNCT
ejpam-5914	66	4	a	a	DET
ejpam-5914	66	5	nonempty	nonempty	NOUN
ejpam-5914	66	6	subset	subset	VERB
ejpam-5914	66	7	g	g	NOUN
ejpam-5914	66	8	of	of	ADP
ejpam-5914	66	9	a	a	DET
ejpam-5914	66	10	sheffer	sheffer	NOUN
ejpam-5914	66	11	stroke	stroke	NOUN
ejpam-5914	66	12	hilbert	hilbert	PROPN
ejpam-5914	66	13	algebra	algebra	PROPN
ejpam-5914	66	14	⟨a	⟨a	PROPN
ejpam-5914	66	15	,	,	PUNCT
ejpam-5914	66	16	|	|	ADV
ejpam-5914	66	17	,	,	PUNCT
ejpam-5914	66	18	0⟩	0⟩	PROPN
ejpam-5914	66	19	is	be	AUX
ejpam-5914	66	20	called	call	VERB
ejpam-5914	66	21	a	a	DET
ejpam-5914	66	22	subalgebra	subalgebra	NOUN
ejpam-5914	66	23	of	of	ADP
ejpam-5914	66	24	a	a	DET
ejpam-5914	66	25	if	if	NOUN
ejpam-5914	66	26	(	(	PUNCT
ejpam-5914	66	27	x|(y|y))|(x|(y|y	x|(y|y))|(x|(y|y	ADJ
ejpam-5914	66	28	)	)	PUNCT
ejpam-5914	66	29	)	)	PUNCT
ejpam-5914	67	1	∈	∈	PROPN
ejpam-5914	67	2	g	g	NOUN
ejpam-5914	67	3	for	for	ADP
ejpam-5914	67	4	all	all	DET
ejpam-5914	67	5	x	x	NOUN
ejpam-5914	67	6	,	,	PUNCT
ejpam-5914	67	7	y	y	PROPN
ejpam-5914	67	8	∈	∈	PROPN
ejpam-5914	67	9	g.	g.	NOUN
ejpam-5914	67	10	definition	definition	NOUN
ejpam-5914	67	11	5	5	NUM
ejpam-5914	67	12	.	.	PUNCT
ejpam-5914	68	1	[	[	X
ejpam-5914	68	2	15	15	NUM
ejpam-5914	68	3	]	]	X
ejpam-5914	68	4	an	an	DET
ejpam-5914	68	5	interval	interval	NOUN
ejpam-5914	68	6	-	-	PUNCT
ejpam-5914	68	7	valued	value	VERB
ejpam-5914	68	8	intuitionistic	intuitionistic	ADJ
ejpam-5914	68	9	fuzzy	fuzzy	ADJ
ejpam-5914	68	10	set	set	NOUN
ejpam-5914	68	11	x	x	PUNCT
ejpam-5914	68	12	over	over	ADP
ejpam-5914	68	13	a	a	DET
ejpam-5914	68	14	nonempty	nonempty	NOUN
ejpam-5914	68	15	set	set	VERB
ejpam-5914	68	16	a	a	PRON
ejpam-5914	68	17	is	be	AUX
ejpam-5914	68	18	an	an	DET
ejpam-5914	68	19	object	object	NOUN
ejpam-5914	68	20	having	have	VERB
ejpam-5914	68	21	the	the	DET
ejpam-5914	68	22	form	form	NOUN
ejpam-5914	68	23	x	x	PUNCT
ejpam-5914	68	24	=	=	NOUN
ejpam-5914	68	25	{	{	PUNCT
ejpam-5914	68	26	⟨x	⟨x	VERB
ejpam-5914	68	27	,	,	PUNCT
ejpam-5914	68	28	µx(x	µx(x	NOUN
ejpam-5914	68	29	)	)	PUNCT
ejpam-5914	68	30	,	,	PUNCT
ejpam-5914	68	31	γx(x)⟩	γx(x)⟩	NUM
ejpam-5914	68	32	:	:	PUNCT
ejpam-5914	68	33	x	x	PUNCT
ejpam-5914	68	34	∈	∈	PROPN
ejpam-5914	68	35	a	a	PRON
ejpam-5914	68	36	}	}	PUNCT
ejpam-5914	68	37	,	,	PUNCT
ejpam-5914	68	38	where	where	SCONJ
ejpam-5914	68	39	µx(x	µx(x	PUNCT
ejpam-5914	68	40	)	)	PUNCT
ejpam-5914	68	41	:	:	PUNCT
ejpam-5914	68	42	a	a	DET
ejpam-5914	68	43	→	→	SYM
ejpam-5914	68	44	d[0	d[0	ADJ
ejpam-5914	68	45	,	,	PUNCT
ejpam-5914	68	46	1	1	NUM
ejpam-5914	68	47	]	]	PUNCT
ejpam-5914	68	48	and	and	CCONJ
ejpam-5914	68	49	γx(x	γx(x	ADV
ejpam-5914	68	50	)	)	PUNCT
ejpam-5914	68	51	:	:	PUNCT
ejpam-5914	68	52	a	a	DET
ejpam-5914	68	53	→	→	SYM
ejpam-5914	68	54	d[0	d[0	ADJ
ejpam-5914	68	55	,	,	PUNCT
ejpam-5914	68	56	1	1	NUM
ejpam-5914	68	57	]	]	PUNCT
ejpam-5914	68	58	and	and	CCONJ
ejpam-5914	68	59	d[0	d[0	ADJ
ejpam-5914	68	60	,	,	PUNCT
ejpam-5914	68	61	1	1	NUM
ejpam-5914	68	62	]	]	PUNCT
ejpam-5914	68	63	is	be	AUX
ejpam-5914	68	64	the	the	DET
ejpam-5914	68	65	set	set	NOUN
ejpam-5914	68	66	of	of	ADP
ejpam-5914	68	67	all	all	DET
ejpam-5914	68	68	intervals	interval	NOUN
ejpam-5914	68	69	of	of	ADP
ejpam-5914	68	70	[	[	X
ejpam-5914	68	71	0	0	NUM
ejpam-5914	68	72	,	,	PUNCT
ejpam-5914	68	73	1	1	NUM
ejpam-5914	68	74	]	]	PUNCT
ejpam-5914	68	75	.	.	PUNCT
ejpam-5914	69	1	the	the	DET
ejpam-5914	69	2	intervals	interval	NOUN
ejpam-5914	69	3	µx(x	µx(x	PUNCT
ejpam-5914	69	4	)	)	PUNCT
ejpam-5914	69	5	and	and	CCONJ
ejpam-5914	69	6	γx(x	γx(x	ADV
ejpam-5914	69	7	)	)	PUNCT
ejpam-5914	69	8	denote	denote	VERB
ejpam-5914	69	9	the	the	DET
ejpam-5914	69	10	intervals	interval	NOUN
ejpam-5914	69	11	of	of	ADP
ejpam-5914	69	12	the	the	DET
ejpam-5914	69	13	degree	degree	NOUN
ejpam-5914	69	14	of	of	ADP
ejpam-5914	69	15	belongingness	belongingness	NOUN
ejpam-5914	69	16	and	and	CCONJ
ejpam-5914	69	17	non	non	ADJ
ejpam-5914	69	18	-	-	NOUN
ejpam-5914	69	19	belongingness	belongingness	NOUN
ejpam-5914	69	20	of	of	ADP
ejpam-5914	69	21	the	the	DET
ejpam-5914	69	22	element	element	NOUN
ejpam-5914	69	23	x	x	PUNCT
ejpam-5914	69	24	to	to	ADP
ejpam-5914	69	25	x	x	PRON
ejpam-5914	69	26	,	,	PUNCT
ejpam-5914	69	27	where	where	SCONJ
ejpam-5914	69	28	µx(x	µx(x	PUNCT
ejpam-5914	69	29	)	)	PUNCT
ejpam-5914	69	30	=	=	PUNCT
ejpam-5914	70	1	[	[	X
ejpam-5914	70	2	µl	µl	ADP
ejpam-5914	70	3	x(x	x(x	NOUN
ejpam-5914	70	4	)	)	PUNCT
ejpam-5914	70	5	,	,	PUNCT
ejpam-5914	70	6	µu	µu	ADP
ejpam-5914	70	7	x(x	x(x	PROPN
ejpam-5914	70	8	)	)	PUNCT
ejpam-5914	70	9	]	]	PUNCT
ejpam-5914	70	10	and	and	CCONJ
ejpam-5914	70	11	γx(x	γx(x	ADV
ejpam-5914	70	12	)	)	PUNCT
ejpam-5914	70	13	=	=	PUNCT
ejpam-5914	71	1	[	[	X
ejpam-5914	71	2	γlx(x	γlx(x	NOUN
ejpam-5914	71	3	)	)	PUNCT
ejpam-5914	71	4	,	,	PUNCT
ejpam-5914	71	5	γux(x	γux(x	PROPN
ejpam-5914	71	6	)	)	PUNCT
ejpam-5914	71	7	]	]	PUNCT
ejpam-5914	71	8	for	for	ADP
ejpam-5914	71	9	all	all	DET
ejpam-5914	71	10	x	x	PROPN
ejpam-5914	71	11	∈	∈	PROPN
ejpam-5914	71	12	a	a	PRON
ejpam-5914	71	13	with	with	ADP
ejpam-5914	71	14	the	the	DET
ejpam-5914	71	15	condition	condition	NOUN
ejpam-5914	71	16	0	0	NUM
ejpam-5914	71	17	≤	≤	NOUN
ejpam-5914	71	18	µl	µl	ADP
ejpam-5914	71	19	x(x	x(x	NOUN
ejpam-5914	71	20	)	)	PUNCT
ejpam-5914	71	21	+	+	CCONJ
ejpam-5914	71	22	γux(x	γux(x	NOUN
ejpam-5914	71	23	)	)	PUNCT
ejpam-5914	71	24	≤	≤	NUM
ejpam-5914	71	25	1	1	NUM
ejpam-5914	71	26	.	.	PUNCT
ejpam-5914	71	27	for	for	ADP
ejpam-5914	71	28	the	the	DET
ejpam-5914	71	29	sake	sake	NOUN
ejpam-5914	71	30	of	of	ADP
ejpam-5914	71	31	simplicity	simplicity	NOUN
ejpam-5914	71	32	,	,	PUNCT
ejpam-5914	71	33	we	we	PRON
ejpam-5914	71	34	shall	shall	AUX
ejpam-5914	71	35	use	use	VERB
ejpam-5914	71	36	the	the	DET
ejpam-5914	71	37	symbol	symbol	NOUN
ejpam-5914	71	38	x	x	PUNCT
ejpam-5914	72	1	=	=	PUNCT
ejpam-5914	72	2	(	(	PUNCT
ejpam-5914	72	3	µx	µx	INTJ
ejpam-5914	72	4	,	,	PUNCT
ejpam-5914	72	5	γx	γx	NOUN
ejpam-5914	72	6	)	)	PUNCT
ejpam-5914	72	7	for	for	ADP
ejpam-5914	72	8	the	the	DET
ejpam-5914	72	9	interval	interval	NOUN
ejpam-5914	72	10	-	-	PUNCT
ejpam-5914	72	11	valued	value	VERB
ejpam-5914	72	12	intuitionistic	intuitionistic	ADJ
ejpam-5914	72	13	fuzzy	fuzzy	ADJ
ejpam-5914	72	14	set	set	NOUN
ejpam-5914	72	15	x	x	SYM
ejpam-5914	72	16	=	=	X
ejpam-5914	72	17	{	{	PUNCT
ejpam-5914	72	18	⟨x	⟨x	VERB
ejpam-5914	72	19	,	,	PUNCT
ejpam-5914	72	20	µx(x	µx(x	NOUN
ejpam-5914	72	21	)	)	PUNCT
ejpam-5914	72	22	,	,	PUNCT
ejpam-5914	72	23	γx(x)⟩	γx(x)⟩	NUM
ejpam-5914	72	24	:	:	PUNCT
ejpam-5914	72	25	x	x	PUNCT
ejpam-5914	72	26	∈	∈	PROPN
ejpam-5914	72	27	a	a	PRON
ejpam-5914	72	28	}	}	PUNCT
ejpam-5914	72	29	.	.	PUNCT
ejpam-5914	73	1	note	note	VERB
ejpam-5914	73	2	that	that	PRON
ejpam-5914	73	3	µx(x	µx(x	PUNCT
ejpam-5914	73	4	)	)	PUNCT
ejpam-5914	73	5	=	=	PUNCT
ejpam-5914	74	1	[	[	X
ejpam-5914	74	2	1−	1−	NUM
ejpam-5914	74	3	µu	µu	ADP
ejpam-5914	74	4	x(x	x(x	PROPN
ejpam-5914	74	5	)	)	PUNCT
ejpam-5914	74	6	,	,	PUNCT
ejpam-5914	74	7	1−	1−	NUM
ejpam-5914	74	8	µl	µl	ADP
ejpam-5914	74	9	x(x	x(x	PROPN
ejpam-5914	74	10	)	)	PUNCT
ejpam-5914	74	11	]	]	PUNCT
ejpam-5914	74	12	and	and	CCONJ
ejpam-5914	74	13	γx(x	γx(x	ADV
ejpam-5914	74	14	)	)	PUNCT
ejpam-5914	74	15	=	=	PUNCT
ejpam-5914	75	1	[	[	X
ejpam-5914	75	2	1−	1−	NUM
ejpam-5914	75	3	γux(x	γux(x	NOUN
ejpam-5914	75	4	)	)	PUNCT
ejpam-5914	75	5	,	,	PUNCT
ejpam-5914	75	6	1−	1−	NUM
ejpam-5914	75	7	γlx(x	γlx(x	NOUN
ejpam-5914	75	8	)	)	PUNCT
ejpam-5914	75	9	]	]	PUNCT
ejpam-5914	75	10	,	,	PUNCT
ejpam-5914	75	11	where	where	SCONJ
ejpam-5914	75	12	[	[	X
ejpam-5914	75	13	µx(x	µx(x	NOUN
ejpam-5914	75	14	)	)	PUNCT
ejpam-5914	75	15	,	,	PUNCT
ejpam-5914	75	16	γx(x	γx(x	ADV
ejpam-5914	75	17	)	)	PUNCT
ejpam-5914	75	18	]	]	PUNCT
ejpam-5914	75	19	represents	represent	VERB
ejpam-5914	75	20	the	the	DET
ejpam-5914	75	21	complement	complement	NOUN
ejpam-5914	75	22	of	of	ADP
ejpam-5914	75	23	x	x	PUNCT
ejpam-5914	75	24	in	in	ADP
ejpam-5914	75	25	x.	x.	PROPN
ejpam-5914	75	26	n.	n.	PROPN
ejpam-5914	75	27	rajesh	rajesh	PROPN
ejpam-5914	75	28	et	et	PROPN
ejpam-5914	76	1	al	al	PROPN
ejpam-5914	76	2	.	.	PUNCT
ejpam-5914	76	3	/	/	SYM
ejpam-5914	76	4	eur	eur	PROPN
ejpam-5914	76	5	.	.	PUNCT
ejpam-5914	77	1	j.	j.	PROPN
ejpam-5914	77	2	pure	pure	PROPN
ejpam-5914	77	3	appl	appl	PROPN
ejpam-5914	77	4	.	.	PROPN
ejpam-5914	77	5	math	math	PROPN
ejpam-5914	77	6	,	,	PUNCT
ejpam-5914	77	7	18	18	NUM
ejpam-5914	77	8	(	(	PUNCT
ejpam-5914	77	9	2	2	NUM
ejpam-5914	77	10	)	)	PUNCT
ejpam-5914	77	11	(	(	PUNCT
ejpam-5914	77	12	2025	2025	NUM
ejpam-5914	77	13	)	)	PUNCT
ejpam-5914	77	14	,	,	PUNCT
ejpam-5914	77	15	5914	5914	NUM
ejpam-5914	77	16	4	4	NUM
ejpam-5914	77	17	of	of	ADP
ejpam-5914	77	18	21	21	NUM
ejpam-5914	77	19	3	3	NUM
ejpam-5914	77	20	.	.	NOUN
ejpam-5914	77	21	length	length	NOUN
ejpam-5914	77	22	of	of	ADP
ejpam-5914	77	23	an	an	DET
ejpam-5914	77	24	interval	interval	NOUN
ejpam-5914	77	25	-	-	PUNCT
ejpam-5914	77	26	valued	value	VERB
ejpam-5914	77	27	fuzzy	fuzzy	ADJ
ejpam-5914	77	28	structure	structure	NOUN
ejpam-5914	77	29	in	in	ADP
ejpam-5914	77	30	sheffer	sheffer	PROPN
ejpam-5914	77	31	stroke	stroke	PROPN
ejpam-5914	77	32	hilbert	hilbert	PROPN
ejpam-5914	77	33	algebras	algebras	PROPN
ejpam-5914	77	34	in	in	ADP
ejpam-5914	77	35	this	this	DET
ejpam-5914	77	36	section	section	NOUN
ejpam-5914	77	37	,	,	PUNCT
ejpam-5914	77	38	we	we	PRON
ejpam-5914	77	39	present	present	VERB
ejpam-5914	77	40	the	the	DET
ejpam-5914	77	41	concept	concept	NOUN
ejpam-5914	77	42	of	of	ADP
ejpam-5914	77	43	the	the	DET
ejpam-5914	77	44	length	length	NOUN
ejpam-5914	77	45	of	of	ADP
ejpam-5914	77	46	an	an	DET
ejpam-5914	77	47	interval	interval	NOUN
ejpam-5914	77	48	-	-	PUNCT
ejpam-5914	77	49	valued	value	VERB
ejpam-5914	77	50	fuzzy	fuzzy	ADJ
ejpam-5914	77	51	structure	structure	NOUN
ejpam-5914	77	52	within	within	ADP
ejpam-5914	77	53	the	the	DET
ejpam-5914	77	54	framework	framework	NOUN
ejpam-5914	77	55	of	of	ADP
ejpam-5914	77	56	sheffer	sheffer	PROPN
ejpam-5914	77	57	stroke	stroke	PROPN
ejpam-5914	77	58	hilbert	hilbert	PROPN
ejpam-5914	77	59	algebras	algebras	PROPN
ejpam-5914	77	60	.	.	PUNCT
ejpam-5914	78	1	we	we	PRON
ejpam-5914	78	2	introduce	introduce	VERB
ejpam-5914	78	3	the	the	DET
ejpam-5914	78	4	notion	notion	NOUN
ejpam-5914	78	5	of	of	ADP
ejpam-5914	78	6	lengthfuzzy	lengthfuzzy	ADJ
ejpam-5914	78	7	subalgebras	subalgebras	PROPN
ejpam-5914	78	8	,	,	PUNCT
ejpam-5914	78	9	which	which	PRON
ejpam-5914	78	10	are	be	AUX
ejpam-5914	78	11	specific	specific	ADJ
ejpam-5914	78	12	to	to	ADP
ejpam-5914	78	13	these	these	DET
ejpam-5914	78	14	algebras	algebra	NOUN
ejpam-5914	78	15	,	,	PUNCT
ejpam-5914	78	16	and	and	CCONJ
ejpam-5914	78	17	explore	explore	VERB
ejpam-5914	78	18	their	their	PRON
ejpam-5914	78	19	fundamental	fundamental	ADJ
ejpam-5914	78	20	properties	property	NOUN
ejpam-5914	78	21	and	and	CCONJ
ejpam-5914	78	22	interrelationships	interrelationship	NOUN
ejpam-5914	78	23	.	.	PUNCT
ejpam-5914	79	1	this	this	DET
ejpam-5914	79	2	analysis	analysis	NOUN
ejpam-5914	79	3	aims	aim	VERB
ejpam-5914	79	4	to	to	PART
ejpam-5914	79	5	deepen	deepen	VERB
ejpam-5914	79	6	the	the	DET
ejpam-5914	79	7	understanding	understanding	NOUN
ejpam-5914	79	8	of	of	ADP
ejpam-5914	79	9	how	how	SCONJ
ejpam-5914	79	10	interval	interval	NOUN
ejpam-5914	79	11	-	-	PUNCT
ejpam-5914	79	12	valued	value	VERB
ejpam-5914	79	13	fuzziness	fuzziness	NOUN
ejpam-5914	79	14	interacts	interact	VERB
ejpam-5914	79	15	with	with	ADP
ejpam-5914	79	16	the	the	DET
ejpam-5914	79	17	algebraic	algebraic	ADJ
ejpam-5914	79	18	operations	operation	NOUN
ejpam-5914	79	19	in	in	ADP
ejpam-5914	79	20	sheffer	sheffer	PROPN
ejpam-5914	79	21	stroke	stroke	PROPN
ejpam-5914	79	22	hilbert	hilbert	PROPN
ejpam-5914	79	23	algebras	algebras	PROPN
ejpam-5914	79	24	,	,	PUNCT
ejpam-5914	79	25	providing	provide	VERB
ejpam-5914	79	26	new	new	ADJ
ejpam-5914	79	27	insights	insight	NOUN
ejpam-5914	79	28	into	into	ADP
ejpam-5914	79	29	their	their	PRON
ejpam-5914	79	30	structural	structural	ADJ
ejpam-5914	79	31	characteristics	characteristic	NOUN
ejpam-5914	79	32	.	.	PUNCT
ejpam-5914	80	1	throughout	throughout	ADP
ejpam-5914	80	2	this	this	DET
ejpam-5914	80	3	discussion	discussion	NOUN
ejpam-5914	80	4	,	,	PUNCT
ejpam-5914	80	5	we	we	PRON
ejpam-5914	80	6	assume	assume	VERB
ejpam-5914	80	7	a	a	DET
ejpam-5914	80	8	=	=	SYM
ejpam-5914	80	9	⟨a	⟨a	NOUN
ejpam-5914	80	10	,	,	PUNCT
ejpam-5914	80	11	|	|	ADV
ejpam-5914	80	12	,	,	PUNCT
ejpam-5914	80	13	0⟩	0⟩	PROPN
ejpam-5914	80	14	to	to	PART
ejpam-5914	80	15	be	be	AUX
ejpam-5914	80	16	a	a	DET
ejpam-5914	80	17	sheffer	sheffer	NOUN
ejpam-5914	80	18	stroke	stroke	NOUN
ejpam-5914	80	19	hilbert	hilbert	PROPN
ejpam-5914	80	20	algebra	algebra	PROPN
ejpam-5914	80	21	,	,	PUNCT
ejpam-5914	80	22	serving	serve	VERB
ejpam-5914	80	23	as	as	ADP
ejpam-5914	80	24	the	the	DET
ejpam-5914	80	25	foundational	foundational	ADJ
ejpam-5914	80	26	structure	structure	NOUN
ejpam-5914	80	27	for	for	ADP
ejpam-5914	80	28	the	the	DET
ejpam-5914	80	29	concepts	concept	NOUN
ejpam-5914	80	30	and	and	CCONJ
ejpam-5914	80	31	results	result	NOUN
ejpam-5914	80	32	developed	develop	VERB
ejpam-5914	80	33	herein	herein	NOUN
ejpam-5914	80	34	.	.	PUNCT
ejpam-5914	81	1	definition	definition	NOUN
ejpam-5914	81	2	6	6	NUM
ejpam-5914	81	3	.	.	PUNCT
ejpam-5914	82	1	given	give	VERB
ejpam-5914	82	2	an	an	DET
ejpam-5914	82	3	interval	interval	NOUN
ejpam-5914	82	4	-	-	PUNCT
ejpam-5914	82	5	valued	value	VERB
ejpam-5914	82	6	fuzzy	fuzzy	ADJ
ejpam-5914	82	7	structure	structure	NOUN
ejpam-5914	82	8	(	(	PUNCT
ejpam-5914	82	9	a	a	PRON
ejpam-5914	82	10	,	,	PUNCT
ejpam-5914	82	11	f̃	f̃	PROPN
ejpam-5914	82	12	)	)	PUNCT
ejpam-5914	82	13	over	over	ADP
ejpam-5914	82	14	a	a	DET
ejpam-5914	82	15	nonempty	nonempty	NOUN
ejpam-5914	82	16	set	set	VERB
ejpam-5914	82	17	a	a	PRON
ejpam-5914	82	18	,	,	PUNCT
ejpam-5914	82	19	we	we	PRON
ejpam-5914	82	20	define	define	VERB
ejpam-5914	82	21	two	two	NUM
ejpam-5914	82	22	fuzzy	fuzzy	ADJ
ejpam-5914	82	23	structures	structure	NOUN
ejpam-5914	82	24	(	(	PUNCT
ejpam-5914	82	25	a	a	PRON
ejpam-5914	82	26	,	,	PUNCT
ejpam-5914	82	27	f̃inf	f̃inf	ADJ
ejpam-5914	82	28	)	)	PUNCT
ejpam-5914	82	29	and	and	CCONJ
ejpam-5914	82	30	(	(	PUNCT
ejpam-5914	82	31	a	a	DET
ejpam-5914	82	32	,	,	PUNCT
ejpam-5914	82	33	f̃sup	f̃sup	NOUN
ejpam-5914	82	34	)	)	PUNCT
ejpam-5914	82	35	in	in	ADP
ejpam-5914	82	36	a	a	PRON
ejpam-5914	82	37	as	as	SCONJ
ejpam-5914	82	38	follows	follow	VERB
ejpam-5914	82	39	:	:	PUNCT
ejpam-5914	82	40	f̃inf	f̃inf	ADP
ejpam-5914	82	41	:	:	PUNCT
ejpam-5914	82	42	a	a	DET
ejpam-5914	82	43	→	→	SYM
ejpam-5914	82	44	[	[	X
ejpam-5914	82	45	0	0	NUM
ejpam-5914	82	46	,	,	PUNCT
ejpam-5914	82	47	1];x	1];x	NUM
ejpam-5914	82	48	7→	7→	NUM
ejpam-5914	82	49	inf{f̃(x	inf{f̃(x	NOUN
ejpam-5914	82	50	)	)	PUNCT
ejpam-5914	82	51	}	}	PUNCT
ejpam-5914	82	52	,	,	PUNCT
ejpam-5914	82	53	and	and	CCONJ
ejpam-5914	82	54	f̃sup	f̃sup	ADJ
ejpam-5914	82	55	:	:	PUNCT
ejpam-5914	82	56	a	a	DET
ejpam-5914	82	57	→	→	SYM
ejpam-5914	82	58	[	[	X
ejpam-5914	82	59	0	0	NUM
ejpam-5914	82	60	,	,	PUNCT
ejpam-5914	82	61	1];x	1];x	NUM
ejpam-5914	82	62	7→	7→	NUM
ejpam-5914	82	63	sup{f̃(x	sup{f̃(x	PROPN
ejpam-5914	82	64	)	)	PUNCT
ejpam-5914	82	65	}	}	PUNCT
ejpam-5914	82	66	.	.	PUNCT
ejpam-5914	83	1	example	example	NOUN
ejpam-5914	84	1	1	1	NUM
ejpam-5914	84	2	.	.	PUNCT
ejpam-5914	85	1	[	[	X
ejpam-5914	85	2	9	9	NUM
ejpam-5914	85	3	]	]	PUNCT
ejpam-5914	85	4	let	let	VERB
ejpam-5914	85	5	a	a	DET
ejpam-5914	85	6	=	=	PUNCT
ejpam-5914	85	7	{	{	PUNCT
ejpam-5914	85	8	0	0	NUM
ejpam-5914	85	9	,	,	PUNCT
ejpam-5914	85	10	u	u	NOUN
ejpam-5914	85	11	,	,	PUNCT
ejpam-5914	85	12	v	v	NOUN
ejpam-5914	85	13	,	,	PUNCT
ejpam-5914	85	14	1	1	NUM
ejpam-5914	85	15	}	}	PUNCT
ejpam-5914	85	16	be	be	AUX
ejpam-5914	85	17	a	a	DET
ejpam-5914	85	18	set	set	NOUN
ejpam-5914	85	19	with	with	ADP
ejpam-5914	85	20	the	the	DET
ejpam-5914	85	21	binary	binary	ADJ
ejpam-5914	85	22	operation	operation	NOUN
ejpam-5914	85	23	|	|	ADV
ejpam-5914	85	24	given	give	VERB
ejpam-5914	85	25	in	in	ADP
ejpam-5914	85	26	the	the	DET
ejpam-5914	85	27	following	follow	VERB
ejpam-5914	85	28	table	table	NOUN
ejpam-5914	85	29	:	:	PUNCT
ejpam-5914	86	1	|	|	ADV
ejpam-5914	86	2	1	1	NUM
ejpam-5914	86	3	u	u	NOUN
ejpam-5914	86	4	v	v	ADP
ejpam-5914	86	5	0	0	NUM
ejpam-5914	86	6	1	1	NUM
ejpam-5914	86	7	0	0	NUM
ejpam-5914	86	8	v	v	NUM
ejpam-5914	86	9	u	u	PROPN
ejpam-5914	86	10	1	1	NUM
ejpam-5914	86	11	u	u	NOUN
ejpam-5914	86	12	v	v	ADP
ejpam-5914	86	13	v	v	NUM
ejpam-5914	86	14	1	1	NUM
ejpam-5914	86	15	1	1	NUM
ejpam-5914	86	16	v	v	NOUN
ejpam-5914	86	17	u	u	NOUN
ejpam-5914	86	18	1	1	NUM
ejpam-5914	86	19	u	u	NOUN
ejpam-5914	86	20	1	1	NUM
ejpam-5914	86	21	0	0	NUM
ejpam-5914	86	22	1	1	NUM
ejpam-5914	86	23	1	1	NUM
ejpam-5914	86	24	1	1	NUM
ejpam-5914	86	25	1	1	NUM
ejpam-5914	86	26	then	then	ADV
ejpam-5914	86	27	(	(	PUNCT
ejpam-5914	86	28	a	a	DET
ejpam-5914	86	29	,	,	PUNCT
ejpam-5914	86	30	|	|	NOUN
ejpam-5914	86	31	)	)	PUNCT
ejpam-5914	86	32	is	be	AUX
ejpam-5914	86	33	a	a	DET
ejpam-5914	86	34	sheffer	sheffer	NOUN
ejpam-5914	86	35	stroke	stroke	NOUN
ejpam-5914	86	36	hilbert	hilbert	PROPN
ejpam-5914	86	37	algebra	algebra	PROPN
ejpam-5914	86	38	.	.	PUNCT
ejpam-5914	87	1	define	define	VERB
ejpam-5914	87	2	an	an	DET
ejpam-5914	87	3	interval	interval	NOUN
ejpam-5914	87	4	-	-	PUNCT
ejpam-5914	87	5	valued	value	VERB
ejpam-5914	87	6	fuzzy	fuzzy	ADJ
ejpam-5914	87	7	structure	structure	NOUN
ejpam-5914	87	8	(	(	PUNCT
ejpam-5914	87	9	a	a	PRON
ejpam-5914	87	10	,	,	PUNCT
ejpam-5914	87	11	f̃	f̃	PROPN
ejpam-5914	87	12	)	)	PUNCT
ejpam-5914	87	13	over	over	ADP
ejpam-5914	87	14	a	a	PRON
ejpam-5914	87	15	by	by	ADP
ejpam-5914	87	16	the	the	DET
ejpam-5914	87	17	table	table	NOUN
ejpam-5914	87	18	below	below	ADV
ejpam-5914	87	19	:	:	PUNCT
ejpam-5914	87	20	a	a	DET
ejpam-5914	87	21	1	1	NUM
ejpam-5914	87	22	u	u	NOUN
ejpam-5914	87	23	v	v	NOUN
ejpam-5914	87	24	0	0	NUM
ejpam-5914	87	25	f̃	f̃	PROPN
ejpam-5914	87	26	{	{	PUNCT
ejpam-5914	87	27	0.3	0.3	NUM
ejpam-5914	87	28	,	,	PUNCT
ejpam-5914	87	29	0.7	0.7	NUM
ejpam-5914	87	30	}	}	PUNCT
ejpam-5914	87	31	[	[	X
ejpam-5914	87	32	0.2	0.2	NUM
ejpam-5914	87	33	,	,	PUNCT
ejpam-5914	87	34	0.4	0.4	NUM
ejpam-5914	87	35	]	]	PUNCT
ejpam-5914	88	1	[	[	X
ejpam-5914	88	2	0.3	0.3	NUM
ejpam-5914	88	3	,	,	PUNCT
ejpam-5914	88	4	0.7	0.7	NUM
ejpam-5914	88	5	]	]	PUNCT
ejpam-5914	89	1	[	[	X
ejpam-5914	89	2	0.1	0.1	NUM
ejpam-5914	89	3	,	,	PUNCT
ejpam-5914	89	4	0.4	0.4	NUM
ejpam-5914	89	5	]	]	PUNCT
ejpam-5914	89	6	then	then	ADV
ejpam-5914	89	7	a	a	DET
ejpam-5914	89	8	1	1	NUM
ejpam-5914	89	9	u	u	NOUN
ejpam-5914	89	10	v	v	ADP
ejpam-5914	89	11	0	0	NUM
ejpam-5914	89	12	f̃inf	f̃inf	ADP
ejpam-5914	89	13	0.3	0.3	NUM
ejpam-5914	89	14	0.2	0.2	NUM
ejpam-5914	89	15	0.3	0.3	NUM
ejpam-5914	89	16	0.1	0.1	NUM
ejpam-5914	89	17	f̃sup	f̃sup	ADJ
ejpam-5914	89	18	0.7	0.7	NUM
ejpam-5914	89	19	0.4	0.4	NUM
ejpam-5914	89	20	0.7	0.7	NUM
ejpam-5914	89	21	0.4	0.4	NUM
ejpam-5914	89	22	definition	definition	NOUN
ejpam-5914	89	23	7	7	NUM
ejpam-5914	89	24	.	.	PUNCT
ejpam-5914	90	1	[	[	X
ejpam-5914	90	2	26	26	NUM
ejpam-5914	90	3	]	]	PUNCT
ejpam-5914	90	4	given	give	VERB
ejpam-5914	90	5	an	an	DET
ejpam-5914	90	6	interval	interval	NOUN
ejpam-5914	90	7	-	-	PUNCT
ejpam-5914	90	8	valued	value	VERB
ejpam-5914	90	9	fuzzy	fuzzy	ADJ
ejpam-5914	90	10	structure	structure	NOUN
ejpam-5914	90	11	(	(	PUNCT
ejpam-5914	90	12	a	a	PRON
ejpam-5914	90	13	,	,	PUNCT
ejpam-5914	90	14	f̃	f̃	PROPN
ejpam-5914	90	15	)	)	PUNCT
ejpam-5914	90	16	over	over	ADP
ejpam-5914	90	17	a	a	PRON
ejpam-5914	90	18	,	,	PUNCT
ejpam-5914	90	19	we	we	PRON
ejpam-5914	90	20	define	define	VERB
ejpam-5914	90	21	a	a	DET
ejpam-5914	90	22	fuzzy	fuzzy	ADJ
ejpam-5914	90	23	structure	structure	NOUN
ejpam-5914	90	24	(	(	PUNCT
ejpam-5914	90	25	a	a	DET
ejpam-5914	90	26	,	,	PUNCT
ejpam-5914	90	27	f̃ℓ	f̃ℓ	NOUN
ejpam-5914	90	28	)	)	PUNCT
ejpam-5914	90	29	in	in	ADP
ejpam-5914	90	30	a	a	PRON
ejpam-5914	90	31	as	as	SCONJ
ejpam-5914	90	32	follows	follow	VERB
ejpam-5914	90	33	:	:	PUNCT
ejpam-5914	90	34	f̃l	f̃l	NUM
ejpam-5914	90	35	:	:	PUNCT
ejpam-5914	90	36	a	a	DET
ejpam-5914	90	37	→	→	SYM
ejpam-5914	90	38	[	[	X
ejpam-5914	90	39	0	0	NUM
ejpam-5914	90	40	,	,	PUNCT
ejpam-5914	90	41	1];x	1];x	NUM
ejpam-5914	90	42	7→	7→	NUM
ejpam-5914	90	43	f̃sup(x)−	f̃sup(x)−	PROPN
ejpam-5914	90	44	f̃inf(x	f̃inf(x	PROPN
ejpam-5914	90	45	)	)	PUNCT
ejpam-5914	90	46	,	,	PUNCT
ejpam-5914	90	47	which	which	PRON
ejpam-5914	90	48	is	be	AUX
ejpam-5914	90	49	called	call	VERB
ejpam-5914	90	50	the	the	DET
ejpam-5914	90	51	length	length	NOUN
ejpam-5914	90	52	of	of	ADP
ejpam-5914	90	53	f̃	f̃	PROPN
ejpam-5914	90	54	.	.	PUNCT
ejpam-5914	91	1	n.	n.	PROPN
ejpam-5914	91	2	rajesh	rajesh	PROPN
ejpam-5914	91	3	et	et	PROPN
ejpam-5914	91	4	al	al	PROPN
ejpam-5914	91	5	.	.	PUNCT
ejpam-5914	91	6	/	/	SYM
ejpam-5914	91	7	eur	eur	PROPN
ejpam-5914	91	8	.	.	PUNCT
ejpam-5914	92	1	j.	j.	PROPN
ejpam-5914	92	2	pure	pure	PROPN
ejpam-5914	92	3	appl	appl	PROPN
ejpam-5914	92	4	.	.	PROPN
ejpam-5914	92	5	math	math	PROPN
ejpam-5914	92	6	,	,	PUNCT
ejpam-5914	92	7	18	18	NUM
ejpam-5914	92	8	(	(	PUNCT
ejpam-5914	92	9	2	2	NUM
ejpam-5914	92	10	)	)	PUNCT
ejpam-5914	92	11	(	(	PUNCT
ejpam-5914	92	12	2025	2025	NUM
ejpam-5914	92	13	)	)	PUNCT
ejpam-5914	92	14	,	,	PUNCT
ejpam-5914	92	15	5914	5914	NUM
ejpam-5914	92	16	5	5	NUM
ejpam-5914	92	17	of	of	ADP
ejpam-5914	92	18	21	21	NUM
ejpam-5914	92	19	example	example	NOUN
ejpam-5914	92	20	2	2	NUM
ejpam-5914	92	21	.	.	X
ejpam-5914	92	22	consider	consider	VERB
ejpam-5914	92	23	example	example	NOUN
ejpam-5914	92	24	1	1	NUM
ejpam-5914	92	25	,	,	PUNCT
ejpam-5914	92	26	we	we	PRON
ejpam-5914	92	27	have	have	VERB
ejpam-5914	92	28	a	a	DET
ejpam-5914	92	29	1	1	NUM
ejpam-5914	92	30	u	u	NOUN
ejpam-5914	92	31	v	v	ADP
ejpam-5914	92	32	0	0	NUM
ejpam-5914	92	33	f̃l	f̃l	PROPN
ejpam-5914	92	34	0.4	0.4	NUM
ejpam-5914	92	35	0.2	0.2	NUM
ejpam-5914	92	36	0.4	0.4	NUM
ejpam-5914	92	37	0.3	0.3	NUM
ejpam-5914	92	38	definition	definition	NOUN
ejpam-5914	92	39	8	8	NUM
ejpam-5914	92	40	.	.	PUNCT
ejpam-5914	93	1	a	a	DET
ejpam-5914	93	2	fuzzy	fuzzy	ADJ
ejpam-5914	93	3	structure	structure	NOUN
ejpam-5914	93	4	(	(	PUNCT
ejpam-5914	93	5	a	a	DET
ejpam-5914	93	6	,	,	PUNCT
ejpam-5914	93	7	f	f	X
ejpam-5914	93	8	)	)	PUNCT
ejpam-5914	93	9	in	in	ADP
ejpam-5914	93	10	a	a	PRON
ejpam-5914	93	11	is	be	AUX
ejpam-5914	93	12	called	call	VERB
ejpam-5914	93	13	(	(	PUNCT
ejpam-5914	93	14	1	1	NUM
ejpam-5914	93	15	)	)	PUNCT
ejpam-5914	93	16	a	a	DET
ejpam-5914	93	17	fuzzy	fuzzy	ADJ
ejpam-5914	93	18	subalgebra	subalgebra	NOUN
ejpam-5914	93	19	of	of	ADP
ejpam-5914	93	20	a	a	DET
ejpam-5914	93	21	with	with	ADP
ejpam-5914	93	22	type	type	NOUN
ejpam-5914	93	23	1	1	NUM
ejpam-5914	93	24	(	(	PUNCT
ejpam-5914	93	25	briefly	briefly	ADV
ejpam-5914	93	26	,	,	PUNCT
ejpam-5914	93	27	1	1	NUM
ejpam-5914	93	28	-	-	PUNCT
ejpam-5914	93	29	fuzzy	fuzzy	ADJ
ejpam-5914	93	30	subalgebra	subalgebra	NOUN
ejpam-5914	93	31	of	of	ADP
ejpam-5914	93	32	a	a	X
ejpam-5914	93	33	)	)	PUNCT
ejpam-5914	93	34	if	if	SCONJ
ejpam-5914	93	35	(	(	PUNCT
ejpam-5914	93	36	∀x	∀x	X
ejpam-5914	93	37	,	,	PUNCT
ejpam-5914	93	38	y	y	PROPN
ejpam-5914	93	39	∈	∈	PROPN
ejpam-5914	93	40	a)(f((x|(y|y))|(x|(y|y	a)(f((x|(y|y))|(x|(y|y	PROPN
ejpam-5914	93	41	)	)	PUNCT
ejpam-5914	93	42	)	)	PUNCT
ejpam-5914	93	43	)	)	PUNCT
ejpam-5914	93	44	≥	≥	PROPN
ejpam-5914	93	45	min{f(x	min{f(x	NOUN
ejpam-5914	93	46	)	)	PUNCT
ejpam-5914	93	47	,	,	PUNCT
ejpam-5914	93	48	f(y	f(y	NOUN
ejpam-5914	93	49	)	)	PUNCT
ejpam-5914	93	50	}	}	PUNCT
ejpam-5914	93	51	)	)	PUNCT
ejpam-5914	93	52	,	,	PUNCT
ejpam-5914	93	53	(	(	PUNCT
ejpam-5914	93	54	2	2	X
ejpam-5914	93	55	)	)	PUNCT
ejpam-5914	93	56	a	a	DET
ejpam-5914	93	57	fuzzy	fuzzy	ADJ
ejpam-5914	93	58	subalgebra	subalgebra	NOUN
ejpam-5914	93	59	of	of	ADP
ejpam-5914	93	60	a	a	DET
ejpam-5914	93	61	with	with	ADP
ejpam-5914	93	62	type	type	NOUN
ejpam-5914	93	63	2	2	NUM
ejpam-5914	93	64	(	(	PUNCT
ejpam-5914	93	65	briefly	briefly	ADV
ejpam-5914	93	66	,	,	PUNCT
ejpam-5914	93	67	2	2	NUM
ejpam-5914	93	68	-	-	PUNCT
ejpam-5914	93	69	fuzzy	fuzzy	ADJ
ejpam-5914	93	70	subalgebra	subalgebra	NOUN
ejpam-5914	93	71	of	of	ADP
ejpam-5914	93	72	a	a	X
ejpam-5914	93	73	)	)	PUNCT
ejpam-5914	93	74	if	if	SCONJ
ejpam-5914	93	75	(	(	PUNCT
ejpam-5914	93	76	∀x	∀x	X
ejpam-5914	93	77	,	,	PUNCT
ejpam-5914	93	78	y	y	PROPN
ejpam-5914	93	79	∈	∈	PROPN
ejpam-5914	93	80	a)(f((x|(y|y))|(x|(y|y	a)(f((x|(y|y))|(x|(y|y	PROPN
ejpam-5914	93	81	)	)	PUNCT
ejpam-5914	93	82	)	)	PUNCT
ejpam-5914	93	83	)	)	PUNCT
ejpam-5914	94	1	≤	≤	NUM
ejpam-5914	94	2	min{f(x	min{f(x	PROPN
ejpam-5914	94	3	)	)	PUNCT
ejpam-5914	94	4	,	,	PUNCT
ejpam-5914	94	5	f(y	f(y	NOUN
ejpam-5914	94	6	)	)	PUNCT
ejpam-5914	94	7	}	}	PUNCT
ejpam-5914	94	8	)	)	PUNCT
ejpam-5914	94	9	,	,	PUNCT
ejpam-5914	94	10	(	(	PUNCT
ejpam-5914	94	11	3	3	X
ejpam-5914	94	12	)	)	PUNCT
ejpam-5914	94	13	a	a	DET
ejpam-5914	94	14	fuzzy	fuzzy	ADJ
ejpam-5914	94	15	subalgebra	subalgebra	NOUN
ejpam-5914	94	16	of	of	ADP
ejpam-5914	94	17	a	a	DET
ejpam-5914	94	18	with	with	ADP
ejpam-5914	94	19	type	type	NOUN
ejpam-5914	94	20	3	3	NUM
ejpam-5914	94	21	(	(	PUNCT
ejpam-5914	94	22	briefly	briefly	ADV
ejpam-5914	94	23	,	,	PUNCT
ejpam-5914	94	24	3	3	NUM
ejpam-5914	94	25	-	-	PUNCT
ejpam-5914	94	26	fuzzy	fuzzy	ADJ
ejpam-5914	94	27	subalgebra	subalgebra	NOUN
ejpam-5914	94	28	of	of	ADP
ejpam-5914	94	29	a	a	X
ejpam-5914	94	30	)	)	PUNCT
ejpam-5914	94	31	if	if	SCONJ
ejpam-5914	94	32	(	(	PUNCT
ejpam-5914	94	33	∀x	∀x	X
ejpam-5914	94	34	,	,	PUNCT
ejpam-5914	94	35	y	y	PROPN
ejpam-5914	94	36	∈	∈	PROPN
ejpam-5914	94	37	a)(f((x|(y|y))|(x|(y|y	a)(f((x|(y|y))|(x|(y|y	PROPN
ejpam-5914	94	38	)	)	PUNCT
ejpam-5914	94	39	)	)	PUNCT
ejpam-5914	94	40	)	)	PUNCT
ejpam-5914	94	41	≥	≥	PROPN
ejpam-5914	94	42	max{f(x	max{f(x	PROPN
ejpam-5914	94	43	)	)	PUNCT
ejpam-5914	94	44	,	,	PUNCT
ejpam-5914	94	45	f(y	f(y	NOUN
ejpam-5914	94	46	)	)	PUNCT
ejpam-5914	94	47	}	}	PUNCT
ejpam-5914	94	48	)	)	PUNCT
ejpam-5914	94	49	,	,	PUNCT
ejpam-5914	94	50	(	(	PUNCT
ejpam-5914	94	51	4	4	X
ejpam-5914	94	52	)	)	PUNCT
ejpam-5914	94	53	a	a	DET
ejpam-5914	94	54	fuzzy	fuzzy	ADJ
ejpam-5914	94	55	subalgebra	subalgebra	NOUN
ejpam-5914	94	56	of	of	ADP
ejpam-5914	94	57	a	a	DET
ejpam-5914	94	58	with	with	ADP
ejpam-5914	94	59	type	type	NOUN
ejpam-5914	94	60	4	4	NUM
ejpam-5914	94	61	(	(	PUNCT
ejpam-5914	94	62	briefly	briefly	ADV
ejpam-5914	94	63	,	,	PUNCT
ejpam-5914	94	64	4	4	NUM
ejpam-5914	94	65	-	-	PUNCT
ejpam-5914	94	66	fuzzy	fuzzy	ADJ
ejpam-5914	94	67	subalgebra	subalgebra	NOUN
ejpam-5914	94	68	of	of	ADP
ejpam-5914	94	69	a	a	X
ejpam-5914	94	70	)	)	PUNCT
ejpam-5914	94	71	if	if	SCONJ
ejpam-5914	94	72	(	(	PUNCT
ejpam-5914	94	73	∀x	∀x	X
ejpam-5914	94	74	,	,	PUNCT
ejpam-5914	94	75	y	y	PROPN
ejpam-5914	94	76	∈	∈	PROPN
ejpam-5914	94	77	a)(f((x|(y|y))|(x|(y|y	a)(f((x|(y|y))|(x|(y|y	PROPN
ejpam-5914	94	78	)	)	PUNCT
ejpam-5914	94	79	)	)	PUNCT
ejpam-5914	94	80	)	)	PUNCT
ejpam-5914	94	81	≤	≤	NUM
ejpam-5914	95	1	max{f(x	max{f(x	PROPN
ejpam-5914	95	2	)	)	PUNCT
ejpam-5914	95	3	,	,	PUNCT
ejpam-5914	95	4	f(y	f(y	NOUN
ejpam-5914	95	5	)	)	PUNCT
ejpam-5914	95	6	}	}	PUNCT
ejpam-5914	95	7	)	)	PUNCT
ejpam-5914	95	8	.	.	PUNCT
ejpam-5914	96	1	example	example	NOUN
ejpam-5914	97	1	3	3	X
ejpam-5914	97	2	.	.	X
ejpam-5914	97	3	consider	consider	VERB
ejpam-5914	97	4	example	example	NOUN
ejpam-5914	97	5	1	1	NUM
ejpam-5914	97	6	,	,	PUNCT
ejpam-5914	97	7	we	we	PRON
ejpam-5914	97	8	have	have	VERB
ejpam-5914	97	9	3	3	NUM
ejpam-5914	97	10	cases	case	NOUN
ejpam-5914	97	11	as	as	SCONJ
ejpam-5914	97	12	follows	follow	VERB
ejpam-5914	97	13	:	:	PUNCT
ejpam-5914	97	14	case	case	NOUN
ejpam-5914	97	15	1	1	NUM
ejpam-5914	97	16	:	:	PUNCT
ejpam-5914	97	17	let	let	VERB
ejpam-5914	97	18	x	x	SYM
ejpam-5914	97	19	=	=	PUNCT
ejpam-5914	97	20	u	u	NOUN
ejpam-5914	97	21	and	and	CCONJ
ejpam-5914	97	22	y	y	PROPN
ejpam-5914	97	23	=	=	PROPN
ejpam-5914	98	1	v.	v.	PROPN
ejpam-5914	98	2	then	then	ADV
ejpam-5914	98	3	f((x|(y|y))|(x|(y|y	f((x|(y|y))|(x|(y|y	ADJ
ejpam-5914	98	4	)	)	PUNCT
ejpam-5914	98	5	)	)	PUNCT
ejpam-5914	98	6	)	)	PUNCT
ejpam-5914	99	1	=	=	SYM
ejpam-5914	99	2	f((u|(v|v))|(u|(v|v	f((u|(v|v))|(u|(v|v	NOUN
ejpam-5914	99	3	)	)	PUNCT
ejpam-5914	99	4	)	)	PUNCT
ejpam-5914	99	5	)	)	PUNCT
ejpam-5914	100	1	=	=	PUNCT
ejpam-5914	100	2	f((u|u)|(u|u	f((u|u)|(u|u	PROPN
ejpam-5914	100	3	)	)	PUNCT
ejpam-5914	100	4	)	)	PUNCT
ejpam-5914	101	1	=	=	SYM
ejpam-5914	101	2	f(v|v	f(v|v	NUM
ejpam-5914	101	3	)	)	PUNCT
ejpam-5914	101	4	=	=	SYM
ejpam-5914	101	5	f(u	f(u	PROPN
ejpam-5914	101	6	)	)	PUNCT
ejpam-5914	101	7	.	.	PUNCT
ejpam-5914	102	1	since	since	SCONJ
ejpam-5914	102	2	f(u	f(u	PROPN
ejpam-5914	102	3	)	)	PUNCT
ejpam-5914	102	4	=	=	PUNCT
ejpam-5914	103	1	[	[	X
ejpam-5914	103	2	0.2	0.2	NUM
ejpam-5914	103	3	,	,	PUNCT
ejpam-5914	103	4	0.4	0.4	NUM
ejpam-5914	103	5	]	]	PUNCT
ejpam-5914	103	6	and	and	CCONJ
ejpam-5914	103	7	f(v	f(v	NOUN
ejpam-5914	103	8	)	)	PUNCT
ejpam-5914	103	9	=	=	PUNCT
ejpam-5914	104	1	[	[	X
ejpam-5914	104	2	0.3	0.3	NUM
ejpam-5914	104	3	,	,	PUNCT
ejpam-5914	104	4	0.7	0.7	NUM
ejpam-5914	104	5	]	]	PUNCT
ejpam-5914	104	6	,	,	PUNCT
ejpam-5914	104	7	it	it	PRON
ejpam-5914	104	8	follows	follow	VERB
ejpam-5914	104	9	that	that	SCONJ
ejpam-5914	104	10	min{f(x	min{f(x	PROPN
ejpam-5914	104	11	)	)	PUNCT
ejpam-5914	104	12	,	,	PUNCT
ejpam-5914	104	13	f(y	f(y	NOUN
ejpam-5914	104	14	)	)	PUNCT
ejpam-5914	104	15	=	=	SYM
ejpam-5914	105	1	min{[0.2	min{[0.2	PROPN
ejpam-5914	105	2	,	,	PUNCT
ejpam-5914	105	3	0.4	0.4	NUM
ejpam-5914	105	4	]	]	PUNCT
ejpam-5914	105	5	,	,	PUNCT
ejpam-5914	105	6	[	[	X
ejpam-5914	105	7	0.3	0.3	NUM
ejpam-5914	105	8	,	,	PUNCT
ejpam-5914	105	9	0.7	0.7	NUM
ejpam-5914	105	10	]	]	PUNCT
ejpam-5914	105	11	}	}	PUNCT
ejpam-5914	105	12	=	=	PUNCT
ejpam-5914	106	1	[	[	X
ejpam-5914	106	2	0.2	0.2	NUM
ejpam-5914	106	3	,	,	PUNCT
ejpam-5914	106	4	0.4	0.4	NUM
ejpam-5914	106	5	]	]	PUNCT
ejpam-5914	106	6	.	.	PUNCT
ejpam-5914	107	1	the	the	DET
ejpam-5914	107	2	condition	condition	NOUN
ejpam-5914	107	3	f((x|(y|y))|(x|(y|y	f((x|(y|y))|(x|(y|y	ADJ
ejpam-5914	107	4	)	)	PUNCT
ejpam-5914	107	5	)	)	PUNCT
ejpam-5914	107	6	)	)	PUNCT
ejpam-5914	108	1	≥	≥	PROPN
ejpam-5914	108	2	min{f(x	min{f(x	NOUN
ejpam-5914	108	3	)	)	PUNCT
ejpam-5914	108	4	,	,	PUNCT
ejpam-5914	108	5	f(y	f(y	NOUN
ejpam-5914	108	6	)	)	PUNCT
ejpam-5914	108	7	}	}	PUNCT
ejpam-5914	108	8	is	be	AUX
ejpam-5914	108	9	satisfied	satisfied	ADJ
ejpam-5914	108	10	,	,	PUNCT
ejpam-5914	108	11	as	as	ADP
ejpam-5914	108	12	the	the	DET
ejpam-5914	108	13	result	result	NOUN
ejpam-5914	108	14	f(u	f(u	PROPN
ejpam-5914	108	15	)	)	PUNCT
ejpam-5914	108	16	=	=	PUNCT
ejpam-5914	109	1	[	[	X
ejpam-5914	109	2	0.2	0.2	NUM
ejpam-5914	109	3	,	,	PUNCT
ejpam-5914	109	4	0.4	0.4	NUM
ejpam-5914	109	5	]	]	PUNCT
ejpam-5914	109	6	is	be	AUX
ejpam-5914	109	7	less	less	ADJ
ejpam-5914	109	8	than	than	ADP
ejpam-5914	109	9	or	or	CCONJ
ejpam-5914	109	10	equal	equal	ADJ
ejpam-5914	109	11	to	to	ADP
ejpam-5914	109	12	min{f(x	min{f(x	NOUN
ejpam-5914	109	13	)	)	PUNCT
ejpam-5914	109	14	,	,	PUNCT
ejpam-5914	109	15	f(y	f(y	NOUN
ejpam-5914	109	16	)	)	PUNCT
ejpam-5914	109	17	}	}	PUNCT
ejpam-5914	109	18	.	.	PUNCT
ejpam-5914	110	1	therefore	therefore	ADV
ejpam-5914	110	2	,	,	PUNCT
ejpam-5914	110	3	(	(	PUNCT
ejpam-5914	110	4	a	a	PRON
ejpam-5914	110	5	,	,	PUNCT
ejpam-5914	110	6	f̃	f̃	PROPN
ejpam-5914	110	7	)	)	PUNCT
ejpam-5914	110	8	over	over	ADP
ejpam-5914	110	9	a	a	DET
ejpam-5914	110	10	forms	form	NOUN
ejpam-5914	110	11	a	a	DET
ejpam-5914	110	12	1	1	NUM
ejpam-5914	110	13	-	-	PUNCT
ejpam-5914	110	14	fuzzy	fuzzy	ADJ
ejpam-5914	110	15	subalgebra	subalgebra	NOUN
ejpam-5914	110	16	of	of	ADP
ejpam-5914	110	17	a.	a.	NOUN
ejpam-5914	110	18	case	case	NOUN
ejpam-5914	110	19	2	2	NUM
ejpam-5914	110	20	:	:	PUNCT
ejpam-5914	110	21	let	let	VERB
ejpam-5914	110	22	x	x	SYM
ejpam-5914	110	23	=	=	SYM
ejpam-5914	110	24	1	1	NUM
ejpam-5914	110	25	and	and	CCONJ
ejpam-5914	111	1	y	y	PROPN
ejpam-5914	111	2	=	=	PUNCT
ejpam-5914	111	3	u.	u.	PROPN
ejpam-5914	111	4	then	then	ADV
ejpam-5914	111	5	f((x|(y|y))|(x|(y|y	f((x|(y|y))|(x|(y|y	ADJ
ejpam-5914	111	6	)	)	PUNCT
ejpam-5914	111	7	)	)	PUNCT
ejpam-5914	111	8	)	)	PUNCT
ejpam-5914	112	1	=	=	SYM
ejpam-5914	112	2	f((1|(u|u))|(1|(u|u	f((1|(u|u))|(1|(u|u	PROPN
ejpam-5914	112	3	)	)	PUNCT
ejpam-5914	112	4	)	)	PUNCT
ejpam-5914	112	5	)	)	PUNCT
ejpam-5914	113	1	=	=	PUNCT
ejpam-5914	113	2	f((1|v)|(1|v	f((1|v)|(1|v	X
ejpam-5914	113	3	)	)	PUNCT
ejpam-5914	113	4	)	)	PUNCT
ejpam-5914	114	1	=	=	SYM
ejpam-5914	114	2	f(u|u	f(u|u	X
ejpam-5914	114	3	)	)	PUNCT
ejpam-5914	114	4	=	=	SYM
ejpam-5914	114	5	f(u	f(u	PROPN
ejpam-5914	114	6	)	)	PUNCT
ejpam-5914	114	7	.	.	PUNCT
ejpam-5914	115	1	given	give	VERB
ejpam-5914	115	2	that	that	DET
ejpam-5914	115	3	f(1	f(1	PROPN
ejpam-5914	115	4	)	)	PUNCT
ejpam-5914	115	5	=	=	PUNCT
ejpam-5914	116	1	[	[	X
ejpam-5914	116	2	0.3	0.3	NUM
ejpam-5914	116	3	,	,	PUNCT
ejpam-5914	116	4	0.7	0.7	NUM
ejpam-5914	116	5	]	]	PUNCT
ejpam-5914	116	6	and	and	CCONJ
ejpam-5914	116	7	f(u	f(u	PROPN
ejpam-5914	116	8	)	)	PUNCT
ejpam-5914	116	9	=	=	PUNCT
ejpam-5914	117	1	[	[	X
ejpam-5914	117	2	0.2	0.2	NUM
ejpam-5914	117	3	,	,	PUNCT
ejpam-5914	117	4	0.4	0.4	NUM
ejpam-5914	117	5	]	]	PUNCT
ejpam-5914	117	6	,	,	PUNCT
ejpam-5914	117	7	we	we	PRON
ejpam-5914	117	8	can	can	AUX
ejpam-5914	117	9	compute	compute	VERB
ejpam-5914	117	10	the	the	DET
ejpam-5914	117	11	minimum	minimum	NOUN
ejpam-5914	117	12	of	of	ADP
ejpam-5914	117	13	the	the	DET
ejpam-5914	117	14	two	two	NUM
ejpam-5914	117	15	fuzzy	fuzzy	ADJ
ejpam-5914	117	16	sets	set	NOUN
ejpam-5914	117	17	:	:	PUNCT
ejpam-5914	117	18	min{f(x	min{f(x	NOUN
ejpam-5914	117	19	)	)	PUNCT
ejpam-5914	117	20	,	,	PUNCT
ejpam-5914	117	21	f(y	f(y	NOUN
ejpam-5914	117	22	)	)	PUNCT
ejpam-5914	117	23	=	=	PUNCT
ejpam-5914	118	1	min{[0.3	min{[0.3	VERB
ejpam-5914	118	2	,	,	PUNCT
ejpam-5914	118	3	0.7	0.7	NUM
ejpam-5914	118	4	]	]	PUNCT
ejpam-5914	118	5	,	,	PUNCT
ejpam-5914	118	6	[	[	X
ejpam-5914	118	7	0.2	0.2	NUM
ejpam-5914	118	8	,	,	PUNCT
ejpam-5914	118	9	0.4	0.4	NUM
ejpam-5914	118	10	]	]	PUNCT
ejpam-5914	118	11	}	}	PUNCT
ejpam-5914	118	12	=	=	PUNCT
ejpam-5914	119	1	[	[	X
ejpam-5914	119	2	0.2	0.2	NUM
ejpam-5914	119	3	,	,	PUNCT
ejpam-5914	119	4	0.4	0.4	NUM
ejpam-5914	119	5	]	]	PUNCT
ejpam-5914	119	6	.	.	PUNCT
ejpam-5914	120	1	the	the	DET
ejpam-5914	120	2	condition	condition	NOUN
ejpam-5914	120	3	f((x|(y|y))|(x|(y|y	f((x|(y|y))|(x|(y|y	ADJ
ejpam-5914	120	4	)	)	PUNCT
ejpam-5914	120	5	)	)	PUNCT
ejpam-5914	120	6	)	)	PUNCT
ejpam-5914	121	1	≤	≤	NUM
ejpam-5914	121	2	min{f(x	min{f(x	PROPN
ejpam-5914	121	3	)	)	PUNCT
ejpam-5914	121	4	,	,	PUNCT
ejpam-5914	121	5	f(y	f(y	NOUN
ejpam-5914	121	6	)	)	PUNCT
ejpam-5914	121	7	}	}	PUNCT
ejpam-5914	121	8	)	)	PUNCT
ejpam-5914	121	9	is	be	AUX
ejpam-5914	121	10	satisfied	satisfied	ADJ
ejpam-5914	121	11	,	,	PUNCT
ejpam-5914	121	12	as	as	ADP
ejpam-5914	121	13	the	the	DET
ejpam-5914	121	14	result	result	NOUN
ejpam-5914	121	15	f(u	f(u	PROPN
ejpam-5914	121	16	)	)	PUNCT
ejpam-5914	121	17	=	=	PUNCT
ejpam-5914	122	1	[	[	X
ejpam-5914	122	2	0.2	0.2	NUM
ejpam-5914	122	3	,	,	PUNCT
ejpam-5914	122	4	0.4	0.4	NUM
ejpam-5914	122	5	]	]	PUNCT
ejpam-5914	122	6	is	be	AUX
ejpam-5914	122	7	indeed	indeed	ADV
ejpam-5914	122	8	less	less	ADJ
ejpam-5914	122	9	than	than	ADP
ejpam-5914	122	10	or	or	CCONJ
ejpam-5914	122	11	equal	equal	ADJ
ejpam-5914	122	12	to	to	ADP
ejpam-5914	122	13	min{f(x	min{f(x	NOUN
ejpam-5914	122	14	)	)	PUNCT
ejpam-5914	122	15	,	,	PUNCT
ejpam-5914	122	16	f(y	f(y	NOUN
ejpam-5914	122	17	)	)	PUNCT
ejpam-5914	122	18	}	}	PUNCT
ejpam-5914	122	19	.	.	PUNCT
ejpam-5914	123	1	therefore	therefore	ADV
ejpam-5914	123	2	,	,	PUNCT
ejpam-5914	123	3	(	(	PUNCT
ejpam-5914	123	4	a	a	PRON
ejpam-5914	123	5	,	,	PUNCT
ejpam-5914	123	6	f̃	f̃	PROPN
ejpam-5914	123	7	)	)	PUNCT
ejpam-5914	123	8	over	over	ADP
ejpam-5914	123	9	a	a	DET
ejpam-5914	123	10	forms	form	NOUN
ejpam-5914	123	11	a	a	DET
ejpam-5914	123	12	2	2	NUM
ejpam-5914	123	13	-	-	PUNCT
ejpam-5914	123	14	fuzzy	fuzzy	ADJ
ejpam-5914	123	15	subalgebra	subalgebra	NOUN
ejpam-5914	123	16	of	of	ADP
ejpam-5914	123	17	a.	a.	NOUN
ejpam-5914	123	18	case	case	NOUN
ejpam-5914	123	19	3	3	NUM
ejpam-5914	123	20	:	:	PUNCT
ejpam-5914	123	21	similarly	similarly	ADV
ejpam-5914	123	22	,	,	PUNCT
ejpam-5914	123	23	by	by	ADP
ejpam-5914	123	24	choosing	choose	VERB
ejpam-5914	123	25	x	x	PUNCT
ejpam-5914	123	26	=	=	SYM
ejpam-5914	123	27	1	1	NUM
ejpam-5914	123	28	and	and	CCONJ
ejpam-5914	123	29	y	y	PROPN
ejpam-5914	123	30	=	=	SYM
ejpam-5914	123	31	u	u	PROPN
ejpam-5914	123	32	,	,	PUNCT
ejpam-5914	123	33	we	we	PRON
ejpam-5914	123	34	have	have	VERB
ejpam-5914	123	35	(	(	PUNCT
ejpam-5914	123	36	a	a	PRON
ejpam-5914	123	37	,	,	PUNCT
ejpam-5914	123	38	f̃	f̃	PROPN
ejpam-5914	123	39	)	)	PUNCT
ejpam-5914	123	40	over	over	ADP
ejpam-5914	123	41	a	a	DET
ejpam-5914	123	42	forms	form	NOUN
ejpam-5914	123	43	a	a	DET
ejpam-5914	123	44	3	3	NUM
ejpam-5914	123	45	-	-	PUNCT
ejpam-5914	123	46	fuzzy	fuzzy	ADJ
ejpam-5914	123	47	subalgebra	subalgebra	NOUN
ejpam-5914	123	48	of	of	ADP
ejpam-5914	123	49	a	a	PRON
ejpam-5914	123	50	,	,	PUNCT
ejpam-5914	123	51	and	and	CCONJ
ejpam-5914	123	52	by	by	ADP
ejpam-5914	123	53	selecting	select	VERB
ejpam-5914	123	54	x	x	X
ejpam-5914	123	55	=	=	SYM
ejpam-5914	123	56	0	0	NUM
ejpam-5914	123	57	and	and	CCONJ
ejpam-5914	123	58	y	y	PROPN
ejpam-5914	123	59	=	=	SYM
ejpam-5914	123	60	v	v	PROPN
ejpam-5914	123	61	,	,	PUNCT
ejpam-5914	123	62	we	we	PRON
ejpam-5914	123	63	have	have	VERB
ejpam-5914	123	64	(	(	PUNCT
ejpam-5914	123	65	a	a	PRON
ejpam-5914	123	66	,	,	PUNCT
ejpam-5914	123	67	f̃	f̃	PROPN
ejpam-5914	123	68	)	)	PUNCT
ejpam-5914	123	69	over	over	ADP
ejpam-5914	123	70	a	a	DET
ejpam-5914	123	71	forms	form	NOUN
ejpam-5914	123	72	a	a	DET
ejpam-5914	123	73	4	4	NUM
ejpam-5914	123	74	-	-	PUNCT
ejpam-5914	123	75	fuzzy	fuzzy	ADJ
ejpam-5914	123	76	subalgebra	subalgebra	NOUN
ejpam-5914	123	77	of	of	ADP
ejpam-5914	123	78	a.	a.	NOUN
ejpam-5914	123	79	definition	definition	NOUN
ejpam-5914	123	80	9	9	NUM
ejpam-5914	123	81	.	.	PUNCT
ejpam-5914	124	1	an	an	DET
ejpam-5914	124	2	interval	interval	NOUN
ejpam-5914	124	3	-	-	PUNCT
ejpam-5914	124	4	valued	value	VERB
ejpam-5914	124	5	fuzzy	fuzzy	ADJ
ejpam-5914	124	6	structure	structure	NOUN
ejpam-5914	124	7	(	(	PUNCT
ejpam-5914	124	8	a	a	PRON
ejpam-5914	124	9	,	,	PUNCT
ejpam-5914	124	10	f̃	f̃	PROPN
ejpam-5914	124	11	)	)	PUNCT
ejpam-5914	124	12	over	over	ADP
ejpam-5914	124	13	a	a	PRON
ejpam-5914	124	14	is	be	AUX
ejpam-5914	124	15	called	call	VERB
ejpam-5914	124	16	a	a	DET
ejpam-5914	124	17	length	length	NOUN
ejpam-5914	124	18	1	1	NUM
ejpam-5914	124	19	-	-	PUNCT
ejpam-5914	124	20	fuzzy	fuzzy	ADJ
ejpam-5914	124	21	(	(	PUNCT
ejpam-5914	124	22	resp	resp	NOUN
ejpam-5914	124	23	.	.	PUNCT
ejpam-5914	124	24	,	,	PUNCT
ejpam-5914	124	25	2	2	NUM
ejpam-5914	124	26	-	-	PUNCT
ejpam-5914	124	27	fuzzy	fuzzy	ADJ
ejpam-5914	124	28	,	,	PUNCT
ejpam-5914	124	29	3	3	NUM
ejpam-5914	124	30	-	-	PUNCT
ejpam-5914	124	31	fuzzy	fuzzy	ADJ
ejpam-5914	124	32	,	,	PUNCT
ejpam-5914	124	33	4	4	NUM
ejpam-5914	124	34	-	-	PUNCT
ejpam-5914	124	35	fuzzy	fuzzy	ADJ
ejpam-5914	124	36	)	)	PUNCT
ejpam-5914	124	37	subalgebra	subalgebra	NOUN
ejpam-5914	124	38	of	of	ADP
ejpam-5914	124	39	a	a	DET
ejpam-5914	124	40	if	if	SCONJ
ejpam-5914	124	41	a	a	DET
ejpam-5914	124	42	fuzzy	fuzzy	ADJ
ejpam-5914	124	43	structure	structure	NOUN
ejpam-5914	124	44	(	(	PUNCT
ejpam-5914	124	45	a	a	DET
ejpam-5914	124	46	,	,	PUNCT
ejpam-5914	124	47	f̃ℓ	f̃ℓ	NOUN
ejpam-5914	124	48	)	)	PUNCT
ejpam-5914	124	49	is	be	AUX
ejpam-5914	124	50	a	a	DET
ejpam-5914	124	51	1	1	NUM
ejpam-5914	124	52	-	-	PUNCT
ejpam-5914	124	53	fuzzy	fuzzy	ADJ
ejpam-5914	124	54	(	(	PUNCT
ejpam-5914	124	55	resp	resp	NOUN
ejpam-5914	124	56	.	.	PUNCT
ejpam-5914	124	57	,	,	PUNCT
ejpam-5914	124	58	2	2	NUM
ejpam-5914	124	59	-	-	PUNCT
ejpam-5914	124	60	fuzzy	fuzzy	ADJ
ejpam-5914	124	61	,	,	PUNCT
ejpam-5914	124	62	3	3	NUM
ejpam-5914	124	63	-	-	PUNCT
ejpam-5914	124	64	fuzzy	fuzzy	ADJ
ejpam-5914	124	65	,	,	PUNCT
ejpam-5914	124	66	4	4	NUM
ejpam-5914	124	67	-	-	PUNCT
ejpam-5914	124	68	fuzzy	fuzzy	ADJ
ejpam-5914	124	69	)	)	PUNCT
ejpam-5914	124	70	subalgebra	subalgebra	NOUN
ejpam-5914	124	71	of	of	ADP
ejpam-5914	124	72	a.	a.	NOUN
ejpam-5914	124	73	proposition	proposition	NOUN
ejpam-5914	124	74	2	2	X
ejpam-5914	124	75	.	.	PUNCT
ejpam-5914	125	1	if	if	SCONJ
ejpam-5914	125	2	(	(	PUNCT
ejpam-5914	125	3	a	a	PRON
ejpam-5914	125	4	,	,	PUNCT
ejpam-5914	125	5	f̃	f̃	PROPN
ejpam-5914	125	6	)	)	PUNCT
ejpam-5914	125	7	is	be	AUX
ejpam-5914	125	8	a	a	DET
ejpam-5914	125	9	length	length	NOUN
ejpam-5914	125	10	k	k	ADJ
ejpam-5914	125	11	-	-	ADJ
ejpam-5914	125	12	fuzzy	fuzzy	ADJ
ejpam-5914	125	13	subalgebra	subalgebra	NOUN
ejpam-5914	125	14	of	of	ADP
ejpam-5914	125	15	a	a	PRON
ejpam-5914	125	16	for	for	ADP
ejpam-5914	125	17	k	k	PROPN
ejpam-5914	125	18	∈	∈	PROPN
ejpam-5914	125	19	{	{	PUNCT
ejpam-5914	125	20	1	1	NUM
ejpam-5914	125	21	,	,	PUNCT
ejpam-5914	125	22	3	3	NUM
ejpam-5914	125	23	}	}	PUNCT
ejpam-5914	125	24	,	,	PUNCT
ejpam-5914	125	25	then	then	ADV
ejpam-5914	125	26	(	(	PUNCT
ejpam-5914	125	27	∀x	∀x	X
ejpam-5914	125	28	∈	∈	PROPN
ejpam-5914	125	29	a)(f̃ℓ(0	a)(f̃ℓ(0	NOUN
ejpam-5914	125	30	)	)	PUNCT
ejpam-5914	125	31	≥	≥	NOUN
ejpam-5914	125	32	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	125	33	)	)	PUNCT
ejpam-5914	125	34	)	)	PUNCT
ejpam-5914	125	35	.	.	PUNCT
ejpam-5914	126	1	(	(	PUNCT
ejpam-5914	126	2	1	1	X
ejpam-5914	126	3	)	)	PUNCT
ejpam-5914	126	4	n.	n.	PROPN
ejpam-5914	126	5	rajesh	rajesh	PROPN
ejpam-5914	126	6	et	et	PROPN
ejpam-5914	126	7	al	al	PROPN
ejpam-5914	126	8	.	.	PUNCT
ejpam-5914	126	9	/	/	SYM
ejpam-5914	126	10	eur	eur	PROPN
ejpam-5914	126	11	.	.	PUNCT
ejpam-5914	127	1	j.	j.	PROPN
ejpam-5914	127	2	pure	pure	PROPN
ejpam-5914	127	3	appl	appl	PROPN
ejpam-5914	127	4	.	.	PROPN
ejpam-5914	127	5	math	math	PROPN
ejpam-5914	127	6	,	,	PUNCT
ejpam-5914	127	7	18	18	NUM
ejpam-5914	127	8	(	(	PUNCT
ejpam-5914	127	9	2	2	NUM
ejpam-5914	127	10	)	)	PUNCT
ejpam-5914	127	11	(	(	PUNCT
ejpam-5914	127	12	2025	2025	NUM
ejpam-5914	127	13	)	)	PUNCT
ejpam-5914	127	14	,	,	PUNCT
ejpam-5914	127	15	5914	5914	NUM
ejpam-5914	127	16	6	6	NUM
ejpam-5914	127	17	of	of	ADP
ejpam-5914	127	18	21	21	NUM
ejpam-5914	127	19	proof	proof	NOUN
ejpam-5914	127	20	.	.	PUNCT
ejpam-5914	128	1	let	let	VERB
ejpam-5914	128	2	(	(	PUNCT
ejpam-5914	128	3	a	a	PRON
ejpam-5914	128	4	,	,	PUNCT
ejpam-5914	128	5	f̃	f̃	PROPN
ejpam-5914	128	6	)	)	PUNCT
ejpam-5914	128	7	be	be	VERB
ejpam-5914	128	8	a	a	DET
ejpam-5914	128	9	length	length	NOUN
ejpam-5914	128	10	1	1	NUM
ejpam-5914	128	11	-	-	PUNCT
ejpam-5914	128	12	fuzzy	fuzzy	ADJ
ejpam-5914	128	13	subalgebra	subalgebra	NOUN
ejpam-5914	128	14	of	of	ADP
ejpam-5914	128	15	a	a	PRON
ejpam-5914	128	16	and	and	CCONJ
ejpam-5914	128	17	x	x	SYM
ejpam-5914	128	18	∈	∈	NOUN
ejpam-5914	128	19	a.	a.	NOUN
ejpam-5914	128	20	then	then	ADV
ejpam-5914	128	21	f̃ℓ(0	f̃ℓ(0	NUM
ejpam-5914	128	22	)	)	PUNCT
ejpam-5914	128	23	=	=	SYM
ejpam-5914	128	24	f̃ℓ(1|1	f̃ℓ(1|1	PROPN
ejpam-5914	128	25	)	)	PUNCT
ejpam-5914	128	26	=	=	SYM
ejpam-5914	128	27	f̃ℓ((x|(x|x))|(x|(x|x	f̃ℓ((x|(x|x))|(x|(x|x	NOUN
ejpam-5914	128	28	)	)	PUNCT
ejpam-5914	128	29	)	)	PUNCT
ejpam-5914	128	30	)	)	PUNCT
ejpam-5914	128	31	≥	≥	PROPN
ejpam-5914	128	32	min{f̃ℓ(x	min{f̃ℓ(x	PROPN
ejpam-5914	128	33	)	)	PUNCT
ejpam-5914	128	34	,	,	PUNCT
ejpam-5914	128	35	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	128	36	)	)	PUNCT
ejpam-5914	128	37	}	}	PUNCT
ejpam-5914	128	38	=	=	SYM
ejpam-5914	128	39	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	128	40	)	)	PUNCT
ejpam-5914	128	41	.	.	PUNCT
ejpam-5914	129	1	let	let	VERB
ejpam-5914	129	2	(	(	PUNCT
ejpam-5914	129	3	a	a	PRON
ejpam-5914	129	4	,	,	PUNCT
ejpam-5914	129	5	f̃	f̃	PROPN
ejpam-5914	129	6	)	)	PUNCT
ejpam-5914	129	7	be	be	VERB
ejpam-5914	129	8	a	a	DET
ejpam-5914	129	9	length	length	NOUN
ejpam-5914	129	10	3	3	NUM
ejpam-5914	129	11	-	-	PUNCT
ejpam-5914	129	12	fuzzy	fuzzy	ADJ
ejpam-5914	129	13	subalgebra	subalgebra	NOUN
ejpam-5914	129	14	of	of	ADP
ejpam-5914	129	15	a.	a.	NOUN
ejpam-5914	129	16	then	then	ADV
ejpam-5914	129	17	f̃ℓ(0	f̃ℓ(0	NUM
ejpam-5914	129	18	)	)	PUNCT
ejpam-5914	129	19	=	=	SYM
ejpam-5914	129	20	f̃ℓ(1|1	f̃ℓ(1|1	PROPN
ejpam-5914	129	21	)	)	PUNCT
ejpam-5914	129	22	=	=	SYM
ejpam-5914	129	23	f̃ℓ((x|(x|x))|(x|(x|x	f̃ℓ((x|(x|x))|(x|(x|x	NOUN
ejpam-5914	129	24	)	)	PUNCT
ejpam-5914	129	25	)	)	PUNCT
ejpam-5914	129	26	)	)	PUNCT
ejpam-5914	129	27	≥	≥	PROPN
ejpam-5914	129	28	max{f̃ℓ(x	max{f̃ℓ(x	PROPN
ejpam-5914	129	29	)	)	PUNCT
ejpam-5914	129	30	,	,	PUNCT
ejpam-5914	129	31	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	129	32	)	)	PUNCT
ejpam-5914	129	33	}	}	PUNCT
ejpam-5914	129	34	=	=	SYM
ejpam-5914	129	35	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	129	36	)	)	PUNCT
ejpam-5914	129	37	.	.	PUNCT
ejpam-5914	130	1	proposition	proposition	NOUN
ejpam-5914	130	2	3	3	NUM
ejpam-5914	130	3	.	.	PUNCT
ejpam-5914	131	1	if	if	SCONJ
ejpam-5914	131	2	(	(	PUNCT
ejpam-5914	131	3	a	a	PRON
ejpam-5914	131	4	,	,	PUNCT
ejpam-5914	131	5	f̃	f̃	PROPN
ejpam-5914	131	6	)	)	PUNCT
ejpam-5914	131	7	is	be	AUX
ejpam-5914	131	8	a	a	DET
ejpam-5914	131	9	length	length	NOUN
ejpam-5914	131	10	k	k	ADJ
ejpam-5914	131	11	-	-	ADJ
ejpam-5914	131	12	fuzzy	fuzzy	ADJ
ejpam-5914	131	13	subalgebra	subalgebra	NOUN
ejpam-5914	131	14	of	of	ADP
ejpam-5914	131	15	a	a	PRON
ejpam-5914	131	16	for	for	ADP
ejpam-5914	131	17	k	k	PROPN
ejpam-5914	131	18	∈	∈	PROPN
ejpam-5914	131	19	{	{	PUNCT
ejpam-5914	131	20	2	2	NUM
ejpam-5914	131	21	,	,	PUNCT
ejpam-5914	131	22	4	4	NUM
ejpam-5914	131	23	}	}	PUNCT
ejpam-5914	131	24	,	,	PUNCT
ejpam-5914	131	25	then	then	ADV
ejpam-5914	131	26	(	(	PUNCT
ejpam-5914	131	27	∀x	∀x	X
ejpam-5914	131	28	∈	∈	PROPN
ejpam-5914	131	29	a)(f̃ℓ(0	a)(f̃ℓ(0	NOUN
ejpam-5914	131	30	)	)	PUNCT
ejpam-5914	131	31	≤	≤	NUM
ejpam-5914	131	32	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	131	33	)	)	PUNCT
ejpam-5914	131	34	)	)	PUNCT
ejpam-5914	131	35	.	.	PUNCT
ejpam-5914	132	1	(	(	PUNCT
ejpam-5914	132	2	2	2	X
ejpam-5914	132	3	)	)	PUNCT
ejpam-5914	132	4	proof	proof	NOUN
ejpam-5914	132	5	.	.	PUNCT
ejpam-5914	133	1	let	let	VERB
ejpam-5914	133	2	(	(	PUNCT
ejpam-5914	133	3	a	a	PRON
ejpam-5914	133	4	,	,	PUNCT
ejpam-5914	133	5	f̃	f̃	PROPN
ejpam-5914	133	6	)	)	PUNCT
ejpam-5914	133	7	be	be	VERB
ejpam-5914	133	8	a	a	DET
ejpam-5914	133	9	length	length	NOUN
ejpam-5914	133	10	2	2	NUM
ejpam-5914	133	11	-	-	PUNCT
ejpam-5914	133	12	fuzzy	fuzzy	ADJ
ejpam-5914	133	13	subalgebra	subalgebra	NOUN
ejpam-5914	133	14	of	of	ADP
ejpam-5914	133	15	a	a	PRON
ejpam-5914	133	16	and	and	CCONJ
ejpam-5914	133	17	x	x	SYM
ejpam-5914	133	18	∈	∈	NOUN
ejpam-5914	133	19	a.	a.	NOUN
ejpam-5914	133	20	then	then	ADV
ejpam-5914	133	21	f̃ℓ(0	f̃ℓ(0	NUM
ejpam-5914	133	22	)	)	PUNCT
ejpam-5914	133	23	=	=	SYM
ejpam-5914	133	24	f̃ℓ(1|1	f̃ℓ(1|1	PROPN
ejpam-5914	133	25	)	)	PUNCT
ejpam-5914	133	26	=	=	SYM
ejpam-5914	133	27	f̃ℓ((x|(x|x))|(x|(x|x	f̃ℓ((x|(x|x))|(x|(x|x	NOUN
ejpam-5914	133	28	)	)	PUNCT
ejpam-5914	133	29	)	)	PUNCT
ejpam-5914	133	30	)	)	PUNCT
ejpam-5914	134	1	≤	≤	NUM
ejpam-5914	134	2	min{f̃ℓ(x	min{f̃ℓ(x	PROPN
ejpam-5914	134	3	)	)	PUNCT
ejpam-5914	134	4	,	,	PUNCT
ejpam-5914	134	5	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	134	6	)	)	PUNCT
ejpam-5914	134	7	}	}	PUNCT
ejpam-5914	134	8	=	=	SYM
ejpam-5914	134	9	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	134	10	)	)	PUNCT
ejpam-5914	134	11	.	.	PUNCT
ejpam-5914	135	1	let	let	VERB
ejpam-5914	135	2	(	(	PUNCT
ejpam-5914	135	3	a	a	PRON
ejpam-5914	135	4	,	,	PUNCT
ejpam-5914	135	5	f̃	f̃	PROPN
ejpam-5914	135	6	)	)	PUNCT
ejpam-5914	135	7	be	be	VERB
ejpam-5914	135	8	a	a	DET
ejpam-5914	135	9	length	length	NOUN
ejpam-5914	135	10	4	4	NUM
ejpam-5914	135	11	-	-	PUNCT
ejpam-5914	135	12	fuzzy	fuzzy	ADJ
ejpam-5914	135	13	subalgebra	subalgebra	NOUN
ejpam-5914	135	14	of	of	ADP
ejpam-5914	135	15	a.	a.	NOUN
ejpam-5914	135	16	then	then	ADV
ejpam-5914	135	17	f̃ℓ(0	f̃ℓ(0	NUM
ejpam-5914	135	18	)	)	PUNCT
ejpam-5914	135	19	=	=	SYM
ejpam-5914	135	20	f̃ℓ(1|1	f̃ℓ(1|1	PROPN
ejpam-5914	135	21	)	)	PUNCT
ejpam-5914	135	22	=	=	SYM
ejpam-5914	135	23	f̃ℓ((x|(x|x))|(x|(x|x	f̃ℓ((x|(x|x))|(x|(x|x	NOUN
ejpam-5914	135	24	)	)	PUNCT
ejpam-5914	135	25	)	)	PUNCT
ejpam-5914	135	26	)	)	PUNCT
ejpam-5914	136	1	≤	≤	NUM
ejpam-5914	136	2	max{f̃ℓ(x	max{f̃ℓ(x	PROPN
ejpam-5914	136	3	)	)	PUNCT
ejpam-5914	136	4	,	,	PUNCT
ejpam-5914	136	5	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	136	6	)	)	PUNCT
ejpam-5914	136	7	}	}	PUNCT
ejpam-5914	136	8	=	=	SYM
ejpam-5914	136	9	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	136	10	)	)	PUNCT
ejpam-5914	136	11	.	.	PUNCT
ejpam-5914	137	1	theorem	theorem	NOUN
ejpam-5914	137	2	1	1	NUM
ejpam-5914	137	3	.	.	PUNCT
ejpam-5914	138	1	every	every	DET
ejpam-5914	138	2	length	length	NOUN
ejpam-5914	138	3	3	3	NUM
ejpam-5914	138	4	-	-	PUNCT
ejpam-5914	138	5	fuzzy	fuzzy	ADJ
ejpam-5914	138	6	subalgebra	subalgebra	NOUN
ejpam-5914	138	7	of	of	ADP
ejpam-5914	138	8	a	a	PRON
ejpam-5914	138	9	is	be	AUX
ejpam-5914	138	10	a	a	DET
ejpam-5914	138	11	length	length	NOUN
ejpam-5914	138	12	1	1	NUM
ejpam-5914	138	13	-	-	PUNCT
ejpam-5914	138	14	fuzzy	fuzzy	ADJ
ejpam-5914	138	15	subalgebra	subalgebra	NOUN
ejpam-5914	138	16	.	.	PUNCT
ejpam-5914	139	1	proof	proof	NOUN
ejpam-5914	139	2	.	.	PUNCT
ejpam-5914	140	1	let	let	VERB
ejpam-5914	140	2	(	(	PUNCT
ejpam-5914	140	3	a	a	PRON
ejpam-5914	140	4	,	,	PUNCT
ejpam-5914	140	5	f̃	f̃	PROPN
ejpam-5914	140	6	)	)	PUNCT
ejpam-5914	140	7	be	be	VERB
ejpam-5914	140	8	a	a	DET
ejpam-5914	140	9	length	length	NOUN
ejpam-5914	140	10	3	3	NUM
ejpam-5914	140	11	-	-	PUNCT
ejpam-5914	140	12	fuzzy	fuzzy	ADJ
ejpam-5914	140	13	subalgebra	subalgebra	NOUN
ejpam-5914	140	14	of	of	ADP
ejpam-5914	140	15	a	a	PRON
ejpam-5914	140	16	and	and	CCONJ
ejpam-5914	140	17	x	x	NOUN
ejpam-5914	140	18	,	,	PUNCT
ejpam-5914	140	19	y	y	PROPN
ejpam-5914	140	20	∈	∈	PROPN
ejpam-5914	140	21	a.	a.	NOUN
ejpam-5914	140	22	then	then	ADV
ejpam-5914	140	23	f̃ℓ((x|(y|y))|(x|(y|y	f̃ℓ((x|(y|y))|(x|(y|y	NUM
ejpam-5914	140	24	)	)	PUNCT
ejpam-5914	140	25	)	)	PUNCT
ejpam-5914	140	26	)	)	PUNCT
ejpam-5914	140	27	≥	≥	PROPN
ejpam-5914	140	28	max{f̃ℓ(x	max{f̃ℓ(x	PROPN
ejpam-5914	140	29	)	)	PUNCT
ejpam-5914	140	30	,	,	PUNCT
ejpam-5914	140	31	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	140	32	)	)	PUNCT
ejpam-5914	140	33	}	}	PUNCT
ejpam-5914	140	34	≥	≥	PROPN
ejpam-5914	140	35	min{f̃ℓ(x	min{f̃ℓ(x	PROPN
ejpam-5914	140	36	)	)	PUNCT
ejpam-5914	140	37	,	,	PUNCT
ejpam-5914	140	38	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	140	39	)	)	PUNCT
ejpam-5914	140	40	}	}	PUNCT
ejpam-5914	140	41	.	.	PUNCT
ejpam-5914	141	1	hence	hence	ADV
ejpam-5914	141	2	,	,	PUNCT
ejpam-5914	141	3	(	(	PUNCT
ejpam-5914	141	4	x	x	X
ejpam-5914	141	5	,	,	PUNCT
ejpam-5914	141	6	f̃	f̃	PROPN
ejpam-5914	141	7	)	)	PUNCT
ejpam-5914	141	8	is	be	AUX
ejpam-5914	141	9	a	a	DET
ejpam-5914	141	10	length	length	NOUN
ejpam-5914	141	11	1	1	NUM
ejpam-5914	141	12	-	-	PUNCT
ejpam-5914	141	13	fuzzy	fuzzy	ADJ
ejpam-5914	141	14	subalgebra	subalgebra	NOUN
ejpam-5914	141	15	of	of	ADP
ejpam-5914	141	16	a.	a.	NOUN
ejpam-5914	141	17	theorem	theorem	NOUN
ejpam-5914	141	18	2	2	NUM
ejpam-5914	141	19	.	.	PUNCT
ejpam-5914	141	20	every	every	DET
ejpam-5914	141	21	length	length	NOUN
ejpam-5914	141	22	2	2	NUM
ejpam-5914	141	23	-	-	PUNCT
ejpam-5914	141	24	fuzzy	fuzzy	ADJ
ejpam-5914	141	25	subalgebra	subalgebra	NOUN
ejpam-5914	141	26	of	of	ADP
ejpam-5914	141	27	a	a	PRON
ejpam-5914	141	28	is	be	AUX
ejpam-5914	141	29	a	a	DET
ejpam-5914	141	30	length	length	NOUN
ejpam-5914	141	31	4	4	NUM
ejpam-5914	141	32	-	-	PUNCT
ejpam-5914	141	33	fuzzy	fuzzy	ADJ
ejpam-5914	141	34	subalgebra	subalgebra	NOUN
ejpam-5914	141	35	.	.	PUNCT
ejpam-5914	142	1	proof	proof	NOUN
ejpam-5914	142	2	.	.	PUNCT
ejpam-5914	143	1	let	let	VERB
ejpam-5914	143	2	(	(	PUNCT
ejpam-5914	143	3	a	a	PRON
ejpam-5914	143	4	,	,	PUNCT
ejpam-5914	143	5	f̃	f̃	PROPN
ejpam-5914	143	6	)	)	PUNCT
ejpam-5914	143	7	be	be	VERB
ejpam-5914	143	8	a	a	DET
ejpam-5914	143	9	length	length	NOUN
ejpam-5914	143	10	2	2	NUM
ejpam-5914	143	11	-	-	PUNCT
ejpam-5914	143	12	fuzzy	fuzzy	ADJ
ejpam-5914	143	13	subalgebra	subalgebra	NOUN
ejpam-5914	143	14	of	of	ADP
ejpam-5914	143	15	a	a	PRON
ejpam-5914	143	16	and	and	CCONJ
ejpam-5914	143	17	x	x	NOUN
ejpam-5914	143	18	,	,	PUNCT
ejpam-5914	143	19	y	y	PROPN
ejpam-5914	143	20	∈	∈	PROPN
ejpam-5914	143	21	a.	a.	NOUN
ejpam-5914	143	22	then	then	ADV
ejpam-5914	143	23	f̃ℓ((x|(y|y))|(x|(y|y	f̃ℓ((x|(y|y))|(x|(y|y	NUM
ejpam-5914	143	24	)	)	PUNCT
ejpam-5914	143	25	)	)	PUNCT
ejpam-5914	143	26	)	)	PUNCT
ejpam-5914	144	1	≤	≤	NUM
ejpam-5914	144	2	min{f̃ℓ(x	min{f̃ℓ(x	PROPN
ejpam-5914	144	3	)	)	PUNCT
ejpam-5914	144	4	,	,	PUNCT
ejpam-5914	144	5	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	144	6	)	)	PUNCT
ejpam-5914	144	7	}	}	PUNCT
ejpam-5914	144	8	≤	≤	NUM
ejpam-5914	144	9	max{f̃ℓ(x	max{f̃ℓ(x	PROPN
ejpam-5914	144	10	)	)	PUNCT
ejpam-5914	144	11	,	,	PUNCT
ejpam-5914	144	12	f̃ℓ(y	f̃ℓ(y	PROPN
ejpam-5914	144	13	)	)	PUNCT
ejpam-5914	144	14	}	}	PUNCT
ejpam-5914	144	15	.	.	PUNCT
ejpam-5914	145	1	hence	hence	ADV
ejpam-5914	145	2	,	,	PUNCT
ejpam-5914	145	3	(	(	PUNCT
ejpam-5914	145	4	a	a	PRON
ejpam-5914	145	5	,	,	PUNCT
ejpam-5914	145	6	f̃	f̃	PROPN
ejpam-5914	145	7	)	)	PUNCT
ejpam-5914	145	8	is	be	AUX
ejpam-5914	145	9	a	a	DET
ejpam-5914	145	10	length	length	NOUN
ejpam-5914	145	11	4	4	NUM
ejpam-5914	145	12	-	-	PUNCT
ejpam-5914	145	13	fuzzy	fuzzy	ADJ
ejpam-5914	145	14	subalgebra	subalgebra	NOUN
ejpam-5914	145	15	of	of	ADP
ejpam-5914	145	16	a.	a.	NOUN
ejpam-5914	145	17	theorem	theorem	NOUN
ejpam-5914	145	18	3	3	NUM
ejpam-5914	145	19	.	.	NOUN
ejpam-5914	145	20	length	length	NOUN
ejpam-5914	145	21	2	2	NUM
ejpam-5914	145	22	-	-	PUNCT
ejpam-5914	145	23	fuzzy	fuzzy	ADJ
ejpam-5914	145	24	subalgebra	subalgebra	NOUN
ejpam-5914	145	25	and	and	CCONJ
ejpam-5914	145	26	length	length	NOUN
ejpam-5914	145	27	3	3	NUM
ejpam-5914	145	28	-	-	PUNCT
ejpam-5914	145	29	fuzzy	fuzzy	ADJ
ejpam-5914	145	30	subalgebra	subalgebra	NOUN
ejpam-5914	145	31	of	of	ADP
ejpam-5914	145	32	a	a	DET
ejpam-5914	145	33	coincide	coincide	NOUN
ejpam-5914	145	34	.	.	PUNCT
ejpam-5914	146	1	proof	proof	NOUN
ejpam-5914	146	2	.	.	PUNCT
ejpam-5914	147	1	it	it	PRON
ejpam-5914	147	2	is	be	AUX
ejpam-5914	147	3	straightforward	straightforward	ADJ
ejpam-5914	147	4	by	by	ADP
ejpam-5914	147	5	theorems	theorem	NOUN
ejpam-5914	147	6	1	1	NUM
ejpam-5914	147	7	and	and	CCONJ
ejpam-5914	147	8	2	2	NUM
ejpam-5914	147	9	.	.	X
ejpam-5914	147	10	theorem	theorem	NOUN
ejpam-5914	147	11	4	4	NUM
ejpam-5914	147	12	.	.	PUNCT
ejpam-5914	147	13	given	give	VERB
ejpam-5914	147	14	a	a	DET
ejpam-5914	147	15	subalgebra	subalgebra	NOUN
ejpam-5914	147	16	s	s	NOUN
ejpam-5914	147	17	of	of	ADP
ejpam-5914	147	18	a	a	PRON
ejpam-5914	147	19	and	and	CCONJ
ejpam-5914	147	20	b1	b1	NOUN
ejpam-5914	147	21	,	,	PUNCT
ejpam-5914	147	22	b2	b2	NOUN
ejpam-5914	147	23	∈	∈	NOUN
ejpam-5914	147	24	d[0	d[0	PROPN
ejpam-5914	147	25	,	,	PUNCT
ejpam-5914	147	26	1	1	NUM
ejpam-5914	147	27	]	]	PUNCT
ejpam-5914	147	28	,	,	PUNCT
ejpam-5914	147	29	let	let	VERB
ejpam-5914	147	30	(	(	PUNCT
ejpam-5914	147	31	a	a	PRON
ejpam-5914	147	32	,	,	PUNCT
ejpam-5914	147	33	f̃	f̃	PROPN
ejpam-5914	147	34	)	)	PUNCT
ejpam-5914	147	35	be	be	VERB
ejpam-5914	147	36	an	an	DET
ejpam-5914	147	37	intervalvalued	intervalvalue	VERB
ejpam-5914	147	38	fuzzy	fuzzy	ADJ
ejpam-5914	147	39	structure	structure	NOUN
ejpam-5914	147	40	over	over	ADP
ejpam-5914	147	41	a	a	DET
ejpam-5914	147	42	given	give	VERB
ejpam-5914	147	43	by	by	ADP
ejpam-5914	147	44	:	:	PUNCT
ejpam-5914	147	45	f̃	f̃	PROPN
ejpam-5914	147	46	:	:	PUNCT
ejpam-5914	147	47	a	a	DET
ejpam-5914	147	48	→	→	SYM
ejpam-5914	147	49	d[0	d[0	ADJ
ejpam-5914	147	50	,	,	PUNCT
ejpam-5914	147	51	1];x	1];x	NUM
ejpam-5914	147	52	7→	7→	NUM
ejpam-5914	147	53	{	{	PUNCT
ejpam-5914	147	54	b2	b2	NOUN
ejpam-5914	147	55	if	if	SCONJ
ejpam-5914	147	56	x	x	PROPN
ejpam-5914	147	57	∈	∈	PROPN
ejpam-5914	147	58	s	s	PART
ejpam-5914	147	59	,	,	PUNCT
ejpam-5914	147	60	b1	b1	VERB
ejpam-5914	147	61	otherwise	otherwise	ADV
ejpam-5914	147	62	.	.	PUNCT
ejpam-5914	148	1	n.	n.	PROPN
ejpam-5914	148	2	rajesh	rajesh	PROPN
ejpam-5914	148	3	et	et	PROPN
ejpam-5914	148	4	al	al	PROPN
ejpam-5914	148	5	.	.	PUNCT
ejpam-5914	148	6	/	/	SYM
ejpam-5914	148	7	eur	eur	PROPN
ejpam-5914	148	8	.	.	PUNCT
ejpam-5914	149	1	j.	j.	PROPN
ejpam-5914	149	2	pure	pure	PROPN
ejpam-5914	149	3	appl	appl	PROPN
ejpam-5914	149	4	.	.	PROPN
ejpam-5914	149	5	math	math	PROPN
ejpam-5914	149	6	,	,	PUNCT
ejpam-5914	149	7	18	18	NUM
ejpam-5914	149	8	(	(	PUNCT
ejpam-5914	149	9	2	2	NUM
ejpam-5914	149	10	)	)	PUNCT
ejpam-5914	149	11	(	(	PUNCT
ejpam-5914	149	12	2025	2025	NUM
ejpam-5914	149	13	)	)	PUNCT
ejpam-5914	149	14	,	,	PUNCT
ejpam-5914	149	15	5914	5914	NUM
ejpam-5914	149	16	7	7	NUM
ejpam-5914	149	17	of	of	ADP
ejpam-5914	149	18	21	21	NUM
ejpam-5914	149	19	(	(	PUNCT
ejpam-5914	149	20	1	1	NUM
ejpam-5914	149	21	)	)	PUNCT
ejpam-5914	149	22	if	if	SCONJ
ejpam-5914	149	23	b1	b1	PROPN
ejpam-5914	149	24	⊂	⊂	PROPN
ejpam-5914	149	25	b2	b2	PROPN
ejpam-5914	149	26	,	,	PUNCT
ejpam-5914	149	27	then	then	ADV
ejpam-5914	149	28	(	(	PUNCT
ejpam-5914	149	29	a	a	PRON
ejpam-5914	149	30	,	,	PUNCT
ejpam-5914	149	31	f̃	f̃	PROPN
ejpam-5914	149	32	)	)	PUNCT
ejpam-5914	149	33	is	be	AUX
ejpam-5914	149	34	a	a	DET
ejpam-5914	149	35	length	length	NOUN
ejpam-5914	149	36	1	1	NUM
ejpam-5914	149	37	-	-	PUNCT
ejpam-5914	149	38	fuzzy	fuzzy	ADJ
ejpam-5914	149	39	subalgebra	subalgebra	NOUN
ejpam-5914	149	40	of	of	ADP
ejpam-5914	149	41	a.	a.	NOUN
ejpam-5914	149	42	(	(	PUNCT
ejpam-5914	149	43	2	2	NUM
ejpam-5914	149	44	)	)	PUNCT
ejpam-5914	149	45	if	if	SCONJ
ejpam-5914	149	46	b2	b2	NOUN
ejpam-5914	149	47	⊂	⊂	PROPN
ejpam-5914	149	48	b1	b1	PROPN
ejpam-5914	149	49	,	,	PUNCT
ejpam-5914	149	50	then	then	ADV
ejpam-5914	149	51	(	(	PUNCT
ejpam-5914	149	52	a	a	PRON
ejpam-5914	149	53	,	,	PUNCT
ejpam-5914	149	54	f̃	f̃	PROPN
ejpam-5914	149	55	)	)	PUNCT
ejpam-5914	149	56	is	be	AUX
ejpam-5914	149	57	a	a	DET
ejpam-5914	149	58	length	length	NOUN
ejpam-5914	149	59	4	4	NUM
ejpam-5914	149	60	-	-	PUNCT
ejpam-5914	149	61	fuzzy	fuzzy	ADJ
ejpam-5914	149	62	subalgebra	subalgebra	NOUN
ejpam-5914	149	63	of	of	ADP
ejpam-5914	149	64	a.	a.	NOUN
ejpam-5914	149	65	proof	proof	NOUN
ejpam-5914	149	66	.	.	PUNCT
ejpam-5914	150	1	if	if	SCONJ
ejpam-5914	150	2	x	x	SYM
ejpam-5914	150	3	∈	∈	PROPN
ejpam-5914	150	4	s	s	NOUN
ejpam-5914	150	5	,	,	PUNCT
ejpam-5914	150	6	then	then	ADV
ejpam-5914	150	7	f̃(x	f̃(x	PROPN
ejpam-5914	150	8	)	)	PUNCT
ejpam-5914	150	9	=	=	SYM
ejpam-5914	150	10	b2	b2	NOUN
ejpam-5914	150	11	.	.	PUNCT
ejpam-5914	151	1	hence	hence	ADV
ejpam-5914	151	2	,	,	PUNCT
ejpam-5914	151	3	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	151	4	)	)	PUNCT
ejpam-5914	151	5	=	=	SYM
ejpam-5914	151	6	f̃sup(x)−	f̃sup(x)−	PROPN
ejpam-5914	151	7	f̃inf(x	f̃inf(x	PROPN
ejpam-5914	151	8	)	)	PUNCT
ejpam-5914	151	9	=	=	SYM
ejpam-5914	151	10	sup	sup	PROPN
ejpam-5914	151	11	f̃(x)−	f̃(x)−	PROPN
ejpam-5914	151	12	inf	inf	PROPN
ejpam-5914	151	13	f̃(x	f̃(x	PROPN
ejpam-5914	151	14	)	)	PUNCT
ejpam-5914	151	15	=	=	SYM
ejpam-5914	152	1	supb2	supb2	NOUN
ejpam-5914	152	2	−	−	PROPN
ejpam-5914	152	3	inf	inf	PROPN
ejpam-5914	152	4	b2	b2	NOUN
ejpam-5914	152	5	.	.	PUNCT
ejpam-5914	153	1	if	if	SCONJ
ejpam-5914	153	2	x	x	PROPN
ejpam-5914	153	3	/∈	/∈	PROPN
ejpam-5914	153	4	s	s	X
ejpam-5914	153	5	,	,	PUNCT
ejpam-5914	153	6	then	then	ADV
ejpam-5914	153	7	f̃(x	f̃(x	PROPN
ejpam-5914	153	8	)	)	PUNCT
ejpam-5914	153	9	=	=	SYM
ejpam-5914	153	10	b1	b1	NOUN
ejpam-5914	153	11	.	.	PUNCT
ejpam-5914	154	1	hence	hence	ADV
ejpam-5914	154	2	,	,	PUNCT
ejpam-5914	154	3	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	154	4	)	)	PUNCT
ejpam-5914	154	5	=	=	SYM
ejpam-5914	154	6	f̃sup(x)−	f̃sup(x)−	PROPN
ejpam-5914	154	7	f̃inf(x	f̃inf(x	PROPN
ejpam-5914	154	8	)	)	PUNCT
ejpam-5914	154	9	=	=	SYM
ejpam-5914	154	10	sup	sup	PROPN
ejpam-5914	154	11	f̃(x)−	f̃(x)−	PROPN
ejpam-5914	154	12	inf	inf	PROPN
ejpam-5914	154	13	f̃(x	f̃(x	PROPN
ejpam-5914	154	14	)	)	PUNCT
ejpam-5914	154	15	=	=	SYM
ejpam-5914	155	1	supb1	supb1	NOUN
ejpam-5914	156	1	−	−	PROPN
ejpam-5914	156	2	inf	inf	PROPN
ejpam-5914	156	3	b1	b1	NOUN
ejpam-5914	156	4	.	.	PUNCT
ejpam-5914	157	1	(	(	PUNCT
ejpam-5914	157	2	1	1	X
ejpam-5914	157	3	)	)	PUNCT
ejpam-5914	157	4	assume	assume	VERB
ejpam-5914	157	5	that	that	SCONJ
ejpam-5914	157	6	b1	b1	PROPN
ejpam-5914	157	7	⊂	⊂	PROPN
ejpam-5914	157	8	b2	b2	PROPN
ejpam-5914	157	9	.	.	PUNCT
ejpam-5914	158	1	then	then	ADV
ejpam-5914	158	2	supb2	supb2	PROPN
ejpam-5914	158	3	−	−	PROPN
ejpam-5914	158	4	inf	inf	PROPN
ejpam-5914	158	5	b2	b2	PROPN
ejpam-5914	158	6	≥	≥	PROPN
ejpam-5914	158	7	supb1	supb1	NOUN
ejpam-5914	159	1	−	−	PROPN
ejpam-5914	159	2	inf	inf	PROPN
ejpam-5914	159	3	b1	b1	NOUN
ejpam-5914	159	4	.	.	PUNCT
ejpam-5914	160	1	case	case	NOUN
ejpam-5914	160	2	1	1	NUM
ejpam-5914	160	3	:	:	PUNCT
ejpam-5914	160	4	let	let	VERB
ejpam-5914	160	5	x	x	PRON
ejpam-5914	160	6	,	,	PUNCT
ejpam-5914	160	7	y	y	PROPN
ejpam-5914	160	8	∈	∈	PROPN
ejpam-5914	160	9	s.	s.	PROPN
ejpam-5914	160	10	then	then	ADV
ejpam-5914	160	11	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	160	12	)	)	PUNCT
ejpam-5914	160	13	=	=	SYM
ejpam-5914	160	14	supb2	supb2	NOUN
ejpam-5914	160	15	−	−	PROPN
ejpam-5914	160	16	inf	inf	NOUN
ejpam-5914	160	17	b2	b2	NOUN
ejpam-5914	160	18	and	and	CCONJ
ejpam-5914	160	19	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	160	20	)	)	PUNCT
ejpam-5914	160	21	=	=	SYM
ejpam-5914	161	1	supb2	supb2	NOUN
ejpam-5914	161	2	−	−	PROPN
ejpam-5914	161	3	inf	inf	PROPN
ejpam-5914	161	4	b2	b2	NOUN
ejpam-5914	161	5	.	.	PUNCT
ejpam-5914	162	1	thus	thus	ADV
ejpam-5914	162	2	,	,	PUNCT
ejpam-5914	162	3	min{f̃ℓ(x	min{f̃ℓ(x	PROPN
ejpam-5914	162	4	)	)	PUNCT
ejpam-5914	162	5	,	,	PUNCT
ejpam-5914	162	6	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	162	7	)	)	PUNCT
ejpam-5914	162	8	}	}	PUNCT
ejpam-5914	162	9	=	=	SYM
ejpam-5914	162	10	supb2	supb2	NOUN
ejpam-5914	162	11	−	−	PROPN
ejpam-5914	162	12	inf	inf	PROPN
ejpam-5914	162	13	b2	b2	NOUN
ejpam-5914	162	14	.	.	PUNCT
ejpam-5914	163	1	since	since	SCONJ
ejpam-5914	163	2	s	s	PROPN
ejpam-5914	163	3	is	be	AUX
ejpam-5914	163	4	a	a	DET
ejpam-5914	163	5	subalgebra	subalgebra	NOUN
ejpam-5914	163	6	of	of	ADP
ejpam-5914	163	7	a	a	PRON
ejpam-5914	163	8	,	,	PUNCT
ejpam-5914	163	9	we	we	PRON
ejpam-5914	163	10	have	have	VERB
ejpam-5914	163	11	(	(	PUNCT
ejpam-5914	163	12	x|(y|y))|(x|(y|y	x|(y|y))|(x|(y|y	ADJ
ejpam-5914	163	13	)	)	PUNCT
ejpam-5914	163	14	)	)	PUNCT
ejpam-5914	164	1	∈	∈	PROPN
ejpam-5914	164	2	s	s	X
ejpam-5914	164	3	and	and	CCONJ
ejpam-5914	164	4	so	so	ADV
ejpam-5914	164	5	f̃ℓ((x|(y|y))|(x|(y|y	f̃ℓ((x|(y|y))|(x|(y|y	ADJ
ejpam-5914	164	6	)	)	PUNCT
ejpam-5914	164	7	)	)	PUNCT
ejpam-5914	164	8	)	)	PUNCT
ejpam-5914	165	1	=	=	SYM
ejpam-5914	165	2	supb2	supb2	NOUN
ejpam-5914	165	3	−	−	PROPN
ejpam-5914	165	4	inf	inf	PROPN
ejpam-5914	165	5	b2	b2	NOUN
ejpam-5914	165	6	.	.	PUNCT
ejpam-5914	166	1	thus	thus	ADV
ejpam-5914	166	2	,	,	PUNCT
ejpam-5914	166	3	f̃ℓ((x|(y|y))|(x|(y|y	f̃ℓ((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	166	4	)	)	PUNCT
ejpam-5914	166	5	)	)	PUNCT
ejpam-5914	166	6	)	)	PUNCT
ejpam-5914	167	1	=	=	SYM
ejpam-5914	167	2	supb2	supb2	NOUN
ejpam-5914	167	3	−	−	PROPN
ejpam-5914	167	4	inf	inf	NOUN
ejpam-5914	167	5	b2	b2	NOUN
ejpam-5914	167	6	=	=	SYM
ejpam-5914	167	7	(	(	PUNCT
ejpam-5914	167	8	≥)min{f̃ℓ(x	≥)min{f̃ℓ(x	NOUN
ejpam-5914	167	9	)	)	PUNCT
ejpam-5914	167	10	,	,	PUNCT
ejpam-5914	167	11	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	167	12	)	)	PUNCT
ejpam-5914	167	13	}	}	PUNCT
ejpam-5914	167	14	.	.	PUNCT
ejpam-5914	168	1	case	case	NOUN
ejpam-5914	168	2	2	2	NUM
ejpam-5914	168	3	:	:	PUNCT
ejpam-5914	168	4	let	let	VERB
ejpam-5914	168	5	x	x	PRON
ejpam-5914	168	6	,	,	PUNCT
ejpam-5914	168	7	y	y	PROPN
ejpam-5914	168	8	/∈	/∈	PUNCT
ejpam-5914	168	9	s.	s.	PROPN
ejpam-5914	169	1	then	then	ADV
ejpam-5914	169	2	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	169	3	)	)	PUNCT
ejpam-5914	170	1	=	=	PUNCT
ejpam-5914	170	2	supb1	supb1	NOUN
ejpam-5914	171	1	−	−	PROPN
ejpam-5914	171	2	inf	inf	PROPN
ejpam-5914	171	3	b1	b1	NOUN
ejpam-5914	171	4	and	and	CCONJ
ejpam-5914	171	5	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	171	6	)	)	PUNCT
ejpam-5914	171	7	=	=	SYM
ejpam-5914	172	1	supb1	supb1	NOUN
ejpam-5914	173	1	−	−	PROPN
ejpam-5914	173	2	inf	inf	PROPN
ejpam-5914	173	3	b1	b1	NOUN
ejpam-5914	173	4	,	,	PUNCT
ejpam-5914	173	5	and	and	CCONJ
ejpam-5914	173	6	so	so	ADV
ejpam-5914	173	7	min{f̃ℓ(x	min{f̃ℓ(x	PROPN
ejpam-5914	173	8	)	)	PUNCT
ejpam-5914	173	9	,	,	PUNCT
ejpam-5914	173	10	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	173	11	)	)	PUNCT
ejpam-5914	173	12	}	}	PUNCT
ejpam-5914	173	13	=	=	SYM
ejpam-5914	173	14	supb1	supb1	NOUN
ejpam-5914	174	1	−	−	PROPN
ejpam-5914	174	2	inf	inf	PROPN
ejpam-5914	174	3	b1	b1	NOUN
ejpam-5914	174	4	.	.	PUNCT
ejpam-5914	175	1	thus	thus	ADV
ejpam-5914	175	2	,	,	PUNCT
ejpam-5914	175	3	f̃ℓ((x|(y|y))|(x|(y|y	f̃ℓ((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	175	4	)	)	PUNCT
ejpam-5914	175	5	)	)	PUNCT
ejpam-5914	175	6	)	)	PUNCT
ejpam-5914	175	7	≥	≥	PROPN
ejpam-5914	175	8	supb1	supb1	NOUN
ejpam-5914	176	1	−	−	PROPN
ejpam-5914	176	2	inf	inf	PROPN
ejpam-5914	176	3	b1	b1	NOUN
ejpam-5914	176	4	=	=	SYM
ejpam-5914	176	5	min{f̃ℓ(x	min{f̃ℓ(x	PROPN
ejpam-5914	176	6	)	)	PUNCT
ejpam-5914	176	7	,	,	PUNCT
ejpam-5914	176	8	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	176	9	)	)	PUNCT
ejpam-5914	176	10	}	}	PUNCT
ejpam-5914	176	11	.	.	PUNCT
ejpam-5914	177	1	case	case	NOUN
ejpam-5914	177	2	3	3	X
ejpam-5914	177	3	:	:	PUNCT
ejpam-5914	177	4	let	let	VERB
ejpam-5914	177	5	x	x	PUNCT
ejpam-5914	177	6	/∈	/∈	PRON
ejpam-5914	177	7	s	s	PART
ejpam-5914	177	8	and	and	CCONJ
ejpam-5914	177	9	y	y	PROPN
ejpam-5914	177	10	∈	∈	PROPN
ejpam-5914	177	11	s.	s.	PROPN
ejpam-5914	177	12	then	then	ADV
ejpam-5914	177	13	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	177	14	)	)	PUNCT
ejpam-5914	177	15	=	=	PUNCT
ejpam-5914	178	1	supb1	supb1	NOUN
ejpam-5914	179	1	−	−	PROPN
ejpam-5914	179	2	inf	inf	PROPN
ejpam-5914	179	3	b1	b1	NOUN
ejpam-5914	179	4	and	and	CCONJ
ejpam-5914	179	5	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	179	6	)	)	PUNCT
ejpam-5914	179	7	=	=	SYM
ejpam-5914	179	8	supb2	supb2	NOUN
ejpam-5914	179	9	−	−	PROPN
ejpam-5914	179	10	inf	inf	PROPN
ejpam-5914	179	11	b2	b2	NOUN
ejpam-5914	179	12	,	,	PUNCT
ejpam-5914	179	13	and	and	CCONJ
ejpam-5914	179	14	so	so	ADV
ejpam-5914	179	15	min{f̃ℓ(x	min{f̃ℓ(x	PROPN
ejpam-5914	179	16	)	)	PUNCT
ejpam-5914	179	17	,	,	PUNCT
ejpam-5914	179	18	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	179	19	)	)	PUNCT
ejpam-5914	179	20	}	}	PUNCT
ejpam-5914	179	21	=	=	SYM
ejpam-5914	179	22	supb1	supb1	NOUN
ejpam-5914	180	1	−	−	PROPN
ejpam-5914	180	2	inf	inf	PROPN
ejpam-5914	180	3	b1	b1	NOUN
ejpam-5914	180	4	.	.	PUNCT
ejpam-5914	181	1	thus	thus	ADV
ejpam-5914	181	2	,	,	PUNCT
ejpam-5914	181	3	f̃ℓ((x|(y|y))|(x|(y|y	f̃ℓ((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	181	4	)	)	PUNCT
ejpam-5914	181	5	)	)	PUNCT
ejpam-5914	181	6	)	)	PUNCT
ejpam-5914	181	7	≥	≥	PROPN
ejpam-5914	181	8	supb1	supb1	NOUN
ejpam-5914	182	1	−	−	PROPN
ejpam-5914	182	2	inf	inf	PROPN
ejpam-5914	182	3	b1	b1	NOUN
ejpam-5914	182	4	=	=	SYM
ejpam-5914	182	5	min{f̃ℓ(x	min{f̃ℓ(x	PROPN
ejpam-5914	182	6	)	)	PUNCT
ejpam-5914	182	7	,	,	PUNCT
ejpam-5914	182	8	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	182	9	)	)	PUNCT
ejpam-5914	182	10	}	}	PUNCT
ejpam-5914	182	11	.	.	PUNCT
ejpam-5914	183	1	case	case	NOUN
ejpam-5914	183	2	4	4	NUM
ejpam-5914	183	3	:	:	PUNCT
ejpam-5914	183	4	let	let	VERB
ejpam-5914	183	5	x	x	PUNCT
ejpam-5914	183	6	∈	∈	PROPN
ejpam-5914	183	7	s	s	X
ejpam-5914	183	8	and	and	CCONJ
ejpam-5914	183	9	y	y	PROPN
ejpam-5914	183	10	/∈	/∈	PUNCT
ejpam-5914	184	1	s.	s.	PROPN
ejpam-5914	184	2	then	then	ADV
ejpam-5914	184	3	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	184	4	)	)	PUNCT
ejpam-5914	184	5	=	=	SYM
ejpam-5914	184	6	supb2	supb2	NOUN
ejpam-5914	184	7	−	−	PROPN
ejpam-5914	184	8	inf	inf	NOUN
ejpam-5914	184	9	b2	b2	NOUN
ejpam-5914	184	10	and	and	CCONJ
ejpam-5914	184	11	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	184	12	)	)	PUNCT
ejpam-5914	184	13	=	=	SYM
ejpam-5914	185	1	supb1	supb1	NOUN
ejpam-5914	186	1	−	−	PROPN
ejpam-5914	186	2	inf	inf	PROPN
ejpam-5914	186	3	b1	b1	NOUN
ejpam-5914	186	4	,	,	PUNCT
ejpam-5914	186	5	and	and	CCONJ
ejpam-5914	186	6	so	so	ADV
ejpam-5914	186	7	min{f̃ℓ(x	min{f̃ℓ(x	PROPN
ejpam-5914	186	8	)	)	PUNCT
ejpam-5914	186	9	,	,	PUNCT
ejpam-5914	186	10	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	186	11	)	)	PUNCT
ejpam-5914	186	12	}	}	PUNCT
ejpam-5914	186	13	=	=	SYM
ejpam-5914	186	14	supb1	supb1	NOUN
ejpam-5914	187	1	−	−	PROPN
ejpam-5914	187	2	inf	inf	PROPN
ejpam-5914	187	3	b1	b1	NOUN
ejpam-5914	187	4	.	.	PUNCT
ejpam-5914	188	1	thus	thus	ADV
ejpam-5914	188	2	,	,	PUNCT
ejpam-5914	188	3	f̃ℓ((x|(y|y))|(x|(y|y	f̃ℓ((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	188	4	)	)	PUNCT
ejpam-5914	188	5	)	)	PUNCT
ejpam-5914	188	6	)	)	PUNCT
ejpam-5914	188	7	≥	≥	PROPN
ejpam-5914	188	8	supb1	supb1	NOUN
ejpam-5914	189	1	−	−	PROPN
ejpam-5914	189	2	inf	inf	PROPN
ejpam-5914	189	3	b1	b1	NOUN
ejpam-5914	189	4	=	=	SYM
ejpam-5914	189	5	min{f̃ℓ(x	min{f̃ℓ(x	PROPN
ejpam-5914	189	6	)	)	PUNCT
ejpam-5914	189	7	,	,	PUNCT
ejpam-5914	189	8	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	189	9	)	)	PUNCT
ejpam-5914	189	10	}	}	PUNCT
ejpam-5914	189	11	.	.	PUNCT
ejpam-5914	190	1	hence	hence	ADV
ejpam-5914	190	2	,	,	PUNCT
ejpam-5914	190	3	f̃ℓ	f̃ℓ	VERB
ejpam-5914	190	4	is	be	AUX
ejpam-5914	190	5	a	a	DET
ejpam-5914	190	6	1	1	NUM
ejpam-5914	190	7	-	-	PUNCT
ejpam-5914	190	8	fuzzy	fuzzy	ADJ
ejpam-5914	190	9	subalgebra	subalgebra	NOUN
ejpam-5914	190	10	of	of	ADP
ejpam-5914	190	11	a	a	PRON
ejpam-5914	190	12	and	and	CCONJ
ejpam-5914	190	13	so	so	ADV
ejpam-5914	190	14	(	(	PUNCT
ejpam-5914	190	15	a	a	PRON
ejpam-5914	190	16	,	,	PUNCT
ejpam-5914	190	17	f̃	f̃	PROPN
ejpam-5914	190	18	)	)	PUNCT
ejpam-5914	190	19	is	be	AUX
ejpam-5914	190	20	a	a	DET
ejpam-5914	190	21	length	length	NOUN
ejpam-5914	190	22	1	1	NUM
ejpam-5914	190	23	-	-	PUNCT
ejpam-5914	190	24	fuzzy	fuzzy	ADJ
ejpam-5914	190	25	subalgebra	subalgebra	NOUN
ejpam-5914	190	26	of	of	ADP
ejpam-5914	190	27	a.	a.	NOUN
ejpam-5914	190	28	(	(	PUNCT
ejpam-5914	190	29	2	2	X
ejpam-5914	190	30	)	)	PUNCT
ejpam-5914	190	31	assume	assume	VERB
ejpam-5914	190	32	that	that	SCONJ
ejpam-5914	190	33	b2	b2	NOUN
ejpam-5914	190	34	⊂	⊂	PROPN
ejpam-5914	190	35	b1	b1	PROPN
ejpam-5914	190	36	.	.	PUNCT
ejpam-5914	191	1	then	then	ADV
ejpam-5914	191	2	supb2	supb2	PROPN
ejpam-5914	191	3	−	−	PROPN
ejpam-5914	191	4	inf	inf	PROPN
ejpam-5914	191	5	b2	b2	NOUN
ejpam-5914	191	6	≤	≤	NUM
ejpam-5914	191	7	supb1	supb1	NOUN
ejpam-5914	192	1	−	−	PROPN
ejpam-5914	192	2	inf	inf	PROPN
ejpam-5914	192	3	b1	b1	NOUN
ejpam-5914	192	4	.	.	PUNCT
ejpam-5914	193	1	case	case	NOUN
ejpam-5914	193	2	1	1	NUM
ejpam-5914	193	3	:	:	PUNCT
ejpam-5914	193	4	let	let	VERB
ejpam-5914	193	5	x	x	PRON
ejpam-5914	193	6	,	,	PUNCT
ejpam-5914	193	7	y	y	PROPN
ejpam-5914	193	8	∈	∈	PROPN
ejpam-5914	193	9	s.	s.	PROPN
ejpam-5914	193	10	then	then	ADV
ejpam-5914	193	11	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	193	12	)	)	PUNCT
ejpam-5914	193	13	=	=	SYM
ejpam-5914	193	14	supb2	supb2	NOUN
ejpam-5914	193	15	−	−	PROPN
ejpam-5914	193	16	inf	inf	NOUN
ejpam-5914	193	17	b2	b2	NOUN
ejpam-5914	193	18	and	and	CCONJ
ejpam-5914	193	19	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	193	20	)	)	PUNCT
ejpam-5914	193	21	=	=	SYM
ejpam-5914	194	1	supb2	supb2	NOUN
ejpam-5914	194	2	−	−	PROPN
ejpam-5914	194	3	inf	inf	PROPN
ejpam-5914	194	4	b2	b2	NOUN
ejpam-5914	194	5	.	.	PUNCT
ejpam-5914	195	1	thus	thus	ADV
ejpam-5914	195	2	,	,	PUNCT
ejpam-5914	195	3	max{f̃ℓ(x	max{f̃ℓ(x	PROPN
ejpam-5914	195	4	)	)	PUNCT
ejpam-5914	195	5	,	,	PUNCT
ejpam-5914	195	6	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	195	7	)	)	PUNCT
ejpam-5914	195	8	}	}	PUNCT
ejpam-5914	195	9	=	=	SYM
ejpam-5914	195	10	supb2	supb2	NOUN
ejpam-5914	195	11	−	−	PROPN
ejpam-5914	195	12	inf	inf	PROPN
ejpam-5914	195	13	b2	b2	NOUN
ejpam-5914	195	14	.	.	PUNCT
ejpam-5914	196	1	since	since	SCONJ
ejpam-5914	196	2	s	s	PROPN
ejpam-5914	196	3	is	be	AUX
ejpam-5914	196	4	a	a	DET
ejpam-5914	196	5	subalgebra	subalgebra	NOUN
ejpam-5914	196	6	of	of	ADP
ejpam-5914	196	7	a	a	PRON
ejpam-5914	196	8	,	,	PUNCT
ejpam-5914	196	9	we	we	PRON
ejpam-5914	196	10	have	have	VERB
ejpam-5914	196	11	(	(	PUNCT
ejpam-5914	196	12	x|(y|y))|(x|(y|y	x|(y|y))|(x|(y|y	ADJ
ejpam-5914	196	13	)	)	PUNCT
ejpam-5914	196	14	)	)	PUNCT
ejpam-5914	197	1	∈	∈	PROPN
ejpam-5914	197	2	s	s	X
ejpam-5914	197	3	and	and	CCONJ
ejpam-5914	197	4	so	so	ADV
ejpam-5914	197	5	f̃ℓ((x|(y|y))|(x|(y|y	f̃ℓ((x|(y|y))|(x|(y|y	ADJ
ejpam-5914	197	6	)	)	PUNCT
ejpam-5914	197	7	)	)	PUNCT
ejpam-5914	197	8	)	)	PUNCT
ejpam-5914	198	1	=	=	SYM
ejpam-5914	198	2	supb2	supb2	NOUN
ejpam-5914	198	3	−	−	PROPN
ejpam-5914	198	4	inf	inf	PROPN
ejpam-5914	198	5	b2	b2	NOUN
ejpam-5914	198	6	.	.	PUNCT
ejpam-5914	199	1	thus	thus	ADV
ejpam-5914	199	2	,	,	PUNCT
ejpam-5914	199	3	f̃ℓ((x|(y|y))|(x|(y|y	f̃ℓ((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	199	4	)	)	PUNCT
ejpam-5914	199	5	)	)	PUNCT
ejpam-5914	199	6	)	)	PUNCT
ejpam-5914	200	1	=	=	SYM
ejpam-5914	200	2	supb2	supb2	NOUN
ejpam-5914	200	3	−	−	PROPN
ejpam-5914	200	4	inf	inf	NOUN
ejpam-5914	200	5	b2	b2	NOUN
ejpam-5914	200	6	=	=	SYM
ejpam-5914	200	7	(	(	PUNCT
ejpam-5914	200	8	≤)max{f̃ℓ(x	≤)max{f̃ℓ(x	NOUN
ejpam-5914	200	9	)	)	PUNCT
ejpam-5914	200	10	,	,	PUNCT
ejpam-5914	200	11	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	200	12	)	)	PUNCT
ejpam-5914	200	13	}	}	PUNCT
ejpam-5914	200	14	.	.	PUNCT
ejpam-5914	201	1	case	case	NOUN
ejpam-5914	201	2	2	2	NUM
ejpam-5914	201	3	:	:	PUNCT
ejpam-5914	201	4	let	let	VERB
ejpam-5914	201	5	x	x	PRON
ejpam-5914	201	6	,	,	PUNCT
ejpam-5914	201	7	y	y	PROPN
ejpam-5914	201	8	/∈	/∈	PUNCT
ejpam-5914	201	9	s.	s.	PROPN
ejpam-5914	202	1	then	then	ADV
ejpam-5914	202	2	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	202	3	)	)	PUNCT
ejpam-5914	203	1	=	=	PUNCT
ejpam-5914	203	2	supb1	supb1	NOUN
ejpam-5914	204	1	−	−	PROPN
ejpam-5914	204	2	inf	inf	PROPN
ejpam-5914	204	3	b1	b1	NOUN
ejpam-5914	204	4	and	and	CCONJ
ejpam-5914	204	5	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	204	6	)	)	PUNCT
ejpam-5914	204	7	=	=	SYM
ejpam-5914	205	1	supb1	supb1	NOUN
ejpam-5914	206	1	−	−	PROPN
ejpam-5914	206	2	inf	inf	PROPN
ejpam-5914	206	3	b1	b1	NOUN
ejpam-5914	206	4	,	,	PUNCT
ejpam-5914	206	5	so	so	SCONJ
ejpam-5914	206	6	max{f̃ℓ(x	max{f̃ℓ(x	PROPN
ejpam-5914	206	7	)	)	PUNCT
ejpam-5914	206	8	,	,	PUNCT
ejpam-5914	206	9	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	206	10	)	)	PUNCT
ejpam-5914	206	11	}	}	PUNCT
ejpam-5914	206	12	=	=	SYM
ejpam-5914	206	13	supb1	supb1	NOUN
ejpam-5914	207	1	−	−	PROPN
ejpam-5914	207	2	inf	inf	PROPN
ejpam-5914	207	3	b1	b1	NOUN
ejpam-5914	207	4	.	.	PUNCT
ejpam-5914	208	1	thus	thus	ADV
ejpam-5914	208	2	,	,	PUNCT
ejpam-5914	208	3	f̃ℓ((x|(y|y))|(x|(y|y	f̃ℓ((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	208	4	)	)	PUNCT
ejpam-5914	208	5	)	)	PUNCT
ejpam-5914	208	6	)	)	PUNCT
ejpam-5914	208	7	≤	≤	NUM
ejpam-5914	209	1	supb1	supb1	NOUN
ejpam-5914	210	1	−	−	PROPN
ejpam-5914	210	2	inf	inf	PROPN
ejpam-5914	210	3	b1	b1	NOUN
ejpam-5914	210	4	=	=	SYM
ejpam-5914	210	5	max{f̃ℓ(x	max{f̃ℓ(x	PROPN
ejpam-5914	210	6	)	)	PUNCT
ejpam-5914	210	7	,	,	PUNCT
ejpam-5914	210	8	f̃ℓ(y	f̃ℓ(y	PROPN
ejpam-5914	210	9	)	)	PUNCT
ejpam-5914	210	10	}	}	PUNCT
ejpam-5914	210	11	.	.	PUNCT
ejpam-5914	211	1	n.	n.	PROPN
ejpam-5914	211	2	rajesh	rajesh	PROPN
ejpam-5914	211	3	et	et	PROPN
ejpam-5914	211	4	al	al	PROPN
ejpam-5914	211	5	.	.	PUNCT
ejpam-5914	211	6	/	/	SYM
ejpam-5914	211	7	eur	eur	PROPN
ejpam-5914	211	8	.	.	PUNCT
ejpam-5914	212	1	j.	j.	PROPN
ejpam-5914	212	2	pure	pure	PROPN
ejpam-5914	212	3	appl	appl	PROPN
ejpam-5914	212	4	.	.	PROPN
ejpam-5914	212	5	math	math	PROPN
ejpam-5914	212	6	,	,	PUNCT
ejpam-5914	212	7	18	18	NUM
ejpam-5914	212	8	(	(	PUNCT
ejpam-5914	212	9	2	2	NUM
ejpam-5914	212	10	)	)	PUNCT
ejpam-5914	212	11	(	(	PUNCT
ejpam-5914	212	12	2025	2025	NUM
ejpam-5914	212	13	)	)	PUNCT
ejpam-5914	212	14	,	,	PUNCT
ejpam-5914	212	15	5914	5914	NUM
ejpam-5914	212	16	8	8	NUM
ejpam-5914	212	17	of	of	ADP
ejpam-5914	212	18	21	21	NUM
ejpam-5914	212	19	case	case	NOUN
ejpam-5914	212	20	3	3	NUM
ejpam-5914	212	21	:	:	PUNCT
ejpam-5914	212	22	let	let	VERB
ejpam-5914	212	23	x	x	PUNCT
ejpam-5914	212	24	/∈	/∈	PRON
ejpam-5914	212	25	s	s	PART
ejpam-5914	212	26	and	and	CCONJ
ejpam-5914	212	27	y	y	PROPN
ejpam-5914	212	28	∈	∈	PROPN
ejpam-5914	212	29	s.	s.	PROPN
ejpam-5914	212	30	then	then	ADV
ejpam-5914	212	31	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	212	32	)	)	PUNCT
ejpam-5914	212	33	=	=	PUNCT
ejpam-5914	213	1	supb1	supb1	NOUN
ejpam-5914	214	1	−	−	PROPN
ejpam-5914	214	2	inf	inf	PROPN
ejpam-5914	214	3	b1	b1	NOUN
ejpam-5914	214	4	and	and	CCONJ
ejpam-5914	214	5	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	214	6	)	)	PUNCT
ejpam-5914	214	7	=	=	SYM
ejpam-5914	214	8	supb2	supb2	NOUN
ejpam-5914	214	9	−	−	PROPN
ejpam-5914	214	10	inf	inf	PROPN
ejpam-5914	214	11	b2	b2	NOUN
ejpam-5914	214	12	,	,	PUNCT
ejpam-5914	214	13	so	so	SCONJ
ejpam-5914	214	14	max{f̃ℓ(x	max{f̃ℓ(x	PROPN
ejpam-5914	214	15	)	)	PUNCT
ejpam-5914	214	16	,	,	PUNCT
ejpam-5914	214	17	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	214	18	)	)	PUNCT
ejpam-5914	214	19	}	}	PUNCT
ejpam-5914	214	20	=	=	SYM
ejpam-5914	214	21	supb1	supb1	NOUN
ejpam-5914	215	1	−	−	PROPN
ejpam-5914	215	2	inf	inf	PROPN
ejpam-5914	215	3	b1	b1	NOUN
ejpam-5914	215	4	.	.	PUNCT
ejpam-5914	216	1	thus	thus	ADV
ejpam-5914	216	2	,	,	PUNCT
ejpam-5914	216	3	f̃ℓ((x|(y|y))|(x|(y|y	f̃ℓ((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	216	4	)	)	PUNCT
ejpam-5914	216	5	)	)	PUNCT
ejpam-5914	216	6	)	)	PUNCT
ejpam-5914	216	7	≤	≤	NUM
ejpam-5914	217	1	supb1	supb1	NOUN
ejpam-5914	218	1	−	−	PROPN
ejpam-5914	218	2	inf	inf	PROPN
ejpam-5914	218	3	b1	b1	NOUN
ejpam-5914	218	4	=	=	SYM
ejpam-5914	218	5	max{f̃ℓ(x	max{f̃ℓ(x	PROPN
ejpam-5914	218	6	)	)	PUNCT
ejpam-5914	218	7	,	,	PUNCT
ejpam-5914	218	8	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	218	9	)	)	PUNCT
ejpam-5914	218	10	}	}	PUNCT
ejpam-5914	218	11	.	.	PUNCT
ejpam-5914	219	1	case	case	NOUN
ejpam-5914	219	2	4	4	NUM
ejpam-5914	219	3	:	:	PUNCT
ejpam-5914	219	4	let	let	VERB
ejpam-5914	219	5	x	x	PUNCT
ejpam-5914	219	6	∈	∈	PROPN
ejpam-5914	219	7	s	s	X
ejpam-5914	219	8	and	and	CCONJ
ejpam-5914	219	9	y	y	PROPN
ejpam-5914	219	10	/∈	/∈	PUNCT
ejpam-5914	220	1	s.	s.	PROPN
ejpam-5914	220	2	then	then	ADV
ejpam-5914	220	3	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	220	4	)	)	PUNCT
ejpam-5914	220	5	=	=	SYM
ejpam-5914	220	6	supb2	supb2	NOUN
ejpam-5914	220	7	−	−	PROPN
ejpam-5914	220	8	inf	inf	NOUN
ejpam-5914	220	9	b2	b2	NOUN
ejpam-5914	220	10	and	and	CCONJ
ejpam-5914	220	11	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	220	12	)	)	PUNCT
ejpam-5914	220	13	=	=	SYM
ejpam-5914	221	1	supb1	supb1	NOUN
ejpam-5914	222	1	−	−	PROPN
ejpam-5914	222	2	inf	inf	PROPN
ejpam-5914	222	3	b1	b1	NOUN
ejpam-5914	222	4	,	,	PUNCT
ejpam-5914	222	5	so	so	SCONJ
ejpam-5914	222	6	max{f̃ℓ(x	max{f̃ℓ(x	PROPN
ejpam-5914	222	7	)	)	PUNCT
ejpam-5914	222	8	,	,	PUNCT
ejpam-5914	222	9	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	222	10	)	)	PUNCT
ejpam-5914	222	11	}	}	PUNCT
ejpam-5914	222	12	=	=	SYM
ejpam-5914	222	13	supb1	supb1	NOUN
ejpam-5914	223	1	−	−	PROPN
ejpam-5914	223	2	inf	inf	PROPN
ejpam-5914	223	3	b1	b1	NOUN
ejpam-5914	223	4	.	.	PUNCT
ejpam-5914	224	1	thus	thus	ADV
ejpam-5914	224	2	,	,	PUNCT
ejpam-5914	224	3	f̃ℓ((x|(y|y))|(x|(y|y	f̃ℓ((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	224	4	)	)	PUNCT
ejpam-5914	224	5	)	)	PUNCT
ejpam-5914	224	6	)	)	PUNCT
ejpam-5914	224	7	≤	≤	NUM
ejpam-5914	225	1	supb1	supb1	NOUN
ejpam-5914	226	1	−	−	PROPN
ejpam-5914	226	2	inf	inf	PROPN
ejpam-5914	226	3	b1	b1	NOUN
ejpam-5914	226	4	=	=	SYM
ejpam-5914	226	5	max{f̃ℓ(x	max{f̃ℓ(x	PROPN
ejpam-5914	226	6	)	)	PUNCT
ejpam-5914	226	7	,	,	PUNCT
ejpam-5914	226	8	f̃ℓ(y	f̃ℓ(y	PROPN
ejpam-5914	226	9	)	)	PUNCT
ejpam-5914	226	10	}	}	PUNCT
ejpam-5914	226	11	.	.	PUNCT
ejpam-5914	227	1	hence	hence	ADV
ejpam-5914	227	2	,	,	PUNCT
ejpam-5914	227	3	f̃ℓ	f̃ℓ	VERB
ejpam-5914	227	4	is	be	AUX
ejpam-5914	227	5	a	a	DET
ejpam-5914	227	6	4	4	NUM
ejpam-5914	227	7	-	-	PUNCT
ejpam-5914	227	8	fuzzy	fuzzy	ADJ
ejpam-5914	227	9	subalgebra	subalgebra	NOUN
ejpam-5914	227	10	of	of	ADP
ejpam-5914	227	11	a	a	PRON
ejpam-5914	227	12	and	and	CCONJ
ejpam-5914	227	13	so	so	ADV
ejpam-5914	227	14	(	(	PUNCT
ejpam-5914	227	15	a	a	PRON
ejpam-5914	227	16	,	,	PUNCT
ejpam-5914	227	17	f̃	f̃	PROPN
ejpam-5914	227	18	)	)	PUNCT
ejpam-5914	227	19	is	be	AUX
ejpam-5914	227	20	a	a	DET
ejpam-5914	227	21	length	length	NOUN
ejpam-5914	227	22	4	4	NUM
ejpam-5914	227	23	-	-	PUNCT
ejpam-5914	227	24	fuzzy	fuzzy	ADJ
ejpam-5914	227	25	subalgebra	subalgebra	NOUN
ejpam-5914	227	26	of	of	ADP
ejpam-5914	227	27	a.	a.	NOUN
ejpam-5914	227	28	definition	definition	NOUN
ejpam-5914	227	29	10	10	NUM
ejpam-5914	227	30	.	.	PUNCT
ejpam-5914	228	1	let	let	AUX
ejpam-5914	228	2	(	(	PUNCT
ejpam-5914	228	3	a	a	PRON
ejpam-5914	228	4	,	,	PUNCT
ejpam-5914	228	5	f	f	X
ejpam-5914	228	6	)	)	PUNCT
ejpam-5914	228	7	be	be	AUX
ejpam-5914	228	8	a	a	DET
ejpam-5914	228	9	fuzzy	fuzzy	ADJ
ejpam-5914	228	10	structure	structure	NOUN
ejpam-5914	228	11	in	in	ADP
ejpam-5914	228	12	a.	a.	NOUN
ejpam-5914	228	13	for	for	ADP
ejpam-5914	228	14	any	any	DET
ejpam-5914	228	15	t	t	NOUN
ejpam-5914	228	16	∈	∈	PROPN
ejpam-5914	229	1	[	[	X
ejpam-5914	229	2	0	0	NUM
ejpam-5914	229	3	,	,	PUNCT
ejpam-5914	229	4	1	1	NUM
ejpam-5914	229	5	]	]	PUNCT
ejpam-5914	229	6	,	,	PUNCT
ejpam-5914	229	7	the	the	DET
ejpam-5914	229	8	sets	set	VERB
ejpam-5914	229	9	u(f	u(f	PROPN
ejpam-5914	229	10	;	;	PUNCT
ejpam-5914	229	11	t	t	X
ejpam-5914	229	12	)	)	PUNCT
ejpam-5914	229	13	=	=	PRON
ejpam-5914	230	1	{	{	PUNCT
ejpam-5914	230	2	x	x	PUNCT
ejpam-5914	230	3	∈	∈	PROPN
ejpam-5914	230	4	a	a	DET
ejpam-5914	230	5	:	:	PUNCT
ejpam-5914	230	6	f(x	f(x	PROPN
ejpam-5914	230	7	)	)	PUNCT
ejpam-5914	230	8	≥	≥	NOUN
ejpam-5914	230	9	t	t	PROPN
ejpam-5914	230	10	}	}	PUNCT
ejpam-5914	230	11	,	,	PUNCT
ejpam-5914	230	12	l(f	l(f	PROPN
ejpam-5914	230	13	;	;	PUNCT
ejpam-5914	230	14	t	t	X
ejpam-5914	230	15	)	)	PUNCT
ejpam-5914	230	16	=	=	PRON
ejpam-5914	231	1	{	{	PUNCT
ejpam-5914	231	2	x	x	PUNCT
ejpam-5914	231	3	∈	∈	PROPN
ejpam-5914	231	4	a	a	DET
ejpam-5914	231	5	:	:	PUNCT
ejpam-5914	231	6	f(x	f(x	PROPN
ejpam-5914	231	7	)	)	PUNCT
ejpam-5914	231	8	≤	≤	NOUN
ejpam-5914	231	9	t	t	PROPN
ejpam-5914	231	10	}	}	PUNCT
ejpam-5914	231	11	,	,	PUNCT
ejpam-5914	231	12	are	be	AUX
ejpam-5914	231	13	called	call	VERB
ejpam-5914	231	14	an	an	DET
ejpam-5914	231	15	upper	upper	ADJ
ejpam-5914	231	16	t	t	NOUN
ejpam-5914	231	17	-	-	PUNCT
ejpam-5914	231	18	level	level	NOUN
ejpam-5914	231	19	subset	subset	NOUN
ejpam-5914	231	20	and	and	CCONJ
ejpam-5914	231	21	a	a	DET
ejpam-5914	231	22	lower	low	ADJ
ejpam-5914	231	23	t	t	NOUN
ejpam-5914	231	24	-	-	PUNCT
ejpam-5914	231	25	level	level	NOUN
ejpam-5914	231	26	subset	subset	NOUN
ejpam-5914	231	27	of	of	ADP
ejpam-5914	231	28	f	f	PROPN
ejpam-5914	231	29	,	,	PUNCT
ejpam-5914	231	30	respectively	respectively	ADV
ejpam-5914	231	31	.	.	PUNCT
ejpam-5914	231	32	example	example	NOUN
ejpam-5914	232	1	4	4	NUM
ejpam-5914	232	2	.	.	X
ejpam-5914	232	3	consider	consider	VERB
ejpam-5914	232	4	example	example	NOUN
ejpam-5914	232	5	2	2	NUM
ejpam-5914	232	6	,	,	PUNCT
ejpam-5914	232	7	and	and	CCONJ
ejpam-5914	232	8	let	let	VERB
ejpam-5914	232	9	t	t	NOUN
ejpam-5914	232	10	=	=	PUNCT
ejpam-5914	232	11	0.3	0.3	NUM
ejpam-5914	232	12	.	.	PUNCT
ejpam-5914	233	1	then	then	ADV
ejpam-5914	233	2	u(f̃l	u(f̃l	NOUN
ejpam-5914	233	3	;	;	PUNCT
ejpam-5914	233	4	0.3	0.3	NUM
ejpam-5914	233	5	)	)	PUNCT
ejpam-5914	233	6	=	=	PRON
ejpam-5914	233	7	{	{	PUNCT
ejpam-5914	233	8	1	1	NUM
ejpam-5914	233	9	,	,	PUNCT
ejpam-5914	233	10	v	v	NOUN
ejpam-5914	233	11	,	,	PUNCT
ejpam-5914	233	12	0	0	NUM
ejpam-5914	233	13	}	}	PUNCT
ejpam-5914	233	14	and	and	CCONJ
ejpam-5914	233	15	l(f̃l	l(f̃l	PROPN
ejpam-5914	233	16	;	;	PUNCT
ejpam-5914	233	17	0.3	0.3	NUM
ejpam-5914	233	18	)	)	PUNCT
ejpam-5914	233	19	=	=	PRON
ejpam-5914	233	20	{	{	PUNCT
ejpam-5914	233	21	u	u	NOUN
ejpam-5914	233	22	,	,	PUNCT
ejpam-5914	233	23	0	0	NUM
ejpam-5914	233	24	}	}	PUNCT
ejpam-5914	233	25	.	.	PUNCT
ejpam-5914	234	1	if	if	SCONJ
ejpam-5914	234	2	t	t	NOUN
ejpam-5914	234	3	=	=	SYM
ejpam-5914	234	4	0.5	0.5	NUM
ejpam-5914	234	5	,	,	PUNCT
ejpam-5914	234	6	then	then	ADV
ejpam-5914	234	7	u(f̃l	u(f̃l	PROPN
ejpam-5914	234	8	;	;	PUNCT
ejpam-5914	234	9	0.5	0.5	NUM
ejpam-5914	234	10	)	)	PUNCT
ejpam-5914	234	11	=	=	NOUN
ejpam-5914	234	12	∅	∅	NOUN
ejpam-5914	234	13	and	and	CCONJ
ejpam-5914	234	14	l(f̃l	l(f̃l	PROPN
ejpam-5914	234	15	;	;	PUNCT
ejpam-5914	234	16	0.5	0.5	NUM
ejpam-5914	234	17	)	)	PUNCT
ejpam-5914	234	18	=	=	NOUN
ejpam-5914	234	19	a.	a.	NOUN
ejpam-5914	234	20	if	if	SCONJ
ejpam-5914	234	21	t	t	PROPN
ejpam-5914	234	22	=	=	SYM
ejpam-5914	234	23	0.1	0.1	NUM
ejpam-5914	234	24	,	,	PUNCT
ejpam-5914	234	25	then	then	ADV
ejpam-5914	234	26	u(f̃l	u(f̃l	PROPN
ejpam-5914	234	27	;	;	PUNCT
ejpam-5914	234	28	0.1	0.1	NUM
ejpam-5914	234	29	)	)	PUNCT
ejpam-5914	234	30	=	=	SYM
ejpam-5914	234	31	a	a	PRON
ejpam-5914	234	32	and	and	CCONJ
ejpam-5914	234	33	l(f̃l	l(f̃l	PROPN
ejpam-5914	234	34	;	;	PUNCT
ejpam-5914	234	35	0.1	0.1	NUM
ejpam-5914	234	36	)	)	PUNCT
ejpam-5914	234	37	=	=	PUNCT
ejpam-5914	234	38	∅.	∅.	NOUN
ejpam-5914	234	39	theorem	theorem	ADJ
ejpam-5914	234	40	5	5	NUM
ejpam-5914	234	41	.	.	PUNCT
ejpam-5914	235	1	an	an	DET
ejpam-5914	235	2	interval	interval	NOUN
ejpam-5914	235	3	-	-	PUNCT
ejpam-5914	235	4	valued	value	VERB
ejpam-5914	235	5	fuzzy	fuzzy	ADJ
ejpam-5914	235	6	structure	structure	NOUN
ejpam-5914	235	7	(	(	PUNCT
ejpam-5914	235	8	a	a	PRON
ejpam-5914	235	9	,	,	PUNCT
ejpam-5914	235	10	f̃	f̃	PROPN
ejpam-5914	235	11	)	)	PUNCT
ejpam-5914	235	12	over	over	ADP
ejpam-5914	235	13	a	a	PRON
ejpam-5914	235	14	is	be	AUX
ejpam-5914	235	15	a	a	DET
ejpam-5914	235	16	length	length	NOUN
ejpam-5914	235	17	1	1	NUM
ejpam-5914	235	18	-	-	PUNCT
ejpam-5914	235	19	fuzzy	fuzzy	ADJ
ejpam-5914	235	20	subalgebra	subalgebra	NOUN
ejpam-5914	235	21	of	of	ADP
ejpam-5914	235	22	a	a	DET
ejpam-5914	235	23	if	if	NOUN
ejpam-5914	235	24	and	and	CCONJ
ejpam-5914	235	25	only	only	ADV
ejpam-5914	235	26	if	if	SCONJ
ejpam-5914	235	27	the	the	DET
ejpam-5914	235	28	set	set	NOUN
ejpam-5914	235	29	u(f̃ℓ	u(f̃ℓ	PROPN
ejpam-5914	235	30	;	;	PUNCT
ejpam-5914	235	31	t	t	PROPN
ejpam-5914	235	32	)	)	PUNCT
ejpam-5914	235	33	is	be	AUX
ejpam-5914	235	34	a	a	DET
ejpam-5914	235	35	subalgebra	subalgebra	NOUN
ejpam-5914	235	36	of	of	ADP
ejpam-5914	235	37	a	a	PRON
ejpam-5914	235	38	for	for	ADP
ejpam-5914	235	39	all	all	DET
ejpam-5914	235	40	t	t	NOUN
ejpam-5914	235	41	∈	∈	PROPN
ejpam-5914	236	1	[	[	X
ejpam-5914	236	2	0	0	NUM
ejpam-5914	236	3	,	,	PUNCT
ejpam-5914	236	4	1	1	NUM
ejpam-5914	236	5	]	]	PUNCT
ejpam-5914	236	6	with	with	ADP
ejpam-5914	236	7	u(f̃ℓ	u(f̃ℓ	PROPN
ejpam-5914	236	8	;	;	PUNCT
ejpam-5914	236	9	t	t	X
ejpam-5914	236	10	)	)	PUNCT
ejpam-5914	236	11	̸=	̸=	PROPN
ejpam-5914	236	12	∅.	∅.	ADP
ejpam-5914	236	13	proof	proof	NOUN
ejpam-5914	236	14	.	.	PUNCT
ejpam-5914	237	1	assume	assume	VERB
ejpam-5914	237	2	that	that	SCONJ
ejpam-5914	237	3	(	(	PUNCT
ejpam-5914	237	4	a	a	PRON
ejpam-5914	237	5	,	,	PUNCT
ejpam-5914	237	6	f̃	f̃	PROPN
ejpam-5914	237	7	)	)	PUNCT
ejpam-5914	237	8	is	be	AUX
ejpam-5914	237	9	a	a	DET
ejpam-5914	237	10	length	length	NOUN
ejpam-5914	237	11	1	1	NUM
ejpam-5914	237	12	-	-	PUNCT
ejpam-5914	237	13	fuzzy	fuzzy	ADJ
ejpam-5914	237	14	subalgebra	subalgebra	NOUN
ejpam-5914	237	15	of	of	ADP
ejpam-5914	237	16	a.	a.	NOUN
ejpam-5914	237	17	let	let	VERB
ejpam-5914	237	18	t	t	PROPN
ejpam-5914	237	19	∈	∈	PROPN
ejpam-5914	238	1	[	[	X
ejpam-5914	238	2	0	0	NUM
ejpam-5914	238	3	,	,	PUNCT
ejpam-5914	238	4	1	1	NUM
ejpam-5914	238	5	]	]	PUNCT
ejpam-5914	238	6	be	be	AUX
ejpam-5914	238	7	such	such	ADJ
ejpam-5914	238	8	that	that	SCONJ
ejpam-5914	238	9	u(f̃ℓ	u(f̃ℓ	PROPN
ejpam-5914	238	10	;	;	PUNCT
ejpam-5914	238	11	t	t	X
ejpam-5914	238	12	)	)	PUNCT
ejpam-5914	238	13	̸=	̸=	PROPN
ejpam-5914	238	14	∅	∅	NOUN
ejpam-5914	238	15	and	and	CCONJ
ejpam-5914	238	16	let	let	VERB
ejpam-5914	238	17	x	x	PRON
ejpam-5914	238	18	,	,	PUNCT
ejpam-5914	238	19	y	y	PROPN
ejpam-5914	238	20	∈	∈	PROPN
ejpam-5914	238	21	u(f̃ℓ	u(f̃ℓ	PROPN
ejpam-5914	238	22	;	;	PUNCT
ejpam-5914	238	23	t	t	PROPN
ejpam-5914	238	24	)	)	PUNCT
ejpam-5914	238	25	.	.	PUNCT
ejpam-5914	239	1	then	then	ADV
ejpam-5914	239	2	f̃ℓ(x	f̃ℓ(x	X
ejpam-5914	239	3	)	)	PUNCT
ejpam-5914	239	4	≥	≥	NOUN
ejpam-5914	239	5	t	t	NOUN
ejpam-5914	239	6	and	and	CCONJ
ejpam-5914	239	7	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	239	8	)	)	PUNCT
ejpam-5914	239	9	≥	≥	NOUN
ejpam-5914	239	10	t.	t.	NOUN
ejpam-5914	239	11	since	since	SCONJ
ejpam-5914	239	12	(	(	PUNCT
ejpam-5914	239	13	a	a	PRON
ejpam-5914	239	14	,	,	PUNCT
ejpam-5914	239	15	f̃	f̃	PROPN
ejpam-5914	239	16	)	)	PUNCT
ejpam-5914	239	17	is	be	AUX
ejpam-5914	239	18	a	a	DET
ejpam-5914	239	19	length	length	NOUN
ejpam-5914	239	20	1	1	NUM
ejpam-5914	239	21	-	-	PUNCT
ejpam-5914	239	22	fuzzy	fuzzy	ADJ
ejpam-5914	239	23	subalgebra	subalgebra	NOUN
ejpam-5914	239	24	of	of	ADP
ejpam-5914	239	25	a	a	PRON
ejpam-5914	239	26	,	,	PUNCT
ejpam-5914	239	27	we	we	PRON
ejpam-5914	239	28	have	have	VERB
ejpam-5914	239	29	f̃ℓ((x|(y|y))|(x|(y|y	f̃ℓ((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	239	30	)	)	PUNCT
ejpam-5914	239	31	)	)	PUNCT
ejpam-5914	239	32	)	)	PUNCT
ejpam-5914	239	33	≥	≥	PROPN
ejpam-5914	239	34	min{f̃ℓ(x	min{f̃ℓ(x	PROPN
ejpam-5914	239	35	)	)	PUNCT
ejpam-5914	239	36	,	,	PUNCT
ejpam-5914	239	37	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	239	38	)	)	PUNCT
ejpam-5914	239	39	}	}	PUNCT
ejpam-5914	239	40	≥	≥	X
ejpam-5914	239	41	t.	t.	PROPN
ejpam-5914	239	42	thus	thus	ADV
ejpam-5914	239	43	,	,	PUNCT
ejpam-5914	239	44	(	(	PUNCT
ejpam-5914	239	45	x|(y|y))|(x|(y|y	x|(y|y))|(x|(y|y	ADJ
ejpam-5914	239	46	)	)	PUNCT
ejpam-5914	239	47	)	)	PUNCT
ejpam-5914	240	1	∈	∈	PROPN
ejpam-5914	240	2	u(f̃ℓ	u(f̃ℓ	PROPN
ejpam-5914	240	3	;	;	PUNCT
ejpam-5914	240	4	t	t	PROPN
ejpam-5914	240	5	)	)	PUNCT
ejpam-5914	240	6	.	.	PUNCT
ejpam-5914	241	1	hence	hence	ADV
ejpam-5914	241	2	,	,	PUNCT
ejpam-5914	241	3	u(f̃ℓ	u(f̃ℓ	PROPN
ejpam-5914	241	4	;	;	PUNCT
ejpam-5914	241	5	t	t	PROPN
ejpam-5914	241	6	)	)	PUNCT
ejpam-5914	241	7	is	be	AUX
ejpam-5914	241	8	a	a	DET
ejpam-5914	241	9	subalgebra	subalgebra	NOUN
ejpam-5914	241	10	of	of	ADP
ejpam-5914	241	11	a.	a.	NOUN
ejpam-5914	241	12	conversely	conversely	ADV
ejpam-5914	241	13	,	,	PUNCT
ejpam-5914	241	14	assume	assume	VERB
ejpam-5914	241	15	that	that	SCONJ
ejpam-5914	241	16	for	for	ADP
ejpam-5914	241	17	all	all	DET
ejpam-5914	241	18	t	t	NOUN
ejpam-5914	241	19	∈	∈	PROPN
ejpam-5914	242	1	[	[	X
ejpam-5914	242	2	0	0	NUM
ejpam-5914	242	3	,	,	PUNCT
ejpam-5914	242	4	1	1	NUM
ejpam-5914	242	5	]	]	PUNCT
ejpam-5914	242	6	,	,	PUNCT
ejpam-5914	242	7	the	the	DET
ejpam-5914	242	8	set	set	NOUN
ejpam-5914	242	9	u(f̃ℓ	u(f̃ℓ	PROPN
ejpam-5914	242	10	;	;	PUNCT
ejpam-5914	242	11	t	t	PROPN
ejpam-5914	242	12	)	)	PUNCT
ejpam-5914	242	13	is	be	AUX
ejpam-5914	242	14	a	a	DET
ejpam-5914	242	15	subalgebra	subalgebra	NOUN
ejpam-5914	242	16	of	of	ADP
ejpam-5914	242	17	a	a	PRON
ejpam-5914	242	18	if	if	SCONJ
ejpam-5914	242	19	u(f̃ℓ	u(f̃ℓ	PROPN
ejpam-5914	242	20	;	;	PUNCT
ejpam-5914	242	21	t	t	X
ejpam-5914	242	22	)	)	PUNCT
ejpam-5914	242	23	̸=	̸=	PROPN
ejpam-5914	242	24	∅.	∅.	ADV
ejpam-5914	242	25	let	let	VERB
ejpam-5914	242	26	x	x	PRON
ejpam-5914	242	27	,	,	PUNCT
ejpam-5914	242	28	y	y	PROPN
ejpam-5914	242	29	∈	∈	PROPN
ejpam-5914	242	30	a.	a.	NOUN
ejpam-5914	242	31	then	then	ADV
ejpam-5914	242	32	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	242	33	)	)	PUNCT
ejpam-5914	242	34	,	,	PUNCT
ejpam-5914	242	35	f̃ℓ(y	f̃ℓ(y	X
ejpam-5914	242	36	)	)	PUNCT
ejpam-5914	242	37	∈	∈	NOUN
ejpam-5914	243	1	[	[	X
ejpam-5914	243	2	0	0	NUM
ejpam-5914	243	3	,	,	PUNCT
ejpam-5914	243	4	1	1	NUM
ejpam-5914	243	5	]	]	PUNCT
ejpam-5914	243	6	.	.	PUNCT
ejpam-5914	244	1	if	if	SCONJ
ejpam-5914	244	2	we	we	PRON
ejpam-5914	244	3	take	take	VERB
ejpam-5914	244	4	t	t	NOUN
ejpam-5914	244	5	=	=	SYM
ejpam-5914	244	6	min{f̃ℓ(x	min{f̃ℓ(x	PROPN
ejpam-5914	244	7	)	)	PUNCT
ejpam-5914	244	8	,	,	PUNCT
ejpam-5914	244	9	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	244	10	)	)	PUNCT
ejpam-5914	244	11	}	}	PUNCT
ejpam-5914	244	12	,	,	PUNCT
ejpam-5914	244	13	then	then	ADV
ejpam-5914	244	14	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	244	15	)	)	PUNCT
ejpam-5914	244	16	≥	≥	NOUN
ejpam-5914	244	17	t	t	NOUN
ejpam-5914	244	18	and	and	CCONJ
ejpam-5914	244	19	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	244	20	)	)	PUNCT
ejpam-5914	244	21	≥	≥	NOUN
ejpam-5914	244	22	t.	t.	NOUN
ejpam-5914	244	23	hence	hence	ADV
ejpam-5914	244	24	,	,	PUNCT
ejpam-5914	244	25	x	x	PRON
ejpam-5914	244	26	,	,	PUNCT
ejpam-5914	244	27	y	y	PROPN
ejpam-5914	244	28	∈	∈	PROPN
ejpam-5914	244	29	u(f̃ℓ	u(f̃ℓ	PROPN
ejpam-5914	244	30	;	;	PUNCT
ejpam-5914	244	31	t	t	X
ejpam-5914	244	32	)	)	PUNCT
ejpam-5914	244	33	̸=	̸=	PROPN
ejpam-5914	244	34	∅.	∅.	VERB
ejpam-5914	244	35	by	by	ADP
ejpam-5914	244	36	assumption	assumption	NOUN
ejpam-5914	244	37	,	,	PUNCT
ejpam-5914	244	38	we	we	PRON
ejpam-5914	244	39	have	have	VERB
ejpam-5914	244	40	u(f̃ℓ	u(f̃ℓ	PROPN
ejpam-5914	244	41	;	;	PUNCT
ejpam-5914	244	42	t	t	PROPN
ejpam-5914	244	43	)	)	PUNCT
ejpam-5914	244	44	is	be	AUX
ejpam-5914	244	45	a	a	DET
ejpam-5914	244	46	subalgebra	subalgebra	NOUN
ejpam-5914	244	47	of	of	ADP
ejpam-5914	244	48	a	a	PRON
ejpam-5914	244	49	,	,	PUNCT
ejpam-5914	244	50	and	and	CCONJ
ejpam-5914	244	51	so	so	ADV
ejpam-5914	244	52	(	(	PUNCT
ejpam-5914	244	53	x|(y|y))|(x|(y|y	x|(y|y))|(x|(y|y	ADJ
ejpam-5914	244	54	)	)	PUNCT
ejpam-5914	244	55	)	)	PUNCT
ejpam-5914	245	1	∈	∈	PROPN
ejpam-5914	245	2	u(f̃ℓ	u(f̃ℓ	PROPN
ejpam-5914	245	3	;	;	PUNCT
ejpam-5914	245	4	t	t	PROPN
ejpam-5914	245	5	)	)	PUNCT
ejpam-5914	245	6	.	.	PUNCT
ejpam-5914	246	1	thus	thus	ADV
ejpam-5914	246	2	,	,	PUNCT
ejpam-5914	246	3	f̃ℓ((x|(y|y))|(x|(y|y	f̃ℓ((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	246	4	)	)	PUNCT
ejpam-5914	246	5	)	)	PUNCT
ejpam-5914	246	6	)	)	PUNCT
ejpam-5914	246	7	≥	≥	PROPN
ejpam-5914	246	8	t	t	NOUN
ejpam-5914	246	9	=	=	SYM
ejpam-5914	246	10	min{f̃ℓ(x	min{f̃ℓ(x	PROPN
ejpam-5914	246	11	)	)	PUNCT
ejpam-5914	246	12	,	,	PUNCT
ejpam-5914	246	13	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	246	14	)	)	PUNCT
ejpam-5914	246	15	}	}	PUNCT
ejpam-5914	246	16	.	.	PUNCT
ejpam-5914	247	1	hence	hence	ADV
ejpam-5914	247	2	,	,	PUNCT
ejpam-5914	247	3	(	(	PUNCT
ejpam-5914	247	4	a	a	DET
ejpam-5914	247	5	,	,	PUNCT
ejpam-5914	247	6	f̃ℓ	f̃ℓ	NOUN
ejpam-5914	247	7	)	)	PUNCT
ejpam-5914	247	8	is	be	AUX
ejpam-5914	247	9	a	a	DET
ejpam-5914	247	10	1	1	NUM
ejpam-5914	247	11	-	-	PUNCT
ejpam-5914	247	12	fuzzy	fuzzy	ADJ
ejpam-5914	247	13	subalgebra	subalgebra	NOUN
ejpam-5914	247	14	of	of	ADP
ejpam-5914	247	15	a	a	PRON
ejpam-5914	247	16	,	,	PUNCT
ejpam-5914	247	17	that	that	ADV
ejpam-5914	247	18	is	is	ADV
ejpam-5914	247	19	,	,	PUNCT
ejpam-5914	247	20	(	(	PUNCT
ejpam-5914	247	21	a	a	PRON
ejpam-5914	247	22	,	,	PUNCT
ejpam-5914	247	23	f̃	f̃	PROPN
ejpam-5914	247	24	)	)	PUNCT
ejpam-5914	247	25	is	be	AUX
ejpam-5914	247	26	a	a	DET
ejpam-5914	247	27	length	length	NOUN
ejpam-5914	247	28	1	1	NUM
ejpam-5914	247	29	-	-	PUNCT
ejpam-5914	247	30	fuzzy	fuzzy	ADJ
ejpam-5914	247	31	subalgebra	subalgebra	NOUN
ejpam-5914	247	32	of	of	ADP
ejpam-5914	247	33	a.	a.	NOUN
ejpam-5914	247	34	corollary	corollary	NOUN
ejpam-5914	247	35	1	1	NUM
ejpam-5914	247	36	.	.	PUNCT
ejpam-5914	248	1	if	if	SCONJ
ejpam-5914	248	2	(	(	PUNCT
ejpam-5914	248	3	a	a	PRON
ejpam-5914	248	4	,	,	PUNCT
ejpam-5914	248	5	f̃	f̃	PROPN
ejpam-5914	248	6	)	)	PUNCT
ejpam-5914	248	7	is	be	AUX
ejpam-5914	248	8	a	a	DET
ejpam-5914	248	9	length	length	NOUN
ejpam-5914	248	10	3	3	NUM
ejpam-5914	248	11	-	-	PUNCT
ejpam-5914	248	12	fuzzy	fuzzy	ADJ
ejpam-5914	248	13	subalgebra	subalgebra	NOUN
ejpam-5914	248	14	of	of	ADP
ejpam-5914	248	15	a	a	PRON
ejpam-5914	248	16	,	,	PUNCT
ejpam-5914	248	17	then	then	ADV
ejpam-5914	248	18	the	the	DET
ejpam-5914	248	19	set	set	NOUN
ejpam-5914	248	20	u(f̃ℓ	u(f̃ℓ	PROPN
ejpam-5914	248	21	;	;	PUNCT
ejpam-5914	248	22	t	t	PROPN
ejpam-5914	248	23	)	)	PUNCT
ejpam-5914	248	24	is	be	AUX
ejpam-5914	248	25	a	a	DET
ejpam-5914	248	26	subalgebra	subalgebra	NOUN
ejpam-5914	248	27	of	of	ADP
ejpam-5914	248	28	a	a	PRON
ejpam-5914	248	29	for	for	ADP
ejpam-5914	248	30	all	all	DET
ejpam-5914	248	31	t	t	NOUN
ejpam-5914	248	32	∈	∈	PROPN
ejpam-5914	249	1	[	[	X
ejpam-5914	249	2	0	0	NUM
ejpam-5914	249	3	,	,	PUNCT
ejpam-5914	249	4	1	1	NUM
ejpam-5914	249	5	]	]	PUNCT
ejpam-5914	249	6	with	with	ADP
ejpam-5914	249	7	u(f̃ℓ	u(f̃ℓ	PROPN
ejpam-5914	249	8	;	;	PUNCT
ejpam-5914	249	9	t	t	X
ejpam-5914	249	10	)	)	PUNCT
ejpam-5914	249	11	̸=	̸=	PROPN
ejpam-5914	249	12	∅.	∅.	ADP
ejpam-5914	249	13	n.	n.	PROPN
ejpam-5914	249	14	rajesh	rajesh	PROPN
ejpam-5914	249	15	et	et	PROPN
ejpam-5914	249	16	al	al	PROPN
ejpam-5914	249	17	.	.	PUNCT
ejpam-5914	249	18	/	/	SYM
ejpam-5914	249	19	eur	eur	PROPN
ejpam-5914	249	20	.	.	PUNCT
ejpam-5914	250	1	j.	j.	PROPN
ejpam-5914	250	2	pure	pure	PROPN
ejpam-5914	250	3	appl	appl	PROPN
ejpam-5914	250	4	.	.	PROPN
ejpam-5914	250	5	math	math	PROPN
ejpam-5914	250	6	,	,	PUNCT
ejpam-5914	250	7	18	18	NUM
ejpam-5914	250	8	(	(	PUNCT
ejpam-5914	250	9	2	2	NUM
ejpam-5914	250	10	)	)	PUNCT
ejpam-5914	250	11	(	(	PUNCT
ejpam-5914	250	12	2025	2025	NUM
ejpam-5914	250	13	)	)	PUNCT
ejpam-5914	250	14	,	,	PUNCT
ejpam-5914	250	15	5914	5914	NUM
ejpam-5914	250	16	9	9	NUM
ejpam-5914	250	17	of	of	ADP
ejpam-5914	250	18	21	21	NUM
ejpam-5914	250	19	proof	proof	NOUN
ejpam-5914	250	20	.	.	PUNCT
ejpam-5914	251	1	it	it	PRON
ejpam-5914	251	2	is	be	AUX
ejpam-5914	251	3	straightforward	straightforward	ADJ
ejpam-5914	251	4	by	by	ADP
ejpam-5914	251	5	theorems	theorem	NOUN
ejpam-5914	251	6	1	1	NUM
ejpam-5914	251	7	and	and	CCONJ
ejpam-5914	251	8	5	5	NUM
ejpam-5914	251	9	.	.	X
ejpam-5914	251	10	theorem	theorem	VERB
ejpam-5914	251	11	6	6	NUM
ejpam-5914	251	12	.	.	PUNCT
ejpam-5914	252	1	an	an	DET
ejpam-5914	252	2	interval	interval	NOUN
ejpam-5914	252	3	-	-	PUNCT
ejpam-5914	252	4	valued	value	VERB
ejpam-5914	252	5	fuzzy	fuzzy	ADJ
ejpam-5914	252	6	structure	structure	NOUN
ejpam-5914	252	7	(	(	PUNCT
ejpam-5914	252	8	a	a	PRON
ejpam-5914	252	9	,	,	PUNCT
ejpam-5914	252	10	f̃	f̃	PROPN
ejpam-5914	252	11	)	)	PUNCT
ejpam-5914	252	12	over	over	ADP
ejpam-5914	252	13	a	a	PRON
ejpam-5914	252	14	is	be	AUX
ejpam-5914	252	15	a	a	DET
ejpam-5914	252	16	length	length	NOUN
ejpam-5914	252	17	4	4	NUM
ejpam-5914	252	18	-	-	PUNCT
ejpam-5914	252	19	fuzzy	fuzzy	ADJ
ejpam-5914	252	20	subalgebra	subalgebra	NOUN
ejpam-5914	252	21	of	of	ADP
ejpam-5914	252	22	a	a	DET
ejpam-5914	252	23	if	if	NOUN
ejpam-5914	252	24	and	and	CCONJ
ejpam-5914	252	25	only	only	ADV
ejpam-5914	252	26	if	if	SCONJ
ejpam-5914	252	27	the	the	DET
ejpam-5914	252	28	set	set	NOUN
ejpam-5914	252	29	l(f̃ℓ	l(f̃ℓ	NOUN
ejpam-5914	252	30	;	;	PUNCT
ejpam-5914	252	31	t	t	PROPN
ejpam-5914	252	32	)	)	PUNCT
ejpam-5914	252	33	is	be	AUX
ejpam-5914	252	34	a	a	DET
ejpam-5914	252	35	subalgebra	subalgebra	NOUN
ejpam-5914	252	36	of	of	ADP
ejpam-5914	252	37	a	a	PRON
ejpam-5914	252	38	for	for	ADP
ejpam-5914	252	39	all	all	DET
ejpam-5914	252	40	t	t	NOUN
ejpam-5914	252	41	∈	∈	PROPN
ejpam-5914	253	1	[	[	X
ejpam-5914	253	2	0	0	NUM
ejpam-5914	253	3	,	,	PUNCT
ejpam-5914	253	4	1	1	NUM
ejpam-5914	253	5	]	]	PUNCT
ejpam-5914	253	6	with	with	ADP
ejpam-5914	253	7	l(f̃ℓ	l(f̃ℓ	NOUN
ejpam-5914	253	8	;	;	PUNCT
ejpam-5914	253	9	t	t	X
ejpam-5914	253	10	)	)	PUNCT
ejpam-5914	253	11	̸=	̸=	PROPN
ejpam-5914	253	12	∅.	∅.	ADP
ejpam-5914	253	13	proof	proof	NOUN
ejpam-5914	253	14	.	.	PUNCT
ejpam-5914	254	1	assume	assume	VERB
ejpam-5914	254	2	that	that	SCONJ
ejpam-5914	254	3	(	(	PUNCT
ejpam-5914	254	4	a	a	PRON
ejpam-5914	254	5	,	,	PUNCT
ejpam-5914	254	6	f̃	f̃	PROPN
ejpam-5914	254	7	)	)	PUNCT
ejpam-5914	254	8	is	be	AUX
ejpam-5914	254	9	a	a	DET
ejpam-5914	254	10	length	length	NOUN
ejpam-5914	254	11	4	4	NUM
ejpam-5914	254	12	-	-	PUNCT
ejpam-5914	254	13	fuzzy	fuzzy	ADJ
ejpam-5914	254	14	subalgebra	subalgebra	NOUN
ejpam-5914	254	15	of	of	ADP
ejpam-5914	254	16	a.	a.	NOUN
ejpam-5914	254	17	let	let	VERB
ejpam-5914	254	18	t	t	PROPN
ejpam-5914	254	19	∈	∈	PROPN
ejpam-5914	255	1	[	[	X
ejpam-5914	255	2	0	0	NUM
ejpam-5914	255	3	,	,	PUNCT
ejpam-5914	255	4	1	1	NUM
ejpam-5914	255	5	]	]	PUNCT
ejpam-5914	255	6	be	be	AUX
ejpam-5914	255	7	such	such	ADJ
ejpam-5914	255	8	that	that	SCONJ
ejpam-5914	255	9	l(f̃ℓ	l(f̃ℓ	NOUN
ejpam-5914	255	10	;	;	PUNCT
ejpam-5914	255	11	t	t	X
ejpam-5914	255	12	)	)	PUNCT
ejpam-5914	255	13	̸=	̸=	PROPN
ejpam-5914	255	14	∅	∅	NOUN
ejpam-5914	255	15	and	and	CCONJ
ejpam-5914	255	16	let	let	VERB
ejpam-5914	255	17	x	x	PRON
ejpam-5914	255	18	,	,	PUNCT
ejpam-5914	255	19	y	y	PROPN
ejpam-5914	255	20	∈	∈	PROPN
ejpam-5914	255	21	l(f̃ℓ	l(f̃ℓ	PROPN
ejpam-5914	255	22	;	;	PUNCT
ejpam-5914	255	23	t	t	PROPN
ejpam-5914	255	24	)	)	PUNCT
ejpam-5914	255	25	.	.	PUNCT
ejpam-5914	256	1	then	then	ADV
ejpam-5914	256	2	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	256	3	)	)	PUNCT
ejpam-5914	256	4	≤	≤	NOUN
ejpam-5914	256	5	t	t	NOUN
ejpam-5914	256	6	and	and	CCONJ
ejpam-5914	256	7	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	256	8	)	)	PUNCT
ejpam-5914	256	9	≤	≤	NOUN
ejpam-5914	256	10	t.	t.	NOUN
ejpam-5914	256	11	since	since	SCONJ
ejpam-5914	256	12	(	(	PUNCT
ejpam-5914	256	13	a	a	PRON
ejpam-5914	256	14	,	,	PUNCT
ejpam-5914	256	15	f̃	f̃	PROPN
ejpam-5914	256	16	)	)	PUNCT
ejpam-5914	256	17	is	be	AUX
ejpam-5914	256	18	a	a	DET
ejpam-5914	256	19	length	length	NOUN
ejpam-5914	256	20	4	4	NUM
ejpam-5914	256	21	-	-	PUNCT
ejpam-5914	256	22	fuzzy	fuzzy	ADJ
ejpam-5914	256	23	subalgebra	subalgebra	NOUN
ejpam-5914	256	24	of	of	ADP
ejpam-5914	256	25	a	a	PRON
ejpam-5914	256	26	,	,	PUNCT
ejpam-5914	256	27	we	we	PRON
ejpam-5914	256	28	have	have	VERB
ejpam-5914	256	29	f̃ℓ((x|(y|y))|(x|(y|y	f̃ℓ((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	256	30	)	)	PUNCT
ejpam-5914	256	31	)	)	PUNCT
ejpam-5914	256	32	)	)	PUNCT
ejpam-5914	257	1	≤	≤	NUM
ejpam-5914	257	2	max{f̃ℓ(x	max{f̃ℓ(x	PROPN
ejpam-5914	257	3	)	)	PUNCT
ejpam-5914	257	4	,	,	PUNCT
ejpam-5914	257	5	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	257	6	)	)	PUNCT
ejpam-5914	257	7	}	}	PUNCT
ejpam-5914	257	8	≤	≤	NOUN
ejpam-5914	257	9	t.	t.	NOUN
ejpam-5914	257	10	thus	thus	ADV
ejpam-5914	257	11	,	,	PUNCT
ejpam-5914	257	12	(	(	PUNCT
ejpam-5914	257	13	x|(y|y))|(x|(y|y	x|(y|y))|(x|(y|y	ADJ
ejpam-5914	257	14	)	)	PUNCT
ejpam-5914	257	15	)	)	PUNCT
ejpam-5914	258	1	∈	∈	PROPN
ejpam-5914	258	2	l(f̃ℓ	l(f̃ℓ	NOUN
ejpam-5914	258	3	;	;	PUNCT
ejpam-5914	258	4	t	t	PROPN
ejpam-5914	258	5	)	)	PUNCT
ejpam-5914	258	6	.	.	PUNCT
ejpam-5914	259	1	hence	hence	ADV
ejpam-5914	259	2	,	,	PUNCT
ejpam-5914	259	3	l(f̃ℓ	l(f̃ℓ	PROPN
ejpam-5914	259	4	;	;	PUNCT
ejpam-5914	259	5	t	t	PROPN
ejpam-5914	259	6	)	)	PUNCT
ejpam-5914	259	7	is	be	AUX
ejpam-5914	259	8	a	a	DET
ejpam-5914	259	9	subalgebra	subalgebra	NOUN
ejpam-5914	259	10	of	of	ADP
ejpam-5914	259	11	a.	a.	NOUN
ejpam-5914	259	12	conversely	conversely	ADV
ejpam-5914	259	13	,	,	PUNCT
ejpam-5914	259	14	assume	assume	VERB
ejpam-5914	259	15	that	that	SCONJ
ejpam-5914	259	16	for	for	ADP
ejpam-5914	259	17	all	all	DET
ejpam-5914	259	18	t	t	NOUN
ejpam-5914	259	19	∈	∈	PROPN
ejpam-5914	260	1	[	[	X
ejpam-5914	260	2	0	0	NUM
ejpam-5914	260	3	,	,	PUNCT
ejpam-5914	260	4	1	1	NUM
ejpam-5914	260	5	]	]	PUNCT
ejpam-5914	260	6	,	,	PUNCT
ejpam-5914	260	7	the	the	DET
ejpam-5914	260	8	set	set	NOUN
ejpam-5914	260	9	l(f̃ℓ	l(f̃ℓ	NOUN
ejpam-5914	260	10	;	;	PUNCT
ejpam-5914	260	11	t	t	PROPN
ejpam-5914	260	12	)	)	PUNCT
ejpam-5914	260	13	is	be	AUX
ejpam-5914	260	14	a	a	DET
ejpam-5914	260	15	subalgebra	subalgebra	NOUN
ejpam-5914	260	16	of	of	ADP
ejpam-5914	260	17	a	a	DET
ejpam-5914	260	18	if	if	SCONJ
ejpam-5914	260	19	l(f̃ℓ	l(f̃ℓ	NOUN
ejpam-5914	260	20	;	;	PUNCT
ejpam-5914	260	21	t	t	X
ejpam-5914	260	22	)	)	PUNCT
ejpam-5914	260	23	̸=	̸=	PROPN
ejpam-5914	260	24	∅.	∅.	ADV
ejpam-5914	260	25	let	let	VERB
ejpam-5914	260	26	x	x	PRON
ejpam-5914	260	27	,	,	PUNCT
ejpam-5914	260	28	y	y	PROPN
ejpam-5914	260	29	∈	∈	PROPN
ejpam-5914	260	30	a.	a.	NOUN
ejpam-5914	260	31	then	then	ADV
ejpam-5914	260	32	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	260	33	)	)	PUNCT
ejpam-5914	260	34	,	,	PUNCT
ejpam-5914	260	35	f̃ℓ(y	f̃ℓ(y	X
ejpam-5914	260	36	)	)	PUNCT
ejpam-5914	260	37	∈	∈	NOUN
ejpam-5914	261	1	[	[	X
ejpam-5914	261	2	0	0	NUM
ejpam-5914	261	3	,	,	PUNCT
ejpam-5914	261	4	1	1	NUM
ejpam-5914	261	5	]	]	PUNCT
ejpam-5914	261	6	.	.	PUNCT
ejpam-5914	262	1	if	if	SCONJ
ejpam-5914	262	2	we	we	PRON
ejpam-5914	262	3	take	take	VERB
ejpam-5914	262	4	t	t	NOUN
ejpam-5914	262	5	=	=	SYM
ejpam-5914	262	6	max{f̃ℓ(x	max{f̃ℓ(x	PROPN
ejpam-5914	262	7	)	)	PUNCT
ejpam-5914	262	8	,	,	PUNCT
ejpam-5914	262	9	f̃ℓ(y	f̃ℓ(y	PROPN
ejpam-5914	262	10	)	)	PUNCT
ejpam-5914	262	11	}	}	PUNCT
ejpam-5914	262	12	.	.	PUNCT
ejpam-5914	263	1	thus	thus	ADV
ejpam-5914	263	2	,	,	PUNCT
ejpam-5914	263	3	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	263	4	)	)	PUNCT
ejpam-5914	263	5	≤	≤	NOUN
ejpam-5914	263	6	t	t	NOUN
ejpam-5914	263	7	and	and	CCONJ
ejpam-5914	263	8	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	263	9	)	)	PUNCT
ejpam-5914	263	10	≤	≤	NOUN
ejpam-5914	263	11	t	t	PROPN
ejpam-5914	263	12	,	,	PUNCT
ejpam-5914	263	13	and	and	CCONJ
ejpam-5914	263	14	so	so	ADV
ejpam-5914	263	15	x	x	NOUN
ejpam-5914	263	16	,	,	PUNCT
ejpam-5914	263	17	y	y	PROPN
ejpam-5914	263	18	∈	∈	PROPN
ejpam-5914	263	19	l(f̃ℓ	l(f̃ℓ	PROPN
ejpam-5914	263	20	;	;	PUNCT
ejpam-5914	263	21	t	t	X
ejpam-5914	263	22	)	)	PUNCT
ejpam-5914	263	23	̸=	̸=	PROPN
ejpam-5914	263	24	∅.	∅.	VERB
ejpam-5914	263	25	by	by	ADP
ejpam-5914	263	26	assumption	assumption	NOUN
ejpam-5914	263	27	,	,	PUNCT
ejpam-5914	263	28	l(f̃ℓ	l(f̃ℓ	PROPN
ejpam-5914	263	29	;	;	PUNCT
ejpam-5914	263	30	t	t	PROPN
ejpam-5914	263	31	)	)	PUNCT
ejpam-5914	263	32	is	be	AUX
ejpam-5914	263	33	a	a	DET
ejpam-5914	263	34	subalgebra	subalgebra	NOUN
ejpam-5914	263	35	of	of	ADP
ejpam-5914	263	36	a	a	PRON
ejpam-5914	263	37	,	,	PUNCT
ejpam-5914	263	38	and	and	CCONJ
ejpam-5914	263	39	so	so	ADV
ejpam-5914	263	40	(	(	PUNCT
ejpam-5914	263	41	x|(y|y))|(x|(y|y	x|(y|y))|(x|(y|y	ADJ
ejpam-5914	263	42	)	)	PUNCT
ejpam-5914	263	43	)	)	PUNCT
ejpam-5914	264	1	∈	∈	PROPN
ejpam-5914	264	2	l(f̃ℓ	l(f̃ℓ	NOUN
ejpam-5914	264	3	;	;	PUNCT
ejpam-5914	264	4	t	t	PROPN
ejpam-5914	264	5	)	)	PUNCT
ejpam-5914	264	6	.	.	PUNCT
ejpam-5914	265	1	thus	thus	ADV
ejpam-5914	265	2	,	,	PUNCT
ejpam-5914	265	3	f̃ℓ((x|(y|y))|(x|(y|y	f̃ℓ((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	265	4	)	)	PUNCT
ejpam-5914	265	5	)	)	PUNCT
ejpam-5914	265	6	)	)	PUNCT
ejpam-5914	265	7	≤	≤	NUM
ejpam-5914	265	8	t	t	NOUN
ejpam-5914	265	9	=	=	SYM
ejpam-5914	265	10	max{f̃ℓ(x	max{f̃ℓ(x	PROPN
ejpam-5914	265	11	)	)	PUNCT
ejpam-5914	265	12	,	,	PUNCT
ejpam-5914	265	13	f̃ℓ(y	f̃ℓ(y	PROPN
ejpam-5914	265	14	)	)	PUNCT
ejpam-5914	265	15	}	}	PUNCT
ejpam-5914	265	16	.	.	PUNCT
ejpam-5914	266	1	hence	hence	ADV
ejpam-5914	266	2	,	,	PUNCT
ejpam-5914	266	3	(	(	PUNCT
ejpam-5914	266	4	a	a	DET
ejpam-5914	266	5	,	,	PUNCT
ejpam-5914	266	6	f̃ℓ	f̃ℓ	NOUN
ejpam-5914	266	7	)	)	PUNCT
ejpam-5914	266	8	is	be	AUX
ejpam-5914	266	9	a	a	DET
ejpam-5914	266	10	4	4	NUM
ejpam-5914	266	11	-	-	PUNCT
ejpam-5914	266	12	fuzzy	fuzzy	ADJ
ejpam-5914	266	13	subalgebra	subalgebra	NOUN
ejpam-5914	266	14	of	of	ADP
ejpam-5914	266	15	a	a	PRON
ejpam-5914	266	16	,	,	PUNCT
ejpam-5914	266	17	that	that	ADV
ejpam-5914	266	18	is	is	ADV
ejpam-5914	266	19	,	,	PUNCT
ejpam-5914	266	20	(	(	PUNCT
ejpam-5914	266	21	a	a	PRON
ejpam-5914	266	22	,	,	PUNCT
ejpam-5914	266	23	f̃	f̃	PROPN
ejpam-5914	266	24	)	)	PUNCT
ejpam-5914	266	25	is	be	AUX
ejpam-5914	266	26	a	a	DET
ejpam-5914	266	27	length	length	NOUN
ejpam-5914	266	28	4	4	NUM
ejpam-5914	266	29	-	-	PUNCT
ejpam-5914	266	30	fuzzy	fuzzy	ADJ
ejpam-5914	266	31	subalgebra	subalgebra	NOUN
ejpam-5914	266	32	of	of	ADP
ejpam-5914	266	33	a.	a.	NOUN
ejpam-5914	266	34	corollary	corollary	NOUN
ejpam-5914	266	35	2	2	NUM
ejpam-5914	266	36	.	.	PUNCT
ejpam-5914	267	1	if	if	SCONJ
ejpam-5914	267	2	(	(	PUNCT
ejpam-5914	267	3	a	a	PRON
ejpam-5914	267	4	,	,	PUNCT
ejpam-5914	267	5	f̃	f̃	PROPN
ejpam-5914	267	6	)	)	PUNCT
ejpam-5914	267	7	is	be	AUX
ejpam-5914	267	8	a	a	DET
ejpam-5914	267	9	length	length	NOUN
ejpam-5914	267	10	2	2	NUM
ejpam-5914	267	11	-	-	PUNCT
ejpam-5914	267	12	fuzzy	fuzzy	ADJ
ejpam-5914	267	13	subalgebra	subalgebra	NOUN
ejpam-5914	267	14	of	of	ADP
ejpam-5914	267	15	a	a	PRON
ejpam-5914	267	16	,	,	PUNCT
ejpam-5914	267	17	then	then	ADV
ejpam-5914	267	18	the	the	DET
ejpam-5914	267	19	set	set	NOUN
ejpam-5914	267	20	l(f̃ℓ	l(f̃ℓ	NOUN
ejpam-5914	267	21	;	;	PUNCT
ejpam-5914	267	22	t	t	PROPN
ejpam-5914	267	23	)	)	PUNCT
ejpam-5914	267	24	is	be	AUX
ejpam-5914	267	25	a	a	DET
ejpam-5914	267	26	subalgebra	subalgebra	NOUN
ejpam-5914	267	27	of	of	ADP
ejpam-5914	267	28	a	a	PRON
ejpam-5914	267	29	for	for	ADP
ejpam-5914	267	30	all	all	DET
ejpam-5914	267	31	t	t	NOUN
ejpam-5914	267	32	∈	∈	PROPN
ejpam-5914	268	1	[	[	X
ejpam-5914	268	2	0	0	NUM
ejpam-5914	268	3	,	,	PUNCT
ejpam-5914	268	4	1	1	NUM
ejpam-5914	268	5	]	]	PUNCT
ejpam-5914	268	6	with	with	ADP
ejpam-5914	268	7	l(f̃ℓ	l(f̃ℓ	NOUN
ejpam-5914	268	8	;	;	PUNCT
ejpam-5914	268	9	t	t	X
ejpam-5914	268	10	)	)	PUNCT
ejpam-5914	268	11	̸=	̸=	PROPN
ejpam-5914	268	12	∅.	∅.	ADP
ejpam-5914	268	13	proof	proof	NOUN
ejpam-5914	268	14	.	.	PUNCT
ejpam-5914	269	1	it	it	PRON
ejpam-5914	269	2	is	be	AUX
ejpam-5914	269	3	straightforward	straightforward	ADJ
ejpam-5914	269	4	by	by	ADP
ejpam-5914	269	5	theorems	theorem	NOUN
ejpam-5914	269	6	2	2	NUM
ejpam-5914	269	7	and	and	CCONJ
ejpam-5914	269	8	6	6	NUM
ejpam-5914	269	9	.	.	PUNCT
ejpam-5914	269	10	theorem	theorem	VERB
ejpam-5914	269	11	7	7	NUM
ejpam-5914	269	12	.	.	PUNCT
ejpam-5914	270	1	if	if	SCONJ
ejpam-5914	270	2	(	(	PUNCT
ejpam-5914	270	3	a	a	PRON
ejpam-5914	270	4	,	,	PUNCT
ejpam-5914	270	5	f̃	f̃	PROPN
ejpam-5914	270	6	)	)	PUNCT
ejpam-5914	270	7	is	be	AUX
ejpam-5914	270	8	a	a	DET
ejpam-5914	270	9	length	length	NOUN
ejpam-5914	270	10	2	2	NUM
ejpam-5914	270	11	-	-	PUNCT
ejpam-5914	270	12	fuzzy	fuzzy	ADJ
ejpam-5914	270	13	subalgebra	subalgebra	NOUN
ejpam-5914	270	14	of	of	ADP
ejpam-5914	270	15	a	a	DET
ejpam-5914	270	16	,	,	PUNCT
ejpam-5914	270	17	then	then	ADV
ejpam-5914	270	18	uℓ(f̃	uℓ(f̃	PROPN
ejpam-5914	270	19	;	;	PUNCT
ejpam-5914	270	20	t	t	X
ejpam-5914	270	21	)	)	PUNCT
ejpam-5914	270	22	c	c	PROPN
ejpam-5914	270	23	is	be	AUX
ejpam-5914	270	24	a	a	DET
ejpam-5914	270	25	subalgebra	subalgebra	NOUN
ejpam-5914	270	26	of	of	ADP
ejpam-5914	270	27	a	a	PRON
ejpam-5914	270	28	for	for	ADP
ejpam-5914	270	29	all	all	DET
ejpam-5914	270	30	t	t	NOUN
ejpam-5914	270	31	∈	∈	PROPN
ejpam-5914	271	1	[	[	X
ejpam-5914	271	2	0	0	NUM
ejpam-5914	271	3	,	,	PUNCT
ejpam-5914	271	4	1	1	NUM
ejpam-5914	271	5	]	]	PUNCT
ejpam-5914	271	6	with	with	ADP
ejpam-5914	271	7	uℓ(f̃	uℓ(f̃	PROPN
ejpam-5914	271	8	;	;	PUNCT
ejpam-5914	271	9	t	t	X
ejpam-5914	271	10	)	)	PUNCT
ejpam-5914	271	11	c	c	PROPN
ejpam-5914	272	1	̸=	̸=	PROPN
ejpam-5914	272	2	∅.	∅.	ADP
ejpam-5914	272	3	proof	proof	NOUN
ejpam-5914	272	4	.	.	PUNCT
ejpam-5914	273	1	assume	assume	VERB
ejpam-5914	273	2	that	that	SCONJ
ejpam-5914	273	3	(	(	PUNCT
ejpam-5914	273	4	a	a	PRON
ejpam-5914	273	5	,	,	PUNCT
ejpam-5914	273	6	f̃	f̃	PROPN
ejpam-5914	273	7	)	)	PUNCT
ejpam-5914	273	8	is	be	AUX
ejpam-5914	273	9	a	a	DET
ejpam-5914	273	10	length	length	NOUN
ejpam-5914	273	11	2	2	NUM
ejpam-5914	273	12	-	-	PUNCT
ejpam-5914	273	13	fuzzy	fuzzy	ADJ
ejpam-5914	273	14	subalgebra	subalgebra	NOUN
ejpam-5914	273	15	of	of	ADP
ejpam-5914	273	16	a	a	PRON
ejpam-5914	273	17	and	and	CCONJ
ejpam-5914	273	18	let	let	VERB
ejpam-5914	273	19	x	x	PRON
ejpam-5914	273	20	,	,	PUNCT
ejpam-5914	273	21	y	y	PROPN
ejpam-5914	273	22	∈	∈	PROPN
ejpam-5914	273	23	a	a	DET
ejpam-5914	273	24	be	be	AUX
ejpam-5914	273	25	such	such	ADJ
ejpam-5914	273	26	that	that	SCONJ
ejpam-5914	273	27	x	x	SYM
ejpam-5914	273	28	∈	∈	PROPN
ejpam-5914	273	29	uℓ(f̃	uℓ(f̃	PROPN
ejpam-5914	273	30	;	;	PUNCT
ejpam-5914	273	31	t	t	X
ejpam-5914	273	32	)	)	PUNCT
ejpam-5914	273	33	c	c	PROPN
ejpam-5914	273	34	and	and	CCONJ
ejpam-5914	273	35	y	y	PROPN
ejpam-5914	273	36	∈	∈	PROPN
ejpam-5914	273	37	uℓ(f̃	uℓ(f̃	PROPN
ejpam-5914	273	38	;	;	PUNCT
ejpam-5914	273	39	t	t	X
ejpam-5914	273	40	)	)	PUNCT
ejpam-5914	273	41	c.	c.	NOUN
ejpam-5914	273	42	this	this	PRON
ejpam-5914	273	43	shows	show	VERB
ejpam-5914	273	44	that	that	SCONJ
ejpam-5914	273	45	f̃ℓ(x	f̃ℓ(x	NOUN
ejpam-5914	273	46	)	)	PUNCT
ejpam-5914	273	47	<	<	X
ejpam-5914	273	48	t	t	NOUN
ejpam-5914	273	49	and	and	CCONJ
ejpam-5914	273	50	f̃ℓ(y	f̃ℓ(y	NUM
ejpam-5914	273	51	)	)	PUNCT
ejpam-5914	273	52	<	<	X
ejpam-5914	274	1	t.	t.	X
ejpam-5914	274	2	this	this	PRON
ejpam-5914	274	3	implies	imply	VERB
ejpam-5914	274	4	that	that	SCONJ
ejpam-5914	274	5	f̃ℓ((x|(y|y))|(x|(y|y	f̃ℓ((x|(y|y))|(x|(y|y	PROPN
ejpam-5914	274	6	)	)	PUNCT
ejpam-5914	274	7	)	)	PUNCT
ejpam-5914	274	8	)	)	PUNCT
ejpam-5914	274	9	≤	≤	NUM
ejpam-5914	274	10	min{f̃ℓ(x	min{f̃ℓ(x	PROPN
ejpam-5914	274	11	)	)	PUNCT
ejpam-5914	274	12	,	,	PUNCT
ejpam-5914	274	13	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	274	14	)	)	PUNCT
ejpam-5914	274	15	}	}	PUNCT
ejpam-5914	274	16	<	<	X
ejpam-5914	274	17	t	t	PROPN
ejpam-5914	274	18	,	,	PUNCT
ejpam-5914	274	19	that	that	ADV
ejpam-5914	274	20	is	is	ADV
ejpam-5914	274	21	,	,	PUNCT
ejpam-5914	274	22	(	(	PUNCT
ejpam-5914	274	23	x|(y|y))|(x|(y|y	x|(y|y))|(x|(y|y	ADJ
ejpam-5914	274	24	)	)	PUNCT
ejpam-5914	274	25	)	)	PUNCT
ejpam-5914	275	1	∈	∈	PROPN
ejpam-5914	275	2	uℓ(f̃	uℓ(f̃	PROPN
ejpam-5914	275	3	;	;	PUNCT
ejpam-5914	275	4	t	t	X
ejpam-5914	275	5	)	)	PUNCT
ejpam-5914	275	6	c.	c.	PROPN
ejpam-5914	275	7	therefore	therefore	ADV
ejpam-5914	275	8	,	,	PUNCT
ejpam-5914	275	9	uℓ(f̃	uℓ(f̃	PROPN
ejpam-5914	275	10	;	;	PUNCT
ejpam-5914	275	11	t	t	X
ejpam-5914	275	12	)	)	PUNCT
ejpam-5914	275	13	c	c	PROPN
ejpam-5914	275	14	is	be	AUX
ejpam-5914	275	15	a	a	DET
ejpam-5914	275	16	subalgebra	subalgebra	NOUN
ejpam-5914	275	17	of	of	ADP
ejpam-5914	275	18	a	a	PRON
ejpam-5914	275	19	for	for	ADP
ejpam-5914	275	20	all	all	DET
ejpam-5914	275	21	t	t	NOUN
ejpam-5914	275	22	∈	∈	PROPN
ejpam-5914	276	1	[	[	X
ejpam-5914	276	2	0	0	NUM
ejpam-5914	276	3	,	,	PUNCT
ejpam-5914	276	4	1	1	NUM
ejpam-5914	276	5	]	]	PUNCT
ejpam-5914	276	6	with	with	ADP
ejpam-5914	276	7	uℓ(f̃	uℓ(f̃	PROPN
ejpam-5914	276	8	;	;	PUNCT
ejpam-5914	276	9	t	t	X
ejpam-5914	276	10	)	)	PUNCT
ejpam-5914	276	11	c	c	PROPN
ejpam-5914	276	12	̸=	̸=	PROPN
ejpam-5914	276	13	∅.	∅.	ADV
ejpam-5914	276	14	theorem	theorem	VERB
ejpam-5914	276	15	8	8	NUM
ejpam-5914	276	16	.	.	PUNCT
ejpam-5914	277	1	if	if	SCONJ
ejpam-5914	277	2	(	(	PUNCT
ejpam-5914	277	3	a	a	PRON
ejpam-5914	277	4	,	,	PUNCT
ejpam-5914	277	5	f̃	f̃	PROPN
ejpam-5914	277	6	)	)	PUNCT
ejpam-5914	277	7	is	be	AUX
ejpam-5914	277	8	a	a	DET
ejpam-5914	277	9	length	length	NOUN
ejpam-5914	277	10	3	3	NUM
ejpam-5914	277	11	-	-	PUNCT
ejpam-5914	277	12	fuzzy	fuzzy	ADJ
ejpam-5914	277	13	subalgebra	subalgebra	NOUN
ejpam-5914	277	14	of	of	ADP
ejpam-5914	277	15	a	a	DET
ejpam-5914	277	16	,	,	PUNCT
ejpam-5914	277	17	then	then	ADV
ejpam-5914	277	18	lℓ(f̃	lℓ(f̃	NOUN
ejpam-5914	277	19	;	;	PUNCT
ejpam-5914	277	20	t	t	X
ejpam-5914	277	21	)	)	PUNCT
ejpam-5914	277	22	c	c	PROPN
ejpam-5914	277	23	is	be	AUX
ejpam-5914	277	24	a	a	DET
ejpam-5914	277	25	subalgebra	subalgebra	NOUN
ejpam-5914	277	26	of	of	ADP
ejpam-5914	277	27	a	a	PRON
ejpam-5914	277	28	for	for	ADP
ejpam-5914	277	29	all	all	DET
ejpam-5914	277	30	t	t	NOUN
ejpam-5914	277	31	∈	∈	PROPN
ejpam-5914	278	1	[	[	X
ejpam-5914	278	2	0	0	NUM
ejpam-5914	278	3	,	,	PUNCT
ejpam-5914	278	4	1	1	NUM
ejpam-5914	278	5	]	]	PUNCT
ejpam-5914	278	6	with	with	ADP
ejpam-5914	278	7	lℓ(f̃	lℓ(f̃	NOUN
ejpam-5914	278	8	;	;	PUNCT
ejpam-5914	278	9	t	t	X
ejpam-5914	278	10	)	)	PUNCT
ejpam-5914	278	11	c	c	PROPN
ejpam-5914	278	12	̸=	̸=	PROPN
ejpam-5914	278	13	∅.	∅.	PRON
ejpam-5914	278	14	proof	proof	NOUN
ejpam-5914	278	15	.	.	PUNCT
ejpam-5914	279	1	it	it	PRON
ejpam-5914	279	2	is	be	AUX
ejpam-5914	279	3	similar	similar	ADJ
ejpam-5914	279	4	to	to	ADP
ejpam-5914	279	5	the	the	DET
ejpam-5914	279	6	proof	proof	NOUN
ejpam-5914	279	7	of	of	ADP
ejpam-5914	279	8	theorem	theorem	ADJ
ejpam-5914	279	9	7	7	NUM
ejpam-5914	279	10	.	.	PUNCT
ejpam-5914	279	11	theorem	theorem	NOUN
ejpam-5914	279	12	9	9	NUM
ejpam-5914	279	13	.	.	PUNCT
ejpam-5914	280	1	if	if	SCONJ
ejpam-5914	280	2	(	(	PUNCT
ejpam-5914	280	3	a	a	PRON
ejpam-5914	280	4	,	,	PUNCT
ejpam-5914	280	5	f̃	f̃	PROPN
ejpam-5914	280	6	)	)	PUNCT
ejpam-5914	280	7	is	be	AUX
ejpam-5914	280	8	an	an	DET
ejpam-5914	280	9	interval	interval	NOUN
ejpam-5914	280	10	-	-	PUNCT
ejpam-5914	280	11	valued	value	VERB
ejpam-5914	280	12	fuzzy	fuzzy	ADJ
ejpam-5914	280	13	structure	structure	NOUN
ejpam-5914	280	14	over	over	ADP
ejpam-5914	280	15	a	a	PRON
ejpam-5914	280	16	in	in	ADP
ejpam-5914	280	17	which	which	PRON
ejpam-5914	280	18	(	(	PUNCT
ejpam-5914	280	19	a	a	PRON
ejpam-5914	280	20	,	,	PUNCT
ejpam-5914	280	21	f̃inf	f̃inf	ADJ
ejpam-5914	280	22	)	)	PUNCT
ejpam-5914	280	23	is	be	AUX
ejpam-5914	280	24	constant	constant	ADJ
ejpam-5914	280	25	and	and	CCONJ
ejpam-5914	280	26	(	(	PUNCT
ejpam-5914	280	27	a	a	DET
ejpam-5914	280	28	,	,	PUNCT
ejpam-5914	280	29	f̃sup	f̃sup	ADJ
ejpam-5914	280	30	)	)	PUNCT
ejpam-5914	280	31	is	be	AUX
ejpam-5914	280	32	a	a	DET
ejpam-5914	280	33	1	1	NUM
ejpam-5914	280	34	-	-	PUNCT
ejpam-5914	280	35	fuzzy	fuzzy	ADJ
ejpam-5914	280	36	subalgebra	subalgebra	NOUN
ejpam-5914	280	37	of	of	ADP
ejpam-5914	280	38	a	a	PRON
ejpam-5914	280	39	,	,	PUNCT
ejpam-5914	280	40	then	then	ADV
ejpam-5914	280	41	(	(	PUNCT
ejpam-5914	280	42	a	a	PRON
ejpam-5914	280	43	,	,	PUNCT
ejpam-5914	280	44	f̃	f̃	PROPN
ejpam-5914	280	45	)	)	PUNCT
ejpam-5914	280	46	is	be	AUX
ejpam-5914	280	47	a	a	DET
ejpam-5914	280	48	length	length	NOUN
ejpam-5914	280	49	1	1	NUM
ejpam-5914	280	50	-	-	PUNCT
ejpam-5914	280	51	fuzzy	fuzzy	ADJ
ejpam-5914	280	52	subalgebra	subalgebra	NOUN
ejpam-5914	280	53	of	of	ADP
ejpam-5914	280	54	a.	a.	PROPN
ejpam-5914	280	55	n.	n.	PROPN
ejpam-5914	280	56	rajesh	rajesh	PROPN
ejpam-5914	280	57	et	et	PROPN
ejpam-5914	280	58	al	al	PROPN
ejpam-5914	280	59	.	.	PUNCT
ejpam-5914	280	60	/	/	SYM
ejpam-5914	280	61	eur	eur	PROPN
ejpam-5914	280	62	.	.	PUNCT
ejpam-5914	281	1	j.	j.	PROPN
ejpam-5914	281	2	pure	pure	PROPN
ejpam-5914	281	3	appl	appl	PROPN
ejpam-5914	281	4	.	.	PROPN
ejpam-5914	281	5	math	math	PROPN
ejpam-5914	281	6	,	,	PUNCT
ejpam-5914	281	7	18	18	NUM
ejpam-5914	281	8	(	(	PUNCT
ejpam-5914	281	9	2	2	NUM
ejpam-5914	281	10	)	)	PUNCT
ejpam-5914	281	11	(	(	PUNCT
ejpam-5914	281	12	2025	2025	NUM
ejpam-5914	281	13	)	)	PUNCT
ejpam-5914	281	14	,	,	PUNCT
ejpam-5914	281	15	5914	5914	NUM
ejpam-5914	281	16	10	10	NUM
ejpam-5914	281	17	of	of	ADP
ejpam-5914	281	18	21	21	NUM
ejpam-5914	281	19	proof	proof	NOUN
ejpam-5914	281	20	.	.	PUNCT
ejpam-5914	282	1	assume	assume	VERB
ejpam-5914	282	2	that	that	SCONJ
ejpam-5914	282	3	(	(	PUNCT
ejpam-5914	282	4	a	a	PRON
ejpam-5914	282	5	,	,	PUNCT
ejpam-5914	282	6	f̃	f̃	PROPN
ejpam-5914	282	7	)	)	PUNCT
ejpam-5914	282	8	is	be	AUX
ejpam-5914	282	9	an	an	DET
ejpam-5914	282	10	interval	interval	NOUN
ejpam-5914	282	11	-	-	PUNCT
ejpam-5914	282	12	valued	value	VERB
ejpam-5914	282	13	fuzzy	fuzzy	ADJ
ejpam-5914	282	14	structure	structure	NOUN
ejpam-5914	282	15	over	over	ADP
ejpam-5914	282	16	a	a	PRON
ejpam-5914	282	17	in	in	ADP
ejpam-5914	282	18	which	which	PRON
ejpam-5914	282	19	(	(	PUNCT
ejpam-5914	282	20	a	a	PRON
ejpam-5914	282	21	,	,	PUNCT
ejpam-5914	282	22	f̃inf	f̃inf	ADJ
ejpam-5914	282	23	)	)	PUNCT
ejpam-5914	282	24	is	be	AUX
ejpam-5914	282	25	constant	constant	ADJ
ejpam-5914	282	26	and	and	CCONJ
ejpam-5914	282	27	(	(	PUNCT
ejpam-5914	282	28	a	a	DET
ejpam-5914	282	29	,	,	PUNCT
ejpam-5914	282	30	f̃sup	f̃sup	ADJ
ejpam-5914	282	31	)	)	PUNCT
ejpam-5914	282	32	is	be	AUX
ejpam-5914	282	33	a	a	DET
ejpam-5914	282	34	1	1	NUM
ejpam-5914	282	35	-	-	PUNCT
ejpam-5914	282	36	fuzzy	fuzzy	ADJ
ejpam-5914	282	37	subalgebra	subalgebra	NOUN
ejpam-5914	282	38	of	of	ADP
ejpam-5914	282	39	a.	a.	NOUN
ejpam-5914	282	40	let	let	VERB
ejpam-5914	282	41	x	x	PRON
ejpam-5914	282	42	,	,	PUNCT
ejpam-5914	282	43	y	y	PROPN
ejpam-5914	282	44	∈	∈	PROPN
ejpam-5914	282	45	a.	a.	NOUN
ejpam-5914	282	46	since	since	SCONJ
ejpam-5914	282	47	(	(	PUNCT
ejpam-5914	282	48	a	a	PRON
ejpam-5914	282	49	,	,	PUNCT
ejpam-5914	282	50	f̃inf	f̃inf	ADJ
ejpam-5914	282	51	)	)	PUNCT
ejpam-5914	282	52	is	be	AUX
ejpam-5914	282	53	constant	constant	ADJ
ejpam-5914	282	54	,	,	PUNCT
ejpam-5914	282	55	we	we	PRON
ejpam-5914	282	56	have	have	VERB
ejpam-5914	282	57	f̃inf(x	f̃inf(x	NOUN
ejpam-5914	282	58	)	)	PUNCT
ejpam-5914	282	59	=	=	SYM
ejpam-5914	282	60	f̃inf(0	f̃inf(0	PROPN
ejpam-5914	282	61	)	)	PUNCT
ejpam-5914	282	62	for	for	SCONJ
ejpam-5914	282	63	all	all	PRON
ejpam-5914	282	64	x	x	SYM
ejpam-5914	282	65	∈	∈	NOUN
ejpam-5914	282	66	a.	a.	NOUN
ejpam-5914	282	67	since	since	SCONJ
ejpam-5914	282	68	(	(	PUNCT
ejpam-5914	282	69	a	a	DET
ejpam-5914	282	70	,	,	PUNCT
ejpam-5914	282	71	f̃sup	f̃sup	ADJ
ejpam-5914	282	72	)	)	PUNCT
ejpam-5914	282	73	is	be	AUX
ejpam-5914	282	74	a	a	DET
ejpam-5914	282	75	1	1	NUM
ejpam-5914	282	76	-	-	PUNCT
ejpam-5914	282	77	fuzzy	fuzzy	ADJ
ejpam-5914	282	78	subalgebra	subalgebra	NOUN
ejpam-5914	282	79	of	of	ADP
ejpam-5914	282	80	a	a	PRON
ejpam-5914	282	81	,	,	PUNCT
ejpam-5914	282	82	we	we	PRON
ejpam-5914	282	83	have	have	VERB
ejpam-5914	282	84	f̃sup((x|(y|y))|(x|(y|y	f̃sup((x|(y|y))|(x|(y|y	VERB
ejpam-5914	282	85	)	)	PUNCT
ejpam-5914	282	86	)	)	PUNCT
ejpam-5914	282	87	)	)	PUNCT
ejpam-5914	282	88	≥	≥	NOUN
ejpam-5914	282	89	min{f̃sup(x	min{f̃sup(x	PROPN
ejpam-5914	282	90	)	)	PUNCT
ejpam-5914	282	91	,	,	PUNCT
ejpam-5914	282	92	f̃sup(y	f̃sup(y	NOUN
ejpam-5914	282	93	)	)	PUNCT
ejpam-5914	282	94	}	}	PUNCT
ejpam-5914	282	95	.	.	PUNCT
ejpam-5914	283	1	thus	thus	ADV
ejpam-5914	283	2	,	,	PUNCT
ejpam-5914	283	3	f̃ℓ((x|(y|y))|(x|(y|y	f̃ℓ((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	283	4	)	)	PUNCT
ejpam-5914	283	5	)	)	PUNCT
ejpam-5914	283	6	)	)	PUNCT
ejpam-5914	284	1	=	=	SYM
ejpam-5914	284	2	f̃sup((x|(y|y))|(x|(y|y)))−	f̃sup((x|(y|y))|(x|(y|y)))−	NOUN
ejpam-5914	284	3	f̃inf((x|(y|y))|(x|(y|y	f̃inf((x|(y|y))|(x|(y|y	ADJ
ejpam-5914	284	4	)	)	PUNCT
ejpam-5914	284	5	)	)	PUNCT
ejpam-5914	284	6	)	)	PUNCT
ejpam-5914	285	1	=	=	PRON
ejpam-5914	285	2	f̃sup((x|(y|y))|(x|(y|y)))−	f̃sup((x|(y|y))|(x|(y|y)))−	PROPN
ejpam-5914	285	3	f̃inf(0	f̃inf(0	PROPN
ejpam-5914	285	4	)	)	PUNCT
ejpam-5914	285	5	≥	≥	NOUN
ejpam-5914	285	6	min{f̃sup(x	min{f̃sup(x	PROPN
ejpam-5914	285	7	)	)	PUNCT
ejpam-5914	285	8	,	,	PUNCT
ejpam-5914	285	9	f̃sup(y	f̃sup(y	NOUN
ejpam-5914	285	10	)	)	PUNCT
ejpam-5914	285	11	}	}	PUNCT
ejpam-5914	285	12	−	−	PROPN
ejpam-5914	285	13	f̃inf(0	f̃inf(0	PROPN
ejpam-5914	285	14	)	)	PUNCT
ejpam-5914	285	15	=	=	SYM
ejpam-5914	285	16	min{f̃sup(x)−	min{f̃sup(x)−	NOUN
ejpam-5914	285	17	f̃inf(0	f̃inf(0	PROPN
ejpam-5914	285	18	)	)	PUNCT
ejpam-5914	285	19	,	,	PUNCT
ejpam-5914	285	20	f̃sup(y)−	f̃sup(y)−	PROPN
ejpam-5914	285	21	f̃inf(0	f̃inf(0	PROPN
ejpam-5914	285	22	)	)	PUNCT
ejpam-5914	285	23	}	}	PUNCT
ejpam-5914	285	24	=	=	PUNCT
ejpam-5914	285	25	min{f̃sup(x)−	min{f̃sup(x)−	PROPN
ejpam-5914	285	26	f̃inf(x	f̃inf(x	PROPN
ejpam-5914	285	27	)	)	PUNCT
ejpam-5914	285	28	,	,	PUNCT
ejpam-5914	285	29	f̃sup(y)−	f̃sup(y)−	PROPN
ejpam-5914	285	30	f̃inf(y	f̃inf(y	NOUN
ejpam-5914	285	31	)	)	PUNCT
ejpam-5914	285	32	}	}	PUNCT
ejpam-5914	285	33	=	=	SYM
ejpam-5914	285	34	min{f̃ℓ(x	min{f̃ℓ(x	PROPN
ejpam-5914	285	35	)	)	PUNCT
ejpam-5914	285	36	,	,	PUNCT
ejpam-5914	285	37	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	285	38	)	)	PUNCT
ejpam-5914	285	39	}	}	PUNCT
ejpam-5914	285	40	.	.	PUNCT
ejpam-5914	286	1	hence	hence	ADV
ejpam-5914	286	2	,	,	PUNCT
ejpam-5914	286	3	(	(	PUNCT
ejpam-5914	286	4	a	a	DET
ejpam-5914	286	5	,	,	PUNCT
ejpam-5914	286	6	f̃ℓ	f̃ℓ	NOUN
ejpam-5914	286	7	)	)	PUNCT
ejpam-5914	286	8	is	be	AUX
ejpam-5914	286	9	a	a	DET
ejpam-5914	286	10	1	1	NUM
ejpam-5914	286	11	-	-	PUNCT
ejpam-5914	286	12	fuzzy	fuzzy	ADJ
ejpam-5914	286	13	subalgebra	subalgebra	NOUN
ejpam-5914	286	14	of	of	ADP
ejpam-5914	286	15	a	a	PRON
ejpam-5914	286	16	,	,	PUNCT
ejpam-5914	286	17	that	that	ADV
ejpam-5914	286	18	is	is	ADV
ejpam-5914	286	19	,	,	PUNCT
ejpam-5914	286	20	(	(	PUNCT
ejpam-5914	286	21	a	a	PRON
ejpam-5914	286	22	,	,	PUNCT
ejpam-5914	286	23	f̃	f̃	PROPN
ejpam-5914	286	24	)	)	PUNCT
ejpam-5914	286	25	is	be	AUX
ejpam-5914	286	26	a	a	DET
ejpam-5914	286	27	length	length	NOUN
ejpam-5914	286	28	1	1	NUM
ejpam-5914	286	29	-	-	PUNCT
ejpam-5914	286	30	fuzzy	fuzzy	ADJ
ejpam-5914	286	31	subalgebra	subalgebra	NOUN
ejpam-5914	286	32	of	of	ADP
ejpam-5914	286	33	a.	a.	NOUN
ejpam-5914	286	34	theorem	theorem	NOUN
ejpam-5914	286	35	10	10	NUM
ejpam-5914	286	36	.	.	PUNCT
ejpam-5914	287	1	if	if	SCONJ
ejpam-5914	287	2	(	(	PUNCT
ejpam-5914	287	3	a	a	PRON
ejpam-5914	287	4	,	,	PUNCT
ejpam-5914	287	5	f̃	f̃	PROPN
ejpam-5914	287	6	)	)	PUNCT
ejpam-5914	287	7	is	be	AUX
ejpam-5914	287	8	an	an	DET
ejpam-5914	287	9	interval	interval	NOUN
ejpam-5914	287	10	-	-	PUNCT
ejpam-5914	287	11	valued	value	VERB
ejpam-5914	287	12	fuzzy	fuzzy	ADJ
ejpam-5914	287	13	structure	structure	NOUN
ejpam-5914	287	14	over	over	ADP
ejpam-5914	287	15	a	a	PRON
ejpam-5914	287	16	in	in	ADP
ejpam-5914	287	17	which	which	PRON
ejpam-5914	287	18	(	(	PUNCT
ejpam-5914	287	19	a	a	PRON
ejpam-5914	287	20	,	,	PUNCT
ejpam-5914	287	21	f̃inf	f̃inf	ADJ
ejpam-5914	287	22	)	)	PUNCT
ejpam-5914	287	23	is	be	AUX
ejpam-5914	287	24	constant	constant	ADJ
ejpam-5914	287	25	and	and	CCONJ
ejpam-5914	287	26	(	(	PUNCT
ejpam-5914	287	27	a	a	DET
ejpam-5914	287	28	,	,	PUNCT
ejpam-5914	287	29	f̃sup	f̃sup	ADJ
ejpam-5914	287	30	)	)	PUNCT
ejpam-5914	287	31	is	be	AUX
ejpam-5914	287	32	a	a	DET
ejpam-5914	287	33	4	4	NUM
ejpam-5914	287	34	-	-	PUNCT
ejpam-5914	287	35	fuzzy	fuzzy	ADJ
ejpam-5914	287	36	subalgebra	subalgebra	NOUN
ejpam-5914	287	37	of	of	ADP
ejpam-5914	287	38	a	a	PRON
ejpam-5914	287	39	,	,	PUNCT
ejpam-5914	287	40	then	then	ADV
ejpam-5914	287	41	(	(	PUNCT
ejpam-5914	287	42	a	a	PRON
ejpam-5914	287	43	,	,	PUNCT
ejpam-5914	287	44	f̃	f̃	PROPN
ejpam-5914	287	45	)	)	PUNCT
ejpam-5914	287	46	is	be	AUX
ejpam-5914	287	47	a	a	DET
ejpam-5914	287	48	length	length	NOUN
ejpam-5914	287	49	4	4	NUM
ejpam-5914	287	50	-	-	PUNCT
ejpam-5914	287	51	fuzzy	fuzzy	ADJ
ejpam-5914	287	52	subalgebra	subalgebra	NOUN
ejpam-5914	287	53	of	of	ADP
ejpam-5914	287	54	a.	a.	NOUN
ejpam-5914	287	55	proof	proof	NOUN
ejpam-5914	287	56	.	.	PUNCT
ejpam-5914	288	1	assume	assume	VERB
ejpam-5914	288	2	that	that	SCONJ
ejpam-5914	288	3	(	(	PUNCT
ejpam-5914	288	4	a	a	PRON
ejpam-5914	288	5	,	,	PUNCT
ejpam-5914	288	6	f̃	f̃	PROPN
ejpam-5914	288	7	)	)	PUNCT
ejpam-5914	288	8	is	be	AUX
ejpam-5914	288	9	an	an	DET
ejpam-5914	288	10	interval	interval	NOUN
ejpam-5914	288	11	-	-	PUNCT
ejpam-5914	288	12	valued	value	VERB
ejpam-5914	288	13	fuzzy	fuzzy	ADJ
ejpam-5914	288	14	structure	structure	NOUN
ejpam-5914	288	15	over	over	ADP
ejpam-5914	288	16	a	a	PRON
ejpam-5914	288	17	in	in	ADP
ejpam-5914	288	18	which	which	PRON
ejpam-5914	288	19	(	(	PUNCT
ejpam-5914	288	20	a	a	PRON
ejpam-5914	288	21	,	,	PUNCT
ejpam-5914	288	22	f̃inf	f̃inf	ADJ
ejpam-5914	288	23	)	)	PUNCT
ejpam-5914	288	24	is	be	AUX
ejpam-5914	288	25	constant	constant	ADJ
ejpam-5914	288	26	and	and	CCONJ
ejpam-5914	288	27	(	(	PUNCT
ejpam-5914	288	28	a	a	DET
ejpam-5914	288	29	,	,	PUNCT
ejpam-5914	288	30	f̃sup	f̃sup	ADJ
ejpam-5914	288	31	)	)	PUNCT
ejpam-5914	288	32	is	be	AUX
ejpam-5914	288	33	a	a	DET
ejpam-5914	288	34	4	4	NUM
ejpam-5914	288	35	-	-	PUNCT
ejpam-5914	288	36	fuzzy	fuzzy	ADJ
ejpam-5914	288	37	subalgebra	subalgebra	NOUN
ejpam-5914	288	38	of	of	ADP
ejpam-5914	288	39	a.	a.	NOUN
ejpam-5914	288	40	let	let	VERB
ejpam-5914	288	41	x	x	PRON
ejpam-5914	288	42	,	,	PUNCT
ejpam-5914	288	43	y	y	PROPN
ejpam-5914	288	44	∈	∈	PROPN
ejpam-5914	288	45	a.	a.	NOUN
ejpam-5914	288	46	since	since	SCONJ
ejpam-5914	288	47	(	(	PUNCT
ejpam-5914	288	48	a	a	PRON
ejpam-5914	288	49	,	,	PUNCT
ejpam-5914	288	50	f̃inf	f̃inf	ADJ
ejpam-5914	288	51	)	)	PUNCT
ejpam-5914	288	52	is	be	AUX
ejpam-5914	288	53	constant	constant	ADJ
ejpam-5914	288	54	,	,	PUNCT
ejpam-5914	288	55	we	we	PRON
ejpam-5914	288	56	have	have	VERB
ejpam-5914	288	57	f̃inf(x	f̃inf(x	NOUN
ejpam-5914	288	58	)	)	PUNCT
ejpam-5914	288	59	=	=	SYM
ejpam-5914	288	60	f̃inf(0	f̃inf(0	PROPN
ejpam-5914	288	61	)	)	PUNCT
ejpam-5914	288	62	for	for	SCONJ
ejpam-5914	288	63	all	all	PRON
ejpam-5914	288	64	x	x	SYM
ejpam-5914	288	65	∈	∈	NOUN
ejpam-5914	288	66	a.	a.	NOUN
ejpam-5914	288	67	since	since	SCONJ
ejpam-5914	288	68	(	(	PUNCT
ejpam-5914	288	69	a	a	DET
ejpam-5914	288	70	,	,	PUNCT
ejpam-5914	288	71	f̃sup	f̃sup	ADJ
ejpam-5914	288	72	)	)	PUNCT
ejpam-5914	288	73	is	be	AUX
ejpam-5914	288	74	a	a	DET
ejpam-5914	288	75	4	4	NUM
ejpam-5914	288	76	-	-	PUNCT
ejpam-5914	288	77	fuzzy	fuzzy	ADJ
ejpam-5914	288	78	subalgebra	subalgebra	NOUN
ejpam-5914	288	79	of	of	ADP
ejpam-5914	288	80	a	a	PRON
ejpam-5914	288	81	,	,	PUNCT
ejpam-5914	288	82	we	we	PRON
ejpam-5914	288	83	have	have	VERB
ejpam-5914	288	84	f̃sup((x|(y|y))|(x|(y|y	f̃sup((x|(y|y))|(x|(y|y	VERB
ejpam-5914	288	85	)	)	PUNCT
ejpam-5914	288	86	)	)	PUNCT
ejpam-5914	288	87	)	)	PUNCT
ejpam-5914	289	1	≤	≤	PROPN
ejpam-5914	289	2	max{f̃sup(x	max{f̃sup(x	PROPN
ejpam-5914	289	3	)	)	PUNCT
ejpam-5914	289	4	,	,	PUNCT
ejpam-5914	289	5	f̃sup(y	f̃sup(y	NOUN
ejpam-5914	289	6	)	)	PUNCT
ejpam-5914	289	7	}	}	PUNCT
ejpam-5914	289	8	.	.	PUNCT
ejpam-5914	290	1	thus	thus	ADV
ejpam-5914	290	2	,	,	PUNCT
ejpam-5914	290	3	f̃ℓ((x|(y|y))|(x|(y|y	f̃ℓ((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	290	4	)	)	PUNCT
ejpam-5914	290	5	)	)	PUNCT
ejpam-5914	290	6	)	)	PUNCT
ejpam-5914	291	1	=	=	SYM
ejpam-5914	291	2	f̃sup((x|(y|y))|(x|(y|y)))−	f̃sup((x|(y|y))|(x|(y|y)))−	NOUN
ejpam-5914	291	3	f̃inf((x|(y|y))|(x|(y|y	f̃inf((x|(y|y))|(x|(y|y	ADJ
ejpam-5914	291	4	)	)	PUNCT
ejpam-5914	291	5	)	)	PUNCT
ejpam-5914	291	6	)	)	PUNCT
ejpam-5914	292	1	=	=	PRON
ejpam-5914	292	2	f̃sup((x|(y|y))|(x|(y|y)))−	f̃sup((x|(y|y))|(x|(y|y)))−	PROPN
ejpam-5914	292	3	f̃inf(0	f̃inf(0	PROPN
ejpam-5914	292	4	)	)	PUNCT
ejpam-5914	292	5	≤	≤	PUNCT
ejpam-5914	292	6	max{f̃sup(x	max{f̃sup(x	PROPN
ejpam-5914	292	7	)	)	PUNCT
ejpam-5914	292	8	,	,	PUNCT
ejpam-5914	292	9	f̃sup(y	f̃sup(y	NOUN
ejpam-5914	292	10	)	)	PUNCT
ejpam-5914	292	11	}	}	PUNCT
ejpam-5914	292	12	−	−	PROPN
ejpam-5914	292	13	f̃inf(0	f̃inf(0	PROPN
ejpam-5914	292	14	)	)	PUNCT
ejpam-5914	292	15	=	=	PROPN
ejpam-5914	292	16	max{f̃sup(x)−	max{f̃sup(x)−	PROPN
ejpam-5914	292	17	f̃inf(0	f̃inf(0	PROPN
ejpam-5914	292	18	)	)	PUNCT
ejpam-5914	292	19	,	,	PUNCT
ejpam-5914	292	20	f̃sup(y)−	f̃sup(y)−	PROPN
ejpam-5914	292	21	f̃inf(0	f̃inf(0	PROPN
ejpam-5914	292	22	)	)	PUNCT
ejpam-5914	292	23	}	}	PUNCT
ejpam-5914	292	24	=	=	PROPN
ejpam-5914	292	25	max{f̃sup(x)−	max{f̃sup(x)−	PROPN
ejpam-5914	292	26	f̃inf(x	f̃inf(x	PROPN
ejpam-5914	292	27	)	)	PUNCT
ejpam-5914	292	28	,	,	PUNCT
ejpam-5914	292	29	f̃sup(y)−	f̃sup(y)−	PROPN
ejpam-5914	292	30	f̃inf(y	f̃inf(y	NOUN
ejpam-5914	292	31	)	)	PUNCT
ejpam-5914	292	32	}	}	PUNCT
ejpam-5914	292	33	=	=	SYM
ejpam-5914	292	34	max{f̃ℓ(x	max{f̃ℓ(x	PROPN
ejpam-5914	292	35	)	)	PUNCT
ejpam-5914	292	36	,	,	PUNCT
ejpam-5914	292	37	f̃ℓ(y	f̃ℓ(y	PROPN
ejpam-5914	292	38	)	)	PUNCT
ejpam-5914	292	39	}	}	PUNCT
ejpam-5914	292	40	.	.	PUNCT
ejpam-5914	293	1	hence	hence	ADV
ejpam-5914	293	2	,	,	PUNCT
ejpam-5914	293	3	(	(	PUNCT
ejpam-5914	293	4	a	a	DET
ejpam-5914	293	5	,	,	PUNCT
ejpam-5914	293	6	f̃ℓ	f̃ℓ	NOUN
ejpam-5914	293	7	)	)	PUNCT
ejpam-5914	293	8	is	be	AUX
ejpam-5914	293	9	a	a	DET
ejpam-5914	293	10	4	4	NUM
ejpam-5914	293	11	-	-	PUNCT
ejpam-5914	293	12	fuzzy	fuzzy	ADJ
ejpam-5914	293	13	subalgebra	subalgebra	NOUN
ejpam-5914	293	14	of	of	ADP
ejpam-5914	293	15	a	a	PRON
ejpam-5914	293	16	,	,	PUNCT
ejpam-5914	293	17	that	that	ADV
ejpam-5914	293	18	is	is	ADV
ejpam-5914	293	19	,	,	PUNCT
ejpam-5914	293	20	(	(	PUNCT
ejpam-5914	293	21	a	a	PRON
ejpam-5914	293	22	,	,	PUNCT
ejpam-5914	293	23	f̃	f̃	PROPN
ejpam-5914	293	24	)	)	PUNCT
ejpam-5914	293	25	is	be	AUX
ejpam-5914	293	26	a	a	DET
ejpam-5914	293	27	length	length	NOUN
ejpam-5914	293	28	4	4	NUM
ejpam-5914	293	29	-	-	PUNCT
ejpam-5914	293	30	fuzzy	fuzzy	ADJ
ejpam-5914	293	31	subalgebra	subalgebra	NOUN
ejpam-5914	293	32	of	of	ADP
ejpam-5914	293	33	a.	a.	NOUN
ejpam-5914	293	34	theorem	theorem	NOUN
ejpam-5914	293	35	11	11	NUM
ejpam-5914	293	36	.	.	PUNCT
ejpam-5914	294	1	if	if	SCONJ
ejpam-5914	294	2	(	(	PUNCT
ejpam-5914	294	3	a	a	PRON
ejpam-5914	294	4	,	,	PUNCT
ejpam-5914	294	5	f̃	f̃	PROPN
ejpam-5914	294	6	)	)	PUNCT
ejpam-5914	294	7	is	be	AUX
ejpam-5914	294	8	an	an	DET
ejpam-5914	294	9	interval	interval	NOUN
ejpam-5914	294	10	-	-	PUNCT
ejpam-5914	294	11	valued	value	VERB
ejpam-5914	294	12	fuzzy	fuzzy	ADJ
ejpam-5914	294	13	structure	structure	NOUN
ejpam-5914	294	14	over	over	ADP
ejpam-5914	294	15	a	a	DET
ejpam-5914	294	16	in	in	ADP
ejpam-5914	294	17	which	which	PRON
ejpam-5914	294	18	(	(	PUNCT
ejpam-5914	294	19	a	a	DET
ejpam-5914	294	20	,	,	PUNCT
ejpam-5914	294	21	f̃sup	f̃sup	ADJ
ejpam-5914	294	22	)	)	PUNCT
ejpam-5914	294	23	is	be	AUX
ejpam-5914	294	24	constant	constant	ADJ
ejpam-5914	294	25	and	and	CCONJ
ejpam-5914	294	26	(	(	PUNCT
ejpam-5914	294	27	a	a	PRON
ejpam-5914	294	28	,	,	PUNCT
ejpam-5914	294	29	f̃inf	f̃inf	ADJ
ejpam-5914	294	30	)	)	PUNCT
ejpam-5914	294	31	is	be	AUX
ejpam-5914	294	32	a	a	DET
ejpam-5914	294	33	4	4	NUM
ejpam-5914	294	34	-	-	PUNCT
ejpam-5914	294	35	fuzzy	fuzzy	ADJ
ejpam-5914	294	36	subalgebra	subalgebra	NOUN
ejpam-5914	294	37	of	of	ADP
ejpam-5914	294	38	a	a	PRON
ejpam-5914	294	39	,	,	PUNCT
ejpam-5914	294	40	then	then	ADV
ejpam-5914	294	41	(	(	PUNCT
ejpam-5914	294	42	a	a	PRON
ejpam-5914	294	43	,	,	PUNCT
ejpam-5914	294	44	f̃	f̃	PROPN
ejpam-5914	294	45	)	)	PUNCT
ejpam-5914	294	46	is	be	AUX
ejpam-5914	294	47	a	a	DET
ejpam-5914	294	48	length	length	NOUN
ejpam-5914	294	49	1	1	NUM
ejpam-5914	294	50	-	-	PUNCT
ejpam-5914	294	51	fuzzy	fuzzy	ADJ
ejpam-5914	294	52	subalgebra	subalgebra	NOUN
ejpam-5914	294	53	of	of	ADP
ejpam-5914	294	54	a.	a.	NOUN
ejpam-5914	294	55	proof	proof	NOUN
ejpam-5914	294	56	.	.	PUNCT
ejpam-5914	295	1	assume	assume	VERB
ejpam-5914	295	2	that	that	SCONJ
ejpam-5914	295	3	(	(	PUNCT
ejpam-5914	295	4	a	a	PRON
ejpam-5914	295	5	,	,	PUNCT
ejpam-5914	295	6	f̃	f̃	PROPN
ejpam-5914	295	7	)	)	PUNCT
ejpam-5914	295	8	is	be	AUX
ejpam-5914	295	9	an	an	DET
ejpam-5914	295	10	interval	interval	NOUN
ejpam-5914	295	11	-	-	PUNCT
ejpam-5914	295	12	valued	value	VERB
ejpam-5914	295	13	fuzzy	fuzzy	ADJ
ejpam-5914	295	14	structure	structure	NOUN
ejpam-5914	295	15	over	over	ADP
ejpam-5914	295	16	a	a	DET
ejpam-5914	295	17	in	in	ADP
ejpam-5914	295	18	which	which	PRON
ejpam-5914	295	19	(	(	PUNCT
ejpam-5914	295	20	a	a	DET
ejpam-5914	295	21	,	,	PUNCT
ejpam-5914	295	22	f̃sup	f̃sup	ADJ
ejpam-5914	295	23	)	)	PUNCT
ejpam-5914	295	24	is	be	AUX
ejpam-5914	295	25	constant	constant	ADJ
ejpam-5914	295	26	and	and	CCONJ
ejpam-5914	295	27	(	(	PUNCT
ejpam-5914	295	28	a	a	PRON
ejpam-5914	295	29	,	,	PUNCT
ejpam-5914	295	30	f̃inf	f̃inf	ADJ
ejpam-5914	295	31	)	)	PUNCT
ejpam-5914	295	32	is	be	AUX
ejpam-5914	295	33	a	a	DET
ejpam-5914	295	34	4	4	NUM
ejpam-5914	295	35	-	-	PUNCT
ejpam-5914	295	36	fuzzy	fuzzy	ADJ
ejpam-5914	295	37	subalgebra	subalgebra	NOUN
ejpam-5914	295	38	of	of	ADP
ejpam-5914	295	39	a.	a.	NOUN
ejpam-5914	295	40	let	let	VERB
ejpam-5914	295	41	x	x	PRON
ejpam-5914	295	42	,	,	PUNCT
ejpam-5914	295	43	y	y	PROPN
ejpam-5914	295	44	∈	∈	PROPN
ejpam-5914	295	45	a.	a.	NOUN
ejpam-5914	295	46	since	since	SCONJ
ejpam-5914	295	47	(	(	PUNCT
ejpam-5914	295	48	a	a	DET
ejpam-5914	295	49	,	,	PUNCT
ejpam-5914	295	50	f̃sup	f̃sup	ADJ
ejpam-5914	295	51	)	)	PUNCT
ejpam-5914	295	52	is	be	AUX
ejpam-5914	295	53	constant	constant	ADJ
ejpam-5914	295	54	,	,	PUNCT
ejpam-5914	295	55	we	we	PRON
ejpam-5914	295	56	have	have	AUX
ejpam-5914	295	57	f̃sup(x	f̃sup(x	VERB
ejpam-5914	295	58	)	)	PUNCT
ejpam-5914	295	59	=	=	SYM
ejpam-5914	295	60	f̃sup(0	f̃sup(0	NOUN
ejpam-5914	295	61	)	)	PUNCT
ejpam-5914	295	62	for	for	SCONJ
ejpam-5914	295	63	all	all	PRON
ejpam-5914	295	64	x	x	SYM
ejpam-5914	295	65	∈	∈	NOUN
ejpam-5914	295	66	a.	a.	NOUN
ejpam-5914	295	67	since	since	SCONJ
ejpam-5914	295	68	(	(	PUNCT
ejpam-5914	295	69	a	a	PRON
ejpam-5914	295	70	,	,	PUNCT
ejpam-5914	295	71	f̃inf	f̃inf	ADJ
ejpam-5914	295	72	)	)	PUNCT
ejpam-5914	295	73	is	be	AUX
ejpam-5914	295	74	a	a	DET
ejpam-5914	295	75	4	4	NUM
ejpam-5914	295	76	-	-	PUNCT
ejpam-5914	295	77	fuzzy	fuzzy	ADJ
ejpam-5914	295	78	n.	n.	PROPN
ejpam-5914	295	79	rajesh	rajesh	PROPN
ejpam-5914	295	80	et	et	PROPN
ejpam-5914	295	81	al	al	PROPN
ejpam-5914	295	82	.	.	PUNCT
ejpam-5914	295	83	/	/	SYM
ejpam-5914	295	84	eur	eur	PROPN
ejpam-5914	295	85	.	.	PUNCT
ejpam-5914	296	1	j.	j.	PROPN
ejpam-5914	296	2	pure	pure	PROPN
ejpam-5914	296	3	appl	appl	PROPN
ejpam-5914	296	4	.	.	PROPN
ejpam-5914	296	5	math	math	PROPN
ejpam-5914	296	6	,	,	PUNCT
ejpam-5914	296	7	18	18	NUM
ejpam-5914	296	8	(	(	PUNCT
ejpam-5914	296	9	2	2	NUM
ejpam-5914	296	10	)	)	PUNCT
ejpam-5914	296	11	(	(	PUNCT
ejpam-5914	296	12	2025	2025	NUM
ejpam-5914	296	13	)	)	PUNCT
ejpam-5914	296	14	,	,	PUNCT
ejpam-5914	296	15	5914	5914	NUM
ejpam-5914	296	16	11	11	NUM
ejpam-5914	296	17	of	of	ADP
ejpam-5914	296	18	21	21	NUM
ejpam-5914	296	19	subalgebra	subalgebra	NOUN
ejpam-5914	296	20	of	of	ADP
ejpam-5914	296	21	a	a	PRON
ejpam-5914	296	22	,	,	PUNCT
ejpam-5914	296	23	we	we	PRON
ejpam-5914	296	24	have	have	AUX
ejpam-5914	296	25	f̃inf((x|(y|y))|(x|(y|y	f̃inf((x|(y|y))|(x|(y|y	VERB
ejpam-5914	296	26	)	)	PUNCT
ejpam-5914	296	27	)	)	PUNCT
ejpam-5914	296	28	)	)	PUNCT
ejpam-5914	297	1	≤	≤	NUM
ejpam-5914	297	2	max{f̃inf(x	max{f̃inf(x	PROPN
ejpam-5914	297	3	)	)	PUNCT
ejpam-5914	297	4	,	,	PUNCT
ejpam-5914	297	5	f̃inf(y	f̃inf(y	NOUN
ejpam-5914	297	6	)	)	PUNCT
ejpam-5914	297	7	}	}	PUNCT
ejpam-5914	297	8	.	.	PUNCT
ejpam-5914	298	1	thus	thus	ADV
ejpam-5914	298	2	,	,	PUNCT
ejpam-5914	298	3	f̃ℓ((x|(y|y))|(x|(y|y	f̃ℓ((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	298	4	)	)	PUNCT
ejpam-5914	298	5	)	)	PUNCT
ejpam-5914	298	6	)	)	PUNCT
ejpam-5914	299	1	=	=	SYM
ejpam-5914	299	2	f̃sup((x|(y|y))|(x|(y|y)))−	f̃sup((x|(y|y))|(x|(y|y)))−	NOUN
ejpam-5914	299	3	f̃inf((x|(y|y))|(x|(y|y	f̃inf((x|(y|y))|(x|(y|y	ADJ
ejpam-5914	299	4	)	)	PUNCT
ejpam-5914	299	5	)	)	PUNCT
ejpam-5914	299	6	)	)	PUNCT
ejpam-5914	300	1	=	=	PUNCT
ejpam-5914	301	1	f̃sup(0)−	f̃sup(0)−	NUM
ejpam-5914	301	2	f̃inf((x|(y|y))|(x|(y|y	f̃inf((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	301	3	)	)	PUNCT
ejpam-5914	301	4	)	)	PUNCT
ejpam-5914	301	5	)	)	PUNCT
ejpam-5914	301	6	≥	≥	PROPN
ejpam-5914	301	7	f̃sup(0)−max{f̃inf(x	f̃sup(0)−max{f̃inf(x	PROPN
ejpam-5914	301	8	)	)	PUNCT
ejpam-5914	301	9	,	,	PUNCT
ejpam-5914	301	10	f̃inf(y	f̃inf(y	NOUN
ejpam-5914	301	11	)	)	PUNCT
ejpam-5914	301	12	}	}	PUNCT
ejpam-5914	301	13	=	=	PROPN
ejpam-5914	301	14	min{f̃sup(0)−	min{f̃sup(0)−	PROPN
ejpam-5914	301	15	f̃inf(x	f̃inf(x	PROPN
ejpam-5914	301	16	)	)	PUNCT
ejpam-5914	301	17	,	,	PUNCT
ejpam-5914	301	18	f̃sup(0)−	f̃sup(0)−	PROPN
ejpam-5914	301	19	f̃inf(y	f̃inf(y	NOUN
ejpam-5914	301	20	)	)	PUNCT
ejpam-5914	301	21	}	}	PUNCT
ejpam-5914	302	1	=	=	PUNCT
ejpam-5914	302	2	min{f̃sup(x)−	min{f̃sup(x)−	PROPN
ejpam-5914	302	3	f̃inf(x	f̃inf(x	PROPN
ejpam-5914	302	4	)	)	PUNCT
ejpam-5914	302	5	,	,	PUNCT
ejpam-5914	302	6	f̃sup(y)−	f̃sup(y)−	PROPN
ejpam-5914	302	7	f̃inf(y	f̃inf(y	NOUN
ejpam-5914	302	8	)	)	PUNCT
ejpam-5914	302	9	}	}	PUNCT
ejpam-5914	302	10	=	=	SYM
ejpam-5914	302	11	min{f̃ℓ(x	min{f̃ℓ(x	PROPN
ejpam-5914	302	12	)	)	PUNCT
ejpam-5914	302	13	,	,	PUNCT
ejpam-5914	302	14	f̃ℓ(y	f̃ℓ(y	NOUN
ejpam-5914	302	15	)	)	PUNCT
ejpam-5914	302	16	}	}	PUNCT
ejpam-5914	302	17	.	.	PUNCT
ejpam-5914	303	1	hence	hence	ADV
ejpam-5914	303	2	,	,	PUNCT
ejpam-5914	303	3	(	(	PUNCT
ejpam-5914	303	4	a	a	DET
ejpam-5914	303	5	,	,	PUNCT
ejpam-5914	303	6	f̃ℓ	f̃ℓ	NOUN
ejpam-5914	303	7	)	)	PUNCT
ejpam-5914	303	8	is	be	AUX
ejpam-5914	303	9	a	a	DET
ejpam-5914	303	10	1	1	NUM
ejpam-5914	303	11	-	-	PUNCT
ejpam-5914	303	12	fuzzy	fuzzy	ADJ
ejpam-5914	303	13	subalgebra	subalgebra	NOUN
ejpam-5914	303	14	of	of	ADP
ejpam-5914	303	15	a	a	PRON
ejpam-5914	303	16	,	,	PUNCT
ejpam-5914	303	17	that	that	ADV
ejpam-5914	303	18	is	is	ADV
ejpam-5914	303	19	,	,	PUNCT
ejpam-5914	303	20	(	(	PUNCT
ejpam-5914	303	21	a	a	PRON
ejpam-5914	303	22	,	,	PUNCT
ejpam-5914	303	23	f̃	f̃	PROPN
ejpam-5914	303	24	)	)	PUNCT
ejpam-5914	303	25	is	be	AUX
ejpam-5914	303	26	a	a	DET
ejpam-5914	303	27	length	length	NOUN
ejpam-5914	303	28	1	1	NUM
ejpam-5914	303	29	-	-	PUNCT
ejpam-5914	303	30	fuzzy	fuzzy	ADJ
ejpam-5914	303	31	subalgebra	subalgebra	NOUN
ejpam-5914	303	32	of	of	ADP
ejpam-5914	303	33	a.	a.	NOUN
ejpam-5914	303	34	theorem	theorem	NOUN
ejpam-5914	303	35	12	12	NUM
ejpam-5914	303	36	.	.	PUNCT
ejpam-5914	304	1	if	if	SCONJ
ejpam-5914	304	2	(	(	PUNCT
ejpam-5914	304	3	a	a	PRON
ejpam-5914	304	4	,	,	PUNCT
ejpam-5914	304	5	f̃	f̃	PROPN
ejpam-5914	304	6	)	)	PUNCT
ejpam-5914	304	7	is	be	AUX
ejpam-5914	304	8	an	an	DET
ejpam-5914	304	9	interval	interval	NOUN
ejpam-5914	304	10	-	-	PUNCT
ejpam-5914	304	11	valued	value	VERB
ejpam-5914	304	12	fuzzy	fuzzy	ADJ
ejpam-5914	304	13	structure	structure	NOUN
ejpam-5914	304	14	over	over	ADP
ejpam-5914	304	15	a	a	DET
ejpam-5914	304	16	in	in	ADP
ejpam-5914	304	17	which	which	PRON
ejpam-5914	304	18	(	(	PUNCT
ejpam-5914	304	19	a	a	DET
ejpam-5914	304	20	,	,	PUNCT
ejpam-5914	304	21	f̃sup	f̃sup	ADJ
ejpam-5914	304	22	)	)	PUNCT
ejpam-5914	304	23	is	be	AUX
ejpam-5914	304	24	constant	constant	ADJ
ejpam-5914	304	25	and	and	CCONJ
ejpam-5914	304	26	(	(	PUNCT
ejpam-5914	304	27	a	a	PRON
ejpam-5914	304	28	,	,	PUNCT
ejpam-5914	304	29	f̃inf	f̃inf	ADJ
ejpam-5914	304	30	)	)	PUNCT
ejpam-5914	304	31	is	be	AUX
ejpam-5914	304	32	a	a	DET
ejpam-5914	304	33	1	1	NUM
ejpam-5914	304	34	-	-	PUNCT
ejpam-5914	304	35	fuzzy	fuzzy	ADJ
ejpam-5914	304	36	subalgebra	subalgebra	NOUN
ejpam-5914	304	37	of	of	ADP
ejpam-5914	304	38	a	a	PRON
ejpam-5914	304	39	,	,	PUNCT
ejpam-5914	304	40	then	then	ADV
ejpam-5914	304	41	(	(	PUNCT
ejpam-5914	304	42	a	a	PRON
ejpam-5914	304	43	,	,	PUNCT
ejpam-5914	304	44	f̃	f̃	PROPN
ejpam-5914	304	45	)	)	PUNCT
ejpam-5914	304	46	is	be	AUX
ejpam-5914	304	47	a	a	DET
ejpam-5914	304	48	length	length	NOUN
ejpam-5914	304	49	4	4	NUM
ejpam-5914	304	50	-	-	PUNCT
ejpam-5914	304	51	fuzzy	fuzzy	ADJ
ejpam-5914	304	52	subalgebra	subalgebra	NOUN
ejpam-5914	304	53	of	of	ADP
ejpam-5914	304	54	a.	a.	NOUN
ejpam-5914	304	55	proof	proof	NOUN
ejpam-5914	304	56	.	.	PUNCT
ejpam-5914	305	1	assume	assume	VERB
ejpam-5914	305	2	that	that	SCONJ
ejpam-5914	305	3	(	(	PUNCT
ejpam-5914	305	4	a	a	PRON
ejpam-5914	305	5	,	,	PUNCT
ejpam-5914	305	6	f̃	f̃	PROPN
ejpam-5914	305	7	)	)	PUNCT
ejpam-5914	305	8	is	be	AUX
ejpam-5914	305	9	an	an	DET
ejpam-5914	305	10	interval	interval	NOUN
ejpam-5914	305	11	-	-	PUNCT
ejpam-5914	305	12	valued	value	VERB
ejpam-5914	305	13	fuzzy	fuzzy	ADJ
ejpam-5914	305	14	structure	structure	NOUN
ejpam-5914	305	15	over	over	ADP
ejpam-5914	305	16	a	a	DET
ejpam-5914	305	17	in	in	ADP
ejpam-5914	305	18	which	which	PRON
ejpam-5914	305	19	(	(	PUNCT
ejpam-5914	305	20	a	a	DET
ejpam-5914	305	21	,	,	PUNCT
ejpam-5914	305	22	f̃sup	f̃sup	ADJ
ejpam-5914	305	23	)	)	PUNCT
ejpam-5914	305	24	is	be	AUX
ejpam-5914	305	25	constant	constant	ADJ
ejpam-5914	305	26	and	and	CCONJ
ejpam-5914	305	27	(	(	PUNCT
ejpam-5914	305	28	a	a	PRON
ejpam-5914	305	29	,	,	PUNCT
ejpam-5914	305	30	f̃inf	f̃inf	ADJ
ejpam-5914	305	31	)	)	PUNCT
ejpam-5914	305	32	is	be	AUX
ejpam-5914	305	33	a	a	DET
ejpam-5914	305	34	1	1	NUM
ejpam-5914	305	35	-	-	PUNCT
ejpam-5914	305	36	fuzzy	fuzzy	ADJ
ejpam-5914	305	37	subalgebra	subalgebra	NOUN
ejpam-5914	305	38	of	of	ADP
ejpam-5914	305	39	a.	a.	NOUN
ejpam-5914	305	40	let	let	VERB
ejpam-5914	305	41	x	x	PRON
ejpam-5914	305	42	,	,	PUNCT
ejpam-5914	305	43	y	y	PROPN
ejpam-5914	305	44	∈	∈	PROPN
ejpam-5914	305	45	a.	a.	NOUN
ejpam-5914	305	46	since	since	SCONJ
ejpam-5914	305	47	(	(	PUNCT
ejpam-5914	305	48	a	a	DET
ejpam-5914	305	49	,	,	PUNCT
ejpam-5914	305	50	f̃sup	f̃sup	ADJ
ejpam-5914	305	51	)	)	PUNCT
ejpam-5914	305	52	is	be	AUX
ejpam-5914	305	53	constant	constant	ADJ
ejpam-5914	305	54	,	,	PUNCT
ejpam-5914	305	55	we	we	PRON
ejpam-5914	305	56	have	have	AUX
ejpam-5914	305	57	f̃sup(x	f̃sup(x	VERB
ejpam-5914	305	58	)	)	PUNCT
ejpam-5914	305	59	=	=	SYM
ejpam-5914	305	60	f̃sup(0	f̃sup(0	NOUN
ejpam-5914	305	61	)	)	PUNCT
ejpam-5914	305	62	for	for	SCONJ
ejpam-5914	305	63	all	all	PRON
ejpam-5914	305	64	x	x	SYM
ejpam-5914	305	65	∈	∈	NOUN
ejpam-5914	305	66	a.	a.	NOUN
ejpam-5914	305	67	since	since	SCONJ
ejpam-5914	305	68	(	(	PUNCT
ejpam-5914	305	69	a	a	PRON
ejpam-5914	305	70	,	,	PUNCT
ejpam-5914	305	71	f̃inf	f̃inf	ADJ
ejpam-5914	305	72	)	)	PUNCT
ejpam-5914	305	73	is	be	AUX
ejpam-5914	305	74	a	a	DET
ejpam-5914	305	75	1	1	NUM
ejpam-5914	305	76	-	-	PUNCT
ejpam-5914	305	77	fuzzy	fuzzy	ADJ
ejpam-5914	305	78	subalgebra	subalgebra	NOUN
ejpam-5914	305	79	of	of	ADP
ejpam-5914	305	80	a	a	PRON
ejpam-5914	305	81	,	,	PUNCT
ejpam-5914	305	82	we	we	PRON
ejpam-5914	305	83	have	have	VERB
ejpam-5914	305	84	f̃inf((x|(y|y))|(x|(y|y	f̃inf((x|(y|y))|(x|(y|y	VERB
ejpam-5914	305	85	)	)	PUNCT
ejpam-5914	305	86	)	)	PUNCT
ejpam-5914	305	87	)	)	PUNCT
ejpam-5914	305	88	≥	≥	PROPN
ejpam-5914	305	89	min{f̃inf(x	min{f̃inf(x	PROPN
ejpam-5914	305	90	)	)	PUNCT
ejpam-5914	305	91	,	,	PUNCT
ejpam-5914	305	92	f̃inf(y	f̃inf(y	NOUN
ejpam-5914	305	93	)	)	PUNCT
ejpam-5914	305	94	}	}	PUNCT
ejpam-5914	305	95	.	.	PUNCT
ejpam-5914	306	1	thus	thus	ADV
ejpam-5914	306	2	,	,	PUNCT
ejpam-5914	306	3	f̃ℓ((x|(y|y))|(x|(y|y	f̃ℓ((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	306	4	)	)	PUNCT
ejpam-5914	306	5	)	)	PUNCT
ejpam-5914	306	6	)	)	PUNCT
ejpam-5914	307	1	=	=	SYM
ejpam-5914	307	2	f̃sup((x|(y|y))|(x|(y|y)))−	f̃sup((x|(y|y))|(x|(y|y)))−	NOUN
ejpam-5914	307	3	f̃inf((x|(y|y))|(x|(y|y	f̃inf((x|(y|y))|(x|(y|y	ADJ
ejpam-5914	307	4	)	)	PUNCT
ejpam-5914	307	5	)	)	PUNCT
ejpam-5914	307	6	)	)	PUNCT
ejpam-5914	308	1	=	=	PUNCT
ejpam-5914	309	1	f̃sup(0)−	f̃sup(0)−	NUM
ejpam-5914	309	2	f̃inf((x|(y|y))|(x|(y|y	f̃inf((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	309	3	)	)	PUNCT
ejpam-5914	309	4	)	)	PUNCT
ejpam-5914	309	5	)	)	PUNCT
ejpam-5914	309	6	≤	≤	NUM
ejpam-5914	309	7	f̃sup(0)−min{f̃inf(x	f̃sup(0)−min{f̃inf(x	NOUN
ejpam-5914	309	8	)	)	PUNCT
ejpam-5914	309	9	,	,	PUNCT
ejpam-5914	309	10	f̃inf(y	f̃inf(y	NOUN
ejpam-5914	309	11	)	)	PUNCT
ejpam-5914	309	12	}	}	PUNCT
ejpam-5914	309	13	=	=	SYM
ejpam-5914	309	14	max{f̃sup(0)−	max{f̃sup(0)−	NOUN
ejpam-5914	309	15	f̃inf(x	f̃inf(x	PROPN
ejpam-5914	309	16	)	)	PUNCT
ejpam-5914	309	17	,	,	PUNCT
ejpam-5914	309	18	f̃sup(0)−	f̃sup(0)−	PROPN
ejpam-5914	309	19	f̃inf(y	f̃inf(y	NOUN
ejpam-5914	309	20	)	)	PUNCT
ejpam-5914	309	21	}	}	PUNCT
ejpam-5914	309	22	=	=	SYM
ejpam-5914	309	23	max{f̃sup(x)−	max{f̃sup(x)−	PROPN
ejpam-5914	309	24	f̃inf(x	f̃inf(x	PROPN
ejpam-5914	309	25	)	)	PUNCT
ejpam-5914	309	26	,	,	PUNCT
ejpam-5914	309	27	f̃sup(y)−	f̃sup(y)−	PROPN
ejpam-5914	309	28	f̃inf(y	f̃inf(y	NOUN
ejpam-5914	309	29	)	)	PUNCT
ejpam-5914	309	30	}	}	PUNCT
ejpam-5914	309	31	=	=	SYM
ejpam-5914	309	32	max{f̃ℓ(x	max{f̃ℓ(x	PROPN
ejpam-5914	309	33	)	)	PUNCT
ejpam-5914	309	34	,	,	PUNCT
ejpam-5914	309	35	f̃ℓ(y	f̃ℓ(y	PROPN
ejpam-5914	309	36	)	)	PUNCT
ejpam-5914	309	37	}	}	PUNCT
ejpam-5914	309	38	.	.	PUNCT
ejpam-5914	310	1	hence	hence	ADV
ejpam-5914	310	2	,	,	PUNCT
ejpam-5914	310	3	(	(	PUNCT
ejpam-5914	310	4	a	a	DET
ejpam-5914	310	5	,	,	PUNCT
ejpam-5914	310	6	f̃ℓ	f̃ℓ	NOUN
ejpam-5914	310	7	)	)	PUNCT
ejpam-5914	310	8	is	be	AUX
ejpam-5914	310	9	a	a	DET
ejpam-5914	310	10	4	4	NUM
ejpam-5914	310	11	-	-	PUNCT
ejpam-5914	310	12	fuzzy	fuzzy	ADJ
ejpam-5914	310	13	subalgebra	subalgebra	NOUN
ejpam-5914	310	14	of	of	ADP
ejpam-5914	310	15	a	a	PRON
ejpam-5914	310	16	,	,	PUNCT
ejpam-5914	310	17	that	that	ADV
ejpam-5914	310	18	is	is	ADV
ejpam-5914	310	19	,	,	PUNCT
ejpam-5914	310	20	(	(	PUNCT
ejpam-5914	310	21	a	a	PRON
ejpam-5914	310	22	,	,	PUNCT
ejpam-5914	310	23	f̃	f̃	PROPN
ejpam-5914	310	24	)	)	PUNCT
ejpam-5914	310	25	is	be	AUX
ejpam-5914	310	26	a	a	DET
ejpam-5914	310	27	length	length	NOUN
ejpam-5914	310	28	4	4	NUM
ejpam-5914	310	29	-	-	PUNCT
ejpam-5914	310	30	fuzzy	fuzzy	ADJ
ejpam-5914	310	31	subalgebra	subalgebra	NOUN
ejpam-5914	310	32	of	of	ADP
ejpam-5914	310	33	a.	a.	NOUN
ejpam-5914	310	34	4	4	NUM
ejpam-5914	310	35	.	.	PUNCT
ejpam-5914	311	1	mean	mean	VERB
ejpam-5914	311	2	of	of	ADP
ejpam-5914	311	3	an	an	DET
ejpam-5914	311	4	interval	interval	NOUN
ejpam-5914	311	5	-	-	PUNCT
ejpam-5914	311	6	valued	value	VERB
ejpam-5914	311	7	fuzzy	fuzzy	ADJ
ejpam-5914	311	8	structure	structure	NOUN
ejpam-5914	311	9	in	in	ADP
ejpam-5914	311	10	sheffer	sheffer	PROPN
ejpam-5914	311	11	stroke	stroke	PROPN
ejpam-5914	311	12	hilbert	hilbert	PROPN
ejpam-5914	311	13	algebras	algebra	VERB
ejpam-5914	311	14	this	this	DET
ejpam-5914	311	15	section	section	NOUN
ejpam-5914	311	16	introduces	introduce	VERB
ejpam-5914	311	17	the	the	DET
ejpam-5914	311	18	concept	concept	NOUN
ejpam-5914	311	19	of	of	ADP
ejpam-5914	311	20	the	the	DET
ejpam-5914	311	21	mean	mean	NOUN
ejpam-5914	311	22	of	of	ADP
ejpam-5914	311	23	an	an	DET
ejpam-5914	311	24	interval	interval	NOUN
ejpam-5914	311	25	-	-	PUNCT
ejpam-5914	311	26	valued	value	VERB
ejpam-5914	311	27	fuzzy	fuzzy	ADJ
ejpam-5914	311	28	structure	structure	NOUN
ejpam-5914	311	29	within	within	ADP
ejpam-5914	311	30	sheffer	sheffer	PROPN
ejpam-5914	311	31	stroke	stroke	PROPN
ejpam-5914	311	32	hilbert	hilbert	PROPN
ejpam-5914	311	33	algebras	algebras	PROPN
ejpam-5914	311	34	,	,	PUNCT
ejpam-5914	311	35	along	along	ADP
ejpam-5914	311	36	with	with	ADP
ejpam-5914	311	37	the	the	DET
ejpam-5914	311	38	corresponding	corresponding	ADJ
ejpam-5914	311	39	notion	notion	NOUN
ejpam-5914	311	40	of	of	ADP
ejpam-5914	311	41	mean	mean	ADJ
ejpam-5914	311	42	-	-	PUNCT
ejpam-5914	311	43	fuzzy	fuzzy	ADJ
ejpam-5914	311	44	subalgebras	subalgebra	NOUN
ejpam-5914	311	45	.	.	PUNCT
ejpam-5914	312	1	the	the	DET
ejpam-5914	312	2	fundamental	fundamental	ADJ
ejpam-5914	312	3	properties	property	NOUN
ejpam-5914	312	4	of	of	ADP
ejpam-5914	312	5	these	these	DET
ejpam-5914	312	6	subalgebras	subalgebra	NOUN
ejpam-5914	312	7	are	be	AUX
ejpam-5914	312	8	examined	examine	VERB
ejpam-5914	312	9	,	,	PUNCT
ejpam-5914	312	10	shedding	shed	VERB
ejpam-5914	312	11	light	light	NOUN
ejpam-5914	312	12	on	on	ADP
ejpam-5914	312	13	their	their	PRON
ejpam-5914	312	14	intrinsic	intrinsic	ADJ
ejpam-5914	312	15	algebraic	algebraic	ADJ
ejpam-5914	312	16	behavior	behavior	NOUN
ejpam-5914	312	17	.	.	PUNCT
ejpam-5914	313	1	we	we	PRON
ejpam-5914	313	2	further	far	ADV
ejpam-5914	313	3	explore	explore	VERB
ejpam-5914	313	4	the	the	DET
ejpam-5914	313	5	connections	connection	NOUN
ejpam-5914	313	6	between	between	ADP
ejpam-5914	313	7	mean	mean	ADJ
ejpam-5914	313	8	-	-	PUNCT
ejpam-5914	313	9	fuzzy	fuzzy	ADJ
ejpam-5914	313	10	subalgebras	subalgebra	NOUN
ejpam-5914	313	11	and	and	CCONJ
ejpam-5914	313	12	classical	classical	ADJ
ejpam-5914	313	13	subalgebras	subalgebra	NOUN
ejpam-5914	313	14	,	,	PUNCT
ejpam-5914	313	15	offering	offer	VERB
ejpam-5914	313	16	a	a	DET
ejpam-5914	313	17	comparative	comparative	ADJ
ejpam-5914	313	18	perspective	perspective	NOUN
ejpam-5914	313	19	on	on	ADP
ejpam-5914	313	20	their	their	PRON
ejpam-5914	313	21	structural	structural	ADJ
ejpam-5914	313	22	interplay	interplay	NOUN
ejpam-5914	313	23	.	.	PUNCT
ejpam-5914	314	1	additionally	additionally	ADV
ejpam-5914	314	2	,	,	PUNCT
ejpam-5914	314	3	the	the	DET
ejpam-5914	314	4	relationships	relationship	NOUN
ejpam-5914	314	5	between	between	ADP
ejpam-5914	314	6	mean	mean	ADJ
ejpam-5914	314	7	-	-	PUNCT
ejpam-5914	314	8	fuzzy	fuzzy	ADJ
ejpam-5914	314	9	subalgebras	subalgebra	NOUN
ejpam-5914	314	10	and	and	CCONJ
ejpam-5914	314	11	various	various	ADJ
ejpam-5914	314	12	level	level	NOUN
ejpam-5914	314	13	subsets	subset	NOUN
ejpam-5914	314	14	—	—	PUNCT
ejpam-5914	314	15	namely	namely	ADV
ejpam-5914	314	16	,	,	PUNCT
ejpam-5914	314	17	upper	upper	ADJ
ejpam-5914	314	18	and	and	CCONJ
ejpam-5914	314	19	lower	low	ADJ
ejpam-5914	314	20	-	-	PUNCT
ejpam-5914	314	21	level	level	NOUN
ejpam-5914	314	22	subsets	subset	NOUN
ejpam-5914	314	23	—	—	PUNCT
ejpam-5914	314	24	of	of	ADP
ejpam-5914	314	25	the	the	DET
ejpam-5914	314	26	mean	mean	NOUN
ejpam-5914	314	27	of	of	ADP
ejpam-5914	314	28	an	an	DET
ejpam-5914	314	29	interval	interval	NOUN
ejpam-5914	314	30	-	-	PUNCT
ejpam-5914	314	31	valued	value	VERB
ejpam-5914	314	32	fuzzy	fuzzy	ADJ
ejpam-5914	314	33	structure	structure	NOUN
ejpam-5914	314	34	are	be	AUX
ejpam-5914	314	35	analyzed	analyze	VERB
ejpam-5914	314	36	,	,	PUNCT
ejpam-5914	314	37	providing	provide	VERB
ejpam-5914	314	38	a	a	DET
ejpam-5914	314	39	comprehensive	comprehensive	ADJ
ejpam-5914	314	40	framework	framework	NOUN
ejpam-5914	314	41	for	for	ADP
ejpam-5914	314	42	understanding	understand	VERB
ejpam-5914	314	43	their	their	PRON
ejpam-5914	314	44	hierarchical	hierarchical	ADJ
ejpam-5914	314	45	and	and	CCONJ
ejpam-5914	314	46	interval	interval	NOUN
ejpam-5914	314	47	-	-	PUNCT
ejpam-5914	314	48	dependent	dependent	ADJ
ejpam-5914	314	49	dynamics	dynamic	NOUN
ejpam-5914	314	50	in	in	ADP
ejpam-5914	314	51	the	the	DET
ejpam-5914	314	52	context	context	NOUN
ejpam-5914	314	53	of	of	ADP
ejpam-5914	314	54	sheffer	sheffer	PROPN
ejpam-5914	314	55	stroke	stroke	PROPN
ejpam-5914	314	56	hilbert	hilbert	PROPN
ejpam-5914	314	57	algebras	algebras	PROPN
ejpam-5914	314	58	.	.	PUNCT
ejpam-5914	315	1	n.	n.	PROPN
ejpam-5914	315	2	rajesh	rajesh	PROPN
ejpam-5914	315	3	et	et	PROPN
ejpam-5914	315	4	al	al	PROPN
ejpam-5914	315	5	.	.	PUNCT
ejpam-5914	315	6	/	/	SYM
ejpam-5914	315	7	eur	eur	PROPN
ejpam-5914	315	8	.	.	PUNCT
ejpam-5914	316	1	j.	j.	PROPN
ejpam-5914	316	2	pure	pure	PROPN
ejpam-5914	316	3	appl	appl	PROPN
ejpam-5914	316	4	.	.	PROPN
ejpam-5914	316	5	math	math	PROPN
ejpam-5914	316	6	,	,	PUNCT
ejpam-5914	316	7	18	18	NUM
ejpam-5914	316	8	(	(	PUNCT
ejpam-5914	316	9	2	2	NUM
ejpam-5914	316	10	)	)	PUNCT
ejpam-5914	316	11	(	(	PUNCT
ejpam-5914	316	12	2025	2025	NUM
ejpam-5914	316	13	)	)	PUNCT
ejpam-5914	316	14	,	,	PUNCT
ejpam-5914	316	15	5914	5914	NUM
ejpam-5914	316	16	12	12	NUM
ejpam-5914	316	17	of	of	ADP
ejpam-5914	316	18	21	21	NUM
ejpam-5914	316	19	definition	definition	NOUN
ejpam-5914	316	20	11	11	NUM
ejpam-5914	316	21	.	.	PUNCT
ejpam-5914	317	1	[	[	X
ejpam-5914	317	2	26	26	NUM
ejpam-5914	317	3	]	]	PUNCT
ejpam-5914	317	4	given	give	VERB
ejpam-5914	317	5	an	an	DET
ejpam-5914	317	6	interval	interval	NOUN
ejpam-5914	317	7	-	-	PUNCT
ejpam-5914	317	8	valued	value	VERB
ejpam-5914	317	9	fuzzy	fuzzy	ADJ
ejpam-5914	317	10	structure	structure	NOUN
ejpam-5914	317	11	(	(	PUNCT
ejpam-5914	317	12	a	a	PRON
ejpam-5914	317	13	,	,	PUNCT
ejpam-5914	317	14	f̃	f̃	PROPN
ejpam-5914	317	15	)	)	PUNCT
ejpam-5914	317	16	over	over	ADP
ejpam-5914	317	17	a	a	PRON
ejpam-5914	317	18	,	,	PUNCT
ejpam-5914	317	19	we	we	PRON
ejpam-5914	317	20	define	define	VERB
ejpam-5914	317	21	a	a	DET
ejpam-5914	317	22	fuzzy	fuzzy	ADJ
ejpam-5914	317	23	structure	structure	NOUN
ejpam-5914	317	24	(	(	PUNCT
ejpam-5914	317	25	a	a	DET
ejpam-5914	317	26	,	,	PUNCT
ejpam-5914	317	27	f̃m	f̃m	NOUN
ejpam-5914	317	28	)	)	PUNCT
ejpam-5914	317	29	in	in	ADP
ejpam-5914	317	30	a	a	PRON
ejpam-5914	317	31	as	as	SCONJ
ejpam-5914	317	32	follows	follow	VERB
ejpam-5914	317	33	:	:	PUNCT
ejpam-5914	317	34	f̃m	f̃m	NOUN
ejpam-5914	317	35	:	:	PUNCT
ejpam-5914	317	36	a	a	DET
ejpam-5914	317	37	→	→	SYM
ejpam-5914	317	38	[	[	X
ejpam-5914	317	39	0	0	NUM
ejpam-5914	317	40	,	,	PUNCT
ejpam-5914	317	41	1];x	1];x	NUM
ejpam-5914	317	42	7→	7→	NUM
ejpam-5914	317	43	f̃sup(x	f̃sup(x	NOUN
ejpam-5914	317	44	)	)	PUNCT
ejpam-5914	317	45	+	+	SYM
ejpam-5914	317	46	f̃inf(x	f̃inf(x	NOUN
ejpam-5914	317	47	)	)	PUNCT
ejpam-5914	317	48	2	2	NUM
ejpam-5914	317	49	,	,	PUNCT
ejpam-5914	317	50	which	which	PRON
ejpam-5914	317	51	is	be	AUX
ejpam-5914	317	52	called	call	VERB
ejpam-5914	317	53	the	the	DET
ejpam-5914	317	54	mean	mean	NOUN
ejpam-5914	317	55	of	of	ADP
ejpam-5914	317	56	f̃	f̃	PROPN
ejpam-5914	317	57	.	.	PUNCT
ejpam-5914	318	1	example	example	NOUN
ejpam-5914	318	2	5	5	NUM
ejpam-5914	318	3	.	.	X
ejpam-5914	319	1	consider	consider	VERB
ejpam-5914	319	2	example	example	NOUN
ejpam-5914	319	3	1	1	NUM
ejpam-5914	319	4	,	,	PUNCT
ejpam-5914	319	5	we	we	PRON
ejpam-5914	319	6	have	have	VERB
ejpam-5914	319	7	a	a	DET
ejpam-5914	319	8	1	1	NUM
ejpam-5914	319	9	u	u	NOUN
ejpam-5914	319	10	v	v	ADP
ejpam-5914	319	11	0	0	NUM
ejpam-5914	320	1	f̃m	f̃m	PROPN
ejpam-5914	320	2	0.5	0.5	NUM
ejpam-5914	320	3	0.3	0.3	NUM
ejpam-5914	320	4	0.5	0.5	NUM
ejpam-5914	320	5	0.25	0.25	NUM
ejpam-5914	320	6	definition	definition	NOUN
ejpam-5914	320	7	12	12	NUM
ejpam-5914	320	8	.	.	PUNCT
ejpam-5914	321	1	an	an	DET
ejpam-5914	321	2	interval	interval	NOUN
ejpam-5914	321	3	-	-	PUNCT
ejpam-5914	321	4	valued	value	VERB
ejpam-5914	321	5	fuzzy	fuzzy	ADJ
ejpam-5914	321	6	structure	structure	NOUN
ejpam-5914	321	7	(	(	PUNCT
ejpam-5914	321	8	a	a	PRON
ejpam-5914	321	9	,	,	PUNCT
ejpam-5914	321	10	f̃	f̃	PROPN
ejpam-5914	321	11	)	)	PUNCT
ejpam-5914	321	12	over	over	ADP
ejpam-5914	321	13	a	a	PRON
ejpam-5914	321	14	is	be	AUX
ejpam-5914	321	15	called	call	VERB
ejpam-5914	321	16	a	a	DET
ejpam-5914	321	17	mean	mean	ADJ
ejpam-5914	321	18	1	1	NUM
ejpam-5914	321	19	-	-	PUNCT
ejpam-5914	321	20	fuzzy	fuzzy	ADJ
ejpam-5914	321	21	(	(	PUNCT
ejpam-5914	321	22	resp	resp	NOUN
ejpam-5914	321	23	.	.	PUNCT
ejpam-5914	321	24	,	,	PUNCT
ejpam-5914	321	25	2	2	NUM
ejpam-5914	321	26	-	-	PUNCT
ejpam-5914	321	27	fuzzy	fuzzy	ADJ
ejpam-5914	321	28	,	,	PUNCT
ejpam-5914	321	29	3	3	NUM
ejpam-5914	321	30	-	-	PUNCT
ejpam-5914	321	31	fuzzy	fuzzy	ADJ
ejpam-5914	321	32	and	and	CCONJ
ejpam-5914	321	33	4	4	NUM
ejpam-5914	321	34	-	-	PUNCT
ejpam-5914	321	35	fuzzy	fuzzy	ADJ
ejpam-5914	321	36	)	)	PUNCT
ejpam-5914	321	37	subalgebra	subalgebra	NOUN
ejpam-5914	321	38	of	of	ADP
ejpam-5914	321	39	a	a	DET
ejpam-5914	321	40	if	if	SCONJ
ejpam-5914	321	41	a	a	DET
ejpam-5914	321	42	fuzzy	fuzzy	ADJ
ejpam-5914	321	43	structure	structure	NOUN
ejpam-5914	321	44	(	(	PUNCT
ejpam-5914	321	45	a	a	DET
ejpam-5914	321	46	,	,	PUNCT
ejpam-5914	321	47	f̃m	f̃m	NOUN
ejpam-5914	321	48	)	)	PUNCT
ejpam-5914	321	49	is	be	AUX
ejpam-5914	321	50	a	a	DET
ejpam-5914	321	51	1	1	NUM
ejpam-5914	321	52	-	-	PUNCT
ejpam-5914	321	53	fuzzy	fuzzy	ADJ
ejpam-5914	321	54	(	(	PUNCT
ejpam-5914	321	55	resp	resp	NOUN
ejpam-5914	321	56	.	.	PUNCT
ejpam-5914	322	1	,	,	PUNCT
ejpam-5914	322	2	2	2	NUM
ejpam-5914	322	3	-	-	PUNCT
ejpam-5914	322	4	fuzzy	fuzzy	ADJ
ejpam-5914	322	5	,	,	PUNCT
ejpam-5914	322	6	3	3	NUM
ejpam-5914	322	7	-	-	PUNCT
ejpam-5914	322	8	fuzzy	fuzzy	ADJ
ejpam-5914	322	9	and	and	CCONJ
ejpam-5914	322	10	4	4	NUM
ejpam-5914	322	11	-	-	PUNCT
ejpam-5914	322	12	fuzzy	fuzzy	ADJ
ejpam-5914	322	13	)	)	PUNCT
ejpam-5914	322	14	subalgebra	subalgebra	NOUN
ejpam-5914	322	15	of	of	ADP
ejpam-5914	322	16	a.	a.	NOUN
ejpam-5914	322	17	proposition	proposition	NOUN
ejpam-5914	322	18	4	4	NUM
ejpam-5914	322	19	.	.	PUNCT
ejpam-5914	323	1	if	if	SCONJ
ejpam-5914	323	2	(	(	PUNCT
ejpam-5914	323	3	a	a	PRON
ejpam-5914	323	4	,	,	PUNCT
ejpam-5914	323	5	f̃	f̃	PROPN
ejpam-5914	323	6	)	)	PUNCT
ejpam-5914	323	7	is	be	AUX
ejpam-5914	323	8	a	a	DET
ejpam-5914	323	9	mean	mean	ADJ
ejpam-5914	323	10	k	k	ADJ
ejpam-5914	323	11	-	-	ADJ
ejpam-5914	323	12	fuzzy	fuzzy	ADJ
ejpam-5914	323	13	subalgebra	subalgebra	NOUN
ejpam-5914	323	14	of	of	ADP
ejpam-5914	323	15	a	a	PRON
ejpam-5914	323	16	for	for	ADP
ejpam-5914	323	17	k	k	PROPN
ejpam-5914	323	18	∈	∈	PROPN
ejpam-5914	323	19	{	{	PUNCT
ejpam-5914	323	20	1	1	NUM
ejpam-5914	323	21	,	,	PUNCT
ejpam-5914	323	22	3	3	NUM
ejpam-5914	323	23	}	}	PUNCT
ejpam-5914	323	24	,	,	PUNCT
ejpam-5914	323	25	then	then	ADV
ejpam-5914	323	26	(	(	PUNCT
ejpam-5914	323	27	∀x	∀x	X
ejpam-5914	323	28	∈	∈	PROPN
ejpam-5914	323	29	a)(f̃m(0	a)(f̃m(0	NOUN
ejpam-5914	323	30	)	)	PUNCT
ejpam-5914	323	31	≥	≥	NOUN
ejpam-5914	323	32	f̃m(x	f̃m(x	NUM
ejpam-5914	323	33	)	)	PUNCT
ejpam-5914	323	34	)	)	PUNCT
ejpam-5914	323	35	.	.	PUNCT
ejpam-5914	324	1	(	(	PUNCT
ejpam-5914	324	2	3	3	X
ejpam-5914	324	3	)	)	PUNCT
ejpam-5914	324	4	proof	proof	NOUN
ejpam-5914	324	5	.	.	PUNCT
ejpam-5914	325	1	let	let	VERB
ejpam-5914	325	2	(	(	PUNCT
ejpam-5914	325	3	a	a	PRON
ejpam-5914	325	4	,	,	PUNCT
ejpam-5914	325	5	f̃	f̃	PROPN
ejpam-5914	325	6	)	)	PUNCT
ejpam-5914	325	7	be	be	VERB
ejpam-5914	325	8	a	a	DET
ejpam-5914	325	9	mean	mean	ADJ
ejpam-5914	325	10	k	k	ADJ
ejpam-5914	325	11	-	-	ADJ
ejpam-5914	325	12	fuzzy	fuzzy	ADJ
ejpam-5914	325	13	subalgebra	subalgebra	NOUN
ejpam-5914	325	14	of	of	ADP
ejpam-5914	325	15	a	a	PRON
ejpam-5914	325	16	for	for	ADP
ejpam-5914	325	17	k	k	PROPN
ejpam-5914	325	18	∈	∈	PROPN
ejpam-5914	325	19	{	{	PUNCT
ejpam-5914	325	20	1	1	NUM
ejpam-5914	325	21	,	,	PUNCT
ejpam-5914	325	22	3	3	NUM
ejpam-5914	325	23	}	}	PUNCT
ejpam-5914	325	24	.	.	PUNCT
ejpam-5914	326	1	then	then	ADV
ejpam-5914	326	2	f̃m(0	f̃m(0	PROPN
ejpam-5914	326	3	)	)	PUNCT
ejpam-5914	326	4	=	=	SYM
ejpam-5914	326	5	f̃sup(0	f̃sup(0	NOUN
ejpam-5914	326	6	)	)	PUNCT
ejpam-5914	327	1	+	+	CCONJ
ejpam-5914	327	2	f̃inf(0	f̃inf(0	NOUN
ejpam-5914	327	3	)	)	PUNCT
ejpam-5914	327	4	2	2	NUM
ejpam-5914	327	5	≥	≥	NOUN
ejpam-5914	327	6	f̃sup(x	f̃sup(x	NOUN
ejpam-5914	327	7	)	)	PUNCT
ejpam-5914	327	8	+	+	SYM
ejpam-5914	327	9	f̃inf(x	f̃inf(x	NOUN
ejpam-5914	327	10	)	)	PUNCT
ejpam-5914	327	11	2	2	NUM
ejpam-5914	327	12	=	=	SYM
ejpam-5914	327	13	f̃m(x	f̃m(x	NUM
ejpam-5914	327	14	)	)	PUNCT
ejpam-5914	327	15	.	.	PUNCT
ejpam-5914	328	1	proposition	proposition	NOUN
ejpam-5914	328	2	5	5	NUM
ejpam-5914	328	3	.	.	PUNCT
ejpam-5914	329	1	if	if	SCONJ
ejpam-5914	329	2	(	(	PUNCT
ejpam-5914	329	3	a	a	PRON
ejpam-5914	329	4	,	,	PUNCT
ejpam-5914	329	5	f̃	f̃	PROPN
ejpam-5914	329	6	)	)	PUNCT
ejpam-5914	329	7	is	be	AUX
ejpam-5914	329	8	a	a	DET
ejpam-5914	329	9	mean	mean	ADJ
ejpam-5914	329	10	k	k	ADJ
ejpam-5914	329	11	-	-	ADJ
ejpam-5914	329	12	fuzzy	fuzzy	ADJ
ejpam-5914	329	13	subalgebra	subalgebra	NOUN
ejpam-5914	329	14	of	of	ADP
ejpam-5914	329	15	a	a	PRON
ejpam-5914	329	16	for	for	ADP
ejpam-5914	329	17	k	k	PROPN
ejpam-5914	329	18	∈	∈	PROPN
ejpam-5914	329	19	{	{	PUNCT
ejpam-5914	329	20	2	2	NUM
ejpam-5914	329	21	,	,	PUNCT
ejpam-5914	329	22	4	4	NUM
ejpam-5914	329	23	}	}	PUNCT
ejpam-5914	329	24	,	,	PUNCT
ejpam-5914	329	25	then	then	ADV
ejpam-5914	329	26	(	(	PUNCT
ejpam-5914	329	27	∀x	∀x	X
ejpam-5914	329	28	∈	∈	PROPN
ejpam-5914	329	29	a)(f̃m(0	a)(f̃m(0	NOUN
ejpam-5914	329	30	)	)	PUNCT
ejpam-5914	329	31	≤	≤	NOUN
ejpam-5914	329	32	f̃m(x	f̃m(x	NUM
ejpam-5914	329	33	)	)	PUNCT
ejpam-5914	329	34	)	)	PUNCT
ejpam-5914	329	35	.	.	PUNCT
ejpam-5914	330	1	(	(	PUNCT
ejpam-5914	330	2	4	4	X
ejpam-5914	330	3	)	)	PUNCT
ejpam-5914	330	4	proof	proof	NOUN
ejpam-5914	330	5	.	.	PUNCT
ejpam-5914	331	1	let	let	VERB
ejpam-5914	331	2	(	(	PUNCT
ejpam-5914	331	3	a	a	PRON
ejpam-5914	331	4	,	,	PUNCT
ejpam-5914	331	5	f̃	f̃	PROPN
ejpam-5914	331	6	)	)	PUNCT
ejpam-5914	331	7	be	be	VERB
ejpam-5914	331	8	a	a	DET
ejpam-5914	331	9	mean	mean	ADJ
ejpam-5914	331	10	k	k	ADJ
ejpam-5914	331	11	-	-	ADJ
ejpam-5914	331	12	fuzzy	fuzzy	ADJ
ejpam-5914	331	13	subalgebra	subalgebra	NOUN
ejpam-5914	331	14	of	of	ADP
ejpam-5914	331	15	a	a	PRON
ejpam-5914	331	16	for	for	ADP
ejpam-5914	331	17	k	k	PROPN
ejpam-5914	331	18	∈	∈	PROPN
ejpam-5914	331	19	{	{	PUNCT
ejpam-5914	331	20	2	2	NUM
ejpam-5914	331	21	,	,	PUNCT
ejpam-5914	331	22	4	4	NUM
ejpam-5914	331	23	}	}	PUNCT
ejpam-5914	331	24	.	.	PUNCT
ejpam-5914	332	1	then	then	ADV
ejpam-5914	332	2	f̃m(0	f̃m(0	PROPN
ejpam-5914	332	3	)	)	PUNCT
ejpam-5914	332	4	=	=	SYM
ejpam-5914	332	5	f̃sup(0	f̃sup(0	NOUN
ejpam-5914	332	6	)	)	PUNCT
ejpam-5914	333	1	+	+	CCONJ
ejpam-5914	333	2	f̃inf(0	f̃inf(0	NOUN
ejpam-5914	333	3	)	)	PUNCT
ejpam-5914	333	4	2	2	NUM
ejpam-5914	333	5	≤	≤	NOUN
ejpam-5914	333	6	f̃sup(x	f̃sup(x	NOUN
ejpam-5914	333	7	)	)	PUNCT
ejpam-5914	333	8	+	+	SYM
ejpam-5914	333	9	f̃inf(x	f̃inf(x	NOUN
ejpam-5914	333	10	)	)	PUNCT
ejpam-5914	333	11	2	2	NUM
ejpam-5914	333	12	=	=	SYM
ejpam-5914	333	13	f̃m(x	f̃m(x	NUM
ejpam-5914	333	14	)	)	PUNCT
ejpam-5914	333	15	.	.	PUNCT
ejpam-5914	334	1	theorem	theorem	VERB
ejpam-5914	334	2	13	13	NUM
ejpam-5914	334	3	.	.	PUNCT
ejpam-5914	335	1	every	every	DET
ejpam-5914	335	2	mean	mean	NOUN
ejpam-5914	335	3	3	3	NUM
ejpam-5914	335	4	-	-	PUNCT
ejpam-5914	335	5	fuzzy	fuzzy	ADJ
ejpam-5914	335	6	subalgebra	subalgebra	NOUN
ejpam-5914	335	7	of	of	ADP
ejpam-5914	335	8	a	a	PRON
ejpam-5914	335	9	is	be	AUX
ejpam-5914	335	10	a	a	DET
ejpam-5914	335	11	mean	mean	ADJ
ejpam-5914	335	12	1	1	NUM
ejpam-5914	335	13	-	-	PUNCT
ejpam-5914	335	14	fuzzy	fuzzy	ADJ
ejpam-5914	335	15	subalgebra	subalgebra	NOUN
ejpam-5914	335	16	.	.	PUNCT
ejpam-5914	336	1	n.	n.	PROPN
ejpam-5914	336	2	rajesh	rajesh	PROPN
ejpam-5914	336	3	et	et	PROPN
ejpam-5914	336	4	al	al	PROPN
ejpam-5914	336	5	.	.	PUNCT
ejpam-5914	336	6	/	/	SYM
ejpam-5914	336	7	eur	eur	PROPN
ejpam-5914	336	8	.	.	PUNCT
ejpam-5914	337	1	j.	j.	PROPN
ejpam-5914	337	2	pure	pure	PROPN
ejpam-5914	337	3	appl	appl	PROPN
ejpam-5914	337	4	.	.	PROPN
ejpam-5914	337	5	math	math	PROPN
ejpam-5914	337	6	,	,	PUNCT
ejpam-5914	337	7	18	18	NUM
ejpam-5914	337	8	(	(	PUNCT
ejpam-5914	337	9	2	2	NUM
ejpam-5914	337	10	)	)	PUNCT
ejpam-5914	337	11	(	(	PUNCT
ejpam-5914	337	12	2025	2025	NUM
ejpam-5914	337	13	)	)	PUNCT
ejpam-5914	337	14	,	,	PUNCT
ejpam-5914	337	15	5914	5914	NUM
ejpam-5914	337	16	13	13	NUM
ejpam-5914	337	17	of	of	ADP
ejpam-5914	337	18	21	21	NUM
ejpam-5914	337	19	proof	proof	NOUN
ejpam-5914	337	20	.	.	PUNCT
ejpam-5914	338	1	let	let	VERB
ejpam-5914	338	2	(	(	PUNCT
ejpam-5914	338	3	a	a	PRON
ejpam-5914	338	4	,	,	PUNCT
ejpam-5914	338	5	f̃	f̃	PROPN
ejpam-5914	338	6	)	)	PUNCT
ejpam-5914	338	7	be	be	VERB
ejpam-5914	338	8	a	a	DET
ejpam-5914	338	9	mean	mean	ADJ
ejpam-5914	338	10	3	3	NUM
ejpam-5914	338	11	-	-	PUNCT
ejpam-5914	338	12	fuzzy	fuzzy	ADJ
ejpam-5914	338	13	subalgebra	subalgebra	NOUN
ejpam-5914	338	14	of	of	ADP
ejpam-5914	338	15	a.	a.	NOUN
ejpam-5914	338	16	then	then	ADV
ejpam-5914	338	17	f̃m((x|(y|y))|(x|(y|y	f̃m((x|(y|y))|(x|(y|y	PROPN
ejpam-5914	338	18	)	)	PUNCT
ejpam-5914	338	19	)	)	PUNCT
ejpam-5914	338	20	)	)	PUNCT
ejpam-5914	339	1	=	=	SYM
ejpam-5914	339	2	f̃sup((x|(y|y))|(x|(y|y	f̃sup((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	339	3	)	)	PUNCT
ejpam-5914	339	4	)	)	PUNCT
ejpam-5914	339	5	)	)	PUNCT
ejpam-5914	340	1	+	+	CCONJ
ejpam-5914	340	2	f̃inf((x|(y|y))|(x|(y|y	f̃inf((x|(y|y))|(x|(y|y	ADJ
ejpam-5914	340	3	)	)	PUNCT
ejpam-5914	340	4	)	)	PUNCT
ejpam-5914	340	5	)	)	PUNCT
ejpam-5914	340	6	2	2	NUM
ejpam-5914	340	7	=	=	SYM
ejpam-5914	340	8	f̃sup((x|(y|y))|(x|(y|y	f̃sup((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	340	9	)	)	PUNCT
ejpam-5914	340	10	)	)	PUNCT
ejpam-5914	340	11	)	)	PUNCT
ejpam-5914	340	12	2	2	NUM
ejpam-5914	340	13	+	+	CCONJ
ejpam-5914	340	14	f̃inf((x|(y|y))|(x|(y|y	f̃inf((x|(y|y))|(x|(y|y	ADJ
ejpam-5914	340	15	)	)	PUNCT
ejpam-5914	340	16	)	)	PUNCT
ejpam-5914	340	17	)	)	PUNCT
ejpam-5914	341	1	2	2	NUM
ejpam-5914	341	2	≥	≥	NOUN
ejpam-5914	341	3	max	max	PROPN
ejpam-5914	341	4	{	{	PUNCT
ejpam-5914	341	5	f̃sup(x	f̃sup(x	NOUN
ejpam-5914	341	6	)	)	PUNCT
ejpam-5914	341	7	2	2	NUM
ejpam-5914	341	8	,	,	PUNCT
ejpam-5914	341	9	f̃sup(y	f̃sup(y	NOUN
ejpam-5914	341	10	)	)	PUNCT
ejpam-5914	341	11	2	2	NUM
ejpam-5914	341	12	}	}	PUNCT
ejpam-5914	341	13	+	+	NOUN
ejpam-5914	341	14	max	max	PROPN
ejpam-5914	341	15	{	{	PUNCT
ejpam-5914	341	16	f̃inf(x	f̃inf(x	PROPN
ejpam-5914	341	17	)	)	PUNCT
ejpam-5914	341	18	2	2	NUM
ejpam-5914	341	19	,	,	PUNCT
ejpam-5914	341	20	f̃inf(y	f̃inf(y	NOUN
ejpam-5914	341	21	)	)	PUNCT
ejpam-5914	341	22	2	2	NUM
ejpam-5914	341	23	}	}	PUNCT
ejpam-5914	341	24	≥	≥	NOUN
ejpam-5914	341	25	min	min	NOUN
ejpam-5914	341	26	{	{	PUNCT
ejpam-5914	341	27	f̃sup(x	f̃sup(x	ADJ
ejpam-5914	341	28	)	)	PUNCT
ejpam-5914	341	29	2	2	NUM
ejpam-5914	341	30	,	,	PUNCT
ejpam-5914	341	31	f̃sup(y	f̃sup(y	NOUN
ejpam-5914	341	32	)	)	PUNCT
ejpam-5914	341	33	2	2	NUM
ejpam-5914	341	34	}	}	PUNCT
ejpam-5914	341	35	+	+	NOUN
ejpam-5914	341	36	min	min	NOUN
ejpam-5914	341	37	{	{	PUNCT
ejpam-5914	341	38	f̃inf(x	f̃inf(x	PROPN
ejpam-5914	341	39	)	)	PUNCT
ejpam-5914	341	40	2	2	NUM
ejpam-5914	341	41	,	,	PUNCT
ejpam-5914	341	42	f̃inf(y	f̃inf(y	NOUN
ejpam-5914	341	43	)	)	PUNCT
ejpam-5914	341	44	2	2	NUM
ejpam-5914	341	45	}	}	PUNCT
ejpam-5914	341	46	=	=	SYM
ejpam-5914	341	47	min	min	NOUN
ejpam-5914	341	48	{	{	PUNCT
ejpam-5914	341	49	f̃sup(x	f̃sup(x	NOUN
ejpam-5914	341	50	)	)	PUNCT
ejpam-5914	341	51	+	+	SYM
ejpam-5914	341	52	f̃inf(x	f̃inf(x	NOUN
ejpam-5914	341	53	)	)	PUNCT
ejpam-5914	341	54	2	2	NUM
ejpam-5914	341	55	,	,	PUNCT
ejpam-5914	341	56	f̃sup(y	f̃sup(y	NOUN
ejpam-5914	341	57	)	)	PUNCT
ejpam-5914	341	58	+	+	SYM
ejpam-5914	341	59	f̃inf(y	f̃inf(y	NOUN
ejpam-5914	341	60	)	)	PUNCT
ejpam-5914	341	61	2	2	NUM
ejpam-5914	341	62	}	}	PUNCT
ejpam-5914	341	63	=	=	SYM
ejpam-5914	341	64	min{f̃m(x	min{f̃m(x	PROPN
ejpam-5914	341	65	)	)	PUNCT
ejpam-5914	341	66	,	,	PUNCT
ejpam-5914	341	67	f̃m(y	f̃m(y	NOUN
ejpam-5914	341	68	)	)	PUNCT
ejpam-5914	341	69	}	}	PUNCT
ejpam-5914	341	70	.	.	PUNCT
ejpam-5914	342	1	hence	hence	ADV
ejpam-5914	342	2	,	,	PUNCT
ejpam-5914	342	3	(	(	PUNCT
ejpam-5914	342	4	a	a	PRON
ejpam-5914	342	5	,	,	PUNCT
ejpam-5914	342	6	f̃	f̃	PROPN
ejpam-5914	342	7	)	)	PUNCT
ejpam-5914	342	8	is	be	AUX
ejpam-5914	342	9	a	a	DET
ejpam-5914	342	10	mean	mean	ADJ
ejpam-5914	342	11	1	1	NUM
ejpam-5914	342	12	-	-	PUNCT
ejpam-5914	342	13	fuzzy	fuzzy	ADJ
ejpam-5914	342	14	subalgebra	subalgebra	NOUN
ejpam-5914	342	15	of	of	ADP
ejpam-5914	342	16	a.	a.	NOUN
ejpam-5914	342	17	theorem	theorem	NOUN
ejpam-5914	342	18	14	14	NUM
ejpam-5914	342	19	.	.	PUNCT
ejpam-5914	343	1	every	every	DET
ejpam-5914	343	2	mean	mean	ADJ
ejpam-5914	343	3	2	2	NUM
ejpam-5914	343	4	-	-	PUNCT
ejpam-5914	343	5	fuzzy	fuzzy	ADJ
ejpam-5914	343	6	subalgebra	subalgebra	NOUN
ejpam-5914	343	7	of	of	ADP
ejpam-5914	343	8	a	a	PRON
ejpam-5914	343	9	is	be	AUX
ejpam-5914	343	10	a	a	DET
ejpam-5914	343	11	mean	mean	ADJ
ejpam-5914	343	12	4	4	NUM
ejpam-5914	343	13	-	-	PUNCT
ejpam-5914	343	14	fuzzy	fuzzy	ADJ
ejpam-5914	343	15	subalgebra	subalgebra	NOUN
ejpam-5914	343	16	.	.	PUNCT
ejpam-5914	344	1	proof	proof	NOUN
ejpam-5914	344	2	.	.	PUNCT
ejpam-5914	345	1	let	let	VERB
ejpam-5914	345	2	(	(	PUNCT
ejpam-5914	345	3	a	a	PRON
ejpam-5914	345	4	,	,	PUNCT
ejpam-5914	345	5	f̃	f̃	PROPN
ejpam-5914	345	6	)	)	PUNCT
ejpam-5914	345	7	be	be	VERB
ejpam-5914	345	8	a	a	DET
ejpam-5914	345	9	mean	mean	ADJ
ejpam-5914	345	10	2	2	NUM
ejpam-5914	345	11	-	-	PUNCT
ejpam-5914	345	12	fuzzy	fuzzy	ADJ
ejpam-5914	345	13	subalgebra	subalgebra	NOUN
ejpam-5914	345	14	of	of	ADP
ejpam-5914	345	15	a.	a.	NOUN
ejpam-5914	345	16	then	then	ADV
ejpam-5914	345	17	f̃m((x|(y|y))|(x|(y|y	f̃m((x|(y|y))|(x|(y|y	PROPN
ejpam-5914	345	18	)	)	PUNCT
ejpam-5914	345	19	)	)	PUNCT
ejpam-5914	345	20	)	)	PUNCT
ejpam-5914	346	1	=	=	SYM
ejpam-5914	346	2	f̃sup((x|(y|y))|(x|(y|y	f̃sup((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	346	3	)	)	PUNCT
ejpam-5914	346	4	)	)	PUNCT
ejpam-5914	346	5	)	)	PUNCT
ejpam-5914	347	1	+	+	CCONJ
ejpam-5914	347	2	f̃inf((x|(y|y))|(x|(y|y	f̃inf((x|(y|y))|(x|(y|y	ADJ
ejpam-5914	347	3	)	)	PUNCT
ejpam-5914	347	4	)	)	PUNCT
ejpam-5914	347	5	)	)	PUNCT
ejpam-5914	347	6	2	2	NUM
ejpam-5914	347	7	=	=	SYM
ejpam-5914	347	8	f̃sup((x|(y|y))|(x|(y|y	f̃sup((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	347	9	)	)	PUNCT
ejpam-5914	347	10	)	)	PUNCT
ejpam-5914	347	11	)	)	PUNCT
ejpam-5914	347	12	2	2	NUM
ejpam-5914	347	13	+	+	CCONJ
ejpam-5914	347	14	f̃inf((x|(y|y))|(x|(y|y	f̃inf((x|(y|y))|(x|(y|y	ADJ
ejpam-5914	347	15	)	)	PUNCT
ejpam-5914	347	16	)	)	PUNCT
ejpam-5914	347	17	)	)	PUNCT
ejpam-5914	347	18	2	2	NUM
ejpam-5914	347	19	≤	≤	NOUN
ejpam-5914	347	20	min	min	NOUN
ejpam-5914	347	21	{	{	PUNCT
ejpam-5914	347	22	f̃sup(x	f̃sup(x	NOUN
ejpam-5914	347	23	)	)	PUNCT
ejpam-5914	347	24	2	2	NUM
ejpam-5914	347	25	,	,	PUNCT
ejpam-5914	347	26	f̃sup(y	f̃sup(y	NOUN
ejpam-5914	347	27	)	)	PUNCT
ejpam-5914	347	28	2	2	NUM
ejpam-5914	347	29	}	}	PUNCT
ejpam-5914	347	30	+	+	NOUN
ejpam-5914	347	31	min	min	NOUN
ejpam-5914	347	32	{	{	PUNCT
ejpam-5914	347	33	f̃inf(x	f̃inf(x	PROPN
ejpam-5914	347	34	)	)	PUNCT
ejpam-5914	347	35	2	2	NUM
ejpam-5914	347	36	,	,	PUNCT
ejpam-5914	347	37	f̃inf(y	f̃inf(y	NOUN
ejpam-5914	347	38	)	)	PUNCT
ejpam-5914	347	39	2	2	NUM
ejpam-5914	347	40	}	}	PUNCT
ejpam-5914	347	41	≤	≤	NUM
ejpam-5914	347	42	max	max	PROPN
ejpam-5914	347	43	{	{	PUNCT
ejpam-5914	347	44	f̃sup(x	f̃sup(x	NOUN
ejpam-5914	347	45	)	)	PUNCT
ejpam-5914	347	46	2	2	NUM
ejpam-5914	347	47	,	,	PUNCT
ejpam-5914	347	48	f̃sup(y	f̃sup(y	NOUN
ejpam-5914	347	49	)	)	PUNCT
ejpam-5914	347	50	2	2	NUM
ejpam-5914	347	51	}	}	PUNCT
ejpam-5914	347	52	+	+	NOUN
ejpam-5914	347	53	max	max	PROPN
ejpam-5914	347	54	{	{	PUNCT
ejpam-5914	347	55	f̃inf(x	f̃inf(x	PROPN
ejpam-5914	347	56	)	)	PUNCT
ejpam-5914	347	57	2	2	NUM
ejpam-5914	347	58	,	,	PUNCT
ejpam-5914	347	59	f̃inf(y	f̃inf(y	NOUN
ejpam-5914	347	60	)	)	PUNCT
ejpam-5914	347	61	2	2	NUM
ejpam-5914	347	62	}	}	PUNCT
ejpam-5914	347	63	=	=	SYM
ejpam-5914	347	64	max	max	PROPN
ejpam-5914	347	65	{	{	PUNCT
ejpam-5914	347	66	f̃sup(x	f̃sup(x	NOUN
ejpam-5914	347	67	)	)	PUNCT
ejpam-5914	347	68	+	+	SYM
ejpam-5914	347	69	f̃inf(x	f̃inf(x	NOUN
ejpam-5914	347	70	)	)	PUNCT
ejpam-5914	347	71	2	2	NUM
ejpam-5914	347	72	,	,	PUNCT
ejpam-5914	347	73	f̃sup(y	f̃sup(y	NOUN
ejpam-5914	347	74	)	)	PUNCT
ejpam-5914	347	75	+	+	SYM
ejpam-5914	347	76	f̃inf(y	f̃inf(y	NOUN
ejpam-5914	347	77	)	)	PUNCT
ejpam-5914	347	78	2	2	NUM
ejpam-5914	347	79	}	}	PUNCT
ejpam-5914	347	80	=	=	SYM
ejpam-5914	347	81	max{f̃m(x	max{f̃m(x	PROPN
ejpam-5914	347	82	)	)	PUNCT
ejpam-5914	347	83	,	,	PUNCT
ejpam-5914	347	84	f̃m(y	f̃m(y	NOUN
ejpam-5914	347	85	)	)	PUNCT
ejpam-5914	347	86	}	}	PUNCT
ejpam-5914	347	87	.	.	PUNCT
ejpam-5914	348	1	hence	hence	ADV
ejpam-5914	348	2	,	,	PUNCT
ejpam-5914	348	3	(	(	PUNCT
ejpam-5914	348	4	a	a	PRON
ejpam-5914	348	5	,	,	PUNCT
ejpam-5914	348	6	f̃	f̃	PROPN
ejpam-5914	348	7	)	)	PUNCT
ejpam-5914	348	8	is	be	AUX
ejpam-5914	348	9	a	a	DET
ejpam-5914	348	10	mean	mean	ADJ
ejpam-5914	348	11	4	4	NUM
ejpam-5914	348	12	-	-	PUNCT
ejpam-5914	348	13	fuzzy	fuzzy	ADJ
ejpam-5914	348	14	subalgebra	subalgebra	NOUN
ejpam-5914	348	15	of	of	ADP
ejpam-5914	348	16	a.	a.	NOUN
ejpam-5914	348	17	theorem	theorem	NOUN
ejpam-5914	348	18	15	15	NUM
ejpam-5914	348	19	.	.	PUNCT
ejpam-5914	349	1	mean	mean	VERB
ejpam-5914	349	2	2	2	NUM
ejpam-5914	349	3	-	-	PUNCT
ejpam-5914	349	4	fuzzy	fuzzy	ADJ
ejpam-5914	349	5	subalgebra	subalgebra	NOUN
ejpam-5914	349	6	and	and	CCONJ
ejpam-5914	349	7	mean	mean	VERB
ejpam-5914	349	8	3	3	NUM
ejpam-5914	349	9	-	-	PUNCT
ejpam-5914	349	10	fuzzy	fuzzy	ADJ
ejpam-5914	349	11	subalgebra	subalgebra	NOUN
ejpam-5914	349	12	of	of	ADP
ejpam-5914	349	13	a	a	DET
ejpam-5914	349	14	coincide	coincide	NOUN
ejpam-5914	349	15	.	.	PUNCT
ejpam-5914	350	1	proof	proof	NOUN
ejpam-5914	350	2	.	.	PUNCT
ejpam-5914	351	1	it	it	PRON
ejpam-5914	351	2	is	be	AUX
ejpam-5914	351	3	straightforward	straightforward	ADJ
ejpam-5914	351	4	by	by	ADP
ejpam-5914	351	5	theorems	theorem	NOUN
ejpam-5914	351	6	13	13	NUM
ejpam-5914	351	7	and	and	CCONJ
ejpam-5914	351	8	14	14	NUM
ejpam-5914	351	9	.	.	PUNCT
ejpam-5914	352	1	theorem	theorem	VERB
ejpam-5914	352	2	16	16	NUM
ejpam-5914	352	3	.	.	PUNCT
ejpam-5914	353	1	given	give	VERB
ejpam-5914	353	2	a	a	DET
ejpam-5914	353	3	subalgebra	subalgebra	NOUN
ejpam-5914	353	4	s	s	NOUN
ejpam-5914	353	5	of	of	ADP
ejpam-5914	353	6	a	a	PRON
ejpam-5914	353	7	and	and	CCONJ
ejpam-5914	353	8	b1	b1	NOUN
ejpam-5914	353	9	,	,	PUNCT
ejpam-5914	353	10	b2	b2	NOUN
ejpam-5914	353	11	∈	∈	NOUN
ejpam-5914	353	12	d[0	d[0	PROPN
ejpam-5914	353	13	,	,	PUNCT
ejpam-5914	353	14	1	1	NUM
ejpam-5914	353	15	]	]	PUNCT
ejpam-5914	353	16	,	,	PUNCT
ejpam-5914	353	17	let	let	VERB
ejpam-5914	353	18	(	(	PUNCT
ejpam-5914	353	19	a	a	PRON
ejpam-5914	353	20	,	,	PUNCT
ejpam-5914	353	21	f̃	f̃	PROPN
ejpam-5914	353	22	)	)	PUNCT
ejpam-5914	353	23	be	be	VERB
ejpam-5914	353	24	an	an	DET
ejpam-5914	353	25	intervalvalued	intervalvalue	VERB
ejpam-5914	353	26	fuzzy	fuzzy	ADJ
ejpam-5914	353	27	structure	structure	NOUN
ejpam-5914	353	28	over	over	ADP
ejpam-5914	353	29	a	a	DET
ejpam-5914	353	30	given	give	VERB
ejpam-5914	353	31	by	by	ADP
ejpam-5914	353	32	f̃	f̃	PROPN
ejpam-5914	353	33	:	:	PUNCT
ejpam-5914	353	34	a	a	DET
ejpam-5914	353	35	→	→	SYM
ejpam-5914	353	36	d[0	d[0	ADJ
ejpam-5914	353	37	,	,	PUNCT
ejpam-5914	353	38	1];x	1];x	NUM
ejpam-5914	353	39	7→	7→	NUM
ejpam-5914	353	40	{	{	PUNCT
ejpam-5914	353	41	b2	b2	NOUN
ejpam-5914	353	42	if	if	SCONJ
ejpam-5914	353	43	x	x	PROPN
ejpam-5914	353	44	∈	∈	PROPN
ejpam-5914	353	45	s	s	PART
ejpam-5914	353	46	,	,	PUNCT
ejpam-5914	353	47	b1	b1	VERB
ejpam-5914	353	48	otherwise	otherwise	ADV
ejpam-5914	353	49	.	.	PUNCT
ejpam-5914	354	1	n.	n.	PROPN
ejpam-5914	354	2	rajesh	rajesh	PROPN
ejpam-5914	354	3	et	et	PROPN
ejpam-5914	354	4	al	al	PROPN
ejpam-5914	354	5	.	.	PUNCT
ejpam-5914	354	6	/	/	SYM
ejpam-5914	354	7	eur	eur	PROPN
ejpam-5914	354	8	.	.	PUNCT
ejpam-5914	355	1	j.	j.	PROPN
ejpam-5914	355	2	pure	pure	PROPN
ejpam-5914	355	3	appl	appl	PROPN
ejpam-5914	355	4	.	.	PROPN
ejpam-5914	355	5	math	math	PROPN
ejpam-5914	355	6	,	,	PUNCT
ejpam-5914	355	7	18	18	NUM
ejpam-5914	355	8	(	(	PUNCT
ejpam-5914	355	9	2	2	NUM
ejpam-5914	355	10	)	)	PUNCT
ejpam-5914	355	11	(	(	PUNCT
ejpam-5914	355	12	2025	2025	NUM
ejpam-5914	355	13	)	)	PUNCT
ejpam-5914	355	14	,	,	PUNCT
ejpam-5914	355	15	5914	5914	NUM
ejpam-5914	355	16	14	14	NUM
ejpam-5914	355	17	of	of	ADP
ejpam-5914	355	18	21	21	NUM
ejpam-5914	355	19	(	(	PUNCT
ejpam-5914	355	20	1	1	NUM
ejpam-5914	355	21	)	)	PUNCT
ejpam-5914	355	22	if	if	SCONJ
ejpam-5914	355	23	supb2	supb2	PROPN
ejpam-5914	355	24	≥	≥	AUX
ejpam-5914	355	25	supb1	supb1	NOUN
ejpam-5914	355	26	and	and	CCONJ
ejpam-5914	355	27	inf	inf	PROPN
ejpam-5914	355	28	b2	b2	PROPN
ejpam-5914	355	29	≥	≥	PROPN
ejpam-5914	355	30	inf	inf	PROPN
ejpam-5914	355	31	b1	b1	NOUN
ejpam-5914	355	32	,	,	PUNCT
ejpam-5914	355	33	then	then	ADV
ejpam-5914	355	34	(	(	PUNCT
ejpam-5914	355	35	a	a	PRON
ejpam-5914	355	36	,	,	PUNCT
ejpam-5914	355	37	f̃	f̃	PROPN
ejpam-5914	355	38	)	)	PUNCT
ejpam-5914	355	39	is	be	AUX
ejpam-5914	355	40	a	a	DET
ejpam-5914	355	41	mean	mean	ADJ
ejpam-5914	355	42	1	1	NUM
ejpam-5914	355	43	-	-	PUNCT
ejpam-5914	355	44	fuzzy	fuzzy	ADJ
ejpam-5914	355	45	subalgebra	subalgebra	NOUN
ejpam-5914	355	46	of	of	ADP
ejpam-5914	355	47	a.	a.	NOUN
ejpam-5914	355	48	(	(	PUNCT
ejpam-5914	355	49	2	2	NUM
ejpam-5914	355	50	)	)	PUNCT
ejpam-5914	355	51	if	if	SCONJ
ejpam-5914	355	52	supb2	supb2	PROPN
ejpam-5914	355	53	≤	≤	X
ejpam-5914	355	54	supb1	supb1	NOUN
ejpam-5914	355	55	and	and	CCONJ
ejpam-5914	355	56	inf	inf	PROPN
ejpam-5914	355	57	b2	b2	PROPN
ejpam-5914	355	58	≤	≤	PROPN
ejpam-5914	355	59	inf	inf	PROPN
ejpam-5914	355	60	b1	b1	NOUN
ejpam-5914	355	61	,	,	PUNCT
ejpam-5914	355	62	then	then	ADV
ejpam-5914	355	63	(	(	PUNCT
ejpam-5914	355	64	a	a	PRON
ejpam-5914	355	65	,	,	PUNCT
ejpam-5914	355	66	f̃	f̃	PROPN
ejpam-5914	355	67	)	)	PUNCT
ejpam-5914	355	68	is	be	AUX
ejpam-5914	355	69	a	a	DET
ejpam-5914	355	70	mean	mean	ADJ
ejpam-5914	355	71	4	4	NUM
ejpam-5914	355	72	-	-	PUNCT
ejpam-5914	355	73	fuzzy	fuzzy	ADJ
ejpam-5914	355	74	subalgebra	subalgebra	NOUN
ejpam-5914	355	75	of	of	ADP
ejpam-5914	355	76	a.	a.	NOUN
ejpam-5914	355	77	proof	proof	NOUN
ejpam-5914	355	78	.	.	PUNCT
ejpam-5914	356	1	if	if	SCONJ
ejpam-5914	356	2	x	x	SYM
ejpam-5914	356	3	∈	∈	PROPN
ejpam-5914	356	4	s	s	NOUN
ejpam-5914	356	5	,	,	PUNCT
ejpam-5914	356	6	then	then	ADV
ejpam-5914	356	7	f̃(x	f̃(x	PROPN
ejpam-5914	356	8	)	)	PUNCT
ejpam-5914	356	9	=	=	SYM
ejpam-5914	356	10	b2	b2	NOUN
ejpam-5914	356	11	and	and	CCONJ
ejpam-5914	356	12	so	so	ADV
ejpam-5914	356	13	f̃m(x	f̃m(x	NUM
ejpam-5914	356	14	)	)	PUNCT
ejpam-5914	356	15	=	=	SYM
ejpam-5914	356	16	f̃sup(x	f̃sup(x	ADJ
ejpam-5914	356	17	)	)	PUNCT
ejpam-5914	356	18	+	+	SYM
ejpam-5914	356	19	f̃inf(x	f̃inf(x	NOUN
ejpam-5914	356	20	)	)	PUNCT
ejpam-5914	356	21	2	2	NUM
ejpam-5914	356	22	=	=	SYM
ejpam-5914	356	23	sup	sup	NOUN
ejpam-5914	356	24	f̃(x	f̃(x	PROPN
ejpam-5914	356	25	)	)	PUNCT
ejpam-5914	356	26	+	+	NUM
ejpam-5914	356	27	inf	inf	PROPN
ejpam-5914	356	28	f̃(x	f̃(x	PROPN
ejpam-5914	356	29	)	)	PUNCT
ejpam-5914	356	30	2	2	NUM
ejpam-5914	356	31	=	=	SYM
ejpam-5914	356	32	supb2	supb2	NOUN
ejpam-5914	356	33	+	+	CCONJ
ejpam-5914	356	34	inf	inf	ADJ
ejpam-5914	356	35	b2	b2	NOUN
ejpam-5914	356	36	2	2	NUM
ejpam-5914	356	37	.	.	PUNCT
ejpam-5914	357	1	if	if	SCONJ
ejpam-5914	357	2	x	x	PROPN
ejpam-5914	357	3	/∈	/∈	PROPN
ejpam-5914	357	4	s	s	X
ejpam-5914	357	5	,	,	PUNCT
ejpam-5914	357	6	then	then	ADV
ejpam-5914	357	7	f̃(x	f̃(x	PROPN
ejpam-5914	357	8	)	)	PUNCT
ejpam-5914	358	1	=	=	SYM
ejpam-5914	358	2	b1	b1	NOUN
ejpam-5914	358	3	and	and	CCONJ
ejpam-5914	358	4	so	so	ADV
ejpam-5914	358	5	f̃m(x	f̃m(x	NUM
ejpam-5914	358	6	)	)	PUNCT
ejpam-5914	358	7	=	=	SYM
ejpam-5914	358	8	f̃sup(x	f̃sup(x	ADJ
ejpam-5914	358	9	)	)	PUNCT
ejpam-5914	358	10	+	+	SYM
ejpam-5914	358	11	f̃inf(x	f̃inf(x	NOUN
ejpam-5914	358	12	)	)	PUNCT
ejpam-5914	358	13	2	2	NUM
ejpam-5914	358	14	=	=	SYM
ejpam-5914	358	15	sup	sup	NOUN
ejpam-5914	358	16	f̃(x	f̃(x	PROPN
ejpam-5914	358	17	)	)	PUNCT
ejpam-5914	358	18	+	+	NUM
ejpam-5914	358	19	inf	inf	PROPN
ejpam-5914	358	20	f̃(x	f̃(x	PROPN
ejpam-5914	358	21	)	)	PUNCT
ejpam-5914	358	22	2	2	NUM
ejpam-5914	358	23	=	=	SYM
ejpam-5914	358	24	supb1	supb1	NOUN
ejpam-5914	358	25	+	+	CCONJ
ejpam-5914	358	26	inf	inf	ADJ
ejpam-5914	358	27	b1	b1	NOUN
ejpam-5914	358	28	2	2	NUM
ejpam-5914	358	29	.	.	PUNCT
ejpam-5914	359	1	(	(	PUNCT
ejpam-5914	359	2	1	1	X
ejpam-5914	359	3	)	)	PUNCT
ejpam-5914	359	4	assume	assume	VERB
ejpam-5914	359	5	that	that	SCONJ
ejpam-5914	359	6	supb2	supb2	PROPN
ejpam-5914	359	7	≥	≥	NUM
ejpam-5914	359	8	supb1	supb1	NOUN
ejpam-5914	359	9	and	and	CCONJ
ejpam-5914	359	10	inf	inf	PROPN
ejpam-5914	359	11	b2	b2	PROPN
ejpam-5914	359	12	≥	≥	PROPN
ejpam-5914	359	13	inf	inf	PROPN
ejpam-5914	359	14	b1	b1	NOUN
ejpam-5914	359	15	.	.	PUNCT
ejpam-5914	360	1	then	then	ADV
ejpam-5914	360	2	supb2	supb2	PROPN
ejpam-5914	360	3	+	+	CCONJ
ejpam-5914	360	4	inf	inf	ADJ
ejpam-5914	360	5	b2	b2	NOUN
ejpam-5914	360	6	2	2	NUM
ejpam-5914	360	7	≥	≥	NOUN
ejpam-5914	360	8	supb1	supb1	NOUN
ejpam-5914	360	9	+	+	CCONJ
ejpam-5914	360	10	inf	inf	ADJ
ejpam-5914	360	11	b1	b1	NOUN
ejpam-5914	360	12	2	2	NUM
ejpam-5914	360	13	.	.	PUNCT
ejpam-5914	361	1	case	case	NOUN
ejpam-5914	361	2	1	1	NUM
ejpam-5914	361	3	:	:	PUNCT
ejpam-5914	361	4	let	let	VERB
ejpam-5914	361	5	x	x	PRON
ejpam-5914	361	6	,	,	PUNCT
ejpam-5914	361	7	y	y	PROPN
ejpam-5914	361	8	∈	∈	PROPN
ejpam-5914	361	9	s.	s.	PROPN
ejpam-5914	361	10	then	then	ADV
ejpam-5914	361	11	f̃m(x	f̃m(x	PUNCT
ejpam-5914	361	12	)	)	PUNCT
ejpam-5914	361	13	=	=	SYM
ejpam-5914	361	14	supb2	supb2	NOUN
ejpam-5914	361	15	+	+	CCONJ
ejpam-5914	361	16	inf	inf	ADJ
ejpam-5914	361	17	b2	b2	NOUN
ejpam-5914	361	18	2	2	NUM
ejpam-5914	361	19	and	and	CCONJ
ejpam-5914	361	20	fm(y	fm(y	NUM
ejpam-5914	361	21	)	)	PUNCT
ejpam-5914	361	22	=	=	SYM
ejpam-5914	361	23	supb2	supb2	NOUN
ejpam-5914	361	24	+	+	CCONJ
ejpam-5914	361	25	inf	inf	ADJ
ejpam-5914	361	26	b2	b2	NOUN
ejpam-5914	361	27	2	2	NUM
ejpam-5914	361	28	.	.	PUNCT
ejpam-5914	362	1	thus	thus	ADV
ejpam-5914	362	2	,	,	PUNCT
ejpam-5914	362	3	min{f̃m(x	min{f̃m(x	PROPN
ejpam-5914	362	4	)	)	PUNCT
ejpam-5914	362	5	,	,	PUNCT
ejpam-5914	362	6	f̃m(y	f̃m(y	NOUN
ejpam-5914	362	7	)	)	PUNCT
ejpam-5914	362	8	}	}	PUNCT
ejpam-5914	362	9	=	=	SYM
ejpam-5914	362	10	supb2	supb2	NOUN
ejpam-5914	362	11	+	+	CCONJ
ejpam-5914	362	12	inf	inf	ADJ
ejpam-5914	362	13	b2	b2	NOUN
ejpam-5914	362	14	2	2	NUM
ejpam-5914	362	15	.	.	PUNCT
ejpam-5914	363	1	since	since	SCONJ
ejpam-5914	363	2	s	s	PROPN
ejpam-5914	363	3	is	be	AUX
ejpam-5914	363	4	a	a	DET
ejpam-5914	363	5	subalgebra	subalgebra	NOUN
ejpam-5914	363	6	of	of	ADP
ejpam-5914	363	7	a	a	PRON
ejpam-5914	363	8	,	,	PUNCT
ejpam-5914	363	9	we	we	PRON
ejpam-5914	363	10	have	have	VERB
ejpam-5914	363	11	(	(	PUNCT
ejpam-5914	363	12	x|(y|y))|(x|(y|y	x|(y|y))|(x|(y|y	ADJ
ejpam-5914	363	13	)	)	PUNCT
ejpam-5914	363	14	)	)	PUNCT
ejpam-5914	364	1	∈	∈	PROPN
ejpam-5914	364	2	s	s	PART
ejpam-5914	364	3	and	and	CCONJ
ejpam-5914	364	4	so	so	ADV
ejpam-5914	364	5	f̃m((x|(y|y))|(x|(y|y	f̃m((x|(y|y))|(x|(y|y	PROPN
ejpam-5914	364	6	)	)	PUNCT
ejpam-5914	364	7	)	)	PUNCT
ejpam-5914	364	8	)	)	PUNCT
ejpam-5914	365	1	=	=	SYM
ejpam-5914	365	2	supb2	supb2	NOUN
ejpam-5914	365	3	+	+	CCONJ
ejpam-5914	365	4	inf	inf	ADJ
ejpam-5914	365	5	b2	b2	NOUN
ejpam-5914	365	6	2	2	NUM
ejpam-5914	365	7	.	.	PUNCT
ejpam-5914	366	1	thus	thus	ADV
ejpam-5914	366	2	,	,	PUNCT
ejpam-5914	366	3	f̃m((x|(y|y))|(x|(y|y	f̃m((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	366	4	)	)	PUNCT
ejpam-5914	366	5	)	)	PUNCT
ejpam-5914	366	6	)	)	PUNCT
ejpam-5914	367	1	=	=	SYM
ejpam-5914	367	2	supb2	supb2	NOUN
ejpam-5914	367	3	+	+	CCONJ
ejpam-5914	367	4	inf	inf	ADJ
ejpam-5914	367	5	b2	b2	NOUN
ejpam-5914	367	6	2	2	NUM
ejpam-5914	367	7	=	=	SYM
ejpam-5914	367	8	(	(	PUNCT
ejpam-5914	367	9	≥)min{f̃m(x	≥)min{f̃m(x	NUM
ejpam-5914	367	10	)	)	PUNCT
ejpam-5914	367	11	,	,	PUNCT
ejpam-5914	367	12	f̃m(y	f̃m(y	NOUN
ejpam-5914	367	13	)	)	PUNCT
ejpam-5914	367	14	}	}	PUNCT
ejpam-5914	367	15	.	.	PUNCT
ejpam-5914	368	1	case	case	NOUN
ejpam-5914	368	2	2	2	NUM
ejpam-5914	368	3	:	:	PUNCT
ejpam-5914	368	4	let	let	VERB
ejpam-5914	368	5	x	x	PRON
ejpam-5914	368	6	,	,	PUNCT
ejpam-5914	368	7	y	y	PROPN
ejpam-5914	368	8	/∈	/∈	PUNCT
ejpam-5914	368	9	s.	s.	PROPN
ejpam-5914	368	10	then	then	ADV
ejpam-5914	368	11	f̃m(x	f̃m(x	PUNCT
ejpam-5914	368	12	)	)	PUNCT
ejpam-5914	369	1	=	=	SYM
ejpam-5914	369	2	supb1	supb1	NOUN
ejpam-5914	369	3	+	+	CCONJ
ejpam-5914	369	4	inf	inf	ADJ
ejpam-5914	369	5	b1	b1	NOUN
ejpam-5914	369	6	2	2	NUM
ejpam-5914	369	7	and	and	CCONJ
ejpam-5914	369	8	f̃m(y	f̃m(y	PRON
ejpam-5914	369	9	)	)	PUNCT
ejpam-5914	370	1	=	=	SYM
ejpam-5914	371	1	supb1	supb1	NOUN
ejpam-5914	372	1	+	+	CCONJ
ejpam-5914	372	2	inf	inf	ADJ
ejpam-5914	372	3	b1	b1	NOUN
ejpam-5914	372	4	2	2	NUM
ejpam-5914	372	5	,	,	PUNCT
ejpam-5914	372	6	so	so	ADV
ejpam-5914	372	7	min{f̃m(x	min{f̃m(x	NOUN
ejpam-5914	372	8	)	)	PUNCT
ejpam-5914	372	9	,	,	PUNCT
ejpam-5914	372	10	f̃m(y	f̃m(y	NOUN
ejpam-5914	372	11	)	)	PUNCT
ejpam-5914	372	12	}	}	PUNCT
ejpam-5914	372	13	=	=	SYM
ejpam-5914	372	14	supb1	supb1	NOUN
ejpam-5914	372	15	+	+	CCONJ
ejpam-5914	372	16	inf	inf	ADJ
ejpam-5914	372	17	b1	b1	NOUN
ejpam-5914	372	18	2	2	NUM
ejpam-5914	372	19	.	.	PUNCT
ejpam-5914	373	1	thus	thus	ADV
ejpam-5914	373	2	,	,	PUNCT
ejpam-5914	373	3	f̃m((x|(y|y))|(x|(y|y	f̃m((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	373	4	)	)	PUNCT
ejpam-5914	373	5	)	)	PUNCT
ejpam-5914	373	6	)	)	PUNCT
ejpam-5914	373	7	≥	≥	PROPN
ejpam-5914	373	8	supb1	supb1	NOUN
ejpam-5914	374	1	+	+	CCONJ
ejpam-5914	374	2	inf	inf	ADJ
ejpam-5914	374	3	b1	b1	NOUN
ejpam-5914	374	4	2	2	NUM
ejpam-5914	374	5	=	=	SYM
ejpam-5914	374	6	min{f̃m(x	min{f̃m(x	PROPN
ejpam-5914	374	7	)	)	PUNCT
ejpam-5914	374	8	,	,	PUNCT
ejpam-5914	374	9	f̃m(y	f̃m(y	NOUN
ejpam-5914	374	10	)	)	PUNCT
ejpam-5914	374	11	}	}	PUNCT
ejpam-5914	374	12	.	.	PUNCT
ejpam-5914	375	1	case	case	NOUN
ejpam-5914	375	2	3	3	X
ejpam-5914	375	3	:	:	PUNCT
ejpam-5914	375	4	let	let	VERB
ejpam-5914	375	5	x	x	PUNCT
ejpam-5914	375	6	/∈	/∈	PRON
ejpam-5914	375	7	s	s	PART
ejpam-5914	375	8	and	and	CCONJ
ejpam-5914	375	9	y	y	PROPN
ejpam-5914	375	10	∈	∈	PROPN
ejpam-5914	375	11	s.	s.	PROPN
ejpam-5914	375	12	then	then	ADV
ejpam-5914	375	13	f̃m(x	f̃m(x	PUNCT
ejpam-5914	375	14	)	)	PUNCT
ejpam-5914	376	1	=	=	SYM
ejpam-5914	376	2	supb1	supb1	NOUN
ejpam-5914	376	3	+	+	CCONJ
ejpam-5914	376	4	inf	inf	ADJ
ejpam-5914	376	5	b1	b1	NOUN
ejpam-5914	376	6	2	2	NUM
ejpam-5914	376	7	and	and	CCONJ
ejpam-5914	376	8	f̃m(y	f̃m(y	PRON
ejpam-5914	376	9	)	)	PUNCT
ejpam-5914	376	10	=	=	SYM
ejpam-5914	376	11	supb2	supb2	NOUN
ejpam-5914	376	12	+	+	CCONJ
ejpam-5914	376	13	inf	inf	ADJ
ejpam-5914	376	14	b2	b2	NOUN
ejpam-5914	376	15	2	2	NUM
ejpam-5914	376	16	,	,	PUNCT
ejpam-5914	376	17	so	so	ADV
ejpam-5914	376	18	min{f̃m(x	min{f̃m(x	NOUN
ejpam-5914	376	19	)	)	PUNCT
ejpam-5914	376	20	,	,	PUNCT
ejpam-5914	376	21	f̃m(y	f̃m(y	NOUN
ejpam-5914	376	22	)	)	PUNCT
ejpam-5914	376	23	}	}	PUNCT
ejpam-5914	377	1	=	=	SYM
ejpam-5914	377	2	supb1	supb1	NOUN
ejpam-5914	377	3	+	+	CCONJ
ejpam-5914	377	4	inf	inf	ADJ
ejpam-5914	377	5	b1	b1	NOUN
ejpam-5914	377	6	2	2	NUM
ejpam-5914	377	7	.	.	PUNCT
ejpam-5914	378	1	thus	thus	ADV
ejpam-5914	378	2	,	,	PUNCT
ejpam-5914	378	3	f̃m((x|(y|y))|(x|(y|y	f̃m((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	378	4	)	)	PUNCT
ejpam-5914	378	5	)	)	PUNCT
ejpam-5914	378	6	)	)	PUNCT
ejpam-5914	378	7	≥	≥	PROPN
ejpam-5914	378	8	supb1	supb1	NOUN
ejpam-5914	379	1	+	+	CCONJ
ejpam-5914	379	2	inf	inf	ADJ
ejpam-5914	379	3	b1	b1	NOUN
ejpam-5914	379	4	2	2	NUM
ejpam-5914	379	5	=	=	SYM
ejpam-5914	379	6	min{f̃m(x	min{f̃m(x	PROPN
ejpam-5914	379	7	)	)	PUNCT
ejpam-5914	379	8	,	,	PUNCT
ejpam-5914	379	9	f̃m(y	f̃m(y	NOUN
ejpam-5914	379	10	)	)	PUNCT
ejpam-5914	379	11	}	}	PUNCT
ejpam-5914	379	12	.	.	PUNCT
ejpam-5914	380	1	case	case	NOUN
ejpam-5914	380	2	4	4	NUM
ejpam-5914	380	3	:	:	PUNCT
ejpam-5914	380	4	let	let	VERB
ejpam-5914	380	5	x	x	PUNCT
ejpam-5914	380	6	∈	∈	PROPN
ejpam-5914	380	7	s	s	X
ejpam-5914	380	8	and	and	CCONJ
ejpam-5914	380	9	y	y	PROPN
ejpam-5914	380	10	/∈	/∈	PUNCT
ejpam-5914	381	1	s.	s.	PROPN
ejpam-5914	381	2	then	then	ADV
ejpam-5914	381	3	f̃m(x	f̃m(x	NUM
ejpam-5914	381	4	)	)	PUNCT
ejpam-5914	382	1	=	=	SYM
ejpam-5914	382	2	supb2	supb2	NOUN
ejpam-5914	382	3	+	+	CCONJ
ejpam-5914	382	4	inf	inf	ADJ
ejpam-5914	382	5	b2	b2	NOUN
ejpam-5914	382	6	2	2	NUM
ejpam-5914	382	7	and	and	CCONJ
ejpam-5914	382	8	f̃m(y	f̃m(y	NUM
ejpam-5914	382	9	)	)	PUNCT
ejpam-5914	383	1	=	=	SYM
ejpam-5914	383	2	supb1	supb1	NOUN
ejpam-5914	383	3	+	+	CCONJ
ejpam-5914	383	4	inf	inf	ADJ
ejpam-5914	383	5	b1	b1	NOUN
ejpam-5914	383	6	2	2	NUM
ejpam-5914	383	7	,	,	PUNCT
ejpam-5914	383	8	so	so	ADV
ejpam-5914	383	9	min{f̃m(x	min{f̃m(x	NOUN
ejpam-5914	383	10	)	)	PUNCT
ejpam-5914	383	11	,	,	PUNCT
ejpam-5914	383	12	f̃m(y	f̃m(y	NOUN
ejpam-5914	383	13	)	)	PUNCT
ejpam-5914	383	14	}	}	PUNCT
ejpam-5914	383	15	=	=	SYM
ejpam-5914	383	16	supb1	supb1	NOUN
ejpam-5914	383	17	+	+	CCONJ
ejpam-5914	383	18	inf	inf	ADJ
ejpam-5914	383	19	b1	b1	NOUN
ejpam-5914	383	20	2	2	NUM
ejpam-5914	383	21	.	.	PUNCT
ejpam-5914	384	1	thus	thus	ADV
ejpam-5914	384	2	,	,	PUNCT
ejpam-5914	384	3	f̃m((x|(y|y))|(x|(y|y	f̃m((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	384	4	)	)	PUNCT
ejpam-5914	384	5	)	)	PUNCT
ejpam-5914	384	6	)	)	PUNCT
ejpam-5914	384	7	≥	≥	PROPN
ejpam-5914	384	8	supb1	supb1	NOUN
ejpam-5914	385	1	+	+	CCONJ
ejpam-5914	385	2	inf	inf	ADJ
ejpam-5914	385	3	b1	b1	NOUN
ejpam-5914	385	4	2	2	NUM
ejpam-5914	385	5	=	=	SYM
ejpam-5914	385	6	min{f̃m(x	min{f̃m(x	PROPN
ejpam-5914	385	7	)	)	PUNCT
ejpam-5914	385	8	,	,	PUNCT
ejpam-5914	385	9	f̃m(y	f̃m(y	NOUN
ejpam-5914	385	10	)	)	PUNCT
ejpam-5914	385	11	}	}	PUNCT
ejpam-5914	385	12	.	.	PUNCT
ejpam-5914	386	1	n.	n.	PROPN
ejpam-5914	386	2	rajesh	rajesh	PROPN
ejpam-5914	386	3	et	et	PROPN
ejpam-5914	386	4	al	al	PROPN
ejpam-5914	386	5	.	.	PUNCT
ejpam-5914	386	6	/	/	SYM
ejpam-5914	386	7	eur	eur	PROPN
ejpam-5914	386	8	.	.	PUNCT
ejpam-5914	387	1	j.	j.	PROPN
ejpam-5914	387	2	pure	pure	PROPN
ejpam-5914	387	3	appl	appl	PROPN
ejpam-5914	387	4	.	.	PROPN
ejpam-5914	387	5	math	math	PROPN
ejpam-5914	387	6	,	,	PUNCT
ejpam-5914	387	7	18	18	NUM
ejpam-5914	387	8	(	(	PUNCT
ejpam-5914	387	9	2	2	NUM
ejpam-5914	387	10	)	)	PUNCT
ejpam-5914	387	11	(	(	PUNCT
ejpam-5914	387	12	2025	2025	NUM
ejpam-5914	387	13	)	)	PUNCT
ejpam-5914	387	14	,	,	PUNCT
ejpam-5914	387	15	5914	5914	NUM
ejpam-5914	387	16	15	15	NUM
ejpam-5914	387	17	of	of	ADP
ejpam-5914	387	18	21	21	NUM
ejpam-5914	387	19	hence	hence	ADV
ejpam-5914	387	20	,	,	PUNCT
ejpam-5914	387	21	f̃m	f̃m	PROPN
ejpam-5914	387	22	is	be	AUX
ejpam-5914	387	23	a	a	DET
ejpam-5914	387	24	1	1	NUM
ejpam-5914	387	25	-	-	PUNCT
ejpam-5914	387	26	fuzzy	fuzzy	ADJ
ejpam-5914	387	27	subalgebra	subalgebra	NOUN
ejpam-5914	387	28	of	of	ADP
ejpam-5914	387	29	a	a	PRON
ejpam-5914	387	30	and	and	CCONJ
ejpam-5914	387	31	so	so	ADV
ejpam-5914	387	32	(	(	PUNCT
ejpam-5914	387	33	a	a	PRON
ejpam-5914	387	34	,	,	PUNCT
ejpam-5914	387	35	f̃	f̃	PROPN
ejpam-5914	387	36	)	)	PUNCT
ejpam-5914	387	37	is	be	AUX
ejpam-5914	387	38	a	a	DET
ejpam-5914	387	39	mean	mean	ADJ
ejpam-5914	387	40	1	1	NUM
ejpam-5914	387	41	-	-	PUNCT
ejpam-5914	387	42	fuzzy	fuzzy	ADJ
ejpam-5914	387	43	subalgebra	subalgebra	NOUN
ejpam-5914	387	44	of	of	ADP
ejpam-5914	387	45	a.	a.	NOUN
ejpam-5914	387	46	(	(	PUNCT
ejpam-5914	387	47	2	2	X
ejpam-5914	387	48	)	)	PUNCT
ejpam-5914	387	49	assume	assume	VERB
ejpam-5914	387	50	that	that	SCONJ
ejpam-5914	387	51	supb2	supb2	PROPN
ejpam-5914	387	52	≤	≤	X
ejpam-5914	387	53	supb1	supb1	NOUN
ejpam-5914	387	54	and	and	CCONJ
ejpam-5914	387	55	inf	inf	PROPN
ejpam-5914	387	56	b2	b2	PROPN
ejpam-5914	387	57	≤	≤	PROPN
ejpam-5914	387	58	inf	inf	NOUN
ejpam-5914	387	59	b1	b1	NOUN
ejpam-5914	387	60	.	.	PUNCT
ejpam-5914	388	1	then	then	ADV
ejpam-5914	388	2	supb2	supb2	PROPN
ejpam-5914	388	3	+	+	CCONJ
ejpam-5914	388	4	inf	inf	ADJ
ejpam-5914	388	5	b2	b2	NOUN
ejpam-5914	388	6	2	2	NUM
ejpam-5914	388	7	≤	≤	NUM
ejpam-5914	388	8	supb1	supb1	NOUN
ejpam-5914	389	1	+	+	CCONJ
ejpam-5914	389	2	inf	inf	ADJ
ejpam-5914	389	3	b1	b1	NOUN
ejpam-5914	389	4	2	2	NUM
ejpam-5914	389	5	.	.	PUNCT
ejpam-5914	389	6	case	case	NOUN
ejpam-5914	389	7	1	1	NUM
ejpam-5914	389	8	:	:	PUNCT
ejpam-5914	389	9	let	let	VERB
ejpam-5914	389	10	x	x	PRON
ejpam-5914	389	11	,	,	PUNCT
ejpam-5914	389	12	y	y	PROPN
ejpam-5914	389	13	∈	∈	PROPN
ejpam-5914	389	14	s.	s.	PROPN
ejpam-5914	389	15	then	then	ADV
ejpam-5914	389	16	f̃m(x	f̃m(x	PUNCT
ejpam-5914	389	17	)	)	PUNCT
ejpam-5914	389	18	=	=	SYM
ejpam-5914	389	19	supb2	supb2	NOUN
ejpam-5914	389	20	+	+	CCONJ
ejpam-5914	389	21	inf	inf	ADJ
ejpam-5914	389	22	b2	b2	NOUN
ejpam-5914	389	23	2	2	NUM
ejpam-5914	389	24	and	and	CCONJ
ejpam-5914	389	25	f̃m(y	f̃m(y	NUM
ejpam-5914	389	26	)	)	PUNCT
ejpam-5914	389	27	=	=	SYM
ejpam-5914	389	28	supb2	supb2	NOUN
ejpam-5914	389	29	+	+	CCONJ
ejpam-5914	389	30	inf	inf	ADJ
ejpam-5914	389	31	b2	b2	NOUN
ejpam-5914	389	32	2	2	NUM
ejpam-5914	389	33	,	,	PUNCT
ejpam-5914	389	34	so	so	ADV
ejpam-5914	389	35	max{f̃m(x	max{f̃m(x	PROPN
ejpam-5914	389	36	)	)	PUNCT
ejpam-5914	389	37	,	,	PUNCT
ejpam-5914	389	38	f̃m(y	f̃m(y	NOUN
ejpam-5914	389	39	)	)	PUNCT
ejpam-5914	389	40	}	}	PUNCT
ejpam-5914	389	41	=	=	SYM
ejpam-5914	389	42	supb2	supb2	NOUN
ejpam-5914	389	43	+	+	CCONJ
ejpam-5914	389	44	inf	inf	ADJ
ejpam-5914	389	45	b2	b2	NOUN
ejpam-5914	389	46	2	2	NUM
ejpam-5914	389	47	.	.	PUNCT
ejpam-5914	390	1	since	since	SCONJ
ejpam-5914	390	2	s	s	PROPN
ejpam-5914	390	3	is	be	AUX
ejpam-5914	390	4	a	a	DET
ejpam-5914	390	5	subalgebra	subalgebra	NOUN
ejpam-5914	390	6	ofa	ofa	PROPN
ejpam-5914	390	7	,	,	PUNCT
ejpam-5914	390	8	we	we	PRON
ejpam-5914	390	9	have	have	VERB
ejpam-5914	390	10	(	(	PUNCT
ejpam-5914	390	11	x|(y|y))|(x|(y|y	x|(y|y))|(x|(y|y	ADJ
ejpam-5914	390	12	)	)	PUNCT
ejpam-5914	390	13	)	)	PUNCT
ejpam-5914	391	1	∈	∈	PROPN
ejpam-5914	391	2	s	s	PART
ejpam-5914	391	3	and	and	CCONJ
ejpam-5914	391	4	so	so	ADV
ejpam-5914	391	5	f̃m((x|(y|y))|(x|(y|y	f̃m((x|(y|y))|(x|(y|y	PROPN
ejpam-5914	391	6	)	)	PUNCT
ejpam-5914	391	7	)	)	PUNCT
ejpam-5914	391	8	)	)	PUNCT
ejpam-5914	392	1	=	=	SYM
ejpam-5914	392	2	supb2	supb2	NOUN
ejpam-5914	392	3	+	+	CCONJ
ejpam-5914	392	4	inf	inf	ADJ
ejpam-5914	392	5	b2	b2	NOUN
ejpam-5914	392	6	2	2	NUM
ejpam-5914	392	7	.	.	PUNCT
ejpam-5914	393	1	thus	thus	ADV
ejpam-5914	393	2	,	,	PUNCT
ejpam-5914	393	3	f̃m((x|(y|y))|(x|(y|y	f̃m((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	393	4	)	)	PUNCT
ejpam-5914	393	5	)	)	PUNCT
ejpam-5914	393	6	)	)	PUNCT
ejpam-5914	394	1	=	=	SYM
ejpam-5914	394	2	supb2	supb2	NOUN
ejpam-5914	394	3	+	+	CCONJ
ejpam-5914	394	4	inf	inf	ADJ
ejpam-5914	394	5	b2	b2	NOUN
ejpam-5914	394	6	2	2	NUM
ejpam-5914	394	7	=	=	SYM
ejpam-5914	394	8	(	(	PUNCT
ejpam-5914	394	9	≤)max{f̃m(x	≤)max{f̃m(x	PROPN
ejpam-5914	394	10	)	)	PUNCT
ejpam-5914	394	11	,	,	PUNCT
ejpam-5914	394	12	f̃m(y	f̃m(y	NOUN
ejpam-5914	394	13	)	)	PUNCT
ejpam-5914	394	14	}	}	PUNCT
ejpam-5914	394	15	.	.	PUNCT
ejpam-5914	395	1	case	case	NOUN
ejpam-5914	395	2	2	2	NUM
ejpam-5914	395	3	:	:	PUNCT
ejpam-5914	395	4	let	let	VERB
ejpam-5914	395	5	x	x	PRON
ejpam-5914	395	6	,	,	PUNCT
ejpam-5914	395	7	y	y	PROPN
ejpam-5914	395	8	/∈	/∈	PUNCT
ejpam-5914	395	9	s.	s.	PROPN
ejpam-5914	395	10	then	then	ADV
ejpam-5914	395	11	f̃m(x	f̃m(x	PUNCT
ejpam-5914	395	12	)	)	PUNCT
ejpam-5914	396	1	=	=	SYM
ejpam-5914	396	2	supb1	supb1	NOUN
ejpam-5914	396	3	+	+	CCONJ
ejpam-5914	396	4	inf	inf	ADJ
ejpam-5914	396	5	b1	b1	NOUN
ejpam-5914	396	6	2	2	NUM
ejpam-5914	396	7	and	and	CCONJ
ejpam-5914	396	8	f̃m(y	f̃m(y	PRON
ejpam-5914	396	9	)	)	PUNCT
ejpam-5914	397	1	=	=	SYM
ejpam-5914	398	1	supb1	supb1	NOUN
ejpam-5914	399	1	+	+	CCONJ
ejpam-5914	399	2	inf	inf	ADJ
ejpam-5914	399	3	b1	b1	NOUN
ejpam-5914	399	4	2	2	NUM
ejpam-5914	399	5	,	,	PUNCT
ejpam-5914	399	6	so	so	ADV
ejpam-5914	399	7	max{f̃m(x	max{f̃m(x	PROPN
ejpam-5914	399	8	)	)	PUNCT
ejpam-5914	399	9	,	,	PUNCT
ejpam-5914	399	10	f̃m(y	f̃m(y	NOUN
ejpam-5914	399	11	)	)	PUNCT
ejpam-5914	399	12	}	}	PUNCT
ejpam-5914	399	13	=	=	SYM
ejpam-5914	399	14	supb1	supb1	NOUN
ejpam-5914	399	15	+	+	CCONJ
ejpam-5914	399	16	inf	inf	ADJ
ejpam-5914	399	17	b1	b1	NOUN
ejpam-5914	399	18	2	2	NUM
ejpam-5914	399	19	.	.	PUNCT
ejpam-5914	400	1	thus	thus	ADV
ejpam-5914	400	2	,	,	PUNCT
ejpam-5914	400	3	f̃m((x|(y|y))|(x|(y|y	f̃m((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	400	4	)	)	PUNCT
ejpam-5914	400	5	)	)	PUNCT
ejpam-5914	400	6	)	)	PUNCT
ejpam-5914	400	7	≤	≤	NUM
ejpam-5914	400	8	supb1	supb1	NOUN
ejpam-5914	400	9	+	+	CCONJ
ejpam-5914	400	10	inf	inf	ADJ
ejpam-5914	400	11	b1	b1	NOUN
ejpam-5914	400	12	2	2	NUM
ejpam-5914	400	13	=	=	SYM
ejpam-5914	400	14	max{f̃m(x	max{f̃m(x	PROPN
ejpam-5914	400	15	)	)	PUNCT
ejpam-5914	400	16	,	,	PUNCT
ejpam-5914	400	17	f̃m(y	f̃m(y	NOUN
ejpam-5914	400	18	)	)	PUNCT
ejpam-5914	400	19	}	}	PUNCT
ejpam-5914	400	20	.	.	PUNCT
ejpam-5914	401	1	case	case	NOUN
ejpam-5914	401	2	3	3	X
ejpam-5914	401	3	:	:	PUNCT
ejpam-5914	401	4	let	let	VERB
ejpam-5914	401	5	x	x	PUNCT
ejpam-5914	401	6	/∈	/∈	PRON
ejpam-5914	401	7	s	s	PART
ejpam-5914	401	8	and	and	CCONJ
ejpam-5914	401	9	y	y	PROPN
ejpam-5914	401	10	∈	∈	PROPN
ejpam-5914	401	11	s.	s.	PROPN
ejpam-5914	401	12	then	then	ADV
ejpam-5914	401	13	f̃m(x	f̃m(x	PUNCT
ejpam-5914	401	14	)	)	PUNCT
ejpam-5914	402	1	=	=	SYM
ejpam-5914	402	2	supb1	supb1	NOUN
ejpam-5914	402	3	+	+	CCONJ
ejpam-5914	402	4	inf	inf	ADJ
ejpam-5914	402	5	b1	b1	NOUN
ejpam-5914	402	6	2	2	NUM
ejpam-5914	402	7	and	and	CCONJ
ejpam-5914	402	8	f̃m(y	f̃m(y	PRON
ejpam-5914	402	9	)	)	PUNCT
ejpam-5914	402	10	=	=	SYM
ejpam-5914	402	11	supb2	supb2	NOUN
ejpam-5914	402	12	+	+	CCONJ
ejpam-5914	402	13	inf	inf	ADJ
ejpam-5914	402	14	b2	b2	NOUN
ejpam-5914	402	15	2	2	NUM
ejpam-5914	402	16	,	,	PUNCT
ejpam-5914	402	17	so	so	ADV
ejpam-5914	402	18	max{f̃m(x	max{f̃m(x	PROPN
ejpam-5914	402	19	)	)	PUNCT
ejpam-5914	402	20	,	,	PUNCT
ejpam-5914	402	21	f̃m(y	f̃m(y	NOUN
ejpam-5914	402	22	)	)	PUNCT
ejpam-5914	402	23	}	}	PUNCT
ejpam-5914	403	1	=	=	SYM
ejpam-5914	403	2	supb1	supb1	NOUN
ejpam-5914	403	3	+	+	CCONJ
ejpam-5914	403	4	inf	inf	ADJ
ejpam-5914	403	5	b1	b1	NOUN
ejpam-5914	403	6	2	2	NUM
ejpam-5914	403	7	.	.	PUNCT
ejpam-5914	404	1	thus	thus	ADV
ejpam-5914	404	2	,	,	PUNCT
ejpam-5914	404	3	f̃m((x|(y|y))|(x|(y|y	f̃m((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	404	4	)	)	PUNCT
ejpam-5914	404	5	)	)	PUNCT
ejpam-5914	404	6	)	)	PUNCT
ejpam-5914	404	7	≤	≤	NUM
ejpam-5914	404	8	supb1	supb1	NOUN
ejpam-5914	404	9	+	+	CCONJ
ejpam-5914	404	10	inf	inf	ADJ
ejpam-5914	404	11	b1	b1	NOUN
ejpam-5914	404	12	2	2	NUM
ejpam-5914	404	13	=	=	SYM
ejpam-5914	404	14	max{f̃m(x	max{f̃m(x	PROPN
ejpam-5914	404	15	)	)	PUNCT
ejpam-5914	404	16	,	,	PUNCT
ejpam-5914	404	17	f̃m(y	f̃m(y	NOUN
ejpam-5914	404	18	)	)	PUNCT
ejpam-5914	404	19	}	}	PUNCT
ejpam-5914	404	20	.	.	PUNCT
ejpam-5914	405	1	case	case	NOUN
ejpam-5914	405	2	4	4	NUM
ejpam-5914	405	3	:	:	PUNCT
ejpam-5914	405	4	let	let	VERB
ejpam-5914	405	5	x	x	PUNCT
ejpam-5914	405	6	∈	∈	PROPN
ejpam-5914	405	7	s	s	X
ejpam-5914	405	8	and	and	CCONJ
ejpam-5914	405	9	y	y	PROPN
ejpam-5914	405	10	/∈	/∈	PUNCT
ejpam-5914	406	1	s.	s.	PROPN
ejpam-5914	406	2	then	then	ADV
ejpam-5914	406	3	f̃m(x	f̃m(x	NUM
ejpam-5914	406	4	)	)	PUNCT
ejpam-5914	407	1	=	=	SYM
ejpam-5914	407	2	supb2	supb2	NOUN
ejpam-5914	407	3	+	+	CCONJ
ejpam-5914	407	4	inf	inf	ADJ
ejpam-5914	407	5	b2	b2	NOUN
ejpam-5914	407	6	2	2	NUM
ejpam-5914	407	7	and	and	CCONJ
ejpam-5914	407	8	f̃m(y	f̃m(y	NUM
ejpam-5914	407	9	)	)	PUNCT
ejpam-5914	408	1	=	=	SYM
ejpam-5914	408	2	supb1	supb1	NOUN
ejpam-5914	408	3	+	+	CCONJ
ejpam-5914	408	4	inf	inf	ADJ
ejpam-5914	408	5	b1	b1	NOUN
ejpam-5914	408	6	2	2	NUM
ejpam-5914	408	7	,	,	PUNCT
ejpam-5914	408	8	so	so	ADV
ejpam-5914	408	9	max{f̃m(x	max{f̃m(x	PROPN
ejpam-5914	408	10	)	)	PUNCT
ejpam-5914	408	11	,	,	PUNCT
ejpam-5914	408	12	f̃m(y	f̃m(y	NOUN
ejpam-5914	408	13	)	)	PUNCT
ejpam-5914	408	14	}	}	PUNCT
ejpam-5914	408	15	=	=	SYM
ejpam-5914	408	16	supb1	supb1	NOUN
ejpam-5914	408	17	+	+	CCONJ
ejpam-5914	408	18	inf	inf	ADJ
ejpam-5914	408	19	b1	b1	NOUN
ejpam-5914	408	20	2	2	NUM
ejpam-5914	408	21	.	.	PUNCT
ejpam-5914	409	1	thus	thus	ADV
ejpam-5914	409	2	,	,	PUNCT
ejpam-5914	409	3	f̃m((x|(y|y))|(x|(y|y	f̃m((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	409	4	)	)	PUNCT
ejpam-5914	409	5	)	)	PUNCT
ejpam-5914	409	6	)	)	PUNCT
ejpam-5914	409	7	≤	≤	NUM
ejpam-5914	409	8	supb1	supb1	NOUN
ejpam-5914	409	9	+	+	CCONJ
ejpam-5914	409	10	inf	inf	ADJ
ejpam-5914	409	11	b1	b1	NOUN
ejpam-5914	409	12	2	2	NUM
ejpam-5914	409	13	=	=	SYM
ejpam-5914	409	14	max{f̃m(x	max{f̃m(x	PROPN
ejpam-5914	409	15	)	)	PUNCT
ejpam-5914	409	16	,	,	PUNCT
ejpam-5914	409	17	f̃m(y	f̃m(y	NOUN
ejpam-5914	409	18	)	)	PUNCT
ejpam-5914	409	19	}	}	PUNCT
ejpam-5914	409	20	.	.	PUNCT
ejpam-5914	410	1	hence	hence	ADV
ejpam-5914	410	2	,	,	PUNCT
ejpam-5914	410	3	f̃m	f̃m	PROPN
ejpam-5914	410	4	is	be	AUX
ejpam-5914	410	5	a	a	DET
ejpam-5914	410	6	4	4	NUM
ejpam-5914	410	7	-	-	PUNCT
ejpam-5914	410	8	fuzzy	fuzzy	ADJ
ejpam-5914	410	9	subalgebra	subalgebra	NOUN
ejpam-5914	410	10	of	of	ADP
ejpam-5914	410	11	a	a	PRON
ejpam-5914	410	12	and	and	CCONJ
ejpam-5914	410	13	so	so	ADV
ejpam-5914	410	14	(	(	PUNCT
ejpam-5914	410	15	a	a	PRON
ejpam-5914	410	16	,	,	PUNCT
ejpam-5914	410	17	f̃	f̃	PROPN
ejpam-5914	410	18	)	)	PUNCT
ejpam-5914	410	19	is	be	AUX
ejpam-5914	410	20	a	a	DET
ejpam-5914	410	21	mean	mean	ADJ
ejpam-5914	410	22	4	4	NUM
ejpam-5914	410	23	-	-	PUNCT
ejpam-5914	410	24	fuzzy	fuzzy	ADJ
ejpam-5914	410	25	subalgebra	subalgebra	NOUN
ejpam-5914	410	26	of	of	ADP
ejpam-5914	410	27	a.	a.	NOUN
ejpam-5914	410	28	theorem	theorem	NOUN
ejpam-5914	410	29	17	17	NUM
ejpam-5914	410	30	.	.	PUNCT
ejpam-5914	411	1	an	an	DET
ejpam-5914	411	2	interval	interval	NOUN
ejpam-5914	411	3	-	-	PUNCT
ejpam-5914	411	4	valued	value	VERB
ejpam-5914	411	5	fuzzy	fuzzy	ADJ
ejpam-5914	411	6	structure	structure	NOUN
ejpam-5914	411	7	(	(	PUNCT
ejpam-5914	411	8	a	a	PRON
ejpam-5914	411	9	,	,	PUNCT
ejpam-5914	411	10	f̃	f̃	PROPN
ejpam-5914	411	11	)	)	PUNCT
ejpam-5914	411	12	over	over	ADP
ejpam-5914	411	13	a	a	PRON
ejpam-5914	411	14	is	be	AUX
ejpam-5914	411	15	a	a	DET
ejpam-5914	411	16	mean	mean	ADJ
ejpam-5914	411	17	1	1	NUM
ejpam-5914	411	18	-	-	PUNCT
ejpam-5914	411	19	fuzzy	fuzzy	ADJ
ejpam-5914	411	20	subalgebra	subalgebra	NOUN
ejpam-5914	411	21	of	of	ADP
ejpam-5914	411	22	a	a	DET
ejpam-5914	411	23	if	if	NOUN
ejpam-5914	411	24	and	and	CCONJ
ejpam-5914	411	25	only	only	ADV
ejpam-5914	411	26	if	if	SCONJ
ejpam-5914	411	27	the	the	DET
ejpam-5914	411	28	set	set	NOUN
ejpam-5914	411	29	u(f̃m	u(f̃m	PROPN
ejpam-5914	411	30	;	;	PUNCT
ejpam-5914	411	31	t	t	PROPN
ejpam-5914	411	32	)	)	PUNCT
ejpam-5914	411	33	is	be	AUX
ejpam-5914	411	34	a	a	DET
ejpam-5914	411	35	subalgebra	subalgebra	NOUN
ejpam-5914	411	36	of	of	ADP
ejpam-5914	411	37	a	a	PRON
ejpam-5914	411	38	for	for	ADP
ejpam-5914	411	39	all	all	DET
ejpam-5914	411	40	t	t	NOUN
ejpam-5914	411	41	∈	∈	PROPN
ejpam-5914	412	1	[	[	X
ejpam-5914	412	2	0	0	NUM
ejpam-5914	412	3	,	,	PUNCT
ejpam-5914	412	4	1	1	NUM
ejpam-5914	412	5	]	]	PUNCT
ejpam-5914	412	6	with	with	ADP
ejpam-5914	412	7	u(f̃m	u(f̃m	ADJ
ejpam-5914	412	8	;	;	PUNCT
ejpam-5914	412	9	t	t	X
ejpam-5914	412	10	)	)	PUNCT
ejpam-5914	412	11	̸=	̸=	PROPN
ejpam-5914	412	12	∅.	∅.	ADP
ejpam-5914	412	13	n.	n.	PROPN
ejpam-5914	412	14	rajesh	rajesh	PROPN
ejpam-5914	412	15	et	et	PROPN
ejpam-5914	412	16	al	al	PROPN
ejpam-5914	412	17	.	.	PUNCT
ejpam-5914	412	18	/	/	SYM
ejpam-5914	412	19	eur	eur	PROPN
ejpam-5914	412	20	.	.	PUNCT
ejpam-5914	413	1	j.	j.	PROPN
ejpam-5914	413	2	pure	pure	PROPN
ejpam-5914	413	3	appl	appl	PROPN
ejpam-5914	413	4	.	.	PROPN
ejpam-5914	413	5	math	math	PROPN
ejpam-5914	413	6	,	,	PUNCT
ejpam-5914	413	7	18	18	NUM
ejpam-5914	413	8	(	(	PUNCT
ejpam-5914	413	9	2	2	NUM
ejpam-5914	413	10	)	)	PUNCT
ejpam-5914	413	11	(	(	PUNCT
ejpam-5914	413	12	2025	2025	NUM
ejpam-5914	413	13	)	)	PUNCT
ejpam-5914	413	14	,	,	PUNCT
ejpam-5914	413	15	5914	5914	NUM
ejpam-5914	413	16	16	16	NUM
ejpam-5914	413	17	of	of	ADP
ejpam-5914	413	18	21	21	NUM
ejpam-5914	413	19	proof	proof	NOUN
ejpam-5914	413	20	.	.	PUNCT
ejpam-5914	414	1	assume	assume	VERB
ejpam-5914	414	2	that	that	SCONJ
ejpam-5914	414	3	(	(	PUNCT
ejpam-5914	414	4	a	a	PRON
ejpam-5914	414	5	,	,	PUNCT
ejpam-5914	414	6	f̃	f̃	PROPN
ejpam-5914	414	7	)	)	PUNCT
ejpam-5914	414	8	is	be	AUX
ejpam-5914	414	9	a	a	DET
ejpam-5914	414	10	mean	mean	ADJ
ejpam-5914	414	11	1	1	NUM
ejpam-5914	414	12	-	-	PUNCT
ejpam-5914	414	13	fuzzy	fuzzy	ADJ
ejpam-5914	414	14	subalgebra	subalgebra	NOUN
ejpam-5914	414	15	of	of	ADP
ejpam-5914	414	16	a.	a.	NOUN
ejpam-5914	414	17	let	let	VERB
ejpam-5914	414	18	t	t	PROPN
ejpam-5914	414	19	∈	∈	PROPN
ejpam-5914	415	1	[	[	X
ejpam-5914	415	2	0	0	NUM
ejpam-5914	415	3	,	,	PUNCT
ejpam-5914	415	4	1	1	NUM
ejpam-5914	415	5	]	]	PUNCT
ejpam-5914	415	6	be	be	AUX
ejpam-5914	415	7	such	such	ADJ
ejpam-5914	415	8	that	that	SCONJ
ejpam-5914	415	9	u(f̃m	u(f̃m	NOUN
ejpam-5914	415	10	;	;	PUNCT
ejpam-5914	415	11	t	t	X
ejpam-5914	415	12	)	)	PUNCT
ejpam-5914	415	13	̸=	̸=	PROPN
ejpam-5914	415	14	∅	∅	NOUN
ejpam-5914	415	15	and	and	CCONJ
ejpam-5914	415	16	let	let	VERB
ejpam-5914	415	17	x	x	PRON
ejpam-5914	415	18	,	,	PUNCT
ejpam-5914	415	19	y	y	PROPN
ejpam-5914	415	20	∈	∈	PROPN
ejpam-5914	415	21	u(f̃m	u(f̃m	NOUN
ejpam-5914	415	22	;	;	PUNCT
ejpam-5914	415	23	t	t	PROPN
ejpam-5914	415	24	)	)	PUNCT
ejpam-5914	415	25	.	.	PUNCT
ejpam-5914	416	1	then	then	ADV
ejpam-5914	416	2	f̃m(x	f̃m(x	NUM
ejpam-5914	416	3	)	)	PUNCT
ejpam-5914	416	4	≥	≥	PROPN
ejpam-5914	416	5	t	t	PROPN
ejpam-5914	416	6	and	and	CCONJ
ejpam-5914	416	7	f̃m(y	f̃m(y	NUM
ejpam-5914	416	8	)	)	PUNCT
ejpam-5914	416	9	≥	≥	NOUN
ejpam-5914	416	10	t.	t.	NOUN
ejpam-5914	416	11	since	since	SCONJ
ejpam-5914	416	12	(	(	PUNCT
ejpam-5914	416	13	a	a	PRON
ejpam-5914	416	14	,	,	PUNCT
ejpam-5914	416	15	f̃	f̃	PROPN
ejpam-5914	416	16	)	)	PUNCT
ejpam-5914	416	17	is	be	AUX
ejpam-5914	416	18	a	a	DET
ejpam-5914	416	19	mean	mean	ADJ
ejpam-5914	416	20	1	1	NUM
ejpam-5914	416	21	-	-	PUNCT
ejpam-5914	416	22	fuzzy	fuzzy	ADJ
ejpam-5914	416	23	subalgebra	subalgebra	NOUN
ejpam-5914	416	24	of	of	ADP
ejpam-5914	416	25	a	a	PRON
ejpam-5914	416	26	,	,	PUNCT
ejpam-5914	416	27	we	we	PRON
ejpam-5914	416	28	have	have	VERB
ejpam-5914	416	29	f̃m((x|(y|y))|(x|(y|y	f̃m((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	416	30	)	)	PUNCT
ejpam-5914	416	31	)	)	PUNCT
ejpam-5914	416	32	)	)	PUNCT
ejpam-5914	416	33	≥	≥	PROPN
ejpam-5914	416	34	min{f̃m(x	min{f̃m(x	PROPN
ejpam-5914	416	35	)	)	PUNCT
ejpam-5914	416	36	,	,	PUNCT
ejpam-5914	416	37	f̃m(y	f̃m(y	NOUN
ejpam-5914	416	38	)	)	PUNCT
ejpam-5914	416	39	}	}	PUNCT
ejpam-5914	416	40	≥	≥	X
ejpam-5914	416	41	t.	t.	PROPN
ejpam-5914	416	42	thus	thus	ADV
ejpam-5914	416	43	,	,	PUNCT
ejpam-5914	416	44	(	(	PUNCT
ejpam-5914	416	45	x|(y|y))|(x|(y|y	x|(y|y))|(x|(y|y	ADJ
ejpam-5914	416	46	)	)	PUNCT
ejpam-5914	416	47	)	)	PUNCT
ejpam-5914	417	1	∈	∈	PROPN
ejpam-5914	417	2	u(f̃m	u(f̃m	NOUN
ejpam-5914	417	3	;	;	PUNCT
ejpam-5914	417	4	t	t	PROPN
ejpam-5914	417	5	)	)	PUNCT
ejpam-5914	417	6	.	.	PUNCT
ejpam-5914	418	1	hence	hence	ADV
ejpam-5914	418	2	,	,	PUNCT
ejpam-5914	418	3	u(f̃m	u(f̃m	PROPN
ejpam-5914	418	4	;	;	PUNCT
ejpam-5914	418	5	t	t	PROPN
ejpam-5914	418	6	)	)	PUNCT
ejpam-5914	418	7	is	be	AUX
ejpam-5914	418	8	a	a	DET
ejpam-5914	418	9	subalgebra	subalgebra	NOUN
ejpam-5914	418	10	of	of	ADP
ejpam-5914	418	11	a.	a.	NOUN
ejpam-5914	418	12	conversely	conversely	ADV
ejpam-5914	418	13	,	,	PUNCT
ejpam-5914	418	14	assume	assume	VERB
ejpam-5914	418	15	that	that	SCONJ
ejpam-5914	418	16	for	for	ADP
ejpam-5914	418	17	all	all	DET
ejpam-5914	418	18	t	t	NOUN
ejpam-5914	418	19	∈	∈	PROPN
ejpam-5914	419	1	[	[	X
ejpam-5914	419	2	0	0	NUM
ejpam-5914	419	3	,	,	PUNCT
ejpam-5914	419	4	1	1	NUM
ejpam-5914	419	5	]	]	PUNCT
ejpam-5914	419	6	,	,	PUNCT
ejpam-5914	419	7	the	the	DET
ejpam-5914	419	8	set	set	NOUN
ejpam-5914	419	9	u(f̃m	u(f̃m	PROPN
ejpam-5914	419	10	;	;	PUNCT
ejpam-5914	419	11	t	t	PROPN
ejpam-5914	419	12	)	)	PUNCT
ejpam-5914	419	13	is	be	AUX
ejpam-5914	419	14	a	a	DET
ejpam-5914	419	15	subalgebra	subalgebra	NOUN
ejpam-5914	419	16	of	of	ADP
ejpam-5914	419	17	a	a	DET
ejpam-5914	419	18	if	if	SCONJ
ejpam-5914	419	19	u(f̃m	u(f̃m	ADJ
ejpam-5914	419	20	;	;	PUNCT
ejpam-5914	419	21	t	t	X
ejpam-5914	419	22	)	)	PUNCT
ejpam-5914	419	23	̸=	̸=	PROPN
ejpam-5914	419	24	∅.	∅.	ADV
ejpam-5914	419	25	let	let	VERB
ejpam-5914	419	26	x	x	PRON
ejpam-5914	419	27	,	,	PUNCT
ejpam-5914	419	28	y	y	PROPN
ejpam-5914	419	29	∈	∈	PROPN
ejpam-5914	419	30	a.	a.	NOUN
ejpam-5914	419	31	then	then	ADV
ejpam-5914	419	32	f̃m(x	f̃m(x	NUM
ejpam-5914	419	33	)	)	PUNCT
ejpam-5914	419	34	,	,	PUNCT
ejpam-5914	419	35	f̃m(y	f̃m(y	NOUN
ejpam-5914	419	36	)	)	PUNCT
ejpam-5914	419	37	∈	∈	PROPN
ejpam-5914	420	1	[	[	X
ejpam-5914	420	2	0	0	NUM
ejpam-5914	420	3	,	,	PUNCT
ejpam-5914	420	4	1	1	NUM
ejpam-5914	420	5	]	]	PUNCT
ejpam-5914	420	6	.	.	PUNCT
ejpam-5914	421	1	choose	choose	VERB
ejpam-5914	421	2	t	t	PROPN
ejpam-5914	421	3	=	=	SYM
ejpam-5914	421	4	min{f̃m(x	min{f̃m(x	PROPN
ejpam-5914	421	5	)	)	PUNCT
ejpam-5914	421	6	,	,	PUNCT
ejpam-5914	421	7	f̃m(y	f̃m(y	NOUN
ejpam-5914	421	8	)	)	PUNCT
ejpam-5914	421	9	}	}	PUNCT
ejpam-5914	421	10	.	.	PUNCT
ejpam-5914	422	1	thus	thus	ADV
ejpam-5914	422	2	,	,	PUNCT
ejpam-5914	422	3	f̃m(x	f̃m(x	NUM
ejpam-5914	422	4	)	)	PUNCT
ejpam-5914	422	5	≥	≥	NOUN
ejpam-5914	422	6	t	t	PROPN
ejpam-5914	422	7	and	and	CCONJ
ejpam-5914	422	8	f̃m(y	f̃m(y	NUM
ejpam-5914	422	9	)	)	PUNCT
ejpam-5914	422	10	≥	≥	PROPN
ejpam-5914	422	11	t.	t.	NOUN
ejpam-5914	423	1	it	it	PRON
ejpam-5914	423	2	follows	follow	VERB
ejpam-5914	423	3	that	that	SCONJ
ejpam-5914	423	4	x	x	SYM
ejpam-5914	423	5	,	,	PUNCT
ejpam-5914	423	6	y	y	PROPN
ejpam-5914	423	7	∈	∈	PROPN
ejpam-5914	423	8	u(f̃m	u(f̃m	NOUN
ejpam-5914	423	9	;	;	PUNCT
ejpam-5914	423	10	t	t	X
ejpam-5914	423	11	)	)	PUNCT
ejpam-5914	423	12	̸=	̸=	PROPN
ejpam-5914	423	13	∅.	∅.	VERB
ejpam-5914	423	14	by	by	ADP
ejpam-5914	423	15	assumption	assumption	NOUN
ejpam-5914	423	16	,	,	PUNCT
ejpam-5914	423	17	we	we	PRON
ejpam-5914	423	18	have	have	VERB
ejpam-5914	423	19	u(f̃m	u(f̃m	ADJ
ejpam-5914	423	20	;	;	PUNCT
ejpam-5914	423	21	t	t	X
ejpam-5914	423	22	)	)	PUNCT
ejpam-5914	423	23	is	be	AUX
ejpam-5914	423	24	a	a	DET
ejpam-5914	423	25	subalgebra	subalgebra	NOUN
ejpam-5914	423	26	of	of	ADP
ejpam-5914	423	27	a	a	PRON
ejpam-5914	423	28	and	and	CCONJ
ejpam-5914	423	29	so	so	ADV
ejpam-5914	423	30	(	(	PUNCT
ejpam-5914	423	31	x|(y|y))|(x|(y|y	x|(y|y))|(x|(y|y	ADJ
ejpam-5914	423	32	)	)	PUNCT
ejpam-5914	423	33	)	)	PUNCT
ejpam-5914	424	1	∈	∈	PROPN
ejpam-5914	424	2	u(f̃m	u(f̃m	NOUN
ejpam-5914	424	3	;	;	PUNCT
ejpam-5914	424	4	t	t	PROPN
ejpam-5914	424	5	)	)	PUNCT
ejpam-5914	424	6	.	.	PUNCT
ejpam-5914	425	1	thus	thus	ADV
ejpam-5914	425	2	,	,	PUNCT
ejpam-5914	425	3	f̃m((x|(y|y))|(x|(y|y	f̃m((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	425	4	)	)	PUNCT
ejpam-5914	425	5	)	)	PUNCT
ejpam-5914	425	6	)	)	PUNCT
ejpam-5914	425	7	≥	≥	X
ejpam-5914	425	8	t	t	NOUN
ejpam-5914	425	9	=	=	SYM
ejpam-5914	425	10	min{f̃m(x	min{f̃m(x	PROPN
ejpam-5914	425	11	)	)	PUNCT
ejpam-5914	425	12	,	,	PUNCT
ejpam-5914	425	13	f̃m(y	f̃m(y	NOUN
ejpam-5914	425	14	)	)	PUNCT
ejpam-5914	425	15	}	}	PUNCT
ejpam-5914	425	16	.	.	PUNCT
ejpam-5914	426	1	hence	hence	ADV
ejpam-5914	426	2	,	,	PUNCT
ejpam-5914	426	3	(	(	PUNCT
ejpam-5914	426	4	a	a	DET
ejpam-5914	426	5	,	,	PUNCT
ejpam-5914	426	6	f̃m	f̃m	NOUN
ejpam-5914	426	7	)	)	PUNCT
ejpam-5914	426	8	is	be	AUX
ejpam-5914	426	9	a	a	DET
ejpam-5914	426	10	1	1	NUM
ejpam-5914	426	11	-	-	PUNCT
ejpam-5914	426	12	fuzzy	fuzzy	ADJ
ejpam-5914	426	13	subalgebra	subalgebra	NOUN
ejpam-5914	426	14	of	of	ADP
ejpam-5914	426	15	a	a	PRON
ejpam-5914	426	16	,	,	PUNCT
ejpam-5914	426	17	that	that	ADV
ejpam-5914	426	18	is	is	ADV
ejpam-5914	426	19	,	,	PUNCT
ejpam-5914	426	20	(	(	PUNCT
ejpam-5914	426	21	a	a	PRON
ejpam-5914	426	22	,	,	PUNCT
ejpam-5914	426	23	f̃	f̃	PROPN
ejpam-5914	426	24	)	)	PUNCT
ejpam-5914	426	25	is	be	AUX
ejpam-5914	426	26	a	a	DET
ejpam-5914	426	27	mean	mean	ADJ
ejpam-5914	426	28	1	1	NUM
ejpam-5914	426	29	-	-	PUNCT
ejpam-5914	426	30	fuzzy	fuzzy	ADJ
ejpam-5914	426	31	subalgebra	subalgebra	NOUN
ejpam-5914	426	32	of	of	ADP
ejpam-5914	426	33	a.	a.	NOUN
ejpam-5914	426	34	corollary	corollary	NOUN
ejpam-5914	427	1	3	3	X
ejpam-5914	427	2	.	.	PUNCT
ejpam-5914	428	1	if	if	SCONJ
ejpam-5914	428	2	(	(	PUNCT
ejpam-5914	428	3	a	a	PRON
ejpam-5914	428	4	,	,	PUNCT
ejpam-5914	428	5	f̃	f̃	PROPN
ejpam-5914	428	6	)	)	PUNCT
ejpam-5914	428	7	is	be	AUX
ejpam-5914	428	8	a	a	DET
ejpam-5914	428	9	mean	mean	ADJ
ejpam-5914	428	10	3	3	NUM
ejpam-5914	428	11	-	-	PUNCT
ejpam-5914	428	12	fuzzy	fuzzy	ADJ
ejpam-5914	428	13	subalgebra	subalgebra	NOUN
ejpam-5914	428	14	of	of	ADP
ejpam-5914	428	15	a	a	PRON
ejpam-5914	428	16	,	,	PUNCT
ejpam-5914	428	17	then	then	ADV
ejpam-5914	428	18	u(f̃m	u(f̃m	ADJ
ejpam-5914	428	19	;	;	PUNCT
ejpam-5914	428	20	t	t	PROPN
ejpam-5914	428	21	)	)	PUNCT
ejpam-5914	428	22	is	be	AUX
ejpam-5914	428	23	a	a	DET
ejpam-5914	428	24	subalgebra	subalgebra	NOUN
ejpam-5914	428	25	of	of	ADP
ejpam-5914	428	26	a	a	PRON
ejpam-5914	428	27	for	for	ADP
ejpam-5914	428	28	all	all	DET
ejpam-5914	428	29	t	t	NOUN
ejpam-5914	428	30	∈	∈	PROPN
ejpam-5914	429	1	[	[	X
ejpam-5914	429	2	0	0	NUM
ejpam-5914	429	3	,	,	PUNCT
ejpam-5914	429	4	1	1	NUM
ejpam-5914	429	5	]	]	PUNCT
ejpam-5914	429	6	with	with	ADP
ejpam-5914	429	7	u(f̃m	u(f̃m	ADJ
ejpam-5914	429	8	;	;	PUNCT
ejpam-5914	429	9	t	t	X
ejpam-5914	429	10	)	)	PUNCT
ejpam-5914	429	11	̸=	̸=	PROPN
ejpam-5914	429	12	∅.	∅.	ADP
ejpam-5914	429	13	proof	proof	NOUN
ejpam-5914	429	14	.	.	PUNCT
ejpam-5914	430	1	it	it	PRON
ejpam-5914	430	2	is	be	AUX
ejpam-5914	430	3	straightforward	straightforward	ADJ
ejpam-5914	430	4	by	by	ADP
ejpam-5914	430	5	theorems	theorem	NOUN
ejpam-5914	430	6	13	13	NUM
ejpam-5914	430	7	and	and	CCONJ
ejpam-5914	430	8	17	17	NUM
ejpam-5914	430	9	.	.	PUNCT
ejpam-5914	431	1	theorem	theorem	VERB
ejpam-5914	431	2	18	18	NUM
ejpam-5914	431	3	.	.	PUNCT
ejpam-5914	432	1	an	an	DET
ejpam-5914	432	2	interval	interval	NOUN
ejpam-5914	432	3	-	-	PUNCT
ejpam-5914	432	4	valued	value	VERB
ejpam-5914	432	5	fuzzy	fuzzy	ADJ
ejpam-5914	432	6	structure	structure	NOUN
ejpam-5914	432	7	(	(	PUNCT
ejpam-5914	432	8	a	a	PRON
ejpam-5914	432	9	,	,	PUNCT
ejpam-5914	432	10	f̃	f̃	PROPN
ejpam-5914	432	11	)	)	PUNCT
ejpam-5914	432	12	over	over	ADP
ejpam-5914	432	13	a	a	PRON
ejpam-5914	432	14	is	be	AUX
ejpam-5914	432	15	a	a	DET
ejpam-5914	432	16	mean	mean	ADJ
ejpam-5914	432	17	4	4	NUM
ejpam-5914	432	18	-	-	PUNCT
ejpam-5914	432	19	fuzzy	fuzzy	ADJ
ejpam-5914	432	20	subalgebra	subalgebra	NOUN
ejpam-5914	432	21	of	of	ADP
ejpam-5914	432	22	a	a	DET
ejpam-5914	432	23	if	if	NOUN
ejpam-5914	432	24	and	and	CCONJ
ejpam-5914	432	25	only	only	ADV
ejpam-5914	432	26	if	if	SCONJ
ejpam-5914	432	27	the	the	DET
ejpam-5914	432	28	set	set	NOUN
ejpam-5914	432	29	l(f̃m	l(f̃m	PROPN
ejpam-5914	432	30	;	;	PUNCT
ejpam-5914	432	31	t	t	X
ejpam-5914	432	32	)	)	PUNCT
ejpam-5914	432	33	is	be	AUX
ejpam-5914	432	34	a	a	DET
ejpam-5914	432	35	subalgebra	subalgebra	NOUN
ejpam-5914	432	36	of	of	ADP
ejpam-5914	432	37	a	a	PRON
ejpam-5914	432	38	for	for	ADP
ejpam-5914	432	39	all	all	DET
ejpam-5914	432	40	t	t	NOUN
ejpam-5914	432	41	∈	∈	PROPN
ejpam-5914	433	1	[	[	X
ejpam-5914	433	2	0	0	NUM
ejpam-5914	433	3	,	,	PUNCT
ejpam-5914	433	4	1	1	NUM
ejpam-5914	433	5	]	]	PUNCT
ejpam-5914	433	6	with	with	ADP
ejpam-5914	433	7	l(f̃m	l(f̃m	PROPN
ejpam-5914	433	8	;	;	PUNCT
ejpam-5914	433	9	t	t	X
ejpam-5914	433	10	)	)	PUNCT
ejpam-5914	433	11	̸=	̸=	PROPN
ejpam-5914	433	12	∅.	∅.	ADP
ejpam-5914	433	13	proof	proof	NOUN
ejpam-5914	433	14	.	.	PUNCT
ejpam-5914	434	1	assume	assume	VERB
ejpam-5914	434	2	that	that	SCONJ
ejpam-5914	434	3	(	(	PUNCT
ejpam-5914	434	4	a	a	PRON
ejpam-5914	434	5	,	,	PUNCT
ejpam-5914	434	6	f̃	f̃	PROPN
ejpam-5914	434	7	)	)	PUNCT
ejpam-5914	434	8	is	be	AUX
ejpam-5914	434	9	a	a	DET
ejpam-5914	434	10	mean	mean	ADJ
ejpam-5914	434	11	4	4	NUM
ejpam-5914	434	12	-	-	PUNCT
ejpam-5914	434	13	fuzzy	fuzzy	ADJ
ejpam-5914	434	14	subalgebra	subalgebra	NOUN
ejpam-5914	434	15	of	of	ADP
ejpam-5914	434	16	a.	a.	NOUN
ejpam-5914	434	17	let	let	VERB
ejpam-5914	434	18	t	t	PROPN
ejpam-5914	434	19	∈	∈	PROPN
ejpam-5914	435	1	[	[	X
ejpam-5914	435	2	0	0	NUM
ejpam-5914	435	3	,	,	PUNCT
ejpam-5914	435	4	1	1	NUM
ejpam-5914	435	5	]	]	PUNCT
ejpam-5914	435	6	be	be	AUX
ejpam-5914	435	7	such	such	ADJ
ejpam-5914	435	8	that	that	SCONJ
ejpam-5914	435	9	l(f̃m	l(f̃m	PROPN
ejpam-5914	435	10	;	;	PUNCT
ejpam-5914	435	11	t	t	X
ejpam-5914	435	12	)	)	PUNCT
ejpam-5914	435	13	̸=	̸=	PROPN
ejpam-5914	435	14	∅	∅	NOUN
ejpam-5914	435	15	and	and	CCONJ
ejpam-5914	435	16	let	let	VERB
ejpam-5914	435	17	x	x	PRON
ejpam-5914	435	18	,	,	PUNCT
ejpam-5914	435	19	y	y	PROPN
ejpam-5914	435	20	∈	∈	PROPN
ejpam-5914	435	21	l(f̃m	l(f̃m	PROPN
ejpam-5914	435	22	;	;	PUNCT
ejpam-5914	435	23	t	t	PROPN
ejpam-5914	435	24	)	)	PUNCT
ejpam-5914	435	25	.	.	PUNCT
ejpam-5914	436	1	then	then	ADV
ejpam-5914	436	2	f̃m(x	f̃m(x	CCONJ
ejpam-5914	436	3	)	)	PUNCT
ejpam-5914	436	4	≤	≤	NOUN
ejpam-5914	436	5	t	t	NOUN
ejpam-5914	436	6	and	and	CCONJ
ejpam-5914	436	7	f̃m(y	f̃m(y	NOUN
ejpam-5914	436	8	)	)	PUNCT
ejpam-5914	436	9	≤	≤	NOUN
ejpam-5914	436	10	t.	t.	NOUN
ejpam-5914	436	11	since	since	SCONJ
ejpam-5914	436	12	(	(	PUNCT
ejpam-5914	436	13	a	a	PRON
ejpam-5914	436	14	,	,	PUNCT
ejpam-5914	436	15	f̃	f̃	PROPN
ejpam-5914	436	16	)	)	PUNCT
ejpam-5914	436	17	is	be	AUX
ejpam-5914	436	18	a	a	DET
ejpam-5914	436	19	mean	mean	ADJ
ejpam-5914	436	20	4	4	NUM
ejpam-5914	436	21	-	-	PUNCT
ejpam-5914	436	22	fuzzy	fuzzy	ADJ
ejpam-5914	436	23	subalgebra	subalgebra	NOUN
ejpam-5914	436	24	of	of	ADP
ejpam-5914	436	25	a	a	PRON
ejpam-5914	436	26	,	,	PUNCT
ejpam-5914	436	27	we	we	PRON
ejpam-5914	436	28	have	have	VERB
ejpam-5914	436	29	f̃m((x|(y|y))|(x|(y|y	f̃m((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	436	30	)	)	PUNCT
ejpam-5914	436	31	)	)	PUNCT
ejpam-5914	436	32	)	)	PUNCT
ejpam-5914	437	1	≤	≤	NUM
ejpam-5914	437	2	max{f̃m(x	max{f̃m(x	PROPN
ejpam-5914	437	3	)	)	PUNCT
ejpam-5914	437	4	,	,	PUNCT
ejpam-5914	437	5	f̃m(y	f̃m(y	NOUN
ejpam-5914	437	6	)	)	PUNCT
ejpam-5914	437	7	}	}	PUNCT
ejpam-5914	437	8	≤	≤	NOUN
ejpam-5914	438	1	t.	t.	NOUN
ejpam-5914	438	2	thus	thus	ADV
ejpam-5914	438	3	,	,	PUNCT
ejpam-5914	438	4	(	(	PUNCT
ejpam-5914	438	5	x|(y|y))|(x|(y|y	x|(y|y))|(x|(y|y	ADJ
ejpam-5914	438	6	)	)	PUNCT
ejpam-5914	438	7	)	)	PUNCT
ejpam-5914	439	1	∈	∈	PROPN
ejpam-5914	439	2	l(f̃m	l(f̃m	PROPN
ejpam-5914	439	3	;	;	PUNCT
ejpam-5914	439	4	t	t	PROPN
ejpam-5914	439	5	)	)	PUNCT
ejpam-5914	439	6	.	.	PUNCT
ejpam-5914	440	1	hence	hence	ADV
ejpam-5914	440	2	,	,	PUNCT
ejpam-5914	440	3	l(f̃m	l(f̃m	PROPN
ejpam-5914	440	4	;	;	PUNCT
ejpam-5914	440	5	t	t	PROPN
ejpam-5914	440	6	)	)	PUNCT
ejpam-5914	440	7	is	be	AUX
ejpam-5914	440	8	a	a	DET
ejpam-5914	440	9	subalgebra	subalgebra	NOUN
ejpam-5914	440	10	of	of	ADP
ejpam-5914	440	11	a.	a.	NOUN
ejpam-5914	440	12	conversely	conversely	ADV
ejpam-5914	440	13	,	,	PUNCT
ejpam-5914	440	14	assume	assume	VERB
ejpam-5914	440	15	that	that	SCONJ
ejpam-5914	440	16	for	for	ADP
ejpam-5914	440	17	all	all	DET
ejpam-5914	440	18	t	t	NOUN
ejpam-5914	440	19	∈	∈	PROPN
ejpam-5914	441	1	[	[	X
ejpam-5914	441	2	0	0	NUM
ejpam-5914	441	3	,	,	PUNCT
ejpam-5914	441	4	1	1	NUM
ejpam-5914	441	5	]	]	PUNCT
ejpam-5914	441	6	,	,	PUNCT
ejpam-5914	441	7	the	the	DET
ejpam-5914	441	8	set	set	NOUN
ejpam-5914	441	9	l(f̃m	l(f̃m	PROPN
ejpam-5914	441	10	;	;	PUNCT
ejpam-5914	441	11	t	t	X
ejpam-5914	441	12	)	)	PUNCT
ejpam-5914	441	13	is	be	AUX
ejpam-5914	441	14	a	a	DET
ejpam-5914	441	15	subalgebra	subalgebra	NOUN
ejpam-5914	441	16	of	of	ADP
ejpam-5914	441	17	a	a	DET
ejpam-5914	441	18	if	if	SCONJ
ejpam-5914	441	19	l(f̃m	l(f̃m	PROPN
ejpam-5914	441	20	;	;	PUNCT
ejpam-5914	441	21	t	t	X
ejpam-5914	441	22	)	)	PUNCT
ejpam-5914	441	23	̸=	̸=	PROPN
ejpam-5914	441	24	∅.	∅.	ADV
ejpam-5914	441	25	let	let	VERB
ejpam-5914	441	26	x	x	PRON
ejpam-5914	441	27	,	,	PUNCT
ejpam-5914	441	28	y	y	PROPN
ejpam-5914	441	29	∈	∈	PROPN
ejpam-5914	441	30	a.	a.	NOUN
ejpam-5914	441	31	then	then	ADV
ejpam-5914	441	32	f̃m(x	f̃m(x	NUM
ejpam-5914	441	33	)	)	PUNCT
ejpam-5914	441	34	,	,	PUNCT
ejpam-5914	441	35	f̃m(y	f̃m(y	NOUN
ejpam-5914	441	36	)	)	PUNCT
ejpam-5914	441	37	∈	∈	PROPN
ejpam-5914	442	1	[	[	X
ejpam-5914	442	2	0	0	NUM
ejpam-5914	442	3	,	,	PUNCT
ejpam-5914	442	4	1	1	NUM
ejpam-5914	442	5	]	]	PUNCT
ejpam-5914	442	6	.	.	PUNCT
ejpam-5914	443	1	choose	choose	VERB
ejpam-5914	443	2	t	t	PROPN
ejpam-5914	443	3	=	=	SYM
ejpam-5914	443	4	max{f̃m(x	max{f̃m(x	PROPN
ejpam-5914	443	5	)	)	PUNCT
ejpam-5914	443	6	,	,	PUNCT
ejpam-5914	443	7	f̃m(y	f̃m(y	NOUN
ejpam-5914	443	8	)	)	PUNCT
ejpam-5914	443	9	}	}	PUNCT
ejpam-5914	443	10	.	.	PUNCT
ejpam-5914	444	1	thus	thus	ADV
ejpam-5914	444	2	,	,	PUNCT
ejpam-5914	444	3	f̃m(x	f̃m(x	CCONJ
ejpam-5914	444	4	)	)	PUNCT
ejpam-5914	444	5	≤	≤	NOUN
ejpam-5914	444	6	t	t	NOUN
ejpam-5914	444	7	and	and	CCONJ
ejpam-5914	444	8	f̃m(y	f̃m(y	NOUN
ejpam-5914	444	9	)	)	PUNCT
ejpam-5914	444	10	≤	≤	NOUN
ejpam-5914	444	11	t	t	PROPN
ejpam-5914	444	12	,	,	PUNCT
ejpam-5914	444	13	and	and	CCONJ
ejpam-5914	444	14	so	so	ADV
ejpam-5914	444	15	x	x	NOUN
ejpam-5914	444	16	,	,	PUNCT
ejpam-5914	444	17	y	y	PROPN
ejpam-5914	444	18	∈	∈	PROPN
ejpam-5914	444	19	l(f̃m	l(f̃m	PROPN
ejpam-5914	444	20	;	;	PUNCT
ejpam-5914	444	21	t	t	X
ejpam-5914	444	22	)	)	PUNCT
ejpam-5914	444	23	̸=	̸=	PROPN
ejpam-5914	444	24	∅.	∅.	VERB
ejpam-5914	444	25	by	by	ADP
ejpam-5914	444	26	assumption	assumption	NOUN
ejpam-5914	444	27	,	,	PUNCT
ejpam-5914	444	28	we	we	PRON
ejpam-5914	444	29	have	have	AUX
ejpam-5914	444	30	l(f̃m	l(f̃m	VERB
ejpam-5914	444	31	;	;	PUNCT
ejpam-5914	444	32	t	t	X
ejpam-5914	444	33	)	)	PUNCT
ejpam-5914	444	34	is	be	AUX
ejpam-5914	444	35	a	a	DET
ejpam-5914	444	36	subalgebra	subalgebra	NOUN
ejpam-5914	444	37	of	of	ADP
ejpam-5914	444	38	a	a	PRON
ejpam-5914	444	39	and	and	CCONJ
ejpam-5914	444	40	so	so	ADV
ejpam-5914	444	41	(	(	PUNCT
ejpam-5914	444	42	x|(y|y))|(x|(y|y	x|(y|y))|(x|(y|y	ADJ
ejpam-5914	444	43	)	)	PUNCT
ejpam-5914	444	44	)	)	PUNCT
ejpam-5914	445	1	∈	∈	PROPN
ejpam-5914	445	2	l(f̃m	l(f̃m	PROPN
ejpam-5914	445	3	;	;	PUNCT
ejpam-5914	445	4	t	t	PROPN
ejpam-5914	445	5	)	)	PUNCT
ejpam-5914	445	6	.	.	PUNCT
ejpam-5914	446	1	thus	thus	ADV
ejpam-5914	446	2	,	,	PUNCT
ejpam-5914	446	3	f̃m((x|(y|y))|(x|(y|y	f̃m((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	446	4	)	)	PUNCT
ejpam-5914	446	5	)	)	PUNCT
ejpam-5914	446	6	)	)	PUNCT
ejpam-5914	446	7	≤	≤	NUM
ejpam-5914	446	8	t	t	NOUN
ejpam-5914	446	9	=	=	SYM
ejpam-5914	446	10	max{f̃m(x	max{f̃m(x	PROPN
ejpam-5914	446	11	)	)	PUNCT
ejpam-5914	446	12	,	,	PUNCT
ejpam-5914	446	13	f̃m(y	f̃m(y	NOUN
ejpam-5914	446	14	)	)	PUNCT
ejpam-5914	446	15	}	}	PUNCT
ejpam-5914	446	16	.	.	PUNCT
ejpam-5914	447	1	hence	hence	ADV
ejpam-5914	447	2	,	,	PUNCT
ejpam-5914	447	3	(	(	PUNCT
ejpam-5914	447	4	a	a	DET
ejpam-5914	447	5	,	,	PUNCT
ejpam-5914	447	6	f̃m	f̃m	NOUN
ejpam-5914	447	7	)	)	PUNCT
ejpam-5914	447	8	is	be	AUX
ejpam-5914	447	9	a	a	DET
ejpam-5914	447	10	4	4	NUM
ejpam-5914	447	11	-	-	PUNCT
ejpam-5914	447	12	fuzzy	fuzzy	ADJ
ejpam-5914	447	13	subalgebra	subalgebra	NOUN
ejpam-5914	447	14	of	of	ADP
ejpam-5914	447	15	a	a	PRON
ejpam-5914	447	16	,	,	PUNCT
ejpam-5914	447	17	that	that	ADV
ejpam-5914	447	18	is	is	ADV
ejpam-5914	447	19	,	,	PUNCT
ejpam-5914	447	20	(	(	PUNCT
ejpam-5914	447	21	a	a	PRON
ejpam-5914	447	22	,	,	PUNCT
ejpam-5914	447	23	f̃	f̃	PROPN
ejpam-5914	447	24	)	)	PUNCT
ejpam-5914	447	25	is	be	AUX
ejpam-5914	447	26	a	a	DET
ejpam-5914	447	27	mean	mean	ADJ
ejpam-5914	447	28	4	4	NUM
ejpam-5914	447	29	-	-	PUNCT
ejpam-5914	447	30	fuzzy	fuzzy	ADJ
ejpam-5914	447	31	subalgebra	subalgebra	NOUN
ejpam-5914	447	32	of	of	ADP
ejpam-5914	447	33	a.	a.	NOUN
ejpam-5914	447	34	corollary	corollary	PROPN
ejpam-5914	447	35	4	4	NUM
ejpam-5914	447	36	.	.	PUNCT
ejpam-5914	448	1	if	if	SCONJ
ejpam-5914	448	2	(	(	PUNCT
ejpam-5914	448	3	a	a	PRON
ejpam-5914	448	4	,	,	PUNCT
ejpam-5914	448	5	f̃	f̃	PROPN
ejpam-5914	448	6	)	)	PUNCT
ejpam-5914	448	7	is	be	AUX
ejpam-5914	448	8	a	a	DET
ejpam-5914	448	9	mean	mean	ADJ
ejpam-5914	448	10	2	2	NUM
ejpam-5914	448	11	-	-	PUNCT
ejpam-5914	448	12	fuzzy	fuzzy	ADJ
ejpam-5914	448	13	subalgebra	subalgebra	NOUN
ejpam-5914	448	14	of	of	ADP
ejpam-5914	448	15	a	a	DET
ejpam-5914	448	16	,	,	PUNCT
ejpam-5914	448	17	then	then	ADV
ejpam-5914	448	18	l(f̃m	l(f̃m	PROPN
ejpam-5914	448	19	;	;	PUNCT
ejpam-5914	448	20	t	t	PROPN
ejpam-5914	448	21	)	)	PUNCT
ejpam-5914	448	22	is	be	AUX
ejpam-5914	448	23	a	a	DET
ejpam-5914	448	24	subalgebra	subalgebra	NOUN
ejpam-5914	448	25	of	of	ADP
ejpam-5914	448	26	a	a	PRON
ejpam-5914	448	27	for	for	ADP
ejpam-5914	448	28	all	all	DET
ejpam-5914	448	29	t	t	NOUN
ejpam-5914	448	30	∈	∈	PROPN
ejpam-5914	449	1	[	[	X
ejpam-5914	449	2	0	0	NUM
ejpam-5914	449	3	,	,	PUNCT
ejpam-5914	449	4	1	1	NUM
ejpam-5914	449	5	]	]	PUNCT
ejpam-5914	449	6	with	with	ADP
ejpam-5914	449	7	l(f̃m	l(f̃m	PROPN
ejpam-5914	449	8	;	;	PUNCT
ejpam-5914	449	9	t	t	X
ejpam-5914	449	10	)	)	PUNCT
ejpam-5914	449	11	̸=	̸=	PROPN
ejpam-5914	449	12	∅.	∅.	ADP
ejpam-5914	449	13	n.	n.	PROPN
ejpam-5914	449	14	rajesh	rajesh	PROPN
ejpam-5914	449	15	et	et	PROPN
ejpam-5914	449	16	al	al	PROPN
ejpam-5914	449	17	.	.	PUNCT
ejpam-5914	449	18	/	/	SYM
ejpam-5914	449	19	eur	eur	PROPN
ejpam-5914	449	20	.	.	PUNCT
ejpam-5914	450	1	j.	j.	PROPN
ejpam-5914	450	2	pure	pure	PROPN
ejpam-5914	450	3	appl	appl	PROPN
ejpam-5914	450	4	.	.	PROPN
ejpam-5914	450	5	math	math	PROPN
ejpam-5914	450	6	,	,	PUNCT
ejpam-5914	450	7	18	18	NUM
ejpam-5914	450	8	(	(	PUNCT
ejpam-5914	450	9	2	2	NUM
ejpam-5914	450	10	)	)	PUNCT
ejpam-5914	450	11	(	(	PUNCT
ejpam-5914	450	12	2025	2025	NUM
ejpam-5914	450	13	)	)	PUNCT
ejpam-5914	450	14	,	,	PUNCT
ejpam-5914	450	15	5914	5914	NUM
ejpam-5914	450	16	17	17	NUM
ejpam-5914	450	17	of	of	ADP
ejpam-5914	450	18	21	21	NUM
ejpam-5914	450	19	proof	proof	NOUN
ejpam-5914	450	20	.	.	PUNCT
ejpam-5914	451	1	it	it	PRON
ejpam-5914	451	2	is	be	AUX
ejpam-5914	451	3	straightforward	straightforward	ADJ
ejpam-5914	451	4	by	by	ADP
ejpam-5914	451	5	theorems	theorem	NOUN
ejpam-5914	451	6	15	15	NUM
ejpam-5914	451	7	and	and	CCONJ
ejpam-5914	451	8	18	18	NUM
ejpam-5914	451	9	.	.	PUNCT
ejpam-5914	451	10	theorem	theorem	NOUN
ejpam-5914	451	11	19	19	NUM
ejpam-5914	451	12	.	.	PUNCT
ejpam-5914	452	1	if	if	SCONJ
ejpam-5914	452	2	(	(	PUNCT
ejpam-5914	452	3	a	a	PRON
ejpam-5914	452	4	,	,	PUNCT
ejpam-5914	452	5	f̃	f̃	PROPN
ejpam-5914	452	6	)	)	PUNCT
ejpam-5914	452	7	is	be	AUX
ejpam-5914	452	8	an	an	DET
ejpam-5914	452	9	interval	interval	NOUN
ejpam-5914	452	10	-	-	PUNCT
ejpam-5914	452	11	valued	value	VERB
ejpam-5914	452	12	fuzzy	fuzzy	ADJ
ejpam-5914	452	13	structure	structure	NOUN
ejpam-5914	452	14	over	over	ADP
ejpam-5914	452	15	a	a	PRON
ejpam-5914	452	16	in	in	ADP
ejpam-5914	452	17	which	which	PRON
ejpam-5914	452	18	(	(	PUNCT
ejpam-5914	452	19	a	a	PRON
ejpam-5914	452	20	,	,	PUNCT
ejpam-5914	452	21	f̃inf	f̃inf	ADJ
ejpam-5914	452	22	)	)	PUNCT
ejpam-5914	452	23	is	be	AUX
ejpam-5914	452	24	constant	constant	ADJ
ejpam-5914	452	25	and	and	CCONJ
ejpam-5914	452	26	(	(	PUNCT
ejpam-5914	452	27	a	a	DET
ejpam-5914	452	28	,	,	PUNCT
ejpam-5914	452	29	f̃sup	f̃sup	ADJ
ejpam-5914	452	30	)	)	PUNCT
ejpam-5914	452	31	is	be	AUX
ejpam-5914	452	32	a	a	DET
ejpam-5914	452	33	1	1	NUM
ejpam-5914	452	34	-	-	PUNCT
ejpam-5914	452	35	fuzzy	fuzzy	ADJ
ejpam-5914	452	36	subalgebra	subalgebra	NOUN
ejpam-5914	452	37	of	of	ADP
ejpam-5914	452	38	a	a	PRON
ejpam-5914	452	39	,	,	PUNCT
ejpam-5914	452	40	then	then	ADV
ejpam-5914	452	41	(	(	PUNCT
ejpam-5914	452	42	a	a	PRON
ejpam-5914	452	43	,	,	PUNCT
ejpam-5914	452	44	f̃	f̃	PROPN
ejpam-5914	452	45	)	)	PUNCT
ejpam-5914	452	46	is	be	AUX
ejpam-5914	452	47	a	a	DET
ejpam-5914	452	48	mean	mean	ADJ
ejpam-5914	452	49	1	1	NUM
ejpam-5914	452	50	-	-	PUNCT
ejpam-5914	452	51	fuzzy	fuzzy	ADJ
ejpam-5914	452	52	subalgebra	subalgebra	NOUN
ejpam-5914	452	53	of	of	ADP
ejpam-5914	452	54	a.	a.	NOUN
ejpam-5914	452	55	proof	proof	NOUN
ejpam-5914	452	56	.	.	PUNCT
ejpam-5914	453	1	assume	assume	VERB
ejpam-5914	453	2	that	that	SCONJ
ejpam-5914	453	3	(	(	PUNCT
ejpam-5914	453	4	a	a	PRON
ejpam-5914	453	5	,	,	PUNCT
ejpam-5914	453	6	f̃	f̃	PROPN
ejpam-5914	453	7	)	)	PUNCT
ejpam-5914	453	8	is	be	AUX
ejpam-5914	453	9	an	an	DET
ejpam-5914	453	10	interval	interval	NOUN
ejpam-5914	453	11	-	-	PUNCT
ejpam-5914	453	12	valued	value	VERB
ejpam-5914	453	13	fuzzy	fuzzy	ADJ
ejpam-5914	453	14	structure	structure	NOUN
ejpam-5914	453	15	over	over	ADP
ejpam-5914	453	16	a	a	PRON
ejpam-5914	453	17	in	in	ADP
ejpam-5914	453	18	which	which	PRON
ejpam-5914	453	19	(	(	PUNCT
ejpam-5914	453	20	a	a	PRON
ejpam-5914	453	21	,	,	PUNCT
ejpam-5914	453	22	f̃inf	f̃inf	ADJ
ejpam-5914	453	23	)	)	PUNCT
ejpam-5914	453	24	is	be	AUX
ejpam-5914	453	25	constant	constant	ADJ
ejpam-5914	453	26	and	and	CCONJ
ejpam-5914	453	27	(	(	PUNCT
ejpam-5914	453	28	a	a	DET
ejpam-5914	453	29	,	,	PUNCT
ejpam-5914	453	30	f̃sup	f̃sup	ADJ
ejpam-5914	453	31	)	)	PUNCT
ejpam-5914	453	32	is	be	AUX
ejpam-5914	453	33	a	a	DET
ejpam-5914	453	34	1	1	NUM
ejpam-5914	453	35	-	-	PUNCT
ejpam-5914	453	36	fuzzy	fuzzy	ADJ
ejpam-5914	453	37	subalgebra	subalgebra	NOUN
ejpam-5914	453	38	of	of	ADP
ejpam-5914	453	39	a.	a.	NOUN
ejpam-5914	453	40	let	let	VERB
ejpam-5914	453	41	x	x	PRON
ejpam-5914	453	42	,	,	PUNCT
ejpam-5914	453	43	y	y	PROPN
ejpam-5914	453	44	∈	∈	PROPN
ejpam-5914	453	45	a.	a.	NOUN
ejpam-5914	453	46	since	since	SCONJ
ejpam-5914	453	47	(	(	PUNCT
ejpam-5914	453	48	a	a	PRON
ejpam-5914	453	49	,	,	PUNCT
ejpam-5914	453	50	f̃inf	f̃inf	ADJ
ejpam-5914	453	51	)	)	PUNCT
ejpam-5914	453	52	is	be	AUX
ejpam-5914	453	53	constant	constant	ADJ
ejpam-5914	453	54	,	,	PUNCT
ejpam-5914	453	55	we	we	PRON
ejpam-5914	453	56	have	have	VERB
ejpam-5914	453	57	f̃inf(x	f̃inf(x	NOUN
ejpam-5914	453	58	)	)	PUNCT
ejpam-5914	453	59	=	=	SYM
ejpam-5914	453	60	f̃inf(0	f̃inf(0	PROPN
ejpam-5914	453	61	)	)	PUNCT
ejpam-5914	453	62	for	for	SCONJ
ejpam-5914	453	63	all	all	PRON
ejpam-5914	453	64	x	x	SYM
ejpam-5914	453	65	∈	∈	NOUN
ejpam-5914	453	66	a.	a.	NOUN
ejpam-5914	453	67	since	since	SCONJ
ejpam-5914	453	68	(	(	PUNCT
ejpam-5914	453	69	a	a	DET
ejpam-5914	453	70	,	,	PUNCT
ejpam-5914	453	71	f̃sup	f̃sup	ADJ
ejpam-5914	453	72	)	)	PUNCT
ejpam-5914	453	73	is	be	AUX
ejpam-5914	453	74	a	a	DET
ejpam-5914	453	75	1	1	NUM
ejpam-5914	453	76	-	-	PUNCT
ejpam-5914	453	77	fuzzy	fuzzy	ADJ
ejpam-5914	453	78	subalgebra	subalgebra	NOUN
ejpam-5914	453	79	of	of	ADP
ejpam-5914	453	80	a	a	PRON
ejpam-5914	453	81	,	,	PUNCT
ejpam-5914	453	82	we	we	PRON
ejpam-5914	453	83	have	have	VERB
ejpam-5914	453	84	f̃sup((x|(y|y))|(x|(y|y	f̃sup((x|(y|y))|(x|(y|y	VERB
ejpam-5914	453	85	)	)	PUNCT
ejpam-5914	453	86	)	)	PUNCT
ejpam-5914	453	87	)	)	PUNCT
ejpam-5914	453	88	≥	≥	NOUN
ejpam-5914	453	89	min{f̃sup(x	min{f̃sup(x	PROPN
ejpam-5914	453	90	)	)	PUNCT
ejpam-5914	453	91	,	,	PUNCT
ejpam-5914	453	92	f̃sup(y	f̃sup(y	NOUN
ejpam-5914	453	93	)	)	PUNCT
ejpam-5914	453	94	}	}	PUNCT
ejpam-5914	453	95	.	.	PUNCT
ejpam-5914	454	1	thus	thus	ADV
ejpam-5914	454	2	,	,	PUNCT
ejpam-5914	454	3	f̃m((x|(y|y))|(x|(y|y	f̃m((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	454	4	)	)	PUNCT
ejpam-5914	454	5	)	)	PUNCT
ejpam-5914	454	6	)	)	PUNCT
ejpam-5914	455	1	=	=	SYM
ejpam-5914	455	2	f̃sup((x|(y|y))|(x|(y|y	f̃sup((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	455	3	)	)	PUNCT
ejpam-5914	455	4	)	)	PUNCT
ejpam-5914	455	5	)	)	PUNCT
ejpam-5914	456	1	+	+	CCONJ
ejpam-5914	456	2	f̃inf((x|(y|y))|(x|(y|y	f̃inf((x|(y|y))|(x|(y|y	ADJ
ejpam-5914	456	3	)	)	PUNCT
ejpam-5914	456	4	)	)	PUNCT
ejpam-5914	456	5	)	)	PUNCT
ejpam-5914	456	6	2	2	NUM
ejpam-5914	456	7	=	=	SYM
ejpam-5914	456	8	f̃sup((x|(y|y))|(x|(y|y	f̃sup((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	456	9	)	)	PUNCT
ejpam-5914	456	10	)	)	PUNCT
ejpam-5914	456	11	)	)	PUNCT
ejpam-5914	456	12	2	2	NUM
ejpam-5914	457	1	+	+	NUM
ejpam-5914	457	2	f̃inf(0	f̃inf(0	NOUN
ejpam-5914	457	3	)	)	PUNCT
ejpam-5914	457	4	2	2	NUM
ejpam-5914	457	5	≥	≥	NOUN
ejpam-5914	457	6	min	min	NOUN
ejpam-5914	457	7	{	{	PUNCT
ejpam-5914	457	8	f̃sup(x	f̃sup(x	ADJ
ejpam-5914	457	9	)	)	PUNCT
ejpam-5914	457	10	2	2	NUM
ejpam-5914	457	11	+	+	SYM
ejpam-5914	457	12	f̃inf(x	f̃inf(x	NOUN
ejpam-5914	457	13	)	)	PUNCT
ejpam-5914	457	14	2	2	NUM
ejpam-5914	457	15	}	}	PUNCT
ejpam-5914	457	16	+	+	NUM
ejpam-5914	457	17	f̃inf(0	f̃inf(0	NOUN
ejpam-5914	457	18	)	)	PUNCT
ejpam-5914	457	19	2	2	NUM
ejpam-5914	457	20	=	=	SYM
ejpam-5914	457	21	min	min	NOUN
ejpam-5914	457	22	{	{	PUNCT
ejpam-5914	457	23	f̃sup(x	f̃sup(x	NOUN
ejpam-5914	457	24	)	)	PUNCT
ejpam-5914	457	25	2	2	NUM
ejpam-5914	457	26	+	+	NUM
ejpam-5914	457	27	f̃inf(0	f̃inf(0	NOUN
ejpam-5914	457	28	)	)	PUNCT
ejpam-5914	457	29	2	2	NUM
ejpam-5914	457	30	,	,	PUNCT
ejpam-5914	457	31	f̃sup(y	f̃sup(y	NOUN
ejpam-5914	457	32	)	)	PUNCT
ejpam-5914	457	33	2	2	NUM
ejpam-5914	457	34	+	+	NUM
ejpam-5914	457	35	f̃inf(0	f̃inf(0	NOUN
ejpam-5914	457	36	)	)	PUNCT
ejpam-5914	457	37	2	2	NUM
ejpam-5914	457	38	}	}	PUNCT
ejpam-5914	457	39	=	=	SYM
ejpam-5914	457	40	min	min	NOUN
ejpam-5914	457	41	{	{	PUNCT
ejpam-5914	457	42	f̃sup(x	f̃sup(x	NOUN
ejpam-5914	457	43	)	)	PUNCT
ejpam-5914	457	44	+	+	SYM
ejpam-5914	457	45	f̃inf(x	f̃inf(x	NOUN
ejpam-5914	457	46	)	)	PUNCT
ejpam-5914	457	47	2	2	NUM
ejpam-5914	457	48	,	,	PUNCT
ejpam-5914	457	49	f̃sup(y	f̃sup(y	NOUN
ejpam-5914	457	50	)	)	PUNCT
ejpam-5914	457	51	+	+	SYM
ejpam-5914	457	52	f̃inf(y	f̃inf(y	NOUN
ejpam-5914	457	53	)	)	PUNCT
ejpam-5914	457	54	2	2	NUM
ejpam-5914	457	55	}	}	PUNCT
ejpam-5914	457	56	=	=	SYM
ejpam-5914	457	57	min{f̃m(x	min{f̃m(x	PROPN
ejpam-5914	457	58	)	)	PUNCT
ejpam-5914	457	59	,	,	PUNCT
ejpam-5914	457	60	f̃m(y	f̃m(y	NOUN
ejpam-5914	457	61	)	)	PUNCT
ejpam-5914	457	62	}	}	PUNCT
ejpam-5914	457	63	.	.	PUNCT
ejpam-5914	458	1	hence	hence	ADV
ejpam-5914	458	2	,	,	PUNCT
ejpam-5914	458	3	(	(	PUNCT
ejpam-5914	458	4	a	a	DET
ejpam-5914	458	5	,	,	PUNCT
ejpam-5914	458	6	f̃m	f̃m	NOUN
ejpam-5914	458	7	)	)	PUNCT
ejpam-5914	458	8	is	be	AUX
ejpam-5914	458	9	a	a	DET
ejpam-5914	458	10	1	1	NUM
ejpam-5914	458	11	-	-	PUNCT
ejpam-5914	458	12	fuzzy	fuzzy	ADJ
ejpam-5914	458	13	subalgebra	subalgebra	NOUN
ejpam-5914	458	14	of	of	ADP
ejpam-5914	458	15	a	a	PRON
ejpam-5914	458	16	,	,	PUNCT
ejpam-5914	458	17	that	that	ADV
ejpam-5914	458	18	is	is	ADV
ejpam-5914	458	19	,	,	PUNCT
ejpam-5914	458	20	(	(	PUNCT
ejpam-5914	458	21	a	a	PRON
ejpam-5914	458	22	,	,	PUNCT
ejpam-5914	458	23	f̃	f̃	PROPN
ejpam-5914	458	24	)	)	PUNCT
ejpam-5914	458	25	is	be	AUX
ejpam-5914	458	26	a	a	DET
ejpam-5914	458	27	mean	mean	ADJ
ejpam-5914	458	28	1	1	NUM
ejpam-5914	458	29	-	-	PUNCT
ejpam-5914	458	30	fuzzy	fuzzy	ADJ
ejpam-5914	458	31	subalgebra	subalgebra	NOUN
ejpam-5914	458	32	of	of	ADP
ejpam-5914	458	33	a.	a.	NOUN
ejpam-5914	458	34	theorem	theorem	NOUN
ejpam-5914	458	35	20	20	NUM
ejpam-5914	458	36	.	.	PUNCT
ejpam-5914	459	1	if	if	SCONJ
ejpam-5914	459	2	(	(	PUNCT
ejpam-5914	459	3	a	a	PRON
ejpam-5914	459	4	,	,	PUNCT
ejpam-5914	459	5	f̃	f̃	PROPN
ejpam-5914	459	6	)	)	PUNCT
ejpam-5914	459	7	is	be	AUX
ejpam-5914	459	8	an	an	DET
ejpam-5914	459	9	interval	interval	NOUN
ejpam-5914	459	10	-	-	PUNCT
ejpam-5914	459	11	valued	value	VERB
ejpam-5914	459	12	fuzzy	fuzzy	ADJ
ejpam-5914	459	13	structure	structure	NOUN
ejpam-5914	459	14	over	over	ADP
ejpam-5914	459	15	a	a	PRON
ejpam-5914	459	16	in	in	ADP
ejpam-5914	459	17	which	which	PRON
ejpam-5914	459	18	(	(	PUNCT
ejpam-5914	459	19	a	a	PRON
ejpam-5914	459	20	,	,	PUNCT
ejpam-5914	459	21	f̃inf	f̃inf	ADJ
ejpam-5914	459	22	)	)	PUNCT
ejpam-5914	459	23	is	be	AUX
ejpam-5914	459	24	constant	constant	ADJ
ejpam-5914	459	25	and	and	CCONJ
ejpam-5914	459	26	(	(	PUNCT
ejpam-5914	459	27	a	a	DET
ejpam-5914	459	28	,	,	PUNCT
ejpam-5914	459	29	f̃sup	f̃sup	ADJ
ejpam-5914	459	30	)	)	PUNCT
ejpam-5914	459	31	is	be	AUX
ejpam-5914	459	32	a	a	DET
ejpam-5914	459	33	4	4	NUM
ejpam-5914	459	34	-	-	PUNCT
ejpam-5914	459	35	fuzzy	fuzzy	ADJ
ejpam-5914	459	36	subalgebra	subalgebra	NOUN
ejpam-5914	459	37	of	of	ADP
ejpam-5914	459	38	a	a	PRON
ejpam-5914	459	39	,	,	PUNCT
ejpam-5914	459	40	then	then	ADV
ejpam-5914	459	41	(	(	PUNCT
ejpam-5914	459	42	a	a	PRON
ejpam-5914	459	43	,	,	PUNCT
ejpam-5914	459	44	f̃	f̃	PROPN
ejpam-5914	459	45	)	)	PUNCT
ejpam-5914	459	46	is	be	AUX
ejpam-5914	459	47	a	a	DET
ejpam-5914	459	48	mean	mean	ADJ
ejpam-5914	459	49	4	4	NUM
ejpam-5914	459	50	-	-	PUNCT
ejpam-5914	459	51	fuzzy	fuzzy	ADJ
ejpam-5914	459	52	subalgebra	subalgebra	NOUN
ejpam-5914	459	53	of	of	ADP
ejpam-5914	459	54	a.	a.	NOUN
ejpam-5914	459	55	proof	proof	NOUN
ejpam-5914	459	56	.	.	PUNCT
ejpam-5914	460	1	assume	assume	VERB
ejpam-5914	460	2	that	that	SCONJ
ejpam-5914	460	3	(	(	PUNCT
ejpam-5914	460	4	a	a	PRON
ejpam-5914	460	5	,	,	PUNCT
ejpam-5914	460	6	f̃	f̃	PROPN
ejpam-5914	460	7	)	)	PUNCT
ejpam-5914	460	8	is	be	AUX
ejpam-5914	460	9	an	an	DET
ejpam-5914	460	10	interval	interval	NOUN
ejpam-5914	460	11	-	-	PUNCT
ejpam-5914	460	12	valued	value	VERB
ejpam-5914	460	13	fuzzy	fuzzy	ADJ
ejpam-5914	460	14	structure	structure	NOUN
ejpam-5914	460	15	over	over	ADP
ejpam-5914	460	16	a	a	PRON
ejpam-5914	460	17	in	in	ADP
ejpam-5914	460	18	which	which	PRON
ejpam-5914	460	19	(	(	PUNCT
ejpam-5914	460	20	a	a	PRON
ejpam-5914	460	21	,	,	PUNCT
ejpam-5914	460	22	f̃inf	f̃inf	ADJ
ejpam-5914	460	23	)	)	PUNCT
ejpam-5914	460	24	is	be	AUX
ejpam-5914	460	25	constant	constant	ADJ
ejpam-5914	460	26	and	and	CCONJ
ejpam-5914	460	27	(	(	PUNCT
ejpam-5914	460	28	a	a	DET
ejpam-5914	460	29	,	,	PUNCT
ejpam-5914	460	30	f̃sup	f̃sup	ADJ
ejpam-5914	460	31	)	)	PUNCT
ejpam-5914	460	32	is	be	AUX
ejpam-5914	460	33	a	a	DET
ejpam-5914	460	34	4	4	NUM
ejpam-5914	460	35	-	-	PUNCT
ejpam-5914	460	36	fuzzy	fuzzy	ADJ
ejpam-5914	460	37	subalgebra	subalgebra	NOUN
ejpam-5914	460	38	of	of	ADP
ejpam-5914	460	39	a.	a.	NOUN
ejpam-5914	460	40	let	let	VERB
ejpam-5914	460	41	x	x	PRON
ejpam-5914	460	42	,	,	PUNCT
ejpam-5914	460	43	y	y	PROPN
ejpam-5914	460	44	∈	∈	PROPN
ejpam-5914	460	45	a.	a.	NOUN
ejpam-5914	460	46	since	since	SCONJ
ejpam-5914	460	47	(	(	PUNCT
ejpam-5914	460	48	a	a	PRON
ejpam-5914	460	49	,	,	PUNCT
ejpam-5914	460	50	f̃inf	f̃inf	ADJ
ejpam-5914	460	51	)	)	PUNCT
ejpam-5914	460	52	is	be	AUX
ejpam-5914	460	53	constant	constant	ADJ
ejpam-5914	460	54	,	,	PUNCT
ejpam-5914	460	55	we	we	PRON
ejpam-5914	460	56	have	have	VERB
ejpam-5914	460	57	f̃inf(x	f̃inf(x	NOUN
ejpam-5914	460	58	)	)	PUNCT
ejpam-5914	460	59	=	=	SYM
ejpam-5914	460	60	f̃inf(0	f̃inf(0	PROPN
ejpam-5914	460	61	)	)	PUNCT
ejpam-5914	460	62	for	for	SCONJ
ejpam-5914	460	63	all	all	PRON
ejpam-5914	460	64	x	x	SYM
ejpam-5914	460	65	∈	∈	NOUN
ejpam-5914	460	66	a.	a.	NOUN
ejpam-5914	460	67	since	since	SCONJ
ejpam-5914	460	68	(	(	PUNCT
ejpam-5914	460	69	a	a	DET
ejpam-5914	460	70	,	,	PUNCT
ejpam-5914	460	71	f̃sup	f̃sup	ADJ
ejpam-5914	460	72	)	)	PUNCT
ejpam-5914	460	73	is	be	AUX
ejpam-5914	460	74	a	a	DET
ejpam-5914	460	75	4	4	NUM
ejpam-5914	460	76	-	-	PUNCT
ejpam-5914	460	77	fuzzy	fuzzy	ADJ
ejpam-5914	460	78	subalgebra	subalgebra	NOUN
ejpam-5914	460	79	of	of	ADP
ejpam-5914	460	80	a	a	PRON
ejpam-5914	460	81	,	,	PUNCT
ejpam-5914	460	82	we	we	PRON
ejpam-5914	460	83	have	have	VERB
ejpam-5914	460	84	f̃sup((x|(y|y))|(x|(y|y	f̃sup((x|(y|y))|(x|(y|y	VERB
ejpam-5914	460	85	)	)	PUNCT
ejpam-5914	460	86	)	)	PUNCT
ejpam-5914	460	87	)	)	PUNCT
ejpam-5914	461	1	≤	≤	PROPN
ejpam-5914	461	2	max{f̃sup(x	max{f̃sup(x	PROPN
ejpam-5914	461	3	)	)	PUNCT
ejpam-5914	461	4	,	,	PUNCT
ejpam-5914	461	5	f̃sup(y	f̃sup(y	NOUN
ejpam-5914	461	6	)	)	PUNCT
ejpam-5914	461	7	}	}	PUNCT
ejpam-5914	461	8	.	.	PUNCT
ejpam-5914	462	1	n.	n.	PROPN
ejpam-5914	462	2	rajesh	rajesh	PROPN
ejpam-5914	462	3	et	et	PROPN
ejpam-5914	462	4	al	al	PROPN
ejpam-5914	462	5	.	.	PUNCT
ejpam-5914	462	6	/	/	SYM
ejpam-5914	462	7	eur	eur	PROPN
ejpam-5914	462	8	.	.	PUNCT
ejpam-5914	463	1	j.	j.	PROPN
ejpam-5914	463	2	pure	pure	PROPN
ejpam-5914	463	3	appl	appl	PROPN
ejpam-5914	463	4	.	.	PROPN
ejpam-5914	463	5	math	math	PROPN
ejpam-5914	463	6	,	,	PUNCT
ejpam-5914	463	7	18	18	NUM
ejpam-5914	463	8	(	(	PUNCT
ejpam-5914	463	9	2	2	NUM
ejpam-5914	463	10	)	)	PUNCT
ejpam-5914	463	11	(	(	PUNCT
ejpam-5914	463	12	2025	2025	NUM
ejpam-5914	463	13	)	)	PUNCT
ejpam-5914	463	14	,	,	PUNCT
ejpam-5914	463	15	5914	5914	NUM
ejpam-5914	463	16	18	18	NUM
ejpam-5914	463	17	of	of	ADP
ejpam-5914	463	18	21	21	NUM
ejpam-5914	463	19	thus	thus	ADV
ejpam-5914	463	20	,	,	PUNCT
ejpam-5914	463	21	f̃m((x|(y|y))|(x|(y|y	f̃m((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	463	22	)	)	PUNCT
ejpam-5914	463	23	)	)	PUNCT
ejpam-5914	463	24	)	)	PUNCT
ejpam-5914	464	1	=	=	SYM
ejpam-5914	464	2	f̃sup((x|(y|y))|(x|(y|y	f̃sup((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	464	3	)	)	PUNCT
ejpam-5914	464	4	)	)	PUNCT
ejpam-5914	464	5	)	)	PUNCT
ejpam-5914	465	1	+	+	CCONJ
ejpam-5914	465	2	f̃inf((x|(y|y))|(x|(y|y	f̃inf((x|(y|y))|(x|(y|y	ADJ
ejpam-5914	465	3	)	)	PUNCT
ejpam-5914	465	4	)	)	PUNCT
ejpam-5914	465	5	)	)	PUNCT
ejpam-5914	465	6	2	2	NUM
ejpam-5914	465	7	=	=	SYM
ejpam-5914	465	8	f̃sup((x|(y|y))|(x|(y|y	f̃sup((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	465	9	)	)	PUNCT
ejpam-5914	465	10	)	)	PUNCT
ejpam-5914	465	11	)	)	PUNCT
ejpam-5914	465	12	2	2	NUM
ejpam-5914	466	1	+	+	NUM
ejpam-5914	466	2	f̃inf(0	f̃inf(0	NOUN
ejpam-5914	466	3	)	)	PUNCT
ejpam-5914	466	4	2	2	NUM
ejpam-5914	466	5	≥	≥	NOUN
ejpam-5914	466	6	min	min	NOUN
ejpam-5914	466	7	{	{	PUNCT
ejpam-5914	466	8	f̃sup(x	f̃sup(x	ADJ
ejpam-5914	466	9	)	)	PUNCT
ejpam-5914	466	10	2	2	NUM
ejpam-5914	466	11	+	+	SYM
ejpam-5914	466	12	f̃inf(x	f̃inf(x	NOUN
ejpam-5914	466	13	)	)	PUNCT
ejpam-5914	466	14	2	2	NUM
ejpam-5914	466	15	}	}	PUNCT
ejpam-5914	466	16	+	+	NUM
ejpam-5914	466	17	f̃inf(0	f̃inf(0	NOUN
ejpam-5914	466	18	)	)	PUNCT
ejpam-5914	466	19	2	2	NUM
ejpam-5914	466	20	=	=	SYM
ejpam-5914	466	21	min	min	NOUN
ejpam-5914	466	22	{	{	PUNCT
ejpam-5914	466	23	f̃sup(x	f̃sup(x	NOUN
ejpam-5914	466	24	)	)	PUNCT
ejpam-5914	466	25	2	2	NUM
ejpam-5914	466	26	+	+	NUM
ejpam-5914	466	27	f̃inf(0	f̃inf(0	NOUN
ejpam-5914	466	28	)	)	PUNCT
ejpam-5914	466	29	2	2	NUM
ejpam-5914	466	30	,	,	PUNCT
ejpam-5914	466	31	f̃sup(y	f̃sup(y	NOUN
ejpam-5914	466	32	)	)	PUNCT
ejpam-5914	466	33	2	2	NUM
ejpam-5914	466	34	+	+	NUM
ejpam-5914	466	35	f̃inf(0	f̃inf(0	NOUN
ejpam-5914	466	36	)	)	PUNCT
ejpam-5914	466	37	2	2	NUM
ejpam-5914	466	38	}	}	PUNCT
ejpam-5914	466	39	=	=	SYM
ejpam-5914	466	40	min	min	NOUN
ejpam-5914	466	41	{	{	PUNCT
ejpam-5914	466	42	f̃sup(x	f̃sup(x	NOUN
ejpam-5914	466	43	)	)	PUNCT
ejpam-5914	466	44	+	+	SYM
ejpam-5914	466	45	f̃inf(x	f̃inf(x	NOUN
ejpam-5914	466	46	)	)	PUNCT
ejpam-5914	466	47	2	2	NUM
ejpam-5914	466	48	,	,	PUNCT
ejpam-5914	466	49	f̃sup(y	f̃sup(y	NOUN
ejpam-5914	466	50	)	)	PUNCT
ejpam-5914	466	51	+	+	SYM
ejpam-5914	466	52	f̃inf(y	f̃inf(y	NOUN
ejpam-5914	466	53	)	)	PUNCT
ejpam-5914	466	54	2	2	NUM
ejpam-5914	466	55	}	}	PUNCT
ejpam-5914	466	56	=	=	SYM
ejpam-5914	466	57	min{f̃m(x	min{f̃m(x	PROPN
ejpam-5914	466	58	)	)	PUNCT
ejpam-5914	466	59	,	,	PUNCT
ejpam-5914	466	60	f̃m(y	f̃m(y	NOUN
ejpam-5914	466	61	)	)	PUNCT
ejpam-5914	466	62	}	}	PUNCT
ejpam-5914	466	63	.	.	PUNCT
ejpam-5914	467	1	hence	hence	ADV
ejpam-5914	467	2	,	,	PUNCT
ejpam-5914	467	3	(	(	PUNCT
ejpam-5914	467	4	a	a	DET
ejpam-5914	467	5	,	,	PUNCT
ejpam-5914	467	6	f̃m	f̃m	NOUN
ejpam-5914	467	7	)	)	PUNCT
ejpam-5914	467	8	is	be	AUX
ejpam-5914	467	9	a	a	DET
ejpam-5914	467	10	4	4	NUM
ejpam-5914	467	11	-	-	PUNCT
ejpam-5914	467	12	fuzzy	fuzzy	ADJ
ejpam-5914	467	13	subalgebra	subalgebra	NOUN
ejpam-5914	467	14	of	of	ADP
ejpam-5914	467	15	a	a	PRON
ejpam-5914	467	16	,	,	PUNCT
ejpam-5914	467	17	that	that	ADV
ejpam-5914	467	18	is	is	ADV
ejpam-5914	467	19	,	,	PUNCT
ejpam-5914	467	20	(	(	PUNCT
ejpam-5914	467	21	a	a	PRON
ejpam-5914	467	22	,	,	PUNCT
ejpam-5914	467	23	f̃	f̃	PROPN
ejpam-5914	467	24	)	)	PUNCT
ejpam-5914	467	25	is	be	AUX
ejpam-5914	467	26	a	a	DET
ejpam-5914	467	27	mean	mean	ADJ
ejpam-5914	467	28	4	4	NUM
ejpam-5914	467	29	-	-	PUNCT
ejpam-5914	467	30	fuzzy	fuzzy	ADJ
ejpam-5914	467	31	subalgebra	subalgebra	NOUN
ejpam-5914	467	32	of	of	ADP
ejpam-5914	467	33	a.	a.	NOUN
ejpam-5914	467	34	theorem	theorem	NOUN
ejpam-5914	467	35	21	21	NUM
ejpam-5914	467	36	.	.	PUNCT
ejpam-5914	468	1	if	if	SCONJ
ejpam-5914	468	2	(	(	PUNCT
ejpam-5914	468	3	a	a	PRON
ejpam-5914	468	4	,	,	PUNCT
ejpam-5914	468	5	f̃	f̃	PROPN
ejpam-5914	468	6	)	)	PUNCT
ejpam-5914	468	7	is	be	AUX
ejpam-5914	468	8	an	an	DET
ejpam-5914	468	9	interval	interval	NOUN
ejpam-5914	468	10	-	-	PUNCT
ejpam-5914	468	11	valued	value	VERB
ejpam-5914	468	12	fuzzy	fuzzy	ADJ
ejpam-5914	468	13	structure	structure	NOUN
ejpam-5914	468	14	over	over	ADP
ejpam-5914	468	15	a	a	DET
ejpam-5914	468	16	in	in	ADP
ejpam-5914	468	17	which	which	PRON
ejpam-5914	468	18	(	(	PUNCT
ejpam-5914	468	19	a	a	DET
ejpam-5914	468	20	,	,	PUNCT
ejpam-5914	468	21	f̃sup	f̃sup	ADJ
ejpam-5914	468	22	)	)	PUNCT
ejpam-5914	468	23	is	be	AUX
ejpam-5914	468	24	constant	constant	ADJ
ejpam-5914	468	25	and	and	CCONJ
ejpam-5914	468	26	(	(	PUNCT
ejpam-5914	468	27	a	a	PRON
ejpam-5914	468	28	,	,	PUNCT
ejpam-5914	468	29	f̃inf	f̃inf	ADJ
ejpam-5914	468	30	)	)	PUNCT
ejpam-5914	468	31	is	be	AUX
ejpam-5914	468	32	a	a	DET
ejpam-5914	468	33	4	4	NUM
ejpam-5914	468	34	-	-	PUNCT
ejpam-5914	468	35	fuzzy	fuzzy	ADJ
ejpam-5914	468	36	subalgebra	subalgebra	NOUN
ejpam-5914	468	37	of	of	ADP
ejpam-5914	468	38	a	a	PRON
ejpam-5914	468	39	,	,	PUNCT
ejpam-5914	468	40	then	then	ADV
ejpam-5914	468	41	(	(	PUNCT
ejpam-5914	468	42	a	a	PRON
ejpam-5914	468	43	,	,	PUNCT
ejpam-5914	468	44	f̃	f̃	PROPN
ejpam-5914	468	45	)	)	PUNCT
ejpam-5914	468	46	is	be	AUX
ejpam-5914	468	47	a	a	DET
ejpam-5914	468	48	mean	mean	ADJ
ejpam-5914	468	49	4	4	NUM
ejpam-5914	468	50	-	-	PUNCT
ejpam-5914	468	51	fuzzy	fuzzy	ADJ
ejpam-5914	468	52	subalgebra	subalgebra	NOUN
ejpam-5914	468	53	of	of	ADP
ejpam-5914	468	54	a.	a.	NOUN
ejpam-5914	468	55	proof	proof	NOUN
ejpam-5914	468	56	.	.	PUNCT
ejpam-5914	469	1	assume	assume	VERB
ejpam-5914	469	2	that	that	SCONJ
ejpam-5914	469	3	(	(	PUNCT
ejpam-5914	469	4	a	a	PRON
ejpam-5914	469	5	,	,	PUNCT
ejpam-5914	469	6	f̃	f̃	PROPN
ejpam-5914	469	7	)	)	PUNCT
ejpam-5914	469	8	is	be	AUX
ejpam-5914	469	9	an	an	DET
ejpam-5914	469	10	interval	interval	NOUN
ejpam-5914	469	11	-	-	PUNCT
ejpam-5914	469	12	valued	value	VERB
ejpam-5914	469	13	fuzzy	fuzzy	ADJ
ejpam-5914	469	14	structure	structure	NOUN
ejpam-5914	469	15	over	over	ADP
ejpam-5914	469	16	a	a	DET
ejpam-5914	469	17	in	in	ADP
ejpam-5914	469	18	which	which	PRON
ejpam-5914	469	19	(	(	PUNCT
ejpam-5914	469	20	a	a	DET
ejpam-5914	469	21	,	,	PUNCT
ejpam-5914	469	22	f̃sup	f̃sup	ADJ
ejpam-5914	469	23	)	)	PUNCT
ejpam-5914	469	24	is	be	AUX
ejpam-5914	469	25	constant	constant	ADJ
ejpam-5914	469	26	and	and	CCONJ
ejpam-5914	469	27	(	(	PUNCT
ejpam-5914	469	28	a	a	PRON
ejpam-5914	469	29	,	,	PUNCT
ejpam-5914	469	30	f̃inf	f̃inf	ADJ
ejpam-5914	469	31	)	)	PUNCT
ejpam-5914	469	32	is	be	AUX
ejpam-5914	469	33	a	a	DET
ejpam-5914	469	34	4	4	NUM
ejpam-5914	469	35	-	-	PUNCT
ejpam-5914	469	36	fuzzy	fuzzy	ADJ
ejpam-5914	469	37	subalgebra	subalgebra	NOUN
ejpam-5914	469	38	of	of	ADP
ejpam-5914	469	39	a.	a.	NOUN
ejpam-5914	469	40	let	let	VERB
ejpam-5914	469	41	x	x	PRON
ejpam-5914	469	42	,	,	PUNCT
ejpam-5914	469	43	y	y	PROPN
ejpam-5914	469	44	∈	∈	PROPN
ejpam-5914	469	45	a.	a.	NOUN
ejpam-5914	469	46	since	since	SCONJ
ejpam-5914	469	47	(	(	PUNCT
ejpam-5914	469	48	a	a	DET
ejpam-5914	469	49	,	,	PUNCT
ejpam-5914	469	50	f̃sup	f̃sup	ADJ
ejpam-5914	469	51	)	)	PUNCT
ejpam-5914	469	52	is	be	AUX
ejpam-5914	469	53	constant	constant	ADJ
ejpam-5914	469	54	,	,	PUNCT
ejpam-5914	469	55	we	we	PRON
ejpam-5914	469	56	have	have	AUX
ejpam-5914	469	57	f̃sup(x	f̃sup(x	VERB
ejpam-5914	469	58	)	)	PUNCT
ejpam-5914	469	59	=	=	SYM
ejpam-5914	469	60	f̃sup(0	f̃sup(0	NOUN
ejpam-5914	469	61	)	)	PUNCT
ejpam-5914	469	62	for	for	SCONJ
ejpam-5914	469	63	all	all	PRON
ejpam-5914	469	64	x	x	SYM
ejpam-5914	469	65	∈	∈	NOUN
ejpam-5914	469	66	a.	a.	NOUN
ejpam-5914	469	67	since	since	SCONJ
ejpam-5914	469	68	(	(	PUNCT
ejpam-5914	469	69	a	a	PRON
ejpam-5914	469	70	,	,	PUNCT
ejpam-5914	469	71	f̃inf	f̃inf	ADJ
ejpam-5914	469	72	)	)	PUNCT
ejpam-5914	469	73	is	be	AUX
ejpam-5914	469	74	a	a	DET
ejpam-5914	469	75	4	4	NUM
ejpam-5914	469	76	-	-	PUNCT
ejpam-5914	469	77	fuzzy	fuzzy	ADJ
ejpam-5914	469	78	subalgebra	subalgebra	NOUN
ejpam-5914	469	79	of	of	ADP
ejpam-5914	469	80	a	a	PRON
ejpam-5914	469	81	,	,	PUNCT
ejpam-5914	469	82	we	we	PRON
ejpam-5914	469	83	have	have	VERB
ejpam-5914	469	84	f̃inf((x|(y|y))|(x|(y|y	f̃inf((x|(y|y))|(x|(y|y	VERB
ejpam-5914	469	85	)	)	PUNCT
ejpam-5914	469	86	)	)	PUNCT
ejpam-5914	469	87	)	)	PUNCT
ejpam-5914	470	1	≤	≤	NUM
ejpam-5914	470	2	max{f̃inf(x	max{f̃inf(x	PROPN
ejpam-5914	470	3	)	)	PUNCT
ejpam-5914	470	4	,	,	PUNCT
ejpam-5914	470	5	f̃inf(y	f̃inf(y	NOUN
ejpam-5914	470	6	)	)	PUNCT
ejpam-5914	470	7	}	}	PUNCT
ejpam-5914	470	8	.	.	PUNCT
ejpam-5914	471	1	thus	thus	ADV
ejpam-5914	471	2	,	,	PUNCT
ejpam-5914	471	3	f̃m((x|(y|y))|(x|(y|y	f̃m((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	471	4	)	)	PUNCT
ejpam-5914	471	5	)	)	PUNCT
ejpam-5914	471	6	)	)	PUNCT
ejpam-5914	472	1	=	=	SYM
ejpam-5914	472	2	f̃sup((x|(y|y))|(x|(y|y	f̃sup((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	472	3	)	)	PUNCT
ejpam-5914	472	4	)	)	PUNCT
ejpam-5914	472	5	)	)	PUNCT
ejpam-5914	473	1	+	+	CCONJ
ejpam-5914	473	2	f̃inf((x|(y|y))|(x|(y|y	f̃inf((x|(y|y))|(x|(y|y	ADJ
ejpam-5914	473	3	)	)	PUNCT
ejpam-5914	473	4	)	)	PUNCT
ejpam-5914	473	5	)	)	PUNCT
ejpam-5914	473	6	2	2	X
ejpam-5914	473	7	=	=	SYM
ejpam-5914	473	8	f̃sup(0	f̃sup(0	NOUN
ejpam-5914	473	9	)	)	PUNCT
ejpam-5914	473	10	+	+	CCONJ
ejpam-5914	473	11	f̃inf((x|(y|y))|(x|(y|y	f̃inf((x|(y|y))|(x|(y|y	ADJ
ejpam-5914	473	12	)	)	PUNCT
ejpam-5914	473	13	)	)	PUNCT
ejpam-5914	473	14	)	)	PUNCT
ejpam-5914	473	15	2	2	X
ejpam-5914	473	16	=	=	SYM
ejpam-5914	473	17	f̃sup(0	f̃sup(0	NOUN
ejpam-5914	473	18	)	)	PUNCT
ejpam-5914	473	19	2	2	NUM
ejpam-5914	474	1	+	+	CCONJ
ejpam-5914	474	2	f̃inf((x|(y|y))|(x|(y|y	f̃inf((x|(y|y))|(x|(y|y	ADJ
ejpam-5914	474	3	)	)	PUNCT
ejpam-5914	474	4	)	)	PUNCT
ejpam-5914	474	5	)	)	PUNCT
ejpam-5914	474	6	2	2	NUM
ejpam-5914	474	7	≤	≤	NOUN
ejpam-5914	474	8	f̃sup(0	f̃sup(0	NOUN
ejpam-5914	474	9	)	)	PUNCT
ejpam-5914	474	10	2	2	NUM
ejpam-5914	475	1	+	+	CCONJ
ejpam-5914	475	2	max	max	PROPN
ejpam-5914	475	3	{	{	PUNCT
ejpam-5914	475	4	f̃sup(x	f̃sup(x	NOUN
ejpam-5914	475	5	)	)	PUNCT
ejpam-5914	475	6	2	2	NUM
ejpam-5914	475	7	,	,	PUNCT
ejpam-5914	475	8	f̃inf(x	f̃inf(x	NOUN
ejpam-5914	475	9	)	)	PUNCT
ejpam-5914	475	10	2	2	NUM
ejpam-5914	475	11	}	}	PUNCT
ejpam-5914	475	12	=	=	SYM
ejpam-5914	475	13	max	max	PROPN
ejpam-5914	475	14	{	{	PUNCT
ejpam-5914	475	15	f̃sup(0	f̃sup(0	NOUN
ejpam-5914	475	16	)	)	PUNCT
ejpam-5914	475	17	2	2	NUM
ejpam-5914	475	18	+	+	SYM
ejpam-5914	475	19	f̃inf(x	f̃inf(x	PROPN
ejpam-5914	475	20	)	)	PUNCT
ejpam-5914	475	21	2	2	NUM
ejpam-5914	475	22	,	,	PUNCT
ejpam-5914	475	23	f̃sup(0	f̃sup(0	NOUN
ejpam-5914	475	24	)	)	PUNCT
ejpam-5914	475	25	2	2	NUM
ejpam-5914	475	26	+	+	SYM
ejpam-5914	475	27	f̃inf(y	f̃inf(y	NOUN
ejpam-5914	475	28	)	)	PUNCT
ejpam-5914	475	29	2	2	NUM
ejpam-5914	475	30	}	}	PUNCT
ejpam-5914	475	31	=	=	SYM
ejpam-5914	475	32	max	max	PROPN
ejpam-5914	475	33	{	{	PUNCT
ejpam-5914	475	34	f̃sup(x	f̃sup(x	NOUN
ejpam-5914	475	35	)	)	PUNCT
ejpam-5914	475	36	+	+	SYM
ejpam-5914	475	37	f̃inf(x	f̃inf(x	NOUN
ejpam-5914	475	38	)	)	PUNCT
ejpam-5914	475	39	2	2	NUM
ejpam-5914	475	40	,	,	PUNCT
ejpam-5914	475	41	f̃sup(y	f̃sup(y	NOUN
ejpam-5914	475	42	)	)	PUNCT
ejpam-5914	476	1	+	+	CCONJ
ejpam-5914	476	2	f̃inf(yx	f̃inf(yx	X
ejpam-5914	476	3	)	)	PUNCT
ejpam-5914	476	4	2	2	NUM
ejpam-5914	476	5	}	}	PUNCT
ejpam-5914	476	6	=	=	SYM
ejpam-5914	476	7	max{f̃m(x	max{f̃m(x	PROPN
ejpam-5914	476	8	)	)	PUNCT
ejpam-5914	476	9	,	,	PUNCT
ejpam-5914	476	10	f̃m(y	f̃m(y	NOUN
ejpam-5914	476	11	)	)	PUNCT
ejpam-5914	476	12	}	}	PUNCT
ejpam-5914	476	13	.	.	PUNCT
ejpam-5914	477	1	n.	n.	PROPN
ejpam-5914	477	2	rajesh	rajesh	PROPN
ejpam-5914	477	3	et	et	PROPN
ejpam-5914	477	4	al	al	PROPN
ejpam-5914	477	5	.	.	PUNCT
ejpam-5914	477	6	/	/	SYM
ejpam-5914	477	7	eur	eur	PROPN
ejpam-5914	477	8	.	.	PUNCT
ejpam-5914	478	1	j.	j.	PROPN
ejpam-5914	478	2	pure	pure	PROPN
ejpam-5914	478	3	appl	appl	PROPN
ejpam-5914	478	4	.	.	PROPN
ejpam-5914	478	5	math	math	PROPN
ejpam-5914	478	6	,	,	PUNCT
ejpam-5914	478	7	18	18	NUM
ejpam-5914	478	8	(	(	PUNCT
ejpam-5914	478	9	2	2	NUM
ejpam-5914	478	10	)	)	PUNCT
ejpam-5914	478	11	(	(	PUNCT
ejpam-5914	478	12	2025	2025	NUM
ejpam-5914	478	13	)	)	PUNCT
ejpam-5914	478	14	,	,	PUNCT
ejpam-5914	478	15	5914	5914	NUM
ejpam-5914	478	16	19	19	NUM
ejpam-5914	478	17	of	of	ADP
ejpam-5914	478	18	21	21	NUM
ejpam-5914	478	19	hence	hence	ADV
ejpam-5914	478	20	,	,	PUNCT
ejpam-5914	478	21	(	(	PUNCT
ejpam-5914	478	22	a	a	DET
ejpam-5914	478	23	,	,	PUNCT
ejpam-5914	478	24	f̃m	f̃m	NOUN
ejpam-5914	478	25	)	)	PUNCT
ejpam-5914	478	26	is	be	AUX
ejpam-5914	478	27	a	a	DET
ejpam-5914	478	28	4	4	NUM
ejpam-5914	478	29	-	-	PUNCT
ejpam-5914	478	30	fuzzy	fuzzy	ADJ
ejpam-5914	478	31	subalgebra	subalgebra	NOUN
ejpam-5914	478	32	of	of	ADP
ejpam-5914	478	33	a	a	PRON
ejpam-5914	478	34	,	,	PUNCT
ejpam-5914	478	35	that	that	ADV
ejpam-5914	478	36	is	is	ADV
ejpam-5914	478	37	,	,	PUNCT
ejpam-5914	478	38	(	(	PUNCT
ejpam-5914	478	39	a	a	PRON
ejpam-5914	478	40	,	,	PUNCT
ejpam-5914	478	41	f̃	f̃	PROPN
ejpam-5914	478	42	)	)	PUNCT
ejpam-5914	478	43	is	be	AUX
ejpam-5914	478	44	a	a	DET
ejpam-5914	478	45	mean	mean	ADJ
ejpam-5914	478	46	4	4	NUM
ejpam-5914	478	47	-	-	PUNCT
ejpam-5914	478	48	fuzzy	fuzzy	ADJ
ejpam-5914	478	49	subalgebra	subalgebra	NOUN
ejpam-5914	478	50	of	of	ADP
ejpam-5914	478	51	a.	a.	NOUN
ejpam-5914	478	52	theorem	theorem	NOUN
ejpam-5914	478	53	22	22	NUM
ejpam-5914	478	54	.	.	PUNCT
ejpam-5914	479	1	if	if	SCONJ
ejpam-5914	479	2	(	(	PUNCT
ejpam-5914	479	3	a	a	PRON
ejpam-5914	479	4	,	,	PUNCT
ejpam-5914	479	5	f̃	f̃	PROPN
ejpam-5914	479	6	)	)	PUNCT
ejpam-5914	479	7	is	be	AUX
ejpam-5914	479	8	an	an	DET
ejpam-5914	479	9	interval	interval	NOUN
ejpam-5914	479	10	-	-	PUNCT
ejpam-5914	479	11	valued	value	VERB
ejpam-5914	479	12	fuzzy	fuzzy	ADJ
ejpam-5914	479	13	structure	structure	NOUN
ejpam-5914	479	14	over	over	ADP
ejpam-5914	479	15	a	a	DET
ejpam-5914	479	16	in	in	ADP
ejpam-5914	479	17	which	which	PRON
ejpam-5914	479	18	(	(	PUNCT
ejpam-5914	479	19	a	a	DET
ejpam-5914	479	20	,	,	PUNCT
ejpam-5914	479	21	f̃sup	f̃sup	ADJ
ejpam-5914	479	22	)	)	PUNCT
ejpam-5914	479	23	is	be	AUX
ejpam-5914	479	24	constant	constant	ADJ
ejpam-5914	479	25	and	and	CCONJ
ejpam-5914	479	26	(	(	PUNCT
ejpam-5914	479	27	a	a	PRON
ejpam-5914	479	28	,	,	PUNCT
ejpam-5914	479	29	f̃inf	f̃inf	ADJ
ejpam-5914	479	30	)	)	PUNCT
ejpam-5914	479	31	is	be	AUX
ejpam-5914	479	32	a	a	DET
ejpam-5914	479	33	1	1	NUM
ejpam-5914	479	34	-	-	PUNCT
ejpam-5914	479	35	fuzzy	fuzzy	ADJ
ejpam-5914	479	36	subalgebra	subalgebra	NOUN
ejpam-5914	479	37	of	of	ADP
ejpam-5914	479	38	a	a	PRON
ejpam-5914	479	39	,	,	PUNCT
ejpam-5914	479	40	then	then	ADV
ejpam-5914	479	41	(	(	PUNCT
ejpam-5914	479	42	a	a	PRON
ejpam-5914	479	43	,	,	PUNCT
ejpam-5914	479	44	f̃	f̃	PROPN
ejpam-5914	479	45	)	)	PUNCT
ejpam-5914	479	46	is	be	AUX
ejpam-5914	479	47	a	a	DET
ejpam-5914	479	48	mean	mean	ADJ
ejpam-5914	479	49	1	1	NUM
ejpam-5914	479	50	-	-	PUNCT
ejpam-5914	479	51	fuzzy	fuzzy	ADJ
ejpam-5914	479	52	subalgebra	subalgebra	NOUN
ejpam-5914	479	53	of	of	ADP
ejpam-5914	479	54	a.	a.	NOUN
ejpam-5914	479	55	proof	proof	NOUN
ejpam-5914	479	56	.	.	PUNCT
ejpam-5914	480	1	assume	assume	VERB
ejpam-5914	480	2	that	that	SCONJ
ejpam-5914	480	3	(	(	PUNCT
ejpam-5914	480	4	a	a	PRON
ejpam-5914	480	5	,	,	PUNCT
ejpam-5914	480	6	f̃	f̃	PROPN
ejpam-5914	480	7	)	)	PUNCT
ejpam-5914	480	8	is	be	AUX
ejpam-5914	480	9	an	an	DET
ejpam-5914	480	10	interval	interval	NOUN
ejpam-5914	480	11	-	-	PUNCT
ejpam-5914	480	12	valued	value	VERB
ejpam-5914	480	13	fuzzy	fuzzy	ADJ
ejpam-5914	480	14	structure	structure	NOUN
ejpam-5914	480	15	over	over	ADP
ejpam-5914	480	16	a	a	DET
ejpam-5914	480	17	in	in	ADP
ejpam-5914	480	18	which	which	PRON
ejpam-5914	480	19	(	(	PUNCT
ejpam-5914	480	20	a	a	DET
ejpam-5914	480	21	,	,	PUNCT
ejpam-5914	480	22	f̃sup	f̃sup	ADJ
ejpam-5914	480	23	)	)	PUNCT
ejpam-5914	480	24	is	be	AUX
ejpam-5914	480	25	constant	constant	ADJ
ejpam-5914	480	26	and	and	CCONJ
ejpam-5914	480	27	(	(	PUNCT
ejpam-5914	480	28	a	a	PRON
ejpam-5914	480	29	,	,	PUNCT
ejpam-5914	480	30	f̃inf	f̃inf	ADJ
ejpam-5914	480	31	)	)	PUNCT
ejpam-5914	480	32	is	be	AUX
ejpam-5914	480	33	a	a	DET
ejpam-5914	480	34	1	1	NUM
ejpam-5914	480	35	-	-	PUNCT
ejpam-5914	480	36	fuzzy	fuzzy	ADJ
ejpam-5914	480	37	subalgebra	subalgebra	NOUN
ejpam-5914	480	38	of	of	ADP
ejpam-5914	480	39	a.	a.	NOUN
ejpam-5914	480	40	let	let	VERB
ejpam-5914	480	41	x	x	PRON
ejpam-5914	480	42	,	,	PUNCT
ejpam-5914	480	43	y	y	PROPN
ejpam-5914	480	44	∈	∈	PROPN
ejpam-5914	480	45	a.	a.	NOUN
ejpam-5914	480	46	since	since	SCONJ
ejpam-5914	480	47	(	(	PUNCT
ejpam-5914	480	48	a	a	DET
ejpam-5914	480	49	,	,	PUNCT
ejpam-5914	480	50	f̃sup	f̃sup	ADJ
ejpam-5914	480	51	)	)	PUNCT
ejpam-5914	480	52	is	be	AUX
ejpam-5914	480	53	constant	constant	ADJ
ejpam-5914	480	54	,	,	PUNCT
ejpam-5914	480	55	we	we	PRON
ejpam-5914	480	56	have	have	AUX
ejpam-5914	480	57	f̃sup(x	f̃sup(x	VERB
ejpam-5914	480	58	)	)	PUNCT
ejpam-5914	480	59	=	=	SYM
ejpam-5914	480	60	f̃sup(0	f̃sup(0	NOUN
ejpam-5914	480	61	)	)	PUNCT
ejpam-5914	480	62	for	for	SCONJ
ejpam-5914	480	63	all	all	PRON
ejpam-5914	480	64	x	x	SYM
ejpam-5914	480	65	∈	∈	NOUN
ejpam-5914	480	66	a.	a.	NOUN
ejpam-5914	480	67	since	since	SCONJ
ejpam-5914	480	68	(	(	PUNCT
ejpam-5914	480	69	a	a	PRON
ejpam-5914	480	70	,	,	PUNCT
ejpam-5914	480	71	f̃inf	f̃inf	ADJ
ejpam-5914	480	72	)	)	PUNCT
ejpam-5914	480	73	is	be	AUX
ejpam-5914	480	74	a	a	DET
ejpam-5914	480	75	1	1	NUM
ejpam-5914	480	76	-	-	PUNCT
ejpam-5914	480	77	fuzzy	fuzzy	ADJ
ejpam-5914	480	78	subalgebra	subalgebra	NOUN
ejpam-5914	480	79	of	of	ADP
ejpam-5914	480	80	a	a	PRON
ejpam-5914	480	81	,	,	PUNCT
ejpam-5914	480	82	we	we	PRON
ejpam-5914	480	83	have	have	VERB
ejpam-5914	480	84	f̃inf((x|(y|y))|(x|(y|y	f̃inf((x|(y|y))|(x|(y|y	VERB
ejpam-5914	480	85	)	)	PUNCT
ejpam-5914	480	86	)	)	PUNCT
ejpam-5914	480	87	)	)	PUNCT
ejpam-5914	480	88	≥	≥	PROPN
ejpam-5914	480	89	min{f̃inf(x	min{f̃inf(x	PROPN
ejpam-5914	480	90	)	)	PUNCT
ejpam-5914	480	91	,	,	PUNCT
ejpam-5914	480	92	f̃inf(y	f̃inf(y	NOUN
ejpam-5914	480	93	)	)	PUNCT
ejpam-5914	480	94	}	}	PUNCT
ejpam-5914	480	95	.	.	PUNCT
ejpam-5914	481	1	thus	thus	ADV
ejpam-5914	481	2	,	,	PUNCT
ejpam-5914	481	3	f̃m((x|(y|y))|(x|(y|y	f̃m((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	481	4	)	)	PUNCT
ejpam-5914	481	5	)	)	PUNCT
ejpam-5914	481	6	)	)	PUNCT
ejpam-5914	482	1	=	=	SYM
ejpam-5914	482	2	f̃sup((x|(y|y))|(x|(y|y	f̃sup((x|(y|y))|(x|(y|y	NOUN
ejpam-5914	482	3	)	)	PUNCT
ejpam-5914	482	4	)	)	PUNCT
ejpam-5914	482	5	)	)	PUNCT
ejpam-5914	483	1	+	+	CCONJ
ejpam-5914	483	2	f̃inf((x|(y|y))|(x|(y|y	f̃inf((x|(y|y))|(x|(y|y	ADJ
ejpam-5914	483	3	)	)	PUNCT
ejpam-5914	483	4	)	)	PUNCT
ejpam-5914	483	5	)	)	PUNCT
ejpam-5914	483	6	2	2	X
ejpam-5914	483	7	=	=	SYM
ejpam-5914	483	8	f̃sup(0	f̃sup(0	NOUN
ejpam-5914	483	9	)	)	PUNCT
ejpam-5914	483	10	+	+	CCONJ
ejpam-5914	483	11	f̃inf((x|(y|y))|(x|(y|y	f̃inf((x|(y|y))|(x|(y|y	ADJ
ejpam-5914	483	12	)	)	PUNCT
ejpam-5914	483	13	)	)	PUNCT
ejpam-5914	483	14	)	)	PUNCT
ejpam-5914	483	15	2	2	X
ejpam-5914	483	16	=	=	SYM
ejpam-5914	483	17	f̃sup(0	f̃sup(0	NOUN
ejpam-5914	483	18	)	)	PUNCT
ejpam-5914	483	19	2	2	NUM
ejpam-5914	484	1	+	+	CCONJ
ejpam-5914	484	2	f̃inf((x|(y|y))|(x|(y|y	f̃inf((x|(y|y))|(x|(y|y	ADJ
ejpam-5914	484	3	)	)	PUNCT
ejpam-5914	484	4	)	)	PUNCT
ejpam-5914	484	5	)	)	PUNCT
ejpam-5914	484	6	2	2	NUM
ejpam-5914	484	7	≥	≥	NOUN
ejpam-5914	484	8	f̃sup(0	f̃sup(0	NOUN
ejpam-5914	484	9	)	)	PUNCT
ejpam-5914	484	10	2	2	NUM
ejpam-5914	484	11	+	+	CCONJ
ejpam-5914	484	12	min	min	NOUN
ejpam-5914	484	13	{	{	PUNCT
ejpam-5914	484	14	f̃inf(x	f̃inf(x	PROPN
ejpam-5914	484	15	)	)	PUNCT
ejpam-5914	484	16	2	2	NUM
ejpam-5914	484	17	,	,	PUNCT
ejpam-5914	484	18	f̃inf(y	f̃inf(y	NOUN
ejpam-5914	484	19	)	)	PUNCT
ejpam-5914	484	20	2	2	NUM
ejpam-5914	484	21	}	}	PUNCT
ejpam-5914	484	22	=	=	SYM
ejpam-5914	484	23	min	min	NOUN
ejpam-5914	484	24	{	{	PUNCT
ejpam-5914	484	25	f̃sup(0	f̃sup(0	NOUN
ejpam-5914	484	26	)	)	PUNCT
ejpam-5914	484	27	2	2	NUM
ejpam-5914	485	1	+	+	CCONJ
ejpam-5914	485	2	f̃sup(x	f̃sup(x	ADJ
ejpam-5914	485	3	)	)	PUNCT
ejpam-5914	485	4	2	2	NUM
ejpam-5914	485	5	,	,	PUNCT
ejpam-5914	485	6	f̃sup(0	f̃sup(0	NOUN
ejpam-5914	485	7	)	)	PUNCT
ejpam-5914	485	8	2	2	NUM
ejpam-5914	485	9	,	,	PUNCT
ejpam-5914	485	10	f̃sup(y	f̃sup(y	NOUN
ejpam-5914	485	11	)	)	PUNCT
ejpam-5914	485	12	2	2	NUM
ejpam-5914	485	13	}	}	PUNCT
ejpam-5914	485	14	=	=	SYM
ejpam-5914	485	15	min	min	NOUN
ejpam-5914	485	16	{	{	PUNCT
ejpam-5914	485	17	f̃sup(0	f̃sup(0	NOUN
ejpam-5914	485	18	)	)	PUNCT
ejpam-5914	486	1	+	+	SYM
ejpam-5914	486	2	f̃sup(x	f̃sup(x	ADJ
ejpam-5914	486	3	)	)	PUNCT
ejpam-5914	486	4	2	2	NUM
ejpam-5914	486	5	,	,	PUNCT
ejpam-5914	486	6	f̃sup(0	f̃sup(0	NOUN
ejpam-5914	486	7	)	)	PUNCT
ejpam-5914	487	1	+	+	CCONJ
ejpam-5914	487	2	f̃sup(y	f̃sup(y	NOUN
ejpam-5914	487	3	)	)	PUNCT
ejpam-5914	487	4	2	2	NUM
ejpam-5914	487	5	}	}	PUNCT
ejpam-5914	487	6	=	=	SYM
ejpam-5914	487	7	min{f̃m(x	min{f̃m(x	PROPN
ejpam-5914	487	8	)	)	PUNCT
ejpam-5914	487	9	,	,	PUNCT
ejpam-5914	487	10	f̃m(y	f̃m(y	NOUN
ejpam-5914	487	11	)	)	PUNCT
ejpam-5914	487	12	}	}	PUNCT
ejpam-5914	487	13	.	.	PUNCT
ejpam-5914	488	1	hence	hence	ADV
ejpam-5914	488	2	,	,	PUNCT
ejpam-5914	488	3	(	(	PUNCT
ejpam-5914	488	4	a	a	DET
ejpam-5914	488	5	,	,	PUNCT
ejpam-5914	488	6	f̃m	f̃m	NOUN
ejpam-5914	488	7	)	)	PUNCT
ejpam-5914	488	8	is	be	AUX
ejpam-5914	488	9	a	a	DET
ejpam-5914	488	10	1	1	NUM
ejpam-5914	488	11	-	-	PUNCT
ejpam-5914	488	12	fuzzy	fuzzy	ADJ
ejpam-5914	488	13	subalgebra	subalgebra	NOUN
ejpam-5914	488	14	of	of	ADP
ejpam-5914	488	15	a	a	PRON
ejpam-5914	488	16	,	,	PUNCT
ejpam-5914	488	17	that	that	ADV
ejpam-5914	488	18	is	is	ADV
ejpam-5914	488	19	,	,	PUNCT
ejpam-5914	488	20	(	(	PUNCT
ejpam-5914	488	21	a	a	PRON
ejpam-5914	488	22	,	,	PUNCT
ejpam-5914	488	23	f̃	f̃	PROPN
ejpam-5914	488	24	)	)	PUNCT
ejpam-5914	488	25	is	be	AUX
ejpam-5914	488	26	a	a	DET
ejpam-5914	488	27	mean	mean	ADJ
ejpam-5914	488	28	1	1	NUM
ejpam-5914	488	29	-	-	PUNCT
ejpam-5914	488	30	fuzzy	fuzzy	ADJ
ejpam-5914	488	31	subalgebra	subalgebra	NOUN
ejpam-5914	488	32	of	of	ADP
ejpam-5914	488	33	a.	a.	NOUN
ejpam-5914	488	34	5	5	NUM
ejpam-5914	488	35	.	.	PUNCT
ejpam-5914	489	1	conclusion	conclusion	NOUN
ejpam-5914	489	2	this	this	DET
ejpam-5914	489	3	study	study	NOUN
ejpam-5914	489	4	advances	advance	VERB
ejpam-5914	489	5	the	the	DET
ejpam-5914	489	6	theoretical	theoretical	ADJ
ejpam-5914	489	7	framework	framework	NOUN
ejpam-5914	489	8	of	of	ADP
ejpam-5914	489	9	sheffer	sheffer	PROPN
ejpam-5914	489	10	stroke	stroke	PROPN
ejpam-5914	489	11	hilbert	hilbert	PROPN
ejpam-5914	489	12	algebras	algebras	PROPN
ejpam-5914	489	13	by	by	ADP
ejpam-5914	489	14	introducing	introduce	VERB
ejpam-5914	489	15	the	the	DET
ejpam-5914	489	16	concepts	concept	NOUN
ejpam-5914	489	17	of	of	ADP
ejpam-5914	489	18	length	length	NOUN
ejpam-5914	489	19	-	-	PUNCT
ejpam-5914	489	20	fuzzy	fuzzy	ADJ
ejpam-5914	489	21	subalgebras	subalgebra	NOUN
ejpam-5914	489	22	and	and	CCONJ
ejpam-5914	489	23	mean	mean	ADJ
ejpam-5914	489	24	-	-	PUNCT
ejpam-5914	489	25	fuzzy	fuzzy	ADJ
ejpam-5914	489	26	subalgebras	subalgebra	NOUN
ejpam-5914	489	27	within	within	ADP
ejpam-5914	489	28	interval	interval	NOUN
ejpam-5914	489	29	-	-	PUNCT
ejpam-5914	489	30	valued	value	VERB
ejpam-5914	489	31	fuzzy	fuzzy	ADJ
ejpam-5914	489	32	structures	structure	NOUN
ejpam-5914	489	33	.	.	PUNCT
ejpam-5914	490	1	these	these	DET
ejpam-5914	490	2	new	new	ADJ
ejpam-5914	490	3	constructs	construct	NOUN
ejpam-5914	490	4	deepen	deepen	VERB
ejpam-5914	490	5	the	the	DET
ejpam-5914	490	6	understanding	understanding	NOUN
ejpam-5914	490	7	of	of	ADP
ejpam-5914	490	8	fuzzy	fuzzy	ADJ
ejpam-5914	490	9	logic	logic	NOUN
ejpam-5914	490	10	in	in	ADP
ejpam-5914	490	11	algebraic	algebraic	ADJ
ejpam-5914	490	12	systems	system	NOUN
ejpam-5914	490	13	,	,	PUNCT
ejpam-5914	490	14	particularly	particularly	ADV
ejpam-5914	490	15	by	by	ADP
ejpam-5914	490	16	elucidating	elucidate	VERB
ejpam-5914	490	17	the	the	DET
ejpam-5914	490	18	interplay	interplay	NOUN
ejpam-5914	490	19	between	between	ADP
ejpam-5914	490	20	fuzzy	fuzzy	ADJ
ejpam-5914	490	21	and	and	CCONJ
ejpam-5914	490	22	traditional	traditional	ADJ
ejpam-5914	490	23	subalgebras	subalgebra	NOUN
ejpam-5914	490	24	.	.	PUNCT
ejpam-5914	491	1	the	the	DET
ejpam-5914	491	2	investigation	investigation	NOUN
ejpam-5914	491	3	reveals	reveal	VERB
ejpam-5914	491	4	key	key	ADJ
ejpam-5914	491	5	properties	property	NOUN
ejpam-5914	491	6	and	and	CCONJ
ejpam-5914	491	7	relationships	relationship	NOUN
ejpam-5914	491	8	,	,	PUNCT
ejpam-5914	491	9	including	include	VERB
ejpam-5914	491	10	their	their	PRON
ejpam-5914	491	11	alignment	alignment	NOUN
ejpam-5914	491	12	with	with	ADP
ejpam-5914	491	13	upper	upper	ADJ
ejpam-5914	491	14	and	and	CCONJ
ejpam-5914	491	15	lower	low	ADJ
ejpam-5914	491	16	-	-	PUNCT
ejpam-5914	491	17	level	level	NOUN
ejpam-5914	491	18	subsets	subset	NOUN
ejpam-5914	491	19	,	,	PUNCT
ejpam-5914	491	20	offering	offer	VERB
ejpam-5914	491	21	a	a	DET
ejpam-5914	491	22	refined	refined	ADJ
ejpam-5914	491	23	perspective	perspective	NOUN
ejpam-5914	491	24	on	on	ADP
ejpam-5914	491	25	the	the	DET
ejpam-5914	491	26	gradations	gradation	NOUN
ejpam-5914	491	27	of	of	ADP
ejpam-5914	491	28	membership	membership	NOUN
ejpam-5914	491	29	functions	function	NOUN
ejpam-5914	491	30	.	.	PUNCT
ejpam-5914	492	1	this	this	DET
ejpam-5914	492	2	framework	framework	NOUN
ejpam-5914	492	3	not	not	PART
ejpam-5914	492	4	only	only	ADV
ejpam-5914	492	5	enriches	enrich	VERB
ejpam-5914	492	6	algebraic	algebraic	PROPN
ejpam-5914	492	7	theory	theory	NOUN
ejpam-5914	492	8	n.	n.	PROPN
ejpam-5914	492	9	rajesh	rajesh	PROPN
ejpam-5914	492	10	et	et	PROPN
ejpam-5914	492	11	al	al	PROPN
ejpam-5914	492	12	.	.	PUNCT
ejpam-5914	492	13	/	/	SYM
ejpam-5914	492	14	eur	eur	PROPN
ejpam-5914	492	15	.	.	PUNCT
ejpam-5914	493	1	j.	j.	PROPN
ejpam-5914	493	2	pure	pure	PROPN
ejpam-5914	493	3	appl	appl	PROPN
ejpam-5914	493	4	.	.	PROPN
ejpam-5914	493	5	math	math	PROPN
ejpam-5914	493	6	,	,	PUNCT
ejpam-5914	493	7	18	18	NUM
ejpam-5914	493	8	(	(	PUNCT
ejpam-5914	493	9	2	2	NUM
ejpam-5914	493	10	)	)	PUNCT
ejpam-5914	493	11	(	(	PUNCT
ejpam-5914	493	12	2025	2025	NUM
ejpam-5914	493	13	)	)	PUNCT
ejpam-5914	493	14	,	,	PUNCT
ejpam-5914	493	15	5914	5914	NUM
ejpam-5914	493	16	20	20	NUM
ejpam-5914	493	17	of	of	ADP
ejpam-5914	493	18	21	21	NUM
ejpam-5914	493	19	but	but	CCONJ
ejpam-5914	493	20	also	also	ADV
ejpam-5914	493	21	underscores	underscore	VERB
ejpam-5914	493	22	the	the	DET
ejpam-5914	493	23	practical	practical	ADJ
ejpam-5914	493	24	relevance	relevance	NOUN
ejpam-5914	493	25	of	of	ADP
ejpam-5914	493	26	fuzzy	fuzzy	ADJ
ejpam-5914	493	27	subalgebras	subalgebra	NOUN
ejpam-5914	493	28	in	in	ADP
ejpam-5914	493	29	fields	field	NOUN
ejpam-5914	493	30	such	such	ADJ
ejpam-5914	493	31	as	as	ADP
ejpam-5914	493	32	logic	logic	NOUN
ejpam-5914	493	33	,	,	PUNCT
ejpam-5914	493	34	computer	computer	NOUN
ejpam-5914	493	35	science	science	NOUN
ejpam-5914	493	36	,	,	PUNCT
ejpam-5914	493	37	and	and	CCONJ
ejpam-5914	493	38	uncertainty	uncertainty	NOUN
ejpam-5914	493	39	modeling	modeling	NOUN
ejpam-5914	493	40	.	.	PUNCT
ejpam-5914	494	1	the	the	DET
ejpam-5914	494	2	findings	finding	NOUN
ejpam-5914	494	3	pave	pave	VERB
ejpam-5914	494	4	the	the	DET
ejpam-5914	494	5	way	way	NOUN
ejpam-5914	494	6	for	for	SCONJ
ejpam-5914	494	7	future	future	ADJ
ejpam-5914	494	8	research	research	NOUN
ejpam-5914	494	9	to	to	PART
ejpam-5914	494	10	explore	explore	VERB
ejpam-5914	494	11	these	these	DET
ejpam-5914	494	12	ideas	idea	NOUN
ejpam-5914	494	13	in	in	ADP
ejpam-5914	494	14	more	more	ADV
ejpam-5914	494	15	complex	complex	ADJ
ejpam-5914	494	16	fuzzy	fuzzy	ADJ
ejpam-5914	494	17	systems	system	NOUN
ejpam-5914	494	18	or	or	CCONJ
ejpam-5914	494	19	adapt	adapt	VERB
ejpam-5914	494	20	them	they	PRON
ejpam-5914	494	21	to	to	ADP
ejpam-5914	494	22	other	other	ADJ
ejpam-5914	494	23	algebraic	algebraic	ADJ
ejpam-5914	494	24	structures	structure	NOUN
ejpam-5914	494	25	,	,	PUNCT
ejpam-5914	494	26	broadening	broaden	VERB
ejpam-5914	494	27	their	their	PRON
ejpam-5914	494	28	applicability	applicability	NOUN
ejpam-5914	494	29	and	and	CCONJ
ejpam-5914	494	30	potential	potential	ADJ
ejpam-5914	494	31	impact	impact	NOUN
ejpam-5914	494	32	across	across	ADP
ejpam-5914	494	33	diverse	diverse	ADJ
ejpam-5914	494	34	domains	domain	NOUN
ejpam-5914	494	35	.	.	PUNCT
ejpam-5914	495	1	acknowledgements	acknowledgement	NOUN
ejpam-5914	495	2	this	this	DET
ejpam-5914	495	3	research	research	NOUN
ejpam-5914	495	4	was	be	AUX
ejpam-5914	495	5	supported	support	VERB
ejpam-5914	495	6	by	by	ADP
ejpam-5914	495	7	university	university	NOUN
ejpam-5914	495	8	of	of	ADP
ejpam-5914	495	9	phayao	phayao	NOUN
ejpam-5914	495	10	and	and	CCONJ
ejpam-5914	495	11	thailand	thailand	PROPN
ejpam-5914	495	12	science	science	PROPN
ejpam-5914	495	13	research	research	PROPN
ejpam-5914	495	14	and	and	CCONJ
ejpam-5914	495	15	innovation	innovation	NOUN
ejpam-5914	495	16	fund	fund	NOUN
ejpam-5914	495	17	(	(	PUNCT
ejpam-5914	495	18	fundamental	fundamental	ADJ
ejpam-5914	495	19	fund	fund	NOUN
ejpam-5914	495	20	2025	2025	NUM
ejpam-5914	495	21	,	,	PUNCT
ejpam-5914	495	22	grant	grant	VERB
ejpam-5914	495	23	no	no	NOUN
ejpam-5914	495	24	.	.	PROPN
ejpam-5914	496	1	5027/2567	5027/2567	NUM
ejpam-5914	496	2	)	)	PUNCT
ejpam-5914	496	3	.	.	PUNCT
ejpam-5914	497	1	references	reference	NOUN
ejpam-5914	497	2	[	[	X
ejpam-5914	497	3	1	1	NUM
ejpam-5914	497	4	]	]	PUNCT
ejpam-5914	497	5	h.	h.	PROPN
ejpam-5914	497	6	m.	m.	PROPN
ejpam-5914	497	7	sheffer	sheffer	PROPN
ejpam-5914	497	8	.	.	PUNCT
ejpam-5914	498	1	a	a	DET
ejpam-5914	498	2	set	set	NOUN
ejpam-5914	498	3	of	of	ADP
ejpam-5914	498	4	five	five	NUM
ejpam-5914	498	5	independent	independent	ADJ
ejpam-5914	498	6	postulates	postulate	NOUN
ejpam-5914	498	7	for	for	ADP
ejpam-5914	498	8	boolean	boolean	ADJ
ejpam-5914	498	9	algebras	algebra	NOUN
ejpam-5914	498	10	,	,	PUNCT
ejpam-5914	498	11	with	with	ADP
ejpam-5914	498	12	application	application	NOUN
ejpam-5914	498	13	to	to	ADP
ejpam-5914	498	14	logical	logical	ADJ
ejpam-5914	498	15	constants	constant	NOUN
ejpam-5914	498	16	.	.	PUNCT
ejpam-5914	499	1	transactions	transaction	NOUN
ejpam-5914	499	2	of	of	ADP
ejpam-5914	499	3	the	the	DET
ejpam-5914	499	4	american	american	PROPN
ejpam-5914	499	5	mathematical	mathematical	PROPN
ejpam-5914	499	6	society	society	NOUN
ejpam-5914	499	7	,	,	PUNCT
ejpam-5914	499	8	14(4):481–488	14(4):481–488	NUM
ejpam-5914	499	9	,	,	PUNCT
ejpam-5914	499	10	1913	1913	NUM
ejpam-5914	499	11	.	.	PUNCT
ejpam-5914	500	1	[	[	X
ejpam-5914	500	2	2	2	NUM
ejpam-5914	500	3	]	]	PUNCT
ejpam-5914	500	4	m.	m.	NOUN
ejpam-5914	500	5	mccune	mccune	PROPN
ejpam-5914	500	6	,	,	PUNCT
ejpam-5914	500	7	r.	r.	PROPN
ejpam-5914	500	8	veroff	veroff	PROPN
ejpam-5914	500	9	,	,	PUNCT
ejpam-5914	500	10	b.	b.	PROPN
ejpam-5914	500	11	fitelson	fitelson	PROPN
ejpam-5914	500	12	,	,	PUNCT
ejpam-5914	500	13	k.	k.	PROPN
ejpam-5914	500	14	harris	harris	PROPN
ejpam-5914	500	15	,	,	PUNCT
ejpam-5914	500	16	a.	a.	NOUN
ejpam-5914	500	17	feist	feist	PROPN
ejpam-5914	500	18	,	,	PUNCT
ejpam-5914	500	19	and	and	CCONJ
ejpam-5914	500	20	l.	l.	PROPN
ejpam-5914	500	21	wos	wos	PROPN
ejpam-5914	500	22	.	.	PUNCT
ejpam-5914	501	1	short	short	ADJ
ejpam-5914	501	2	single	single	ADJ
ejpam-5914	501	3	axioms	axiom	NOUN
ejpam-5914	501	4	for	for	ADP
ejpam-5914	501	5	boolean	boolean	ADJ
ejpam-5914	501	6	algebra	algebra	NOUN
ejpam-5914	501	7	.	.	PUNCT
ejpam-5914	502	1	journal	journal	NOUN
ejpam-5914	502	2	of	of	ADP
ejpam-5914	502	3	automated	automate	VERB
ejpam-5914	502	4	reasoning	reasoning	NOUN
ejpam-5914	502	5	,	,	PUNCT
ejpam-5914	502	6	29:1–16	29:1–16	NUM
ejpam-5914	502	7	,	,	PUNCT
ejpam-5914	502	8	2002	2002	NUM
ejpam-5914	502	9	.	.	PUNCT
ejpam-5914	503	1	[	[	X
ejpam-5914	503	2	3	3	NUM
ejpam-5914	503	3	]	]	X
ejpam-5914	503	4	i.	i.	NOUN
ejpam-5914	503	5	chajad	chajad	PROPN
ejpam-5914	503	6	.	.	PUNCT
ejpam-5914	504	1	sheffer	sheffer	PROPN
ejpam-5914	504	2	operation	operation	NOUN
ejpam-5914	504	3	in	in	ADP
ejpam-5914	504	4	ortholattices	ortholattice	NOUN
ejpam-5914	504	5	.	.	PUNCT
ejpam-5914	505	1	acta	acta	PROPN
ejpam-5914	505	2	universitatis	universitatis	PROPN
ejpam-5914	505	3	palackianae	palackianae	PROPN
ejpam-5914	505	4	olomucensis	olomucensis	NOUN
ejpam-5914	505	5	,	,	PUNCT
ejpam-5914	505	6	facultas	faculta	NOUN
ejpam-5914	505	7	rerum	rerum	PROPN
ejpam-5914	505	8	naturalium	naturalium	NOUN
ejpam-5914	505	9	,	,	PUNCT
ejpam-5914	505	10	mathematica	mathematica	PROPN
ejpam-5914	505	11	,	,	PUNCT
ejpam-5914	505	12	44(1):19–23	44(1):19–23	NUM
ejpam-5914	505	13	,	,	PUNCT
ejpam-5914	505	14	2005	2005	NUM
ejpam-5914	505	15	.	.	PUNCT
ejpam-5914	506	1	[	[	X
ejpam-5914	506	2	4	4	NUM
ejpam-5914	506	3	]	]	X
ejpam-5914	506	4	i.	i.	NOUN
ejpam-5914	506	5	chajda	chajda	PROPN
ejpam-5914	506	6	,	,	PUNCT
ejpam-5914	506	7	r.	r.	PROPN
ejpam-5914	506	8	halas̆	halas̆	PROPN
ejpam-5914	506	9	,	,	PUNCT
ejpam-5914	506	10	and	and	CCONJ
ejpam-5914	506	11	h.	h.	PROPN
ejpam-5914	506	12	länger	länger	PROPN
ejpam-5914	506	13	.	.	PUNCT
ejpam-5914	507	1	operations	operation	NOUN
ejpam-5914	507	2	and	and	CCONJ
ejpam-5914	507	3	structures	structure	NOUN
ejpam-5914	507	4	derived	derive	VERB
ejpam-5914	507	5	from	from	ADP
ejpam-5914	507	6	nonassociative	nonassociative	ADJ
ejpam-5914	507	7	mv	mv	PROPN
ejpam-5914	507	8	-	-	PUNCT
ejpam-5914	507	9	algebras	algebra	NOUN
ejpam-5914	507	10	.	.	PUNCT
ejpam-5914	507	11	soft	soft	ADJ
ejpam-5914	507	12	computing	computing	NOUN
ejpam-5914	507	13	,	,	PUNCT
ejpam-5914	507	14	23:3935–3944	23:3935–3944	NUM
ejpam-5914	507	15	,	,	PUNCT
ejpam-5914	507	16	2019	2019	NUM
ejpam-5914	507	17	.	.	PUNCT
ejpam-5914	508	1	[	[	X
ejpam-5914	508	2	5	5	NUM
ejpam-5914	508	3	]	]	PUNCT
ejpam-5914	508	4	i.	i.	NOUN
ejpam-5914	508	5	senturk	senturk	PROPN
ejpam-5914	508	6	.	.	PUNCT
ejpam-5914	509	1	riečan	riečan	NOUN
ejpam-5914	509	2	and	and	CCONJ
ejpam-5914	509	3	bosbach	bosbach	ADJ
ejpam-5914	509	4	state	state	NOUN
ejpam-5914	509	5	operators	operator	NOUN
ejpam-5914	509	6	on	on	ADP
ejpam-5914	509	7	sheffer	sheffer	PROPN
ejpam-5914	509	8	stroke	stroke	NOUN
ejpam-5914	509	9	mtl	mtl	PROPN
ejpam-5914	509	10	-	-	PUNCT
ejpam-5914	509	11	algebras	algebras	PROPN
ejpam-5914	509	12	.	.	PUNCT
ejpam-5914	510	1	bulletin	bulletin	NOUN
ejpam-5914	510	2	of	of	ADP
ejpam-5914	510	3	international	international	ADJ
ejpam-5914	510	4	mathematical	mathematical	ADJ
ejpam-5914	510	5	virtual	virtual	PROPN
ejpam-5914	510	6	institute	institute	PROPN
ejpam-5914	510	7	,	,	PUNCT
ejpam-5914	510	8	12(1):181–193	12(1):181–193	PROPN
ejpam-5914	510	9	,	,	PUNCT
ejpam-5914	510	10	2022	2022	NUM
ejpam-5914	510	11	.	.	PUNCT
ejpam-5914	511	1	[	[	X
ejpam-5914	511	2	6	6	NUM
ejpam-5914	511	3	]	]	PUNCT
ejpam-5914	511	4	l.	l.	PROPN
ejpam-5914	511	5	henkin	henkin	PROPN
ejpam-5914	511	6	.	.	PUNCT
ejpam-5914	512	1	an	an	DET
ejpam-5914	512	2	algebraic	algebraic	ADJ
ejpam-5914	512	3	characterization	characterization	NOUN
ejpam-5914	512	4	of	of	ADP
ejpam-5914	512	5	quantifiers	quantifier	NOUN
ejpam-5914	512	6	.	.	PUNCT
ejpam-5914	513	1	fundamenta	fundamenta	PROPN
ejpam-5914	513	2	mathematicae	mathematicae	PROPN
ejpam-5914	513	3	,	,	PUNCT
ejpam-5914	513	4	37(1):63–74	37(1):63–74	NUM
ejpam-5914	513	5	,	,	PUNCT
ejpam-5914	513	6	1950	1950	NUM
ejpam-5914	513	7	.	.	PUNCT
ejpam-5914	514	1	[	[	X
ejpam-5914	514	2	7	7	NUM
ejpam-5914	514	3	]	]	PUNCT
ejpam-5914	514	4	a.	a.	NOUN
ejpam-5914	514	5	monteiro	monteiro	PROPN
ejpam-5914	514	6	.	.	PUNCT
ejpam-5914	515	1	lectures	lecture	NOUN
ejpam-5914	515	2	on	on	ADP
ejpam-5914	515	3	hilbert	hilbert	NOUN
ejpam-5914	515	4	and	and	CCONJ
ejpam-5914	515	5	tarski	tarski	PROPN
ejpam-5914	515	6	algebras	algebra	NOUN
ejpam-5914	515	7	.	.	PUNCT
ejpam-5914	516	1	insitituto	insitituto	PROPN
ejpam-5914	516	2	de	de	PROPN
ejpam-5914	516	3	mathemática	mathemática	PROPN
ejpam-5914	516	4	,	,	PUNCT
ejpam-5914	516	5	universuidad	universuidad	PROPN
ejpam-5914	516	6	nacional	nacional	PROPN
ejpam-5914	516	7	del	del	PROPN
ejpam-5914	516	8	sur	sur	PROPN
ejpam-5914	516	9	,	,	PUNCT
ejpam-5914	516	10	bah́ıa	bah́ıa	X
ejpam-5914	516	11	blanca	blanca	PROPN
ejpam-5914	516	12	,	,	PUNCT
ejpam-5914	516	13	argentina	argentina	PROPN
ejpam-5914	516	14	,	,	PUNCT
ejpam-5914	516	15	1960	1960	NUM
ejpam-5914	516	16	.	.	PUNCT
ejpam-5914	517	1	[	[	X
ejpam-5914	517	2	8	8	NUM
ejpam-5914	517	3	]	]	PUNCT
ejpam-5914	517	4	a.	a.	NOUN
ejpam-5914	517	5	diego	diego	PROPN
ejpam-5914	517	6	.	.	PUNCT
ejpam-5914	518	1	sur	sur	PROPN
ejpam-5914	518	2	les	les	PROPN
ejpam-5914	518	3	algèbres	algèbre	NOUN
ejpam-5914	518	4	de	de	X
ejpam-5914	518	5	hilbert	hilbert	NOUN
ejpam-5914	518	6	.	.	PUNCT
ejpam-5914	519	1	gauthier	gauthier	PROPN
ejpam-5914	519	2	-	-	PUNCT
ejpam-5914	519	3	villars	villars	PROPN
ejpam-5914	519	4	,	,	PUNCT
ejpam-5914	519	5	paris	paris	PROPN
ejpam-5914	519	6	,	,	PUNCT
ejpam-5914	519	7	1966	1966	NUM
ejpam-5914	519	8	.	.	PUNCT
ejpam-5914	520	1	[	[	X
ejpam-5914	520	2	9	9	NUM
ejpam-5914	520	3	]	]	PUNCT
ejpam-5914	520	4	t.	t.	NOUN
ejpam-5914	520	5	oner	oner	NOUN
ejpam-5914	520	6	,	,	PUNCT
ejpam-5914	520	7	t.	t.	PROPN
ejpam-5914	520	8	katican	katican	PROPN
ejpam-5914	520	9	,	,	PUNCT
ejpam-5914	520	10	and	and	CCONJ
ejpam-5914	520	11	a.	a.	PROPN
ejpam-5914	520	12	borumand	borumand	PROPN
ejpam-5914	520	13	saeid	saeid	PROPN
ejpam-5914	520	14	.	.	PUNCT
ejpam-5914	521	1	relation	relation	NOUN
ejpam-5914	521	2	between	between	ADP
ejpam-5914	521	3	sheffer	sheffer	PROPN
ejpam-5914	521	4	stroke	stroke	PROPN
ejpam-5914	521	5	and	and	CCONJ
ejpam-5914	521	6	hilbert	hilbert	PROPN
ejpam-5914	521	7	algebras	algebras	PROPN
ejpam-5914	521	8	.	.	PUNCT
ejpam-5914	521	9	categories	category	NOUN
ejpam-5914	521	10	and	and	CCONJ
ejpam-5914	521	11	general	general	ADJ
ejpam-5914	521	12	algebraic	algebraic	ADJ
ejpam-5914	521	13	structures	structure	NOUN
ejpam-5914	521	14	with	with	ADP
ejpam-5914	521	15	applications	application	NOUN
ejpam-5914	521	16	,	,	PUNCT
ejpam-5914	521	17	14(1):245–268	14(1):245–268	NUM
ejpam-5914	521	18	,	,	PUNCT
ejpam-5914	521	19	2021	2021	NUM
ejpam-5914	521	20	.	.	PUNCT
ejpam-5914	522	1	[	[	X
ejpam-5914	522	2	10	10	NUM
ejpam-5914	522	3	]	]	PUNCT
ejpam-5914	522	4	t.	t.	PROPN
ejpam-5914	522	5	katican	katican	PROPN
ejpam-5914	522	6	and	and	CCONJ
ejpam-5914	522	7	h.	h.	PROPN
ejpam-5914	522	8	bordbar	bordbar	PROPN
ejpam-5914	522	9	.	.	PUNCT
ejpam-5914	523	1	sheffer	sheffer	PROPN
ejpam-5914	523	2	stroke	stroke	PROPN
ejpam-5914	523	3	hilbert	hilbert	PROPN
ejpam-5914	523	4	algebras	algebras	PROPN
ejpam-5914	523	5	stabilizing	stabilize	VERB
ejpam-5914	523	6	by	by	ADP
ejpam-5914	523	7	ideals	ideal	NOUN
ejpam-5914	523	8	.	.	PUNCT
ejpam-5914	524	1	axioms	axiom	NOUN
ejpam-5914	524	2	,	,	PUNCT
ejpam-5914	524	3	13(2):97	13(2):97	NUM
ejpam-5914	524	4	,	,	PUNCT
ejpam-5914	524	5	2024	2024	NUM
ejpam-5914	524	6	.	.	PUNCT
ejpam-5914	525	1	[	[	X
ejpam-5914	525	2	11	11	NUM
ejpam-5914	525	3	]	]	PUNCT
ejpam-5914	525	4	l.	l.	PROPN
ejpam-5914	525	5	a.	a.	PROPN
ejpam-5914	525	6	zadeh	zadeh	PROPN
ejpam-5914	525	7	.	.	PUNCT
ejpam-5914	525	8	fuzzy	fuzzy	ADJ
ejpam-5914	525	9	sets	set	NOUN
ejpam-5914	525	10	.	.	PUNCT
ejpam-5914	526	1	information	information	NOUN
ejpam-5914	526	2	and	and	CCONJ
ejpam-5914	526	3	control	control	NOUN
ejpam-5914	526	4	,	,	PUNCT
ejpam-5914	526	5	8(3):338–353	8(3):338–353	NUM
ejpam-5914	526	6	,	,	PUNCT
ejpam-5914	526	7	1965	1965	NUM
ejpam-5914	526	8	.	.	PUNCT
ejpam-5914	527	1	[	[	X
ejpam-5914	527	2	12	12	NUM
ejpam-5914	527	3	]	]	PUNCT
ejpam-5914	527	4	k.	k.	PROPN
ejpam-5914	527	5	t.	t.	PROPN
ejpam-5914	527	6	atanassov	atanassov	PROPN
ejpam-5914	527	7	.	.	PUNCT
ejpam-5914	528	1	intuitionistic	intuitionistic	ADJ
ejpam-5914	528	2	fuzzy	fuzzy	ADJ
ejpam-5914	528	3	sets	set	NOUN
ejpam-5914	528	4	.	.	PUNCT
ejpam-5914	529	1	fuzzy	fuzzy	ADJ
ejpam-5914	529	2	sets	set	NOUN
ejpam-5914	529	3	and	and	CCONJ
ejpam-5914	529	4	systems	system	NOUN
ejpam-5914	529	5	,	,	PUNCT
ejpam-5914	529	6	20(1):87–96	20(1):87–96	NUM
ejpam-5914	529	7	,	,	PUNCT
ejpam-5914	529	8	1986	1986	NUM
ejpam-5914	529	9	.	.	PUNCT
ejpam-5914	530	1	[	[	X
ejpam-5914	530	2	13	13	NUM
ejpam-5914	530	3	]	]	PUNCT
ejpam-5914	530	4	j.	j.	PROPN
ejpam-5914	530	5	a.	a.	PROPN
ejpam-5914	530	6	goguen	goguen	PROPN
ejpam-5914	530	7	.	.	PUNCT
ejpam-5914	531	1	l	l	ADJ
ejpam-5914	531	2	-	-	ADJ
ejpam-5914	531	3	fuzzy	fuzzy	ADJ
ejpam-5914	531	4	sets	set	NOUN
ejpam-5914	531	5	.	.	PUNCT
ejpam-5914	532	1	journal	journal	NOUN
ejpam-5914	532	2	of	of	ADP
ejpam-5914	532	3	mathematical	mathematical	ADJ
ejpam-5914	532	4	analysis	analysis	NOUN
ejpam-5914	532	5	and	and	CCONJ
ejpam-5914	532	6	applications	application	NOUN
ejpam-5914	532	7	,	,	PUNCT
ejpam-5914	532	8	18(1):145–174	18(1):145–174	NUM
ejpam-5914	532	9	,	,	PUNCT
ejpam-5914	532	10	1967	1967	NUM
ejpam-5914	532	11	.	.	PUNCT
ejpam-5914	533	1	[	[	X
ejpam-5914	533	2	14	14	NUM
ejpam-5914	533	3	]	]	PUNCT
ejpam-5914	533	4	m.	m.	NOUN
ejpam-5914	533	5	mizumoto	mizumoto	NOUN
ejpam-5914	533	6	and	and	CCONJ
ejpam-5914	533	7	k.	k.	PROPN
ejpam-5914	533	8	tanaka	tanaka	PROPN
ejpam-5914	533	9	.	.	PUNCT
ejpam-5914	534	1	some	some	DET
ejpam-5914	534	2	properties	property	NOUN
ejpam-5914	534	3	of	of	ADP
ejpam-5914	534	4	fuzzy	fuzzy	ADJ
ejpam-5914	534	5	sets	set	NOUN
ejpam-5914	534	6	of	of	ADP
ejpam-5914	534	7	type	type	NOUN
ejpam-5914	534	8	2	2	NUM
ejpam-5914	534	9	.	.	PUNCT
ejpam-5914	534	10	information	information	NOUN
ejpam-5914	534	11	and	and	CCONJ
ejpam-5914	534	12	control	control	NOUN
ejpam-5914	534	13	,	,	PUNCT
ejpam-5914	534	14	31(4):312–340	31(4):312–340	NUM
ejpam-5914	534	15	,	,	PUNCT
ejpam-5914	534	16	1976	1976	NUM
ejpam-5914	534	17	.	.	PUNCT
ejpam-5914	535	1	[	[	X
ejpam-5914	535	2	15	15	NUM
ejpam-5914	535	3	]	]	PUNCT
ejpam-5914	535	4	k.	k.	PROPN
ejpam-5914	535	5	t.	t.	PROPN
ejpam-5914	535	6	atanassov	atanassov	PROPN
ejpam-5914	535	7	and	and	CCONJ
ejpam-5914	535	8	g.	g.	PROPN
ejpam-5914	535	9	gargov	gargov	PROPN
ejpam-5914	535	10	.	.	PUNCT
ejpam-5914	536	1	interval	interval	NOUN
ejpam-5914	536	2	valued	value	VERB
ejpam-5914	536	3	intuitionistic	intuitionistic	ADJ
ejpam-5914	536	4	fuzzy	fuzzy	ADJ
ejpam-5914	536	5	sets	set	NOUN
ejpam-5914	536	6	.	.	PUNCT
ejpam-5914	537	1	fuzzy	fuzzy	ADJ
ejpam-5914	537	2	sets	set	NOUN
ejpam-5914	537	3	and	and	CCONJ
ejpam-5914	537	4	systems	system	NOUN
ejpam-5914	537	5	,	,	PUNCT
ejpam-5914	537	6	31(3):343–349	31(3):343–349	NUM
ejpam-5914	537	7	,	,	PUNCT
ejpam-5914	537	8	1989	1989	NUM
ejpam-5914	537	9	.	.	PUNCT
ejpam-5914	538	1	[	[	X
ejpam-5914	538	2	16	16	NUM
ejpam-5914	538	3	]	]	X
ejpam-5914	538	4	s.	s.	PROPN
ejpam-5914	538	5	sebastian	sebastian	PROPN
ejpam-5914	538	6	and	and	CCONJ
ejpam-5914	538	7	t.	t.	PROPN
ejpam-5914	538	8	v.	v.	PROPN
ejpam-5914	538	9	ramakrishnan	ramakrishnan	PROPN
ejpam-5914	538	10	.	.	PUNCT
ejpam-5914	539	1	multi	multi	ADJ
ejpam-5914	539	2	-	-	ADJ
ejpam-5914	539	3	fuzzy	fuzzy	ADJ
ejpam-5914	539	4	sets	set	NOUN
ejpam-5914	539	5	:	:	PUNCT
ejpam-5914	539	6	an	an	DET
ejpam-5914	539	7	extension	extension	NOUN
ejpam-5914	539	8	of	of	ADP
ejpam-5914	539	9	fuzzy	fuzzy	ADJ
ejpam-5914	539	10	sets	set	NOUN
ejpam-5914	539	11	.	.	PUNCT
ejpam-5914	540	1	fuzzy	fuzzy	ADJ
ejpam-5914	540	2	information	information	NOUN
ejpam-5914	540	3	and	and	CCONJ
ejpam-5914	540	4	engineering	engineering	NOUN
ejpam-5914	540	5	,	,	PUNCT
ejpam-5914	540	6	3(1):35–43	3(1):35–43	NUM
ejpam-5914	540	7	,	,	PUNCT
ejpam-5914	540	8	2011	2011	NUM
ejpam-5914	540	9	.	.	PUNCT
ejpam-5914	541	1	n.	n.	PROPN
ejpam-5914	541	2	rajesh	rajesh	PROPN
ejpam-5914	541	3	et	et	PROPN
ejpam-5914	541	4	al	al	PROPN
ejpam-5914	541	5	.	.	PUNCT
ejpam-5914	541	6	/	/	SYM
ejpam-5914	541	7	eur	eur	PROPN
ejpam-5914	541	8	.	.	PUNCT
ejpam-5914	542	1	j.	j.	PROPN
ejpam-5914	542	2	pure	pure	PROPN
ejpam-5914	542	3	appl	appl	PROPN
ejpam-5914	542	4	.	.	PROPN
ejpam-5914	542	5	math	math	PROPN
ejpam-5914	542	6	,	,	PUNCT
ejpam-5914	542	7	18	18	NUM
ejpam-5914	542	8	(	(	PUNCT
ejpam-5914	542	9	2	2	NUM
ejpam-5914	542	10	)	)	PUNCT
ejpam-5914	542	11	(	(	PUNCT
ejpam-5914	542	12	2025	2025	NUM
ejpam-5914	542	13	)	)	PUNCT
ejpam-5914	542	14	,	,	PUNCT
ejpam-5914	542	15	5914	5914	NUM
ejpam-5914	542	16	21	21	NUM
ejpam-5914	542	17	of	of	ADP
ejpam-5914	542	18	21	21	NUM
ejpam-5914	543	1	[	[	X
ejpam-5914	543	2	17	17	NUM
ejpam-5914	543	3	]	]	PUNCT
ejpam-5914	543	4	k.	k.	PROPN
ejpam-5914	543	5	j.	j.	PROPN
ejpam-5914	543	6	lee	lee	PROPN
ejpam-5914	543	7	.	.	PUNCT
ejpam-5914	544	1	bipolar	bipolar	ADJ
ejpam-5914	544	2	-	-	PUNCT
ejpam-5914	544	3	valued	value	VERB
ejpam-5914	544	4	fuzzy	fuzzy	ADJ
ejpam-5914	544	5	sets	set	NOUN
ejpam-5914	544	6	and	and	CCONJ
ejpam-5914	544	7	their	their	PRON
ejpam-5914	544	8	operations	operation	NOUN
ejpam-5914	544	9	.	.	PUNCT
ejpam-5914	545	1	in	in	ADP
ejpam-5914	545	2	proceedings	proceeding	NOUN
ejpam-5914	545	3	of	of	ADP
ejpam-5914	545	4	international	international	ADJ
ejpam-5914	545	5	conference	conference	NOUN
ejpam-5914	545	6	on	on	ADP
ejpam-5914	545	7	intelligent	intelligent	ADJ
ejpam-5914	545	8	technologies	technology	NOUN
ejpam-5914	545	9	,	,	PUNCT
ejpam-5914	545	10	pages	page	NOUN
ejpam-5914	545	11	307–312	307–312	NUM
ejpam-5914	545	12	.	.	PUNCT
ejpam-5914	545	13	bangkok	bangkok	PROPN
ejpam-5914	545	14	,	,	PUNCT
ejpam-5914	545	15	2000	2000	NUM
ejpam-5914	545	16	.	.	PUNCT
ejpam-5914	546	1	[	[	X
ejpam-5914	546	2	18	18	NUM
ejpam-5914	546	3	]	]	PUNCT
ejpam-5914	546	4	j.	j.	PROPN
ejpam-5914	546	5	chen	chen	PROPN
ejpam-5914	546	6	,	,	PUNCT
ejpam-5914	546	7	s.	s.	PROPN
ejpam-5914	546	8	li	li	PROPN
ejpam-5914	546	9	,	,	PUNCT
ejpam-5914	546	10	s.	s.	PROPN
ejpam-5914	546	11	ma	ma	PROPN
ejpam-5914	546	12	,	,	PUNCT
ejpam-5914	546	13	and	and	CCONJ
ejpam-5914	546	14	x.	x.	PROPN
ejpam-5914	546	15	wang	wang	PROPN
ejpam-5914	546	16	.	.	PUNCT
ejpam-5914	547	1	m	m	PROPN
ejpam-5914	547	2	-	-	ADJ
ejpam-5914	547	3	polar	polar	ADJ
ejpam-5914	547	4	fuzzy	fuzzy	ADJ
ejpam-5914	547	5	sets	set	NOUN
ejpam-5914	547	6	:	:	PUNCT
ejpam-5914	547	7	an	an	DET
ejpam-5914	547	8	extension	extension	NOUN
ejpam-5914	547	9	of	of	ADP
ejpam-5914	547	10	bipolar	bipolar	ADJ
ejpam-5914	547	11	fuzzy	fuzzy	ADJ
ejpam-5914	547	12	sets	set	NOUN
ejpam-5914	547	13	.	.	PUNCT
ejpam-5914	548	1	scientific	scientific	ADJ
ejpam-5914	548	2	world	world	NOUN
ejpam-5914	548	3	journal	journal	NOUN
ejpam-5914	548	4	,	,	PUNCT
ejpam-5914	548	5	2014	2014	NUM
ejpam-5914	548	6	:	:	PUNCT
ejpam-5914	548	7	article	article	NOUN
ejpam-5914	548	8	i	i	PROPN
ejpam-5914	548	9	d	d	PROPN
ejpam-5914	548	10	416530	416530	NUM
ejpam-5914	548	11	,	,	PUNCT
ejpam-5914	548	12	8	8	NUM
ejpam-5914	548	13	pages	page	NOUN
ejpam-5914	548	14	,	,	PUNCT
ejpam-5914	548	15	2014	2014	NUM
ejpam-5914	548	16	.	.	PUNCT
ejpam-5914	549	1	[	[	X
ejpam-5914	549	2	19	19	NUM
ejpam-5914	549	3	]	]	X
ejpam-5914	549	4	f.	f.	PROPN
ejpam-5914	549	5	smarandache	smarandache	PROPN
ejpam-5914	549	6	.	.	PUNCT
ejpam-5914	550	1	neutrosophy	neutrosophy	NOUN
ejpam-5914	550	2	,	,	PUNCT
ejpam-5914	550	3	neutrosophic	neutrosophic	ADJ
ejpam-5914	550	4	probability	probability	NOUN
ejpam-5914	550	5	,	,	PUNCT
ejpam-5914	550	6	set	set	NOUN
ejpam-5914	550	7	and	and	CCONJ
ejpam-5914	550	8	logic	logic	NOUN
ejpam-5914	550	9	.	.	PUNCT
ejpam-5914	551	1	in	in	ADP
ejpam-5914	551	2	proquest	proquest	PROPN
ejpam-5914	551	3	information	information	NOUN
ejpam-5914	551	4	and	and	CCONJ
ejpam-5914	551	5	learning	learning	PROPN
ejpam-5914	551	6	,	,	PUNCT
ejpam-5914	551	7	ann	ann	PROPN
ejpam-5914	551	8	arbor	arbor	PROPN
ejpam-5914	551	9	,	,	PUNCT
ejpam-5914	551	10	michigan	michigan	PROPN
ejpam-5914	551	11	,	,	PUNCT
ejpam-5914	551	12	usa	usa	PROPN
ejpam-5914	551	13	,	,	PUNCT
ejpam-5914	551	14	1998	1998	NUM
ejpam-5914	551	15	.	.	PUNCT
ejpam-5914	552	1	[	[	X
ejpam-5914	552	2	20	20	NUM
ejpam-5914	552	3	]	]	PUNCT
ejpam-5914	552	4	f.	f.	PROPN
ejpam-5914	552	5	smarandache	smarandache	PROPN
ejpam-5914	552	6	.	.	PUNCT
ejpam-5914	553	1	neutrosophic	neutrosophic	PROPN
ejpam-5914	553	2	set	set	NOUN
ejpam-5914	553	3	—	—	PUNCT
ejpam-5914	553	4	a	a	DET
ejpam-5914	553	5	generalization	generalization	NOUN
ejpam-5914	553	6	of	of	ADP
ejpam-5914	553	7	the	the	DET
ejpam-5914	553	8	intuitionistic	intuitionistic	ADJ
ejpam-5914	553	9	fuzzy	fuzzy	ADJ
ejpam-5914	553	10	set	set	NOUN
ejpam-5914	553	11	.	.	PUNCT
ejpam-5914	554	1	in	in	ADP
ejpam-5914	554	2	2006	2006	NUM
ejpam-5914	554	3	international	international	ADJ
ejpam-5914	554	4	conference	conference	NOUN
ejpam-5914	554	5	on	on	ADP
ejpam-5914	554	6	granular	granular	ADJ
ejpam-5914	554	7	computing	computing	NOUN
ejpam-5914	554	8	,	,	PUNCT
ejpam-5914	554	9	atlanta	atlanta	PROPN
ejpam-5914	554	10	,	,	PUNCT
ejpam-5914	554	11	ga	ga	PROPN
ejpam-5914	554	12	,	,	PUNCT
ejpam-5914	554	13	usa	usa	PROPN
ejpam-5914	554	14	,	,	PUNCT
ejpam-5914	554	15	pages	page	NOUN
ejpam-5914	554	16	38–42	38–42	NUM
ejpam-5914	554	17	.	.	PUNCT
ejpam-5914	554	18	ieee	ieee	PROPN
ejpam-5914	554	19	,	,	PUNCT
ejpam-5914	554	20	2006	2006	NUM
ejpam-5914	554	21	.	.	PUNCT
ejpam-5914	555	1	[	[	X
ejpam-5914	555	2	21	21	NUM
ejpam-5914	555	3	]	]	X
ejpam-5914	555	4	r.	r.	PROPN
ejpam-5914	555	5	a.	a.	PROPN
ejpam-5914	555	6	borzooei	borzooei	PROPN
ejpam-5914	555	7	,	,	PUNCT
ejpam-5914	555	8	g.	g.	PROPN
ejpam-5914	555	9	r.	r.	PROPN
ejpam-5914	555	10	rezaei	rezaei	PROPN
ejpam-5914	555	11	,	,	PUNCT
ejpam-5914	555	12	and	and	CCONJ
ejpam-5914	555	13	y.	y.	PROPN
ejpam-5914	555	14	b.	b.	PROPN
ejpam-5914	555	15	jun	jun	PROPN
ejpam-5914	555	16	.	.	PROPN
ejpam-5914	555	17	fuzzy	fuzzy	ADJ
ejpam-5914	555	18	weak	weak	ADJ
ejpam-5914	555	19	filters	filter	NOUN
ejpam-5914	555	20	of	of	ADP
ejpam-5914	555	21	sheffer	sheffer	PROPN
ejpam-5914	555	22	stroke	stroke	PROPN
ejpam-5914	555	23	hilbert	hilbert	PROPN
ejpam-5914	555	24	algebras	algebras	PROPN
ejpam-5914	555	25	.	.	PUNCT
ejpam-5914	556	1	annales	annales	PROPN
ejpam-5914	556	2	mathematicae	mathematicae	PROPN
ejpam-5914	556	3	silesianae	silesianae	PROPN
ejpam-5914	556	4	,	,	PUNCT
ejpam-5914	556	5	37(2):185–203	37(2):185–203	PROPN
ejpam-5914	556	6	,	,	PUNCT
ejpam-5914	556	7	2023	2023	NUM
ejpam-5914	556	8	.	.	PUNCT
ejpam-5914	557	1	[	[	X
ejpam-5914	557	2	22	22	NUM
ejpam-5914	557	3	]	]	PUNCT
ejpam-5914	557	4	h.	h.	PROPN
ejpam-5914	557	5	s.	s.	PROPN
ejpam-5914	557	6	kim	kim	PROPN
ejpam-5914	557	7	,	,	PUNCT
ejpam-5914	557	8	s.	s.	PROPN
ejpam-5914	557	9	z.	z.	PROPN
ejpam-5914	557	10	song	song	PROPN
ejpam-5914	557	11	,	,	PUNCT
ejpam-5914	557	12	s.	s.	PROPN
ejpam-5914	557	13	s.	s.	PROPN
ejpam-5914	557	14	ahn	ahn	PROPN
ejpam-5914	557	15	,	,	PUNCT
ejpam-5914	557	16	and	and	CCONJ
ejpam-5914	557	17	y.	y.	PROPN
ejpam-5914	557	18	b.	b.	PROPN
ejpam-5914	557	19	jun	jun	PROPN
ejpam-5914	557	20	.	.	PROPN
ejpam-5914	557	21	deductive	deductive	ADJ
ejpam-5914	557	22	systems	system	NOUN
ejpam-5914	557	23	and	and	CCONJ
ejpam-5914	557	24	filters	filter	NOUN
ejpam-5914	557	25	of	of	ADP
ejpam-5914	557	26	sheffer	sheffer	PROPN
ejpam-5914	557	27	stroke	stroke	PROPN
ejpam-5914	557	28	hilbert	hilbert	PROPN
ejpam-5914	557	29	algebras	algebras	PROPN
ejpam-5914	557	30	based	base	VERB
ejpam-5914	557	31	on	on	ADP
ejpam-5914	557	32	the	the	DET
ejpam-5914	557	33	bipolar	bipolar	ADV
ejpam-5914	557	34	-	-	PUNCT
ejpam-5914	557	35	valued	value	VERB
ejpam-5914	557	36	fuzzy	fuzzy	ADJ
ejpam-5914	557	37	set	set	NOUN
ejpam-5914	557	38	environment	environment	NOUN
ejpam-5914	557	39	.	.	PUNCT
ejpam-5914	558	1	journal	journal	PROPN
ejpam-5914	558	2	of	of	ADP
ejpam-5914	558	3	computational	computational	ADJ
ejpam-5914	558	4	analysis	analysis	NOUN
ejpam-5914	558	5	and	and	CCONJ
ejpam-5914	558	6	applications	application	NOUN
ejpam-5914	558	7	,	,	PUNCT
ejpam-5914	558	8	32(1):192–210	32(1):192–210	NUM
ejpam-5914	558	9	,	,	PUNCT
ejpam-5914	558	10	2024	2024	NUM
ejpam-5914	558	11	.	.	PUNCT
ejpam-5914	559	1	[	[	X
ejpam-5914	559	2	23	23	NUM
ejpam-5914	559	3	]	]	PUNCT
ejpam-5914	559	4	t.	t.	NOUN
ejpam-5914	559	5	oner	oner	NOUN
ejpam-5914	559	6	,	,	PUNCT
ejpam-5914	559	7	t.	t.	PROPN
ejpam-5914	559	8	katican	katican	PROPN
ejpam-5914	559	9	,	,	PUNCT
ejpam-5914	559	10	and	and	CCONJ
ejpam-5914	559	11	a.	a.	PROPN
ejpam-5914	559	12	borumand	borumand	PROPN
ejpam-5914	559	13	saeid	saeid	PROPN
ejpam-5914	559	14	.	.	PUNCT
ejpam-5914	560	1	fuzzy	fuzzy	ADJ
ejpam-5914	560	2	filters	filter	NOUN
ejpam-5914	560	3	of	of	ADP
ejpam-5914	560	4	sheffer	sheffer	PROPN
ejpam-5914	560	5	stroke	stroke	PROPN
ejpam-5914	560	6	hilbert	hilbert	PROPN
ejpam-5914	560	7	algebras	algebras	PROPN
ejpam-5914	560	8	.	.	PUNCT
ejpam-5914	561	1	journal	journal	PROPN
ejpam-5914	561	2	of	of	ADP
ejpam-5914	561	3	intelligent	intelligent	ADJ
ejpam-5914	561	4	and	and	CCONJ
ejpam-5914	561	5	fuzzy	fuzzy	ADJ
ejpam-5914	561	6	systems	system	NOUN
ejpam-5914	561	7	,	,	PUNCT
ejpam-5914	561	8	40(1):759–772	40(1):759–772	NOUN
ejpam-5914	561	9	,	,	PUNCT
ejpam-5914	561	10	2021	2021	NUM
ejpam-5914	561	11	.	.	PUNCT
ejpam-5914	562	1	[	[	X
ejpam-5914	562	2	24	24	NUM
ejpam-5914	562	3	]	]	X
ejpam-5914	562	4	n.	n.	PROPN
ejpam-5914	562	5	rajesh	rajesh	PROPN
ejpam-5914	562	6	,	,	PUNCT
ejpam-5914	562	7	t.	t.	PROPN
ejpam-5914	562	8	oner	oner	NOUN
ejpam-5914	562	9	,	,	PUNCT
ejpam-5914	562	10	a.	a.	NOUN
ejpam-5914	562	11	iampan	iampan	PROPN
ejpam-5914	562	12	,	,	PUNCT
ejpam-5914	562	13	and	and	CCONJ
ejpam-5914	562	14	i.	i.	PROPN
ejpam-5914	562	15	senturk	senturk	PROPN
ejpam-5914	562	16	.	.	PUNCT
ejpam-5914	563	1	on	on	ADP
ejpam-5914	563	2	length	length	NOUN
ejpam-5914	563	3	and	and	CCONJ
ejpam-5914	563	4	mean	mean	VERB
ejpam-5914	563	5	fuzzy	fuzzy	ADJ
ejpam-5914	563	6	ideals	ideal	NOUN
ejpam-5914	563	7	of	of	ADP
ejpam-5914	563	8	sheffer	sheffer	PROPN
ejpam-5914	563	9	stroke	stroke	PROPN
ejpam-5914	563	10	hilbert	hilbert	PROPN
ejpam-5914	563	11	algebras	algebras	PROPN
ejpam-5914	563	12	.	.	PUNCT
ejpam-5914	564	1	european	european	PROPN
ejpam-5914	564	2	journal	journal	PROPN
ejpam-5914	564	3	of	of	ADP
ejpam-5914	564	4	pure	pure	ADJ
ejpam-5914	564	5	and	and	CCONJ
ejpam-5914	564	6	applied	applied	ADJ
ejpam-5914	564	7	mathematics	mathematic	NOUN
ejpam-5914	564	8	,	,	PUNCT
ejpam-5914	564	9	18(1):5779	18(1):5779	NUM
ejpam-5914	564	10	,	,	PUNCT
ejpam-5914	564	11	2025	2025	NUM
ejpam-5914	564	12	.	.	PUNCT
ejpam-5914	565	1	[	[	X
ejpam-5914	565	2	25	25	NUM
ejpam-5914	565	3	]	]	X
ejpam-5914	565	4	y.	y.	PROPN
ejpam-5914	565	5	b.	b.	PROPN
ejpam-5914	565	6	jun	jun	PROPN
ejpam-5914	565	7	.	.	PROPN
ejpam-5914	566	1	commutative	commutative	PROPN
ejpam-5914	566	2	hilbert	hilbert	PROPN
ejpam-5914	566	3	algebras	algebras	PROPN
ejpam-5914	566	4	.	.	PUNCT
ejpam-5914	567	1	soochow	soochow	PROPN
ejpam-5914	567	2	journal	journal	PROPN
ejpam-5914	567	3	of	of	ADP
ejpam-5914	567	4	mathematics	mathematic	NOUN
ejpam-5914	567	5	,	,	PUNCT
ejpam-5914	567	6	22(4):477–484	22(4):477–484	NUM
ejpam-5914	567	7	,	,	PUNCT
ejpam-5914	567	8	1996	1996	NUM
ejpam-5914	567	9	.	.	PUNCT
ejpam-5914	568	1	[	[	X
ejpam-5914	568	2	26	26	NUM
ejpam-5914	568	3	]	]	X
ejpam-5914	568	4	n.	n.	NOUN
ejpam-5914	568	5	tacha	tacha	PROPN
ejpam-5914	568	6	,	,	PUNCT
ejpam-5914	568	7	p.	p.	PROPN
ejpam-5914	568	8	phayapsiang	phayapsiang	PROPN
ejpam-5914	568	9	,	,	PUNCT
ejpam-5914	568	10	and	and	CCONJ
ejpam-5914	568	11	a.	a.	NOUN
ejpam-5914	568	12	iampan	iampan	PROPN
ejpam-5914	568	13	.	.	PUNCT
ejpam-5914	569	1	length	length	NOUN
ejpam-5914	569	2	and	and	CCONJ
ejpam-5914	569	3	mean	mean	VERB
ejpam-5914	569	4	fuzzy	fuzzy	ADJ
ejpam-5914	569	5	up	up	ADP
ejpam-5914	569	6	-	-	PUNCT
ejpam-5914	569	7	subalgebras	subalgebra	NOUN
ejpam-5914	569	8	of	of	ADP
ejpam-5914	569	9	up	up	ADP
ejpam-5914	569	10	-	-	PUNCT
ejpam-5914	569	11	algebras	algebras	X
ejpam-5914	569	12	.	.	PUNCT
ejpam-5914	570	1	caspian	caspian	PROPN
ejpam-5914	570	2	journal	journal	PROPN
ejpam-5914	570	3	of	of	ADP
ejpam-5914	570	4	mathematical	mathematical	ADJ
ejpam-5914	570	5	sciences	sciences	PROPN
ejpam-5914	570	6	,	,	PUNCT
ejpam-5914	570	7	11(1):264–302	11(1):264–302	NUM
ejpam-5914	570	8	,	,	PUNCT
ejpam-5914	570	9	2022	2022	NUM
ejpam-5914	570	10	.	.	PUNCT