id	sid	tid	token	lemma	pos
ejpam-5779	1	1	european	european	PROPN
ejpam-5779	1	2	journal	journal	PROPN
ejpam-5779	1	3	of	of	ADP
ejpam-5779	1	4	pure	pure	ADJ
ejpam-5779	1	5	and	and	CCONJ
ejpam-5779	1	6	applied	applied	ADJ
ejpam-5779	1	7	mathematics	mathematic	NOUN
ejpam-5779	1	8	2025	2025	NUM
ejpam-5779	1	9	,	,	PUNCT
ejpam-5779	1	10	vol	vol	NOUN
ejpam-5779	1	11	.	.	PROPN
ejpam-5779	1	12	18	18	NUM
ejpam-5779	1	13	,	,	PUNCT
ejpam-5779	1	14	issue	issue	NOUN
ejpam-5779	1	15	1	1	NUM
ejpam-5779	1	16	,	,	PUNCT
ejpam-5779	1	17	article	article	NOUN
ejpam-5779	1	18	number	number	NOUN
ejpam-5779	1	19	5779	5779	NUM
ejpam-5779	1	20	issn	issn	PROPN
ejpam-5779	1	21	1307	1307	NUM
ejpam-5779	1	22	-	-	SYM
ejpam-5779	1	23	5543	5543	NUM
ejpam-5779	1	24	–	–	PUNCT
ejpam-5779	1	25	ejpam.com	ejpam.com	X
ejpam-5779	1	26	published	publish	VERB
ejpam-5779	1	27	by	by	ADP
ejpam-5779	1	28	new	new	PROPN
ejpam-5779	1	29	york	york	PROPN
ejpam-5779	1	30	business	business	PROPN
ejpam-5779	1	31	global	global	ADJ
ejpam-5779	1	32	1	1	NUM
ejpam-5779	1	33	on	on	ADP
ejpam-5779	1	34	length	length	NOUN
ejpam-5779	1	35	and	and	CCONJ
ejpam-5779	1	36	mean	mean	VERB
ejpam-5779	1	37	fuzzy	fuzzy	ADJ
ejpam-5779	1	38	ideals	ideal	NOUN
ejpam-5779	1	39	of	of	ADP
ejpam-5779	1	40	sheffer	sheffer	ADJ
ejpam-5779	1	41	stroke2	stroke2	ADJ
ejpam-5779	1	42	hilbert	hilbert	PROPN
ejpam-5779	1	43	algebras3	algebras3	PROPN
ejpam-5779	1	44	neelamegarajan	neelamegarajan	PROPN
ejpam-5779	1	45	rajesh1	rajesh1	PROPN
ejpam-5779	1	46	,	,	PUNCT
ejpam-5779	1	47	tahsin	tahsin	PROPN
ejpam-5779	1	48	oner2	oner2	VERB
ejpam-5779	1	49	,	,	PUNCT
ejpam-5779	1	50	aiyared	aiyare	VERB
ejpam-5779	1	51	iampan3,∗	iampan3,∗	ADJ
ejpam-5779	1	52	,	,	PUNCT
ejpam-5779	1	53	ibrahim	ibrahim	PROPN
ejpam-5779	1	54	senturk24	senturk24	PROPN
ejpam-5779	1	55	1	1	NUM
ejpam-5779	1	56	department	department	NOUN
ejpam-5779	1	57	of	of	ADP
ejpam-5779	1	58	mathematics	mathematic	NOUN
ejpam-5779	1	59	,	,	PUNCT
ejpam-5779	1	60	rajah	rajah	NOUN
ejpam-5779	1	61	serfoji	serfoji	ADJ
ejpam-5779	1	62	government	government	NOUN
ejpam-5779	1	63	college	college	NOUN
ejpam-5779	1	64	,	,	PUNCT
ejpam-5779	1	65	thanjavur-613005	thanjavur-613005	NOUN
ejpam-5779	1	66	,	,	PUNCT
ejpam-5779	1	67	tamil5	tamil5	NOUN
ejpam-5779	1	68	nadu	nadu	NOUN
ejpam-5779	1	69	,	,	PUNCT
ejpam-5779	1	70	india6	india6	ADJ
ejpam-5779	1	71	2	2	NUM
ejpam-5779	1	72	department	department	NOUN
ejpam-5779	1	73	of	of	ADP
ejpam-5779	1	74	mathematics	mathematic	NOUN
ejpam-5779	1	75	,	,	PUNCT
ejpam-5779	1	76	faculty	faculty	NOUN
ejpam-5779	1	77	of	of	ADP
ejpam-5779	1	78	science	science	NOUN
ejpam-5779	1	79	,	,	PUNCT
ejpam-5779	1	80	ege	ege	PROPN
ejpam-5779	1	81	university	university	NOUN
ejpam-5779	1	82	,	,	PUNCT
ejpam-5779	1	83	35100	35100	NUM
ejpam-5779	1	84	izmir	izmir	NOUN
ejpam-5779	1	85	,	,	PUNCT
ejpam-5779	1	86	turkey7	turkey7	PROPN
ejpam-5779	1	87	3	3	NUM
ejpam-5779	1	88	department	department	NOUN
ejpam-5779	1	89	of	of	ADP
ejpam-5779	1	90	mathematics	mathematic	NOUN
ejpam-5779	1	91	,	,	PUNCT
ejpam-5779	1	92	school	school	NOUN
ejpam-5779	1	93	of	of	ADP
ejpam-5779	1	94	science	science	NOUN
ejpam-5779	1	95	,	,	PUNCT
ejpam-5779	1	96	university	university	NOUN
ejpam-5779	1	97	of	of	ADP
ejpam-5779	1	98	phayao	phayao	NOUN
ejpam-5779	1	99	,	,	PUNCT
ejpam-5779	1	100	mae	mae	PROPN
ejpam-5779	1	101	ka	ka	PROPN
ejpam-5779	1	102	,	,	PUNCT
ejpam-5779	1	103	mueang,8	mueang,8	NOUN
ejpam-5779	1	104	phayao	phayao	VERB
ejpam-5779	1	105	56000	56000	NUM
ejpam-5779	1	106	,	,	PUNCT
ejpam-5779	1	107	thailand9	thailand9	NOUN
ejpam-5779	1	108	10	10	NUM
ejpam-5779	1	109	abstract	abstract	ADJ
ejpam-5779	1	110	.	.	PUNCT
ejpam-5779	2	1	this	this	DET
ejpam-5779	2	2	paper	paper	NOUN
ejpam-5779	2	3	presents	present	VERB
ejpam-5779	2	4	a	a	DET
ejpam-5779	2	5	detailed	detailed	ADJ
ejpam-5779	2	6	exploration	exploration	NOUN
ejpam-5779	2	7	of	of	ADP
ejpam-5779	2	8	sheffer	sheffer	PROPN
ejpam-5779	2	9	stroke	stroke	PROPN
ejpam-5779	2	10	hilbert	hilbert	PROPN
ejpam-5779	2	11	algebras	algebras	PROPN
ejpam-5779	2	12	,	,	PUNCT
ejpam-5779	2	13	introducing	introduce	VERB
ejpam-5779	2	14	the	the	DET
ejpam-5779	2	15	innovative	innovative	ADJ
ejpam-5779	2	16	concepts	concept	NOUN
ejpam-5779	2	17	of	of	ADP
ejpam-5779	2	18	length	length	NOUN
ejpam-5779	2	19	fuzzy	fuzzy	ADJ
ejpam-5779	2	20	ideals	ideal	NOUN
ejpam-5779	2	21	and	and	CCONJ
ejpam-5779	2	22	mean	mean	VERB
ejpam-5779	2	23	fuzzy	fuzzy	ADJ
ejpam-5779	2	24	ideals	ideal	NOUN
ejpam-5779	2	25	within	within	ADP
ejpam-5779	2	26	an	an	DET
ejpam-5779	2	27	interval	interval	NOUN
ejpam-5779	2	28	-	-	PUNCT
ejpam-5779	2	29	valued	value	VERB
ejpam-5779	2	30	fuzzy	fuzzy	ADJ
ejpam-5779	2	31	framework	framework	NOUN
ejpam-5779	2	32	.	.	PUNCT
ejpam-5779	3	1	these	these	DET
ejpam-5779	3	2	new	new	ADJ
ejpam-5779	3	3	constructs	construct	NOUN
ejpam-5779	3	4	extend	extend	VERB
ejpam-5779	3	5	classical	classical	ADJ
ejpam-5779	3	6	ideal	ideal	NOUN
ejpam-5779	3	7	theory	theory	NOUN
ejpam-5779	3	8	by	by	ADP
ejpam-5779	3	9	incorporating	incorporate	VERB
ejpam-5779	3	10	fuzzy	fuzzy	ADJ
ejpam-5779	3	11	logic	logic	NOUN
ejpam-5779	3	12	,	,	PUNCT
ejpam-5779	3	13	providing	provide	VERB
ejpam-5779	3	14	precise	precise	ADJ
ejpam-5779	3	15	mathematical	mathematical	ADJ
ejpam-5779	3	16	tools	tool	NOUN
ejpam-5779	3	17	to	to	PART
ejpam-5779	3	18	analyze	analyze	VERB
ejpam-5779	3	19	and	and	CCONJ
ejpam-5779	3	20	measure	measure	VERB
ejpam-5779	3	21	membership	membership	NOUN
ejpam-5779	3	22	gradations	gradation	NOUN
ejpam-5779	3	23	.	.	PUNCT
ejpam-5779	4	1	specifically	specifically	ADV
ejpam-5779	4	2	,	,	PUNCT
ejpam-5779	4	3	the	the	DET
ejpam-5779	4	4	study	study	NOUN
ejpam-5779	4	5	establishes	establish	VERB
ejpam-5779	4	6	critical	critical	ADJ
ejpam-5779	4	7	relationships	relationship	NOUN
ejpam-5779	4	8	between	between	ADP
ejpam-5779	4	9	length	length	NOUN
ejpam-5779	4	10	fuzzy	fuzzy	ADJ
ejpam-5779	4	11	ideals	ideal	NOUN
ejpam-5779	4	12	and	and	CCONJ
ejpam-5779	4	13	mean	mean	VERB
ejpam-5779	4	14	fuzzy	fuzzy	ADJ
ejpam-5779	4	15	ideals	ideal	NOUN
ejpam-5779	4	16	,	,	PUNCT
ejpam-5779	4	17	their	their	PRON
ejpam-5779	4	18	hierarchical	hierarchical	ADJ
ejpam-5779	4	19	subsets	subset	NOUN
ejpam-5779	4	20	,	,	PUNCT
ejpam-5779	4	21	and	and	CCONJ
ejpam-5779	4	22	their	their	PRON
ejpam-5779	4	23	implications	implication	NOUN
ejpam-5779	4	24	for	for	ADP
ejpam-5779	4	25	algebraic	algebraic	ADJ
ejpam-5779	4	26	consistency	consistency	NOUN
ejpam-5779	4	27	and	and	CCONJ
ejpam-5779	4	28	computational	computational	ADJ
ejpam-5779	4	29	logic	logic	NOUN
ejpam-5779	4	30	.	.	PUNCT
ejpam-5779	5	1	key	key	ADJ
ejpam-5779	5	2	findings	finding	NOUN
ejpam-5779	5	3	demonstrate	demonstrate	VERB
ejpam-5779	5	4	that	that	SCONJ
ejpam-5779	5	5	length	length	NOUN
ejpam-5779	5	6	fuzzy	fuzzy	ADJ
ejpam-5779	5	7	ideals	ideal	NOUN
ejpam-5779	5	8	align	align	VERB
ejpam-5779	5	9	closely	closely	ADV
ejpam-5779	5	10	with	with	ADP
ejpam-5779	5	11	interval	interval	NOUN
ejpam-5779	5	12	-	-	PUNCT
ejpam-5779	5	13	valued	value	VERB
ejpam-5779	5	14	fuzzy	fuzzy	ADJ
ejpam-5779	5	15	subsets	subset	NOUN
ejpam-5779	5	16	,	,	PUNCT
ejpam-5779	5	17	while	while	SCONJ
ejpam-5779	5	18	mean	mean	VERB
ejpam-5779	5	19	fuzzy	fuzzy	ADJ
ejpam-5779	5	20	ideals	ideal	NOUN
ejpam-5779	5	21	offer	offer	VERB
ejpam-5779	5	22	a	a	DET
ejpam-5779	5	23	unique	unique	ADJ
ejpam-5779	5	24	averaging	averaging	NOUN
ejpam-5779	5	25	perspective	perspective	NOUN
ejpam-5779	5	26	for	for	ADP
ejpam-5779	5	27	understanding	understand	VERB
ejpam-5779	5	28	ideal	ideal	ADJ
ejpam-5779	5	29	structures	structure	NOUN
ejpam-5779	5	30	.	.	PUNCT
ejpam-5779	6	1	these	these	DET
ejpam-5779	6	2	contributions	contribution	NOUN
ejpam-5779	6	3	significantly	significantly	ADV
ejpam-5779	6	4	advance	advance	VERB
ejpam-5779	6	5	the	the	DET
ejpam-5779	6	6	field	field	NOUN
ejpam-5779	6	7	of	of	ADP
ejpam-5779	6	8	fuzzy	fuzzy	ADJ
ejpam-5779	6	9	algebra	algebra	NOUN
ejpam-5779	6	10	,	,	PUNCT
ejpam-5779	6	11	offering	offer	VERB
ejpam-5779	6	12	theoretical	theoretical	ADJ
ejpam-5779	6	13	insights	insight	NOUN
ejpam-5779	6	14	and	and	CCONJ
ejpam-5779	6	15	potential	potential	ADJ
ejpam-5779	6	16	applications	application	NOUN
ejpam-5779	6	17	in	in	ADP
ejpam-5779	6	18	computational	computational	ADJ
ejpam-5779	6	19	logic	logic	NOUN
ejpam-5779	6	20	,	,	PUNCT
ejpam-5779	6	21	uncertainty	uncertainty	NOUN
ejpam-5779	6	22	modeling	modeling	NOUN
ejpam-5779	6	23	,	,	PUNCT
ejpam-5779	6	24	and	and	CCONJ
ejpam-5779	6	25	algorithmic	algorithmic	ADJ
ejpam-5779	6	26	design	design	NOUN
ejpam-5779	6	27	.	.	PUNCT
ejpam-5779	7	1	2020	2020	NUM
ejpam-5779	7	2	mathematics	mathematic	NOUN
ejpam-5779	7	3	subject	subject	NOUN
ejpam-5779	7	4	classifications	classification	NOUN
ejpam-5779	7	5	:	:	PUNCT
ejpam-5779	7	6	20n05	20n05	NUM
ejpam-5779	7	7	,	,	PUNCT
ejpam-5779	7	8	94d05	94d05	NUM
ejpam-5779	7	9	,	,	PUNCT
ejpam-5779	7	10	03e7211	03e7211	PROPN
ejpam-5779	7	11	key	key	ADJ
ejpam-5779	7	12	words	word	NOUN
ejpam-5779	7	13	and	and	CCONJ
ejpam-5779	7	14	phrases	phrase	NOUN
ejpam-5779	7	15	:	:	PUNCT
ejpam-5779	7	16	sheffer	sheffer	NOUN
ejpam-5779	7	17	stroke	stroke	PROPN
ejpam-5779	7	18	hilbert	hilbert	PROPN
ejpam-5779	7	19	algebra	algebra	PROPN
ejpam-5779	7	20	(	(	PUNCT
ejpam-5779	7	21	sha	sha	PROPN
ejpam-5779	7	22	)	)	PUNCT
ejpam-5779	7	23	,	,	PUNCT
ejpam-5779	7	24	ideal	ideal	ADJ
ejpam-5779	7	25	,	,	PUNCT
ejpam-5779	7	26	length	length	NOUN
ejpam-5779	7	27	fuzzy	fuzzy	ADJ
ejpam-5779	7	28	ideal	ideal	NOUN
ejpam-5779	7	29	,	,	PUNCT
ejpam-5779	7	30	mean12	mean12	X
ejpam-5779	7	31	fuzzy	fuzzy	ADJ
ejpam-5779	7	32	ideal13	ideal13	PROPN
ejpam-5779	7	33	14	14	NUM
ejpam-5779	7	34	1	1	NUM
ejpam-5779	7	35	.	.	PUNCT
ejpam-5779	7	36	introduction15	introduction15	PROPN
ejpam-5779	8	1	hilbert	hilbert	PROPN
ejpam-5779	8	2	algebras	algebras	PROPN
ejpam-5779	8	3	,	,	PUNCT
ejpam-5779	8	4	often	often	ADV
ejpam-5779	8	5	referred	refer	VERB
ejpam-5779	8	6	to	to	ADP
ejpam-5779	8	7	as	as	ADP
ejpam-5779	8	8	implicative	implicative	ADJ
ejpam-5779	8	9	algebras	algebra	NOUN
ejpam-5779	8	10	,	,	PUNCT
ejpam-5779	8	11	are	be	AUX
ejpam-5779	8	12	algebraic	algebraic	ADJ
ejpam-5779	8	13	structures16	structures16	ADJ
ejpam-5779	8	14	that	that	PRON
ejpam-5779	8	15	extend	extend	VERB
ejpam-5779	8	16	the	the	DET
ejpam-5779	8	17	classical	classical	ADJ
ejpam-5779	8	18	operations	operation	NOUN
ejpam-5779	8	19	of	of	ADP
ejpam-5779	8	20	logic	logic	NOUN
ejpam-5779	8	21	.	.	PUNCT
ejpam-5779	9	1	these	these	DET
ejpam-5779	9	2	algebras	algebra	NOUN
ejpam-5779	9	3	are	be	AUX
ejpam-5779	9	4	typically	typically	ADV
ejpam-5779	9	5	defined	define	VERB
ejpam-5779	9	6	by	by	ADP
ejpam-5779	9	7	a17	a17	PROPN
ejpam-5779	9	8	set	set	NOUN
ejpam-5779	9	9	of	of	ADP
ejpam-5779	9	10	axioms	axiom	NOUN
ejpam-5779	9	11	involving	involve	VERB
ejpam-5779	9	12	a	a	DET
ejpam-5779	9	13	binary	binary	ADJ
ejpam-5779	9	14	operation	operation	NOUN
ejpam-5779	9	15	,	,	PUNCT
ejpam-5779	9	16	the	the	DET
ejpam-5779	9	17	sheffer	sheffer	NOUN
ejpam-5779	9	18	stroke	stroke	NOUN
ejpam-5779	9	19	,	,	PUNCT
ejpam-5779	9	20	which	which	PRON
ejpam-5779	9	21	is	be	AUX
ejpam-5779	9	22	a	a	DET
ejpam-5779	9	23	generalization18	generalization18	NOUN
ejpam-5779	9	24	of	of	ADP
ejpam-5779	9	25	the	the	DET
ejpam-5779	9	26	nand	nand	NOUN
ejpam-5779	9	27	operation	operation	NOUN
ejpam-5779	9	28	in	in	ADP
ejpam-5779	9	29	propositional	propositional	ADJ
ejpam-5779	9	30	logic	logic	NOUN
ejpam-5779	9	31	.	.	PUNCT
ejpam-5779	10	1	the	the	DET
ejpam-5779	10	2	study	study	NOUN
ejpam-5779	10	3	of	of	ADP
ejpam-5779	10	4	hilbert	hilbert	PROPN
ejpam-5779	10	5	algebras	algebras	PROPN
ejpam-5779	10	6	is	be	AUX
ejpam-5779	10	7	integral19	integral19	NOUN
ejpam-5779	10	8	to	to	ADP
ejpam-5779	10	9	understanding	understand	VERB
ejpam-5779	10	10	non	non	ADJ
ejpam-5779	10	11	-	-	ADJ
ejpam-5779	10	12	classical	classical	ADJ
ejpam-5779	10	13	logics	logic	NOUN
ejpam-5779	10	14	,	,	PUNCT
ejpam-5779	10	15	modal	modal	NOUN
ejpam-5779	10	16	logics	logic	NOUN
ejpam-5779	10	17	,	,	PUNCT
ejpam-5779	10	18	and	and	CCONJ
ejpam-5779	10	19	lattice	lattice	PROPN
ejpam-5779	10	20	theory	theory	NOUN
ejpam-5779	10	21	,	,	PUNCT
ejpam-5779	10	22	offering	offer	VERB
ejpam-5779	10	23	essential20	essential20	NOUN
ejpam-5779	10	24	insights	insight	NOUN
ejpam-5779	10	25	into	into	ADP
ejpam-5779	10	26	the	the	DET
ejpam-5779	10	27	foundational	foundational	ADJ
ejpam-5779	10	28	structure	structure	NOUN
ejpam-5779	10	29	of	of	ADP
ejpam-5779	10	30	logical	logical	ADJ
ejpam-5779	10	31	systems	system	NOUN
ejpam-5779	10	32	[	[	X
ejpam-5779	10	33	3].21	3].21	NUM
ejpam-5779	10	34	the	the	DET
ejpam-5779	10	35	sheffer	sheffer	NOUN
ejpam-5779	10	36	stroke	stroke	NOUN
ejpam-5779	10	37	is	be	AUX
ejpam-5779	10	38	a	a	DET
ejpam-5779	10	39	fundamental	fundamental	ADJ
ejpam-5779	10	40	element	element	NOUN
ejpam-5779	10	41	in	in	ADP
ejpam-5779	10	42	both	both	CCONJ
ejpam-5779	10	43	classical	classical	ADJ
ejpam-5779	10	44	and	and	CCONJ
ejpam-5779	10	45	non	non	ADJ
ejpam-5779	10	46	-	-	ADJ
ejpam-5779	10	47	classical	classical	ADJ
ejpam-5779	10	48	logic22	logic22	NOUN
ejpam-5779	10	49	due	due	ADP
ejpam-5779	10	50	to	to	ADP
ejpam-5779	10	51	its	its	PRON
ejpam-5779	10	52	property	property	NOUN
ejpam-5779	10	53	as	as	ADP
ejpam-5779	10	54	a	a	DET
ejpam-5779	10	55	functionally	functionally	ADV
ejpam-5779	10	56	complete	complete	ADJ
ejpam-5779	10	57	operation	operation	NOUN
ejpam-5779	10	58	[	[	X
ejpam-5779	10	59	13	13	NUM
ejpam-5779	10	60	]	]	PUNCT
ejpam-5779	10	61	.	.	PUNCT
ejpam-5779	11	1	this	this	PRON
ejpam-5779	11	2	means	mean	VERB
ejpam-5779	11	3	it	it	PRON
ejpam-5779	11	4	can	can	AUX
ejpam-5779	11	5	operate	operate	VERB
ejpam-5779	11	6	by23	by23	PROPN
ejpam-5779	11	7	∗corresponding	∗corresponding	NOUN
ejpam-5779	11	8	author	author	NOUN
ejpam-5779	11	9	.	.	PUNCT
ejpam-5779	12	1	doi	doi	NOUN
ejpam-5779	12	2	:	:	PUNCT
ejpam-5779	12	3	https://doi.org/10.29020/nybg.ejpam.v18i1.5779	https://doi.org/10.29020/nybg.ejpam.v18i1.5779	NOUN
ejpam-5779	12	4	email	email	NOUN
ejpam-5779	12	5	addresses	address	VERB
ejpam-5779	12	6	:	:	PUNCT
ejpam-5779	12	7	nrajesh	nrajesh	PROPN
ejpam-5779	12	8	topology@yahoo.co.in	topology@yahoo.co.in	PROPN
ejpam-5779	12	9	(	(	PUNCT
ejpam-5779	12	10	n.	n.	PROPN
ejpam-5779	12	11	rajesh	rajesh	PROPN
ejpam-5779	12	12	)	)	PUNCT
ejpam-5779	12	13	,	,	PUNCT
ejpam-5779	12	14	tahsin.oner@ege.edu.tr	tahsin.oner@ege.edu.tr	NOUN
ejpam-5779	12	15	(	(	PUNCT
ejpam-5779	12	16	t.	t.	NOUN
ejpam-5779	12	17	oner	oner	PROPN
ejpam-5779	12	18	)	)	PUNCT
ejpam-5779	12	19	,	,	PUNCT
ejpam-5779	12	20	aiyared.ia@up.ac.th	aiyared.ia@up.ac.th	NOUN
ejpam-5779	12	21	(	(	PUNCT
ejpam-5779	12	22	a.	a.	NOUN
ejpam-5779	12	23	iampan	iampan	PROPN
ejpam-5779	12	24	)	)	PUNCT
ejpam-5779	12	25	,	,	PUNCT
ejpam-5779	12	26	ibrahim.senturk@ege.edu.tr	ibrahim.senturk@ege.edu.tr	PROPN
ejpam-5779	12	27	(	(	PUNCT
ejpam-5779	12	28	i.	i.	PROPN
ejpam-5779	12	29	senturk	senturk	PROPN
ejpam-5779	12	30	)	)	PUNCT
ejpam-5779	12	31	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5779	12	32	1	1	NUM
ejpam-5779	12	33	copyright	copyright	NOUN
ejpam-5779	12	34	:	:	PUNCT
ejpam-5779	13	1	©	©	PROPN
ejpam-5779	13	2	2025	2025	NUM
ejpam-5779	13	3	the	the	DET
ejpam-5779	13	4	author(s	author(s	NOUN
ejpam-5779	13	5	)	)	PUNCT
ejpam-5779	13	6	.	.	PUNCT
ejpam-5779	14	1	(	(	PUNCT
ejpam-5779	14	2	cc	cc	NOUN
ejpam-5779	14	3	by	by	ADP
ejpam-5779	14	4	-	-	PUNCT
ejpam-5779	14	5	nc	nc	PROPN
ejpam-5779	14	6	4.0	4.0	NUM
ejpam-5779	14	7	)	)	PUNCT
ejpam-5779	14	8	n.	n.	PROPN
ejpam-5779	14	9	rajesh	rajesh	PROPN
ejpam-5779	14	10	,	,	PUNCT
ejpam-5779	14	11	t.	t.	PROPN
ejpam-5779	14	12	oner	oner	NOUN
ejpam-5779	14	13	,	,	PUNCT
ejpam-5779	14	14	a.	a.	NOUN
ejpam-5779	14	15	iampan	iampan	PROPN
ejpam-5779	14	16	,	,	PUNCT
ejpam-5779	14	17	i.	i.	PROPN
ejpam-5779	14	18	senturk	senturk	PROPN
ejpam-5779	14	19	/	/	SYM
ejpam-5779	14	20	eur	eur	PROPN
ejpam-5779	14	21	.	.	PUNCT
ejpam-5779	15	1	j.	j.	PROPN
ejpam-5779	15	2	pure	pure	PROPN
ejpam-5779	15	3	appl	appl	PROPN
ejpam-5779	15	4	.	.	PROPN
ejpam-5779	15	5	math	math	PROPN
ejpam-5779	15	6	,	,	PUNCT
ejpam-5779	15	7	18	18	NUM
ejpam-5779	15	8	(	(	PUNCT
ejpam-5779	15	9	1	1	NUM
ejpam-5779	15	10	)	)	PUNCT
ejpam-5779	15	11	(	(	PUNCT
ejpam-5779	15	12	2025	2025	NUM
ejpam-5779	15	13	)	)	PUNCT
ejpam-5779	15	14	,	,	PUNCT
ejpam-5779	15	15	5779	5779	NUM
ejpam-5779	15	16	2	2	NUM
ejpam-5779	15	17	of	of	ADP
ejpam-5779	15	18	18	18	NUM
ejpam-5779	15	19	itself	itself	PRON
ejpam-5779	15	20	without	without	ADP
ejpam-5779	15	21	requiring	require	VERB
ejpam-5779	15	22	any	any	DET
ejpam-5779	15	23	other	other	ADJ
ejpam-5779	15	24	logical	logical	ADJ
ejpam-5779	15	25	operators	operator	NOUN
ejpam-5779	15	26	to	to	PART
ejpam-5779	15	27	form	form	VERB
ejpam-5779	15	28	a	a	DET
ejpam-5779	15	29	comprehensive	comprehensive	ADJ
ejpam-5779	15	30	logical	logical	ADJ
ejpam-5779	15	31	system.24	system.24	NOUN
ejpam-5779	15	32	in	in	ADP
ejpam-5779	15	33	simpler	simple	ADJ
ejpam-5779	15	34	terms	term	NOUN
ejpam-5779	15	35	,	,	PUNCT
ejpam-5779	15	36	every	every	DET
ejpam-5779	15	37	logical	logical	ADJ
ejpam-5779	15	38	axiom	axiom	NOUN
ejpam-5779	15	39	can	can	AUX
ejpam-5779	15	40	be	be	AUX
ejpam-5779	15	41	restated	restate	VERB
ejpam-5779	15	42	using	use	VERB
ejpam-5779	15	43	just	just	ADV
ejpam-5779	15	44	the	the	DET
ejpam-5779	15	45	sheffer	sheffer	NOUN
ejpam-5779	15	46	stroke	stroke	NOUN
ejpam-5779	15	47	.	.	PUNCT
ejpam-5779	16	1	this25	this25	NOUN
ejpam-5779	16	2	capability	capability	NOUN
ejpam-5779	16	3	greatly	greatly	ADV
ejpam-5779	16	4	simplifies	simplify	VERB
ejpam-5779	16	5	the	the	DET
ejpam-5779	16	6	manipulation	manipulation	NOUN
ejpam-5779	16	7	and	and	CCONJ
ejpam-5779	16	8	control	control	NOUN
ejpam-5779	16	9	of	of	ADP
ejpam-5779	16	10	various	various	ADJ
ejpam-5779	16	11	properties	property	NOUN
ejpam-5779	16	12	within26	within26	VERB
ejpam-5779	16	13	the	the	DET
ejpam-5779	16	14	logical	logical	ADJ
ejpam-5779	16	15	system	system	NOUN
ejpam-5779	16	16	it	it	PRON
ejpam-5779	16	17	creates	create	VERB
ejpam-5779	16	18	.	.	PUNCT
ejpam-5779	17	1	moreover	moreover	ADV
ejpam-5779	17	2	,	,	PUNCT
ejpam-5779	17	3	it	it	PRON
ejpam-5779	17	4	is	be	AUX
ejpam-5779	17	5	noteworthy	noteworthy	ADJ
ejpam-5779	17	6	that	that	SCONJ
ejpam-5779	17	7	the	the	DET
ejpam-5779	17	8	axioms	axiom	NOUN
ejpam-5779	17	9	of	of	ADP
ejpam-5779	17	10	boolean27	boolean27	PROPN
ejpam-5779	17	11	algebra	algebra	PROPN
ejpam-5779	17	12	,	,	PUNCT
ejpam-5779	17	13	which	which	PRON
ejpam-5779	17	14	correspond	correspond	VERB
ejpam-5779	17	15	to	to	ADP
ejpam-5779	17	16	classical	classical	ADJ
ejpam-5779	17	17	propositional	propositional	ADJ
ejpam-5779	17	18	logic	logic	NOUN
ejpam-5779	17	19	,	,	PUNCT
ejpam-5779	17	20	can	can	AUX
ejpam-5779	17	21	be	be	AUX
ejpam-5779	17	22	entirely	entirely	ADV
ejpam-5779	17	23	represented28	represented28	NOUN
ejpam-5779	17	24	using	use	VERB
ejpam-5779	17	25	the	the	DET
ejpam-5779	17	26	sheffer	sheffer	NOUN
ejpam-5779	17	27	stroke	stroke	NOUN
ejpam-5779	17	28	.	.	PUNCT
ejpam-5779	18	1	this	this	DET
ejpam-5779	18	2	highlights	highlight	VERB
ejpam-5779	18	3	the	the	DET
ejpam-5779	18	4	sheffer	sheffer	PROPN
ejpam-5779	18	5	stroke	stroke	NOUN
ejpam-5779	18	6	’s	’s	PART
ejpam-5779	18	7	foundational	foundational	ADJ
ejpam-5779	18	8	importance	importance	NOUN
ejpam-5779	18	9	and29	and29	VERB
ejpam-5779	18	10	its	its	PRON
ejpam-5779	18	11	versatility	versatility	NOUN
ejpam-5779	18	12	within	within	ADP
ejpam-5779	18	13	both	both	CCONJ
ejpam-5779	18	14	logical	logical	ADJ
ejpam-5779	18	15	and	and	CCONJ
ejpam-5779	18	16	algebraic	algebraic	PROPN
ejpam-5779	18	17	systems.30	systems.30	VERB
ejpam-5779	18	18	the	the	DET
ejpam-5779	18	19	sheffer	sheffer	NOUN
ejpam-5779	18	20	stroke	stroke	NOUN
ejpam-5779	18	21	has	have	AUX
ejpam-5779	18	22	been	be	AUX
ejpam-5779	18	23	utilized	utilize	VERB
ejpam-5779	18	24	in	in	ADP
ejpam-5779	18	25	various	various	ADJ
ejpam-5779	18	26	algebraic	algebraic	ADJ
ejpam-5779	18	27	structures	structure	NOUN
ejpam-5779	18	28	,	,	PUNCT
ejpam-5779	18	29	such	such	ADJ
ejpam-5779	18	30	as	as	ADP
ejpam-5779	18	31	boolean31	boolean31	NOUN
ejpam-5779	18	32	algebras	algebra	NOUN
ejpam-5779	18	33	,	,	PUNCT
ejpam-5779	18	34	basic	basic	ADJ
ejpam-5779	18	35	algebras	algebra	NOUN
ejpam-5779	18	36	,	,	PUNCT
ejpam-5779	18	37	mv	mv	PROPN
ejpam-5779	18	38	-	-	PUNCT
ejpam-5779	18	39	algebras	algebra	NOUN
ejpam-5779	18	40	,	,	PUNCT
ejpam-5779	18	41	bck	bck	NOUN
ejpam-5779	18	42	-	-	PUNCT
ejpam-5779	18	43	algebras	algebra	NOUN
ejpam-5779	18	44	,	,	PUNCT
ejpam-5779	18	45	mtl	mtl	PROPN
ejpam-5779	18	46	-	-	PUNCT
ejpam-5779	18	47	algebras	algebras	PROPN
ejpam-5779	18	48	and	and	CCONJ
ejpam-5779	18	49	ortholattices,32	ortholattices,32	VERB
ejpam-5779	18	50	among	among	ADP
ejpam-5779	18	51	others	other	NOUN
ejpam-5779	18	52	,	,	PUNCT
ejpam-5779	18	53	and	and	CCONJ
ejpam-5779	18	54	is	be	AUX
ejpam-5779	18	55	also	also	ADV
ejpam-5779	18	56	explored	explore	VERB
ejpam-5779	18	57	within	within	ADP
ejpam-5779	18	58	fuzzy	fuzzy	ADJ
ejpam-5779	18	59	contexts	contexts	NOUN
ejpam-5779	18	60	(	(	PUNCT
ejpam-5779	18	61	see	see	VERB
ejpam-5779	18	62	[	[	X
ejpam-5779	18	63	1	1	NUM
ejpam-5779	18	64	,	,	PUNCT
ejpam-5779	18	65	4–7	4–7	NOUN
ejpam-5779	18	66	,	,	PUNCT
ejpam-5779	18	67	9–12	9–12	PROPN
ejpam-5779	18	68	]	]	PUNCT
ejpam-5779	18	69	)	)	PUNCT
ejpam-5779	18	70	.	.	PUNCT
ejpam-5779	19	1	in	in	ADP
ejpam-5779	19	2	2021,33	2021,33	NUM
ejpam-5779	19	3	oner	oner	NOUN
ejpam-5779	19	4	et	et	PROPN
ejpam-5779	19	5	al	al	PROPN
ejpam-5779	19	6	.	.	PUNCT
ejpam-5779	20	1	[	[	X
ejpam-5779	20	2	6	6	NUM
ejpam-5779	20	3	]	]	PUNCT
ejpam-5779	20	4	extended	extend	VERB
ejpam-5779	20	5	the	the	DET
ejpam-5779	20	6	sheffer	sheffer	NOUN
ejpam-5779	20	7	stroke	stroke	NOUN
ejpam-5779	20	8	to	to	ADP
ejpam-5779	20	9	hilbert	hilbert	PROPN
ejpam-5779	20	10	algebras	algebras	PROPN
ejpam-5779	20	11	,	,	PUNCT
ejpam-5779	20	12	defining	define	VERB
ejpam-5779	20	13	the	the	DET
ejpam-5779	20	14	sheffer	sheffer	NOUN
ejpam-5779	20	15	stroke34	stroke34	PROPN
ejpam-5779	20	16	hilbert	hilbert	PROPN
ejpam-5779	20	17	algebra	algebra	PROPN
ejpam-5779	20	18	and	and	CCONJ
ejpam-5779	20	19	studying	study	VERB
ejpam-5779	20	20	its	its	PRON
ejpam-5779	20	21	various	various	ADJ
ejpam-5779	20	22	properties	property	NOUN
ejpam-5779	20	23	.	.	PUNCT
ejpam-5779	21	1	in	in	ADP
ejpam-5779	21	2	[	[	X
ejpam-5779	21	3	5	5	NUM
ejpam-5779	21	4	]	]	PUNCT
ejpam-5779	21	5	,	,	PUNCT
ejpam-5779	21	6	they	they	PRON
ejpam-5779	21	7	introduced	introduce	VERB
ejpam-5779	21	8	the	the	DET
ejpam-5779	21	9	concepts35	concepts35	NOUN
ejpam-5779	21	10	of	of	ADP
ejpam-5779	21	11	a	a	DET
ejpam-5779	21	12	deductive	deductive	ADJ
ejpam-5779	21	13	system	system	NOUN
ejpam-5779	21	14	and	and	CCONJ
ejpam-5779	21	15	filter	filter	NOUN
ejpam-5779	21	16	for	for	ADP
ejpam-5779	21	17	sheffer	sheffer	PROPN
ejpam-5779	21	18	stroke	stroke	PROPN
ejpam-5779	21	19	hilbert	hilbert	PROPN
ejpam-5779	21	20	algebras	algebras	PROPN
ejpam-5779	21	21	and	and	CCONJ
ejpam-5779	21	22	explored	explore	VERB
ejpam-5779	21	23	their36	their36	NOUN
ejpam-5779	21	24	fuzzification	fuzzification	NOUN
ejpam-5779	21	25	.	.	PUNCT
ejpam-5779	22	1	additionally	additionally	ADV
ejpam-5779	22	2	,	,	PUNCT
ejpam-5779	22	3	oner	oner	NOUN
ejpam-5779	22	4	et	et	PROPN
ejpam-5779	22	5	al	al	PROPN
ejpam-5779	22	6	.	.	PUNCT
ejpam-5779	23	1	[	[	X
ejpam-5779	23	2	6	6	NUM
ejpam-5779	23	3	]	]	PUNCT
ejpam-5779	23	4	presented	present	VERB
ejpam-5779	23	5	the	the	DET
ejpam-5779	23	6	idea	idea	NOUN
ejpam-5779	23	7	of	of	ADP
ejpam-5779	23	8	an	an	DET
ejpam-5779	23	9	ideal	ideal	NOUN
ejpam-5779	23	10	in	in	ADP
ejpam-5779	23	11	sheffer	sheffer	NOUN
ejpam-5779	23	12	stroke37	stroke37	NOUN
ejpam-5779	23	13	hilbert	hilbert	PROPN
ejpam-5779	23	14	algebras	algebras	PROPN
ejpam-5779	23	15	and	and	CCONJ
ejpam-5779	23	16	analyzed	analyze	VERB
ejpam-5779	23	17	its	its	PRON
ejpam-5779	23	18	properties.38	properties.38	NOUN
ejpam-5779	23	19	the	the	DET
ejpam-5779	23	20	field	field	NOUN
ejpam-5779	23	21	of	of	ADP
ejpam-5779	23	22	fuzzy	fuzzy	ADJ
ejpam-5779	23	23	logic	logic	NOUN
ejpam-5779	23	24	,	,	PUNCT
ejpam-5779	23	25	introduced	introduce	VERB
ejpam-5779	23	26	by	by	ADP
ejpam-5779	23	27	[	[	X
ejpam-5779	23	28	15	15	NUM
ejpam-5779	23	29	]	]	PUNCT
ejpam-5779	23	30	,	,	PUNCT
ejpam-5779	23	31	broadens	broaden	VERB
ejpam-5779	23	32	classical	classical	ADJ
ejpam-5779	23	33	logic	logic	NOUN
ejpam-5779	23	34	by	by	ADP
ejpam-5779	23	35	incorporating39	incorporating39	ADJ
ejpam-5779	23	36	truth	truth	NOUN
ejpam-5779	23	37	values	value	NOUN
ejpam-5779	23	38	that	that	PRON
ejpam-5779	23	39	range	range	VERB
ejpam-5779	23	40	continuously	continuously	ADV
ejpam-5779	23	41	between	between	ADP
ejpam-5779	23	42	0	0	NUM
ejpam-5779	23	43	and	and	CCONJ
ejpam-5779	23	44	1	1	NUM
ejpam-5779	23	45	,	,	PUNCT
ejpam-5779	23	46	rather	rather	ADV
ejpam-5779	23	47	than	than	ADP
ejpam-5779	23	48	being	be	AUX
ejpam-5779	23	49	restricted	restrict	VERB
ejpam-5779	23	50	to40	to40	PROPN
ejpam-5779	23	51	binary	binary	ADJ
ejpam-5779	23	52	true	true	ADJ
ejpam-5779	23	53	/	/	SYM
ejpam-5779	23	54	false	false	ADJ
ejpam-5779	23	55	values	value	NOUN
ejpam-5779	23	56	.	.	PUNCT
ejpam-5779	24	1	this	this	DET
ejpam-5779	24	2	flexibility	flexibility	NOUN
ejpam-5779	24	3	makes	make	VERB
ejpam-5779	24	4	fuzzy	fuzzy	ADJ
ejpam-5779	24	5	logic	logic	NOUN
ejpam-5779	24	6	particularly	particularly	ADV
ejpam-5779	24	7	useful	useful	ADJ
ejpam-5779	24	8	in	in	ADP
ejpam-5779	24	9	scenarios41	scenarios41	NOUN
ejpam-5779	24	10	involving	involve	VERB
ejpam-5779	24	11	uncertainty	uncertainty	NOUN
ejpam-5779	24	12	and	and	CCONJ
ejpam-5779	24	13	imprecision	imprecision	NOUN
ejpam-5779	24	14	.	.	PUNCT
ejpam-5779	25	1	integrating	integrate	VERB
ejpam-5779	25	2	fuzzy	fuzzy	ADJ
ejpam-5779	25	3	logic	logic	NOUN
ejpam-5779	25	4	with	with	ADP
ejpam-5779	25	5	hilbert	hilbert	PROPN
ejpam-5779	25	6	algebras	algebras	PROPN
ejpam-5779	25	7	results42	results42	NOUN
ejpam-5779	25	8	in	in	ADP
ejpam-5779	25	9	the	the	DET
ejpam-5779	25	10	concept	concept	NOUN
ejpam-5779	25	11	of	of	ADP
ejpam-5779	25	12	fuzzy	fuzzy	ADJ
ejpam-5779	25	13	ideals	ideal	NOUN
ejpam-5779	25	14	,	,	PUNCT
ejpam-5779	25	15	where	where	SCONJ
ejpam-5779	25	16	the	the	DET
ejpam-5779	25	17	elements	element	NOUN
ejpam-5779	25	18	of	of	ADP
ejpam-5779	25	19	an	an	DET
ejpam-5779	25	20	ideal	ideal	NOUN
ejpam-5779	25	21	can	can	AUX
ejpam-5779	25	22	have	have	VERB
ejpam-5779	25	23	varying	vary	VERB
ejpam-5779	25	24	degrees	degree	NOUN
ejpam-5779	25	25	of43	of43	PROPN
ejpam-5779	25	26	membership	membership	NOUN
ejpam-5779	25	27	rather	rather	ADV
ejpam-5779	25	28	than	than	ADP
ejpam-5779	25	29	being	be	AUX
ejpam-5779	25	30	limited	limit	VERB
ejpam-5779	25	31	to	to	ADP
ejpam-5779	25	32	crisp	crisp	ADJ
ejpam-5779	25	33	values	value	NOUN
ejpam-5779	25	34	.	.	PUNCT
ejpam-5779	26	1	this	this	DET
ejpam-5779	26	2	extension	extension	NOUN
ejpam-5779	26	3	offers	offer	VERB
ejpam-5779	26	4	a	a	DET
ejpam-5779	26	5	more	more	ADJ
ejpam-5779	26	6	refined44	refined44	NOUN
ejpam-5779	26	7	approach	approach	NOUN
ejpam-5779	26	8	to	to	ADP
ejpam-5779	26	9	analyzing	analyze	VERB
ejpam-5779	26	10	the	the	DET
ejpam-5779	26	11	algebraic	algebraic	ADJ
ejpam-5779	26	12	properties	property	NOUN
ejpam-5779	26	13	of	of	ADP
ejpam-5779	26	14	hilbert	hilbert	PROPN
ejpam-5779	26	15	algebras	algebras	PROPN
ejpam-5779	27	1	[	[	X
ejpam-5779	27	2	2].45	2].45	NUM
ejpam-5779	27	3	a	a	DET
ejpam-5779	27	4	recent	recent	ADJ
ejpam-5779	27	5	innovation	innovation	NOUN
ejpam-5779	27	6	in	in	ADP
ejpam-5779	27	7	the	the	DET
ejpam-5779	27	8	theory	theory	NOUN
ejpam-5779	27	9	of	of	ADP
ejpam-5779	27	10	fuzzy	fuzzy	ADJ
ejpam-5779	27	11	ideals	ideal	NOUN
ejpam-5779	27	12	is	be	AUX
ejpam-5779	27	13	the	the	DET
ejpam-5779	27	14	introduction	introduction	NOUN
ejpam-5779	27	15	of	of	ADP
ejpam-5779	27	16	length	length	NOUN
ejpam-5779	27	17	-	-	PUNCT
ejpam-5779	27	18	fuzzy46	fuzzy46	NOUN
ejpam-5779	27	19	ideals	ideal	NOUN
ejpam-5779	27	20	.	.	PUNCT
ejpam-5779	28	1	this	this	DET
ejpam-5779	28	2	concept	concept	NOUN
ejpam-5779	28	3	enhances	enhance	VERB
ejpam-5779	28	4	the	the	DET
ejpam-5779	28	5	classical	classical	ADJ
ejpam-5779	28	6	definition	definition	NOUN
ejpam-5779	28	7	of	of	ADP
ejpam-5779	28	8	an	an	DET
ejpam-5779	28	9	ideal	ideal	NOUN
ejpam-5779	28	10	in	in	ADP
ejpam-5779	28	11	sheffer	sheffer	PROPN
ejpam-5779	28	12	stroke	stroke	PROPN
ejpam-5779	28	13	hilbert	hilbert	PROPN
ejpam-5779	28	14	al-47	al-47	PROPN
ejpam-5779	28	15	gebras	gebras	PROPN
ejpam-5779	28	16	by	by	ADP
ejpam-5779	28	17	associating	associate	VERB
ejpam-5779	28	18	a	a	DET
ejpam-5779	28	19	fuzzy	fuzzy	ADJ
ejpam-5779	28	20	function	function	NOUN
ejpam-5779	28	21	that	that	PRON
ejpam-5779	28	22	measures	measure	VERB
ejpam-5779	28	23	the	the	DET
ejpam-5779	28	24	“	"	PUNCT
ejpam-5779	28	25	length	length	NOUN
ejpam-5779	28	26	”	"	PUNCT
ejpam-5779	28	27	or	or	CCONJ
ejpam-5779	28	28	degree	degree	NOUN
ejpam-5779	28	29	of	of	ADP
ejpam-5779	28	30	membership48	membership48	NOUN
ejpam-5779	28	31	of	of	ADP
ejpam-5779	28	32	elements	element	NOUN
ejpam-5779	28	33	within	within	ADP
ejpam-5779	28	34	an	an	DET
ejpam-5779	28	35	ideal	ideal	NOUN
ejpam-5779	28	36	.	.	PUNCT
ejpam-5779	29	1	this	this	DET
ejpam-5779	29	2	new	new	ADJ
ejpam-5779	29	3	perspective	perspective	NOUN
ejpam-5779	29	4	provides	provide	VERB
ejpam-5779	29	5	a	a	DET
ejpam-5779	29	6	more	more	ADV
ejpam-5779	29	7	nuanced	nuanced	ADJ
ejpam-5779	29	8	understanding49	understanding49	NOUN
ejpam-5779	29	9	of	of	ADP
ejpam-5779	29	10	the	the	DET
ejpam-5779	29	11	structure	structure	NOUN
ejpam-5779	29	12	and	and	CCONJ
ejpam-5779	29	13	behavior	behavior	NOUN
ejpam-5779	29	14	of	of	ADP
ejpam-5779	29	15	these	these	DET
ejpam-5779	29	16	algebras	algebra	NOUN
ejpam-5779	29	17	,	,	PUNCT
ejpam-5779	29	18	enriching	enrich	VERB
ejpam-5779	29	19	the	the	DET
ejpam-5779	29	20	classical	classical	ADJ
ejpam-5779	29	21	theory	theory	NOUN
ejpam-5779	29	22	with	with	ADP
ejpam-5779	29	23	elements50	elements50	NOUN
ejpam-5779	29	24	of	of	ADP
ejpam-5779	29	25	fuzzy	fuzzy	ADJ
ejpam-5779	29	26	logic	logic	NOUN
ejpam-5779	29	27	[	[	X
ejpam-5779	29	28	8	8	NUM
ejpam-5779	29	29	]	]	PUNCT
ejpam-5779	29	30	.	.	PUNCT
ejpam-5779	30	1	the	the	DET
ejpam-5779	30	2	application	application	NOUN
ejpam-5779	30	3	of	of	ADP
ejpam-5779	30	4	length	length	NOUN
ejpam-5779	30	5	-	-	PUNCT
ejpam-5779	30	6	fuzzy	fuzzy	ADJ
ejpam-5779	30	7	ideals	ideal	NOUN
ejpam-5779	30	8	allows	allow	VERB
ejpam-5779	30	9	for	for	ADP
ejpam-5779	30	10	a	a	DET
ejpam-5779	30	11	more	more	ADV
ejpam-5779	30	12	refined	refined	ADJ
ejpam-5779	30	13	analysis51	analysis51	NOUN
ejpam-5779	30	14	of	of	ADP
ejpam-5779	30	15	ideals	ideal	NOUN
ejpam-5779	30	16	with	with	ADP
ejpam-5779	30	17	fuzzy	fuzzy	ADJ
ejpam-5779	30	18	characteristics	characteristic	NOUN
ejpam-5779	30	19	,	,	PUNCT
ejpam-5779	30	20	enabling	enable	VERB
ejpam-5779	30	21	better	well	ADJ
ejpam-5779	30	22	modeling	modeling	NOUN
ejpam-5779	30	23	of	of	ADP
ejpam-5779	30	24	systems	system	NOUN
ejpam-5779	30	25	with	with	ADP
ejpam-5779	30	26	inherent52	inherent52	NOUN
ejpam-5779	30	27	uncertainty	uncertainty	NOUN
ejpam-5779	30	28	or	or	CCONJ
ejpam-5779	30	29	imprecision	imprecision	NOUN
ejpam-5779	30	30	.	.	PUNCT
ejpam-5779	31	1	by	by	ADP
ejpam-5779	31	2	using	use	VERB
ejpam-5779	31	3	fuzzy	fuzzy	ADJ
ejpam-5779	31	4	functions	function	NOUN
ejpam-5779	31	5	to	to	PART
ejpam-5779	31	6	measure	measure	VERB
ejpam-5779	31	7	membership	membership	NOUN
ejpam-5779	31	8	degrees	degree	NOUN
ejpam-5779	31	9	,	,	PUNCT
ejpam-5779	31	10	this53	this53	NOUN
ejpam-5779	31	11	approach	approach	NOUN
ejpam-5779	31	12	is	be	AUX
ejpam-5779	31	13	applicable	applicable	ADJ
ejpam-5779	31	14	in	in	ADP
ejpam-5779	31	15	decision	decision	NOUN
ejpam-5779	31	16	-	-	PUNCT
ejpam-5779	31	17	making	make	VERB
ejpam-5779	31	18	processes	process	NOUN
ejpam-5779	31	19	under	under	ADP
ejpam-5779	31	20	ambiguity	ambiguity	NOUN
ejpam-5779	31	21	,	,	PUNCT
ejpam-5779	31	22	the	the	DET
ejpam-5779	31	23	design	design	NOUN
ejpam-5779	31	24	of	of	ADP
ejpam-5779	31	25	algo-54	algo-54	PROPN
ejpam-5779	31	26	rithms	rithm	NOUN
ejpam-5779	31	27	for	for	ADP
ejpam-5779	31	28	complex	complex	ADJ
ejpam-5779	31	29	computations	computation	NOUN
ejpam-5779	31	30	,	,	PUNCT
ejpam-5779	31	31	and	and	CCONJ
ejpam-5779	31	32	the	the	DET
ejpam-5779	31	33	study	study	NOUN
ejpam-5779	31	34	of	of	ADP
ejpam-5779	31	35	structures	structure	NOUN
ejpam-5779	31	36	in	in	ADP
ejpam-5779	31	37	systems	system	NOUN
ejpam-5779	31	38	with	with	ADP
ejpam-5779	31	39	incomplete55	incomplete55	ADJ
ejpam-5779	31	40	or	or	CCONJ
ejpam-5779	31	41	vague	vague	ADJ
ejpam-5779	31	42	data	datum	NOUN
ejpam-5779	31	43	.	.	PUNCT
ejpam-5779	32	1	integrating	integrate	VERB
ejpam-5779	32	2	fuzzy	fuzzy	ADJ
ejpam-5779	32	3	logic	logic	NOUN
ejpam-5779	32	4	into	into	ADP
ejpam-5779	32	5	classical	classical	ADJ
ejpam-5779	32	6	theory	theory	NOUN
ejpam-5779	32	7	not	not	PART
ejpam-5779	32	8	only	only	ADV
ejpam-5779	32	9	deepens	deepen	VERB
ejpam-5779	32	10	its	its	PRON
ejpam-5779	32	11	theoretical56	theoretical56	ADJ
ejpam-5779	32	12	base	base	NOUN
ejpam-5779	32	13	but	but	CCONJ
ejpam-5779	32	14	also	also	ADV
ejpam-5779	32	15	extends	extend	VERB
ejpam-5779	32	16	its	its	PRON
ejpam-5779	32	17	applicability	applicability	NOUN
ejpam-5779	32	18	to	to	ADP
ejpam-5779	32	19	fields	field	NOUN
ejpam-5779	32	20	such	such	ADJ
ejpam-5779	32	21	as	as	ADP
ejpam-5779	32	22	computer	computer	NOUN
ejpam-5779	32	23	science	science	NOUN
ejpam-5779	32	24	,	,	PUNCT
ejpam-5779	32	25	engineering	engineering	NOUN
ejpam-5779	32	26	,	,	PUNCT
ejpam-5779	32	27	and57	and57	ADJ
ejpam-5779	32	28	areas	area	NOUN
ejpam-5779	32	29	involving	involve	VERB
ejpam-5779	32	30	uncertain	uncertain	ADJ
ejpam-5779	32	31	or	or	CCONJ
ejpam-5779	32	32	imprecise	imprecise	ADJ
ejpam-5779	32	33	information	information	NOUN
ejpam-5779	32	34	processing.58	processing.58	VERB
ejpam-5779	32	35	this	this	DET
ejpam-5779	32	36	paper	paper	NOUN
ejpam-5779	32	37	examines	examine	VERB
ejpam-5779	32	38	the	the	DET
ejpam-5779	32	39	properties	property	NOUN
ejpam-5779	32	40	and	and	CCONJ
ejpam-5779	32	41	characteristics	characteristic	NOUN
ejpam-5779	32	42	of	of	ADP
ejpam-5779	32	43	length	length	NOUN
ejpam-5779	32	44	-	-	PUNCT
ejpam-5779	32	45	fuzzy	fuzzy	ADJ
ejpam-5779	32	46	ideals	ideal	NOUN
ejpam-5779	32	47	in	in	ADP
ejpam-5779	32	48	sheffer59	sheffer59	PROPN
ejpam-5779	32	49	stroke	stroke	PROPN
ejpam-5779	32	50	hilbert	hilbert	PROPN
ejpam-5779	32	51	algebras	algebras	PROPN
ejpam-5779	32	52	.	.	PUNCT
ejpam-5779	33	1	by	by	ADP
ejpam-5779	33	2	investigating	investigate	VERB
ejpam-5779	33	3	these	these	DET
ejpam-5779	33	4	properties	property	NOUN
ejpam-5779	33	5	,	,	PUNCT
ejpam-5779	33	6	the	the	DET
ejpam-5779	33	7	goal	goal	NOUN
ejpam-5779	33	8	is	be	AUX
ejpam-5779	33	9	to	to	PART
ejpam-5779	33	10	provide	provide	VERB
ejpam-5779	33	11	fresh	fresh	ADJ
ejpam-5779	33	12	per-60	per-60	NOUN
ejpam-5779	33	13	spectives	spective	VERB
ejpam-5779	33	14	on	on	ADP
ejpam-5779	33	15	the	the	DET
ejpam-5779	33	16	theoretical	theoretical	ADJ
ejpam-5779	33	17	foundation	foundation	NOUN
ejpam-5779	33	18	of	of	ADP
ejpam-5779	33	19	hilbert	hilbert	PROPN
ejpam-5779	33	20	algebras	algebras	PROPN
ejpam-5779	33	21	and	and	CCONJ
ejpam-5779	33	22	their	their	PRON
ejpam-5779	33	23	potential	potential	ADJ
ejpam-5779	33	24	applications61	applications61	NOUN
ejpam-5779	33	25	in	in	ADP
ejpam-5779	33	26	fields	field	NOUN
ejpam-5779	33	27	such	such	ADJ
ejpam-5779	33	28	as	as	ADP
ejpam-5779	33	29	logic	logic	NOUN
ejpam-5779	33	30	,	,	PUNCT
ejpam-5779	33	31	computer	computer	NOUN
ejpam-5779	33	32	science	science	NOUN
ejpam-5779	33	33	,	,	PUNCT
ejpam-5779	33	34	and	and	CCONJ
ejpam-5779	33	35	beyond	beyond	ADP
ejpam-5779	33	36	.	.	PUNCT
ejpam-5779	34	1	the	the	DET
ejpam-5779	34	2	concepts	concept	NOUN
ejpam-5779	34	3	of	of	ADP
ejpam-5779	34	4	length	length	NOUN
ejpam-5779	34	5	fuzzy	fuzzy	ADJ
ejpam-5779	34	6	ideals62	ideals62	NOUN
ejpam-5779	34	7	and	and	CCONJ
ejpam-5779	34	8	mean	mean	VERB
ejpam-5779	34	9	fuzzy	fuzzy	ADJ
ejpam-5779	34	10	ideals	ideal	NOUN
ejpam-5779	34	11	are	be	AUX
ejpam-5779	34	12	introduced	introduce	VERB
ejpam-5779	34	13	in	in	ADP
ejpam-5779	34	14	the	the	DET
ejpam-5779	34	15	context	context	NOUN
ejpam-5779	34	16	of	of	ADP
ejpam-5779	34	17	sheffer	sheffer	PROPN
ejpam-5779	34	18	stroke	stroke	PROPN
ejpam-5779	34	19	hilbert	hilbert	PROPN
ejpam-5779	34	20	algebras	algebras	PROPN
ejpam-5779	34	21	,	,	PUNCT
ejpam-5779	34	22	and63	and63	VERB
ejpam-5779	34	23	their	their	PRON
ejpam-5779	34	24	properties	property	NOUN
ejpam-5779	34	25	are	be	AUX
ejpam-5779	34	26	analyzed	analyze	VERB
ejpam-5779	34	27	.	.	PUNCT
ejpam-5779	35	1	the	the	DET
ejpam-5779	35	2	paper	paper	NOUN
ejpam-5779	35	3	further	far	ADV
ejpam-5779	35	4	explores	explore	VERB
ejpam-5779	35	5	the	the	DET
ejpam-5779	35	6	relationships	relationship	NOUN
ejpam-5779	35	7	between	between	ADP
ejpam-5779	35	8	length64	length64	VERB
ejpam-5779	35	9	fuzzy	fuzzy	ADJ
ejpam-5779	35	10	ideals	ideal	NOUN
ejpam-5779	35	11	(	(	PUNCT
ejpam-5779	35	12	and	and	CCONJ
ejpam-5779	35	13	mean	mean	VERB
ejpam-5779	35	14	fuzzy	fuzzy	ADJ
ejpam-5779	35	15	ideals	ideal	NOUN
ejpam-5779	35	16	)	)	PUNCT
ejpam-5779	35	17	and	and	CCONJ
ejpam-5779	35	18	traditional	traditional	ADJ
ejpam-5779	35	19	ideals	ideal	NOUN
ejpam-5779	35	20	.	.	PUNCT
ejpam-5779	36	1	additionally	additionally	ADV
ejpam-5779	36	2	,	,	PUNCT
ejpam-5779	36	3	it	it	PRON
ejpam-5779	36	4	discusses	discuss	VERB
ejpam-5779	36	5	how65	how65	ADV
ejpam-5779	36	6	length	length	NOUN
ejpam-5779	36	7	fuzzy	fuzzy	ADJ
ejpam-5779	36	8	ideals	ideal	NOUN
ejpam-5779	36	9	(	(	PUNCT
ejpam-5779	36	10	and	and	CCONJ
ejpam-5779	36	11	mean	mean	VERB
ejpam-5779	36	12	fuzzy	fuzzy	ADJ
ejpam-5779	36	13	ideals	ideal	NOUN
ejpam-5779	36	14	)	)	PUNCT
ejpam-5779	36	15	are	be	AUX
ejpam-5779	36	16	related	relate	VERB
ejpam-5779	36	17	to	to	ADP
ejpam-5779	36	18	upper	upper	ADJ
ejpam-5779	36	19	and	and	CCONJ
ejpam-5779	36	20	lower	low	ADJ
ejpam-5779	36	21	level	level	NOUN
ejpam-5779	36	22	subsets66	subsets66	NOUN
ejpam-5779	36	23	n.	n.	PROPN
ejpam-5779	36	24	rajesh	rajesh	PROPN
ejpam-5779	36	25	,	,	PUNCT
ejpam-5779	36	26	t.	t.	PROPN
ejpam-5779	36	27	oner	oner	NOUN
ejpam-5779	36	28	,	,	PUNCT
ejpam-5779	36	29	a.	a.	NOUN
ejpam-5779	36	30	iampan	iampan	PROPN
ejpam-5779	36	31	,	,	PUNCT
ejpam-5779	36	32	i.	i.	PROPN
ejpam-5779	36	33	senturk	senturk	PROPN
ejpam-5779	36	34	/	/	SYM
ejpam-5779	36	35	eur	eur	PROPN
ejpam-5779	36	36	.	.	PUNCT
ejpam-5779	37	1	j.	j.	PROPN
ejpam-5779	37	2	pure	pure	PROPN
ejpam-5779	37	3	appl	appl	PROPN
ejpam-5779	37	4	.	.	PROPN
ejpam-5779	37	5	math	math	PROPN
ejpam-5779	37	6	,	,	PUNCT
ejpam-5779	37	7	18	18	NUM
ejpam-5779	37	8	(	(	PUNCT
ejpam-5779	37	9	1	1	NUM
ejpam-5779	37	10	)	)	PUNCT
ejpam-5779	37	11	(	(	PUNCT
ejpam-5779	37	12	2025	2025	NUM
ejpam-5779	37	13	)	)	PUNCT
ejpam-5779	37	14	,	,	PUNCT
ejpam-5779	37	15	5779	5779	NUM
ejpam-5779	37	16	3	3	NUM
ejpam-5779	37	17	of	of	ADP
ejpam-5779	37	18	18	18	NUM
ejpam-5779	37	19	based	base	VERB
ejpam-5779	37	20	on	on	ADP
ejpam-5779	37	21	the	the	DET
ejpam-5779	37	22	length	length	NOUN
ejpam-5779	37	23	(	(	PUNCT
ejpam-5779	37	24	or	or	CCONJ
ejpam-5779	37	25	mean	mean	VERB
ejpam-5779	37	26	)	)	PUNCT
ejpam-5779	37	27	of	of	ADP
ejpam-5779	37	28	a	a	DET
ejpam-5779	37	29	fuzzy	fuzzy	ADJ
ejpam-5779	37	30	structure	structure	NOUN
ejpam-5779	37	31	within	within	ADP
ejpam-5779	37	32	sheffer	sheffer	PROPN
ejpam-5779	37	33	stroke	stroke	PROPN
ejpam-5779	37	34	hilbert	hilbert	PROPN
ejpam-5779	37	35	algebras.67	algebras.67	PROPN
ejpam-5779	37	36	2	2	NUM
ejpam-5779	37	37	.	.	PUNCT
ejpam-5779	37	38	preliminaries68	preliminaries68	PROPN
ejpam-5779	37	39	sheffer	sheffer	PROPN
ejpam-5779	37	40	stroke	stroke	PROPN
ejpam-5779	37	41	hilbert	hilbert	PROPN
ejpam-5779	37	42	algebras	algebras	PROPN
ejpam-5779	37	43	represent	represent	VERB
ejpam-5779	37	44	an	an	DET
ejpam-5779	37	45	important	important	ADJ
ejpam-5779	37	46	algebraic	algebraic	ADJ
ejpam-5779	37	47	system	system	NOUN
ejpam-5779	37	48	in	in	ADP
ejpam-5779	37	49	the	the	DET
ejpam-5779	37	50	study69	study69	NOUN
ejpam-5779	37	51	of	of	ADP
ejpam-5779	37	52	logic	logic	NOUN
ejpam-5779	37	53	and	and	CCONJ
ejpam-5779	37	54	lattice	lattice	PROPN
ejpam-5779	37	55	theory	theory	NOUN
ejpam-5779	37	56	.	.	PUNCT
ejpam-5779	38	1	these	these	DET
ejpam-5779	38	2	algebras	algebra	NOUN
ejpam-5779	38	3	are	be	AUX
ejpam-5779	38	4	characterized	characterize	VERB
ejpam-5779	38	5	by	by	ADP
ejpam-5779	38	6	the	the	DET
ejpam-5779	38	7	inclusion	inclusion	NOUN
ejpam-5779	38	8	of	of	ADP
ejpam-5779	38	9	the	the	DET
ejpam-5779	38	10	sheffer70	sheffer70	NOUN
ejpam-5779	38	11	stroke	stroke	PROPN
ejpam-5779	38	12	(	(	PUNCT
ejpam-5779	38	13	nand	nand	NOUN
ejpam-5779	38	14	)	)	PUNCT
ejpam-5779	38	15	operation	operation	NOUN
ejpam-5779	38	16	,	,	PUNCT
ejpam-5779	38	17	a	a	DET
ejpam-5779	38	18	fundamental	fundamental	ADJ
ejpam-5779	38	19	logical	logical	ADJ
ejpam-5779	38	20	connectives	connective	NOUN
ejpam-5779	38	21	in	in	ADP
ejpam-5779	38	22	boolean	boolean	ADJ
ejpam-5779	38	23	algebra	algebra	NOUN
ejpam-5779	38	24	.	.	PUNCT
ejpam-5779	39	1	by71	by71	PROPN
ejpam-5779	39	2	extending	extend	VERB
ejpam-5779	39	3	classical	classical	ADJ
ejpam-5779	39	4	hilbert	hilbert	NOUN
ejpam-5779	39	5	algebras	algebra	NOUN
ejpam-5779	39	6	with	with	ADP
ejpam-5779	39	7	this	this	DET
ejpam-5779	39	8	operation	operation	NOUN
ejpam-5779	39	9	,	,	PUNCT
ejpam-5779	39	10	sheffer	sheffer	NOUN
ejpam-5779	39	11	stroke	stroke	PROPN
ejpam-5779	39	12	hilbert	hilbert	PROPN
ejpam-5779	39	13	algebras72	algebras72	NOUN
ejpam-5779	39	14	provide	provide	VERB
ejpam-5779	39	15	a	a	DET
ejpam-5779	39	16	powerful	powerful	ADJ
ejpam-5779	39	17	framework	framework	NOUN
ejpam-5779	39	18	for	for	ADP
ejpam-5779	39	19	investigating	investigate	VERB
ejpam-5779	39	20	logical	logical	ADJ
ejpam-5779	39	21	structures	structure	NOUN
ejpam-5779	39	22	,	,	PUNCT
ejpam-5779	39	23	with	with	ADP
ejpam-5779	39	24	applications	application	NOUN
ejpam-5779	39	25	in73	in73	PROPN
ejpam-5779	39	26	fuzzy	fuzzy	ADJ
ejpam-5779	39	27	logic	logic	NOUN
ejpam-5779	39	28	,	,	PUNCT
ejpam-5779	39	29	decision	decision	NOUN
ejpam-5779	39	30	-	-	PUNCT
ejpam-5779	39	31	making	making	NOUN
ejpam-5779	39	32	,	,	PUNCT
ejpam-5779	39	33	and	and	CCONJ
ejpam-5779	39	34	computational	computational	ADJ
ejpam-5779	39	35	theory	theory	NOUN
ejpam-5779	39	36	.	.	PUNCT
ejpam-5779	40	1	their	their	PRON
ejpam-5779	40	2	study	study	NOUN
ejpam-5779	40	3	enhances	enhance	VERB
ejpam-5779	40	4	both	both	PRON
ejpam-5779	40	5	the74	the74	PROPN
ejpam-5779	40	6	theoretical	theoretical	ADJ
ejpam-5779	40	7	understanding	understanding	NOUN
ejpam-5779	40	8	of	of	ADP
ejpam-5779	40	9	algebraic	algebraic	ADJ
ejpam-5779	40	10	systems	system	NOUN
ejpam-5779	40	11	and	and	CCONJ
ejpam-5779	40	12	their	their	PRON
ejpam-5779	40	13	practical	practical	ADJ
ejpam-5779	40	14	applications	application	NOUN
ejpam-5779	40	15	in	in	ADP
ejpam-5779	40	16	modeling75	modeling75	NOUN
ejpam-5779	40	17	uncertainty	uncertainty	NOUN
ejpam-5779	40	18	and	and	CCONJ
ejpam-5779	40	19	imprecision.76	imprecision.76	ADJ
ejpam-5779	40	20	definition	definition	NOUN
ejpam-5779	40	21	1	1	NUM
ejpam-5779	40	22	.	.	PUNCT
ejpam-5779	41	1	[	[	X
ejpam-5779	41	2	13	13	NUM
ejpam-5779	41	3	]	]	PUNCT
ejpam-5779	41	4	the	the	DET
ejpam-5779	41	5	operation	operation	NOUN
ejpam-5779	41	6	|	|	ADV
ejpam-5779	41	7	in	in	ADP
ejpam-5779	41	8	a	a	DET
ejpam-5779	41	9	groupoid	groupoid	NOUN
ejpam-5779	41	10	a	a	X
ejpam-5779	41	11	=	=	X
ejpam-5779	41	12	(	(	PUNCT
ejpam-5779	41	13	a	a	DET
ejpam-5779	41	14	,	,	PUNCT
ejpam-5779	41	15	|	|	NOUN
ejpam-5779	41	16	)	)	PUNCT
ejpam-5779	41	17	is	be	AUX
ejpam-5779	41	18	referred	refer	VERB
ejpam-5779	41	19	to	to	ADP
ejpam-5779	41	20	as	as	ADP
ejpam-5779	41	21	the	the	DET
ejpam-5779	41	22	sheffer77	sheffer77	NOUN
ejpam-5779	41	23	stroke	stroke	NOUN
ejpam-5779	41	24	or	or	CCONJ
ejpam-5779	41	25	sheffer	sheffer	VERB
ejpam-5779	41	26	operation	operation	NOUN
ejpam-5779	41	27	if	if	SCONJ
ejpam-5779	41	28	it	it	PRON
ejpam-5779	41	29	satisfies	satisfy	VERB
ejpam-5779	41	30	the	the	DET
ejpam-5779	41	31	following	follow	VERB
ejpam-5779	41	32	condition	condition	NOUN
ejpam-5779	41	33	:	:	PUNCT
ejpam-5779	41	34	for	for	ADP
ejpam-5779	41	35	all	all	DET
ejpam-5779	41	36	c	c	NOUN
ejpam-5779	41	37	,	,	PUNCT
ejpam-5779	41	38	b	b	NOUN
ejpam-5779	41	39	,	,	PUNCT
ejpam-5779	41	40	d	d	PROPN
ejpam-5779	41	41	∈	∈	PROPN
ejpam-5779	41	42	a,78	a,78	PROPN
ejpam-5779	41	43	(	(	PUNCT
ejpam-5779	41	44	s1	s1	PROPN
ejpam-5779	41	45	)	)	PUNCT
ejpam-5779	41	46	c|b	c|b	NOUN
ejpam-5779	41	47	=	=	SYM
ejpam-5779	41	48	b|c	b|c	PROPN
ejpam-5779	41	49	,	,	PUNCT
ejpam-5779	41	50	(	(	PUNCT
ejpam-5779	41	51	s2	s2	PROPN
ejpam-5779	41	52	)	)	PUNCT
ejpam-5779	41	53	(	(	PUNCT
ejpam-5779	41	54	c|c)|(c|b	c|c)|(c|b	NOUN
ejpam-5779	41	55	)	)	PUNCT
ejpam-5779	41	56	=	=	SYM
ejpam-5779	41	57	b	b	PROPN
ejpam-5779	41	58	,	,	PUNCT
ejpam-5779	41	59	(	(	PUNCT
ejpam-5779	41	60	s3	s3	PROPN
ejpam-5779	41	61	)	)	PUNCT
ejpam-5779	41	62	c|((b|d)|(b|d	c|((b|d)|(b|d	NUM
ejpam-5779	41	63	)	)	PUNCT
ejpam-5779	41	64	)	)	PUNCT
ejpam-5779	42	1	=	=	SYM
ejpam-5779	42	2	(	(	PUNCT
ejpam-5779	42	3	(	(	PUNCT
ejpam-5779	42	4	c|b)|(c|b))|b	c|b)|(c|b))|b	PROPN
ejpam-5779	42	5	,	,	PUNCT
ejpam-5779	42	6	(	(	PUNCT
ejpam-5779	42	7	s4	s4	PROPN
ejpam-5779	42	8	)	)	PUNCT
ejpam-5779	42	9	(	(	PUNCT
ejpam-5779	42	10	c|((c|c)|(b|b)))|(c|((c|c)|(b|b	c|((c|c)|(b|b)))|(c|((c|c)|(b|b	NOUN
ejpam-5779	42	11	)	)	PUNCT
ejpam-5779	42	12	)	)	PUNCT
ejpam-5779	42	13	)	)	PUNCT
ejpam-5779	43	1	=	=	SYM
ejpam-5779	43	2	c.	c.	NOUN
ejpam-5779	43	3	to	to	PART
ejpam-5779	43	4	improve	improve	VERB
ejpam-5779	43	5	the	the	DET
ejpam-5779	43	6	clarity	clarity	NOUN
ejpam-5779	43	7	of	of	ADP
ejpam-5779	43	8	this	this	DET
ejpam-5779	43	9	manuscript	manuscript	NOUN
ejpam-5779	43	10	on	on	ADP
ejpam-5779	43	11	sheffer	sheffer	PROPN
ejpam-5779	43	12	stroke	stroke	PROPN
ejpam-5779	43	13	hilbert	hilbert	PROPN
ejpam-5779	43	14	algebras	algebras	PROPN
ejpam-5779	43	15	,	,	PUNCT
ejpam-5779	43	16	we	we	PRON
ejpam-5779	43	17	will79	will79	VERB
ejpam-5779	43	18	adopt	adopt	VERB
ejpam-5779	43	19	the	the	DET
ejpam-5779	43	20	following	follow	VERB
ejpam-5779	43	21	notation	notation	NOUN
ejpam-5779	43	22	throughout:80	throughout:80	SYM
ejpam-5779	43	23	p|(q|q	p|(q|q	PROPN
ejpam-5779	43	24	)	)	PUNCT
ejpam-5779	43	25	=	=	SYM
ejpam-5779	43	26	pq	pq	PROPN
ejpam-5779	43	27	.	.	PUNCT
ejpam-5779	43	28	definition	definition	NOUN
ejpam-5779	43	29	2	2	NUM
ejpam-5779	43	30	.	.	PUNCT
ejpam-5779	44	1	[	[	X
ejpam-5779	44	2	6	6	NUM
ejpam-5779	44	3	]	]	PUNCT
ejpam-5779	44	4	a	a	DET
ejpam-5779	44	5	sheffer	sheffer	NOUN
ejpam-5779	44	6	stroke	stroke	NOUN
ejpam-5779	44	7	hilbert	hilbert	PROPN
ejpam-5779	44	8	algebra	algebra	PROPN
ejpam-5779	44	9	(	(	PUNCT
ejpam-5779	44	10	abbreviated	abbreviate	VERB
ejpam-5779	44	11	sha	sha	PROPN
ejpam-5779	44	12	)	)	PUNCT
ejpam-5779	44	13	refers	refer	VERB
ejpam-5779	44	14	to	to	ADP
ejpam-5779	44	15	a	a	DET
ejpam-5779	44	16	groupoid81	groupoid81	NOUN
ejpam-5779	44	17	a	a	PRON
ejpam-5779	44	18	=	=	PUNCT
ejpam-5779	44	19	(	(	PUNCT
ejpam-5779	44	20	a	a	PRON
ejpam-5779	44	21	,	,	PUNCT
ejpam-5779	44	22	|	|	ADV
ejpam-5779	44	23	,	,	PUNCT
ejpam-5779	44	24	0	0	NUM
ejpam-5779	44	25	)	)	PUNCT
ejpam-5779	44	26	equipped	equip	VERB
ejpam-5779	44	27	with	with	ADP
ejpam-5779	44	28	a	a	DET
ejpam-5779	44	29	sheffer	sheffer	NOUN
ejpam-5779	44	30	stroke	stroke	NOUN
ejpam-5779	44	31	operation	operation	NOUN
ejpam-5779	44	32	|	|	ADV
ejpam-5779	44	33	and	and	CCONJ
ejpam-5779	44	34	0	0	NUM
ejpam-5779	44	35	is	be	AUX
ejpam-5779	44	36	the	the	DET
ejpam-5779	44	37	fixed	fix	VERB
ejpam-5779	44	38	element	element	NOUN
ejpam-5779	44	39	in	in	ADP
ejpam-5779	44	40	a,82	a,82	PRON
ejpam-5779	44	41	and	and	CCONJ
ejpam-5779	44	42	it	it	PRON
ejpam-5779	44	43	must	must	AUX
ejpam-5779	44	44	satisfy	satisfy	VERB
ejpam-5779	44	45	the	the	DET
ejpam-5779	44	46	following	following	ADJ
ejpam-5779	44	47	conditions	condition	NOUN
ejpam-5779	44	48	:	:	PUNCT
ejpam-5779	44	49	for	for	ADP
ejpam-5779	44	50	all	all	PRON
ejpam-5779	44	51	p	p	NOUN
ejpam-5779	44	52	,	,	PUNCT
ejpam-5779	44	53	q	q	ADJ
ejpam-5779	44	54	,	,	PUNCT
ejpam-5779	44	55	r	r	NOUN
ejpam-5779	44	56	∈	∈	NOUN
ejpam-5779	44	57	a,83	a,83	X
ejpam-5779	44	58	(	(	PUNCT
ejpam-5779	44	59	1	1	X
ejpam-5779	44	60	)	)	PUNCT
ejpam-5779	44	61	(	(	PUNCT
ejpam-5779	44	62	p|(qr|pq))|((pq)(pr)|(pq)(pr	p|(qr|pq))|((pq)(pr)|(pq)(pr	PROPN
ejpam-5779	44	63	)	)	PUNCT
ejpam-5779	44	64	)	)	PUNCT
ejpam-5779	45	1	=	=	PUNCT
ejpam-5779	45	2	pp,84	pp,84	X
ejpam-5779	45	3	(	(	PUNCT
ejpam-5779	45	4	2	2	NUM
ejpam-5779	45	5	)	)	PUNCT
ejpam-5779	45	6	pq	pq	NOUN
ejpam-5779	45	7	=	=	PUNCT
ejpam-5779	45	8	qp	qp	PROPN
ejpam-5779	45	9	⇒	⇒	NOUN
ejpam-5779	45	10	p	p	X
ejpam-5779	45	11	=	=	PUNCT
ejpam-5779	45	12	q.85	q.85	PROPN
ejpam-5779	45	13	proposition	proposition	NOUN
ejpam-5779	45	14	1	1	NUM
ejpam-5779	45	15	.	.	PUNCT
ejpam-5779	46	1	[	[	X
ejpam-5779	46	2	6	6	NUM
ejpam-5779	46	3	]	]	PUNCT
ejpam-5779	46	4	let	let	VERB
ejpam-5779	46	5	a	a	PRON
ejpam-5779	46	6	=	=	X
ejpam-5779	46	7	(	(	PUNCT
ejpam-5779	46	8	a	a	PRON
ejpam-5779	46	9	,	,	PUNCT
ejpam-5779	46	10	|	|	INTJ
ejpam-5779	46	11	,	,	PUNCT
ejpam-5779	46	12	0	0	NUM
ejpam-5779	46	13	)	)	PUNCT
ejpam-5779	46	14	be	be	AUX
ejpam-5779	46	15	an	an	DET
ejpam-5779	46	16	sha	sha	PROPN
ejpam-5779	46	17	.	.	PUNCT
ejpam-5779	47	1	then	then	ADV
ejpam-5779	47	2	the	the	DET
ejpam-5779	47	3	binary	binary	PROPN
ejpam-5779	47	4	relation	relation	PROPN
ejpam-5779	47	5	p	p	PROPN
ejpam-5779	47	6	≤	≤	PROPN
ejpam-5779	47	7	q	q	NOUN
ejpam-5779	48	1	if	if	SCONJ
ejpam-5779	48	2	and86	and86	PROPN
ejpam-5779	49	1	only	only	ADV
ejpam-5779	49	2	if	if	SCONJ
ejpam-5779	49	3	pq	pq	NOUN
ejpam-5779	49	4	=	=	NOUN
ejpam-5779	49	5	1	1	NUM
ejpam-5779	49	6	is	be	AUX
ejpam-5779	49	7	a	a	DET
ejpam-5779	49	8	partial	partial	ADJ
ejpam-5779	49	9	order	order	NOUN
ejpam-5779	49	10	on	on	ADP
ejpam-5779	49	11	a.87	a.87	NOUN
ejpam-5779	49	12	definition	definition	NOUN
ejpam-5779	49	13	3	3	NUM
ejpam-5779	49	14	.	.	PUNCT
ejpam-5779	50	1	[	[	X
ejpam-5779	50	2	6	6	NUM
ejpam-5779	50	3	]	]	PUNCT
ejpam-5779	50	4	let	let	VERB
ejpam-5779	50	5	a	a	PRON
ejpam-5779	50	6	=	=	X
ejpam-5779	50	7	(	(	PUNCT
ejpam-5779	50	8	a	a	PRON
ejpam-5779	50	9	,	,	PUNCT
ejpam-5779	50	10	|	|	INTJ
ejpam-5779	50	11	,	,	PUNCT
ejpam-5779	50	12	0	0	NUM
ejpam-5779	50	13	)	)	PUNCT
ejpam-5779	50	14	be	be	AUX
ejpam-5779	50	15	an	an	DET
ejpam-5779	50	16	sha	sha	PROPN
ejpam-5779	50	17	.	.	PUNCT
ejpam-5779	51	1	a	a	DET
ejpam-5779	51	2	nonempty	nonempty	NOUN
ejpam-5779	51	3	subset	subset	VERB
ejpam-5779	51	4	g	g	NOUN
ejpam-5779	51	5	of	of	ADP
ejpam-5779	51	6	a	a	PRON
ejpam-5779	51	7	is	be	AUX
ejpam-5779	51	8	called	call	VERB
ejpam-5779	51	9	an88	an88	PROPN
ejpam-5779	51	10	ideal	ideal	NOUN
ejpam-5779	51	11	of	of	ADP
ejpam-5779	51	12	a	a	DET
ejpam-5779	51	13	if	if	NOUN
ejpam-5779	51	14	for	for	ADP
ejpam-5779	51	15	all	all	DET
ejpam-5779	51	16	p	p	NOUN
ejpam-5779	51	17	,	,	PUNCT
ejpam-5779	51	18	q	q	PROPN
ejpam-5779	51	19	∈	∈	PROPN
ejpam-5779	51	20	a,89	a,89	ADJ
ejpam-5779	51	21	(	(	PUNCT
ejpam-5779	51	22	1	1	NUM
ejpam-5779	51	23	)	)	PUNCT
ejpam-5779	51	24	0	0	NUM
ejpam-5779	52	1	∈	∈	NOUN
ejpam-5779	52	2	g,90	g,90	X
ejpam-5779	52	3	(	(	PUNCT
ejpam-5779	52	4	2	2	X
ejpam-5779	52	5	)	)	PUNCT
ejpam-5779	52	6	pq	pq	NOUN
ejpam-5779	52	7	∈	∈	PROPN
ejpam-5779	52	8	g	g	PROPN
ejpam-5779	52	9	and	and	CCONJ
ejpam-5779	52	10	q	q	NOUN
ejpam-5779	52	11	∈	∈	PROPN
ejpam-5779	52	12	g	g	PROPN
ejpam-5779	52	13	⇒	⇒	VERB
ejpam-5779	52	14	p	p	PROPN
ejpam-5779	52	15	∈	∈	PROPN
ejpam-5779	52	16	g.91	g.91	PROPN
ejpam-5779	52	17	n.	n.	PROPN
ejpam-5779	52	18	rajesh	rajesh	PROPN
ejpam-5779	52	19	,	,	PUNCT
ejpam-5779	52	20	t.	t.	PROPN
ejpam-5779	52	21	oner	oner	NOUN
ejpam-5779	52	22	,	,	PUNCT
ejpam-5779	52	23	a.	a.	NOUN
ejpam-5779	52	24	iampan	iampan	PROPN
ejpam-5779	52	25	,	,	PUNCT
ejpam-5779	52	26	i.	i.	PROPN
ejpam-5779	52	27	senturk	senturk	PROPN
ejpam-5779	52	28	/	/	SYM
ejpam-5779	52	29	eur	eur	PROPN
ejpam-5779	52	30	.	.	PUNCT
ejpam-5779	53	1	j.	j.	PROPN
ejpam-5779	53	2	pure	pure	PROPN
ejpam-5779	53	3	appl	appl	PROPN
ejpam-5779	53	4	.	.	PROPN
ejpam-5779	53	5	math	math	PROPN
ejpam-5779	53	6	,	,	PUNCT
ejpam-5779	53	7	18	18	NUM
ejpam-5779	53	8	(	(	PUNCT
ejpam-5779	53	9	1	1	NUM
ejpam-5779	53	10	)	)	PUNCT
ejpam-5779	53	11	(	(	PUNCT
ejpam-5779	53	12	2025	2025	NUM
ejpam-5779	53	13	)	)	PUNCT
ejpam-5779	53	14	,	,	PUNCT
ejpam-5779	53	15	5779	5779	NUM
ejpam-5779	53	16	4	4	NUM
ejpam-5779	53	17	of	of	ADP
ejpam-5779	53	18	18	18	NUM
ejpam-5779	53	19	3	3	NUM
ejpam-5779	53	20	.	.	NOUN
ejpam-5779	53	21	length	length	NOUN
ejpam-5779	53	22	fuzzy	fuzzy	ADJ
ejpam-5779	53	23	ideals	ideal	NOUN
ejpam-5779	53	24	of	of	ADP
ejpam-5779	53	25	sheffer	sheffer	PROPN
ejpam-5779	53	26	stroke	stroke	PROPN
ejpam-5779	53	27	hilbert	hilbert	PROPN
ejpam-5779	53	28	algebras92	algebras92	PROPN
ejpam-5779	53	29	this	this	DET
ejpam-5779	53	30	paper	paper	NOUN
ejpam-5779	53	31	introduces	introduce	VERB
ejpam-5779	53	32	the	the	DET
ejpam-5779	53	33	concept	concept	NOUN
ejpam-5779	53	34	of	of	ADP
ejpam-5779	53	35	length	length	NOUN
ejpam-5779	53	36	fuzzy	fuzzy	ADJ
ejpam-5779	53	37	ideals	ideal	NOUN
ejpam-5779	53	38	in	in	ADP
ejpam-5779	53	39	sheffer	sheffer	PROPN
ejpam-5779	53	40	stroke	stroke	PROPN
ejpam-5779	53	41	hilbert93	hilbert93	ADV
ejpam-5779	53	42	algebras	algebra	VERB
ejpam-5779	53	43	and	and	CCONJ
ejpam-5779	53	44	examines	examine	VERB
ejpam-5779	53	45	their	their	PRON
ejpam-5779	53	46	associated	associated	ADJ
ejpam-5779	53	47	properties	property	NOUN
ejpam-5779	53	48	.	.	PUNCT
ejpam-5779	54	1	it	it	PRON
ejpam-5779	54	2	establishes	establish	VERB
ejpam-5779	54	3	the	the	DET
ejpam-5779	54	4	connections	connection	NOUN
ejpam-5779	54	5	between94	between94	VERB
ejpam-5779	54	6	length	length	NOUN
ejpam-5779	54	7	fuzzy	fuzzy	ADJ
ejpam-5779	54	8	ideals	ideal	NOUN
ejpam-5779	54	9	and	and	CCONJ
ejpam-5779	54	10	conventional	conventional	ADJ
ejpam-5779	54	11	ideals	ideal	NOUN
ejpam-5779	54	12	.	.	PUNCT
ejpam-5779	55	1	furthermore	furthermore	ADV
ejpam-5779	55	2	,	,	PUNCT
ejpam-5779	55	3	it	it	PRON
ejpam-5779	55	4	explores	explore	VERB
ejpam-5779	55	5	the	the	DET
ejpam-5779	55	6	relationships95	relationships95	NOUN
ejpam-5779	55	7	between	between	ADP
ejpam-5779	55	8	length	length	NOUN
ejpam-5779	55	9	fuzzy	fuzzy	ADJ
ejpam-5779	55	10	ideals	ideal	NOUN
ejpam-5779	55	11	and	and	CCONJ
ejpam-5779	55	12	the	the	DET
ejpam-5779	55	13	upper	upper	ADJ
ejpam-5779	55	14	and	and	CCONJ
ejpam-5779	55	15	lower	low	ADJ
ejpam-5779	55	16	level	level	NOUN
ejpam-5779	55	17	subsets	subset	NOUN
ejpam-5779	55	18	of	of	ADP
ejpam-5779	55	19	the	the	DET
ejpam-5779	55	20	length	length	NOUN
ejpam-5779	55	21	in	in	ADP
ejpam-5779	55	22	an96	an96	PROPN
ejpam-5779	55	23	interval	interval	NOUN
ejpam-5779	55	24	-	-	PUNCT
ejpam-5779	55	25	valued	value	VERB
ejpam-5779	55	26	fuzzy	fuzzy	ADJ
ejpam-5779	55	27	structure	structure	NOUN
ejpam-5779	55	28	within	within	ADP
ejpam-5779	55	29	sheffer	sheffer	PROPN
ejpam-5779	55	30	stroke	stroke	PROPN
ejpam-5779	55	31	hilbert	hilbert	PROPN
ejpam-5779	55	32	algebras.97	algebras.97	PROPN
ejpam-5779	55	33	from	from	ADP
ejpam-5779	55	34	now	now	ADV
ejpam-5779	55	35	on	on	ADV
ejpam-5779	55	36	,	,	PUNCT
ejpam-5779	55	37	unless	unless	SCONJ
ejpam-5779	55	38	stated	state	VERB
ejpam-5779	55	39	otherwise	otherwise	ADV
ejpam-5779	55	40	,	,	PUNCT
ejpam-5779	55	41	we	we	PRON
ejpam-5779	55	42	denote	denote	VERB
ejpam-5779	55	43	an	an	DET
ejpam-5779	55	44	sha	sha	NOUN
ejpam-5779	55	45	by	by	ADP
ejpam-5779	55	46	a	a	DET
ejpam-5779	55	47	=	=	X
ejpam-5779	55	48	(	(	PUNCT
ejpam-5779	55	49	a	a	PRON
ejpam-5779	55	50	,	,	PUNCT
ejpam-5779	55	51	|	|	ADV
ejpam-5779	55	52	,	,	PUNCT
ejpam-5779	55	53	0).98	0).98	NUM
ejpam-5779	55	54	definition	definition	NOUN
ejpam-5779	55	55	4	4	NUM
ejpam-5779	55	56	.	.	PUNCT
ejpam-5779	56	1	a	a	DET
ejpam-5779	56	2	fuzzy	fuzzy	ADJ
ejpam-5779	56	3	structure	structure	NOUN
ejpam-5779	56	4	(	(	PUNCT
ejpam-5779	56	5	a	a	DET
ejpam-5779	56	6	,	,	PUNCT
ejpam-5779	56	7	f	f	NOUN
ejpam-5779	56	8	)	)	PUNCT
ejpam-5779	56	9	of	of	ADP
ejpam-5779	56	10	a	a	PRON
ejpam-5779	56	11	is	be	AUX
ejpam-5779	56	12	defined	define	VERB
ejpam-5779	56	13	as:99	as:99	PROPN
ejpam-5779	56	14	(	(	PUNCT
ejpam-5779	56	15	1	1	NUM
ejpam-5779	56	16	)	)	PUNCT
ejpam-5779	56	17	a	a	DET
ejpam-5779	56	18	fuzzy	fuzzy	ADJ
ejpam-5779	56	19	ideal	ideal	NOUN
ejpam-5779	56	20	of	of	ADP
ejpam-5779	56	21	a	a	PRON
ejpam-5779	56	22	of	of	ADP
ejpam-5779	56	23	type	type	NOUN
ejpam-5779	56	24	1	1	NUM
ejpam-5779	56	25	(	(	PUNCT
ejpam-5779	56	26	simply	simply	ADV
ejpam-5779	56	27	a	a	DET
ejpam-5779	56	28	1	1	NUM
ejpam-5779	56	29	-	-	PUNCT
ejpam-5779	56	30	fuzzy	fuzzy	ADJ
ejpam-5779	56	31	ideal	ideal	NOUN
ejpam-5779	56	32	of	of	ADP
ejpam-5779	56	33	a	a	DET
ejpam-5779	56	34	)	)	PUNCT
ejpam-5779	56	35	if100	if100	NOUN
ejpam-5779	56	36	(	(	PUNCT
ejpam-5779	56	37	∀p	∀p	NOUN
ejpam-5779	56	38	∈	∈	PROPN
ejpam-5779	56	39	a)(f(0	a)(f(0	PROPN
ejpam-5779	56	40	)	)	PUNCT
ejpam-5779	56	41	≥	≥	NOUN
ejpam-5779	56	42	f(p	f(p	NOUN
ejpam-5779	56	43	)	)	PUNCT
ejpam-5779	56	44	)	)	PUNCT
ejpam-5779	56	45	,	,	PUNCT
ejpam-5779	56	46	(	(	PUNCT
ejpam-5779	56	47	1	1	X
ejpam-5779	56	48	)	)	PUNCT
ejpam-5779	56	49	(	(	PUNCT
ejpam-5779	56	50	∀p	∀p	X
ejpam-5779	56	51	,	,	PUNCT
ejpam-5779	56	52	q	q	PROPN
ejpam-5779	56	53	∈	∈	PROPN
ejpam-5779	56	54	a)(f(p	a)(f(p	X
ejpam-5779	56	55	)	)	PUNCT
ejpam-5779	56	56	≥	≥	NOUN
ejpam-5779	56	57	min{f(pq	min{f(pq	NOUN
ejpam-5779	56	58	)	)	PUNCT
ejpam-5779	56	59	,	,	PUNCT
ejpam-5779	56	60	f(q	f(q	PROPN
ejpam-5779	56	61	)	)	PUNCT
ejpam-5779	56	62	}	}	PUNCT
ejpam-5779	56	63	)	)	PUNCT
ejpam-5779	56	64	.	.	PUNCT
ejpam-5779	57	1	(	(	PUNCT
ejpam-5779	57	2	2	2	X
ejpam-5779	57	3	)	)	PUNCT
ejpam-5779	57	4	(	(	PUNCT
ejpam-5779	57	5	2	2	X
ejpam-5779	57	6	)	)	PUNCT
ejpam-5779	57	7	a	a	DET
ejpam-5779	57	8	fuzzy	fuzzy	ADJ
ejpam-5779	57	9	ideal	ideal	NOUN
ejpam-5779	57	10	of	of	ADP
ejpam-5779	57	11	a	a	PRON
ejpam-5779	57	12	of	of	ADP
ejpam-5779	57	13	type	type	NOUN
ejpam-5779	57	14	2	2	NUM
ejpam-5779	57	15	(	(	PUNCT
ejpam-5779	57	16	simply	simply	ADV
ejpam-5779	57	17	a	a	DET
ejpam-5779	57	18	2	2	NUM
ejpam-5779	57	19	-	-	PUNCT
ejpam-5779	57	20	fuzzy	fuzzy	ADJ
ejpam-5779	57	21	ideal	ideal	NOUN
ejpam-5779	57	22	of	of	ADP
ejpam-5779	57	23	a	a	X
ejpam-5779	57	24	)	)	PUNCT
ejpam-5779	58	1	if101	if101	PROPN
ejpam-5779	58	2	(	(	PUNCT
ejpam-5779	58	3	∀p	∀p	NOUN
ejpam-5779	58	4	∈	∈	PROPN
ejpam-5779	58	5	a)(f(0	a)(f(0	NOUN
ejpam-5779	58	6	)	)	PUNCT
ejpam-5779	58	7	≤	≤	NOUN
ejpam-5779	58	8	f(p	f(p	NOUN
ejpam-5779	58	9	)	)	PUNCT
ejpam-5779	58	10	)	)	PUNCT
ejpam-5779	58	11	,	,	PUNCT
ejpam-5779	58	12	(	(	PUNCT
ejpam-5779	58	13	3	3	X
ejpam-5779	58	14	)	)	PUNCT
ejpam-5779	58	15	(	(	PUNCT
ejpam-5779	58	16	∀p	∀p	X
ejpam-5779	58	17	,	,	PUNCT
ejpam-5779	58	18	q	q	PROPN
ejpam-5779	58	19	∈	∈	PROPN
ejpam-5779	58	20	a)(f(p	a)(f(p	PUNCT
ejpam-5779	58	21	)	)	PUNCT
ejpam-5779	58	22	≤	≤	NUM
ejpam-5779	58	23	min{f(pq	min{f(pq	NOUN
ejpam-5779	58	24	)	)	PUNCT
ejpam-5779	58	25	,	,	PUNCT
ejpam-5779	58	26	f(q	f(q	PROPN
ejpam-5779	58	27	)	)	PUNCT
ejpam-5779	58	28	}	}	PUNCT
ejpam-5779	58	29	)	)	PUNCT
ejpam-5779	58	30	.	.	PUNCT
ejpam-5779	59	1	(	(	PUNCT
ejpam-5779	59	2	4	4	X
ejpam-5779	59	3	)	)	PUNCT
ejpam-5779	59	4	(	(	PUNCT
ejpam-5779	59	5	3	3	X
ejpam-5779	59	6	)	)	PUNCT
ejpam-5779	59	7	a	a	DET
ejpam-5779	59	8	fuzzy	fuzzy	ADJ
ejpam-5779	59	9	ideal	ideal	NOUN
ejpam-5779	59	10	of	of	ADP
ejpam-5779	59	11	a	a	PRON
ejpam-5779	59	12	of	of	ADP
ejpam-5779	59	13	type	type	NOUN
ejpam-5779	59	14	3	3	NUM
ejpam-5779	59	15	(	(	PUNCT
ejpam-5779	59	16	simply	simply	ADV
ejpam-5779	59	17	a	a	DET
ejpam-5779	59	18	3	3	NUM
ejpam-5779	59	19	-	-	PUNCT
ejpam-5779	59	20	fuzzy	fuzzy	ADJ
ejpam-5779	59	21	ideal	ideal	NOUN
ejpam-5779	59	22	of	of	ADP
ejpam-5779	59	23	a	a	PRON
ejpam-5779	59	24	)	)	PUNCT
ejpam-5779	59	25	if102	if102	PROPN
ejpam-5779	59	26	(	(	PUNCT
ejpam-5779	59	27	∀p	∀p	NOUN
ejpam-5779	59	28	∈	∈	PROPN
ejpam-5779	59	29	a)(f(0	a)(f(0	PROPN
ejpam-5779	59	30	)	)	PUNCT
ejpam-5779	59	31	≥	≥	NOUN
ejpam-5779	59	32	f(p	f(p	NOUN
ejpam-5779	59	33	)	)	PUNCT
ejpam-5779	59	34	)	)	PUNCT
ejpam-5779	59	35	,	,	PUNCT
ejpam-5779	59	36	(	(	PUNCT
ejpam-5779	59	37	5	5	NUM
ejpam-5779	59	38	)	)	PUNCT
ejpam-5779	59	39	(	(	PUNCT
ejpam-5779	59	40	∀p	∀p	X
ejpam-5779	59	41	,	,	PUNCT
ejpam-5779	59	42	q	q	PROPN
ejpam-5779	59	43	∈	∈	PROPN
ejpam-5779	59	44	a)(f(p	a)(f(p	X
ejpam-5779	59	45	)	)	PUNCT
ejpam-5779	59	46	≥	≥	NOUN
ejpam-5779	59	47	max{f(pq	max{f(pq	PROPN
ejpam-5779	59	48	)	)	PUNCT
ejpam-5779	59	49	,	,	PUNCT
ejpam-5779	59	50	f(q	f(q	PROPN
ejpam-5779	59	51	)	)	PUNCT
ejpam-5779	59	52	}	}	PUNCT
ejpam-5779	59	53	)	)	PUNCT
ejpam-5779	59	54	.	.	PUNCT
ejpam-5779	60	1	(	(	PUNCT
ejpam-5779	60	2	6	6	NUM
ejpam-5779	60	3	)	)	PUNCT
ejpam-5779	60	4	(	(	PUNCT
ejpam-5779	60	5	4	4	X
ejpam-5779	60	6	)	)	PUNCT
ejpam-5779	60	7	a	a	DET
ejpam-5779	60	8	fuzzy	fuzzy	ADJ
ejpam-5779	60	9	ideal	ideal	NOUN
ejpam-5779	60	10	of	of	ADP
ejpam-5779	60	11	a	a	PRON
ejpam-5779	60	12	of	of	ADP
ejpam-5779	60	13	type	type	NOUN
ejpam-5779	60	14	4	4	NUM
ejpam-5779	60	15	(	(	PUNCT
ejpam-5779	60	16	simply	simply	ADV
ejpam-5779	60	17	a	a	DET
ejpam-5779	60	18	4	4	NUM
ejpam-5779	60	19	-	-	PUNCT
ejpam-5779	60	20	fuzzy	fuzzy	ADJ
ejpam-5779	60	21	ideal	ideal	NOUN
ejpam-5779	60	22	of	of	ADP
ejpam-5779	60	23	a	a	PRON
ejpam-5779	60	24	)	)	PUNCT
ejpam-5779	60	25	if103	if103	PROPN
ejpam-5779	60	26	(	(	PUNCT
ejpam-5779	60	27	∀p	∀p	NOUN
ejpam-5779	60	28	∈	∈	PROPN
ejpam-5779	60	29	a)(f(0	a)(f(0	NOUN
ejpam-5779	60	30	)	)	PUNCT
ejpam-5779	60	31	≤	≤	NOUN
ejpam-5779	60	32	f(p	f(p	NOUN
ejpam-5779	60	33	)	)	PUNCT
ejpam-5779	60	34	)	)	PUNCT
ejpam-5779	60	35	,	,	PUNCT
ejpam-5779	60	36	(	(	PUNCT
ejpam-5779	60	37	7	7	X
ejpam-5779	60	38	)	)	PUNCT
ejpam-5779	60	39	(	(	PUNCT
ejpam-5779	60	40	∀p	∀p	X
ejpam-5779	60	41	,	,	PUNCT
ejpam-5779	60	42	q	q	PROPN
ejpam-5779	60	43	∈	∈	PROPN
ejpam-5779	60	44	a)(f(p	a)(f(p	PUNCT
ejpam-5779	60	45	)	)	PUNCT
ejpam-5779	60	46	≤	≤	NUM
ejpam-5779	60	47	max{f(pq	max{f(pq	PROPN
ejpam-5779	60	48	)	)	PUNCT
ejpam-5779	60	49	,	,	PUNCT
ejpam-5779	60	50	f(q	f(q	PROPN
ejpam-5779	60	51	)	)	PUNCT
ejpam-5779	60	52	}	}	PUNCT
ejpam-5779	60	53	)	)	PUNCT
ejpam-5779	60	54	.	.	PUNCT
ejpam-5779	61	1	(	(	PUNCT
ejpam-5779	61	2	8)	8)	NUM
ejpam-5779	61	3	definition	definition	NOUN
ejpam-5779	61	4	5	5	NUM
ejpam-5779	61	5	.	.	PUNCT
ejpam-5779	62	1	[	[	X
ejpam-5779	62	2	14	14	NUM
ejpam-5779	62	3	]	]	PUNCT
ejpam-5779	62	4	given	give	VERB
ejpam-5779	62	5	an	an	DET
ejpam-5779	62	6	interval	interval	NOUN
ejpam-5779	62	7	-	-	PUNCT
ejpam-5779	62	8	valued	value	VERB
ejpam-5779	62	9	fuzzy	fuzzy	ADJ
ejpam-5779	62	10	structure	structure	NOUN
ejpam-5779	62	11	(	(	PUNCT
ejpam-5779	62	12	a	a	PRON
ejpam-5779	62	13	,	,	PUNCT
ejpam-5779	62	14	f̃	f̃	PROPN
ejpam-5779	62	15	)	)	PUNCT
ejpam-5779	62	16	over	over	ADP
ejpam-5779	62	17	a	a	PRON
ejpam-5779	62	18	,	,	PUNCT
ejpam-5779	62	19	we	we	PRON
ejpam-5779	62	20	define	define	VERB
ejpam-5779	62	21	a	a	DET
ejpam-5779	62	22	fuzzy	fuzzy	ADJ
ejpam-5779	62	23	structure	structure	NOUN
ejpam-5779	62	24	(	(	PUNCT
ejpam-5779	62	25	a	a	DET
ejpam-5779	62	26	,	,	PUNCT
ejpam-5779	62	27	f̃l	f̃l	PROPN
ejpam-5779	62	28	)	)	PUNCT
ejpam-5779	62	29	on	on	ADP
ejpam-5779	62	30	a	a	PRON
ejpam-5779	62	31	as	as	SCONJ
ejpam-5779	62	32	follows	follow	VERB
ejpam-5779	62	33	:	:	PUNCT
ejpam-5779	62	34	f̃l	f̃l	NUM
ejpam-5779	62	35	:	:	PUNCT
ejpam-5779	62	36	a	a	DET
ejpam-5779	62	37	→	→	SYM
ejpam-5779	62	38	[	[	X
ejpam-5779	62	39	0	0	NUM
ejpam-5779	62	40	,	,	PUNCT
ejpam-5779	62	41	1	1	NUM
ejpam-5779	62	42	]	]	PUNCT
ejpam-5779	62	43	;	;	PUNCT
ejpam-5779	62	44	p	p	PROPN
ejpam-5779	62	45	7→	7→	NUM
ejpam-5779	62	46	f̃sup(p)−	f̃sup(p)−	PROPN
ejpam-5779	62	47	f̃inf(p	f̃inf(p	NOUN
ejpam-5779	62	48	)	)	PUNCT
ejpam-5779	62	49	,	,	PUNCT
ejpam-5779	62	50	which	which	PRON
ejpam-5779	62	51	is	be	AUX
ejpam-5779	62	52	referred	refer	VERB
ejpam-5779	62	53	to	to	ADP
ejpam-5779	62	54	as	as	ADP
ejpam-5779	62	55	the	the	DET
ejpam-5779	62	56	length	length	NOUN
ejpam-5779	62	57	of	of	ADP
ejpam-5779	62	58	f̃	f̃	PROPN
ejpam-5779	62	59	.104	.104	NUM
ejpam-5779	62	60	definition	definition	NOUN
ejpam-5779	62	61	6	6	NUM
ejpam-5779	62	62	.	.	PUNCT
ejpam-5779	63	1	an	an	DET
ejpam-5779	63	2	interval	interval	NOUN
ejpam-5779	63	3	-	-	PUNCT
ejpam-5779	63	4	valued	value	VERB
ejpam-5779	63	5	fuzzy	fuzzy	ADJ
ejpam-5779	63	6	structure	structure	NOUN
ejpam-5779	63	7	(	(	PUNCT
ejpam-5779	63	8	a	a	PRON
ejpam-5779	63	9	,	,	PUNCT
ejpam-5779	63	10	f̃	f̃	PROPN
ejpam-5779	63	11	)	)	PUNCT
ejpam-5779	63	12	over	over	ADP
ejpam-5779	63	13	a	a	PRON
ejpam-5779	63	14	is	be	AUX
ejpam-5779	63	15	referred	refer	VERB
ejpam-5779	63	16	to	to	ADP
ejpam-5779	63	17	as	as	ADP
ejpam-5779	63	18	a	a	DET
ejpam-5779	63	19	length105	length105	PROPN
ejpam-5779	63	20	1	1	NUM
ejpam-5779	63	21	-	-	PUNCT
ejpam-5779	63	22	fuzzy	fuzzy	ADJ
ejpam-5779	63	23	(	(	PUNCT
ejpam-5779	63	24	resp	resp	NOUN
ejpam-5779	63	25	.	.	PUNCT
ejpam-5779	63	26	,	,	PUNCT
ejpam-5779	63	27	2	2	NUM
ejpam-5779	63	28	-	-	PUNCT
ejpam-5779	63	29	fuzzy	fuzzy	ADJ
ejpam-5779	63	30	,	,	PUNCT
ejpam-5779	63	31	3	3	NUM
ejpam-5779	63	32	-	-	PUNCT
ejpam-5779	63	33	fuzzy	fuzzy	ADJ
ejpam-5779	63	34	,	,	PUNCT
ejpam-5779	63	35	4	4	NUM
ejpam-5779	63	36	-	-	PUNCT
ejpam-5779	63	37	fuzzy	fuzzy	ADJ
ejpam-5779	63	38	)	)	PUNCT
ejpam-5779	63	39	ideal	ideal	NOUN
ejpam-5779	63	40	of	of	ADP
ejpam-5779	63	41	a	a	PRON
ejpam-5779	63	42	if	if	SCONJ
ejpam-5779	63	43	the	the	DET
ejpam-5779	63	44	fuzzy	fuzzy	ADJ
ejpam-5779	63	45	structure	structure	NOUN
ejpam-5779	63	46	(	(	PUNCT
ejpam-5779	63	47	a	a	DET
ejpam-5779	63	48	,	,	PUNCT
ejpam-5779	63	49	f̃l	f̃l	PROPN
ejpam-5779	63	50	)	)	PUNCT
ejpam-5779	63	51	is	be	AUX
ejpam-5779	63	52	a	a	DET
ejpam-5779	63	53	1	1	NUM
ejpam-5779	63	54	-	-	PUNCT
ejpam-5779	63	55	fuzzy106	fuzzy106	NOUN
ejpam-5779	63	56	(	(	PUNCT
ejpam-5779	63	57	resp	resp	NOUN
ejpam-5779	63	58	.	.	PUNCT
ejpam-5779	63	59	,	,	PUNCT
ejpam-5779	63	60	2	2	NUM
ejpam-5779	63	61	-	-	PUNCT
ejpam-5779	63	62	fuzzy	fuzzy	ADJ
ejpam-5779	63	63	,	,	PUNCT
ejpam-5779	63	64	3	3	NUM
ejpam-5779	63	65	-	-	PUNCT
ejpam-5779	63	66	fuzzy	fuzzy	ADJ
ejpam-5779	63	67	,	,	PUNCT
ejpam-5779	63	68	4	4	NUM
ejpam-5779	63	69	-	-	PUNCT
ejpam-5779	63	70	fuzzy	fuzzy	ADJ
ejpam-5779	63	71	)	)	PUNCT
ejpam-5779	63	72	ideal	ideal	NOUN
ejpam-5779	63	73	of	of	ADP
ejpam-5779	63	74	a.107	a.107	DET
ejpam-5779	63	75	proposition	proposition	NOUN
ejpam-5779	63	76	2	2	NUM
ejpam-5779	63	77	.	.	PUNCT
ejpam-5779	63	78	given	give	VERB
ejpam-5779	63	79	an	an	DET
ejpam-5779	63	80	interval	interval	NOUN
ejpam-5779	63	81	-	-	PUNCT
ejpam-5779	63	82	valued	value	VERB
ejpam-5779	63	83	fuzzy	fuzzy	ADJ
ejpam-5779	63	84	structure	structure	NOUN
ejpam-5779	63	85	(	(	PUNCT
ejpam-5779	63	86	a	a	PRON
ejpam-5779	63	87	,	,	PUNCT
ejpam-5779	63	88	f̃	f̃	PROPN
ejpam-5779	63	89	)	)	PUNCT
ejpam-5779	63	90	on	on	ADP
ejpam-5779	63	91	a	a	PRON
ejpam-5779	63	92	,	,	PUNCT
ejpam-5779	63	93	the	the	DET
ejpam-5779	63	94	following	follow	VERB
ejpam-5779	63	95	state-108	state-108	ADJ
ejpam-5779	63	96	ments	ment	NOUN
ejpam-5779	63	97	hold.109	hold.109	PROPN
ejpam-5779	63	98	n.	n.	PROPN
ejpam-5779	63	99	rajesh	rajesh	PROPN
ejpam-5779	63	100	,	,	PUNCT
ejpam-5779	63	101	t.	t.	PROPN
ejpam-5779	63	102	oner	oner	NOUN
ejpam-5779	63	103	,	,	PUNCT
ejpam-5779	63	104	a.	a.	NOUN
ejpam-5779	63	105	iampan	iampan	PROPN
ejpam-5779	63	106	,	,	PUNCT
ejpam-5779	63	107	i.	i.	PROPN
ejpam-5779	63	108	senturk	senturk	PROPN
ejpam-5779	63	109	/	/	SYM
ejpam-5779	63	110	eur	eur	PROPN
ejpam-5779	63	111	.	.	PUNCT
ejpam-5779	64	1	j.	j.	PROPN
ejpam-5779	64	2	pure	pure	PROPN
ejpam-5779	64	3	appl	appl	PROPN
ejpam-5779	64	4	.	.	PROPN
ejpam-5779	64	5	math	math	PROPN
ejpam-5779	64	6	,	,	PUNCT
ejpam-5779	64	7	18	18	NUM
ejpam-5779	64	8	(	(	PUNCT
ejpam-5779	64	9	1	1	NUM
ejpam-5779	64	10	)	)	PUNCT
ejpam-5779	64	11	(	(	PUNCT
ejpam-5779	64	12	2025	2025	NUM
ejpam-5779	64	13	)	)	PUNCT
ejpam-5779	64	14	,	,	PUNCT
ejpam-5779	64	15	5779	5779	NUM
ejpam-5779	64	16	5	5	NUM
ejpam-5779	64	17	of	of	ADP
ejpam-5779	64	18	18	18	NUM
ejpam-5779	64	19	(	(	PUNCT
ejpam-5779	64	20	1	1	NUM
ejpam-5779	64	21	)	)	PUNCT
ejpam-5779	64	22	if	if	SCONJ
ejpam-5779	64	23	(	(	PUNCT
ejpam-5779	64	24	a	a	PRON
ejpam-5779	64	25	,	,	PUNCT
ejpam-5779	64	26	f̃	f̃	PROPN
ejpam-5779	64	27	)	)	PUNCT
ejpam-5779	64	28	is	be	AUX
ejpam-5779	64	29	a	a	DET
ejpam-5779	64	30	length	length	NOUN
ejpam-5779	64	31	k	k	ADJ
ejpam-5779	64	32	-	-	PUNCT
ejpam-5779	64	33	fuzzy	fuzzy	ADJ
ejpam-5779	64	34	ideal	ideal	NOUN
ejpam-5779	64	35	of	of	ADP
ejpam-5779	64	36	a	a	PRON
ejpam-5779	64	37	for	for	ADP
ejpam-5779	64	38	k	k	PROPN
ejpam-5779	64	39	∈	∈	PROPN
ejpam-5779	64	40	{	{	PUNCT
ejpam-5779	64	41	1	1	NUM
ejpam-5779	64	42	,	,	PUNCT
ejpam-5779	64	43	3	3	NUM
ejpam-5779	64	44	}	}	PUNCT
ejpam-5779	64	45	,	,	PUNCT
ejpam-5779	64	46	then	then	ADV
ejpam-5779	64	47	(	(	PUNCT
ejpam-5779	64	48	∀p	∀p	X
ejpam-5779	64	49	,	,	PUNCT
ejpam-5779	64	50	q	q	PUNCT
ejpam-5779	64	51	∈	∈	NOUN
ejpam-5779	64	52	a)(p	a)(p	X
ejpam-5779	64	53	≤	≤	PUNCT
ejpam-5779	65	1	p	p	NOUN
ejpam-5779	65	2	⇒	⇒	NOUN
ejpam-5779	65	3	f̃l(p	f̃l(p	PROPN
ejpam-5779	65	4	)	)	PUNCT
ejpam-5779	65	5	≥	≥	NOUN
ejpam-5779	65	6	f̃l(p	f̃l(p	NOUN
ejpam-5779	65	7	)	)	PUNCT
ejpam-5779	65	8	)	)	PUNCT
ejpam-5779	65	9	.	.	PUNCT
ejpam-5779	66	1	(	(	PUNCT
ejpam-5779	66	2	2	2	X
ejpam-5779	66	3	)	)	PUNCT
ejpam-5779	66	4	if	if	SCONJ
ejpam-5779	66	5	(	(	PUNCT
ejpam-5779	66	6	a	a	PRON
ejpam-5779	66	7	,	,	PUNCT
ejpam-5779	66	8	f̃	f̃	PROPN
ejpam-5779	66	9	)	)	PUNCT
ejpam-5779	66	10	is	be	AUX
ejpam-5779	66	11	a	a	DET
ejpam-5779	66	12	length	length	NOUN
ejpam-5779	66	13	k	k	ADJ
ejpam-5779	66	14	-	-	PUNCT
ejpam-5779	66	15	fuzzy	fuzzy	ADJ
ejpam-5779	66	16	ideal	ideal	NOUN
ejpam-5779	66	17	of	of	ADP
ejpam-5779	66	18	a	a	PRON
ejpam-5779	66	19	for	for	ADP
ejpam-5779	66	20	k	k	PROPN
ejpam-5779	66	21	∈	∈	PROPN
ejpam-5779	66	22	{	{	PUNCT
ejpam-5779	66	23	2	2	NUM
ejpam-5779	66	24	,	,	PUNCT
ejpam-5779	66	25	4	4	NUM
ejpam-5779	66	26	}	}	PUNCT
ejpam-5779	66	27	,	,	PUNCT
ejpam-5779	66	28	then	then	ADV
ejpam-5779	66	29	(	(	PUNCT
ejpam-5779	66	30	∀p	∀p	X
ejpam-5779	66	31	,	,	PUNCT
ejpam-5779	66	32	q	q	PUNCT
ejpam-5779	66	33	∈	∈	NOUN
ejpam-5779	66	34	a)(p	a)(p	VERB
ejpam-5779	66	35	≤	≤	X
ejpam-5779	66	36	q	q	PART
ejpam-5779	66	37	⇒	⇒	NOUN
ejpam-5779	66	38	f̃l(p	f̃l(p	PROPN
ejpam-5779	66	39	)	)	PUNCT
ejpam-5779	66	40	≤	≤	NOUN
ejpam-5779	66	41	f̃l(q	f̃l(q	PROPN
ejpam-5779	66	42	)	)	PUNCT
ejpam-5779	66	43	)	)	PUNCT
ejpam-5779	66	44	.	.	PUNCT
ejpam-5779	67	1	proof	proof	NOUN
ejpam-5779	67	2	.	.	PUNCT
ejpam-5779	68	1	let	let	VERB
ejpam-5779	68	2	p	p	PRON
ejpam-5779	68	3	,	,	PUNCT
ejpam-5779	68	4	q	q	ADJ
ejpam-5779	68	5	∈	∈	PROPN
ejpam-5779	68	6	a	a	PRON
ejpam-5779	68	7	be	be	AUX
ejpam-5779	68	8	such	such	ADJ
ejpam-5779	68	9	that	that	SCONJ
ejpam-5779	68	10	p	p	PROPN
ejpam-5779	68	11	≤	≤	X
ejpam-5779	68	12	q.	q.	NOUN
ejpam-5779	69	1	if	if	SCONJ
ejpam-5779	69	2	(	(	PUNCT
ejpam-5779	69	3	a	a	PRON
ejpam-5779	69	4	,	,	PUNCT
ejpam-5779	69	5	f̃	f̃	PROPN
ejpam-5779	69	6	)	)	PUNCT
ejpam-5779	69	7	is	be	AUX
ejpam-5779	69	8	a	a	DET
ejpam-5779	69	9	length	length	NOUN
ejpam-5779	69	10	k	k	ADJ
ejpam-5779	69	11	-	-	PUNCT
ejpam-5779	69	12	fuzzy	fuzzy	ADJ
ejpam-5779	69	13	ideal	ideal	NOUN
ejpam-5779	69	14	of	of	ADP
ejpam-5779	69	15	a	a	DET
ejpam-5779	69	16	for110	for110	PROPN
ejpam-5779	69	17	k	k	PROPN
ejpam-5779	69	18	∈	∈	PROPN
ejpam-5779	69	19	{	{	PUNCT
ejpam-5779	69	20	1	1	NUM
ejpam-5779	69	21	,	,	PUNCT
ejpam-5779	69	22	3	3	NUM
ejpam-5779	69	23	}	}	PUNCT
ejpam-5779	69	24	,	,	PUNCT
ejpam-5779	69	25	then111	then111	PROPN
ejpam-5779	69	26	f̃l(p	f̃l(p	PROPN
ejpam-5779	69	27	)	)	PUNCT
ejpam-5779	69	28	≥	≥	NOUN
ejpam-5779	69	29	min{f̃l(pq	min{f̃l(pq	NOUN
ejpam-5779	69	30	)	)	PUNCT
ejpam-5779	69	31	,	,	PUNCT
ejpam-5779	69	32	f̃l(q	f̃l(q	PROPN
ejpam-5779	69	33	)	)	PUNCT
ejpam-5779	69	34	}	}	PUNCT
ejpam-5779	69	35	=	=	SYM
ejpam-5779	69	36	min{f̃l(0	min{f̃l(0	PROPN
ejpam-5779	69	37	)	)	PUNCT
ejpam-5779	69	38	,	,	PUNCT
ejpam-5779	69	39	f̃l(q	f̃l(q	PROPN
ejpam-5779	69	40	)	)	PUNCT
ejpam-5779	69	41	}	}	PUNCT
ejpam-5779	69	42	=	=	SYM
ejpam-5779	69	43	f̃l(q	f̃l(q	X
ejpam-5779	69	44	)	)	PUNCT
ejpam-5779	69	45	and112	and112	ADJ
ejpam-5779	69	46	f̃l(p	f̃l(p	NOUN
ejpam-5779	69	47	)	)	PUNCT
ejpam-5779	69	48	≤	≤	NUM
ejpam-5779	69	49	max{f̃l(pq	max{f̃l(pq	NOUN
ejpam-5779	69	50	)	)	PUNCT
ejpam-5779	69	51	,	,	PUNCT
ejpam-5779	69	52	f̃l(q	f̃l(q	PROPN
ejpam-5779	69	53	)	)	PUNCT
ejpam-5779	69	54	}	}	PUNCT
ejpam-5779	69	55	=	=	SYM
ejpam-5779	69	56	max{f̃l(0	max{f̃l(0	NOUN
ejpam-5779	69	57	)	)	PUNCT
ejpam-5779	69	58	,	,	PUNCT
ejpam-5779	69	59	f̃l(q	f̃l(q	PROPN
ejpam-5779	69	60	)	)	PUNCT
ejpam-5779	69	61	}	}	PUNCT
ejpam-5779	69	62	=	=	PUNCT
ejpam-5779	69	63	f̃l(q	f̃l(q	PROPN
ejpam-5779	69	64	)	)	PUNCT
ejpam-5779	69	65	.	.	PUNCT
ejpam-5779	70	1	if	if	SCONJ
ejpam-5779	70	2	(	(	PUNCT
ejpam-5779	70	3	a	a	PRON
ejpam-5779	70	4	,	,	PUNCT
ejpam-5779	70	5	f̃	f̃	PROPN
ejpam-5779	70	6	)	)	PUNCT
ejpam-5779	70	7	is	be	AUX
ejpam-5779	70	8	a	a	DET
ejpam-5779	70	9	length	length	NOUN
ejpam-5779	70	10	k	k	ADJ
ejpam-5779	70	11	-	-	PUNCT
ejpam-5779	70	12	fuzzy	fuzzy	ADJ
ejpam-5779	70	13	ideal	ideal	NOUN
ejpam-5779	70	14	of	of	ADP
ejpam-5779	70	15	a	a	PRON
ejpam-5779	70	16	for	for	ADP
ejpam-5779	70	17	k	k	PROPN
ejpam-5779	70	18	∈	∈	PROPN
ejpam-5779	70	19	{	{	PUNCT
ejpam-5779	70	20	2	2	NUM
ejpam-5779	70	21	,	,	PUNCT
ejpam-5779	70	22	4	4	NUM
ejpam-5779	70	23	}	}	PUNCT
ejpam-5779	70	24	,	,	PUNCT
ejpam-5779	70	25	then113	then113	PROPN
ejpam-5779	70	26	f̃l(p	f̃l(p	PROPN
ejpam-5779	70	27	)	)	PUNCT
ejpam-5779	70	28	≥	≥	NOUN
ejpam-5779	70	29	min{f̃l(pq	min{f̃l(pq	NOUN
ejpam-5779	70	30	)	)	PUNCT
ejpam-5779	70	31	,	,	PUNCT
ejpam-5779	70	32	f̃l(q	f̃l(q	PROPN
ejpam-5779	70	33	)	)	PUNCT
ejpam-5779	70	34	}	}	PUNCT
ejpam-5779	70	35	=	=	SYM
ejpam-5779	70	36	min{f̃l(0	min{f̃l(0	PROPN
ejpam-5779	70	37	)	)	PUNCT
ejpam-5779	70	38	,	,	PUNCT
ejpam-5779	70	39	f̃l(q	f̃l(q	PROPN
ejpam-5779	70	40	)	)	PUNCT
ejpam-5779	70	41	}	}	PUNCT
ejpam-5779	71	1	=	=	SYM
ejpam-5779	71	2	f̃l(q	f̃l(q	X
ejpam-5779	71	3	)	)	PUNCT
ejpam-5779	71	4	and114	and114	PROPN
ejpam-5779	71	5	f̃l(p	f̃l(p	NOUN
ejpam-5779	71	6	)	)	PUNCT
ejpam-5779	71	7	≤	≤	NUM
ejpam-5779	71	8	max{f̃l(pq	max{f̃l(pq	NOUN
ejpam-5779	71	9	)	)	PUNCT
ejpam-5779	71	10	,	,	PUNCT
ejpam-5779	71	11	f̃l(q	f̃l(q	PROPN
ejpam-5779	71	12	)	)	PUNCT
ejpam-5779	71	13	}	}	PUNCT
ejpam-5779	71	14	=	=	SYM
ejpam-5779	71	15	max{f̃l(0	max{f̃l(0	NOUN
ejpam-5779	71	16	)	)	PUNCT
ejpam-5779	71	17	,	,	PUNCT
ejpam-5779	71	18	f̃l(q	f̃l(q	PROPN
ejpam-5779	71	19	)	)	PUNCT
ejpam-5779	71	20	}	}	PUNCT
ejpam-5779	71	21	=	=	PUNCT
ejpam-5779	71	22	f̃l(q	f̃l(q	PROPN
ejpam-5779	71	23	)	)	PUNCT
ejpam-5779	71	24	.	.	PUNCT
ejpam-5779	72	1	theorem	theorem	NOUN
ejpam-5779	72	2	1	1	NUM
ejpam-5779	72	3	.	.	X
ejpam-5779	73	1	for	for	ADP
ejpam-5779	73	2	any	any	DET
ejpam-5779	73	3	interval	interval	NOUN
ejpam-5779	73	4	-	-	PUNCT
ejpam-5779	73	5	valued	value	VERB
ejpam-5779	73	6	fuzzy	fuzzy	ADJ
ejpam-5779	73	7	structure	structure	NOUN
ejpam-5779	73	8	(	(	PUNCT
ejpam-5779	73	9	a	a	PRON
ejpam-5779	73	10	,	,	PUNCT
ejpam-5779	73	11	f̃	f̃	PROPN
ejpam-5779	73	12	)	)	PUNCT
ejpam-5779	73	13	on	on	ADP
ejpam-5779	73	14	a	a	PRON
ejpam-5779	73	15	,	,	PUNCT
ejpam-5779	73	16	the	the	DET
ejpam-5779	73	17	following	follow	VERB
ejpam-5779	73	18	assertions115	assertions115	PROPN
ejpam-5779	73	19	are	be	AUX
ejpam-5779	73	20	true:116	true:116	DET
ejpam-5779	73	21	(	(	PUNCT
ejpam-5779	73	22	1	1	NUM
ejpam-5779	73	23	)	)	PUNCT
ejpam-5779	73	24	every	every	DET
ejpam-5779	73	25	length	length	NOUN
ejpam-5779	73	26	3	3	NUM
ejpam-5779	73	27	-	-	PUNCT
ejpam-5779	73	28	fuzzy	fuzzy	ADJ
ejpam-5779	73	29	ideal	ideal	NOUN
ejpam-5779	73	30	of	of	ADP
ejpam-5779	73	31	a	a	PRON
ejpam-5779	73	32	is	be	AUX
ejpam-5779	73	33	also	also	ADV
ejpam-5779	73	34	a	a	DET
ejpam-5779	73	35	length	length	NOUN
ejpam-5779	73	36	1	1	NUM
ejpam-5779	73	37	-	-	PUNCT
ejpam-5779	73	38	fuzzy	fuzzy	ADJ
ejpam-5779	73	39	ideal	ideal	NOUN
ejpam-5779	73	40	of	of	ADP
ejpam-5779	73	41	a.117	a.117	NUM
ejpam-5779	73	42	(	(	PUNCT
ejpam-5779	73	43	2	2	NUM
ejpam-5779	73	44	)	)	PUNCT
ejpam-5779	73	45	every	every	DET
ejpam-5779	73	46	length	length	NOUN
ejpam-5779	73	47	2	2	NUM
ejpam-5779	73	48	-	-	PUNCT
ejpam-5779	73	49	fuzzy	fuzzy	ADJ
ejpam-5779	73	50	ideal	ideal	NOUN
ejpam-5779	73	51	of	of	ADP
ejpam-5779	73	52	a	a	PRON
ejpam-5779	73	53	is	be	AUX
ejpam-5779	73	54	also	also	ADV
ejpam-5779	73	55	a	a	DET
ejpam-5779	73	56	length	length	NOUN
ejpam-5779	73	57	4	4	NUM
ejpam-5779	73	58	-	-	PUNCT
ejpam-5779	73	59	fuzzy	fuzzy	ADJ
ejpam-5779	73	60	ideal	ideal	NOUN
ejpam-5779	73	61	of	of	ADP
ejpam-5779	73	62	a.118	a.118	VERB
ejpam-5779	73	63	proof	proof	NOUN
ejpam-5779	73	64	.	.	PUNCT
ejpam-5779	74	1	(	(	PUNCT
ejpam-5779	74	2	1	1	X
ejpam-5779	74	3	)	)	PUNCT
ejpam-5779	74	4	let	let	VERB
ejpam-5779	74	5	(	(	PUNCT
ejpam-5779	74	6	a	a	PRON
ejpam-5779	74	7	,	,	PUNCT
ejpam-5779	74	8	f̃	f̃	PROPN
ejpam-5779	74	9	)	)	PUNCT
ejpam-5779	74	10	be	be	VERB
ejpam-5779	74	11	a	a	DET
ejpam-5779	74	12	length	length	NOUN
ejpam-5779	74	13	3	3	NUM
ejpam-5779	74	14	-	-	PUNCT
ejpam-5779	74	15	fuzzy	fuzzy	ADJ
ejpam-5779	74	16	ideal	ideal	NOUN
ejpam-5779	74	17	of	of	ADP
ejpam-5779	74	18	a	a	PRON
ejpam-5779	74	19	and	and	CCONJ
ejpam-5779	74	20	p	p	NOUN
ejpam-5779	74	21	,	,	PUNCT
ejpam-5779	74	22	q	q	PROPN
ejpam-5779	74	23	∈	∈	PROPN
ejpam-5779	74	24	a.	a.	NOUN
ejpam-5779	74	25	then119	then119	PROPN
ejpam-5779	74	26	f̃l(p	f̃l(p	PROPN
ejpam-5779	74	27	)	)	PUNCT
ejpam-5779	74	28	≥	≥	NOUN
ejpam-5779	74	29	max{f̃l(pq	max{f̃l(pq	NOUN
ejpam-5779	74	30	)	)	PUNCT
ejpam-5779	74	31	,	,	PUNCT
ejpam-5779	74	32	f̃l(q	f̃l(q	PROPN
ejpam-5779	74	33	)	)	PUNCT
ejpam-5779	74	34	}	}	PUNCT
ejpam-5779	74	35	≥	≥	X
ejpam-5779	74	36	min{f̃l(pq	min{f̃l(pq	NOUN
ejpam-5779	74	37	)	)	PUNCT
ejpam-5779	74	38	,	,	PUNCT
ejpam-5779	74	39	f̃l(q	f̃l(q	PROPN
ejpam-5779	74	40	)	)	PUNCT
ejpam-5779	74	41	}	}	PUNCT
ejpam-5779	74	42	.	.	PUNCT
ejpam-5779	75	1	hence	hence	ADV
ejpam-5779	75	2	,	,	PUNCT
ejpam-5779	75	3	(	(	PUNCT
ejpam-5779	75	4	a	a	PRON
ejpam-5779	75	5	,	,	PUNCT
ejpam-5779	75	6	f̃	f̃	PROPN
ejpam-5779	75	7	)	)	PUNCT
ejpam-5779	75	8	is	be	AUX
ejpam-5779	75	9	a	a	DET
ejpam-5779	75	10	length	length	NOUN
ejpam-5779	75	11	1	1	NUM
ejpam-5779	75	12	-	-	PUNCT
ejpam-5779	75	13	fuzzy	fuzzy	ADJ
ejpam-5779	75	14	ideal	ideal	NOUN
ejpam-5779	75	15	of	of	ADP
ejpam-5779	75	16	a.120	a.120	ADJ
ejpam-5779	75	17	(	(	PUNCT
ejpam-5779	75	18	2	2	NUM
ejpam-5779	75	19	)	)	PUNCT
ejpam-5779	75	20	let	let	VERB
ejpam-5779	75	21	(	(	PUNCT
ejpam-5779	75	22	a	a	PRON
ejpam-5779	75	23	,	,	PUNCT
ejpam-5779	75	24	f̃	f̃	PROPN
ejpam-5779	75	25	)	)	PUNCT
ejpam-5779	75	26	be	be	VERB
ejpam-5779	75	27	a	a	DET
ejpam-5779	75	28	length	length	NOUN
ejpam-5779	75	29	2	2	NUM
ejpam-5779	75	30	-	-	PUNCT
ejpam-5779	75	31	fuzzy	fuzzy	ADJ
ejpam-5779	75	32	ideal	ideal	NOUN
ejpam-5779	75	33	of	of	ADP
ejpam-5779	75	34	a	a	PRON
ejpam-5779	75	35	and	and	CCONJ
ejpam-5779	75	36	p	p	NOUN
ejpam-5779	75	37	,	,	PUNCT
ejpam-5779	75	38	q	q	PROPN
ejpam-5779	75	39	∈	∈	PROPN
ejpam-5779	75	40	a.	a.	NOUN
ejpam-5779	75	41	then121	then121	PROPN
ejpam-5779	75	42	f̃l(p	f̃l(p	NUM
ejpam-5779	75	43	)	)	PUNCT
ejpam-5779	75	44	≤	≤	NOUN
ejpam-5779	75	45	min{f̃l(pq	min{f̃l(pq	NOUN
ejpam-5779	75	46	)	)	PUNCT
ejpam-5779	75	47	,	,	PUNCT
ejpam-5779	75	48	f̃l(q	f̃l(q	PROPN
ejpam-5779	75	49	)	)	PUNCT
ejpam-5779	75	50	}	}	PUNCT
ejpam-5779	75	51	≤	≤	NUM
ejpam-5779	75	52	max{f̃l(pq	max{f̃l(pq	NOUN
ejpam-5779	75	53	)	)	PUNCT
ejpam-5779	75	54	,	,	PUNCT
ejpam-5779	75	55	f̃l(q	f̃l(q	PROPN
ejpam-5779	75	56	)	)	PUNCT
ejpam-5779	75	57	}	}	PUNCT
ejpam-5779	75	58	.	.	PUNCT
ejpam-5779	76	1	hence	hence	ADV
ejpam-5779	76	2	,	,	PUNCT
ejpam-5779	76	3	(	(	PUNCT
ejpam-5779	76	4	a	a	PRON
ejpam-5779	76	5	,	,	PUNCT
ejpam-5779	76	6	f̃	f̃	PROPN
ejpam-5779	76	7	)	)	PUNCT
ejpam-5779	76	8	is	be	AUX
ejpam-5779	76	9	a	a	DET
ejpam-5779	76	10	length	length	NOUN
ejpam-5779	76	11	4	4	NUM
ejpam-5779	76	12	-	-	PUNCT
ejpam-5779	76	13	fuzzy	fuzzy	ADJ
ejpam-5779	76	14	ideal	ideal	NOUN
ejpam-5779	76	15	of	of	ADP
ejpam-5779	76	16	a.122	a.122	PROPN
ejpam-5779	76	17	n.	n.	PROPN
ejpam-5779	76	18	rajesh	rajesh	PROPN
ejpam-5779	76	19	,	,	PUNCT
ejpam-5779	76	20	t.	t.	PROPN
ejpam-5779	76	21	oner	oner	NOUN
ejpam-5779	76	22	,	,	PUNCT
ejpam-5779	76	23	a.	a.	NOUN
ejpam-5779	76	24	iampan	iampan	PROPN
ejpam-5779	76	25	,	,	PUNCT
ejpam-5779	76	26	i.	i.	PROPN
ejpam-5779	76	27	senturk	senturk	PROPN
ejpam-5779	76	28	/	/	SYM
ejpam-5779	76	29	eur	eur	PROPN
ejpam-5779	76	30	.	.	PUNCT
ejpam-5779	77	1	j.	j.	PROPN
ejpam-5779	77	2	pure	pure	PROPN
ejpam-5779	77	3	appl	appl	PROPN
ejpam-5779	77	4	.	.	PROPN
ejpam-5779	77	5	math	math	PROPN
ejpam-5779	77	6	,	,	PUNCT
ejpam-5779	77	7	18	18	NUM
ejpam-5779	77	8	(	(	PUNCT
ejpam-5779	77	9	1	1	NUM
ejpam-5779	77	10	)	)	PUNCT
ejpam-5779	77	11	(	(	PUNCT
ejpam-5779	77	12	2025	2025	NUM
ejpam-5779	77	13	)	)	PUNCT
ejpam-5779	77	14	,	,	PUNCT
ejpam-5779	77	15	5779	5779	NUM
ejpam-5779	77	16	6	6	NUM
ejpam-5779	77	17	of	of	ADP
ejpam-5779	77	18	18	18	NUM
ejpam-5779	77	19	theorem	theorem	NOUN
ejpam-5779	77	20	2	2	NUM
ejpam-5779	77	21	.	.	PUNCT
ejpam-5779	77	22	given	give	VERB
ejpam-5779	77	23	an	an	DET
ejpam-5779	77	24	ideal	ideal	NOUN
ejpam-5779	77	25	s	s	NOUN
ejpam-5779	77	26	of	of	ADP
ejpam-5779	77	27	a	a	PRON
ejpam-5779	77	28	and	and	CCONJ
ejpam-5779	77	29	b1	b1	NOUN
ejpam-5779	77	30	,	,	PUNCT
ejpam-5779	77	31	b2	b2	NOUN
ejpam-5779	77	32	∈	∈	NOUN
ejpam-5779	77	33	p	p	X
ejpam-5779	77	34	(	(	PUNCT
ejpam-5779	77	35	[	[	X
ejpam-5779	77	36	0	0	NUM
ejpam-5779	77	37	,	,	PUNCT
ejpam-5779	77	38	1	1	NUM
ejpam-5779	77	39	]	]	NUM
ejpam-5779	77	40	)	)	PUNCT
ejpam-5779	77	41	,	,	PUNCT
ejpam-5779	77	42	let	let	VERB
ejpam-5779	77	43	(	(	PUNCT
ejpam-5779	77	44	a	a	PRON
ejpam-5779	77	45	,	,	PUNCT
ejpam-5779	77	46	f̃	f̃	PROPN
ejpam-5779	77	47	)	)	PUNCT
ejpam-5779	77	48	be	be	VERB
ejpam-5779	77	49	an	an	DET
ejpam-5779	77	50	interval	interval	NOUN
ejpam-5779	77	51	-	-	PUNCT
ejpam-5779	77	52	valued	value	VERB
ejpam-5779	77	53	fuzzy	fuzzy	ADJ
ejpam-5779	77	54	structure	structure	NOUN
ejpam-5779	77	55	over	over	ADP
ejpam-5779	77	56	a	a	DET
ejpam-5779	77	57	given	give	VERB
ejpam-5779	77	58	by	by	ADP
ejpam-5779	77	59	f̃	f̃	PROPN
ejpam-5779	77	60	:	:	PUNCT
ejpam-5779	77	61	a	a	DET
ejpam-5779	77	62	→	→	X
ejpam-5779	77	63	p	p	X
ejpam-5779	77	64	(	(	PUNCT
ejpam-5779	77	65	[	[	X
ejpam-5779	77	66	0	0	NUM
ejpam-5779	77	67	,	,	PUNCT
ejpam-5779	77	68	1	1	NUM
ejpam-5779	77	69	]	]	NUM
ejpam-5779	77	70	)	)	PUNCT
ejpam-5779	77	71	;	;	PUNCT
ejpam-5779	77	72	p	p	PROPN
ejpam-5779	77	73	7→	7→	PROPN
ejpam-5779	77	74	{	{	PUNCT
ejpam-5779	77	75	b2	b2	NOUN
ejpam-5779	77	76	if	if	SCONJ
ejpam-5779	77	77	p	p	PROPN
ejpam-5779	77	78	∈	∈	PROPN
ejpam-5779	77	79	s	s	PART
ejpam-5779	77	80	,	,	PUNCT
ejpam-5779	77	81	b1	b1	NOUN
ejpam-5779	77	82	otherwise	otherwise	ADV
ejpam-5779	77	83	.	.	PUNCT
ejpam-5779	78	1	(	(	PUNCT
ejpam-5779	78	2	1	1	X
ejpam-5779	78	3	)	)	PUNCT
ejpam-5779	78	4	if	if	SCONJ
ejpam-5779	78	5	b1	b1	PROPN
ejpam-5779	78	6	⊂	⊂	PROPN
ejpam-5779	78	7	b2	b2	PROPN
ejpam-5779	78	8	,	,	PUNCT
ejpam-5779	78	9	then	then	ADV
ejpam-5779	78	10	(	(	PUNCT
ejpam-5779	78	11	a	a	PRON
ejpam-5779	78	12	,	,	PUNCT
ejpam-5779	78	13	f̃	f̃	PROPN
ejpam-5779	78	14	)	)	PUNCT
ejpam-5779	78	15	is	be	AUX
ejpam-5779	78	16	a	a	DET
ejpam-5779	78	17	length	length	NOUN
ejpam-5779	78	18	1	1	NUM
ejpam-5779	78	19	-	-	PUNCT
ejpam-5779	78	20	fuzzy	fuzzy	ADJ
ejpam-5779	78	21	ideal	ideal	NOUN
ejpam-5779	78	22	of	of	ADP
ejpam-5779	78	23	a.123	a.123	NOUN
ejpam-5779	78	24	(	(	PUNCT
ejpam-5779	78	25	2	2	NUM
ejpam-5779	78	26	)	)	PUNCT
ejpam-5779	78	27	if	if	SCONJ
ejpam-5779	78	28	b2	b2	NOUN
ejpam-5779	78	29	⊂	⊂	PROPN
ejpam-5779	78	30	b1	b1	PROPN
ejpam-5779	78	31	,	,	PUNCT
ejpam-5779	78	32	then	then	ADV
ejpam-5779	78	33	(	(	PUNCT
ejpam-5779	78	34	a	a	PRON
ejpam-5779	78	35	,	,	PUNCT
ejpam-5779	78	36	f̃	f̃	PROPN
ejpam-5779	78	37	)	)	PUNCT
ejpam-5779	78	38	is	be	AUX
ejpam-5779	78	39	a	a	DET
ejpam-5779	78	40	length	length	NOUN
ejpam-5779	78	41	4	4	NUM
ejpam-5779	78	42	-	-	PUNCT
ejpam-5779	78	43	fuzzy	fuzzy	ADJ
ejpam-5779	78	44	ideal	ideal	NOUN
ejpam-5779	78	45	of	of	ADP
ejpam-5779	78	46	a.124	a.124	NOUN
ejpam-5779	78	47	proof	proof	NOUN
ejpam-5779	78	48	.	.	PUNCT
ejpam-5779	79	1	if	if	SCONJ
ejpam-5779	79	2	p	p	PROPN
ejpam-5779	79	3	∈	∈	PROPN
ejpam-5779	79	4	s	s	NOUN
ejpam-5779	79	5	,	,	PUNCT
ejpam-5779	79	6	then	then	ADV
ejpam-5779	79	7	f̃(p	f̃(p	NOUN
ejpam-5779	79	8	)	)	PUNCT
ejpam-5779	79	9	=	=	SYM
ejpam-5779	79	10	b2	b2	NOUN
ejpam-5779	79	11	and	and	CCONJ
ejpam-5779	79	12	so	so	ADV
ejpam-5779	79	13	f̃l(p	f̃l(p	ADJ
ejpam-5779	79	14	)	)	PUNCT
ejpam-5779	79	15	=	=	SYM
ejpam-5779	79	16	f̃sup(p)−	f̃sup(p)−	PROPN
ejpam-5779	79	17	f̃inf(p	f̃inf(p	NOUN
ejpam-5779	79	18	)	)	PUNCT
ejpam-5779	79	19	=	=	SYM
ejpam-5779	79	20	sup	sup	NOUN
ejpam-5779	79	21	f̃(p)−	f̃(p)−	ADJ
ejpam-5779	79	22	inf	inf	NOUN
ejpam-5779	79	23	f̃(p	f̃(p	PROPN
ejpam-5779	79	24	)	)	PUNCT
ejpam-5779	79	25	=	=	SYM
ejpam-5779	79	26	supb2	supb2	NOUN
ejpam-5779	79	27	−	−	PROPN
ejpam-5779	79	28	inf	inf	PROPN
ejpam-5779	79	29	b2	b2	NOUN
ejpam-5779	79	30	.	.	PUNCT
ejpam-5779	80	1	if	if	SCONJ
ejpam-5779	80	2	p	p	X
ejpam-5779	80	3	/∈	/∈	PUNCT
ejpam-5779	80	4	s	s	X
ejpam-5779	80	5	,	,	PUNCT
ejpam-5779	80	6	then	then	ADV
ejpam-5779	80	7	f̃(p	f̃(p	NOUN
ejpam-5779	80	8	)	)	PUNCT
ejpam-5779	80	9	=	=	SYM
ejpam-5779	80	10	b1	b1	NOUN
ejpam-5779	80	11	and	and	CCONJ
ejpam-5779	80	12	so	so	ADV
ejpam-5779	80	13	f̃l(p	f̃l(p	ADJ
ejpam-5779	80	14	)	)	PUNCT
ejpam-5779	80	15	=	=	SYM
ejpam-5779	80	16	f̃sup(p)−	f̃sup(p)−	PROPN
ejpam-5779	80	17	f̃inf(p	f̃inf(p	NOUN
ejpam-5779	80	18	)	)	PUNCT
ejpam-5779	80	19	=	=	SYM
ejpam-5779	80	20	sup	sup	NOUN
ejpam-5779	80	21	f̃(p)−	f̃(p)−	ADJ
ejpam-5779	80	22	inf	inf	NOUN
ejpam-5779	80	23	f̃(p	f̃(p	NOUN
ejpam-5779	80	24	)	)	PUNCT
ejpam-5779	80	25	=	=	SYM
ejpam-5779	81	1	supb1	supb1	NOUN
ejpam-5779	82	1	−	−	PROPN
ejpam-5779	82	2	inf	inf	PROPN
ejpam-5779	82	3	b1	b1	NOUN
ejpam-5779	82	4	.	.	PUNCT
ejpam-5779	83	1	(	(	PUNCT
ejpam-5779	83	2	1	1	X
ejpam-5779	83	3	)	)	PUNCT
ejpam-5779	83	4	assume	assume	VERB
ejpam-5779	83	5	that	that	SCONJ
ejpam-5779	83	6	b1	b1	PROPN
ejpam-5779	83	7	⊂	⊂	PROPN
ejpam-5779	83	8	b2	b2	PROPN
ejpam-5779	83	9	.	.	PUNCT
ejpam-5779	84	1	then	then	ADV
ejpam-5779	84	2	supb2	supb2	PROPN
ejpam-5779	84	3	−	−	PROPN
ejpam-5779	84	4	inf	inf	PROPN
ejpam-5779	84	5	b2	b2	PROPN
ejpam-5779	84	6	≥	≥	PROPN
ejpam-5779	84	7	supb1	supb1	NOUN
ejpam-5779	85	1	−	−	PROPN
ejpam-5779	85	2	inf	inf	PROPN
ejpam-5779	85	3	b1	b1	NOUN
ejpam-5779	85	4	.	.	PUNCT
ejpam-5779	86	1	since	since	SCONJ
ejpam-5779	86	2	0	0	NUM
ejpam-5779	86	3	∈	∈	PROPN
ejpam-5779	86	4	i,125	i,125	NOUN
ejpam-5779	86	5	f̃l(0	f̃l(0	NOUN
ejpam-5779	86	6	)	)	PUNCT
ejpam-5779	86	7	=	=	SYM
ejpam-5779	86	8	f̃sup(0)−	f̃sup(0)−	NUM
ejpam-5779	86	9	f̃inf(0	f̃inf(0	PROPN
ejpam-5779	86	10	)	)	PUNCT
ejpam-5779	86	11	=	=	SYM
ejpam-5779	86	12	supb2	supb2	NOUN
ejpam-5779	86	13	−	−	PROPN
ejpam-5779	86	14	inf	inf	PROPN
ejpam-5779	86	15	b2	b2	PROPN
ejpam-5779	86	16	≥	≥	NOUN
ejpam-5779	86	17	f̃l(p	f̃l(p	PUNCT
ejpam-5779	86	18	)	)	PUNCT
ejpam-5779	86	19	for	for	ADP
ejpam-5779	86	20	all	all	PRON
ejpam-5779	86	21	p	p	PROPN
ejpam-5779	86	22	∈	∈	NOUN
ejpam-5779	86	23	a.126	a.126	NOUN
ejpam-5779	86	24	case	case	NOUN
ejpam-5779	86	25	1	1	NUM
ejpam-5779	86	26	:	:	PUNCT
ejpam-5779	86	27	let	let	VERB
ejpam-5779	86	28	pq	pq	INTJ
ejpam-5779	86	29	,	,	PUNCT
ejpam-5779	86	30	q	q	PROPN
ejpam-5779	86	31	∈	∈	PROPN
ejpam-5779	86	32	s.	s.	PROPN
ejpam-5779	86	33	then	then	ADV
ejpam-5779	86	34	f̃l(p	f̃l(p	PROPN
ejpam-5779	86	35	q	q	NOUN
ejpam-5779	86	36	)	)	PUNCT
ejpam-5779	86	37	=	=	SYM
ejpam-5779	86	38	supb2	supb2	NOUN
ejpam-5779	87	1	−	−	PROPN
ejpam-5779	87	2	inf	inf	NOUN
ejpam-5779	87	3	b2	b2	NOUN
ejpam-5779	87	4	and	and	CCONJ
ejpam-5779	87	5	f̃l(q	f̃l(q	PRON
ejpam-5779	87	6	)	)	PUNCT
ejpam-5779	87	7	=	=	SYM
ejpam-5779	88	1	supb2	supb2	NOUN
ejpam-5779	88	2	−	−	PROPN
ejpam-5779	88	3	inf	inf	PROPN
ejpam-5779	88	4	b2.127	b2.127	PROPN
ejpam-5779	88	5	thus	thus	ADV
ejpam-5779	88	6	,	,	PUNCT
ejpam-5779	88	7	min{f̃l(pq	min{f̃l(pq	NOUN
ejpam-5779	88	8	)	)	PUNCT
ejpam-5779	88	9	,	,	PUNCT
ejpam-5779	88	10	f̃l(q	f̃l(q	PROPN
ejpam-5779	88	11	)	)	PUNCT
ejpam-5779	88	12	}	}	PUNCT
ejpam-5779	88	13	=	=	SYM
ejpam-5779	88	14	supb2	supb2	NOUN
ejpam-5779	88	15	−	−	PROPN
ejpam-5779	88	16	inf	inf	PROPN
ejpam-5779	88	17	b2	b2	NOUN
ejpam-5779	88	18	.	.	PUNCT
ejpam-5779	89	1	since	since	SCONJ
ejpam-5779	89	2	s	s	PROPN
ejpam-5779	89	3	is	be	AUX
ejpam-5779	89	4	an	an	DET
ejpam-5779	89	5	ideal	ideal	NOUN
ejpam-5779	89	6	of	of	ADP
ejpam-5779	89	7	a	a	PRON
ejpam-5779	89	8	,	,	PUNCT
ejpam-5779	89	9	p	p	PROPN
ejpam-5779	89	10	∈	∈	PROPN
ejpam-5779	89	11	s	s	X
ejpam-5779	89	12	and	and	CCONJ
ejpam-5779	89	13	so128	so128	ADJ
ejpam-5779	89	14	f̃l(p	f̃l(p	NOUN
ejpam-5779	89	15	)	)	PUNCT
ejpam-5779	89	16	=	=	SYM
ejpam-5779	89	17	supb2	supb2	NOUN
ejpam-5779	89	18	−	−	PROPN
ejpam-5779	89	19	inf	inf	PROPN
ejpam-5779	89	20	b2	b2	NOUN
ejpam-5779	89	21	.	.	PUNCT
ejpam-5779	90	1	thus	thus	ADV
ejpam-5779	90	2	,	,	PUNCT
ejpam-5779	90	3	f̃l(p	f̃l(p	ADJ
ejpam-5779	90	4	)	)	PUNCT
ejpam-5779	90	5	=	=	SYM
ejpam-5779	90	6	supb2	supb2	NOUN
ejpam-5779	90	7	−	−	PROPN
ejpam-5779	90	8	inf	inf	PROPN
ejpam-5779	90	9	b2	b2	NOUN
ejpam-5779	90	10	=	=	SYM
ejpam-5779	90	11	min{f̃l(pq	min{f̃l(pq	NOUN
ejpam-5779	90	12	)	)	PUNCT
ejpam-5779	90	13	,	,	PUNCT
ejpam-5779	90	14	f̃l(q)}.129	f̃l(q)}.129	PROPN
ejpam-5779	90	15	case	case	NOUN
ejpam-5779	90	16	2	2	NUM
ejpam-5779	90	17	:	:	PUNCT
ejpam-5779	90	18	let	let	VERB
ejpam-5779	90	19	pq	pq	INTJ
ejpam-5779	90	20	,	,	PUNCT
ejpam-5779	90	21	q	q	PROPN
ejpam-5779	90	22	/∈	/∈	PUNCT
ejpam-5779	91	1	s.	s.	PROPN
ejpam-5779	91	2	then	then	ADV
ejpam-5779	91	3	f̃l(p	f̃l(p	PROPN
ejpam-5779	91	4	q	q	NOUN
ejpam-5779	91	5	)	)	PUNCT
ejpam-5779	91	6	=	=	SYM
ejpam-5779	91	7	supb1	supb1	NOUN
ejpam-5779	92	1	−	−	PROPN
ejpam-5779	92	2	inf	inf	PROPN
ejpam-5779	92	3	b1	b1	NOUN
ejpam-5779	92	4	and	and	CCONJ
ejpam-5779	92	5	f̃l(q	f̃l(q	PRON
ejpam-5779	92	6	)	)	PUNCT
ejpam-5779	92	7	=	=	SYM
ejpam-5779	93	1	supb1	supb1	NOUN
ejpam-5779	94	1	−	−	PROPN
ejpam-5779	94	2	inf	inf	PROPN
ejpam-5779	94	3	b1	b1	NOUN
ejpam-5779	94	4	,	,	PUNCT
ejpam-5779	94	5	so130	so130	PROPN
ejpam-5779	94	6	min{f̃l(pq	min{f̃l(pq	NOUN
ejpam-5779	94	7	)	)	PUNCT
ejpam-5779	94	8	,	,	PUNCT
ejpam-5779	94	9	f̃l(q	f̃l(q	PROPN
ejpam-5779	94	10	)	)	PUNCT
ejpam-5779	94	11	}	}	PUNCT
ejpam-5779	94	12	=	=	SYM
ejpam-5779	94	13	supb1	supb1	NOUN
ejpam-5779	95	1	−	−	PROPN
ejpam-5779	95	2	inf	inf	PROPN
ejpam-5779	95	3	b1	b1	NOUN
ejpam-5779	95	4	.	.	PUNCT
ejpam-5779	96	1	thus	thus	ADV
ejpam-5779	96	2	,	,	PUNCT
ejpam-5779	96	3	f̃l(p	f̃l(p	PROPN
ejpam-5779	96	4	)	)	PUNCT
ejpam-5779	96	5	≥	≥	NOUN
ejpam-5779	96	6	supb1	supb1	NOUN
ejpam-5779	97	1	−	−	PROPN
ejpam-5779	97	2	inf	inf	PROPN
ejpam-5779	97	3	b1	b1	NOUN
ejpam-5779	97	4	=	=	SYM
ejpam-5779	97	5	min{f̃l(pq	min{f̃l(pq	NOUN
ejpam-5779	97	6	)	)	PUNCT
ejpam-5779	97	7	,	,	PUNCT
ejpam-5779	97	8	f̃l(q)}.131	f̃l(q)}.131	NOUN
ejpam-5779	97	9	case	case	NOUN
ejpam-5779	97	10	3	3	X
ejpam-5779	97	11	:	:	PUNCT
ejpam-5779	97	12	let	let	VERB
ejpam-5779	97	13	pq	pq	INTJ
ejpam-5779	97	14	/∈	/∈	PUNCT
ejpam-5779	97	15	s	s	PROPN
ejpam-5779	97	16	and	and	CCONJ
ejpam-5779	97	17	q	q	PROPN
ejpam-5779	97	18	∈	∈	PROPN
ejpam-5779	97	19	s.	s.	PROPN
ejpam-5779	97	20	then	then	ADV
ejpam-5779	97	21	f̃l(p	f̃l(p	PROPN
ejpam-5779	97	22	q	q	NOUN
ejpam-5779	97	23	)	)	PUNCT
ejpam-5779	97	24	=	=	SYM
ejpam-5779	97	25	supb1	supb1	NOUN
ejpam-5779	98	1	−	−	PROPN
ejpam-5779	98	2	inf	inf	PROPN
ejpam-5779	98	3	b1	b1	NOUN
ejpam-5779	98	4	and	and	CCONJ
ejpam-5779	98	5	f̃l(q	f̃l(q	PRON
ejpam-5779	98	6	)	)	PUNCT
ejpam-5779	98	7	=	=	SYM
ejpam-5779	98	8	supb2	supb2	PROPN
ejpam-5779	98	9	−132	−132	PROPN
ejpam-5779	98	10	inf	inf	PROPN
ejpam-5779	98	11	b2	b2	NOUN
ejpam-5779	98	12	,	,	PUNCT
ejpam-5779	98	13	so	so	ADV
ejpam-5779	98	14	min{f̃l(pq	min{f̃l(pq	ADJ
ejpam-5779	98	15	)	)	PUNCT
ejpam-5779	98	16	,	,	PUNCT
ejpam-5779	98	17	f̃l(q	f̃l(q	PROPN
ejpam-5779	98	18	)	)	PUNCT
ejpam-5779	98	19	}	}	PUNCT
ejpam-5779	98	20	=	=	SYM
ejpam-5779	98	21	supb1−inf	supb1−inf	NOUN
ejpam-5779	98	22	b1	b1	NOUN
ejpam-5779	98	23	.	.	PUNCT
ejpam-5779	99	1	thus	thus	ADV
ejpam-5779	99	2	,	,	PUNCT
ejpam-5779	99	3	f̃l(p	f̃l(p	PROPN
ejpam-5779	99	4	)	)	PUNCT
ejpam-5779	99	5	≥	≥	NOUN
ejpam-5779	99	6	supb1−inf	supb1−inf	NOUN
ejpam-5779	99	7	b1	b1	NOUN
ejpam-5779	99	8	=	=	SYM
ejpam-5779	99	9	min{f̃l(pq	min{f̃l(pq	NOUN
ejpam-5779	99	10	)	)	PUNCT
ejpam-5779	99	11	,	,	PUNCT
ejpam-5779	99	12	f̃l(q)}.133	f̃l(q)}.133	PROPN
ejpam-5779	99	13	case	case	NOUN
ejpam-5779	99	14	4	4	NUM
ejpam-5779	99	15	:	:	PUNCT
ejpam-5779	99	16	let	let	VERB
ejpam-5779	99	17	pq	pq	INTJ
ejpam-5779	99	18	∈	∈	PROPN
ejpam-5779	99	19	s	s	PART
ejpam-5779	99	20	and	and	CCONJ
ejpam-5779	99	21	q	q	PROPN
ejpam-5779	99	22	/∈	/∈	PUNCT
ejpam-5779	100	1	s.	s.	PROPN
ejpam-5779	100	2	then	then	ADV
ejpam-5779	100	3	f̃l(p	f̃l(p	PROPN
ejpam-5779	100	4	q	q	NOUN
ejpam-5779	100	5	)	)	PUNCT
ejpam-5779	100	6	=	=	SYM
ejpam-5779	100	7	supb2	supb2	NOUN
ejpam-5779	101	1	−	−	PROPN
ejpam-5779	101	2	inf	inf	NOUN
ejpam-5779	101	3	b2	b2	NOUN
ejpam-5779	101	4	and	and	CCONJ
ejpam-5779	101	5	f̃l(q	f̃l(q	PRON
ejpam-5779	101	6	)	)	PUNCT
ejpam-5779	101	7	=	=	SYM
ejpam-5779	101	8	supb1	supb1	PROPN
ejpam-5779	101	9	−134	−134	PROPN
ejpam-5779	101	10	inf	inf	PROPN
ejpam-5779	101	11	b1	b1	PROPN
ejpam-5779	101	12	,	,	PUNCT
ejpam-5779	101	13	so	so	ADV
ejpam-5779	101	14	min{f̃l(pq	min{f̃l(pq	ADJ
ejpam-5779	101	15	)	)	PUNCT
ejpam-5779	101	16	,	,	PUNCT
ejpam-5779	101	17	f̃l(q	f̃l(q	PROPN
ejpam-5779	101	18	)	)	PUNCT
ejpam-5779	101	19	}	}	PUNCT
ejpam-5779	101	20	=	=	SYM
ejpam-5779	101	21	supb1−inf	supb1−inf	NOUN
ejpam-5779	101	22	b1	b1	NOUN
ejpam-5779	101	23	.	.	PUNCT
ejpam-5779	102	1	thus	thus	ADV
ejpam-5779	102	2	,	,	PUNCT
ejpam-5779	102	3	f̃l(p	f̃l(p	PROPN
ejpam-5779	102	4	)	)	PUNCT
ejpam-5779	102	5	≥	≥	NOUN
ejpam-5779	102	6	supb1−inf	supb1−inf	NOUN
ejpam-5779	102	7	b1	b1	NOUN
ejpam-5779	102	8	=	=	SYM
ejpam-5779	102	9	min{f̃l(pq	min{f̃l(pq	NOUN
ejpam-5779	102	10	)	)	PUNCT
ejpam-5779	102	11	,	,	PUNCT
ejpam-5779	102	12	f̃l(q)}.135	f̃l(q)}.135	VERB
ejpam-5779	102	13	hence	hence	ADV
ejpam-5779	102	14	,	,	PUNCT
ejpam-5779	102	15	f̃l	f̃l	PROPN
ejpam-5779	102	16	is	be	AUX
ejpam-5779	102	17	a	a	DET
ejpam-5779	102	18	1	1	NUM
ejpam-5779	102	19	-	-	PUNCT
ejpam-5779	102	20	fuzzy	fuzzy	ADJ
ejpam-5779	102	21	ideal	ideal	NOUN
ejpam-5779	102	22	of	of	ADP
ejpam-5779	102	23	a	a	PRON
ejpam-5779	102	24	and	and	CCONJ
ejpam-5779	102	25	so	so	ADV
ejpam-5779	102	26	(	(	PUNCT
ejpam-5779	102	27	a	a	PRON
ejpam-5779	102	28	,	,	PUNCT
ejpam-5779	102	29	f̃	f̃	PROPN
ejpam-5779	102	30	)	)	PUNCT
ejpam-5779	102	31	is	be	AUX
ejpam-5779	102	32	a	a	DET
ejpam-5779	102	33	length	length	NOUN
ejpam-5779	102	34	1	1	NUM
ejpam-5779	102	35	-	-	PUNCT
ejpam-5779	102	36	fuzzy	fuzzy	ADJ
ejpam-5779	102	37	ideal	ideal	NOUN
ejpam-5779	102	38	of	of	ADP
ejpam-5779	102	39	a.136	a.136	PROPN
ejpam-5779	102	40	(	(	PUNCT
ejpam-5779	102	41	2	2	NUM
ejpam-5779	102	42	)	)	PUNCT
ejpam-5779	102	43	assume	assume	VERB
ejpam-5779	102	44	that	that	SCONJ
ejpam-5779	102	45	b2	b2	NOUN
ejpam-5779	102	46	⊂	⊂	PROPN
ejpam-5779	102	47	b1	b1	PROPN
ejpam-5779	102	48	.	.	PUNCT
ejpam-5779	103	1	then	then	ADV
ejpam-5779	103	2	supb2	supb2	PROPN
ejpam-5779	103	3	−	−	PROPN
ejpam-5779	103	4	inf	inf	PROPN
ejpam-5779	103	5	b2	b2	NOUN
ejpam-5779	103	6	≤	≤	NUM
ejpam-5779	103	7	supb1	supb1	NOUN
ejpam-5779	104	1	−	−	PROPN
ejpam-5779	104	2	inf	inf	PROPN
ejpam-5779	104	3	b1	b1	NOUN
ejpam-5779	104	4	.	.	PUNCT
ejpam-5779	105	1	since	since	SCONJ
ejpam-5779	105	2	0	0	NUM
ejpam-5779	105	3	∈	∈	PROPN
ejpam-5779	105	4	i,137	i,137	NOUN
ejpam-5779	105	5	f̃l(0	f̃l(0	PROPN
ejpam-5779	105	6	)	)	PUNCT
ejpam-5779	105	7	=	=	SYM
ejpam-5779	105	8	f̃sup(0)−	f̃sup(0)−	NUM
ejpam-5779	105	9	f̃inf(0	f̃inf(0	PROPN
ejpam-5779	105	10	)	)	PUNCT
ejpam-5779	105	11	=	=	SYM
ejpam-5779	105	12	supb2	supb2	NOUN
ejpam-5779	105	13	−	−	PROPN
ejpam-5779	105	14	inf	inf	PROPN
ejpam-5779	105	15	b2	b2	NOUN
ejpam-5779	105	16	≤	≤	NUM
ejpam-5779	105	17	f̃l(p	f̃l(p	NOUN
ejpam-5779	105	18	)	)	PUNCT
ejpam-5779	105	19	for	for	ADP
ejpam-5779	105	20	all	all	PRON
ejpam-5779	105	21	p	p	PROPN
ejpam-5779	105	22	∈	∈	NOUN
ejpam-5779	105	23	a.138	a.138	NOUN
ejpam-5779	105	24	case	case	NOUN
ejpam-5779	105	25	1	1	NUM
ejpam-5779	105	26	:	:	PUNCT
ejpam-5779	105	27	let	let	VERB
ejpam-5779	105	28	pq	pq	INTJ
ejpam-5779	105	29	,	,	PUNCT
ejpam-5779	105	30	q	q	PROPN
ejpam-5779	105	31	∈	∈	PROPN
ejpam-5779	105	32	s.	s.	PROPN
ejpam-5779	105	33	then	then	ADV
ejpam-5779	105	34	f̃l(p	f̃l(p	PROPN
ejpam-5779	105	35	q	q	NOUN
ejpam-5779	105	36	)	)	PUNCT
ejpam-5779	106	1	=	=	SYM
ejpam-5779	106	2	supb2	supb2	NOUN
ejpam-5779	106	3	−	−	PROPN
ejpam-5779	106	4	inf	inf	NOUN
ejpam-5779	106	5	b2	b2	NOUN
ejpam-5779	106	6	and	and	CCONJ
ejpam-5779	106	7	f̃l(q	f̃l(q	PRON
ejpam-5779	106	8	)	)	PUNCT
ejpam-5779	106	9	=	=	SYM
ejpam-5779	106	10	supb2	supb2	NOUN
ejpam-5779	107	1	−	−	PROPN
ejpam-5779	107	2	inf	inf	PROPN
ejpam-5779	107	3	b2.139	b2.139	PROPN
ejpam-5779	107	4	thus	thus	ADV
ejpam-5779	107	5	,	,	PUNCT
ejpam-5779	107	6	max{f̃l(pq	max{f̃l(pq	NOUN
ejpam-5779	107	7	)	)	PUNCT
ejpam-5779	107	8	,	,	PUNCT
ejpam-5779	107	9	f̃l(q	f̃l(q	PROPN
ejpam-5779	107	10	)	)	PUNCT
ejpam-5779	107	11	}	}	PUNCT
ejpam-5779	107	12	=	=	SYM
ejpam-5779	107	13	supb2	supb2	NOUN
ejpam-5779	107	14	−	−	PROPN
ejpam-5779	107	15	inf	inf	PROPN
ejpam-5779	107	16	b2	b2	NOUN
ejpam-5779	107	17	.	.	PUNCT
ejpam-5779	108	1	since	since	SCONJ
ejpam-5779	108	2	s	s	PROPN
ejpam-5779	108	3	is	be	AUX
ejpam-5779	108	4	an	an	DET
ejpam-5779	108	5	ideal	ideal	NOUN
ejpam-5779	108	6	of	of	ADP
ejpam-5779	108	7	a	a	PRON
ejpam-5779	108	8	,	,	PUNCT
ejpam-5779	108	9	x	x	SYM
ejpam-5779	108	10	∈	∈	NOUN
ejpam-5779	108	11	s	s	X
ejpam-5779	108	12	and	and	CCONJ
ejpam-5779	108	13	so140	so140	ADJ
ejpam-5779	108	14	f̃l(p	f̃l(p	NOUN
ejpam-5779	108	15	)	)	PUNCT
ejpam-5779	108	16	=	=	SYM
ejpam-5779	108	17	supb2	supb2	NOUN
ejpam-5779	108	18	−	−	PROPN
ejpam-5779	108	19	inf	inf	PROPN
ejpam-5779	108	20	b2	b2	NOUN
ejpam-5779	108	21	.	.	PUNCT
ejpam-5779	109	1	thus	thus	ADV
ejpam-5779	109	2	,	,	PUNCT
ejpam-5779	109	3	f̃l(p	f̃l(p	ADJ
ejpam-5779	109	4	)	)	PUNCT
ejpam-5779	109	5	=	=	SYM
ejpam-5779	109	6	supb2	supb2	NOUN
ejpam-5779	109	7	−	−	PROPN
ejpam-5779	109	8	inf	inf	NOUN
ejpam-5779	109	9	b2	b2	NOUN
ejpam-5779	109	10	=	=	SYM
ejpam-5779	109	11	max{f̃l(pq	max{f̃l(pq	NOUN
ejpam-5779	109	12	)	)	PUNCT
ejpam-5779	109	13	,	,	PUNCT
ejpam-5779	109	14	f̃l(q)}.141	f̃l(q)}.141	NOUN
ejpam-5779	109	15	case	case	NOUN
ejpam-5779	109	16	2	2	NUM
ejpam-5779	109	17	:	:	PUNCT
ejpam-5779	109	18	let	let	VERB
ejpam-5779	109	19	pq	pq	INTJ
ejpam-5779	109	20	,	,	PUNCT
ejpam-5779	109	21	q	q	PROPN
ejpam-5779	109	22	/∈	/∈	PUNCT
ejpam-5779	109	23	s.	s.	PROPN
ejpam-5779	109	24	then	then	ADV
ejpam-5779	109	25	f̃l(p	f̃l(p	PROPN
ejpam-5779	109	26	q	q	NOUN
ejpam-5779	109	27	)	)	PUNCT
ejpam-5779	109	28	=	=	SYM
ejpam-5779	109	29	supb1	supb1	NOUN
ejpam-5779	110	1	−	−	PROPN
ejpam-5779	110	2	inf	inf	PROPN
ejpam-5779	110	3	b1	b1	NOUN
ejpam-5779	110	4	and	and	CCONJ
ejpam-5779	110	5	f̃l(q	f̃l(q	PRON
ejpam-5779	110	6	)	)	PUNCT
ejpam-5779	110	7	=	=	SYM
ejpam-5779	111	1	supb1	supb1	NOUN
ejpam-5779	112	1	−	−	PROPN
ejpam-5779	112	2	inf	inf	PROPN
ejpam-5779	112	3	b1	b1	NOUN
ejpam-5779	112	4	,	,	PUNCT
ejpam-5779	112	5	so142	so142	PROPN
ejpam-5779	112	6	max{f̃l(pq	max{f̃l(pq	NOUN
ejpam-5779	112	7	)	)	PUNCT
ejpam-5779	112	8	,	,	PUNCT
ejpam-5779	112	9	f̃l(q	f̃l(q	PROPN
ejpam-5779	112	10	)	)	PUNCT
ejpam-5779	112	11	}	}	PUNCT
ejpam-5779	112	12	=	=	SYM
ejpam-5779	112	13	supb1	supb1	NOUN
ejpam-5779	113	1	−	−	PROPN
ejpam-5779	113	2	inf	inf	PROPN
ejpam-5779	113	3	b1	b1	NOUN
ejpam-5779	113	4	.	.	PUNCT
ejpam-5779	114	1	thus	thus	ADV
ejpam-5779	114	2	,	,	PUNCT
ejpam-5779	114	3	f̃l(p	f̃l(p	NOUN
ejpam-5779	114	4	)	)	PUNCT
ejpam-5779	114	5	≤	≤	NUM
ejpam-5779	115	1	supb1	supb1	NOUN
ejpam-5779	116	1	−	−	PROPN
ejpam-5779	116	2	inf	inf	PROPN
ejpam-5779	116	3	b1	b1	NOUN
ejpam-5779	116	4	=	=	SYM
ejpam-5779	116	5	max{f̃l(pq	max{f̃l(pq	NOUN
ejpam-5779	116	6	)	)	PUNCT
ejpam-5779	116	7	,	,	PUNCT
ejpam-5779	116	8	f̃l(q)}.143	f̃l(q)}.143	DET
ejpam-5779	116	9	case	case	NOUN
ejpam-5779	116	10	3	3	X
ejpam-5779	116	11	:	:	PUNCT
ejpam-5779	116	12	let	let	VERB
ejpam-5779	116	13	pq	pq	INTJ
ejpam-5779	116	14	/∈	/∈	PUNCT
ejpam-5779	116	15	s	s	PROPN
ejpam-5779	116	16	and	and	CCONJ
ejpam-5779	116	17	q	q	PROPN
ejpam-5779	116	18	∈	∈	PROPN
ejpam-5779	116	19	s.	s.	PROPN
ejpam-5779	116	20	then	then	ADV
ejpam-5779	116	21	f̃l(p	f̃l(p	PROPN
ejpam-5779	116	22	q	q	NOUN
ejpam-5779	116	23	)	)	PUNCT
ejpam-5779	116	24	)	)	PUNCT
ejpam-5779	117	1	=	=	PUNCT
ejpam-5779	117	2	supb1	supb1	NOUN
ejpam-5779	118	1	−	−	PROPN
ejpam-5779	118	2	inf	inf	PROPN
ejpam-5779	118	3	b1	b1	NOUN
ejpam-5779	118	4	and	and	CCONJ
ejpam-5779	118	5	f̃l(q	f̃l(q	PRON
ejpam-5779	118	6	)	)	PUNCT
ejpam-5779	118	7	=	=	NOUN
ejpam-5779	118	8	144	144	NUM
ejpam-5779	118	9	supb2	supb2	NOUN
ejpam-5779	118	10	−	−	PROPN
ejpam-5779	118	11	inf	inf	PROPN
ejpam-5779	118	12	b2	b2	NOUN
ejpam-5779	118	13	,	,	PUNCT
ejpam-5779	118	14	so	so	SCONJ
ejpam-5779	118	15	max{f̃l(pq	max{f̃l(pq	NOUN
ejpam-5779	118	16	)	)	PUNCT
ejpam-5779	118	17	,	,	PUNCT
ejpam-5779	118	18	f̃l(q	f̃l(q	PROPN
ejpam-5779	118	19	)	)	PUNCT
ejpam-5779	118	20	}	}	PUNCT
ejpam-5779	118	21	=	=	SYM
ejpam-5779	118	22	supb1	supb1	NOUN
ejpam-5779	119	1	−	−	PROPN
ejpam-5779	119	2	inf	inf	PROPN
ejpam-5779	119	3	b1	b1	NOUN
ejpam-5779	119	4	.	.	PUNCT
ejpam-5779	120	1	thus	thus	ADV
ejpam-5779	120	2	,	,	PUNCT
ejpam-5779	120	3	f̃l(p	f̃l(p	NOUN
ejpam-5779	120	4	)	)	PUNCT
ejpam-5779	120	5	≤	≤	NUM
ejpam-5779	121	1	supb1	supb1	NOUN
ejpam-5779	122	1	−	−	PROPN
ejpam-5779	122	2	inf	inf	PROPN
ejpam-5779	122	3	b1	b1	NOUN
ejpam-5779	122	4	=	=	NOUN
ejpam-5779	122	5	145	145	NUM
ejpam-5779	122	6	max{f̃l(pq	max{f̃l(pq	NOUN
ejpam-5779	122	7	)	)	PUNCT
ejpam-5779	122	8	,	,	PUNCT
ejpam-5779	122	9	f̃l(q)}.146	f̃l(q)}.146	PROPN
ejpam-5779	122	10	case	case	NOUN
ejpam-5779	122	11	4	4	NUM
ejpam-5779	122	12	:	:	PUNCT
ejpam-5779	122	13	let	let	VERB
ejpam-5779	122	14	pq	pq	INTJ
ejpam-5779	122	15	∈	∈	PROPN
ejpam-5779	122	16	s	s	PART
ejpam-5779	122	17	and	and	CCONJ
ejpam-5779	122	18	q	q	PROPN
ejpam-5779	122	19	/∈	/∈	PUNCT
ejpam-5779	123	1	s.	s.	PROPN
ejpam-5779	123	2	then	then	ADV
ejpam-5779	123	3	f̃l(p	f̃l(p	PROPN
ejpam-5779	123	4	q	q	NOUN
ejpam-5779	123	5	)	)	PUNCT
ejpam-5779	123	6	=	=	SYM
ejpam-5779	123	7	supb2	supb2	NOUN
ejpam-5779	124	1	−	−	PROPN
ejpam-5779	124	2	inf	inf	NOUN
ejpam-5779	124	3	b2	b2	NOUN
ejpam-5779	124	4	and	and	CCONJ
ejpam-5779	124	5	f̃l(q	f̃l(q	PRON
ejpam-5779	124	6	)	)	PUNCT
ejpam-5779	125	1	=	=	SYM
ejpam-5779	125	2	supb1	supb1	NOUN
ejpam-5779	126	1	−147	−147	PROPN
ejpam-5779	126	2	inf	inf	PROPN
ejpam-5779	126	3	b1	b1	PROPN
ejpam-5779	126	4	,	,	PUNCT
ejpam-5779	126	5	so	so	ADV
ejpam-5779	126	6	max{f̃l(pq	max{f̃l(pq	NOUN
ejpam-5779	126	7	)	)	PUNCT
ejpam-5779	126	8	,	,	PUNCT
ejpam-5779	126	9	f̃l(q	f̃l(q	PROPN
ejpam-5779	126	10	)	)	PUNCT
ejpam-5779	126	11	}	}	PUNCT
ejpam-5779	126	12	=	=	SYM
ejpam-5779	126	13	supb1−inf	supb1−inf	NOUN
ejpam-5779	126	14	b1	b1	NOUN
ejpam-5779	126	15	.	.	PUNCT
ejpam-5779	127	1	thus	thus	ADV
ejpam-5779	127	2	,	,	PUNCT
ejpam-5779	127	3	f̃l(p	f̃l(p	NOUN
ejpam-5779	127	4	)	)	PUNCT
ejpam-5779	127	5	≤	≤	NUM
ejpam-5779	127	6	supb1−inf	supb1−inf	NOUN
ejpam-5779	127	7	b1	b1	NOUN
ejpam-5779	127	8	=	=	SYM
ejpam-5779	127	9	max{f̃l(pq	max{f̃l(pq	NOUN
ejpam-5779	127	10	)	)	PUNCT
ejpam-5779	127	11	,	,	PUNCT
ejpam-5779	127	12	f̃l(q)}.148	f̃l(q)}.148	NOUN
ejpam-5779	127	13	hence	hence	ADV
ejpam-5779	127	14	,	,	PUNCT
ejpam-5779	127	15	f̃l	f̃l	PROPN
ejpam-5779	127	16	is	be	AUX
ejpam-5779	127	17	a	a	DET
ejpam-5779	127	18	4	4	NUM
ejpam-5779	127	19	-	-	PUNCT
ejpam-5779	127	20	fuzzy	fuzzy	ADJ
ejpam-5779	127	21	ideal	ideal	NOUN
ejpam-5779	127	22	of	of	ADP
ejpam-5779	127	23	a	a	PRON
ejpam-5779	127	24	and	and	CCONJ
ejpam-5779	127	25	so	so	ADV
ejpam-5779	127	26	(	(	PUNCT
ejpam-5779	127	27	a	a	PRON
ejpam-5779	127	28	,	,	PUNCT
ejpam-5779	127	29	f̃	f̃	PROPN
ejpam-5779	127	30	)	)	PUNCT
ejpam-5779	127	31	is	be	AUX
ejpam-5779	127	32	a	a	DET
ejpam-5779	127	33	length	length	NOUN
ejpam-5779	127	34	4	4	NUM
ejpam-5779	127	35	-	-	PUNCT
ejpam-5779	127	36	fuzzy	fuzzy	ADJ
ejpam-5779	127	37	ideal	ideal	NOUN
ejpam-5779	127	38	of	of	ADP
ejpam-5779	127	39	a.149	a.149	DET
ejpam-5779	127	40	n.	n.	PROPN
ejpam-5779	127	41	rajesh	rajesh	PROPN
ejpam-5779	127	42	,	,	PUNCT
ejpam-5779	127	43	t.	t.	PROPN
ejpam-5779	127	44	oner	oner	NOUN
ejpam-5779	127	45	,	,	PUNCT
ejpam-5779	127	46	a.	a.	NOUN
ejpam-5779	127	47	iampan	iampan	PROPN
ejpam-5779	127	48	,	,	PUNCT
ejpam-5779	127	49	i.	i.	PROPN
ejpam-5779	127	50	senturk	senturk	PROPN
ejpam-5779	127	51	/	/	SYM
ejpam-5779	127	52	eur	eur	PROPN
ejpam-5779	127	53	.	.	PUNCT
ejpam-5779	128	1	j.	j.	PROPN
ejpam-5779	128	2	pure	pure	PROPN
ejpam-5779	128	3	appl	appl	PROPN
ejpam-5779	128	4	.	.	PROPN
ejpam-5779	128	5	math	math	PROPN
ejpam-5779	128	6	,	,	PUNCT
ejpam-5779	128	7	18	18	NUM
ejpam-5779	128	8	(	(	PUNCT
ejpam-5779	128	9	1	1	NUM
ejpam-5779	128	10	)	)	PUNCT
ejpam-5779	128	11	(	(	PUNCT
ejpam-5779	128	12	2025	2025	NUM
ejpam-5779	128	13	)	)	PUNCT
ejpam-5779	128	14	,	,	PUNCT
ejpam-5779	128	15	5779	5779	NUM
ejpam-5779	128	16	7	7	NUM
ejpam-5779	128	17	of	of	ADP
ejpam-5779	128	18	18	18	NUM
ejpam-5779	128	19	definition	definition	NOUN
ejpam-5779	128	20	7	7	NUM
ejpam-5779	128	21	.	.	PUNCT
ejpam-5779	129	1	let	let	AUX
ejpam-5779	129	2	(	(	PUNCT
ejpam-5779	129	3	a	a	PRON
ejpam-5779	129	4	,	,	PUNCT
ejpam-5779	129	5	f	f	X
ejpam-5779	129	6	)	)	PUNCT
ejpam-5779	129	7	be	be	AUX
ejpam-5779	129	8	a	a	DET
ejpam-5779	129	9	fuzzy	fuzzy	ADJ
ejpam-5779	129	10	structure	structure	NOUN
ejpam-5779	129	11	in	in	ADP
ejpam-5779	129	12	a.	a.	NOUN
ejpam-5779	129	13	for	for	ADP
ejpam-5779	129	14	any	any	DET
ejpam-5779	129	15	t	t	NOUN
ejpam-5779	129	16	∈	∈	PROPN
ejpam-5779	130	1	[	[	X
ejpam-5779	130	2	0	0	NUM
ejpam-5779	130	3	,	,	PUNCT
ejpam-5779	130	4	1	1	NUM
ejpam-5779	130	5	]	]	PUNCT
ejpam-5779	130	6	,	,	PUNCT
ejpam-5779	130	7	the	the	DET
ejpam-5779	130	8	sets	set	NOUN
ejpam-5779	130	9	u(f	u(f	NOUN
ejpam-5779	130	10	,	,	PUNCT
ejpam-5779	130	11	t	t	NOUN
ejpam-5779	130	12	)	)	PUNCT
ejpam-5779	130	13	=	=	PRON
ejpam-5779	131	1	{	{	PUNCT
ejpam-5779	131	2	p	p	X
ejpam-5779	131	3	∈	∈	PROPN
ejpam-5779	131	4	a	a	DET
ejpam-5779	131	5	:	:	PUNCT
ejpam-5779	131	6	f(p	f(p	PROPN
ejpam-5779	131	7	)	)	PUNCT
ejpam-5779	131	8	≥	≥	NOUN
ejpam-5779	131	9	t	t	PROPN
ejpam-5779	131	10	}	}	PUNCT
ejpam-5779	131	11	,	,	PUNCT
ejpam-5779	131	12	l(f	l(f	PROPN
ejpam-5779	131	13	,	,	PUNCT
ejpam-5779	131	14	t	t	PROPN
ejpam-5779	131	15	)	)	PUNCT
ejpam-5779	131	16	=	=	PRON
ejpam-5779	132	1	{	{	PUNCT
ejpam-5779	132	2	p	p	X
ejpam-5779	132	3	∈	∈	PROPN
ejpam-5779	132	4	a	a	DET
ejpam-5779	132	5	:	:	PUNCT
ejpam-5779	132	6	f(p	f(p	PROPN
ejpam-5779	132	7	)	)	PUNCT
ejpam-5779	132	8	≤	≤	NOUN
ejpam-5779	132	9	t	t	PROPN
ejpam-5779	132	10	}	}	PUNCT
ejpam-5779	132	11	,	,	PUNCT
ejpam-5779	132	12	are	be	AUX
ejpam-5779	132	13	called	call	VERB
ejpam-5779	132	14	upper	upper	ADJ
ejpam-5779	132	15	t	t	NOUN
ejpam-5779	132	16	-	-	PUNCT
ejpam-5779	132	17	level	level	NOUN
ejpam-5779	132	18	subset	subset	NOUN
ejpam-5779	132	19	and	and	CCONJ
ejpam-5779	132	20	lower	low	ADJ
ejpam-5779	132	21	t	t	NOUN
ejpam-5779	132	22	-	-	PUNCT
ejpam-5779	132	23	level	level	NOUN
ejpam-5779	132	24	subset	subset	NOUN
ejpam-5779	132	25	of	of	ADP
ejpam-5779	132	26	f	f	PROPN
ejpam-5779	132	27	,	,	PUNCT
ejpam-5779	132	28	respectively.150	respectively.150	NOUN
ejpam-5779	132	29	theorem	theorem	NOUN
ejpam-5779	132	30	3	3	NUM
ejpam-5779	132	31	.	.	PUNCT
ejpam-5779	132	32	an	an	DET
ejpam-5779	132	33	interval	interval	NOUN
ejpam-5779	132	34	-	-	PUNCT
ejpam-5779	132	35	valued	value	VERB
ejpam-5779	132	36	fuzzy	fuzzy	ADJ
ejpam-5779	132	37	structure	structure	NOUN
ejpam-5779	132	38	(	(	PUNCT
ejpam-5779	132	39	a	a	PRON
ejpam-5779	132	40	,	,	PUNCT
ejpam-5779	132	41	f̃	f̃	PROPN
ejpam-5779	132	42	)	)	PUNCT
ejpam-5779	132	43	over	over	ADP
ejpam-5779	132	44	a	a	PRON
ejpam-5779	132	45	is	be	AUX
ejpam-5779	132	46	a	a	DET
ejpam-5779	132	47	length	length	NOUN
ejpam-5779	132	48	1	1	NUM
ejpam-5779	132	49	-	-	PUNCT
ejpam-5779	132	50	fuzzy	fuzzy	ADJ
ejpam-5779	132	51	ideal	ideal	NOUN
ejpam-5779	132	52	of151	of151	PROPN
ejpam-5779	132	53	a	a	DET
ejpam-5779	132	54	if	if	NOUN
ejpam-5779	133	1	and	and	CCONJ
ejpam-5779	133	2	only	only	ADV
ejpam-5779	133	3	if	if	SCONJ
ejpam-5779	133	4	the	the	DET
ejpam-5779	133	5	set	set	NOUN
ejpam-5779	133	6	u(f̃l	u(f̃l	PROPN
ejpam-5779	133	7	,	,	PUNCT
ejpam-5779	133	8	t	t	PROPN
ejpam-5779	133	9	)	)	PUNCT
ejpam-5779	133	10	is	be	AUX
ejpam-5779	133	11	an	an	DET
ejpam-5779	133	12	ideal	ideal	NOUN
ejpam-5779	133	13	of	of	ADP
ejpam-5779	133	14	a	a	PRON
ejpam-5779	133	15	for	for	ADP
ejpam-5779	133	16	all	all	DET
ejpam-5779	133	17	t	t	NOUN
ejpam-5779	133	18	∈	∈	PROPN
ejpam-5779	134	1	[	[	X
ejpam-5779	134	2	0	0	NUM
ejpam-5779	134	3	,	,	PUNCT
ejpam-5779	134	4	1	1	NUM
ejpam-5779	134	5	]	]	PUNCT
ejpam-5779	134	6	with	with	ADP
ejpam-5779	134	7	u(f̃l	u(f̃l	PROPN
ejpam-5779	134	8	,	,	PUNCT
ejpam-5779	134	9	t	t	PROPN
ejpam-5779	134	10	)	)	PUNCT
ejpam-5779	134	11	̸=	̸=	PROPN
ejpam-5779	134	12	∅.152	∅.152	NUM
ejpam-5779	134	13	proof	proof	NOUN
ejpam-5779	134	14	.	.	PUNCT
ejpam-5779	135	1	assume	assume	VERB
ejpam-5779	135	2	that	that	SCONJ
ejpam-5779	135	3	an	an	DET
ejpam-5779	135	4	interval	interval	NOUN
ejpam-5779	135	5	-	-	PUNCT
ejpam-5779	135	6	valued	value	VERB
ejpam-5779	135	7	fuzzy	fuzzy	ADJ
ejpam-5779	135	8	structure	structure	NOUN
ejpam-5779	135	9	(	(	PUNCT
ejpam-5779	135	10	a	a	PRON
ejpam-5779	135	11	,	,	PUNCT
ejpam-5779	135	12	f̃	f̃	PROPN
ejpam-5779	135	13	)	)	PUNCT
ejpam-5779	135	14	over	over	ADP
ejpam-5779	135	15	a	a	PRON
ejpam-5779	135	16	is	be	AUX
ejpam-5779	135	17	a	a	DET
ejpam-5779	135	18	length	length	NOUN
ejpam-5779	135	19	1	1	NUM
ejpam-5779	135	20	-	-	PUNCT
ejpam-5779	135	21	fuzzy153	fuzzy153	VERB
ejpam-5779	135	22	ideal	ideal	NOUN
ejpam-5779	135	23	of	of	ADP
ejpam-5779	135	24	a	a	PRON
ejpam-5779	135	25	and	and	CCONJ
ejpam-5779	135	26	let	let	VERB
ejpam-5779	135	27	t	t	X
ejpam-5779	135	28	∈	∈	PROPN
ejpam-5779	136	1	[	[	X
ejpam-5779	136	2	0	0	NUM
ejpam-5779	136	3	,	,	PUNCT
ejpam-5779	136	4	1	1	NUM
ejpam-5779	136	5	]	]	PUNCT
ejpam-5779	136	6	be	be	AUX
ejpam-5779	136	7	such	such	ADJ
ejpam-5779	136	8	that	that	SCONJ
ejpam-5779	136	9	u(f̃	u(f̃	PROPN
ejpam-5779	136	10	,	,	PUNCT
ejpam-5779	136	11	t	t	PROPN
ejpam-5779	136	12	)	)	PUNCT
ejpam-5779	136	13	is	be	AUX
ejpam-5779	136	14	nonempty	nonempty	ADJ
ejpam-5779	136	15	.	.	PUNCT
ejpam-5779	137	1	obviously	obviously	ADV
ejpam-5779	137	2	,	,	PUNCT
ejpam-5779	137	3	0	0	NUM
ejpam-5779	137	4	∈	∈	PROPN
ejpam-5779	137	5	u(f̃	u(f̃	PROPN
ejpam-5779	137	6	,	,	PUNCT
ejpam-5779	137	7	t	t	PROPN
ejpam-5779	137	8	)	)	PUNCT
ejpam-5779	137	9	.	.	PUNCT
ejpam-5779	138	1	let154	let154	PROPN
ejpam-5779	138	2	p	p	PRON
ejpam-5779	138	3	,	,	PUNCT
ejpam-5779	138	4	q	q	PROPN
ejpam-5779	138	5	∈	∈	PROPN
ejpam-5779	138	6	a	a	DET
ejpam-5779	138	7	be	be	AUX
ejpam-5779	138	8	such	such	ADJ
ejpam-5779	138	9	that	that	SCONJ
ejpam-5779	138	10	pq	pq	PROPN
ejpam-5779	138	11	∈	∈	PROPN
ejpam-5779	138	12	u(f̃	u(f̃	PROPN
ejpam-5779	138	13	,	,	PUNCT
ejpam-5779	138	14	t	t	PROPN
ejpam-5779	138	15	)	)	PUNCT
ejpam-5779	138	16	and	and	CCONJ
ejpam-5779	138	17	q	q	ADJ
ejpam-5779	138	18	∈	∈	PROPN
ejpam-5779	138	19	u(f̃	u(f̃	PROPN
ejpam-5779	138	20	,	,	PUNCT
ejpam-5779	138	21	t	t	PROPN
ejpam-5779	138	22	)	)	PUNCT
ejpam-5779	138	23	.	.	PUNCT
ejpam-5779	139	1	then	then	ADV
ejpam-5779	139	2	f̃l(p	f̃l(p	PROPN
ejpam-5779	139	3	q	q	NOUN
ejpam-5779	139	4	)	)	PUNCT
ejpam-5779	139	5	≥	≥	NOUN
ejpam-5779	139	6	t	t	NOUN
ejpam-5779	139	7	and	and	CCONJ
ejpam-5779	139	8	f̃l(q	f̃l(q	PRON
ejpam-5779	139	9	)	)	PUNCT
ejpam-5779	139	10	≥	≥	NOUN
ejpam-5779	139	11	t	t	PROPN
ejpam-5779	139	12	,	,	PUNCT
ejpam-5779	139	13	which155	which155	PROPN
ejpam-5779	139	14	imply	imply	VERB
ejpam-5779	139	15	from	from	ADP
ejpam-5779	139	16	(	(	PUNCT
ejpam-5779	139	17	2	2	NUM
ejpam-5779	139	18	)	)	PUNCT
ejpam-5779	139	19	that	that	DET
ejpam-5779	139	20	f̃l(p	f̃l(p	NOUN
ejpam-5779	139	21	)	)	PUNCT
ejpam-5779	139	22	≥	≥	NOUN
ejpam-5779	139	23	min{f̃l(pq	min{f̃l(pq	NOUN
ejpam-5779	139	24	)	)	PUNCT
ejpam-5779	139	25	,	,	PUNCT
ejpam-5779	139	26	f̃l(q	f̃l(q	PROPN
ejpam-5779	139	27	)	)	PUNCT
ejpam-5779	139	28	}	}	PUNCT
ejpam-5779	139	29	≥	≥	NOUN
ejpam-5779	139	30	t.	t.	NOUN
ejpam-5779	139	31	hence	hence	ADV
ejpam-5779	139	32	,	,	PUNCT
ejpam-5779	139	33	p	p	PROPN
ejpam-5779	139	34	∈	∈	PROPN
ejpam-5779	139	35	u(f̃	u(f̃	PROPN
ejpam-5779	139	36	,	,	PUNCT
ejpam-5779	139	37	t	t	PROPN
ejpam-5779	139	38	)	)	PUNCT
ejpam-5779	139	39	,	,	PUNCT
ejpam-5779	139	40	and	and	CCONJ
ejpam-5779	139	41	therefore156	therefore156	PROPN
ejpam-5779	139	42	u(f̃	u(f̃	PROPN
ejpam-5779	139	43	,	,	PUNCT
ejpam-5779	139	44	t	t	PROPN
ejpam-5779	139	45	)	)	PUNCT
ejpam-5779	139	46	is	be	AUX
ejpam-5779	139	47	an	an	DET
ejpam-5779	139	48	ideal	ideal	NOUN
ejpam-5779	139	49	of	of	ADP
ejpam-5779	139	50	a.157	a.157	NOUN
ejpam-5779	139	51	conversely	conversely	ADV
ejpam-5779	139	52	,	,	PUNCT
ejpam-5779	139	53	suppose	suppose	VERB
ejpam-5779	139	54	that	that	SCONJ
ejpam-5779	139	55	u(f̃l	u(f̃l	PROPN
ejpam-5779	139	56	,	,	PUNCT
ejpam-5779	139	57	t	t	PROPN
ejpam-5779	139	58	)	)	PUNCT
ejpam-5779	139	59	is	be	AUX
ejpam-5779	139	60	an	an	DET
ejpam-5779	139	61	ideal	ideal	NOUN
ejpam-5779	139	62	of	of	ADP
ejpam-5779	139	63	a	a	PRON
ejpam-5779	139	64	for	for	ADP
ejpam-5779	139	65	all	all	DET
ejpam-5779	139	66	t	t	NOUN
ejpam-5779	139	67	∈	∈	PROPN
ejpam-5779	140	1	[	[	X
ejpam-5779	140	2	0	0	NUM
ejpam-5779	140	3	,	,	PUNCT
ejpam-5779	140	4	1	1	NUM
ejpam-5779	140	5	]	]	PUNCT
ejpam-5779	140	6	with	with	ADP
ejpam-5779	140	7	u(f̃l	u(f̃l	PROPN
ejpam-5779	140	8	,	,	PUNCT
ejpam-5779	140	9	t	t	PROPN
ejpam-5779	140	10	)	)	PUNCT
ejpam-5779	140	11	̸=	̸=	PROPN
ejpam-5779	140	12	∅.	∅.	ADP
ejpam-5779	140	13	if158	if158	PROPN
ejpam-5779	140	14	f̃l(0	f̃l(0	NOUN
ejpam-5779	140	15	)	)	PUNCT
ejpam-5779	140	16	<	<	X
ejpam-5779	140	17	f̃l(k	f̃l(k	X
ejpam-5779	140	18	)	)	PUNCT
ejpam-5779	140	19	for	for	ADP
ejpam-5779	140	20	some	some	DET
ejpam-5779	140	21	k	k	PROPN
ejpam-5779	140	22	∈	∈	PROPN
ejpam-5779	140	23	a	a	PRON
ejpam-5779	140	24	,	,	PUNCT
ejpam-5779	140	25	then	then	ADV
ejpam-5779	140	26	k	k	PROPN
ejpam-5779	140	27	∈	∈	PROPN
ejpam-5779	140	28	u(f̃l	u(f̃l	PROPN
ejpam-5779	140	29	,	,	PUNCT
ejpam-5779	140	30	f̃l(k	f̃l(k	NOUN
ejpam-5779	140	31	)	)	PUNCT
ejpam-5779	140	32	)	)	PUNCT
ejpam-5779	140	33	and	and	CCONJ
ejpam-5779	140	34	hence	hence	ADV
ejpam-5779	140	35	u(f̃l	u(f̃l	PROPN
ejpam-5779	140	36	,	,	PUNCT
ejpam-5779	140	37	f̃l(k	f̃l(k	NOUN
ejpam-5779	140	38	)	)	PUNCT
ejpam-5779	140	39	)	)	PUNCT
ejpam-5779	140	40	is	be	AUX
ejpam-5779	140	41	an	an	DET
ejpam-5779	140	42	ideal	ideal	NOUN
ejpam-5779	140	43	of	of	ADP
ejpam-5779	140	44	a.159	a.159	NOUN
ejpam-5779	140	45	thus	thus	ADV
ejpam-5779	140	46	,	,	PUNCT
ejpam-5779	140	47	0	0	NUM
ejpam-5779	140	48	∈	∈	PROPN
ejpam-5779	140	49	u(f̃l	u(f̃l	PROPN
ejpam-5779	140	50	,	,	PUNCT
ejpam-5779	140	51	f̃l(k	f̃l(k	NOUN
ejpam-5779	140	52	)	)	PUNCT
ejpam-5779	140	53	)	)	PUNCT
ejpam-5779	140	54	,	,	PUNCT
ejpam-5779	140	55	and	and	CCONJ
ejpam-5779	140	56	so	so	ADV
ejpam-5779	140	57	f̃l(0	f̃l(0	PROPN
ejpam-5779	140	58	)	)	PUNCT
ejpam-5779	140	59	≥	≥	NOUN
ejpam-5779	140	60	f̃l(k	f̃l(k	ADV
ejpam-5779	140	61	)	)	PUNCT
ejpam-5779	140	62	.	.	PUNCT
ejpam-5779	141	1	this	this	PRON
ejpam-5779	141	2	is	be	AUX
ejpam-5779	141	3	a	a	DET
ejpam-5779	141	4	contradiction	contradiction	NOUN
ejpam-5779	141	5	,	,	PUNCT
ejpam-5779	141	6	and	and	CCONJ
ejpam-5779	141	7	thus	thus	ADV
ejpam-5779	141	8	f̃l(0	f̃l(0	NOUN
ejpam-5779	141	9	)	)	PUNCT
ejpam-5779	141	10	≥	≥	NOUN
ejpam-5779	141	11	f̃l(p)160	f̃l(p)160	NUM
ejpam-5779	141	12	for	for	ADP
ejpam-5779	141	13	all	all	DET
ejpam-5779	141	14	p	p	PROPN
ejpam-5779	141	15	∈	∈	PROPN
ejpam-5779	141	16	a.	a.	NOUN
ejpam-5779	141	17	assume	assume	VERB
ejpam-5779	141	18	that	that	SCONJ
ejpam-5779	141	19	there	there	PRON
ejpam-5779	141	20	exist	exist	VERB
ejpam-5779	141	21	k	k	PROPN
ejpam-5779	141	22	,	,	PUNCT
ejpam-5779	141	23	l	l	PROPN
ejpam-5779	141	24	∈	∈	PROPN
ejpam-5779	141	25	a	a	DET
ejpam-5779	141	26	such	such	ADJ
ejpam-5779	141	27	that	that	SCONJ
ejpam-5779	141	28	f̃l(k	f̃l(k	NOUN
ejpam-5779	141	29	)	)	PUNCT
ejpam-5779	141	30	<	<	X
ejpam-5779	141	31	min{f̃l(kl	min{f̃l(kl	NUM
ejpam-5779	141	32	)	)	PUNCT
ejpam-5779	141	33	,	,	PUNCT
ejpam-5779	141	34	f̃l(l)}.161	f̃l(l)}.161	X
ejpam-5779	141	35	taking	take	VERB
ejpam-5779	141	36	t	t	NOUN
ejpam-5779	141	37	=	=	SYM
ejpam-5779	141	38	min{f̃l(kl	min{f̃l(kl	PROPN
ejpam-5779	141	39	)	)	PUNCT
ejpam-5779	141	40	,	,	PUNCT
ejpam-5779	141	41	f̃l(l	f̃l(l	NUM
ejpam-5779	141	42	)	)	PUNCT
ejpam-5779	141	43	}	}	PUNCT
ejpam-5779	141	44	implies	imply	VERB
ejpam-5779	141	45	that	that	SCONJ
ejpam-5779	141	46	k	k	PROPN
ejpam-5779	141	47	∈	∈	PROPN
ejpam-5779	141	48	u(f̃l	u(f̃l	PROPN
ejpam-5779	141	49	,	,	PUNCT
ejpam-5779	141	50	t	t	PROPN
ejpam-5779	141	51	)	)	PUNCT
ejpam-5779	141	52	.	.	PUNCT
ejpam-5779	142	1	since	since	SCONJ
ejpam-5779	142	2	u(f̃l	u(f̃l	PROPN
ejpam-5779	142	3	,	,	PUNCT
ejpam-5779	142	4	t	t	PROPN
ejpam-5779	142	5	)	)	PUNCT
ejpam-5779	142	6	is	be	AUX
ejpam-5779	142	7	an	an	DET
ejpam-5779	142	8	ideal	ideal	NOUN
ejpam-5779	142	9	of	of	ADP
ejpam-5779	142	10	a,162	a,162	NOUN
ejpam-5779	142	11	a	a	DET
ejpam-5779	142	12	∈	∈	PROPN
ejpam-5779	142	13	u(f̃l	u(f̃l	PROPN
ejpam-5779	142	14	,	,	PUNCT
ejpam-5779	142	15	t	t	PROPN
ejpam-5779	142	16	)	)	PUNCT
ejpam-5779	142	17	.	.	PUNCT
ejpam-5779	143	1	hence	hence	ADV
ejpam-5779	143	2	,	,	PUNCT
ejpam-5779	143	3	f̃l(k	f̃l(k	ADV
ejpam-5779	143	4	)	)	PUNCT
ejpam-5779	143	5	≥	≥	NOUN
ejpam-5779	143	6	t	t	NOUN
ejpam-5779	143	7	=	=	PUNCT
ejpam-5779	143	8	min{f̃l(kl	min{f̃l(kl	PROPN
ejpam-5779	143	9	)	)	PUNCT
ejpam-5779	143	10	,	,	PUNCT
ejpam-5779	143	11	f̃l(l	f̃l(l	NOUN
ejpam-5779	143	12	)	)	PUNCT
ejpam-5779	143	13	}	}	PUNCT
ejpam-5779	143	14	,	,	PUNCT
ejpam-5779	143	15	which	which	PRON
ejpam-5779	143	16	is	be	AUX
ejpam-5779	143	17	a	a	DET
ejpam-5779	143	18	contradiction	contradiction	NOUN
ejpam-5779	143	19	.	.	PUNCT
ejpam-5779	144	1	hence,163	hence,163	VERB
ejpam-5779	144	2	f̃l(p	f̃l(p	PROPN
ejpam-5779	144	3	)	)	PUNCT
ejpam-5779	144	4	≥	≥	NOUN
ejpam-5779	144	5	min{f̃l(pq	min{f̃l(pq	NOUN
ejpam-5779	144	6	)	)	PUNCT
ejpam-5779	144	7	,	,	PUNCT
ejpam-5779	144	8	f̃l(q	f̃l(q	PROPN
ejpam-5779	144	9	)	)	PUNCT
ejpam-5779	144	10	}	}	PUNCT
ejpam-5779	144	11	for	for	ADP
ejpam-5779	144	12	all	all	DET
ejpam-5779	144	13	p	p	NOUN
ejpam-5779	144	14	,	,	PUNCT
ejpam-5779	144	15	q	q	PROPN
ejpam-5779	144	16	∈	∈	PROPN
ejpam-5779	144	17	a.	a.	NOUN
ejpam-5779	144	18	therefore	therefore	ADV
ejpam-5779	144	19	,	,	PUNCT
ejpam-5779	144	20	(	(	PUNCT
ejpam-5779	144	21	a	a	PRON
ejpam-5779	144	22	,	,	PUNCT
ejpam-5779	144	23	f̃	f̃	PROPN
ejpam-5779	144	24	)	)	PUNCT
ejpam-5779	144	25	is	be	AUX
ejpam-5779	144	26	a	a	DET
ejpam-5779	144	27	length	length	NOUN
ejpam-5779	144	28	1	1	NUM
ejpam-5779	144	29	-	-	PUNCT
ejpam-5779	144	30	fuzzy	fuzzy	ADJ
ejpam-5779	144	31	ideal	ideal	NOUN
ejpam-5779	144	32	of	of	ADP
ejpam-5779	144	33	a.164	a.164	NOUN
ejpam-5779	144	34	corollary	corollary	ADJ
ejpam-5779	144	35	1	1	NUM
ejpam-5779	144	36	.	.	PUNCT
ejpam-5779	145	1	if	if	SCONJ
ejpam-5779	145	2	(	(	PUNCT
ejpam-5779	145	3	a	a	PRON
ejpam-5779	145	4	,	,	PUNCT
ejpam-5779	145	5	f̃	f̃	PROPN
ejpam-5779	145	6	)	)	PUNCT
ejpam-5779	145	7	is	be	AUX
ejpam-5779	145	8	a	a	DET
ejpam-5779	145	9	length	length	NOUN
ejpam-5779	145	10	3	3	NUM
ejpam-5779	145	11	-	-	PUNCT
ejpam-5779	145	12	fuzzy	fuzzy	ADJ
ejpam-5779	145	13	ideal	ideal	NOUN
ejpam-5779	145	14	of	of	ADP
ejpam-5779	145	15	a	a	PRON
ejpam-5779	145	16	,	,	PUNCT
ejpam-5779	145	17	then	then	ADV
ejpam-5779	145	18	the	the	DET
ejpam-5779	145	19	set	set	NOUN
ejpam-5779	145	20	u(f̃l	u(f̃l	PROPN
ejpam-5779	145	21	,	,	PUNCT
ejpam-5779	145	22	t	t	PROPN
ejpam-5779	145	23	)	)	PUNCT
ejpam-5779	145	24	is	be	AUX
ejpam-5779	145	25	an	an	DET
ejpam-5779	145	26	ideal	ideal	NOUN
ejpam-5779	145	27	of	of	ADP
ejpam-5779	145	28	a165	a165	PROPN
ejpam-5779	145	29	for	for	ADP
ejpam-5779	145	30	all	all	DET
ejpam-5779	145	31	t	t	NOUN
ejpam-5779	145	32	∈	∈	PROPN
ejpam-5779	146	1	[	[	X
ejpam-5779	146	2	0	0	NUM
ejpam-5779	146	3	,	,	PUNCT
ejpam-5779	146	4	1	1	NUM
ejpam-5779	146	5	]	]	PUNCT
ejpam-5779	146	6	with	with	ADP
ejpam-5779	146	7	u(f̃l	u(f̃l	PROPN
ejpam-5779	146	8	,	,	PUNCT
ejpam-5779	146	9	t	t	PROPN
ejpam-5779	146	10	)	)	PUNCT
ejpam-5779	146	11	̸=	̸=	PROPN
ejpam-5779	146	12	∅.166	∅.166	ADJ
ejpam-5779	146	13	proof	proof	NOUN
ejpam-5779	146	14	.	.	PUNCT
ejpam-5779	147	1	it	it	PRON
ejpam-5779	147	2	is	be	AUX
ejpam-5779	147	3	straightforward	straightforward	ADJ
ejpam-5779	147	4	by	by	ADP
ejpam-5779	147	5	theorems	theorem	NOUN
ejpam-5779	147	6	1	1	NUM
ejpam-5779	147	7	and	and	CCONJ
ejpam-5779	147	8	3.167	3.167	NUM
ejpam-5779	147	9	theorem	theorem	NOUN
ejpam-5779	147	10	4	4	NUM
ejpam-5779	147	11	.	.	PUNCT
ejpam-5779	148	1	an	an	DET
ejpam-5779	148	2	interval	interval	NOUN
ejpam-5779	148	3	-	-	PUNCT
ejpam-5779	148	4	valued	value	VERB
ejpam-5779	148	5	fuzzy	fuzzy	ADJ
ejpam-5779	148	6	structure	structure	NOUN
ejpam-5779	148	7	(	(	PUNCT
ejpam-5779	148	8	a	a	PRON
ejpam-5779	148	9	,	,	PUNCT
ejpam-5779	148	10	f̃	f̃	PROPN
ejpam-5779	148	11	)	)	PUNCT
ejpam-5779	148	12	over	over	ADP
ejpam-5779	148	13	a	a	PRON
ejpam-5779	148	14	is	be	AUX
ejpam-5779	148	15	a	a	DET
ejpam-5779	148	16	length	length	NOUN
ejpam-5779	148	17	4	4	NUM
ejpam-5779	148	18	-	-	PUNCT
ejpam-5779	148	19	fuzzy	fuzzy	ADJ
ejpam-5779	148	20	ideal	ideal	NOUN
ejpam-5779	148	21	of168	of168	ADP
ejpam-5779	148	22	a	a	DET
ejpam-5779	148	23	if	if	NOUN
ejpam-5779	148	24	and	and	CCONJ
ejpam-5779	148	25	only	only	ADV
ejpam-5779	148	26	if	if	SCONJ
ejpam-5779	148	27	the	the	DET
ejpam-5779	148	28	set	set	NOUN
ejpam-5779	148	29	l(f̃l	l(f̃l	PROPN
ejpam-5779	148	30	,	,	PUNCT
ejpam-5779	148	31	t	t	PROPN
ejpam-5779	148	32	)	)	PUNCT
ejpam-5779	148	33	is	be	AUX
ejpam-5779	148	34	an	an	DET
ejpam-5779	148	35	ideal	ideal	NOUN
ejpam-5779	148	36	of	of	ADP
ejpam-5779	148	37	a	a	PRON
ejpam-5779	148	38	for	for	ADP
ejpam-5779	148	39	all	all	DET
ejpam-5779	148	40	t	t	NOUN
ejpam-5779	148	41	∈	∈	PROPN
ejpam-5779	149	1	[	[	X
ejpam-5779	149	2	0	0	NUM
ejpam-5779	149	3	,	,	PUNCT
ejpam-5779	149	4	1	1	NUM
ejpam-5779	149	5	]	]	PUNCT
ejpam-5779	149	6	with	with	ADP
ejpam-5779	149	7	l(f̃l	l(f̃l	PROPN
ejpam-5779	149	8	,	,	PUNCT
ejpam-5779	149	9	t	t	PROPN
ejpam-5779	149	10	)	)	PUNCT
ejpam-5779	149	11	̸=	̸=	PROPN
ejpam-5779	149	12	∅.169	∅.169	PROPN
ejpam-5779	149	13	proof	proof	NOUN
ejpam-5779	149	14	.	.	PUNCT
ejpam-5779	150	1	assume	assume	VERB
ejpam-5779	150	2	that	that	SCONJ
ejpam-5779	150	3	an	an	DET
ejpam-5779	150	4	interval	interval	NOUN
ejpam-5779	150	5	-	-	PUNCT
ejpam-5779	150	6	valued	value	VERB
ejpam-5779	150	7	fuzzy	fuzzy	ADJ
ejpam-5779	150	8	structure	structure	NOUN
ejpam-5779	150	9	(	(	PUNCT
ejpam-5779	150	10	a	a	PRON
ejpam-5779	150	11	,	,	PUNCT
ejpam-5779	150	12	f̃	f̃	PROPN
ejpam-5779	150	13	)	)	PUNCT
ejpam-5779	150	14	over	over	ADP
ejpam-5779	150	15	a	a	PRON
ejpam-5779	150	16	is	be	AUX
ejpam-5779	150	17	a	a	DET
ejpam-5779	150	18	length	length	NOUN
ejpam-5779	150	19	4	4	NUM
ejpam-5779	150	20	-	-	PUNCT
ejpam-5779	150	21	fuzzy170	fuzzy170	NOUN
ejpam-5779	150	22	ideal	ideal	NOUN
ejpam-5779	150	23	of	of	ADP
ejpam-5779	150	24	a	a	PRON
ejpam-5779	150	25	and	and	CCONJ
ejpam-5779	150	26	let	let	VERB
ejpam-5779	150	27	t	t	X
ejpam-5779	150	28	∈	∈	PROPN
ejpam-5779	151	1	[	[	X
ejpam-5779	151	2	0	0	NUM
ejpam-5779	151	3	,	,	PUNCT
ejpam-5779	151	4	1	1	NUM
ejpam-5779	151	5	]	]	PUNCT
ejpam-5779	151	6	be	be	AUX
ejpam-5779	151	7	such	such	ADJ
ejpam-5779	151	8	that	that	SCONJ
ejpam-5779	151	9	l(f̃	l(f̃	PROPN
ejpam-5779	151	10	,	,	PUNCT
ejpam-5779	151	11	t	t	PROPN
ejpam-5779	151	12	)	)	PUNCT
ejpam-5779	151	13	is	be	AUX
ejpam-5779	151	14	nonempty	nonempty	ADJ
ejpam-5779	151	15	.	.	PUNCT
ejpam-5779	152	1	obviously	obviously	ADV
ejpam-5779	152	2	,	,	PUNCT
ejpam-5779	152	3	0	0	NUM
ejpam-5779	152	4	∈	∈	PROPN
ejpam-5779	152	5	l(f̃	l(f̃	PROPN
ejpam-5779	152	6	,	,	PUNCT
ejpam-5779	152	7	t	t	PROPN
ejpam-5779	152	8	)	)	PUNCT
ejpam-5779	152	9	.	.	PUNCT
ejpam-5779	153	1	let171	let171	PROPN
ejpam-5779	153	2	p	p	NOUN
ejpam-5779	153	3	,	,	PUNCT
ejpam-5779	153	4	q	q	PROPN
ejpam-5779	153	5	∈	∈	PROPN
ejpam-5779	153	6	a	a	DET
ejpam-5779	153	7	be	be	AUX
ejpam-5779	153	8	such	such	ADJ
ejpam-5779	153	9	that	that	DET
ejpam-5779	153	10	pq	pq	PROPN
ejpam-5779	153	11	∈	∈	PROPN
ejpam-5779	153	12	l(f̃	l(f̃	PROPN
ejpam-5779	153	13	,	,	PUNCT
ejpam-5779	153	14	t	t	PROPN
ejpam-5779	153	15	)	)	PUNCT
ejpam-5779	153	16	and	and	CCONJ
ejpam-5779	153	17	q	q	NOUN
ejpam-5779	153	18	∈	∈	PROPN
ejpam-5779	153	19	l(f̃	l(f̃	PROPN
ejpam-5779	153	20	,	,	PUNCT
ejpam-5779	153	21	t	t	PROPN
ejpam-5779	153	22	)	)	PUNCT
ejpam-5779	153	23	.	.	PUNCT
ejpam-5779	154	1	then	then	ADV
ejpam-5779	154	2	f̃l(p	f̃l(p	PROPN
ejpam-5779	154	3	q	q	NOUN
ejpam-5779	154	4	)	)	PUNCT
ejpam-5779	154	5	≤	≤	NOUN
ejpam-5779	154	6	t	t	NOUN
ejpam-5779	154	7	and	and	CCONJ
ejpam-5779	154	8	f̃l(q	f̃l(q	X
ejpam-5779	154	9	)	)	PUNCT
ejpam-5779	154	10	≤	≤	NOUN
ejpam-5779	154	11	t	t	PROPN
ejpam-5779	154	12	,	,	PUNCT
ejpam-5779	154	13	which172	which172	PROPN
ejpam-5779	154	14	imply	imply	VERB
ejpam-5779	154	15	from	from	ADP
ejpam-5779	154	16	(	(	PUNCT
ejpam-5779	154	17	8)	8)	NUM
ejpam-5779	154	18	that	that	DET
ejpam-5779	154	19	f̃l(p	f̃l(p	NOUN
ejpam-5779	154	20	)	)	PUNCT
ejpam-5779	154	21	≤	≤	NOUN
ejpam-5779	154	22	min{f̃l(pq	min{f̃l(pq	NOUN
ejpam-5779	154	23	)	)	PUNCT
ejpam-5779	154	24	,	,	PUNCT
ejpam-5779	154	25	f̃l(q	f̃l(q	PROPN
ejpam-5779	154	26	)	)	PUNCT
ejpam-5779	154	27	}	}	PUNCT
ejpam-5779	154	28	≤	≤	NUM
ejpam-5779	155	1	t.	t.	NOUN
ejpam-5779	155	2	hence	hence	ADV
ejpam-5779	155	3	,	,	PUNCT
ejpam-5779	155	4	p	p	NOUN
ejpam-5779	155	5	∈	∈	PROPN
ejpam-5779	155	6	l(f̃	l(f̃	PROPN
ejpam-5779	155	7	,	,	PUNCT
ejpam-5779	155	8	t	t	PROPN
ejpam-5779	155	9	)	)	PUNCT
ejpam-5779	155	10	,	,	PUNCT
ejpam-5779	155	11	and	and	CCONJ
ejpam-5779	155	12	therefore173	therefore173	PROPN
ejpam-5779	155	13	l(f̃	l(f̃	PROPN
ejpam-5779	155	14	,	,	PUNCT
ejpam-5779	155	15	t	t	PROPN
ejpam-5779	155	16	)	)	PUNCT
ejpam-5779	155	17	is	be	AUX
ejpam-5779	155	18	an	an	DET
ejpam-5779	155	19	ideal	ideal	NOUN
ejpam-5779	155	20	of	of	ADP
ejpam-5779	155	21	a.174	a.174	ADV
ejpam-5779	155	22	conversely	conversely	ADV
ejpam-5779	155	23	,	,	PUNCT
ejpam-5779	155	24	suppose	suppose	VERB
ejpam-5779	155	25	that	that	SCONJ
ejpam-5779	155	26	l(f̃l	l(f̃l	PROPN
ejpam-5779	155	27	,	,	PUNCT
ejpam-5779	155	28	t	t	PROPN
ejpam-5779	155	29	)	)	PUNCT
ejpam-5779	155	30	is	be	AUX
ejpam-5779	155	31	an	an	DET
ejpam-5779	155	32	ideal	ideal	NOUN
ejpam-5779	155	33	of	of	ADP
ejpam-5779	155	34	a	a	PRON
ejpam-5779	155	35	for	for	ADP
ejpam-5779	155	36	all	all	DET
ejpam-5779	155	37	t	t	NOUN
ejpam-5779	155	38	∈	∈	PROPN
ejpam-5779	156	1	[	[	X
ejpam-5779	156	2	0	0	NUM
ejpam-5779	156	3	,	,	PUNCT
ejpam-5779	156	4	1	1	NUM
ejpam-5779	156	5	]	]	PUNCT
ejpam-5779	156	6	with	with	ADP
ejpam-5779	156	7	l(f̃l	l(f̃l	PROPN
ejpam-5779	156	8	,	,	PUNCT
ejpam-5779	156	9	t	t	PROPN
ejpam-5779	156	10	)	)	PUNCT
ejpam-5779	156	11	̸=	̸=	PROPN
ejpam-5779	156	12	∅.	∅.	ADP
ejpam-5779	156	13	if175	if175	PROPN
ejpam-5779	156	14	f̃l(0	f̃l(0	PROPN
ejpam-5779	156	15	)	)	PUNCT
ejpam-5779	156	16	>	>	X
ejpam-5779	157	1	f̃l(k	f̃l(k	ADV
ejpam-5779	157	2	)	)	PUNCT
ejpam-5779	157	3	for	for	ADP
ejpam-5779	157	4	some	some	DET
ejpam-5779	157	5	k	k	PROPN
ejpam-5779	157	6	∈	∈	PROPN
ejpam-5779	157	7	a	a	PRON
ejpam-5779	157	8	,	,	PUNCT
ejpam-5779	157	9	then	then	ADV
ejpam-5779	157	10	k	k	PROPN
ejpam-5779	157	11	∈	∈	PROPN
ejpam-5779	157	12	l(f̃l	l(f̃l	PROPN
ejpam-5779	157	13	,	,	PUNCT
ejpam-5779	157	14	f̃l(k	f̃l(k	NOUN
ejpam-5779	157	15	)	)	PUNCT
ejpam-5779	157	16	)	)	PUNCT
ejpam-5779	157	17	and	and	CCONJ
ejpam-5779	157	18	hence	hence	ADV
ejpam-5779	157	19	l(f̃l	l(f̃l	PROPN
ejpam-5779	157	20	,	,	PUNCT
ejpam-5779	157	21	f̃l(k	f̃l(k	NOUN
ejpam-5779	157	22	)	)	PUNCT
ejpam-5779	157	23	)	)	PUNCT
ejpam-5779	157	24	is	be	AUX
ejpam-5779	157	25	an	an	DET
ejpam-5779	157	26	ideal	ideal	NOUN
ejpam-5779	157	27	of	of	ADP
ejpam-5779	157	28	a.176	a.176	NOUN
ejpam-5779	157	29	thus	thus	ADV
ejpam-5779	157	30	,	,	PUNCT
ejpam-5779	157	31	0	0	NUM
ejpam-5779	157	32	∈	∈	PROPN
ejpam-5779	157	33	l(f̃l	l(f̃l	PROPN
ejpam-5779	157	34	,	,	PUNCT
ejpam-5779	157	35	f̃l(k	f̃l(k	NOUN
ejpam-5779	157	36	)	)	PUNCT
ejpam-5779	157	37	)	)	PUNCT
ejpam-5779	157	38	,	,	PUNCT
ejpam-5779	157	39	and	and	CCONJ
ejpam-5779	157	40	so	so	ADV
ejpam-5779	157	41	f̃l(0	f̃l(0	PROPN
ejpam-5779	157	42	)	)	PUNCT
ejpam-5779	157	43	≤	≤	NOUN
ejpam-5779	157	44	f̃l(k	f̃l(k	ADV
ejpam-5779	157	45	)	)	PUNCT
ejpam-5779	157	46	.	.	PUNCT
ejpam-5779	158	1	this	this	PRON
ejpam-5779	158	2	is	be	AUX
ejpam-5779	158	3	a	a	DET
ejpam-5779	158	4	contradiction	contradiction	NOUN
ejpam-5779	158	5	,	,	PUNCT
ejpam-5779	158	6	and	and	CCONJ
ejpam-5779	158	7	thus	thus	ADV
ejpam-5779	158	8	f̃l(0	f̃l(0	NOUN
ejpam-5779	158	9	)	)	PUNCT
ejpam-5779	158	10	≤	≤	NUM
ejpam-5779	158	11	f̃l(p)177	f̃l(p)177	NOUN
ejpam-5779	158	12	for	for	ADP
ejpam-5779	158	13	all	all	DET
ejpam-5779	158	14	p	p	PROPN
ejpam-5779	158	15	∈	∈	PROPN
ejpam-5779	158	16	a.	a.	NOUN
ejpam-5779	158	17	assume	assume	VERB
ejpam-5779	158	18	that	that	SCONJ
ejpam-5779	158	19	there	there	PRON
ejpam-5779	158	20	exist	exist	VERB
ejpam-5779	158	21	k	k	PROPN
ejpam-5779	158	22	,	,	PUNCT
ejpam-5779	158	23	l	l	PROPN
ejpam-5779	158	24	∈	∈	PROPN
ejpam-5779	158	25	a	a	DET
ejpam-5779	158	26	such	such	ADJ
ejpam-5779	158	27	that	that	SCONJ
ejpam-5779	158	28	f̃l(k	f̃l(k	NOUN
ejpam-5779	158	29	)	)	PUNCT
ejpam-5779	158	30	>	>	X
ejpam-5779	158	31	max{f̃l(kl	max{f̃l(kl	PROPN
ejpam-5779	158	32	)	)	PUNCT
ejpam-5779	158	33	,	,	PUNCT
ejpam-5779	158	34	f̃l(l)}.178	f̃l(l)}.178	PROPN
ejpam-5779	158	35	taking	take	VERB
ejpam-5779	158	36	t	t	PROPN
ejpam-5779	158	37	=	=	SYM
ejpam-5779	158	38	max{f̃l(kl	max{f̃l(kl	NOUN
ejpam-5779	158	39	)	)	PUNCT
ejpam-5779	158	40	,	,	PUNCT
ejpam-5779	158	41	f̃l(l	f̃l(l	NOUN
ejpam-5779	158	42	)	)	PUNCT
ejpam-5779	158	43	}	}	PUNCT
ejpam-5779	158	44	implies	imply	VERB
ejpam-5779	158	45	that	that	SCONJ
ejpam-5779	158	46	k	k	PROPN
ejpam-5779	158	47	∈	∈	PROPN
ejpam-5779	158	48	l(f̃l	l(f̃l	PROPN
ejpam-5779	158	49	,	,	PUNCT
ejpam-5779	158	50	t	t	PROPN
ejpam-5779	158	51	)	)	PUNCT
ejpam-5779	158	52	.	.	PUNCT
ejpam-5779	159	1	since	since	SCONJ
ejpam-5779	159	2	l(f̃l	l(f̃l	PROPN
ejpam-5779	159	3	,	,	PUNCT
ejpam-5779	159	4	t	t	PROPN
ejpam-5779	159	5	)	)	PUNCT
ejpam-5779	159	6	is	be	AUX
ejpam-5779	159	7	an	an	DET
ejpam-5779	159	8	ideal	ideal	NOUN
ejpam-5779	159	9	of	of	ADP
ejpam-5779	159	10	a,179	a,179	NUM
ejpam-5779	159	11	k	k	PROPN
ejpam-5779	159	12	∈	∈	PROPN
ejpam-5779	159	13	l(f̃l	l(f̃l	PROPN
ejpam-5779	159	14	,	,	PUNCT
ejpam-5779	159	15	t	t	PROPN
ejpam-5779	159	16	)	)	PUNCT
ejpam-5779	159	17	.	.	PUNCT
ejpam-5779	160	1	hence	hence	ADV
ejpam-5779	160	2	,	,	PUNCT
ejpam-5779	160	3	f̃l(k	f̃l(k	ADV
ejpam-5779	160	4	)	)	PUNCT
ejpam-5779	160	5	≤	≤	NOUN
ejpam-5779	160	6	t	t	NOUN
ejpam-5779	160	7	=	=	SYM
ejpam-5779	160	8	max{f̃l(kl	max{f̃l(kl	NOUN
ejpam-5779	160	9	)	)	PUNCT
ejpam-5779	160	10	,	,	PUNCT
ejpam-5779	160	11	f̃l(l	f̃l(l	NOUN
ejpam-5779	160	12	)	)	PUNCT
ejpam-5779	160	13	}	}	PUNCT
ejpam-5779	160	14	,	,	PUNCT
ejpam-5779	160	15	which	which	PRON
ejpam-5779	160	16	is	be	AUX
ejpam-5779	160	17	a	a	DET
ejpam-5779	160	18	contradiction	contradiction	NOUN
ejpam-5779	160	19	.	.	PUNCT
ejpam-5779	161	1	hence,180	hence,180	NOUN
ejpam-5779	161	2	f̃l(p	f̃l(p	NOUN
ejpam-5779	161	3	)	)	PUNCT
ejpam-5779	161	4	≤	≤	NUM
ejpam-5779	161	5	max{f̃l(pq	max{f̃l(pq	NOUN
ejpam-5779	161	6	)	)	PUNCT
ejpam-5779	161	7	,	,	PUNCT
ejpam-5779	161	8	f̃l(q	f̃l(q	PROPN
ejpam-5779	161	9	)	)	PUNCT
ejpam-5779	161	10	}	}	PUNCT
ejpam-5779	161	11	for	for	ADP
ejpam-5779	161	12	all	all	DET
ejpam-5779	161	13	p	p	NOUN
ejpam-5779	161	14	,	,	PUNCT
ejpam-5779	161	15	q	q	PROPN
ejpam-5779	161	16	∈	∈	PROPN
ejpam-5779	161	17	a.	a.	NOUN
ejpam-5779	161	18	therefore	therefore	ADV
ejpam-5779	161	19	,	,	PUNCT
ejpam-5779	161	20	(	(	PUNCT
ejpam-5779	161	21	a	a	PRON
ejpam-5779	161	22	,	,	PUNCT
ejpam-5779	161	23	f̃	f̃	PROPN
ejpam-5779	161	24	)	)	PUNCT
ejpam-5779	161	25	is	be	AUX
ejpam-5779	161	26	a	a	DET
ejpam-5779	161	27	length	length	NOUN
ejpam-5779	161	28	4	4	NUM
ejpam-5779	161	29	-	-	PUNCT
ejpam-5779	161	30	fuzzy	fuzzy	ADJ
ejpam-5779	161	31	ideal	ideal	NOUN
ejpam-5779	161	32	of	of	ADP
ejpam-5779	161	33	a.181	a.181	PROPN
ejpam-5779	161	34	n.	n.	PROPN
ejpam-5779	161	35	rajesh	rajesh	PROPN
ejpam-5779	161	36	,	,	PUNCT
ejpam-5779	161	37	t.	t.	PROPN
ejpam-5779	161	38	oner	oner	NOUN
ejpam-5779	161	39	,	,	PUNCT
ejpam-5779	161	40	a.	a.	NOUN
ejpam-5779	161	41	iampan	iampan	PROPN
ejpam-5779	161	42	,	,	PUNCT
ejpam-5779	161	43	i.	i.	PROPN
ejpam-5779	161	44	senturk	senturk	PROPN
ejpam-5779	161	45	/	/	SYM
ejpam-5779	161	46	eur	eur	PROPN
ejpam-5779	161	47	.	.	PUNCT
ejpam-5779	162	1	j.	j.	PROPN
ejpam-5779	162	2	pure	pure	PROPN
ejpam-5779	162	3	appl	appl	PROPN
ejpam-5779	162	4	.	.	PROPN
ejpam-5779	162	5	math	math	PROPN
ejpam-5779	162	6	,	,	PUNCT
ejpam-5779	162	7	18	18	NUM
ejpam-5779	162	8	(	(	PUNCT
ejpam-5779	162	9	1	1	NUM
ejpam-5779	162	10	)	)	PUNCT
ejpam-5779	162	11	(	(	PUNCT
ejpam-5779	162	12	2025	2025	NUM
ejpam-5779	162	13	)	)	PUNCT
ejpam-5779	162	14	,	,	PUNCT
ejpam-5779	162	15	5779	5779	NUM
ejpam-5779	162	16	8	8	NUM
ejpam-5779	162	17	of	of	ADP
ejpam-5779	162	18	18	18	NUM
ejpam-5779	162	19	corollary	corollary	ADJ
ejpam-5779	162	20	2	2	NUM
ejpam-5779	162	21	.	.	PUNCT
ejpam-5779	163	1	if	if	SCONJ
ejpam-5779	163	2	(	(	PUNCT
ejpam-5779	163	3	a	a	PRON
ejpam-5779	163	4	,	,	PUNCT
ejpam-5779	163	5	f̃	f̃	PROPN
ejpam-5779	163	6	)	)	PUNCT
ejpam-5779	163	7	is	be	AUX
ejpam-5779	163	8	a	a	DET
ejpam-5779	163	9	length	length	NOUN
ejpam-5779	163	10	2	2	NUM
ejpam-5779	163	11	-	-	PUNCT
ejpam-5779	163	12	fuzzy	fuzzy	ADJ
ejpam-5779	163	13	ideal	ideal	NOUN
ejpam-5779	163	14	of	of	ADP
ejpam-5779	163	15	a	a	PRON
ejpam-5779	163	16	,	,	PUNCT
ejpam-5779	163	17	then	then	ADV
ejpam-5779	163	18	the	the	DET
ejpam-5779	163	19	set	set	ADJ
ejpam-5779	163	20	l(f̃l	l(f̃l	PROPN
ejpam-5779	163	21	,	,	PUNCT
ejpam-5779	163	22	t	t	PROPN
ejpam-5779	163	23	)	)	PUNCT
ejpam-5779	163	24	is	be	AUX
ejpam-5779	163	25	an	an	DET
ejpam-5779	163	26	ideal	ideal	NOUN
ejpam-5779	163	27	of	of	ADP
ejpam-5779	163	28	a182	a182	PROPN
ejpam-5779	163	29	for	for	ADP
ejpam-5779	163	30	all	all	DET
ejpam-5779	163	31	t	t	NOUN
ejpam-5779	163	32	∈	∈	PROPN
ejpam-5779	164	1	[	[	X
ejpam-5779	164	2	0	0	NUM
ejpam-5779	164	3	,	,	PUNCT
ejpam-5779	164	4	1	1	NUM
ejpam-5779	164	5	]	]	PUNCT
ejpam-5779	164	6	with	with	ADP
ejpam-5779	164	7	l(f̃l	l(f̃l	PROPN
ejpam-5779	164	8	,	,	PUNCT
ejpam-5779	164	9	t	t	PROPN
ejpam-5779	164	10	)	)	PUNCT
ejpam-5779	164	11	̸=	̸=	PROPN
ejpam-5779	164	12	∅.183	∅.183	VERB
ejpam-5779	164	13	proof	proof	NOUN
ejpam-5779	164	14	.	.	PUNCT
ejpam-5779	165	1	it	it	PRON
ejpam-5779	165	2	is	be	AUX
ejpam-5779	165	3	straightforward	straightforward	ADJ
ejpam-5779	165	4	by	by	ADP
ejpam-5779	165	5	theorems	theorem	NOUN
ejpam-5779	165	6	1	1	NUM
ejpam-5779	165	7	and	and	CCONJ
ejpam-5779	165	8	4.184	4.184	NUM
ejpam-5779	165	9	theorem	theorem	VERB
ejpam-5779	165	10	5	5	NUM
ejpam-5779	165	11	.	.	PUNCT
ejpam-5779	166	1	if	if	SCONJ
ejpam-5779	166	2	(	(	PUNCT
ejpam-5779	166	3	a	a	PRON
ejpam-5779	166	4	,	,	PUNCT
ejpam-5779	166	5	f̃	f̃	PROPN
ejpam-5779	166	6	)	)	PUNCT
ejpam-5779	166	7	is	be	AUX
ejpam-5779	166	8	an	an	DET
ejpam-5779	166	9	interval	interval	NOUN
ejpam-5779	166	10	-	-	PUNCT
ejpam-5779	166	11	valued	value	VERB
ejpam-5779	166	12	fuzzy	fuzzy	ADJ
ejpam-5779	166	13	structure	structure	NOUN
ejpam-5779	166	14	over	over	ADP
ejpam-5779	166	15	a	a	DET
ejpam-5779	166	16	in	in	ADP
ejpam-5779	166	17	which	which	PRON
ejpam-5779	166	18	(	(	PUNCT
ejpam-5779	166	19	a	a	PRON
ejpam-5779	166	20	,	,	PUNCT
ejpam-5779	166	21	f̃inf	f̃inf	ADJ
ejpam-5779	166	22	)	)	PUNCT
ejpam-5779	166	23	is185	is185	NOUN
ejpam-5779	166	24	constant	constant	ADJ
ejpam-5779	166	25	and	and	CCONJ
ejpam-5779	166	26	(	(	PUNCT
ejpam-5779	166	27	a	a	DET
ejpam-5779	166	28	,	,	PUNCT
ejpam-5779	166	29	f̃sup	f̃sup	ADJ
ejpam-5779	166	30	)	)	PUNCT
ejpam-5779	166	31	is	be	AUX
ejpam-5779	166	32	a	a	DET
ejpam-5779	166	33	1	1	NUM
ejpam-5779	166	34	-	-	PUNCT
ejpam-5779	166	35	fuzzy	fuzzy	ADJ
ejpam-5779	166	36	ideal	ideal	NOUN
ejpam-5779	166	37	of	of	ADP
ejpam-5779	166	38	a	a	PRON
ejpam-5779	166	39	,	,	PUNCT
ejpam-5779	166	40	then	then	ADV
ejpam-5779	166	41	(	(	PUNCT
ejpam-5779	166	42	a	a	PRON
ejpam-5779	166	43	,	,	PUNCT
ejpam-5779	166	44	f̃	f̃	PROPN
ejpam-5779	166	45	)	)	PUNCT
ejpam-5779	166	46	is	be	AUX
ejpam-5779	166	47	a	a	DET
ejpam-5779	166	48	length	length	NOUN
ejpam-5779	166	49	1	1	NUM
ejpam-5779	166	50	-	-	PUNCT
ejpam-5779	166	51	fuzzy	fuzzy	ADJ
ejpam-5779	166	52	ideal	ideal	NOUN
ejpam-5779	166	53	of	of	ADP
ejpam-5779	166	54	a.186	a.186	ADJ
ejpam-5779	166	55	proof	proof	NOUN
ejpam-5779	166	56	.	.	PUNCT
ejpam-5779	167	1	assume	assume	VERB
ejpam-5779	167	2	that	that	SCONJ
ejpam-5779	167	3	(	(	PUNCT
ejpam-5779	167	4	a	a	PRON
ejpam-5779	167	5	,	,	PUNCT
ejpam-5779	167	6	f̃	f̃	PROPN
ejpam-5779	167	7	)	)	PUNCT
ejpam-5779	167	8	is	be	AUX
ejpam-5779	167	9	an	an	DET
ejpam-5779	167	10	interval	interval	NOUN
ejpam-5779	167	11	-	-	PUNCT
ejpam-5779	167	12	valued	value	VERB
ejpam-5779	167	13	fuzzy	fuzzy	ADJ
ejpam-5779	167	14	structure	structure	NOUN
ejpam-5779	167	15	over	over	ADP
ejpam-5779	167	16	a	a	PRON
ejpam-5779	167	17	in	in	ADP
ejpam-5779	167	18	which	which	PRON
ejpam-5779	167	19	(	(	PUNCT
ejpam-5779	167	20	a	a	PRON
ejpam-5779	167	21	,	,	PUNCT
ejpam-5779	167	22	f̃inf)187	f̃inf)187	NUM
ejpam-5779	167	23	is	be	AUX
ejpam-5779	167	24	constant	constant	ADJ
ejpam-5779	167	25	and	and	CCONJ
ejpam-5779	167	26	(	(	PUNCT
ejpam-5779	167	27	a	a	DET
ejpam-5779	167	28	,	,	PUNCT
ejpam-5779	167	29	f̃sup	f̃sup	ADJ
ejpam-5779	167	30	)	)	PUNCT
ejpam-5779	167	31	is	be	AUX
ejpam-5779	167	32	a	a	DET
ejpam-5779	167	33	1	1	NUM
ejpam-5779	167	34	-	-	PUNCT
ejpam-5779	167	35	fuzzy	fuzzy	ADJ
ejpam-5779	167	36	ideal	ideal	NOUN
ejpam-5779	167	37	of	of	ADP
ejpam-5779	167	38	a.	a.	NOUN
ejpam-5779	167	39	let	let	VERB
ejpam-5779	167	40	p	p	PRON
ejpam-5779	167	41	,	,	PUNCT
ejpam-5779	167	42	q	q	PROPN
ejpam-5779	167	43	∈	∈	PROPN
ejpam-5779	167	44	a.	a.	NOUN
ejpam-5779	167	45	since	since	SCONJ
ejpam-5779	167	46	(	(	PUNCT
ejpam-5779	167	47	a	a	PRON
ejpam-5779	167	48	,	,	PUNCT
ejpam-5779	167	49	f̃inf	f̃inf	ADJ
ejpam-5779	167	50	)	)	PUNCT
ejpam-5779	167	51	is	be	AUX
ejpam-5779	167	52	constant,188	constant,188	X
ejpam-5779	167	53	f̃inf(p	f̃inf(p	ADJ
ejpam-5779	167	54	)	)	PUNCT
ejpam-5779	168	1	=	=	SYM
ejpam-5779	168	2	f̃inf(0	f̃inf(0	PROPN
ejpam-5779	168	3	)	)	PUNCT
ejpam-5779	168	4	for	for	ADP
ejpam-5779	168	5	all	all	DET
ejpam-5779	168	6	p	p	PROPN
ejpam-5779	168	7	∈	∈	PROPN
ejpam-5779	168	8	a.	a.	NOUN
ejpam-5779	168	9	since	since	SCONJ
ejpam-5779	168	10	(	(	PUNCT
ejpam-5779	168	11	a	a	DET
ejpam-5779	168	12	,	,	PUNCT
ejpam-5779	168	13	f̃sup	f̃sup	ADJ
ejpam-5779	168	14	)	)	PUNCT
ejpam-5779	168	15	is	be	AUX
ejpam-5779	168	16	a	a	DET
ejpam-5779	168	17	1	1	NUM
ejpam-5779	168	18	-	-	PUNCT
ejpam-5779	168	19	fuzzy	fuzzy	ADJ
ejpam-5779	168	20	ideal	ideal	NOUN
ejpam-5779	168	21	of	of	ADP
ejpam-5779	168	22	a,189	a,189	PROPN
ejpam-5779	168	23	(	(	PUNCT
ejpam-5779	168	24	∀p	∀p	NOUN
ejpam-5779	168	25	∈	∈	NOUN
ejpam-5779	168	26	a)(f̃sup(0	a)(f̃sup(0	NOUN
ejpam-5779	168	27	)	)	PUNCT
ejpam-5779	168	28	≥	≥	NOUN
ejpam-5779	168	29	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	168	30	)	)	PUNCT
ejpam-5779	168	31	)	)	PUNCT
ejpam-5779	168	32	,	,	PUNCT
ejpam-5779	168	33	(	(	PUNCT
ejpam-5779	168	34	9	9	NUM
ejpam-5779	168	35	)	)	PUNCT
ejpam-5779	168	36	(	(	PUNCT
ejpam-5779	168	37	∀p	∀p	X
ejpam-5779	168	38	,	,	PUNCT
ejpam-5779	168	39	q	q	PUNCT
ejpam-5779	168	40	∈	∈	PROPN
ejpam-5779	168	41	a)(f̃sup(p	a)(f̃sup(p	NOUN
ejpam-5779	168	42	)	)	PUNCT
ejpam-5779	168	43	≥	≥	NOUN
ejpam-5779	168	44	min{f̃sup(pq	min{f̃sup(pq	PROPN
ejpam-5779	168	45	)	)	PUNCT
ejpam-5779	168	46	,	,	PUNCT
ejpam-5779	168	47	f̃sup(q	f̃sup(q	NOUN
ejpam-5779	168	48	)	)	PUNCT
ejpam-5779	168	49	}	}	PUNCT
ejpam-5779	168	50	)	)	PUNCT
ejpam-5779	168	51	.	.	PUNCT
ejpam-5779	169	1	(	(	PUNCT
ejpam-5779	169	2	10	10	NUM
ejpam-5779	169	3	)	)	PUNCT
ejpam-5779	169	4	let	let	VERB
ejpam-5779	169	5	p	p	PRON
ejpam-5779	169	6	∈	∈	PROPN
ejpam-5779	169	7	a.	a.	NOUN
ejpam-5779	169	8	then190	then190	PROPN
ejpam-5779	169	9	f̃l(0	f̃l(0	PROPN
ejpam-5779	169	10	)	)	PUNCT
ejpam-5779	170	1	=	=	SYM
ejpam-5779	170	2	f̃sup(0)−	f̃sup(0)−	NUM
ejpam-5779	170	3	f̃inf(0	f̃inf(0	PROPN
ejpam-5779	170	4	)	)	PUNCT
ejpam-5779	170	5	≥	≥	PART
ejpam-5779	170	6	f̃sup(p)−	f̃sup(p)−	PROPN
ejpam-5779	170	7	f̃inf(0	f̃inf(0	PROPN
ejpam-5779	170	8	)	)	PUNCT
ejpam-5779	170	9	=	=	SYM
ejpam-5779	170	10	f̃sup(p)−	f̃sup(p)−	PROPN
ejpam-5779	170	11	f̃inf(p	f̃inf(p	NOUN
ejpam-5779	170	12	)	)	PUNCT
ejpam-5779	170	13	=	=	SYM
ejpam-5779	170	14	f̃l(p	f̃l(p	PROPN
ejpam-5779	170	15	)	)	PUNCT
ejpam-5779	170	16	.	.	PUNCT
ejpam-5779	171	1	let	let	VERB
ejpam-5779	171	2	p	p	PRON
ejpam-5779	171	3	,	,	PUNCT
ejpam-5779	171	4	q	q	PROPN
ejpam-5779	171	5	∈	∈	PROPN
ejpam-5779	171	6	a.	a.	NOUN
ejpam-5779	171	7	then191	then191	PROPN
ejpam-5779	171	8	f̃l(p	f̃l(p	NOUN
ejpam-5779	171	9	)	)	PUNCT
ejpam-5779	171	10	=	=	SYM
ejpam-5779	171	11	f̃sup(p)−	f̃sup(p)−	PROPN
ejpam-5779	171	12	f̃inf(p	f̃inf(p	NOUN
ejpam-5779	171	13	)	)	PUNCT
ejpam-5779	172	1	=	=	SYM
ejpam-5779	172	2	f̃sup(p)−	f̃sup(p)−	PROPN
ejpam-5779	172	3	f̃inf(0	f̃inf(0	PROPN
ejpam-5779	172	4	)	)	PUNCT
ejpam-5779	172	5	≥	≥	NOUN
ejpam-5779	172	6	min{f̃sup(pq	min{f̃sup(pq	PROPN
ejpam-5779	172	7	)	)	PUNCT
ejpam-5779	172	8	,	,	PUNCT
ejpam-5779	172	9	f̃sup(q	f̃sup(q	NOUN
ejpam-5779	172	10	)	)	PUNCT
ejpam-5779	172	11	}	}	PUNCT
ejpam-5779	172	12	−	−	ADP
ejpam-5779	172	13	f̃inf(0	f̃inf(0	PROPN
ejpam-5779	172	14	)	)	PUNCT
ejpam-5779	172	15	=	=	SYM
ejpam-5779	172	16	min{f̃sup(pq)−	min{f̃sup(pq)−	NUM
ejpam-5779	172	17	f̃inf(0	f̃inf(0	PROPN
ejpam-5779	172	18	)	)	PUNCT
ejpam-5779	172	19	,	,	PUNCT
ejpam-5779	172	20	f̃sup(q)−	f̃sup(q)−	PROPN
ejpam-5779	172	21	f̃inf(0	f̃inf(0	PROPN
ejpam-5779	172	22	)	)	PUNCT
ejpam-5779	172	23	}	}	PUNCT
ejpam-5779	172	24	=	=	PUNCT
ejpam-5779	173	1	min{f̃sup(pq)−	min{f̃sup(pq)−	NUM
ejpam-5779	173	2	f̃inf(p	f̃inf(p	X
ejpam-5779	173	3	q	q	NOUN
ejpam-5779	173	4	)	)	PUNCT
ejpam-5779	173	5	,	,	PUNCT
ejpam-5779	173	6	f̃sup(q)−	f̃sup(q)−	PROPN
ejpam-5779	173	7	f̃inf(q	f̃inf(q	PROPN
ejpam-5779	173	8	)	)	PUNCT
ejpam-5779	173	9	}	}	PUNCT
ejpam-5779	173	10	=	=	SYM
ejpam-5779	173	11	min{f̃l(pq	min{f̃l(pq	NOUN
ejpam-5779	173	12	)	)	PUNCT
ejpam-5779	173	13	,	,	PUNCT
ejpam-5779	173	14	f̃l(q	f̃l(q	PROPN
ejpam-5779	173	15	)	)	PUNCT
ejpam-5779	173	16	}	}	PUNCT
ejpam-5779	173	17	.	.	PUNCT
ejpam-5779	174	1	hence	hence	ADV
ejpam-5779	174	2	,	,	PUNCT
ejpam-5779	174	3	(	(	PUNCT
ejpam-5779	174	4	a	a	DET
ejpam-5779	174	5	,	,	PUNCT
ejpam-5779	174	6	f̃l	f̃l	PROPN
ejpam-5779	174	7	)	)	PUNCT
ejpam-5779	174	8	is	be	AUX
ejpam-5779	174	9	a	a	DET
ejpam-5779	174	10	1	1	NUM
ejpam-5779	174	11	-	-	PUNCT
ejpam-5779	174	12	fuzzy	fuzzy	ADJ
ejpam-5779	174	13	ideal	ideal	NOUN
ejpam-5779	174	14	of	of	ADP
ejpam-5779	174	15	a	a	PRON
ejpam-5779	174	16	,	,	PUNCT
ejpam-5779	174	17	that	that	ADV
ejpam-5779	174	18	is	is	ADV
ejpam-5779	174	19	,	,	PUNCT
ejpam-5779	174	20	(	(	PUNCT
ejpam-5779	174	21	a	a	PRON
ejpam-5779	174	22	,	,	PUNCT
ejpam-5779	174	23	f̃	f̃	PROPN
ejpam-5779	174	24	)	)	PUNCT
ejpam-5779	174	25	is	be	AUX
ejpam-5779	174	26	a	a	DET
ejpam-5779	174	27	length	length	NOUN
ejpam-5779	174	28	1	1	NUM
ejpam-5779	174	29	-	-	PUNCT
ejpam-5779	174	30	fuzzy	fuzzy	ADJ
ejpam-5779	174	31	ideal	ideal	NOUN
ejpam-5779	174	32	of	of	ADP
ejpam-5779	174	33	a.192	a.192	NOUN
ejpam-5779	174	34	theorem	theorem	VERB
ejpam-5779	174	35	6	6	NUM
ejpam-5779	174	36	.	.	PUNCT
ejpam-5779	175	1	if	if	SCONJ
ejpam-5779	175	2	(	(	PUNCT
ejpam-5779	175	3	a	a	PRON
ejpam-5779	175	4	,	,	PUNCT
ejpam-5779	175	5	f̃	f̃	PROPN
ejpam-5779	175	6	)	)	PUNCT
ejpam-5779	175	7	is	be	AUX
ejpam-5779	175	8	an	an	DET
ejpam-5779	175	9	interval	interval	NOUN
ejpam-5779	175	10	-	-	PUNCT
ejpam-5779	175	11	valued	value	VERB
ejpam-5779	175	12	fuzzy	fuzzy	ADJ
ejpam-5779	175	13	structure	structure	NOUN
ejpam-5779	175	14	over	over	ADP
ejpam-5779	175	15	a	a	DET
ejpam-5779	175	16	in	in	ADP
ejpam-5779	175	17	which	which	PRON
ejpam-5779	175	18	(	(	PUNCT
ejpam-5779	175	19	a	a	PRON
ejpam-5779	175	20	,	,	PUNCT
ejpam-5779	175	21	f̃inf	f̃inf	ADJ
ejpam-5779	175	22	)	)	PUNCT
ejpam-5779	175	23	is193	is193	NOUN
ejpam-5779	175	24	constant	constant	ADJ
ejpam-5779	175	25	and	and	CCONJ
ejpam-5779	175	26	(	(	PUNCT
ejpam-5779	175	27	a	a	DET
ejpam-5779	175	28	,	,	PUNCT
ejpam-5779	175	29	f̃sup	f̃sup	ADJ
ejpam-5779	175	30	)	)	PUNCT
ejpam-5779	175	31	is	be	AUX
ejpam-5779	175	32	a	a	DET
ejpam-5779	175	33	4	4	NUM
ejpam-5779	175	34	-	-	PUNCT
ejpam-5779	175	35	fuzzy	fuzzy	ADJ
ejpam-5779	175	36	ideal	ideal	NOUN
ejpam-5779	175	37	of	of	ADP
ejpam-5779	175	38	a	a	PRON
ejpam-5779	175	39	,	,	PUNCT
ejpam-5779	175	40	then	then	ADV
ejpam-5779	175	41	(	(	PUNCT
ejpam-5779	175	42	a	a	PRON
ejpam-5779	175	43	,	,	PUNCT
ejpam-5779	175	44	f̃	f̃	PROPN
ejpam-5779	175	45	)	)	PUNCT
ejpam-5779	175	46	is	be	AUX
ejpam-5779	175	47	a	a	DET
ejpam-5779	175	48	length	length	NOUN
ejpam-5779	175	49	4	4	NUM
ejpam-5779	175	50	-	-	PUNCT
ejpam-5779	175	51	fuzzy	fuzzy	ADJ
ejpam-5779	175	52	ideal	ideal	NOUN
ejpam-5779	175	53	of	of	ADP
ejpam-5779	175	54	a.194	a.194	PROPN
ejpam-5779	175	55	proof	proof	NOUN
ejpam-5779	175	56	.	.	PUNCT
ejpam-5779	176	1	assume	assume	VERB
ejpam-5779	176	2	that	that	SCONJ
ejpam-5779	176	3	(	(	PUNCT
ejpam-5779	176	4	a	a	PRON
ejpam-5779	176	5	,	,	PUNCT
ejpam-5779	176	6	f̃	f̃	PROPN
ejpam-5779	176	7	)	)	PUNCT
ejpam-5779	176	8	is	be	AUX
ejpam-5779	176	9	an	an	DET
ejpam-5779	176	10	interval	interval	NOUN
ejpam-5779	176	11	-	-	PUNCT
ejpam-5779	176	12	valued	value	VERB
ejpam-5779	176	13	fuzzy	fuzzy	ADJ
ejpam-5779	176	14	structure	structure	NOUN
ejpam-5779	176	15	over	over	ADP
ejpam-5779	176	16	a	a	PRON
ejpam-5779	176	17	in	in	ADP
ejpam-5779	176	18	which	which	PRON
ejpam-5779	176	19	(	(	PUNCT
ejpam-5779	176	20	a	a	X
ejpam-5779	176	21	,	,	PUNCT
ejpam-5779	176	22	f̃inf)195	f̃inf)195	NOUN
ejpam-5779	176	23	is	be	AUX
ejpam-5779	176	24	constant	constant	ADJ
ejpam-5779	176	25	and	and	CCONJ
ejpam-5779	176	26	(	(	PUNCT
ejpam-5779	176	27	a	a	DET
ejpam-5779	176	28	,	,	PUNCT
ejpam-5779	176	29	f̃sup	f̃sup	ADJ
ejpam-5779	176	30	)	)	PUNCT
ejpam-5779	176	31	is	be	AUX
ejpam-5779	176	32	a	a	DET
ejpam-5779	176	33	4	4	NUM
ejpam-5779	176	34	-	-	PUNCT
ejpam-5779	176	35	fuzzy	fuzzy	ADJ
ejpam-5779	176	36	ideal	ideal	NOUN
ejpam-5779	176	37	of	of	ADP
ejpam-5779	176	38	a.	a.	NOUN
ejpam-5779	176	39	let	let	VERB
ejpam-5779	176	40	p	p	PRON
ejpam-5779	176	41	,	,	PUNCT
ejpam-5779	176	42	q	q	PROPN
ejpam-5779	176	43	∈	∈	PROPN
ejpam-5779	176	44	a.	a.	NOUN
ejpam-5779	176	45	since	since	SCONJ
ejpam-5779	176	46	(	(	PUNCT
ejpam-5779	176	47	a	a	PRON
ejpam-5779	176	48	,	,	PUNCT
ejpam-5779	176	49	f̃inf	f̃inf	ADJ
ejpam-5779	176	50	)	)	PUNCT
ejpam-5779	176	51	is	be	AUX
ejpam-5779	176	52	constant,196	constant,196	NOUN
ejpam-5779	177	1	we	we	PRON
ejpam-5779	177	2	have	have	VERB
ejpam-5779	177	3	f̃inf(p	f̃inf(p	NOUN
ejpam-5779	177	4	)	)	PUNCT
ejpam-5779	177	5	=	=	SYM
ejpam-5779	177	6	f̃inf(0	f̃inf(0	PROPN
ejpam-5779	177	7	)	)	PUNCT
ejpam-5779	177	8	for	for	ADP
ejpam-5779	177	9	all	all	DET
ejpam-5779	177	10	p	p	PROPN
ejpam-5779	177	11	∈	∈	PROPN
ejpam-5779	177	12	a.	a.	NOUN
ejpam-5779	177	13	since	since	SCONJ
ejpam-5779	177	14	(	(	PUNCT
ejpam-5779	177	15	a	a	DET
ejpam-5779	177	16	,	,	PUNCT
ejpam-5779	177	17	f̃sup	f̃sup	ADJ
ejpam-5779	177	18	)	)	PUNCT
ejpam-5779	177	19	is	be	AUX
ejpam-5779	177	20	a	a	DET
ejpam-5779	177	21	4	4	NUM
ejpam-5779	177	22	-	-	PUNCT
ejpam-5779	177	23	fuzzy	fuzzy	ADJ
ejpam-5779	177	24	ideal	ideal	NOUN
ejpam-5779	177	25	of	of	ADP
ejpam-5779	177	26	a	a	DET
ejpam-5779	177	27	,	,	PUNCT
ejpam-5779	177	28	we	we	PRON
ejpam-5779	177	29	have197	have197	PROPN
ejpam-5779	177	30	(	(	PUNCT
ejpam-5779	177	31	∀p	∀p	NOUN
ejpam-5779	177	32	∈	∈	NOUN
ejpam-5779	177	33	a)(f̃sup(0	a)(f̃sup(0	NOUN
ejpam-5779	177	34	)	)	PUNCT
ejpam-5779	177	35	≤	≤	NUM
ejpam-5779	177	36	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	177	37	)	)	PUNCT
ejpam-5779	177	38	)	)	PUNCT
ejpam-5779	177	39	,	,	PUNCT
ejpam-5779	177	40	(	(	PUNCT
ejpam-5779	177	41	11	11	NUM
ejpam-5779	177	42	)	)	PUNCT
ejpam-5779	177	43	(	(	PUNCT
ejpam-5779	177	44	∀p	∀p	X
ejpam-5779	177	45	,	,	PUNCT
ejpam-5779	177	46	q	q	PUNCT
ejpam-5779	177	47	∈	∈	PROPN
ejpam-5779	177	48	a)(f̃sup(p	a)(f̃sup(p	NOUN
ejpam-5779	177	49	)	)	PUNCT
ejpam-5779	177	50	≤	≤	NOUN
ejpam-5779	177	51	max{f̃sup(pq	max{f̃sup(pq	NOUN
ejpam-5779	177	52	)	)	PUNCT
ejpam-5779	177	53	,	,	PUNCT
ejpam-5779	177	54	f̃sup(q	f̃sup(q	NOUN
ejpam-5779	177	55	)	)	PUNCT
ejpam-5779	177	56	}	}	PUNCT
ejpam-5779	177	57	)	)	PUNCT
ejpam-5779	177	58	.	.	PUNCT
ejpam-5779	178	1	(	(	PUNCT
ejpam-5779	178	2	12	12	NUM
ejpam-5779	178	3	)	)	PUNCT
ejpam-5779	178	4	n.	n.	PROPN
ejpam-5779	178	5	rajesh	rajesh	PROPN
ejpam-5779	178	6	,	,	PUNCT
ejpam-5779	178	7	t.	t.	PROPN
ejpam-5779	178	8	oner	oner	NOUN
ejpam-5779	178	9	,	,	PUNCT
ejpam-5779	178	10	a.	a.	NOUN
ejpam-5779	178	11	iampan	iampan	PROPN
ejpam-5779	178	12	,	,	PUNCT
ejpam-5779	178	13	i.	i.	PROPN
ejpam-5779	178	14	senturk	senturk	PROPN
ejpam-5779	178	15	/	/	SYM
ejpam-5779	178	16	eur	eur	PROPN
ejpam-5779	178	17	.	.	PUNCT
ejpam-5779	179	1	j.	j.	PROPN
ejpam-5779	179	2	pure	pure	PROPN
ejpam-5779	179	3	appl	appl	PROPN
ejpam-5779	179	4	.	.	PROPN
ejpam-5779	179	5	math	math	PROPN
ejpam-5779	179	6	,	,	PUNCT
ejpam-5779	179	7	18	18	NUM
ejpam-5779	179	8	(	(	PUNCT
ejpam-5779	179	9	1	1	NUM
ejpam-5779	179	10	)	)	PUNCT
ejpam-5779	179	11	(	(	PUNCT
ejpam-5779	179	12	2025	2025	NUM
ejpam-5779	179	13	)	)	PUNCT
ejpam-5779	179	14	,	,	PUNCT
ejpam-5779	179	15	5779	5779	NUM
ejpam-5779	179	16	9	9	NUM
ejpam-5779	179	17	of	of	ADP
ejpam-5779	179	18	18	18	NUM
ejpam-5779	179	19	let	let	VERB
ejpam-5779	179	20	p	p	PROPN
ejpam-5779	179	21	∈	∈	PROPN
ejpam-5779	179	22	a.	a.	NOUN
ejpam-5779	179	23	then198	then198	PROPN
ejpam-5779	179	24	f̃l(0	f̃l(0	PROPN
ejpam-5779	179	25	)	)	PUNCT
ejpam-5779	180	1	=	=	SYM
ejpam-5779	180	2	f̃sup(0)−	f̃sup(0)−	NUM
ejpam-5779	180	3	f̃inf(0	f̃inf(0	PROPN
ejpam-5779	180	4	)	)	PUNCT
ejpam-5779	180	5	≤	≤	NUM
ejpam-5779	180	6	f̃sup(p)−	f̃sup(p)−	PROPN
ejpam-5779	180	7	f̃inf(0	f̃inf(0	PROPN
ejpam-5779	180	8	)	)	PUNCT
ejpam-5779	180	9	=	=	SYM
ejpam-5779	180	10	f̃sup(p)−	f̃sup(p)−	PROPN
ejpam-5779	180	11	f̃inf(p	f̃inf(p	NOUN
ejpam-5779	180	12	)	)	PUNCT
ejpam-5779	180	13	=	=	SYM
ejpam-5779	180	14	f̃l(p	f̃l(p	PROPN
ejpam-5779	180	15	)	)	PUNCT
ejpam-5779	180	16	.	.	PUNCT
ejpam-5779	181	1	let	let	VERB
ejpam-5779	181	2	p	p	PRON
ejpam-5779	181	3	,	,	PUNCT
ejpam-5779	181	4	q	q	PROPN
ejpam-5779	181	5	∈	∈	PROPN
ejpam-5779	181	6	a.	a.	NOUN
ejpam-5779	181	7	then199	then199	PROPN
ejpam-5779	181	8	f̃l(p	f̃l(p	PROPN
ejpam-5779	181	9	)	)	PUNCT
ejpam-5779	182	1	=	=	SYM
ejpam-5779	182	2	f̃sup(p)−	f̃sup(p)−	PROPN
ejpam-5779	182	3	f̃inf(p	f̃inf(p	NOUN
ejpam-5779	182	4	)	)	PUNCT
ejpam-5779	182	5	=	=	SYM
ejpam-5779	183	1	f̃sup(p)−	f̃sup(p)−	PROPN
ejpam-5779	183	2	f̃inf(0	f̃inf(0	PROPN
ejpam-5779	183	3	)	)	PUNCT
ejpam-5779	183	4	≤	≤	NUM
ejpam-5779	183	5	max{f̃sup(pq	max{f̃sup(pq	NOUN
ejpam-5779	183	6	)	)	PUNCT
ejpam-5779	183	7	,	,	PUNCT
ejpam-5779	183	8	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	183	9	)	)	PUNCT
ejpam-5779	183	10	}	}	PUNCT
ejpam-5779	183	11	−	−	ADP
ejpam-5779	183	12	f̃inf(0	f̃inf(0	PROPN
ejpam-5779	183	13	)	)	PUNCT
ejpam-5779	183	14	=	=	PUNCT
ejpam-5779	183	15	max{f̃sup(pq)−	max{f̃sup(pq)−	NUM
ejpam-5779	183	16	f̃inf(0	f̃inf(0	PROPN
ejpam-5779	183	17	)	)	PUNCT
ejpam-5779	183	18	,	,	PUNCT
ejpam-5779	183	19	f̃sup(q)−	f̃sup(q)−	PROPN
ejpam-5779	183	20	f̃inf(0	f̃inf(0	PROPN
ejpam-5779	183	21	)	)	PUNCT
ejpam-5779	183	22	}	}	PUNCT
ejpam-5779	183	23	=	=	PUNCT
ejpam-5779	183	24	max{f̃sup(pq)−	max{f̃sup(pq)−	NUM
ejpam-5779	183	25	f̃inf(p	f̃inf(p	PROPN
ejpam-5779	183	26	q	q	PROPN
ejpam-5779	183	27	)	)	PUNCT
ejpam-5779	183	28	,	,	PUNCT
ejpam-5779	183	29	f̃sup(q)−	f̃sup(q)−	PROPN
ejpam-5779	183	30	f̃inf(q	f̃inf(q	PROPN
ejpam-5779	183	31	)	)	PUNCT
ejpam-5779	183	32	}	}	PUNCT
ejpam-5779	183	33	=	=	SYM
ejpam-5779	183	34	max{f̃l(pq	max{f̃l(pq	NOUN
ejpam-5779	183	35	)	)	PUNCT
ejpam-5779	183	36	,	,	PUNCT
ejpam-5779	183	37	f̃l(q	f̃l(q	PROPN
ejpam-5779	183	38	)	)	PUNCT
ejpam-5779	183	39	}	}	PUNCT
ejpam-5779	183	40	.	.	PUNCT
ejpam-5779	184	1	hence	hence	ADV
ejpam-5779	184	2	,	,	PUNCT
ejpam-5779	184	3	(	(	PUNCT
ejpam-5779	184	4	a	a	DET
ejpam-5779	184	5	,	,	PUNCT
ejpam-5779	184	6	f̃l	f̃l	PROPN
ejpam-5779	184	7	)	)	PUNCT
ejpam-5779	184	8	is	be	AUX
ejpam-5779	184	9	a	a	DET
ejpam-5779	184	10	4	4	NUM
ejpam-5779	184	11	-	-	PUNCT
ejpam-5779	184	12	fuzzy	fuzzy	ADJ
ejpam-5779	184	13	ideal	ideal	NOUN
ejpam-5779	184	14	of	of	ADP
ejpam-5779	184	15	a	a	PRON
ejpam-5779	184	16	,	,	PUNCT
ejpam-5779	184	17	that	that	ADV
ejpam-5779	184	18	is	is	ADV
ejpam-5779	184	19	,	,	PUNCT
ejpam-5779	184	20	(	(	PUNCT
ejpam-5779	184	21	a	a	PRON
ejpam-5779	184	22	,	,	PUNCT
ejpam-5779	184	23	f̃	f̃	PROPN
ejpam-5779	184	24	)	)	PUNCT
ejpam-5779	184	25	is	be	AUX
ejpam-5779	184	26	a	a	DET
ejpam-5779	184	27	length	length	NOUN
ejpam-5779	184	28	4	4	NUM
ejpam-5779	184	29	-	-	PUNCT
ejpam-5779	184	30	fuzzy	fuzzy	ADJ
ejpam-5779	184	31	ideal	ideal	NOUN
ejpam-5779	184	32	of	of	ADP
ejpam-5779	184	33	a.200	a.200	PRON
ejpam-5779	184	34	corollary	corollary	ADJ
ejpam-5779	184	35	3	3	NUM
ejpam-5779	184	36	.	.	PUNCT
ejpam-5779	185	1	if	if	SCONJ
ejpam-5779	185	2	(	(	PUNCT
ejpam-5779	185	3	a	a	PRON
ejpam-5779	185	4	,	,	PUNCT
ejpam-5779	185	5	f̃	f̃	PROPN
ejpam-5779	185	6	)	)	PUNCT
ejpam-5779	185	7	is	be	AUX
ejpam-5779	185	8	an	an	DET
ejpam-5779	185	9	interval	interval	NOUN
ejpam-5779	185	10	-	-	PUNCT
ejpam-5779	185	11	valued	value	VERB
ejpam-5779	185	12	fuzzy	fuzzy	ADJ
ejpam-5779	185	13	structure	structure	NOUN
ejpam-5779	185	14	over	over	ADP
ejpam-5779	185	15	a	a	PRON
ejpam-5779	185	16	in	in	ADP
ejpam-5779	185	17	which	which	PRON
ejpam-5779	185	18	(	(	PUNCT
ejpam-5779	185	19	a	a	PRON
ejpam-5779	185	20	,	,	PUNCT
ejpam-5779	185	21	f̃inf	f̃inf	ADJ
ejpam-5779	185	22	)	)	PUNCT
ejpam-5779	185	23	is201	is201	ADJ
ejpam-5779	185	24	constant	constant	ADJ
ejpam-5779	185	25	and	and	CCONJ
ejpam-5779	185	26	(	(	PUNCT
ejpam-5779	185	27	a	a	DET
ejpam-5779	185	28	,	,	PUNCT
ejpam-5779	185	29	f̃sup	f̃sup	ADJ
ejpam-5779	185	30	)	)	PUNCT
ejpam-5779	185	31	is	be	AUX
ejpam-5779	185	32	a	a	DET
ejpam-5779	185	33	2	2	NUM
ejpam-5779	185	34	-	-	PUNCT
ejpam-5779	185	35	fuzzy	fuzzy	ADJ
ejpam-5779	185	36	ideal	ideal	NOUN
ejpam-5779	185	37	of	of	ADP
ejpam-5779	185	38	a	a	PRON
ejpam-5779	185	39	,	,	PUNCT
ejpam-5779	185	40	then	then	ADV
ejpam-5779	185	41	(	(	PUNCT
ejpam-5779	185	42	a	a	PRON
ejpam-5779	185	43	,	,	PUNCT
ejpam-5779	185	44	f̃	f̃	PROPN
ejpam-5779	185	45	)	)	PUNCT
ejpam-5779	185	46	is	be	AUX
ejpam-5779	185	47	a	a	DET
ejpam-5779	185	48	length	length	NOUN
ejpam-5779	185	49	4	4	NUM
ejpam-5779	185	50	-	-	PUNCT
ejpam-5779	185	51	fuzzy	fuzzy	ADJ
ejpam-5779	185	52	ideal	ideal	NOUN
ejpam-5779	185	53	of	of	ADP
ejpam-5779	185	54	a.202	a.202	PROPN
ejpam-5779	185	55	proof	proof	NOUN
ejpam-5779	185	56	.	.	PUNCT
ejpam-5779	186	1	it	it	PRON
ejpam-5779	186	2	is	be	AUX
ejpam-5779	186	3	straightforward	straightforward	ADJ
ejpam-5779	186	4	by	by	ADP
ejpam-5779	186	5	theorems	theorem	NOUN
ejpam-5779	186	6	1	1	NUM
ejpam-5779	186	7	and	and	CCONJ
ejpam-5779	186	8	6.203	6.203	NUM
ejpam-5779	186	9	corollary	corollary	ADJ
ejpam-5779	186	10	4	4	NUM
ejpam-5779	186	11	.	.	PUNCT
ejpam-5779	187	1	for	for	ADP
ejpam-5779	187	2	j	j	PROPN
ejpam-5779	187	3	∈	∈	PROPN
ejpam-5779	187	4	{	{	PUNCT
ejpam-5779	187	5	2	2	NUM
ejpam-5779	187	6	,	,	PUNCT
ejpam-5779	187	7	4	4	NUM
ejpam-5779	187	8	}	}	PUNCT
ejpam-5779	187	9	,	,	PUNCT
ejpam-5779	187	10	every	every	DET
ejpam-5779	187	11	(	(	PUNCT
ejpam-5779	187	12	2(3	2(3	NUM
ejpam-5779	187	13	)	)	PUNCT
ejpam-5779	187	14	,	,	PUNCT
ejpam-5779	187	15	j)-hyperfuzzy	j)-hyperfuzzy	X
ejpam-5779	187	16	ideal	ideal	NOUN
ejpam-5779	187	17	of	of	ADP
ejpam-5779	187	18	a	a	PRON
ejpam-5779	187	19	is	be	AUX
ejpam-5779	187	20	a	a	DET
ejpam-5779	187	21	length	length	NOUN
ejpam-5779	187	22	4	4	NUM
ejpam-5779	187	23	-	-	PUNCT
ejpam-5779	187	24	fuzzy	fuzzy	ADJ
ejpam-5779	187	25	ideal.204	ideal.204	ADJ
ejpam-5779	187	26	proof	proof	NOUN
ejpam-5779	187	27	.	.	PUNCT
ejpam-5779	188	1	it	it	PRON
ejpam-5779	188	2	is	be	AUX
ejpam-5779	188	3	straightforward	straightforward	ADJ
ejpam-5779	188	4	by	by	ADP
ejpam-5779	188	5	theorem	theorem	ADJ
ejpam-5779	188	6	6	6	NUM
ejpam-5779	188	7	and	and	CCONJ
ejpam-5779	188	8	corollary	corollary	ADJ
ejpam-5779	188	9	3.205	3.205	NUM
ejpam-5779	188	10	theorem	theorem	NOUN
ejpam-5779	188	11	7	7	NUM
ejpam-5779	188	12	.	.	PUNCT
ejpam-5779	189	1	if	if	SCONJ
ejpam-5779	189	2	(	(	PUNCT
ejpam-5779	189	3	a	a	PRON
ejpam-5779	189	4	,	,	PUNCT
ejpam-5779	189	5	f̃	f̃	PROPN
ejpam-5779	189	6	)	)	PUNCT
ejpam-5779	189	7	is	be	AUX
ejpam-5779	189	8	an	an	DET
ejpam-5779	189	9	interval	interval	NOUN
ejpam-5779	189	10	-	-	PUNCT
ejpam-5779	189	11	valued	value	VERB
ejpam-5779	189	12	fuzzy	fuzzy	ADJ
ejpam-5779	189	13	structure	structure	NOUN
ejpam-5779	189	14	over	over	ADP
ejpam-5779	189	15	a	a	DET
ejpam-5779	189	16	in	in	ADP
ejpam-5779	189	17	which	which	PRON
ejpam-5779	189	18	(	(	PUNCT
ejpam-5779	189	19	a	a	PRON
ejpam-5779	189	20	,	,	PUNCT
ejpam-5779	189	21	f̃sup	f̃sup	ADJ
ejpam-5779	189	22	)	)	PUNCT
ejpam-5779	189	23	is206	is206	PROPN
ejpam-5779	189	24	constant	constant	ADJ
ejpam-5779	189	25	and	and	CCONJ
ejpam-5779	189	26	(	(	PUNCT
ejpam-5779	189	27	a	a	PRON
ejpam-5779	189	28	,	,	PUNCT
ejpam-5779	189	29	f̃inf	f̃inf	ADJ
ejpam-5779	189	30	)	)	PUNCT
ejpam-5779	189	31	is	be	AUX
ejpam-5779	189	32	a	a	DET
ejpam-5779	189	33	4	4	NUM
ejpam-5779	189	34	-	-	PUNCT
ejpam-5779	189	35	fuzzy	fuzzy	ADJ
ejpam-5779	189	36	ideal	ideal	NOUN
ejpam-5779	189	37	of	of	ADP
ejpam-5779	189	38	a	a	PRON
ejpam-5779	189	39	,	,	PUNCT
ejpam-5779	189	40	then	then	ADV
ejpam-5779	189	41	(	(	PUNCT
ejpam-5779	189	42	a	a	PRON
ejpam-5779	189	43	,	,	PUNCT
ejpam-5779	189	44	f̃	f̃	PROPN
ejpam-5779	189	45	)	)	PUNCT
ejpam-5779	189	46	is	be	AUX
ejpam-5779	189	47	a	a	DET
ejpam-5779	189	48	length	length	NOUN
ejpam-5779	189	49	1	1	NUM
ejpam-5779	189	50	-	-	PUNCT
ejpam-5779	189	51	fuzzy	fuzzy	ADJ
ejpam-5779	189	52	ideal	ideal	NOUN
ejpam-5779	189	53	of	of	ADP
ejpam-5779	189	54	a.207	a.207	NOUN
ejpam-5779	189	55	proof	proof	NOUN
ejpam-5779	189	56	.	.	PUNCT
ejpam-5779	190	1	assume	assume	VERB
ejpam-5779	190	2	that	that	SCONJ
ejpam-5779	190	3	(	(	PUNCT
ejpam-5779	190	4	a	a	PRON
ejpam-5779	190	5	,	,	PUNCT
ejpam-5779	190	6	f̃	f̃	PROPN
ejpam-5779	190	7	)	)	PUNCT
ejpam-5779	190	8	is	be	AUX
ejpam-5779	190	9	an	an	DET
ejpam-5779	190	10	interval	interval	NOUN
ejpam-5779	190	11	-	-	PUNCT
ejpam-5779	190	12	valued	value	VERB
ejpam-5779	190	13	fuzzy	fuzzy	ADJ
ejpam-5779	190	14	structure	structure	NOUN
ejpam-5779	190	15	over	over	ADP
ejpam-5779	190	16	a	a	PRON
ejpam-5779	190	17	in	in	ADP
ejpam-5779	190	18	which208	which208	PROPN
ejpam-5779	190	19	(	(	PUNCT
ejpam-5779	190	20	a	a	DET
ejpam-5779	190	21	,	,	PUNCT
ejpam-5779	190	22	f̃sup	f̃sup	ADJ
ejpam-5779	190	23	)	)	PUNCT
ejpam-5779	190	24	is	be	AUX
ejpam-5779	190	25	constant	constant	ADJ
ejpam-5779	190	26	and	and	CCONJ
ejpam-5779	190	27	(	(	PUNCT
ejpam-5779	190	28	a	a	PRON
ejpam-5779	190	29	,	,	PUNCT
ejpam-5779	190	30	f̃inf	f̃inf	ADJ
ejpam-5779	190	31	)	)	PUNCT
ejpam-5779	190	32	is	be	AUX
ejpam-5779	190	33	a	a	DET
ejpam-5779	190	34	4	4	NUM
ejpam-5779	190	35	-	-	PUNCT
ejpam-5779	190	36	fuzzy	fuzzy	ADJ
ejpam-5779	190	37	ideal	ideal	NOUN
ejpam-5779	190	38	of	of	ADP
ejpam-5779	190	39	a.	a.	NOUN
ejpam-5779	190	40	let	let	VERB
ejpam-5779	190	41	p	p	PRON
ejpam-5779	190	42	,	,	PUNCT
ejpam-5779	190	43	q	q	PROPN
ejpam-5779	190	44	∈	∈	PROPN
ejpam-5779	190	45	a.	a.	NOUN
ejpam-5779	190	46	since	since	SCONJ
ejpam-5779	190	47	(	(	PUNCT
ejpam-5779	190	48	a	a	DET
ejpam-5779	190	49	,	,	PUNCT
ejpam-5779	190	50	f̃sup	f̃sup	ADJ
ejpam-5779	190	51	)	)	PUNCT
ejpam-5779	190	52	is209	is209	VERB
ejpam-5779	190	53	constant	constant	ADJ
ejpam-5779	190	54	,	,	PUNCT
ejpam-5779	190	55	we	we	PRON
ejpam-5779	190	56	have	have	VERB
ejpam-5779	190	57	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	190	58	)	)	PUNCT
ejpam-5779	191	1	=	=	SYM
ejpam-5779	191	2	f̃sup(0	f̃sup(0	NOUN
ejpam-5779	191	3	)	)	PUNCT
ejpam-5779	191	4	for	for	ADP
ejpam-5779	191	5	all	all	DET
ejpam-5779	191	6	p	p	PROPN
ejpam-5779	191	7	∈	∈	PROPN
ejpam-5779	191	8	a.	a.	NOUN
ejpam-5779	191	9	since	since	SCONJ
ejpam-5779	191	10	(	(	PUNCT
ejpam-5779	191	11	a	a	PRON
ejpam-5779	191	12	,	,	PUNCT
ejpam-5779	191	13	f̃inf	f̃inf	ADJ
ejpam-5779	191	14	)	)	PUNCT
ejpam-5779	191	15	is	be	AUX
ejpam-5779	191	16	a	a	DET
ejpam-5779	191	17	4	4	NUM
ejpam-5779	191	18	-	-	PUNCT
ejpam-5779	191	19	fuzzy	fuzzy	ADJ
ejpam-5779	191	20	ideal	ideal	NOUN
ejpam-5779	191	21	of	of	ADP
ejpam-5779	191	22	a,210	a,210	NOUN
ejpam-5779	191	23	we	we	PRON
ejpam-5779	191	24	have211	have211	PROPN
ejpam-5779	191	25	(	(	PUNCT
ejpam-5779	191	26	∀p	∀p	NOUN
ejpam-5779	191	27	∈	∈	PROPN
ejpam-5779	191	28	a)(f̃inf(0	a)(f̃inf(0	PROPN
ejpam-5779	191	29	)	)	PUNCT
ejpam-5779	191	30	≤	≤	NOUN
ejpam-5779	191	31	f̃inf(p	f̃inf(p	NOUN
ejpam-5779	191	32	)	)	PUNCT
ejpam-5779	191	33	)	)	PUNCT
ejpam-5779	191	34	,	,	PUNCT
ejpam-5779	191	35	(	(	PUNCT
ejpam-5779	191	36	13	13	NUM
ejpam-5779	191	37	)	)	PUNCT
ejpam-5779	191	38	(	(	PUNCT
ejpam-5779	191	39	∀p	∀p	X
ejpam-5779	191	40	,	,	PUNCT
ejpam-5779	191	41	q	q	PROPN
ejpam-5779	191	42	∈	∈	PROPN
ejpam-5779	191	43	a)(f̃inf(p	a)(f̃inf(p	PROPN
ejpam-5779	191	44	)	)	PUNCT
ejpam-5779	191	45	≤	≤	NOUN
ejpam-5779	191	46	max{f̃inf(pq	max{f̃inf(pq	NOUN
ejpam-5779	191	47	)	)	PUNCT
ejpam-5779	191	48	,	,	PUNCT
ejpam-5779	191	49	f̃inf(q	f̃inf(q	PROPN
ejpam-5779	191	50	)	)	PUNCT
ejpam-5779	191	51	}	}	PUNCT
ejpam-5779	191	52	)	)	PUNCT
ejpam-5779	191	53	.	.	PUNCT
ejpam-5779	192	1	(	(	PUNCT
ejpam-5779	192	2	14	14	NUM
ejpam-5779	192	3	)	)	PUNCT
ejpam-5779	192	4	let	let	VERB
ejpam-5779	192	5	p	p	PRON
ejpam-5779	192	6	∈	∈	PROPN
ejpam-5779	192	7	a.	a.	NOUN
ejpam-5779	192	8	then212	then212	PROPN
ejpam-5779	192	9	f̃l(0	f̃l(0	PROPN
ejpam-5779	192	10	)	)	PUNCT
ejpam-5779	192	11	=	=	SYM
ejpam-5779	192	12	f̃sup(0)−	f̃sup(0)−	NUM
ejpam-5779	192	13	f̃inf(0	f̃inf(0	PROPN
ejpam-5779	192	14	)	)	PUNCT
ejpam-5779	192	15	≥	≥	NOUN
ejpam-5779	192	16	f̃sup(0)−	f̃sup(0)−	NUM
ejpam-5779	192	17	f̃inf(p	f̃inf(p	NOUN
ejpam-5779	192	18	)	)	PUNCT
ejpam-5779	192	19	=	=	SYM
ejpam-5779	193	1	f̃sup(p)−	f̃sup(p)−	PROPN
ejpam-5779	193	2	f̃inf(p	f̃inf(p	NOUN
ejpam-5779	193	3	)	)	PUNCT
ejpam-5779	193	4	=	=	SYM
ejpam-5779	193	5	f̃l(p	f̃l(p	PROPN
ejpam-5779	193	6	)	)	PUNCT
ejpam-5779	193	7	.	.	PUNCT
ejpam-5779	194	1	n.	n.	PROPN
ejpam-5779	194	2	rajesh	rajesh	PROPN
ejpam-5779	194	3	,	,	PUNCT
ejpam-5779	194	4	t.	t.	PROPN
ejpam-5779	194	5	oner	oner	NOUN
ejpam-5779	194	6	,	,	PUNCT
ejpam-5779	194	7	a.	a.	NOUN
ejpam-5779	194	8	iampan	iampan	PROPN
ejpam-5779	194	9	,	,	PUNCT
ejpam-5779	194	10	i.	i.	PROPN
ejpam-5779	194	11	senturk	senturk	PROPN
ejpam-5779	194	12	/	/	SYM
ejpam-5779	194	13	eur	eur	PROPN
ejpam-5779	194	14	.	.	PUNCT
ejpam-5779	195	1	j.	j.	PROPN
ejpam-5779	195	2	pure	pure	PROPN
ejpam-5779	195	3	appl	appl	PROPN
ejpam-5779	195	4	.	.	PROPN
ejpam-5779	195	5	math	math	PROPN
ejpam-5779	195	6	,	,	PUNCT
ejpam-5779	195	7	18	18	NUM
ejpam-5779	195	8	(	(	PUNCT
ejpam-5779	195	9	1	1	NUM
ejpam-5779	195	10	)	)	PUNCT
ejpam-5779	195	11	(	(	PUNCT
ejpam-5779	195	12	2025	2025	NUM
ejpam-5779	195	13	)	)	PUNCT
ejpam-5779	195	14	,	,	PUNCT
ejpam-5779	195	15	5779	5779	NUM
ejpam-5779	195	16	10	10	NUM
ejpam-5779	195	17	of	of	ADP
ejpam-5779	195	18	18	18	NUM
ejpam-5779	195	19	let	let	VERB
ejpam-5779	195	20	p	p	PRON
ejpam-5779	195	21	,	,	PUNCT
ejpam-5779	195	22	q	q	PROPN
ejpam-5779	195	23	∈	∈	PROPN
ejpam-5779	195	24	a.	a.	NOUN
ejpam-5779	195	25	then213	then213	PROPN
ejpam-5779	195	26	f̃l(p	f̃l(p	PROPN
ejpam-5779	195	27	)	)	PUNCT
ejpam-5779	196	1	=	=	SYM
ejpam-5779	196	2	f̃sup(p)−	f̃sup(p)−	PROPN
ejpam-5779	196	3	f̃inf(p	f̃inf(p	NOUN
ejpam-5779	196	4	)	)	PUNCT
ejpam-5779	197	1	=	=	SYM
ejpam-5779	197	2	f̃sup(0)−	f̃sup(0)−	PROPN
ejpam-5779	197	3	f̃inf(p	f̃inf(p	PROPN
ejpam-5779	197	4	)	)	PUNCT
ejpam-5779	197	5	≥	≥	NOUN
ejpam-5779	197	6	f̃sup(0)−max{f̃inf(pq	f̃sup(0)−max{f̃inf(pq	ADJ
ejpam-5779	197	7	)	)	PUNCT
ejpam-5779	197	8	,	,	PUNCT
ejpam-5779	197	9	f̃inf(q	f̃inf(q	PROPN
ejpam-5779	197	10	)	)	PUNCT
ejpam-5779	197	11	}	}	PUNCT
ejpam-5779	197	12	=	=	SYM
ejpam-5779	197	13	min{f̃sup(0)−	min{f̃sup(0)−	PROPN
ejpam-5779	197	14	f̃inf(p	f̃inf(p	PROPN
ejpam-5779	197	15	q	q	PROPN
ejpam-5779	197	16	)	)	PUNCT
ejpam-5779	197	17	,	,	PUNCT
ejpam-5779	197	18	f̃sup(0)−	f̃sup(0)−	PROPN
ejpam-5779	197	19	f̃inf(q	f̃inf(q	PROPN
ejpam-5779	197	20	)	)	PUNCT
ejpam-5779	197	21	}	}	PUNCT
ejpam-5779	197	22	=	=	PUNCT
ejpam-5779	198	1	min{f̃sup(pq)−	min{f̃sup(pq)−	NUM
ejpam-5779	198	2	f̃inf(p	f̃inf(p	X
ejpam-5779	198	3	q	q	NOUN
ejpam-5779	198	4	)	)	PUNCT
ejpam-5779	198	5	,	,	PUNCT
ejpam-5779	198	6	f̃sup(q)−	f̃sup(q)−	PROPN
ejpam-5779	198	7	f̃inf(q	f̃inf(q	PROPN
ejpam-5779	198	8	)	)	PUNCT
ejpam-5779	198	9	}	}	PUNCT
ejpam-5779	198	10	=	=	SYM
ejpam-5779	198	11	min{f̃l(pq	min{f̃l(pq	NOUN
ejpam-5779	198	12	)	)	PUNCT
ejpam-5779	198	13	,	,	PUNCT
ejpam-5779	198	14	f̃l(q	f̃l(q	PROPN
ejpam-5779	198	15	)	)	PUNCT
ejpam-5779	198	16	}	}	PUNCT
ejpam-5779	198	17	.	.	PUNCT
ejpam-5779	199	1	hence	hence	ADV
ejpam-5779	199	2	,	,	PUNCT
ejpam-5779	199	3	(	(	PUNCT
ejpam-5779	199	4	a	a	DET
ejpam-5779	199	5	,	,	PUNCT
ejpam-5779	199	6	f̃l	f̃l	PROPN
ejpam-5779	199	7	)	)	PUNCT
ejpam-5779	199	8	is	be	AUX
ejpam-5779	199	9	a	a	DET
ejpam-5779	199	10	1	1	NUM
ejpam-5779	199	11	-	-	PUNCT
ejpam-5779	199	12	fuzzy	fuzzy	ADJ
ejpam-5779	199	13	ideal	ideal	NOUN
ejpam-5779	199	14	of	of	ADP
ejpam-5779	199	15	a	a	PRON
ejpam-5779	199	16	,	,	PUNCT
ejpam-5779	199	17	that	that	ADV
ejpam-5779	199	18	is	is	ADV
ejpam-5779	199	19	,	,	PUNCT
ejpam-5779	199	20	(	(	PUNCT
ejpam-5779	199	21	a	a	PRON
ejpam-5779	199	22	,	,	PUNCT
ejpam-5779	199	23	f̃	f̃	PROPN
ejpam-5779	199	24	)	)	PUNCT
ejpam-5779	199	25	is	be	AUX
ejpam-5779	199	26	a	a	DET
ejpam-5779	199	27	length	length	NOUN
ejpam-5779	199	28	1	1	NUM
ejpam-5779	199	29	-	-	PUNCT
ejpam-5779	199	30	fuzzy	fuzzy	ADJ
ejpam-5779	199	31	ideal	ideal	NOUN
ejpam-5779	199	32	of	of	ADP
ejpam-5779	199	33	a.214	a.214	ADJ
ejpam-5779	199	34	corollary	corollary	ADJ
ejpam-5779	199	35	5	5	NUM
ejpam-5779	199	36	.	.	PUNCT
ejpam-5779	200	1	if	if	SCONJ
ejpam-5779	200	2	(	(	PUNCT
ejpam-5779	200	3	a	a	PRON
ejpam-5779	200	4	,	,	PUNCT
ejpam-5779	200	5	f̃	f̃	PROPN
ejpam-5779	200	6	)	)	PUNCT
ejpam-5779	200	7	is	be	AUX
ejpam-5779	200	8	an	an	DET
ejpam-5779	200	9	interval	interval	NOUN
ejpam-5779	200	10	-	-	PUNCT
ejpam-5779	200	11	valued	value	VERB
ejpam-5779	200	12	fuzzy	fuzzy	ADJ
ejpam-5779	200	13	structure	structure	NOUN
ejpam-5779	200	14	over	over	ADP
ejpam-5779	200	15	a	a	PRON
ejpam-5779	200	16	in	in	ADP
ejpam-5779	200	17	which	which	PRON
ejpam-5779	200	18	(	(	PUNCT
ejpam-5779	200	19	a	a	DET
ejpam-5779	200	20	,	,	PUNCT
ejpam-5779	200	21	f̃sup	f̃sup	ADJ
ejpam-5779	200	22	)	)	PUNCT
ejpam-5779	201	1	is215	is215	VERB
ejpam-5779	201	2	constant	constant	ADJ
ejpam-5779	201	3	and	and	CCONJ
ejpam-5779	201	4	(	(	PUNCT
ejpam-5779	201	5	a	a	PRON
ejpam-5779	201	6	,	,	PUNCT
ejpam-5779	201	7	f̃inf	f̃inf	ADJ
ejpam-5779	201	8	)	)	PUNCT
ejpam-5779	201	9	is	be	AUX
ejpam-5779	201	10	a	a	DET
ejpam-5779	201	11	2	2	NUM
ejpam-5779	201	12	-	-	PUNCT
ejpam-5779	201	13	fuzzy	fuzzy	ADJ
ejpam-5779	201	14	ideal	ideal	NOUN
ejpam-5779	201	15	of	of	ADP
ejpam-5779	201	16	a	a	PRON
ejpam-5779	201	17	,	,	PUNCT
ejpam-5779	201	18	then	then	ADV
ejpam-5779	201	19	(	(	PUNCT
ejpam-5779	201	20	a	a	PRON
ejpam-5779	201	21	,	,	PUNCT
ejpam-5779	201	22	f̃	f̃	PROPN
ejpam-5779	201	23	)	)	PUNCT
ejpam-5779	201	24	is	be	AUX
ejpam-5779	201	25	a	a	DET
ejpam-5779	201	26	length	length	NOUN
ejpam-5779	201	27	1	1	NUM
ejpam-5779	201	28	-	-	PUNCT
ejpam-5779	201	29	fuzzy	fuzzy	ADJ
ejpam-5779	201	30	ideal	ideal	NOUN
ejpam-5779	201	31	of	of	ADP
ejpam-5779	201	32	a.216	a.216	NOUN
ejpam-5779	201	33	proof	proof	NOUN
ejpam-5779	201	34	.	.	PUNCT
ejpam-5779	202	1	it	it	PRON
ejpam-5779	202	2	is	be	AUX
ejpam-5779	202	3	straightforward	straightforward	ADJ
ejpam-5779	202	4	by	by	ADP
ejpam-5779	202	5	theorems	theorem	NOUN
ejpam-5779	202	6	1	1	NUM
ejpam-5779	202	7	and	and	CCONJ
ejpam-5779	202	8	7.217	7.217	NUM
ejpam-5779	202	9	corollary	corollary	ADJ
ejpam-5779	202	10	6	6	NUM
ejpam-5779	202	11	.	.	PUNCT
ejpam-5779	203	1	for	for	ADP
ejpam-5779	203	2	i	i	PRON
ejpam-5779	203	3	∈	∈	PROPN
ejpam-5779	203	4	{	{	PUNCT
ejpam-5779	203	5	2	2	NUM
ejpam-5779	203	6	,	,	PUNCT
ejpam-5779	203	7	4	4	NUM
ejpam-5779	203	8	}	}	PUNCT
ejpam-5779	203	9	,	,	PUNCT
ejpam-5779	203	10	every	every	PRON
ejpam-5779	203	11	(	(	PUNCT
ejpam-5779	203	12	i	i	NOUN
ejpam-5779	203	13	,	,	PUNCT
ejpam-5779	203	14	2(3))-hyperfuzzy	2(3))-hyperfuzzy	NUM
ejpam-5779	203	15	ideal	ideal	NOUN
ejpam-5779	203	16	of	of	ADP
ejpam-5779	203	17	a	a	PRON
ejpam-5779	203	18	is	be	AUX
ejpam-5779	203	19	a	a	DET
ejpam-5779	203	20	length	length	NOUN
ejpam-5779	203	21	1	1	NUM
ejpam-5779	203	22	-	-	PUNCT
ejpam-5779	203	23	fuzzy	fuzzy	ADJ
ejpam-5779	203	24	ideal.218	ideal.218	NOUN
ejpam-5779	203	25	proof	proof	NOUN
ejpam-5779	203	26	.	.	PUNCT
ejpam-5779	204	1	it	it	PRON
ejpam-5779	204	2	is	be	AUX
ejpam-5779	204	3	straightforward	straightforward	ADJ
ejpam-5779	204	4	by	by	ADP
ejpam-5779	204	5	theorem	theorem	ADJ
ejpam-5779	204	6	7	7	NUM
ejpam-5779	204	7	and	and	CCONJ
ejpam-5779	204	8	corollary	corollary	ADJ
ejpam-5779	204	9	5.219	5.219	NUM
ejpam-5779	204	10	theorem	theorem	NOUN
ejpam-5779	204	11	8	8	NUM
ejpam-5779	204	12	.	.	PUNCT
ejpam-5779	205	1	if	if	SCONJ
ejpam-5779	205	2	(	(	PUNCT
ejpam-5779	205	3	a	a	PRON
ejpam-5779	205	4	,	,	PUNCT
ejpam-5779	205	5	f̃	f̃	PROPN
ejpam-5779	205	6	)	)	PUNCT
ejpam-5779	205	7	is	be	AUX
ejpam-5779	205	8	an	an	DET
ejpam-5779	205	9	interval	interval	NOUN
ejpam-5779	205	10	-	-	PUNCT
ejpam-5779	205	11	valued	value	VERB
ejpam-5779	205	12	fuzzy	fuzzy	ADJ
ejpam-5779	205	13	structure	structure	NOUN
ejpam-5779	205	14	over	over	ADP
ejpam-5779	205	15	a	a	DET
ejpam-5779	205	16	in	in	ADP
ejpam-5779	205	17	which	which	PRON
ejpam-5779	205	18	(	(	PUNCT
ejpam-5779	205	19	a	a	DET
ejpam-5779	205	20	,	,	PUNCT
ejpam-5779	205	21	f̃sup	f̃sup	ADJ
ejpam-5779	205	22	)	)	PUNCT
ejpam-5779	205	23	is220	is220	VERB
ejpam-5779	205	24	constant	constant	ADJ
ejpam-5779	205	25	and	and	CCONJ
ejpam-5779	205	26	(	(	PUNCT
ejpam-5779	205	27	a	a	PRON
ejpam-5779	205	28	,	,	PUNCT
ejpam-5779	205	29	f̃inf	f̃inf	ADJ
ejpam-5779	205	30	)	)	PUNCT
ejpam-5779	205	31	is	be	AUX
ejpam-5779	205	32	a	a	DET
ejpam-5779	205	33	1	1	NUM
ejpam-5779	205	34	-	-	PUNCT
ejpam-5779	205	35	fuzzy	fuzzy	ADJ
ejpam-5779	205	36	ideal	ideal	NOUN
ejpam-5779	205	37	of	of	ADP
ejpam-5779	205	38	a	a	PRON
ejpam-5779	205	39	,	,	PUNCT
ejpam-5779	205	40	then	then	ADV
ejpam-5779	205	41	(	(	PUNCT
ejpam-5779	205	42	a	a	PRON
ejpam-5779	205	43	,	,	PUNCT
ejpam-5779	205	44	f̃	f̃	PROPN
ejpam-5779	205	45	)	)	PUNCT
ejpam-5779	205	46	is	be	AUX
ejpam-5779	205	47	a	a	DET
ejpam-5779	205	48	length	length	NOUN
ejpam-5779	205	49	4	4	NUM
ejpam-5779	205	50	-	-	PUNCT
ejpam-5779	205	51	fuzzy	fuzzy	ADJ
ejpam-5779	205	52	ideal	ideal	NOUN
ejpam-5779	205	53	of	of	ADP
ejpam-5779	205	54	a.221	a.221	NOUN
ejpam-5779	205	55	proof	proof	NOUN
ejpam-5779	205	56	.	.	PUNCT
ejpam-5779	206	1	assume	assume	VERB
ejpam-5779	206	2	that	that	SCONJ
ejpam-5779	206	3	(	(	PUNCT
ejpam-5779	206	4	a	a	PRON
ejpam-5779	206	5	,	,	PUNCT
ejpam-5779	206	6	f̃	f̃	PROPN
ejpam-5779	206	7	)	)	PUNCT
ejpam-5779	206	8	is	be	AUX
ejpam-5779	206	9	an	an	DET
ejpam-5779	206	10	interval	interval	NOUN
ejpam-5779	206	11	-	-	PUNCT
ejpam-5779	206	12	valued	value	VERB
ejpam-5779	206	13	fuzzy	fuzzy	ADJ
ejpam-5779	206	14	structure	structure	NOUN
ejpam-5779	206	15	over	over	ADP
ejpam-5779	206	16	a	a	PRON
ejpam-5779	206	17	in	in	ADP
ejpam-5779	206	18	which222	which222	PROPN
ejpam-5779	206	19	(	(	PUNCT
ejpam-5779	206	20	a	a	DET
ejpam-5779	206	21	,	,	PUNCT
ejpam-5779	206	22	f̃sup	f̃sup	ADJ
ejpam-5779	206	23	)	)	PUNCT
ejpam-5779	206	24	is	be	AUX
ejpam-5779	206	25	constant	constant	ADJ
ejpam-5779	206	26	and	and	CCONJ
ejpam-5779	206	27	(	(	PUNCT
ejpam-5779	206	28	a	a	PRON
ejpam-5779	206	29	,	,	PUNCT
ejpam-5779	206	30	f̃inf	f̃inf	ADJ
ejpam-5779	206	31	)	)	PUNCT
ejpam-5779	206	32	is	be	AUX
ejpam-5779	206	33	a	a	DET
ejpam-5779	206	34	1	1	NUM
ejpam-5779	206	35	-	-	PUNCT
ejpam-5779	206	36	fuzzy	fuzzy	ADJ
ejpam-5779	206	37	ideal	ideal	NOUN
ejpam-5779	206	38	of	of	ADP
ejpam-5779	206	39	a.	a.	NOUN
ejpam-5779	206	40	let	let	VERB
ejpam-5779	206	41	p	p	PRON
ejpam-5779	206	42	,	,	PUNCT
ejpam-5779	206	43	q	q	PROPN
ejpam-5779	206	44	∈	∈	PROPN
ejpam-5779	206	45	a.	a.	NOUN
ejpam-5779	206	46	since	since	SCONJ
ejpam-5779	206	47	(	(	PUNCT
ejpam-5779	206	48	a	a	DET
ejpam-5779	206	49	,	,	PUNCT
ejpam-5779	206	50	f̃sup	f̃sup	ADJ
ejpam-5779	206	51	)	)	PUNCT
ejpam-5779	206	52	is223	is223	NOUN
ejpam-5779	206	53	constant	constant	ADJ
ejpam-5779	206	54	,	,	PUNCT
ejpam-5779	206	55	we	we	PRON
ejpam-5779	206	56	have	have	VERB
ejpam-5779	206	57	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	206	58	)	)	PUNCT
ejpam-5779	206	59	=	=	SYM
ejpam-5779	206	60	f̃sup(0	f̃sup(0	NOUN
ejpam-5779	206	61	)	)	PUNCT
ejpam-5779	206	62	for	for	ADP
ejpam-5779	206	63	all	all	DET
ejpam-5779	206	64	p	p	PROPN
ejpam-5779	206	65	∈	∈	PROPN
ejpam-5779	206	66	a.	a.	NOUN
ejpam-5779	206	67	since	since	SCONJ
ejpam-5779	206	68	(	(	PUNCT
ejpam-5779	206	69	a	a	PRON
ejpam-5779	206	70	,	,	PUNCT
ejpam-5779	206	71	f̃inf	f̃inf	ADJ
ejpam-5779	206	72	)	)	PUNCT
ejpam-5779	206	73	is	be	AUX
ejpam-5779	206	74	a	a	DET
ejpam-5779	206	75	1	1	NUM
ejpam-5779	206	76	-	-	PUNCT
ejpam-5779	206	77	fuzzy	fuzzy	ADJ
ejpam-5779	206	78	ideal	ideal	NOUN
ejpam-5779	206	79	of	of	ADP
ejpam-5779	206	80	a,224	a,224	PROPN
ejpam-5779	207	1	we	we	PRON
ejpam-5779	207	2	have225	have225	PROPN
ejpam-5779	207	3	(	(	PUNCT
ejpam-5779	207	4	∀p	∀p	NOUN
ejpam-5779	207	5	∈	∈	PROPN
ejpam-5779	207	6	a)(f̃inf(0	a)(f̃inf(0	PROPN
ejpam-5779	207	7	)	)	PUNCT
ejpam-5779	207	8	≥	≥	NOUN
ejpam-5779	207	9	f̃inf(p	f̃inf(p	NOUN
ejpam-5779	207	10	)	)	PUNCT
ejpam-5779	207	11	)	)	PUNCT
ejpam-5779	207	12	,	,	PUNCT
ejpam-5779	207	13	(	(	PUNCT
ejpam-5779	207	14	15	15	NUM
ejpam-5779	207	15	)	)	PUNCT
ejpam-5779	207	16	(	(	PUNCT
ejpam-5779	207	17	∀p	∀p	X
ejpam-5779	207	18	,	,	PUNCT
ejpam-5779	207	19	q	q	PROPN
ejpam-5779	207	20	∈	∈	PROPN
ejpam-5779	207	21	a)(f̃inf(p	a)(f̃inf(p	PROPN
ejpam-5779	207	22	)	)	PUNCT
ejpam-5779	207	23	≥	≥	NOUN
ejpam-5779	207	24	min{f̃inf(pq	min{f̃inf(pq	NOUN
ejpam-5779	207	25	)	)	PUNCT
ejpam-5779	207	26	,	,	PUNCT
ejpam-5779	207	27	f̃inf(q	f̃inf(q	PROPN
ejpam-5779	207	28	)	)	PUNCT
ejpam-5779	207	29	}	}	PUNCT
ejpam-5779	207	30	)	)	PUNCT
ejpam-5779	207	31	.	.	PUNCT
ejpam-5779	208	1	(	(	PUNCT
ejpam-5779	208	2	16	16	X
ejpam-5779	208	3	)	)	PUNCT
ejpam-5779	208	4	let	let	VERB
ejpam-5779	208	5	p	p	PRON
ejpam-5779	208	6	∈	∈	PROPN
ejpam-5779	208	7	a.	a.	NOUN
ejpam-5779	208	8	then226	then226	PROPN
ejpam-5779	208	9	f̃l(0	f̃l(0	PROPN
ejpam-5779	208	10	)	)	PUNCT
ejpam-5779	208	11	=	=	SYM
ejpam-5779	209	1	f̃sup(0)−	f̃sup(0)−	NUM
ejpam-5779	209	2	f̃inf(0	f̃inf(0	PROPN
ejpam-5779	209	3	)	)	PUNCT
ejpam-5779	209	4	≤	≤	NOUN
ejpam-5779	209	5	f̃sup(0)−	f̃sup(0)−	NUM
ejpam-5779	209	6	f̃inf(p	f̃inf(p	NOUN
ejpam-5779	209	7	)	)	PUNCT
ejpam-5779	209	8	=	=	SYM
ejpam-5779	209	9	f̃sup(p)−	f̃sup(p)−	PROPN
ejpam-5779	209	10	f̃inf(p	f̃inf(p	NOUN
ejpam-5779	209	11	)	)	PUNCT
ejpam-5779	209	12	=	=	SYM
ejpam-5779	209	13	f̃l(p	f̃l(p	PROPN
ejpam-5779	209	14	)	)	PUNCT
ejpam-5779	209	15	.	.	PUNCT
ejpam-5779	210	1	let	let	VERB
ejpam-5779	210	2	p	p	PRON
ejpam-5779	210	3	,	,	PUNCT
ejpam-5779	210	4	q	q	PROPN
ejpam-5779	210	5	∈	∈	NOUN
ejpam-5779	210	6	a.	a.	NOUN
ejpam-5779	210	7	then227	then227	PROPN
ejpam-5779	210	8	f̃l(p	f̃l(p	PROPN
ejpam-5779	210	9	)	)	PUNCT
ejpam-5779	211	1	=	=	SYM
ejpam-5779	211	2	f̃sup(p)−	f̃sup(p)−	PROPN
ejpam-5779	211	3	f̃inf(p	f̃inf(p	NOUN
ejpam-5779	211	4	)	)	PUNCT
ejpam-5779	212	1	=	=	SYM
ejpam-5779	212	2	f̃sup(0)−	f̃sup(0)−	PROPN
ejpam-5779	212	3	f̃inf(p	f̃inf(p	ADJ
ejpam-5779	212	4	)	)	PUNCT
ejpam-5779	212	5	≤	≤	NUM
ejpam-5779	212	6	f̃sup(0)−min{f̃pinf(pq	f̃sup(0)−min{f̃pinf(pq	NOUN
ejpam-5779	212	7	)	)	PUNCT
ejpam-5779	212	8	,	,	PUNCT
ejpam-5779	212	9	f̃inf(q	f̃inf(q	PROPN
ejpam-5779	212	10	)	)	PUNCT
ejpam-5779	212	11	}	}	PUNCT
ejpam-5779	212	12	=	=	SYM
ejpam-5779	212	13	max{f̃sup(0)−	max{f̃sup(0)−	X
ejpam-5779	212	14	f̃inf(p	f̃inf(p	PROPN
ejpam-5779	212	15	q	q	PROPN
ejpam-5779	212	16	)	)	PUNCT
ejpam-5779	212	17	,	,	PUNCT
ejpam-5779	212	18	f̃sup(0)−	f̃sup(0)−	PROPN
ejpam-5779	212	19	f̃inf(q	f̃inf(q	PROPN
ejpam-5779	212	20	)	)	PUNCT
ejpam-5779	212	21	}	}	PUNCT
ejpam-5779	212	22	=	=	PUNCT
ejpam-5779	212	23	max{f̃sup(pq)−	max{f̃sup(pq)−	NUM
ejpam-5779	212	24	f̃inf(p	f̃inf(p	PROPN
ejpam-5779	212	25	q	q	PROPN
ejpam-5779	212	26	)	)	PUNCT
ejpam-5779	212	27	,	,	PUNCT
ejpam-5779	212	28	f̃sup(qq)−	f̃sup(qq)−	PROPN
ejpam-5779	212	29	f̃inf(q	f̃inf(q	PROPN
ejpam-5779	212	30	)	)	PUNCT
ejpam-5779	212	31	}	}	PUNCT
ejpam-5779	212	32	=	=	SYM
ejpam-5779	212	33	max{f̃l(pq	max{f̃l(pq	NOUN
ejpam-5779	212	34	)	)	PUNCT
ejpam-5779	212	35	,	,	PUNCT
ejpam-5779	212	36	f̃l(q	f̃l(q	PROPN
ejpam-5779	212	37	)	)	PUNCT
ejpam-5779	212	38	}	}	PUNCT
ejpam-5779	212	39	.	.	PUNCT
ejpam-5779	213	1	hence	hence	ADV
ejpam-5779	213	2	,	,	PUNCT
ejpam-5779	213	3	(	(	PUNCT
ejpam-5779	213	4	a	a	DET
ejpam-5779	213	5	,	,	PUNCT
ejpam-5779	213	6	f̃l	f̃l	PROPN
ejpam-5779	213	7	)	)	PUNCT
ejpam-5779	213	8	is	be	AUX
ejpam-5779	213	9	a	a	DET
ejpam-5779	213	10	4	4	NUM
ejpam-5779	213	11	-	-	PUNCT
ejpam-5779	213	12	fuzzy	fuzzy	ADJ
ejpam-5779	213	13	ideal	ideal	NOUN
ejpam-5779	213	14	of	of	ADP
ejpam-5779	213	15	a	a	PRON
ejpam-5779	213	16	,	,	PUNCT
ejpam-5779	213	17	that	that	ADV
ejpam-5779	213	18	is	is	ADV
ejpam-5779	213	19	,	,	PUNCT
ejpam-5779	213	20	(	(	PUNCT
ejpam-5779	213	21	a	a	PRON
ejpam-5779	213	22	,	,	PUNCT
ejpam-5779	213	23	f̃	f̃	PROPN
ejpam-5779	213	24	)	)	PUNCT
ejpam-5779	213	25	is	be	AUX
ejpam-5779	213	26	a	a	DET
ejpam-5779	213	27	length	length	NOUN
ejpam-5779	213	28	4	4	NUM
ejpam-5779	213	29	-	-	PUNCT
ejpam-5779	213	30	fuzzy	fuzzy	ADJ
ejpam-5779	213	31	ideal	ideal	NOUN
ejpam-5779	213	32	of	of	ADP
ejpam-5779	213	33	a.228	a.228	PROPN
ejpam-5779	213	34	n.	n.	PROPN
ejpam-5779	213	35	rajesh	rajesh	PROPN
ejpam-5779	213	36	,	,	PUNCT
ejpam-5779	213	37	t.	t.	PROPN
ejpam-5779	213	38	oner	oner	NOUN
ejpam-5779	213	39	,	,	PUNCT
ejpam-5779	213	40	a.	a.	NOUN
ejpam-5779	213	41	iampan	iampan	PROPN
ejpam-5779	213	42	,	,	PUNCT
ejpam-5779	213	43	i.	i.	PROPN
ejpam-5779	213	44	senturk	senturk	PROPN
ejpam-5779	213	45	/	/	SYM
ejpam-5779	213	46	eur	eur	PROPN
ejpam-5779	213	47	.	.	PUNCT
ejpam-5779	214	1	j.	j.	PROPN
ejpam-5779	214	2	pure	pure	PROPN
ejpam-5779	214	3	appl	appl	PROPN
ejpam-5779	214	4	.	.	PROPN
ejpam-5779	214	5	math	math	PROPN
ejpam-5779	214	6	,	,	PUNCT
ejpam-5779	214	7	18	18	NUM
ejpam-5779	214	8	(	(	PUNCT
ejpam-5779	214	9	1	1	NUM
ejpam-5779	214	10	)	)	PUNCT
ejpam-5779	214	11	(	(	PUNCT
ejpam-5779	214	12	2025	2025	NUM
ejpam-5779	214	13	)	)	PUNCT
ejpam-5779	214	14	,	,	PUNCT
ejpam-5779	214	15	5779	5779	NUM
ejpam-5779	214	16	11	11	NUM
ejpam-5779	214	17	of	of	ADP
ejpam-5779	214	18	18	18	NUM
ejpam-5779	214	19	4	4	NUM
ejpam-5779	214	20	.	.	PUNCT
ejpam-5779	215	1	mean	mean	VERB
ejpam-5779	215	2	fuzzy	fuzzy	ADJ
ejpam-5779	215	3	ideals	ideal	NOUN
ejpam-5779	215	4	of	of	ADP
ejpam-5779	215	5	sheffer	sheffer	PROPN
ejpam-5779	215	6	stroke	stroke	PROPN
ejpam-5779	215	7	hilbert	hilbert	PROPN
ejpam-5779	215	8	algebras229	algebras229	PROPN
ejpam-5779	215	9	in	in	ADP
ejpam-5779	215	10	this	this	DET
ejpam-5779	215	11	section	section	NOUN
ejpam-5779	215	12	,	,	PUNCT
ejpam-5779	215	13	we	we	PRON
ejpam-5779	215	14	introduce	introduce	VERB
ejpam-5779	215	15	the	the	DET
ejpam-5779	215	16	concept	concept	NOUN
ejpam-5779	215	17	of	of	ADP
ejpam-5779	215	18	the	the	DET
ejpam-5779	215	19	mean	mean	NOUN
ejpam-5779	215	20	of	of	ADP
ejpam-5779	215	21	an	an	DET
ejpam-5779	215	22	interval	interval	NOUN
ejpam-5779	215	23	-	-	PUNCT
ejpam-5779	215	24	valued	value	VERB
ejpam-5779	215	25	fuzzy230	fuzzy230	PROPN
ejpam-5779	215	26	structure	structure	NOUN
ejpam-5779	215	27	within	within	ADP
ejpam-5779	215	28	sheffer	sheffer	PROPN
ejpam-5779	215	29	stroke	stroke	PROPN
ejpam-5779	215	30	hilbert	hilbert	PROPN
ejpam-5779	215	31	algebras	algebras	PROPN
ejpam-5779	215	32	.	.	PUNCT
ejpam-5779	216	1	we	we	PRON
ejpam-5779	216	2	also	also	ADV
ejpam-5779	216	3	define	define	VERB
ejpam-5779	216	4	the	the	DET
ejpam-5779	216	5	notion	notion	NOUN
ejpam-5779	216	6	of	of	ADP
ejpam-5779	216	7	mean	mean	ADJ
ejpam-5779	216	8	fuzzy231	fuzzy231	PROPN
ejpam-5779	216	9	ideals	ideal	NOUN
ejpam-5779	216	10	in	in	ADP
ejpam-5779	216	11	these	these	DET
ejpam-5779	216	12	algebras	algebra	NOUN
ejpam-5779	216	13	and	and	CCONJ
ejpam-5779	216	14	investigate	investigate	VERB
ejpam-5779	216	15	their	their	PRON
ejpam-5779	216	16	related	related	ADJ
ejpam-5779	216	17	properties	property	NOUN
ejpam-5779	216	18	.	.	PUNCT
ejpam-5779	217	1	furthermore	furthermore	ADV
ejpam-5779	217	2	,	,	PUNCT
ejpam-5779	217	3	we	we	PRON
ejpam-5779	217	4	establish232	establish232	VERB
ejpam-5779	217	5	the	the	DET
ejpam-5779	217	6	relationships	relationship	NOUN
ejpam-5779	217	7	between	between	ADP
ejpam-5779	217	8	mean	mean	ADJ
ejpam-5779	217	9	fuzzy	fuzzy	ADJ
ejpam-5779	217	10	ideals	ideal	NOUN
ejpam-5779	217	11	and	and	CCONJ
ejpam-5779	217	12	traditional	traditional	ADJ
ejpam-5779	217	13	fuzzy	fuzzy	ADJ
ejpam-5779	217	14	ideals.233	ideals.233	VERB
ejpam-5779	217	15	definition	definition	NOUN
ejpam-5779	217	16	8	8	NUM
ejpam-5779	217	17	.	.	PUNCT
ejpam-5779	218	1	[	[	X
ejpam-5779	218	2	14	14	NUM
ejpam-5779	218	3	]	]	PUNCT
ejpam-5779	218	4	given	give	VERB
ejpam-5779	218	5	an	an	DET
ejpam-5779	218	6	interval	interval	NOUN
ejpam-5779	218	7	-	-	PUNCT
ejpam-5779	218	8	valued	value	VERB
ejpam-5779	218	9	fuzzy	fuzzy	ADJ
ejpam-5779	218	10	structure	structure	NOUN
ejpam-5779	218	11	(	(	PUNCT
ejpam-5779	218	12	a	a	PRON
ejpam-5779	218	13	,	,	PUNCT
ejpam-5779	218	14	f̃	f̃	PROPN
ejpam-5779	218	15	)	)	PUNCT
ejpam-5779	218	16	over	over	ADP
ejpam-5779	218	17	a	a	PRON
ejpam-5779	218	18	,	,	PUNCT
ejpam-5779	218	19	we	we	PRON
ejpam-5779	218	20	define	define	VERB
ejpam-5779	218	21	a	a	DET
ejpam-5779	218	22	fuzzy	fuzzy	ADJ
ejpam-5779	218	23	structure	structure	NOUN
ejpam-5779	218	24	(	(	PUNCT
ejpam-5779	218	25	a	a	DET
ejpam-5779	218	26	,	,	PUNCT
ejpam-5779	218	27	f̃m	f̃m	NOUN
ejpam-5779	218	28	)	)	PUNCT
ejpam-5779	218	29	in	in	ADP
ejpam-5779	218	30	a	a	PRON
ejpam-5779	218	31	as	as	SCONJ
ejpam-5779	218	32	follows	follow	VERB
ejpam-5779	218	33	:	:	PUNCT
ejpam-5779	218	34	f̃m	f̃m	NOUN
ejpam-5779	218	35	:	:	PUNCT
ejpam-5779	218	36	a	a	DET
ejpam-5779	218	37	→	→	SYM
ejpam-5779	218	38	[	[	X
ejpam-5779	218	39	0	0	NUM
ejpam-5779	218	40	,	,	PUNCT
ejpam-5779	218	41	1	1	NUM
ejpam-5779	218	42	]	]	PUNCT
ejpam-5779	218	43	;	;	PUNCT
ejpam-5779	218	44	p	p	PROPN
ejpam-5779	218	45	7→	7→	NUM
ejpam-5779	218	46	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	218	47	)	)	PUNCT
ejpam-5779	218	48	+	+	SYM
ejpam-5779	218	49	f̃inf(p	f̃inf(p	X
ejpam-5779	218	50	)	)	PUNCT
ejpam-5779	218	51	2	2	NUM
ejpam-5779	218	52	,	,	PUNCT
ejpam-5779	218	53	which	which	PRON
ejpam-5779	218	54	is	be	AUX
ejpam-5779	218	55	called	call	VERB
ejpam-5779	218	56	the	the	DET
ejpam-5779	218	57	mean	mean	NOUN
ejpam-5779	218	58	of	of	ADP
ejpam-5779	218	59	f̃	f̃	PROPN
ejpam-5779	218	60	.234	.234	NUM
ejpam-5779	218	61	definition	definition	NOUN
ejpam-5779	218	62	9	9	NUM
ejpam-5779	218	63	.	.	PUNCT
ejpam-5779	219	1	an	an	DET
ejpam-5779	219	2	interval	interval	NOUN
ejpam-5779	219	3	-	-	PUNCT
ejpam-5779	219	4	valued	value	VERB
ejpam-5779	219	5	fuzzy	fuzzy	ADJ
ejpam-5779	219	6	structure	structure	NOUN
ejpam-5779	219	7	(	(	PUNCT
ejpam-5779	219	8	a	a	PRON
ejpam-5779	219	9	,	,	PUNCT
ejpam-5779	219	10	f̃	f̃	PROPN
ejpam-5779	219	11	)	)	PUNCT
ejpam-5779	219	12	over	over	ADP
ejpam-5779	219	13	a	a	PRON
ejpam-5779	219	14	is	be	AUX
ejpam-5779	219	15	called	call	VERB
ejpam-5779	219	16	a	a	DET
ejpam-5779	219	17	mean	mean	ADJ
ejpam-5779	219	18	1	1	NUM
ejpam-5779	219	19	-	-	PUNCT
ejpam-5779	219	20	fuzzy235	fuzzy235	ADJ
ejpam-5779	219	21	(	(	PUNCT
ejpam-5779	219	22	resp	resp	NOUN
ejpam-5779	219	23	.	.	PUNCT
ejpam-5779	219	24	,	,	PUNCT
ejpam-5779	219	25	2	2	NUM
ejpam-5779	219	26	-	-	PUNCT
ejpam-5779	219	27	fuzzy	fuzzy	ADJ
ejpam-5779	219	28	,	,	PUNCT
ejpam-5779	219	29	3	3	NUM
ejpam-5779	219	30	-	-	PUNCT
ejpam-5779	219	31	fuzzy	fuzzy	ADJ
ejpam-5779	219	32	and	and	CCONJ
ejpam-5779	219	33	4	4	NUM
ejpam-5779	219	34	-	-	PUNCT
ejpam-5779	219	35	fuzzy	fuzzy	ADJ
ejpam-5779	219	36	)	)	PUNCT
ejpam-5779	219	37	ideal	ideal	NOUN
ejpam-5779	219	38	of	of	ADP
ejpam-5779	219	39	a	a	PRON
ejpam-5779	219	40	if	if	SCONJ
ejpam-5779	219	41	the	the	DET
ejpam-5779	219	42	fuzzy	fuzzy	ADJ
ejpam-5779	219	43	structure	structure	NOUN
ejpam-5779	219	44	(	(	PUNCT
ejpam-5779	219	45	a	a	DET
ejpam-5779	219	46	,	,	PUNCT
ejpam-5779	219	47	f̃m	f̃m	NOUN
ejpam-5779	219	48	)	)	PUNCT
ejpam-5779	219	49	is	be	AUX
ejpam-5779	219	50	a	a	DET
ejpam-5779	219	51	1	1	NUM
ejpam-5779	219	52	-	-	PUNCT
ejpam-5779	219	53	fuzzy236	fuzzy236	PROPN
ejpam-5779	219	54	(	(	PUNCT
ejpam-5779	219	55	resp	resp	NOUN
ejpam-5779	219	56	.	.	PUNCT
ejpam-5779	219	57	,	,	PUNCT
ejpam-5779	219	58	2	2	NUM
ejpam-5779	219	59	-	-	PUNCT
ejpam-5779	219	60	fuzzy	fuzzy	ADJ
ejpam-5779	219	61	,	,	PUNCT
ejpam-5779	219	62	3	3	NUM
ejpam-5779	219	63	-	-	PUNCT
ejpam-5779	219	64	fuzzy	fuzzy	ADJ
ejpam-5779	219	65	and	and	CCONJ
ejpam-5779	219	66	4	4	NUM
ejpam-5779	219	67	-	-	PUNCT
ejpam-5779	219	68	fuzzy	fuzzy	ADJ
ejpam-5779	219	69	)	)	PUNCT
ejpam-5779	219	70	ideal	ideal	NOUN
ejpam-5779	219	71	of	of	ADP
ejpam-5779	219	72	a.237	a.237	PROPN
ejpam-5779	219	73	proposition	proposition	NOUN
ejpam-5779	219	74	3	3	X
ejpam-5779	219	75	.	.	PUNCT
ejpam-5779	220	1	if	if	SCONJ
ejpam-5779	220	2	(	(	PUNCT
ejpam-5779	220	3	a	a	PRON
ejpam-5779	220	4	,	,	PUNCT
ejpam-5779	220	5	f̃	f̃	PROPN
ejpam-5779	220	6	)	)	PUNCT
ejpam-5779	220	7	is	be	AUX
ejpam-5779	220	8	a	a	DET
ejpam-5779	220	9	mean	mean	ADJ
ejpam-5779	220	10	k	k	ADJ
ejpam-5779	220	11	-	-	ADJ
ejpam-5779	220	12	fuzzy	fuzzy	ADJ
ejpam-5779	220	13	ideal	ideal	NOUN
ejpam-5779	220	14	of	of	ADP
ejpam-5779	220	15	a	a	PRON
ejpam-5779	220	16	for	for	ADP
ejpam-5779	220	17	k	k	PROPN
ejpam-5779	220	18	=	=	SYM
ejpam-5779	220	19	1	1	NUM
ejpam-5779	220	20	,	,	PUNCT
ejpam-5779	220	21	3	3	NUM
ejpam-5779	220	22	,	,	PUNCT
ejpam-5779	220	23	then	then	ADV
ejpam-5779	220	24	(	(	PUNCT
ejpam-5779	220	25	∀p	∀p	NOUN
ejpam-5779	220	26	∈	∈	PROPN
ejpam-5779	220	27	a)(f̃m(0	a)(f̃m(0	NOUN
ejpam-5779	220	28	)	)	PUNCT
ejpam-5779	220	29	≥	≥	NOUN
ejpam-5779	220	30	f̃m(p	f̃m(p	NUM
ejpam-5779	220	31	)	)	PUNCT
ejpam-5779	220	32	)	)	PUNCT
ejpam-5779	220	33	.	.	PUNCT
ejpam-5779	221	1	proof	proof	NOUN
ejpam-5779	221	2	.	.	PUNCT
ejpam-5779	222	1	let	let	VERB
ejpam-5779	222	2	(	(	PUNCT
ejpam-5779	222	3	a	a	PRON
ejpam-5779	222	4	,	,	PUNCT
ejpam-5779	222	5	f̃	f̃	PROPN
ejpam-5779	222	6	)	)	PUNCT
ejpam-5779	222	7	be	be	VERB
ejpam-5779	222	8	a	a	DET
ejpam-5779	222	9	mean	mean	ADJ
ejpam-5779	222	10	k	k	ADJ
ejpam-5779	222	11	-	-	ADJ
ejpam-5779	222	12	fuzzy	fuzzy	ADJ
ejpam-5779	222	13	ideal	ideal	NOUN
ejpam-5779	222	14	of	of	ADP
ejpam-5779	222	15	a	a	PRON
ejpam-5779	222	16	for	for	ADP
ejpam-5779	222	17	k	k	PROPN
ejpam-5779	222	18	=	=	SYM
ejpam-5779	222	19	1	1	NUM
ejpam-5779	222	20	,	,	PUNCT
ejpam-5779	222	21	3	3	NUM
ejpam-5779	222	22	and	and	CCONJ
ejpam-5779	222	23	p	p	PROPN
ejpam-5779	222	24	∈	∈	PROPN
ejpam-5779	222	25	a.	a.	NOUN
ejpam-5779	222	26	then238	then238	VERB
ejpam-5779	222	27	f̃m(0	f̃m(0	PROPN
ejpam-5779	222	28	)	)	PUNCT
ejpam-5779	222	29	=	=	SYM
ejpam-5779	222	30	f̃sup(0	f̃sup(0	NOUN
ejpam-5779	222	31	)	)	PUNCT
ejpam-5779	223	1	+	+	CCONJ
ejpam-5779	223	2	f̃inf(0	f̃inf(0	NOUN
ejpam-5779	223	3	)	)	PUNCT
ejpam-5779	223	4	2	2	NUM
ejpam-5779	223	5	≥	≥	NOUN
ejpam-5779	223	6	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	223	7	)	)	PUNCT
ejpam-5779	224	1	+	+	SYM
ejpam-5779	224	2	f̃inf(p	f̃inf(p	X
ejpam-5779	224	3	)	)	PUNCT
ejpam-5779	224	4	2	2	NUM
ejpam-5779	224	5	=	=	SYM
ejpam-5779	224	6	f̃m(p	f̃m(p	NUM
ejpam-5779	224	7	)	)	PUNCT
ejpam-5779	224	8	.	.	PUNCT
ejpam-5779	225	1	proposition	proposition	NOUN
ejpam-5779	225	2	4	4	NUM
ejpam-5779	225	3	.	.	PUNCT
ejpam-5779	226	1	if	if	SCONJ
ejpam-5779	226	2	(	(	PUNCT
ejpam-5779	226	3	a	a	PRON
ejpam-5779	226	4	,	,	PUNCT
ejpam-5779	226	5	f̃	f̃	PROPN
ejpam-5779	226	6	)	)	PUNCT
ejpam-5779	226	7	is	be	AUX
ejpam-5779	226	8	a	a	DET
ejpam-5779	226	9	mean	mean	ADJ
ejpam-5779	226	10	k	k	ADJ
ejpam-5779	226	11	-	-	ADJ
ejpam-5779	226	12	fuzzy	fuzzy	ADJ
ejpam-5779	226	13	ideal	ideal	NOUN
ejpam-5779	226	14	of	of	ADP
ejpam-5779	226	15	a	a	PRON
ejpam-5779	226	16	for	for	ADP
ejpam-5779	226	17	k	k	PROPN
ejpam-5779	226	18	=	=	SYM
ejpam-5779	226	19	2	2	NUM
ejpam-5779	226	20	,	,	PUNCT
ejpam-5779	226	21	4	4	NUM
ejpam-5779	226	22	,	,	PUNCT
ejpam-5779	226	23	then	then	ADV
ejpam-5779	226	24	(	(	PUNCT
ejpam-5779	226	25	∀p	∀p	NOUN
ejpam-5779	226	26	∈	∈	NOUN
ejpam-5779	226	27	a)(f̃m(0	a)(f̃m(0	NOUN
ejpam-5779	226	28	)	)	PUNCT
ejpam-5779	226	29	≤	≤	NOUN
ejpam-5779	226	30	f̃m(p	f̃m(p	NUM
ejpam-5779	226	31	)	)	PUNCT
ejpam-5779	226	32	)	)	PUNCT
ejpam-5779	226	33	.	.	PUNCT
ejpam-5779	227	1	proof	proof	NOUN
ejpam-5779	227	2	.	.	PUNCT
ejpam-5779	228	1	let	let	VERB
ejpam-5779	228	2	(	(	PUNCT
ejpam-5779	228	3	a	a	PRON
ejpam-5779	228	4	,	,	PUNCT
ejpam-5779	228	5	f̃	f̃	PROPN
ejpam-5779	228	6	)	)	PUNCT
ejpam-5779	228	7	be	be	VERB
ejpam-5779	228	8	a	a	DET
ejpam-5779	228	9	mean	mean	ADJ
ejpam-5779	228	10	k	k	ADJ
ejpam-5779	228	11	-	-	ADJ
ejpam-5779	228	12	fuzzy	fuzzy	ADJ
ejpam-5779	228	13	ideal	ideal	NOUN
ejpam-5779	228	14	of	of	ADP
ejpam-5779	228	15	a	a	PRON
ejpam-5779	228	16	for	for	ADP
ejpam-5779	228	17	k	k	PROPN
ejpam-5779	228	18	=	=	SYM
ejpam-5779	228	19	2	2	NUM
ejpam-5779	228	20	,	,	PUNCT
ejpam-5779	228	21	4	4	NUM
ejpam-5779	228	22	and	and	CCONJ
ejpam-5779	228	23	p	p	PROPN
ejpam-5779	228	24	∈	∈	PROPN
ejpam-5779	228	25	a.	a.	NOUN
ejpam-5779	228	26	then239	then239	PROPN
ejpam-5779	228	27	f̃m(0	f̃m(0	PROPN
ejpam-5779	228	28	)	)	PUNCT
ejpam-5779	228	29	=	=	SYM
ejpam-5779	228	30	f̃sup(0	f̃sup(0	NOUN
ejpam-5779	228	31	)	)	PUNCT
ejpam-5779	229	1	+	+	CCONJ
ejpam-5779	229	2	f̃inf(0	f̃inf(0	NOUN
ejpam-5779	229	3	)	)	PUNCT
ejpam-5779	229	4	2	2	NUM
ejpam-5779	229	5	≤	≤	NUM
ejpam-5779	229	6	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	229	7	)	)	PUNCT
ejpam-5779	230	1	+	+	SYM
ejpam-5779	230	2	f̃inf(p	f̃inf(p	X
ejpam-5779	230	3	)	)	PUNCT
ejpam-5779	230	4	2	2	NUM
ejpam-5779	230	5	=	=	SYM
ejpam-5779	230	6	f̃m(p	f̃m(p	NUM
ejpam-5779	230	7	)	)	PUNCT
ejpam-5779	230	8	.	.	PUNCT
ejpam-5779	231	1	theorem	theorem	VERB
ejpam-5779	231	2	9	9	NUM
ejpam-5779	231	3	.	.	PUNCT
ejpam-5779	232	1	every	every	DET
ejpam-5779	232	2	mean	mean	NOUN
ejpam-5779	232	3	3	3	NUM
ejpam-5779	232	4	-	-	PUNCT
ejpam-5779	232	5	fuzzy	fuzzy	ADJ
ejpam-5779	232	6	ideal	ideal	NOUN
ejpam-5779	232	7	of	of	ADP
ejpam-5779	232	8	a	a	PRON
ejpam-5779	232	9	is	be	AUX
ejpam-5779	232	10	a	a	DET
ejpam-5779	232	11	mean	mean	ADJ
ejpam-5779	232	12	1	1	NUM
ejpam-5779	232	13	-	-	PUNCT
ejpam-5779	232	14	fuzzy	fuzzy	ADJ
ejpam-5779	232	15	ideal	ideal	NOUN
ejpam-5779	232	16	of	of	ADP
ejpam-5779	232	17	a.240	a.240	PROPN
ejpam-5779	232	18	n.	n.	PROPN
ejpam-5779	232	19	rajesh	rajesh	PROPN
ejpam-5779	232	20	,	,	PUNCT
ejpam-5779	232	21	t.	t.	PROPN
ejpam-5779	232	22	oner	oner	NOUN
ejpam-5779	232	23	,	,	PUNCT
ejpam-5779	232	24	a.	a.	NOUN
ejpam-5779	232	25	iampan	iampan	PROPN
ejpam-5779	232	26	,	,	PUNCT
ejpam-5779	232	27	i.	i.	PROPN
ejpam-5779	232	28	senturk	senturk	PROPN
ejpam-5779	232	29	/	/	SYM
ejpam-5779	232	30	eur	eur	PROPN
ejpam-5779	232	31	.	.	PUNCT
ejpam-5779	233	1	j.	j.	PROPN
ejpam-5779	233	2	pure	pure	PROPN
ejpam-5779	233	3	appl	appl	PROPN
ejpam-5779	233	4	.	.	PROPN
ejpam-5779	233	5	math	math	PROPN
ejpam-5779	233	6	,	,	PUNCT
ejpam-5779	233	7	18	18	NUM
ejpam-5779	233	8	(	(	PUNCT
ejpam-5779	233	9	1	1	NUM
ejpam-5779	233	10	)	)	PUNCT
ejpam-5779	233	11	(	(	PUNCT
ejpam-5779	233	12	2025	2025	NUM
ejpam-5779	233	13	)	)	PUNCT
ejpam-5779	233	14	,	,	PUNCT
ejpam-5779	233	15	5779	5779	NUM
ejpam-5779	233	16	12	12	NUM
ejpam-5779	233	17	of	of	ADP
ejpam-5779	233	18	18	18	NUM
ejpam-5779	233	19	proof	proof	NOUN
ejpam-5779	233	20	.	.	PUNCT
ejpam-5779	234	1	let	let	VERB
ejpam-5779	234	2	(	(	PUNCT
ejpam-5779	234	3	a	a	PRON
ejpam-5779	234	4	,	,	PUNCT
ejpam-5779	234	5	f̃	f̃	PROPN
ejpam-5779	234	6	)	)	PUNCT
ejpam-5779	234	7	be	be	VERB
ejpam-5779	234	8	a	a	DET
ejpam-5779	234	9	mean	mean	ADJ
ejpam-5779	234	10	3	3	NUM
ejpam-5779	234	11	-	-	PUNCT
ejpam-5779	234	12	fuzzy	fuzzy	ADJ
ejpam-5779	234	13	ideal	ideal	NOUN
ejpam-5779	234	14	of	of	ADP
ejpam-5779	234	15	a	a	PRON
ejpam-5779	234	16	and	and	CCONJ
ejpam-5779	234	17	p	p	NOUN
ejpam-5779	234	18	,	,	PUNCT
ejpam-5779	234	19	q	q	PROPN
ejpam-5779	234	20	∈	∈	PROPN
ejpam-5779	234	21	a.	a.	NOUN
ejpam-5779	234	22	then241	then241	PROPN
ejpam-5779	234	23	f̃m(p	f̃m(p	NUM
ejpam-5779	234	24	)	)	PUNCT
ejpam-5779	234	25	=	=	SYM
ejpam-5779	234	26	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	234	27	)	)	PUNCT
ejpam-5779	235	1	+	+	SYM
ejpam-5779	235	2	f̃inf(p	f̃inf(p	X
ejpam-5779	235	3	)	)	PUNCT
ejpam-5779	235	4	2	2	NUM
ejpam-5779	235	5	=	=	SYM
ejpam-5779	235	6	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	235	7	)	)	PUNCT
ejpam-5779	235	8	2	2	NUM
ejpam-5779	236	1	+	+	SYM
ejpam-5779	236	2	f̃inf(p	f̃inf(p	X
ejpam-5779	236	3	)	)	PUNCT
ejpam-5779	236	4	2	2	NUM
ejpam-5779	236	5	≥	≥	NOUN
ejpam-5779	236	6	max	max	PROPN
ejpam-5779	236	7	{	{	PUNCT
ejpam-5779	236	8	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	236	9	q	q	NOUN
ejpam-5779	236	10	)	)	PUNCT
ejpam-5779	236	11	2	2	NUM
ejpam-5779	236	12	,	,	PUNCT
ejpam-5779	236	13	f̃sup(q	f̃sup(q	NOUN
ejpam-5779	236	14	)	)	PUNCT
ejpam-5779	236	15	2	2	NUM
ejpam-5779	236	16	}	}	PUNCT
ejpam-5779	236	17	+	+	NOUN
ejpam-5779	236	18	max	max	PROPN
ejpam-5779	236	19	{	{	PUNCT
ejpam-5779	236	20	f̃inf(p	f̃inf(p	X
ejpam-5779	236	21	q	q	PROPN
ejpam-5779	236	22	)	)	PUNCT
ejpam-5779	236	23	2	2	NUM
ejpam-5779	236	24	,	,	PUNCT
ejpam-5779	236	25	f̃inf(q	f̃inf(q	PROPN
ejpam-5779	236	26	)	)	PUNCT
ejpam-5779	236	27	2	2	NUM
ejpam-5779	236	28	}	}	PUNCT
ejpam-5779	236	29	≥	≥	NOUN
ejpam-5779	236	30	min	min	NOUN
ejpam-5779	236	31	{	{	PUNCT
ejpam-5779	236	32	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	236	33	q	q	NOUN
ejpam-5779	236	34	)	)	PUNCT
ejpam-5779	236	35	2	2	NUM
ejpam-5779	236	36	,	,	PUNCT
ejpam-5779	236	37	f̃sup(q	f̃sup(q	NOUN
ejpam-5779	236	38	)	)	PUNCT
ejpam-5779	236	39	2	2	NUM
ejpam-5779	236	40	}	}	PUNCT
ejpam-5779	236	41	+	+	PROPN
ejpam-5779	236	42	min	min	NOUN
ejpam-5779	236	43	{	{	PUNCT
ejpam-5779	236	44	f̃inf(p	f̃inf(p	X
ejpam-5779	236	45	q	q	SYM
ejpam-5779	236	46	)	)	PUNCT
ejpam-5779	236	47	2	2	NUM
ejpam-5779	236	48	,	,	PUNCT
ejpam-5779	236	49	f̃inf(q	f̃inf(q	PROPN
ejpam-5779	236	50	)	)	PUNCT
ejpam-5779	236	51	2	2	NUM
ejpam-5779	236	52	}	}	PUNCT
ejpam-5779	236	53	=	=	SYM
ejpam-5779	236	54	min	min	NOUN
ejpam-5779	236	55	{	{	PUNCT
ejpam-5779	236	56	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	236	57	q	q	NOUN
ejpam-5779	236	58	)	)	PUNCT
ejpam-5779	236	59	+	+	NUM
ejpam-5779	236	60	f̃inf(p	f̃inf(p	X
ejpam-5779	236	61	q	q	NOUN
ejpam-5779	236	62	)	)	PUNCT
ejpam-5779	236	63	2	2	NUM
ejpam-5779	236	64	,	,	PUNCT
ejpam-5779	236	65	f̃sup(q	f̃sup(q	NOUN
ejpam-5779	236	66	)	)	PUNCT
ejpam-5779	236	67	+	+	CCONJ
ejpam-5779	236	68	f̃inf(q	f̃inf(q	X
ejpam-5779	236	69	)	)	PUNCT
ejpam-5779	236	70	2	2	NUM
ejpam-5779	236	71	}	}	PUNCT
ejpam-5779	236	72	=	=	SYM
ejpam-5779	236	73	min{f̃m(pq	min{f̃m(pq	NOUN
ejpam-5779	236	74	)	)	PUNCT
ejpam-5779	236	75	,	,	PUNCT
ejpam-5779	236	76	f̃m(q	f̃m(q	NOUN
ejpam-5779	236	77	)	)	PUNCT
ejpam-5779	236	78	}	}	PUNCT
ejpam-5779	236	79	.	.	PUNCT
ejpam-5779	237	1	hence	hence	ADV
ejpam-5779	237	2	,	,	PUNCT
ejpam-5779	237	3	(	(	PUNCT
ejpam-5779	237	4	a	a	PRON
ejpam-5779	237	5	,	,	PUNCT
ejpam-5779	237	6	f̃	f̃	PROPN
ejpam-5779	237	7	)	)	PUNCT
ejpam-5779	237	8	is	be	AUX
ejpam-5779	237	9	a	a	DET
ejpam-5779	237	10	mean	mean	ADJ
ejpam-5779	237	11	1	1	NUM
ejpam-5779	237	12	-	-	PUNCT
ejpam-5779	237	13	fuzzy	fuzzy	ADJ
ejpam-5779	237	14	ideal	ideal	NOUN
ejpam-5779	237	15	of	of	ADP
ejpam-5779	237	16	a.242	a.242	NOUN
ejpam-5779	237	17	theorem	theorem	VERB
ejpam-5779	237	18	10	10	NUM
ejpam-5779	237	19	.	.	PUNCT
ejpam-5779	238	1	every	every	DET
ejpam-5779	238	2	mean	mean	ADJ
ejpam-5779	238	3	2	2	NUM
ejpam-5779	238	4	-	-	PUNCT
ejpam-5779	238	5	fuzzy	fuzzy	ADJ
ejpam-5779	238	6	ideal	ideal	NOUN
ejpam-5779	238	7	of	of	ADP
ejpam-5779	238	8	a	a	PRON
ejpam-5779	238	9	is	be	AUX
ejpam-5779	238	10	a	a	DET
ejpam-5779	238	11	mean	mean	ADJ
ejpam-5779	238	12	4	4	NUM
ejpam-5779	238	13	-	-	PUNCT
ejpam-5779	238	14	fuzzy	fuzzy	ADJ
ejpam-5779	238	15	ideal	ideal	NOUN
ejpam-5779	238	16	of	of	ADP
ejpam-5779	238	17	a.243	a.243	NOUN
ejpam-5779	238	18	proof	proof	NOUN
ejpam-5779	238	19	.	.	PUNCT
ejpam-5779	239	1	let	let	VERB
ejpam-5779	239	2	(	(	PUNCT
ejpam-5779	239	3	a	a	PRON
ejpam-5779	239	4	,	,	PUNCT
ejpam-5779	239	5	f̃	f̃	PROPN
ejpam-5779	239	6	)	)	PUNCT
ejpam-5779	239	7	be	be	VERB
ejpam-5779	239	8	a	a	DET
ejpam-5779	239	9	mean	mean	ADJ
ejpam-5779	239	10	2	2	NUM
ejpam-5779	239	11	-	-	PUNCT
ejpam-5779	239	12	fuzzy	fuzzy	ADJ
ejpam-5779	239	13	ideal	ideal	NOUN
ejpam-5779	239	14	of	of	ADP
ejpam-5779	239	15	a	a	PRON
ejpam-5779	239	16	and	and	CCONJ
ejpam-5779	239	17	p	p	NOUN
ejpam-5779	239	18	,	,	PUNCT
ejpam-5779	239	19	q	q	PROPN
ejpam-5779	239	20	∈	∈	PROPN
ejpam-5779	239	21	a.	a.	NOUN
ejpam-5779	239	22	then244	then244	PROPN
ejpam-5779	239	23	f̃m(p	f̃m(p	NUM
ejpam-5779	239	24	)	)	PUNCT
ejpam-5779	239	25	=	=	SYM
ejpam-5779	239	26	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	239	27	)	)	PUNCT
ejpam-5779	240	1	+	+	SYM
ejpam-5779	240	2	f̃inf(p	f̃inf(p	X
ejpam-5779	240	3	)	)	PUNCT
ejpam-5779	240	4	2	2	NUM
ejpam-5779	240	5	=	=	SYM
ejpam-5779	240	6	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	240	7	)	)	PUNCT
ejpam-5779	240	8	2	2	NUM
ejpam-5779	241	1	+	+	SYM
ejpam-5779	241	2	f̃inf(p	f̃inf(p	X
ejpam-5779	241	3	)	)	PUNCT
ejpam-5779	241	4	2	2	NUM
ejpam-5779	241	5	≤	≤	NOUN
ejpam-5779	241	6	min	min	NOUN
ejpam-5779	241	7	{	{	PUNCT
ejpam-5779	241	8	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	241	9	q	q	NOUN
ejpam-5779	241	10	)	)	PUNCT
ejpam-5779	241	11	2	2	NUM
ejpam-5779	241	12	,	,	PUNCT
ejpam-5779	241	13	f̃sup(q	f̃sup(q	NOUN
ejpam-5779	241	14	)	)	PUNCT
ejpam-5779	241	15	2	2	NUM
ejpam-5779	241	16	}	}	PUNCT
ejpam-5779	241	17	+	+	PROPN
ejpam-5779	241	18	min	min	NOUN
ejpam-5779	241	19	{	{	PUNCT
ejpam-5779	241	20	f̃inf(p	f̃inf(p	X
ejpam-5779	241	21	q	q	SYM
ejpam-5779	241	22	)	)	PUNCT
ejpam-5779	241	23	2	2	NUM
ejpam-5779	241	24	,	,	PUNCT
ejpam-5779	241	25	f̃inf(q	f̃inf(q	PROPN
ejpam-5779	241	26	)	)	PUNCT
ejpam-5779	241	27	2	2	NUM
ejpam-5779	241	28	}	}	PUNCT
ejpam-5779	241	29	≤	≤	NUM
ejpam-5779	241	30	max	max	PROPN
ejpam-5779	241	31	{	{	PUNCT
ejpam-5779	241	32	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	241	33	q	q	NOUN
ejpam-5779	241	34	)	)	PUNCT
ejpam-5779	241	35	2	2	NUM
ejpam-5779	241	36	,	,	PUNCT
ejpam-5779	241	37	f̃sup(q	f̃sup(q	NOUN
ejpam-5779	241	38	)	)	PUNCT
ejpam-5779	241	39	2	2	NUM
ejpam-5779	241	40	}	}	PUNCT
ejpam-5779	241	41	+	+	NOUN
ejpam-5779	241	42	max	max	PROPN
ejpam-5779	241	43	{	{	PUNCT
ejpam-5779	241	44	f̃inf(p	f̃inf(p	X
ejpam-5779	241	45	q	q	PROPN
ejpam-5779	241	46	)	)	PUNCT
ejpam-5779	241	47	2	2	NUM
ejpam-5779	241	48	,	,	PUNCT
ejpam-5779	241	49	f̃inf(q	f̃inf(q	PROPN
ejpam-5779	241	50	)	)	PUNCT
ejpam-5779	241	51	2	2	NUM
ejpam-5779	241	52	}	}	PUNCT
ejpam-5779	241	53	=	=	SYM
ejpam-5779	241	54	max	max	X
ejpam-5779	241	55	{	{	PUNCT
ejpam-5779	241	56	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	241	57	q	q	NOUN
ejpam-5779	241	58	)	)	PUNCT
ejpam-5779	241	59	+	+	NUM
ejpam-5779	241	60	f̃inf(p	f̃inf(p	X
ejpam-5779	241	61	q	q	NOUN
ejpam-5779	241	62	)	)	PUNCT
ejpam-5779	241	63	2	2	NUM
ejpam-5779	241	64	,	,	PUNCT
ejpam-5779	241	65	f̃sup(q	f̃sup(q	NOUN
ejpam-5779	241	66	)	)	PUNCT
ejpam-5779	241	67	+	+	CCONJ
ejpam-5779	241	68	f̃inf(q	f̃inf(q	X
ejpam-5779	241	69	)	)	PUNCT
ejpam-5779	241	70	2	2	NUM
ejpam-5779	241	71	}	}	PUNCT
ejpam-5779	241	72	=	=	SYM
ejpam-5779	241	73	max{f̃m(pq	max{f̃m(pq	NOUN
ejpam-5779	241	74	)	)	PUNCT
ejpam-5779	241	75	,	,	PUNCT
ejpam-5779	241	76	f̃m(q	f̃m(q	NOUN
ejpam-5779	241	77	)	)	PUNCT
ejpam-5779	241	78	}	}	PUNCT
ejpam-5779	241	79	.	.	PUNCT
ejpam-5779	242	1	hence	hence	ADV
ejpam-5779	242	2	,	,	PUNCT
ejpam-5779	242	3	(	(	PUNCT
ejpam-5779	242	4	a	a	PRON
ejpam-5779	242	5	,	,	PUNCT
ejpam-5779	242	6	f̃	f̃	PROPN
ejpam-5779	242	7	)	)	PUNCT
ejpam-5779	242	8	is	be	AUX
ejpam-5779	242	9	a	a	DET
ejpam-5779	242	10	mean	mean	ADJ
ejpam-5779	242	11	4	4	NUM
ejpam-5779	242	12	-	-	PUNCT
ejpam-5779	242	13	fuzzy	fuzzy	ADJ
ejpam-5779	242	14	ideal	ideal	NOUN
ejpam-5779	242	15	of	of	ADP
ejpam-5779	242	16	a.245	a.245	PROPN
ejpam-5779	242	17	theorem	theorem	VERB
ejpam-5779	242	18	11	11	NUM
ejpam-5779	242	19	.	.	PUNCT
ejpam-5779	243	1	mean	mean	VERB
ejpam-5779	243	2	2	2	NUM
ejpam-5779	243	3	-	-	PUNCT
ejpam-5779	243	4	fuzzy	fuzzy	ADJ
ejpam-5779	243	5	ideal	ideal	NOUN
ejpam-5779	243	6	and	and	CCONJ
ejpam-5779	243	7	mean	mean	VERB
ejpam-5779	243	8	3	3	NUM
ejpam-5779	243	9	-	-	PUNCT
ejpam-5779	243	10	fuzzy	fuzzy	ADJ
ejpam-5779	243	11	ideal	ideal	NOUN
ejpam-5779	243	12	of	of	ADP
ejpam-5779	243	13	a	a	DET
ejpam-5779	243	14	coincide.246	coincide.246	PROPN
ejpam-5779	243	15	proof	proof	NOUN
ejpam-5779	243	16	.	.	PUNCT
ejpam-5779	244	1	it	it	PRON
ejpam-5779	244	2	is	be	AUX
ejpam-5779	244	3	straightforward	straightforward	ADJ
ejpam-5779	244	4	by	by	ADP
ejpam-5779	244	5	theorems	theorem	NOUN
ejpam-5779	244	6	9	9	NUM
ejpam-5779	244	7	and	and	CCONJ
ejpam-5779	244	8	10.247	10.247	NUM
ejpam-5779	244	9	theorem	theorem	NOUN
ejpam-5779	244	10	12	12	NUM
ejpam-5779	244	11	.	.	PUNCT
ejpam-5779	245	1	given	give	VERB
ejpam-5779	245	2	an	an	DET
ejpam-5779	245	3	ideal	ideal	NOUN
ejpam-5779	245	4	s	s	NOUN
ejpam-5779	245	5	of	of	ADP
ejpam-5779	245	6	a	a	PRON
ejpam-5779	245	7	and	and	CCONJ
ejpam-5779	245	8	b1	b1	NOUN
ejpam-5779	245	9	,	,	PUNCT
ejpam-5779	245	10	b2	b2	NOUN
ejpam-5779	245	11	∈	∈	NOUN
ejpam-5779	245	12	p	p	X
ejpam-5779	245	13	(	(	PUNCT
ejpam-5779	245	14	[	[	X
ejpam-5779	245	15	0	0	NUM
ejpam-5779	245	16	,	,	PUNCT
ejpam-5779	245	17	1	1	NUM
ejpam-5779	245	18	]	]	NUM
ejpam-5779	245	19	)	)	PUNCT
ejpam-5779	245	20	,	,	PUNCT
ejpam-5779	245	21	let	let	VERB
ejpam-5779	245	22	(	(	PUNCT
ejpam-5779	245	23	a	a	PRON
ejpam-5779	245	24	,	,	PUNCT
ejpam-5779	245	25	f̃	f̃	PROPN
ejpam-5779	245	26	)	)	PUNCT
ejpam-5779	245	27	be	be	VERB
ejpam-5779	245	28	an	an	DET
ejpam-5779	245	29	intervalvalued	intervalvalue	VERB
ejpam-5779	245	30	fuzzy	fuzzy	ADJ
ejpam-5779	245	31	structure	structure	NOUN
ejpam-5779	245	32	over	over	ADP
ejpam-5779	245	33	a	a	DET
ejpam-5779	245	34	given	give	VERB
ejpam-5779	245	35	by	by	ADP
ejpam-5779	245	36	f̃	f̃	PROPN
ejpam-5779	245	37	:	:	PUNCT
ejpam-5779	245	38	a	a	DET
ejpam-5779	245	39	→	→	X
ejpam-5779	245	40	p	p	X
ejpam-5779	245	41	(	(	PUNCT
ejpam-5779	245	42	[	[	X
ejpam-5779	245	43	0	0	NUM
ejpam-5779	245	44	,	,	PUNCT
ejpam-5779	245	45	1	1	NUM
ejpam-5779	245	46	]	]	NUM
ejpam-5779	245	47	)	)	PUNCT
ejpam-5779	245	48	;	;	PUNCT
ejpam-5779	245	49	p	p	PROPN
ejpam-5779	245	50	7→	7→	PROPN
ejpam-5779	245	51	{	{	PUNCT
ejpam-5779	245	52	b2	b2	NOUN
ejpam-5779	245	53	,	,	PUNCT
ejpam-5779	245	54	if	if	SCONJ
ejpam-5779	245	55	p	p	PRON
ejpam-5779	245	56	∈	∈	PROPN
ejpam-5779	245	57	s	s	VERB
ejpam-5779	245	58	b1	b1	NOUN
ejpam-5779	245	59	,	,	PUNCT
ejpam-5779	245	60	otherwise	otherwise	ADV
ejpam-5779	245	61	.	.	PUNCT
ejpam-5779	246	1	n.	n.	PROPN
ejpam-5779	246	2	rajesh	rajesh	PROPN
ejpam-5779	246	3	,	,	PUNCT
ejpam-5779	246	4	t.	t.	PROPN
ejpam-5779	246	5	oner	oner	NOUN
ejpam-5779	246	6	,	,	PUNCT
ejpam-5779	246	7	a.	a.	NOUN
ejpam-5779	246	8	iampan	iampan	PROPN
ejpam-5779	246	9	,	,	PUNCT
ejpam-5779	246	10	i.	i.	PROPN
ejpam-5779	246	11	senturk	senturk	PROPN
ejpam-5779	246	12	/	/	SYM
ejpam-5779	246	13	eur	eur	PROPN
ejpam-5779	246	14	.	.	PUNCT
ejpam-5779	247	1	j.	j.	PROPN
ejpam-5779	247	2	pure	pure	PROPN
ejpam-5779	247	3	appl	appl	PROPN
ejpam-5779	247	4	.	.	PROPN
ejpam-5779	247	5	math	math	PROPN
ejpam-5779	247	6	,	,	PUNCT
ejpam-5779	247	7	18	18	NUM
ejpam-5779	247	8	(	(	PUNCT
ejpam-5779	247	9	1	1	NUM
ejpam-5779	247	10	)	)	PUNCT
ejpam-5779	247	11	(	(	PUNCT
ejpam-5779	247	12	2025	2025	NUM
ejpam-5779	247	13	)	)	PUNCT
ejpam-5779	247	14	,	,	PUNCT
ejpam-5779	247	15	5779	5779	NUM
ejpam-5779	247	16	13	13	NUM
ejpam-5779	247	17	of	of	ADP
ejpam-5779	247	18	18	18	NUM
ejpam-5779	247	19	(	(	PUNCT
ejpam-5779	247	20	1	1	NUM
ejpam-5779	247	21	)	)	PUNCT
ejpam-5779	247	22	if	if	SCONJ
ejpam-5779	247	23	supb2	supb2	PROPN
ejpam-5779	247	24	≥	≥	AUX
ejpam-5779	247	25	supb1	supb1	NOUN
ejpam-5779	247	26	and	and	CCONJ
ejpam-5779	247	27	inf	inf	PROPN
ejpam-5779	247	28	b2	b2	PROPN
ejpam-5779	247	29	≥	≥	PROPN
ejpam-5779	247	30	inf	inf	PROPN
ejpam-5779	247	31	b1	b1	NOUN
ejpam-5779	247	32	,	,	PUNCT
ejpam-5779	247	33	then	then	ADV
ejpam-5779	247	34	(	(	PUNCT
ejpam-5779	247	35	a	a	PRON
ejpam-5779	247	36	,	,	PUNCT
ejpam-5779	247	37	f̃	f̃	PROPN
ejpam-5779	247	38	)	)	PUNCT
ejpam-5779	247	39	is	be	AUX
ejpam-5779	247	40	a	a	DET
ejpam-5779	247	41	mean	mean	ADJ
ejpam-5779	247	42	1	1	NUM
ejpam-5779	247	43	-	-	PUNCT
ejpam-5779	247	44	fuzzy	fuzzy	ADJ
ejpam-5779	247	45	ideal	ideal	NOUN
ejpam-5779	247	46	of	of	ADP
ejpam-5779	247	47	a.248	a.248	PROPN
ejpam-5779	247	48	(	(	PUNCT
ejpam-5779	247	49	2	2	X
ejpam-5779	247	50	)	)	PUNCT
ejpam-5779	247	51	if	if	SCONJ
ejpam-5779	247	52	supb2	supb2	PROPN
ejpam-5779	247	53	≤	≤	X
ejpam-5779	247	54	supb1	supb1	NOUN
ejpam-5779	247	55	and	and	CCONJ
ejpam-5779	247	56	inf	inf	PROPN
ejpam-5779	247	57	b2	b2	PROPN
ejpam-5779	247	58	≤	≤	PROPN
ejpam-5779	247	59	inf	inf	PROPN
ejpam-5779	247	60	b1	b1	NOUN
ejpam-5779	247	61	,	,	PUNCT
ejpam-5779	247	62	then	then	ADV
ejpam-5779	247	63	(	(	PUNCT
ejpam-5779	247	64	a	a	PRON
ejpam-5779	247	65	,	,	PUNCT
ejpam-5779	247	66	f̃	f̃	PROPN
ejpam-5779	247	67	)	)	PUNCT
ejpam-5779	247	68	is	be	AUX
ejpam-5779	247	69	a	a	DET
ejpam-5779	247	70	mean	mean	ADJ
ejpam-5779	247	71	4	4	NUM
ejpam-5779	247	72	-	-	PUNCT
ejpam-5779	247	73	fuzzy	fuzzy	ADJ
ejpam-5779	247	74	ideal	ideal	NOUN
ejpam-5779	247	75	of	of	ADP
ejpam-5779	247	76	a.249	a.249	NOUN
ejpam-5779	247	77	proof	proof	NOUN
ejpam-5779	247	78	.	.	PUNCT
ejpam-5779	248	1	if	if	SCONJ
ejpam-5779	248	2	p	p	PROPN
ejpam-5779	248	3	∈	∈	PROPN
ejpam-5779	248	4	s	s	NOUN
ejpam-5779	248	5	,	,	PUNCT
ejpam-5779	248	6	then	then	ADV
ejpam-5779	248	7	f̃(p	f̃(p	NOUN
ejpam-5779	248	8	)	)	PUNCT
ejpam-5779	248	9	=	=	SYM
ejpam-5779	248	10	b2	b2	NOUN
ejpam-5779	248	11	and	and	CCONJ
ejpam-5779	248	12	so	so	ADV
ejpam-5779	248	13	f̃m(p	f̃m(p	NUM
ejpam-5779	248	14	)	)	PUNCT
ejpam-5779	248	15	=	=	SYM
ejpam-5779	248	16	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	248	17	)	)	PUNCT
ejpam-5779	248	18	+	+	SYM
ejpam-5779	248	19	f̃inf(p	f̃inf(p	X
ejpam-5779	248	20	)	)	PUNCT
ejpam-5779	248	21	2	2	NUM
ejpam-5779	248	22	=	=	SYM
ejpam-5779	248	23	sup	sup	NOUN
ejpam-5779	248	24	f̃(p	f̃(p	NOUN
ejpam-5779	248	25	)	)	PUNCT
ejpam-5779	248	26	+	+	NUM
ejpam-5779	248	27	inf	inf	ADJ
ejpam-5779	248	28	f̃(p	f̃(p	NOUN
ejpam-5779	248	29	)	)	PUNCT
ejpam-5779	248	30	2	2	NUM
ejpam-5779	248	31	=	=	SYM
ejpam-5779	248	32	supb2	supb2	NOUN
ejpam-5779	248	33	+	+	CCONJ
ejpam-5779	248	34	inf	inf	ADJ
ejpam-5779	248	35	b2	b2	NOUN
ejpam-5779	248	36	2	2	NUM
ejpam-5779	248	37	.	.	PUNCT
ejpam-5779	249	1	if	if	SCONJ
ejpam-5779	249	2	p	p	X
ejpam-5779	249	3	/∈	/∈	PUNCT
ejpam-5779	249	4	s	s	X
ejpam-5779	249	5	,	,	PUNCT
ejpam-5779	249	6	then	then	ADV
ejpam-5779	249	7	f̃(p	f̃(p	NOUN
ejpam-5779	249	8	)	)	PUNCT
ejpam-5779	249	9	=	=	SYM
ejpam-5779	249	10	b1	b1	NOUN
ejpam-5779	249	11	and	and	CCONJ
ejpam-5779	249	12	so	so	ADV
ejpam-5779	249	13	f̃m(p	f̃m(p	NUM
ejpam-5779	249	14	)	)	PUNCT
ejpam-5779	249	15	=	=	SYM
ejpam-5779	249	16	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	249	17	)	)	PUNCT
ejpam-5779	249	18	+	+	SYM
ejpam-5779	249	19	f̃inf(p	f̃inf(p	X
ejpam-5779	249	20	)	)	PUNCT
ejpam-5779	249	21	2	2	NUM
ejpam-5779	249	22	=	=	SYM
ejpam-5779	249	23	sup	sup	NOUN
ejpam-5779	249	24	f̃(p	f̃(p	NOUN
ejpam-5779	249	25	)	)	PUNCT
ejpam-5779	250	1	+	+	NUM
ejpam-5779	250	2	inf	inf	ADJ
ejpam-5779	250	3	f̃(p	f̃(p	NOUN
ejpam-5779	250	4	)	)	PUNCT
ejpam-5779	250	5	2	2	NUM
ejpam-5779	250	6	=	=	SYM
ejpam-5779	250	7	supb1	supb1	NOUN
ejpam-5779	250	8	+	+	CCONJ
ejpam-5779	250	9	inf	inf	ADJ
ejpam-5779	250	10	b1	b1	NOUN
ejpam-5779	250	11	2	2	NUM
ejpam-5779	250	12	.	.	PUNCT
ejpam-5779	251	1	(	(	PUNCT
ejpam-5779	251	2	1	1	X
ejpam-5779	251	3	)	)	PUNCT
ejpam-5779	251	4	assume	assume	VERB
ejpam-5779	251	5	that	that	SCONJ
ejpam-5779	251	6	supb2	supb2	PROPN
ejpam-5779	251	7	≥	≥	NUM
ejpam-5779	251	8	supb1	supb1	NOUN
ejpam-5779	251	9	and	and	CCONJ
ejpam-5779	251	10	inf	inf	PROPN
ejpam-5779	251	11	b2	b2	PROPN
ejpam-5779	251	12	≥	≥	PROPN
ejpam-5779	251	13	inf	inf	PROPN
ejpam-5779	251	14	b1	b1	NOUN
ejpam-5779	251	15	.	.	PUNCT
ejpam-5779	252	1	then	then	ADV
ejpam-5779	252	2	supb2	supb2	PROPN
ejpam-5779	252	3	+	+	CCONJ
ejpam-5779	252	4	inf	inf	ADJ
ejpam-5779	252	5	b2	b2	NOUN
ejpam-5779	252	6	2	2	NUM
ejpam-5779	252	7	≥	≥	NOUN
ejpam-5779	252	8	supb1	supb1	NOUN
ejpam-5779	252	9	+	+	CCONJ
ejpam-5779	252	10	inf	inf	ADJ
ejpam-5779	252	11	b1	b1	NOUN
ejpam-5779	252	12	2	2	NUM
ejpam-5779	252	13	.	.	PUNCT
ejpam-5779	253	1	case	case	NOUN
ejpam-5779	253	2	1	1	NUM
ejpam-5779	253	3	:	:	PUNCT
ejpam-5779	253	4	let	let	VERB
ejpam-5779	253	5	pq	pq	INTJ
ejpam-5779	253	6	,	,	PUNCT
ejpam-5779	253	7	q	q	PROPN
ejpam-5779	253	8	∈	∈	PROPN
ejpam-5779	253	9	s.	s.	PROPN
ejpam-5779	253	10	then	then	ADV
ejpam-5779	253	11	f̃m(pq	f̃m(pq	PROPN
ejpam-5779	253	12	)	)	PUNCT
ejpam-5779	253	13	=	=	SYM
ejpam-5779	253	14	supb2	supb2	NOUN
ejpam-5779	253	15	+	+	CCONJ
ejpam-5779	253	16	inf	inf	ADJ
ejpam-5779	253	17	b2	b2	NOUN
ejpam-5779	253	18	2	2	NUM
ejpam-5779	253	19	and	and	CCONJ
ejpam-5779	253	20	fm(q	fm(q	NUM
ejpam-5779	253	21	)	)	PUNCT
ejpam-5779	253	22	=	=	SYM
ejpam-5779	253	23	supb2	supb2	NOUN
ejpam-5779	253	24	+	+	CCONJ
ejpam-5779	253	25	inf	inf	ADJ
ejpam-5779	253	26	b2	b2	NOUN
ejpam-5779	253	27	2	2	NUM
ejpam-5779	253	28	.250	.250	NUM
ejpam-5779	253	29	thus	thus	ADV
ejpam-5779	253	30	,	,	PUNCT
ejpam-5779	253	31	min{f̃m(pq	min{f̃m(pq	NOUN
ejpam-5779	253	32	)	)	PUNCT
ejpam-5779	253	33	,	,	PUNCT
ejpam-5779	253	34	f̃m(q	f̃m(q	NOUN
ejpam-5779	253	35	)	)	PUNCT
ejpam-5779	253	36	}	}	PUNCT
ejpam-5779	253	37	=	=	SYM
ejpam-5779	253	38	supb2	supb2	NOUN
ejpam-5779	253	39	+	+	CCONJ
ejpam-5779	253	40	inf	inf	ADJ
ejpam-5779	253	41	b2	b2	NOUN
ejpam-5779	253	42	2	2	NUM
ejpam-5779	253	43	.	.	PUNCT
ejpam-5779	254	1	since	since	SCONJ
ejpam-5779	254	2	s	s	PROPN
ejpam-5779	254	3	is	be	AUX
ejpam-5779	254	4	an	an	DET
ejpam-5779	254	5	ideal	ideal	NOUN
ejpam-5779	254	6	of	of	ADP
ejpam-5779	254	7	a	a	PRON
ejpam-5779	254	8	,	,	PUNCT
ejpam-5779	254	9	we	we	PRON
ejpam-5779	254	10	have	have	VERB
ejpam-5779	254	11	p	p	NOUN
ejpam-5779	254	12	∈	∈	PROPN
ejpam-5779	254	13	s	s	PART
ejpam-5779	254	14	and251	and251	NOUN
ejpam-5779	254	15	so	so	ADV
ejpam-5779	254	16	f̃m(p	f̃m(p	NUM
ejpam-5779	254	17	)	)	PUNCT
ejpam-5779	254	18	=	=	SYM
ejpam-5779	254	19	supb2	supb2	NOUN
ejpam-5779	254	20	+	+	CCONJ
ejpam-5779	254	21	inf	inf	ADJ
ejpam-5779	254	22	b2	b2	NOUN
ejpam-5779	254	23	2	2	NUM
ejpam-5779	254	24	.	.	PUNCT
ejpam-5779	255	1	thus	thus	ADV
ejpam-5779	255	2	,	,	PUNCT
ejpam-5779	255	3	f̃m(p	f̃m(p	NUM
ejpam-5779	255	4	)	)	PUNCT
ejpam-5779	255	5	=	=	SYM
ejpam-5779	255	6	supb2	supb2	NOUN
ejpam-5779	255	7	+	+	CCONJ
ejpam-5779	255	8	inf	inf	ADJ
ejpam-5779	255	9	b2	b2	NOUN
ejpam-5779	255	10	2	2	NUM
ejpam-5779	255	11	=	=	SYM
ejpam-5779	255	12	min{f̃m(pq	min{f̃m(pq	NOUN
ejpam-5779	255	13	)	)	PUNCT
ejpam-5779	255	14	,	,	PUNCT
ejpam-5779	255	15	f̃m(q)}.252	f̃m(q)}.252	NOUN
ejpam-5779	255	16	case	case	NOUN
ejpam-5779	255	17	2	2	NUM
ejpam-5779	255	18	:	:	PUNCT
ejpam-5779	255	19	let	let	VERB
ejpam-5779	255	20	pq	pq	INTJ
ejpam-5779	255	21	,	,	PUNCT
ejpam-5779	255	22	q	q	PROPN
ejpam-5779	255	23	/∈	/∈	PUNCT
ejpam-5779	256	1	s.	s.	PROPN
ejpam-5779	256	2	then	then	ADV
ejpam-5779	256	3	f̃m(pq	f̃m(pq	PROPN
ejpam-5779	256	4	)	)	PUNCT
ejpam-5779	257	1	=	=	SYM
ejpam-5779	257	2	supb1	supb1	NOUN
ejpam-5779	257	3	+	+	CCONJ
ejpam-5779	257	4	inf	inf	ADJ
ejpam-5779	257	5	b1	b1	NOUN
ejpam-5779	257	6	2	2	NUM
ejpam-5779	257	7	and	and	CCONJ
ejpam-5779	257	8	f̃m(q	f̃m(q	NOUN
ejpam-5779	257	9	)	)	PUNCT
ejpam-5779	257	10	=	=	SYM
ejpam-5779	258	1	supb1	supb1	NOUN
ejpam-5779	258	2	+	+	CCONJ
ejpam-5779	258	3	inf	inf	ADJ
ejpam-5779	258	4	b1	b1	NOUN
ejpam-5779	258	5	2	2	NUM
ejpam-5779	258	6	,	,	PUNCT
ejpam-5779	258	7	so253	so253	PROPN
ejpam-5779	258	8	min{f̃m(pq	min{f̃m(pq	PROPN
ejpam-5779	258	9	)	)	PUNCT
ejpam-5779	258	10	,	,	PUNCT
ejpam-5779	258	11	f̃m(q	f̃m(q	NOUN
ejpam-5779	258	12	)	)	PUNCT
ejpam-5779	258	13	}	}	PUNCT
ejpam-5779	258	14	=	=	SYM
ejpam-5779	259	1	supb1	supb1	NOUN
ejpam-5779	259	2	+	+	CCONJ
ejpam-5779	259	3	inf	inf	ADJ
ejpam-5779	259	4	b1	b1	NOUN
ejpam-5779	259	5	2	2	NUM
ejpam-5779	259	6	.	.	PUNCT
ejpam-5779	260	1	thus	thus	ADV
ejpam-5779	260	2	,	,	PUNCT
ejpam-5779	260	3	f̃m(p	f̃m(p	NUM
ejpam-5779	260	4	)	)	PUNCT
ejpam-5779	260	5	≥	≥	NOUN
ejpam-5779	260	6	supb1	supb1	NOUN
ejpam-5779	261	1	+	+	CCONJ
ejpam-5779	261	2	inf	inf	ADJ
ejpam-5779	261	3	b1	b1	NOUN
ejpam-5779	261	4	2	2	NUM
ejpam-5779	261	5	=	=	SYM
ejpam-5779	261	6	min{f̃m(pq	min{f̃m(pq	NOUN
ejpam-5779	261	7	)	)	PUNCT
ejpam-5779	261	8	,	,	PUNCT
ejpam-5779	261	9	f̃m(q)}.254	f̃m(q)}.254	NOUN
ejpam-5779	261	10	255	255	NUM
ejpam-5779	261	11	case	case	NOUN
ejpam-5779	261	12	3	3	X
ejpam-5779	261	13	:	:	PUNCT
ejpam-5779	261	14	let	let	VERB
ejpam-5779	261	15	pq	pq	INTJ
ejpam-5779	261	16	/∈	/∈	PUNCT
ejpam-5779	261	17	s	s	PROPN
ejpam-5779	261	18	and	and	CCONJ
ejpam-5779	261	19	q	q	PROPN
ejpam-5779	261	20	∈	∈	PROPN
ejpam-5779	261	21	s.	s.	PROPN
ejpam-5779	261	22	then	then	ADV
ejpam-5779	261	23	f̃m(pq	f̃m(pq	PROPN
ejpam-5779	261	24	)	)	PUNCT
ejpam-5779	261	25	=	=	SYM
ejpam-5779	261	26	supb1	supb1	NOUN
ejpam-5779	261	27	+	+	CCONJ
ejpam-5779	261	28	inf	inf	ADJ
ejpam-5779	261	29	b1	b1	NOUN
ejpam-5779	261	30	2	2	NUM
ejpam-5779	261	31	and	and	CCONJ
ejpam-5779	261	32	f̃m(q	f̃m(q	NUM
ejpam-5779	261	33	)	)	PUNCT
ejpam-5779	261	34	=	=	SYM
ejpam-5779	261	35	256	256	NUM
ejpam-5779	261	36	supb2	supb2	NOUN
ejpam-5779	261	37	+	+	CCONJ
ejpam-5779	261	38	inf	inf	ADJ
ejpam-5779	261	39	b2	b2	NOUN
ejpam-5779	261	40	2	2	NUM
ejpam-5779	261	41	,	,	PUNCT
ejpam-5779	261	42	so	so	ADV
ejpam-5779	261	43	min{f̃m(pq	min{f̃m(pq	NOUN
ejpam-5779	261	44	)	)	PUNCT
ejpam-5779	261	45	,	,	PUNCT
ejpam-5779	261	46	f̃m(q	f̃m(q	NOUN
ejpam-5779	261	47	)	)	PUNCT
ejpam-5779	261	48	}	}	PUNCT
ejpam-5779	261	49	=	=	SYM
ejpam-5779	262	1	supb1	supb1	NOUN
ejpam-5779	262	2	+	+	CCONJ
ejpam-5779	262	3	inf	inf	ADJ
ejpam-5779	262	4	b1	b1	NOUN
ejpam-5779	262	5	2	2	NUM
ejpam-5779	262	6	.	.	PUNCT
ejpam-5779	263	1	thus	thus	ADV
ejpam-5779	263	2	,	,	PUNCT
ejpam-5779	263	3	f̃m(p	f̃m(p	NUM
ejpam-5779	263	4	)	)	PUNCT
ejpam-5779	263	5	≥	≥	NOUN
ejpam-5779	263	6	supb1	supb1	NOUN
ejpam-5779	264	1	+	+	CCONJ
ejpam-5779	264	2	inf	inf	ADJ
ejpam-5779	264	3	b1	b1	NOUN
ejpam-5779	264	4	2	2	NUM
ejpam-5779	264	5	=	=	SYM
ejpam-5779	264	6	257	257	NUM
ejpam-5779	264	7	min{f̃m(pq	min{f̃m(pq	NOUN
ejpam-5779	264	8	)	)	PUNCT
ejpam-5779	264	9	,	,	PUNCT
ejpam-5779	264	10	f̃m(q)}.258	f̃m(q)}.258	PROPN
ejpam-5779	264	11	259	259	NUM
ejpam-5779	264	12	case	case	NOUN
ejpam-5779	264	13	4	4	NUM
ejpam-5779	264	14	:	:	PUNCT
ejpam-5779	264	15	let	let	VERB
ejpam-5779	264	16	pq	pq	INTJ
ejpam-5779	264	17	∈	∈	PROPN
ejpam-5779	264	18	s	s	PART
ejpam-5779	264	19	and	and	CCONJ
ejpam-5779	264	20	q	q	PROPN
ejpam-5779	264	21	/∈	/∈	PUNCT
ejpam-5779	265	1	s.	s.	PROPN
ejpam-5779	265	2	then	then	ADV
ejpam-5779	265	3	f̃m(pq	f̃m(pq	PROPN
ejpam-5779	265	4	)	)	PUNCT
ejpam-5779	265	5	=	=	SYM
ejpam-5779	265	6	supb2	supb2	NOUN
ejpam-5779	265	7	+	+	CCONJ
ejpam-5779	265	8	inf	inf	ADJ
ejpam-5779	265	9	b2	b2	NOUN
ejpam-5779	265	10	2	2	NUM
ejpam-5779	265	11	and	and	CCONJ
ejpam-5779	265	12	f̃m(q	f̃m(q	NUM
ejpam-5779	265	13	)	)	PUNCT
ejpam-5779	266	1	=	=	SYM
ejpam-5779	266	2	260	260	NUM
ejpam-5779	266	3	supb1	supb1	NOUN
ejpam-5779	266	4	+	+	CCONJ
ejpam-5779	266	5	inf	inf	ADJ
ejpam-5779	266	6	b1	b1	NOUN
ejpam-5779	266	7	2	2	NUM
ejpam-5779	266	8	,	,	PUNCT
ejpam-5779	266	9	so	so	ADV
ejpam-5779	266	10	min{f̃m(pq	min{f̃m(pq	NOUN
ejpam-5779	266	11	)	)	PUNCT
ejpam-5779	266	12	,	,	PUNCT
ejpam-5779	266	13	f̃m(q	f̃m(q	NOUN
ejpam-5779	266	14	)	)	PUNCT
ejpam-5779	266	15	}	}	PUNCT
ejpam-5779	266	16	=	=	SYM
ejpam-5779	267	1	supb1	supb1	NOUN
ejpam-5779	267	2	+	+	CCONJ
ejpam-5779	267	3	inf	inf	ADJ
ejpam-5779	267	4	b1	b1	NOUN
ejpam-5779	267	5	2	2	NUM
ejpam-5779	267	6	.	.	PUNCT
ejpam-5779	268	1	thus	thus	ADV
ejpam-5779	268	2	,	,	PUNCT
ejpam-5779	268	3	f̃m(p	f̃m(p	NUM
ejpam-5779	268	4	)	)	PUNCT
ejpam-5779	268	5	≥	≥	NOUN
ejpam-5779	268	6	supb1	supb1	NOUN
ejpam-5779	269	1	+	+	CCONJ
ejpam-5779	269	2	inf	inf	ADJ
ejpam-5779	269	3	b1	b1	NOUN
ejpam-5779	269	4	2	2	NUM
ejpam-5779	269	5	=	=	SYM
ejpam-5779	269	6	261	261	NUM
ejpam-5779	269	7	min{f̃m(pq	min{f̃m(pq	NOUN
ejpam-5779	269	8	)	)	PUNCT
ejpam-5779	269	9	,	,	PUNCT
ejpam-5779	269	10	f̃m(q)}.262	f̃m(q)}.262	PROPN
ejpam-5779	269	11	hence	hence	ADV
ejpam-5779	269	12	,	,	PUNCT
ejpam-5779	269	13	f̃m	f̃m	PROPN
ejpam-5779	269	14	is	be	AUX
ejpam-5779	269	15	a	a	DET
ejpam-5779	269	16	1	1	NUM
ejpam-5779	269	17	-	-	PUNCT
ejpam-5779	269	18	fuzzy	fuzzy	ADJ
ejpam-5779	269	19	ideal	ideal	NOUN
ejpam-5779	269	20	of	of	ADP
ejpam-5779	269	21	a	a	PRON
ejpam-5779	269	22	and	and	CCONJ
ejpam-5779	269	23	so	so	ADV
ejpam-5779	269	24	(	(	PUNCT
ejpam-5779	269	25	a	a	PRON
ejpam-5779	269	26	,	,	PUNCT
ejpam-5779	269	27	f̃	f̃	PROPN
ejpam-5779	269	28	)	)	PUNCT
ejpam-5779	269	29	is	be	AUX
ejpam-5779	269	30	a	a	DET
ejpam-5779	269	31	mean	mean	ADJ
ejpam-5779	269	32	1	1	NUM
ejpam-5779	269	33	-	-	PUNCT
ejpam-5779	269	34	fuzzy	fuzzy	ADJ
ejpam-5779	269	35	ideal	ideal	NOUN
ejpam-5779	269	36	of	of	ADP
ejpam-5779	269	37	a.263	a.263	PROPN
ejpam-5779	269	38	(	(	PUNCT
ejpam-5779	269	39	2	2	X
ejpam-5779	269	40	)	)	PUNCT
ejpam-5779	269	41	assume	assume	VERB
ejpam-5779	269	42	that	that	SCONJ
ejpam-5779	269	43	supb2	supb2	PROPN
ejpam-5779	269	44	≤	≤	X
ejpam-5779	269	45	supb1	supb1	NOUN
ejpam-5779	269	46	and	and	CCONJ
ejpam-5779	269	47	inf	inf	PROPN
ejpam-5779	269	48	b2	b2	PROPN
ejpam-5779	269	49	≤	≤	PROPN
ejpam-5779	269	50	inf	inf	NOUN
ejpam-5779	269	51	b1	b1	NOUN
ejpam-5779	269	52	.	.	PUNCT
ejpam-5779	270	1	then	then	ADV
ejpam-5779	270	2	supb2	supb2	PROPN
ejpam-5779	270	3	+	+	CCONJ
ejpam-5779	270	4	inf	inf	ADJ
ejpam-5779	270	5	b2	b2	NOUN
ejpam-5779	270	6	2	2	NUM
ejpam-5779	270	7	≤	≤	NUM
ejpam-5779	270	8	supb1	supb1	NOUN
ejpam-5779	271	1	+	+	CCONJ
ejpam-5779	271	2	inf	inf	ADJ
ejpam-5779	271	3	b1	b1	NOUN
ejpam-5779	271	4	2	2	NUM
ejpam-5779	271	5	.	.	PUNCT
ejpam-5779	271	6	case	case	NOUN
ejpam-5779	271	7	1	1	NUM
ejpam-5779	271	8	:	:	PUNCT
ejpam-5779	271	9	let	let	VERB
ejpam-5779	271	10	pq	pq	INTJ
ejpam-5779	271	11	,	,	PUNCT
ejpam-5779	271	12	q	q	PROPN
ejpam-5779	271	13	∈	∈	PROPN
ejpam-5779	271	14	s.	s.	PROPN
ejpam-5779	271	15	then	then	ADV
ejpam-5779	271	16	f̃m(pq	f̃m(pq	PROPN
ejpam-5779	271	17	)	)	PUNCT
ejpam-5779	271	18	=	=	SYM
ejpam-5779	271	19	supb2	supb2	NOUN
ejpam-5779	271	20	+	+	CCONJ
ejpam-5779	271	21	inf	inf	ADJ
ejpam-5779	271	22	b2	b2	NOUN
ejpam-5779	271	23	2	2	NUM
ejpam-5779	271	24	and	and	CCONJ
ejpam-5779	271	25	f̃m(q	f̃m(q	NUM
ejpam-5779	271	26	)	)	PUNCT
ejpam-5779	271	27	=	=	SYM
ejpam-5779	271	28	supb2	supb2	NOUN
ejpam-5779	271	29	+	+	CCONJ
ejpam-5779	271	30	inf	inf	ADJ
ejpam-5779	271	31	b2	b2	NOUN
ejpam-5779	271	32	2	2	NUM
ejpam-5779	271	33	,	,	PUNCT
ejpam-5779	271	34	264	264	NUM
ejpam-5779	271	35	so	so	ADV
ejpam-5779	271	36	max{f̃m(pq	max{f̃m(pq	ADJ
ejpam-5779	271	37	)	)	PUNCT
ejpam-5779	271	38	,	,	PUNCT
ejpam-5779	271	39	f̃m(q	f̃m(q	NOUN
ejpam-5779	271	40	)	)	PUNCT
ejpam-5779	271	41	}	}	PUNCT
ejpam-5779	271	42	=	=	SYM
ejpam-5779	271	43	supb2	supb2	NOUN
ejpam-5779	271	44	+	+	CCONJ
ejpam-5779	271	45	inf	inf	ADJ
ejpam-5779	271	46	b2	b2	NOUN
ejpam-5779	271	47	2	2	NUM
ejpam-5779	271	48	.	.	PUNCT
ejpam-5779	272	1	since	since	SCONJ
ejpam-5779	272	2	s	s	PROPN
ejpam-5779	272	3	is	be	AUX
ejpam-5779	272	4	an	an	DET
ejpam-5779	272	5	ideal	ideal	NOUN
ejpam-5779	272	6	of	of	ADP
ejpam-5779	272	7	a	a	PRON
ejpam-5779	272	8	,	,	PUNCT
ejpam-5779	272	9	we	we	PRON
ejpam-5779	272	10	have	have	VERB
ejpam-5779	272	11	p	p	NOUN
ejpam-5779	272	12	∈	∈	PROPN
ejpam-5779	272	13	s	s	PART
ejpam-5779	272	14	and	and	CCONJ
ejpam-5779	272	15	so265	so265	ADJ
ejpam-5779	272	16	f̃m(p	f̃m(p	NUM
ejpam-5779	272	17	)	)	PUNCT
ejpam-5779	272	18	=	=	SYM
ejpam-5779	272	19	supb2	supb2	NOUN
ejpam-5779	272	20	+	+	CCONJ
ejpam-5779	272	21	inf	inf	ADJ
ejpam-5779	272	22	b2	b2	NOUN
ejpam-5779	272	23	2	2	NUM
ejpam-5779	272	24	.	.	PUNCT
ejpam-5779	273	1	thus	thus	ADV
ejpam-5779	273	2	,	,	PUNCT
ejpam-5779	273	3	f̃m(p	f̃m(p	NUM
ejpam-5779	273	4	)	)	PUNCT
ejpam-5779	273	5	=	=	SYM
ejpam-5779	273	6	supb2	supb2	NOUN
ejpam-5779	273	7	+	+	CCONJ
ejpam-5779	273	8	inf	inf	ADJ
ejpam-5779	273	9	b2	b2	NOUN
ejpam-5779	273	10	2	2	NUM
ejpam-5779	273	11	=	=	SYM
ejpam-5779	273	12	max{f̃m(pq	max{f̃m(pq	NOUN
ejpam-5779	273	13	)	)	PUNCT
ejpam-5779	273	14	,	,	PUNCT
ejpam-5779	273	15	f̃m(q)}.266	f̃m(q)}.266	PROPN
ejpam-5779	273	16	n.	n.	PROPN
ejpam-5779	273	17	rajesh	rajesh	PROPN
ejpam-5779	273	18	,	,	PUNCT
ejpam-5779	273	19	t.	t.	PROPN
ejpam-5779	273	20	oner	oner	NOUN
ejpam-5779	273	21	,	,	PUNCT
ejpam-5779	273	22	a.	a.	NOUN
ejpam-5779	273	23	iampan	iampan	PROPN
ejpam-5779	273	24	,	,	PUNCT
ejpam-5779	273	25	i.	i.	PROPN
ejpam-5779	273	26	senturk	senturk	PROPN
ejpam-5779	273	27	/	/	SYM
ejpam-5779	273	28	eur	eur	PROPN
ejpam-5779	273	29	.	.	PUNCT
ejpam-5779	274	1	j.	j.	PROPN
ejpam-5779	274	2	pure	pure	PROPN
ejpam-5779	274	3	appl	appl	PROPN
ejpam-5779	274	4	.	.	PROPN
ejpam-5779	274	5	math	math	PROPN
ejpam-5779	274	6	,	,	PUNCT
ejpam-5779	274	7	18	18	NUM
ejpam-5779	274	8	(	(	PUNCT
ejpam-5779	274	9	1	1	NUM
ejpam-5779	274	10	)	)	PUNCT
ejpam-5779	274	11	(	(	PUNCT
ejpam-5779	274	12	2025	2025	NUM
ejpam-5779	274	13	)	)	PUNCT
ejpam-5779	274	14	,	,	PUNCT
ejpam-5779	274	15	5779	5779	NUM
ejpam-5779	274	16	14	14	NUM
ejpam-5779	274	17	of	of	ADP
ejpam-5779	274	18	18	18	NUM
ejpam-5779	274	19	case	case	NOUN
ejpam-5779	274	20	2	2	NUM
ejpam-5779	274	21	:	:	PUNCT
ejpam-5779	274	22	let	let	VERB
ejpam-5779	274	23	pq	pq	INTJ
ejpam-5779	274	24	,	,	PUNCT
ejpam-5779	274	25	q	q	PROPN
ejpam-5779	274	26	/∈	/∈	PUNCT
ejpam-5779	275	1	s.	s.	PROPN
ejpam-5779	275	2	then	then	ADV
ejpam-5779	275	3	f̃m(pq	f̃m(pq	PROPN
ejpam-5779	275	4	)	)	PUNCT
ejpam-5779	276	1	=	=	SYM
ejpam-5779	276	2	supb1	supb1	NOUN
ejpam-5779	276	3	+	+	CCONJ
ejpam-5779	276	4	inf	inf	ADJ
ejpam-5779	276	5	b1	b1	NOUN
ejpam-5779	276	6	2	2	NUM
ejpam-5779	276	7	and	and	CCONJ
ejpam-5779	276	8	f̃m(q	f̃m(q	NOUN
ejpam-5779	276	9	)	)	PUNCT
ejpam-5779	276	10	=	=	SYM
ejpam-5779	277	1	supb1	supb1	NOUN
ejpam-5779	277	2	+	+	CCONJ
ejpam-5779	277	3	inf	inf	ADJ
ejpam-5779	277	4	b1	b1	NOUN
ejpam-5779	277	5	2	2	NUM
ejpam-5779	277	6	,	,	PUNCT
ejpam-5779	277	7	so267	so267	PROPN
ejpam-5779	277	8	max{f̃m(pq	max{f̃m(pq	NOUN
ejpam-5779	277	9	)	)	PUNCT
ejpam-5779	277	10	,	,	PUNCT
ejpam-5779	277	11	f̃m(q	f̃m(q	NOUN
ejpam-5779	277	12	)	)	PUNCT
ejpam-5779	277	13	}	}	PUNCT
ejpam-5779	277	14	=	=	SYM
ejpam-5779	278	1	supb1	supb1	NOUN
ejpam-5779	278	2	+	+	CCONJ
ejpam-5779	278	3	inf	inf	ADJ
ejpam-5779	278	4	b1	b1	NOUN
ejpam-5779	278	5	2	2	NUM
ejpam-5779	278	6	.	.	PUNCT
ejpam-5779	279	1	thus	thus	ADV
ejpam-5779	279	2	,	,	PUNCT
ejpam-5779	279	3	f̃m(p	f̃m(p	NUM
ejpam-5779	279	4	)	)	PUNCT
ejpam-5779	279	5	≤	≤	NUM
ejpam-5779	280	1	supb1	supb1	NOUN
ejpam-5779	281	1	+	+	CCONJ
ejpam-5779	281	2	inf	inf	ADJ
ejpam-5779	281	3	b1	b1	NOUN
ejpam-5779	281	4	2	2	NUM
ejpam-5779	281	5	=	=	SYM
ejpam-5779	281	6	max{f̃m(pq	max{f̃m(pq	NOUN
ejpam-5779	281	7	)	)	PUNCT
ejpam-5779	281	8	,	,	PUNCT
ejpam-5779	281	9	f̃m(q)}.268	f̃m(q)}.268	PROPN
ejpam-5779	281	10	case	case	NOUN
ejpam-5779	281	11	3	3	X
ejpam-5779	281	12	:	:	PUNCT
ejpam-5779	281	13	let	let	VERB
ejpam-5779	281	14	pq	pq	INTJ
ejpam-5779	281	15	/∈	/∈	PUNCT
ejpam-5779	281	16	s	s	PROPN
ejpam-5779	281	17	and	and	CCONJ
ejpam-5779	281	18	q	q	PROPN
ejpam-5779	281	19	∈	∈	PROPN
ejpam-5779	281	20	s.	s.	PROPN
ejpam-5779	281	21	then	then	ADV
ejpam-5779	281	22	f̃m(pq	f̃m(pq	PROPN
ejpam-5779	281	23	)	)	PUNCT
ejpam-5779	281	24	=	=	SYM
ejpam-5779	281	25	supb1	supb1	NOUN
ejpam-5779	281	26	+	+	CCONJ
ejpam-5779	281	27	inf	inf	ADJ
ejpam-5779	281	28	b1	b1	NOUN
ejpam-5779	281	29	2	2	NUM
ejpam-5779	281	30	and	and	CCONJ
ejpam-5779	281	31	f̃m(q	f̃m(q	VERB
ejpam-5779	281	32	)	)	PUNCT
ejpam-5779	281	33	=	=	SYM
ejpam-5779	281	34	269	269	NUM
ejpam-5779	281	35	supb2	supb2	NOUN
ejpam-5779	281	36	+	+	CCONJ
ejpam-5779	281	37	inf	inf	ADJ
ejpam-5779	281	38	b2	b2	NOUN
ejpam-5779	281	39	2	2	NUM
ejpam-5779	281	40	,	,	PUNCT
ejpam-5779	281	41	so	so	ADV
ejpam-5779	281	42	max{f̃m(pq	max{f̃m(pq	ADJ
ejpam-5779	281	43	)	)	PUNCT
ejpam-5779	281	44	,	,	PUNCT
ejpam-5779	281	45	f̃m(q	f̃m(q	NOUN
ejpam-5779	281	46	)	)	PUNCT
ejpam-5779	281	47	}	}	PUNCT
ejpam-5779	281	48	=	=	SYM
ejpam-5779	282	1	supb1	supb1	NOUN
ejpam-5779	282	2	+	+	CCONJ
ejpam-5779	282	3	inf	inf	ADJ
ejpam-5779	282	4	b1	b1	NOUN
ejpam-5779	282	5	2	2	NUM
ejpam-5779	282	6	.	.	PUNCT
ejpam-5779	283	1	thus	thus	ADV
ejpam-5779	283	2	,	,	PUNCT
ejpam-5779	283	3	f̃m(p	f̃m(p	NUM
ejpam-5779	283	4	)	)	PUNCT
ejpam-5779	283	5	≤	≤	NUM
ejpam-5779	284	1	supb1	supb1	NOUN
ejpam-5779	285	1	+	+	CCONJ
ejpam-5779	285	2	inf	inf	ADJ
ejpam-5779	285	3	b1	b1	NOUN
ejpam-5779	285	4	2	2	NUM
ejpam-5779	285	5	=	=	SYM
ejpam-5779	285	6	270	270	NUM
ejpam-5779	285	7	max{f̃m(pq	max{f̃m(pq	NOUN
ejpam-5779	285	8	)	)	PUNCT
ejpam-5779	285	9	)	)	PUNCT
ejpam-5779	285	10	,	,	PUNCT
ejpam-5779	285	11	f̃m(q)}.271	f̃m(q)}.271	NOUN
ejpam-5779	285	12	case	case	NOUN
ejpam-5779	285	13	4	4	NUM
ejpam-5779	285	14	:	:	PUNCT
ejpam-5779	285	15	let	let	VERB
ejpam-5779	285	16	pq	pq	INTJ
ejpam-5779	285	17	∈	∈	PROPN
ejpam-5779	285	18	s	s	PART
ejpam-5779	285	19	and	and	CCONJ
ejpam-5779	285	20	q	q	PROPN
ejpam-5779	285	21	/∈	/∈	PUNCT
ejpam-5779	286	1	s.	s.	PROPN
ejpam-5779	286	2	then	then	ADV
ejpam-5779	286	3	f̃m(pq	f̃m(pq	PROPN
ejpam-5779	286	4	)	)	PUNCT
ejpam-5779	286	5	=	=	SYM
ejpam-5779	286	6	supb2	supb2	NOUN
ejpam-5779	286	7	+	+	CCONJ
ejpam-5779	286	8	inf	inf	ADJ
ejpam-5779	286	9	b2	b2	NOUN
ejpam-5779	286	10	2	2	NUM
ejpam-5779	286	11	and	and	CCONJ
ejpam-5779	286	12	f̃m(q	f̃m(q	NOUN
ejpam-5779	286	13	)	)	PUNCT
ejpam-5779	287	1	=	=	NOUN
ejpam-5779	287	2	272	272	NUM
ejpam-5779	287	3	supb1	supb1	NOUN
ejpam-5779	287	4	+	+	CCONJ
ejpam-5779	287	5	inf	inf	ADJ
ejpam-5779	287	6	b1	b1	NOUN
ejpam-5779	287	7	2	2	NUM
ejpam-5779	287	8	,	,	PUNCT
ejpam-5779	287	9	so	so	ADV
ejpam-5779	287	10	max{f̃m(pq	max{f̃m(pq	ADJ
ejpam-5779	287	11	)	)	PUNCT
ejpam-5779	287	12	,	,	PUNCT
ejpam-5779	287	13	f̃m(q	f̃m(q	NOUN
ejpam-5779	287	14	)	)	PUNCT
ejpam-5779	287	15	}	}	PUNCT
ejpam-5779	287	16	=	=	SYM
ejpam-5779	288	1	supb1	supb1	NOUN
ejpam-5779	288	2	+	+	CCONJ
ejpam-5779	288	3	inf	inf	ADJ
ejpam-5779	288	4	b1	b1	NOUN
ejpam-5779	288	5	2	2	NUM
ejpam-5779	288	6	.	.	PUNCT
ejpam-5779	289	1	thus	thus	ADV
ejpam-5779	289	2	,	,	PUNCT
ejpam-5779	289	3	f̃m(p	f̃m(p	NUM
ejpam-5779	289	4	)	)	PUNCT
ejpam-5779	289	5	≤	≤	NUM
ejpam-5779	290	1	supb1	supb1	NOUN
ejpam-5779	291	1	+	+	CCONJ
ejpam-5779	291	2	inf	inf	ADJ
ejpam-5779	291	3	b1	b1	NOUN
ejpam-5779	291	4	2	2	NUM
ejpam-5779	291	5	=	=	SYM
ejpam-5779	291	6	273	273	NUM
ejpam-5779	291	7	max{f̃m(pq	max{f̃m(pq	NOUN
ejpam-5779	291	8	)	)	PUNCT
ejpam-5779	291	9	,	,	PUNCT
ejpam-5779	291	10	f̃m(q)}.274	f̃m(q)}.274	PROPN
ejpam-5779	291	11	hence	hence	ADV
ejpam-5779	291	12	,	,	PUNCT
ejpam-5779	291	13	f̃m	f̃m	PROPN
ejpam-5779	291	14	is	be	AUX
ejpam-5779	291	15	a	a	DET
ejpam-5779	291	16	4	4	NUM
ejpam-5779	291	17	-	-	PUNCT
ejpam-5779	291	18	fuzzy	fuzzy	ADJ
ejpam-5779	291	19	ideal	ideal	NOUN
ejpam-5779	291	20	of	of	ADP
ejpam-5779	291	21	a	a	PRON
ejpam-5779	291	22	and	and	CCONJ
ejpam-5779	291	23	so	so	ADV
ejpam-5779	291	24	(	(	PUNCT
ejpam-5779	291	25	a	a	PRON
ejpam-5779	291	26	,	,	PUNCT
ejpam-5779	291	27	f̃	f̃	PROPN
ejpam-5779	291	28	)	)	PUNCT
ejpam-5779	291	29	is	be	AUX
ejpam-5779	291	30	a	a	DET
ejpam-5779	291	31	mean	mean	ADJ
ejpam-5779	291	32	4	4	NUM
ejpam-5779	291	33	-	-	PUNCT
ejpam-5779	291	34	fuzzy	fuzzy	ADJ
ejpam-5779	291	35	ideal	ideal	NOUN
ejpam-5779	291	36	of	of	ADP
ejpam-5779	291	37	a.275	a.275	PROPN
ejpam-5779	291	38	theorem	theorem	VERB
ejpam-5779	291	39	13	13	NUM
ejpam-5779	291	40	.	.	PUNCT
ejpam-5779	292	1	an	an	DET
ejpam-5779	292	2	interval	interval	NOUN
ejpam-5779	292	3	-	-	PUNCT
ejpam-5779	292	4	valued	value	VERB
ejpam-5779	292	5	fuzzy	fuzzy	ADJ
ejpam-5779	292	6	structure	structure	NOUN
ejpam-5779	292	7	(	(	PUNCT
ejpam-5779	292	8	a	a	DET
ejpam-5779	292	9	,	,	PUNCT
ejpam-5779	292	10	f̃m	f̃m	NOUN
ejpam-5779	292	11	)	)	PUNCT
ejpam-5779	292	12	over	over	ADP
ejpam-5779	292	13	a	a	PRON
ejpam-5779	292	14	is	be	AUX
ejpam-5779	292	15	a	a	DET
ejpam-5779	292	16	mean	mean	ADJ
ejpam-5779	292	17	1	1	NUM
ejpam-5779	292	18	-	-	PUNCT
ejpam-5779	292	19	fuzzy	fuzzy	ADJ
ejpam-5779	292	20	ideal276	ideal276	PROPN
ejpam-5779	292	21	of	of	ADP
ejpam-5779	292	22	a	a	DET
ejpam-5779	292	23	if	if	NOUN
ejpam-5779	292	24	and	and	CCONJ
ejpam-5779	292	25	only	only	ADV
ejpam-5779	292	26	if	if	SCONJ
ejpam-5779	292	27	the	the	DET
ejpam-5779	292	28	set	set	NOUN
ejpam-5779	292	29	u(f̃m	u(f̃m	PROPN
ejpam-5779	292	30	,	,	PUNCT
ejpam-5779	292	31	t	t	PROPN
ejpam-5779	292	32	)	)	PUNCT
ejpam-5779	292	33	is	be	AUX
ejpam-5779	292	34	an	an	DET
ejpam-5779	292	35	ideal	ideal	NOUN
ejpam-5779	292	36	of	of	ADP
ejpam-5779	292	37	a	a	PRON
ejpam-5779	292	38	for	for	ADP
ejpam-5779	292	39	all	all	DET
ejpam-5779	292	40	t	t	NOUN
ejpam-5779	292	41	∈	∈	PROPN
ejpam-5779	293	1	[	[	X
ejpam-5779	293	2	0	0	NUM
ejpam-5779	293	3	,	,	PUNCT
ejpam-5779	293	4	1	1	NUM
ejpam-5779	293	5	]	]	PUNCT
ejpam-5779	293	6	with	with	ADP
ejpam-5779	293	7	u(f̃m	u(f̃m	PROPN
ejpam-5779	293	8	,	,	PUNCT
ejpam-5779	293	9	t	t	PROPN
ejpam-5779	293	10	)	)	PUNCT
ejpam-5779	293	11	̸=	̸=	PROPN
ejpam-5779	293	12	∅.277	∅.277	NOUN
ejpam-5779	293	13	proof	proof	NOUN
ejpam-5779	293	14	.	.	PUNCT
ejpam-5779	294	1	assume	assume	VERB
ejpam-5779	294	2	that	that	SCONJ
ejpam-5779	294	3	an	an	DET
ejpam-5779	294	4	interval	interval	NOUN
ejpam-5779	294	5	-	-	PUNCT
ejpam-5779	294	6	valued	value	VERB
ejpam-5779	294	7	fuzzy	fuzzy	ADJ
ejpam-5779	294	8	structure	structure	NOUN
ejpam-5779	294	9	(	(	PUNCT
ejpam-5779	294	10	a	a	DET
ejpam-5779	294	11	,	,	PUNCT
ejpam-5779	294	12	f̃m	f̃m	NOUN
ejpam-5779	294	13	)	)	PUNCT
ejpam-5779	294	14	over	over	ADP
ejpam-5779	294	15	a	a	PRON
ejpam-5779	294	16	is	be	AUX
ejpam-5779	294	17	a	a	DET
ejpam-5779	294	18	mean	mean	ADJ
ejpam-5779	294	19	1	1	NUM
ejpam-5779	294	20	-	-	PUNCT
ejpam-5779	294	21	fuzzy278	fuzzy278	ADJ
ejpam-5779	294	22	ideal	ideal	NOUN
ejpam-5779	294	23	of	of	ADP
ejpam-5779	294	24	a	a	PRON
ejpam-5779	294	25	and	and	CCONJ
ejpam-5779	294	26	let	let	VERB
ejpam-5779	294	27	t	t	X
ejpam-5779	294	28	∈	∈	PROPN
ejpam-5779	295	1	[	[	X
ejpam-5779	295	2	0	0	NUM
ejpam-5779	295	3	,	,	PUNCT
ejpam-5779	295	4	1	1	NUM
ejpam-5779	295	5	]	]	PUNCT
ejpam-5779	295	6	be	be	AUX
ejpam-5779	295	7	such	such	ADJ
ejpam-5779	295	8	that	that	SCONJ
ejpam-5779	295	9	u(f̃m	u(f̃m	ADJ
ejpam-5779	295	10	,	,	PUNCT
ejpam-5779	295	11	t	t	PROPN
ejpam-5779	295	12	)	)	PUNCT
ejpam-5779	295	13	is	be	AUX
ejpam-5779	295	14	nonempty	nonempty	ADJ
ejpam-5779	295	15	.	.	PUNCT
ejpam-5779	296	1	obviously	obviously	ADV
ejpam-5779	296	2	,	,	PUNCT
ejpam-5779	296	3	0	0	NUM
ejpam-5779	296	4	∈	∈	PROPN
ejpam-5779	296	5	u(f̃m	u(f̃m	NOUN
ejpam-5779	296	6	,	,	PUNCT
ejpam-5779	296	7	t).279	t).279	VERB
ejpam-5779	296	8	let	let	VERB
ejpam-5779	296	9	p	p	NOUN
ejpam-5779	296	10	,	,	PUNCT
ejpam-5779	296	11	q	q	ADJ
ejpam-5779	296	12	∈	∈	PROPN
ejpam-5779	296	13	a	a	PRON
ejpam-5779	296	14	be	be	AUX
ejpam-5779	296	15	such	such	ADJ
ejpam-5779	296	16	that	that	PRON
ejpam-5779	296	17	pq	pq	PROPN
ejpam-5779	296	18	∈	∈	PROPN
ejpam-5779	296	19	u(f̃m	u(f̃m	PROPN
ejpam-5779	296	20	,	,	PUNCT
ejpam-5779	296	21	t	t	PROPN
ejpam-5779	296	22	)	)	PUNCT
ejpam-5779	296	23	and	and	CCONJ
ejpam-5779	296	24	q	q	PROPN
ejpam-5779	296	25	∈	∈	PROPN
ejpam-5779	296	26	u(f̃m	u(f̃m	NOUN
ejpam-5779	296	27	,	,	PUNCT
ejpam-5779	296	28	t	t	PROPN
ejpam-5779	296	29	)	)	PUNCT
ejpam-5779	296	30	.	.	PUNCT
ejpam-5779	297	1	then	then	ADV
ejpam-5779	297	2	f̃m(pq	f̃m(pq	PROPN
ejpam-5779	297	3	)	)	PUNCT
ejpam-5779	297	4	≥	≥	NOUN
ejpam-5779	297	5	t	t	PROPN
ejpam-5779	297	6	and	and	CCONJ
ejpam-5779	297	7	f̃m(q	f̃m(q	PROPN
ejpam-5779	297	8	)	)	PUNCT
ejpam-5779	297	9	≥	≥	NOUN
ejpam-5779	297	10	t,280	t,280	NOUN
ejpam-5779	297	11	which	which	PRON
ejpam-5779	297	12	imply	imply	VERB
ejpam-5779	297	13	from	from	ADP
ejpam-5779	297	14	(	(	PUNCT
ejpam-5779	297	15	2	2	NUM
ejpam-5779	297	16	)	)	PUNCT
ejpam-5779	297	17	that	that	DET
ejpam-5779	297	18	f̃m(p	f̃m(p	NUM
ejpam-5779	297	19	)	)	PUNCT
ejpam-5779	297	20	≥	≥	NOUN
ejpam-5779	297	21	min{f̃m(pq	min{f̃m(pq	NOUN
ejpam-5779	297	22	)	)	PUNCT
ejpam-5779	297	23	,	,	PUNCT
ejpam-5779	297	24	f̃m(q	f̃m(q	NOUN
ejpam-5779	297	25	)	)	PUNCT
ejpam-5779	297	26	}	}	PUNCT
ejpam-5779	297	27	≥	≥	NOUN
ejpam-5779	297	28	t.	t.	NOUN
ejpam-5779	297	29	hence	hence	ADV
ejpam-5779	297	30	,	,	PUNCT
ejpam-5779	297	31	p	p	PROPN
ejpam-5779	297	32	∈	∈	PROPN
ejpam-5779	297	33	u(f̃m	u(f̃m	NOUN
ejpam-5779	297	34	,	,	PUNCT
ejpam-5779	297	35	t	t	PROPN
ejpam-5779	297	36	)	)	PUNCT
ejpam-5779	297	37	,	,	PUNCT
ejpam-5779	297	38	and281	and281	VERB
ejpam-5779	297	39	therefore	therefore	ADV
ejpam-5779	297	40	u(f̃m	u(f̃m	PROPN
ejpam-5779	297	41	,	,	PUNCT
ejpam-5779	297	42	t	t	PROPN
ejpam-5779	297	43	)	)	PUNCT
ejpam-5779	297	44	is	be	AUX
ejpam-5779	297	45	an	an	DET
ejpam-5779	297	46	ideal	ideal	NOUN
ejpam-5779	297	47	of	of	ADP
ejpam-5779	297	48	a.282	a.282	ADJ
ejpam-5779	297	49	conversely	conversely	ADV
ejpam-5779	297	50	,	,	PUNCT
ejpam-5779	297	51	suppose	suppose	VERB
ejpam-5779	297	52	that	that	SCONJ
ejpam-5779	297	53	u(f̃m	u(f̃m	PROPN
ejpam-5779	297	54	,	,	PUNCT
ejpam-5779	297	55	t	t	PROPN
ejpam-5779	297	56	)	)	PUNCT
ejpam-5779	297	57	is	be	AUX
ejpam-5779	297	58	an	an	DET
ejpam-5779	297	59	ideal	ideal	NOUN
ejpam-5779	297	60	of	of	ADP
ejpam-5779	297	61	a	a	PRON
ejpam-5779	297	62	for	for	ADP
ejpam-5779	297	63	all	all	DET
ejpam-5779	297	64	t	t	NOUN
ejpam-5779	297	65	∈	∈	PROPN
ejpam-5779	298	1	[	[	X
ejpam-5779	298	2	0	0	NUM
ejpam-5779	298	3	,	,	PUNCT
ejpam-5779	298	4	1	1	NUM
ejpam-5779	298	5	]	]	PUNCT
ejpam-5779	298	6	with	with	ADP
ejpam-5779	298	7	u(f̃m	u(f̃m	PROPN
ejpam-5779	298	8	,	,	PUNCT
ejpam-5779	298	9	t	t	PROPN
ejpam-5779	298	10	)	)	PUNCT
ejpam-5779	298	11	̸=	̸=	PROPN
ejpam-5779	298	12	∅.283	∅.283	NOUN
ejpam-5779	298	13	if	if	SCONJ
ejpam-5779	298	14	f̃m(0	f̃m(0	PROPN
ejpam-5779	298	15	)	)	PUNCT
ejpam-5779	298	16	<	<	X
ejpam-5779	298	17	f̃m(k	f̃m(k	PROPN
ejpam-5779	298	18	)	)	PUNCT
ejpam-5779	298	19	for	for	ADP
ejpam-5779	298	20	some	some	DET
ejpam-5779	298	21	k	k	PROPN
ejpam-5779	298	22	∈	∈	PROPN
ejpam-5779	298	23	a	a	PRON
ejpam-5779	298	24	,	,	PUNCT
ejpam-5779	298	25	then	then	ADV
ejpam-5779	298	26	k	k	PROPN
ejpam-5779	298	27	∈	∈	PROPN
ejpam-5779	298	28	u(f̃m	u(f̃m	NOUN
ejpam-5779	298	29	,	,	PUNCT
ejpam-5779	298	30	f̃m(k	f̃m(k	PROPN
ejpam-5779	298	31	)	)	PUNCT
ejpam-5779	298	32	)	)	PUNCT
ejpam-5779	298	33	and	and	CCONJ
ejpam-5779	298	34	hence	hence	ADV
ejpam-5779	298	35	u(f̃m	u(f̃m	ADJ
ejpam-5779	298	36	,	,	PUNCT
ejpam-5779	298	37	f̃m(k	f̃m(k	PROPN
ejpam-5779	298	38	)	)	PUNCT
ejpam-5779	298	39	)	)	PUNCT
ejpam-5779	298	40	is	be	AUX
ejpam-5779	298	41	an284	an284	PROPN
ejpam-5779	298	42	ideal	ideal	NOUN
ejpam-5779	298	43	of	of	ADP
ejpam-5779	298	44	a.	a.	NOUN
ejpam-5779	298	45	thus	thus	ADV
ejpam-5779	298	46	,	,	PUNCT
ejpam-5779	298	47	0	0	NUM
ejpam-5779	298	48	∈	∈	PROPN
ejpam-5779	298	49	u(f̃m	u(f̃m	NOUN
ejpam-5779	298	50	,	,	PUNCT
ejpam-5779	298	51	f̃m(k	f̃m(k	PROPN
ejpam-5779	298	52	)	)	PUNCT
ejpam-5779	298	53	)	)	PUNCT
ejpam-5779	298	54	,	,	PUNCT
ejpam-5779	298	55	and	and	CCONJ
ejpam-5779	298	56	so	so	ADV
ejpam-5779	298	57	f̃m(0	f̃m(0	PROPN
ejpam-5779	298	58	)	)	PUNCT
ejpam-5779	298	59	≥	≥	NOUN
ejpam-5779	298	60	f̃m(k	f̃m(k	PROPN
ejpam-5779	298	61	)	)	PUNCT
ejpam-5779	298	62	.	.	PUNCT
ejpam-5779	299	1	this	this	PRON
ejpam-5779	299	2	is	be	AUX
ejpam-5779	299	3	a	a	DET
ejpam-5779	299	4	contradiction,285	contradiction,285	PROPN
ejpam-5779	299	5	and	and	CCONJ
ejpam-5779	299	6	thus	thus	ADV
ejpam-5779	299	7	f̃m(0	f̃m(0	PROPN
ejpam-5779	299	8	)	)	PUNCT
ejpam-5779	299	9	≥	≥	NOUN
ejpam-5779	299	10	f̃m(p	f̃m(p	NUM
ejpam-5779	299	11	)	)	PUNCT
ejpam-5779	299	12	for	for	ADP
ejpam-5779	299	13	all	all	DET
ejpam-5779	299	14	p	p	PROPN
ejpam-5779	299	15	∈	∈	PROPN
ejpam-5779	299	16	a.	a.	NOUN
ejpam-5779	299	17	assume	assume	VERB
ejpam-5779	299	18	that	that	SCONJ
ejpam-5779	299	19	there	there	PRON
ejpam-5779	299	20	exist	exist	VERB
ejpam-5779	299	21	k	k	PROPN
ejpam-5779	299	22	,	,	PUNCT
ejpam-5779	299	23	l	l	PROPN
ejpam-5779	299	24	∈	∈	PROPN
ejpam-5779	299	25	a	a	DET
ejpam-5779	299	26	such	such	ADJ
ejpam-5779	299	27	that286	that286	PROPN
ejpam-5779	299	28	f̃m(k	f̃m(k	NOUN
ejpam-5779	299	29	)	)	PUNCT
ejpam-5779	299	30	<	<	X
ejpam-5779	299	31	min{f̃m(kl	min{f̃m(kl	NOUN
ejpam-5779	299	32	)	)	PUNCT
ejpam-5779	299	33	,	,	PUNCT
ejpam-5779	299	34	f̃m(l	f̃m(l	NUM
ejpam-5779	299	35	)	)	PUNCT
ejpam-5779	299	36	}	}	PUNCT
ejpam-5779	299	37	.	.	PUNCT
ejpam-5779	300	1	taking	take	VERB
ejpam-5779	300	2	t	t	NOUN
ejpam-5779	300	3	=	=	SYM
ejpam-5779	300	4	min{f̃m(kl	min{f̃m(kl	NOUN
ejpam-5779	300	5	)	)	PUNCT
ejpam-5779	300	6	,	,	PUNCT
ejpam-5779	300	7	f̃m(l	f̃m(l	NUM
ejpam-5779	300	8	)	)	PUNCT
ejpam-5779	300	9	}	}	PUNCT
ejpam-5779	300	10	implies	imply	VERB
ejpam-5779	300	11	that	that	SCONJ
ejpam-5779	300	12	k	k	PROPN
ejpam-5779	300	13	∈	∈	PROPN
ejpam-5779	300	14	u(f̃	u(f̃	PROPN
ejpam-5779	300	15	,	,	PUNCT
ejpam-5779	300	16	t	t	PROPN
ejpam-5779	300	17	)	)	PUNCT
ejpam-5779	300	18	.	.	PUNCT
ejpam-5779	301	1	since287	since287	PROPN
ejpam-5779	301	2	u(f̃m	u(f̃m	ADJ
ejpam-5779	301	3	,	,	PUNCT
ejpam-5779	301	4	t	t	PROPN
ejpam-5779	301	5	)	)	PUNCT
ejpam-5779	301	6	is	be	AUX
ejpam-5779	301	7	an	an	DET
ejpam-5779	301	8	ideal	ideal	NOUN
ejpam-5779	301	9	of	of	ADP
ejpam-5779	301	10	a	a	PRON
ejpam-5779	301	11	,	,	PUNCT
ejpam-5779	301	12	we	we	PRON
ejpam-5779	301	13	have	have	VERB
ejpam-5779	301	14	k	k	PROPN
ejpam-5779	301	15	∈	∈	PROPN
ejpam-5779	301	16	u(f̃	u(f̃	PROPN
ejpam-5779	301	17	,	,	PUNCT
ejpam-5779	301	18	t	t	PROPN
ejpam-5779	301	19	)	)	PUNCT
ejpam-5779	301	20	.	.	PUNCT
ejpam-5779	302	1	hence	hence	ADV
ejpam-5779	302	2	,	,	PUNCT
ejpam-5779	302	3	f̃m(k	f̃m(k	PROPN
ejpam-5779	302	4	)	)	PUNCT
ejpam-5779	302	5	≥	≥	NOUN
ejpam-5779	302	6	t	t	NOUN
ejpam-5779	302	7	=	=	PUNCT
ejpam-5779	302	8	min{f̃m(kl	min{f̃m(kl	NOUN
ejpam-5779	302	9	)	)	PUNCT
ejpam-5779	302	10	,	,	PUNCT
ejpam-5779	302	11	f̃m(k)},288	f̃m(k)},288	NOUN
ejpam-5779	302	12	which	which	PRON
ejpam-5779	302	13	is	be	AUX
ejpam-5779	302	14	a	a	DET
ejpam-5779	302	15	contradiction	contradiction	NOUN
ejpam-5779	302	16	.	.	PUNCT
ejpam-5779	303	1	hence	hence	ADV
ejpam-5779	303	2	,	,	PUNCT
ejpam-5779	303	3	f̃m(p	f̃m(p	PROPN
ejpam-5779	303	4	)	)	PUNCT
ejpam-5779	303	5	≥	≥	NOUN
ejpam-5779	303	6	min{f̃m(pq	min{f̃m(pq	NOUN
ejpam-5779	303	7	)	)	PUNCT
ejpam-5779	303	8	,	,	PUNCT
ejpam-5779	303	9	f̃m(q	f̃m(q	NOUN
ejpam-5779	303	10	)	)	PUNCT
ejpam-5779	303	11	}	}	PUNCT
ejpam-5779	303	12	for	for	ADP
ejpam-5779	303	13	all	all	DET
ejpam-5779	303	14	p	p	NOUN
ejpam-5779	303	15	,	,	PUNCT
ejpam-5779	303	16	q	q	PROPN
ejpam-5779	304	1	∈	∈	NOUN
ejpam-5779	304	2	a.	a.	NOUN
ejpam-5779	304	3	therefore,289	therefore,289	PROPN
ejpam-5779	304	4	(	(	PUNCT
ejpam-5779	304	5	a	a	PRON
ejpam-5779	304	6	,	,	PUNCT
ejpam-5779	304	7	f̃m	f̃m	NOUN
ejpam-5779	304	8	)	)	PUNCT
ejpam-5779	304	9	is	be	AUX
ejpam-5779	304	10	a	a	DET
ejpam-5779	304	11	mean	mean	ADJ
ejpam-5779	304	12	1	1	NUM
ejpam-5779	304	13	-	-	PUNCT
ejpam-5779	304	14	fuzzy	fuzzy	ADJ
ejpam-5779	304	15	ideal	ideal	NOUN
ejpam-5779	304	16	of	of	ADP
ejpam-5779	304	17	a.290	a.290	NUM
ejpam-5779	304	18	corollary	corollary	ADJ
ejpam-5779	304	19	7	7	NUM
ejpam-5779	304	20	.	.	PUNCT
ejpam-5779	305	1	if	if	SCONJ
ejpam-5779	305	2	(	(	PUNCT
ejpam-5779	305	3	a	a	PRON
ejpam-5779	305	4	,	,	PUNCT
ejpam-5779	305	5	f̃	f̃	PROPN
ejpam-5779	305	6	)	)	PUNCT
ejpam-5779	305	7	is	be	AUX
ejpam-5779	305	8	a	a	DET
ejpam-5779	305	9	mean	mean	ADJ
ejpam-5779	305	10	3	3	NUM
ejpam-5779	305	11	-	-	PUNCT
ejpam-5779	305	12	fuzzy	fuzzy	ADJ
ejpam-5779	305	13	ideal	ideal	NOUN
ejpam-5779	305	14	of	of	ADP
ejpam-5779	305	15	a	a	DET
ejpam-5779	305	16	,	,	PUNCT
ejpam-5779	305	17	then	then	ADV
ejpam-5779	305	18	u(f̃m	u(f̃m	ADJ
ejpam-5779	305	19	,	,	PUNCT
ejpam-5779	305	20	t	t	PROPN
ejpam-5779	305	21	)	)	PUNCT
ejpam-5779	305	22	is	be	AUX
ejpam-5779	305	23	an	an	DET
ejpam-5779	305	24	ideal	ideal	NOUN
ejpam-5779	305	25	of	of	ADP
ejpam-5779	305	26	a	a	PRON
ejpam-5779	305	27	for	for	ADP
ejpam-5779	305	28	all291	all291	PROPN
ejpam-5779	305	29	t	t	PROPN
ejpam-5779	305	30	∈	∈	PROPN
ejpam-5779	306	1	[	[	X
ejpam-5779	306	2	0	0	NUM
ejpam-5779	306	3	,	,	PUNCT
ejpam-5779	306	4	1	1	NUM
ejpam-5779	306	5	]	]	PUNCT
ejpam-5779	306	6	with	with	ADP
ejpam-5779	306	7	u(f̃m	u(f̃m	PROPN
ejpam-5779	306	8	,	,	PUNCT
ejpam-5779	306	9	t	t	PROPN
ejpam-5779	306	10	)	)	PUNCT
ejpam-5779	306	11	̸=	̸=	PROPN
ejpam-5779	306	12	∅.292	∅.292	VERB
ejpam-5779	306	13	proof	proof	NOUN
ejpam-5779	306	14	.	.	PUNCT
ejpam-5779	307	1	it	it	PRON
ejpam-5779	307	2	is	be	AUX
ejpam-5779	307	3	straightforward	straightforward	ADJ
ejpam-5779	307	4	by	by	ADP
ejpam-5779	307	5	theorems	theorem	NOUN
ejpam-5779	307	6	9	9	NUM
ejpam-5779	307	7	and	and	CCONJ
ejpam-5779	307	8	13.293	13.293	NUM
ejpam-5779	307	9	theorem	theorem	VERB
ejpam-5779	307	10	14	14	NUM
ejpam-5779	307	11	.	.	PUNCT
ejpam-5779	308	1	an	an	DET
ejpam-5779	308	2	interval	interval	NOUN
ejpam-5779	308	3	-	-	PUNCT
ejpam-5779	308	4	valued	value	VERB
ejpam-5779	308	5	fuzzy	fuzzy	ADJ
ejpam-5779	308	6	structure	structure	NOUN
ejpam-5779	308	7	(	(	PUNCT
ejpam-5779	308	8	a	a	PRON
ejpam-5779	308	9	,	,	PUNCT
ejpam-5779	308	10	f̃	f̃	PROPN
ejpam-5779	308	11	)	)	PUNCT
ejpam-5779	308	12	over	over	ADP
ejpam-5779	308	13	a	a	PRON
ejpam-5779	308	14	is	be	AUX
ejpam-5779	308	15	a	a	DET
ejpam-5779	308	16	mean	mean	ADJ
ejpam-5779	308	17	4	4	NUM
ejpam-5779	308	18	-	-	PUNCT
ejpam-5779	308	19	fuzzy	fuzzy	ADJ
ejpam-5779	308	20	ideal	ideal	NOUN
ejpam-5779	308	21	of294	of294	PRON
ejpam-5779	309	1	a	a	DET
ejpam-5779	309	2	if	if	NOUN
ejpam-5779	309	3	and	and	CCONJ
ejpam-5779	309	4	only	only	ADV
ejpam-5779	309	5	if	if	SCONJ
ejpam-5779	309	6	the	the	DET
ejpam-5779	309	7	set	set	NOUN
ejpam-5779	309	8	l(f̃m	l(f̃m	PROPN
ejpam-5779	309	9	,	,	PUNCT
ejpam-5779	309	10	t	t	PROPN
ejpam-5779	309	11	)	)	PUNCT
ejpam-5779	309	12	is	be	AUX
ejpam-5779	309	13	an	an	DET
ejpam-5779	309	14	ideal	ideal	NOUN
ejpam-5779	309	15	of	of	ADP
ejpam-5779	309	16	a	a	PRON
ejpam-5779	309	17	for	for	ADP
ejpam-5779	309	18	all	all	DET
ejpam-5779	309	19	t	t	NOUN
ejpam-5779	309	20	∈	∈	PROPN
ejpam-5779	310	1	[	[	X
ejpam-5779	310	2	0	0	NUM
ejpam-5779	310	3	,	,	PUNCT
ejpam-5779	310	4	1	1	NUM
ejpam-5779	310	5	]	]	PUNCT
ejpam-5779	310	6	with	with	ADP
ejpam-5779	310	7	l(f̃m	l(f̃m	PROPN
ejpam-5779	310	8	,	,	PUNCT
ejpam-5779	310	9	t	t	PROPN
ejpam-5779	310	10	)	)	PUNCT
ejpam-5779	310	11	̸=	̸=	PROPN
ejpam-5779	310	12	∅.295	∅.295	NOUN
ejpam-5779	310	13	proof	proof	NOUN
ejpam-5779	310	14	.	.	PUNCT
ejpam-5779	311	1	assume	assume	VERB
ejpam-5779	311	2	that	that	SCONJ
ejpam-5779	311	3	an	an	DET
ejpam-5779	311	4	interval	interval	NOUN
ejpam-5779	311	5	-	-	PUNCT
ejpam-5779	311	6	valued	value	VERB
ejpam-5779	311	7	fuzzy	fuzzy	ADJ
ejpam-5779	311	8	structure	structure	NOUN
ejpam-5779	311	9	(	(	PUNCT
ejpam-5779	311	10	a	a	DET
ejpam-5779	311	11	,	,	PUNCT
ejpam-5779	311	12	f̃m	f̃m	NOUN
ejpam-5779	311	13	)	)	PUNCT
ejpam-5779	311	14	over	over	ADP
ejpam-5779	311	15	a	a	PRON
ejpam-5779	311	16	is	be	AUX
ejpam-5779	311	17	a	a	DET
ejpam-5779	311	18	mean	mean	ADJ
ejpam-5779	311	19	4	4	NUM
ejpam-5779	311	20	-	-	PUNCT
ejpam-5779	311	21	fuzzy296	fuzzy296	NOUN
ejpam-5779	311	22	ideal	ideal	NOUN
ejpam-5779	311	23	of	of	ADP
ejpam-5779	311	24	a	a	PRON
ejpam-5779	311	25	and	and	CCONJ
ejpam-5779	311	26	let	let	VERB
ejpam-5779	311	27	t	t	X
ejpam-5779	311	28	∈	∈	PROPN
ejpam-5779	312	1	[	[	X
ejpam-5779	312	2	0	0	NUM
ejpam-5779	312	3	,	,	PUNCT
ejpam-5779	312	4	1	1	NUM
ejpam-5779	312	5	]	]	PUNCT
ejpam-5779	312	6	be	be	AUX
ejpam-5779	312	7	such	such	ADJ
ejpam-5779	312	8	that	that	SCONJ
ejpam-5779	312	9	l(f̃m	l(f̃m	PROPN
ejpam-5779	312	10	,	,	PUNCT
ejpam-5779	312	11	t	t	PROPN
ejpam-5779	312	12	)	)	PUNCT
ejpam-5779	312	13	is	be	AUX
ejpam-5779	312	14	nonempty	nonempty	ADJ
ejpam-5779	312	15	.	.	PUNCT
ejpam-5779	313	1	obviously	obviously	ADV
ejpam-5779	313	2	,	,	PUNCT
ejpam-5779	313	3	0	0	NUM
ejpam-5779	313	4	∈	∈	PROPN
ejpam-5779	313	5	l(f̃m	l(f̃m	PROPN
ejpam-5779	313	6	,	,	PUNCT
ejpam-5779	313	7	t).297	t).297	PRON
ejpam-5779	313	8	let	let	VERB
ejpam-5779	313	9	p	p	PRON
ejpam-5779	313	10	,	,	PUNCT
ejpam-5779	313	11	q	q	ADJ
ejpam-5779	313	12	∈	∈	PROPN
ejpam-5779	313	13	a	a	PRON
ejpam-5779	313	14	be	be	AUX
ejpam-5779	313	15	such	such	ADJ
ejpam-5779	313	16	that	that	PRON
ejpam-5779	313	17	pq	pq	PROPN
ejpam-5779	313	18	∈	∈	PROPN
ejpam-5779	313	19	l(f̃m	l(f̃m	PROPN
ejpam-5779	313	20	,	,	PUNCT
ejpam-5779	313	21	t	t	PROPN
ejpam-5779	313	22	)	)	PUNCT
ejpam-5779	313	23	and	and	CCONJ
ejpam-5779	313	24	q	q	PROPN
ejpam-5779	313	25	∈	∈	PROPN
ejpam-5779	313	26	l(f̃m	l(f̃m	PROPN
ejpam-5779	313	27	,	,	PUNCT
ejpam-5779	313	28	t	t	PROPN
ejpam-5779	313	29	)	)	PUNCT
ejpam-5779	313	30	.	.	PUNCT
ejpam-5779	314	1	then	then	ADV
ejpam-5779	314	2	f̃m(pq	f̃m(pq	PROPN
ejpam-5779	314	3	)	)	PUNCT
ejpam-5779	314	4	≤	≤	NOUN
ejpam-5779	314	5	t	t	NOUN
ejpam-5779	314	6	and	and	CCONJ
ejpam-5779	314	7	f̃m(q	f̃m(q	VERB
ejpam-5779	314	8	)	)	PUNCT
ejpam-5779	314	9	≤	≤	NOUN
ejpam-5779	314	10	t,298	t,298	NOUN
ejpam-5779	314	11	which	which	PRON
ejpam-5779	314	12	imply	imply	VERB
ejpam-5779	314	13	from	from	ADP
ejpam-5779	314	14	(	(	PUNCT
ejpam-5779	314	15	8)	8)	NUM
ejpam-5779	314	16	that	that	DET
ejpam-5779	314	17	f̃m(p	f̃m(p	NOUN
ejpam-5779	314	18	)	)	PUNCT
ejpam-5779	314	19	≤	≤	NOUN
ejpam-5779	314	20	max{f̃m(pq	max{f̃m(pq	NOUN
ejpam-5779	314	21	)	)	PUNCT
ejpam-5779	314	22	,	,	PUNCT
ejpam-5779	314	23	f̃m(q	f̃m(q	NOUN
ejpam-5779	314	24	)	)	PUNCT
ejpam-5779	314	25	}	}	PUNCT
ejpam-5779	314	26	≤	≤	NUM
ejpam-5779	315	1	t.	t.	NOUN
ejpam-5779	315	2	hence	hence	ADV
ejpam-5779	315	3	,	,	PUNCT
ejpam-5779	315	4	p	p	PROPN
ejpam-5779	315	5	∈	∈	PROPN
ejpam-5779	315	6	l(f̃m	l(f̃m	PROPN
ejpam-5779	315	7	,	,	PUNCT
ejpam-5779	315	8	t	t	PROPN
ejpam-5779	315	9	)	)	PUNCT
ejpam-5779	315	10	,	,	PUNCT
ejpam-5779	315	11	and299	and299	PROPN
ejpam-5779	315	12	therefore	therefore	ADV
ejpam-5779	315	13	l(f̃m	l(f̃m	PROPN
ejpam-5779	315	14	,	,	PUNCT
ejpam-5779	315	15	t	t	PROPN
ejpam-5779	315	16	)	)	PUNCT
ejpam-5779	315	17	is	be	AUX
ejpam-5779	315	18	an	an	DET
ejpam-5779	315	19	ideal	ideal	NOUN
ejpam-5779	315	20	of	of	ADP
ejpam-5779	315	21	a.300	a.300	PROPN
ejpam-5779	315	22	n.	n.	PROPN
ejpam-5779	315	23	rajesh	rajesh	PROPN
ejpam-5779	315	24	,	,	PUNCT
ejpam-5779	315	25	t.	t.	PROPN
ejpam-5779	315	26	oner	oner	NOUN
ejpam-5779	315	27	,	,	PUNCT
ejpam-5779	315	28	a.	a.	NOUN
ejpam-5779	315	29	iampan	iampan	PROPN
ejpam-5779	315	30	,	,	PUNCT
ejpam-5779	315	31	i.	i.	PROPN
ejpam-5779	315	32	senturk	senturk	PROPN
ejpam-5779	315	33	/	/	SYM
ejpam-5779	315	34	eur	eur	PROPN
ejpam-5779	315	35	.	.	PUNCT
ejpam-5779	316	1	j.	j.	PROPN
ejpam-5779	316	2	pure	pure	PROPN
ejpam-5779	316	3	appl	appl	PROPN
ejpam-5779	316	4	.	.	PROPN
ejpam-5779	316	5	math	math	PROPN
ejpam-5779	316	6	,	,	PUNCT
ejpam-5779	316	7	18	18	NUM
ejpam-5779	316	8	(	(	PUNCT
ejpam-5779	316	9	1	1	NUM
ejpam-5779	316	10	)	)	PUNCT
ejpam-5779	316	11	(	(	PUNCT
ejpam-5779	316	12	2025	2025	NUM
ejpam-5779	316	13	)	)	PUNCT
ejpam-5779	316	14	,	,	PUNCT
ejpam-5779	316	15	5779	5779	NUM
ejpam-5779	316	16	15	15	NUM
ejpam-5779	316	17	of	of	ADP
ejpam-5779	316	18	18	18	NUM
ejpam-5779	316	19	conversely	conversely	ADV
ejpam-5779	316	20	,	,	PUNCT
ejpam-5779	316	21	suppose	suppose	VERB
ejpam-5779	316	22	that	that	SCONJ
ejpam-5779	316	23	l(f̃m	l(f̃m	PROPN
ejpam-5779	316	24	,	,	PUNCT
ejpam-5779	316	25	t	t	PROPN
ejpam-5779	316	26	)	)	PUNCT
ejpam-5779	316	27	is	be	AUX
ejpam-5779	316	28	an	an	DET
ejpam-5779	316	29	ideal	ideal	NOUN
ejpam-5779	316	30	of	of	ADP
ejpam-5779	316	31	a	a	PRON
ejpam-5779	316	32	for	for	ADP
ejpam-5779	316	33	all	all	DET
ejpam-5779	316	34	t	t	NOUN
ejpam-5779	316	35	∈	∈	PROPN
ejpam-5779	317	1	[	[	X
ejpam-5779	317	2	0	0	NUM
ejpam-5779	317	3	,	,	PUNCT
ejpam-5779	317	4	1	1	NUM
ejpam-5779	317	5	]	]	PUNCT
ejpam-5779	317	6	with	with	ADP
ejpam-5779	317	7	l(f̃m	l(f̃m	PROPN
ejpam-5779	317	8	,	,	PUNCT
ejpam-5779	317	9	t	t	PROPN
ejpam-5779	317	10	)	)	PUNCT
ejpam-5779	317	11	̸=	̸=	PROPN
ejpam-5779	317	12	∅.301	∅.301	NOUN
ejpam-5779	317	13	if	if	SCONJ
ejpam-5779	317	14	f̃m(0	f̃m(0	PROPN
ejpam-5779	317	15	)	)	PUNCT
ejpam-5779	317	16	>	>	X
ejpam-5779	317	17	f̃m(k	f̃m(k	PROPN
ejpam-5779	317	18	)	)	PUNCT
ejpam-5779	317	19	for	for	ADP
ejpam-5779	317	20	some	some	DET
ejpam-5779	317	21	k	k	PROPN
ejpam-5779	317	22	∈	∈	PROPN
ejpam-5779	317	23	a	a	PRON
ejpam-5779	317	24	,	,	PUNCT
ejpam-5779	317	25	then	then	ADV
ejpam-5779	317	26	k	k	PROPN
ejpam-5779	317	27	∈	∈	PROPN
ejpam-5779	317	28	l(f̃m	l(f̃m	PROPN
ejpam-5779	317	29	,	,	PUNCT
ejpam-5779	317	30	f̃m(k	f̃m(k	PROPN
ejpam-5779	317	31	)	)	PUNCT
ejpam-5779	317	32	)	)	PUNCT
ejpam-5779	317	33	and	and	CCONJ
ejpam-5779	317	34	hence	hence	ADV
ejpam-5779	317	35	l(f̃m	l(f̃m	PROPN
ejpam-5779	317	36	,	,	PUNCT
ejpam-5779	317	37	f̃m(k	f̃m(k	PROPN
ejpam-5779	317	38	)	)	PUNCT
ejpam-5779	317	39	)	)	PUNCT
ejpam-5779	317	40	is	be	AUX
ejpam-5779	317	41	an302	an302	ADJ
ejpam-5779	317	42	ideal	ideal	NOUN
ejpam-5779	317	43	of	of	ADP
ejpam-5779	317	44	a.	a.	NOUN
ejpam-5779	317	45	thus	thus	ADV
ejpam-5779	317	46	,	,	PUNCT
ejpam-5779	317	47	0	0	NUM
ejpam-5779	317	48	∈	∈	PROPN
ejpam-5779	317	49	l(f̃m	l(f̃m	PROPN
ejpam-5779	317	50	,	,	PUNCT
ejpam-5779	317	51	f̃m(k	f̃m(k	PROPN
ejpam-5779	317	52	)	)	PUNCT
ejpam-5779	317	53	)	)	PUNCT
ejpam-5779	317	54	,	,	PUNCT
ejpam-5779	317	55	and	and	CCONJ
ejpam-5779	317	56	so	so	ADV
ejpam-5779	317	57	f̃m(0	f̃m(0	PROPN
ejpam-5779	317	58	)	)	PUNCT
ejpam-5779	317	59	≤	≤	NOUN
ejpam-5779	317	60	f̃m(k	f̃m(k	NUM
ejpam-5779	317	61	)	)	PUNCT
ejpam-5779	317	62	.	.	PUNCT
ejpam-5779	318	1	this	this	PRON
ejpam-5779	318	2	is	be	AUX
ejpam-5779	318	3	a	a	DET
ejpam-5779	318	4	contradiction	contradiction	NOUN
ejpam-5779	318	5	,	,	PUNCT
ejpam-5779	318	6	and303	and303	ADJ
ejpam-5779	318	7	thus	thus	ADV
ejpam-5779	318	8	f̃m(0	f̃m(0	PROPN
ejpam-5779	318	9	)	)	PUNCT
ejpam-5779	318	10	≤	≤	NOUN
ejpam-5779	318	11	f̃m(p	f̃m(p	NUM
ejpam-5779	318	12	)	)	PUNCT
ejpam-5779	318	13	for	for	ADP
ejpam-5779	318	14	all	all	DET
ejpam-5779	318	15	p	p	PROPN
ejpam-5779	318	16	∈	∈	PROPN
ejpam-5779	318	17	a.	a.	NOUN
ejpam-5779	318	18	assume	assume	VERB
ejpam-5779	318	19	that	that	SCONJ
ejpam-5779	318	20	there	there	PRON
ejpam-5779	318	21	exist	exist	VERB
ejpam-5779	318	22	k	k	PROPN
ejpam-5779	318	23	,	,	PUNCT
ejpam-5779	318	24	l	l	PROPN
ejpam-5779	318	25	∈	∈	PROPN
ejpam-5779	318	26	a	a	DET
ejpam-5779	318	27	such	such	ADJ
ejpam-5779	318	28	that	that	DET
ejpam-5779	318	29	f̃m(k	f̃m(k	NOUN
ejpam-5779	318	30	)	)	PUNCT
ejpam-5779	318	31	>	>	PUNCT
ejpam-5779	319	1	304	304	NUM
ejpam-5779	319	2	max{f̃m(kl	max{f̃m(kl	NOUN
ejpam-5779	319	3	)	)	PUNCT
ejpam-5779	319	4	)	)	PUNCT
ejpam-5779	319	5	,	,	PUNCT
ejpam-5779	319	6	f̃m(l	f̃m(l	NUM
ejpam-5779	319	7	)	)	PUNCT
ejpam-5779	319	8	}	}	PUNCT
ejpam-5779	319	9	.	.	PUNCT
ejpam-5779	320	1	taking	take	VERB
ejpam-5779	320	2	t	t	NOUN
ejpam-5779	320	3	=	=	SYM
ejpam-5779	320	4	max{f̃m(kl	max{f̃m(kl	NOUN
ejpam-5779	320	5	)	)	PUNCT
ejpam-5779	320	6	,	,	PUNCT
ejpam-5779	320	7	f̃m(l	f̃m(l	NUM
ejpam-5779	320	8	)	)	PUNCT
ejpam-5779	320	9	}	}	PUNCT
ejpam-5779	320	10	implies	imply	VERB
ejpam-5779	320	11	that	that	SCONJ
ejpam-5779	320	12	k	k	PROPN
ejpam-5779	320	13	∈	∈	PROPN
ejpam-5779	320	14	l(f̃m	l(f̃m	PROPN
ejpam-5779	320	15	,	,	PUNCT
ejpam-5779	320	16	t	t	PROPN
ejpam-5779	320	17	)	)	PUNCT
ejpam-5779	320	18	.	.	PUNCT
ejpam-5779	321	1	since305	since305	PROPN
ejpam-5779	321	2	l(f̃m	l(f̃m	PROPN
ejpam-5779	321	3	,	,	PUNCT
ejpam-5779	321	4	t	t	PROPN
ejpam-5779	321	5	)	)	PUNCT
ejpam-5779	321	6	is	be	AUX
ejpam-5779	321	7	an	an	DET
ejpam-5779	321	8	ideal	ideal	NOUN
ejpam-5779	321	9	of	of	ADP
ejpam-5779	321	10	a	a	PRON
ejpam-5779	321	11	,	,	PUNCT
ejpam-5779	321	12	we	we	PRON
ejpam-5779	321	13	have	have	VERB
ejpam-5779	321	14	k	k	PROPN
ejpam-5779	321	15	∈	∈	PROPN
ejpam-5779	321	16	l(f̃m	l(f̃m	PROPN
ejpam-5779	321	17	,	,	PUNCT
ejpam-5779	321	18	t	t	PROPN
ejpam-5779	321	19	)	)	PUNCT
ejpam-5779	321	20	.	.	PUNCT
ejpam-5779	322	1	hence	hence	ADV
ejpam-5779	322	2	,	,	PUNCT
ejpam-5779	322	3	f̃m(k	f̃m(k	PROPN
ejpam-5779	322	4	)	)	PUNCT
ejpam-5779	322	5	≤	≤	NOUN
ejpam-5779	322	6	t	t	NOUN
ejpam-5779	322	7	=	=	SYM
ejpam-5779	322	8	max{f̃m(kl	max{f̃m(kl	NOUN
ejpam-5779	322	9	)	)	PUNCT
ejpam-5779	322	10	,	,	PUNCT
ejpam-5779	322	11	f̃m(l)},306	f̃m(l)},306	NOUN
ejpam-5779	322	12	which	which	PRON
ejpam-5779	322	13	is	be	AUX
ejpam-5779	322	14	a	a	DET
ejpam-5779	322	15	contradiction	contradiction	NOUN
ejpam-5779	322	16	.	.	PUNCT
ejpam-5779	323	1	hence	hence	ADV
ejpam-5779	323	2	,	,	PUNCT
ejpam-5779	323	3	f̃m(p	f̃m(p	NOUN
ejpam-5779	323	4	)	)	PUNCT
ejpam-5779	323	5	≤	≤	NOUN
ejpam-5779	323	6	max{f̃m(pq	max{f̃m(pq	NOUN
ejpam-5779	323	7	)	)	PUNCT
ejpam-5779	323	8	,	,	PUNCT
ejpam-5779	323	9	f̃m(q	f̃m(q	NOUN
ejpam-5779	323	10	)	)	PUNCT
ejpam-5779	323	11	}	}	PUNCT
ejpam-5779	323	12	for	for	ADP
ejpam-5779	323	13	all	all	DET
ejpam-5779	323	14	p	p	NOUN
ejpam-5779	323	15	,	,	PUNCT
ejpam-5779	323	16	q	q	PROPN
ejpam-5779	323	17	∈	∈	PROPN
ejpam-5779	323	18	a.	a.	NOUN
ejpam-5779	323	19	therefore,307	therefore,307	PROPN
ejpam-5779	324	1	(	(	PUNCT
ejpam-5779	324	2	a	a	DET
ejpam-5779	324	3	,	,	PUNCT
ejpam-5779	324	4	f̃m	f̃m	NOUN
ejpam-5779	324	5	)	)	PUNCT
ejpam-5779	324	6	is	be	AUX
ejpam-5779	324	7	a	a	DET
ejpam-5779	324	8	mean	mean	ADJ
ejpam-5779	324	9	4	4	NUM
ejpam-5779	324	10	-	-	PUNCT
ejpam-5779	324	11	fuzzy	fuzzy	ADJ
ejpam-5779	324	12	ideal	ideal	NOUN
ejpam-5779	324	13	of	of	ADP
ejpam-5779	324	14	a.308	a.308	ADJ
ejpam-5779	324	15	corollary	corollary	ADJ
ejpam-5779	324	16	8	8	NUM
ejpam-5779	324	17	.	.	PUNCT
ejpam-5779	325	1	if	if	SCONJ
ejpam-5779	325	2	(	(	PUNCT
ejpam-5779	325	3	a	a	PRON
ejpam-5779	325	4	,	,	PUNCT
ejpam-5779	325	5	f̃	f̃	PROPN
ejpam-5779	325	6	)	)	PUNCT
ejpam-5779	325	7	is	be	AUX
ejpam-5779	325	8	a	a	DET
ejpam-5779	325	9	mean	mean	ADJ
ejpam-5779	325	10	2	2	NUM
ejpam-5779	325	11	-	-	PUNCT
ejpam-5779	325	12	fuzzy	fuzzy	ADJ
ejpam-5779	325	13	ideal	ideal	NOUN
ejpam-5779	325	14	of	of	ADP
ejpam-5779	325	15	a	a	PRON
ejpam-5779	325	16	,	,	PUNCT
ejpam-5779	325	17	then	then	ADV
ejpam-5779	325	18	l(f̃m	l(f̃m	PROPN
ejpam-5779	325	19	,	,	PUNCT
ejpam-5779	325	20	t	t	PROPN
ejpam-5779	325	21	)	)	PUNCT
ejpam-5779	325	22	is	be	AUX
ejpam-5779	325	23	an	an	DET
ejpam-5779	325	24	ideal	ideal	NOUN
ejpam-5779	325	25	of	of	ADP
ejpam-5779	325	26	a	a	PRON
ejpam-5779	325	27	for	for	ADP
ejpam-5779	325	28	all309	all309	PROPN
ejpam-5779	325	29	t	t	PROPN
ejpam-5779	325	30	∈	∈	PROPN
ejpam-5779	326	1	[	[	X
ejpam-5779	326	2	0	0	NUM
ejpam-5779	326	3	,	,	PUNCT
ejpam-5779	326	4	1	1	NUM
ejpam-5779	326	5	]	]	PUNCT
ejpam-5779	326	6	with	with	ADP
ejpam-5779	326	7	l(f̃m	l(f̃m	PROPN
ejpam-5779	326	8	,	,	PUNCT
ejpam-5779	326	9	t	t	PROPN
ejpam-5779	326	10	)	)	PUNCT
ejpam-5779	326	11	̸=	̸=	PROPN
ejpam-5779	326	12	∅.310	∅.310	NOUN
ejpam-5779	326	13	proof	proof	NOUN
ejpam-5779	326	14	.	.	PUNCT
ejpam-5779	327	1	it	it	PRON
ejpam-5779	327	2	is	be	AUX
ejpam-5779	327	3	straightforward	straightforward	ADJ
ejpam-5779	327	4	by	by	ADP
ejpam-5779	327	5	theorems	theorem	NOUN
ejpam-5779	327	6	1	1	NUM
ejpam-5779	327	7	and	and	CCONJ
ejpam-5779	327	8	14.311	14.311	NUM
ejpam-5779	327	9	theorem	theorem	NOUN
ejpam-5779	327	10	15	15	NUM
ejpam-5779	327	11	.	.	PUNCT
ejpam-5779	328	1	if	if	SCONJ
ejpam-5779	328	2	(	(	PUNCT
ejpam-5779	328	3	a	a	PRON
ejpam-5779	328	4	,	,	PUNCT
ejpam-5779	328	5	f̃	f̃	PROPN
ejpam-5779	328	6	)	)	PUNCT
ejpam-5779	328	7	is	be	AUX
ejpam-5779	328	8	an	an	DET
ejpam-5779	328	9	interval	interval	NOUN
ejpam-5779	328	10	-	-	PUNCT
ejpam-5779	328	11	valued	value	VERB
ejpam-5779	328	12	fuzzy	fuzzy	ADJ
ejpam-5779	328	13	structure	structure	NOUN
ejpam-5779	328	14	over	over	ADP
ejpam-5779	328	15	a	a	PRON
ejpam-5779	328	16	in	in	ADP
ejpam-5779	328	17	which	which	PRON
ejpam-5779	328	18	(	(	PUNCT
ejpam-5779	328	19	a	a	PRON
ejpam-5779	328	20	,	,	PUNCT
ejpam-5779	328	21	f̃inf	f̃inf	ADJ
ejpam-5779	328	22	)	)	PUNCT
ejpam-5779	329	1	is312	is312	PROPN
ejpam-5779	329	2	constant	constant	ADJ
ejpam-5779	329	3	and	and	CCONJ
ejpam-5779	329	4	(	(	PUNCT
ejpam-5779	329	5	a	a	DET
ejpam-5779	329	6	,	,	PUNCT
ejpam-5779	329	7	f̃sup	f̃sup	ADJ
ejpam-5779	329	8	)	)	PUNCT
ejpam-5779	329	9	is	be	AUX
ejpam-5779	329	10	a	a	DET
ejpam-5779	329	11	1	1	NUM
ejpam-5779	329	12	-	-	PUNCT
ejpam-5779	329	13	fuzzy	fuzzy	ADJ
ejpam-5779	329	14	ideal	ideal	NOUN
ejpam-5779	329	15	of	of	ADP
ejpam-5779	329	16	a	a	PRON
ejpam-5779	329	17	,	,	PUNCT
ejpam-5779	329	18	then	then	ADV
ejpam-5779	329	19	(	(	PUNCT
ejpam-5779	329	20	a	a	PRON
ejpam-5779	329	21	,	,	PUNCT
ejpam-5779	329	22	f̃	f̃	PROPN
ejpam-5779	329	23	)	)	PUNCT
ejpam-5779	329	24	is	be	AUX
ejpam-5779	329	25	a	a	DET
ejpam-5779	329	26	mean	mean	ADJ
ejpam-5779	329	27	1	1	NUM
ejpam-5779	329	28	-	-	PUNCT
ejpam-5779	329	29	fuzzy	fuzzy	ADJ
ejpam-5779	329	30	ideal	ideal	NOUN
ejpam-5779	329	31	of	of	ADP
ejpam-5779	329	32	a.313	a.313	NOUN
ejpam-5779	329	33	proof	proof	NOUN
ejpam-5779	329	34	.	.	PUNCT
ejpam-5779	330	1	assume	assume	VERB
ejpam-5779	330	2	that	that	SCONJ
ejpam-5779	330	3	(	(	PUNCT
ejpam-5779	330	4	a	a	PRON
ejpam-5779	330	5	,	,	PUNCT
ejpam-5779	330	6	f̃	f̃	PROPN
ejpam-5779	330	7	)	)	PUNCT
ejpam-5779	330	8	is	be	AUX
ejpam-5779	330	9	an	an	DET
ejpam-5779	330	10	interval	interval	NOUN
ejpam-5779	330	11	-	-	PUNCT
ejpam-5779	330	12	valued	value	VERB
ejpam-5779	330	13	fuzzy	fuzzy	ADJ
ejpam-5779	330	14	structure	structure	NOUN
ejpam-5779	330	15	over	over	ADP
ejpam-5779	330	16	a	a	PRON
ejpam-5779	330	17	in	in	ADP
ejpam-5779	330	18	which	which	PRON
ejpam-5779	330	19	(	(	PUNCT
ejpam-5779	330	20	a	a	X
ejpam-5779	330	21	,	,	PUNCT
ejpam-5779	330	22	f̃inf)314	f̃inf)314	PROPN
ejpam-5779	330	23	is	be	AUX
ejpam-5779	330	24	constant	constant	ADJ
ejpam-5779	330	25	and	and	CCONJ
ejpam-5779	330	26	(	(	PUNCT
ejpam-5779	330	27	a	a	DET
ejpam-5779	330	28	,	,	PUNCT
ejpam-5779	330	29	f̃sup	f̃sup	ADJ
ejpam-5779	330	30	)	)	PUNCT
ejpam-5779	330	31	is	be	AUX
ejpam-5779	330	32	a	a	DET
ejpam-5779	330	33	1	1	NUM
ejpam-5779	330	34	-	-	PUNCT
ejpam-5779	330	35	fuzzy	fuzzy	ADJ
ejpam-5779	330	36	ideal	ideal	NOUN
ejpam-5779	330	37	of	of	ADP
ejpam-5779	330	38	a.	a.	NOUN
ejpam-5779	330	39	let	let	VERB
ejpam-5779	330	40	p	p	PRON
ejpam-5779	330	41	,	,	PUNCT
ejpam-5779	330	42	q	q	PROPN
ejpam-5779	330	43	∈	∈	PROPN
ejpam-5779	330	44	a.	a.	NOUN
ejpam-5779	330	45	since	since	SCONJ
ejpam-5779	330	46	(	(	PUNCT
ejpam-5779	330	47	a	a	PRON
ejpam-5779	330	48	,	,	PUNCT
ejpam-5779	330	49	f̃inf	f̃inf	ADJ
ejpam-5779	330	50	)	)	PUNCT
ejpam-5779	330	51	is	be	AUX
ejpam-5779	330	52	constant,315	constant,315	PROPN
ejpam-5779	330	53	we	we	PRON
ejpam-5779	330	54	have	have	VERB
ejpam-5779	330	55	f̃inf(p	f̃inf(p	NOUN
ejpam-5779	330	56	)	)	PUNCT
ejpam-5779	331	1	=	=	SYM
ejpam-5779	331	2	f̃inf(0	f̃inf(0	PROPN
ejpam-5779	331	3	)	)	PUNCT
ejpam-5779	331	4	for	for	ADP
ejpam-5779	331	5	all	all	DET
ejpam-5779	331	6	p	p	PROPN
ejpam-5779	331	7	∈	∈	PROPN
ejpam-5779	331	8	a.	a.	NOUN
ejpam-5779	331	9	since	since	SCONJ
ejpam-5779	331	10	(	(	PUNCT
ejpam-5779	331	11	a	a	DET
ejpam-5779	331	12	,	,	PUNCT
ejpam-5779	331	13	f̃sup	f̃sup	ADJ
ejpam-5779	331	14	)	)	PUNCT
ejpam-5779	331	15	is	be	AUX
ejpam-5779	331	16	a	a	DET
ejpam-5779	331	17	1	1	NUM
ejpam-5779	331	18	-	-	PUNCT
ejpam-5779	331	19	fuzzy	fuzzy	ADJ
ejpam-5779	331	20	ideal	ideal	NOUN
ejpam-5779	331	21	of	of	ADP
ejpam-5779	331	22	a	a	PRON
ejpam-5779	331	23	,	,	PUNCT
ejpam-5779	331	24	we	we	PRON
ejpam-5779	331	25	have316	have316	PROPN
ejpam-5779	331	26	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	331	27	)	)	PUNCT
ejpam-5779	331	28	≥	≥	NOUN
ejpam-5779	331	29	min{f̃sup(p	min{f̃sup(p	NOUN
ejpam-5779	331	30	)	)	PUNCT
ejpam-5779	331	31	,	,	PUNCT
ejpam-5779	331	32	f̃sup(q	f̃sup(q	NOUN
ejpam-5779	331	33	)	)	PUNCT
ejpam-5779	331	34	}	}	PUNCT
ejpam-5779	331	35	.	.	PUNCT
ejpam-5779	332	1	thus,317	thus,317	PROPN
ejpam-5779	332	2	f̃m(p	f̃m(p	NUM
ejpam-5779	332	3	)	)	PUNCT
ejpam-5779	332	4	=	=	SYM
ejpam-5779	332	5	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	332	6	)	)	PUNCT
ejpam-5779	332	7	+	+	SYM
ejpam-5779	332	8	f̃inf(p	f̃inf(p	X
ejpam-5779	332	9	)	)	PUNCT
ejpam-5779	332	10	2	2	NUM
ejpam-5779	332	11	=	=	SYM
ejpam-5779	332	12	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	332	13	)	)	PUNCT
ejpam-5779	332	14	2	2	NUM
ejpam-5779	333	1	+	+	NUM
ejpam-5779	333	2	f̃inf(0	f̃inf(0	NOUN
ejpam-5779	333	3	)	)	PUNCT
ejpam-5779	333	4	2	2	NUM
ejpam-5779	333	5	≥	≥	NOUN
ejpam-5779	333	6	min	min	NOUN
ejpam-5779	333	7	{	{	PUNCT
ejpam-5779	333	8	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	333	9	q	q	NOUN
ejpam-5779	333	10	)	)	PUNCT
ejpam-5779	333	11	2	2	NUM
ejpam-5779	333	12	+	+	CCONJ
ejpam-5779	333	13	f̃inf(q	f̃inf(q	X
ejpam-5779	333	14	)	)	PUNCT
ejpam-5779	333	15	2	2	NUM
ejpam-5779	333	16	}	}	PUNCT
ejpam-5779	333	17	+	+	NUM
ejpam-5779	333	18	f̃inf(0	f̃inf(0	NOUN
ejpam-5779	333	19	)	)	PUNCT
ejpam-5779	333	20	2	2	NUM
ejpam-5779	333	21	=	=	SYM
ejpam-5779	333	22	min	min	PROPN
ejpam-5779	333	23	{	{	PUNCT
ejpam-5779	333	24	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	333	25	q	q	NOUN
ejpam-5779	333	26	)	)	PUNCT
ejpam-5779	333	27	2	2	NUM
ejpam-5779	333	28	+	+	NUM
ejpam-5779	333	29	f̃inf(0	f̃inf(0	NOUN
ejpam-5779	333	30	)	)	PUNCT
ejpam-5779	333	31	2	2	NUM
ejpam-5779	333	32	,	,	PUNCT
ejpam-5779	333	33	f̃sup(q	f̃sup(q	NOUN
ejpam-5779	333	34	)	)	PUNCT
ejpam-5779	333	35	2	2	NUM
ejpam-5779	333	36	+	+	NUM
ejpam-5779	333	37	f̃inf(0	f̃inf(0	NOUN
ejpam-5779	333	38	)	)	PUNCT
ejpam-5779	333	39	2	2	NUM
ejpam-5779	333	40	}	}	PUNCT
ejpam-5779	333	41	=	=	SYM
ejpam-5779	333	42	min	min	NOUN
ejpam-5779	333	43	{	{	PUNCT
ejpam-5779	333	44	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	333	45	q	q	NOUN
ejpam-5779	333	46	)	)	PUNCT
ejpam-5779	333	47	)	)	PUNCT
ejpam-5779	334	1	+	+	PUNCT
ejpam-5779	334	2	f̃inf(p	f̃inf(p	X
ejpam-5779	334	3	)	)	PUNCT
ejpam-5779	334	4	2	2	NUM
ejpam-5779	334	5	,	,	PUNCT
ejpam-5779	334	6	f̃sup(q	f̃sup(q	NOUN
ejpam-5779	334	7	)	)	PUNCT
ejpam-5779	335	1	+	+	CCONJ
ejpam-5779	335	2	f̃inf(q	f̃inf(q	X
ejpam-5779	335	3	)	)	PUNCT
ejpam-5779	335	4	2	2	NUM
ejpam-5779	335	5	}	}	PUNCT
ejpam-5779	335	6	=	=	SYM
ejpam-5779	335	7	min{f̃m(pq	min{f̃m(pq	NOUN
ejpam-5779	335	8	)	)	PUNCT
ejpam-5779	335	9	,	,	PUNCT
ejpam-5779	335	10	f̃m(q	f̃m(q	NOUN
ejpam-5779	335	11	)	)	PUNCT
ejpam-5779	335	12	}	}	PUNCT
ejpam-5779	335	13	.	.	PUNCT
ejpam-5779	336	1	hence	hence	ADV
ejpam-5779	336	2	,	,	PUNCT
ejpam-5779	336	3	(	(	PUNCT
ejpam-5779	336	4	a	a	DET
ejpam-5779	336	5	,	,	PUNCT
ejpam-5779	336	6	f̃m	f̃m	NOUN
ejpam-5779	336	7	)	)	PUNCT
ejpam-5779	336	8	is	be	AUX
ejpam-5779	336	9	a	a	DET
ejpam-5779	336	10	1	1	NUM
ejpam-5779	336	11	-	-	PUNCT
ejpam-5779	336	12	fuzzy	fuzzy	ADJ
ejpam-5779	336	13	ideal	ideal	NOUN
ejpam-5779	336	14	of	of	ADP
ejpam-5779	336	15	a	a	PRON
ejpam-5779	336	16	,	,	PUNCT
ejpam-5779	336	17	that	that	ADV
ejpam-5779	336	18	is	is	ADV
ejpam-5779	336	19	,	,	PUNCT
ejpam-5779	336	20	(	(	PUNCT
ejpam-5779	336	21	a	a	PRON
ejpam-5779	336	22	,	,	PUNCT
ejpam-5779	336	23	f̃	f̃	PROPN
ejpam-5779	336	24	)	)	PUNCT
ejpam-5779	336	25	is	be	AUX
ejpam-5779	336	26	a	a	DET
ejpam-5779	336	27	mean	mean	ADJ
ejpam-5779	336	28	1	1	NUM
ejpam-5779	336	29	-	-	PUNCT
ejpam-5779	336	30	fuzzy	fuzzy	ADJ
ejpam-5779	336	31	ideal	ideal	NOUN
ejpam-5779	336	32	of	of	ADP
ejpam-5779	336	33	a.318	a.318	NOUN
ejpam-5779	336	34	theorem	theorem	VERB
ejpam-5779	336	35	16	16	NUM
ejpam-5779	336	36	.	.	PUNCT
ejpam-5779	337	1	if	if	SCONJ
ejpam-5779	337	2	(	(	PUNCT
ejpam-5779	337	3	a	a	PRON
ejpam-5779	337	4	,	,	PUNCT
ejpam-5779	337	5	f̃	f̃	PROPN
ejpam-5779	337	6	)	)	PUNCT
ejpam-5779	337	7	is	be	AUX
ejpam-5779	337	8	an	an	DET
ejpam-5779	337	9	interval	interval	NOUN
ejpam-5779	337	10	-	-	PUNCT
ejpam-5779	337	11	valued	value	VERB
ejpam-5779	337	12	fuzzy	fuzzy	ADJ
ejpam-5779	337	13	structure	structure	NOUN
ejpam-5779	337	14	over	over	ADP
ejpam-5779	337	15	a	a	PRON
ejpam-5779	337	16	in	in	ADP
ejpam-5779	337	17	which	which	PRON
ejpam-5779	337	18	(	(	PUNCT
ejpam-5779	337	19	a	a	PRON
ejpam-5779	337	20	,	,	PUNCT
ejpam-5779	337	21	f̃inf	f̃inf	ADJ
ejpam-5779	337	22	)	)	PUNCT
ejpam-5779	337	23	is319	is319	PROPN
ejpam-5779	337	24	constant	constant	ADJ
ejpam-5779	337	25	and	and	CCONJ
ejpam-5779	337	26	(	(	PUNCT
ejpam-5779	337	27	a	a	DET
ejpam-5779	337	28	,	,	PUNCT
ejpam-5779	337	29	f̃sup	f̃sup	ADJ
ejpam-5779	337	30	)	)	PUNCT
ejpam-5779	337	31	is	be	AUX
ejpam-5779	337	32	a	a	DET
ejpam-5779	337	33	4	4	NUM
ejpam-5779	337	34	-	-	PUNCT
ejpam-5779	337	35	fuzzy	fuzzy	ADJ
ejpam-5779	337	36	ideal	ideal	NOUN
ejpam-5779	337	37	of	of	ADP
ejpam-5779	337	38	a	a	PRON
ejpam-5779	337	39	,	,	PUNCT
ejpam-5779	337	40	then	then	ADV
ejpam-5779	337	41	(	(	PUNCT
ejpam-5779	337	42	a	a	PRON
ejpam-5779	337	43	,	,	PUNCT
ejpam-5779	337	44	f̃	f̃	PROPN
ejpam-5779	337	45	)	)	PUNCT
ejpam-5779	337	46	is	be	AUX
ejpam-5779	337	47	a	a	DET
ejpam-5779	337	48	mean	mean	ADJ
ejpam-5779	337	49	4	4	NUM
ejpam-5779	337	50	-	-	PUNCT
ejpam-5779	337	51	fuzzy	fuzzy	ADJ
ejpam-5779	337	52	ideal	ideal	NOUN
ejpam-5779	337	53	of	of	ADP
ejpam-5779	337	54	a.320	a.320	NOUN
ejpam-5779	337	55	proof	proof	NOUN
ejpam-5779	337	56	.	.	PUNCT
ejpam-5779	338	1	assume	assume	VERB
ejpam-5779	338	2	that	that	SCONJ
ejpam-5779	338	3	(	(	PUNCT
ejpam-5779	338	4	a	a	PRON
ejpam-5779	338	5	,	,	PUNCT
ejpam-5779	338	6	f̃	f̃	PROPN
ejpam-5779	338	7	)	)	PUNCT
ejpam-5779	338	8	is	be	AUX
ejpam-5779	338	9	an	an	DET
ejpam-5779	338	10	interval	interval	NOUN
ejpam-5779	338	11	-	-	PUNCT
ejpam-5779	338	12	valued	value	VERB
ejpam-5779	338	13	fuzzy	fuzzy	ADJ
ejpam-5779	338	14	structure	structure	NOUN
ejpam-5779	338	15	over	over	ADP
ejpam-5779	338	16	a	a	PRON
ejpam-5779	338	17	in	in	ADP
ejpam-5779	338	18	which	which	PRON
ejpam-5779	338	19	(	(	PUNCT
ejpam-5779	338	20	a	a	X
ejpam-5779	338	21	,	,	PUNCT
ejpam-5779	338	22	f̃inf)321	f̃inf)321	PRON
ejpam-5779	338	23	is	be	AUX
ejpam-5779	338	24	constant	constant	ADJ
ejpam-5779	338	25	and	and	CCONJ
ejpam-5779	338	26	(	(	PUNCT
ejpam-5779	338	27	a	a	DET
ejpam-5779	338	28	,	,	PUNCT
ejpam-5779	338	29	f̃sup	f̃sup	ADJ
ejpam-5779	338	30	)	)	PUNCT
ejpam-5779	338	31	is	be	AUX
ejpam-5779	338	32	a	a	DET
ejpam-5779	338	33	4	4	NUM
ejpam-5779	338	34	-	-	PUNCT
ejpam-5779	338	35	fuzzy	fuzzy	ADJ
ejpam-5779	338	36	ideal	ideal	NOUN
ejpam-5779	338	37	of	of	ADP
ejpam-5779	338	38	a.	a.	NOUN
ejpam-5779	338	39	let	let	VERB
ejpam-5779	338	40	p	p	PRON
ejpam-5779	338	41	,	,	PUNCT
ejpam-5779	338	42	q	q	PROPN
ejpam-5779	338	43	∈	∈	PROPN
ejpam-5779	338	44	a.	a.	NOUN
ejpam-5779	338	45	since	since	SCONJ
ejpam-5779	338	46	(	(	PUNCT
ejpam-5779	338	47	a	a	PRON
ejpam-5779	338	48	,	,	PUNCT
ejpam-5779	338	49	f̃inf	f̃inf	ADJ
ejpam-5779	338	50	)	)	PUNCT
ejpam-5779	338	51	is	be	AUX
ejpam-5779	338	52	constant,322	constant,322	PROPN
ejpam-5779	338	53	n.	n.	PROPN
ejpam-5779	338	54	rajesh	rajesh	PROPN
ejpam-5779	338	55	,	,	PUNCT
ejpam-5779	338	56	t.	t.	PROPN
ejpam-5779	338	57	oner	oner	NOUN
ejpam-5779	338	58	,	,	PUNCT
ejpam-5779	338	59	a.	a.	NOUN
ejpam-5779	338	60	iampan	iampan	PROPN
ejpam-5779	338	61	,	,	PUNCT
ejpam-5779	338	62	i.	i.	PROPN
ejpam-5779	338	63	senturk	senturk	PROPN
ejpam-5779	338	64	/	/	SYM
ejpam-5779	338	65	eur	eur	PROPN
ejpam-5779	338	66	.	.	PUNCT
ejpam-5779	339	1	j.	j.	PROPN
ejpam-5779	339	2	pure	pure	PROPN
ejpam-5779	339	3	appl	appl	PROPN
ejpam-5779	339	4	.	.	PROPN
ejpam-5779	339	5	math	math	PROPN
ejpam-5779	339	6	,	,	PUNCT
ejpam-5779	339	7	18	18	NUM
ejpam-5779	339	8	(	(	PUNCT
ejpam-5779	339	9	1	1	NUM
ejpam-5779	339	10	)	)	PUNCT
ejpam-5779	339	11	(	(	PUNCT
ejpam-5779	339	12	2025	2025	NUM
ejpam-5779	339	13	)	)	PUNCT
ejpam-5779	339	14	,	,	PUNCT
ejpam-5779	339	15	5779	5779	NUM
ejpam-5779	339	16	16	16	NUM
ejpam-5779	339	17	of	of	ADP
ejpam-5779	339	18	18	18	NUM
ejpam-5779	339	19	we	we	PRON
ejpam-5779	339	20	have	have	VERB
ejpam-5779	339	21	f̃inf(p	f̃inf(p	NOUN
ejpam-5779	339	22	)	)	PUNCT
ejpam-5779	339	23	=	=	SYM
ejpam-5779	339	24	f̃inf(0	f̃inf(0	PROPN
ejpam-5779	339	25	)	)	PUNCT
ejpam-5779	339	26	for	for	ADP
ejpam-5779	339	27	all	all	DET
ejpam-5779	339	28	p	p	PROPN
ejpam-5779	339	29	∈	∈	PROPN
ejpam-5779	339	30	a.	a.	NOUN
ejpam-5779	339	31	since	since	SCONJ
ejpam-5779	339	32	(	(	PUNCT
ejpam-5779	339	33	a	a	DET
ejpam-5779	339	34	,	,	PUNCT
ejpam-5779	339	35	f̃sup	f̃sup	ADJ
ejpam-5779	339	36	)	)	PUNCT
ejpam-5779	339	37	is	be	AUX
ejpam-5779	339	38	a	a	DET
ejpam-5779	339	39	4	4	NUM
ejpam-5779	339	40	-	-	PUNCT
ejpam-5779	339	41	fuzzy	fuzzy	ADJ
ejpam-5779	339	42	ideal	ideal	NOUN
ejpam-5779	339	43	of	of	ADP
ejpam-5779	339	44	a	a	PRON
ejpam-5779	339	45	,	,	PUNCT
ejpam-5779	339	46	we	we	PRON
ejpam-5779	339	47	have323	have323	PROPN
ejpam-5779	339	48	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	339	49	)	)	PUNCT
ejpam-5779	339	50	≤	≤	NUM
ejpam-5779	339	51	max{f̃sup(p	max{f̃sup(p	NOUN
ejpam-5779	339	52	)	)	PUNCT
ejpam-5779	339	53	,	,	PUNCT
ejpam-5779	339	54	f̃sup(q	f̃sup(q	NOUN
ejpam-5779	339	55	)	)	PUNCT
ejpam-5779	339	56	}	}	PUNCT
ejpam-5779	339	57	.	.	PUNCT
ejpam-5779	340	1	thus,324	thus,324	PROPN
ejpam-5779	340	2	f̃m(p	f̃m(p	NUM
ejpam-5779	340	3	)	)	PUNCT
ejpam-5779	340	4	=	=	SYM
ejpam-5779	340	5	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	340	6	)	)	PUNCT
ejpam-5779	340	7	+	+	SYM
ejpam-5779	340	8	f̃inf(p	f̃inf(p	X
ejpam-5779	340	9	)	)	PUNCT
ejpam-5779	340	10	2	2	NUM
ejpam-5779	340	11	=	=	SYM
ejpam-5779	340	12	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	340	13	)	)	PUNCT
ejpam-5779	340	14	2	2	NUM
ejpam-5779	340	15	+	+	NUM
ejpam-5779	340	16	f̃inf(0	f̃inf(0	NOUN
ejpam-5779	340	17	)	)	PUNCT
ejpam-5779	340	18	2	2	NUM
ejpam-5779	340	19	≥	≥	NOUN
ejpam-5779	340	20	min	min	NOUN
ejpam-5779	340	21	{	{	PUNCT
ejpam-5779	340	22	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	340	23	q	q	NOUN
ejpam-5779	340	24	)	)	PUNCT
ejpam-5779	340	25	2	2	NUM
ejpam-5779	340	26	+	+	CCONJ
ejpam-5779	340	27	f̃inf(q	f̃inf(q	X
ejpam-5779	340	28	)	)	PUNCT
ejpam-5779	340	29	2	2	NUM
ejpam-5779	340	30	}	}	PUNCT
ejpam-5779	340	31	+	+	NUM
ejpam-5779	340	32	f̃inf(0	f̃inf(0	NOUN
ejpam-5779	340	33	)	)	PUNCT
ejpam-5779	340	34	2	2	NUM
ejpam-5779	340	35	=	=	SYM
ejpam-5779	340	36	min	min	PROPN
ejpam-5779	340	37	{	{	PUNCT
ejpam-5779	340	38	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	340	39	q	q	NOUN
ejpam-5779	340	40	)	)	PUNCT
ejpam-5779	340	41	2	2	NUM
ejpam-5779	340	42	+	+	NUM
ejpam-5779	340	43	f̃inf(0	f̃inf(0	NOUN
ejpam-5779	340	44	)	)	PUNCT
ejpam-5779	340	45	2	2	NUM
ejpam-5779	340	46	,	,	PUNCT
ejpam-5779	340	47	f̃sup(q	f̃sup(q	NOUN
ejpam-5779	340	48	)	)	PUNCT
ejpam-5779	340	49	2	2	NUM
ejpam-5779	340	50	+	+	NUM
ejpam-5779	340	51	f̃inf(0	f̃inf(0	NOUN
ejpam-5779	340	52	)	)	PUNCT
ejpam-5779	340	53	2	2	NUM
ejpam-5779	340	54	}	}	PUNCT
ejpam-5779	340	55	=	=	SYM
ejpam-5779	340	56	min	min	NOUN
ejpam-5779	340	57	{	{	PUNCT
ejpam-5779	340	58	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	340	59	q	q	NOUN
ejpam-5779	340	60	)	)	PUNCT
ejpam-5779	340	61	+	+	NUM
ejpam-5779	340	62	f̃inf(p	f̃inf(p	X
ejpam-5779	340	63	q	q	NOUN
ejpam-5779	340	64	)	)	PUNCT
ejpam-5779	340	65	2	2	NUM
ejpam-5779	340	66	,	,	PUNCT
ejpam-5779	340	67	f̃sup(q	f̃sup(q	NOUN
ejpam-5779	340	68	)	)	PUNCT
ejpam-5779	340	69	+	+	CCONJ
ejpam-5779	340	70	f̃inf(q	f̃inf(q	X
ejpam-5779	340	71	)	)	PUNCT
ejpam-5779	340	72	2	2	NUM
ejpam-5779	340	73	}	}	PUNCT
ejpam-5779	340	74	=	=	SYM
ejpam-5779	340	75	min{f̃m(pq	min{f̃m(pq	NOUN
ejpam-5779	340	76	)	)	PUNCT
ejpam-5779	340	77	,	,	PUNCT
ejpam-5779	340	78	f̃m(q	f̃m(q	NOUN
ejpam-5779	340	79	)	)	PUNCT
ejpam-5779	340	80	}	}	PUNCT
ejpam-5779	340	81	.	.	PUNCT
ejpam-5779	341	1	hence	hence	ADV
ejpam-5779	341	2	,	,	PUNCT
ejpam-5779	341	3	(	(	PUNCT
ejpam-5779	341	4	a	a	DET
ejpam-5779	341	5	,	,	PUNCT
ejpam-5779	341	6	f̃m	f̃m	NOUN
ejpam-5779	341	7	)	)	PUNCT
ejpam-5779	341	8	is	be	AUX
ejpam-5779	341	9	a	a	DET
ejpam-5779	341	10	4	4	NUM
ejpam-5779	341	11	-	-	PUNCT
ejpam-5779	341	12	fuzzy	fuzzy	ADJ
ejpam-5779	341	13	ideal	ideal	NOUN
ejpam-5779	341	14	of	of	ADP
ejpam-5779	341	15	a	a	PRON
ejpam-5779	341	16	,	,	PUNCT
ejpam-5779	341	17	that	that	ADV
ejpam-5779	341	18	is	is	ADV
ejpam-5779	341	19	,	,	PUNCT
ejpam-5779	341	20	(	(	PUNCT
ejpam-5779	341	21	a	a	PRON
ejpam-5779	341	22	,	,	PUNCT
ejpam-5779	341	23	f̃	f̃	PROPN
ejpam-5779	341	24	)	)	PUNCT
ejpam-5779	341	25	is	be	AUX
ejpam-5779	341	26	a	a	DET
ejpam-5779	341	27	mean	mean	ADJ
ejpam-5779	341	28	4	4	NUM
ejpam-5779	341	29	-	-	PUNCT
ejpam-5779	341	30	fuzzy	fuzzy	ADJ
ejpam-5779	341	31	ideal	ideal	NOUN
ejpam-5779	341	32	of	of	ADP
ejpam-5779	341	33	a.325	a.325	PROPN
ejpam-5779	341	34	theorem	theorem	VERB
ejpam-5779	341	35	17	17	NUM
ejpam-5779	341	36	.	.	PUNCT
ejpam-5779	342	1	if	if	SCONJ
ejpam-5779	342	2	(	(	PUNCT
ejpam-5779	342	3	a	a	PRON
ejpam-5779	342	4	,	,	PUNCT
ejpam-5779	342	5	f̃	f̃	PROPN
ejpam-5779	342	6	)	)	PUNCT
ejpam-5779	342	7	is	be	AUX
ejpam-5779	342	8	an	an	DET
ejpam-5779	342	9	interval	interval	NOUN
ejpam-5779	342	10	-	-	PUNCT
ejpam-5779	342	11	valued	value	VERB
ejpam-5779	342	12	fuzzy	fuzzy	ADJ
ejpam-5779	342	13	structure	structure	NOUN
ejpam-5779	342	14	over	over	ADP
ejpam-5779	342	15	a	a	PRON
ejpam-5779	342	16	in	in	ADP
ejpam-5779	342	17	which	which	PRON
ejpam-5779	342	18	(	(	PUNCT
ejpam-5779	342	19	a	a	DET
ejpam-5779	342	20	,	,	PUNCT
ejpam-5779	342	21	f̃sup	f̃sup	ADJ
ejpam-5779	342	22	)	)	PUNCT
ejpam-5779	342	23	is326	is326	NOUN
ejpam-5779	342	24	constant	constant	ADJ
ejpam-5779	342	25	and	and	CCONJ
ejpam-5779	342	26	(	(	PUNCT
ejpam-5779	342	27	a	a	PRON
ejpam-5779	342	28	,	,	PUNCT
ejpam-5779	342	29	f̃inf	f̃inf	ADJ
ejpam-5779	342	30	)	)	PUNCT
ejpam-5779	342	31	is	be	AUX
ejpam-5779	342	32	a	a	DET
ejpam-5779	342	33	4	4	NUM
ejpam-5779	342	34	-	-	PUNCT
ejpam-5779	342	35	fuzzy	fuzzy	ADJ
ejpam-5779	342	36	ideal	ideal	NOUN
ejpam-5779	342	37	of	of	ADP
ejpam-5779	342	38	a	a	PRON
ejpam-5779	342	39	,	,	PUNCT
ejpam-5779	342	40	then	then	ADV
ejpam-5779	342	41	(	(	PUNCT
ejpam-5779	342	42	a	a	PRON
ejpam-5779	342	43	,	,	PUNCT
ejpam-5779	342	44	f̃	f̃	PROPN
ejpam-5779	342	45	)	)	PUNCT
ejpam-5779	342	46	is	be	AUX
ejpam-5779	342	47	a	a	DET
ejpam-5779	342	48	mean	mean	ADJ
ejpam-5779	342	49	4	4	NUM
ejpam-5779	342	50	-	-	PUNCT
ejpam-5779	342	51	fuzzy	fuzzy	ADJ
ejpam-5779	342	52	ideal	ideal	NOUN
ejpam-5779	342	53	of	of	ADP
ejpam-5779	342	54	a.327	a.327	NOUN
ejpam-5779	342	55	proof	proof	NOUN
ejpam-5779	342	56	.	.	PUNCT
ejpam-5779	343	1	assume	assume	VERB
ejpam-5779	343	2	that	that	SCONJ
ejpam-5779	343	3	(	(	PUNCT
ejpam-5779	343	4	a	a	PRON
ejpam-5779	343	5	,	,	PUNCT
ejpam-5779	343	6	f̃	f̃	PROPN
ejpam-5779	343	7	)	)	PUNCT
ejpam-5779	343	8	is	be	AUX
ejpam-5779	343	9	an	an	DET
ejpam-5779	343	10	interval	interval	NOUN
ejpam-5779	343	11	-	-	PUNCT
ejpam-5779	343	12	valued	value	VERB
ejpam-5779	343	13	fuzzy	fuzzy	ADJ
ejpam-5779	343	14	structure	structure	NOUN
ejpam-5779	343	15	over	over	ADP
ejpam-5779	343	16	a	a	PRON
ejpam-5779	343	17	in	in	ADP
ejpam-5779	343	18	which328	which328	PROPN
ejpam-5779	343	19	(	(	PUNCT
ejpam-5779	343	20	a	a	DET
ejpam-5779	343	21	,	,	PUNCT
ejpam-5779	343	22	f̃sup	f̃sup	ADJ
ejpam-5779	343	23	)	)	PUNCT
ejpam-5779	343	24	is	be	AUX
ejpam-5779	343	25	constant	constant	ADJ
ejpam-5779	343	26	and	and	CCONJ
ejpam-5779	343	27	(	(	PUNCT
ejpam-5779	343	28	a	a	PRON
ejpam-5779	343	29	,	,	PUNCT
ejpam-5779	343	30	f̃inf	f̃inf	ADJ
ejpam-5779	343	31	)	)	PUNCT
ejpam-5779	343	32	is	be	AUX
ejpam-5779	343	33	a	a	DET
ejpam-5779	343	34	4	4	NUM
ejpam-5779	343	35	-	-	PUNCT
ejpam-5779	343	36	fuzzy	fuzzy	ADJ
ejpam-5779	343	37	ideal	ideal	NOUN
ejpam-5779	343	38	of	of	ADP
ejpam-5779	343	39	a.	a.	NOUN
ejpam-5779	343	40	let	let	VERB
ejpam-5779	343	41	p	p	PRON
ejpam-5779	343	42	,	,	PUNCT
ejpam-5779	343	43	q	q	PROPN
ejpam-5779	343	44	∈	∈	PROPN
ejpam-5779	343	45	a.	a.	NOUN
ejpam-5779	343	46	since	since	SCONJ
ejpam-5779	343	47	(	(	PUNCT
ejpam-5779	343	48	a	a	DET
ejpam-5779	343	49	,	,	PUNCT
ejpam-5779	343	50	f̃sup	f̃sup	ADJ
ejpam-5779	343	51	)	)	PUNCT
ejpam-5779	344	1	is329	is329	PROPN
ejpam-5779	344	2	constant	constant	ADJ
ejpam-5779	344	3	,	,	PUNCT
ejpam-5779	344	4	we	we	PRON
ejpam-5779	344	5	have	have	VERB
ejpam-5779	344	6	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	344	7	)	)	PUNCT
ejpam-5779	344	8	=	=	SYM
ejpam-5779	344	9	f̃sup(0	f̃sup(0	NOUN
ejpam-5779	344	10	)	)	PUNCT
ejpam-5779	344	11	for	for	ADP
ejpam-5779	344	12	all	all	DET
ejpam-5779	344	13	p	p	PROPN
ejpam-5779	344	14	∈	∈	PROPN
ejpam-5779	344	15	a.	a.	NOUN
ejpam-5779	344	16	since	since	SCONJ
ejpam-5779	344	17	(	(	PUNCT
ejpam-5779	344	18	a	a	PRON
ejpam-5779	344	19	,	,	PUNCT
ejpam-5779	344	20	f̃inf	f̃inf	ADJ
ejpam-5779	344	21	)	)	PUNCT
ejpam-5779	344	22	is	be	AUX
ejpam-5779	344	23	a	a	DET
ejpam-5779	344	24	4	4	NUM
ejpam-5779	344	25	-	-	PUNCT
ejpam-5779	344	26	fuzzy	fuzzy	ADJ
ejpam-5779	344	27	ideal	ideal	NOUN
ejpam-5779	344	28	of	of	ADP
ejpam-5779	344	29	a,330	a,330	NUM
ejpam-5779	344	30	we	we	PRON
ejpam-5779	344	31	have	have	VERB
ejpam-5779	344	32	f̃inf(p	f̃inf(p	NOUN
ejpam-5779	344	33	)	)	PUNCT
ejpam-5779	344	34	≤	≤	NOUN
ejpam-5779	345	1	max{f̃inf(p	max{f̃inf(p	PROPN
ejpam-5779	345	2	)	)	PUNCT
ejpam-5779	345	3	,	,	PUNCT
ejpam-5779	345	4	f̃inf(q	f̃inf(q	PROPN
ejpam-5779	345	5	)	)	PUNCT
ejpam-5779	345	6	}	}	PUNCT
ejpam-5779	345	7	.	.	PUNCT
ejpam-5779	346	1	thus,331	thus,331	PROPN
ejpam-5779	346	2	f̃m(p	f̃m(p	NUM
ejpam-5779	346	3	)	)	PUNCT
ejpam-5779	346	4	=	=	SYM
ejpam-5779	346	5	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	346	6	)	)	PUNCT
ejpam-5779	346	7	+	+	SYM
ejpam-5779	346	8	f̃inf(p	f̃inf(p	X
ejpam-5779	346	9	)	)	PUNCT
ejpam-5779	346	10	2	2	NUM
ejpam-5779	346	11	=	=	SYM
ejpam-5779	346	12	f̃sup(0	f̃sup(0	NOUN
ejpam-5779	346	13	)	)	PUNCT
ejpam-5779	346	14	+	+	SYM
ejpam-5779	346	15	f̃inf(p	f̃inf(p	X
ejpam-5779	346	16	)	)	PUNCT
ejpam-5779	346	17	2	2	NUM
ejpam-5779	346	18	=	=	SYM
ejpam-5779	346	19	f̃sup(0	f̃sup(0	NOUN
ejpam-5779	346	20	)	)	PUNCT
ejpam-5779	346	21	2	2	NUM
ejpam-5779	346	22	+	+	SYM
ejpam-5779	346	23	f̃inf(p	f̃inf(p	X
ejpam-5779	346	24	)	)	PUNCT
ejpam-5779	346	25	2	2	NUM
ejpam-5779	346	26	≤	≤	NOUN
ejpam-5779	346	27	f̃sup(0	f̃sup(0	NOUN
ejpam-5779	346	28	)	)	PUNCT
ejpam-5779	346	29	2	2	NUM
ejpam-5779	347	1	+	+	CCONJ
ejpam-5779	347	2	max	max	PROPN
ejpam-5779	347	3	{	{	PUNCT
ejpam-5779	347	4	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	347	5	q	q	NOUN
ejpam-5779	347	6	)	)	PUNCT
ejpam-5779	347	7	2	2	NUM
ejpam-5779	347	8	,	,	PUNCT
ejpam-5779	347	9	f̃inf(q	f̃inf(q	PROPN
ejpam-5779	347	10	)	)	PUNCT
ejpam-5779	347	11	2	2	NUM
ejpam-5779	347	12	}	}	PUNCT
ejpam-5779	347	13	=	=	SYM
ejpam-5779	347	14	max	max	PROPN
ejpam-5779	347	15	{	{	PUNCT
ejpam-5779	347	16	f̃sup(0	f̃sup(0	NOUN
ejpam-5779	347	17	)	)	PUNCT
ejpam-5779	347	18	2	2	NUM
ejpam-5779	347	19	+	+	NUM
ejpam-5779	347	20	f̃inf(p	f̃inf(p	X
ejpam-5779	347	21	q	q	NOUN
ejpam-5779	347	22	)	)	PUNCT
ejpam-5779	347	23	2	2	NUM
ejpam-5779	347	24	,	,	PUNCT
ejpam-5779	347	25	f̃sup(0	f̃sup(0	NOUN
ejpam-5779	347	26	)	)	PUNCT
ejpam-5779	347	27	2	2	NUM
ejpam-5779	348	1	+	+	CCONJ
ejpam-5779	348	2	f̃inf(q	f̃inf(q	X
ejpam-5779	348	3	)	)	PUNCT
ejpam-5779	348	4	2	2	NUM
ejpam-5779	348	5	}	}	PUNCT
ejpam-5779	348	6	=	=	SYM
ejpam-5779	348	7	max	max	X
ejpam-5779	348	8	{	{	PUNCT
ejpam-5779	348	9	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	348	10	q	q	NOUN
ejpam-5779	348	11	)	)	PUNCT
ejpam-5779	348	12	)	)	PUNCT
ejpam-5779	349	1	+	+	CCONJ
ejpam-5779	349	2	f̃inf(p	f̃inf(p	X
ejpam-5779	349	3	q	q	NOUN
ejpam-5779	349	4	)	)	PUNCT
ejpam-5779	349	5	2	2	NUM
ejpam-5779	349	6	,	,	PUNCT
ejpam-5779	349	7	f̃sup(q	f̃sup(q	NOUN
ejpam-5779	349	8	)	)	PUNCT
ejpam-5779	349	9	+	+	CCONJ
ejpam-5779	349	10	f̃inf(q	f̃inf(q	X
ejpam-5779	349	11	)	)	PUNCT
ejpam-5779	349	12	2	2	NUM
ejpam-5779	349	13	}	}	PUNCT
ejpam-5779	349	14	=	=	SYM
ejpam-5779	349	15	max{f̃m(pq	max{f̃m(pq	NOUN
ejpam-5779	349	16	)	)	PUNCT
ejpam-5779	349	17	,	,	PUNCT
ejpam-5779	349	18	f̃m(q	f̃m(q	NOUN
ejpam-5779	349	19	)	)	PUNCT
ejpam-5779	349	20	}	}	PUNCT
ejpam-5779	349	21	.	.	PUNCT
ejpam-5779	350	1	hence	hence	ADV
ejpam-5779	350	2	,	,	PUNCT
ejpam-5779	350	3	(	(	PUNCT
ejpam-5779	350	4	a	a	DET
ejpam-5779	350	5	,	,	PUNCT
ejpam-5779	350	6	f̃m	f̃m	NOUN
ejpam-5779	350	7	)	)	PUNCT
ejpam-5779	350	8	is	be	AUX
ejpam-5779	350	9	a	a	DET
ejpam-5779	350	10	4	4	NUM
ejpam-5779	350	11	-	-	PUNCT
ejpam-5779	350	12	fuzzy	fuzzy	ADJ
ejpam-5779	350	13	ideal	ideal	NOUN
ejpam-5779	350	14	of	of	ADP
ejpam-5779	350	15	a	a	PRON
ejpam-5779	350	16	,	,	PUNCT
ejpam-5779	350	17	that	that	ADV
ejpam-5779	350	18	is	is	ADV
ejpam-5779	350	19	,	,	PUNCT
ejpam-5779	350	20	(	(	PUNCT
ejpam-5779	350	21	a	a	PRON
ejpam-5779	350	22	,	,	PUNCT
ejpam-5779	350	23	f̃	f̃	PROPN
ejpam-5779	350	24	)	)	PUNCT
ejpam-5779	350	25	is	be	AUX
ejpam-5779	350	26	a	a	DET
ejpam-5779	350	27	mean	mean	ADJ
ejpam-5779	350	28	4	4	NUM
ejpam-5779	350	29	-	-	PUNCT
ejpam-5779	350	30	fuzzy	fuzzy	ADJ
ejpam-5779	350	31	ideal	ideal	NOUN
ejpam-5779	350	32	of	of	ADP
ejpam-5779	350	33	a.332	a.332	PROPN
ejpam-5779	350	34	n.	n.	PROPN
ejpam-5779	350	35	rajesh	rajesh	PROPN
ejpam-5779	350	36	,	,	PUNCT
ejpam-5779	350	37	t.	t.	PROPN
ejpam-5779	350	38	oner	oner	NOUN
ejpam-5779	350	39	,	,	PUNCT
ejpam-5779	350	40	a.	a.	NOUN
ejpam-5779	350	41	iampan	iampan	PROPN
ejpam-5779	350	42	,	,	PUNCT
ejpam-5779	350	43	i.	i.	PROPN
ejpam-5779	350	44	senturk	senturk	PROPN
ejpam-5779	350	45	/	/	SYM
ejpam-5779	350	46	eur	eur	PROPN
ejpam-5779	350	47	.	.	PUNCT
ejpam-5779	351	1	j.	j.	PROPN
ejpam-5779	351	2	pure	pure	PROPN
ejpam-5779	351	3	appl	appl	PROPN
ejpam-5779	351	4	.	.	PROPN
ejpam-5779	351	5	math	math	PROPN
ejpam-5779	351	6	,	,	PUNCT
ejpam-5779	351	7	18	18	NUM
ejpam-5779	351	8	(	(	PUNCT
ejpam-5779	351	9	1	1	NUM
ejpam-5779	351	10	)	)	PUNCT
ejpam-5779	351	11	(	(	PUNCT
ejpam-5779	351	12	2025	2025	NUM
ejpam-5779	351	13	)	)	PUNCT
ejpam-5779	351	14	,	,	PUNCT
ejpam-5779	351	15	5779	5779	NUM
ejpam-5779	351	16	17	17	NUM
ejpam-5779	351	17	of	of	ADP
ejpam-5779	351	18	18	18	NUM
ejpam-5779	351	19	theorem	theorem	ADJ
ejpam-5779	351	20	18	18	NUM
ejpam-5779	351	21	.	.	PUNCT
ejpam-5779	352	1	if	if	SCONJ
ejpam-5779	352	2	(	(	PUNCT
ejpam-5779	352	3	a	a	PRON
ejpam-5779	352	4	,	,	PUNCT
ejpam-5779	352	5	f̃	f̃	PROPN
ejpam-5779	352	6	)	)	PUNCT
ejpam-5779	352	7	is	be	AUX
ejpam-5779	352	8	an	an	DET
ejpam-5779	352	9	interval	interval	NOUN
ejpam-5779	352	10	-	-	PUNCT
ejpam-5779	352	11	valued	value	VERB
ejpam-5779	352	12	fuzzy	fuzzy	ADJ
ejpam-5779	352	13	structure	structure	NOUN
ejpam-5779	352	14	over	over	ADP
ejpam-5779	352	15	a	a	PRON
ejpam-5779	352	16	in	in	ADP
ejpam-5779	352	17	which	which	PRON
ejpam-5779	352	18	(	(	PUNCT
ejpam-5779	352	19	a	a	DET
ejpam-5779	352	20	,	,	PUNCT
ejpam-5779	352	21	f̃sup	f̃sup	ADJ
ejpam-5779	352	22	)	)	PUNCT
ejpam-5779	353	1	is333	is333	NOUN
ejpam-5779	353	2	constant	constant	ADJ
ejpam-5779	353	3	and	and	CCONJ
ejpam-5779	353	4	(	(	PUNCT
ejpam-5779	353	5	a	a	PRON
ejpam-5779	353	6	,	,	PUNCT
ejpam-5779	353	7	f̃inf	f̃inf	ADJ
ejpam-5779	353	8	)	)	PUNCT
ejpam-5779	353	9	is	be	AUX
ejpam-5779	353	10	a	a	DET
ejpam-5779	353	11	1	1	NUM
ejpam-5779	353	12	-	-	PUNCT
ejpam-5779	353	13	fuzzy	fuzzy	ADJ
ejpam-5779	353	14	ideal	ideal	NOUN
ejpam-5779	353	15	of	of	ADP
ejpam-5779	353	16	a	a	PRON
ejpam-5779	353	17	,	,	PUNCT
ejpam-5779	353	18	then	then	ADV
ejpam-5779	353	19	(	(	PUNCT
ejpam-5779	353	20	a	a	PRON
ejpam-5779	353	21	,	,	PUNCT
ejpam-5779	353	22	f̃	f̃	PROPN
ejpam-5779	353	23	)	)	PUNCT
ejpam-5779	353	24	is	be	AUX
ejpam-5779	353	25	a	a	DET
ejpam-5779	353	26	mean	mean	ADJ
ejpam-5779	353	27	1	1	NUM
ejpam-5779	353	28	-	-	PUNCT
ejpam-5779	353	29	fuzzy	fuzzy	ADJ
ejpam-5779	353	30	ideal	ideal	NOUN
ejpam-5779	353	31	of	of	ADP
ejpam-5779	353	32	a.334	a.334	NOUN
ejpam-5779	353	33	proof	proof	NOUN
ejpam-5779	353	34	.	.	PUNCT
ejpam-5779	354	1	assume	assume	VERB
ejpam-5779	354	2	that	that	SCONJ
ejpam-5779	354	3	(	(	PUNCT
ejpam-5779	354	4	a	a	PRON
ejpam-5779	354	5	,	,	PUNCT
ejpam-5779	354	6	f̃	f̃	PROPN
ejpam-5779	354	7	)	)	PUNCT
ejpam-5779	354	8	is	be	AUX
ejpam-5779	354	9	an	an	DET
ejpam-5779	354	10	interval	interval	NOUN
ejpam-5779	354	11	-	-	PUNCT
ejpam-5779	354	12	valued	value	VERB
ejpam-5779	354	13	fuzzy	fuzzy	ADJ
ejpam-5779	354	14	structure	structure	NOUN
ejpam-5779	354	15	over	over	ADP
ejpam-5779	354	16	a	a	PRON
ejpam-5779	354	17	in	in	ADP
ejpam-5779	354	18	which335	which335	PROPN
ejpam-5779	354	19	(	(	PUNCT
ejpam-5779	354	20	a	a	DET
ejpam-5779	354	21	,	,	PUNCT
ejpam-5779	354	22	f̃sup	f̃sup	ADJ
ejpam-5779	354	23	)	)	PUNCT
ejpam-5779	354	24	is	be	AUX
ejpam-5779	354	25	constant	constant	ADJ
ejpam-5779	354	26	and	and	CCONJ
ejpam-5779	354	27	(	(	PUNCT
ejpam-5779	354	28	a	a	PRON
ejpam-5779	354	29	,	,	PUNCT
ejpam-5779	354	30	f̃inf	f̃inf	ADJ
ejpam-5779	354	31	)	)	PUNCT
ejpam-5779	354	32	is	be	AUX
ejpam-5779	354	33	a	a	DET
ejpam-5779	354	34	1	1	NUM
ejpam-5779	354	35	-	-	PUNCT
ejpam-5779	354	36	fuzzy	fuzzy	ADJ
ejpam-5779	354	37	ideal	ideal	NOUN
ejpam-5779	354	38	of	of	ADP
ejpam-5779	354	39	a.	a.	NOUN
ejpam-5779	354	40	let	let	VERB
ejpam-5779	354	41	p	p	PRON
ejpam-5779	354	42	,	,	PUNCT
ejpam-5779	354	43	q	q	PROPN
ejpam-5779	354	44	∈	∈	PROPN
ejpam-5779	354	45	a.	a.	NOUN
ejpam-5779	354	46	since	since	SCONJ
ejpam-5779	354	47	(	(	PUNCT
ejpam-5779	354	48	a	a	DET
ejpam-5779	354	49	,	,	PUNCT
ejpam-5779	354	50	f̃sup	f̃sup	ADJ
ejpam-5779	354	51	)	)	PUNCT
ejpam-5779	355	1	is336	is336	PROPN
ejpam-5779	355	2	constant	constant	ADJ
ejpam-5779	355	3	,	,	PUNCT
ejpam-5779	355	4	we	we	PRON
ejpam-5779	355	5	have	have	VERB
ejpam-5779	355	6	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	355	7	)	)	PUNCT
ejpam-5779	356	1	=	=	SYM
ejpam-5779	356	2	f̃sup(0	f̃sup(0	NOUN
ejpam-5779	356	3	)	)	PUNCT
ejpam-5779	356	4	for	for	ADP
ejpam-5779	356	5	all	all	DET
ejpam-5779	356	6	p	p	PROPN
ejpam-5779	356	7	∈	∈	PROPN
ejpam-5779	356	8	a.	a.	NOUN
ejpam-5779	356	9	since	since	SCONJ
ejpam-5779	356	10	(	(	PUNCT
ejpam-5779	356	11	a	a	PRON
ejpam-5779	356	12	,	,	PUNCT
ejpam-5779	356	13	f̃inf	f̃inf	ADJ
ejpam-5779	356	14	)	)	PUNCT
ejpam-5779	356	15	is	be	AUX
ejpam-5779	356	16	a	a	DET
ejpam-5779	356	17	1	1	NUM
ejpam-5779	356	18	-	-	PUNCT
ejpam-5779	356	19	fuzzy	fuzzy	ADJ
ejpam-5779	356	20	ideal	ideal	NOUN
ejpam-5779	356	21	of	of	ADP
ejpam-5779	356	22	a,337	a,337	NOUN
ejpam-5779	356	23	we	we	PRON
ejpam-5779	356	24	obtain	obtain	VERB
ejpam-5779	356	25	f̃inf(p	f̃inf(p	X
ejpam-5779	356	26	)	)	PUNCT
ejpam-5779	356	27	≥	≥	PROPN
ejpam-5779	356	28	min{f̃inf(p	min{f̃inf(p	PROPN
ejpam-5779	356	29	)	)	PUNCT
ejpam-5779	356	30	,	,	PUNCT
ejpam-5779	356	31	f̃inf(q	f̃inf(q	PROPN
ejpam-5779	356	32	)	)	PUNCT
ejpam-5779	356	33	}	}	PUNCT
ejpam-5779	356	34	.	.	PUNCT
ejpam-5779	357	1	thus,338	thus,338	NOUN
ejpam-5779	357	2	f̃m(p	f̃m(p	NUM
ejpam-5779	357	3	)	)	PUNCT
ejpam-5779	357	4	=	=	SYM
ejpam-5779	357	5	f̃sup(p	f̃sup(p	NOUN
ejpam-5779	357	6	)	)	PUNCT
ejpam-5779	358	1	+	+	SYM
ejpam-5779	358	2	f̃inf(p	f̃inf(p	X
ejpam-5779	358	3	)	)	PUNCT
ejpam-5779	358	4	2	2	NUM
ejpam-5779	358	5	=	=	SYM
ejpam-5779	358	6	f̃sup(0	f̃sup(0	NOUN
ejpam-5779	358	7	)	)	PUNCT
ejpam-5779	358	8	+	+	SYM
ejpam-5779	358	9	f̃inf(p	f̃inf(p	X
ejpam-5779	358	10	)	)	PUNCT
ejpam-5779	358	11	2	2	NUM
ejpam-5779	358	12	=	=	SYM
ejpam-5779	358	13	f̃sup(0	f̃sup(0	NOUN
ejpam-5779	358	14	)	)	PUNCT
ejpam-5779	358	15	2	2	NUM
ejpam-5779	358	16	+	+	SYM
ejpam-5779	358	17	f̃inf(p	f̃inf(p	X
ejpam-5779	358	18	)	)	PUNCT
ejpam-5779	358	19	2	2	NUM
ejpam-5779	358	20	≥	≥	NOUN
ejpam-5779	358	21	f̃sup(0	f̃sup(0	NOUN
ejpam-5779	358	22	)	)	PUNCT
ejpam-5779	358	23	2	2	NUM
ejpam-5779	358	24	+	+	CCONJ
ejpam-5779	358	25	min	min	NOUN
ejpam-5779	358	26	{	{	PUNCT
ejpam-5779	358	27	f̃inf(p	f̃inf(p	X
ejpam-5779	358	28	q	q	SYM
ejpam-5779	358	29	)	)	PUNCT
ejpam-5779	358	30	2	2	NUM
ejpam-5779	358	31	,	,	PUNCT
ejpam-5779	358	32	f̃inf(q	f̃inf(q	PROPN
ejpam-5779	358	33	)	)	PUNCT
ejpam-5779	358	34	2	2	NUM
ejpam-5779	358	35	}	}	PUNCT
ejpam-5779	358	36	=	=	SYM
ejpam-5779	358	37	min	min	NOUN
ejpam-5779	358	38	{	{	PUNCT
ejpam-5779	358	39	f̃sup(0	f̃sup(0	NOUN
ejpam-5779	358	40	)	)	PUNCT
ejpam-5779	358	41	2	2	NUM
ejpam-5779	358	42	+	+	NUM
ejpam-5779	358	43	f̃sup(p	f̃sup(p	PRON
ejpam-5779	358	44	q	q	NOUN
ejpam-5779	358	45	)	)	PUNCT
ejpam-5779	358	46	2	2	NUM
ejpam-5779	358	47	,	,	PUNCT
ejpam-5779	358	48	f̃sup(0	f̃sup(0	NOUN
ejpam-5779	358	49	)	)	PUNCT
ejpam-5779	358	50	2	2	NUM
ejpam-5779	358	51	,	,	PUNCT
ejpam-5779	358	52	f̃sup(q	f̃sup(q	NOUN
ejpam-5779	358	53	)	)	PUNCT
ejpam-5779	358	54	2	2	NUM
ejpam-5779	358	55	}	}	PUNCT
ejpam-5779	358	56	=	=	SYM
ejpam-5779	358	57	min	min	NOUN
ejpam-5779	358	58	{	{	PUNCT
ejpam-5779	358	59	f̃sup(0	f̃sup(0	NOUN
ejpam-5779	358	60	)	)	PUNCT
ejpam-5779	358	61	+	+	NUM
ejpam-5779	358	62	f̃sup(p	f̃sup(p	PRON
ejpam-5779	358	63	q	q	NOUN
ejpam-5779	358	64	)	)	PUNCT
ejpam-5779	358	65	2	2	NUM
ejpam-5779	358	66	,	,	PUNCT
ejpam-5779	358	67	f̃sup(0	f̃sup(0	NOUN
ejpam-5779	358	68	)	)	PUNCT
ejpam-5779	358	69	+	+	NUM
ejpam-5779	358	70	f̃sup(q	f̃sup(q	NOUN
ejpam-5779	358	71	)	)	PUNCT
ejpam-5779	358	72	2	2	NUM
ejpam-5779	358	73	}	}	PUNCT
ejpam-5779	358	74	=	=	SYM
ejpam-5779	358	75	min{f̃m(pq	min{f̃m(pq	NOUN
ejpam-5779	358	76	)	)	PUNCT
ejpam-5779	358	77	,	,	PUNCT
ejpam-5779	358	78	f̃m(q	f̃m(q	NOUN
ejpam-5779	358	79	)	)	PUNCT
ejpam-5779	358	80	}	}	PUNCT
ejpam-5779	358	81	.	.	PUNCT
ejpam-5779	359	1	hence	hence	ADV
ejpam-5779	359	2	,	,	PUNCT
ejpam-5779	359	3	(	(	PUNCT
ejpam-5779	359	4	a	a	DET
ejpam-5779	359	5	,	,	PUNCT
ejpam-5779	359	6	f̃m	f̃m	NOUN
ejpam-5779	359	7	)	)	PUNCT
ejpam-5779	359	8	is	be	AUX
ejpam-5779	359	9	a	a	DET
ejpam-5779	359	10	1	1	NUM
ejpam-5779	359	11	-	-	PUNCT
ejpam-5779	359	12	fuzzy	fuzzy	ADJ
ejpam-5779	359	13	ideal	ideal	NOUN
ejpam-5779	359	14	of	of	ADP
ejpam-5779	359	15	a	a	PRON
ejpam-5779	359	16	,	,	PUNCT
ejpam-5779	359	17	that	that	ADV
ejpam-5779	359	18	is	is	ADV
ejpam-5779	359	19	,	,	PUNCT
ejpam-5779	359	20	(	(	PUNCT
ejpam-5779	359	21	a	a	PRON
ejpam-5779	359	22	,	,	PUNCT
ejpam-5779	359	23	f̃	f̃	PROPN
ejpam-5779	359	24	)	)	PUNCT
ejpam-5779	359	25	is	be	AUX
ejpam-5779	359	26	a	a	DET
ejpam-5779	359	27	mean	mean	ADJ
ejpam-5779	359	28	1	1	NUM
ejpam-5779	359	29	-	-	PUNCT
ejpam-5779	359	30	fuzzy	fuzzy	ADJ
ejpam-5779	359	31	ideal	ideal	NOUN
ejpam-5779	359	32	of	of	ADP
ejpam-5779	359	33	a.339	a.339	NUM
ejpam-5779	359	34	5	5	NUM
ejpam-5779	359	35	.	.	PUNCT
ejpam-5779	359	36	conclusion340	conclusion340	NOUN
ejpam-5779	360	1	this	this	DET
ejpam-5779	360	2	study	study	NOUN
ejpam-5779	360	3	extends	extend	VERB
ejpam-5779	360	4	the	the	DET
ejpam-5779	360	5	theoretical	theoretical	ADJ
ejpam-5779	360	6	foundation	foundation	NOUN
ejpam-5779	360	7	of	of	ADP
ejpam-5779	360	8	sheffer	sheffer	PROPN
ejpam-5779	360	9	stroke	stroke	PROPN
ejpam-5779	360	10	hilbert	hilbert	PROPN
ejpam-5779	360	11	algebras	algebras	PROPN
ejpam-5779	360	12	by341	by341	PROPN
ejpam-5779	360	13	introducing	introduce	VERB
ejpam-5779	360	14	and	and	CCONJ
ejpam-5779	360	15	analyzing	analyze	VERB
ejpam-5779	360	16	the	the	DET
ejpam-5779	360	17	notions	notion	NOUN
ejpam-5779	360	18	of	of	ADP
ejpam-5779	360	19	length	length	NOUN
ejpam-5779	360	20	fuzzy	fuzzy	ADJ
ejpam-5779	360	21	ideals	ideal	NOUN
ejpam-5779	360	22	and	and	CCONJ
ejpam-5779	360	23	mean	mean	VERB
ejpam-5779	360	24	fuzzy	fuzzy	ADJ
ejpam-5779	360	25	ideals	ideal	NOUN
ejpam-5779	360	26	within342	within342	PROPN
ejpam-5779	360	27	an	an	DET
ejpam-5779	360	28	interval	interval	NOUN
ejpam-5779	360	29	-	-	PUNCT
ejpam-5779	360	30	valued	value	VERB
ejpam-5779	360	31	fuzzy	fuzzy	ADJ
ejpam-5779	360	32	structure	structure	NOUN
ejpam-5779	360	33	.	.	PUNCT
ejpam-5779	361	1	by	by	ADP
ejpam-5779	361	2	defining	define	VERB
ejpam-5779	361	3	these	these	DET
ejpam-5779	361	4	concepts	concept	NOUN
ejpam-5779	361	5	,	,	PUNCT
ejpam-5779	361	6	the	the	DET
ejpam-5779	361	7	research	research	NOUN
ejpam-5779	361	8	provides	provide	VERB
ejpam-5779	361	9	a343	a343	PRON
ejpam-5779	361	10	more	more	ADV
ejpam-5779	361	11	nuanced	nuanced	ADJ
ejpam-5779	361	12	understanding	understanding	NOUN
ejpam-5779	361	13	of	of	ADP
ejpam-5779	361	14	fuzzy	fuzzy	ADJ
ejpam-5779	361	15	logic	logic	NOUN
ejpam-5779	361	16	applications	application	NOUN
ejpam-5779	361	17	in	in	ADP
ejpam-5779	361	18	algebraic	algebraic	ADJ
ejpam-5779	361	19	structures	structure	NOUN
ejpam-5779	361	20	,	,	PUNCT
ejpam-5779	361	21	empha-344	empha-344	NOUN
ejpam-5779	361	22	sizing	size	VERB
ejpam-5779	361	23	the	the	DET
ejpam-5779	361	24	relationships	relationship	NOUN
ejpam-5779	361	25	between	between	ADP
ejpam-5779	361	26	fuzzy	fuzzy	ADJ
ejpam-5779	361	27	ideals	ideal	NOUN
ejpam-5779	361	28	and	and	CCONJ
ejpam-5779	361	29	traditional	traditional	ADJ
ejpam-5779	361	30	ideals	ideal	NOUN
ejpam-5779	361	31	.	.	PUNCT
ejpam-5779	362	1	the	the	DET
ejpam-5779	362	2	characterizations345	characterizations345	PROPN
ejpam-5779	362	3	and	and	CCONJ
ejpam-5779	362	4	properties	property	NOUN
ejpam-5779	362	5	of	of	ADP
ejpam-5779	362	6	length	length	NOUN
ejpam-5779	362	7	fuzzy	fuzzy	ADJ
ejpam-5779	362	8	ideals	ideal	NOUN
ejpam-5779	362	9	and	and	CCONJ
ejpam-5779	362	10	mean	mean	VERB
ejpam-5779	362	11	fuzzy	fuzzy	ADJ
ejpam-5779	362	12	ideals	ideal	NOUN
ejpam-5779	362	13	demonstrate	demonstrate	VERB
ejpam-5779	362	14	their	their	PRON
ejpam-5779	362	15	alignment346	alignment346	PRON
ejpam-5779	362	16	with	with	ADP
ejpam-5779	362	17	upper	upper	ADJ
ejpam-5779	362	18	and	and	CCONJ
ejpam-5779	362	19	lower	low	ADJ
ejpam-5779	362	20	level	level	NOUN
ejpam-5779	362	21	subsets	subset	NOUN
ejpam-5779	362	22	,	,	PUNCT
ejpam-5779	362	23	offering	offer	VERB
ejpam-5779	362	24	a	a	DET
ejpam-5779	362	25	framework	framework	NOUN
ejpam-5779	362	26	to	to	PART
ejpam-5779	362	27	explore	explore	VERB
ejpam-5779	362	28	the	the	DET
ejpam-5779	362	29	gradations	gradation	NOUN
ejpam-5779	362	30	of347	of347	PROPN
ejpam-5779	362	31	membership	membership	NOUN
ejpam-5779	362	32	functions	function	NOUN
ejpam-5779	362	33	.	.	PUNCT
ejpam-5779	363	1	furthermore	furthermore	ADV
ejpam-5779	363	2	,	,	PUNCT
ejpam-5779	363	3	the	the	DET
ejpam-5779	363	4	findings	finding	NOUN
ejpam-5779	363	5	highlight	highlight	VERB
ejpam-5779	363	6	the	the	DET
ejpam-5779	363	7	potential	potential	NOUN
ejpam-5779	363	8	of	of	ADP
ejpam-5779	363	9	these	these	DET
ejpam-5779	363	10	fuzzy348	fuzzy348	PROPN
ejpam-5779	363	11	constructs	construct	NOUN
ejpam-5779	363	12	in	in	ADP
ejpam-5779	363	13	bridging	bridge	VERB
ejpam-5779	363	14	algebraic	algebraic	ADJ
ejpam-5779	363	15	theory	theory	NOUN
ejpam-5779	363	16	with	with	ADP
ejpam-5779	363	17	practical	practical	ADJ
ejpam-5779	363	18	applications	application	NOUN
ejpam-5779	363	19	in	in	ADP
ejpam-5779	363	20	logic	logic	NOUN
ejpam-5779	363	21	,	,	PUNCT
ejpam-5779	363	22	computer	computer	NOUN
ejpam-5779	363	23	sci-349	sci-349	NOUN
ejpam-5779	363	24	ence	ence	NOUN
ejpam-5779	363	25	,	,	PUNCT
ejpam-5779	363	26	and	and	CCONJ
ejpam-5779	363	27	uncertainty	uncertainty	NOUN
ejpam-5779	363	28	modeling	modeling	NOUN
ejpam-5779	363	29	.	.	PUNCT
ejpam-5779	364	1	future	future	ADJ
ejpam-5779	364	2	studies	study	NOUN
ejpam-5779	364	3	could	could	AUX
ejpam-5779	364	4	investigate	investigate	VERB
ejpam-5779	364	5	the	the	DET
ejpam-5779	364	6	applicability	applicability	NOUN
ejpam-5779	364	7	of	of	ADP
ejpam-5779	364	8	these350	these350	PROPN
ejpam-5779	364	9	ideas	idea	NOUN
ejpam-5779	364	10	in	in	ADP
ejpam-5779	364	11	more	more	ADV
ejpam-5779	364	12	complex	complex	ADJ
ejpam-5779	364	13	fuzzy	fuzzy	ADJ
ejpam-5779	364	14	systems	system	NOUN
ejpam-5779	364	15	or	or	CCONJ
ejpam-5779	364	16	extend	extend	VERB
ejpam-5779	364	17	the	the	DET
ejpam-5779	364	18	analysis	analysis	NOUN
ejpam-5779	364	19	to	to	ADP
ejpam-5779	364	20	other	other	ADJ
ejpam-5779	364	21	algebraic	algebraic	ADJ
ejpam-5779	364	22	frameworks,351	frameworks,351	NOUN
ejpam-5779	364	23	broadening	broaden	VERB
ejpam-5779	364	24	the	the	DET
ejpam-5779	364	25	impact	impact	NOUN
ejpam-5779	364	26	and	and	CCONJ
ejpam-5779	364	27	utility	utility	NOUN
ejpam-5779	364	28	of	of	ADP
ejpam-5779	364	29	these	these	DET
ejpam-5779	364	30	innovative	innovative	ADJ
ejpam-5779	364	31	concepts.352	concepts.352	NOUN
ejpam-5779	364	32	acknowledgements353	acknowledgements353	PROPN
ejpam-5779	364	33	this	this	DET
ejpam-5779	364	34	research	research	NOUN
ejpam-5779	364	35	was	be	AUX
ejpam-5779	364	36	supported	support	VERB
ejpam-5779	364	37	by	by	ADP
ejpam-5779	364	38	university	university	NOUN
ejpam-5779	364	39	of	of	ADP
ejpam-5779	364	40	phayao	phayao	NOUN
ejpam-5779	364	41	and	and	CCONJ
ejpam-5779	364	42	thailand	thailand	PROPN
ejpam-5779	364	43	science	science	PROPN
ejpam-5779	364	44	research354	research354	PROPN
ejpam-5779	364	45	and	and	CCONJ
ejpam-5779	364	46	innovation	innovation	NOUN
ejpam-5779	364	47	fund	fund	NOUN
ejpam-5779	364	48	(	(	PUNCT
ejpam-5779	364	49	fundamental	fundamental	ADJ
ejpam-5779	364	50	fund	fund	NOUN
ejpam-5779	364	51	2025	2025	NUM
ejpam-5779	364	52	,	,	PUNCT
ejpam-5779	364	53	grant	grant	VERB
ejpam-5779	364	54	no	no	NOUN
ejpam-5779	364	55	.	.	PUNCT
ejpam-5779	365	1	5027/2567).355	5027/2567).355	NUM
ejpam-5779	365	2	n.	n.	PROPN
ejpam-5779	365	3	rajesh	rajesh	PROPN
ejpam-5779	365	4	,	,	PUNCT
ejpam-5779	365	5	t.	t.	PROPN
ejpam-5779	365	6	oner	oner	NOUN
ejpam-5779	365	7	,	,	PUNCT
ejpam-5779	365	8	a.	a.	NOUN
ejpam-5779	365	9	iampan	iampan	PROPN
ejpam-5779	365	10	,	,	PUNCT
ejpam-5779	365	11	i.	i.	PROPN
ejpam-5779	365	12	senturk	senturk	PROPN
ejpam-5779	365	13	/	/	SYM
ejpam-5779	365	14	eur	eur	PROPN
ejpam-5779	365	15	.	.	PUNCT
ejpam-5779	366	1	j.	j.	PROPN
ejpam-5779	366	2	pure	pure	PROPN
ejpam-5779	366	3	appl	appl	PROPN
ejpam-5779	366	4	.	.	PROPN
ejpam-5779	366	5	math	math	PROPN
ejpam-5779	366	6	,	,	PUNCT
ejpam-5779	366	7	18	18	NUM
ejpam-5779	366	8	(	(	PUNCT
ejpam-5779	366	9	1	1	NUM
ejpam-5779	366	10	)	)	PUNCT
ejpam-5779	366	11	(	(	PUNCT
ejpam-5779	366	12	2025	2025	NUM
ejpam-5779	366	13	)	)	PUNCT
ejpam-5779	366	14	,	,	PUNCT
ejpam-5779	366	15	5779	5779	NUM
ejpam-5779	366	16	18	18	NUM
ejpam-5779	366	17	of	of	ADP
ejpam-5779	366	18	18	18	NUM
ejpam-5779	366	19	references356	references356	NOUN
ejpam-5779	366	20	[	[	X
ejpam-5779	366	21	1	1	NUM
ejpam-5779	366	22	]	]	PUNCT
ejpam-5779	366	23	i.	i.	NOUN
ejpam-5779	366	24	chajad	chajad	PROPN
ejpam-5779	366	25	.	.	PUNCT
ejpam-5779	367	1	sheffer	sheffer	PROPN
ejpam-5779	367	2	operation	operation	NOUN
ejpam-5779	367	3	in	in	ADP
ejpam-5779	367	4	ortholattices	ortholattice	NOUN
ejpam-5779	367	5	.	.	PUNCT
ejpam-5779	368	1	acta	acta	PROPN
ejpam-5779	368	2	universitatis	universitatis	PROPN
ejpam-5779	368	3	palackianae	palackianae	PROPN
ejpam-5779	368	4	olomu-357	olomu-357	VERB
ejpam-5779	368	5	censis	censis	NOUN
ejpam-5779	368	6	.	.	PUNCT
ejpam-5779	369	1	facultas	facultas	PROPN
ejpam-5779	369	2	rerum	rerum	PROPN
ejpam-5779	369	3	naturalium	naturalium	PROPN
ejpam-5779	369	4	.	.	PUNCT
ejpam-5779	370	1	mathematica	mathematica	PROPN
ejpam-5779	370	2	,	,	PUNCT
ejpam-5779	370	3	44(1):19–23	44(1):19–23	NUM
ejpam-5779	370	4	,	,	PUNCT
ejpam-5779	370	5	2005.358	2005.358	NUM
ejpam-5779	370	6	[	[	X
ejpam-5779	370	7	2	2	NUM
ejpam-5779	370	8	]	]	X
ejpam-5779	370	9	d.	d.	PROPN
ejpam-5779	370	10	dubois	dubois	PROPN
ejpam-5779	370	11	and	and	CCONJ
ejpam-5779	370	12	h.	h.	PROPN
ejpam-5779	370	13	prade	prade	PROPN
ejpam-5779	370	14	.	.	PUNCT
ejpam-5779	371	1	fuzzy	fuzzy	ADJ
ejpam-5779	371	2	sets	set	NOUN
ejpam-5779	371	3	and	and	CCONJ
ejpam-5779	371	4	systems	system	NOUN
ejpam-5779	371	5	:	:	PUNCT
ejpam-5779	371	6	theory	theory	NOUN
ejpam-5779	371	7	and	and	CCONJ
ejpam-5779	371	8	applications	application	NOUN
ejpam-5779	371	9	.	.	PUNCT
ejpam-5779	372	1	academic359	academic359	VERB
ejpam-5779	372	2	press	press	NOUN
ejpam-5779	372	3	,	,	PUNCT
ejpam-5779	372	4	1980.360	1980.360	NUM
ejpam-5779	372	5	[	[	X
ejpam-5779	372	6	3	3	NUM
ejpam-5779	372	7	]	]	PUNCT
ejpam-5779	372	8	w.	w.	PROPN
ejpam-5779	372	9	hodges	hodges	PROPN
ejpam-5779	372	10	.	.	PUNCT
ejpam-5779	373	1	model	model	PROPN
ejpam-5779	373	2	theory	theory	PROPN
ejpam-5779	373	3	:	:	PUNCT
ejpam-5779	373	4	an	an	DET
ejpam-5779	373	5	introduction	introduction	NOUN
ejpam-5779	373	6	.	.	PUNCT
ejpam-5779	374	1	cambridge	cambridge	PROPN
ejpam-5779	374	2	university	university	PROPN
ejpam-5779	374	3	press	press	NOUN
ejpam-5779	374	4	,	,	PUNCT
ejpam-5779	374	5	2003.361	2003.361	NUM
ejpam-5779	375	1	[	[	X
ejpam-5779	375	2	4	4	NUM
ejpam-5779	375	3	]	]	X
ejpam-5779	375	4	v.	v.	CCONJ
ejpam-5779	375	5	kozarkiewicz	kozarkiewicz	PROPN
ejpam-5779	375	6	and	and	CCONJ
ejpam-5779	375	7	a.	a.	PROPN
ejpam-5779	375	8	grabowski	grabowski	PROPN
ejpam-5779	375	9	.	.	PUNCT
ejpam-5779	376	1	axiomatization	axiomatization	NOUN
ejpam-5779	376	2	of	of	ADP
ejpam-5779	376	3	boolean	boolean	ADJ
ejpam-5779	376	4	algebras	algebra	NOUN
ejpam-5779	376	5	based	base	VERB
ejpam-5779	376	6	on362	on362	PROPN
ejpam-5779	376	7	sheffer	sheffer	PROPN
ejpam-5779	376	8	stroke	stroke	PROPN
ejpam-5779	376	9	.	.	PUNCT
ejpam-5779	377	1	formalized	formalize	VERB
ejpam-5779	377	2	mathematics	mathematic	NOUN
ejpam-5779	377	3	,	,	PUNCT
ejpam-5779	377	4	12(3):355–361	12(3):355–361	NUM
ejpam-5779	377	5	,	,	PUNCT
ejpam-5779	377	6	2004.363	2004.363	NUM
ejpam-5779	377	7	[	[	X
ejpam-5779	377	8	5	5	NUM
ejpam-5779	377	9	]	]	PUNCT
ejpam-5779	377	10	t.	t.	NOUN
ejpam-5779	377	11	oner	oner	NOUN
ejpam-5779	377	12	,	,	PUNCT
ejpam-5779	377	13	t.	t.	PROPN
ejpam-5779	377	14	katican	katican	PROPN
ejpam-5779	377	15	,	,	PUNCT
ejpam-5779	377	16	and	and	CCONJ
ejpam-5779	377	17	a.	a.	PROPN
ejpam-5779	377	18	b.	b.	PROPN
ejpam-5779	377	19	saeid	saeid	PROPN
ejpam-5779	377	20	.	.	PUNCT
ejpam-5779	378	1	fuzzy	fuzzy	ADJ
ejpam-5779	378	2	filters	filter	NOUN
ejpam-5779	378	3	of	of	ADP
ejpam-5779	378	4	sheffer	sheffer	PROPN
ejpam-5779	378	5	stroke	stroke	PROPN
ejpam-5779	378	6	hilbert	hilbert	PROPN
ejpam-5779	378	7	algebras.364	algebras.364	PROPN
ejpam-5779	378	8	journal	journal	NOUN
ejpam-5779	378	9	of	of	ADP
ejpam-5779	378	10	intelligent	intelligent	ADJ
ejpam-5779	378	11	and	and	CCONJ
ejpam-5779	378	12	fuzzy	fuzzy	ADJ
ejpam-5779	378	13	systems	system	NOUN
ejpam-5779	378	14	,	,	PUNCT
ejpam-5779	378	15	40(1):759–772	40(1):759–772	NOUN
ejpam-5779	378	16	,	,	PUNCT
ejpam-5779	378	17	2021.365	2021.365	NUM
ejpam-5779	379	1	[	[	X
ejpam-5779	379	2	6	6	NUM
ejpam-5779	379	3	]	]	PUNCT
ejpam-5779	379	4	t.	t.	NOUN
ejpam-5779	379	5	oner	oner	NOUN
ejpam-5779	379	6	,	,	PUNCT
ejpam-5779	379	7	t.	t.	PROPN
ejpam-5779	379	8	katican	katican	PROPN
ejpam-5779	379	9	,	,	PUNCT
ejpam-5779	379	10	and	and	CCONJ
ejpam-5779	379	11	a.	a.	PROPN
ejpam-5779	379	12	b.	b.	PROPN
ejpam-5779	379	13	saeid	saeid	PROPN
ejpam-5779	379	14	.	.	PUNCT
ejpam-5779	380	1	relation	relation	NOUN
ejpam-5779	380	2	between	between	ADP
ejpam-5779	380	3	sheffer	sheffer	PROPN
ejpam-5779	380	4	stroke	stroke	NOUN
ejpam-5779	380	5	and	and	CCONJ
ejpam-5779	380	6	hilbert366	hilbert366	PROPN
ejpam-5779	380	7	algebras	algebras	PROPN
ejpam-5779	380	8	.	.	PUNCT
ejpam-5779	380	9	categories	category	NOUN
ejpam-5779	380	10	and	and	CCONJ
ejpam-5779	380	11	general	general	ADJ
ejpam-5779	380	12	algebraic	algebraic	ADJ
ejpam-5779	380	13	structures	structure	NOUN
ejpam-5779	380	14	with	with	ADP
ejpam-5779	380	15	applications	application	NOUN
ejpam-5779	380	16	,	,	PUNCT
ejpam-5779	380	17	14(1):245–367	14(1):245–367	PROPN
ejpam-5779	380	18	268	268	NUM
ejpam-5779	380	19	,	,	PUNCT
ejpam-5779	380	20	2021.368	2021.368	NUM
ejpam-5779	381	1	[	[	X
ejpam-5779	381	2	7	7	X
ejpam-5779	381	3	]	]	PUNCT
ejpam-5779	381	4	t.	t.	NOUN
ejpam-5779	381	5	oner	oner	NOUN
ejpam-5779	381	6	,	,	PUNCT
ejpam-5779	381	7	t.	t.	PROPN
ejpam-5779	381	8	katican	katican	PROPN
ejpam-5779	381	9	,	,	PUNCT
ejpam-5779	381	10	and	and	CCONJ
ejpam-5779	381	11	a.	a.	PROPN
ejpam-5779	381	12	b.	b.	PROPN
ejpam-5779	381	13	saeid	saeid	PROPN
ejpam-5779	381	14	.	.	PUNCT
ejpam-5779	382	1	class	class	NOUN
ejpam-5779	382	2	of	of	ADP
ejpam-5779	382	3	sheffer	sheffer	PROPN
ejpam-5779	382	4	stroke	stroke	NOUN
ejpam-5779	382	5	bck	bck	PROPN
ejpam-5779	382	6	-	-	PUNCT
ejpam-5779	382	7	algebras	algebras	PROPN
ejpam-5779	382	8	.	.	PUNCT
ejpam-5779	382	9	analele369	analele369	PROPN
ejpam-5779	382	10	stiintifice	stiintifice	PROPN
ejpam-5779	382	11	ale	ale	NOUN
ejpam-5779	382	12	universitatii	universitatii	PROPN
ejpam-5779	382	13	ovidius	ovidius	PROPN
ejpam-5779	382	14	constanta	constanta	PROPN
ejpam-5779	382	15	,	,	PUNCT
ejpam-5779	382	16	30(1):247–269	30(1):247–269	NOUN
ejpam-5779	382	17	,	,	PUNCT
ejpam-5779	382	18	2022.370	2022.370	NUM
ejpam-5779	382	19	[	[	X
ejpam-5779	382	20	8	8	NUM
ejpam-5779	382	21	]	]	PUNCT
ejpam-5779	382	22	t.	t.	NOUN
ejpam-5779	382	23	oner	oner	NOUN
ejpam-5779	382	24	,	,	PUNCT
ejpam-5779	382	25	t.	t.	PROPN
ejpam-5779	382	26	katican	katican	PROPN
ejpam-5779	382	27	,	,	PUNCT
ejpam-5779	382	28	and	and	CCONJ
ejpam-5779	382	29	a.	a.	PROPN
ejpam-5779	382	30	b.	b.	PROPN
ejpam-5779	382	31	saeid	saeid	PROPN
ejpam-5779	382	32	.	.	PUNCT
ejpam-5779	383	1	fuzzy	fuzzy	ADJ
ejpam-5779	383	2	ideals	ideal	NOUN
ejpam-5779	383	3	of	of	ADP
ejpam-5779	383	4	sheffer	sheffer	PROPN
ejpam-5779	383	5	stroke	stroke	PROPN
ejpam-5779	383	6	hilbert	hilbert	PROPN
ejpam-5779	383	7	algebras.371	algebras.371	PROPN
ejpam-5779	383	8	proceedings	proceeding	NOUN
ejpam-5779	383	9	of	of	ADP
ejpam-5779	383	10	the	the	DET
ejpam-5779	383	11	national	national	PROPN
ejpam-5779	383	12	academy	academy	PROPN
ejpam-5779	383	13	of	of	ADP
ejpam-5779	383	14	sciences	sciences	PROPN
ejpam-5779	383	15	,	,	PUNCT
ejpam-5779	383	16	india	india	PROPN
ejpam-5779	383	17	section	section	PROPN
ejpam-5779	383	18	a	a	PRON
ejpam-5779	383	19	:	:	PUNCT
ejpam-5779	383	20	physical	physical	ADJ
ejpam-5779	383	21	sciences,372	sciences,372	PROPN
ejpam-5779	383	22	93:85–94	93:85–94	NUM
ejpam-5779	383	23	,	,	PUNCT
ejpam-5779	383	24	2023.373	2023.373	NUM
ejpam-5779	383	25	[	[	X
ejpam-5779	383	26	9	9	NUM
ejpam-5779	383	27	]	]	PUNCT
ejpam-5779	383	28	i.	i.	NOUN
ejpam-5779	383	29	senturk	senturk	PROPN
ejpam-5779	383	30	.	.	PUNCT
ejpam-5779	384	1	a	a	DET
ejpam-5779	384	2	bridge	bridge	NOUN
ejpam-5779	384	3	construction	construction	NOUN
ejpam-5779	384	4	from	from	ADP
ejpam-5779	384	5	sheffer	sheffer	PROPN
ejpam-5779	384	6	stroke	stroke	NOUN
ejpam-5779	384	7	basic	basic	ADJ
ejpam-5779	384	8	algebras	algebra	NOUN
ejpam-5779	384	9	to	to	ADP
ejpam-5779	384	10	mtl	mtl	PROPN
ejpam-5779	384	11	-	-	PUNCT
ejpam-5779	384	12	algebras.374	algebras.374	PROPN
ejpam-5779	384	13	journal	journal	NOUN
ejpam-5779	384	14	of	of	ADP
ejpam-5779	384	15	balıkesir	balıkesir	PROPN
ejpam-5779	384	16	university	university	PROPN
ejpam-5779	384	17	of	of	ADP
ejpam-5779	384	18	science	science	NOUN
ejpam-5779	384	19	and	and	CCONJ
ejpam-5779	384	20	technology	technology	NOUN
ejpam-5779	384	21	,	,	PUNCT
ejpam-5779	384	22	22(1):193–203	22(1):193–203	PROPN
ejpam-5779	384	23	,	,	PUNCT
ejpam-5779	384	24	2020.375	2020.375	PROPN
ejpam-5779	384	25	[	[	X
ejpam-5779	384	26	10	10	NUM
ejpam-5779	384	27	]	]	X
ejpam-5779	384	28	i.	i.	NOUN
ejpam-5779	384	29	senturk	senturk	PROPN
ejpam-5779	384	30	and	and	CCONJ
ejpam-5779	384	31	t.	t.	PROPN
ejpam-5779	384	32	oner	oner	NOUN
ejpam-5779	384	33	.	.	PUNCT
ejpam-5779	385	1	the	the	DET
ejpam-5779	385	2	sheffer	sheffer	PROPN
ejpam-5779	385	3	stroke	stroke	NOUN
ejpam-5779	385	4	operation	operation	NOUN
ejpam-5779	385	5	reducts	reduct	NOUN
ejpam-5779	385	6	of	of	ADP
ejpam-5779	385	7	basic	basic	ADJ
ejpam-5779	385	8	algebras	algebra	NOUN
ejpam-5779	385	9	.	.	PUNCT
ejpam-5779	386	1	open376	open376	PROPN
ejpam-5779	386	2	mathematics	mathematics	PROPN
ejpam-5779	386	3	,	,	PUNCT
ejpam-5779	386	4	15:926–935	15:926–935	NUM
ejpam-5779	386	5	,	,	PUNCT
ejpam-5779	386	6	2017.377	2017.377	NUM
ejpam-5779	387	1	[	[	X
ejpam-5779	387	2	11	11	NUM
ejpam-5779	387	3	]	]	PUNCT
ejpam-5779	387	4	i.	i.	NOUN
ejpam-5779	387	5	senturk	senturk	PROPN
ejpam-5779	387	6	and	and	CCONJ
ejpam-5779	387	7	t.	t.	PROPN
ejpam-5779	387	8	oner	oner	NOUN
ejpam-5779	387	9	.	.	PUNCT
ejpam-5779	388	1	a	a	DET
ejpam-5779	388	2	construction	construction	NOUN
ejpam-5779	388	3	of	of	ADP
ejpam-5779	388	4	very	very	ADV
ejpam-5779	388	5	true	true	ADJ
ejpam-5779	388	6	operator	operator	NOUN
ejpam-5779	388	7	on	on	ADP
ejpam-5779	388	8	sheffer	sheffer	NOUN
ejpam-5779	388	9	stroke	stroke	PROPN
ejpam-5779	388	10	mtl-378	mtl-378	PROPN
ejpam-5779	388	11	algebras	algebras	PROPN
ejpam-5779	388	12	.	.	PUNCT
ejpam-5779	389	1	international	international	ADJ
ejpam-5779	389	2	journal	journal	PROPN
ejpam-5779	389	3	of	of	ADP
ejpam-5779	389	4	maps	map	NOUN
ejpam-5779	389	5	in	in	ADP
ejpam-5779	389	6	mathematics	mathematic	NOUN
ejpam-5779	389	7	,	,	PUNCT
ejpam-5779	389	8	4(2):93–106	4(2):93–106	NUM
ejpam-5779	389	9	,	,	PUNCT
ejpam-5779	389	10	2021.379	2021.379	PROPN
ejpam-5779	389	11	[	[	X
ejpam-5779	389	12	12	12	NUM
ejpam-5779	389	13	]	]	PUNCT
ejpam-5779	389	14	i.	i.	NOUN
ejpam-5779	389	15	senturk	senturk	PROPN
ejpam-5779	389	16	,	,	PUNCT
ejpam-5779	389	17	t.	t.	PROPN
ejpam-5779	389	18	oner	oner	NOUN
ejpam-5779	389	19	,	,	PUNCT
ejpam-5779	389	20	and	and	CCONJ
ejpam-5779	389	21	a.	a.	PROPN
ejpam-5779	389	22	b.	b.	PROPN
ejpam-5779	389	23	saeid	saeid	PROPN
ejpam-5779	389	24	.	.	PUNCT
ejpam-5779	390	1	congruences	congruence	NOUN
ejpam-5779	390	2	of	of	ADP
ejpam-5779	390	3	sheffer	sheffer	NOUN
ejpam-5779	390	4	stroke	stroke	NOUN
ejpam-5779	390	5	basic	basic	ADJ
ejpam-5779	390	6	algebras.380	algebras.380	NOUN
ejpam-5779	390	7	analele	analele	ADP
ejpam-5779	390	8	stiintifice	stiintifice	NOUN
ejpam-5779	390	9	ale	ale	PROPN
ejpam-5779	390	10	universitatii	universitatii	PROPN
ejpam-5779	390	11	ovidius	ovidius	PROPN
ejpam-5779	390	12	constanta	constanta	PROPN
ejpam-5779	390	13	,	,	PUNCT
ejpam-5779	390	14	seria	seria	PROPN
ejpam-5779	390	15	matematica	matematica	PROPN
ejpam-5779	390	16	,	,	PUNCT
ejpam-5779	390	17	28(2):209–381	28(2):209–381	PROPN
ejpam-5779	390	18	228	228	NUM
ejpam-5779	390	19	,	,	PUNCT
ejpam-5779	390	20	2020.382	2020.382	NUM
ejpam-5779	390	21	[	[	X
ejpam-5779	390	22	13	13	NUM
ejpam-5779	390	23	]	]	X
ejpam-5779	390	24	h.	h.	PROPN
ejpam-5779	390	25	m.	m.	PROPN
ejpam-5779	390	26	sheffer	sheffer	PROPN
ejpam-5779	390	27	.	.	PUNCT
ejpam-5779	391	1	a	a	DET
ejpam-5779	391	2	set	set	NOUN
ejpam-5779	391	3	of	of	ADP
ejpam-5779	391	4	five	five	NUM
ejpam-5779	391	5	independent	independent	ADJ
ejpam-5779	391	6	postulates	postulate	NOUN
ejpam-5779	391	7	for	for	ADP
ejpam-5779	391	8	boolean	boolean	ADJ
ejpam-5779	391	9	algebras	algebra	NOUN
ejpam-5779	391	10	,	,	PUNCT
ejpam-5779	391	11	with	with	ADP
ejpam-5779	391	12	ap-383	ap-383	PROPN
ejpam-5779	391	13	plication	plication	NOUN
ejpam-5779	391	14	to	to	ADP
ejpam-5779	391	15	logical	logical	ADJ
ejpam-5779	391	16	constants	constant	NOUN
ejpam-5779	391	17	.	.	PUNCT
ejpam-5779	392	1	transactions	transaction	NOUN
ejpam-5779	392	2	of	of	ADP
ejpam-5779	392	3	the	the	DET
ejpam-5779	392	4	american	american	PROPN
ejpam-5779	392	5	mathematical	mathematical	PROPN
ejpam-5779	392	6	society,384	society,384	PROPN
ejpam-5779	392	7	14(4):481–488	14(4):481–488	PROPN
ejpam-5779	392	8	,	,	PUNCT
ejpam-5779	392	9	1913.385	1913.385	NUM
ejpam-5779	393	1	[	[	X
ejpam-5779	393	2	14	14	NUM
ejpam-5779	393	3	]	]	X
ejpam-5779	393	4	n.	n.	NOUN
ejpam-5779	393	5	tacha	tacha	PROPN
ejpam-5779	393	6	,	,	PUNCT
ejpam-5779	393	7	p.	p.	PROPN
ejpam-5779	393	8	phayapsiang	phayapsiang	PROPN
ejpam-5779	393	9	,	,	PUNCT
ejpam-5779	393	10	and	and	CCONJ
ejpam-5779	393	11	a.	a.	NOUN
ejpam-5779	393	12	iampan	iampan	PROPN
ejpam-5779	393	13	.	.	PUNCT
ejpam-5779	394	1	length	length	NOUN
ejpam-5779	394	2	and	and	CCONJ
ejpam-5779	394	3	mean	mean	VERB
ejpam-5779	394	4	fuzzy	fuzzy	ADJ
ejpam-5779	394	5	up	up	ADP
ejpam-5779	394	6	-	-	PUNCT
ejpam-5779	394	7	subalgebras386	subalgebras386	NOUN
ejpam-5779	394	8	of	of	ADP
ejpam-5779	394	9	up	up	ADP
ejpam-5779	394	10	-	-	PUNCT
ejpam-5779	394	11	algebras	algebras	X
ejpam-5779	394	12	.	.	PUNCT
ejpam-5779	395	1	caspian	caspian	PROPN
ejpam-5779	395	2	journal	journal	PROPN
ejpam-5779	395	3	of	of	ADP
ejpam-5779	395	4	mathematical	mathematical	ADJ
ejpam-5779	395	5	sciences	sciences	PROPN
ejpam-5779	395	6	,	,	PUNCT
ejpam-5779	395	7	11(1):264–302	11(1):264–302	NUM
ejpam-5779	395	8	,	,	PUNCT
ejpam-5779	396	1	2022.387	2022.387	NUM
ejpam-5779	396	2	[	[	X
ejpam-5779	396	3	15	15	NUM
ejpam-5779	396	4	]	]	X
ejpam-5779	396	5	l.	l.	PROPN
ejpam-5779	396	6	a.	a.	PROPN
ejpam-5779	396	7	zadeh	zadeh	PROPN
ejpam-5779	396	8	.	.	PUNCT
ejpam-5779	397	1	fuzzy	fuzzy	ADJ
ejpam-5779	397	2	sets	set	NOUN
ejpam-5779	397	3	.	.	PUNCT
ejpam-5779	398	1	information	information	NOUN
ejpam-5779	398	2	and	and	CCONJ
ejpam-5779	398	3	control	control	NOUN
ejpam-5779	398	4	,	,	PUNCT
ejpam-5779	398	5	8(3):338–353	8(3):338–353	NUM
ejpam-5779	398	6	,	,	PUNCT
ejpam-5779	398	7	1965.388	1965.388	NUM