id	sid	tid	token	lemma	pos
ejpam-5298	1	1	european	european	PROPN
ejpam-5298	1	2	journal	journal	PROPN
ejpam-5298	1	3	of	of	ADP
ejpam-5298	1	4	pure	pure	ADJ
ejpam-5298	1	5	and	and	CCONJ
ejpam-5298	1	6	applied	apply	VERB
ejpam-5298	1	7	mathematics	mathematic	NOUN
ejpam-5298	1	8	vol	vol	NOUN
ejpam-5298	1	9	.	.	PROPN
ejpam-5298	2	1	17	17	NUM
ejpam-5298	2	2	,	,	PUNCT
ejpam-5298	2	3	no	no	INTJ
ejpam-5298	2	4	.	.	NOUN
ejpam-5298	2	5	4	4	NUM
ejpam-5298	2	6	,	,	PUNCT
ejpam-5298	2	7	2024	2024	NUM
ejpam-5298	2	8	,	,	PUNCT
ejpam-5298	2	9	2726	2726	NUM
ejpam-5298	2	10	-	-	SYM
ejpam-5298	2	11	2737	2737	NUM
ejpam-5298	2	12	issn	issn	PROPN
ejpam-5298	2	13	1307	1307	NUM
ejpam-5298	2	14	-	-	SYM
ejpam-5298	2	15	5543	5543	NUM
ejpam-5298	2	16	–	–	PUNCT
ejpam-5298	2	17	ejpam.com	ejpam.com	X
ejpam-5298	2	18	published	publish	VERB
ejpam-5298	2	19	by	by	ADP
ejpam-5298	2	20	new	new	PROPN
ejpam-5298	2	21	york	york	PROPN
ejpam-5298	2	22	business	business	PROPN
ejpam-5298	2	23	global	global	PROPN
ejpam-5298	2	24	on	on	ADP
ejpam-5298	2	25	multipliers	multiplier	NOUN
ejpam-5298	2	26	of	of	ADP
ejpam-5298	2	27	hilbert	hilbert	PROPN
ejpam-5298	2	28	algebras	algebras	PROPN
ejpam-5298	2	29	aiyared	aiyare	VERB
ejpam-5298	2	30	iampan1,∗	iampan1,∗	NOUN
ejpam-5298	2	31	,	,	PUNCT
ejpam-5298	2	32	neelamegarajan	neelamegarajan	NOUN
ejpam-5298	2	33	rajesh2	rajesh2	PROPN
ejpam-5298	2	34	1	1	NUM
ejpam-5298	2	35	department	department	NOUN
ejpam-5298	2	36	of	of	ADP
ejpam-5298	2	37	mathematics	mathematic	NOUN
ejpam-5298	2	38	,	,	PUNCT
ejpam-5298	2	39	school	school	NOUN
ejpam-5298	2	40	of	of	ADP
ejpam-5298	2	41	science	science	NOUN
ejpam-5298	2	42	,	,	PUNCT
ejpam-5298	2	43	university	university	NOUN
ejpam-5298	2	44	of	of	ADP
ejpam-5298	2	45	phayao	phayao	NOUN
ejpam-5298	2	46	,	,	PUNCT
ejpam-5298	2	47	mae	mae	PROPN
ejpam-5298	2	48	ka	ka	PROPN
ejpam-5298	2	49	,	,	PUNCT
ejpam-5298	2	50	mueang	mueang	PROPN
ejpam-5298	2	51	,	,	PUNCT
ejpam-5298	2	52	phayao	phayao	NOUN
ejpam-5298	2	53	56000	56000	NUM
ejpam-5298	2	54	,	,	PUNCT
ejpam-5298	2	55	thailand	thailand	PROPN
ejpam-5298	2	56	2	2	NUM
ejpam-5298	2	57	department	department	NOUN
ejpam-5298	2	58	of	of	ADP
ejpam-5298	2	59	mathematics	mathematic	NOUN
ejpam-5298	2	60	,	,	PUNCT
ejpam-5298	2	61	rajah	rajah	NOUN
ejpam-5298	2	62	serfoji	serfoji	ADJ
ejpam-5298	2	63	government	government	NOUN
ejpam-5298	2	64	college	college	NOUN
ejpam-5298	2	65	(	(	PUNCT
ejpam-5298	2	66	affiliated	affiliate	VERB
ejpam-5298	2	67	to	to	PART
ejpam-5298	2	68	bharathidasan	bharathidasan	VERB
ejpam-5298	2	69	university	university	NOUN
ejpam-5298	2	70	)	)	PUNCT
ejpam-5298	2	71	,	,	PUNCT
ejpam-5298	2	72	thanjavur-613005	thanjavur-613005	NOUN
ejpam-5298	2	73	,	,	PUNCT
ejpam-5298	2	74	tamilnadu	tamilnadu	ADJ
ejpam-5298	2	75	,	,	PUNCT
ejpam-5298	2	76	india	india	PROPN
ejpam-5298	2	77	abstract	abstract	NOUN
ejpam-5298	2	78	.	.	PUNCT
ejpam-5298	3	1	this	this	DET
ejpam-5298	3	2	paper	paper	NOUN
ejpam-5298	3	3	explores	explore	VERB
ejpam-5298	3	4	the	the	DET
ejpam-5298	3	5	concept	concept	NOUN
ejpam-5298	3	6	of	of	ADP
ejpam-5298	3	7	multipliers	multiplier	NOUN
ejpam-5298	3	8	in	in	ADP
ejpam-5298	3	9	hilbert	hilbert	PROPN
ejpam-5298	3	10	algebras	algebras	PROPN
ejpam-5298	3	11	,	,	PUNCT
ejpam-5298	3	12	unveiling	unveil	VERB
ejpam-5298	3	13	various	various	ADJ
ejpam-5298	3	14	intriguing	intriguing	ADJ
ejpam-5298	3	15	properties	property	NOUN
ejpam-5298	3	16	.	.	PUNCT
ejpam-5298	4	1	we	we	PRON
ejpam-5298	4	2	delve	delve	VERB
ejpam-5298	4	3	into	into	ADP
ejpam-5298	4	4	the	the	DET
ejpam-5298	4	5	intricate	intricate	ADJ
ejpam-5298	4	6	relationships	relationship	NOUN
ejpam-5298	4	7	between	between	ADP
ejpam-5298	4	8	fixed	fix	VERB
ejpam-5298	4	9	sets	set	NOUN
ejpam-5298	4	10	,	,	PUNCT
ejpam-5298	4	11	kernels	kernel	NOUN
ejpam-5298	4	12	,	,	PUNCT
ejpam-5298	4	13	and	and	CCONJ
ejpam-5298	4	14	near	near	ADP
ejpam-5298	4	15	filters	filter	NOUN
ejpam-5298	4	16	of	of	ADP
ejpam-5298	4	17	multipliers	multiplier	NOUN
ejpam-5298	4	18	,	,	PUNCT
ejpam-5298	4	19	shedding	shed	VERB
ejpam-5298	4	20	light	light	NOUN
ejpam-5298	4	21	on	on	ADP
ejpam-5298	4	22	their	their	PRON
ejpam-5298	4	23	interconnected	interconnected	ADJ
ejpam-5298	4	24	roles	role	NOUN
ejpam-5298	4	25	within	within	ADP
ejpam-5298	4	26	hilbert	hilbert	PROPN
ejpam-5298	4	27	algebras	algebras	PROPN
ejpam-5298	4	28	.	.	PUNCT
ejpam-5298	5	1	this	this	DET
ejpam-5298	5	2	investigation	investigation	NOUN
ejpam-5298	5	3	aims	aim	VERB
ejpam-5298	5	4	to	to	PART
ejpam-5298	5	5	provide	provide	VERB
ejpam-5298	5	6	a	a	DET
ejpam-5298	5	7	deeper	deep	ADJ
ejpam-5298	5	8	understanding	understanding	NOUN
ejpam-5298	5	9	of	of	ADP
ejpam-5298	5	10	algebraic	algebraic	ADJ
ejpam-5298	5	11	structures	structure	NOUN
ejpam-5298	5	12	and	and	CCONJ
ejpam-5298	5	13	their	their	PRON
ejpam-5298	5	14	dynamic	dynamic	ADJ
ejpam-5298	5	15	interactions	interaction	NOUN
ejpam-5298	5	16	.	.	PUNCT
ejpam-5298	6	1	2020	2020	NUM
ejpam-5298	6	2	mathematics	mathematic	NOUN
ejpam-5298	6	3	subject	subject	NOUN
ejpam-5298	6	4	classifications	classification	NOUN
ejpam-5298	6	5	:	:	PUNCT
ejpam-5298	6	6	03g25	03g25	NUM
ejpam-5298	6	7	,	,	PUNCT
ejpam-5298	6	8	43a22	43a22	NUM
ejpam-5298	6	9	key	key	ADJ
ejpam-5298	6	10	words	word	NOUN
ejpam-5298	6	11	and	and	CCONJ
ejpam-5298	6	12	phrases	phrase	NOUN
ejpam-5298	6	13	:	:	PUNCT
ejpam-5298	6	14	hilbert	hilbert	NOUN
ejpam-5298	6	15	algebra	algebra	PROPN
ejpam-5298	6	16	,	,	PUNCT
ejpam-5298	6	17	multiplier	multipli	ADJ
ejpam-5298	6	18	,	,	PUNCT
ejpam-5298	6	19	kernel	kernel	PROPN
ejpam-5298	6	20	,	,	PUNCT
ejpam-5298	6	21	fixed	fix	VERB
ejpam-5298	6	22	set	set	NOUN
ejpam-5298	6	23	,	,	PUNCT
ejpam-5298	6	24	near	near	ADP
ejpam-5298	6	25	filter	filter	NOUN
ejpam-5298	6	26	1	1	NUM
ejpam-5298	6	27	.	.	PUNCT
ejpam-5298	6	28	introduction	introduction	NOUN
ejpam-5298	6	29	the	the	DET
ejpam-5298	6	30	study	study	NOUN
ejpam-5298	6	31	of	of	ADP
ejpam-5298	6	32	multipliers	multiplier	NOUN
ejpam-5298	6	33	has	have	AUX
ejpam-5298	6	34	attracted	attract	VERB
ejpam-5298	6	35	considerable	considerable	ADJ
ejpam-5298	6	36	attention	attention	NOUN
ejpam-5298	6	37	in	in	ADP
ejpam-5298	6	38	various	various	ADJ
ejpam-5298	6	39	branches	branch	NOUN
ejpam-5298	6	40	of	of	ADP
ejpam-5298	6	41	algebra	algebra	PROPN
ejpam-5298	6	42	,	,	PUNCT
ejpam-5298	6	43	demonstrating	demonstrate	VERB
ejpam-5298	6	44	its	its	PRON
ejpam-5298	6	45	versatility	versatility	NOUN
ejpam-5298	6	46	and	and	CCONJ
ejpam-5298	6	47	relevance	relevance	NOUN
ejpam-5298	6	48	across	across	ADP
ejpam-5298	6	49	different	different	ADJ
ejpam-5298	6	50	algebraic	algebraic	ADJ
ejpam-5298	6	51	systems	system	NOUN
ejpam-5298	6	52	.	.	PUNCT
ejpam-5298	7	1	in	in	ADP
ejpam-5298	7	2	1986	1986	NUM
ejpam-5298	7	3	,	,	PUNCT
ejpam-5298	7	4	cirulis	ciruli	VERB
ejpam-5298	8	1	[	[	X
ejpam-5298	8	2	5	5	NUM
ejpam-5298	8	3	]	]	PUNCT
ejpam-5298	8	4	was	be	AUX
ejpam-5298	8	5	among	among	ADP
ejpam-5298	8	6	the	the	DET
ejpam-5298	8	7	pioneers	pioneer	NOUN
ejpam-5298	8	8	,	,	PUNCT
ejpam-5298	8	9	examining	examine	VERB
ejpam-5298	8	10	multipliers	multiplier	NOUN
ejpam-5298	8	11	within	within	ADP
ejpam-5298	8	12	the	the	DET
ejpam-5298	8	13	framework	framework	NOUN
ejpam-5298	8	14	of	of	ADP
ejpam-5298	8	15	implicative	implicative	ADJ
ejpam-5298	8	16	algebras	algebra	NOUN
ejpam-5298	8	17	.	.	PUNCT
ejpam-5298	9	1	his	his	PRON
ejpam-5298	9	2	work	work	NOUN
ejpam-5298	9	3	set	set	VERB
ejpam-5298	9	4	the	the	DET
ejpam-5298	9	5	stage	stage	NOUN
ejpam-5298	9	6	for	for	ADP
ejpam-5298	9	7	further	further	ADJ
ejpam-5298	9	8	inquiries	inquiry	NOUN
ejpam-5298	9	9	into	into	ADP
ejpam-5298	9	10	how	how	SCONJ
ejpam-5298	9	11	multipliers	multiplier	NOUN
ejpam-5298	9	12	interact	interact	VERB
ejpam-5298	9	13	with	with	ADP
ejpam-5298	9	14	logical	logical	ADJ
ejpam-5298	9	15	operations	operation	NOUN
ejpam-5298	9	16	in	in	ADP
ejpam-5298	9	17	these	these	DET
ejpam-5298	9	18	structures	structure	NOUN
ejpam-5298	9	19	.	.	PUNCT
ejpam-5298	10	1	subsequently	subsequently	ADV
ejpam-5298	10	2	,	,	PUNCT
ejpam-5298	10	3	in	in	ADP
ejpam-5298	10	4	2011	2011	NUM
ejpam-5298	10	5	,	,	PUNCT
ejpam-5298	10	6	kim	kim	PROPN
ejpam-5298	11	1	[	[	X
ejpam-5298	11	2	16	16	NUM
ejpam-5298	11	3	]	]	PUNCT
ejpam-5298	11	4	extended	extend	VERB
ejpam-5298	11	5	the	the	DET
ejpam-5298	11	6	concept	concept	NOUN
ejpam-5298	11	7	to	to	PART
ejpam-5298	11	8	be	be	AUX
ejpam-5298	11	9	-	-	PUNCT
ejpam-5298	11	10	algebras	algebras	X
ejpam-5298	11	11	,	,	PUNCT
ejpam-5298	11	12	providing	provide	VERB
ejpam-5298	11	13	a	a	DET
ejpam-5298	11	14	new	new	ADJ
ejpam-5298	11	15	perspective	perspective	NOUN
ejpam-5298	11	16	emphasising	emphasise	VERB
ejpam-5298	11	17	multipliers	multiplier	NOUN
ejpam-5298	11	18	’	'	PUNCT
ejpam-5298	11	19	role	role	NOUN
ejpam-5298	11	20	in	in	ADP
ejpam-5298	11	21	non	non	ADJ
ejpam-5298	11	22	-	-	ADJ
ejpam-5298	11	23	classical	classical	ADJ
ejpam-5298	11	24	algebraic	algebraic	ADJ
ejpam-5298	11	25	settings	setting	NOUN
ejpam-5298	11	26	.	.	PUNCT
ejpam-5298	12	1	this	this	DET
ejpam-5298	12	2	expansion	expansion	NOUN
ejpam-5298	12	3	continued	continue	VERB
ejpam-5298	12	4	when	when	SCONJ
ejpam-5298	12	5	chaudhry	chaudhry	PROPN
ejpam-5298	12	6	and	and	CCONJ
ejpam-5298	12	7	ali	ali	PROPN
ejpam-5298	12	8	[	[	X
ejpam-5298	12	9	4	4	NUM
ejpam-5298	12	10	]	]	PUNCT
ejpam-5298	12	11	introduced	introduce	VERB
ejpam-5298	12	12	multipliers	multiplier	NOUN
ejpam-5298	12	13	in	in	ADP
ejpam-5298	12	14	d	d	NOUN
ejpam-5298	12	15	-	-	PUNCT
ejpam-5298	12	16	algebras	algebras	PROPN
ejpam-5298	12	17	in	in	ADP
ejpam-5298	12	18	2012	2012	NUM
ejpam-5298	12	19	,	,	PUNCT
ejpam-5298	12	20	shedding	shed	VERB
ejpam-5298	12	21	light	light	NOUN
ejpam-5298	12	22	on	on	ADP
ejpam-5298	12	23	their	their	PRON
ejpam-5298	12	24	function	function	NOUN
ejpam-5298	12	25	in	in	ADP
ejpam-5298	12	26	algebras	algebra	NOUN
ejpam-5298	12	27	characterized	characterize	VERB
ejpam-5298	12	28	by	by	ADP
ejpam-5298	12	29	a	a	DET
ejpam-5298	12	30	specific	specific	ADJ
ejpam-5298	12	31	set	set	NOUN
ejpam-5298	12	32	of	of	ADP
ejpam-5298	12	33	order	order	NOUN
ejpam-5298	12	34	and	and	CCONJ
ejpam-5298	12	35	combination	combination	NOUN
ejpam-5298	12	36	rules	rule	NOUN
ejpam-5298	12	37	.	.	PUNCT
ejpam-5298	13	1	in	in	ADP
ejpam-5298	13	2	2013	2013	NUM
ejpam-5298	13	3	,	,	PUNCT
ejpam-5298	13	4	kim	kim	PROPN
ejpam-5298	13	5	and	and	CCONJ
ejpam-5298	13	6	lim	lim	PROPN
ejpam-5298	13	7	[	[	X
ejpam-5298	13	8	17	17	NUM
ejpam-5298	13	9	]	]	PUNCT
ejpam-5298	13	10	brought	bring	VERB
ejpam-5298	13	11	the	the	DET
ejpam-5298	13	12	concept	concept	NOUN
ejpam-5298	13	13	of	of	ADP
ejpam-5298	13	14	multipliers	multiplier	NOUN
ejpam-5298	13	15	into	into	ADP
ejpam-5298	13	16	the	the	DET
ejpam-5298	13	17	realm	realm	NOUN
ejpam-5298	13	18	of	of	ADP
ejpam-5298	13	19	bcc	bcc	PROPN
ejpam-5298	13	20	-	-	PUNCT
ejpam-5298	13	21	algebras	algebras	X
ejpam-5298	13	22	,	,	PUNCT
ejpam-5298	13	23	exploring	explore	VERB
ejpam-5298	13	24	their	their	PRON
ejpam-5298	13	25	implications	implication	NOUN
ejpam-5298	13	26	in	in	ADP
ejpam-5298	13	27	algebras	algebra	NOUN
ejpam-5298	13	28	defined	define	VERB
ejpam-5298	13	29	by	by	ADP
ejpam-5298	13	30	bounded	bound	VERB
ejpam-5298	13	31	commutative	commutative	ADJ
ejpam-5298	13	32	cancellative	cancellative	ADJ
ejpam-5298	13	33	properties	property	NOUN
ejpam-5298	13	34	.	.	PUNCT
ejpam-5298	14	1	around	around	ADP
ejpam-5298	14	2	the	the	DET
ejpam-5298	14	3	same	same	ADJ
ejpam-5298	14	4	time	time	NOUN
ejpam-5298	14	5	,	,	PUNCT
ejpam-5298	14	6	lee	lee	PROPN
ejpam-5298	14	7	and	and	CCONJ
ejpam-5298	14	8	kim	kim	PROPN
ejpam-5298	15	1	[	[	X
ejpam-5298	15	2	18	18	NUM
ejpam-5298	15	3	]	]	PUNCT
ejpam-5298	15	4	focused	focus	VERB
ejpam-5298	15	5	on	on	ADP
ejpam-5298	15	6	subtraction	subtraction	NOUN
ejpam-5298	15	7	algebras	algebra	NOUN
ejpam-5298	15	8	,	,	PUNCT
ejpam-5298	15	9	where	where	SCONJ
ejpam-5298	15	10	they	they	PRON
ejpam-5298	15	11	defined	define	VERB
ejpam-5298	15	12	multipliers	multiplier	NOUN
ejpam-5298	15	13	that	that	PRON
ejpam-5298	15	14	operate	operate	VERB
ejpam-5298	15	15	within	within	ADP
ejpam-5298	15	16	algebraic	algebraic	ADJ
ejpam-5298	15	17	structures	structure	NOUN
ejpam-5298	15	18	built	build	VERB
ejpam-5298	15	19	on	on	ADP
ejpam-5298	15	20	the	the	DET
ejpam-5298	15	21	notion	notion	NOUN
ejpam-5298	15	22	of	of	ADP
ejpam-5298	15	23	difference	difference	NOUN
ejpam-5298	15	24	rather	rather	ADV
ejpam-5298	15	25	than	than	ADP
ejpam-5298	15	26	addition	addition	NOUN
ejpam-5298	15	27	.	.	PUNCT
ejpam-5298	16	1	in	in	ADP
ejpam-5298	16	2	2014	2014	NUM
ejpam-5298	16	3	,	,	PUNCT
ejpam-5298	16	4	khorami	khorami	NOUN
ejpam-5298	16	5	and	and	CCONJ
ejpam-5298	16	6	saeid	saeid	PROPN
ejpam-5298	16	7	[	[	X
ejpam-5298	16	8	15	15	NUM
ejpam-5298	16	9	]	]	PUNCT
ejpam-5298	16	10	introduced	introduce	VERB
ejpam-5298	16	11	the	the	DET
ejpam-5298	16	12	concept	concept	NOUN
ejpam-5298	16	13	of	of	ADP
ejpam-5298	16	14	multipliers	multiplier	NOUN
ejpam-5298	16	15	in	in	ADP
ejpam-5298	16	16	bl	bl	NOUN
ejpam-5298	16	17	-	-	PUNCT
ejpam-5298	16	18	algebras	algebras	PROPN
ejpam-5298	16	19	and	and	CCONJ
ejpam-5298	16	20	conducted	conduct	VERB
ejpam-5298	16	21	an	an	DET
ejpam-5298	16	22	in	in	ADP
ejpam-5298	16	23	-	-	PUNCT
ejpam-5298	16	24	depth	depth	NOUN
ejpam-5298	16	25	study	study	NOUN
ejpam-5298	16	26	on	on	ADP
ejpam-5298	16	27	the	the	DET
ejpam-5298	16	28	connections	connection	NOUN
ejpam-5298	16	29	between	between	ADP
ejpam-5298	16	30	multipliers	multiplier	NOUN
ejpam-5298	16	31	and	and	CCONJ
ejpam-5298	16	32	specific	specific	ADJ
ejpam-5298	16	33	mappings	mapping	NOUN
ejpam-5298	16	34	,	,	PUNCT
ejpam-5298	16	35	such	such	ADJ
ejpam-5298	16	36	as	as	ADP
ejpam-5298	16	37	closure	closure	NOUN
ejpam-5298	16	38	operators	operator	NOUN
ejpam-5298	16	39	,	,	PUNCT
ejpam-5298	16	40	homomorphisms	homomorphism	NOUN
ejpam-5298	16	41	,	,	PUNCT
ejpam-5298	16	42	and	and	CCONJ
ejpam-5298	16	43	(	(	PUNCT
ejpam-5298	16	44	⊙,∨)-derivations	⊙,∨)-derivation	NOUN
ejpam-5298	16	45	∗corresponding	∗corresponding	NOUN
ejpam-5298	16	46	author	author	NOUN
ejpam-5298	16	47	.	.	PUNCT
ejpam-5298	17	1	doi	doi	NOUN
ejpam-5298	17	2	:	:	PUNCT
ejpam-5298	17	3	https://doi.org/10.29020/nybg.ejpam.v17i4.5298	https://doi.org/10.29020/nybg.ejpam.v17i4.5298	NUM
ejpam-5298	17	4	email	email	NOUN
ejpam-5298	17	5	addresses	address	NOUN
ejpam-5298	17	6	:	:	PUNCT
ejpam-5298	18	1	aiyared.ia@up.ac.th	aiyared.ia@up.ac.th	NOUN
ejpam-5298	18	2	(	(	PUNCT
ejpam-5298	18	3	a.	a.	NOUN
ejpam-5298	18	4	iampan	iampan	PROPN
ejpam-5298	18	5	)	)	PUNCT
ejpam-5298	18	6	,	,	PUNCT
ejpam-5298	18	7	nrajesh	nrajesh	PROPN
ejpam-5298	18	8	topology@yahoo.co.in	topology@yahoo.co.in	PROPN
ejpam-5298	18	9	(	(	PUNCT
ejpam-5298	18	10	n.	n.	PROPN
ejpam-5298	18	11	rajesh	rajesh	PROPN
ejpam-5298	18	12	)	)	PUNCT
ejpam-5298	18	13	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5298	18	14	2726	2726	NUM
ejpam-5298	18	15	copyright	copyright	NOUN
ejpam-5298	18	16	:	:	PUNCT
ejpam-5298	18	17	©	©	PROPN
ejpam-5298	18	18	2024	2024	NUM
ejpam-5298	18	19	the	the	DET
ejpam-5298	18	20	author(s	author(s	NOUN
ejpam-5298	18	21	)	)	PUNCT
ejpam-5298	18	22	.	.	PUNCT
ejpam-5298	19	1	(	(	PUNCT
ejpam-5298	19	2	cc	cc	NOUN
ejpam-5298	19	3	by	by	ADP
ejpam-5298	19	4	-	-	PUNCT
ejpam-5298	19	5	nc	nc	PROPN
ejpam-5298	19	6	4.0	4.0	NUM
ejpam-5298	19	7	)	)	PUNCT
ejpam-5298	19	8	a.	a.	NOUN
ejpam-5298	19	9	iampan	iampan	PROPN
ejpam-5298	19	10	,	,	PUNCT
ejpam-5298	19	11	n.	n.	PROPN
ejpam-5298	19	12	rajesh	rajesh	PROPN
ejpam-5298	19	13	/	/	SYM
ejpam-5298	19	14	eur	eur	PROPN
ejpam-5298	19	15	.	.	PUNCT
ejpam-5298	20	1	j.	j.	PROPN
ejpam-5298	20	2	pure	pure	PROPN
ejpam-5298	20	3	appl	appl	PROPN
ejpam-5298	20	4	.	.	PROPN
ejpam-5298	20	5	math	math	PROPN
ejpam-5298	20	6	,	,	PUNCT
ejpam-5298	20	7	17	17	NUM
ejpam-5298	20	8	(	(	PUNCT
ejpam-5298	20	9	4	4	NUM
ejpam-5298	20	10	)	)	PUNCT
ejpam-5298	20	11	(	(	PUNCT
ejpam-5298	20	12	2024	2024	NUM
ejpam-5298	20	13	)	)	PUNCT
ejpam-5298	20	14	,	,	PUNCT
ejpam-5298	20	15	2726	2726	NUM
ejpam-5298	20	16	-	-	SYM
ejpam-5298	20	17	2737	2737	NUM
ejpam-5298	20	18	2727	2727	NUM
ejpam-5298	20	19	within	within	ADP
ejpam-5298	20	20	bl	bl	PROPN
ejpam-5298	20	21	-	-	PUNCT
ejpam-5298	20	22	algebras	algebras	PROPN
ejpam-5298	20	23	.	.	PUNCT
ejpam-5298	21	1	a	a	DET
ejpam-5298	21	2	significant	significant	ADJ
ejpam-5298	21	3	leap	leap	NOUN
ejpam-5298	21	4	occurred	occur	VERB
ejpam-5298	21	5	in	in	ADP
ejpam-5298	21	6	2021	2021	NUM
ejpam-5298	21	7	when	when	SCONJ
ejpam-5298	21	8	iampan	iampan	NOUN
ejpam-5298	21	9	[	[	X
ejpam-5298	21	10	10	10	NUM
ejpam-5298	21	11	]	]	PUNCT
ejpam-5298	21	12	introduced	introduce	VERB
ejpam-5298	21	13	a	a	DET
ejpam-5298	21	14	more	more	ADV
ejpam-5298	21	15	comprehensive	comprehensive	ADJ
ejpam-5298	21	16	categorization	categorization	NOUN
ejpam-5298	21	17	of	of	ADP
ejpam-5298	21	18	multipliers	multiplier	NOUN
ejpam-5298	21	19	within	within	ADP
ejpam-5298	21	20	up	up	ADP
ejpam-5298	21	21	-	-	PUNCT
ejpam-5298	21	22	algebras	algebras	X
ejpam-5298	21	23	.	.	PUNCT
ejpam-5298	22	1	his	his	PRON
ejpam-5298	22	2	work	work	NOUN
ejpam-5298	22	3	distinguished	distinguish	VERB
ejpam-5298	22	4	between	between	ADP
ejpam-5298	22	5	left	left	ADJ
ejpam-5298	22	6	multipliers	multiplier	NOUN
ejpam-5298	22	7	,	,	PUNCT
ejpam-5298	22	8	right	right	ADJ
ejpam-5298	22	9	multipliers	multiplier	NOUN
ejpam-5298	22	10	,	,	PUNCT
ejpam-5298	22	11	anti	anti	ADJ
ejpam-5298	22	12	-	-	ADJ
ejpam-5298	22	13	left	left	ADJ
ejpam-5298	22	14	multipliers	multiplier	NOUN
ejpam-5298	22	15	,	,	PUNCT
ejpam-5298	22	16	and	and	CCONJ
ejpam-5298	22	17	anti	anti	ADJ
ejpam-5298	22	18	-	-	ADJ
ejpam-5298	22	19	right	right	ADJ
ejpam-5298	22	20	multipliers	multiplier	NOUN
ejpam-5298	22	21	,	,	PUNCT
ejpam-5298	22	22	each	each	PRON
ejpam-5298	22	23	playing	play	VERB
ejpam-5298	22	24	a	a	DET
ejpam-5298	22	25	distinct	distinct	ADJ
ejpam-5298	22	26	role	role	NOUN
ejpam-5298	22	27	in	in	ADP
ejpam-5298	22	28	governing	govern	VERB
ejpam-5298	22	29	the	the	DET
ejpam-5298	22	30	structural	structural	ADJ
ejpam-5298	22	31	dynamics	dynamic	NOUN
ejpam-5298	22	32	of	of	ADP
ejpam-5298	22	33	these	these	DET
ejpam-5298	22	34	algebras	algebra	NOUN
ejpam-5298	22	35	.	.	PUNCT
ejpam-5298	23	1	this	this	DET
ejpam-5298	23	2	categorization	categorization	NOUN
ejpam-5298	23	3	offered	offer	VERB
ejpam-5298	23	4	a	a	DET
ejpam-5298	23	5	more	more	ADV
ejpam-5298	23	6	nuanced	nuanced	ADJ
ejpam-5298	23	7	understanding	understanding	NOUN
ejpam-5298	23	8	of	of	ADP
ejpam-5298	23	9	how	how	SCONJ
ejpam-5298	23	10	multipliers	multiplier	NOUN
ejpam-5298	23	11	can	can	AUX
ejpam-5298	23	12	be	be	AUX
ejpam-5298	23	13	applied	apply	VERB
ejpam-5298	23	14	,	,	PUNCT
ejpam-5298	23	15	expanding	expand	VERB
ejpam-5298	23	16	their	their	PRON
ejpam-5298	23	17	utility	utility	NOUN
ejpam-5298	23	18	across	across	ADP
ejpam-5298	23	19	different	different	ADJ
ejpam-5298	23	20	branches	branch	NOUN
ejpam-5298	23	21	of	of	ADP
ejpam-5298	23	22	algebra	algebra	PROPN
ejpam-5298	23	23	.	.	PUNCT
ejpam-5298	24	1	the	the	DET
ejpam-5298	24	2	concept	concept	NOUN
ejpam-5298	24	3	of	of	ADP
ejpam-5298	24	4	hilbert	hilbert	PROPN
ejpam-5298	24	5	algebras	algebras	PROPN
ejpam-5298	24	6	emerged	emerge	VERB
ejpam-5298	24	7	in	in	ADP
ejpam-5298	24	8	the	the	DET
ejpam-5298	24	9	early	early	ADJ
ejpam-5298	24	10	1950s	1950s	NUM
ejpam-5298	24	11	,	,	PUNCT
ejpam-5298	24	12	introduced	introduce	VERB
ejpam-5298	24	13	by	by	ADP
ejpam-5298	24	14	henkin	henkin	PROPN
ejpam-5298	25	1	[	[	X
ejpam-5298	25	2	9	9	NUM
ejpam-5298	25	3	]	]	PUNCT
ejpam-5298	25	4	as	as	ADP
ejpam-5298	25	5	a	a	DET
ejpam-5298	25	6	tool	tool	NOUN
ejpam-5298	25	7	for	for	ADP
ejpam-5298	25	8	analyzing	analyze	VERB
ejpam-5298	25	9	implications	implication	NOUN
ejpam-5298	25	10	in	in	ADP
ejpam-5298	25	11	intuitionistic	intuitionistic	ADJ
ejpam-5298	25	12	and	and	CCONJ
ejpam-5298	25	13	other	other	ADJ
ejpam-5298	25	14	non	non	ADJ
ejpam-5298	25	15	-	-	ADJ
ejpam-5298	25	16	classical	classical	ADJ
ejpam-5298	25	17	logic	logic	NOUN
ejpam-5298	25	18	.	.	PUNCT
ejpam-5298	26	1	by	by	ADP
ejpam-5298	26	2	the	the	DET
ejpam-5298	26	3	1960s	1960	NOUN
ejpam-5298	26	4	,	,	PUNCT
ejpam-5298	26	5	the	the	DET
ejpam-5298	26	6	algebraic	algebraic	ADJ
ejpam-5298	26	7	foundations	foundation	NOUN
ejpam-5298	26	8	of	of	ADP
ejpam-5298	26	9	these	these	DET
ejpam-5298	26	10	structures	structure	NOUN
ejpam-5298	26	11	gained	gain	VERB
ejpam-5298	26	12	prominence	prominence	NOUN
ejpam-5298	26	13	,	,	PUNCT
ejpam-5298	26	14	particularly	particularly	ADV
ejpam-5298	26	15	through	through	ADP
ejpam-5298	26	16	the	the	DET
ejpam-5298	26	17	work	work	NOUN
ejpam-5298	26	18	of	of	ADP
ejpam-5298	26	19	diego	diego	PROPN
ejpam-5298	27	1	[	[	X
ejpam-5298	27	2	6	6	NUM
ejpam-5298	27	3	]	]	PUNCT
ejpam-5298	27	4	,	,	PUNCT
ejpam-5298	27	5	who	who	PRON
ejpam-5298	27	6	demonstrated	demonstrate	VERB
ejpam-5298	27	7	that	that	SCONJ
ejpam-5298	27	8	hilbert	hilbert	PROPN
ejpam-5298	27	9	algebras	algebras	PROPN
ejpam-5298	27	10	constitute	constitute	VERB
ejpam-5298	27	11	a	a	DET
ejpam-5298	27	12	locally	locally	ADV
ejpam-5298	27	13	finite	finite	ADJ
ejpam-5298	27	14	variety	variety	PROPN
ejpam-5298	27	15	.	.	PUNCT
ejpam-5298	28	1	diego	diego	PROPN
ejpam-5298	28	2	’s	’s	PART
ejpam-5298	28	3	contributions	contribution	NOUN
ejpam-5298	28	4	marked	mark	VERB
ejpam-5298	28	5	a	a	DET
ejpam-5298	28	6	significant	significant	ADJ
ejpam-5298	28	7	milestone	milestone	NOUN
ejpam-5298	28	8	,	,	PUNCT
ejpam-5298	28	9	grounding	ground	VERB
ejpam-5298	28	10	these	these	DET
ejpam-5298	28	11	algebras	algebra	NOUN
ejpam-5298	28	12	in	in	ADP
ejpam-5298	28	13	formal	formal	ADJ
ejpam-5298	28	14	algebraic	algebraic	ADJ
ejpam-5298	28	15	theory	theory	NOUN
ejpam-5298	28	16	.	.	PUNCT
ejpam-5298	29	1	subsequent	subsequent	ADJ
ejpam-5298	29	2	research	research	NOUN
ejpam-5298	29	3	further	far	ADV
ejpam-5298	29	4	enriched	enrich	VERB
ejpam-5298	29	5	the	the	DET
ejpam-5298	29	6	understanding	understanding	NOUN
ejpam-5298	29	7	of	of	ADP
ejpam-5298	29	8	hilbert	hilbert	PROPN
ejpam-5298	29	9	algebras	algebras	PROPN
ejpam-5298	29	10	.	.	PUNCT
ejpam-5298	30	1	busneag	busneag	PROPN
ejpam-5298	31	1	[	[	X
ejpam-5298	31	2	1	1	NUM
ejpam-5298	31	3	,	,	PUNCT
ejpam-5298	31	4	2	2	NUM
ejpam-5298	31	5	]	]	PUNCT
ejpam-5298	31	6	and	and	CCONJ
ejpam-5298	31	7	jun	jun	PROPN
ejpam-5298	32	1	[	[	X
ejpam-5298	32	2	13	13	NUM
ejpam-5298	32	3	]	]	PUNCT
ejpam-5298	32	4	extended	extend	VERB
ejpam-5298	32	5	the	the	DET
ejpam-5298	32	6	study	study	NOUN
ejpam-5298	32	7	by	by	ADP
ejpam-5298	32	8	identifying	identify	VERB
ejpam-5298	32	9	filters	filter	NOUN
ejpam-5298	32	10	that	that	PRON
ejpam-5298	32	11	form	form	VERB
ejpam-5298	32	12	deductive	deductive	ADJ
ejpam-5298	32	13	systems	system	NOUN
ejpam-5298	32	14	within	within	ADP
ejpam-5298	32	15	these	these	DET
ejpam-5298	32	16	algebras	algebra	NOUN
ejpam-5298	32	17	.	.	PUNCT
ejpam-5298	33	1	their	their	PRON
ejpam-5298	33	2	work	work	NOUN
ejpam-5298	33	3	highlighted	highlight	VERB
ejpam-5298	33	4	the	the	DET
ejpam-5298	33	5	logical	logical	ADJ
ejpam-5298	33	6	and	and	CCONJ
ejpam-5298	33	7	algebraic	algebraic	ADJ
ejpam-5298	33	8	interplay	interplay	NOUN
ejpam-5298	33	9	embedded	embed	VERB
ejpam-5298	33	10	in	in	ADP
ejpam-5298	33	11	these	these	DET
ejpam-5298	33	12	structures	structure	NOUN
ejpam-5298	33	13	.	.	PUNCT
ejpam-5298	34	1	additionally	additionally	ADV
ejpam-5298	34	2	,	,	PUNCT
ejpam-5298	34	3	dudek	dudek	PROPN
ejpam-5298	35	1	[	[	X
ejpam-5298	35	2	7	7	X
ejpam-5298	35	3	]	]	PUNCT
ejpam-5298	35	4	introduced	introduce	VERB
ejpam-5298	35	5	a	a	DET
ejpam-5298	35	6	novel	novel	ADJ
ejpam-5298	35	7	perspective	perspective	NOUN
ejpam-5298	35	8	by	by	ADP
ejpam-5298	35	9	exploring	explore	VERB
ejpam-5298	35	10	the	the	DET
ejpam-5298	35	11	fuzzification	fuzzification	NOUN
ejpam-5298	35	12	of	of	ADP
ejpam-5298	35	13	subalgebras	subalgebras	PROPN
ejpam-5298	35	14	and	and	CCONJ
ejpam-5298	35	15	deductive	deductive	ADJ
ejpam-5298	35	16	systems	system	NOUN
ejpam-5298	35	17	,	,	PUNCT
ejpam-5298	35	18	broadening	broaden	VERB
ejpam-5298	35	19	the	the	DET
ejpam-5298	35	20	scope	scope	NOUN
ejpam-5298	35	21	of	of	ADP
ejpam-5298	35	22	hilbert	hilbert	PROPN
ejpam-5298	35	23	algebra	algebra	PROPN
ejpam-5298	35	24	applications	application	NOUN
ejpam-5298	35	25	in	in	ADP
ejpam-5298	35	26	the	the	DET
ejpam-5298	35	27	study	study	NOUN
ejpam-5298	35	28	of	of	ADP
ejpam-5298	35	29	uncertainty	uncertainty	NOUN
ejpam-5298	35	30	and	and	CCONJ
ejpam-5298	35	31	graded	grade	VERB
ejpam-5298	35	32	logic	logic	NOUN
ejpam-5298	35	33	.	.	PUNCT
ejpam-5298	36	1	2	2	X
ejpam-5298	36	2	.	.	X
ejpam-5298	36	3	preliminaries	preliminary	NOUN
ejpam-5298	36	4	understanding	understand	VERB
ejpam-5298	36	5	the	the	DET
ejpam-5298	36	6	foundational	foundational	ADJ
ejpam-5298	36	7	structures	structure	NOUN
ejpam-5298	36	8	within	within	ADP
ejpam-5298	36	9	mathematical	mathematical	ADJ
ejpam-5298	36	10	logic	logic	NOUN
ejpam-5298	36	11	and	and	CCONJ
ejpam-5298	36	12	algebra	algebra	NOUN
ejpam-5298	36	13	is	be	AUX
ejpam-5298	36	14	pivotal	pivotal	ADJ
ejpam-5298	36	15	for	for	ADP
ejpam-5298	36	16	advancing	advance	VERB
ejpam-5298	36	17	both	both	CCONJ
ejpam-5298	36	18	theoretical	theoretical	ADJ
ejpam-5298	36	19	insights	insight	NOUN
ejpam-5298	36	20	and	and	CCONJ
ejpam-5298	36	21	practical	practical	ADJ
ejpam-5298	36	22	applications	application	NOUN
ejpam-5298	36	23	.	.	PUNCT
ejpam-5298	37	1	hilbert	hilbert	PROPN
ejpam-5298	37	2	algebras	algebras	PROPN
ejpam-5298	37	3	,	,	PUNCT
ejpam-5298	37	4	named	name	VERB
ejpam-5298	37	5	after	after	ADP
ejpam-5298	37	6	the	the	DET
ejpam-5298	37	7	eminent	eminent	ADJ
ejpam-5298	37	8	mathematician	mathematician	NOUN
ejpam-5298	37	9	david	david	PROPN
ejpam-5298	37	10	hilbert	hilbert	PROPN
ejpam-5298	37	11	,	,	PUNCT
ejpam-5298	37	12	form	form	VERB
ejpam-5298	37	13	a	a	DET
ejpam-5298	37	14	crucial	crucial	ADJ
ejpam-5298	37	15	part	part	NOUN
ejpam-5298	37	16	of	of	ADP
ejpam-5298	37	17	this	this	DET
ejpam-5298	37	18	framework	framework	NOUN
ejpam-5298	37	19	.	.	PUNCT
ejpam-5298	38	1	these	these	DET
ejpam-5298	38	2	algebras	algebra	NOUN
ejpam-5298	38	3	not	not	PART
ejpam-5298	38	4	only	only	ADV
ejpam-5298	38	5	provide	provide	VERB
ejpam-5298	38	6	a	a	DET
ejpam-5298	38	7	robust	robust	ADJ
ejpam-5298	38	8	structure	structure	NOUN
ejpam-5298	38	9	for	for	ADP
ejpam-5298	38	10	exploring	explore	VERB
ejpam-5298	38	11	logical	logical	ADJ
ejpam-5298	38	12	connectives	connective	NOUN
ejpam-5298	38	13	and	and	CCONJ
ejpam-5298	38	14	implications	implication	NOUN
ejpam-5298	38	15	but	but	CCONJ
ejpam-5298	38	16	also	also	ADV
ejpam-5298	38	17	serve	serve	VERB
ejpam-5298	38	18	as	as	ADP
ejpam-5298	38	19	a	a	DET
ejpam-5298	38	20	bridge	bridge	NOUN
ejpam-5298	38	21	between	between	ADP
ejpam-5298	38	22	logic	logic	NOUN
ejpam-5298	38	23	and	and	CCONJ
ejpam-5298	38	24	algebraic	algebraic	ADJ
ejpam-5298	38	25	operations	operation	NOUN
ejpam-5298	38	26	.	.	PUNCT
ejpam-5298	39	1	their	their	PRON
ejpam-5298	39	2	unique	unique	ADJ
ejpam-5298	39	3	properties	property	NOUN
ejpam-5298	39	4	and	and	CCONJ
ejpam-5298	39	5	the	the	DET
ejpam-5298	39	6	relationships	relationship	NOUN
ejpam-5298	39	7	they	they	PRON
ejpam-5298	39	8	encapsulate	encapsulate	VERB
ejpam-5298	39	9	are	be	AUX
ejpam-5298	39	10	instrumental	instrumental	ADJ
ejpam-5298	39	11	in	in	ADP
ejpam-5298	39	12	various	various	ADJ
ejpam-5298	39	13	fields	field	NOUN
ejpam-5298	39	14	,	,	PUNCT
ejpam-5298	39	15	including	include	VERB
ejpam-5298	39	16	proof	proof	NOUN
ejpam-5298	39	17	theory	theory	NOUN
ejpam-5298	39	18	,	,	PUNCT
ejpam-5298	39	19	model	model	NOUN
ejpam-5298	39	20	theory	theory	NOUN
ejpam-5298	39	21	,	,	PUNCT
ejpam-5298	39	22	and	and	CCONJ
ejpam-5298	39	23	lattice	lattice	PROPN
ejpam-5298	39	24	theory	theory	NOUN
ejpam-5298	39	25	.	.	PUNCT
ejpam-5298	40	1	by	by	ADP
ejpam-5298	40	2	delving	delve	VERB
ejpam-5298	40	3	into	into	ADP
ejpam-5298	40	4	the	the	DET
ejpam-5298	40	5	notion	notion	NOUN
ejpam-5298	40	6	of	of	ADP
ejpam-5298	40	7	hilbert	hilbert	PROPN
ejpam-5298	40	8	algebras	algebras	PROPN
ejpam-5298	40	9	,	,	PUNCT
ejpam-5298	40	10	we	we	PRON
ejpam-5298	40	11	can	can	AUX
ejpam-5298	40	12	uncover	uncover	VERB
ejpam-5298	40	13	the	the	DET
ejpam-5298	40	14	intricate	intricate	ADJ
ejpam-5298	40	15	web	web	NOUN
ejpam-5298	40	16	of	of	ADP
ejpam-5298	40	17	connections	connection	NOUN
ejpam-5298	40	18	that	that	PRON
ejpam-5298	40	19	underpin	underpin	VERB
ejpam-5298	40	20	logical	logical	ADJ
ejpam-5298	40	21	deduction	deduction	NOUN
ejpam-5298	40	22	and	and	CCONJ
ejpam-5298	40	23	algebraic	algebraic	ADJ
ejpam-5298	40	24	manipulation	manipulation	NOUN
ejpam-5298	40	25	,	,	PUNCT
ejpam-5298	40	26	paving	pave	VERB
ejpam-5298	40	27	the	the	DET
ejpam-5298	40	28	way	way	NOUN
ejpam-5298	40	29	for	for	ADP
ejpam-5298	40	30	deeper	deep	ADJ
ejpam-5298	40	31	mathematical	mathematical	ADJ
ejpam-5298	40	32	discoveries	discovery	NOUN
ejpam-5298	40	33	.	.	PUNCT
ejpam-5298	41	1	definition	definition	NOUN
ejpam-5298	41	2	1	1	NUM
ejpam-5298	41	3	.	.	PUNCT
ejpam-5298	42	1	[	[	X
ejpam-5298	42	2	6	6	NUM
ejpam-5298	42	3	]	]	PUNCT
ejpam-5298	42	4	a	a	DET
ejpam-5298	42	5	hilbert	hilbert	NOUN
ejpam-5298	42	6	algebra	algebra	NOUN
ejpam-5298	42	7	is	be	AUX
ejpam-5298	42	8	a	a	DET
ejpam-5298	42	9	triplet	triplet	NOUN
ejpam-5298	42	10	with	with	ADP
ejpam-5298	42	11	the	the	DET
ejpam-5298	42	12	formula	formula	NOUN
ejpam-5298	42	13	h	h	NOUN
ejpam-5298	42	14	=	=	PUNCT
ejpam-5298	42	15	(	(	PUNCT
ejpam-5298	42	16	h	h	NOUN
ejpam-5298	42	17	,	,	PUNCT
ejpam-5298	42	18	·	·	PUNCT
ejpam-5298	42	19	,	,	PUNCT
ejpam-5298	42	20	1	1	NUM
ejpam-5298	42	21	)	)	PUNCT
ejpam-5298	42	22	,	,	PUNCT
ejpam-5298	42	23	where	where	SCONJ
ejpam-5298	42	24	h	h	NOUN
ejpam-5298	42	25	is	be	AUX
ejpam-5298	42	26	a	a	DET
ejpam-5298	42	27	nonempty	nonempty	ADJ
ejpam-5298	42	28	set	set	VERB
ejpam-5298	42	29	,	,	PUNCT
ejpam-5298	42	30	·	·	PUNCT
ejpam-5298	42	31	is	be	AUX
ejpam-5298	42	32	a	a	DET
ejpam-5298	42	33	binary	binary	ADJ
ejpam-5298	42	34	operation	operation	NOUN
ejpam-5298	42	35	,	,	PUNCT
ejpam-5298	42	36	and	and	CCONJ
ejpam-5298	42	37	1	1	NUM
ejpam-5298	42	38	is	be	AUX
ejpam-5298	42	39	a	a	DET
ejpam-5298	42	40	fixed	fix	VERB
ejpam-5298	42	41	member	member	NOUN
ejpam-5298	42	42	of	of	ADP
ejpam-5298	42	43	h	h	NOUN
ejpam-5298	42	44	that	that	PRON
ejpam-5298	42	45	is	be	AUX
ejpam-5298	42	46	true	true	ADJ
ejpam-5298	42	47	according	accord	VERB
ejpam-5298	42	48	to	to	ADP
ejpam-5298	42	49	the	the	DET
ejpam-5298	42	50	axioms	axiom	NOUN
ejpam-5298	42	51	stated	state	VERB
ejpam-5298	42	52	below	below	ADV
ejpam-5298	42	53	:	:	PUNCT
ejpam-5298	42	54	(	(	PUNCT
ejpam-5298	42	55	1	1	X
ejpam-5298	42	56	)	)	PUNCT
ejpam-5298	42	57	(	(	PUNCT
ejpam-5298	42	58	∀x	∀x	X
ejpam-5298	42	59	,	,	PUNCT
ejpam-5298	42	60	y	y	PROPN
ejpam-5298	42	61	∈	∈	PROPN
ejpam-5298	42	62	h)(x	h)(x	NOUN
ejpam-5298	42	63	·	·	PUNCT
ejpam-5298	42	64	(	(	PUNCT
ejpam-5298	42	65	y	y	PROPN
ejpam-5298	42	66	·	·	PUNCT
ejpam-5298	42	67	x	x	X
ejpam-5298	42	68	)	)	PUNCT
ejpam-5298	42	69	=	=	SYM
ejpam-5298	42	70	1	1	NUM
ejpam-5298	42	71	)	)	PUNCT
ejpam-5298	42	72	,	,	PUNCT
ejpam-5298	42	73	(	(	PUNCT
ejpam-5298	42	74	2	2	X
ejpam-5298	42	75	)	)	PUNCT
ejpam-5298	42	76	(	(	PUNCT
ejpam-5298	42	77	∀x	∀x	X
ejpam-5298	42	78	,	,	PUNCT
ejpam-5298	42	79	y	y	PROPN
ejpam-5298	42	80	,	,	PUNCT
ejpam-5298	42	81	z	z	PROPN
ejpam-5298	42	82	∈	∈	PROPN
ejpam-5298	42	83	h)((x	h)((x	NOUN
ejpam-5298	42	84	·	·	PUNCT
ejpam-5298	42	85	(	(	PUNCT
ejpam-5298	42	86	y	y	PROPN
ejpam-5298	42	87	·	·	PUNCT
ejpam-5298	42	88	z	z	NOUN
ejpam-5298	42	89	)	)	PUNCT
ejpam-5298	42	90	)	)	PUNCT
ejpam-5298	42	91	·	·	PUNCT
ejpam-5298	43	1	(	(	PUNCT
ejpam-5298	43	2	(	(	PUNCT
ejpam-5298	43	3	x	x	SYM
ejpam-5298	43	4	·	·	PUNCT
ejpam-5298	43	5	y	y	X
ejpam-5298	43	6	)	)	PUNCT
ejpam-5298	43	7	·	·	PUNCT
ejpam-5298	44	1	(	(	PUNCT
ejpam-5298	44	2	x	x	X
ejpam-5298	44	3	·	·	PUNCT
ejpam-5298	44	4	z	z	NOUN
ejpam-5298	44	5	)	)	PUNCT
ejpam-5298	44	6	)	)	PUNCT
ejpam-5298	45	1	=	=	SYM
ejpam-5298	45	2	1	1	X
ejpam-5298	45	3	)	)	PUNCT
ejpam-5298	45	4	,	,	PUNCT
ejpam-5298	45	5	(	(	PUNCT
ejpam-5298	45	6	3	3	X
ejpam-5298	45	7	)	)	PUNCT
ejpam-5298	45	8	(	(	PUNCT
ejpam-5298	45	9	∀x	∀x	X
ejpam-5298	45	10	,	,	PUNCT
ejpam-5298	45	11	y	y	PROPN
ejpam-5298	45	12	∈	∈	PROPN
ejpam-5298	45	13	h)(x	h)(x	NOUN
ejpam-5298	45	14	·	·	PUNCT
ejpam-5298	45	15	y	y	SYM
ejpam-5298	45	16	=	=	SYM
ejpam-5298	45	17	1	1	NUM
ejpam-5298	45	18	,	,	PUNCT
ejpam-5298	45	19	y	y	PROPN
ejpam-5298	45	20	·	·	PUNCT
ejpam-5298	45	21	x	x	PUNCT
ejpam-5298	45	22	=	=	SYM
ejpam-5298	45	23	1	1	NUM
ejpam-5298	45	24	⇒	⇒	NOUN
ejpam-5298	45	25	x	x	PUNCT
ejpam-5298	45	26	=	=	SYM
ejpam-5298	45	27	y	y	PROPN
ejpam-5298	45	28	)	)	PUNCT
ejpam-5298	45	29	.	.	PUNCT
ejpam-5298	46	1	a.	a.	PROPN
ejpam-5298	46	2	iampan	iampan	PROPN
ejpam-5298	46	3	,	,	PUNCT
ejpam-5298	46	4	n.	n.	PROPN
ejpam-5298	46	5	rajesh	rajesh	PROPN
ejpam-5298	46	6	/	/	SYM
ejpam-5298	46	7	eur	eur	PROPN
ejpam-5298	46	8	.	.	PUNCT
ejpam-5298	47	1	j.	j.	PROPN
ejpam-5298	47	2	pure	pure	PROPN
ejpam-5298	47	3	appl	appl	PROPN
ejpam-5298	47	4	.	.	PROPN
ejpam-5298	47	5	math	math	PROPN
ejpam-5298	47	6	,	,	PUNCT
ejpam-5298	47	7	17	17	NUM
ejpam-5298	47	8	(	(	PUNCT
ejpam-5298	47	9	4	4	NUM
ejpam-5298	47	10	)	)	PUNCT
ejpam-5298	47	11	(	(	PUNCT
ejpam-5298	47	12	2024	2024	NUM
ejpam-5298	47	13	)	)	PUNCT
ejpam-5298	47	14	,	,	PUNCT
ejpam-5298	47	15	2726	2726	NUM
ejpam-5298	47	16	-	-	SYM
ejpam-5298	47	17	2737	2737	NUM
ejpam-5298	47	18	2728	2728	NUM
ejpam-5298	47	19	example	example	NOUN
ejpam-5298	47	20	1	1	NUM
ejpam-5298	47	21	.	.	PUNCT
ejpam-5298	48	1	[	[	X
ejpam-5298	48	2	11	11	NUM
ejpam-5298	48	3	]	]	PUNCT
ejpam-5298	48	4	let	let	VERB
ejpam-5298	48	5	h	h	NOUN
ejpam-5298	48	6	=	=	PRON
ejpam-5298	48	7	{	{	PUNCT
ejpam-5298	48	8	1	1	NUM
ejpam-5298	48	9	,	,	PUNCT
ejpam-5298	48	10	α	α	NOUN
ejpam-5298	48	11	,	,	PUNCT
ejpam-5298	48	12	β	β	X
ejpam-5298	48	13	,	,	PUNCT
ejpam-5298	48	14	γ	γ	PROPN
ejpam-5298	48	15	,	,	PUNCT
ejpam-5298	48	16	ϵ	ϵ	X
ejpam-5298	48	17	}	}	PUNCT
ejpam-5298	48	18	with	with	ADP
ejpam-5298	48	19	the	the	DET
ejpam-5298	48	20	following	follow	VERB
ejpam-5298	48	21	cayley	cayley	ADJ
ejpam-5298	48	22	table	table	NOUN
ejpam-5298	48	23	:	:	PUNCT
ejpam-5298	48	24	·	·	PUNCT
ejpam-5298	48	25	1	1	NUM
ejpam-5298	48	26	α	α	NOUN
ejpam-5298	48	27	β	β	X
ejpam-5298	48	28	γ	γ	X
ejpam-5298	48	29	ϵ	ϵ	PROPN
ejpam-5298	48	30	1	1	NUM
ejpam-5298	48	31	1	1	NUM
ejpam-5298	48	32	α	α	NOUN
ejpam-5298	48	33	β	β	X
ejpam-5298	48	34	γ	γ	X
ejpam-5298	48	35	ϵ	ϵ	PROPN
ejpam-5298	48	36	α	α	PROPN
ejpam-5298	48	37	1	1	NUM
ejpam-5298	48	38	1	1	NUM
ejpam-5298	48	39	β	β	X
ejpam-5298	48	40	γ	γ	X
ejpam-5298	48	41	ϵ	ϵ	X
ejpam-5298	48	42	β	β	X
ejpam-5298	48	43	1	1	NUM
ejpam-5298	48	44	α	α	NOUN
ejpam-5298	48	45	1	1	NUM
ejpam-5298	48	46	γ	γ	PROPN
ejpam-5298	48	47	γ	γ	X
ejpam-5298	48	48	γ	γ	X
ejpam-5298	48	49	1	1	NUM
ejpam-5298	48	50	1	1	NUM
ejpam-5298	48	51	β	β	NOUN
ejpam-5298	48	52	1	1	NUM
ejpam-5298	48	53	β	β	NOUN
ejpam-5298	48	54	ϵ	ϵ	NOUN
ejpam-5298	48	55	1	1	NUM
ejpam-5298	48	56	1	1	NUM
ejpam-5298	48	57	1	1	NUM
ejpam-5298	48	58	1	1	NUM
ejpam-5298	48	59	1	1	NUM
ejpam-5298	48	60	then	then	ADV
ejpam-5298	48	61	h	h	NOUN
ejpam-5298	48	62	=	=	PUNCT
ejpam-5298	48	63	(	(	PUNCT
ejpam-5298	48	64	h	h	NOUN
ejpam-5298	48	65	,	,	PUNCT
ejpam-5298	48	66	·	·	PUNCT
ejpam-5298	48	67	,	,	PUNCT
ejpam-5298	48	68	1	1	NUM
ejpam-5298	48	69	)	)	PUNCT
ejpam-5298	48	70	is	be	AUX
ejpam-5298	48	71	a	a	DET
ejpam-5298	48	72	hilbert	hilbert	NOUN
ejpam-5298	48	73	algebra	algebra	NOUN
ejpam-5298	48	74	.	.	PUNCT
ejpam-5298	49	1	in	in	ADP
ejpam-5298	49	2	[	[	X
ejpam-5298	49	3	7	7	NUM
ejpam-5298	49	4	]	]	PUNCT
ejpam-5298	49	5	,	,	PUNCT
ejpam-5298	49	6	the	the	DET
ejpam-5298	49	7	following	follow	VERB
ejpam-5298	49	8	conclusion	conclusion	NOUN
ejpam-5298	49	9	was	be	AUX
ejpam-5298	49	10	established	establish	VERB
ejpam-5298	49	11	.	.	PUNCT
ejpam-5298	50	1	lemma	lemma	PROPN
ejpam-5298	50	2	1	1	X
ejpam-5298	50	3	.	.	PUNCT
ejpam-5298	51	1	let	let	VERB
ejpam-5298	51	2	h	h	NOUN
ejpam-5298	51	3	=	=	PUNCT
ejpam-5298	51	4	(	(	PUNCT
ejpam-5298	51	5	h	h	NOUN
ejpam-5298	51	6	,	,	PUNCT
ejpam-5298	51	7	·	·	PUNCT
ejpam-5298	51	8	,	,	PUNCT
ejpam-5298	51	9	1	1	X
ejpam-5298	51	10	)	)	PUNCT
ejpam-5298	51	11	be	be	AUX
ejpam-5298	51	12	a	a	DET
ejpam-5298	51	13	hilbert	hilbert	NOUN
ejpam-5298	51	14	algebra	algebra	NOUN
ejpam-5298	51	15	.	.	PUNCT
ejpam-5298	52	1	then	then	ADV
ejpam-5298	52	2	(	(	PUNCT
ejpam-5298	52	3	1	1	X
ejpam-5298	52	4	)	)	PUNCT
ejpam-5298	52	5	(	(	PUNCT
ejpam-5298	52	6	∀x	∀x	X
ejpam-5298	52	7	∈	∈	PROPN
ejpam-5298	52	8	h)(x	h)(x	NOUN
ejpam-5298	52	9	·	·	PUNCT
ejpam-5298	52	10	x	x	SYM
ejpam-5298	52	11	=	=	SYM
ejpam-5298	52	12	1	1	NUM
ejpam-5298	52	13	)	)	PUNCT
ejpam-5298	52	14	,	,	PUNCT
ejpam-5298	52	15	(	(	PUNCT
ejpam-5298	52	16	2	2	X
ejpam-5298	52	17	)	)	PUNCT
ejpam-5298	52	18	(	(	PUNCT
ejpam-5298	52	19	∀x	∀x	X
ejpam-5298	52	20	∈	∈	PROPN
ejpam-5298	52	21	h)(1	h)(1	X
ejpam-5298	52	22	·	·	PUNCT
ejpam-5298	52	23	x	x	PUNCT
ejpam-5298	52	24	=	=	PUNCT
ejpam-5298	52	25	x	x	NOUN
ejpam-5298	52	26	)	)	PUNCT
ejpam-5298	52	27	,	,	PUNCT
ejpam-5298	52	28	(	(	PUNCT
ejpam-5298	52	29	3	3	X
ejpam-5298	52	30	)	)	PUNCT
ejpam-5298	52	31	(	(	PUNCT
ejpam-5298	52	32	∀x	∀x	X
ejpam-5298	52	33	∈	∈	PROPN
ejpam-5298	52	34	h)(x	h)(x	NOUN
ejpam-5298	52	35	·	·	PUNCT
ejpam-5298	52	36	1	1	NUM
ejpam-5298	52	37	=	=	SYM
ejpam-5298	52	38	1	1	NUM
ejpam-5298	52	39	)	)	PUNCT
ejpam-5298	52	40	,	,	PUNCT
ejpam-5298	52	41	(	(	PUNCT
ejpam-5298	52	42	4	4	X
ejpam-5298	52	43	)	)	PUNCT
ejpam-5298	52	44	(	(	PUNCT
ejpam-5298	52	45	∀x	∀x	X
ejpam-5298	52	46	,	,	PUNCT
ejpam-5298	52	47	y	y	PROPN
ejpam-5298	52	48	,	,	PUNCT
ejpam-5298	52	49	z	z	NOUN
ejpam-5298	52	50	∈	∈	PROPN
ejpam-5298	52	51	h)(x	h)(x	NOUN
ejpam-5298	52	52	·	·	PUNCT
ejpam-5298	52	53	(	(	PUNCT
ejpam-5298	52	54	y	y	PROPN
ejpam-5298	52	55	·	·	PUNCT
ejpam-5298	52	56	z	z	X
ejpam-5298	52	57	)	)	PUNCT
ejpam-5298	52	58	=	=	SYM
ejpam-5298	52	59	y	y	PROPN
ejpam-5298	52	60	·	·	PUNCT
ejpam-5298	52	61	(	(	PUNCT
ejpam-5298	52	62	x	x	X
ejpam-5298	52	63	·	·	PUNCT
ejpam-5298	52	64	z	z	NOUN
ejpam-5298	52	65	)	)	PUNCT
ejpam-5298	52	66	)	)	PUNCT
ejpam-5298	52	67	,	,	PUNCT
ejpam-5298	52	68	(	(	PUNCT
ejpam-5298	52	69	5	5	X
ejpam-5298	52	70	)	)	PUNCT
ejpam-5298	52	71	(	(	PUNCT
ejpam-5298	52	72	∀x	∀x	X
ejpam-5298	52	73	,	,	PUNCT
ejpam-5298	52	74	y	y	PROPN
ejpam-5298	52	75	,	,	PUNCT
ejpam-5298	52	76	z	z	PROPN
ejpam-5298	52	77	∈	∈	PROPN
ejpam-5298	52	78	h)((x	h)((x	NOUN
ejpam-5298	52	79	·	·	PUNCT
ejpam-5298	52	80	z	z	X
ejpam-5298	52	81	)	)	PUNCT
ejpam-5298	52	82	·	·	PUNCT
ejpam-5298	52	83	(	(	PUNCT
ejpam-5298	52	84	(	(	PUNCT
ejpam-5298	52	85	z	z	NOUN
ejpam-5298	52	86	·	·	PUNCT
ejpam-5298	52	87	y	y	X
ejpam-5298	52	88	)	)	PUNCT
ejpam-5298	52	89	·	·	PUNCT
ejpam-5298	52	90	(	(	PUNCT
ejpam-5298	52	91	x	x	X
ejpam-5298	52	92	·	·	PUNCT
ejpam-5298	52	93	y	y	X
ejpam-5298	52	94	)	)	PUNCT
ejpam-5298	52	95	)	)	PUNCT
ejpam-5298	53	1	=	=	SYM
ejpam-5298	53	2	1	1	NUM
ejpam-5298	53	3	)	)	PUNCT
ejpam-5298	53	4	.	.	PUNCT
ejpam-5298	54	1	in	in	ADP
ejpam-5298	54	2	a	a	DET
ejpam-5298	54	3	hilbert	hilbert	NOUN
ejpam-5298	54	4	algebra	algebra	NOUN
ejpam-5298	54	5	h	h	NOUN
ejpam-5298	54	6	=	=	PUNCT
ejpam-5298	54	7	(	(	PUNCT
ejpam-5298	54	8	h	h	NOUN
ejpam-5298	54	9	,	,	PUNCT
ejpam-5298	54	10	·	·	PUNCT
ejpam-5298	54	11	,	,	PUNCT
ejpam-5298	54	12	1	1	NUM
ejpam-5298	54	13	)	)	PUNCT
ejpam-5298	54	14	,	,	PUNCT
ejpam-5298	54	15	the	the	DET
ejpam-5298	54	16	binary	binary	PROPN
ejpam-5298	54	17	relation	relation	PROPN
ejpam-5298	54	18	≤	≤	NUM
ejpam-5298	54	19	is	be	AUX
ejpam-5298	54	20	defined	define	VERB
ejpam-5298	54	21	by	by	ADP
ejpam-5298	54	22	(	(	PUNCT
ejpam-5298	54	23	∀x	∀x	NUM
ejpam-5298	54	24	,	,	PUNCT
ejpam-5298	54	25	y	y	PROPN
ejpam-5298	54	26	∈	∈	PROPN
ejpam-5298	54	27	h)(x	h)(x	PROPN
ejpam-5298	54	28	≤	≤	PROPN
ejpam-5298	54	29	y	y	PROPN
ejpam-5298	54	30	⇔	⇔	PROPN
ejpam-5298	54	31	x	x	PROPN
ejpam-5298	54	32	·	·	PUNCT
ejpam-5298	54	33	y	y	SYM
ejpam-5298	54	34	=	=	SYM
ejpam-5298	54	35	1	1	NUM
ejpam-5298	54	36	)	)	PUNCT
ejpam-5298	54	37	,	,	PUNCT
ejpam-5298	54	38	which	which	PRON
ejpam-5298	54	39	is	be	AUX
ejpam-5298	54	40	a	a	DET
ejpam-5298	54	41	partial	partial	ADJ
ejpam-5298	54	42	order	order	NOUN
ejpam-5298	54	43	on	on	ADP
ejpam-5298	54	44	h	h	NOUN
ejpam-5298	54	45	with	with	ADP
ejpam-5298	54	46	1	1	NUM
ejpam-5298	54	47	as	as	ADP
ejpam-5298	54	48	the	the	DET
ejpam-5298	54	49	largest	large	ADJ
ejpam-5298	54	50	element	element	NOUN
ejpam-5298	54	51	.	.	PUNCT
ejpam-5298	55	1	partial	partial	ADJ
ejpam-5298	55	2	order	order	NOUN
ejpam-5298	55	3	in	in	ADP
ejpam-5298	55	4	hilbert	hilbert	PROPN
ejpam-5298	55	5	algebra	algebra	PROPN
ejpam-5298	55	6	is	be	AUX
ejpam-5298	55	7	essential	essential	ADJ
ejpam-5298	55	8	for	for	ADP
ejpam-5298	55	9	studying	study	VERB
ejpam-5298	55	10	logical	logical	ADJ
ejpam-5298	55	11	structures	structure	NOUN
ejpam-5298	55	12	such	such	ADJ
ejpam-5298	55	13	as	as	ADP
ejpam-5298	55	14	lattice	lattice	ADJ
ejpam-5298	55	15	formations	formation	NOUN
ejpam-5298	55	16	and	and	CCONJ
ejpam-5298	55	17	their	their	PRON
ejpam-5298	55	18	applications	application	NOUN
ejpam-5298	55	19	in	in	ADP
ejpam-5298	55	20	model	model	NOUN
ejpam-5298	55	21	theory	theory	NOUN
ejpam-5298	55	22	.	.	PUNCT
ejpam-5298	56	1	it	it	PRON
ejpam-5298	56	2	enhances	enhance	VERB
ejpam-5298	56	3	proof	proof	NOUN
ejpam-5298	56	4	systems	system	NOUN
ejpam-5298	56	5	by	by	ADP
ejpam-5298	56	6	verifying	verify	VERB
ejpam-5298	56	7	entailment	entailment	NOUN
ejpam-5298	56	8	and	and	CCONJ
ejpam-5298	56	9	inference	inference	NOUN
ejpam-5298	56	10	theorems	theorem	NOUN
ejpam-5298	56	11	,	,	PUNCT
ejpam-5298	56	12	thereby	thereby	ADV
ejpam-5298	56	13	improving	improve	VERB
ejpam-5298	56	14	logical	logical	ADJ
ejpam-5298	56	15	deduction	deduction	NOUN
ejpam-5298	56	16	processes	process	NOUN
ejpam-5298	56	17	.	.	PUNCT
ejpam-5298	57	1	additionally	additionally	ADV
ejpam-5298	57	2	,	,	PUNCT
ejpam-5298	57	3	it	it	PRON
ejpam-5298	57	4	facilitates	facilitate	VERB
ejpam-5298	57	5	the	the	DET
ejpam-5298	57	6	analysis	analysis	NOUN
ejpam-5298	57	7	of	of	ADP
ejpam-5298	57	8	complex	complex	ADJ
ejpam-5298	57	9	algebraic	algebraic	ADJ
ejpam-5298	57	10	structures	structure	NOUN
ejpam-5298	57	11	like	like	ADP
ejpam-5298	57	12	boolean	boolean	ADJ
ejpam-5298	57	13	algebras	algebra	NOUN
ejpam-5298	57	14	,	,	PUNCT
ejpam-5298	57	15	offering	offer	VERB
ejpam-5298	57	16	insights	insight	NOUN
ejpam-5298	57	17	into	into	ADP
ejpam-5298	57	18	their	their	PRON
ejpam-5298	57	19	hierarchical	hierarchical	ADJ
ejpam-5298	57	20	relationships	relationship	NOUN
ejpam-5298	57	21	.	.	PUNCT
ejpam-5298	58	1	these	these	DET
ejpam-5298	58	2	applications	application	NOUN
ejpam-5298	58	3	underscore	underscore	VERB
ejpam-5298	58	4	the	the	DET
ejpam-5298	58	5	significance	significance	NOUN
ejpam-5298	58	6	of	of	ADP
ejpam-5298	58	7	partial	partial	ADJ
ejpam-5298	58	8	order	order	NOUN
ejpam-5298	58	9	in	in	ADP
ejpam-5298	58	10	comprehending	comprehend	VERB
ejpam-5298	58	11	and	and	CCONJ
ejpam-5298	58	12	investigating	investigate	VERB
ejpam-5298	58	13	mathematical	mathematical	ADJ
ejpam-5298	58	14	and	and	CCONJ
ejpam-5298	58	15	logical	logical	ADJ
ejpam-5298	58	16	frameworks	framework	NOUN
ejpam-5298	58	17	within	within	ADP
ejpam-5298	58	18	hilbert	hilbert	PROPN
ejpam-5298	58	19	algebras	algebras	PROPN
ejpam-5298	58	20	.	.	PUNCT
ejpam-5298	59	1	definition	definition	NOUN
ejpam-5298	59	2	2	2	NUM
ejpam-5298	59	3	.	.	PUNCT
ejpam-5298	60	1	[	[	X
ejpam-5298	60	2	14	14	NUM
ejpam-5298	60	3	]	]	PUNCT
ejpam-5298	60	4	a	a	DET
ejpam-5298	60	5	nonempty	nonempty	NOUN
ejpam-5298	60	6	subset	subset	VERB
ejpam-5298	60	7	d	d	NOUN
ejpam-5298	60	8	of	of	ADP
ejpam-5298	60	9	a	a	DET
ejpam-5298	60	10	hilbert	hilbert	NOUN
ejpam-5298	60	11	algebra	algebra	NOUN
ejpam-5298	60	12	h	h	NOUN
ejpam-5298	60	13	=	=	PUNCT
ejpam-5298	60	14	(	(	PUNCT
ejpam-5298	60	15	h	h	NOUN
ejpam-5298	60	16	,	,	PUNCT
ejpam-5298	60	17	·	·	PUNCT
ejpam-5298	60	18	,	,	PUNCT
ejpam-5298	60	19	1	1	NUM
ejpam-5298	60	20	)	)	PUNCT
ejpam-5298	60	21	is	be	AUX
ejpam-5298	60	22	called	call	VERB
ejpam-5298	60	23	a	a	DET
ejpam-5298	60	24	subalgebra	subalgebra	NOUN
ejpam-5298	60	25	of	of	ADP
ejpam-5298	60	26	h	h	NOUN
ejpam-5298	60	27	if	if	SCONJ
ejpam-5298	60	28	x	x	X
ejpam-5298	60	29	·	·	PUNCT
ejpam-5298	60	30	y	y	X
ejpam-5298	60	31	∈	∈	PROPN
ejpam-5298	60	32	d	d	NOUN
ejpam-5298	60	33	for	for	ADP
ejpam-5298	60	34	all	all	DET
ejpam-5298	60	35	x	x	NOUN
ejpam-5298	60	36	,	,	PUNCT
ejpam-5298	60	37	y	y	PROPN
ejpam-5298	60	38	∈	∈	PROPN
ejpam-5298	60	39	d.	d.	PROPN
ejpam-5298	60	40	definition	definition	NOUN
ejpam-5298	60	41	3	3	NUM
ejpam-5298	60	42	.	.	PUNCT
ejpam-5298	61	1	[	[	X
ejpam-5298	61	2	3	3	X
ejpam-5298	61	3	]	]	PUNCT
ejpam-5298	61	4	a	a	DET
ejpam-5298	61	5	nonempty	nonempty	NOUN
ejpam-5298	61	6	subset	subset	VERB
ejpam-5298	61	7	d	d	NOUN
ejpam-5298	61	8	of	of	ADP
ejpam-5298	61	9	a	a	DET
ejpam-5298	61	10	hilbert	hilbert	NOUN
ejpam-5298	61	11	algebra	algebra	NOUN
ejpam-5298	61	12	h	h	NOUN
ejpam-5298	61	13	=	=	PUNCT
ejpam-5298	61	14	(	(	PUNCT
ejpam-5298	61	15	h	h	NOUN
ejpam-5298	61	16	,	,	PUNCT
ejpam-5298	61	17	·	·	PUNCT
ejpam-5298	61	18	,	,	PUNCT
ejpam-5298	61	19	1	1	NUM
ejpam-5298	61	20	)	)	PUNCT
ejpam-5298	61	21	is	be	AUX
ejpam-5298	61	22	called	call	VERB
ejpam-5298	61	23	an	an	DET
ejpam-5298	61	24	ideal	ideal	NOUN
ejpam-5298	61	25	of	of	ADP
ejpam-5298	61	26	h	h	NOUN
ejpam-5298	61	27	if	if	SCONJ
ejpam-5298	61	28	the	the	DET
ejpam-5298	61	29	following	follow	VERB
ejpam-5298	61	30	conditions	condition	NOUN
ejpam-5298	61	31	hold	hold	VERB
ejpam-5298	61	32	:	:	PUNCT
ejpam-5298	61	33	(	(	PUNCT
ejpam-5298	61	34	1	1	X
ejpam-5298	61	35	)	)	SYM
ejpam-5298	61	36	1	1	NUM
ejpam-5298	61	37	∈	∈	NOUN
ejpam-5298	61	38	d	d	NOUN
ejpam-5298	61	39	,	,	PUNCT
ejpam-5298	61	40	(	(	PUNCT
ejpam-5298	61	41	2	2	NUM
ejpam-5298	61	42	)	)	PUNCT
ejpam-5298	61	43	(	(	PUNCT
ejpam-5298	61	44	∀x	∀x	X
ejpam-5298	61	45	,	,	PUNCT
ejpam-5298	61	46	y	y	PROPN
ejpam-5298	61	47	∈	∈	PROPN
ejpam-5298	61	48	h)(y	h)(y	PUNCT
ejpam-5298	61	49	∈	∈	PROPN
ejpam-5298	61	50	d	d	NOUN
ejpam-5298	61	51	⇒	⇒	NOUN
ejpam-5298	61	52	x	x	X
ejpam-5298	61	53	·	·	PUNCT
ejpam-5298	61	54	y	y	X
ejpam-5298	61	55	∈	∈	PROPN
ejpam-5298	61	56	d	d	PROPN
ejpam-5298	61	57	)	)	PUNCT
ejpam-5298	61	58	,	,	PUNCT
ejpam-5298	61	59	(	(	PUNCT
ejpam-5298	61	60	3	3	X
ejpam-5298	61	61	)	)	PUNCT
ejpam-5298	61	62	(	(	PUNCT
ejpam-5298	61	63	∀x	∀x	X
ejpam-5298	61	64	,	,	PUNCT
ejpam-5298	61	65	y1	y1	X
ejpam-5298	61	66	,	,	PUNCT
ejpam-5298	61	67	y2	y2	PROPN
ejpam-5298	61	68	∈	∈	PROPN
ejpam-5298	61	69	h)(y1	h)(y1	NOUN
ejpam-5298	61	70	,	,	PUNCT
ejpam-5298	61	71	y2	y2	NOUN
ejpam-5298	61	72	∈	∈	PROPN
ejpam-5298	61	73	d	d	X
ejpam-5298	61	74	⇒	⇒	NOUN
ejpam-5298	61	75	(	(	PUNCT
ejpam-5298	61	76	y1	y1	INTJ
ejpam-5298	61	77	·	·	PUNCT
ejpam-5298	61	78	(	(	PUNCT
ejpam-5298	61	79	y2	y2	INTJ
ejpam-5298	61	80	·	·	PUNCT
ejpam-5298	61	81	x	x	X
ejpam-5298	61	82	)	)	PUNCT
ejpam-5298	61	83	)	)	PUNCT
ejpam-5298	61	84	·	·	PUNCT
ejpam-5298	62	1	x	x	PUNCT
ejpam-5298	62	2	∈	∈	PROPN
ejpam-5298	62	3	d	d	NOUN
ejpam-5298	62	4	)	)	PUNCT
ejpam-5298	62	5	.	.	PUNCT
ejpam-5298	63	1	a.	a.	PROPN
ejpam-5298	63	2	iampan	iampan	PROPN
ejpam-5298	63	3	,	,	PUNCT
ejpam-5298	63	4	n.	n.	PROPN
ejpam-5298	63	5	rajesh	rajesh	PROPN
ejpam-5298	63	6	/	/	SYM
ejpam-5298	63	7	eur	eur	PROPN
ejpam-5298	63	8	.	.	PUNCT
ejpam-5298	64	1	j.	j.	PROPN
ejpam-5298	64	2	pure	pure	PROPN
ejpam-5298	64	3	appl	appl	PROPN
ejpam-5298	64	4	.	.	PROPN
ejpam-5298	64	5	math	math	PROPN
ejpam-5298	64	6	,	,	PUNCT
ejpam-5298	64	7	17	17	NUM
ejpam-5298	64	8	(	(	PUNCT
ejpam-5298	64	9	4	4	NUM
ejpam-5298	64	10	)	)	PUNCT
ejpam-5298	64	11	(	(	PUNCT
ejpam-5298	64	12	2024	2024	NUM
ejpam-5298	64	13	)	)	PUNCT
ejpam-5298	64	14	,	,	PUNCT
ejpam-5298	64	15	2726	2726	NUM
ejpam-5298	64	16	-	-	SYM
ejpam-5298	64	17	2737	2737	NUM
ejpam-5298	64	18	2729	2729	NUM
ejpam-5298	64	19	definition	definition	NOUN
ejpam-5298	64	20	4	4	NUM
ejpam-5298	64	21	.	.	PUNCT
ejpam-5298	65	1	[	[	X
ejpam-5298	65	2	8	8	X
ejpam-5298	65	3	]	]	PUNCT
ejpam-5298	65	4	a	a	DET
ejpam-5298	65	5	nonempty	nonempty	NOUN
ejpam-5298	65	6	subset	subset	VERB
ejpam-5298	65	7	d	d	NOUN
ejpam-5298	65	8	of	of	ADP
ejpam-5298	65	9	a	a	DET
ejpam-5298	65	10	hilbert	hilbert	NOUN
ejpam-5298	65	11	algebra	algebra	NOUN
ejpam-5298	65	12	h	h	NOUN
ejpam-5298	65	13	=	=	PUNCT
ejpam-5298	65	14	(	(	PUNCT
ejpam-5298	65	15	h	h	NOUN
ejpam-5298	65	16	,	,	PUNCT
ejpam-5298	65	17	·	·	PUNCT
ejpam-5298	65	18	,	,	PUNCT
ejpam-5298	65	19	1	1	NUM
ejpam-5298	65	20	)	)	PUNCT
ejpam-5298	65	21	is	be	AUX
ejpam-5298	65	22	called	call	VERB
ejpam-5298	65	23	a	a	DET
ejpam-5298	65	24	deductive	deductive	ADJ
ejpam-5298	65	25	system	system	NOUN
ejpam-5298	65	26	(	(	PUNCT
ejpam-5298	65	27	or	or	CCONJ
ejpam-5298	65	28	implication	implication	NOUN
ejpam-5298	65	29	filter	filter	NOUN
ejpam-5298	65	30	or	or	CCONJ
ejpam-5298	65	31	simply	simply	ADV
ejpam-5298	65	32	filter	filter	NOUN
ejpam-5298	65	33	)	)	PUNCT
ejpam-5298	65	34	of	of	ADP
ejpam-5298	65	35	h	h	NOUN
ejpam-5298	65	36	if	if	SCONJ
ejpam-5298	65	37	the	the	DET
ejpam-5298	65	38	following	follow	VERB
ejpam-5298	65	39	conditions	condition	NOUN
ejpam-5298	65	40	hold	hold	VERB
ejpam-5298	65	41	:	:	PUNCT
ejpam-5298	66	1	(	(	PUNCT
ejpam-5298	66	2	1	1	X
ejpam-5298	66	3	)	)	SYM
ejpam-5298	66	4	1	1	NUM
ejpam-5298	66	5	∈	∈	NOUN
ejpam-5298	66	6	d	d	NOUN
ejpam-5298	66	7	,	,	PUNCT
ejpam-5298	66	8	(	(	PUNCT
ejpam-5298	66	9	2	2	NUM
ejpam-5298	66	10	)	)	PUNCT
ejpam-5298	66	11	(	(	PUNCT
ejpam-5298	66	12	∀x	∀x	X
ejpam-5298	66	13	,	,	PUNCT
ejpam-5298	66	14	y	y	PROPN
ejpam-5298	66	15	∈	∈	PROPN
ejpam-5298	66	16	h)(x	h)(x	PROPN
ejpam-5298	66	17	·	·	PUNCT
ejpam-5298	66	18	y	y	X
ejpam-5298	66	19	,	,	PUNCT
ejpam-5298	66	20	x	x	SYM
ejpam-5298	66	21	∈	∈	PROPN
ejpam-5298	66	22	d	d	X
ejpam-5298	66	23	⇒	⇒	NOUN
ejpam-5298	66	24	y	y	PROPN
ejpam-5298	66	25	∈	∈	PROPN
ejpam-5298	66	26	d	d	PROPN
ejpam-5298	66	27	)	)	PUNCT
ejpam-5298	66	28	.	.	PUNCT
ejpam-5298	67	1	3	3	X
ejpam-5298	67	2	.	.	X
ejpam-5298	67	3	multipliers	multiplier	NOUN
ejpam-5298	67	4	of	of	ADP
ejpam-5298	67	5	hilbert	hilbert	PROPN
ejpam-5298	67	6	algebras	algebras	PROPN
ejpam-5298	67	7	in	in	ADP
ejpam-5298	67	8	this	this	DET
ejpam-5298	67	9	section	section	NOUN
ejpam-5298	67	10	,	,	PUNCT
ejpam-5298	67	11	we	we	PRON
ejpam-5298	67	12	present	present	VERB
ejpam-5298	67	13	the	the	DET
ejpam-5298	67	14	concepts	concept	NOUN
ejpam-5298	67	15	of	of	ADP
ejpam-5298	67	16	left	left	ADJ
ejpam-5298	67	17	multipliers	multiplier	NOUN
ejpam-5298	67	18	,	,	PUNCT
ejpam-5298	67	19	right	right	ADJ
ejpam-5298	67	20	multipliers	multiplier	NOUN
ejpam-5298	67	21	,	,	PUNCT
ejpam-5298	67	22	anti	anti	ADJ
ejpam-5298	67	23	-	-	ADJ
ejpam-5298	67	24	left	left	ADJ
ejpam-5298	67	25	multipliers	multiplier	NOUN
ejpam-5298	67	26	,	,	PUNCT
ejpam-5298	67	27	and	and	CCONJ
ejpam-5298	67	28	anti	anti	ADJ
ejpam-5298	67	29	-	-	ADJ
ejpam-5298	67	30	right	right	ADJ
ejpam-5298	67	31	multipliers	multiplier	NOUN
ejpam-5298	67	32	within	within	ADP
ejpam-5298	67	33	hilbert	hilbert	PROPN
ejpam-5298	67	34	algebras	algebras	PROPN
ejpam-5298	67	35	,	,	PUNCT
ejpam-5298	67	36	along	along	ADP
ejpam-5298	67	37	with	with	ADP
ejpam-5298	67	38	the	the	DET
ejpam-5298	67	39	notion	notion	NOUN
ejpam-5298	67	40	of	of	ADP
ejpam-5298	67	41	near	near	ADJ
ejpam-5298	67	42	filters	filter	NOUN
ejpam-5298	67	43	.	.	PUNCT
ejpam-5298	68	1	we	we	PRON
ejpam-5298	68	2	also	also	ADV
ejpam-5298	68	3	examine	examine	VERB
ejpam-5298	68	4	the	the	DET
ejpam-5298	68	5	intricate	intricate	ADJ
ejpam-5298	68	6	relationship	relationship	NOUN
ejpam-5298	68	7	between	between	ADP
ejpam-5298	68	8	right	right	ADJ
ejpam-5298	68	9	multipliers	multiplier	NOUN
ejpam-5298	68	10	and	and	CCONJ
ejpam-5298	68	11	near	near	ADJ
ejpam-5298	68	12	filters	filter	NOUN
ejpam-5298	68	13	,	,	PUNCT
ejpam-5298	68	14	shedding	shed	VERB
ejpam-5298	68	15	light	light	NOUN
ejpam-5298	68	16	on	on	ADP
ejpam-5298	68	17	how	how	SCONJ
ejpam-5298	68	18	these	these	DET
ejpam-5298	68	19	elements	element	NOUN
ejpam-5298	68	20	interact	interact	VERB
ejpam-5298	68	21	within	within	ADP
ejpam-5298	68	22	the	the	DET
ejpam-5298	68	23	algebraic	algebraic	ADJ
ejpam-5298	68	24	framework	framework	NOUN
ejpam-5298	68	25	.	.	PUNCT
ejpam-5298	69	1	henceforth	henceforth	ADV
ejpam-5298	69	2	,	,	PUNCT
ejpam-5298	69	3	unless	unless	SCONJ
ejpam-5298	69	4	stated	state	VERB
ejpam-5298	69	5	otherwise	otherwise	ADV
ejpam-5298	69	6	,	,	PUNCT
ejpam-5298	69	7	we	we	PRON
ejpam-5298	69	8	will	will	AUX
ejpam-5298	69	9	consider	consider	VERB
ejpam-5298	69	10	h	h	NOUN
ejpam-5298	69	11	as	as	ADP
ejpam-5298	69	12	a	a	DET
ejpam-5298	69	13	hilbert	hilbert	NOUN
ejpam-5298	69	14	algebra	algebra	NOUN
ejpam-5298	69	15	denoted	denote	VERB
ejpam-5298	69	16	by	by	ADP
ejpam-5298	69	17	h	h	NOUN
ejpam-5298	69	18	=	=	SYM
ejpam-5298	69	19	(	(	PUNCT
ejpam-5298	69	20	h	h	NOUN
ejpam-5298	69	21	,	,	PUNCT
ejpam-5298	69	22	·	·	PUNCT
ejpam-5298	69	23	,	,	PUNCT
ejpam-5298	69	24	1	1	NUM
ejpam-5298	69	25	)	)	PUNCT
ejpam-5298	69	26	.	.	PUNCT
ejpam-5298	70	1	definition	definition	NOUN
ejpam-5298	70	2	5	5	NUM
ejpam-5298	70	3	.	.	PUNCT
ejpam-5298	71	1	a	a	DET
ejpam-5298	71	2	self	self	NOUN
ejpam-5298	71	3	-	-	PUNCT
ejpam-5298	71	4	map	map	NOUN
ejpam-5298	71	5	m	m	NOUN
ejpam-5298	71	6	of	of	ADP
ejpam-5298	71	7	h	h	NOUN
ejpam-5298	71	8	is	be	AUX
ejpam-5298	71	9	called	call	VERB
ejpam-5298	71	10	(	(	PUNCT
ejpam-5298	71	11	1	1	NUM
ejpam-5298	71	12	)	)	PUNCT
ejpam-5298	71	13	a	a	DET
ejpam-5298	71	14	left	left	ADJ
ejpam-5298	71	15	multiplier	multiplier	ADV
ejpam-5298	71	16	if	if	SCONJ
ejpam-5298	71	17	m(x	m(x	PROPN
ejpam-5298	71	18	·	·	PUNCT
ejpam-5298	71	19	y	y	X
ejpam-5298	71	20	)	)	PUNCT
ejpam-5298	71	21	=	=	SYM
ejpam-5298	71	22	m(x	m(x	PROPN
ejpam-5298	71	23	)	)	PUNCT
ejpam-5298	71	24	·	·	PUNCT
ejpam-5298	72	1	y	y	PROPN
ejpam-5298	72	2	for	for	ADP
ejpam-5298	72	3	all	all	DET
ejpam-5298	72	4	x	x	NOUN
ejpam-5298	72	5	,	,	PUNCT
ejpam-5298	72	6	y	y	PROPN
ejpam-5298	72	7	∈	∈	PROPN
ejpam-5298	72	8	h	h	NOUN
ejpam-5298	72	9	,	,	PUNCT
ejpam-5298	72	10	(	(	PUNCT
ejpam-5298	72	11	2	2	X
ejpam-5298	72	12	)	)	PUNCT
ejpam-5298	72	13	a	a	DET
ejpam-5298	72	14	right	right	NOUN
ejpam-5298	72	15	multiplier	multiplier	ADV
ejpam-5298	72	16	if	if	SCONJ
ejpam-5298	72	17	m(x	m(x	PROPN
ejpam-5298	72	18	·	·	PUNCT
ejpam-5298	72	19	y	y	X
ejpam-5298	72	20	)	)	PUNCT
ejpam-5298	72	21	=	=	SYM
ejpam-5298	72	22	x	x	SYM
ejpam-5298	72	23	·	·	PUNCT
ejpam-5298	72	24	m(y	m(y	NOUN
ejpam-5298	72	25	)	)	PUNCT
ejpam-5298	72	26	for	for	ADP
ejpam-5298	72	27	all	all	DET
ejpam-5298	72	28	x	x	NOUN
ejpam-5298	72	29	,	,	PUNCT
ejpam-5298	72	30	y	y	PROPN
ejpam-5298	72	31	∈	∈	PROPN
ejpam-5298	72	32	h	h	NOUN
ejpam-5298	72	33	,	,	PUNCT
ejpam-5298	72	34	(	(	PUNCT
ejpam-5298	72	35	3	3	X
ejpam-5298	72	36	)	)	PUNCT
ejpam-5298	72	37	an	an	DET
ejpam-5298	72	38	anti	anti	ADJ
ejpam-5298	72	39	-	-	ADJ
ejpam-5298	72	40	left	left	ADJ
ejpam-5298	72	41	multiplier	multipli	ADJ
ejpam-5298	72	42	if	if	SCONJ
ejpam-5298	72	43	m(x	m(x	PROPN
ejpam-5298	72	44	·	·	PUNCT
ejpam-5298	72	45	y	y	X
ejpam-5298	72	46	)	)	PUNCT
ejpam-5298	72	47	=	=	SYM
ejpam-5298	72	48	y	y	PROPN
ejpam-5298	72	49	·	·	PUNCT
ejpam-5298	72	50	m(x	m(x	PROPN
ejpam-5298	72	51	)	)	PUNCT
ejpam-5298	72	52	for	for	ADP
ejpam-5298	72	53	all	all	DET
ejpam-5298	72	54	x	x	NOUN
ejpam-5298	72	55	,	,	PUNCT
ejpam-5298	72	56	y	y	PROPN
ejpam-5298	72	57	∈	∈	PROPN
ejpam-5298	72	58	h	h	NOUN
ejpam-5298	72	59	,	,	PUNCT
ejpam-5298	72	60	(	(	PUNCT
ejpam-5298	72	61	4	4	X
ejpam-5298	72	62	)	)	PUNCT
ejpam-5298	72	63	an	an	DET
ejpam-5298	72	64	anti	anti	ADJ
ejpam-5298	72	65	-	-	ADJ
ejpam-5298	72	66	right	right	ADJ
ejpam-5298	72	67	multiplier	multiplier	ADV
ejpam-5298	72	68	if	if	SCONJ
ejpam-5298	72	69	m(x	m(x	PROPN
ejpam-5298	72	70	·	·	PUNCT
ejpam-5298	72	71	y	y	X
ejpam-5298	72	72	)	)	PUNCT
ejpam-5298	72	73	=	=	SYM
ejpam-5298	72	74	m(y	m(y	NOUN
ejpam-5298	72	75	)	)	PUNCT
ejpam-5298	72	76	·	·	PUNCT
ejpam-5298	73	1	x	x	PUNCT
ejpam-5298	73	2	for	for	ADP
ejpam-5298	73	3	all	all	DET
ejpam-5298	73	4	x	x	NOUN
ejpam-5298	73	5	,	,	PUNCT
ejpam-5298	73	6	y	y	PROPN
ejpam-5298	73	7	∈	∈	PROPN
ejpam-5298	73	8	h.	h.	PROPN
ejpam-5298	73	9	definition	definition	NOUN
ejpam-5298	73	10	6	6	NUM
ejpam-5298	73	11	.	.	PUNCT
ejpam-5298	73	12	define	define	VERB
ejpam-5298	73	13	a	a	DET
ejpam-5298	73	14	self	self	NOUN
ejpam-5298	73	15	-	-	PUNCT
ejpam-5298	73	16	map	map	NOUN
ejpam-5298	73	17	ih	ih	X
ejpam-5298	73	18	:	:	PUNCT
ejpam-5298	73	19	h	h	NOUN
ejpam-5298	73	20	→	→	SYM
ejpam-5298	73	21	h	h	NOUN
ejpam-5298	73	22	by	by	ADV
ejpam-5298	73	23	,	,	PUNCT
ejpam-5298	73	24	for	for	ADP
ejpam-5298	73	25	any	any	DET
ejpam-5298	73	26	x	x	SYM
ejpam-5298	73	27	∈	∈	PROPN
ejpam-5298	73	28	h	h	NOUN
ejpam-5298	73	29	,	,	PUNCT
ejpam-5298	73	30	ih(x	ih(x	X
ejpam-5298	73	31	)	)	PUNCT
ejpam-5298	74	1	=	=	PUNCT
ejpam-5298	75	1	x.	x.	NOUN
ejpam-5298	75	2	then	then	ADV
ejpam-5298	75	3	ih	ih	PROPN
ejpam-5298	75	4	is	be	AUX
ejpam-5298	75	5	a	a	DET
ejpam-5298	75	6	left	left	ADJ
ejpam-5298	75	7	multiplier	multipli	ADJ
ejpam-5298	75	8	and	and	CCONJ
ejpam-5298	75	9	a	a	DET
ejpam-5298	75	10	right	right	ADJ
ejpam-5298	75	11	multiplier	multipli	ADJ
ejpam-5298	75	12	of	of	ADP
ejpam-5298	75	13	h.	h.	NOUN
ejpam-5298	75	14	proposition	proposition	PROPN
ejpam-5298	75	15	1	1	X
ejpam-5298	75	16	.	.	PUNCT
ejpam-5298	76	1	let	let	VERB
ejpam-5298	76	2	m	m	PRON
ejpam-5298	76	3	be	be	AUX
ejpam-5298	76	4	a	a	DET
ejpam-5298	76	5	self	self	NOUN
ejpam-5298	76	6	-	-	PUNCT
ejpam-5298	76	7	map	map	NOUN
ejpam-5298	76	8	of	of	ADP
ejpam-5298	76	9	h.	h.	PROPN
ejpam-5298	76	10	then	then	ADV
ejpam-5298	76	11	the	the	DET
ejpam-5298	76	12	following	following	ADJ
ejpam-5298	76	13	statements	statement	NOUN
ejpam-5298	76	14	hold	hold	VERB
ejpam-5298	76	15	:	:	PUNCT
ejpam-5298	76	16	(	(	PUNCT
ejpam-5298	76	17	1	1	X
ejpam-5298	76	18	)	)	PUNCT
ejpam-5298	77	1	m	m	VERB
ejpam-5298	77	2	is	be	AUX
ejpam-5298	77	3	a	a	DET
ejpam-5298	77	4	left	left	ADJ
ejpam-5298	77	5	multiplier	multipli	ADJ
ejpam-5298	77	6	h	h	NOUN
ejpam-5298	78	1	if	if	SCONJ
ejpam-5298	79	1	and	and	CCONJ
ejpam-5298	79	2	only	only	ADV
ejpam-5298	79	3	if	if	SCONJ
ejpam-5298	79	4	m	m	VERB
ejpam-5298	79	5	=	=	VERB
ejpam-5298	79	6	ih	ih	NOUN
ejpam-5298	79	7	,	,	PUNCT
ejpam-5298	79	8	(	(	PUNCT
ejpam-5298	79	9	2	2	X
ejpam-5298	79	10	)	)	PUNCT
ejpam-5298	79	11	if	if	SCONJ
ejpam-5298	79	12	m	m	NOUN
ejpam-5298	79	13	is	be	AUX
ejpam-5298	79	14	an	an	DET
ejpam-5298	79	15	anti	anti	ADJ
ejpam-5298	79	16	-	-	ADJ
ejpam-5298	79	17	left	left	ADJ
ejpam-5298	79	18	multiplier	multipli	ADJ
ejpam-5298	79	19	of	of	ADP
ejpam-5298	79	20	h	h	NOUN
ejpam-5298	79	21	,	,	PUNCT
ejpam-5298	79	22	then	then	ADV
ejpam-5298	79	23	m	m	VERB
ejpam-5298	79	24	is	be	AUX
ejpam-5298	79	25	constant	constant	ADJ
ejpam-5298	79	26	,	,	PUNCT
ejpam-5298	79	27	(	(	PUNCT
ejpam-5298	79	28	3	3	X
ejpam-5298	79	29	)	)	PUNCT
ejpam-5298	79	30	m	m	VERB
ejpam-5298	79	31	is	be	AUX
ejpam-5298	79	32	an	an	DET
ejpam-5298	79	33	anti	anti	ADJ
ejpam-5298	79	34	-	-	ADJ
ejpam-5298	79	35	right	right	ADJ
ejpam-5298	79	36	multiplier	multipli	ADJ
ejpam-5298	79	37	of	of	ADP
ejpam-5298	79	38	h	h	NOUN
ejpam-5298	80	1	if	if	SCONJ
ejpam-5298	81	1	and	and	CCONJ
ejpam-5298	81	2	only	only	ADV
ejpam-5298	81	3	if	if	SCONJ
ejpam-5298	81	4	h	h	NOUN
ejpam-5298	81	5	=	=	PUNCT
ejpam-5298	81	6	{	{	PUNCT
ejpam-5298	81	7	1	1	NUM
ejpam-5298	81	8	}	}	PUNCT
ejpam-5298	81	9	.	.	PUNCT
ejpam-5298	82	1	proof	proof	NOUN
ejpam-5298	82	2	.	.	PUNCT
ejpam-5298	83	1	(	(	PUNCT
ejpam-5298	83	2	1	1	X
ejpam-5298	83	3	)	)	PUNCT
ejpam-5298	83	4	assume	assume	VERB
ejpam-5298	83	5	that	that	SCONJ
ejpam-5298	83	6	m	m	PROPN
ejpam-5298	83	7	is	be	AUX
ejpam-5298	83	8	a	a	DET
ejpam-5298	83	9	left	left	ADJ
ejpam-5298	83	10	multiplier	multipli	ADJ
ejpam-5298	83	11	of	of	ADP
ejpam-5298	83	12	h.	h.	PROPN
ejpam-5298	83	13	then	then	ADV
ejpam-5298	83	14	m(1	m(1	PROPN
ejpam-5298	83	15	)	)	PUNCT
ejpam-5298	84	1	=	=	PUNCT
ejpam-5298	84	2	m(1	m(1	NOUN
ejpam-5298	84	3	·	·	PUNCT
ejpam-5298	84	4	1	1	NUM
ejpam-5298	84	5	)	)	PUNCT
ejpam-5298	84	6	(	(	PUNCT
ejpam-5298	84	7	lemma	lemma	PROPN
ejpam-5298	84	8	1	1	NUM
ejpam-5298	84	9	(	(	PUNCT
ejpam-5298	84	10	1	1	NUM
ejpam-5298	84	11	)	)	PUNCT
ejpam-5298	84	12	)	)	PUNCT
ejpam-5298	85	1	=	=	SYM
ejpam-5298	86	1	m(1	m(1	NOUN
ejpam-5298	86	2	)	)	PUNCT
ejpam-5298	86	3	·	·	PUNCT
ejpam-5298	86	4	1	1	NUM
ejpam-5298	86	5	(	(	PUNCT
ejpam-5298	86	6	left	leave	VERB
ejpam-5298	86	7	multiplier	multiplier	ADV
ejpam-5298	86	8	)	)	PUNCT
ejpam-5298	86	9	=	=	SYM
ejpam-5298	87	1	1	1	X
ejpam-5298	87	2	.	.	PUNCT
ejpam-5298	88	1	(	(	PUNCT
ejpam-5298	88	2	lemma	lemma	PROPN
ejpam-5298	88	3	1	1	NUM
ejpam-5298	88	4	(	(	PUNCT
ejpam-5298	88	5	3	3	NUM
ejpam-5298	88	6	)	)	PUNCT
ejpam-5298	88	7	)	)	PUNCT
ejpam-5298	88	8	let	let	VERB
ejpam-5298	88	9	x	x	SYM
ejpam-5298	88	10	∈	∈	PROPN
ejpam-5298	88	11	h.	h.	PROPN
ejpam-5298	88	12	then	then	ADV
ejpam-5298	88	13	m(x	m(x	X
ejpam-5298	88	14	)	)	PUNCT
ejpam-5298	89	1	=	=	PUNCT
ejpam-5298	90	1	m(1	m(1	NOUN
ejpam-5298	90	2	·	·	PUNCT
ejpam-5298	90	3	x	x	X
ejpam-5298	90	4	)	)	PUNCT
ejpam-5298	90	5	(	(	PUNCT
ejpam-5298	90	6	lemma	lemma	PROPN
ejpam-5298	90	7	1	1	NUM
ejpam-5298	90	8	(	(	PUNCT
ejpam-5298	90	9	2	2	NUM
ejpam-5298	90	10	)	)	PUNCT
ejpam-5298	90	11	)	)	PUNCT
ejpam-5298	90	12	=	=	SYM
ejpam-5298	91	1	m(1	m(1	NOUN
ejpam-5298	91	2	)	)	PUNCT
ejpam-5298	91	3	·	·	PUNCT
ejpam-5298	92	1	x	x	X
ejpam-5298	92	2	(	(	PUNCT
ejpam-5298	92	3	left	leave	VERB
ejpam-5298	92	4	multiplier	multiplier	ADV
ejpam-5298	92	5	)	)	PUNCT
ejpam-5298	92	6	a.	a.	NOUN
ejpam-5298	92	7	iampan	iampan	PROPN
ejpam-5298	92	8	,	,	PUNCT
ejpam-5298	92	9	n.	n.	PROPN
ejpam-5298	92	10	rajesh	rajesh	PROPN
ejpam-5298	92	11	/	/	SYM
ejpam-5298	92	12	eur	eur	PROPN
ejpam-5298	92	13	.	.	PUNCT
ejpam-5298	93	1	j.	j.	PROPN
ejpam-5298	93	2	pure	pure	PROPN
ejpam-5298	93	3	appl	appl	PROPN
ejpam-5298	93	4	.	.	PROPN
ejpam-5298	93	5	math	math	PROPN
ejpam-5298	93	6	,	,	PUNCT
ejpam-5298	93	7	17	17	NUM
ejpam-5298	93	8	(	(	PUNCT
ejpam-5298	93	9	4	4	NUM
ejpam-5298	93	10	)	)	PUNCT
ejpam-5298	93	11	(	(	PUNCT
ejpam-5298	93	12	2024	2024	NUM
ejpam-5298	93	13	)	)	PUNCT
ejpam-5298	93	14	,	,	PUNCT
ejpam-5298	93	15	2726	2726	NUM
ejpam-5298	93	16	-	-	SYM
ejpam-5298	93	17	2737	2737	NUM
ejpam-5298	93	18	2730	2730	NUM
ejpam-5298	93	19	=	=	SYM
ejpam-5298	94	1	1	1	NUM
ejpam-5298	94	2	·	·	PUNCT
ejpam-5298	94	3	x	x	SYM
ejpam-5298	94	4	(	(	PUNCT
ejpam-5298	94	5	m(1	m(1	NOUN
ejpam-5298	94	6	)	)	PUNCT
ejpam-5298	94	7	=	=	SYM
ejpam-5298	94	8	1	1	X
ejpam-5298	94	9	)	)	PUNCT
ejpam-5298	94	10	=	=	PUNCT
ejpam-5298	94	11	x.	x.	NOUN
ejpam-5298	94	12	(	(	PUNCT
ejpam-5298	94	13	lemma	lemma	PROPN
ejpam-5298	94	14	1	1	NUM
ejpam-5298	94	15	(	(	PUNCT
ejpam-5298	94	16	2	2	NUM
ejpam-5298	94	17	)	)	PUNCT
ejpam-5298	94	18	)	)	PUNCT
ejpam-5298	94	19	hence	hence	ADV
ejpam-5298	94	20	,	,	PUNCT
ejpam-5298	94	21	m	m	VERB
ejpam-5298	94	22	=	=	ADJ
ejpam-5298	94	23	ih	ih	X
ejpam-5298	94	24	.	.	PUNCT
ejpam-5298	95	1	the	the	DET
ejpam-5298	95	2	converse	converse	NOUN
ejpam-5298	95	3	is	be	AUX
ejpam-5298	95	4	obvious	obvious	ADJ
ejpam-5298	95	5	.	.	PUNCT
ejpam-5298	96	1	(	(	PUNCT
ejpam-5298	96	2	2	2	X
ejpam-5298	96	3	)	)	PUNCT
ejpam-5298	96	4	assume	assume	VERB
ejpam-5298	96	5	that	that	SCONJ
ejpam-5298	96	6	m	m	PROPN
ejpam-5298	96	7	is	be	AUX
ejpam-5298	96	8	an	an	DET
ejpam-5298	96	9	anti	anti	ADJ
ejpam-5298	96	10	-	-	ADJ
ejpam-5298	96	11	left	left	ADJ
ejpam-5298	96	12	multiplier	multipli	ADJ
ejpam-5298	96	13	of	of	ADP
ejpam-5298	96	14	h.	h.	PROPN
ejpam-5298	96	15	let	let	VERB
ejpam-5298	96	16	x	x	SYM
ejpam-5298	96	17	∈	∈	PROPN
ejpam-5298	96	18	h.	h.	PROPN
ejpam-5298	96	19	then	then	ADV
ejpam-5298	96	20	m(1	m(1	PROPN
ejpam-5298	96	21	)	)	PUNCT
ejpam-5298	97	1	=	=	SYM
ejpam-5298	97	2	m(x	m(x	X
ejpam-5298	97	3	·	·	PUNCT
ejpam-5298	97	4	1	1	X
ejpam-5298	97	5	)	)	PUNCT
ejpam-5298	97	6	(	(	PUNCT
ejpam-5298	97	7	lemma	lemma	PROPN
ejpam-5298	97	8	1	1	NUM
ejpam-5298	97	9	(	(	PUNCT
ejpam-5298	97	10	3	3	NUM
ejpam-5298	97	11	)	)	PUNCT
ejpam-5298	97	12	)	)	PUNCT
ejpam-5298	97	13	=	=	SYM
ejpam-5298	97	14	1	1	X
ejpam-5298	97	15	·	·	SYM
ejpam-5298	97	16	m(x	m(x	X
ejpam-5298	97	17	)	)	PUNCT
ejpam-5298	97	18	(	(	PUNCT
ejpam-5298	97	19	anti	anti	ADJ
ejpam-5298	97	20	-	-	ADJ
ejpam-5298	97	21	left	left	ADJ
ejpam-5298	97	22	multiplier	multiplier	ADV
ejpam-5298	97	23	)	)	PUNCT
ejpam-5298	97	24	=	=	SYM
ejpam-5298	97	25	m(x	m(x	PROPN
ejpam-5298	97	26	)	)	PUNCT
ejpam-5298	97	27	.	.	PUNCT
ejpam-5298	98	1	(	(	PUNCT
ejpam-5298	98	2	lemma	lemma	PROPN
ejpam-5298	98	3	1	1	NUM
ejpam-5298	98	4	(	(	PUNCT
ejpam-5298	98	5	2	2	NUM
ejpam-5298	98	6	)	)	PUNCT
ejpam-5298	98	7	)	)	PUNCT
ejpam-5298	98	8	hence	hence	ADV
ejpam-5298	98	9	,	,	PUNCT
ejpam-5298	98	10	m	m	VERB
ejpam-5298	98	11	is	be	AUX
ejpam-5298	98	12	constant	constant	ADJ
ejpam-5298	98	13	.	.	PUNCT
ejpam-5298	99	1	(	(	PUNCT
ejpam-5298	99	2	3	3	X
ejpam-5298	99	3	)	)	PUNCT
ejpam-5298	99	4	assume	assume	VERB
ejpam-5298	99	5	that	that	SCONJ
ejpam-5298	99	6	m	m	PROPN
ejpam-5298	99	7	is	be	AUX
ejpam-5298	99	8	an	an	DET
ejpam-5298	99	9	anti	anti	ADJ
ejpam-5298	99	10	-	-	ADJ
ejpam-5298	99	11	right	right	ADJ
ejpam-5298	99	12	multiplier	multipli	ADJ
ejpam-5298	99	13	of	of	ADP
ejpam-5298	99	14	h.	h.	PROPN
ejpam-5298	99	15	then	then	ADV
ejpam-5298	99	16	m(1	m(1	PROPN
ejpam-5298	99	17	)	)	PUNCT
ejpam-5298	100	1	=	=	PUNCT
ejpam-5298	100	2	m(1	m(1	NOUN
ejpam-5298	100	3	·	·	PUNCT
ejpam-5298	100	4	1	1	NUM
ejpam-5298	100	5	)	)	PUNCT
ejpam-5298	100	6	(	(	PUNCT
ejpam-5298	100	7	lemma	lemma	PROPN
ejpam-5298	100	8	1	1	NUM
ejpam-5298	100	9	(	(	PUNCT
ejpam-5298	100	10	1	1	NUM
ejpam-5298	100	11	)	)	PUNCT
ejpam-5298	100	12	)	)	PUNCT
ejpam-5298	101	1	=	=	SYM
ejpam-5298	102	1	m(1	m(1	NOUN
ejpam-5298	102	2	)	)	PUNCT
ejpam-5298	102	3	·	·	PUNCT
ejpam-5298	102	4	1	1	NUM
ejpam-5298	102	5	(	(	PUNCT
ejpam-5298	102	6	anti	anti	ADJ
ejpam-5298	102	7	-	-	ADJ
ejpam-5298	102	8	right	right	ADJ
ejpam-5298	102	9	multiplier	multipli	ADJ
ejpam-5298	102	10	)	)	PUNCT
ejpam-5298	102	11	=	=	SYM
ejpam-5298	103	1	1	1	X
ejpam-5298	103	2	.	.	PUNCT
ejpam-5298	104	1	(	(	PUNCT
ejpam-5298	104	2	lemma	lemma	PROPN
ejpam-5298	104	3	1	1	NUM
ejpam-5298	104	4	(	(	PUNCT
ejpam-5298	104	5	3	3	NUM
ejpam-5298	104	6	)	)	PUNCT
ejpam-5298	104	7	)	)	PUNCT
ejpam-5298	104	8	let	let	VERB
ejpam-5298	104	9	x	x	SYM
ejpam-5298	104	10	∈	∈	PROPN
ejpam-5298	104	11	h.	h.	NOUN
ejpam-5298	104	12	then	then	ADV
ejpam-5298	104	13	x	x	X
ejpam-5298	104	14	=	=	SYM
ejpam-5298	104	15	1	1	NUM
ejpam-5298	104	16	·	·	PUNCT
ejpam-5298	104	17	x	x	X
ejpam-5298	104	18	(	(	PUNCT
ejpam-5298	104	19	lemma	lemma	PROPN
ejpam-5298	104	20	1	1	NUM
ejpam-5298	104	21	(	(	PUNCT
ejpam-5298	104	22	2	2	NUM
ejpam-5298	104	23	)	)	PUNCT
ejpam-5298	104	24	)	)	PUNCT
ejpam-5298	105	1	=	=	SYM
ejpam-5298	105	2	m(1	m(1	NOUN
ejpam-5298	105	3	)	)	PUNCT
ejpam-5298	105	4	·	·	PUNCT
ejpam-5298	106	1	x	x	X
ejpam-5298	106	2	(	(	PUNCT
ejpam-5298	106	3	m(1	m(1	NOUN
ejpam-5298	106	4	)	)	PUNCT
ejpam-5298	106	5	=	=	SYM
ejpam-5298	106	6	1	1	X
ejpam-5298	106	7	)	)	PUNCT
ejpam-5298	106	8	=	=	NOUN
ejpam-5298	106	9	m(x	m(x	PROPN
ejpam-5298	106	10	·	·	PUNCT
ejpam-5298	106	11	1	1	X
ejpam-5298	106	12	)	)	PUNCT
ejpam-5298	106	13	(	(	PUNCT
ejpam-5298	106	14	anti	anti	ADJ
ejpam-5298	106	15	-	-	ADJ
ejpam-5298	106	16	right	right	ADJ
ejpam-5298	106	17	multiplier	multipli	ADJ
ejpam-5298	106	18	)	)	PUNCT
ejpam-5298	106	19	=	=	SYM
ejpam-5298	107	1	m(1	m(1	NOUN
ejpam-5298	107	2	)	)	PUNCT
ejpam-5298	107	3	(	(	PUNCT
ejpam-5298	107	4	lemma	lemma	PROPN
ejpam-5298	107	5	1	1	NUM
ejpam-5298	107	6	(	(	PUNCT
ejpam-5298	107	7	3	3	NUM
ejpam-5298	107	8	)	)	PUNCT
ejpam-5298	107	9	)	)	PUNCT
ejpam-5298	107	10	=	=	SYM
ejpam-5298	108	1	1	1	X
ejpam-5298	108	2	.	.	PUNCT
ejpam-5298	108	3	(	(	PUNCT
ejpam-5298	108	4	m(1	m(1	NOUN
ejpam-5298	108	5	)	)	PUNCT
ejpam-5298	108	6	=	=	SYM
ejpam-5298	108	7	1	1	X
ejpam-5298	108	8	)	)	PUNCT
ejpam-5298	108	9	hence	hence	ADV
ejpam-5298	108	10	,	,	PUNCT
ejpam-5298	108	11	h	h	NOUN
ejpam-5298	108	12	=	=	PUNCT
ejpam-5298	108	13	{	{	PUNCT
ejpam-5298	108	14	1	1	NUM
ejpam-5298	108	15	}	}	PUNCT
ejpam-5298	108	16	.	.	PUNCT
ejpam-5298	109	1	the	the	DET
ejpam-5298	109	2	converse	converse	NOUN
ejpam-5298	109	3	is	be	AUX
ejpam-5298	109	4	obvious	obvious	ADJ
ejpam-5298	109	5	.	.	PUNCT
ejpam-5298	109	6	example	example	NOUN
ejpam-5298	110	1	2	2	NUM
ejpam-5298	110	2	.	.	PUNCT
ejpam-5298	111	1	let	let	VERB
ejpam-5298	111	2	h	h	NOUN
ejpam-5298	111	3	=	=	PUNCT
ejpam-5298	111	4	{	{	PUNCT
ejpam-5298	111	5	1	1	NUM
ejpam-5298	111	6	,	,	PUNCT
ejpam-5298	111	7	2	2	NUM
ejpam-5298	111	8	,	,	PUNCT
ejpam-5298	111	9	3	3	NUM
ejpam-5298	111	10	,	,	PUNCT
ejpam-5298	111	11	4	4	NUM
ejpam-5298	111	12	}	}	PUNCT
ejpam-5298	111	13	with	with	ADP
ejpam-5298	111	14	a	a	DET
ejpam-5298	111	15	fixed	fix	VERB
ejpam-5298	111	16	element	element	NOUN
ejpam-5298	111	17	1	1	NUM
ejpam-5298	111	18	and	and	CCONJ
ejpam-5298	111	19	a	a	DET
ejpam-5298	111	20	binary	binary	ADJ
ejpam-5298	111	21	operation	operation	NOUN
ejpam-5298	111	22	·	·	PUNCT
ejpam-5298	111	23	defined	define	VERB
ejpam-5298	111	24	by	by	ADP
ejpam-5298	111	25	the	the	DET
ejpam-5298	111	26	following	following	ADJ
ejpam-5298	111	27	cayley	cayley	ADJ
ejpam-5298	111	28	table	table	NOUN
ejpam-5298	111	29	:	:	PUNCT
ejpam-5298	111	30	·	·	PUNCT
ejpam-5298	111	31	1	1	NUM
ejpam-5298	111	32	2	2	NUM
ejpam-5298	111	33	3	3	NUM
ejpam-5298	111	34	4	4	NUM
ejpam-5298	111	35	1	1	NUM
ejpam-5298	111	36	1	1	NUM
ejpam-5298	111	37	2	2	NUM
ejpam-5298	111	38	3	3	NUM
ejpam-5298	111	39	4	4	NUM
ejpam-5298	111	40	2	2	NUM
ejpam-5298	111	41	1	1	NUM
ejpam-5298	111	42	1	1	NUM
ejpam-5298	111	43	3	3	NUM
ejpam-5298	111	44	4	4	NUM
ejpam-5298	111	45	3	3	NUM
ejpam-5298	111	46	1	1	NUM
ejpam-5298	111	47	2	2	NUM
ejpam-5298	111	48	1	1	NUM
ejpam-5298	111	49	4	4	NUM
ejpam-5298	111	50	4	4	NUM
ejpam-5298	111	51	1	1	NUM
ejpam-5298	111	52	2	2	NUM
ejpam-5298	111	53	3	3	NUM
ejpam-5298	111	54	1	1	NUM
ejpam-5298	111	55	then	then	ADV
ejpam-5298	111	56	h	h	NOUN
ejpam-5298	112	1	=	=	PUNCT
ejpam-5298	113	1	(	(	PUNCT
ejpam-5298	113	2	h	h	NOUN
ejpam-5298	113	3	,	,	PUNCT
ejpam-5298	113	4	·	·	PUNCT
ejpam-5298	113	5	,	,	PUNCT
ejpam-5298	113	6	1	1	NUM
ejpam-5298	113	7	)	)	PUNCT
ejpam-5298	113	8	is	be	AUX
ejpam-5298	113	9	a	a	DET
ejpam-5298	113	10	hilbert	hilbert	NOUN
ejpam-5298	113	11	algebra	algebra	NOUN
ejpam-5298	113	12	.	.	PUNCT
ejpam-5298	114	1	we	we	PRON
ejpam-5298	114	2	define	define	VERB
ejpam-5298	114	3	a	a	DET
ejpam-5298	114	4	self	self	NOUN
ejpam-5298	114	5	-	-	PUNCT
ejpam-5298	114	6	map	map	NOUN
ejpam-5298	114	7	m	m	NOUN
ejpam-5298	114	8	:	:	PUNCT
ejpam-5298	114	9	h	h	PROPN
ejpam-5298	114	10	→	→	SYM
ejpam-5298	114	11	h	h	NOUN
ejpam-5298	114	12	as	as	SCONJ
ejpam-5298	114	13	follows	follow	VERB
ejpam-5298	114	14	:	:	PUNCT
ejpam-5298	114	15	m(x	m(x	X
ejpam-5298	114	16	)	)	PUNCT
ejpam-5298	115	1	=	=	PRON
ejpam-5298	115	2	{	{	PUNCT
ejpam-5298	115	3	1	1	NUM
ejpam-5298	115	4	if	if	SCONJ
ejpam-5298	115	5	x	x	SYM
ejpam-5298	115	6	=	=	SYM
ejpam-5298	115	7	1	1	NUM
ejpam-5298	115	8	,	,	PUNCT
ejpam-5298	115	9	2	2	NUM
ejpam-5298	115	10	x	x	SYM
ejpam-5298	115	11	if	if	SCONJ
ejpam-5298	115	12	x	x	X
ejpam-5298	115	13	=	=	SYM
ejpam-5298	115	14	3	3	NUM
ejpam-5298	115	15	,	,	PUNCT
ejpam-5298	115	16	4	4	NUM
ejpam-5298	115	17	.	.	PUNCT
ejpam-5298	116	1	in	in	ADP
ejpam-5298	116	2	this	this	DET
ejpam-5298	116	3	case	case	NOUN
ejpam-5298	116	4	,	,	PUNCT
ejpam-5298	116	5	m	m	VERB
ejpam-5298	116	6	serves	serve	VERB
ejpam-5298	116	7	as	as	ADP
ejpam-5298	116	8	a	a	DET
ejpam-5298	116	9	right	right	ADJ
ejpam-5298	116	10	multiplier	multipli	ADJ
ejpam-5298	116	11	of	of	ADP
ejpam-5298	116	12	h	h	NOUN
ejpam-5298	116	13	,	,	PUNCT
ejpam-5298	116	14	while	while	SCONJ
ejpam-5298	116	15	it	it	PRON
ejpam-5298	116	16	does	do	AUX
ejpam-5298	116	17	not	not	PART
ejpam-5298	116	18	qualify	qualify	VERB
ejpam-5298	116	19	as	as	ADP
ejpam-5298	116	20	a	a	DET
ejpam-5298	116	21	left	left	ADJ
ejpam-5298	116	22	multiplier	multipli	ADJ
ejpam-5298	116	23	,	,	PUNCT
ejpam-5298	116	24	an	an	DET
ejpam-5298	116	25	anti	anti	ADJ
ejpam-5298	116	26	-	-	ADJ
ejpam-5298	116	27	left	left	ADJ
ejpam-5298	116	28	multiplier	multiplier	ADV
ejpam-5298	116	29	,	,	PUNCT
ejpam-5298	116	30	or	or	CCONJ
ejpam-5298	116	31	an	an	DET
ejpam-5298	116	32	anti	anti	ADJ
ejpam-5298	116	33	-	-	ADJ
ejpam-5298	116	34	right	right	ADJ
ejpam-5298	116	35	multiplier	multipli	ADJ
ejpam-5298	116	36	of	of	ADP
ejpam-5298	116	37	h.	h.	PROPN
ejpam-5298	116	38	a.	a.	PROPN
ejpam-5298	116	39	iampan	iampan	PROPN
ejpam-5298	116	40	,	,	PUNCT
ejpam-5298	116	41	n.	n.	PROPN
ejpam-5298	116	42	rajesh	rajesh	PROPN
ejpam-5298	116	43	/	/	SYM
ejpam-5298	116	44	eur	eur	PROPN
ejpam-5298	116	45	.	.	PUNCT
ejpam-5298	117	1	j.	j.	PROPN
ejpam-5298	117	2	pure	pure	PROPN
ejpam-5298	117	3	appl	appl	PROPN
ejpam-5298	117	4	.	.	PROPN
ejpam-5298	117	5	math	math	PROPN
ejpam-5298	117	6	,	,	PUNCT
ejpam-5298	117	7	17	17	NUM
ejpam-5298	117	8	(	(	PUNCT
ejpam-5298	117	9	4	4	NUM
ejpam-5298	117	10	)	)	PUNCT
ejpam-5298	117	11	(	(	PUNCT
ejpam-5298	117	12	2024	2024	NUM
ejpam-5298	117	13	)	)	PUNCT
ejpam-5298	117	14	,	,	PUNCT
ejpam-5298	117	15	2726	2726	NUM
ejpam-5298	117	16	-	-	SYM
ejpam-5298	117	17	2737	2737	NUM
ejpam-5298	117	18	2731	2731	NUM
ejpam-5298	117	19	following	follow	VERB
ejpam-5298	117	20	this	this	PRON
ejpam-5298	118	1	,	,	PUNCT
ejpam-5298	118	2	we	we	PRON
ejpam-5298	118	3	will	will	AUX
ejpam-5298	118	4	focus	focus	VERB
ejpam-5298	118	5	specifically	specifically	ADV
ejpam-5298	118	6	on	on	ADP
ejpam-5298	118	7	right	right	ADJ
ejpam-5298	118	8	multipliers	multiplier	NOUN
ejpam-5298	118	9	of	of	ADP
ejpam-5298	118	10	h	h	NOUN
ejpam-5298	118	11	,	,	PUNCT
ejpam-5298	118	12	referring	refer	VERB
ejpam-5298	118	13	to	to	ADP
ejpam-5298	118	14	them	they	PRON
ejpam-5298	118	15	simply	simply	ADV
ejpam-5298	118	16	as	as	ADP
ejpam-5298	118	17	multipliers	multiplier	NOUN
ejpam-5298	118	18	for	for	ADP
ejpam-5298	118	19	brevity	brevity	NOUN
ejpam-5298	118	20	.	.	PUNCT
ejpam-5298	119	1	proposition	proposition	NOUN
ejpam-5298	119	2	2	2	NUM
ejpam-5298	119	3	.	.	PUNCT
ejpam-5298	120	1	a	a	DET
ejpam-5298	120	2	multiplier	multipli	ADJ
ejpam-5298	120	3	m	m	NOUN
ejpam-5298	120	4	of	of	ADP
ejpam-5298	120	5	h	h	NOUN
ejpam-5298	120	6	is	be	AUX
ejpam-5298	120	7	ih	ih	NOUN
ejpam-5298	120	8	if	if	SCONJ
ejpam-5298	120	9	and	and	CCONJ
ejpam-5298	120	10	only	only	ADV
ejpam-5298	120	11	if	if	SCONJ
ejpam-5298	120	12	m	m	NOUN
ejpam-5298	120	13	is	be	AUX
ejpam-5298	120	14	a	a	DET
ejpam-5298	120	15	left	left	ADJ
ejpam-5298	120	16	multiplier	multipli	ADJ
ejpam-5298	120	17	of	of	ADP
ejpam-5298	120	18	h.	h.	NOUN
ejpam-5298	120	19	proof	proof	NOUN
ejpam-5298	120	20	.	.	PUNCT
ejpam-5298	121	1	it	it	PRON
ejpam-5298	121	2	is	be	AUX
ejpam-5298	121	3	done	do	VERB
ejpam-5298	121	4	in	in	ADP
ejpam-5298	121	5	proposition	proposition	NOUN
ejpam-5298	121	6	1	1	NUM
ejpam-5298	121	7	(	(	PUNCT
ejpam-5298	121	8	1	1	NUM
ejpam-5298	121	9	)	)	PUNCT
ejpam-5298	121	10	.	.	PUNCT
ejpam-5298	122	1	example	example	NOUN
ejpam-5298	123	1	3	3	NUM
ejpam-5298	123	2	.	.	X
ejpam-5298	123	3	from	from	ADP
ejpam-5298	123	4	example	example	NOUN
ejpam-5298	123	5	1	1	NUM
ejpam-5298	123	6	,	,	PUNCT
ejpam-5298	123	7	we	we	PRON
ejpam-5298	123	8	define	define	VERB
ejpam-5298	123	9	a	a	DET
ejpam-5298	123	10	self	self	NOUN
ejpam-5298	123	11	-	-	PUNCT
ejpam-5298	123	12	map	map	NOUN
ejpam-5298	123	13	m	m	NOUN
ejpam-5298	123	14	:	:	PUNCT
ejpam-5298	123	15	h	h	PROPN
ejpam-5298	123	16	→	→	SYM
ejpam-5298	123	17	h	h	NOUN
ejpam-5298	123	18	as	as	SCONJ
ejpam-5298	123	19	follows	follow	VERB
ejpam-5298	123	20	:	:	PUNCT
ejpam-5298	123	21	m(1	m(1	X
ejpam-5298	123	22	)	)	PUNCT
ejpam-5298	123	23	=	=	SYM
ejpam-5298	123	24	1,m(α	1,m(α	PROPN
ejpam-5298	123	25	)	)	PUNCT
ejpam-5298	123	26	=	=	SYM
ejpam-5298	124	1	β	β	X
ejpam-5298	124	2	,	,	PUNCT
ejpam-5298	124	3	m(β	m(β	PROPN
ejpam-5298	124	4	)	)	PUNCT
ejpam-5298	124	5	=	=	SYM
ejpam-5298	125	1	γ	γ	X
ejpam-5298	125	2	,	,	PUNCT
ejpam-5298	125	3	m(ϵ	m(ϵ	X
ejpam-5298	125	4	)	)	PUNCT
ejpam-5298	125	5	=	=	SYM
ejpam-5298	125	6	ϵ,m(γ	ϵ,m(γ	NOUN
ejpam-5298	125	7	)	)	PUNCT
ejpam-5298	125	8	=	=	SYM
ejpam-5298	126	1	α	α	X
ejpam-5298	126	2	.	.	PUNCT
ejpam-5298	127	1	then	then	ADV
ejpam-5298	127	2	m	m	PROPN
ejpam-5298	127	3	is	be	AUX
ejpam-5298	127	4	a	a	DET
ejpam-5298	127	5	multiplier	multipli	ADJ
ejpam-5298	127	6	of	of	ADP
ejpam-5298	127	7	h.	h.	NOUN
ejpam-5298	127	8	since	since	SCONJ
ejpam-5298	127	9	m2(α	m2(α	PROPN
ejpam-5298	127	10	)	)	PUNCT
ejpam-5298	127	11	=	=	PUNCT
ejpam-5298	127	12	m(m(α	m(m(α	PROPN
ejpam-5298	127	13	)	)	PUNCT
ejpam-5298	127	14	)	)	PUNCT
ejpam-5298	128	1	=	=	SYM
ejpam-5298	128	2	m(β	m(β	X
ejpam-5298	128	3	)	)	PUNCT
ejpam-5298	128	4	=	=	PUNCT
ejpam-5298	129	1	γ	γ	X
ejpam-5298	129	2	̸=	̸=	PROPN
ejpam-5298	129	3	β	β	X
ejpam-5298	129	4	=	=	SYM
ejpam-5298	129	5	m(α	m(α	PROPN
ejpam-5298	129	6	)	)	PUNCT
ejpam-5298	129	7	,	,	PUNCT
ejpam-5298	129	8	we	we	PRON
ejpam-5298	129	9	have	have	VERB
ejpam-5298	129	10	m2	m2	PROPN
ejpam-5298	129	11	̸=	̸=	PROPN
ejpam-5298	129	12	m.	m.	NOUN
ejpam-5298	129	13	example	example	NOUN
ejpam-5298	129	14	3	3	NUM
ejpam-5298	129	15	shows	show	VERB
ejpam-5298	129	16	that	that	SCONJ
ejpam-5298	129	17	if	if	SCONJ
ejpam-5298	129	18	m	m	NOUN
ejpam-5298	129	19	is	be	AUX
ejpam-5298	129	20	a	a	DET
ejpam-5298	129	21	multiplier	multipli	ADJ
ejpam-5298	129	22	of	of	ADP
ejpam-5298	129	23	h	h	NOUN
ejpam-5298	129	24	,	,	PUNCT
ejpam-5298	129	25	then	then	ADV
ejpam-5298	129	26	m2	m2	PROPN
ejpam-5298	129	27	=	=	PROPN
ejpam-5298	129	28	m	m	PROPN
ejpam-5298	129	29	,	,	PUNCT
ejpam-5298	129	30	which	which	PRON
ejpam-5298	129	31	is	be	AUX
ejpam-5298	129	32	not	not	PART
ejpam-5298	129	33	valid	valid	ADJ
ejpam-5298	129	34	in	in	ADP
ejpam-5298	129	35	general	general	ADJ
ejpam-5298	129	36	.	.	PUNCT
ejpam-5298	130	1	the	the	DET
ejpam-5298	130	2	following	follow	VERB
ejpam-5298	130	3	proposition	proposition	NOUN
ejpam-5298	130	4	illustrates	illustrate	VERB
ejpam-5298	130	5	the	the	DET
ejpam-5298	130	6	relationship	relationship	NOUN
ejpam-5298	130	7	of	of	ADP
ejpam-5298	130	8	these	these	DET
ejpam-5298	130	9	conditions	condition	NOUN
ejpam-5298	130	10	.	.	PUNCT
ejpam-5298	131	1	proposition	proposition	NOUN
ejpam-5298	131	2	3	3	NUM
ejpam-5298	131	3	.	.	PUNCT
ejpam-5298	132	1	a	a	DET
ejpam-5298	132	2	multiplier	multipli	ADJ
ejpam-5298	132	3	m	m	NOUN
ejpam-5298	132	4	of	of	ADP
ejpam-5298	132	5	h	h	NOUN
ejpam-5298	132	6	is	be	AUX
ejpam-5298	132	7	ih	ih	NOUN
ejpam-5298	132	8	if	if	SCONJ
ejpam-5298	132	9	and	and	CCONJ
ejpam-5298	132	10	only	only	ADV
ejpam-5298	132	11	if	if	SCONJ
ejpam-5298	132	12	the	the	DET
ejpam-5298	132	13	following	follow	VERB
ejpam-5298	132	14	statements	statement	NOUN
ejpam-5298	132	15	hold	hold	VERB
ejpam-5298	132	16	:	:	PUNCT
ejpam-5298	132	17	(	(	PUNCT
ejpam-5298	132	18	1	1	X
ejpam-5298	132	19	)	)	PUNCT
ejpam-5298	132	20	m2	m2	PROPN
ejpam-5298	132	21	=	=	SYM
ejpam-5298	132	22	m	m	PROPN
ejpam-5298	132	23	,	,	PUNCT
ejpam-5298	132	24	(	(	PUNCT
ejpam-5298	132	25	2	2	X
ejpam-5298	132	26	)	)	PUNCT
ejpam-5298	132	27	m(x	m(x	PROPN
ejpam-5298	132	28	·	·	PUNCT
ejpam-5298	133	1	y	y	X
ejpam-5298	133	2	)	)	PUNCT
ejpam-5298	133	3	=	=	SYM
ejpam-5298	133	4	m(x	m(x	PROPN
ejpam-5298	133	5	)	)	PUNCT
ejpam-5298	133	6	·	·	PUNCT
ejpam-5298	133	7	m(y	m(y	NOUN
ejpam-5298	133	8	)	)	PUNCT
ejpam-5298	133	9	for	for	ADP
ejpam-5298	133	10	all	all	DET
ejpam-5298	133	11	x	x	NOUN
ejpam-5298	133	12	,	,	PUNCT
ejpam-5298	133	13	y	y	PROPN
ejpam-5298	133	14	∈	∈	PROPN
ejpam-5298	133	15	h	h	NOUN
ejpam-5298	133	16	,	,	PUNCT
ejpam-5298	133	17	(	(	PUNCT
ejpam-5298	133	18	3	3	X
ejpam-5298	133	19	)	)	PUNCT
ejpam-5298	133	20	m2(x	m2(x	PROPN
ejpam-5298	133	21	)	)	PUNCT
ejpam-5298	133	22	·	·	PUNCT
ejpam-5298	134	1	y	y	SYM
ejpam-5298	134	2	=	=	SYM
ejpam-5298	134	3	m(x	m(x	PROPN
ejpam-5298	134	4	)	)	PUNCT
ejpam-5298	134	5	·	·	PUNCT
ejpam-5298	134	6	m(y	m(y	NOUN
ejpam-5298	134	7	)	)	PUNCT
ejpam-5298	134	8	for	for	ADP
ejpam-5298	134	9	all	all	DET
ejpam-5298	134	10	x	x	NOUN
ejpam-5298	134	11	,	,	PUNCT
ejpam-5298	134	12	y	y	PROPN
ejpam-5298	134	13	∈	∈	PROPN
ejpam-5298	134	14	h.	h.	NOUN
ejpam-5298	134	15	proof	proof	NOUN
ejpam-5298	134	16	.	.	PUNCT
ejpam-5298	135	1	the	the	DET
ejpam-5298	135	2	condition	condition	NOUN
ejpam-5298	135	3	for	for	ADP
ejpam-5298	135	4	necessity	necessity	NOUN
ejpam-5298	135	5	is	be	AUX
ejpam-5298	135	6	obvious	obvious	ADJ
ejpam-5298	135	7	.	.	PUNCT
ejpam-5298	136	1	conversely	conversely	ADV
ejpam-5298	136	2	,	,	PUNCT
ejpam-5298	136	3	assume	assume	VERB
ejpam-5298	136	4	that	that	SCONJ
ejpam-5298	136	5	(	(	PUNCT
ejpam-5298	136	6	1	1	NUM
ejpam-5298	136	7	)	)	PUNCT
ejpam-5298	136	8	,	,	PUNCT
ejpam-5298	136	9	(	(	PUNCT
ejpam-5298	136	10	2	2	NUM
ejpam-5298	136	11	)	)	PUNCT
ejpam-5298	136	12	,	,	PUNCT
ejpam-5298	136	13	and	and	CCONJ
ejpam-5298	136	14	(	(	PUNCT
ejpam-5298	136	15	3	3	X
ejpam-5298	136	16	)	)	PUNCT
ejpam-5298	136	17	hold	hold	NOUN
ejpam-5298	136	18	.	.	PUNCT
ejpam-5298	137	1	then	then	ADV
ejpam-5298	137	2	for	for	ADP
ejpam-5298	137	3	x	x	X
ejpam-5298	137	4	,	,	PUNCT
ejpam-5298	137	5	y	y	PROPN
ejpam-5298	137	6	∈	∈	PROPN
ejpam-5298	137	7	h	h	NOUN
ejpam-5298	137	8	,	,	PUNCT
ejpam-5298	137	9	we	we	PRON
ejpam-5298	137	10	have	have	VERB
ejpam-5298	137	11	m(x	m(x	PROPN
ejpam-5298	137	12	·	·	PUNCT
ejpam-5298	137	13	y	y	X
ejpam-5298	137	14	)	)	PUNCT
ejpam-5298	137	15	=	=	SYM
ejpam-5298	137	16	m(x	m(x	PROPN
ejpam-5298	137	17	)	)	PUNCT
ejpam-5298	137	18	·	·	PUNCT
ejpam-5298	137	19	m(y	m(y	X
ejpam-5298	137	20	)	)	PUNCT
ejpam-5298	137	21	=	=	SYM
ejpam-5298	137	22	m2(x	m2(x	PROPN
ejpam-5298	137	23	)	)	PUNCT
ejpam-5298	137	24	·	·	PUNCT
ejpam-5298	137	25	y	y	SYM
ejpam-5298	137	26	=	=	SYM
ejpam-5298	137	27	m(x	m(x	PROPN
ejpam-5298	137	28	)	)	PUNCT
ejpam-5298	137	29	·	·	PUNCT
ejpam-5298	138	1	y.	y.	PROPN
ejpam-5298	138	2	thus	thus	ADV
ejpam-5298	138	3	,	,	PUNCT
ejpam-5298	138	4	m	m	PROPN
ejpam-5298	138	5	is	be	AUX
ejpam-5298	138	6	a	a	DET
ejpam-5298	138	7	left	left	ADJ
ejpam-5298	138	8	multiplier	multipli	ADJ
ejpam-5298	138	9	of	of	ADP
ejpam-5298	138	10	h.	h.	NOUN
ejpam-5298	138	11	by	by	ADP
ejpam-5298	138	12	proposition	proposition	NOUN
ejpam-5298	138	13	2	2	NUM
ejpam-5298	138	14	,	,	PUNCT
ejpam-5298	138	15	we	we	PRON
ejpam-5298	138	16	have	have	VERB
ejpam-5298	138	17	m	m	NOUN
ejpam-5298	138	18	=	=	ADJ
ejpam-5298	138	19	ih	ih	X
ejpam-5298	138	20	.	.	PUNCT
ejpam-5298	139	1	proposition	proposition	NOUN
ejpam-5298	139	2	4	4	NUM
ejpam-5298	139	3	.	.	PUNCT
ejpam-5298	140	1	let	let	VERB
ejpam-5298	140	2	m	m	PRON
ejpam-5298	140	3	be	be	AUX
ejpam-5298	140	4	a	a	DET
ejpam-5298	140	5	multiplier	multipli	ADJ
ejpam-5298	140	6	of	of	ADP
ejpam-5298	140	7	h.	h.	PROPN
ejpam-5298	140	8	then	then	ADV
ejpam-5298	140	9	m(m(x	m(m(x	PROPN
ejpam-5298	140	10	)	)	PUNCT
ejpam-5298	140	11	·	·	PUNCT
ejpam-5298	141	1	x	x	X
ejpam-5298	141	2	)	)	PUNCT
ejpam-5298	141	3	=	=	SYM
ejpam-5298	141	4	1	1	NUM
ejpam-5298	141	5	for	for	ADP
ejpam-5298	141	6	all	all	DET
ejpam-5298	141	7	x	x	SYM
ejpam-5298	141	8	∈	∈	PROPN
ejpam-5298	141	9	h.	h.	NOUN
ejpam-5298	141	10	proof	proof	NOUN
ejpam-5298	141	11	.	.	PUNCT
ejpam-5298	142	1	by	by	ADP
ejpam-5298	142	2	lemma	lemma	PROPN
ejpam-5298	142	3	1	1	NUM
ejpam-5298	142	4	(	(	PUNCT
ejpam-5298	142	5	1	1	NUM
ejpam-5298	142	6	)	)	PUNCT
ejpam-5298	142	7	,	,	PUNCT
ejpam-5298	142	8	we	we	PRON
ejpam-5298	142	9	have	have	VERB
ejpam-5298	142	10	m(m(x	m(m(x	PROPN
ejpam-5298	142	11	)	)	PUNCT
ejpam-5298	142	12	·	·	PUNCT
ejpam-5298	143	1	x	x	X
ejpam-5298	143	2	)	)	PUNCT
ejpam-5298	143	3	=	=	SYM
ejpam-5298	143	4	m(x	m(x	PROPN
ejpam-5298	143	5	)	)	PUNCT
ejpam-5298	143	6	·	·	PUNCT
ejpam-5298	143	7	m(x	m(x	X
ejpam-5298	143	8	)	)	PUNCT
ejpam-5298	143	9	=	=	SYM
ejpam-5298	143	10	1	1	NUM
ejpam-5298	143	11	for	for	ADP
ejpam-5298	143	12	all	all	DET
ejpam-5298	143	13	x	x	SYM
ejpam-5298	143	14	∈	∈	PROPN
ejpam-5298	143	15	h.	h.	NOUN
ejpam-5298	143	16	definition	definition	NOUN
ejpam-5298	143	17	7	7	NUM
ejpam-5298	143	18	.	.	PUNCT
ejpam-5298	144	1	a	a	DET
ejpam-5298	144	2	self	self	NOUN
ejpam-5298	144	3	-	-	PUNCT
ejpam-5298	144	4	map	map	NOUN
ejpam-5298	144	5	m	m	NOUN
ejpam-5298	144	6	of	of	ADP
ejpam-5298	144	7	h	h	NOUN
ejpam-5298	144	8	is	be	AUX
ejpam-5298	144	9	said	say	VERB
ejpam-5298	144	10	to	to	PART
ejpam-5298	144	11	be	be	AUX
ejpam-5298	144	12	regular	regular	ADJ
ejpam-5298	144	13	if	if	SCONJ
ejpam-5298	144	14	m(1	m(1	NOUN
ejpam-5298	144	15	)	)	PUNCT
ejpam-5298	144	16	=	=	SYM
ejpam-5298	144	17	1	1	X
ejpam-5298	144	18	.	.	X
ejpam-5298	144	19	proposition	proposition	NOUN
ejpam-5298	144	20	5	5	NUM
ejpam-5298	144	21	.	.	PUNCT
ejpam-5298	145	1	every	every	DET
ejpam-5298	145	2	multiplier	multipli	ADJ
ejpam-5298	145	3	of	of	ADP
ejpam-5298	145	4	h	h	NOUN
ejpam-5298	145	5	is	be	AUX
ejpam-5298	145	6	regular	regular	ADJ
ejpam-5298	145	7	.	.	PUNCT
ejpam-5298	146	1	proof	proof	NOUN
ejpam-5298	146	2	.	.	PUNCT
ejpam-5298	147	1	let	let	VERB
ejpam-5298	147	2	m	m	PRON
ejpam-5298	147	3	be	be	AUX
ejpam-5298	147	4	a	a	DET
ejpam-5298	147	5	multiplier	multipli	ADJ
ejpam-5298	147	6	of	of	ADP
ejpam-5298	147	7	h.	h.	PROPN
ejpam-5298	147	8	then	then	ADV
ejpam-5298	147	9	,	,	PUNCT
ejpam-5298	147	10	by	by	ADP
ejpam-5298	147	11	lemma	lemma	PROPN
ejpam-5298	147	12	1	1	NUM
ejpam-5298	147	13	(	(	PUNCT
ejpam-5298	147	14	3	3	NUM
ejpam-5298	147	15	)	)	PUNCT
ejpam-5298	147	16	,	,	PUNCT
ejpam-5298	147	17	we	we	PRON
ejpam-5298	147	18	have	have	VERB
ejpam-5298	147	19	m(1	m(1	NOUN
ejpam-5298	147	20	)	)	PUNCT
ejpam-5298	148	1	=	=	SYM
ejpam-5298	148	2	m(x	m(x	X
ejpam-5298	148	3	·	·	PUNCT
ejpam-5298	149	1	1	1	X
ejpam-5298	149	2	)	)	PUNCT
ejpam-5298	149	3	=	=	PUNCT
ejpam-5298	149	4	x	x	SYM
ejpam-5298	149	5	·	·	PUNCT
ejpam-5298	149	6	m(1	m(1	NOUN
ejpam-5298	149	7	)	)	PUNCT
ejpam-5298	149	8	for	for	ADP
ejpam-5298	149	9	all	all	DET
ejpam-5298	149	10	x	x	SYM
ejpam-5298	149	11	∈	∈	PROPN
ejpam-5298	149	12	h.	h.	NOUN
ejpam-5298	149	13	by	by	ADP
ejpam-5298	149	14	lemma	lemma	PROPN
ejpam-5298	149	15	1	1	NUM
ejpam-5298	149	16	(	(	PUNCT
ejpam-5298	149	17	1	1	NUM
ejpam-5298	149	18	)	)	PUNCT
ejpam-5298	149	19	,	,	PUNCT
ejpam-5298	149	20	we	we	PRON
ejpam-5298	149	21	have	have	VERB
ejpam-5298	149	22	m(1	m(1	NOUN
ejpam-5298	149	23	)	)	PUNCT
ejpam-5298	149	24	=	=	SYM
ejpam-5298	149	25	m(1	m(1	NOUN
ejpam-5298	149	26	)	)	PUNCT
ejpam-5298	149	27	·	·	PUNCT
ejpam-5298	150	1	m(1	m(1	NOUN
ejpam-5298	150	2	)	)	PUNCT
ejpam-5298	150	3	=	=	SYM
ejpam-5298	150	4	1	1	X
ejpam-5298	150	5	.	.	PUNCT
ejpam-5298	151	1	hence	hence	ADV
ejpam-5298	151	2	,	,	PUNCT
ejpam-5298	151	3	m	m	VERB
ejpam-5298	151	4	is	be	AUX
ejpam-5298	151	5	regular	regular	ADJ
ejpam-5298	151	6	.	.	PUNCT
ejpam-5298	152	1	proposition	proposition	NOUN
ejpam-5298	152	2	6	6	NUM
ejpam-5298	152	3	.	.	PUNCT
ejpam-5298	153	1	let	let	VERB
ejpam-5298	153	2	m	m	PRON
ejpam-5298	153	3	be	be	AUX
ejpam-5298	153	4	a	a	DET
ejpam-5298	153	5	multiplier	multipli	ADJ
ejpam-5298	153	6	of	of	ADP
ejpam-5298	153	7	h.	h.	NOUN
ejpam-5298	153	8	then	then	ADV
ejpam-5298	153	9	the	the	DET
ejpam-5298	153	10	following	following	ADJ
ejpam-5298	153	11	statements	statement	NOUN
ejpam-5298	153	12	hold	hold	VERB
ejpam-5298	153	13	:	:	PUNCT
ejpam-5298	153	14	(	(	PUNCT
ejpam-5298	153	15	1	1	X
ejpam-5298	153	16	)	)	PUNCT
ejpam-5298	153	17	m(1	m(1	NOUN
ejpam-5298	153	18	)	)	PUNCT
ejpam-5298	154	1	=	=	SYM
ejpam-5298	154	2	1	1	NUM
ejpam-5298	154	3	,	,	PUNCT
ejpam-5298	154	4	(	(	PUNCT
ejpam-5298	154	5	2	2	X
ejpam-5298	154	6	)	)	PUNCT
ejpam-5298	154	7	x	x	SYM
ejpam-5298	154	8	≤	≤	ADJ
ejpam-5298	154	9	m(x	m(x	PROPN
ejpam-5298	154	10	)	)	PUNCT
ejpam-5298	154	11	for	for	ADP
ejpam-5298	154	12	all	all	DET
ejpam-5298	154	13	x	x	SYM
ejpam-5298	154	14	∈	∈	PROPN
ejpam-5298	154	15	h	h	NOUN
ejpam-5298	154	16	,	,	PUNCT
ejpam-5298	154	17	(	(	PUNCT
ejpam-5298	154	18	3	3	X
ejpam-5298	154	19	)	)	PUNCT
ejpam-5298	154	20	for	for	ADP
ejpam-5298	154	21	any	any	DET
ejpam-5298	154	22	x	x	NOUN
ejpam-5298	154	23	,	,	PUNCT
ejpam-5298	154	24	y	y	PROPN
ejpam-5298	154	25	∈	∈	PROPN
ejpam-5298	154	26	h	h	NOUN
ejpam-5298	154	27	,	,	PUNCT
ejpam-5298	154	28	if	if	SCONJ
ejpam-5298	154	29	x	x	ADP
ejpam-5298	154	30	≤	≤	NOUN
ejpam-5298	154	31	y	y	NOUN
ejpam-5298	154	32	,	,	PUNCT
ejpam-5298	154	33	then	then	ADV
ejpam-5298	154	34	x	x	X
ejpam-5298	154	35	≤	≤	PROPN
ejpam-5298	154	36	m(y	m(y	NOUN
ejpam-5298	154	37	)	)	PUNCT
ejpam-5298	154	38	,	,	PUNCT
ejpam-5298	154	39	(	(	PUNCT
ejpam-5298	154	40	4	4	X
ejpam-5298	154	41	)	)	PUNCT
ejpam-5298	154	42	if	if	SCONJ
ejpam-5298	154	43	m	m	NOUN
ejpam-5298	154	44	is	be	AUX
ejpam-5298	154	45	injective	injective	ADJ
ejpam-5298	154	46	,	,	PUNCT
ejpam-5298	154	47	then	then	ADV
ejpam-5298	154	48	m	m	VERB
ejpam-5298	154	49	=	=	ADJ
ejpam-5298	154	50	ih	ih	X
ejpam-5298	154	51	.	.	PUNCT
ejpam-5298	154	52	a.	a.	PROPN
ejpam-5298	154	53	iampan	iampan	PROPN
ejpam-5298	154	54	,	,	PUNCT
ejpam-5298	154	55	n.	n.	PROPN
ejpam-5298	154	56	rajesh	rajesh	PROPN
ejpam-5298	154	57	/	/	SYM
ejpam-5298	154	58	eur	eur	PROPN
ejpam-5298	154	59	.	.	PUNCT
ejpam-5298	155	1	j.	j.	PROPN
ejpam-5298	155	2	pure	pure	PROPN
ejpam-5298	155	3	appl	appl	PROPN
ejpam-5298	155	4	.	.	PROPN
ejpam-5298	155	5	math	math	PROPN
ejpam-5298	155	6	,	,	PUNCT
ejpam-5298	155	7	17	17	NUM
ejpam-5298	155	8	(	(	PUNCT
ejpam-5298	155	9	4	4	NUM
ejpam-5298	155	10	)	)	PUNCT
ejpam-5298	155	11	(	(	PUNCT
ejpam-5298	155	12	2024	2024	NUM
ejpam-5298	155	13	)	)	PUNCT
ejpam-5298	155	14	,	,	PUNCT
ejpam-5298	155	15	2726	2726	NUM
ejpam-5298	155	16	-	-	SYM
ejpam-5298	155	17	2737	2737	NUM
ejpam-5298	155	18	2732	2732	NUM
ejpam-5298	155	19	proof	proof	NOUN
ejpam-5298	155	20	.	.	PUNCT
ejpam-5298	156	1	(	(	PUNCT
ejpam-5298	156	2	1	1	X
ejpam-5298	156	3	)	)	PUNCT
ejpam-5298	156	4	it	it	PRON
ejpam-5298	156	5	is	be	AUX
ejpam-5298	156	6	done	do	VERB
ejpam-5298	156	7	in	in	ADP
ejpam-5298	156	8	proposition	proposition	NOUN
ejpam-5298	156	9	5	5	NUM
ejpam-5298	156	10	.	.	PUNCT
ejpam-5298	157	1	(	(	PUNCT
ejpam-5298	157	2	2	2	X
ejpam-5298	157	3	)	)	PUNCT
ejpam-5298	157	4	let	let	VERB
ejpam-5298	157	5	x	x	SYM
ejpam-5298	157	6	∈	∈	PROPN
ejpam-5298	157	7	h.	h.	PROPN
ejpam-5298	157	8	then	then	ADV
ejpam-5298	157	9	,	,	PUNCT
ejpam-5298	157	10	by	by	ADP
ejpam-5298	157	11	(	(	PUNCT
ejpam-5298	157	12	1	1	NUM
ejpam-5298	157	13	)	)	PUNCT
ejpam-5298	157	14	and	and	CCONJ
ejpam-5298	157	15	lemma	lemma	PROPN
ejpam-5298	157	16	1	1	NUM
ejpam-5298	157	17	(	(	PUNCT
ejpam-5298	157	18	1	1	NUM
ejpam-5298	157	19	)	)	PUNCT
ejpam-5298	157	20	,	,	PUNCT
ejpam-5298	157	21	we	we	PRON
ejpam-5298	157	22	have	have	VERB
ejpam-5298	157	23	1	1	NUM
ejpam-5298	157	24	=	=	SYM
ejpam-5298	157	25	m(1	m(1	NOUN
ejpam-5298	157	26	)	)	PUNCT
ejpam-5298	158	1	=	=	SYM
ejpam-5298	158	2	m(x	m(x	X
ejpam-5298	158	3	·	·	PUNCT
ejpam-5298	158	4	x	x	X
ejpam-5298	158	5	)	)	PUNCT
ejpam-5298	158	6	=	=	SYM
ejpam-5298	158	7	x	x	SYM
ejpam-5298	158	8	·	·	PUNCT
ejpam-5298	158	9	m(x	m(x	NOUN
ejpam-5298	158	10	)	)	PUNCT
ejpam-5298	158	11	,	,	PUNCT
ejpam-5298	158	12	that	that	ADV
ejpam-5298	158	13	is	is	ADV
ejpam-5298	158	14	,	,	PUNCT
ejpam-5298	158	15	x	x	SYM
ejpam-5298	158	16	≤	≤	ADJ
ejpam-5298	158	17	m(x	m(x	PROPN
ejpam-5298	158	18	)	)	PUNCT
ejpam-5298	158	19	.	.	PUNCT
ejpam-5298	159	1	(	(	PUNCT
ejpam-5298	159	2	3	3	X
ejpam-5298	159	3	)	)	PUNCT
ejpam-5298	159	4	let	let	VERB
ejpam-5298	159	5	x	x	PRON
ejpam-5298	159	6	,	,	PUNCT
ejpam-5298	159	7	y	y	PROPN
ejpam-5298	159	8	∈	∈	PROPN
ejpam-5298	159	9	h	h	NOUN
ejpam-5298	159	10	be	be	AUX
ejpam-5298	159	11	such	such	ADJ
ejpam-5298	159	12	that	that	SCONJ
ejpam-5298	159	13	x	x	X
ejpam-5298	159	14	≤	≤	X
ejpam-5298	159	15	y.	y.	NOUN
ejpam-5298	159	16	then	then	ADV
ejpam-5298	159	17	x	x	X
ejpam-5298	159	18	·	·	PUNCT
ejpam-5298	159	19	y	y	X
ejpam-5298	159	20	=	=	SYM
ejpam-5298	159	21	1	1	X
ejpam-5298	159	22	.	.	PUNCT
ejpam-5298	159	23	by	by	ADP
ejpam-5298	159	24	(	(	PUNCT
ejpam-5298	159	25	1	1	NUM
ejpam-5298	159	26	)	)	PUNCT
ejpam-5298	159	27	,	,	PUNCT
ejpam-5298	159	28	we	we	PRON
ejpam-5298	159	29	have	have	VERB
ejpam-5298	159	30	1	1	NUM
ejpam-5298	159	31	=	=	SYM
ejpam-5298	159	32	m(1	m(1	NOUN
ejpam-5298	159	33	)	)	PUNCT
ejpam-5298	160	1	=	=	SYM
ejpam-5298	160	2	m(x	m(x	PROPN
ejpam-5298	160	3	·	·	PUNCT
ejpam-5298	160	4	y	y	X
ejpam-5298	160	5	)	)	PUNCT
ejpam-5298	160	6	=	=	SYM
ejpam-5298	160	7	x	x	SYM
ejpam-5298	160	8	·	·	PUNCT
ejpam-5298	160	9	m(y	m(y	NUM
ejpam-5298	160	10	)	)	PUNCT
ejpam-5298	160	11	,	,	PUNCT
ejpam-5298	160	12	that	that	ADV
ejpam-5298	160	13	is	is	ADV
ejpam-5298	160	14	,	,	PUNCT
ejpam-5298	160	15	x	x	PUNCT
ejpam-5298	160	16	≤	≤	PROPN
ejpam-5298	160	17	m(y	m(y	NOUN
ejpam-5298	160	18	)	)	PUNCT
ejpam-5298	160	19	.	.	PUNCT
ejpam-5298	161	1	(	(	PUNCT
ejpam-5298	161	2	4	4	X
ejpam-5298	161	3	)	)	PUNCT
ejpam-5298	161	4	assume	assume	VERB
ejpam-5298	161	5	that	that	SCONJ
ejpam-5298	161	6	m	m	NOUN
ejpam-5298	161	7	is	be	AUX
ejpam-5298	161	8	injective	injective	ADJ
ejpam-5298	161	9	.	.	PUNCT
ejpam-5298	162	1	let	let	VERB
ejpam-5298	162	2	x	x	SYM
ejpam-5298	162	3	∈	∈	PROPN
ejpam-5298	162	4	h.	h.	PROPN
ejpam-5298	162	5	by	by	ADP
ejpam-5298	162	6	(	(	PUNCT
ejpam-5298	162	7	1	1	X
ejpam-5298	162	8	)	)	PUNCT
ejpam-5298	162	9	and	and	CCONJ
ejpam-5298	162	10	lemma	lemma	PROPN
ejpam-5298	162	11	1	1	NUM
ejpam-5298	162	12	(	(	PUNCT
ejpam-5298	162	13	1	1	NUM
ejpam-5298	162	14	)	)	PUNCT
ejpam-5298	162	15	,	,	PUNCT
ejpam-5298	162	16	we	we	PRON
ejpam-5298	162	17	have	have	VERB
ejpam-5298	162	18	m(m(x	m(m(x	PROPN
ejpam-5298	162	19	)	)	PUNCT
ejpam-5298	162	20	·	·	PUNCT
ejpam-5298	163	1	x	x	X
ejpam-5298	163	2	)	)	PUNCT
ejpam-5298	163	3	=	=	SYM
ejpam-5298	163	4	m(x	m(x	PROPN
ejpam-5298	163	5	)	)	PUNCT
ejpam-5298	163	6	·	·	PUNCT
ejpam-5298	163	7	m(x	m(x	X
ejpam-5298	163	8	)	)	PUNCT
ejpam-5298	163	9	=	=	SYM
ejpam-5298	163	10	1	1	NUM
ejpam-5298	163	11	=	=	SYM
ejpam-5298	163	12	m(1	m(1	NOUN
ejpam-5298	163	13	)	)	PUNCT
ejpam-5298	163	14	.	.	PUNCT
ejpam-5298	164	1	since	since	SCONJ
ejpam-5298	164	2	m	m	PROPN
ejpam-5298	164	3	is	be	AUX
ejpam-5298	164	4	injective	injective	ADJ
ejpam-5298	164	5	,	,	PUNCT
ejpam-5298	164	6	m(x	m(x	PROPN
ejpam-5298	164	7	)	)	PUNCT
ejpam-5298	164	8	·	·	PUNCT
ejpam-5298	165	1	x	x	PUNCT
ejpam-5298	165	2	=	=	SYM
ejpam-5298	165	3	1	1	X
ejpam-5298	165	4	.	.	PUNCT
ejpam-5298	165	5	then	then	ADV
ejpam-5298	165	6	m(x	m(x	NOUN
ejpam-5298	165	7	)	)	PUNCT
ejpam-5298	165	8	≤	≤	NUM
ejpam-5298	165	9	x.	x.	NOUN
ejpam-5298	166	1	it	it	PRON
ejpam-5298	166	2	follows	follow	VERB
ejpam-5298	166	3	from	from	ADP
ejpam-5298	166	4	(	(	PUNCT
ejpam-5298	166	5	2	2	NUM
ejpam-5298	166	6	)	)	PUNCT
ejpam-5298	166	7	and	and	CCONJ
ejpam-5298	166	8	definition	definition	NOUN
ejpam-5298	166	9	1	1	NUM
ejpam-5298	166	10	(	(	PUNCT
ejpam-5298	166	11	3	3	NUM
ejpam-5298	166	12	)	)	PUNCT
ejpam-5298	166	13	that	that	DET
ejpam-5298	166	14	m(x	m(x	X
ejpam-5298	166	15	)	)	PUNCT
ejpam-5298	167	1	=	=	SYM
ejpam-5298	167	2	x	x	X
ejpam-5298	167	3	,	,	PUNCT
ejpam-5298	167	4	so	so	ADV
ejpam-5298	167	5	m	m	VERB
ejpam-5298	167	6	=	=	ADJ
ejpam-5298	167	7	ih	ih	X
ejpam-5298	167	8	.	.	PUNCT
ejpam-5298	168	1	definition	definition	NOUN
ejpam-5298	168	2	8	8	NUM
ejpam-5298	168	3	.	.	PUNCT
ejpam-5298	169	1	a	a	DET
ejpam-5298	169	2	self	self	NOUN
ejpam-5298	169	3	-	-	PUNCT
ejpam-5298	169	4	map	map	NOUN
ejpam-5298	169	5	m	m	NOUN
ejpam-5298	169	6	of	of	ADP
ejpam-5298	169	7	h	h	NOUN
ejpam-5298	169	8	is	be	AUX
ejpam-5298	169	9	said	say	VERB
ejpam-5298	169	10	to	to	PART
ejpam-5298	169	11	be	be	AUX
ejpam-5298	169	12	nonexpansive	nonexpansive	ADJ
ejpam-5298	169	13	if	if	SCONJ
ejpam-5298	169	14	m(x	m(x	NOUN
ejpam-5298	169	15	)	)	PUNCT
ejpam-5298	169	16	≤	≤	NUM
ejpam-5298	169	17	x	x	PUNCT
ejpam-5298	169	18	for	for	ADP
ejpam-5298	169	19	all	all	DET
ejpam-5298	169	20	x	x	SYM
ejpam-5298	169	21	∈	∈	PROPN
ejpam-5298	169	22	h.	h.	PROPN
ejpam-5298	169	23	example	example	NOUN
ejpam-5298	170	1	4	4	NUM
ejpam-5298	170	2	.	.	PUNCT
ejpam-5298	171	1	[	[	X
ejpam-5298	171	2	12	12	NUM
ejpam-5298	171	3	]	]	PUNCT
ejpam-5298	171	4	let	let	VERB
ejpam-5298	171	5	h	h	NOUN
ejpam-5298	171	6	=	=	PRON
ejpam-5298	171	7	{	{	PUNCT
ejpam-5298	171	8	1	1	NUM
ejpam-5298	171	9	,	,	PUNCT
ejpam-5298	171	10	α	α	NOUN
ejpam-5298	171	11	,	,	PUNCT
ejpam-5298	171	12	β	β	X
ejpam-5298	171	13	,	,	PUNCT
ejpam-5298	171	14	γ	γ	PROPN
ejpam-5298	171	15	,	,	PUNCT
ejpam-5298	171	16	ϵ	ϵ	X
ejpam-5298	171	17	}	}	PUNCT
ejpam-5298	171	18	with	with	ADP
ejpam-5298	171	19	the	the	DET
ejpam-5298	171	20	following	follow	VERB
ejpam-5298	171	21	cayley	cayley	ADJ
ejpam-5298	171	22	table	table	NOUN
ejpam-5298	171	23	:	:	PUNCT
ejpam-5298	171	24	·	·	PUNCT
ejpam-5298	171	25	1	1	NUM
ejpam-5298	171	26	α	α	NOUN
ejpam-5298	171	27	β	β	X
ejpam-5298	171	28	γ	γ	X
ejpam-5298	171	29	ϵ	ϵ	PROPN
ejpam-5298	171	30	1	1	NUM
ejpam-5298	171	31	1	1	NUM
ejpam-5298	171	32	α	α	NOUN
ejpam-5298	171	33	β	β	X
ejpam-5298	171	34	γ	γ	X
ejpam-5298	171	35	ϵ	ϵ	PROPN
ejpam-5298	171	36	α	α	PROPN
ejpam-5298	171	37	1	1	NUM
ejpam-5298	171	38	1	1	NUM
ejpam-5298	171	39	β	β	X
ejpam-5298	171	40	γ	γ	X
ejpam-5298	171	41	ϵ	ϵ	X
ejpam-5298	171	42	β	β	X
ejpam-5298	171	43	1	1	NUM
ejpam-5298	171	44	α	α	NOUN
ejpam-5298	171	45	1	1	NUM
ejpam-5298	171	46	γ	γ	PROPN
ejpam-5298	171	47	γ	γ	X
ejpam-5298	171	48	γ	γ	X
ejpam-5298	171	49	1	1	NUM
ejpam-5298	171	50	1	1	NUM
ejpam-5298	171	51	β	β	NOUN
ejpam-5298	171	52	1	1	NUM
ejpam-5298	171	53	β	β	NOUN
ejpam-5298	171	54	ϵ	ϵ	NOUN
ejpam-5298	171	55	1	1	NUM
ejpam-5298	171	56	1	1	NUM
ejpam-5298	171	57	1	1	NUM
ejpam-5298	171	58	1	1	NUM
ejpam-5298	171	59	1	1	NUM
ejpam-5298	171	60	then	then	ADV
ejpam-5298	171	61	h	h	NOUN
ejpam-5298	172	1	=	=	PUNCT
ejpam-5298	172	2	(	(	PUNCT
ejpam-5298	172	3	h	h	NOUN
ejpam-5298	172	4	,	,	PUNCT
ejpam-5298	172	5	·	·	PUNCT
ejpam-5298	172	6	,	,	PUNCT
ejpam-5298	172	7	1	1	NUM
ejpam-5298	172	8	)	)	PUNCT
ejpam-5298	172	9	is	be	AUX
ejpam-5298	172	10	a	a	DET
ejpam-5298	172	11	hilbert	hilbert	NOUN
ejpam-5298	172	12	algebra	algebra	NOUN
ejpam-5298	172	13	.	.	PUNCT
ejpam-5298	173	1	we	we	PRON
ejpam-5298	173	2	define	define	VERB
ejpam-5298	173	3	a	a	DET
ejpam-5298	173	4	self	self	NOUN
ejpam-5298	173	5	-	-	PUNCT
ejpam-5298	173	6	map	map	NOUN
ejpam-5298	173	7	m	m	NOUN
ejpam-5298	173	8	:	:	PUNCT
ejpam-5298	173	9	h	h	PROPN
ejpam-5298	173	10	→	→	SYM
ejpam-5298	173	11	h	h	NOUN
ejpam-5298	173	12	as	as	SCONJ
ejpam-5298	173	13	follows	follow	VERB
ejpam-5298	173	14	:	:	PUNCT
ejpam-5298	173	15	m(1	m(1	NOUN
ejpam-5298	173	16	)	)	PUNCT
ejpam-5298	173	17	=	=	SYM
ejpam-5298	173	18	β	β	X
ejpam-5298	173	19	,	,	PUNCT
ejpam-5298	173	20	m(α	m(α	PROPN
ejpam-5298	173	21	)	)	PUNCT
ejpam-5298	173	22	=	=	SYM
ejpam-5298	173	23	α	α	PROPN
ejpam-5298	173	24	,	,	PUNCT
ejpam-5298	173	25	m(β	m(β	PROPN
ejpam-5298	173	26	)	)	PUNCT
ejpam-5298	173	27	=	=	SYM
ejpam-5298	173	28	β	β	X
ejpam-5298	173	29	,	,	PUNCT
ejpam-5298	173	30	m(γ	m(γ	NOUN
ejpam-5298	173	31	)	)	PUNCT
ejpam-5298	173	32	=	=	SYM
ejpam-5298	173	33	γ	γ	NOUN
ejpam-5298	173	34	and	and	CCONJ
ejpam-5298	173	35	m(ϵ	m(ϵ	NOUN
ejpam-5298	173	36	)	)	PUNCT
ejpam-5298	173	37	=	=	VERB
ejpam-5298	174	1	ϵ.	ϵ.	NOUN
ejpam-5298	174	2	then	then	ADV
ejpam-5298	174	3	m	m	PROPN
ejpam-5298	174	4	is	be	AUX
ejpam-5298	174	5	nonexpansive	nonexpansive	ADJ
ejpam-5298	174	6	.	.	PUNCT
ejpam-5298	175	1	since	since	SCONJ
ejpam-5298	175	2	m(1	m(1	NOUN
ejpam-5298	175	3	)	)	PUNCT
ejpam-5298	175	4	=	=	PUNCT
ejpam-5298	175	5	β	β	X
ejpam-5298	175	6	̸=	̸=	PROPN
ejpam-5298	175	7	1	1	NUM
ejpam-5298	175	8	,	,	PUNCT
ejpam-5298	175	9	it	it	PRON
ejpam-5298	175	10	follows	follow	VERB
ejpam-5298	175	11	from	from	ADP
ejpam-5298	175	12	proposition	proposition	NOUN
ejpam-5298	175	13	5	5	NUM
ejpam-5298	175	14	that	that	PRON
ejpam-5298	175	15	m	m	VERB
ejpam-5298	175	16	is	be	AUX
ejpam-5298	175	17	not	not	PART
ejpam-5298	175	18	a	a	DET
ejpam-5298	175	19	multiplier	multipli	ADJ
ejpam-5298	175	20	of	of	ADP
ejpam-5298	175	21	h.	h.	NOUN
ejpam-5298	175	22	proposition	proposition	NOUN
ejpam-5298	175	23	7	7	X
ejpam-5298	175	24	.	.	PUNCT
ejpam-5298	176	1	if	if	SCONJ
ejpam-5298	176	2	m	m	NOUN
ejpam-5298	176	3	is	be	AUX
ejpam-5298	176	4	a	a	DET
ejpam-5298	176	5	nonexpansive	nonexpansive	ADJ
ejpam-5298	176	6	multiplier	multipli	ADJ
ejpam-5298	176	7	of	of	ADP
ejpam-5298	176	8	h	h	NOUN
ejpam-5298	176	9	,	,	PUNCT
ejpam-5298	176	10	then	then	ADV
ejpam-5298	176	11	m	m	VERB
ejpam-5298	176	12	=	=	ADJ
ejpam-5298	176	13	ih	ih	X
ejpam-5298	176	14	.	.	PUNCT
ejpam-5298	177	1	proof	proof	NOUN
ejpam-5298	177	2	.	.	PUNCT
ejpam-5298	178	1	by	by	ADP
ejpam-5298	178	2	assumption	assumption	NOUN
ejpam-5298	178	3	,	,	PUNCT
ejpam-5298	178	4	proposition	proposition	NOUN
ejpam-5298	178	5	6	6	NUM
ejpam-5298	178	6	(	(	PUNCT
ejpam-5298	178	7	2	2	NUM
ejpam-5298	178	8	)	)	PUNCT
ejpam-5298	178	9	,	,	PUNCT
ejpam-5298	178	10	and	and	CCONJ
ejpam-5298	178	11	definition	definition	NOUN
ejpam-5298	178	12	1	1	NUM
ejpam-5298	178	13	(	(	PUNCT
ejpam-5298	178	14	3	3	NUM
ejpam-5298	178	15	)	)	PUNCT
ejpam-5298	178	16	,	,	PUNCT
ejpam-5298	178	17	we	we	PRON
ejpam-5298	178	18	have	have	VERB
ejpam-5298	178	19	m(x	m(x	NOUN
ejpam-5298	178	20	)	)	PUNCT
ejpam-5298	179	1	=	=	PUNCT
ejpam-5298	180	1	x	x	PUNCT
ejpam-5298	180	2	for	for	ADP
ejpam-5298	180	3	all	all	DET
ejpam-5298	180	4	x	x	SYM
ejpam-5298	180	5	∈	∈	PROPN
ejpam-5298	180	6	h.	h.	NOUN
ejpam-5298	180	7	hence	hence	ADV
ejpam-5298	180	8	,	,	PUNCT
ejpam-5298	180	9	m	m	VERB
ejpam-5298	180	10	=	=	ADJ
ejpam-5298	180	11	ih	ih	X
ejpam-5298	180	12	.	.	PUNCT
ejpam-5298	181	1	proposition	proposition	NOUN
ejpam-5298	181	2	8	8	NUM
ejpam-5298	181	3	.	.	PUNCT
ejpam-5298	182	1	let	let	VERB
ejpam-5298	182	2	m1	m1	PROPN
ejpam-5298	182	3	and	and	CCONJ
ejpam-5298	182	4	m2	m2	PROPN
ejpam-5298	182	5	be	be	VERB
ejpam-5298	182	6	two	two	NUM
ejpam-5298	182	7	multipliers	multiplier	NOUN
ejpam-5298	182	8	of	of	ADP
ejpam-5298	182	9	h.	h.	PROPN
ejpam-5298	182	10	then	then	ADV
ejpam-5298	182	11	m1	m1	PROPN
ejpam-5298	182	12	◦	◦	NOUN
ejpam-5298	182	13	m2	m2	PROPN
ejpam-5298	182	14	is	be	AUX
ejpam-5298	182	15	a	a	DET
ejpam-5298	182	16	multiplier	multipli	ADJ
ejpam-5298	182	17	of	of	ADP
ejpam-5298	182	18	h.	h.	NOUN
ejpam-5298	182	19	proof	proof	NOUN
ejpam-5298	182	20	.	.	PUNCT
ejpam-5298	183	1	let	let	VERB
ejpam-5298	183	2	x	x	PRON
ejpam-5298	183	3	,	,	PUNCT
ejpam-5298	183	4	y	y	PROPN
ejpam-5298	183	5	∈	∈	PROPN
ejpam-5298	183	6	h.	h.	PROPN
ejpam-5298	183	7	then	then	ADV
ejpam-5298	183	8	(	(	PUNCT
ejpam-5298	183	9	m1	m1	PROPN
ejpam-5298	183	10	◦	◦	PROPN
ejpam-5298	183	11	m2)(x	m2)(x	PROPN
ejpam-5298	183	12	·	·	PUNCT
ejpam-5298	183	13	y	y	X
ejpam-5298	183	14	)	)	PUNCT
ejpam-5298	183	15	=	=	SYM
ejpam-5298	183	16	m1(m2(x	m1(m2(x	NOUN
ejpam-5298	183	17	·	·	PUNCT
ejpam-5298	183	18	y	y	X
ejpam-5298	183	19	)	)	PUNCT
ejpam-5298	183	20	)	)	PUNCT
ejpam-5298	184	1	=	=	PUNCT
ejpam-5298	184	2	m1(x	m1(x	X
ejpam-5298	184	3	·	·	SYM
ejpam-5298	184	4	m2(y	m2(y	NOUN
ejpam-5298	184	5	)	)	PUNCT
ejpam-5298	184	6	)	)	PUNCT
ejpam-5298	185	1	=	=	PUNCT
ejpam-5298	185	2	x	x	SYM
ejpam-5298	185	3	·	·	SYM
ejpam-5298	185	4	m1(m2(y	m1(m2(y	NOUN
ejpam-5298	185	5	)	)	PUNCT
ejpam-5298	185	6	)	)	PUNCT
ejpam-5298	186	1	=	=	PUNCT
ejpam-5298	186	2	x	x	SYM
ejpam-5298	186	3	·	·	PUNCT
ejpam-5298	186	4	(	(	PUNCT
ejpam-5298	186	5	m1	m1	PROPN
ejpam-5298	186	6	◦	◦	PROPN
ejpam-5298	186	7	m2)(y	m2)(y	PROPN
ejpam-5298	186	8	)	)	PUNCT
ejpam-5298	186	9	.	.	PUNCT
ejpam-5298	187	1	hence	hence	ADV
ejpam-5298	187	2	,	,	PUNCT
ejpam-5298	187	3	m1	m1	PROPN
ejpam-5298	187	4	◦	◦	NOUN
ejpam-5298	187	5	m2	m2	PROPN
ejpam-5298	187	6	is	be	AUX
ejpam-5298	187	7	a	a	DET
ejpam-5298	187	8	multiplier	multipli	ADJ
ejpam-5298	187	9	of	of	ADP
ejpam-5298	187	10	h.	h.	PROPN
ejpam-5298	187	11	corollary	corollary	PROPN
ejpam-5298	187	12	1	1	PROPN
ejpam-5298	187	13	.	.	PUNCT
ejpam-5298	188	1	let	let	VERB
ejpam-5298	188	2	m	m	PRON
ejpam-5298	188	3	be	be	AUX
ejpam-5298	188	4	a	a	DET
ejpam-5298	188	5	multiplier	multipli	ADJ
ejpam-5298	188	6	of	of	ADP
ejpam-5298	188	7	h.	h.	PROPN
ejpam-5298	188	8	then	then	ADV
ejpam-5298	188	9	mn	mn	PROPN
ejpam-5298	188	10	is	be	AUX
ejpam-5298	188	11	a	a	DET
ejpam-5298	188	12	multiplier	multipli	ADJ
ejpam-5298	188	13	of	of	ADP
ejpam-5298	188	14	h	h	NOUN
ejpam-5298	188	15	for	for	ADP
ejpam-5298	188	16	all	all	DET
ejpam-5298	188	17	positive	positive	ADJ
ejpam-5298	188	18	integer	integer	NOUN
ejpam-5298	188	19	n.	n.	NOUN
ejpam-5298	188	20	let	let	VERB
ejpam-5298	188	21	(	(	PUNCT
ejpam-5298	188	22	h	h	NOUN
ejpam-5298	188	23	,	,	PUNCT
ejpam-5298	188	24	·	·	PUNCT
ejpam-5298	188	25	,	,	PUNCT
ejpam-5298	188	26	1h	1h	NUM
ejpam-5298	188	27	)	)	PUNCT
ejpam-5298	188	28	and	and	CCONJ
ejpam-5298	188	29	(	(	PUNCT
ejpam-5298	188	30	k	k	NOUN
ejpam-5298	188	31	,	,	PUNCT
ejpam-5298	188	32	∗	∗	NOUN
ejpam-5298	188	33	,	,	PUNCT
ejpam-5298	188	34	1k	1k	NUM
ejpam-5298	188	35	)	)	PUNCT
ejpam-5298	188	36	be	be	VERB
ejpam-5298	188	37	hilbert	hilbert	NOUN
ejpam-5298	188	38	algebras	algebras	PROPN
ejpam-5298	188	39	.	.	PUNCT
ejpam-5298	189	1	then	then	ADV
ejpam-5298	189	2	(	(	PUNCT
ejpam-5298	189	3	h×k	h×k	PROPN
ejpam-5298	189	4	,	,	PUNCT
ejpam-5298	189	5	⋄	⋄	PROPN
ejpam-5298	189	6	,	,	PUNCT
ejpam-5298	189	7	(	(	PUNCT
ejpam-5298	189	8	1h	1h	NUM
ejpam-5298	189	9	,	,	PUNCT
ejpam-5298	189	10	1k	1k	NUM
ejpam-5298	189	11	)	)	PUNCT
ejpam-5298	189	12	)	)	PUNCT
ejpam-5298	189	13	is	be	AUX
ejpam-5298	189	14	a	a	DET
ejpam-5298	189	15	hilbert	hilbert	NOUN
ejpam-5298	189	16	algebra	algebra	NOUN
ejpam-5298	189	17	defined	define	VERB
ejpam-5298	189	18	by	by	ADP
ejpam-5298	189	19	(	(	PUNCT
ejpam-5298	189	20	a	a	DET
ejpam-5298	189	21	,	,	PUNCT
ejpam-5298	189	22	b	b	NOUN
ejpam-5298	189	23	)	)	PUNCT
ejpam-5298	189	24	⋄	⋄	NOUN
ejpam-5298	189	25	(	(	PUNCT
ejpam-5298	189	26	c	c	X
ejpam-5298	189	27	,	,	PUNCT
ejpam-5298	189	28	d	d	NOUN
ejpam-5298	189	29	)	)	PUNCT
ejpam-5298	189	30	=	=	SYM
ejpam-5298	189	31	(	(	PUNCT
ejpam-5298	189	32	a	a	DET
ejpam-5298	189	33	·	·	PUNCT
ejpam-5298	189	34	c	c	X
ejpam-5298	189	35	,	,	PUNCT
ejpam-5298	189	36	b	b	PROPN
ejpam-5298	189	37	∗	∗	X
ejpam-5298	189	38	d	d	NOUN
ejpam-5298	189	39	)	)	PUNCT
ejpam-5298	189	40	for	for	ADP
ejpam-5298	189	41	all	all	DET
ejpam-5298	189	42	a	a	PRON
ejpam-5298	189	43	,	,	PUNCT
ejpam-5298	189	44	c	c	PROPN
ejpam-5298	189	45	∈	∈	PROPN
ejpam-5298	189	46	h	h	NOUN
ejpam-5298	189	47	and	and	CCONJ
ejpam-5298	189	48	b	b	NOUN
ejpam-5298	189	49	,	,	PUNCT
ejpam-5298	189	50	d	d	PROPN
ejpam-5298	189	51	∈	∈	PROPN
ejpam-5298	189	52	k.	k.	PROPN
ejpam-5298	189	53	a.	a.	PROPN
ejpam-5298	189	54	iampan	iampan	PROPN
ejpam-5298	189	55	,	,	PUNCT
ejpam-5298	189	56	n.	n.	PROPN
ejpam-5298	189	57	rajesh	rajesh	PROPN
ejpam-5298	189	58	/	/	SYM
ejpam-5298	189	59	eur	eur	PROPN
ejpam-5298	189	60	.	.	PUNCT
ejpam-5298	190	1	j.	j.	PROPN
ejpam-5298	190	2	pure	pure	PROPN
ejpam-5298	190	3	appl	appl	PROPN
ejpam-5298	190	4	.	.	PROPN
ejpam-5298	190	5	math	math	PROPN
ejpam-5298	190	6	,	,	PUNCT
ejpam-5298	190	7	17	17	NUM
ejpam-5298	190	8	(	(	PUNCT
ejpam-5298	190	9	4	4	NUM
ejpam-5298	190	10	)	)	PUNCT
ejpam-5298	190	11	(	(	PUNCT
ejpam-5298	190	12	2024	2024	NUM
ejpam-5298	190	13	)	)	PUNCT
ejpam-5298	190	14	,	,	PUNCT
ejpam-5298	190	15	2726	2726	NUM
ejpam-5298	190	16	-	-	SYM
ejpam-5298	190	17	2737	2737	NUM
ejpam-5298	190	18	2733	2733	NUM
ejpam-5298	190	19	proposition	proposition	NOUN
ejpam-5298	190	20	9	9	NUM
ejpam-5298	190	21	.	.	PUNCT
ejpam-5298	191	1	let	let	VERB
ejpam-5298	191	2	h	h	NOUN
ejpam-5298	191	3	=	=	PUNCT
ejpam-5298	191	4	(	(	PUNCT
ejpam-5298	191	5	h	h	NOUN
ejpam-5298	191	6	,	,	PUNCT
ejpam-5298	191	7	·	·	PUNCT
ejpam-5298	191	8	,	,	PUNCT
ejpam-5298	191	9	1h	1h	NUM
ejpam-5298	191	10	)	)	PUNCT
ejpam-5298	191	11	and	and	CCONJ
ejpam-5298	191	12	k	k	NOUN
ejpam-5298	191	13	=	=	PRON
ejpam-5298	191	14	(	(	PUNCT
ejpam-5298	191	15	k	k	NOUN
ejpam-5298	191	16	,	,	PUNCT
ejpam-5298	191	17	∗	∗	NOUN
ejpam-5298	191	18	,	,	PUNCT
ejpam-5298	191	19	1k	1k	NUM
ejpam-5298	191	20	)	)	PUNCT
ejpam-5298	191	21	be	be	VERB
ejpam-5298	191	22	hilbert	hilbert	PROPN
ejpam-5298	191	23	algebras	algebras	PROPN
ejpam-5298	191	24	.	.	PUNCT
ejpam-5298	192	1	define	define	VERB
ejpam-5298	192	2	the	the	DET
ejpam-5298	192	3	self	self	NOUN
ejpam-5298	192	4	-	-	PUNCT
ejpam-5298	192	5	map	map	NOUN
ejpam-5298	192	6	m	m	NOUN
ejpam-5298	192	7	of	of	ADP
ejpam-5298	192	8	h×k	h×k	PROPN
ejpam-5298	192	9	by	by	ADP
ejpam-5298	192	10	,	,	PUNCT
ejpam-5298	192	11	for	for	ADP
ejpam-5298	192	12	any	any	DET
ejpam-5298	192	13	(	(	PUNCT
ejpam-5298	192	14	x	x	NOUN
ejpam-5298	192	15	,	,	PUNCT
ejpam-5298	192	16	y	y	NOUN
ejpam-5298	192	17	)	)	PUNCT
ejpam-5298	192	18	∈	∈	NOUN
ejpam-5298	192	19	h×k	h×k	PROPN
ejpam-5298	192	20	,	,	PUNCT
ejpam-5298	192	21	m(x	m(x	PROPN
ejpam-5298	192	22	,	,	PUNCT
ejpam-5298	192	23	y	y	NOUN
ejpam-5298	192	24	)	)	PUNCT
ejpam-5298	192	25	=	=	SYM
ejpam-5298	192	26	(	(	PUNCT
ejpam-5298	192	27	x	x	X
ejpam-5298	192	28	,	,	PUNCT
ejpam-5298	192	29	1k	1k	NUM
ejpam-5298	192	30	)	)	PUNCT
ejpam-5298	192	31	.	.	PUNCT
ejpam-5298	193	1	then	then	ADV
ejpam-5298	193	2	m	m	PROPN
ejpam-5298	193	3	is	be	AUX
ejpam-5298	193	4	a	a	DET
ejpam-5298	193	5	multiplier	multipli	ADJ
ejpam-5298	193	6	of	of	ADP
ejpam-5298	193	7	h×k	h×k	NOUN
ejpam-5298	193	8	.	.	PUNCT
ejpam-5298	194	1	proof	proof	NOUN
ejpam-5298	194	2	.	.	PUNCT
ejpam-5298	195	1	let	let	VERB
ejpam-5298	195	2	(	(	PUNCT
ejpam-5298	195	3	x1	x1	ADJ
ejpam-5298	195	4	,	,	PUNCT
ejpam-5298	195	5	x2	x2	PROPN
ejpam-5298	195	6	)	)	PUNCT
ejpam-5298	195	7	,	,	PUNCT
ejpam-5298	195	8	(	(	PUNCT
ejpam-5298	195	9	y1	y1	INTJ
ejpam-5298	195	10	,	,	PUNCT
ejpam-5298	195	11	y2	y2	NOUN
ejpam-5298	195	12	)	)	PUNCT
ejpam-5298	195	13	∈	∈	PROPN
ejpam-5298	195	14	h	h	NOUN
ejpam-5298	195	15	×k	×k	PROPN
ejpam-5298	195	16	.	.	PUNCT
ejpam-5298	196	1	then	then	ADV
ejpam-5298	196	2	m((x1	m((x1	PROPN
ejpam-5298	196	3	,	,	PUNCT
ejpam-5298	196	4	x2	x2	PROPN
ejpam-5298	196	5	)	)	PUNCT
ejpam-5298	196	6	⋄	⋄	PROPN
ejpam-5298	196	7	(	(	PUNCT
ejpam-5298	196	8	y1	y1	INTJ
ejpam-5298	196	9	,	,	PUNCT
ejpam-5298	196	10	y2	y2	PROPN
ejpam-5298	196	11	)	)	PUNCT
ejpam-5298	196	12	)	)	PUNCT
ejpam-5298	197	1	=	=	X
ejpam-5298	197	2	m(x1	m(x1	NOUN
ejpam-5298	197	3	·	·	PUNCT
ejpam-5298	197	4	y1	y1	INTJ
ejpam-5298	197	5	,	,	PUNCT
ejpam-5298	197	6	x2	x2	PROPN
ejpam-5298	197	7	∗	∗	NOUN
ejpam-5298	197	8	y2	y2	NOUN
ejpam-5298	197	9	)	)	PUNCT
ejpam-5298	197	10	=	=	SYM
ejpam-5298	198	1	(	(	PUNCT
ejpam-5298	198	2	x1	x1	PROPN
ejpam-5298	198	3	·	·	PUNCT
ejpam-5298	198	4	y1	y1	INTJ
ejpam-5298	198	5	,	,	PUNCT
ejpam-5298	198	6	1k	1k	NUM
ejpam-5298	198	7	)	)	PUNCT
ejpam-5298	198	8	=	=	SYM
ejpam-5298	199	1	(	(	PUNCT
ejpam-5298	199	2	x1	x1	PROPN
ejpam-5298	199	3	·	·	PUNCT
ejpam-5298	199	4	y1	y1	INTJ
ejpam-5298	199	5	,	,	PUNCT
ejpam-5298	199	6	x2	x2	PROPN
ejpam-5298	199	7	∗	∗	NOUN
ejpam-5298	199	8	1k	1k	NUM
ejpam-5298	199	9	)	)	PUNCT
ejpam-5298	199	10	=	=	SYM
ejpam-5298	200	1	(	(	PUNCT
ejpam-5298	200	2	x1	x1	PROPN
ejpam-5298	200	3	,	,	PUNCT
ejpam-5298	200	4	x2	x2	PROPN
ejpam-5298	200	5	)	)	PUNCT
ejpam-5298	200	6	⋄	⋄	PROPN
ejpam-5298	200	7	(	(	PUNCT
ejpam-5298	200	8	y1	y1	INTJ
ejpam-5298	200	9	,	,	PUNCT
ejpam-5298	200	10	1k	1k	NUM
ejpam-5298	200	11	)	)	PUNCT
ejpam-5298	200	12	=	=	SYM
ejpam-5298	200	13	(	(	PUNCT
ejpam-5298	200	14	x1	x1	PROPN
ejpam-5298	200	15	,	,	PUNCT
ejpam-5298	200	16	x2	x2	PROPN
ejpam-5298	200	17	)	)	PUNCT
ejpam-5298	200	18	⋄m(y1	⋄m(y1	X
ejpam-5298	200	19	,	,	PUNCT
ejpam-5298	200	20	y2	y2	PROPN
ejpam-5298	200	21	)	)	PUNCT
ejpam-5298	200	22	.	.	PUNCT
ejpam-5298	201	1	hence	hence	ADV
ejpam-5298	201	2	,	,	PUNCT
ejpam-5298	201	3	m	m	VERB
ejpam-5298	201	4	is	be	AUX
ejpam-5298	201	5	a	a	DET
ejpam-5298	201	6	multiplier	multipli	ADJ
ejpam-5298	201	7	of	of	ADP
ejpam-5298	201	8	h×k	h×k	PROPN
ejpam-5298	201	9	.	.	PUNCT
ejpam-5298	202	1	proposition	proposition	NOUN
ejpam-5298	202	2	10	10	NUM
ejpam-5298	202	3	.	.	PUNCT
ejpam-5298	203	1	let	let	VERB
ejpam-5298	203	2	h	h	NOUN
ejpam-5298	203	3	=	=	PUNCT
ejpam-5298	203	4	(	(	PUNCT
ejpam-5298	203	5	h	h	NOUN
ejpam-5298	203	6	,	,	PUNCT
ejpam-5298	203	7	·	·	PUNCT
ejpam-5298	203	8	,	,	PUNCT
ejpam-5298	203	9	1h	1h	NUM
ejpam-5298	203	10	)	)	PUNCT
ejpam-5298	203	11	and	and	CCONJ
ejpam-5298	203	12	k	k	NOUN
ejpam-5298	203	13	=	=	PRON
ejpam-5298	203	14	(	(	PUNCT
ejpam-5298	203	15	k	k	NOUN
ejpam-5298	203	16	,	,	PUNCT
ejpam-5298	203	17	∗	∗	NOUN
ejpam-5298	203	18	,	,	PUNCT
ejpam-5298	203	19	1k	1k	NUM
ejpam-5298	203	20	)	)	PUNCT
ejpam-5298	203	21	be	be	VERB
ejpam-5298	203	22	hilbert	hilbert	PROPN
ejpam-5298	203	23	algebras	algebras	PROPN
ejpam-5298	203	24	.	.	PUNCT
ejpam-5298	204	1	define	define	VERB
ejpam-5298	204	2	the	the	DET
ejpam-5298	204	3	self	self	NOUN
ejpam-5298	204	4	-	-	PUNCT
ejpam-5298	204	5	map	map	NOUN
ejpam-5298	204	6	m	m	NOUN
ejpam-5298	204	7	of	of	ADP
ejpam-5298	204	8	h×k	h×k	PROPN
ejpam-5298	204	9	by	by	ADP
ejpam-5298	204	10	,	,	PUNCT
ejpam-5298	204	11	for	for	ADP
ejpam-5298	204	12	any	any	DET
ejpam-5298	204	13	(	(	PUNCT
ejpam-5298	204	14	x	x	NOUN
ejpam-5298	204	15	,	,	PUNCT
ejpam-5298	204	16	y	y	NOUN
ejpam-5298	204	17	)	)	PUNCT
ejpam-5298	204	18	∈	∈	NOUN
ejpam-5298	204	19	h×k	h×k	PROPN
ejpam-5298	204	20	,	,	PUNCT
ejpam-5298	204	21	m(x	m(x	PROPN
ejpam-5298	204	22	,	,	PUNCT
ejpam-5298	204	23	y	y	NOUN
ejpam-5298	204	24	)	)	PUNCT
ejpam-5298	204	25	=	=	SYM
ejpam-5298	204	26	(	(	PUNCT
ejpam-5298	204	27	1h	1h	NUM
ejpam-5298	204	28	,	,	PUNCT
ejpam-5298	204	29	y	y	PROPN
ejpam-5298	204	30	)	)	PUNCT
ejpam-5298	204	31	.	.	PUNCT
ejpam-5298	205	1	then	then	ADV
ejpam-5298	205	2	m	m	PROPN
ejpam-5298	205	3	is	be	AUX
ejpam-5298	205	4	a	a	DET
ejpam-5298	205	5	multiplier	multipli	ADJ
ejpam-5298	205	6	of	of	ADP
ejpam-5298	205	7	h×k	h×k	NOUN
ejpam-5298	205	8	.	.	PUNCT
ejpam-5298	206	1	proof	proof	NOUN
ejpam-5298	206	2	.	.	PUNCT
ejpam-5298	207	1	let	let	VERB
ejpam-5298	207	2	(	(	PUNCT
ejpam-5298	207	3	x1	x1	ADJ
ejpam-5298	207	4	,	,	PUNCT
ejpam-5298	207	5	x2	x2	PROPN
ejpam-5298	207	6	)	)	PUNCT
ejpam-5298	207	7	,	,	PUNCT
ejpam-5298	207	8	(	(	PUNCT
ejpam-5298	207	9	y1	y1	INTJ
ejpam-5298	207	10	,	,	PUNCT
ejpam-5298	207	11	y2	y2	NOUN
ejpam-5298	207	12	)	)	PUNCT
ejpam-5298	207	13	∈	∈	PROPN
ejpam-5298	207	14	h	h	NOUN
ejpam-5298	207	15	×k	×k	PROPN
ejpam-5298	207	16	.	.	PUNCT
ejpam-5298	208	1	then	then	ADV
ejpam-5298	208	2	m((x1	m((x1	PROPN
ejpam-5298	208	3	,	,	PUNCT
ejpam-5298	208	4	x2	x2	PROPN
ejpam-5298	208	5	)	)	PUNCT
ejpam-5298	208	6	⋄	⋄	PROPN
ejpam-5298	208	7	(	(	PUNCT
ejpam-5298	208	8	y1	y1	INTJ
ejpam-5298	208	9	,	,	PUNCT
ejpam-5298	208	10	y2	y2	PROPN
ejpam-5298	208	11	)	)	PUNCT
ejpam-5298	208	12	)	)	PUNCT
ejpam-5298	209	1	=	=	X
ejpam-5298	209	2	m(x1	m(x1	NOUN
ejpam-5298	209	3	·	·	PUNCT
ejpam-5298	209	4	y1	y1	INTJ
ejpam-5298	209	5	,	,	PUNCT
ejpam-5298	209	6	x2	x2	PROPN
ejpam-5298	209	7	∗	∗	NOUN
ejpam-5298	209	8	y2	y2	NOUN
ejpam-5298	209	9	)	)	PUNCT
ejpam-5298	209	10	=	=	PRON
ejpam-5298	209	11	(	(	PUNCT
ejpam-5298	209	12	1h	1h	NUM
ejpam-5298	209	13	,	,	PUNCT
ejpam-5298	209	14	x2	x2	PROPN
ejpam-5298	209	15	∗	∗	NOUN
ejpam-5298	209	16	y2	y2	NOUN
ejpam-5298	209	17	)	)	PUNCT
ejpam-5298	209	18	=	=	SYM
ejpam-5298	209	19	(	(	PUNCT
ejpam-5298	209	20	x1	x1	PROPN
ejpam-5298	209	21	·	·	PUNCT
ejpam-5298	209	22	1h	1h	NUM
ejpam-5298	209	23	,	,	PUNCT
ejpam-5298	209	24	x2	x2	PROPN
ejpam-5298	209	25	∗	∗	NOUN
ejpam-5298	209	26	y2	y2	NOUN
ejpam-5298	209	27	)	)	PUNCT
ejpam-5298	209	28	=	=	PRON
ejpam-5298	209	29	(	(	PUNCT
ejpam-5298	209	30	x1	x1	PROPN
ejpam-5298	209	31	,	,	PUNCT
ejpam-5298	209	32	x2	x2	PROPN
ejpam-5298	209	33	)	)	PUNCT
ejpam-5298	209	34	⋄	⋄	PROPN
ejpam-5298	209	35	(	(	PUNCT
ejpam-5298	209	36	1h	1h	NUM
ejpam-5298	209	37	,	,	PUNCT
ejpam-5298	209	38	y2	y2	NOUN
ejpam-5298	209	39	)	)	PUNCT
ejpam-5298	209	40	=	=	PRON
ejpam-5298	209	41	(	(	PUNCT
ejpam-5298	209	42	x1	x1	PROPN
ejpam-5298	209	43	,	,	PUNCT
ejpam-5298	209	44	x2	x2	PROPN
ejpam-5298	209	45	)	)	PUNCT
ejpam-5298	209	46	⋄m(y1	⋄m(y1	X
ejpam-5298	209	47	,	,	PUNCT
ejpam-5298	209	48	y2	y2	PROPN
ejpam-5298	209	49	)	)	PUNCT
ejpam-5298	209	50	.	.	PUNCT
ejpam-5298	210	1	hence	hence	ADV
ejpam-5298	210	2	,	,	PUNCT
ejpam-5298	210	3	m	m	VERB
ejpam-5298	210	4	is	be	AUX
ejpam-5298	210	5	a	a	DET
ejpam-5298	210	6	multiplier	multipli	ADJ
ejpam-5298	210	7	of	of	ADP
ejpam-5298	210	8	h×k	h×k	PROPN
ejpam-5298	210	9	.	.	PUNCT
ejpam-5298	211	1	definition	definition	NOUN
ejpam-5298	211	2	9	9	NUM
ejpam-5298	211	3	.	.	PUNCT
ejpam-5298	212	1	let	let	VERB
ejpam-5298	212	2	m	m	PRON
ejpam-5298	212	3	be	be	AUX
ejpam-5298	212	4	a	a	DET
ejpam-5298	212	5	self	self	NOUN
ejpam-5298	212	6	-	-	PUNCT
ejpam-5298	212	7	map	map	NOUN
ejpam-5298	212	8	of	of	ADP
ejpam-5298	212	9	h.	h.	NOUN
ejpam-5298	212	10	define	define	VERB
ejpam-5298	212	11	the	the	DET
ejpam-5298	212	12	fixed	fix	VERB
ejpam-5298	212	13	set	set	NOUN
ejpam-5298	212	14	fixm(h	fixm(h	NOUN
ejpam-5298	212	15	)	)	PUNCT
ejpam-5298	212	16	and	and	CCONJ
ejpam-5298	212	17	the	the	DET
ejpam-5298	212	18	kernel	kernel	PROPN
ejpam-5298	212	19	kerm(h	kerm(h	PROPN
ejpam-5298	212	20	)	)	PUNCT
ejpam-5298	212	21	of	of	ADP
ejpam-5298	212	22	m	m	PRON
ejpam-5298	212	23	by	by	ADP
ejpam-5298	212	24	fixm(h	fixm(h	NOUN
ejpam-5298	212	25	)	)	PUNCT
ejpam-5298	212	26	=	=	PRON
ejpam-5298	213	1	{	{	PUNCT
ejpam-5298	213	2	x	x	PUNCT
ejpam-5298	213	3	∈	∈	PROPN
ejpam-5298	213	4	h	h	NOUN
ejpam-5298	213	5	:	:	PUNCT
ejpam-5298	213	6	m(x	m(x	X
ejpam-5298	213	7	)	)	PUNCT
ejpam-5298	213	8	=	=	SYM
ejpam-5298	214	1	x	x	X
ejpam-5298	214	2	}	}	PUNCT
ejpam-5298	214	3	and	and	CCONJ
ejpam-5298	214	4	kerm(h	kerm(h	NOUN
ejpam-5298	214	5	)	)	PUNCT
ejpam-5298	214	6	=	=	PRON
ejpam-5298	214	7	{	{	PUNCT
ejpam-5298	214	8	x	x	PUNCT
ejpam-5298	214	9	∈	∈	PROPN
ejpam-5298	214	10	h	h	NOUN
ejpam-5298	214	11	:	:	PUNCT
ejpam-5298	214	12	m(x	m(x	X
ejpam-5298	214	13	)	)	PUNCT
ejpam-5298	214	14	=	=	PUNCT
ejpam-5298	215	1	1	1	NUM
ejpam-5298	215	2	}	}	PUNCT
ejpam-5298	215	3	.	.	PUNCT
ejpam-5298	216	1	remark	remark	NOUN
ejpam-5298	216	2	1	1	NUM
ejpam-5298	216	3	.	.	PUNCT
ejpam-5298	217	1	let	let	VERB
ejpam-5298	217	2	m	m	PRON
ejpam-5298	217	3	be	be	AUX
ejpam-5298	217	4	a	a	DET
ejpam-5298	217	5	multiplier	multipli	ADJ
ejpam-5298	217	6	of	of	ADP
ejpam-5298	217	7	h.	h.	NOUN
ejpam-5298	217	8	by	by	ADP
ejpam-5298	217	9	proposition	proposition	NOUN
ejpam-5298	217	10	5	5	NUM
ejpam-5298	217	11	,	,	PUNCT
ejpam-5298	217	12	we	we	PRON
ejpam-5298	217	13	have	have	VERB
ejpam-5298	217	14	m(1	m(1	NOUN
ejpam-5298	217	15	)	)	PUNCT
ejpam-5298	217	16	=	=	SYM
ejpam-5298	218	1	1	1	X
ejpam-5298	218	2	.	.	PUNCT
ejpam-5298	219	1	then	then	ADV
ejpam-5298	219	2	1	1	NUM
ejpam-5298	219	3	∈	∈	NOUN
ejpam-5298	219	4	fixm(h	fixm(h	NOUN
ejpam-5298	219	5	)	)	PUNCT
ejpam-5298	219	6	̸=	̸=	PROPN
ejpam-5298	219	7	∅	∅	NOUN
ejpam-5298	219	8	and	and	CCONJ
ejpam-5298	219	9	1	1	NUM
ejpam-5298	219	10	∈	∈	PROPN
ejpam-5298	219	11	kerm(h	kerm(h	NOUN
ejpam-5298	219	12	)	)	PUNCT
ejpam-5298	219	13	̸=	̸=	PROPN
ejpam-5298	219	14	∅.	∅.	PRON
ejpam-5298	219	15	proposition	proposition	NOUN
ejpam-5298	219	16	11	11	NUM
ejpam-5298	219	17	.	.	PUNCT
ejpam-5298	220	1	let	let	VERB
ejpam-5298	220	2	m	m	PRON
ejpam-5298	220	3	be	be	AUX
ejpam-5298	220	4	a	a	DET
ejpam-5298	220	5	multiplier	multipli	ADJ
ejpam-5298	220	6	of	of	ADP
ejpam-5298	220	7	h.	h.	NOUN
ejpam-5298	221	1	then	then	ADV
ejpam-5298	221	2	m	m	VERB
ejpam-5298	221	3	is	be	AUX
ejpam-5298	221	4	injective	injective	ADJ
ejpam-5298	221	5	if	if	SCONJ
ejpam-5298	221	6	and	and	CCONJ
ejpam-5298	221	7	only	only	ADV
ejpam-5298	221	8	if	if	SCONJ
ejpam-5298	221	9	kerm(h	kerm(h	PROPN
ejpam-5298	221	10	)	)	PUNCT
ejpam-5298	221	11	=	=	PUNCT
ejpam-5298	221	12	{	{	PUNCT
ejpam-5298	221	13	1	1	NUM
ejpam-5298	221	14	}	}	PUNCT
ejpam-5298	221	15	.	.	PUNCT
ejpam-5298	222	1	proof	proof	NOUN
ejpam-5298	222	2	.	.	PUNCT
ejpam-5298	223	1	if	if	SCONJ
ejpam-5298	223	2	m	m	NOUN
ejpam-5298	223	3	is	be	AUX
ejpam-5298	223	4	injective	injective	ADJ
ejpam-5298	223	5	,	,	PUNCT
ejpam-5298	223	6	it	it	PRON
ejpam-5298	223	7	is	be	AUX
ejpam-5298	223	8	done	do	VERB
ejpam-5298	223	9	in	in	ADP
ejpam-5298	223	10	proposition	proposition	NOUN
ejpam-5298	223	11	6	6	NUM
ejpam-5298	223	12	(	(	PUNCT
ejpam-5298	223	13	4	4	NUM
ejpam-5298	223	14	)	)	PUNCT
ejpam-5298	223	15	.	.	PUNCT
ejpam-5298	224	1	conversely	conversely	ADV
ejpam-5298	224	2	,	,	PUNCT
ejpam-5298	224	3	assume	assume	VERB
ejpam-5298	224	4	that	that	SCONJ
ejpam-5298	224	5	kerm(h	kerm(h	NOUN
ejpam-5298	224	6	)	)	PUNCT
ejpam-5298	224	7	=	=	PUNCT
ejpam-5298	224	8	{	{	PUNCT
ejpam-5298	224	9	1	1	NUM
ejpam-5298	224	10	}	}	PUNCT
ejpam-5298	224	11	.	.	PUNCT
ejpam-5298	225	1	then	then	ADV
ejpam-5298	225	2	m(1	m(1	NUM
ejpam-5298	225	3	)	)	PUNCT
ejpam-5298	225	4	=	=	SYM
ejpam-5298	226	1	1	1	X
ejpam-5298	226	2	.	.	PUNCT
ejpam-5298	226	3	let	let	VERB
ejpam-5298	226	4	a	a	PRON
ejpam-5298	226	5	,	,	PUNCT
ejpam-5298	226	6	b	b	X
ejpam-5298	226	7	∈	∈	PROPN
ejpam-5298	226	8	h	h	NOUN
ejpam-5298	226	9	be	be	AUX
ejpam-5298	226	10	such	such	ADJ
ejpam-5298	226	11	that	that	SCONJ
ejpam-5298	226	12	m(a	m(a	NOUN
ejpam-5298	226	13	)	)	PUNCT
ejpam-5298	226	14	=	=	SYM
ejpam-5298	226	15	m(b	m(b	NOUN
ejpam-5298	226	16	)	)	PUNCT
ejpam-5298	226	17	.	.	PUNCT
ejpam-5298	227	1	then	then	ADV
ejpam-5298	227	2	m(a	m(a	PROPN
ejpam-5298	227	3	·	·	PUNCT
ejpam-5298	227	4	b	b	X
ejpam-5298	227	5	)	)	PUNCT
ejpam-5298	227	6	=	=	SYM
ejpam-5298	227	7	a	a	DET
ejpam-5298	227	8	·	·	SYM
ejpam-5298	227	9	m(b	m(b	NOUN
ejpam-5298	227	10	)	)	PUNCT
ejpam-5298	227	11	(	(	PUNCT
ejpam-5298	227	12	multiplier	multiplier	X
ejpam-5298	227	13	)	)	PUNCT
ejpam-5298	227	14	a.	a.	NOUN
ejpam-5298	227	15	iampan	iampan	PROPN
ejpam-5298	227	16	,	,	PUNCT
ejpam-5298	227	17	n.	n.	PROPN
ejpam-5298	227	18	rajesh	rajesh	PROPN
ejpam-5298	227	19	/	/	SYM
ejpam-5298	227	20	eur	eur	PROPN
ejpam-5298	227	21	.	.	PUNCT
ejpam-5298	228	1	j.	j.	PROPN
ejpam-5298	228	2	pure	pure	PROPN
ejpam-5298	228	3	appl	appl	PROPN
ejpam-5298	228	4	.	.	PROPN
ejpam-5298	228	5	math	math	PROPN
ejpam-5298	228	6	,	,	PUNCT
ejpam-5298	228	7	17	17	NUM
ejpam-5298	228	8	(	(	PUNCT
ejpam-5298	228	9	4	4	NUM
ejpam-5298	228	10	)	)	PUNCT
ejpam-5298	228	11	(	(	PUNCT
ejpam-5298	228	12	2024	2024	NUM
ejpam-5298	228	13	)	)	PUNCT
ejpam-5298	228	14	,	,	PUNCT
ejpam-5298	228	15	2726	2726	NUM
ejpam-5298	228	16	-	-	SYM
ejpam-5298	228	17	2737	2737	NUM
ejpam-5298	228	18	2734	2734	NUM
ejpam-5298	228	19	=	=	SYM
ejpam-5298	228	20	a	a	DET
ejpam-5298	228	21	·	·	SYM
ejpam-5298	228	22	m(a	m(a	NOUN
ejpam-5298	228	23	)	)	PUNCT
ejpam-5298	228	24	(	(	PUNCT
ejpam-5298	228	25	m(a	m(a	PROPN
ejpam-5298	228	26	)	)	PUNCT
ejpam-5298	228	27	=	=	SYM
ejpam-5298	228	28	m(b	m(b	NOUN
ejpam-5298	228	29	)	)	PUNCT
ejpam-5298	228	30	)	)	PUNCT
ejpam-5298	229	1	=	=	PUNCT
ejpam-5298	230	1	m(a	m(a	NOUN
ejpam-5298	230	2	·	·	PUNCT
ejpam-5298	230	3	a	a	X
ejpam-5298	230	4	)	)	PUNCT
ejpam-5298	230	5	(	(	PUNCT
ejpam-5298	230	6	multiplier	multiplier	X
ejpam-5298	230	7	)	)	PUNCT
ejpam-5298	230	8	=	=	SYM
ejpam-5298	230	9	m(1	m(1	NOUN
ejpam-5298	230	10	)	)	PUNCT
ejpam-5298	230	11	(	(	PUNCT
ejpam-5298	230	12	lemma	lemma	PROPN
ejpam-5298	230	13	1	1	NUM
ejpam-5298	230	14	(	(	PUNCT
ejpam-5298	230	15	1	1	NUM
ejpam-5298	230	16	)	)	PUNCT
ejpam-5298	230	17	)	)	PUNCT
ejpam-5298	230	18	=	=	SYM
ejpam-5298	230	19	1	1	NUM
ejpam-5298	230	20	,	,	PUNCT
ejpam-5298	230	21	(	(	PUNCT
ejpam-5298	230	22	m(1	m(1	NOUN
ejpam-5298	230	23	)	)	PUNCT
ejpam-5298	230	24	=	=	SYM
ejpam-5298	230	25	1	1	X
ejpam-5298	230	26	)	)	PUNCT
ejpam-5298	230	27	m(b	m(b	X
ejpam-5298	230	28	·	·	PUNCT
ejpam-5298	230	29	a	a	X
ejpam-5298	230	30	)	)	PUNCT
ejpam-5298	230	31	=	=	SYM
ejpam-5298	230	32	b	b	X
ejpam-5298	230	33	·	·	PUNCT
ejpam-5298	230	34	m(a	m(a	NOUN
ejpam-5298	230	35	)	)	PUNCT
ejpam-5298	230	36	(	(	PUNCT
ejpam-5298	230	37	multiplier	multiplier	X
ejpam-5298	230	38	)	)	PUNCT
ejpam-5298	230	39	=	=	SYM
ejpam-5298	230	40	b	b	X
ejpam-5298	230	41	·	·	SYM
ejpam-5298	230	42	m(b	m(b	NOUN
ejpam-5298	230	43	)	)	PUNCT
ejpam-5298	230	44	(	(	PUNCT
ejpam-5298	230	45	m(a	m(a	PROPN
ejpam-5298	230	46	)	)	PUNCT
ejpam-5298	230	47	=	=	SYM
ejpam-5298	230	48	m(b	m(b	NOUN
ejpam-5298	230	49	)	)	PUNCT
ejpam-5298	230	50	)	)	PUNCT
ejpam-5298	231	1	=	=	SYM
ejpam-5298	231	2	m(b	m(b	X
ejpam-5298	231	3	·	·	PUNCT
ejpam-5298	231	4	b	b	X
ejpam-5298	231	5	)	)	PUNCT
ejpam-5298	231	6	(	(	PUNCT
ejpam-5298	231	7	multiplier	multiplier	X
ejpam-5298	231	8	)	)	PUNCT
ejpam-5298	231	9	=	=	SYM
ejpam-5298	231	10	m(1	m(1	NOUN
ejpam-5298	231	11	)	)	PUNCT
ejpam-5298	231	12	(	(	PUNCT
ejpam-5298	231	13	lemma	lemma	PROPN
ejpam-5298	231	14	1	1	NUM
ejpam-5298	231	15	(	(	PUNCT
ejpam-5298	231	16	1	1	NUM
ejpam-5298	231	17	)	)	PUNCT
ejpam-5298	231	18	)	)	PUNCT
ejpam-5298	231	19	=	=	SYM
ejpam-5298	232	1	1	1	X
ejpam-5298	232	2	.	.	PUNCT
ejpam-5298	232	3	(	(	PUNCT
ejpam-5298	232	4	m(1	m(1	NOUN
ejpam-5298	232	5	)	)	PUNCT
ejpam-5298	232	6	=	=	SYM
ejpam-5298	232	7	1	1	X
ejpam-5298	232	8	)	)	PUNCT
ejpam-5298	232	9	thus	thus	ADV
ejpam-5298	232	10	,	,	PUNCT
ejpam-5298	232	11	a	a	DET
ejpam-5298	232	12	·	·	SYM
ejpam-5298	232	13	b	b	NOUN
ejpam-5298	232	14	,	,	PUNCT
ejpam-5298	232	15	b	b	PROPN
ejpam-5298	232	16	·	·	PUNCT
ejpam-5298	232	17	a	a	DET
ejpam-5298	232	18	∈	∈	PROPN
ejpam-5298	232	19	kerm(h	kerm(h	NOUN
ejpam-5298	232	20	)	)	PUNCT
ejpam-5298	232	21	=	=	PUNCT
ejpam-5298	232	22	{	{	PUNCT
ejpam-5298	232	23	1	1	NUM
ejpam-5298	232	24	}	}	PUNCT
ejpam-5298	232	25	,	,	PUNCT
ejpam-5298	232	26	so	so	SCONJ
ejpam-5298	232	27	a	a	DET
ejpam-5298	232	28	·	·	PUNCT
ejpam-5298	232	29	b	b	X
ejpam-5298	232	30	=	=	SYM
ejpam-5298	232	31	1	1	NUM
ejpam-5298	232	32	and	and	CCONJ
ejpam-5298	232	33	b	b	PROPN
ejpam-5298	232	34	·	·	PUNCT
ejpam-5298	232	35	a	a	DET
ejpam-5298	232	36	=	=	ADJ
ejpam-5298	232	37	1	1	X
ejpam-5298	232	38	.	.	PUNCT
ejpam-5298	233	1	by	by	ADP
ejpam-5298	233	2	definition	definition	NOUN
ejpam-5298	233	3	1	1	NUM
ejpam-5298	233	4	(	(	PUNCT
ejpam-5298	233	5	3	3	NUM
ejpam-5298	233	6	)	)	PUNCT
ejpam-5298	233	7	,	,	PUNCT
ejpam-5298	233	8	we	we	PRON
ejpam-5298	233	9	have	have	VERB
ejpam-5298	233	10	a	a	DET
ejpam-5298	233	11	=	=	X
ejpam-5298	233	12	b.	b.	PROPN
ejpam-5298	233	13	hence	hence	ADV
ejpam-5298	233	14	,	,	PUNCT
ejpam-5298	233	15	m	m	VERB
ejpam-5298	233	16	is	be	AUX
ejpam-5298	233	17	injective	injective	ADJ
ejpam-5298	233	18	.	.	PUNCT
ejpam-5298	234	1	proposition	proposition	NOUN
ejpam-5298	234	2	12	12	NUM
ejpam-5298	234	3	.	.	PUNCT
ejpam-5298	235	1	let	let	VERB
ejpam-5298	235	2	m	m	PRON
ejpam-5298	235	3	be	be	AUX
ejpam-5298	235	4	a	a	DET
ejpam-5298	235	5	multiplier	multipli	ADJ
ejpam-5298	235	6	of	of	ADP
ejpam-5298	235	7	h.	h.	NOUN
ejpam-5298	236	1	if	if	SCONJ
ejpam-5298	236	2	x	x	PROPN
ejpam-5298	236	3	∈	∈	PROPN
ejpam-5298	236	4	fixm(h	fixm(h	NOUN
ejpam-5298	236	5	)	)	PUNCT
ejpam-5298	236	6	,	,	PUNCT
ejpam-5298	236	7	then	then	ADV
ejpam-5298	236	8	x	x	PART
ejpam-5298	236	9	∈	∈	PROPN
ejpam-5298	236	10	fixmn(h	fixmn(h	PROPN
ejpam-5298	236	11	)	)	PUNCT
ejpam-5298	236	12	for	for	ADP
ejpam-5298	236	13	all	all	DET
ejpam-5298	236	14	positive	positive	ADJ
ejpam-5298	236	15	integer	integer	NOUN
ejpam-5298	236	16	n.	n.	NOUN
ejpam-5298	236	17	proof	proof	NOUN
ejpam-5298	236	18	.	.	PUNCT
ejpam-5298	237	1	let	let	VERB
ejpam-5298	237	2	n	n	PRON
ejpam-5298	237	3	be	be	AUX
ejpam-5298	237	4	a	a	DET
ejpam-5298	237	5	positive	positive	ADJ
ejpam-5298	237	6	integer	integer	NOUN
ejpam-5298	237	7	such	such	ADJ
ejpam-5298	237	8	that	that	SCONJ
ejpam-5298	237	9	x	x	SYM
ejpam-5298	237	10	∈	∈	PROPN
ejpam-5298	237	11	fixmn(h	fixmn(h	PROPN
ejpam-5298	237	12	)	)	PUNCT
ejpam-5298	237	13	.	.	PUNCT
ejpam-5298	238	1	then	then	ADV
ejpam-5298	238	2	mn(x	mn(x	PUNCT
ejpam-5298	238	3	)	)	PUNCT
ejpam-5298	238	4	=	=	PUNCT
ejpam-5298	238	5	x.	x.	NOUN
ejpam-5298	238	6	thus	thus	ADV
ejpam-5298	238	7	,	,	PUNCT
ejpam-5298	238	8	mn+1(x	mn+1(x	PROPN
ejpam-5298	238	9	)	)	PUNCT
ejpam-5298	238	10	=	=	PUNCT
ejpam-5298	239	1	(	(	PUNCT
ejpam-5298	239	2	mn	mn	PROPN
ejpam-5298	239	3	◦	◦	PROPN
ejpam-5298	239	4	m)(x	m)(x	NOUN
ejpam-5298	239	5	)	)	PUNCT
ejpam-5298	240	1	=	=	SYM
ejpam-5298	240	2	mn(m(x	mn(m(x	NOUN
ejpam-5298	240	3	)	)	PUNCT
ejpam-5298	240	4	)	)	PUNCT
ejpam-5298	241	1	=	=	SYM
ejpam-5298	241	2	mn(x	mn(x	X
ejpam-5298	241	3	)	)	PUNCT
ejpam-5298	241	4	=	=	SYM
ejpam-5298	242	1	x.	x.	NOUN
ejpam-5298	243	1	so	so	ADV
ejpam-5298	243	2	,	,	PUNCT
ejpam-5298	243	3	x	x	PROPN
ejpam-5298	243	4	∈	∈	PROPN
ejpam-5298	243	5	fixmn+1(h	fixmn+1(h	PROPN
ejpam-5298	243	6	)	)	PUNCT
ejpam-5298	243	7	.	.	PUNCT
ejpam-5298	244	1	hence	hence	ADV
ejpam-5298	244	2	,	,	PUNCT
ejpam-5298	244	3	x	x	X
ejpam-5298	244	4	∈	∈	PROPN
ejpam-5298	244	5	fixmn(h	fixmn(h	PROPN
ejpam-5298	244	6	)	)	PUNCT
ejpam-5298	244	7	for	for	ADP
ejpam-5298	244	8	all	all	DET
ejpam-5298	244	9	positive	positive	ADJ
ejpam-5298	244	10	integer	integer	NOUN
ejpam-5298	244	11	n.	n.	NOUN
ejpam-5298	244	12	proposition	proposition	NOUN
ejpam-5298	244	13	13	13	NUM
ejpam-5298	244	14	.	.	PUNCT
ejpam-5298	245	1	if	if	SCONJ
ejpam-5298	245	2	m	m	NOUN
ejpam-5298	245	3	is	be	AUX
ejpam-5298	245	4	a	a	DET
ejpam-5298	245	5	multiplier	multipli	ADJ
ejpam-5298	245	6	of	of	ADP
ejpam-5298	245	7	h	h	NOUN
ejpam-5298	245	8	,	,	PUNCT
ejpam-5298	245	9	then	then	ADV
ejpam-5298	245	10	fixm(h	fixm(h	NOUN
ejpam-5298	245	11	)	)	PUNCT
ejpam-5298	245	12	is	be	AUX
ejpam-5298	245	13	a	a	DET
ejpam-5298	245	14	subalgebra	subalgebra	NOUN
ejpam-5298	245	15	of	of	ADP
ejpam-5298	245	16	h.	h.	NOUN
ejpam-5298	245	17	proof	proof	NOUN
ejpam-5298	245	18	.	.	PUNCT
ejpam-5298	246	1	by	by	ADP
ejpam-5298	246	2	proposition	proposition	NOUN
ejpam-5298	246	3	6	6	NUM
ejpam-5298	246	4	(	(	PUNCT
ejpam-5298	246	5	1	1	NUM
ejpam-5298	246	6	)	)	PUNCT
ejpam-5298	246	7	,	,	PUNCT
ejpam-5298	246	8	we	we	PRON
ejpam-5298	246	9	have	have	VERB
ejpam-5298	246	10	m(1	m(1	NOUN
ejpam-5298	246	11	)	)	PUNCT
ejpam-5298	246	12	=	=	SYM
ejpam-5298	246	13	1	1	NUM
ejpam-5298	246	14	and	and	CCONJ
ejpam-5298	246	15	so	so	ADV
ejpam-5298	246	16	1	1	NUM
ejpam-5298	246	17	∈	∈	NOUN
ejpam-5298	246	18	fixm(h	fixm(h	NOUN
ejpam-5298	246	19	)	)	PUNCT
ejpam-5298	246	20	̸=	̸=	PROPN
ejpam-5298	246	21	∅.	∅.	ADV
ejpam-5298	246	22	let	let	VERB
ejpam-5298	246	23	x	x	PRON
ejpam-5298	246	24	,	,	PUNCT
ejpam-5298	246	25	y	y	PROPN
ejpam-5298	246	26	∈	∈	PROPN
ejpam-5298	246	27	fixm(h	fixm(h	PROPN
ejpam-5298	246	28	)	)	PUNCT
ejpam-5298	246	29	̸=	̸=	PROPN
ejpam-5298	246	30	∅.	∅.	ADP
ejpam-5298	246	31	then	then	ADV
ejpam-5298	246	32	m(x	m(x	X
ejpam-5298	246	33	)	)	PUNCT
ejpam-5298	246	34	=	=	SYM
ejpam-5298	247	1	x	x	PROPN
ejpam-5298	247	2	and	and	CCONJ
ejpam-5298	247	3	m(y	m(y	NOUN
ejpam-5298	247	4	)	)	PUNCT
ejpam-5298	248	1	=	=	SYM
ejpam-5298	248	2	y	y	PROPN
ejpam-5298	248	3	,	,	PUNCT
ejpam-5298	248	4	so	so	ADV
ejpam-5298	248	5	m(x	m(x	PROPN
ejpam-5298	248	6	·	·	PUNCT
ejpam-5298	248	7	y	y	X
ejpam-5298	248	8	)	)	PUNCT
ejpam-5298	248	9	=	=	PUNCT
ejpam-5298	248	10	x	x	SYM
ejpam-5298	248	11	·	·	PUNCT
ejpam-5298	248	12	m(y	m(y	NUM
ejpam-5298	248	13	)	)	PUNCT
ejpam-5298	248	14	=	=	SYM
ejpam-5298	249	1	x	x	PUNCT
ejpam-5298	249	2	·	·	PUNCT
ejpam-5298	249	3	y.	y.	PROPN
ejpam-5298	249	4	thus	thus	ADV
ejpam-5298	249	5	,	,	PUNCT
ejpam-5298	249	6	x	x	X
ejpam-5298	249	7	·	·	PUNCT
ejpam-5298	249	8	y	y	PROPN
ejpam-5298	249	9	∈	∈	PROPN
ejpam-5298	249	10	fixm(h	fixm(h	PROPN
ejpam-5298	249	11	)	)	PUNCT
ejpam-5298	249	12	̸=	̸=	PROPN
ejpam-5298	249	13	∅.	∅.	PRON
ejpam-5298	249	14	hence	hence	ADV
ejpam-5298	249	15	,	,	PUNCT
ejpam-5298	249	16	fixm(h	fixm(h	PROPN
ejpam-5298	249	17	)	)	PUNCT
ejpam-5298	249	18	is	be	AUX
ejpam-5298	249	19	a	a	DET
ejpam-5298	249	20	subalgebra	subalgebra	NOUN
ejpam-5298	249	21	of	of	ADP
ejpam-5298	249	22	h.	h.	NOUN
ejpam-5298	249	23	proposition	proposition	PROPN
ejpam-5298	249	24	14	14	NUM
ejpam-5298	249	25	.	.	PUNCT
ejpam-5298	250	1	if	if	SCONJ
ejpam-5298	250	2	m	m	NOUN
ejpam-5298	250	3	is	be	AUX
ejpam-5298	250	4	a	a	DET
ejpam-5298	250	5	multiplier	multipli	ADJ
ejpam-5298	250	6	of	of	ADP
ejpam-5298	250	7	h	h	NOUN
ejpam-5298	250	8	,	,	PUNCT
ejpam-5298	250	9	then	then	ADV
ejpam-5298	250	10	kerm(h	kerm(h	PROPN
ejpam-5298	250	11	)	)	PUNCT
ejpam-5298	250	12	is	be	AUX
ejpam-5298	250	13	a	a	DET
ejpam-5298	250	14	subalgebra	subalgebra	NOUN
ejpam-5298	250	15	of	of	ADP
ejpam-5298	250	16	h.	h.	NOUN
ejpam-5298	250	17	proof	proof	NOUN
ejpam-5298	250	18	.	.	PUNCT
ejpam-5298	251	1	by	by	ADP
ejpam-5298	251	2	proposition	proposition	NOUN
ejpam-5298	251	3	6	6	NUM
ejpam-5298	251	4	(	(	PUNCT
ejpam-5298	251	5	1	1	NUM
ejpam-5298	251	6	)	)	PUNCT
ejpam-5298	251	7	,	,	PUNCT
ejpam-5298	251	8	we	we	PRON
ejpam-5298	251	9	have	have	VERB
ejpam-5298	251	10	m(1	m(1	NOUN
ejpam-5298	251	11	)	)	PUNCT
ejpam-5298	251	12	=	=	SYM
ejpam-5298	251	13	1	1	NUM
ejpam-5298	251	14	and	and	CCONJ
ejpam-5298	251	15	so	so	ADV
ejpam-5298	251	16	1	1	NUM
ejpam-5298	251	17	∈	∈	PROPN
ejpam-5298	251	18	kerm(h	kerm(h	NOUN
ejpam-5298	251	19	)	)	PUNCT
ejpam-5298	251	20	̸=	̸=	PROPN
ejpam-5298	251	21	∅.	∅.	ADV
ejpam-5298	251	22	let	let	VERB
ejpam-5298	251	23	x	x	PRON
ejpam-5298	251	24	,	,	PUNCT
ejpam-5298	251	25	y	y	PROPN
ejpam-5298	251	26	∈	∈	PROPN
ejpam-5298	251	27	kerm(h	kerm(h	PROPN
ejpam-5298	251	28	)	)	PUNCT
ejpam-5298	251	29	.	.	PUNCT
ejpam-5298	252	1	then	then	ADV
ejpam-5298	252	2	m(x	m(x	X
ejpam-5298	252	3	)	)	PUNCT
ejpam-5298	252	4	=	=	SYM
ejpam-5298	252	5	1	1	NUM
ejpam-5298	252	6	and	and	CCONJ
ejpam-5298	252	7	m(y	m(y	NOUN
ejpam-5298	252	8	)	)	PUNCT
ejpam-5298	252	9	=	=	SYM
ejpam-5298	252	10	1	1	NUM
ejpam-5298	252	11	,	,	PUNCT
ejpam-5298	252	12	it	it	PRON
ejpam-5298	252	13	follows	follow	VERB
ejpam-5298	252	14	from	from	ADP
ejpam-5298	252	15	lemma	lemma	PROPN
ejpam-5298	252	16	1	1	NUM
ejpam-5298	252	17	(	(	PUNCT
ejpam-5298	252	18	3	3	NUM
ejpam-5298	252	19	)	)	PUNCT
ejpam-5298	252	20	that	that	SCONJ
ejpam-5298	252	21	m(x	m(x	PROPN
ejpam-5298	252	22	·	·	PUNCT
ejpam-5298	252	23	y	y	X
ejpam-5298	252	24	)	)	PUNCT
ejpam-5298	252	25	=	=	SYM
ejpam-5298	252	26	x	x	SYM
ejpam-5298	252	27	·	·	PUNCT
ejpam-5298	252	28	m(y	m(y	X
ejpam-5298	252	29	)	)	PUNCT
ejpam-5298	252	30	=	=	SYM
ejpam-5298	253	1	x	x	PUNCT
ejpam-5298	253	2	·	·	PUNCT
ejpam-5298	253	3	1	1	NUM
ejpam-5298	253	4	=	=	SYM
ejpam-5298	253	5	1	1	NUM
ejpam-5298	253	6	.	.	PUNCT
ejpam-5298	254	1	thus	thus	ADV
ejpam-5298	254	2	,	,	PUNCT
ejpam-5298	254	3	x	x	X
ejpam-5298	254	4	·	·	PUNCT
ejpam-5298	254	5	y	y	X
ejpam-5298	254	6	∈	∈	PROPN
ejpam-5298	254	7	kerm(h	kerm(h	PROPN
ejpam-5298	254	8	)	)	PUNCT
ejpam-5298	254	9	.	.	PUNCT
ejpam-5298	255	1	hence	hence	ADV
ejpam-5298	255	2	,	,	PUNCT
ejpam-5298	255	3	kerm(h	kerm(h	PROPN
ejpam-5298	255	4	)	)	PUNCT
ejpam-5298	255	5	is	be	AUX
ejpam-5298	255	6	a	a	DET
ejpam-5298	255	7	subalgebra	subalgebra	NOUN
ejpam-5298	255	8	of	of	ADP
ejpam-5298	255	9	h.	h.	PROPN
ejpam-5298	255	10	definition	definition	NOUN
ejpam-5298	255	11	10	10	NUM
ejpam-5298	255	12	.	.	PUNCT
ejpam-5298	256	1	a	a	DET
ejpam-5298	256	2	nonempty	nonempty	ADJ
ejpam-5298	256	3	subset	subset	VERB
ejpam-5298	256	4	s	s	NOUN
ejpam-5298	256	5	of	of	ADP
ejpam-5298	256	6	a	a	DET
ejpam-5298	256	7	hilbert	hilbert	NOUN
ejpam-5298	256	8	algebra	algebra	NOUN
ejpam-5298	256	9	h	h	NOUN
ejpam-5298	256	10	=	=	PUNCT
ejpam-5298	256	11	(	(	PUNCT
ejpam-5298	256	12	h	h	NOUN
ejpam-5298	256	13	,	,	PUNCT
ejpam-5298	256	14	·	·	PUNCT
ejpam-5298	256	15	,	,	PUNCT
ejpam-5298	256	16	1	1	NUM
ejpam-5298	256	17	)	)	PUNCT
ejpam-5298	256	18	is	be	AUX
ejpam-5298	256	19	called	call	VERB
ejpam-5298	256	20	a	a	DET
ejpam-5298	256	21	near	near	ADJ
ejpam-5298	256	22	filter	filter	NOUN
ejpam-5298	256	23	of	of	ADP
ejpam-5298	256	24	h	h	NOUN
ejpam-5298	256	25	if	if	SCONJ
ejpam-5298	256	26	it	it	PRON
ejpam-5298	256	27	satisfies	satisfy	VERB
ejpam-5298	256	28	the	the	DET
ejpam-5298	256	29	following	follow	VERB
ejpam-5298	256	30	properties	property	NOUN
ejpam-5298	256	31	:	:	PUNCT
ejpam-5298	256	32	(	(	PUNCT
ejpam-5298	256	33	1	1	X
ejpam-5298	256	34	)	)	SYM
ejpam-5298	256	35	1	1	NUM
ejpam-5298	256	36	∈	∈	PROPN
ejpam-5298	256	37	s	s	NOUN
ejpam-5298	256	38	,	,	PUNCT
ejpam-5298	256	39	(	(	PUNCT
ejpam-5298	256	40	2	2	NUM
ejpam-5298	256	41	)	)	PUNCT
ejpam-5298	256	42	(	(	PUNCT
ejpam-5298	256	43	∀x	∀x	X
ejpam-5298	256	44	,	,	PUNCT
ejpam-5298	256	45	y	y	PROPN
ejpam-5298	256	46	∈	∈	PROPN
ejpam-5298	256	47	h)(y	h)(y	PUNCT
ejpam-5298	256	48	∈	∈	PROPN
ejpam-5298	256	49	s	s	PART
ejpam-5298	256	50	⇒	⇒	NOUN
ejpam-5298	256	51	x	x	X
ejpam-5298	256	52	·	·	PUNCT
ejpam-5298	256	53	y	y	X
ejpam-5298	256	54	∈	∈	PROPN
ejpam-5298	256	55	s	s	PART
ejpam-5298	256	56	)	)	PUNCT
ejpam-5298	256	57	.	.	PUNCT
ejpam-5298	257	1	a.	a.	PROPN
ejpam-5298	257	2	iampan	iampan	PROPN
ejpam-5298	257	3	,	,	PUNCT
ejpam-5298	257	4	n.	n.	PROPN
ejpam-5298	257	5	rajesh	rajesh	PROPN
ejpam-5298	257	6	/	/	SYM
ejpam-5298	257	7	eur	eur	PROPN
ejpam-5298	257	8	.	.	PUNCT
ejpam-5298	258	1	j.	j.	PROPN
ejpam-5298	258	2	pure	pure	PROPN
ejpam-5298	258	3	appl	appl	PROPN
ejpam-5298	258	4	.	.	PROPN
ejpam-5298	258	5	math	math	PROPN
ejpam-5298	258	6	,	,	PUNCT
ejpam-5298	258	7	17	17	NUM
ejpam-5298	258	8	(	(	PUNCT
ejpam-5298	258	9	4	4	NUM
ejpam-5298	258	10	)	)	PUNCT
ejpam-5298	258	11	(	(	PUNCT
ejpam-5298	258	12	2024	2024	NUM
ejpam-5298	258	13	)	)	PUNCT
ejpam-5298	258	14	,	,	PUNCT
ejpam-5298	258	15	2726	2726	NUM
ejpam-5298	258	16	-	-	SYM
ejpam-5298	258	17	2737	2737	NUM
ejpam-5298	258	18	2735	2735	NUM
ejpam-5298	258	19	we	we	PRON
ejpam-5298	258	20	see	see	VERB
ejpam-5298	258	21	that	that	SCONJ
ejpam-5298	258	22	near	near	ADJ
ejpam-5298	258	23	filters	filter	NOUN
ejpam-5298	258	24	are	be	AUX
ejpam-5298	258	25	generalizations	generalization	NOUN
ejpam-5298	258	26	of	of	ADP
ejpam-5298	258	27	ideals	ideal	NOUN
ejpam-5298	258	28	,	,	PUNCT
ejpam-5298	258	29	and	and	CCONJ
ejpam-5298	258	30	subalgebras	subalgebras	PROPN
ejpam-5298	258	31	are	be	AUX
ejpam-5298	258	32	generalizations	generalization	NOUN
ejpam-5298	258	33	of	of	ADP
ejpam-5298	258	34	near	near	ADJ
ejpam-5298	258	35	filters	filter	NOUN
ejpam-5298	258	36	.	.	PUNCT
ejpam-5298	259	1	theorem	theorem	NOUN
ejpam-5298	259	2	1	1	NUM
ejpam-5298	259	3	.	.	PUNCT
ejpam-5298	260	1	let	let	VERB
ejpam-5298	260	2	m	m	PRON
ejpam-5298	260	3	be	be	AUX
ejpam-5298	260	4	a	a	DET
ejpam-5298	260	5	multiplier	multipli	ADJ
ejpam-5298	260	6	of	of	ADP
ejpam-5298	260	7	h.	h.	NOUN
ejpam-5298	260	8	then	then	ADV
ejpam-5298	260	9	the	the	DET
ejpam-5298	260	10	following	following	ADJ
ejpam-5298	260	11	statements	statement	NOUN
ejpam-5298	260	12	hold	hold	VERB
ejpam-5298	260	13	:	:	PUNCT
ejpam-5298	260	14	(	(	PUNCT
ejpam-5298	260	15	1	1	X
ejpam-5298	260	16	)	)	PUNCT
ejpam-5298	260	17	if	if	SCONJ
ejpam-5298	260	18	s	s	NOUN
ejpam-5298	260	19	is	be	AUX
ejpam-5298	260	20	a	a	DET
ejpam-5298	260	21	near	near	ADJ
ejpam-5298	260	22	filter	filter	NOUN
ejpam-5298	260	23	of	of	ADP
ejpam-5298	260	24	h	h	NOUN
ejpam-5298	260	25	,	,	PUNCT
ejpam-5298	260	26	then	then	ADV
ejpam-5298	260	27	m(s	m(s	PROPN
ejpam-5298	260	28	)	)	PUNCT
ejpam-5298	260	29	is	be	AUX
ejpam-5298	260	30	a	a	DET
ejpam-5298	260	31	near	near	ADJ
ejpam-5298	260	32	filter	filter	NOUN
ejpam-5298	260	33	of	of	ADP
ejpam-5298	260	34	h	h	NOUN
ejpam-5298	260	35	,	,	PUNCT
ejpam-5298	260	36	(	(	PUNCT
ejpam-5298	260	37	2	2	X
ejpam-5298	260	38	)	)	PUNCT
ejpam-5298	260	39	if	if	SCONJ
ejpam-5298	260	40	s	s	NOUN
ejpam-5298	260	41	is	be	AUX
ejpam-5298	260	42	a	a	DET
ejpam-5298	260	43	near	near	ADJ
ejpam-5298	260	44	filter	filter	NOUN
ejpam-5298	260	45	of	of	ADP
ejpam-5298	260	46	h	h	NOUN
ejpam-5298	260	47	,	,	PUNCT
ejpam-5298	260	48	then	then	ADV
ejpam-5298	260	49	m−1(s	m−1(s	PROPN
ejpam-5298	260	50	)	)	PUNCT
ejpam-5298	260	51	is	be	AUX
ejpam-5298	260	52	a	a	DET
ejpam-5298	260	53	near	near	ADJ
ejpam-5298	260	54	filter	filter	NOUN
ejpam-5298	260	55	of	of	ADP
ejpam-5298	260	56	h.	h.	NOUN
ejpam-5298	260	57	proof	proof	NOUN
ejpam-5298	260	58	.	.	PUNCT
ejpam-5298	261	1	(	(	PUNCT
ejpam-5298	261	2	1	1	X
ejpam-5298	261	3	)	)	PUNCT
ejpam-5298	261	4	assume	assume	VERB
ejpam-5298	261	5	that	that	SCONJ
ejpam-5298	261	6	s	s	VERB
ejpam-5298	261	7	is	be	AUX
ejpam-5298	261	8	a	a	DET
ejpam-5298	261	9	near	near	ADJ
ejpam-5298	261	10	filter	filter	NOUN
ejpam-5298	261	11	of	of	ADP
ejpam-5298	261	12	h.	h.	NOUN
ejpam-5298	261	13	since	since	SCONJ
ejpam-5298	261	14	1	1	NUM
ejpam-5298	261	15	∈	∈	PROPN
ejpam-5298	261	16	s	s	NOUN
ejpam-5298	261	17	and	and	CCONJ
ejpam-5298	261	18	m(1	m(1	NOUN
ejpam-5298	261	19	)	)	PUNCT
ejpam-5298	261	20	=	=	SYM
ejpam-5298	261	21	1	1	NUM
ejpam-5298	261	22	,	,	PUNCT
ejpam-5298	261	23	we	we	PRON
ejpam-5298	261	24	have	have	VERB
ejpam-5298	261	25	1	1	NUM
ejpam-5298	261	26	=	=	SYM
ejpam-5298	261	27	m(1	m(1	PROPN
ejpam-5298	261	28	)	)	PUNCT
ejpam-5298	261	29	∈	∈	PROPN
ejpam-5298	261	30	m(s	m(s	PROPN
ejpam-5298	261	31	)	)	PUNCT
ejpam-5298	261	32	.	.	PUNCT
ejpam-5298	262	1	let	let	VERB
ejpam-5298	262	2	x	x	SYM
ejpam-5298	262	3	∈	∈	PROPN
ejpam-5298	262	4	h	h	NOUN
ejpam-5298	262	5	and	and	CCONJ
ejpam-5298	262	6	y	y	PROPN
ejpam-5298	262	7	∈	∈	PROPN
ejpam-5298	262	8	m(s	m(s	PROPN
ejpam-5298	262	9	)	)	PUNCT
ejpam-5298	262	10	.	.	PUNCT
ejpam-5298	263	1	then	then	ADV
ejpam-5298	263	2	y	y	PROPN
ejpam-5298	263	3	=	=	SYM
ejpam-5298	263	4	m(s	m(s	PROPN
ejpam-5298	263	5	)	)	PUNCT
ejpam-5298	263	6	for	for	ADP
ejpam-5298	263	7	some	some	DET
ejpam-5298	263	8	s	s	PART
ejpam-5298	263	9	∈	∈	PROPN
ejpam-5298	263	10	s	s	NOUN
ejpam-5298	263	11	,	,	PUNCT
ejpam-5298	263	12	so	so	ADV
ejpam-5298	263	13	x	x	SYM
ejpam-5298	263	14	·	·	PUNCT
ejpam-5298	264	1	y	y	X
ejpam-5298	264	2	=	=	PUNCT
ejpam-5298	264	3	x	x	SYM
ejpam-5298	264	4	·	·	PUNCT
ejpam-5298	264	5	m(s	m(s	PROPN
ejpam-5298	264	6	)	)	PUNCT
ejpam-5298	264	7	=	=	SYM
ejpam-5298	264	8	m(x	m(x	PROPN
ejpam-5298	264	9	·	·	PUNCT
ejpam-5298	264	10	s	s	X
ejpam-5298	264	11	)	)	PUNCT
ejpam-5298	264	12	∈	∈	PROPN
ejpam-5298	264	13	m(s	m(s	PROPN
ejpam-5298	264	14	)	)	PUNCT
ejpam-5298	264	15	since	since	SCONJ
ejpam-5298	264	16	x	x	X
ejpam-5298	264	17	·	·	PUNCT
ejpam-5298	264	18	s	s	PART
ejpam-5298	264	19	∈	∈	PROPN
ejpam-5298	264	20	s.	s.	PROPN
ejpam-5298	264	21	hence	hence	ADV
ejpam-5298	264	22	,	,	PUNCT
ejpam-5298	264	23	m(s	m(s	PROPN
ejpam-5298	264	24	)	)	PUNCT
ejpam-5298	264	25	is	be	AUX
ejpam-5298	264	26	a	a	DET
ejpam-5298	264	27	near	near	ADJ
ejpam-5298	264	28	filter	filter	NOUN
ejpam-5298	264	29	of	of	ADP
ejpam-5298	264	30	h.	h.	PROPN
ejpam-5298	264	31	(	(	PUNCT
ejpam-5298	264	32	2	2	X
ejpam-5298	264	33	)	)	PUNCT
ejpam-5298	264	34	assume	assume	VERB
ejpam-5298	264	35	that	that	SCONJ
ejpam-5298	264	36	s	s	VERB
ejpam-5298	264	37	is	be	AUX
ejpam-5298	264	38	a	a	DET
ejpam-5298	264	39	near	near	ADJ
ejpam-5298	264	40	filter	filter	NOUN
ejpam-5298	264	41	of	of	ADP
ejpam-5298	264	42	h.	h.	NOUN
ejpam-5298	264	43	since	since	SCONJ
ejpam-5298	264	44	m(1	m(1	PROPN
ejpam-5298	264	45	)	)	PUNCT
ejpam-5298	264	46	=	=	SYM
ejpam-5298	264	47	1	1	NUM
ejpam-5298	264	48	∈	∈	NOUN
ejpam-5298	264	49	s	s	NOUN
ejpam-5298	264	50	,	,	PUNCT
ejpam-5298	264	51	we	we	PRON
ejpam-5298	264	52	have	have	VERB
ejpam-5298	264	53	1	1	NUM
ejpam-5298	264	54	∈	∈	NOUN
ejpam-5298	264	55	m−1(s	m−1(	NOUN
ejpam-5298	264	56	)	)	PUNCT
ejpam-5298	264	57	.	.	PUNCT
ejpam-5298	265	1	let	let	VERB
ejpam-5298	265	2	x	x	SYM
ejpam-5298	265	3	∈	∈	PROPN
ejpam-5298	265	4	h	h	NOUN
ejpam-5298	265	5	and	and	CCONJ
ejpam-5298	265	6	y	y	PROPN
ejpam-5298	265	7	∈	∈	PROPN
ejpam-5298	265	8	m−1(s	m−1(s	PROPN
ejpam-5298	265	9	)	)	PUNCT
ejpam-5298	265	10	.	.	PUNCT
ejpam-5298	266	1	then	then	ADV
ejpam-5298	266	2	m(y	m(y	NOUN
ejpam-5298	266	3	)	)	PUNCT
ejpam-5298	266	4	∈	∈	PROPN
ejpam-5298	266	5	s	s	NOUN
ejpam-5298	266	6	,	,	PUNCT
ejpam-5298	266	7	so	so	ADV
ejpam-5298	266	8	m(x	m(x	PROPN
ejpam-5298	266	9	·	·	PUNCT
ejpam-5298	267	1	y	y	X
ejpam-5298	267	2	)	)	PUNCT
ejpam-5298	267	3	=	=	SYM
ejpam-5298	267	4	x	x	SYM
ejpam-5298	267	5	·	·	PUNCT
ejpam-5298	267	6	m(y	m(y	NOUN
ejpam-5298	267	7	)	)	PUNCT
ejpam-5298	267	8	∈	∈	PROPN
ejpam-5298	267	9	s.	s.	PROPN
ejpam-5298	268	1	thus	thus	ADV
ejpam-5298	268	2	,	,	PUNCT
ejpam-5298	268	3	x	x	X
ejpam-5298	268	4	·	·	PUNCT
ejpam-5298	268	5	y	y	SYM
ejpam-5298	268	6	∈	∈	PROPN
ejpam-5298	268	7	m−1(s	m−1(s	PROPN
ejpam-5298	268	8	)	)	PUNCT
ejpam-5298	268	9	.	.	PUNCT
ejpam-5298	269	1	hence	hence	ADV
ejpam-5298	269	2	,	,	PUNCT
ejpam-5298	269	3	m−1(s	m−1(s	PROPN
ejpam-5298	269	4	)	)	PUNCT
ejpam-5298	269	5	is	be	AUX
ejpam-5298	269	6	a	a	DET
ejpam-5298	269	7	near	near	ADJ
ejpam-5298	269	8	filter	filter	NOUN
ejpam-5298	269	9	of	of	ADP
ejpam-5298	269	10	h.	h.	PROPN
ejpam-5298	269	11	theorem	theorem	PROPN
ejpam-5298	269	12	2	2	X
ejpam-5298	269	13	.	.	PUNCT
ejpam-5298	270	1	let	let	VERB
ejpam-5298	270	2	m	m	PRON
ejpam-5298	270	3	be	be	AUX
ejpam-5298	270	4	a	a	DET
ejpam-5298	270	5	multiplier	multipli	ADJ
ejpam-5298	270	6	of	of	ADP
ejpam-5298	270	7	h.	h.	NOUN
ejpam-5298	270	8	then	then	ADV
ejpam-5298	270	9	the	the	DET
ejpam-5298	270	10	following	following	ADJ
ejpam-5298	270	11	statements	statement	NOUN
ejpam-5298	270	12	hold	hold	VERB
ejpam-5298	270	13	:	:	PUNCT
ejpam-5298	270	14	(	(	PUNCT
ejpam-5298	270	15	1	1	X
ejpam-5298	270	16	)	)	PUNCT
ejpam-5298	270	17	kerm(h	kerm(h	NOUN
ejpam-5298	270	18	)	)	PUNCT
ejpam-5298	270	19	is	be	AUX
ejpam-5298	270	20	a	a	DET
ejpam-5298	270	21	near	near	ADJ
ejpam-5298	270	22	filter	filter	NOUN
ejpam-5298	270	23	of	of	ADP
ejpam-5298	270	24	h	h	NOUN
ejpam-5298	270	25	,	,	PUNCT
ejpam-5298	270	26	(	(	PUNCT
ejpam-5298	270	27	2	2	X
ejpam-5298	270	28	)	)	PUNCT
ejpam-5298	270	29	fixm(h	fixm(h	NOUN
ejpam-5298	270	30	)	)	PUNCT
ejpam-5298	270	31	is	be	AUX
ejpam-5298	270	32	a	a	DET
ejpam-5298	270	33	near	near	ADJ
ejpam-5298	270	34	filter	filter	NOUN
ejpam-5298	270	35	of	of	ADP
ejpam-5298	270	36	h	h	NOUN
ejpam-5298	270	37	,	,	PUNCT
ejpam-5298	270	38	(	(	PUNCT
ejpam-5298	270	39	3	3	NUM
ejpam-5298	270	40	)	)	PUNCT
ejpam-5298	270	41	im(m	im(m	NOUN
ejpam-5298	270	42	)	)	PUNCT
ejpam-5298	270	43	is	be	AUX
ejpam-5298	270	44	a	a	DET
ejpam-5298	270	45	near	near	ADJ
ejpam-5298	270	46	filter	filter	NOUN
ejpam-5298	270	47	of	of	ADP
ejpam-5298	270	48	h.	h.	NOUN
ejpam-5298	270	49	proof	proof	NOUN
ejpam-5298	270	50	.	.	PUNCT
ejpam-5298	271	1	(	(	PUNCT
ejpam-5298	271	2	1	1	X
ejpam-5298	271	3	)	)	PUNCT
ejpam-5298	271	4	since	since	SCONJ
ejpam-5298	271	5	{	{	PUNCT
ejpam-5298	271	6	1	1	NUM
ejpam-5298	271	7	}	}	PUNCT
ejpam-5298	271	8	is	be	AUX
ejpam-5298	271	9	a	a	DET
ejpam-5298	271	10	near	near	ADJ
ejpam-5298	271	11	filter	filter	NOUN
ejpam-5298	271	12	of	of	ADP
ejpam-5298	271	13	h	h	NOUN
ejpam-5298	271	14	,	,	PUNCT
ejpam-5298	271	15	it	it	PRON
ejpam-5298	271	16	follows	follow	VERB
ejpam-5298	271	17	from	from	ADP
ejpam-5298	271	18	theorem	theorem	ADJ
ejpam-5298	271	19	1	1	NUM
ejpam-5298	271	20	(	(	PUNCT
ejpam-5298	271	21	2	2	NUM
ejpam-5298	271	22	)	)	PUNCT
ejpam-5298	271	23	that	that	DET
ejpam-5298	271	24	kerm(h	kerm(h	NOUN
ejpam-5298	271	25	)	)	PUNCT
ejpam-5298	271	26	=	=	SYM
ejpam-5298	272	1	m−1({1	m−1({1	NOUN
ejpam-5298	272	2	}	}	PUNCT
ejpam-5298	272	3	)	)	PUNCT
ejpam-5298	272	4	is	be	AUX
ejpam-5298	272	5	a	a	DET
ejpam-5298	272	6	near	near	ADJ
ejpam-5298	272	7	filter	filter	NOUN
ejpam-5298	272	8	of	of	ADP
ejpam-5298	272	9	h.	h.	PROPN
ejpam-5298	272	10	(	(	PUNCT
ejpam-5298	272	11	2	2	NUM
ejpam-5298	272	12	)	)	PUNCT
ejpam-5298	272	13	since	since	SCONJ
ejpam-5298	272	14	m(1	m(1	NOUN
ejpam-5298	272	15	)	)	PUNCT
ejpam-5298	272	16	=	=	SYM
ejpam-5298	272	17	1	1	NUM
ejpam-5298	272	18	,	,	PUNCT
ejpam-5298	272	19	we	we	PRON
ejpam-5298	272	20	have	have	VERB
ejpam-5298	272	21	1	1	NUM
ejpam-5298	272	22	∈	∈	NOUN
ejpam-5298	272	23	fixm(h	fixm(h	NOUN
ejpam-5298	272	24	)	)	PUNCT
ejpam-5298	272	25	.	.	PUNCT
ejpam-5298	273	1	let	let	VERB
ejpam-5298	273	2	x	x	SYM
ejpam-5298	273	3	∈	∈	PROPN
ejpam-5298	273	4	h	h	NOUN
ejpam-5298	273	5	and	and	CCONJ
ejpam-5298	273	6	y	y	PROPN
ejpam-5298	273	7	∈	∈	PROPN
ejpam-5298	273	8	fixm(h	fixm(h	PROPN
ejpam-5298	273	9	)	)	PUNCT
ejpam-5298	273	10	.	.	PUNCT
ejpam-5298	274	1	then	then	ADV
ejpam-5298	274	2	m(y	m(y	NOUN
ejpam-5298	274	3	)	)	PUNCT
ejpam-5298	275	1	=	=	SYM
ejpam-5298	275	2	y	y	PROPN
ejpam-5298	275	3	,	,	PUNCT
ejpam-5298	275	4	so	so	ADV
ejpam-5298	275	5	m(x	m(x	PROPN
ejpam-5298	275	6	·	·	PUNCT
ejpam-5298	276	1	y	y	X
ejpam-5298	276	2	)	)	PUNCT
ejpam-5298	276	3	=	=	SYM
ejpam-5298	276	4	x	x	SYM
ejpam-5298	276	5	·	·	PUNCT
ejpam-5298	276	6	m(y	m(y	X
ejpam-5298	276	7	)	)	PUNCT
ejpam-5298	277	1	=	=	SYM
ejpam-5298	277	2	x	x	PUNCT
ejpam-5298	277	3	·	·	PUNCT
ejpam-5298	277	4	y.	y.	PROPN
ejpam-5298	277	5	thus	thus	ADV
ejpam-5298	277	6	,	,	PUNCT
ejpam-5298	277	7	x	x	X
ejpam-5298	277	8	·	·	PUNCT
ejpam-5298	277	9	y	y	PROPN
ejpam-5298	277	10	∈	∈	PROPN
ejpam-5298	277	11	fixm(h	fixm(h	PROPN
ejpam-5298	277	12	)	)	PUNCT
ejpam-5298	277	13	.	.	PUNCT
ejpam-5298	278	1	hence	hence	ADV
ejpam-5298	278	2	,	,	PUNCT
ejpam-5298	278	3	fixm(h	fixm(h	PROPN
ejpam-5298	278	4	)	)	PUNCT
ejpam-5298	278	5	is	be	AUX
ejpam-5298	278	6	a	a	DET
ejpam-5298	278	7	near	near	ADJ
ejpam-5298	278	8	filter	filter	NOUN
ejpam-5298	278	9	of	of	ADP
ejpam-5298	278	10	h.	h.	PROPN
ejpam-5298	278	11	(	(	PUNCT
ejpam-5298	278	12	3	3	NUM
ejpam-5298	278	13	)	)	PUNCT
ejpam-5298	278	14	since	since	SCONJ
ejpam-5298	278	15	h	h	NOUN
ejpam-5298	278	16	is	be	AUX
ejpam-5298	278	17	a	a	DET
ejpam-5298	278	18	near	near	ADJ
ejpam-5298	278	19	filter	filter	NOUN
ejpam-5298	278	20	of	of	ADP
ejpam-5298	278	21	h	h	NOUN
ejpam-5298	278	22	,	,	PUNCT
ejpam-5298	278	23	it	it	PRON
ejpam-5298	278	24	follows	follow	VERB
ejpam-5298	278	25	from	from	ADP
ejpam-5298	278	26	theorem	theorem	ADJ
ejpam-5298	278	27	1	1	NUM
ejpam-5298	278	28	(	(	PUNCT
ejpam-5298	278	29	1	1	NUM
ejpam-5298	278	30	)	)	PUNCT
ejpam-5298	278	31	that	that	PRON
ejpam-5298	278	32	im(m	im(m	VERB
ejpam-5298	278	33	)	)	PUNCT
ejpam-5298	278	34	=	=	SYM
ejpam-5298	278	35	m(h	m(h	NOUN
ejpam-5298	278	36	)	)	PUNCT
ejpam-5298	278	37	is	be	AUX
ejpam-5298	278	38	a	a	DET
ejpam-5298	278	39	near	near	ADJ
ejpam-5298	278	40	filter	filter	NOUN
ejpam-5298	278	41	of	of	ADP
ejpam-5298	278	42	h.	h.	PROPN
ejpam-5298	278	43	since	since	SCONJ
ejpam-5298	278	44	subalgebras	subalgebras	PROPN
ejpam-5298	278	45	are	be	AUX
ejpam-5298	278	46	generalizations	generalization	NOUN
ejpam-5298	278	47	of	of	ADP
ejpam-5298	278	48	near	near	ADJ
ejpam-5298	278	49	filters	filter	NOUN
ejpam-5298	278	50	,	,	PUNCT
ejpam-5298	278	51	we	we	PRON
ejpam-5298	278	52	can	can	AUX
ejpam-5298	278	53	replace	replace	VERB
ejpam-5298	278	54	a	a	DET
ejpam-5298	278	55	near	near	ADJ
ejpam-5298	278	56	filter	filter	NOUN
ejpam-5298	278	57	with	with	ADP
ejpam-5298	278	58	a	a	DET
ejpam-5298	278	59	subalgebra	subalgebra	NOUN
ejpam-5298	278	60	in	in	ADP
ejpam-5298	278	61	theorem	theorem	NOUN
ejpam-5298	278	62	2	2	NUM
ejpam-5298	278	63	.	.	NOUN
ejpam-5298	278	64	4	4	NUM
ejpam-5298	278	65	.	.	X
ejpam-5298	278	66	conclusion	conclusion	NOUN
ejpam-5298	278	67	in	in	ADP
ejpam-5298	278	68	this	this	DET
ejpam-5298	278	69	study	study	NOUN
ejpam-5298	278	70	,	,	PUNCT
ejpam-5298	278	71	we	we	PRON
ejpam-5298	278	72	extensively	extensively	ADV
ejpam-5298	278	73	investigated	investigate	VERB
ejpam-5298	278	74	the	the	DET
ejpam-5298	278	75	four	four	NUM
ejpam-5298	278	76	key	key	ADJ
ejpam-5298	278	77	notions	notion	NOUN
ejpam-5298	278	78	of	of	ADP
ejpam-5298	278	79	left	left	ADJ
ejpam-5298	278	80	multipliers	multiplier	NOUN
ejpam-5298	278	81	,	,	PUNCT
ejpam-5298	278	82	right	right	ADJ
ejpam-5298	278	83	multipliers	multiplier	NOUN
ejpam-5298	278	84	,	,	PUNCT
ejpam-5298	278	85	anti	anti	ADJ
ejpam-5298	278	86	-	-	ADJ
ejpam-5298	278	87	left	left	ADJ
ejpam-5298	278	88	multipliers	multiplier	NOUN
ejpam-5298	278	89	,	,	PUNCT
ejpam-5298	278	90	and	and	CCONJ
ejpam-5298	278	91	anti	anti	ADJ
ejpam-5298	278	92	-	-	ADJ
ejpam-5298	278	93	right	right	ADJ
ejpam-5298	278	94	multipliers	multiplier	NOUN
ejpam-5298	278	95	within	within	ADP
ejpam-5298	278	96	the	the	DET
ejpam-5298	278	97	context	context	NOUN
ejpam-5298	278	98	of	of	ADP
ejpam-5298	278	99	hilbert	hilbert	PROPN
ejpam-5298	278	100	algebras	algebras	PROPN
ejpam-5298	278	101	.	.	PUNCT
ejpam-5298	279	1	our	our	PRON
ejpam-5298	279	2	findings	finding	NOUN
ejpam-5298	279	3	revealed	reveal	VERB
ejpam-5298	279	4	that	that	SCONJ
ejpam-5298	279	5	the	the	DET
ejpam-5298	279	6	identity	identity	NOUN
ejpam-5298	279	7	function	function	NOUN
ejpam-5298	279	8	is	be	AUX
ejpam-5298	279	9	the	the	DET
ejpam-5298	279	10	only	only	ADJ
ejpam-5298	279	11	valid	valid	ADJ
ejpam-5298	279	12	left	leave	VERB
ejpam-5298	279	13	multiplier	multiplier	ADV
ejpam-5298	279	14	,	,	PUNCT
ejpam-5298	279	15	the	the	DET
ejpam-5298	279	16	anti	anti	ADJ
ejpam-5298	279	17	-	-	ADJ
ejpam-5298	279	18	left	left	ADJ
ejpam-5298	279	19	multiplier	multipli	ADJ
ejpam-5298	279	20	remains	remain	VERB
ejpam-5298	279	21	constant	constant	ADJ
ejpam-5298	279	22	,	,	PUNCT
ejpam-5298	279	23	and	and	CCONJ
ejpam-5298	279	24	the	the	DET
ejpam-5298	279	25	anti	anti	ADJ
ejpam-5298	279	26	-	-	ADJ
ejpam-5298	279	27	right	right	ADJ
ejpam-5298	279	28	multiplier	multipli	ADJ
ejpam-5298	279	29	exists	exist	VERB
ejpam-5298	279	30	solely	solely	ADV
ejpam-5298	279	31	if	if	SCONJ
ejpam-5298	279	32	the	the	DET
ejpam-5298	279	33	algebra	algebra	NOUN
ejpam-5298	279	34	is	be	AUX
ejpam-5298	279	35	a	a	DET
ejpam-5298	279	36	singleton	singleton	NOUN
ejpam-5298	279	37	.	.	PUNCT
ejpam-5298	280	1	we	we	PRON
ejpam-5298	280	2	further	far	ADV
ejpam-5298	280	3	identified	identify	VERB
ejpam-5298	280	4	the	the	DET
ejpam-5298	280	5	conditions	condition	NOUN
ejpam-5298	280	6	under	under	ADP
ejpam-5298	280	7	which	which	PRON
ejpam-5298	280	8	a	a	DET
ejpam-5298	280	9	right	right	NOUN
ejpam-5298	280	10	multiplier	multiplier	ADV
ejpam-5298	280	11	becomes	become	VERB
ejpam-5298	280	12	the	the	DET
ejpam-5298	280	13	identity	identity	NOUN
ejpam-5298	280	14	function	function	NOUN
ejpam-5298	280	15	,	,	PUNCT
ejpam-5298	280	16	as	as	SCONJ
ejpam-5298	280	17	detailed	detailed	ADJ
ejpam-5298	280	18	in	in	ADP
ejpam-5298	280	19	proposition	proposition	NOUN
ejpam-5298	280	20	3	3	NUM
ejpam-5298	280	21	.	.	PUNCT
ejpam-5298	281	1	moreover	moreover	ADV
ejpam-5298	281	2	,	,	PUNCT
ejpam-5298	281	3	we	we	PRON
ejpam-5298	281	4	developed	develop	VERB
ejpam-5298	281	5	a	a	DET
ejpam-5298	281	6	specific	specific	ADJ
ejpam-5298	281	7	right	right	ADJ
ejpam-5298	281	8	multiplier	multipli	ADJ
ejpam-5298	281	9	for	for	ADP
ejpam-5298	281	10	the	the	DET
ejpam-5298	281	11	cartesian	cartesian	ADJ
ejpam-5298	281	12	products	product	NOUN
ejpam-5298	281	13	of	of	ADP
ejpam-5298	281	14	hilbert	hilbert	PROPN
ejpam-5298	281	15	algebras	algebras	PROPN
ejpam-5298	281	16	.	.	PUNCT
ejpam-5298	282	1	lastly	lastly	ADV
ejpam-5298	282	2	,	,	PUNCT
ejpam-5298	282	3	we	we	PRON
ejpam-5298	282	4	demonstrated	demonstrate	VERB
ejpam-5298	282	5	that	that	SCONJ
ejpam-5298	282	6	the	the	DET
ejpam-5298	282	7	sets	set	NOUN
ejpam-5298	282	8	kerm(h	kerm(h	PROPN
ejpam-5298	282	9	)	)	PUNCT
ejpam-5298	282	10	,	,	PUNCT
ejpam-5298	282	11	fixm(h	fixm(h	NOUN
ejpam-5298	282	12	)	)	PUNCT
ejpam-5298	282	13	,	,	PUNCT
ejpam-5298	282	14	and	and	CCONJ
ejpam-5298	282	15	im(m	im(m	PUNCT
ejpam-5298	282	16	)	)	PUNCT
ejpam-5298	282	17	associated	associate	VERB
ejpam-5298	282	18	with	with	ADP
ejpam-5298	282	19	a	a	DET
ejpam-5298	282	20	right	right	ADJ
ejpam-5298	282	21	multiplier	multipli	ADJ
ejpam-5298	282	22	m	m	VERB
ejpam-5298	282	23	on	on	ADP
ejpam-5298	282	24	a	a	DET
ejpam-5298	282	25	hilbert	hilbert	NOUN
ejpam-5298	282	26	algebra	algebra	NOUN
ejpam-5298	282	27	h	h	NOUN
ejpam-5298	282	28	form	form	VERB
ejpam-5298	282	29	a	a	DET
ejpam-5298	282	30	near	near	ADJ
ejpam-5298	282	31	filter	filter	NOUN
ejpam-5298	282	32	.	.	PUNCT
ejpam-5298	283	1	references	reference	NOUN
ejpam-5298	283	2	2736	2736	NUM
ejpam-5298	283	3	acknowledgements	acknowledgement	NOUN
ejpam-5298	283	4	this	this	DET
ejpam-5298	283	5	research	research	NOUN
ejpam-5298	283	6	was	be	AUX
ejpam-5298	283	7	supported	support	VERB
ejpam-5298	283	8	by	by	ADP
ejpam-5298	283	9	university	university	NOUN
ejpam-5298	283	10	of	of	ADP
ejpam-5298	283	11	phayao	phayao	NOUN
ejpam-5298	283	12	and	and	CCONJ
ejpam-5298	283	13	thailand	thailand	PROPN
ejpam-5298	283	14	science	science	PROPN
ejpam-5298	283	15	research	research	PROPN
ejpam-5298	283	16	and	and	CCONJ
ejpam-5298	283	17	innovation	innovation	NOUN
ejpam-5298	283	18	fund	fund	NOUN
ejpam-5298	283	19	(	(	PUNCT
ejpam-5298	283	20	fundamental	fundamental	ADJ
ejpam-5298	283	21	fund	fund	NOUN
ejpam-5298	283	22	2025	2025	NUM
ejpam-5298	283	23	,	,	PUNCT
ejpam-5298	283	24	grant	grant	VERB
ejpam-5298	283	25	no	no	NOUN
ejpam-5298	283	26	.	.	PROPN
ejpam-5298	283	27	5027/2567	5027/2567	NUM
ejpam-5298	283	28	)	)	PUNCT
ejpam-5298	283	29	.	.	PUNCT
ejpam-5298	284	1	references	reference	NOUN
ejpam-5298	284	2	[	[	X
ejpam-5298	284	3	1	1	NUM
ejpam-5298	284	4	]	]	X
ejpam-5298	284	5	d.	d.	PROPN
ejpam-5298	284	6	busneag	busneag	PROPN
ejpam-5298	284	7	.	.	PUNCT
ejpam-5298	285	1	a	a	DET
ejpam-5298	285	2	note	note	NOUN
ejpam-5298	285	3	on	on	ADP
ejpam-5298	285	4	deductive	deductive	ADJ
ejpam-5298	285	5	systems	system	NOUN
ejpam-5298	285	6	of	of	ADP
ejpam-5298	285	7	a	a	DET
ejpam-5298	285	8	hilbert	hilbert	NOUN
ejpam-5298	285	9	algebra	algebra	NOUN
ejpam-5298	285	10	.	.	PUNCT
ejpam-5298	286	1	kobe	kobe	PROPN
ejpam-5298	286	2	j.	j.	PROPN
ejpam-5298	286	3	math	math	PROPN
ejpam-5298	286	4	.	.	PROPN
ejpam-5298	286	5	,	,	PUNCT
ejpam-5298	286	6	2:29–35	2:29–35	NUM
ejpam-5298	286	7	,	,	PUNCT
ejpam-5298	286	8	1985	1985	NUM
ejpam-5298	286	9	.	.	PUNCT
ejpam-5298	287	1	[	[	X
ejpam-5298	287	2	2	2	NUM
ejpam-5298	287	3	]	]	X
ejpam-5298	287	4	d.	d.	PROPN
ejpam-5298	287	5	busneag	busneag	PROPN
ejpam-5298	287	6	.	.	PUNCT
ejpam-5298	288	1	hilbert	hilbert	PROPN
ejpam-5298	288	2	algebras	algebras	PROPN
ejpam-5298	288	3	of	of	ADP
ejpam-5298	288	4	fractions	fraction	NOUN
ejpam-5298	288	5	and	and	CCONJ
ejpam-5298	288	6	maximal	maximal	ADJ
ejpam-5298	288	7	hilbert	hilbert	NOUN
ejpam-5298	288	8	algebras	algebra	NOUN
ejpam-5298	288	9	of	of	ADP
ejpam-5298	288	10	quotients	quotient	NOUN
ejpam-5298	288	11	.	.	PUNCT
ejpam-5298	289	1	kobe	kobe	PROPN
ejpam-5298	289	2	j.	j.	PROPN
ejpam-5298	289	3	math	math	PROPN
ejpam-5298	289	4	.	.	PUNCT
ejpam-5298	289	5	,	,	PUNCT
ejpam-5298	289	6	5:161–172	5:161–172	NOUN
ejpam-5298	289	7	,	,	PUNCT
ejpam-5298	289	8	1988	1988	NUM
ejpam-5298	289	9	.	.	PUNCT
ejpam-5298	290	1	[	[	X
ejpam-5298	290	2	3	3	NUM
ejpam-5298	290	3	]	]	X
ejpam-5298	290	4	i.	i.	NOUN
ejpam-5298	290	5	chajda	chajda	PROPN
ejpam-5298	290	6	and	and	CCONJ
ejpam-5298	290	7	r.	r.	PROPN
ejpam-5298	290	8	halaš.	halaš.	PROPN
ejpam-5298	290	9	congruences	congruence	NOUN
ejpam-5298	290	10	and	and	CCONJ
ejpam-5298	290	11	ideals	ideal	NOUN
ejpam-5298	290	12	in	in	ADP
ejpam-5298	290	13	hilbert	hilbert	PROPN
ejpam-5298	290	14	algebras	algebras	PROPN
ejpam-5298	290	15	.	.	PUNCT
ejpam-5298	291	1	kyungpook	kyungpook	PROPN
ejpam-5298	291	2	math	math	PROPN
ejpam-5298	291	3	.	.	PUNCT
ejpam-5298	292	1	j.	j.	PROPN
ejpam-5298	292	2	,	,	PUNCT
ejpam-5298	292	3	39(2):429–432	39(2):429–432	PROPN
ejpam-5298	292	4	,	,	PUNCT
ejpam-5298	292	5	1999	1999	NUM
ejpam-5298	292	6	.	.	PUNCT
ejpam-5298	293	1	[	[	X
ejpam-5298	293	2	4	4	X
ejpam-5298	293	3	]	]	PUNCT
ejpam-5298	293	4	m.	m.	NOUN
ejpam-5298	293	5	a.	a.	PROPN
ejpam-5298	293	6	chaudhry	chaudhry	PROPN
ejpam-5298	293	7	and	and	CCONJ
ejpam-5298	293	8	f.	f.	PROPN
ejpam-5298	293	9	ali	ali	PROPN
ejpam-5298	293	10	.	.	PROPN
ejpam-5298	293	11	multipliers	multiplier	NOUN
ejpam-5298	293	12	in	in	ADP
ejpam-5298	293	13	d	d	PROPN
ejpam-5298	293	14	-	-	PUNCT
ejpam-5298	293	15	algebras	algebras	PROPN
ejpam-5298	293	16	.	.	PUNCT
ejpam-5298	294	1	world	world	PROPN
ejpam-5298	294	2	appl	appl	PROPN
ejpam-5298	294	3	.	.	PUNCT
ejpam-5298	295	1	sci	sci	PROPN
ejpam-5298	295	2	.	.	PUNCT
ejpam-5298	295	3	j.	j.	PROPN
ejpam-5298	295	4	,	,	PUNCT
ejpam-5298	295	5	18(11):1649–1653	18(11):1649–1653	NUM
ejpam-5298	295	6	,	,	PUNCT
ejpam-5298	295	7	2012	2012	NUM
ejpam-5298	295	8	.	.	PUNCT
ejpam-5298	296	1	[	[	X
ejpam-5298	296	2	5	5	X
ejpam-5298	296	3	]	]	PUNCT
ejpam-5298	296	4	j.	j.	PROPN
ejpam-5298	296	5	cirulis	cirulis	PROPN
ejpam-5298	296	6	.	.	PUNCT
ejpam-5298	297	1	multipliers	multiplier	NOUN
ejpam-5298	297	2	in	in	ADP
ejpam-5298	297	3	implicative	implicative	ADJ
ejpam-5298	297	4	algebras	algebra	NOUN
ejpam-5298	297	5	.	.	PUNCT
ejpam-5298	298	1	bull	bull	NOUN
ejpam-5298	298	2	.	.	PUNCT
ejpam-5298	299	1	sect	sect	NOUN
ejpam-5298	299	2	.	.	PUNCT
ejpam-5298	300	1	logic	logic	NOUN
ejpam-5298	300	2	,	,	PUNCT
ejpam-5298	300	3	15(4):152–157	15(4):152–157	NOUN
ejpam-5298	300	4	,	,	PUNCT
ejpam-5298	300	5	1986	1986	NUM
ejpam-5298	300	6	.	.	PUNCT
ejpam-5298	301	1	[	[	X
ejpam-5298	301	2	6	6	NUM
ejpam-5298	301	3	]	]	PUNCT
ejpam-5298	301	4	a.	a.	NOUN
ejpam-5298	301	5	diego	diego	PROPN
ejpam-5298	301	6	.	.	PUNCT
ejpam-5298	302	1	sur	sur	PROPN
ejpam-5298	302	2	les	les	PROPN
ejpam-5298	302	3	algébres	algébres	PROPN
ejpam-5298	302	4	de	de	X
ejpam-5298	302	5	hilbert	hilbert	PROPN
ejpam-5298	302	6	.	.	PUNCT
ejpam-5298	303	1	collection	collection	PROPN
ejpam-5298	303	2	de	de	X
ejpam-5298	303	3	logique	logique	X
ejpam-5298	303	4	math	math	PROPN
ejpam-5298	303	5	.	.	PUNCT
ejpam-5298	304	1	ser	ser	PROPN
ejpam-5298	304	2	.	.	PUNCT
ejpam-5298	305	1	a	a	DET
ejpam-5298	305	2	(	(	PUNCT
ejpam-5298	305	3	ed	ed	NOUN
ejpam-5298	305	4	.	.	PUNCT
ejpam-5298	305	5	hermann	hermann	PROPN
ejpam-5298	305	6	,	,	PUNCT
ejpam-5298	305	7	paris	paris	PROPN
ejpam-5298	305	8	)	)	PUNCT
ejpam-5298	305	9	,	,	PUNCT
ejpam-5298	305	10	21:1–52	21:1–52	NUM
ejpam-5298	305	11	,	,	PUNCT
ejpam-5298	305	12	1966	1966	NUM
ejpam-5298	305	13	.	.	PUNCT
ejpam-5298	306	1	[	[	X
ejpam-5298	306	2	7	7	X
ejpam-5298	306	3	]	]	PUNCT
ejpam-5298	306	4	w.	w.	PROPN
ejpam-5298	306	5	a.	a.	PROPN
ejpam-5298	306	6	dudek	dudek	PROPN
ejpam-5298	306	7	.	.	PUNCT
ejpam-5298	307	1	on	on	ADP
ejpam-5298	307	2	fuzzification	fuzzification	NOUN
ejpam-5298	307	3	in	in	ADP
ejpam-5298	307	4	hilbert	hilbert	PROPN
ejpam-5298	307	5	algebras	algebras	PROPN
ejpam-5298	307	6	.	.	PUNCT
ejpam-5298	308	1	contrib	contrib	PROPN
ejpam-5298	308	2	.	.	PUNCT
ejpam-5298	308	3	gen	gen	PROPN
ejpam-5298	308	4	.	.	PROPN
ejpam-5298	308	5	algebra	algebra	PROPN
ejpam-5298	308	6	,	,	PUNCT
ejpam-5298	308	7	11:77–83	11:77–83	NUM
ejpam-5298	308	8	,	,	PUNCT
ejpam-5298	308	9	1999	1999	NUM
ejpam-5298	308	10	.	.	PUNCT
ejpam-5298	309	1	[	[	X
ejpam-5298	309	2	8	8	NUM
ejpam-5298	309	3	]	]	X
ejpam-5298	309	4	w.	w.	PROPN
ejpam-5298	309	5	a.	a.	PROPN
ejpam-5298	309	6	dudek	dudek	PROPN
ejpam-5298	309	7	.	.	PUNCT
ejpam-5298	310	1	on	on	ADP
ejpam-5298	310	2	ideals	ideal	NOUN
ejpam-5298	310	3	in	in	ADP
ejpam-5298	310	4	hilbert	hilbert	PROPN
ejpam-5298	310	5	algebras	algebras	PROPN
ejpam-5298	310	6	.	.	PUNCT
ejpam-5298	311	1	acta	acta	PROPN
ejpam-5298	311	2	universitatis	universitatis	PROPN
ejpam-5298	311	3	palackianae	palackianae	VERB
ejpam-5298	311	4	olomuciensis	olomuciensis	PROPN
ejpam-5298	311	5	fac	fac	PROPN
ejpam-5298	311	6	.	.	PUNCT
ejpam-5298	312	1	rer	rer	PROPN
ejpam-5298	312	2	.	.	PUNCT
ejpam-5298	313	1	nat	nat	PROPN
ejpam-5298	313	2	.	.	PUNCT
ejpam-5298	314	1	ser	ser	PROPN
ejpam-5298	314	2	.	.	PUNCT
ejpam-5298	314	3	math	math	PROPN
ejpam-5298	314	4	.	.	PUNCT
ejpam-5298	314	5	,	,	PUNCT
ejpam-5298	315	1	38:31–34	38:31–34	NUM
ejpam-5298	315	2	,	,	PUNCT
ejpam-5298	315	3	1999	1999	NUM
ejpam-5298	315	4	.	.	PUNCT
ejpam-5298	316	1	[	[	X
ejpam-5298	316	2	9	9	NUM
ejpam-5298	316	3	]	]	X
ejpam-5298	316	4	l.	l.	PROPN
ejpam-5298	316	5	henkin	henkin	PROPN
ejpam-5298	316	6	.	.	PUNCT
ejpam-5298	317	1	an	an	DET
ejpam-5298	317	2	algebraic	algebraic	ADJ
ejpam-5298	317	3	characterization	characterization	NOUN
ejpam-5298	317	4	of	of	ADP
ejpam-5298	317	5	quantifiers	quantifier	NOUN
ejpam-5298	317	6	.	.	PUNCT
ejpam-5298	318	1	fund	fund	NOUN
ejpam-5298	318	2	.	.	PUNCT
ejpam-5298	319	1	math	math	NOUN
ejpam-5298	319	2	.	.	PUNCT
ejpam-5298	319	3	,	,	PUNCT
ejpam-5298	320	1	37:63–74	37:63–74	NUM
ejpam-5298	320	2	,	,	PUNCT
ejpam-5298	320	3	1950	1950	NUM
ejpam-5298	320	4	.	.	PUNCT
ejpam-5298	321	1	[	[	X
ejpam-5298	321	2	10	10	NUM
ejpam-5298	321	3	]	]	PUNCT
ejpam-5298	321	4	a.	a.	NOUN
ejpam-5298	321	5	iampan	iampan	PROPN
ejpam-5298	321	6	.	.	PUNCT
ejpam-5298	322	1	multipliers	multiplier	NOUN
ejpam-5298	322	2	and	and	CCONJ
ejpam-5298	322	3	near	near	ADP
ejpam-5298	322	4	up	up	ADP
ejpam-5298	322	5	-	-	PUNCT
ejpam-5298	322	6	filters	filter	NOUN
ejpam-5298	322	7	of	of	ADP
ejpam-5298	322	8	up	up	ADP
ejpam-5298	322	9	-	-	PUNCT
ejpam-5298	322	10	algebras	algebras	X
ejpam-5298	322	11	.	.	PUNCT
ejpam-5298	323	1	j.	j.	PROPN
ejpam-5298	323	2	discrete	discrete	PROPN
ejpam-5298	323	3	math	math	PROPN
ejpam-5298	323	4	.	.	PUNCT
ejpam-5298	324	1	sci	sci	PROPN
ejpam-5298	324	2	.	.	PUNCT
ejpam-5298	324	3	cryptography	cryptography	PROPN
ejpam-5298	324	4	,	,	PUNCT
ejpam-5298	324	5	24(3):667–680	24(3):667–680	PROPN
ejpam-5298	324	6	,	,	PUNCT
ejpam-5298	324	7	2021	2021	NUM
ejpam-5298	324	8	.	.	PUNCT
ejpam-5298	325	1	[	[	X
ejpam-5298	325	2	11	11	NUM
ejpam-5298	325	3	]	]	PUNCT
ejpam-5298	325	4	a.	a.	NOUN
ejpam-5298	325	5	iampan	iampan	PROPN
ejpam-5298	325	6	,	,	PUNCT
ejpam-5298	325	7	p.	p.	PROPN
ejpam-5298	325	8	jayaraman	jayaraman	NOUN
ejpam-5298	325	9	,	,	PUNCT
ejpam-5298	325	10	s.	s.	PROPN
ejpam-5298	325	11	d.	d.	PROPN
ejpam-5298	325	12	sudha	sudha	PROPN
ejpam-5298	325	13	,	,	PUNCT
ejpam-5298	325	14	and	and	CCONJ
ejpam-5298	325	15	n.	n.	PROPN
ejpam-5298	325	16	rajesh	rajesh	PROPN
ejpam-5298	325	17	.	.	PUNCT
ejpam-5298	326	1	interval	interval	NOUN
ejpam-5298	326	2	-	-	PUNCT
ejpam-5298	326	3	valued	value	VERB
ejpam-5298	326	4	neutrosophic	neutrosophic	ADJ
ejpam-5298	326	5	ideals	ideal	NOUN
ejpam-5298	326	6	of	of	ADP
ejpam-5298	326	7	hilbert	hilbert	PROPN
ejpam-5298	326	8	algebras	algebras	PROPN
ejpam-5298	326	9	.	.	PUNCT
ejpam-5298	327	1	int	int	NOUN
ejpam-5298	327	2	.	.	PUNCT
ejpam-5298	328	1	j.	j.	PROPN
ejpam-5298	328	2	neutrosophic	neutrosophic	PROPN
ejpam-5298	328	3	sci	sci	PROPN
ejpam-5298	328	4	.	.	PROPN
ejpam-5298	328	5	,	,	PUNCT
ejpam-5298	328	6	18(4):223–237	18(4):223–237	NUM
ejpam-5298	328	7	,	,	PUNCT
ejpam-5298	328	8	2022	2022	NUM
ejpam-5298	328	9	.	.	PUNCT
ejpam-5298	329	1	[	[	X
ejpam-5298	329	2	12	12	NUM
ejpam-5298	329	3	]	]	PUNCT
ejpam-5298	329	4	a.	a.	NOUN
ejpam-5298	329	5	iampan	iampan	PROPN
ejpam-5298	329	6	,	,	PUNCT
ejpam-5298	329	7	n.	n.	PROPN
ejpam-5298	329	8	rajesh	rajesh	PROPN
ejpam-5298	329	9	,	,	PUNCT
ejpam-5298	329	10	and	and	CCONJ
ejpam-5298	329	11	b.	b.	PROPN
ejpam-5298	329	12	brundha	brundha	PROPN
ejpam-5298	329	13	.	.	PUNCT
ejpam-5298	330	1	neutrosophic	neutrosophic	PROPN
ejpam-5298	330	2	set	set	PROPN
ejpam-5298	330	3	theory	theory	NOUN
ejpam-5298	330	4	applied	apply	VERB
ejpam-5298	330	5	to	to	ADP
ejpam-5298	330	6	hilbert	hilbert	PROPN
ejpam-5298	330	7	algebras	algebras	PROPN
ejpam-5298	330	8	.	.	PUNCT
ejpam-5298	331	1	int	int	NOUN
ejpam-5298	331	2	.	.	PUNCT
ejpam-5298	332	1	j.	j.	PROPN
ejpam-5298	332	2	neutrosophic	neutrosophic	PROPN
ejpam-5298	332	3	sci	sci	PROPN
ejpam-5298	332	4	.	.	PROPN
ejpam-5298	332	5	,	,	PUNCT
ejpam-5298	332	6	21(4):84–93	21(4):84–93	NUM
ejpam-5298	332	7	,	,	PUNCT
ejpam-5298	332	8	2023	2023	NUM
ejpam-5298	332	9	.	.	PUNCT
ejpam-5298	333	1	[	[	X
ejpam-5298	333	2	13	13	NUM
ejpam-5298	333	3	]	]	X
ejpam-5298	333	4	y.	y.	PROPN
ejpam-5298	333	5	b.	b.	PROPN
ejpam-5298	333	6	jun	jun	PROPN
ejpam-5298	333	7	.	.	PROPN
ejpam-5298	333	8	deductive	deductive	ADJ
ejpam-5298	333	9	systems	system	NOUN
ejpam-5298	333	10	of	of	ADP
ejpam-5298	333	11	hilbert	hilbert	PROPN
ejpam-5298	333	12	algebras	algebras	PROPN
ejpam-5298	333	13	.	.	PUNCT
ejpam-5298	334	1	math	math	PROPN
ejpam-5298	334	2	.	.	PUNCT
ejpam-5298	335	1	japon	japon	PROPN
ejpam-5298	335	2	.	.	PROPN
ejpam-5298	335	3	,	,	PUNCT
ejpam-5298	335	4	43:51–54	43:51–54	NUM
ejpam-5298	335	5	,	,	PUNCT
ejpam-5298	335	6	1996	1996	NUM
ejpam-5298	335	7	.	.	PUNCT
ejpam-5298	336	1	[	[	X
ejpam-5298	336	2	14	14	NUM
ejpam-5298	336	3	]	]	X
ejpam-5298	336	4	y.	y.	PROPN
ejpam-5298	336	5	b.	b.	PROPN
ejpam-5298	336	6	jun	jun	PROPN
ejpam-5298	336	7	,	,	PUNCT
ejpam-5298	336	8	j.	j.	PROPN
ejpam-5298	336	9	w.	w.	PROPN
ejpam-5298	336	10	nam	nam	PROPN
ejpam-5298	336	11	,	,	PUNCT
ejpam-5298	336	12	and	and	CCONJ
ejpam-5298	336	13	s.	s.	PROPN
ejpam-5298	336	14	m.	m.	PROPN
ejpam-5298	336	15	hong	hong	PROPN
ejpam-5298	336	16	.	.	PUNCT
ejpam-5298	337	1	a	a	DET
ejpam-5298	337	2	note	note	NOUN
ejpam-5298	337	3	on	on	ADP
ejpam-5298	337	4	hilbert	hilbert	PROPN
ejpam-5298	337	5	algebras	algebras	PROPN
ejpam-5298	337	6	.	.	PUNCT
ejpam-5298	338	1	pusan	pusan	PROPN
ejpam-5298	338	2	kyongnam	kyongnam	PROPN
ejpam-5298	338	3	math	math	PROPN
ejpam-5298	338	4	.	.	PUNCT
ejpam-5298	339	1	j.	j.	PROPN
ejpam-5298	339	2	,	,	PUNCT
ejpam-5298	339	3	10(2):279–285	10(2):279–285	PROPN
ejpam-5298	339	4	,	,	PUNCT
ejpam-5298	339	5	1994	1994	NUM
ejpam-5298	339	6	.	.	PUNCT
ejpam-5298	340	1	[	[	X
ejpam-5298	340	2	15	15	NUM
ejpam-5298	340	3	]	]	X
ejpam-5298	340	4	r.	r.	PROPN
ejpam-5298	340	5	t.	t.	PROPN
ejpam-5298	340	6	khorami	khorami	PROPN
ejpam-5298	340	7	and	and	CCONJ
ejpam-5298	340	8	a.	a.	PROPN
ejpam-5298	340	9	b.	b.	PROPN
ejpam-5298	340	10	saeid	saeid	PROPN
ejpam-5298	340	11	.	.	PUNCT
ejpam-5298	341	1	multiplier	multiplier	ADV
ejpam-5298	341	2	in	in	ADP
ejpam-5298	341	3	bl	bl	NOUN
ejpam-5298	341	4	-	-	PUNCT
ejpam-5298	341	5	algebras	algebras	PROPN
ejpam-5298	341	6	.	.	PUNCT
ejpam-5298	342	1	iran	iran	PROPN
ejpam-5298	342	2	.	.	PUNCT
ejpam-5298	343	1	j.	j.	PROPN
ejpam-5298	343	2	sci	sci	PROPN
ejpam-5298	343	3	.	.	PROPN
ejpam-5298	343	4	,	,	PUNCT
ejpam-5298	343	5	38(2):95–103	38(2):95–103	NUM
ejpam-5298	343	6	,	,	PUNCT
ejpam-5298	343	7	2014	2014	NUM
ejpam-5298	343	8	.	.	PUNCT
ejpam-5298	344	1	references	reference	NOUN
ejpam-5298	344	2	2737	2737	NUM
ejpam-5298	345	1	[	[	X
ejpam-5298	345	2	16	16	NUM
ejpam-5298	345	3	]	]	PUNCT
ejpam-5298	346	1	k.	k.	PROPN
ejpam-5298	346	2	h.	h.	PROPN
ejpam-5298	346	3	kim	kim	PROPN
ejpam-5298	346	4	.	.	PUNCT
ejpam-5298	347	1	multipliers	multiplier	NOUN
ejpam-5298	347	2	in	in	ADP
ejpam-5298	347	3	be	be	NOUN
ejpam-5298	347	4	-	-	PUNCT
ejpam-5298	347	5	algebras	algebra	NOUN
ejpam-5298	347	6	.	.	PUNCT
ejpam-5298	348	1	int	int	NOUN
ejpam-5298	348	2	.	.	PUNCT
ejpam-5298	349	1	math	math	NOUN
ejpam-5298	349	2	.	.	PUNCT
ejpam-5298	350	1	forum	forum	PROPN
ejpam-5298	350	2	,	,	PUNCT
ejpam-5298	350	3	6(17):815–820	6(17):815–820	NOUN
ejpam-5298	350	4	,	,	PUNCT
ejpam-5298	350	5	2011	2011	NUM
ejpam-5298	350	6	.	.	PUNCT
ejpam-5298	351	1	[	[	X
ejpam-5298	351	2	17	17	NUM
ejpam-5298	351	3	]	]	PUNCT
ejpam-5298	351	4	k.	k.	PROPN
ejpam-5298	351	5	h.	h.	PROPN
ejpam-5298	351	6	kim	kim	PROPN
ejpam-5298	351	7	and	and	CCONJ
ejpam-5298	351	8	h.	h.	PROPN
ejpam-5298	351	9	j.	j.	PROPN
ejpam-5298	351	10	lim	lim	PROPN
ejpam-5298	351	11	.	.	PUNCT
ejpam-5298	352	1	on	on	ADP
ejpam-5298	352	2	multipliers	multiplier	NOUN
ejpam-5298	352	3	of	of	ADP
ejpam-5298	352	4	bcc	bcc	PROPN
ejpam-5298	352	5	-	-	PUNCT
ejpam-5298	352	6	algebras	algebras	PROPN
ejpam-5298	352	7	.	.	PUNCT
ejpam-5298	353	1	honam	honam	PROPN
ejpam-5298	353	2	math	math	PROPN
ejpam-5298	353	3	.	.	PUNCT
ejpam-5298	354	1	j.	j.	PROPN
ejpam-5298	354	2	,	,	PUNCT
ejpam-5298	354	3	35(2):201–210	35(2):201–210	PROPN
ejpam-5298	354	4	,	,	PUNCT
ejpam-5298	354	5	2013	2013	NUM
ejpam-5298	354	6	.	.	PUNCT
ejpam-5298	355	1	[	[	X
ejpam-5298	355	2	18	18	NUM
ejpam-5298	355	3	]	]	X
ejpam-5298	355	4	s.	s.	PROPN
ejpam-5298	355	5	d.	d.	PROPN
ejpam-5298	355	6	lee	lee	PROPN
ejpam-5298	355	7	and	and	CCONJ
ejpam-5298	355	8	k.	k.	PROPN
ejpam-5298	355	9	h.	h.	PROPN
ejpam-5298	355	10	kim	kim	PROPN
ejpam-5298	355	11	.	.	PUNCT
ejpam-5298	356	1	a	a	DET
ejpam-5298	356	2	note	note	NOUN
ejpam-5298	356	3	on	on	ADP
ejpam-5298	356	4	multipliers	multiplier	NOUN
ejpam-5298	356	5	of	of	ADP
ejpam-5298	356	6	subtraction	subtraction	NOUN
ejpam-5298	356	7	algebras	algebra	NOUN
ejpam-5298	356	8	.	.	PUNCT
ejpam-5298	357	1	hacet	hacet	PROPN
ejpam-5298	357	2	.	.	PUNCT
ejpam-5298	358	1	j.	j.	PROPN
ejpam-5298	358	2	math	math	PROPN
ejpam-5298	358	3	.	.	PUNCT
ejpam-5298	359	1	stat	stat	PROPN
ejpam-5298	359	2	.	.	PUNCT
ejpam-5298	359	3	,	,	PUNCT
ejpam-5298	359	4	42(2):165–171	42(2):165–171	PROPN
ejpam-5298	359	5	,	,	PUNCT
ejpam-5298	359	6	2013	2013	NUM
ejpam-5298	359	7	.	.	PUNCT