id	sid	tid	token	lemma	pos
ejpam-6496	1	1	european	european	PROPN
ejpam-6496	1	2	journal	journal	PROPN
ejpam-6496	1	3	of	of	ADP
ejpam-6496	1	4	pure	pure	ADJ
ejpam-6496	1	5	and	and	CCONJ
ejpam-6496	1	6	applied	applied	ADJ
ejpam-6496	1	7	mathematics	mathematic	NOUN
ejpam-6496	1	8	2025	2025	NUM
ejpam-6496	1	9	,	,	PUNCT
ejpam-6496	1	10	vol	vol	NOUN
ejpam-6496	1	11	.	.	PROPN
ejpam-6496	1	12	18	18	NUM
ejpam-6496	1	13	,	,	PUNCT
ejpam-6496	1	14	issue	issue	NOUN
ejpam-6496	1	15	3	3	NUM
ejpam-6496	1	16	,	,	PUNCT
ejpam-6496	1	17	article	article	NOUN
ejpam-6496	1	18	number	number	NOUN
ejpam-6496	1	19	6496	6496	NUM
ejpam-6496	1	20	issn	issn	VERB
ejpam-6496	1	21	1307	1307	NUM
ejpam-6496	1	22	-	-	SYM
ejpam-6496	1	23	5543	5543	NUM
ejpam-6496	1	24	–	–	PUNCT
ejpam-6496	1	25	ejpam.com	ejpam.com	X
ejpam-6496	1	26	published	publish	VERB
ejpam-6496	1	27	by	by	ADP
ejpam-6496	1	28	new	new	PROPN
ejpam-6496	1	29	york	york	PROPN
ejpam-6496	1	30	business	business	PROPN
ejpam-6496	1	31	global	global	VERB
ejpam-6496	1	32	some	some	DET
ejpam-6496	1	33	properties	property	NOUN
ejpam-6496	1	34	of	of	ADP
ejpam-6496	1	35	differentiable	differentiable	NOUN
ejpam-6496	1	36	and	and	CCONJ
ejpam-6496	1	37	riemann	riemann	PROPN
ejpam-6496	1	38	integrable	integrable	ADJ
ejpam-6496	1	39	functions	function	NOUN
ejpam-6496	1	40	via	via	ADP
ejpam-6496	1	41	δ	δ	PROPN
ejpam-6496	1	42	-	-	PUNCT
ejpam-6496	1	43	fine	fine	ADJ
ejpam-6496	1	44	tagged	tag	VERB
ejpam-6496	1	45	partitions	partition	NOUN
ejpam-6496	1	46	sirinya	sirinya	PROPN
ejpam-6496	1	47	prongjit1	prongjit1	PROPN
ejpam-6496	1	48	,	,	PUNCT
ejpam-6496	1	49	patarawadee	patarawadee	PROPN
ejpam-6496	1	50	prasertsang1,∗	prasertsang1,∗	NOUN
ejpam-6496	1	51	1	1	NUM
ejpam-6496	1	52	department	department	NOUN
ejpam-6496	1	53	of	of	ADP
ejpam-6496	1	54	general	general	ADJ
ejpam-6496	1	55	science	science	NOUN
ejpam-6496	1	56	,	,	PUNCT
ejpam-6496	1	57	faculty	faculty	NOUN
ejpam-6496	1	58	of	of	ADP
ejpam-6496	1	59	science	science	NOUN
ejpam-6496	1	60	and	and	CCONJ
ejpam-6496	1	61	engineering	engineering	NOUN
ejpam-6496	1	62	,	,	PUNCT
ejpam-6496	1	63	kasetsart	kasetsart	PROPN
ejpam-6496	1	64	university	university	PROPN
ejpam-6496	1	65	,	,	PUNCT
ejpam-6496	1	66	chalermphrakiat	chalermphrakiat	PROPN
ejpam-6496	1	67	sakon	sakon	PROPN
ejpam-6496	1	68	nakhon	nakhon	PROPN
ejpam-6496	1	69	province	province	PROPN
ejpam-6496	1	70	campus	campus	PROPN
ejpam-6496	1	71	,	,	PUNCT
ejpam-6496	1	72	sakon	sakon	PROPN
ejpam-6496	1	73	nakhon	nakhon	PROPN
ejpam-6496	1	74	,	,	PUNCT
ejpam-6496	1	75	thailand	thailand	PROPN
ejpam-6496	1	76	abstract	abstract	PROPN
ejpam-6496	1	77	.	.	PUNCT
ejpam-6496	2	1	the	the	DET
ejpam-6496	2	2	concept	concept	NOUN
ejpam-6496	2	3	of	of	ADP
ejpam-6496	2	4	δ	δ	PROPN
ejpam-6496	2	5	-	-	PUNCT
ejpam-6496	2	6	fine	fine	ADJ
ejpam-6496	2	7	tagged	tag	VERB
ejpam-6496	2	8	partitions	partition	NOUN
ejpam-6496	2	9	is	be	AUX
ejpam-6496	2	10	used	use	VERB
ejpam-6496	2	11	to	to	PART
ejpam-6496	2	12	simplify	simplify	VERB
ejpam-6496	2	13	and	and	CCONJ
ejpam-6496	2	14	unify	unify	VERB
ejpam-6496	2	15	proofs	proof	NOUN
ejpam-6496	2	16	of	of	ADP
ejpam-6496	2	17	various	various	ADJ
ejpam-6496	2	18	theorems	theorem	NOUN
ejpam-6496	2	19	in	in	ADP
ejpam-6496	2	20	elementary	elementary	ADJ
ejpam-6496	2	21	real	real	ADJ
ejpam-6496	2	22	analysis	analysis	NOUN
ejpam-6496	2	23	.	.	PUNCT
ejpam-6496	3	1	this	this	DET
ejpam-6496	3	2	paper	paper	NOUN
ejpam-6496	3	3	is	be	AUX
ejpam-6496	3	4	a	a	DET
ejpam-6496	3	5	continuation	continuation	NOUN
ejpam-6496	3	6	of	of	ADP
ejpam-6496	3	7	the	the	DET
ejpam-6496	3	8	author	author	NOUN
ejpam-6496	3	9	’s	’s	PART
ejpam-6496	3	10	project	project	NOUN
ejpam-6496	3	11	to	to	PART
ejpam-6496	3	12	present	present	VERB
ejpam-6496	3	13	some	some	DET
ejpam-6496	3	14	properties	property	NOUN
ejpam-6496	3	15	of	of	ADP
ejpam-6496	3	16	differentiable	differentiable	ADJ
ejpam-6496	3	17	functions	function	NOUN
ejpam-6496	3	18	via	via	ADP
ejpam-6496	3	19	δ	δ	PROPN
ejpam-6496	3	20	-	-	PUNCT
ejpam-6496	3	21	fine	fine	ADJ
ejpam-6496	3	22	tagged	tag	VERB
ejpam-6496	3	23	partitions	partition	NOUN
ejpam-6496	3	24	.	.	PUNCT
ejpam-6496	4	1	some	some	DET
ejpam-6496	4	2	basic	basic	ADJ
ejpam-6496	4	3	theorems	theorem	NOUN
ejpam-6496	4	4	concerning	concern	VERB
ejpam-6496	4	5	riemann	riemann	PROPN
ejpam-6496	4	6	integrable	integrable	ADJ
ejpam-6496	4	7	functions	function	NOUN
ejpam-6496	4	8	are	be	AUX
ejpam-6496	4	9	also	also	ADV
ejpam-6496	4	10	reproved	reprove	VERB
ejpam-6496	4	11	by	by	ADP
ejpam-6496	4	12	this	this	DET
ejpam-6496	4	13	concept	concept	NOUN
ejpam-6496	4	14	.	.	PUNCT
ejpam-6496	5	1	2020	2020	NUM
ejpam-6496	5	2	mathematics	mathematic	NOUN
ejpam-6496	5	3	subject	subject	NOUN
ejpam-6496	5	4	classifications	classification	NOUN
ejpam-6496	5	5	:	:	PUNCT
ejpam-6496	5	6	26a24	26a24	NUM
ejpam-6496	5	7	,	,	PUNCT
ejpam-6496	5	8	26a42	26a42	NUM
ejpam-6496	5	9	key	key	ADJ
ejpam-6496	5	10	words	word	NOUN
ejpam-6496	5	11	and	and	CCONJ
ejpam-6496	5	12	phrases	phrase	NOUN
ejpam-6496	5	13	:	:	PUNCT
ejpam-6496	5	14	δ	δ	NOUN
ejpam-6496	5	15	-	-	PUNCT
ejpam-6496	5	16	fine	fine	ADJ
ejpam-6496	5	17	tagged	tag	VERB
ejpam-6496	5	18	partitions	partition	NOUN
ejpam-6496	5	19	,	,	PUNCT
ejpam-6496	5	20	differentiable	differentiable	ADJ
ejpam-6496	5	21	function	function	NOUN
ejpam-6496	5	22	,	,	PUNCT
ejpam-6496	5	23	full	full	ADJ
ejpam-6496	5	24	covers	cover	NOUN
ejpam-6496	5	25	,	,	PUNCT
ejpam-6496	5	26	riemann	riemann	PROPN
ejpam-6496	5	27	integrable	integrable	ADJ
ejpam-6496	5	28	function	function	NOUN
ejpam-6496	5	29	1	1	NUM
ejpam-6496	5	30	.	.	PUNCT
ejpam-6496	5	31	introduction	introduction	NOUN
ejpam-6496	5	32	the	the	DET
ejpam-6496	5	33	concept	concept	NOUN
ejpam-6496	5	34	of	of	ADP
ejpam-6496	5	35	full	full	ADJ
ejpam-6496	5	36	covers	cover	NOUN
ejpam-6496	5	37	was	be	AUX
ejpam-6496	5	38	introduced	introduce	VERB
ejpam-6496	5	39	to	to	PART
ejpam-6496	5	40	simplify	simplify	VERB
ejpam-6496	5	41	and	and	CCONJ
ejpam-6496	5	42	unify	unify	VERB
ejpam-6496	5	43	the	the	DET
ejpam-6496	5	44	proofs	proof	NOUN
ejpam-6496	5	45	of	of	ADP
ejpam-6496	5	46	various	various	ADJ
ejpam-6496	5	47	theorems	theorem	NOUN
ejpam-6496	5	48	in	in	ADP
ejpam-6496	5	49	real	real	ADJ
ejpam-6496	5	50	analysis	analysis	NOUN
ejpam-6496	5	51	by	by	ADP
ejpam-6496	5	52	botsko	botsko	ADJ
ejpam-6496	6	1	[	[	X
ejpam-6496	6	2	1	1	NUM
ejpam-6496	6	3	]	]	PUNCT
ejpam-6496	6	4	in	in	ADP
ejpam-6496	6	5	1987	1987	NUM
ejpam-6496	6	6	.	.	PUNCT
ejpam-6496	7	1	depending	depend	VERB
ejpam-6496	7	2	upon	upon	SCONJ
ejpam-6496	7	3	thomson	thomson	PROPN
ejpam-6496	7	4	’s	’s	PART
ejpam-6496	7	5	lemma	lemma	PROPN
ejpam-6496	7	6	which	which	PRON
ejpam-6496	7	7	ensures	ensure	VERB
ejpam-6496	7	8	that	that	SCONJ
ejpam-6496	7	9	every	every	DET
ejpam-6496	7	10	full	full	ADJ
ejpam-6496	7	11	cover	cover	NOUN
ejpam-6496	7	12	c	c	NOUN
ejpam-6496	7	13	of	of	ADP
ejpam-6496	7	14	a	a	DET
ejpam-6496	7	15	closed	closed	ADJ
ejpam-6496	7	16	interval	interval	NOUN
ejpam-6496	7	17	[	[	PUNCT
ejpam-6496	7	18	a	a	PROPN
ejpam-6496	7	19	,	,	PUNCT
ejpam-6496	7	20	b	b	NOUN
ejpam-6496	7	21	]	]	PUNCT
ejpam-6496	7	22	must	must	AUX
ejpam-6496	7	23	contain	contain	VERB
ejpam-6496	7	24	a	a	DET
ejpam-6496	7	25	partition	partition	NOUN
ejpam-6496	7	26	of	of	ADP
ejpam-6496	7	27	[	[	PUNCT
ejpam-6496	7	28	a	a	PROPN
ejpam-6496	7	29	,	,	PUNCT
ejpam-6496	7	30	b	b	NOUN
ejpam-6496	7	31	]	]	X
ejpam-6496	7	32	,	,	PUNCT
ejpam-6496	7	33	this	this	DET
ejpam-6496	7	34	concept	concept	NOUN
ejpam-6496	7	35	bring	bring	VERB
ejpam-6496	7	36	harmony	harmony	NOUN
ejpam-6496	7	37	into	into	ADP
ejpam-6496	7	38	the	the	DET
ejpam-6496	7	39	proofs	proof	NOUN
ejpam-6496	7	40	of	of	ADP
ejpam-6496	7	41	diverse	diverse	ADJ
ejpam-6496	7	42	theorems	theorem	NOUN
ejpam-6496	7	43	.	.	PUNCT
ejpam-6496	8	1	later	later	ADV
ejpam-6496	8	2	in	in	ADP
ejpam-6496	8	3	1989	1989	NUM
ejpam-6496	8	4	,	,	PUNCT
ejpam-6496	8	5	botsko	botsko	ADJ
ejpam-6496	8	6	[	[	X
ejpam-6496	8	7	2	2	NUM
ejpam-6496	8	8	]	]	PUNCT
ejpam-6496	8	9	published	publish	VERB
ejpam-6496	8	10	the	the	DET
ejpam-6496	8	11	second	second	ADJ
ejpam-6496	8	12	paper	paper	NOUN
ejpam-6496	8	13	on	on	ADP
ejpam-6496	8	14	this	this	DET
ejpam-6496	8	15	topic	topic	NOUN
ejpam-6496	8	16	concerning	concern	VERB
ejpam-6496	8	17	some	some	DET
ejpam-6496	8	18	harder	hard	ADJ
ejpam-6496	8	19	theorems	theorem	NOUN
ejpam-6496	8	20	than	than	ADP
ejpam-6496	8	21	the	the	DET
ejpam-6496	8	22	previous	previous	ADJ
ejpam-6496	8	23	study	study	NOUN
ejpam-6496	8	24	.	.	PUNCT
ejpam-6496	9	1	moreover	moreover	ADV
ejpam-6496	9	2	,	,	PUNCT
ejpam-6496	9	3	klaimon	klaimon	ADJ
ejpam-6496	9	4	[	[	X
ejpam-6496	9	5	3	3	NUM
ejpam-6496	9	6	]	]	PUNCT
ejpam-6496	9	7	as	as	ADV
ejpam-6496	9	8	well	well	ADV
ejpam-6496	9	9	as	as	ADP
ejpam-6496	9	10	zangara	zangara	NOUN
ejpam-6496	9	11	and	and	CCONJ
ejpam-6496	9	12	marafino	marafino	NOUN
ejpam-6496	9	13	[	[	X
ejpam-6496	9	14	4	4	X
ejpam-6496	9	15	]	]	PUNCT
ejpam-6496	9	16	also	also	ADV
ejpam-6496	9	17	use	use	VERB
ejpam-6496	9	18	this	this	DET
ejpam-6496	9	19	concept	concept	NOUN
ejpam-6496	9	20	to	to	PART
ejpam-6496	9	21	prove	prove	VERB
ejpam-6496	9	22	in	in	ADP
ejpam-6496	9	23	a	a	DET
ejpam-6496	9	24	unified	unified	ADJ
ejpam-6496	9	25	style	style	NOUN
ejpam-6496	9	26	for	for	ADP
ejpam-6496	9	27	many	many	ADJ
ejpam-6496	9	28	other	other	ADJ
ejpam-6496	9	29	theorems	theorem	NOUN
ejpam-6496	9	30	in	in	ADP
ejpam-6496	9	31	real	real	ADJ
ejpam-6496	9	32	analysis	analysis	NOUN
ejpam-6496	9	33	.	.	PUNCT
ejpam-6496	10	1	by	by	ADP
ejpam-6496	10	2	analyzing	analyze	VERB
ejpam-6496	10	3	proofs	proof	NOUN
ejpam-6496	10	4	in	in	ADP
ejpam-6496	10	5	botsko	botsko	PROPN
ejpam-6496	10	6	’s	’s	PART
ejpam-6496	10	7	works	work	NOUN
ejpam-6496	10	8	,	,	PUNCT
ejpam-6496	10	9	partitions	partition	NOUN
ejpam-6496	10	10	extracted	extract	VERB
ejpam-6496	10	11	from	from	ADP
ejpam-6496	10	12	full	full	ADJ
ejpam-6496	10	13	covers	cover	NOUN
ejpam-6496	10	14	should	should	AUX
ejpam-6496	10	15	be	be	AUX
ejpam-6496	10	16	the	the	DET
ejpam-6496	10	17	keys	key	NOUN
ejpam-6496	10	18	in	in	ADP
ejpam-6496	10	19	establishing	establish	VERB
ejpam-6496	10	20	the	the	DET
ejpam-6496	10	21	unified	unified	ADJ
ejpam-6496	10	22	treatments	treatment	NOUN
ejpam-6496	10	23	of	of	ADP
ejpam-6496	10	24	various	various	ADJ
ejpam-6496	10	25	theorems	theorem	NOUN
ejpam-6496	10	26	.	.	PUNCT
ejpam-6496	11	1	in	in	ADP
ejpam-6496	11	2	1998	1998	NUM
ejpam-6496	11	3	,	,	PUNCT
ejpam-6496	11	4	gordon	gordon	PROPN
ejpam-6496	11	5	[	[	X
ejpam-6496	11	6	5	5	NUM
ejpam-6496	11	7	]	]	PUNCT
ejpam-6496	11	8	showed	show	VERB
ejpam-6496	11	9	alternate	alternate	ADJ
ejpam-6496	11	10	approach	approach	NOUN
ejpam-6496	11	11	to	to	ADP
ejpam-6496	11	12	several	several	ADJ
ejpam-6496	11	13	well	well	ADV
ejpam-6496	11	14	known	know	VERB
ejpam-6496	11	15	theorems	theorem	NOUN
ejpam-6496	11	16	in	in	ADP
ejpam-6496	11	17	elementary	elementary	ADJ
ejpam-6496	11	18	real	real	ADJ
ejpam-6496	11	19	analysis	analysis	NOUN
ejpam-6496	11	20	by	by	ADP
ejpam-6496	11	21	using	use	VERB
ejpam-6496	11	22	δ	δ	PROPN
ejpam-6496	11	23	fine	fine	ADJ
ejpam-6496	11	24	tagged	tag	VERB
ejpam-6496	11	25	partitions	partition	NOUN
ejpam-6496	11	26	instead	instead	ADV
ejpam-6496	11	27	of	of	ADP
ejpam-6496	11	28	using	use	VERB
ejpam-6496	11	29	the	the	DET
ejpam-6496	11	30	full	full	ADJ
ejpam-6496	11	31	covering	covering	NOUN
ejpam-6496	11	32	.	.	PUNCT
ejpam-6496	12	1	prongjit	prongjit	ADV
ejpam-6496	12	2	and	and	CCONJ
ejpam-6496	12	3	sodsiri	sodsiri	NOUN
ejpam-6496	13	1	[	[	X
ejpam-6496	13	2	6	6	NUM
ejpam-6496	13	3	,	,	PUNCT
ejpam-6496	13	4	7	7	NUM
ejpam-6496	13	5	]	]	PUNCT
ejpam-6496	13	6	published	publish	VERB
ejpam-6496	13	7	in	in	ADP
ejpam-6496	13	8	2014	2014	NUM
ejpam-6496	13	9	presented	present	VERB
ejpam-6496	13	10	how	how	SCONJ
ejpam-6496	13	11	δ	δ	NOUN
ejpam-6496	13	12	-	-	PUNCT
ejpam-6496	13	13	fine	fine	ADJ
ejpam-6496	13	14	tagged	tag	VERB
ejpam-6496	13	15	partitions	partition	NOUN
ejpam-6496	13	16	could	could	AUX
ejpam-6496	13	17	be	be	AUX
ejpam-6496	13	18	used	use	VERB
ejpam-6496	13	19	to	to	PART
ejpam-6496	13	20	replace	replace	VERB
ejpam-6496	13	21	partitions	partition	NOUN
ejpam-6496	13	22	taking	take	VERB
ejpam-6496	13	23	from	from	ADP
ejpam-6496	13	24	full	full	ADJ
ejpam-6496	13	25	covers	cover	NOUN
ejpam-6496	13	26	,	,	PUNCT
ejpam-6496	13	27	and	and	CCONJ
ejpam-6496	13	28	these	these	DET
ejpam-6496	13	29	papers	paper	NOUN
ejpam-6496	13	30	reproved	reprove	VERB
ejpam-6496	13	31	almost	almost	ADV
ejpam-6496	13	32	theorems	theorem	NOUN
ejpam-6496	13	33	discussed	discuss	VERB
ejpam-6496	13	34	in	in	ADP
ejpam-6496	13	35	[	[	X
ejpam-6496	13	36	2–4	2–4	NUM
ejpam-6496	13	37	]	]	PUNCT
ejpam-6496	13	38	.	.	PUNCT
ejpam-6496	14	1	furthermore	furthermore	ADV
ejpam-6496	14	2	,	,	PUNCT
ejpam-6496	14	3	zheng	zheng	PROPN
ejpam-6496	14	4	and	and	CCONJ
ejpam-6496	14	5	shi	shi	PROPN
ejpam-6496	15	1	[	[	X
ejpam-6496	15	2	8	8	NUM
ejpam-6496	15	3	]	]	PUNCT
ejpam-6496	15	4	have	have	AUX
ejpam-6496	15	5	provided	provide	VERB
ejpam-6496	15	6	notion	notion	NOUN
ejpam-6496	15	7	of	of	ADP
ejpam-6496	15	8	dyadic	dyadic	ADJ
ejpam-6496	15	9	partitions	partition	NOUN
ejpam-6496	15	10	in	in	ADP
ejpam-6496	15	11	showing	show	VERB
ejpam-6496	15	12	alternative	alternative	ADJ
ejpam-6496	15	13	unified	unified	ADJ
ejpam-6496	15	14	proofs	proof	NOUN
ejpam-6496	15	15	of	of	ADP
ejpam-6496	15	16	theorems	theorem	NOUN
ejpam-6496	15	17	in	in	ADP
ejpam-6496	15	18	real	real	ADJ
ejpam-6496	15	19	analysis	analysis	NOUN
ejpam-6496	15	20	as	as	ADP
ejpam-6496	15	21	distinct	distinct	ADJ
ejpam-6496	15	22	from	from	ADP
ejpam-6496	15	23	the	the	DET
ejpam-6496	15	24	δ	δ	PROPN
ejpam-6496	15	25	-	-	PUNCT
ejpam-6496	15	26	fine	fine	ADJ
ejpam-6496	15	27	tagged	tag	VERB
ejpam-6496	15	28	partition	partition	NOUN
ejpam-6496	15	29	concept	concept	NOUN
ejpam-6496	15	30	.	.	PUNCT
ejpam-6496	16	1	∗corresponding	∗corresponde	VERB
ejpam-6496	16	2	author	author	NOUN
ejpam-6496	16	3	.	.	PUNCT
ejpam-6496	17	1	doi	doi	NOUN
ejpam-6496	17	2	:	:	PUNCT
ejpam-6496	17	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6496	https://doi.org/10.29020/nybg.ejpam.v18i3.6496	PROPN
ejpam-6496	17	4	email	email	NOUN
ejpam-6496	17	5	addresses	address	NOUN
ejpam-6496	17	6	:	:	PUNCT
ejpam-6496	17	7	sirinya.pr@ku.th	sirinya.pr@ku.th	PRON
ejpam-6496	17	8	(	(	PUNCT
ejpam-6496	17	9	s.	s.	PROPN
ejpam-6496	17	10	prongjit	prongjit	ADV
ejpam-6496	17	11	)	)	PUNCT
ejpam-6496	17	12	,	,	PUNCT
ejpam-6496	17	13	patarawadee.s@ku.th	patarawadee.s@ku.th	PROPN
ejpam-6496	17	14	(	(	PUNCT
ejpam-6496	17	15	p.	p.	PROPN
ejpam-6496	17	16	prasertsang	prasertsang	PROPN
ejpam-6496	17	17	)	)	PUNCT
ejpam-6496	17	18	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6496	18	1	1	1	NUM
ejpam-6496	18	2	copyright	copyright	NOUN
ejpam-6496	18	3	:	:	PUNCT
ejpam-6496	18	4	©	©	PROPN
ejpam-6496	18	5	2025	2025	NUM
ejpam-6496	18	6	the	the	DET
ejpam-6496	18	7	author(s	author(s	NOUN
ejpam-6496	18	8	)	)	PUNCT
ejpam-6496	18	9	.	.	PUNCT
ejpam-6496	19	1	(	(	PUNCT
ejpam-6496	19	2	cc	cc	NOUN
ejpam-6496	19	3	by	by	ADP
ejpam-6496	19	4	-	-	PUNCT
ejpam-6496	19	5	nc	nc	PROPN
ejpam-6496	19	6	4.0	4.0	NUM
ejpam-6496	19	7	)	)	PUNCT
ejpam-6496	19	8	s.	s.	PROPN
ejpam-6496	20	1	prongjit	prongjit	ADV
ejpam-6496	20	2	,	,	PUNCT
ejpam-6496	20	3	p.	p.	PROPN
ejpam-6496	20	4	prasertsang	prasertsang	PROPN
ejpam-6496	20	5	/	/	SYM
ejpam-6496	20	6	eur	eur	PROPN
ejpam-6496	20	7	.	.	PUNCT
ejpam-6496	21	1	j.	j.	PROPN
ejpam-6496	21	2	pure	pure	PROPN
ejpam-6496	21	3	appl	appl	PROPN
ejpam-6496	21	4	.	.	PROPN
ejpam-6496	21	5	math	math	PROPN
ejpam-6496	21	6	,	,	PUNCT
ejpam-6496	21	7	18	18	NUM
ejpam-6496	21	8	(	(	PUNCT
ejpam-6496	21	9	3	3	NUM
ejpam-6496	21	10	)	)	PUNCT
ejpam-6496	21	11	(	(	PUNCT
ejpam-6496	21	12	2025	2025	NUM
ejpam-6496	21	13	)	)	PUNCT
ejpam-6496	21	14	,	,	PUNCT
ejpam-6496	21	15	6496	6496	NUM
ejpam-6496	21	16	2	2	NUM
ejpam-6496	21	17	of	of	ADP
ejpam-6496	21	18	10	10	NUM
ejpam-6496	21	19	this	this	DET
ejpam-6496	21	20	research	research	NOUN
ejpam-6496	21	21	continue	continue	VERB
ejpam-6496	21	22	to	to	ADP
ejpam-6496	21	23	the	the	DET
ejpam-6496	21	24	results	result	NOUN
ejpam-6496	21	25	from	from	ADP
ejpam-6496	21	26	[	[	X
ejpam-6496	21	27	6	6	NUM
ejpam-6496	21	28	]	]	PUNCT
ejpam-6496	21	29	,	,	PUNCT
ejpam-6496	21	30	which	which	PRON
ejpam-6496	21	31	shines	shine	VERB
ejpam-6496	21	32	us	we	PRON
ejpam-6496	21	33	to	to	PART
ejpam-6496	21	34	draw	draw	VERB
ejpam-6496	21	35	some	some	DET
ejpam-6496	21	36	properties	property	NOUN
ejpam-6496	21	37	of	of	ADP
ejpam-6496	21	38	differentiable	differentiable	ADJ
ejpam-6496	21	39	functions	function	NOUN
ejpam-6496	21	40	to	to	PART
ejpam-6496	21	41	study	study	VERB
ejpam-6496	21	42	via	via	ADP
ejpam-6496	21	43	δ	δ	PROPN
ejpam-6496	21	44	-	-	PUNCT
ejpam-6496	21	45	fine	fine	ADJ
ejpam-6496	21	46	tagged	tag	VERB
ejpam-6496	21	47	partitions	partition	NOUN
ejpam-6496	21	48	.	.	PUNCT
ejpam-6496	22	1	some	some	PRON
ejpam-6496	22	2	of	of	ADP
ejpam-6496	22	3	these	these	DET
ejpam-6496	22	4	properties	property	NOUN
ejpam-6496	22	5	too	too	ADV
ejpam-6496	22	6	can	can	AUX
ejpam-6496	22	7	be	be	AUX
ejpam-6496	22	8	applied	apply	VERB
ejpam-6496	22	9	to	to	PART
ejpam-6496	22	10	reveal	reveal	VERB
ejpam-6496	22	11	another	another	DET
ejpam-6496	22	12	view	view	NOUN
ejpam-6496	22	13	of	of	ADP
ejpam-6496	22	14	mean	mean	ADJ
ejpam-6496	22	15	value	value	NOUN
ejpam-6496	22	16	theorem	theorem	NOUN
ejpam-6496	22	17	and	and	CCONJ
ejpam-6496	22	18	cauchy	cauchy	PROPN
ejpam-6496	22	19	mean	mean	NOUN
ejpam-6496	22	20	value	value	NOUN
ejpam-6496	22	21	theorem	theorem	VERB
ejpam-6496	22	22	.	.	PUNCT
ejpam-6496	23	1	besides	besides	ADV
ejpam-6496	23	2	,	,	PUNCT
ejpam-6496	23	3	we	we	PRON
ejpam-6496	23	4	also	also	ADV
ejpam-6496	23	5	study	study	VERB
ejpam-6496	23	6	some	some	DET
ejpam-6496	23	7	basic	basic	ADJ
ejpam-6496	23	8	properties	property	NOUN
ejpam-6496	23	9	of	of	ADP
ejpam-6496	23	10	riemann	riemann	PROPN
ejpam-6496	23	11	integrable	integrable	ADJ
ejpam-6496	23	12	functions	function	NOUN
ejpam-6496	23	13	via	via	ADP
ejpam-6496	23	14	δ	δ	PROPN
ejpam-6496	23	15	-	-	PUNCT
ejpam-6496	23	16	fine	fine	ADJ
ejpam-6496	23	17	tagged	tag	VERB
ejpam-6496	23	18	partitions	partition	NOUN
ejpam-6496	23	19	that	that	PRON
ejpam-6496	23	20	deal	deal	VERB
ejpam-6496	23	21	with	with	ADP
ejpam-6496	23	22	the	the	DET
ejpam-6496	23	23	cauchy	cauchy	ADJ
ejpam-6496	23	24	criterion	criterion	NOUN
ejpam-6496	23	25	for	for	ADP
ejpam-6496	23	26	riemann	riemann	PROPN
ejpam-6496	23	27	integrability	integrability	PROPN
ejpam-6496	23	28	version	version	NOUN
ejpam-6496	23	29	formulated	formulate	VERB
ejpam-6496	23	30	by	by	ADP
ejpam-6496	23	31	gordon	gordon	PROPN
ejpam-6496	23	32	(	(	PUNCT
ejpam-6496	23	33	see	see	VERB
ejpam-6496	23	34	[	[	X
ejpam-6496	23	35	5	5	NUM
ejpam-6496	23	36	]	]	NUM
ejpam-6496	23	37	)	)	PUNCT
ejpam-6496	23	38	.	.	PUNCT
ejpam-6496	24	1	2	2	X
ejpam-6496	24	2	.	.	X
ejpam-6496	24	3	preliminaries	preliminary	NOUN
ejpam-6496	24	4	throughout	throughout	ADP
ejpam-6496	24	5	this	this	DET
ejpam-6496	24	6	paper	paper	NOUN
ejpam-6496	24	7	,	,	PUNCT
ejpam-6496	24	8	we	we	PRON
ejpam-6496	24	9	assume	assume	VERB
ejpam-6496	24	10	that	that	SCONJ
ejpam-6496	24	11	a	a	DET
ejpam-6496	24	12	,	,	PUNCT
ejpam-6496	24	13	b	b	X
ejpam-6496	24	14	∈	∈	NOUN
ejpam-6496	24	15	r	r	NOUN
ejpam-6496	24	16	with	with	ADP
ejpam-6496	24	17	a	a	DET
ejpam-6496	24	18	<	<	X
ejpam-6496	24	19	b.	b.	NOUN
ejpam-6496	24	20	we	we	PRON
ejpam-6496	24	21	denote	denote	VERB
ejpam-6496	24	22	]	]	PUNCT
ejpam-6496	24	23	a	a	X
ejpam-6496	24	24	,	,	PUNCT
ejpam-6496	24	25	b	b	X
ejpam-6496	24	26	[	[	PUNCT
ejpam-6496	24	27	for	for	ADP
ejpam-6496	24	28	an	an	DET
ejpam-6496	24	29	open	open	ADJ
ejpam-6496	24	30	interval	interval	NOUN
ejpam-6496	24	31	,	,	PUNCT
ejpam-6496	24	32	while	while	SCONJ
ejpam-6496	24	33	[	[	PUNCT
ejpam-6496	24	34	a	a	X
ejpam-6496	24	35	,	,	PUNCT
ejpam-6496	24	36	b	b	NOUN
ejpam-6496	24	37	]	]	PUNCT
ejpam-6496	24	38	is	be	AUX
ejpam-6496	24	39	denoted	denote	VERB
ejpam-6496	24	40	for	for	ADP
ejpam-6496	24	41	a	a	DET
ejpam-6496	24	42	closed	closed	ADJ
ejpam-6496	24	43	interval	interval	NOUN
ejpam-6496	24	44	as	as	ADP
ejpam-6496	24	45	usual	usual	ADJ
ejpam-6496	24	46	.	.	PUNCT
ejpam-6496	25	1	2.1	2.1	NUM
ejpam-6496	25	2	.	.	PUNCT
ejpam-6496	26	1	δ	δ	NOUN
ejpam-6496	26	2	-	-	PUNCT
ejpam-6496	26	3	fine	fine	ADJ
ejpam-6496	26	4	tagged	tag	VERB
ejpam-6496	26	5	partitions	partition	NOUN
ejpam-6496	26	6	definition	definition	NOUN
ejpam-6496	26	7	1	1	NUM
ejpam-6496	26	8	.	.	PUNCT
ejpam-6496	27	1	[	[	X
ejpam-6496	27	2	9	9	NUM
ejpam-6496	27	3	]	]	X
ejpam-6496	27	4	a	a	DET
ejpam-6496	27	5	partition	partition	NOUN
ejpam-6496	27	6	of	of	ADP
ejpam-6496	27	7	a	a	DET
ejpam-6496	27	8	closed	closed	ADJ
ejpam-6496	27	9	interval	interval	NOUN
ejpam-6496	27	10	[	[	PUNCT
ejpam-6496	27	11	a	a	PROPN
ejpam-6496	27	12	,	,	PUNCT
ejpam-6496	27	13	b	b	NOUN
ejpam-6496	27	14	]	]	PUNCT
ejpam-6496	27	15	is	be	AUX
ejpam-6496	27	16	a	a	DET
ejpam-6496	27	17	finite	finite	ADJ
ejpam-6496	27	18	collection	collection	NOUN
ejpam-6496	27	19	of	of	ADP
ejpam-6496	27	20	closed	closed	ADJ
ejpam-6496	27	21	intervals	interval	NOUN
ejpam-6496	27	22	{	{	PUNCT
ejpam-6496	28	1	[	[	X
ejpam-6496	28	2	xi−1	xi−1	PROPN
ejpam-6496	28	3	,	,	PUNCT
ejpam-6496	28	4	xi	xi	X
ejpam-6496	28	5	]	]	PUNCT
ejpam-6496	28	6	:	:	PUNCT
ejpam-6496	28	7	i	i	NOUN
ejpam-6496	28	8	=	=	NOUN
ejpam-6496	28	9	1	1	NUM
ejpam-6496	28	10	,	,	PUNCT
ejpam-6496	28	11	.	.	PUNCT
ejpam-6496	28	12	.	.	PUNCT
ejpam-6496	29	1	.	.	PUNCT
ejpam-6496	30	1	,	,	PUNCT
ejpam-6496	31	1	n	n	CCONJ
ejpam-6496	31	2	}	}	PUNCT
ejpam-6496	31	3	,	,	PUNCT
ejpam-6496	32	1	where	where	SCONJ
ejpam-6496	32	2	a	a	PRON
ejpam-6496	32	3	=	=	X
ejpam-6496	32	4	x0	x0	PROPN
ejpam-6496	32	5	<	<	X
ejpam-6496	32	6	x1	x1	X
ejpam-6496	32	7	<	<	X
ejpam-6496	32	8	·	·	PUNCT
ejpam-6496	32	9	·	·	PUNCT
ejpam-6496	32	10	·	·	PUNCT
ejpam-6496	32	11	<	<	X
ejpam-6496	32	12	xn	xn	PUNCT
ejpam-6496	32	13	=	=	SYM
ejpam-6496	32	14	b.	b.	PROPN
ejpam-6496	32	15	definition	definition	NOUN
ejpam-6496	32	16	2	2	NUM
ejpam-6496	32	17	.	.	PUNCT
ejpam-6496	33	1	[	[	X
ejpam-6496	33	2	9	9	NUM
ejpam-6496	33	3	]	]	PUNCT
ejpam-6496	33	4	a	a	DET
ejpam-6496	33	5	tagged	tag	VERB
ejpam-6496	33	6	partition	partition	NOUN
ejpam-6496	33	7	p	p	NOUN
ejpam-6496	33	8	of	of	ADP
ejpam-6496	33	9	[	[	PUNCT
ejpam-6496	33	10	a	a	PROPN
ejpam-6496	33	11	,	,	PUNCT
ejpam-6496	33	12	b	b	NOUN
ejpam-6496	33	13	]	]	PUNCT
ejpam-6496	33	14	is	be	AUX
ejpam-6496	33	15	a	a	DET
ejpam-6496	33	16	finite	finite	ADJ
ejpam-6496	33	17	collection	collection	NOUN
ejpam-6496	33	18	of	of	ADP
ejpam-6496	33	19	order	order	NOUN
ejpam-6496	33	20	pairs	pair	VERB
ejpam-6496	33	21	p	p	NOUN
ejpam-6496	33	22	=	=	X
ejpam-6496	33	23	{	{	PUNCT
ejpam-6496	33	24	(	(	PUNCT
ejpam-6496	33	25	ti	ti	NOUN
ejpam-6496	33	26	,	,	PUNCT
ejpam-6496	33	27	[	[	X
ejpam-6496	33	28	xi−1	xi−1	PROPN
ejpam-6496	33	29	,	,	PUNCT
ejpam-6496	33	30	xi	xi	X
ejpam-6496	33	31	]	]	PUNCT
ejpam-6496	33	32	)	)	PUNCT
ejpam-6496	33	33	:	:	PUNCT
ejpam-6496	34	1	i	i	NOUN
ejpam-6496	34	2	=	=	NOUN
ejpam-6496	34	3	1	1	NUM
ejpam-6496	34	4	,	,	PUNCT
ejpam-6496	34	5	.	.	PUNCT
ejpam-6496	34	6	.	.	PUNCT
ejpam-6496	34	7	.	.	PUNCT
ejpam-6496	34	8	,	,	PUNCT
ejpam-6496	34	9	n	n	CCONJ
ejpam-6496	34	10	}	}	PUNCT
ejpam-6496	34	11	,	,	PUNCT
ejpam-6496	34	12	where	where	SCONJ
ejpam-6496	34	13	a	a	PRON
ejpam-6496	34	14	=	=	X
ejpam-6496	34	15	x0	x0	PROPN
ejpam-6496	34	16	<	<	X
ejpam-6496	34	17	x1	x1	X
ejpam-6496	34	18	<	<	X
ejpam-6496	34	19	·	·	PUNCT
ejpam-6496	34	20	·	·	PUNCT
ejpam-6496	34	21	·	·	PUNCT
ejpam-6496	35	1	<	<	X
ejpam-6496	35	2	xn	xn	PUNCT
ejpam-6496	35	3	=	=	SYM
ejpam-6496	35	4	b	b	PROPN
ejpam-6496	35	5	and	and	CCONJ
ejpam-6496	35	6	each	each	DET
ejpam-6496	35	7	ti	ti	PROPN
ejpam-6496	35	8	∈	∈	PROPN
ejpam-6496	35	9	[	[	X
ejpam-6496	35	10	xi−1	xi−1	PROPN
ejpam-6496	35	11	,	,	PUNCT
ejpam-6496	35	12	xi	xi	X
ejpam-6496	35	13	]	]	PUNCT
ejpam-6496	35	14	.	.	PUNCT
ejpam-6496	36	1	definition	definition	NOUN
ejpam-6496	36	2	3	3	NUM
ejpam-6496	36	3	.	.	PUNCT
ejpam-6496	37	1	[	[	X
ejpam-6496	37	2	9	9	NUM
ejpam-6496	37	3	]	]	PUNCT
ejpam-6496	37	4	a	a	DET
ejpam-6496	37	5	gauge	gauge	NOUN
ejpam-6496	37	6	on	on	ADP
ejpam-6496	37	7	[	[	PUNCT
ejpam-6496	37	8	a	a	PROPN
ejpam-6496	37	9	,	,	PUNCT
ejpam-6496	37	10	b	b	NOUN
ejpam-6496	37	11	]	]	PUNCT
ejpam-6496	37	12	is	be	AUX
ejpam-6496	37	13	a	a	DET
ejpam-6496	37	14	strictly	strictly	ADV
ejpam-6496	37	15	positive	positive	ADJ
ejpam-6496	37	16	real	real	ADJ
ejpam-6496	37	17	valued	value	VERB
ejpam-6496	37	18	function	function	NOUN
ejpam-6496	37	19	defined	define	VERB
ejpam-6496	37	20	on	on	ADP
ejpam-6496	37	21	[	[	PUNCT
ejpam-6496	37	22	a	a	PROPN
ejpam-6496	37	23	,	,	PUNCT
ejpam-6496	37	24	b	b	NOUN
ejpam-6496	37	25	]	]	X
ejpam-6496	37	26	.	.	PUNCT
ejpam-6496	38	1	if	if	SCONJ
ejpam-6496	38	2	δ	δ	PROPN
ejpam-6496	38	3	is	be	AUX
ejpam-6496	38	4	a	a	DET
ejpam-6496	38	5	gauge	gauge	NOUN
ejpam-6496	38	6	on	on	ADP
ejpam-6496	38	7	[	[	PUNCT
ejpam-6496	38	8	a	a	PROPN
ejpam-6496	38	9	,	,	PUNCT
ejpam-6496	38	10	b	b	NOUN
ejpam-6496	38	11	]	]	X
ejpam-6496	38	12	,	,	PUNCT
ejpam-6496	38	13	then	then	ADV
ejpam-6496	38	14	a	a	DET
ejpam-6496	38	15	tagged	tag	VERB
ejpam-6496	38	16	partition	partition	NOUN
ejpam-6496	38	17	p	p	NOUN
ejpam-6496	38	18	=	=	X
ejpam-6496	38	19	{	{	PUNCT
ejpam-6496	38	20	(	(	PUNCT
ejpam-6496	38	21	ti	ti	NOUN
ejpam-6496	38	22	,	,	PUNCT
ejpam-6496	38	23	[	[	X
ejpam-6496	38	24	xi−1	xi−1	PROPN
ejpam-6496	38	25	,	,	PUNCT
ejpam-6496	38	26	xi	xi	X
ejpam-6496	38	27	]	]	PUNCT
ejpam-6496	38	28	)	)	PUNCT
ejpam-6496	38	29	:	:	PUNCT
ejpam-6496	38	30	i	i	NOUN
ejpam-6496	38	31	=	=	NOUN
ejpam-6496	38	32	1	1	NUM
ejpam-6496	38	33	,	,	PUNCT
ejpam-6496	38	34	.	.	PUNCT
ejpam-6496	38	35	.	.	PUNCT
ejpam-6496	38	36	.	.	PUNCT
ejpam-6496	39	1	,	,	PUNCT
ejpam-6496	39	2	n	n	CCONJ
ejpam-6496	39	3	}	}	PUNCT
ejpam-6496	39	4	is	be	AUX
ejpam-6496	39	5	said	say	VERB
ejpam-6496	39	6	to	to	PART
ejpam-6496	39	7	be	be	AUX
ejpam-6496	39	8	a	a	DET
ejpam-6496	39	9	δ	δ	NOUN
ejpam-6496	39	10	-	-	PUNCT
ejpam-6496	39	11	fine	fine	ADJ
ejpam-6496	39	12	tagged	tag	VERB
ejpam-6496	39	13	partition	partition	NOUN
ejpam-6496	39	14	of	of	ADP
ejpam-6496	39	15	[	[	PUNCT
ejpam-6496	39	16	a	a	PROPN
ejpam-6496	39	17	,	,	PUNCT
ejpam-6496	39	18	b	b	NOUN
ejpam-6496	39	19	]	]	X
ejpam-6496	40	1	if	if	SCONJ
ejpam-6496	40	2	and	and	CCONJ
ejpam-6496	40	3	only	only	ADV
ejpam-6496	40	4	if	if	SCONJ
ejpam-6496	40	5	[	[	X
ejpam-6496	40	6	xi−1	xi−1	PROPN
ejpam-6496	40	7	,	,	PUNCT
ejpam-6496	40	8	xi	xi	X
ejpam-6496	40	9	]	]	PUNCT
ejpam-6496	41	1	⊂	⊂	X
ejpam-6496	41	2	]	]	PUNCT
ejpam-6496	41	3	ti	ti	X
ejpam-6496	41	4	−	−	PROPN
ejpam-6496	41	5	δ(ti	δ(ti	PROPN
ejpam-6496	41	6	)	)	PUNCT
ejpam-6496	41	7	,	,	PUNCT
ejpam-6496	41	8	ti	ti	X
ejpam-6496	41	9	+	+	SYM
ejpam-6496	41	10	δ(ti	δ(ti	NOUN
ejpam-6496	41	11	)	)	PUNCT
ejpam-6496	41	12	[	[	PUNCT
ejpam-6496	41	13	for	for	ADP
ejpam-6496	41	14	all	all	DET
ejpam-6496	41	15	i	i	PRON
ejpam-6496	41	16	=	=	NOUN
ejpam-6496	41	17	1	1	NUM
ejpam-6496	41	18	,	,	PUNCT
ejpam-6496	41	19	.	.	PUNCT
ejpam-6496	41	20	.	.	PUNCT
ejpam-6496	41	21	.	.	PUNCT
ejpam-6496	42	1	,	,	PUNCT
ejpam-6496	42	2	n.	n.	NOUN
ejpam-6496	42	3	figure	figure	NOUN
ejpam-6496	42	4	1	1	NUM
ejpam-6496	42	5	shows	show	VERB
ejpam-6496	42	6	an	an	DET
ejpam-6496	42	7	illustration	illustration	NOUN
ejpam-6496	42	8	of	of	ADP
ejpam-6496	42	9	a	a	DET
ejpam-6496	42	10	portion	portion	NOUN
ejpam-6496	42	11	of	of	ADP
ejpam-6496	42	12	some	some	DET
ejpam-6496	42	13	δ	δ	NOUN
ejpam-6496	42	14	-	-	PUNCT
ejpam-6496	42	15	fine	fine	ADJ
ejpam-6496	42	16	tagged	tag	VERB
ejpam-6496	42	17	partition	partition	NOUN
ejpam-6496	42	18	.	.	PUNCT
ejpam-6496	43	1	x0	x0	PROPN
ejpam-6496	43	2	.	.	PUNCT
ejpam-6496	43	3	.	.	PUNCT
ejpam-6496	43	4	.	.	PUNCT
ejpam-6496	44	1	xi−2	xi−2	PROPN
ejpam-6496	44	2	xi−1	xi−1	PROPN
ejpam-6496	44	3	xi	xi	PROPN
ejpam-6496	44	4	xi+1	xi+1	PROPN
ejpam-6496	44	5	.	.	PUNCT
ejpam-6496	44	6	.	.	PUNCT
ejpam-6496	44	7	.	.	PUNCT
ejpam-6496	45	1	xnti−1	xnti−1	PROPN
ejpam-6496	45	2	ti−1	ti−1	PROPN
ejpam-6496	45	3	−	−	PROPN
ejpam-6496	45	4	δ(ti−1	δ(ti−1	NUM
ejpam-6496	45	5	)	)	PUNCT
ejpam-6496	45	6	ti−1	ti−1	NOUN
ejpam-6496	45	7	+	+	CCONJ
ejpam-6496	45	8	δ(ti−1	δ(ti−1	NOUN
ejpam-6496	45	9	)	)	PUNCT
ejpam-6496	45	10	ti	ti	NOUN
ejpam-6496	45	11	ti	ti	NOUN
ejpam-6496	45	12	−	−	PROPN
ejpam-6496	45	13	δ(ti	δ(ti	PROPN
ejpam-6496	45	14	)	)	PUNCT
ejpam-6496	45	15	ti	ti	NOUN
ejpam-6496	45	16	+	+	CCONJ
ejpam-6496	45	17	δ(ti	δ(ti	NOUN
ejpam-6496	45	18	)	)	PUNCT
ejpam-6496	45	19	ti+1	ti+1	ADV
ejpam-6496	45	20	ti+1	ti+1	ADV
ejpam-6496	45	21	−	−	NOUN
ejpam-6496	45	22	δ(ti+1	δ(ti+1	PROPN
ejpam-6496	45	23	)	)	PUNCT
ejpam-6496	45	24	ti+1	ti+1	NOUN
ejpam-6496	46	1	+	+	CCONJ
ejpam-6496	46	2	δ(ti+1	δ(ti+1	ADJ
ejpam-6496	46	3	)	)	PUNCT
ejpam-6496	46	4	figure	figure	NOUN
ejpam-6496	46	5	1	1	NUM
ejpam-6496	46	6	:	:	PUNCT
ejpam-6496	46	7	a	a	DET
ejpam-6496	46	8	portion	portion	NOUN
ejpam-6496	46	9	of	of	ADP
ejpam-6496	46	10	some	some	DET
ejpam-6496	46	11	δ	δ	NOUN
ejpam-6496	46	12	-	-	PUNCT
ejpam-6496	46	13	fine	fine	ADJ
ejpam-6496	46	14	tagged	tag	VERB
ejpam-6496	46	15	partition	partition	NOUN
ejpam-6496	46	16	of	of	ADP
ejpam-6496	46	17	[	[	X
ejpam-6496	46	18	a	a	X
ejpam-6496	46	19	,	,	PUNCT
ejpam-6496	46	20	b	b	NOUN
ejpam-6496	46	21	]	]	X
ejpam-6496	46	22	s.	s.	PROPN
ejpam-6496	46	23	prongjit	prongjit	PROPN
ejpam-6496	46	24	,	,	PUNCT
ejpam-6496	46	25	p.	p.	PROPN
ejpam-6496	46	26	prasertsang	prasertsang	PROPN
ejpam-6496	46	27	/	/	SYM
ejpam-6496	46	28	eur	eur	PROPN
ejpam-6496	46	29	.	.	PUNCT
ejpam-6496	47	1	j.	j.	PROPN
ejpam-6496	47	2	pure	pure	PROPN
ejpam-6496	47	3	appl	appl	PROPN
ejpam-6496	47	4	.	.	PROPN
ejpam-6496	47	5	math	math	PROPN
ejpam-6496	47	6	,	,	PUNCT
ejpam-6496	47	7	18	18	NUM
ejpam-6496	47	8	(	(	PUNCT
ejpam-6496	47	9	3	3	NUM
ejpam-6496	47	10	)	)	PUNCT
ejpam-6496	47	11	(	(	PUNCT
ejpam-6496	47	12	2025	2025	NUM
ejpam-6496	47	13	)	)	PUNCT
ejpam-6496	47	14	,	,	PUNCT
ejpam-6496	47	15	6496	6496	NUM
ejpam-6496	47	16	3	3	NUM
ejpam-6496	47	17	of	of	ADP
ejpam-6496	47	18	10	10	NUM
ejpam-6496	47	19	the	the	DET
ejpam-6496	47	20	following	follow	VERB
ejpam-6496	47	21	lemma	lemma	PROPN
ejpam-6496	47	22	is	be	AUX
ejpam-6496	47	23	the	the	DET
ejpam-6496	47	24	backbone	backbone	NOUN
ejpam-6496	47	25	of	of	ADP
ejpam-6496	47	26	proving	prove	VERB
ejpam-6496	47	27	several	several	ADJ
ejpam-6496	47	28	theorems	theorem	NOUN
ejpam-6496	47	29	in	in	ADP
ejpam-6496	47	30	a	a	DET
ejpam-6496	47	31	unified	unified	ADJ
ejpam-6496	47	32	style	style	NOUN
ejpam-6496	47	33	.	.	PUNCT
ejpam-6496	48	1	it	it	PRON
ejpam-6496	48	2	assures	assure	VERB
ejpam-6496	48	3	the	the	DET
ejpam-6496	48	4	existence	existence	NOUN
ejpam-6496	48	5	of	of	ADP
ejpam-6496	48	6	a	a	DET
ejpam-6496	48	7	δ	δ	NOUN
ejpam-6496	48	8	-	-	PUNCT
ejpam-6496	48	9	fine	fine	ADJ
ejpam-6496	48	10	tagged	tag	VERB
ejpam-6496	48	11	partition	partition	NOUN
ejpam-6496	48	12	on	on	ADP
ejpam-6496	48	13	[	[	PUNCT
ejpam-6496	48	14	a	a	PROPN
ejpam-6496	48	15	,	,	PUNCT
ejpam-6496	48	16	b	b	NOUN
ejpam-6496	48	17	]	]	PUNCT
ejpam-6496	48	18	for	for	ADP
ejpam-6496	48	19	every	every	DET
ejpam-6496	48	20	given	give	VERB
ejpam-6496	48	21	gauge	gauge	ADJ
ejpam-6496	48	22	δ	δ	PROPN
ejpam-6496	48	23	on	on	ADP
ejpam-6496	48	24	[	[	PUNCT
ejpam-6496	48	25	a	a	PROPN
ejpam-6496	48	26	,	,	PUNCT
ejpam-6496	48	27	b	b	NOUN
ejpam-6496	48	28	]	]	PUNCT
ejpam-6496	48	29	.	.	PUNCT
ejpam-6496	49	1	lemma	lemma	PROPN
ejpam-6496	49	2	1	1	NUM
ejpam-6496	49	3	(	(	PUNCT
ejpam-6496	49	4	cousin	cousin	PROPN
ejpam-6496	49	5	’s	’s	PART
ejpam-6496	49	6	lemma	lemma	PROPN
ejpam-6496	49	7	)	)	PUNCT
ejpam-6496	49	8	.	.	PUNCT
ejpam-6496	50	1	[	[	X
ejpam-6496	50	2	9	9	NUM
ejpam-6496	50	3	]	]	PUNCT
ejpam-6496	50	4	for	for	ADP
ejpam-6496	50	5	every	every	DET
ejpam-6496	50	6	gauge	gauge	NOUN
ejpam-6496	50	7	δ	δ	PROPN
ejpam-6496	50	8	on	on	ADP
ejpam-6496	50	9	[	[	PUNCT
ejpam-6496	50	10	a	a	PROPN
ejpam-6496	50	11	,	,	PUNCT
ejpam-6496	50	12	b	b	NOUN
ejpam-6496	50	13	]	]	X
ejpam-6496	50	14	,	,	PUNCT
ejpam-6496	50	15	there	there	PRON
ejpam-6496	50	16	exists	exist	VERB
ejpam-6496	50	17	a	a	DET
ejpam-6496	50	18	δ	δ	NOUN
ejpam-6496	50	19	-	-	PUNCT
ejpam-6496	50	20	fine	fine	ADJ
ejpam-6496	50	21	tagged	tag	VERB
ejpam-6496	50	22	partition	partition	NOUN
ejpam-6496	50	23	of	of	ADP
ejpam-6496	50	24	[	[	PUNCT
ejpam-6496	50	25	a	a	PROPN
ejpam-6496	50	26	,	,	PUNCT
ejpam-6496	50	27	b	b	NOUN
ejpam-6496	50	28	]	]	X
ejpam-6496	50	29	.	.	PUNCT
ejpam-6496	51	1	2.2	2.2	NUM
ejpam-6496	51	2	.	.	PUNCT
ejpam-6496	51	3	basic	basic	ADJ
ejpam-6496	51	4	knowledge	knowledge	NOUN
ejpam-6496	51	5	from	from	ADP
ejpam-6496	51	6	real	real	ADJ
ejpam-6496	51	7	analysis	analysis	NOUN
ejpam-6496	51	8	theorem	theorem	VERB
ejpam-6496	51	9	1	1	NUM
ejpam-6496	51	10	(	(	PUNCT
ejpam-6496	51	11	maximum	maximum	ADJ
ejpam-6496	51	12	-	-	PUNCT
ejpam-6496	51	13	minimum	minimum	ADJ
ejpam-6496	51	14	theorem	theorem	NOUN
ejpam-6496	51	15	)	)	PUNCT
ejpam-6496	51	16	.	.	PUNCT
ejpam-6496	52	1	[	[	X
ejpam-6496	52	2	9	9	X
ejpam-6496	52	3	]	]	X
ejpam-6496	52	4	if	if	SCONJ
ejpam-6496	52	5	f	f	PROPN
ejpam-6496	52	6	:	:	PUNCT
ejpam-6496	52	7	[	[	PUNCT
ejpam-6496	52	8	a	a	X
ejpam-6496	52	9	,	,	PUNCT
ejpam-6496	52	10	b	b	NOUN
ejpam-6496	52	11	]	]	PUNCT
ejpam-6496	52	12	→	→	PUNCT
ejpam-6496	52	13	r	r	NOUN
ejpam-6496	52	14	is	be	AUX
ejpam-6496	52	15	continuous	continuous	ADJ
ejpam-6496	52	16	on	on	ADP
ejpam-6496	52	17	[	[	PUNCT
ejpam-6496	52	18	a	a	PROPN
ejpam-6496	52	19	,	,	PUNCT
ejpam-6496	52	20	b	b	NOUN
ejpam-6496	52	21	]	]	X
ejpam-6496	52	22	,	,	PUNCT
ejpam-6496	52	23	then	then	ADV
ejpam-6496	52	24	f	f	PROPN
ejpam-6496	52	25	has	have	VERB
ejpam-6496	52	26	an	an	DET
ejpam-6496	52	27	absolute	absolute	ADJ
ejpam-6496	52	28	maximum	maximum	NOUN
ejpam-6496	52	29	and	and	CCONJ
ejpam-6496	52	30	an	an	DET
ejpam-6496	52	31	absulute	absulute	NOUN
ejpam-6496	52	32	minimum	minimum	NOUN
ejpam-6496	52	33	on	on	ADP
ejpam-6496	52	34	[	[	PUNCT
ejpam-6496	52	35	a	a	PROPN
ejpam-6496	52	36	,	,	PUNCT
ejpam-6496	52	37	b	b	NOUN
ejpam-6496	52	38	]	]	PUNCT
ejpam-6496	52	39	.	.	PUNCT
ejpam-6496	53	1	theorem	theorem	ADJ
ejpam-6496	53	2	2	2	NUM
ejpam-6496	53	3	(	(	PUNCT
ejpam-6496	53	4	interior	interior	ADJ
ejpam-6496	53	5	extremum	extremum	ADJ
ejpam-6496	53	6	theorem	theorem	VERB
ejpam-6496	53	7	)	)	PUNCT
ejpam-6496	53	8	.	.	PUNCT
ejpam-6496	54	1	[	[	X
ejpam-6496	54	2	9	9	X
ejpam-6496	54	3	]	]	PUNCT
ejpam-6496	54	4	let	let	VERB
ejpam-6496	54	5	c	c	PRON
ejpam-6496	54	6	be	be	AUX
ejpam-6496	54	7	an	an	DET
ejpam-6496	54	8	interior	interior	ADJ
ejpam-6496	54	9	point	point	NOUN
ejpam-6496	54	10	of	of	ADP
ejpam-6496	54	11	an	an	DET
ejpam-6496	54	12	interval	interval	NOUN
ejpam-6496	54	13	i	i	PRON
ejpam-6496	54	14	at	at	ADP
ejpam-6496	54	15	which	which	PRON
ejpam-6496	54	16	f	f	X
ejpam-6496	54	17	:	:	PUNCT
ejpam-6496	54	18	i	i	PRON
ejpam-6496	54	19	→	→	PUNCT
ejpam-6496	54	20	r	r	NOUN
ejpam-6496	54	21	has	have	VERB
ejpam-6496	54	22	a	a	DET
ejpam-6496	54	23	relative	relative	ADJ
ejpam-6496	54	24	extremum	extremum	NOUN
ejpam-6496	54	25	.	.	PUNCT
ejpam-6496	55	1	if	if	SCONJ
ejpam-6496	55	2	the	the	DET
ejpam-6496	55	3	derivative	derivative	NOUN
ejpam-6496	55	4	of	of	ADP
ejpam-6496	55	5	f	f	PROPN
ejpam-6496	55	6	at	at	ADP
ejpam-6496	55	7	c	c	PROPN
ejpam-6496	55	8	exists	exist	VERB
ejpam-6496	55	9	,	,	PUNCT
ejpam-6496	55	10	then	then	ADV
ejpam-6496	55	11	f	f	PROPN
ejpam-6496	55	12	′(c	′(c	NOUN
ejpam-6496	55	13	)	)	PUNCT
ejpam-6496	56	1	=	=	SYM
ejpam-6496	56	2	0	0	X
ejpam-6496	56	3	.	.	PUNCT
ejpam-6496	56	4	definition	definition	NOUN
ejpam-6496	56	5	4	4	NUM
ejpam-6496	56	6	.	.	PUNCT
ejpam-6496	57	1	[	[	X
ejpam-6496	57	2	9	9	NUM
ejpam-6496	57	3	]	]	X
ejpam-6496	57	4	let	let	VERB
ejpam-6496	57	5	{	{	PUNCT
ejpam-6496	57	6	[	[	PUNCT
ejpam-6496	57	7	cj−1	cj−1	NOUN
ejpam-6496	57	8	,	,	PUNCT
ejpam-6496	57	9	cj	cj	NOUN
ejpam-6496	57	10	]	]	PUNCT
ejpam-6496	57	11	:	:	PUNCT
ejpam-6496	58	1	j	j	X
ejpam-6496	58	2	=	=	SYM
ejpam-6496	58	3	1	1	NUM
ejpam-6496	58	4	,	,	PUNCT
ejpam-6496	58	5	.	.	PUNCT
ejpam-6496	58	6	.	.	PUNCT
ejpam-6496	58	7	.	.	PUNCT
ejpam-6496	59	1	,	,	PUNCT
ejpam-6496	59	2	l	l	NOUN
ejpam-6496	59	3	}	}	PUNCT
ejpam-6496	59	4	be	be	AUX
ejpam-6496	59	5	a	a	DET
ejpam-6496	59	6	partition	partition	NOUN
ejpam-6496	59	7	of	of	ADP
ejpam-6496	59	8	[	[	PUNCT
ejpam-6496	59	9	a	a	PROPN
ejpam-6496	59	10	,	,	PUNCT
ejpam-6496	59	11	b	b	NOUN
ejpam-6496	59	12	]	]	X
ejpam-6496	59	13	,	,	PUNCT
ejpam-6496	59	14	and	and	CCONJ
ejpam-6496	59	15	let	let	VERB
ejpam-6496	59	16	each	each	DET
ejpam-6496	59	17	kj	kj	PROPN
ejpam-6496	59	18	∈	∈	PROPN
ejpam-6496	59	19	r.	r.	PROPN
ejpam-6496	59	20	let	let	VERB
ejpam-6496	59	21	φ	φ	PROPN
ejpam-6496	59	22	:	:	PUNCT
ejpam-6496	59	23	[	[	PUNCT
ejpam-6496	59	24	a	a	X
ejpam-6496	59	25	,	,	PUNCT
ejpam-6496	59	26	b	b	NOUN
ejpam-6496	59	27	]	]	PUNCT
ejpam-6496	59	28	→	→	PUNCT
ejpam-6496	59	29	r	r	NOUN
ejpam-6496	59	30	be	be	VERB
ejpam-6496	59	31	such	such	ADJ
ejpam-6496	59	32	that	that	SCONJ
ejpam-6496	59	33	φ	φ	PROPN
ejpam-6496	59	34	=	=	SYM
ejpam-6496	60	1	l∑	l∑	PROPN
ejpam-6496	61	1	j=1	j=1	PROPN
ejpam-6496	61	2	kjφjj	kjφjj	ADV
ejpam-6496	61	3	,	,	PUNCT
ejpam-6496	61	4	for	for	ADP
ejpam-6496	61	5	each	each	DET
ejpam-6496	61	6	j	j	PROPN
ejpam-6496	61	7	=	=	SYM
ejpam-6496	61	8	1	1	NUM
ejpam-6496	61	9	,	,	PUNCT
ejpam-6496	61	10	.	.	PUNCT
ejpam-6496	61	11	.	.	PUNCT
ejpam-6496	61	12	.	.	PUNCT
ejpam-6496	62	1	,	,	PUNCT
ejpam-6496	62	2	l	l	NOUN
ejpam-6496	62	3	,	,	PUNCT
ejpam-6496	62	4	jj	jj	PROPN
ejpam-6496	62	5	:	:	PUNCT
ejpam-6496	63	1	=	=	SYM
ejpam-6496	63	2	[	[	PUNCT
ejpam-6496	63	3	cj−1	cj−1	NOUN
ejpam-6496	63	4	,	,	PUNCT
ejpam-6496	63	5	cj	cj	NOUN
ejpam-6496	63	6	]	]	PUNCT
ejpam-6496	63	7	and	and	CCONJ
ejpam-6496	63	8	φjj	φjj	X
ejpam-6496	63	9	(	(	PUNCT
ejpam-6496	63	10	x	x	NOUN
ejpam-6496	63	11	)	)	PUNCT
ejpam-6496	63	12	=	=	SYM
ejpam-6496	63	13	{	{	PUNCT
ejpam-6496	63	14	1	1	NUM
ejpam-6496	63	15	if	if	SCONJ
ejpam-6496	63	16	cj−1	cj−1	NOUN
ejpam-6496	63	17	≤	≤	NUM
ejpam-6496	63	18	x	x	X
ejpam-6496	63	19	<	<	X
ejpam-6496	63	20	cj	cj	X
ejpam-6496	63	21	,	,	PUNCT
ejpam-6496	63	22	0	0	NUM
ejpam-6496	63	23	elsewhere	elsewhere	ADV
ejpam-6496	63	24	,	,	PUNCT
ejpam-6496	63	25	except	except	SCONJ
ejpam-6496	63	26	for	for	ADP
ejpam-6496	63	27	the	the	DET
ejpam-6496	63	28	last	last	ADJ
ejpam-6496	63	29	φjl	φjl	NOUN
ejpam-6496	63	30	that	that	SCONJ
ejpam-6496	63	31	φjl(x	φjl(x	PROPN
ejpam-6496	63	32	)	)	PUNCT
ejpam-6496	63	33	=	=	NOUN
ejpam-6496	63	34	{	{	PUNCT
ejpam-6496	63	35	1	1	NUM
ejpam-6496	63	36	if	if	SCONJ
ejpam-6496	63	37	cl−1	cl−1	VERB
ejpam-6496	63	38	≤	≤	NUM
ejpam-6496	63	39	x	x	SYM
ejpam-6496	63	40	≤	≤	NUM
ejpam-6496	63	41	cl	cl	NOUN
ejpam-6496	63	42	,	,	PUNCT
ejpam-6496	63	43	0	0	PUNCT
ejpam-6496	63	44	elsewhere	elsewhere	ADV
ejpam-6496	63	45	.	.	PUNCT
ejpam-6496	64	1	we	we	PRON
ejpam-6496	64	2	say	say	VERB
ejpam-6496	64	3	that	that	SCONJ
ejpam-6496	64	4	φ	φ	PROPN
ejpam-6496	64	5	is	be	AUX
ejpam-6496	64	6	a	a	DET
ejpam-6496	64	7	step	step	NOUN
ejpam-6496	64	8	function	function	NOUN
ejpam-6496	64	9	on	on	ADP
ejpam-6496	64	10	[	[	PUNCT
ejpam-6496	64	11	a	a	PROPN
ejpam-6496	64	12	,	,	PUNCT
ejpam-6496	64	13	b	b	NOUN
ejpam-6496	64	14	]	]	X
ejpam-6496	64	15	.	.	PUNCT
ejpam-6496	65	1	an	an	DET
ejpam-6496	65	2	example	example	NOUN
ejpam-6496	65	3	of	of	ADP
ejpam-6496	65	4	a	a	DET
ejpam-6496	65	5	step	step	NOUN
ejpam-6496	65	6	function	function	NOUN
ejpam-6496	65	7	on	on	ADP
ejpam-6496	65	8	[	[	PUNCT
ejpam-6496	65	9	1	1	NUM
ejpam-6496	65	10	,	,	PUNCT
ejpam-6496	65	11	4	4	NUM
ejpam-6496	65	12	]	]	PUNCT
ejpam-6496	65	13	is	be	AUX
ejpam-6496	65	14	illustrated	illustrate	VERB
ejpam-6496	65	15	in	in	ADP
ejpam-6496	65	16	figure	figure	NOUN
ejpam-6496	65	17	2	2	NUM
ejpam-6496	65	18	.	.	PUNCT
ejpam-6496	65	19	definition	definition	NOUN
ejpam-6496	65	20	5	5	NUM
ejpam-6496	65	21	.	.	PUNCT
ejpam-6496	66	1	[	[	X
ejpam-6496	66	2	5	5	NUM
ejpam-6496	66	3	]	]	PUNCT
ejpam-6496	66	4	a	a	DET
ejpam-6496	66	5	function	function	NOUN
ejpam-6496	66	6	f	f	PROPN
ejpam-6496	66	7	is	be	AUX
ejpam-6496	66	8	riemann	riemann	PROPN
ejpam-6496	66	9	integrable	integrable	ADJ
ejpam-6496	66	10	on	on	ADP
ejpam-6496	66	11	[	[	PUNCT
ejpam-6496	66	12	a	a	PROPN
ejpam-6496	66	13	,	,	PUNCT
ejpam-6496	66	14	b	b	NOUN
ejpam-6496	66	15	]	]	X
ejpam-6496	67	1	if	if	SCONJ
ejpam-6496	67	2	and	and	CCONJ
ejpam-6496	67	3	only	only	ADV
ejpam-6496	67	4	if	if	SCONJ
ejpam-6496	67	5	for	for	ADP
ejpam-6496	67	6	each	each	DET
ejpam-6496	67	7	ε	ε	PROPN
ejpam-6496	67	8	>	>	X
ejpam-6496	67	9	0	0	PUNCT
ejpam-6496	68	1	there	there	PRON
ejpam-6496	68	2	exists	exist	VERB
ejpam-6496	68	3	a	a	DET
ejpam-6496	68	4	partition	partition	NOUN
ejpam-6496	68	5	{	{	PUNCT
ejpam-6496	68	6	[	[	X
ejpam-6496	68	7	xi−1	xi−1	PROPN
ejpam-6496	68	8	,	,	PUNCT
ejpam-6496	68	9	xi	xi	X
ejpam-6496	68	10	]	]	PUNCT
ejpam-6496	68	11	:	:	PUNCT
ejpam-6496	68	12	i	i	NOUN
ejpam-6496	68	13	=	=	NOUN
ejpam-6496	68	14	1	1	NUM
ejpam-6496	68	15	,	,	PUNCT
ejpam-6496	68	16	.	.	PUNCT
ejpam-6496	68	17	.	.	PUNCT
ejpam-6496	69	1	.	.	PUNCT
ejpam-6496	70	1	,	,	PUNCT
ejpam-6496	70	2	n	n	CCONJ
ejpam-6496	70	3	}	}	PUNCT
ejpam-6496	70	4	of	of	ADP
ejpam-6496	70	5	[	[	PUNCT
ejpam-6496	70	6	a	a	PROPN
ejpam-6496	70	7	,	,	PUNCT
ejpam-6496	70	8	b	b	NOUN
ejpam-6496	70	9	]	]	PUNCT
ejpam-6496	70	10	such	such	ADJ
ejpam-6496	70	11	that	that	SCONJ
ejpam-6496	70	12	n∑	n∑	PROPN
ejpam-6496	70	13	i=1	i=1	PROPN
ejpam-6496	70	14	ω(f	ω(f	ADJ
ejpam-6496	70	15	,	,	PUNCT
ejpam-6496	70	16	[	[	X
ejpam-6496	70	17	xi−1	xi−1	PROPN
ejpam-6496	70	18	,	,	PUNCT
ejpam-6496	70	19	xi	xi	X
ejpam-6496	70	20	]	]	X
ejpam-6496	70	21	)	)	PUNCT
ejpam-6496	70	22	(	(	PUNCT
ejpam-6496	70	23	xi	xi	X
ejpam-6496	70	24	−	−	PROPN
ejpam-6496	70	25	xi−1	xi−1	PROPN
ejpam-6496	70	26	)	)	PUNCT
ejpam-6496	70	27	<	<	X
ejpam-6496	70	28	ε	ε	PROPN
ejpam-6496	70	29	,	,	PUNCT
ejpam-6496	70	30	where	where	SCONJ
ejpam-6496	70	31	ω(f	ω(f	ADJ
ejpam-6496	70	32	,	,	PUNCT
ejpam-6496	70	33	[	[	PUNCT
ejpam-6496	70	34	c	c	X
ejpam-6496	70	35	,	,	PUNCT
ejpam-6496	70	36	d	d	NOUN
ejpam-6496	70	37	]	]	X
ejpam-6496	70	38	)	)	PUNCT
ejpam-6496	71	1	=	=	SYM
ejpam-6496	71	2	sup{|f(t)−	sup{|f(t)−	PROPN
ejpam-6496	72	1	f(s)|	f(s)|	NOUN
ejpam-6496	73	1	:	:	PUNCT
ejpam-6496	74	1	s	s	X
ejpam-6496	74	2	,	,	PUNCT
ejpam-6496	74	3	t	t	PROPN
ejpam-6496	74	4	∈	∈	PROPN
ejpam-6496	74	5	[	[	PUNCT
ejpam-6496	74	6	c	c	X
ejpam-6496	74	7	,	,	PUNCT
ejpam-6496	74	8	d	d	X
ejpam-6496	74	9	]	]	X
ejpam-6496	74	10	}	}	PUNCT
ejpam-6496	74	11	.	.	PUNCT
ejpam-6496	75	1	3	3	X
ejpam-6496	75	2	.	.	X
ejpam-6496	75	3	proving	prove	VERB
ejpam-6496	75	4	theorems	theorem	NOUN
ejpam-6496	75	5	via	via	ADP
ejpam-6496	75	6	δ	δ	PROPN
ejpam-6496	75	7	-	-	PUNCT
ejpam-6496	75	8	fine	fine	ADJ
ejpam-6496	75	9	tagged	tag	VERB
ejpam-6496	75	10	partitions	partition	NOUN
ejpam-6496	75	11	3.1	3.1	NUM
ejpam-6496	75	12	.	.	PUNCT
ejpam-6496	75	13	differentiation	differentiation	NOUN
ejpam-6496	75	14	via	via	ADP
ejpam-6496	75	15	δ	δ	PROPN
ejpam-6496	75	16	-	-	PUNCT
ejpam-6496	75	17	fine	fine	ADJ
ejpam-6496	75	18	tagged	tag	VERB
ejpam-6496	75	19	partitions	partition	NOUN
ejpam-6496	75	20	the	the	DET
ejpam-6496	75	21	following	follow	VERB
ejpam-6496	75	22	lemma	lemma	PROPN
ejpam-6496	75	23	provides	provide	VERB
ejpam-6496	75	24	important	important	ADJ
ejpam-6496	75	25	results	result	NOUN
ejpam-6496	75	26	for	for	ADP
ejpam-6496	75	27	defining	define	VERB
ejpam-6496	75	28	a	a	DET
ejpam-6496	75	29	gauge	gauge	ADJ
ejpam-6496	75	30	δ	δ	NOUN
ejpam-6496	75	31	in	in	ADP
ejpam-6496	75	32	proving	prove	VERB
ejpam-6496	75	33	theorem	theorem	ADJ
ejpam-6496	75	34	3	3	NUM
ejpam-6496	75	35	.	.	PUNCT
ejpam-6496	75	36	theorem	theorem	ADJ
ejpam-6496	75	37	4	4	NUM
ejpam-6496	75	38	arise	arise	VERB
ejpam-6496	75	39	from	from	ADP
ejpam-6496	75	40	theorem	theorem	NOUN
ejpam-6496	75	41	3	3	NUM
ejpam-6496	75	42	by	by	ADP
ejpam-6496	75	43	extending	extend	VERB
ejpam-6496	75	44	an	an	DET
ejpam-6496	75	45	open	open	ADJ
ejpam-6496	75	46	interval	interval	NOUN
ejpam-6496	75	47	]	]	PUNCT
ejpam-6496	75	48	a	a	X
ejpam-6496	75	49	,	,	PUNCT
ejpam-6496	75	50	b	b	X
ejpam-6496	75	51	[	[	PUNCT
ejpam-6496	75	52	to	to	ADP
ejpam-6496	75	53	a	a	DET
ejpam-6496	75	54	closed	closed	ADJ
ejpam-6496	75	55	interval	interval	NOUN
ejpam-6496	75	56	[	[	PUNCT
ejpam-6496	75	57	a	a	PROPN
ejpam-6496	75	58	,	,	PUNCT
ejpam-6496	75	59	b	b	NOUN
ejpam-6496	75	60	]	]	PUNCT
ejpam-6496	75	61	.	.	PUNCT
ejpam-6496	76	1	applying	apply	VERB
ejpam-6496	76	2	theorem	theorem	NOUN
ejpam-6496	76	3	4	4	NUM
ejpam-6496	76	4	,	,	PUNCT
ejpam-6496	76	5	the	the	DET
ejpam-6496	76	6	reproving	reproving	NOUN
ejpam-6496	76	7	of	of	ADP
ejpam-6496	76	8	the	the	DET
ejpam-6496	76	9	mean	mean	ADJ
ejpam-6496	76	10	value	value	NOUN
ejpam-6496	76	11	theorem	theorem	NOUN
ejpam-6496	76	12	and	and	CCONJ
ejpam-6496	76	13	the	the	DET
ejpam-6496	76	14	cauchy	cauchy	PROPN
ejpam-6496	76	15	mean	mean	NOUN
ejpam-6496	76	16	value	value	NOUN
ejpam-6496	76	17	theorem	theorem	NOUN
ejpam-6496	76	18	are	be	AUX
ejpam-6496	76	19	obtained	obtain	VERB
ejpam-6496	76	20	.	.	PUNCT
ejpam-6496	77	1	in	in	ADP
ejpam-6496	77	2	addition	addition	NOUN
ejpam-6496	77	3	,	,	PUNCT
ejpam-6496	77	4	corollary	corollary	ADJ
ejpam-6496	77	5	1	1	NUM
ejpam-6496	77	6	is	be	AUX
ejpam-6496	77	7	also	also	ADV
ejpam-6496	77	8	presented	present	VERB
ejpam-6496	77	9	,	,	PUNCT
ejpam-6496	77	10	for	for	ADP
ejpam-6496	77	11	this	this	DET
ejpam-6496	77	12	result	result	NOUN
ejpam-6496	77	13	and	and	CCONJ
ejpam-6496	77	14	theorem	theorem	VERB
ejpam-6496	77	15	3	3	NUM
ejpam-6496	77	16	are	be	AUX
ejpam-6496	77	17	rely	rely	ADJ
ejpam-6496	77	18	upon	upon	SCONJ
ejpam-6496	77	19	each	each	DET
ejpam-6496	77	20	other	other	ADJ
ejpam-6496	77	21	.	.	PUNCT
ejpam-6496	78	1	s.	s.	PROPN
ejpam-6496	78	2	prongjit	prongjit	ADV
ejpam-6496	78	3	,	,	PUNCT
ejpam-6496	78	4	p.	p.	PROPN
ejpam-6496	78	5	prasertsang	prasertsang	PROPN
ejpam-6496	78	6	/	/	SYM
ejpam-6496	78	7	eur	eur	PROPN
ejpam-6496	78	8	.	.	PUNCT
ejpam-6496	79	1	j.	j.	PROPN
ejpam-6496	79	2	pure	pure	PROPN
ejpam-6496	79	3	appl	appl	PROPN
ejpam-6496	79	4	.	.	PROPN
ejpam-6496	79	5	math	math	PROPN
ejpam-6496	79	6	,	,	PUNCT
ejpam-6496	79	7	18	18	NUM
ejpam-6496	79	8	(	(	PUNCT
ejpam-6496	79	9	3	3	NUM
ejpam-6496	79	10	)	)	PUNCT
ejpam-6496	79	11	(	(	PUNCT
ejpam-6496	79	12	2025	2025	NUM
ejpam-6496	79	13	)	)	PUNCT
ejpam-6496	79	14	,	,	PUNCT
ejpam-6496	79	15	6496	6496	NUM
ejpam-6496	79	16	4	4	NUM
ejpam-6496	79	17	of	of	ADP
ejpam-6496	79	18	10	10	NUM
ejpam-6496	79	19	x	x	SYM
ejpam-6496	79	20	y	y	PROPN
ejpam-6496	79	21	k4	k4	PROPN
ejpam-6496	79	22	k2	k2	PROPN
ejpam-6496	79	23	k1	k1	PROPN
ejpam-6496	79	24	k5	k5	PROPN
ejpam-6496	79	25	k3	k3	VERB
ejpam-6496	79	26	1	1	NUM
ejpam-6496	79	27	=	=	SYM
ejpam-6496	79	28	c0	c0	PROPN
ejpam-6496	79	29	c5	c5	PROPN
ejpam-6496	79	30	=	=	PROPN
ejpam-6496	79	31	4c1	4c1	NUM
ejpam-6496	79	32	c2	c2	PROPN
ejpam-6496	79	33	c3	c3	PROPN
ejpam-6496	79	34	c4	c4	PROPN
ejpam-6496	79	35	j1	j1	PROPN
ejpam-6496	79	36	j2	j2	PROPN
ejpam-6496	79	37	j3	j3	PROPN
ejpam-6496	79	38	j4	j4	PROPN
ejpam-6496	79	39	j5	j5	PROPN
ejpam-6496	79	40	figure	figure	NOUN
ejpam-6496	79	41	2	2	NUM
ejpam-6496	79	42	:	:	PUNCT
ejpam-6496	79	43	a	a	DET
ejpam-6496	79	44	step	step	NOUN
ejpam-6496	79	45	function	function	NOUN
ejpam-6496	79	46	φ	φ	NOUN
ejpam-6496	79	47	=	=	SYM
ejpam-6496	79	48	5∑	5∑	NUM
ejpam-6496	79	49	j=1	j=1	PROPN
ejpam-6496	79	50	kjφjj	kjφjj	ADV
ejpam-6496	79	51	on	on	ADP
ejpam-6496	79	52	the	the	DET
ejpam-6496	79	53	closed	closed	ADJ
ejpam-6496	79	54	interval	interval	NOUN
ejpam-6496	79	55	[	[	PUNCT
ejpam-6496	79	56	1	1	NUM
ejpam-6496	79	57	,	,	PUNCT
ejpam-6496	79	58	4	4	NUM
ejpam-6496	79	59	]	]	PUNCT
ejpam-6496	79	60	lemma	lemma	PROPN
ejpam-6496	79	61	2	2	X
ejpam-6496	79	62	.	.	PUNCT
ejpam-6496	79	63	suppose	suppose	VERB
ejpam-6496	79	64	that	that	SCONJ
ejpam-6496	79	65	f	f	PROPN
ejpam-6496	79	66	is	be	AUX
ejpam-6496	79	67	differentiable	differentiable	ADJ
ejpam-6496	79	68	on	on	ADP
ejpam-6496	79	69	]	]	PUNCT
ejpam-6496	79	70	a	a	X
ejpam-6496	79	71	,	,	PUNCT
ejpam-6496	79	72	b	b	NOUN
ejpam-6496	79	73	[	[	PUNCT
ejpam-6496	79	74	and	and	CCONJ
ejpam-6496	80	1	that	that	SCONJ
ejpam-6496	80	2	f	f	PROPN
ejpam-6496	80	3	′(x	′(x	NOUN
ejpam-6496	80	4	)	)	PUNCT
ejpam-6496	80	5	̸=	̸=	PROPN
ejpam-6496	80	6	0	0	NUM
ejpam-6496	80	7	for	for	ADP
ejpam-6496	80	8	all	all	DET
ejpam-6496	80	9	x	x	SYM
ejpam-6496	80	10	∈	∈	PROPN
ejpam-6496	80	11	]	]	PUNCT
ejpam-6496	80	12	a	a	X
ejpam-6496	80	13	,	,	PUNCT
ejpam-6496	80	14	b	b	X
ejpam-6496	80	15	[	[	PUNCT
ejpam-6496	80	16	.	.	PUNCT
ejpam-6496	81	1	(	(	PUNCT
ejpam-6496	81	2	a	a	X
ejpam-6496	81	3	)	)	PUNCT
ejpam-6496	81	4	if	if	SCONJ
ejpam-6496	81	5	f	f	PROPN
ejpam-6496	81	6	′(x	′(x	PROPN
ejpam-6496	81	7	)	)	PUNCT
ejpam-6496	81	8	>	>	X
ejpam-6496	81	9	0	0	NUM
ejpam-6496	81	10	,	,	PUNCT
ejpam-6496	81	11	then	then	ADV
ejpam-6496	81	12	there	there	PRON
ejpam-6496	81	13	exists	exist	VERB
ejpam-6496	81	14	a	a	DET
ejpam-6496	81	15	real	real	ADJ
ejpam-6496	81	16	number	number	NOUN
ejpam-6496	81	17	ζ	ζ	NOUN
ejpam-6496	81	18	>	>	X
ejpam-6496	81	19	0	0	NUM
ejpam-6496	82	1	such	such	ADJ
ejpam-6496	82	2	that	that	SCONJ
ejpam-6496	82	3	f	f	PROPN
ejpam-6496	82	4	is	be	AUX
ejpam-6496	82	5	strictly	strictly	ADV
ejpam-6496	82	6	increasing	increase	VERB
ejpam-6496	82	7	on	on	ADP
ejpam-6496	82	8	]	]	PUNCT
ejpam-6496	82	9	x−	x−	PROPN
ejpam-6496	82	10	ζ	ζ	PROPN
ejpam-6496	82	11	,	,	PUNCT
ejpam-6496	82	12	x+	x+	ADJ
ejpam-6496	82	13	ζ	ζ	NOUN
ejpam-6496	82	14	[	[	PUNCT
ejpam-6496	82	15	.	.	PUNCT
ejpam-6496	83	1	(	(	PUNCT
ejpam-6496	83	2	b	b	X
ejpam-6496	83	3	)	)	PUNCT
ejpam-6496	83	4	if	if	SCONJ
ejpam-6496	83	5	f	f	PROPN
ejpam-6496	83	6	′(x	′(x	PROPN
ejpam-6496	83	7	)	)	PUNCT
ejpam-6496	83	8	<	<	X
ejpam-6496	83	9	0	0	NUM
ejpam-6496	83	10	,	,	PUNCT
ejpam-6496	83	11	then	then	ADV
ejpam-6496	83	12	there	there	PRON
ejpam-6496	83	13	exists	exist	VERB
ejpam-6496	83	14	a	a	DET
ejpam-6496	83	15	real	real	ADJ
ejpam-6496	83	16	number	number	NOUN
ejpam-6496	83	17	ξ	ξ	X
ejpam-6496	83	18	>	>	X
ejpam-6496	83	19	0	0	NUM
ejpam-6496	83	20	such	such	ADJ
ejpam-6496	83	21	that	that	SCONJ
ejpam-6496	83	22	f	f	PROPN
ejpam-6496	83	23	is	be	AUX
ejpam-6496	83	24	strictly	strictly	ADV
ejpam-6496	83	25	decreasing	decrease	VERB
ejpam-6496	83	26	on	on	ADP
ejpam-6496	83	27	]	]	X
ejpam-6496	83	28	x−	x−	PROPN
ejpam-6496	83	29	ξ	ξ	PROPN
ejpam-6496	83	30	,	,	PUNCT
ejpam-6496	83	31	x+	x+	X
ejpam-6496	83	32	ξ	ξ	X
ejpam-6496	83	33	[	[	PUNCT
ejpam-6496	83	34	.	.	PUNCT
ejpam-6496	84	1	proof	proof	NOUN
ejpam-6496	84	2	.	.	PUNCT
ejpam-6496	85	1	(	(	PUNCT
ejpam-6496	85	2	a	a	X
ejpam-6496	85	3	)	)	PUNCT
ejpam-6496	85	4	since	since	SCONJ
ejpam-6496	85	5	f	f	PROPN
ejpam-6496	85	6	′(x	′(x	PROPN
ejpam-6496	85	7	)	)	PUNCT
ejpam-6496	85	8	>	>	X
ejpam-6496	85	9	0	0	NUM
ejpam-6496	85	10	,	,	PUNCT
ejpam-6496	85	11	then	then	ADV
ejpam-6496	85	12	there	there	PRON
ejpam-6496	85	13	exists	exist	VERB
ejpam-6496	85	14	a	a	DET
ejpam-6496	85	15	real	real	ADJ
ejpam-6496	85	16	number	number	NOUN
ejpam-6496	85	17	ζ	ζ	NOUN
ejpam-6496	85	18	>	>	X
ejpam-6496	85	19	0	0	NUM
ejpam-6496	86	1	such	such	ADJ
ejpam-6496	86	2	that	that	SCONJ
ejpam-6496	86	3	]	]	X
ejpam-6496	86	4	x−	x−	PROPN
ejpam-6496	86	5	ζ	ζ	PROPN
ejpam-6496	86	6	,	,	PUNCT
ejpam-6496	86	7	x+	x+	ADJ
ejpam-6496	86	8	ζ	ζ	X
ejpam-6496	86	9	[	[	PUNCT
ejpam-6496	86	10	⊆	⊆	NUM
ejpam-6496	86	11	]	]	PUNCT
ejpam-6496	86	12	a	a	DET
ejpam-6496	86	13	,	,	PUNCT
ejpam-6496	86	14	b	b	X
ejpam-6496	86	15	[	[	PUNCT
ejpam-6496	86	16	,	,	PUNCT
ejpam-6496	86	17	and	and	CCONJ
ejpam-6496	86	18	f(y)−	f(y)−	PROPN
ejpam-6496	86	19	f(x	f(x	PROPN
ejpam-6496	86	20	)	)	PUNCT
ejpam-6496	86	21	y	y	PROPN
ejpam-6496	87	1	−	−	PROPN
ejpam-6496	87	2	x	x	SYM
ejpam-6496	87	3	>	>	X
ejpam-6496	87	4	0	0	NUM
ejpam-6496	88	1	for	for	ADP
ejpam-6496	88	2	all	all	DET
ejpam-6496	88	3	y	y	PROPN
ejpam-6496	88	4	∈	∈	PROPN
ejpam-6496	88	5	]	]	PUNCT
ejpam-6496	88	6	x−	x−	PROPN
ejpam-6496	88	7	ζ	ζ	PROPN
ejpam-6496	88	8	,	,	PUNCT
ejpam-6496	88	9	x+	x+	ADJ
ejpam-6496	88	10	ζ	ζ	NOUN
ejpam-6496	88	11	[	[	PUNCT
ejpam-6496	88	12	with	with	ADP
ejpam-6496	88	13	y	y	PROPN
ejpam-6496	88	14	̸=	̸=	PROPN
ejpam-6496	88	15	x.	x.	NOUN
ejpam-6496	88	16	claim	claim	VERB
ejpam-6496	88	17	that	that	SCONJ
ejpam-6496	88	18	f	f	PROPN
ejpam-6496	88	19	is	be	AUX
ejpam-6496	88	20	strictly	strictly	ADV
ejpam-6496	88	21	increasing	increase	VERB
ejpam-6496	88	22	on	on	ADP
ejpam-6496	88	23	]	]	PUNCT
ejpam-6496	88	24	x−	x−	PROPN
ejpam-6496	88	25	ζ	ζ	PROPN
ejpam-6496	88	26	,	,	PUNCT
ejpam-6496	88	27	x+	x+	ADJ
ejpam-6496	88	28	ζ	ζ	NOUN
ejpam-6496	88	29	[	[	PUNCT
ejpam-6496	88	30	.	.	PUNCT
ejpam-6496	89	1	let	let	VERB
ejpam-6496	89	2	s	s	NOUN
ejpam-6496	89	3	,	,	PUNCT
ejpam-6496	89	4	t	t	PROPN
ejpam-6496	89	5	∈	∈	PROPN
ejpam-6496	89	6	]	]	PUNCT
ejpam-6496	89	7	x−	x−	PROPN
ejpam-6496	89	8	ζ	ζ	PROPN
ejpam-6496	89	9	,	,	PUNCT
ejpam-6496	89	10	x+	x+	ADJ
ejpam-6496	89	11	ζ	ζ	NOUN
ejpam-6496	89	12	[	[	PUNCT
ejpam-6496	89	13	with	with	ADP
ejpam-6496	89	14	s	s	PRON
ejpam-6496	89	15	<	<	X
ejpam-6496	89	16	t	t	PROPN
ejpam-6496	89	17	,	,	PUNCT
ejpam-6496	89	18	then	then	ADV
ejpam-6496	89	19	we	we	PRON
ejpam-6496	89	20	have	have	VERB
ejpam-6496	89	21	three	three	NUM
ejpam-6496	89	22	cases	case	NOUN
ejpam-6496	89	23	to	to	PART
ejpam-6496	89	24	consider	consider	VERB
ejpam-6496	89	25	.	.	PUNCT
ejpam-6496	90	1	case	case	NOUN
ejpam-6496	90	2	1	1	NUM
ejpam-6496	90	3	:	:	PUNCT
ejpam-6496	90	4	s	s	VERB
ejpam-6496	90	5	≤	≤	NUM
ejpam-6496	90	6	x	x	PUNCT
ejpam-6496	90	7	<	<	X
ejpam-6496	90	8	t	t	NOUN
ejpam-6496	90	9	or	or	CCONJ
ejpam-6496	90	10	s	s	X
ejpam-6496	90	11	<	<	X
ejpam-6496	90	12	x	x	X
ejpam-6496	90	13	≤	≤	ADJ
ejpam-6496	90	14	t.	t.	NOUN
ejpam-6496	90	15	it	it	PRON
ejpam-6496	90	16	is	be	AUX
ejpam-6496	90	17	clear	clear	ADJ
ejpam-6496	90	18	that	that	SCONJ
ejpam-6496	90	19	f(s	f(s	ADV
ejpam-6496	90	20	)	)	PUNCT
ejpam-6496	90	21	<	<	X
ejpam-6496	90	22	f(t	f(t	NOUN
ejpam-6496	90	23	)	)	PUNCT
ejpam-6496	90	24	.	.	PUNCT
ejpam-6496	91	1	case	case	NOUN
ejpam-6496	91	2	2	2	NUM
ejpam-6496	91	3	:	:	PUNCT
ejpam-6496	91	4	s	s	AUX
ejpam-6496	91	5	<	<	X
ejpam-6496	91	6	t	t	X
ejpam-6496	91	7	<	<	X
ejpam-6496	91	8	x.	x.	NOUN
ejpam-6496	91	9	suppose	suppose	VERB
ejpam-6496	91	10	to	to	ADP
ejpam-6496	91	11	the	the	DET
ejpam-6496	91	12	contrary	contrary	NOUN
ejpam-6496	91	13	that	that	PRON
ejpam-6496	91	14	f(s	f(s	ADV
ejpam-6496	91	15	)	)	PUNCT
ejpam-6496	91	16	≥	≥	NOUN
ejpam-6496	91	17	f(t	f(t	PROPN
ejpam-6496	91	18	)	)	PUNCT
ejpam-6496	91	19	.	.	PUNCT
ejpam-6496	92	1	by	by	ADP
ejpam-6496	92	2	theorem	theorem	NOUN
ejpam-6496	92	3	1	1	NUM
ejpam-6496	92	4	,	,	PUNCT
ejpam-6496	92	5	the	the	DET
ejpam-6496	92	6	function	function	NOUN
ejpam-6496	92	7	f	f	PROPN
ejpam-6496	92	8	has	have	VERB
ejpam-6496	92	9	an	an	DET
ejpam-6496	92	10	absolute	absolute	ADJ
ejpam-6496	92	11	minimum	minimum	NOUN
ejpam-6496	92	12	at	at	ADP
ejpam-6496	92	13	some	some	DET
ejpam-6496	92	14	point	point	NOUN
ejpam-6496	92	15	c	c	NOUN
ejpam-6496	92	16	∈	∈	PROPN
ejpam-6496	92	17	]	]	X
ejpam-6496	92	18	s	s	X
ejpam-6496	92	19	,	,	PUNCT
ejpam-6496	92	20	x	x	X
ejpam-6496	92	21	[	[	PUNCT
ejpam-6496	92	22	.	.	PUNCT
ejpam-6496	93	1	since	since	SCONJ
ejpam-6496	93	2	c	c	PROPN
ejpam-6496	93	3	is	be	AUX
ejpam-6496	93	4	an	an	DET
ejpam-6496	93	5	interior	interior	ADJ
ejpam-6496	93	6	point	point	NOUN
ejpam-6496	93	7	of	of	ADP
ejpam-6496	93	8	[	[	PUNCT
ejpam-6496	93	9	s	s	X
ejpam-6496	93	10	,	,	PUNCT
ejpam-6496	93	11	x	x	SYM
ejpam-6496	93	12	]	]	PUNCT
ejpam-6496	93	13	and	and	CCONJ
ejpam-6496	93	14	by	by	ADP
ejpam-6496	93	15	theorem	theorem	NOUN
ejpam-6496	93	16	2	2	NUM
ejpam-6496	93	17	,	,	PUNCT
ejpam-6496	93	18	it	it	PRON
ejpam-6496	93	19	follows	follow	VERB
ejpam-6496	93	20	that	that	SCONJ
ejpam-6496	93	21	f	f	PROPN
ejpam-6496	93	22	′(c	′(c	NOUN
ejpam-6496	93	23	)	)	PUNCT
ejpam-6496	93	24	=	=	SYM
ejpam-6496	93	25	0	0	NUM
ejpam-6496	93	26	which	which	PRON
ejpam-6496	93	27	contradicts	contradict	VERB
ejpam-6496	93	28	by	by	ADP
ejpam-6496	93	29	the	the	DET
ejpam-6496	93	30	assumption	assumption	NOUN
ejpam-6496	93	31	of	of	ADP
ejpam-6496	93	32	the	the	DET
ejpam-6496	93	33	lemma	lemma	PROPN
ejpam-6496	93	34	.	.	PUNCT
ejpam-6496	94	1	thus	thus	ADV
ejpam-6496	94	2	,	,	PUNCT
ejpam-6496	94	3	f(s	f(s	ADV
ejpam-6496	94	4	)	)	PUNCT
ejpam-6496	94	5	<	<	X
ejpam-6496	94	6	f(t	f(t	NOUN
ejpam-6496	94	7	)	)	PUNCT
ejpam-6496	94	8	.	.	PUNCT
ejpam-6496	95	1	case	case	NOUN
ejpam-6496	95	2	3	3	NUM
ejpam-6496	95	3	:	:	PUNCT
ejpam-6496	95	4	x	x	PUNCT
ejpam-6496	95	5	<	<	X
ejpam-6496	95	6	s	s	X
ejpam-6496	95	7	<	<	X
ejpam-6496	95	8	t.	t.	NOUN
ejpam-6496	95	9	the	the	DET
ejpam-6496	95	10	proof	proof	NOUN
ejpam-6496	95	11	is	be	AUX
ejpam-6496	95	12	similar	similar	ADJ
ejpam-6496	95	13	to	to	ADP
ejpam-6496	95	14	case	case	NOUN
ejpam-6496	95	15	2	2	NUM
ejpam-6496	95	16	.	.	PUNCT
ejpam-6496	96	1	therefore	therefore	ADV
ejpam-6496	96	2	,	,	PUNCT
ejpam-6496	96	3	f	f	PROPN
ejpam-6496	96	4	is	be	AUX
ejpam-6496	96	5	strictly	strictly	ADV
ejpam-6496	96	6	increasing	increase	VERB
ejpam-6496	96	7	on	on	ADP
ejpam-6496	96	8	]	]	PUNCT
ejpam-6496	96	9	x−	x−	PROPN
ejpam-6496	96	10	ζ	ζ	PROPN
ejpam-6496	96	11	,	,	PUNCT
ejpam-6496	96	12	x+	x+	ADJ
ejpam-6496	96	13	ζ	ζ	NOUN
ejpam-6496	96	14	[	[	PUNCT
ejpam-6496	96	15	as	as	SCONJ
ejpam-6496	96	16	claimed	claim	VERB
ejpam-6496	96	17	.	.	PUNCT
ejpam-6496	97	1	(	(	PUNCT
ejpam-6496	97	2	b	b	X
ejpam-6496	97	3	)	)	PUNCT
ejpam-6496	97	4	similar	similar	ADJ
ejpam-6496	97	5	to	to	ADP
ejpam-6496	97	6	the	the	DET
ejpam-6496	97	7	proof	proof	NOUN
ejpam-6496	97	8	of	of	ADP
ejpam-6496	97	9	(	(	PUNCT
ejpam-6496	97	10	a	a	NOUN
ejpam-6496	97	11	)	)	PUNCT
ejpam-6496	97	12	.	.	PUNCT
ejpam-6496	98	1	theorem	theorem	NOUN
ejpam-6496	98	2	3	3	X
ejpam-6496	98	3	.	.	PUNCT
ejpam-6496	98	4	suppose	suppose	VERB
ejpam-6496	98	5	that	that	SCONJ
ejpam-6496	98	6	f	f	PROPN
ejpam-6496	98	7	is	be	AUX
ejpam-6496	98	8	differentiable	differentiable	ADJ
ejpam-6496	98	9	on	on	ADP
ejpam-6496	98	10	]	]	PUNCT
ejpam-6496	98	11	a	a	X
ejpam-6496	98	12	,	,	PUNCT
ejpam-6496	98	13	b	b	X
ejpam-6496	98	14	[	[	PUNCT
ejpam-6496	98	15	.	.	PUNCT
ejpam-6496	99	1	if	if	SCONJ
ejpam-6496	99	2	f	f	PROPN
ejpam-6496	99	3	′(x	′(x	NOUN
ejpam-6496	99	4	)	)	PUNCT
ejpam-6496	99	5	̸=	̸=	PROPN
ejpam-6496	99	6	0	0	NUM
ejpam-6496	99	7	for	for	ADP
ejpam-6496	99	8	all	all	DET
ejpam-6496	99	9	x	x	SYM
ejpam-6496	99	10	∈	∈	PROPN
ejpam-6496	99	11	]	]	PUNCT
ejpam-6496	99	12	a	a	X
ejpam-6496	99	13	,	,	PUNCT
ejpam-6496	99	14	b	b	X
ejpam-6496	99	15	[	[	PUNCT
ejpam-6496	99	16	,	,	PUNCT
ejpam-6496	99	17	then	then	ADV
ejpam-6496	99	18	f	f	PROPN
ejpam-6496	99	19	is	be	AUX
ejpam-6496	99	20	strictly	strictly	ADV
ejpam-6496	99	21	monotone	monotone	ADJ
ejpam-6496	99	22	on	on	ADP
ejpam-6496	99	23	]	]	PUNCT
ejpam-6496	99	24	a	a	X
ejpam-6496	99	25	,	,	PUNCT
ejpam-6496	99	26	b	b	X
ejpam-6496	99	27	[	[	PUNCT
ejpam-6496	99	28	.	.	PUNCT
ejpam-6496	100	1	proof	proof	NOUN
ejpam-6496	100	2	.	.	PUNCT
ejpam-6496	101	1	let	let	VERB
ejpam-6496	101	2	c	c	X
ejpam-6496	101	3	,	,	PUNCT
ejpam-6496	101	4	d	d	PROPN
ejpam-6496	101	5	∈	∈	PROPN
ejpam-6496	101	6	]	]	PUNCT
ejpam-6496	101	7	a	a	X
ejpam-6496	101	8	,	,	PUNCT
ejpam-6496	101	9	b	b	X
ejpam-6496	101	10	[	[	PUNCT
ejpam-6496	101	11	with	with	ADP
ejpam-6496	101	12	c	c	PROPN
ejpam-6496	101	13	<	<	X
ejpam-6496	101	14	d.	d.	PROPN
ejpam-6496	101	15	define	define	VERB
ejpam-6496	101	16	a	a	DET
ejpam-6496	101	17	gauge	gauge	NOUN
ejpam-6496	101	18	δ	δ	NOUN
ejpam-6496	101	19	:	:	PUNCT
ejpam-6496	101	20	[	[	PUNCT
ejpam-6496	101	21	c	c	X
ejpam-6496	101	22	,	,	PUNCT
ejpam-6496	101	23	d	d	NOUN
ejpam-6496	101	24	]	]	X
ejpam-6496	101	25	→	→	X
ejpam-6496	101	26	r+	r+	NOUN
ejpam-6496	101	27	as	as	SCONJ
ejpam-6496	101	28	follows	follow	VERB
ejpam-6496	101	29	:	:	PUNCT
ejpam-6496	101	30	let	let	VERB
ejpam-6496	101	31	x	x	X
ejpam-6496	101	32	∈	∈	PROPN
ejpam-6496	101	33	[	[	PUNCT
ejpam-6496	101	34	c	c	X
ejpam-6496	101	35	,	,	PUNCT
ejpam-6496	101	36	d	d	X
ejpam-6496	101	37	]	]	PUNCT
ejpam-6496	101	38	be	be	AUX
ejpam-6496	101	39	fixed	fix	VERB
ejpam-6496	101	40	.	.	PUNCT
ejpam-6496	102	1	by	by	ADP
ejpam-6496	102	2	lemma	lemma	PROPN
ejpam-6496	102	3	2	2	NUM
ejpam-6496	102	4	,	,	PUNCT
ejpam-6496	102	5	there	there	PRON
ejpam-6496	102	6	exists	exist	VERB
ejpam-6496	102	7	a	a	DET
ejpam-6496	102	8	δx	δx	NOUN
ejpam-6496	102	9	>	>	X
ejpam-6496	102	10	0	0	NUM
ejpam-6496	102	11	such	such	ADJ
ejpam-6496	102	12	that	that	SCONJ
ejpam-6496	102	13	f	f	PROPN
ejpam-6496	102	14	is	be	AUX
ejpam-6496	102	15	either	either	CCONJ
ejpam-6496	102	16	strictly	strictly	ADV
ejpam-6496	102	17	increasing	increase	VERB
ejpam-6496	102	18	or	or	CCONJ
ejpam-6496	102	19	strictly	strictly	ADV
ejpam-6496	102	20	decreasing	decrease	VERB
ejpam-6496	102	21	on	on	ADP
ejpam-6496	102	22	]	]	X
ejpam-6496	102	23	x−	x−	PROPN
ejpam-6496	102	24	δx	δx	PROPN
ejpam-6496	102	25	,	,	PUNCT
ejpam-6496	102	26	x+	x+	ADJ
ejpam-6496	102	27	δx	δx	PROPN
ejpam-6496	102	28	[	[	PUNCT
ejpam-6496	102	29	.	.	PUNCT
ejpam-6496	103	1	we	we	PRON
ejpam-6496	103	2	define	define	VERB
ejpam-6496	103	3	δ(x	δ(x	PROPN
ejpam-6496	103	4	)	)	PUNCT
ejpam-6496	103	5	:	:	PUNCT
ejpam-6496	103	6	=	=	NUM
ejpam-6496	103	7	δx	δx	NOUN
ejpam-6496	103	8	,	,	PUNCT
ejpam-6496	103	9	and	and	CCONJ
ejpam-6496	103	10	let	let	VERB
ejpam-6496	103	11	p	p	NOUN
ejpam-6496	103	12	:	:	PUNCT
ejpam-6496	103	13	=	=	SYM
ejpam-6496	103	14	{	{	PUNCT
ejpam-6496	103	15	(	(	PUNCT
ejpam-6496	103	16	ti	ti	NOUN
ejpam-6496	103	17	,	,	PUNCT
ejpam-6496	103	18	[	[	X
ejpam-6496	103	19	xi−1	xi−1	PROPN
ejpam-6496	103	20	,	,	PUNCT
ejpam-6496	103	21	xi	xi	X
ejpam-6496	103	22	]	]	PUNCT
ejpam-6496	103	23	)	)	PUNCT
ejpam-6496	103	24	:	:	PUNCT
ejpam-6496	104	1	i	i	NOUN
ejpam-6496	104	2	=	=	NOUN
ejpam-6496	104	3	1	1	NUM
ejpam-6496	104	4	,	,	PUNCT
ejpam-6496	104	5	.	.	PUNCT
ejpam-6496	104	6	.	.	PUNCT
ejpam-6496	104	7	.	.	PUNCT
ejpam-6496	105	1	,	,	PUNCT
ejpam-6496	105	2	n	n	CCONJ
ejpam-6496	105	3	}	}	PUNCT
ejpam-6496	105	4	be	be	AUX
ejpam-6496	105	5	a	a	DET
ejpam-6496	105	6	δ	δ	NOUN
ejpam-6496	105	7	-	-	PUNCT
ejpam-6496	105	8	fine	fine	ADJ
ejpam-6496	105	9	tagged	tag	VERB
ejpam-6496	105	10	partition	partition	NOUN
ejpam-6496	105	11	of	of	ADP
ejpam-6496	105	12	[	[	PUNCT
ejpam-6496	105	13	c	c	X
ejpam-6496	105	14	,	,	PUNCT
ejpam-6496	105	15	d	d	NOUN
ejpam-6496	105	16	]	]	PUNCT
ejpam-6496	105	17	.	.	PUNCT
ejpam-6496	106	1	note	note	VERB
ejpam-6496	106	2	that	that	SCONJ
ejpam-6496	106	3	f	f	PROPN
ejpam-6496	106	4	is	be	AUX
ejpam-6496	106	5	either	either	CCONJ
ejpam-6496	106	6	s.	s.	PROPN
ejpam-6496	106	7	prongjit	prongjit	PROPN
ejpam-6496	106	8	,	,	PUNCT
ejpam-6496	106	9	p.	p.	PROPN
ejpam-6496	106	10	prasertsang	prasertsang	PROPN
ejpam-6496	106	11	/	/	SYM
ejpam-6496	106	12	eur	eur	PROPN
ejpam-6496	106	13	.	.	PUNCT
ejpam-6496	107	1	j.	j.	PROPN
ejpam-6496	107	2	pure	pure	PROPN
ejpam-6496	107	3	appl	appl	PROPN
ejpam-6496	107	4	.	.	PROPN
ejpam-6496	107	5	math	math	PROPN
ejpam-6496	107	6	,	,	PUNCT
ejpam-6496	107	7	18	18	NUM
ejpam-6496	107	8	(	(	PUNCT
ejpam-6496	107	9	3	3	NUM
ejpam-6496	107	10	)	)	PUNCT
ejpam-6496	107	11	(	(	PUNCT
ejpam-6496	107	12	2025	2025	NUM
ejpam-6496	107	13	)	)	PUNCT
ejpam-6496	107	14	,	,	PUNCT
ejpam-6496	107	15	6496	6496	NUM
ejpam-6496	107	16	5	5	NUM
ejpam-6496	107	17	of	of	ADP
ejpam-6496	107	18	10	10	NUM
ejpam-6496	107	19	strictly	strictly	ADV
ejpam-6496	107	20	increasing	increase	VERB
ejpam-6496	107	21	or	or	CCONJ
ejpam-6496	107	22	strictly	strictly	ADV
ejpam-6496	107	23	decreasing	decrease	VERB
ejpam-6496	107	24	on	on	ADP
ejpam-6496	107	25	each	each	DET
ejpam-6496	107	26	[	[	X
ejpam-6496	107	27	xi−1	xi−1	PROPN
ejpam-6496	107	28	,	,	PUNCT
ejpam-6496	107	29	xi	xi	X
ejpam-6496	107	30	]	]	PUNCT
ejpam-6496	107	31	,	,	PUNCT
ejpam-6496	107	32	and	and	CCONJ
ejpam-6496	107	33	now	now	ADV
ejpam-6496	107	34	we	we	PRON
ejpam-6496	107	35	have	have	VERB
ejpam-6496	107	36	two	two	NUM
ejpam-6496	107	37	cases	case	NOUN
ejpam-6496	107	38	to	to	PART
ejpam-6496	107	39	consider	consider	VERB
ejpam-6496	107	40	.	.	PUNCT
ejpam-6496	108	1	case	case	NOUN
ejpam-6496	108	2	1	1	NUM
ejpam-6496	108	3	:	:	PUNCT
ejpam-6496	108	4	f	f	X
ejpam-6496	108	5	is	be	AUX
ejpam-6496	108	6	strictly	strictly	ADV
ejpam-6496	108	7	increasing	increase	VERB
ejpam-6496	108	8	on	on	ADP
ejpam-6496	108	9	[	[	X
ejpam-6496	108	10	x0	x0	PROPN
ejpam-6496	108	11	,	,	PUNCT
ejpam-6496	108	12	x1	x1	PROPN
ejpam-6496	108	13	]	]	X
ejpam-6496	108	14	.	.	PUNCT
ejpam-6496	109	1	to	to	PART
ejpam-6496	109	2	show	show	VERB
ejpam-6496	109	3	that	that	SCONJ
ejpam-6496	109	4	the	the	DET
ejpam-6496	109	5	function	function	NOUN
ejpam-6496	109	6	f	f	PROPN
ejpam-6496	109	7	is	be	AUX
ejpam-6496	109	8	strictly	strictly	ADV
ejpam-6496	109	9	increasing	increase	VERB
ejpam-6496	109	10	on	on	ADP
ejpam-6496	109	11	[	[	PUNCT
ejpam-6496	109	12	c	c	X
ejpam-6496	109	13	,	,	PUNCT
ejpam-6496	109	14	d	d	NOUN
ejpam-6496	109	15	]	]	PUNCT
ejpam-6496	109	16	.	.	PUNCT
ejpam-6496	110	1	suppose	suppose	VERB
ejpam-6496	110	2	to	to	ADP
ejpam-6496	110	3	the	the	DET
ejpam-6496	110	4	contrary	contrary	NOUN
ejpam-6496	110	5	that	that	SCONJ
ejpam-6496	110	6	f	f	PROPN
ejpam-6496	110	7	is	be	AUX
ejpam-6496	110	8	strictly	strictly	ADV
ejpam-6496	110	9	decreasing	decrease	VERB
ejpam-6496	110	10	on	on	ADP
ejpam-6496	110	11	[	[	X
ejpam-6496	110	12	x1	x1	PROPN
ejpam-6496	110	13	,	,	PUNCT
ejpam-6496	110	14	x2	x2	PROPN
ejpam-6496	110	15	]	]	X
ejpam-6496	110	16	.	.	PUNCT
ejpam-6496	111	1	this	this	PRON
ejpam-6496	111	2	leads	lead	VERB
ejpam-6496	111	3	f	f	NOUN
ejpam-6496	111	4	to	to	PART
ejpam-6496	111	5	have	have	VERB
ejpam-6496	111	6	a	a	DET
ejpam-6496	111	7	relative	relative	ADJ
ejpam-6496	111	8	maximum	maximum	NOUN
ejpam-6496	111	9	at	at	ADP
ejpam-6496	111	10	the	the	DET
ejpam-6496	111	11	interior	interior	ADJ
ejpam-6496	111	12	point	point	NOUN
ejpam-6496	111	13	x1	x1	PROPN
ejpam-6496	111	14	∈	∈	PROPN
ejpam-6496	111	15	]	]	X
ejpam-6496	111	16	x0	x0	PROPN
ejpam-6496	111	17	,	,	PUNCT
ejpam-6496	111	18	x2	x2	PROPN
ejpam-6496	111	19	[	[	PUNCT
ejpam-6496	111	20	.	.	PUNCT
ejpam-6496	112	1	hence	hence	ADV
ejpam-6496	112	2	,	,	PUNCT
ejpam-6496	112	3	f	f	PROPN
ejpam-6496	112	4	′(x1	′(x1	PROPN
ejpam-6496	112	5	)	)	PUNCT
ejpam-6496	113	1	=	=	SYM
ejpam-6496	113	2	0	0	NUM
ejpam-6496	113	3	and	and	CCONJ
ejpam-6496	113	4	a	a	DET
ejpam-6496	113	5	contradiction	contradiction	NOUN
ejpam-6496	113	6	is	be	AUX
ejpam-6496	113	7	obtained	obtain	VERB
ejpam-6496	113	8	now	now	ADV
ejpam-6496	113	9	.	.	PUNCT
ejpam-6496	114	1	continuing	continue	VERB
ejpam-6496	114	2	the	the	DET
ejpam-6496	114	3	process	process	NOUN
ejpam-6496	114	4	,	,	PUNCT
ejpam-6496	114	5	we	we	PRON
ejpam-6496	114	6	can	can	AUX
ejpam-6496	114	7	conclude	conclude	VERB
ejpam-6496	114	8	that	that	SCONJ
ejpam-6496	114	9	f	f	PROPN
ejpam-6496	114	10	is	be	AUX
ejpam-6496	114	11	strictly	strictly	ADV
ejpam-6496	114	12	increasing	increase	VERB
ejpam-6496	114	13	on	on	ADP
ejpam-6496	114	14	[	[	PUNCT
ejpam-6496	114	15	c	c	X
ejpam-6496	114	16	,	,	PUNCT
ejpam-6496	114	17	d	d	NOUN
ejpam-6496	114	18	]	]	PUNCT
ejpam-6496	114	19	.	.	PUNCT
ejpam-6496	115	1	case	case	NOUN
ejpam-6496	115	2	2	2	NUM
ejpam-6496	115	3	:	:	PUNCT
ejpam-6496	115	4	f	f	X
ejpam-6496	115	5	is	be	AUX
ejpam-6496	115	6	strictly	strictly	ADV
ejpam-6496	115	7	decreasing	decrease	VERB
ejpam-6496	115	8	on	on	ADP
ejpam-6496	115	9	[	[	X
ejpam-6496	115	10	x0	x0	PROPN
ejpam-6496	115	11	,	,	PUNCT
ejpam-6496	115	12	x1	x1	PROPN
ejpam-6496	115	13	]	]	X
ejpam-6496	115	14	.	.	PUNCT
ejpam-6496	116	1	similarly	similarly	ADV
ejpam-6496	116	2	,	,	PUNCT
ejpam-6496	116	3	we	we	PRON
ejpam-6496	116	4	can	can	AUX
ejpam-6496	116	5	prove	prove	VERB
ejpam-6496	116	6	that	that	SCONJ
ejpam-6496	116	7	f	f	PROPN
ejpam-6496	116	8	is	be	AUX
ejpam-6496	116	9	strictly	strictly	ADV
ejpam-6496	116	10	decreasing	decrease	VERB
ejpam-6496	116	11	on	on	ADP
ejpam-6496	116	12	[	[	PUNCT
ejpam-6496	116	13	c	c	X
ejpam-6496	116	14	,	,	PUNCT
ejpam-6496	116	15	d	d	NOUN
ejpam-6496	116	16	]	]	X
ejpam-6496	116	17	.	.	PUNCT
ejpam-6496	117	1	these	these	DET
ejpam-6496	117	2	two	two	NUM
ejpam-6496	117	3	cases	case	NOUN
ejpam-6496	117	4	give	give	VERB
ejpam-6496	117	5	the	the	DET
ejpam-6496	117	6	conclusion	conclusion	NOUN
ejpam-6496	117	7	that	that	SCONJ
ejpam-6496	117	8	f	f	PROPN
ejpam-6496	117	9	is	be	AUX
ejpam-6496	117	10	either	either	CCONJ
ejpam-6496	117	11	strictly	strictly	ADV
ejpam-6496	117	12	increasing	increase	VERB
ejpam-6496	117	13	or	or	CCONJ
ejpam-6496	117	14	strictly	strictly	ADV
ejpam-6496	117	15	decreasing	decrease	VERB
ejpam-6496	117	16	on	on	ADP
ejpam-6496	117	17	[	[	PUNCT
ejpam-6496	117	18	c	c	X
ejpam-6496	117	19	,	,	PUNCT
ejpam-6496	117	20	d	d	NOUN
ejpam-6496	117	21	]	]	X
ejpam-6496	117	22	.	.	PUNCT
ejpam-6496	118	1	therefore	therefore	ADV
ejpam-6496	118	2	,	,	PUNCT
ejpam-6496	118	3	f	f	PROPN
ejpam-6496	118	4	is	be	AUX
ejpam-6496	118	5	strictly	strictly	ADV
ejpam-6496	118	6	monotone	monotone	ADJ
ejpam-6496	118	7	on	on	ADP
ejpam-6496	118	8	[	[	PUNCT
ejpam-6496	118	9	c	c	X
ejpam-6496	118	10	,	,	PUNCT
ejpam-6496	118	11	d	d	NOUN
ejpam-6496	118	12	]	]	X
ejpam-6496	118	13	.	.	PUNCT
ejpam-6496	119	1	since	since	SCONJ
ejpam-6496	119	2	c	c	PROPN
ejpam-6496	119	3	and	and	CCONJ
ejpam-6496	119	4	d	d	PROPN
ejpam-6496	119	5	are	be	AUX
ejpam-6496	119	6	arbitrary	arbitrary	ADJ
ejpam-6496	119	7	elements	element	NOUN
ejpam-6496	119	8	of	of	ADP
ejpam-6496	119	9	]	]	X
ejpam-6496	119	10	a	a	X
ejpam-6496	119	11	,	,	PUNCT
ejpam-6496	119	12	b	b	X
ejpam-6496	119	13	[	[	PUNCT
ejpam-6496	119	14	,	,	PUNCT
ejpam-6496	119	15	so	so	CCONJ
ejpam-6496	119	16	f	f	PROPN
ejpam-6496	119	17	is	be	AUX
ejpam-6496	119	18	strictly	strictly	ADV
ejpam-6496	119	19	monotone	monotone	ADJ
ejpam-6496	119	20	on	on	ADP
ejpam-6496	119	21	]	]	PUNCT
ejpam-6496	119	22	a	a	X
ejpam-6496	119	23	,	,	PUNCT
ejpam-6496	119	24	b	b	X
ejpam-6496	119	25	[	[	PUNCT
ejpam-6496	119	26	.	.	PUNCT
ejpam-6496	120	1	the	the	DET
ejpam-6496	120	2	proof	proof	NOUN
ejpam-6496	120	3	is	be	AUX
ejpam-6496	120	4	completed	complete	VERB
ejpam-6496	120	5	.	.	PUNCT
ejpam-6496	121	1	corollary	corollary	ADJ
ejpam-6496	121	2	1	1	PROPN
ejpam-6496	121	3	.	.	PUNCT
ejpam-6496	121	4	suppose	suppose	VERB
ejpam-6496	121	5	that	that	SCONJ
ejpam-6496	121	6	f	f	PROPN
ejpam-6496	121	7	is	be	AUX
ejpam-6496	121	8	differentiable	differentiable	ADJ
ejpam-6496	121	9	on	on	ADP
ejpam-6496	121	10	]	]	PUNCT
ejpam-6496	121	11	a	a	X
ejpam-6496	121	12	,	,	PUNCT
ejpam-6496	121	13	b	b	X
ejpam-6496	121	14	[	[	PUNCT
ejpam-6496	121	15	.	.	PUNCT
ejpam-6496	122	1	if	if	SCONJ
ejpam-6496	122	2	f	f	PROPN
ejpam-6496	122	3	′(x	′(x	NOUN
ejpam-6496	122	4	)	)	PUNCT
ejpam-6496	122	5	̸=	̸=	PROPN
ejpam-6496	122	6	0	0	NUM
ejpam-6496	122	7	for	for	ADP
ejpam-6496	122	8	all	all	DET
ejpam-6496	122	9	x	x	SYM
ejpam-6496	122	10	∈	∈	PROPN
ejpam-6496	122	11	]	]	PUNCT
ejpam-6496	122	12	a	a	X
ejpam-6496	122	13	,	,	PUNCT
ejpam-6496	122	14	b	b	X
ejpam-6496	122	15	[	[	PUNCT
ejpam-6496	122	16	,	,	PUNCT
ejpam-6496	122	17	then	then	ADV
ejpam-6496	122	18	either	either	CCONJ
ejpam-6496	122	19	f	f	PROPN
ejpam-6496	122	20	′(x	′(x	PROPN
ejpam-6496	122	21	)	)	PUNCT
ejpam-6496	122	22	>	>	X
ejpam-6496	122	23	0	0	PUNCT
ejpam-6496	123	1	for	for	ADP
ejpam-6496	123	2	all	all	DET
ejpam-6496	123	3	x	x	SYM
ejpam-6496	123	4	∈	∈	PROPN
ejpam-6496	123	5	]	]	PUNCT
ejpam-6496	123	6	a	a	X
ejpam-6496	123	7	,	,	PUNCT
ejpam-6496	123	8	b	b	NOUN
ejpam-6496	123	9	[	[	PUNCT
ejpam-6496	123	10	or	or	CCONJ
ejpam-6496	123	11	f	f	PROPN
ejpam-6496	123	12	′(x	′(x	NOUN
ejpam-6496	123	13	)	)	PUNCT
ejpam-6496	123	14	<	<	X
ejpam-6496	123	15	0	0	PUNCT
ejpam-6496	123	16	for	for	ADP
ejpam-6496	123	17	all	all	DET
ejpam-6496	123	18	x	x	SYM
ejpam-6496	123	19	∈	∈	PROPN
ejpam-6496	123	20	]	]	PUNCT
ejpam-6496	123	21	a	a	X
ejpam-6496	123	22	,	,	PUNCT
ejpam-6496	123	23	b	b	X
ejpam-6496	123	24	[	[	PUNCT
ejpam-6496	123	25	.	.	PUNCT
ejpam-6496	123	26	proof	proof	NOUN
ejpam-6496	123	27	.	.	PUNCT
ejpam-6496	124	1	from	from	ADP
ejpam-6496	124	2	theorem	theorem	ADJ
ejpam-6496	124	3	3	3	NUM
ejpam-6496	124	4	,	,	PUNCT
ejpam-6496	124	5	we	we	PRON
ejpam-6496	124	6	have	have	VERB
ejpam-6496	124	7	two	two	NUM
ejpam-6496	124	8	cases	case	NOUN
ejpam-6496	124	9	to	to	PART
ejpam-6496	124	10	consider	consider	VERB
ejpam-6496	124	11	.	.	PUNCT
ejpam-6496	125	1	case	case	NOUN
ejpam-6496	125	2	1	1	NUM
ejpam-6496	125	3	:	:	PUNCT
ejpam-6496	125	4	f	f	X
ejpam-6496	125	5	is	be	AUX
ejpam-6496	125	6	strictly	strictly	ADV
ejpam-6496	125	7	increasing	increase	VERB
ejpam-6496	125	8	on	on	ADP
ejpam-6496	125	9	]	]	PUNCT
ejpam-6496	125	10	a	a	X
ejpam-6496	125	11	,	,	PUNCT
ejpam-6496	125	12	b	b	X
ejpam-6496	125	13	[	[	PUNCT
ejpam-6496	125	14	.	.	PUNCT
ejpam-6496	126	1	let	let	VERB
ejpam-6496	126	2	x	x	SYM
ejpam-6496	126	3	∈	∈	PROPN
ejpam-6496	126	4	]	]	PUNCT
ejpam-6496	126	5	a	a	X
ejpam-6496	126	6	,	,	PUNCT
ejpam-6496	126	7	b	b	X
ejpam-6496	126	8	[	[	PUNCT
ejpam-6496	126	9	be	be	AUX
ejpam-6496	126	10	fixed	fix	VERB
ejpam-6496	126	11	.	.	PUNCT
ejpam-6496	127	1	since	since	SCONJ
ejpam-6496	127	2	f(y)−f(x	f(y)−f(x	NOUN
ejpam-6496	127	3	)	)	PUNCT
ejpam-6496	127	4	y−x	y−x	PROPN
ejpam-6496	127	5	>	>	X
ejpam-6496	127	6	0	0	NUM
ejpam-6496	128	1	for	for	ADP
ejpam-6496	128	2	every	every	DET
ejpam-6496	128	3	y	y	PROPN
ejpam-6496	128	4	∈	∈	PROPN
ejpam-6496	128	5	]	]	PUNCT
ejpam-6496	128	6	a	a	X
ejpam-6496	128	7	,	,	PUNCT
ejpam-6496	128	8	b	b	X
ejpam-6496	128	9	[	[	PUNCT
ejpam-6496	128	10	with	with	ADP
ejpam-6496	128	11	y	y	PROPN
ejpam-6496	128	12	̸=	̸=	PROPN
ejpam-6496	128	13	x	x	NUM
ejpam-6496	128	14	,	,	PUNCT
ejpam-6496	128	15	then	then	ADV
ejpam-6496	128	16	f	f	PROPN
ejpam-6496	128	17	′(x	′(x	PROPN
ejpam-6496	128	18	)	)	PUNCT
ejpam-6496	128	19	=	=	VERB
ejpam-6496	129	1	lim	lim	PROPN
ejpam-6496	129	2	y→x	y→x	PROPN
ejpam-6496	129	3	f(y)−	f(y)−	PROPN
ejpam-6496	129	4	f(x	f(x	PROPN
ejpam-6496	129	5	)	)	PUNCT
ejpam-6496	129	6	y	y	PROPN
ejpam-6496	129	7	−	−	PROPN
ejpam-6496	129	8	x	x	SYM
ejpam-6496	129	9	≥	≥	NOUN
ejpam-6496	129	10	0	0	NUM
ejpam-6496	129	11	.	.	PUNCT
ejpam-6496	130	1	hence	hence	ADV
ejpam-6496	130	2	,	,	PUNCT
ejpam-6496	130	3	we	we	PRON
ejpam-6496	130	4	conclude	conclude	VERB
ejpam-6496	130	5	that	that	SCONJ
ejpam-6496	130	6	f	f	PROPN
ejpam-6496	130	7	′(x	′(x	PROPN
ejpam-6496	130	8	)	)	PUNCT
ejpam-6496	130	9	>	>	X
ejpam-6496	130	10	0	0	X
ejpam-6496	130	11	.	.	PUNCT
ejpam-6496	130	12	case	case	NOUN
ejpam-6496	130	13	2	2	NUM
ejpam-6496	130	14	:	:	PUNCT
ejpam-6496	130	15	f	f	X
ejpam-6496	130	16	is	be	AUX
ejpam-6496	130	17	strictly	strictly	ADV
ejpam-6496	130	18	decreasing	decrease	VERB
ejpam-6496	130	19	on	on	ADP
ejpam-6496	130	20	]	]	PUNCT
ejpam-6496	130	21	a	a	X
ejpam-6496	130	22	,	,	PUNCT
ejpam-6496	130	23	b	b	X
ejpam-6496	130	24	[	[	PUNCT
ejpam-6496	130	25	.	.	PUNCT
ejpam-6496	131	1	we	we	PRON
ejpam-6496	131	2	can	can	AUX
ejpam-6496	131	3	prove	prove	VERB
ejpam-6496	131	4	in	in	ADP
ejpam-6496	131	5	the	the	DET
ejpam-6496	131	6	same	same	ADJ
ejpam-6496	131	7	manner	manner	NOUN
ejpam-6496	131	8	that	that	PRON
ejpam-6496	131	9	f	f	PROPN
ejpam-6496	131	10	′(x	′(x	PROPN
ejpam-6496	131	11	)	)	PUNCT
ejpam-6496	131	12	<	<	X
ejpam-6496	131	13	0	0	PUNCT
ejpam-6496	131	14	for	for	ADP
ejpam-6496	131	15	all	all	DET
ejpam-6496	131	16	x	x	SYM
ejpam-6496	131	17	∈	∈	PROPN
ejpam-6496	131	18	]	]	PUNCT
ejpam-6496	131	19	a	a	X
ejpam-6496	131	20	,	,	PUNCT
ejpam-6496	131	21	b	b	X
ejpam-6496	131	22	[	[	PUNCT
ejpam-6496	131	23	.	.	PUNCT
ejpam-6496	132	1	theorem	theorem	NOUN
ejpam-6496	132	2	4	4	NUM
ejpam-6496	132	3	.	.	PUNCT
ejpam-6496	132	4	suppose	suppose	VERB
ejpam-6496	132	5	that	that	SCONJ
ejpam-6496	132	6	f	f	PROPN
ejpam-6496	132	7	is	be	AUX
ejpam-6496	132	8	continuous	continuous	ADJ
ejpam-6496	132	9	on	on	ADP
ejpam-6496	132	10	[	[	PUNCT
ejpam-6496	132	11	a	a	PROPN
ejpam-6496	132	12	,	,	PUNCT
ejpam-6496	132	13	b	b	NOUN
ejpam-6496	132	14	]	]	PUNCT
ejpam-6496	132	15	and	and	CCONJ
ejpam-6496	132	16	differentiable	differentiable	VERB
ejpam-6496	132	17	on	on	ADP
ejpam-6496	132	18	]	]	PUNCT
ejpam-6496	132	19	a	a	X
ejpam-6496	132	20	,	,	PUNCT
ejpam-6496	132	21	b	b	X
ejpam-6496	132	22	[	[	PUNCT
ejpam-6496	132	23	.	.	PUNCT
ejpam-6496	133	1	if	if	SCONJ
ejpam-6496	133	2	f	f	PROPN
ejpam-6496	133	3	′(x	′(x	NOUN
ejpam-6496	133	4	)	)	PUNCT
ejpam-6496	133	5	̸=	̸=	PROPN
ejpam-6496	133	6	0	0	NUM
ejpam-6496	133	7	for	for	ADP
ejpam-6496	133	8	all	all	DET
ejpam-6496	133	9	x	x	SYM
ejpam-6496	133	10	∈	∈	PROPN
ejpam-6496	133	11	]	]	PUNCT
ejpam-6496	133	12	a	a	X
ejpam-6496	133	13	,	,	PUNCT
ejpam-6496	133	14	b	b	X
ejpam-6496	133	15	[	[	PUNCT
ejpam-6496	133	16	,	,	PUNCT
ejpam-6496	133	17	then	then	ADV
ejpam-6496	133	18	f	f	PROPN
ejpam-6496	133	19	is	be	AUX
ejpam-6496	133	20	strictly	strictly	ADV
ejpam-6496	133	21	monotone	monotone	ADJ
ejpam-6496	133	22	on	on	ADP
ejpam-6496	133	23	[	[	PUNCT
ejpam-6496	133	24	a	a	PROPN
ejpam-6496	133	25	,	,	PUNCT
ejpam-6496	133	26	b	b	NOUN
ejpam-6496	133	27	]	]	PUNCT
ejpam-6496	133	28	.	.	PUNCT
ejpam-6496	134	1	proof	proof	NOUN
ejpam-6496	134	2	.	.	PUNCT
ejpam-6496	135	1	from	from	ADP
ejpam-6496	135	2	theorem	theorem	ADJ
ejpam-6496	135	3	3	3	NUM
ejpam-6496	135	4	,	,	PUNCT
ejpam-6496	135	5	let	let	VERB
ejpam-6496	135	6	us	we	PRON
ejpam-6496	135	7	consider	consider	VERB
ejpam-6496	135	8	as	as	SCONJ
ejpam-6496	135	9	follows	follow	VERB
ejpam-6496	135	10	:	:	PUNCT
ejpam-6496	135	11	case	case	NOUN
ejpam-6496	135	12	1	1	NUM
ejpam-6496	135	13	:	:	PUNCT
ejpam-6496	135	14	f	f	X
ejpam-6496	135	15	is	be	AUX
ejpam-6496	135	16	strictly	strictly	ADV
ejpam-6496	135	17	increasing	increase	VERB
ejpam-6496	135	18	on	on	ADP
ejpam-6496	135	19	]	]	PUNCT
ejpam-6496	135	20	a	a	X
ejpam-6496	135	21	,	,	PUNCT
ejpam-6496	135	22	b	b	X
ejpam-6496	135	23	[	[	PUNCT
ejpam-6496	135	24	.	.	PUNCT
ejpam-6496	136	1	let	let	VERB
ejpam-6496	136	2	c	c	X
ejpam-6496	136	3	,	,	PUNCT
ejpam-6496	136	4	s	s	X
ejpam-6496	136	5	,	,	PUNCT
ejpam-6496	136	6	d	d	PROPN
ejpam-6496	136	7	∈	∈	PROPN
ejpam-6496	136	8	]	]	PUNCT
ejpam-6496	136	9	a	a	X
ejpam-6496	136	10	,	,	PUNCT
ejpam-6496	136	11	b	b	X
ejpam-6496	136	12	[	[	PUNCT
ejpam-6496	136	13	be	be	AUX
ejpam-6496	136	14	such	such	ADJ
ejpam-6496	136	15	that	that	SCONJ
ejpam-6496	136	16	c	c	PROPN
ejpam-6496	136	17	<	<	X
ejpam-6496	136	18	s	s	X
ejpam-6496	136	19	<	<	X
ejpam-6496	136	20	d.	d.	PROPN
ejpam-6496	136	21	since	since	SCONJ
ejpam-6496	136	22	f	f	PROPN
ejpam-6496	136	23	is	be	AUX
ejpam-6496	136	24	continuous	continuous	ADJ
ejpam-6496	136	25	at	at	ADP
ejpam-6496	136	26	a	a	PRON
ejpam-6496	136	27	and	and	CCONJ
ejpam-6496	136	28	f(y	f(y	NOUN
ejpam-6496	136	29	)	)	PUNCT
ejpam-6496	136	30	<	<	X
ejpam-6496	136	31	f(c	f(c	PROPN
ejpam-6496	136	32	)	)	PUNCT
ejpam-6496	136	33	for	for	ADP
ejpam-6496	136	34	every	every	DET
ejpam-6496	136	35	y	y	PROPN
ejpam-6496	136	36	∈	∈	PROPN
ejpam-6496	136	37	]	]	PUNCT
ejpam-6496	136	38	a	a	X
ejpam-6496	136	39	,	,	PUNCT
ejpam-6496	136	40	c	c	PROPN
ejpam-6496	136	41	[	[	PUNCT
ejpam-6496	136	42	,	,	PUNCT
ejpam-6496	136	43	then	then	ADV
ejpam-6496	136	44	f(a	f(a	PROPN
ejpam-6496	136	45	)	)	PUNCT
ejpam-6496	137	1	=	=	SYM
ejpam-6496	137	2	lim	lim	PROPN
ejpam-6496	137	3	y→a+	y→a+	PROPN
ejpam-6496	137	4	f(y	f(y	PROPN
ejpam-6496	137	5	)	)	PUNCT
ejpam-6496	137	6	≤	≤	NUM
ejpam-6496	137	7	f(c	f(c	PROPN
ejpam-6496	137	8	)	)	PUNCT
ejpam-6496	137	9	<	<	X
ejpam-6496	137	10	f(s	f(	NOUN
ejpam-6496	137	11	)	)	PUNCT
ejpam-6496	137	12	.	.	PUNCT
ejpam-6496	138	1	likewise	likewise	ADV
ejpam-6496	138	2	,	,	PUNCT
ejpam-6496	138	3	since	since	SCONJ
ejpam-6496	138	4	f	f	PROPN
ejpam-6496	138	5	is	be	AUX
ejpam-6496	138	6	continuous	continuous	ADJ
ejpam-6496	138	7	at	at	ADP
ejpam-6496	138	8	b	b	NOUN
ejpam-6496	138	9	and	and	CCONJ
ejpam-6496	138	10	f(d	f(d	PROPN
ejpam-6496	138	11	)	)	PUNCT
ejpam-6496	138	12	<	<	X
ejpam-6496	138	13	f(y	f(y	NOUN
ejpam-6496	138	14	)	)	PUNCT
ejpam-6496	138	15	for	for	ADP
ejpam-6496	138	16	all	all	DET
ejpam-6496	138	17	y	y	PROPN
ejpam-6496	138	18	∈	∈	PROPN
ejpam-6496	138	19	]	]	PUNCT
ejpam-6496	138	20	d	d	X
ejpam-6496	138	21	,	,	PUNCT
ejpam-6496	138	22	b	b	PROPN
ejpam-6496	138	23	[	[	PUNCT
ejpam-6496	138	24	,	,	PUNCT
ejpam-6496	138	25	then	then	ADV
ejpam-6496	138	26	f(b	f(b	PROPN
ejpam-6496	138	27	)	)	PUNCT
ejpam-6496	138	28	=	=	PROPN
ejpam-6496	138	29	lim	lim	PROPN
ejpam-6496	138	30	y→b−	y→b−	PROPN
ejpam-6496	138	31	f(y	f(y	PROPN
ejpam-6496	138	32	)	)	PUNCT
ejpam-6496	138	33	≥	≥	NOUN
ejpam-6496	138	34	f(d	f(d	PROPN
ejpam-6496	138	35	)	)	PUNCT
ejpam-6496	138	36	>	>	X
ejpam-6496	138	37	f(s	f(	NOUN
ejpam-6496	138	38	)	)	PUNCT
ejpam-6496	138	39	.	.	PUNCT
ejpam-6496	139	1	this	this	PRON
ejpam-6496	139	2	give	give	VERB
ejpam-6496	139	3	us	we	PRON
ejpam-6496	139	4	f(a	f(a	NOUN
ejpam-6496	139	5	)	)	PUNCT
ejpam-6496	139	6	<	<	X
ejpam-6496	139	7	f(s	f(	NOUN
ejpam-6496	139	8	)	)	PUNCT
ejpam-6496	139	9	<	<	X
ejpam-6496	139	10	f(b	f(b	PROPN
ejpam-6496	139	11	)	)	PUNCT
ejpam-6496	139	12	for	for	ADP
ejpam-6496	139	13	all	all	DET
ejpam-6496	139	14	s	s	PROPN
ejpam-6496	139	15	∈	∈	X
ejpam-6496	139	16	]	]	PUNCT
ejpam-6496	139	17	a	a	X
ejpam-6496	139	18	,	,	PUNCT
ejpam-6496	139	19	b	b	X
ejpam-6496	139	20	[	[	PUNCT
ejpam-6496	139	21	.	.	PUNCT
ejpam-6496	140	1	consequently	consequently	ADV
ejpam-6496	140	2	,	,	PUNCT
ejpam-6496	140	3	we	we	PRON
ejpam-6496	140	4	can	can	AUX
ejpam-6496	140	5	prove	prove	VERB
ejpam-6496	140	6	that	that	SCONJ
ejpam-6496	140	7	f	f	PROPN
ejpam-6496	140	8	is	be	AUX
ejpam-6496	140	9	strictly	strictly	ADV
ejpam-6496	140	10	increasing	increase	VERB
ejpam-6496	140	11	on	on	ADP
ejpam-6496	140	12	[	[	PUNCT
ejpam-6496	140	13	a	a	PROPN
ejpam-6496	140	14	,	,	PUNCT
ejpam-6496	140	15	b	b	NOUN
ejpam-6496	140	16	]	]	PUNCT
ejpam-6496	140	17	.	.	PUNCT
ejpam-6496	141	1	case	case	NOUN
ejpam-6496	141	2	2	2	NUM
ejpam-6496	141	3	:	:	PUNCT
ejpam-6496	141	4	f	f	X
ejpam-6496	141	5	is	be	AUX
ejpam-6496	141	6	strictly	strictly	ADV
ejpam-6496	141	7	decreasing	decrease	VERB
ejpam-6496	141	8	on	on	ADP
ejpam-6496	141	9	]	]	PUNCT
ejpam-6496	141	10	a	a	X
ejpam-6496	141	11	,	,	PUNCT
ejpam-6496	141	12	b	b	X
ejpam-6496	141	13	[	[	PUNCT
ejpam-6496	141	14	.	.	PUNCT
ejpam-6496	141	15	similar	similar	ADJ
ejpam-6496	141	16	to	to	ADP
ejpam-6496	141	17	case	case	NOUN
ejpam-6496	141	18	1	1	NUM
ejpam-6496	141	19	,	,	PUNCT
ejpam-6496	141	20	we	we	PRON
ejpam-6496	141	21	can	can	AUX
ejpam-6496	141	22	conclude	conclude	VERB
ejpam-6496	141	23	that	that	SCONJ
ejpam-6496	141	24	f	f	PROPN
ejpam-6496	141	25	s.	s.	PROPN
ejpam-6496	141	26	prongjit	prongjit	PROPN
ejpam-6496	141	27	,	,	PUNCT
ejpam-6496	141	28	p.	p.	PROPN
ejpam-6496	141	29	prasertsang	prasertsang	PROPN
ejpam-6496	141	30	/	/	SYM
ejpam-6496	141	31	eur	eur	PROPN
ejpam-6496	141	32	.	.	PUNCT
ejpam-6496	142	1	j.	j.	PROPN
ejpam-6496	142	2	pure	pure	PROPN
ejpam-6496	142	3	appl	appl	PROPN
ejpam-6496	142	4	.	.	PROPN
ejpam-6496	142	5	math	math	PROPN
ejpam-6496	142	6	,	,	PUNCT
ejpam-6496	142	7	18	18	NUM
ejpam-6496	142	8	(	(	PUNCT
ejpam-6496	142	9	3	3	NUM
ejpam-6496	142	10	)	)	PUNCT
ejpam-6496	142	11	(	(	PUNCT
ejpam-6496	142	12	2025	2025	NUM
ejpam-6496	142	13	)	)	PUNCT
ejpam-6496	142	14	,	,	PUNCT
ejpam-6496	142	15	6496	6496	NUM
ejpam-6496	142	16	6	6	NUM
ejpam-6496	142	17	of	of	ADP
ejpam-6496	142	18	10	10	NUM
ejpam-6496	142	19	is	be	AUX
ejpam-6496	142	20	strictly	strictly	ADV
ejpam-6496	142	21	decreasing	decrease	VERB
ejpam-6496	142	22	on	on	ADP
ejpam-6496	142	23	[	[	PUNCT
ejpam-6496	142	24	a	a	PROPN
ejpam-6496	142	25	,	,	PUNCT
ejpam-6496	142	26	b	b	NOUN
ejpam-6496	142	27	]	]	PUNCT
ejpam-6496	142	28	.	.	PUNCT
ejpam-6496	143	1	by	by	ADP
ejpam-6496	143	2	applying	apply	VERB
ejpam-6496	143	3	theorem	theorem	NOUN
ejpam-6496	143	4	4	4	NUM
ejpam-6496	143	5	,	,	PUNCT
ejpam-6496	143	6	the	the	DET
ejpam-6496	143	7	proving	proving	NOUN
ejpam-6496	143	8	of	of	ADP
ejpam-6496	143	9	two	two	NUM
ejpam-6496	143	10	following	follow	VERB
ejpam-6496	143	11	well	well	ADV
ejpam-6496	143	12	known	know	VERB
ejpam-6496	143	13	theorems	theorem	NOUN
ejpam-6496	143	14	(	(	PUNCT
ejpam-6496	143	15	see	see	VERB
ejpam-6496	143	16	[	[	X
ejpam-6496	143	17	9	9	NUM
ejpam-6496	143	18	]	]	PUNCT
ejpam-6496	143	19	)	)	PUNCT
ejpam-6496	143	20	give	give	VERB
ejpam-6496	143	21	a	a	DET
ejpam-6496	143	22	different	different	ADJ
ejpam-6496	143	23	view	view	NOUN
ejpam-6496	143	24	of	of	ADP
ejpam-6496	143	25	these	these	DET
ejpam-6496	143	26	theorems	theorem	NOUN
ejpam-6496	143	27	.	.	PUNCT
ejpam-6496	144	1	theorem	theorem	NOUN
ejpam-6496	144	2	5	5	NUM
ejpam-6496	144	3	(	(	PUNCT
ejpam-6496	144	4	mean	mean	NOUN
ejpam-6496	144	5	value	value	NOUN
ejpam-6496	144	6	theorem	theorem	NOUN
ejpam-6496	144	7	)	)	PUNCT
ejpam-6496	144	8	.	.	PUNCT
ejpam-6496	145	1	let	let	VERB
ejpam-6496	145	2	f	f	PRON
ejpam-6496	145	3	be	be	AUX
ejpam-6496	145	4	continuous	continuous	ADJ
ejpam-6496	145	5	on	on	ADP
ejpam-6496	145	6	[	[	PUNCT
ejpam-6496	145	7	a	a	PROPN
ejpam-6496	145	8	,	,	PUNCT
ejpam-6496	145	9	b	b	NOUN
ejpam-6496	145	10	]	]	PUNCT
ejpam-6496	145	11	and	and	CCONJ
ejpam-6496	145	12	differentiable	differentiable	VERB
ejpam-6496	145	13	on	on	ADP
ejpam-6496	145	14	]	]	PUNCT
ejpam-6496	145	15	a	a	X
ejpam-6496	145	16	,	,	PUNCT
ejpam-6496	145	17	b	b	X
ejpam-6496	145	18	[	[	PUNCT
ejpam-6496	145	19	.	.	PUNCT
ejpam-6496	146	1	then	then	ADV
ejpam-6496	146	2	,	,	PUNCT
ejpam-6496	146	3	there	there	PRON
ejpam-6496	146	4	exists	exist	VERB
ejpam-6496	146	5	an	an	DET
ejpam-6496	146	6	element	element	NOUN
ejpam-6496	146	7	c	c	NOUN
ejpam-6496	146	8	∈	∈	PROPN
ejpam-6496	146	9	]	]	PUNCT
ejpam-6496	146	10	a	a	X
ejpam-6496	146	11	,	,	PUNCT
ejpam-6496	146	12	b	b	NOUN
ejpam-6496	146	13	[	[	PUNCT
ejpam-6496	146	14	which	which	DET
ejpam-6496	146	15	f(b)−	f(b)−	PROPN
ejpam-6496	146	16	f(a	f(a	PROPN
ejpam-6496	146	17	)	)	PUNCT
ejpam-6496	147	1	=	=	SYM
ejpam-6496	147	2	f	f	PROPN
ejpam-6496	147	3	′(c)(b−	′(c)(b−	NOUN
ejpam-6496	147	4	a	a	X
ejpam-6496	147	5	)	)	PUNCT
ejpam-6496	147	6	.	.	PUNCT
ejpam-6496	148	1	proof	proof	NOUN
ejpam-6496	148	2	.	.	PUNCT
ejpam-6496	149	1	let	let	VERB
ejpam-6496	149	2	g	g	NOUN
ejpam-6496	149	3	:	:	PUNCT
ejpam-6496	149	4	[	[	PUNCT
ejpam-6496	149	5	a	a	X
ejpam-6496	149	6	,	,	PUNCT
ejpam-6496	149	7	b	b	NOUN
ejpam-6496	149	8	]	]	PUNCT
ejpam-6496	149	9	→	→	SYM
ejpam-6496	149	10	r	r	NOUN
ejpam-6496	149	11	defined	define	VERB
ejpam-6496	149	12	by	by	ADP
ejpam-6496	149	13	g(x	g(x	NOUN
ejpam-6496	149	14	)	)	PUNCT
ejpam-6496	149	15	:	:	PUNCT
ejpam-6496	150	1	=	=	PUNCT
ejpam-6496	150	2	f(x)−	f(x)−	PROPN
ejpam-6496	150	3	f(b)−	f(b)−	PROPN
ejpam-6496	150	4	f(a	f(a	PROPN
ejpam-6496	150	5	)	)	PUNCT
ejpam-6496	150	6	b−	b−	PROPN
ejpam-6496	150	7	a	a	PRON
ejpam-6496	150	8	x.	x.	NOUN
ejpam-6496	150	9	suppose	suppose	VERB
ejpam-6496	150	10	to	to	ADP
ejpam-6496	150	11	the	the	DET
ejpam-6496	150	12	contrary	contrary	NOUN
ejpam-6496	150	13	that	that	PRON
ejpam-6496	150	14	g′(x	g′(x	AUX
ejpam-6496	150	15	)	)	PUNCT
ejpam-6496	150	16	̸=	̸=	NOUN
ejpam-6496	150	17	0	0	NUM
ejpam-6496	150	18	for	for	ADP
ejpam-6496	150	19	all	all	DET
ejpam-6496	150	20	x	x	SYM
ejpam-6496	150	21	∈	∈	PROPN
ejpam-6496	150	22	]	]	PUNCT
ejpam-6496	150	23	a	a	X
ejpam-6496	150	24	,	,	PUNCT
ejpam-6496	150	25	b	b	X
ejpam-6496	150	26	[	[	PUNCT
ejpam-6496	150	27	.	.	PUNCT
ejpam-6496	151	1	since	since	SCONJ
ejpam-6496	151	2	g	g	PROPN
ejpam-6496	151	3	satisfies	satisfy	VERB
ejpam-6496	151	4	the	the	DET
ejpam-6496	151	5	hypotheses	hypothesis	NOUN
ejpam-6496	151	6	of	of	ADP
ejpam-6496	151	7	theorem	theorem	NOUN
ejpam-6496	151	8	4	4	NUM
ejpam-6496	151	9	,	,	PUNCT
ejpam-6496	151	10	then	then	ADV
ejpam-6496	151	11	either	either	CCONJ
ejpam-6496	151	12	g(a	g(a	PROPN
ejpam-6496	151	13	)	)	PUNCT
ejpam-6496	152	1	<	<	X
ejpam-6496	152	2	g(b	g(b	X
ejpam-6496	152	3	)	)	PUNCT
ejpam-6496	152	4	or	or	CCONJ
ejpam-6496	152	5	g(a	g(a	PROPN
ejpam-6496	152	6	)	)	PUNCT
ejpam-6496	152	7	>	>	X
ejpam-6496	153	1	g(b	g(b	PROPN
ejpam-6496	153	2	)	)	PUNCT
ejpam-6496	153	3	which	which	PRON
ejpam-6496	153	4	contradicts	contradict	VERB
ejpam-6496	153	5	the	the	DET
ejpam-6496	153	6	fact	fact	NOUN
ejpam-6496	154	1	that	that	SCONJ
ejpam-6496	154	2	g(a	g(a	PROPN
ejpam-6496	154	3	)	)	PUNCT
ejpam-6496	154	4	=	=	PUNCT
ejpam-6496	154	5	g(b	g(b	NOUN
ejpam-6496	154	6	)	)	PUNCT
ejpam-6496	154	7	.	.	PUNCT
ejpam-6496	155	1	thus	thus	ADV
ejpam-6496	155	2	,	,	PUNCT
ejpam-6496	155	3	there	there	PRON
ejpam-6496	155	4	must	must	AUX
ejpam-6496	155	5	be	be	AUX
ejpam-6496	155	6	some	some	DET
ejpam-6496	155	7	c	c	NOUN
ejpam-6496	155	8	∈	∈	PROPN
ejpam-6496	155	9	]	]	PUNCT
ejpam-6496	155	10	a	a	X
ejpam-6496	155	11	,	,	PUNCT
ejpam-6496	155	12	b	b	X
ejpam-6496	155	13	[	[	PUNCT
ejpam-6496	155	14	such	such	ADJ
ejpam-6496	155	15	that	that	SCONJ
ejpam-6496	155	16	g′(c	g′(c	VERB
ejpam-6496	155	17	)	)	PUNCT
ejpam-6496	155	18	=	=	SYM
ejpam-6496	155	19	0	0	X
ejpam-6496	155	20	.	.	PUNCT
ejpam-6496	156	1	actually	actually	ADV
ejpam-6496	156	2	,	,	PUNCT
ejpam-6496	156	3	0	0	X
ejpam-6496	156	4	=	=	SYM
ejpam-6496	156	5	g′(c	g′(c	PROPN
ejpam-6496	156	6	)	)	PUNCT
ejpam-6496	156	7	=	=	SYM
ejpam-6496	156	8	f	f	X
ejpam-6496	156	9	′(c)−	′(c)−	VERB
ejpam-6496	156	10	f(b)−	f(b)−	PROPN
ejpam-6496	156	11	f(a	f(a	PROPN
ejpam-6496	156	12	)	)	PUNCT
ejpam-6496	156	13	b−	b−	PROPN
ejpam-6496	156	14	a	a	PRON
ejpam-6496	156	15	.	.	PUNCT
ejpam-6496	157	1	hence	hence	ADV
ejpam-6496	157	2	,	,	PUNCT
ejpam-6496	157	3	f(b)−	f(b)−	PROPN
ejpam-6496	157	4	f(a	f(a	PROPN
ejpam-6496	157	5	)	)	PUNCT
ejpam-6496	157	6	=	=	SYM
ejpam-6496	157	7	f	f	PROPN
ejpam-6496	157	8	′(c)(b−	′(c)(b−	NOUN
ejpam-6496	157	9	a	a	PRON
ejpam-6496	157	10	)	)	PUNCT
ejpam-6496	157	11	.	.	PUNCT
ejpam-6496	158	1	theorem	theorem	NOUN
ejpam-6496	158	2	6	6	NUM
ejpam-6496	158	3	(	(	PUNCT
ejpam-6496	158	4	cauchy	cauchy	NOUN
ejpam-6496	158	5	mean	mean	NOUN
ejpam-6496	158	6	value	value	NOUN
ejpam-6496	158	7	theorem	theorem	VERB
ejpam-6496	158	8	)	)	PUNCT
ejpam-6496	158	9	.	.	PUNCT
ejpam-6496	159	1	let	let	VERB
ejpam-6496	159	2	f	f	PROPN
ejpam-6496	159	3	and	and	CCONJ
ejpam-6496	159	4	g	g	PROPN
ejpam-6496	159	5	be	be	VERB
ejpam-6496	159	6	continuous	continuous	ADJ
ejpam-6496	159	7	on	on	ADP
ejpam-6496	159	8	[	[	PUNCT
ejpam-6496	159	9	a	a	PROPN
ejpam-6496	159	10	,	,	PUNCT
ejpam-6496	159	11	b	b	NOUN
ejpam-6496	159	12	]	]	PUNCT
ejpam-6496	159	13	and	and	CCONJ
ejpam-6496	159	14	differentiable	differentiable	VERB
ejpam-6496	159	15	on	on	ADP
ejpam-6496	159	16	]	]	PUNCT
ejpam-6496	159	17	a	a	X
ejpam-6496	159	18	,	,	PUNCT
ejpam-6496	159	19	b	b	X
ejpam-6496	159	20	[	[	PUNCT
ejpam-6496	159	21	.	.	PUNCT
ejpam-6496	160	1	if	if	SCONJ
ejpam-6496	160	2	g′(x	g′(x	NOUN
ejpam-6496	160	3	)	)	PUNCT
ejpam-6496	160	4	̸=	̸=	NOUN
ejpam-6496	160	5	0	0	NUM
ejpam-6496	160	6	for	for	ADP
ejpam-6496	160	7	all	all	PRON
ejpam-6496	160	8	]	]	X
ejpam-6496	160	9	a	a	DET
ejpam-6496	160	10	,	,	PUNCT
ejpam-6496	160	11	b	b	X
ejpam-6496	160	12	[	[	PUNCT
ejpam-6496	160	13	,	,	PUNCT
ejpam-6496	160	14	then	then	ADV
ejpam-6496	160	15	there	there	PRON
ejpam-6496	160	16	exists	exist	VERB
ejpam-6496	160	17	c	c	PROPN
ejpam-6496	160	18	∈	∈	PROPN
ejpam-6496	160	19	]	]	PUNCT
ejpam-6496	160	20	a	a	X
ejpam-6496	160	21	,	,	PUNCT
ejpam-6496	160	22	b	b	X
ejpam-6496	160	23	[	[	PUNCT
ejpam-6496	160	24	such	such	ADJ
ejpam-6496	160	25	that	that	DET
ejpam-6496	160	26	f(b)−	f(b)−	PROPN
ejpam-6496	160	27	f(a	f(a	PROPN
ejpam-6496	160	28	)	)	PUNCT
ejpam-6496	160	29	g(b)−	g(b)−	PROPN
ejpam-6496	160	30	g(a	g(a	PROPN
ejpam-6496	160	31	)	)	PUNCT
ejpam-6496	161	1	=	=	SYM
ejpam-6496	161	2	f	f	PROPN
ejpam-6496	161	3	′(c	′(c	NOUN
ejpam-6496	161	4	)	)	PUNCT
ejpam-6496	161	5	g′(c	g′(c	NOUN
ejpam-6496	161	6	)	)	PUNCT
ejpam-6496	161	7	.	.	PUNCT
ejpam-6496	162	1	proof	proof	NOUN
ejpam-6496	162	2	.	.	PUNCT
ejpam-6496	163	1	since	since	SCONJ
ejpam-6496	163	2	g′(x	g′(x	NOUN
ejpam-6496	163	3	)	)	PUNCT
ejpam-6496	163	4	̸=	̸=	PROPN
ejpam-6496	163	5	0	0	NUM
ejpam-6496	163	6	for	for	ADP
ejpam-6496	163	7	all	all	DET
ejpam-6496	163	8	x	x	SYM
ejpam-6496	163	9	∈	∈	PROPN
ejpam-6496	163	10	]	]	PUNCT
ejpam-6496	163	11	a	a	X
ejpam-6496	163	12	,	,	PUNCT
ejpam-6496	163	13	b	b	X
ejpam-6496	163	14	[	[	PUNCT
ejpam-6496	163	15	,	,	PUNCT
ejpam-6496	163	16	then	then	ADV
ejpam-6496	163	17	it	it	PRON
ejpam-6496	163	18	follows	follow	VERB
ejpam-6496	163	19	from	from	ADP
ejpam-6496	163	20	theorem	theorem	ADJ
ejpam-6496	163	21	4	4	NUM
ejpam-6496	163	22	that	that	PRON
ejpam-6496	163	23	g(a	g(a	PROPN
ejpam-6496	163	24	)	)	PUNCT
ejpam-6496	163	25	̸=	̸=	PROPN
ejpam-6496	163	26	g(b	g(b	NOUN
ejpam-6496	163	27	)	)	PUNCT
ejpam-6496	163	28	.	.	PUNCT
ejpam-6496	164	1	so	so	ADV
ejpam-6496	164	2	we	we	PRON
ejpam-6496	164	3	can	can	AUX
ejpam-6496	164	4	define	define	VERB
ejpam-6496	164	5	h	h	NOUN
ejpam-6496	164	6	:	:	PUNCT
ejpam-6496	164	7	[	[	PUNCT
ejpam-6496	164	8	a	a	X
ejpam-6496	164	9	,	,	PUNCT
ejpam-6496	164	10	b	b	NOUN
ejpam-6496	164	11	]	]	PUNCT
ejpam-6496	164	12	→	→	PUNCT
ejpam-6496	164	13	r	r	NOUN
ejpam-6496	164	14	by	by	ADP
ejpam-6496	164	15	h(x	h(x	PROPN
ejpam-6496	164	16	)	)	PUNCT
ejpam-6496	164	17	:	:	PUNCT
ejpam-6496	164	18	=	=	PUNCT
ejpam-6496	164	19	f(b)−	f(b)−	PROPN
ejpam-6496	164	20	f(a	f(a	PROPN
ejpam-6496	164	21	)	)	PUNCT
ejpam-6496	164	22	g(b)−	g(b)−	PROPN
ejpam-6496	164	23	g(a	g(a	PROPN
ejpam-6496	164	24	)	)	PUNCT
ejpam-6496	164	25	g(x)−	g(x)−	PROPN
ejpam-6496	164	26	f(x	f(x	PROPN
ejpam-6496	164	27	)	)	PUNCT
ejpam-6496	164	28	.	.	PUNCT
ejpam-6496	165	1	suppose	suppose	VERB
ejpam-6496	165	2	to	to	ADP
ejpam-6496	165	3	the	the	DET
ejpam-6496	165	4	contrary	contrary	NOUN
ejpam-6496	165	5	that	that	SCONJ
ejpam-6496	165	6	h′(x	h′(x	VERB
ejpam-6496	165	7	)	)	PUNCT
ejpam-6496	165	8	̸=	̸=	PROPN
ejpam-6496	165	9	0	0	NUM
ejpam-6496	165	10	for	for	ADP
ejpam-6496	165	11	all	all	DET
ejpam-6496	165	12	x	x	SYM
ejpam-6496	165	13	∈	∈	PROPN
ejpam-6496	165	14	]	]	PUNCT
ejpam-6496	165	15	a	a	X
ejpam-6496	165	16	,	,	PUNCT
ejpam-6496	165	17	b	b	X
ejpam-6496	165	18	[	[	PUNCT
ejpam-6496	165	19	.	.	PUNCT
ejpam-6496	166	1	we	we	PRON
ejpam-6496	166	2	can	can	AUX
ejpam-6496	166	3	prove	prove	VERB
ejpam-6496	166	4	by	by	ADP
ejpam-6496	166	5	the	the	DET
ejpam-6496	166	6	same	same	ADJ
ejpam-6496	166	7	manner	manner	NOUN
ejpam-6496	166	8	as	as	SCONJ
ejpam-6496	166	9	theorem	theorem	VERB
ejpam-6496	166	10	5	5	NUM
ejpam-6496	166	11	that	that	SCONJ
ejpam-6496	166	12	the	the	DET
ejpam-6496	166	13	supposition	supposition	NOUN
ejpam-6496	166	14	leads	lead	VERB
ejpam-6496	166	15	to	to	ADP
ejpam-6496	166	16	h(a	h(a	PROPN
ejpam-6496	166	17	)	)	PUNCT
ejpam-6496	166	18	̸=	̸=	PROPN
ejpam-6496	166	19	h(b	h(b	PROPN
ejpam-6496	166	20	)	)	PUNCT
ejpam-6496	166	21	,	,	PUNCT
ejpam-6496	166	22	which	which	PRON
ejpam-6496	166	23	is	be	AUX
ejpam-6496	166	24	a	a	DET
ejpam-6496	166	25	contradiction	contradiction	NOUN
ejpam-6496	166	26	.	.	PUNCT
ejpam-6496	167	1	hence	hence	ADV
ejpam-6496	167	2	,	,	PUNCT
ejpam-6496	167	3	there	there	PRON
ejpam-6496	167	4	must	must	AUX
ejpam-6496	167	5	be	be	AUX
ejpam-6496	167	6	some	some	DET
ejpam-6496	167	7	c	c	NOUN
ejpam-6496	167	8	∈	∈	PROPN
ejpam-6496	167	9	]	]	PUNCT
ejpam-6496	167	10	a	a	X
ejpam-6496	167	11	,	,	PUNCT
ejpam-6496	167	12	b	b	X
ejpam-6496	167	13	[	[	PUNCT
ejpam-6496	167	14	such	such	ADJ
ejpam-6496	167	15	that	that	SCONJ
ejpam-6496	167	16	h′(c	h′(c	NOUN
ejpam-6496	167	17	)	)	PUNCT
ejpam-6496	167	18	=	=	SYM
ejpam-6496	167	19	0	0	X
ejpam-6496	167	20	.	.	PUNCT
ejpam-6496	168	1	actually	actually	ADV
ejpam-6496	168	2	,	,	PUNCT
ejpam-6496	168	3	0	0	NUM
ejpam-6496	168	4	=	=	SYM
ejpam-6496	168	5	h′(c	h′(c	PROPN
ejpam-6496	168	6	)	)	PUNCT
ejpam-6496	168	7	=	=	SYM
ejpam-6496	168	8	f(b)−	f(b)−	PROPN
ejpam-6496	168	9	f(a	f(a	PROPN
ejpam-6496	168	10	)	)	PUNCT
ejpam-6496	168	11	g(b)−	g(b)−	PROPN
ejpam-6496	168	12	g(a	g(a	PROPN
ejpam-6496	168	13	)	)	PUNCT
ejpam-6496	168	14	g′(c)−	g′(c)−	NOUN
ejpam-6496	168	15	f	f	NOUN
ejpam-6496	168	16	′(c	′(c	NOUN
ejpam-6496	168	17	)	)	PUNCT
ejpam-6496	168	18	.	.	PUNCT
ejpam-6496	169	1	therefore	therefore	ADV
ejpam-6496	169	2	,	,	PUNCT
ejpam-6496	169	3	f(b)−f(a	f(b)−f(a	NOUN
ejpam-6496	169	4	)	)	PUNCT
ejpam-6496	169	5	g(b)−g(a	g(b)−g(a	PUNCT
ejpam-6496	169	6	)	)	PUNCT
ejpam-6496	170	1	=	=	SYM
ejpam-6496	170	2	f	f	PROPN
ejpam-6496	170	3	′(c	′(c	NOUN
ejpam-6496	170	4	)	)	PUNCT
ejpam-6496	170	5	g′(c	g′(c	NOUN
ejpam-6496	170	6	)	)	PUNCT
ejpam-6496	170	7	.	.	PUNCT
ejpam-6496	171	1	s.	s.	PROPN
ejpam-6496	171	2	prongjit	prongjit	ADV
ejpam-6496	171	3	,	,	PUNCT
ejpam-6496	171	4	p.	p.	PROPN
ejpam-6496	171	5	prasertsang	prasertsang	PROPN
ejpam-6496	171	6	/	/	SYM
ejpam-6496	171	7	eur	eur	PROPN
ejpam-6496	171	8	.	.	PUNCT
ejpam-6496	172	1	j.	j.	PROPN
ejpam-6496	172	2	pure	pure	PROPN
ejpam-6496	172	3	appl	appl	PROPN
ejpam-6496	172	4	.	.	PROPN
ejpam-6496	172	5	math	math	PROPN
ejpam-6496	172	6	,	,	PUNCT
ejpam-6496	172	7	18	18	NUM
ejpam-6496	172	8	(	(	PUNCT
ejpam-6496	172	9	3	3	NUM
ejpam-6496	172	10	)	)	PUNCT
ejpam-6496	172	11	(	(	PUNCT
ejpam-6496	172	12	2025	2025	NUM
ejpam-6496	172	13	)	)	PUNCT
ejpam-6496	172	14	,	,	PUNCT
ejpam-6496	172	15	6496	6496	NUM
ejpam-6496	172	16	7	7	NUM
ejpam-6496	172	17	of	of	ADP
ejpam-6496	172	18	10	10	NUM
ejpam-6496	172	19	3.2	3.2	NUM
ejpam-6496	172	20	.	.	PUNCT
ejpam-6496	173	1	riemann	riemann	PROPN
ejpam-6496	173	2	integral	integral	PROPN
ejpam-6496	173	3	via	via	ADP
ejpam-6496	173	4	δ	δ	PROPN
ejpam-6496	173	5	-	-	PUNCT
ejpam-6496	173	6	fine	fine	ADJ
ejpam-6496	173	7	tagged	tag	VERB
ejpam-6496	173	8	partitions	partition	NOUN
ejpam-6496	173	9	in	in	ADP
ejpam-6496	173	10	this	this	DET
ejpam-6496	173	11	section	section	NOUN
ejpam-6496	173	12	,	,	PUNCT
ejpam-6496	173	13	we	we	PRON
ejpam-6496	173	14	deal	deal	VERB
ejpam-6496	173	15	with	with	ADP
ejpam-6496	173	16	some	some	DET
ejpam-6496	173	17	basic	basic	ADJ
ejpam-6496	173	18	properties	property	NOUN
ejpam-6496	173	19	of	of	ADP
ejpam-6496	173	20	riemann	riemann	PROPN
ejpam-6496	173	21	integrable	integrable	ADJ
ejpam-6496	173	22	functions	function	NOUN
ejpam-6496	173	23	.	.	PUNCT
ejpam-6496	174	1	definition	definition	NOUN
ejpam-6496	174	2	5	5	NUM
ejpam-6496	174	3	is	be	AUX
ejpam-6496	174	4	a	a	DET
ejpam-6496	174	5	version	version	NOUN
ejpam-6496	174	6	of	of	ADP
ejpam-6496	174	7	cauchy	cauchy	ADJ
ejpam-6496	174	8	criterion	criterion	NOUN
ejpam-6496	174	9	for	for	ADP
ejpam-6496	174	10	riemann	riemann	PROPN
ejpam-6496	174	11	integrability	integrability	PROPN
ejpam-6496	174	12	formulated	formulate	VERB
ejpam-6496	174	13	by	by	ADP
ejpam-6496	174	14	gordon	gordon	PROPN
ejpam-6496	175	1	[	[	X
ejpam-6496	175	2	5	5	NUM
ejpam-6496	175	3	]	]	PUNCT
ejpam-6496	175	4	that	that	SCONJ
ejpam-6496	175	5	we	we	PRON
ejpam-6496	175	6	shall	shall	AUX
ejpam-6496	175	7	use	use	VERB
ejpam-6496	175	8	in	in	ADP
ejpam-6496	175	9	this	this	DET
ejpam-6496	175	10	paper	paper	NOUN
ejpam-6496	175	11	.	.	PUNCT
ejpam-6496	176	1	theorem	theorem	VERB
ejpam-6496	176	2	7	7	NUM
ejpam-6496	176	3	.	.	PUNCT
ejpam-6496	177	1	if	if	SCONJ
ejpam-6496	177	2	f	f	PROPN
ejpam-6496	177	3	:	:	PUNCT
ejpam-6496	177	4	[	[	PUNCT
ejpam-6496	177	5	a	a	X
ejpam-6496	177	6	,	,	PUNCT
ejpam-6496	177	7	b	b	NOUN
ejpam-6496	177	8	]	]	PUNCT
ejpam-6496	177	9	→	→	PUNCT
ejpam-6496	177	10	r	r	NOUN
ejpam-6496	177	11	is	be	AUX
ejpam-6496	177	12	monotone	monotone	ADJ
ejpam-6496	177	13	on	on	ADP
ejpam-6496	177	14	[	[	PUNCT
ejpam-6496	177	15	a	a	PROPN
ejpam-6496	177	16	,	,	PUNCT
ejpam-6496	177	17	b	b	NOUN
ejpam-6496	177	18	]	]	X
ejpam-6496	177	19	,	,	PUNCT
ejpam-6496	177	20	then	then	ADV
ejpam-6496	177	21	f	f	PROPN
ejpam-6496	177	22	is	be	AUX
ejpam-6496	177	23	riemann	riemann	PROPN
ejpam-6496	177	24	integrable	integrable	ADJ
ejpam-6496	177	25	on	on	ADP
ejpam-6496	177	26	[	[	PUNCT
ejpam-6496	177	27	a	a	PROPN
ejpam-6496	177	28	,	,	PUNCT
ejpam-6496	177	29	b	b	NOUN
ejpam-6496	177	30	]	]	PUNCT
ejpam-6496	177	31	.	.	PUNCT
ejpam-6496	178	1	proof	proof	NOUN
ejpam-6496	178	2	.	.	PUNCT
ejpam-6496	179	1	assume	assume	VERB
ejpam-6496	179	2	that	that	SCONJ
ejpam-6496	179	3	f	f	PROPN
ejpam-6496	179	4	is	be	AUX
ejpam-6496	179	5	increasing	increase	VERB
ejpam-6496	179	6	on	on	ADP
ejpam-6496	179	7	[	[	PUNCT
ejpam-6496	179	8	a	a	PROPN
ejpam-6496	179	9	,	,	PUNCT
ejpam-6496	179	10	b	b	NOUN
ejpam-6496	179	11	]	]	X
ejpam-6496	179	12	,	,	PUNCT
ejpam-6496	179	13	and	and	CCONJ
ejpam-6496	179	14	let	let	VERB
ejpam-6496	179	15	ε	ε	PROPN
ejpam-6496	179	16	>	>	X
ejpam-6496	179	17	0	0	NUM
ejpam-6496	179	18	be	be	AUX
ejpam-6496	179	19	fixed	fix	VERB
ejpam-6496	179	20	.	.	PUNCT
ejpam-6496	180	1	define	define	VERB
ejpam-6496	180	2	a	a	DET
ejpam-6496	180	3	gauge	gauge	NOUN
ejpam-6496	180	4	δ	δ	NOUN
ejpam-6496	180	5	:	:	PUNCT
ejpam-6496	180	6	[	[	PUNCT
ejpam-6496	180	7	a	a	X
ejpam-6496	180	8	,	,	PUNCT
ejpam-6496	180	9	b	b	NOUN
ejpam-6496	180	10	]	]	PUNCT
ejpam-6496	180	11	→	→	SYM
ejpam-6496	180	12	r+	r+	NOUN
ejpam-6496	180	13	by	by	ADP
ejpam-6496	180	14	δ(x	δ(x	PROPN
ejpam-6496	180	15	)	)	PUNCT
ejpam-6496	181	1	:	:	PUNCT
ejpam-6496	181	2	=	=	PUNCT
ejpam-6496	181	3	ε	ε	AUX
ejpam-6496	181	4	.	.	PUNCT
ejpam-6496	181	5	let	let	VERB
ejpam-6496	181	6	p	p	NOUN
ejpam-6496	181	7	:	:	PUNCT
ejpam-6496	181	8	=	=	SYM
ejpam-6496	181	9	{	{	PUNCT
ejpam-6496	181	10	(	(	PUNCT
ejpam-6496	181	11	ti	ti	NOUN
ejpam-6496	181	12	,	,	PUNCT
ejpam-6496	181	13	[	[	X
ejpam-6496	181	14	xi−1	xi−1	PROPN
ejpam-6496	181	15	,	,	PUNCT
ejpam-6496	181	16	xi	xi	X
ejpam-6496	181	17	]	]	PUNCT
ejpam-6496	181	18	)	)	PUNCT
ejpam-6496	181	19	:	:	PUNCT
ejpam-6496	182	1	i	i	NOUN
ejpam-6496	182	2	=	=	NOUN
ejpam-6496	182	3	1	1	NUM
ejpam-6496	182	4	,	,	PUNCT
ejpam-6496	182	5	.	.	PUNCT
ejpam-6496	182	6	.	.	PUNCT
ejpam-6496	182	7	.	.	PUNCT
ejpam-6496	183	1	,	,	PUNCT
ejpam-6496	183	2	n	n	CCONJ
ejpam-6496	183	3	}	}	PUNCT
ejpam-6496	183	4	be	be	AUX
ejpam-6496	183	5	a	a	DET
ejpam-6496	183	6	δ	δ	NOUN
ejpam-6496	183	7	-	-	PUNCT
ejpam-6496	183	8	fine	fine	ADJ
ejpam-6496	183	9	tagged	tag	VERB
ejpam-6496	183	10	partition	partition	NOUN
ejpam-6496	183	11	of	of	ADP
ejpam-6496	183	12	[	[	PUNCT
ejpam-6496	183	13	a	a	PROPN
ejpam-6496	183	14	,	,	PUNCT
ejpam-6496	183	15	b	b	NOUN
ejpam-6496	183	16	]	]	X
ejpam-6496	183	17	.	.	PUNCT
ejpam-6496	184	1	for	for	ADP
ejpam-6496	184	2	each	each	DET
ejpam-6496	184	3	i	i	NOUN
ejpam-6496	184	4	=	=	NOUN
ejpam-6496	184	5	1	1	NUM
ejpam-6496	184	6	,	,	PUNCT
ejpam-6496	184	7	.	.	PUNCT
ejpam-6496	184	8	.	.	PUNCT
ejpam-6496	184	9	.	.	PUNCT
ejpam-6496	185	1	,	,	PUNCT
ejpam-6496	185	2	n	n	CCONJ
ejpam-6496	185	3	,	,	PUNCT
ejpam-6496	185	4	we	we	PRON
ejpam-6496	185	5	have	have	VERB
ejpam-6496	185	6	ω(f	ω(f	ADJ
ejpam-6496	185	7	,	,	PUNCT
ejpam-6496	185	8	[	[	X
ejpam-6496	185	9	xi−1	xi−1	PROPN
ejpam-6496	185	10	,	,	PUNCT
ejpam-6496	185	11	xi	xi	X
ejpam-6496	185	12	]	]	PUNCT
ejpam-6496	185	13	)	)	PUNCT
ejpam-6496	185	14	=	=	SYM
ejpam-6496	185	15	sup{|f(s)−	sup{|f(s)−	PROPN
ejpam-6496	186	1	f(t)|	f(t)|	ADJ
ejpam-6496	186	2	:	:	PUNCT
ejpam-6496	186	3	s	s	X
ejpam-6496	186	4	,	,	PUNCT
ejpam-6496	186	5	t	t	PROPN
ejpam-6496	186	6	∈	∈	PROPN
ejpam-6496	187	1	[	[	X
ejpam-6496	187	2	xi−1	xi−1	PROPN
ejpam-6496	187	3	,	,	PUNCT
ejpam-6496	187	4	xi	xi	X
ejpam-6496	187	5	]	]	PUNCT
ejpam-6496	187	6	}	}	PUNCT
ejpam-6496	187	7	=	=	SYM
ejpam-6496	187	8	f(xi)−	f(xi)−	NOUN
ejpam-6496	187	9	f(xi−1	f(xi−1	PROPN
ejpam-6496	187	10	)	)	PUNCT
ejpam-6496	187	11	,	,	PUNCT
ejpam-6496	187	12	and	and	CCONJ
ejpam-6496	187	13	[	[	X
ejpam-6496	187	14	xi−1	xi−1	PROPN
ejpam-6496	187	15	,	,	PUNCT
ejpam-6496	187	16	xi	xi	X
ejpam-6496	187	17	]	]	PUNCT
ejpam-6496	187	18	⊂	⊂	X
ejpam-6496	187	19	]	]	PUNCT
ejpam-6496	187	20	ti	ti	X
ejpam-6496	187	21	−	−	PROPN
ejpam-6496	187	22	δ(ti	δ(ti	PROPN
ejpam-6496	187	23	)	)	PUNCT
ejpam-6496	187	24	,	,	PUNCT
ejpam-6496	187	25	ti	ti	X
ejpam-6496	187	26	+	+	SYM
ejpam-6496	187	27	δ(ti	δ(ti	NOUN
ejpam-6496	187	28	)	)	PUNCT
ejpam-6496	187	29	[	[	PUNCT
ejpam-6496	187	30	which	which	PRON
ejpam-6496	187	31	implies	imply	VERB
ejpam-6496	187	32	xi	xi	X
ejpam-6496	187	33	−	−	PROPN
ejpam-6496	187	34	xi−1	xi−1	PROPN
ejpam-6496	187	35	<	<	X
ejpam-6496	187	36	2ε	2ε	NUM
ejpam-6496	187	37	.	.	PUNCT
ejpam-6496	188	1	then	then	ADV
ejpam-6496	188	2	,	,	PUNCT
ejpam-6496	188	3	we	we	PRON
ejpam-6496	188	4	have	have	VERB
ejpam-6496	188	5	a	a	DET
ejpam-6496	188	6	partition	partition	NOUN
ejpam-6496	188	7	{	{	PUNCT
ejpam-6496	188	8	[	[	X
ejpam-6496	188	9	xi−1	xi−1	PROPN
ejpam-6496	188	10	,	,	PUNCT
ejpam-6496	188	11	xi	xi	X
ejpam-6496	188	12	]	]	PUNCT
ejpam-6496	188	13	:	:	PUNCT
ejpam-6496	188	14	i	i	NOUN
ejpam-6496	188	15	=	=	NOUN
ejpam-6496	188	16	1	1	NUM
ejpam-6496	188	17	,	,	PUNCT
ejpam-6496	188	18	.	.	PUNCT
ejpam-6496	188	19	.	.	PUNCT
ejpam-6496	189	1	.	.	PUNCT
ejpam-6496	190	1	,	,	PUNCT
ejpam-6496	190	2	n	n	CCONJ
ejpam-6496	190	3	}	}	PUNCT
ejpam-6496	190	4	of	of	ADP
ejpam-6496	190	5	[	[	PUNCT
ejpam-6496	190	6	a	a	PROPN
ejpam-6496	190	7	,	,	PUNCT
ejpam-6496	190	8	b	b	NOUN
ejpam-6496	190	9	]	]	PUNCT
ejpam-6496	190	10	which	which	PRON
ejpam-6496	190	11	is	be	AUX
ejpam-6496	190	12	extracted	extract	VERB
ejpam-6496	190	13	from	from	ADP
ejpam-6496	190	14	p	p	PRON
ejpam-6496	190	15	such	such	ADJ
ejpam-6496	190	16	that	that	SCONJ
ejpam-6496	190	17	n∑	n∑	PROPN
ejpam-6496	190	18	i=1	i=1	PROPN
ejpam-6496	190	19	ω(f	ω(f	ADJ
ejpam-6496	190	20	,	,	PUNCT
ejpam-6496	190	21	[	[	X
ejpam-6496	190	22	xi−1	xi−1	PROPN
ejpam-6496	190	23	,	,	PUNCT
ejpam-6496	190	24	xi	xi	X
ejpam-6496	190	25	]	]	X
ejpam-6496	190	26	)	)	PUNCT
ejpam-6496	190	27	(	(	PUNCT
ejpam-6496	190	28	xi	xi	X
ejpam-6496	190	29	−	−	PROPN
ejpam-6496	190	30	xi−1	xi−1	PROPN
ejpam-6496	190	31	)	)	PUNCT
ejpam-6496	190	32	<	<	X
ejpam-6496	191	1	n∑	n∑	X
ejpam-6496	191	2	i=1	i=1	PROPN
ejpam-6496	191	3	(	(	PUNCT
ejpam-6496	191	4	f(xi)−	f(xi)−	NOUN
ejpam-6496	191	5	f(xi−1))2ε	f(xi−1))2ε	PROPN
ejpam-6496	191	6	=	=	SYM
ejpam-6496	191	7	2ε(f(b)−	2ε(f(b)−	NUM
ejpam-6496	191	8	f(a	f(a	NOUN
ejpam-6496	191	9	)	)	PUNCT
ejpam-6496	191	10	)	)	PUNCT
ejpam-6496	191	11	.	.	PUNCT
ejpam-6496	192	1	since	since	SCONJ
ejpam-6496	192	2	ε	ε	PROPN
ejpam-6496	192	3	is	be	AUX
ejpam-6496	192	4	arbitrary	arbitrary	ADJ
ejpam-6496	192	5	,	,	PUNCT
ejpam-6496	192	6	so	so	CCONJ
ejpam-6496	192	7	f	f	PROPN
ejpam-6496	192	8	is	be	AUX
ejpam-6496	192	9	riemann	riemann	PROPN
ejpam-6496	192	10	integrable	integrable	ADJ
ejpam-6496	192	11	on	on	ADP
ejpam-6496	192	12	[	[	PUNCT
ejpam-6496	192	13	a	a	PROPN
ejpam-6496	192	14	,	,	PUNCT
ejpam-6496	192	15	b	b	NOUN
ejpam-6496	192	16	]	]	X
ejpam-6496	192	17	.	.	PUNCT
ejpam-6496	193	1	we	we	PRON
ejpam-6496	193	2	can	can	AUX
ejpam-6496	193	3	prove	prove	VERB
ejpam-6496	193	4	in	in	ADP
ejpam-6496	193	5	the	the	DET
ejpam-6496	193	6	same	same	ADJ
ejpam-6496	193	7	manner	manner	NOUN
ejpam-6496	193	8	whence	whence	SCONJ
ejpam-6496	193	9	f	f	PROPN
ejpam-6496	193	10	is	be	AUX
ejpam-6496	193	11	decreasing	decrease	VERB
ejpam-6496	193	12	on	on	ADP
ejpam-6496	193	13	[	[	PUNCT
ejpam-6496	193	14	a	a	PROPN
ejpam-6496	193	15	,	,	PUNCT
ejpam-6496	193	16	b	b	NOUN
ejpam-6496	193	17	]	]	PUNCT
ejpam-6496	193	18	.	.	PUNCT
ejpam-6496	194	1	lemma	lemma	PROPN
ejpam-6496	194	2	3	3	X
ejpam-6496	194	3	.	.	PUNCT
ejpam-6496	195	1	let	let	VERB
ejpam-6496	195	2	f	f	NOUN
ejpam-6496	195	3	:	:	PUNCT
ejpam-6496	195	4	[	[	PUNCT
ejpam-6496	195	5	a	a	X
ejpam-6496	195	6	,	,	PUNCT
ejpam-6496	195	7	b	b	NOUN
ejpam-6496	195	8	]	]	PUNCT
ejpam-6496	195	9	→	→	PUNCT
ejpam-6496	195	10	r	r	NOUN
ejpam-6496	195	11	be	be	AUX
ejpam-6496	195	12	a	a	DET
ejpam-6496	195	13	step	step	NOUN
ejpam-6496	195	14	function	function	NOUN
ejpam-6496	195	15	such	such	ADJ
ejpam-6496	195	16	that	that	SCONJ
ejpam-6496	195	17	f	f	PROPN
ejpam-6496	195	18	=	=	PUNCT
ejpam-6496	195	19	l∑	l∑	PROPN
ejpam-6496	195	20	j=1	j=1	PROPN
ejpam-6496	195	21	kjφjj	kjφjj	ADV
ejpam-6496	195	22	.	.	PUNCT
ejpam-6496	196	1	if	if	SCONJ
ejpam-6496	196	2	p	p	X
ejpam-6496	196	3	:	:	PUNCT
ejpam-6496	196	4	=	=	SYM
ejpam-6496	196	5	{	{	PUNCT
ejpam-6496	197	1	[	[	X
ejpam-6496	197	2	xi−1	xi−1	PROPN
ejpam-6496	197	3	,	,	PUNCT
ejpam-6496	197	4	xi	xi	X
ejpam-6496	197	5	]	]	PUNCT
ejpam-6496	197	6	:	:	PUNCT
ejpam-6496	197	7	i	i	NOUN
ejpam-6496	197	8	=	=	NOUN
ejpam-6496	197	9	1	1	NUM
ejpam-6496	197	10	,	,	PUNCT
ejpam-6496	197	11	.	.	PUNCT
ejpam-6496	197	12	.	.	PUNCT
ejpam-6496	198	1	.	.	PUNCT
ejpam-6496	199	1	,	,	PUNCT
ejpam-6496	199	2	n	n	CCONJ
ejpam-6496	199	3	}	}	PUNCT
ejpam-6496	199	4	is	be	AUX
ejpam-6496	199	5	a	a	DET
ejpam-6496	199	6	partition	partition	NOUN
ejpam-6496	199	7	of	of	ADP
ejpam-6496	199	8	[	[	PUNCT
ejpam-6496	199	9	a	a	PROPN
ejpam-6496	199	10	,	,	PUNCT
ejpam-6496	199	11	b	b	NOUN
ejpam-6496	199	12	]	]	X
ejpam-6496	199	13	,	,	PUNCT
ejpam-6496	199	14	then	then	ADV
ejpam-6496	199	15	the	the	DET
ejpam-6496	199	16	number	number	NOUN
ejpam-6496	199	17	of	of	ADP
ejpam-6496	199	18	elements	element	NOUN
ejpam-6496	199	19	in	in	ADP
ejpam-6496	199	20	p	p	NOUN
ejpam-6496	199	21	which	which	PRON
ejpam-6496	199	22	are	be	AUX
ejpam-6496	199	23	not	not	PART
ejpam-6496	199	24	contained	contain	VERB
ejpam-6496	199	25	in	in	ADP
ejpam-6496	199	26	any	any	DET
ejpam-6496	199	27	open	open	ADJ
ejpam-6496	199	28	interval	interval	NOUN
ejpam-6496	199	29	]	]	PUNCT
ejpam-6496	199	30	cj−1	cj−1	NOUN
ejpam-6496	199	31	,	,	PUNCT
ejpam-6496	199	32	cj	cj	NOUN
ejpam-6496	199	33	[	[	PUNCT
ejpam-6496	199	34	is	be	AUX
ejpam-6496	199	35	not	not	PART
ejpam-6496	199	36	greater	great	ADJ
ejpam-6496	199	37	than	than	ADP
ejpam-6496	199	38	2l	2l	NUM
ejpam-6496	199	39	.	.	PUNCT
ejpam-6496	200	1	(	(	PUNCT
ejpam-6496	200	2	recall	recall	VERB
ejpam-6496	200	3	from	from	ADP
ejpam-6496	200	4	definition	definition	NOUN
ejpam-6496	200	5	4	4	NUM
ejpam-6496	200	6	that	that	PRON
ejpam-6496	200	7	jj	jj	VERB
ejpam-6496	200	8	:	:	PUNCT
ejpam-6496	200	9	=	=	SYM
ejpam-6496	200	10	[	[	PUNCT
ejpam-6496	200	11	cj−1	cj−1	NOUN
ejpam-6496	200	12	,	,	PUNCT
ejpam-6496	200	13	cj	cj	X
ejpam-6496	200	14	]	]	PUNCT
ejpam-6496	200	15	.	.	PUNCT
ejpam-6496	200	16	)	)	PUNCT
ejpam-6496	201	1	proof	proof	NOUN
ejpam-6496	201	2	.	.	PUNCT
ejpam-6496	202	1	let	let	VERB
ejpam-6496	202	2	c	c	NOUN
ejpam-6496	202	3	:	:	PUNCT
ejpam-6496	202	4	=	=	SYM
ejpam-6496	202	5	{	{	PUNCT
ejpam-6496	202	6	i	i	NOUN
ejpam-6496	202	7	∈	∈	PROPN
ejpam-6496	203	1	p	p	X
ejpam-6496	203	2	:	:	PUNCT
ejpam-6496	203	3	i	i	PRON
ejpam-6496	203	4	̸⊆	̸⊆	X
ejpam-6496	203	5	]	]	PUNCT
ejpam-6496	204	1	cj−1	cj−1	NOUN
ejpam-6496	204	2	,	,	PUNCT
ejpam-6496	204	3	cj	cj	NOUN
ejpam-6496	204	4	[	[	PUNCT
ejpam-6496	204	5	for	for	ADP
ejpam-6496	204	6	all	all	DET
ejpam-6496	204	7	j	j	NOUN
ejpam-6496	204	8	=	=	SYM
ejpam-6496	204	9	1	1	NUM
ejpam-6496	204	10	,	,	PUNCT
ejpam-6496	204	11	.	.	PUNCT
ejpam-6496	204	12	.	.	PUNCT
ejpam-6496	204	13	.	.	PUNCT
ejpam-6496	205	1	,	,	PUNCT
ejpam-6496	205	2	l	l	NOUN
ejpam-6496	205	3	}	}	PUNCT
ejpam-6496	205	4	.	.	PUNCT
ejpam-6496	206	1	note	note	VERB
ejpam-6496	206	2	that	that	SCONJ
ejpam-6496	206	3	if	if	SCONJ
ejpam-6496	206	4	[	[	X
ejpam-6496	206	5	α	α	X
ejpam-6496	206	6	,	,	PUNCT
ejpam-6496	206	7	β	β	X
ejpam-6496	206	8	]	]	PUNCT
ejpam-6496	206	9	∈	∈	PROPN
ejpam-6496	206	10	c	c	X
ejpam-6496	206	11	,	,	PUNCT
ejpam-6496	206	12	then	then	ADV
ejpam-6496	206	13	there	there	PRON
ejpam-6496	206	14	must	must	AUX
ejpam-6496	206	15	be	be	AUX
ejpam-6496	206	16	at	at	ADV
ejpam-6496	206	17	least	least	ADV
ejpam-6496	206	18	one	one	NUM
ejpam-6496	206	19	point	point	NOUN
ejpam-6496	206	20	cj	cj	X
ejpam-6496	206	21	∈	∈	PROPN
ejpam-6496	206	22	[	[	X
ejpam-6496	206	23	α	α	X
ejpam-6496	206	24	,	,	PUNCT
ejpam-6496	206	25	β	β	X
ejpam-6496	206	26	]	]	PUNCT
ejpam-6496	206	27	for	for	ADP
ejpam-6496	206	28	some	some	DET
ejpam-6496	206	29	j	j	PROPN
ejpam-6496	206	30	=	=	SYM
ejpam-6496	206	31	0	0	NUM
ejpam-6496	206	32	,	,	PUNCT
ejpam-6496	206	33	1	1	NUM
ejpam-6496	206	34	,	,	PUNCT
ejpam-6496	206	35	.	.	PUNCT
ejpam-6496	206	36	.	.	PUNCT
ejpam-6496	207	1	.	.	PUNCT
ejpam-6496	208	1	,	,	PUNCT
ejpam-6496	208	2	l.	l.	PROPN
ejpam-6496	208	3	since	since	SCONJ
ejpam-6496	208	4	c	c	PROPN
ejpam-6496	208	5	is	be	AUX
ejpam-6496	208	6	the	the	DET
ejpam-6496	208	7	set	set	NOUN
ejpam-6496	208	8	of	of	ADP
ejpam-6496	208	9	nonoverlapping	nonoverlapping	NOUN
ejpam-6496	208	10	closed	closed	ADJ
ejpam-6496	208	11	intervals	interval	NOUN
ejpam-6496	208	12	,	,	PUNCT
ejpam-6496	208	13	then	then	ADV
ejpam-6496	208	14	each	each	DET
ejpam-6496	208	15	cj	cj	NOUN
ejpam-6496	208	16	can	can	AUX
ejpam-6496	208	17	be	be	AUX
ejpam-6496	208	18	attributed	attribute	VERB
ejpam-6496	208	19	to	to	PART
ejpam-6496	208	20	belong	belong	VERB
ejpam-6496	208	21	in	in	ADP
ejpam-6496	208	22	any	any	DET
ejpam-6496	208	23	elements	element	NOUN
ejpam-6496	208	24	of	of	ADP
ejpam-6496	208	25	c	c	NOUN
ejpam-6496	208	26	by	by	ADP
ejpam-6496	208	27	separating	separate	VERB
ejpam-6496	208	28	into	into	ADP
ejpam-6496	208	29	two	two	NUM
ejpam-6496	208	30	cases	case	NOUN
ejpam-6496	208	31	as	as	SCONJ
ejpam-6496	208	32	follows	follow	VERB
ejpam-6496	208	33	.	.	PUNCT
ejpam-6496	209	1	case	case	NOUN
ejpam-6496	209	2	1	1	NUM
ejpam-6496	209	3	:	:	PUNCT
ejpam-6496	209	4	j	j	PROPN
ejpam-6496	209	5	=	=	SYM
ejpam-6496	209	6	0	0	PROPN
ejpam-6496	209	7	,	,	PUNCT
ejpam-6496	209	8	l.	l.	NOUN
ejpam-6496	210	1	it	it	PRON
ejpam-6496	210	2	is	be	AUX
ejpam-6496	210	3	clear	clear	ADJ
ejpam-6496	210	4	that	that	SCONJ
ejpam-6496	210	5	c0	c0	NOUN
ejpam-6496	210	6	and	and	CCONJ
ejpam-6496	210	7	cl	cl	NOUN
ejpam-6496	210	8	can	can	AUX
ejpam-6496	210	9	only	only	ADV
ejpam-6496	210	10	be	be	AUX
ejpam-6496	210	11	contained	contain	VERB
ejpam-6496	210	12	in	in	ADP
ejpam-6496	210	13	[	[	X
ejpam-6496	210	14	x0	x0	PROPN
ejpam-6496	210	15	,	,	PUNCT
ejpam-6496	210	16	x1	x1	PROPN
ejpam-6496	210	17	]	]	PUNCT
ejpam-6496	210	18	and	and	CCONJ
ejpam-6496	211	1	[	[	X
ejpam-6496	211	2	xn−1	xn−1	PROPN
ejpam-6496	211	3	,	,	PUNCT
ejpam-6496	211	4	xn	xn	PROPN
ejpam-6496	211	5	]	]	PUNCT
ejpam-6496	211	6	respectively	respectively	ADV
ejpam-6496	211	7	.	.	PUNCT
ejpam-6496	212	1	then	then	ADV
ejpam-6496	212	2	we	we	PRON
ejpam-6496	212	3	get	get	VERB
ejpam-6496	212	4	exactly	exactly	ADV
ejpam-6496	212	5	two	two	NUM
ejpam-6496	212	6	elements	element	NOUN
ejpam-6496	212	7	of	of	ADP
ejpam-6496	212	8	c	c	PROPN
ejpam-6496	212	9	from	from	ADP
ejpam-6496	212	10	this	this	PRON
ejpam-6496	212	11	.	.	PUNCT
ejpam-6496	213	1	case	case	NOUN
ejpam-6496	213	2	2	2	NUM
ejpam-6496	213	3	:	:	PUNCT
ejpam-6496	213	4	j	j	PROPN
ejpam-6496	213	5	=	=	SYM
ejpam-6496	213	6	1	1	NUM
ejpam-6496	213	7	,	,	PUNCT
ejpam-6496	213	8	.	.	PUNCT
ejpam-6496	213	9	.	.	PUNCT
ejpam-6496	213	10	.	.	PUNCT
ejpam-6496	214	1	,	,	PUNCT
ejpam-6496	215	1	l	l	NOUN
ejpam-6496	215	2	−	−	NOUN
ejpam-6496	215	3	1	1	X
ejpam-6496	215	4	.	.	PUNCT
ejpam-6496	216	1	since	since	SCONJ
ejpam-6496	216	2	cj	cj	NOUN
ejpam-6496	216	3	may	may	AUX
ejpam-6496	216	4	be	be	AUX
ejpam-6496	216	5	an	an	DET
ejpam-6496	216	6	endpoint	endpoint	NOUN
ejpam-6496	216	7	of	of	ADP
ejpam-6496	216	8	two	two	NUM
ejpam-6496	216	9	adjacent	adjacent	ADJ
ejpam-6496	216	10	elements	element	NOUN
ejpam-6496	216	11	of	of	ADP
ejpam-6496	216	12	c	c	NOUN
ejpam-6496	216	13	,	,	PUNCT
ejpam-6496	216	14	then	then	ADV
ejpam-6496	216	15	cj	cj	PROPN
ejpam-6496	216	16	can	can	AUX
ejpam-6496	216	17	only	only	ADV
ejpam-6496	216	18	be	be	AUX
ejpam-6496	216	19	contained	contain	VERB
ejpam-6496	216	20	in	in	ADP
ejpam-6496	216	21	at	at	ADP
ejpam-6496	216	22	most	most	ADJ
ejpam-6496	216	23	two	two	NUM
ejpam-6496	216	24	elements	element	NOUN
ejpam-6496	216	25	of	of	ADP
ejpam-6496	216	26	c.	c.	NOUN
ejpam-6496	216	27	this	this	PRON
ejpam-6496	216	28	give	give	VERB
ejpam-6496	216	29	us	we	PRON
ejpam-6496	216	30	at	at	ADP
ejpam-6496	216	31	most	most	ADJ
ejpam-6496	216	32	2(l−	2(l−	NUM
ejpam-6496	216	33	1	1	NUM
ejpam-6496	216	34	)	)	PUNCT
ejpam-6496	216	35	elements	element	NOUN
ejpam-6496	216	36	of	of	ADP
ejpam-6496	216	37	c	c	PROPN
ejpam-6496	216	38	to	to	PART
ejpam-6496	216	39	count	count	VERB
ejpam-6496	216	40	.	.	PUNCT
ejpam-6496	217	1	consequently	consequently	ADV
ejpam-6496	217	2	,	,	PUNCT
ejpam-6496	217	3	|c|	|c|	PROPN
ejpam-6496	217	4	≤	≤	NUM
ejpam-6496	217	5	2l	2l	NUM
ejpam-6496	217	6	.	.	PUNCT
ejpam-6496	218	1	theorem	theorem	VERB
ejpam-6496	218	2	8	8	NUM
ejpam-6496	218	3	.	.	PUNCT
ejpam-6496	219	1	if	if	SCONJ
ejpam-6496	219	2	f	f	PROPN
ejpam-6496	219	3	:	:	PUNCT
ejpam-6496	219	4	[	[	PUNCT
ejpam-6496	219	5	a	a	X
ejpam-6496	219	6	,	,	PUNCT
ejpam-6496	219	7	b	b	NOUN
ejpam-6496	219	8	]	]	PUNCT
ejpam-6496	219	9	→	→	PUNCT
ejpam-6496	219	10	r	r	NOUN
ejpam-6496	219	11	is	be	AUX
ejpam-6496	219	12	a	a	DET
ejpam-6496	219	13	step	step	NOUN
ejpam-6496	219	14	function	function	NOUN
ejpam-6496	219	15	,	,	PUNCT
ejpam-6496	219	16	then	then	ADV
ejpam-6496	219	17	f	f	PROPN
ejpam-6496	219	18	is	be	AUX
ejpam-6496	219	19	riemann	riemann	PROPN
ejpam-6496	219	20	integrable	integrable	ADJ
ejpam-6496	219	21	on	on	ADP
ejpam-6496	219	22	[	[	PUNCT
ejpam-6496	219	23	a	a	PROPN
ejpam-6496	219	24	,	,	PUNCT
ejpam-6496	219	25	b	b	NOUN
ejpam-6496	219	26	]	]	PUNCT
ejpam-6496	219	27	.	.	PUNCT
ejpam-6496	220	1	s.	s.	PROPN
ejpam-6496	220	2	prongjit	prongjit	ADV
ejpam-6496	220	3	,	,	PUNCT
ejpam-6496	220	4	p.	p.	PROPN
ejpam-6496	220	5	prasertsang	prasertsang	PROPN
ejpam-6496	220	6	/	/	SYM
ejpam-6496	220	7	eur	eur	PROPN
ejpam-6496	220	8	.	.	PUNCT
ejpam-6496	221	1	j.	j.	PROPN
ejpam-6496	221	2	pure	pure	PROPN
ejpam-6496	221	3	appl	appl	PROPN
ejpam-6496	221	4	.	.	PROPN
ejpam-6496	221	5	math	math	PROPN
ejpam-6496	221	6	,	,	PUNCT
ejpam-6496	221	7	18	18	NUM
ejpam-6496	221	8	(	(	PUNCT
ejpam-6496	221	9	3	3	NUM
ejpam-6496	221	10	)	)	PUNCT
ejpam-6496	221	11	(	(	PUNCT
ejpam-6496	221	12	2025	2025	NUM
ejpam-6496	221	13	)	)	PUNCT
ejpam-6496	221	14	,	,	PUNCT
ejpam-6496	221	15	6496	6496	NUM
ejpam-6496	221	16	8	8	NUM
ejpam-6496	221	17	of	of	ADP
ejpam-6496	221	18	10	10	NUM
ejpam-6496	221	19	proof	proof	NOUN
ejpam-6496	221	20	.	.	PUNCT
ejpam-6496	222	1	let	let	VERB
ejpam-6496	222	2	f	f	PRON
ejpam-6496	222	3	be	be	AUX
ejpam-6496	222	4	a	a	DET
ejpam-6496	222	5	step	step	NOUN
ejpam-6496	222	6	function	function	NOUN
ejpam-6496	222	7	on	on	ADP
ejpam-6496	222	8	[	[	PUNCT
ejpam-6496	222	9	a	a	PROPN
ejpam-6496	222	10	,	,	PUNCT
ejpam-6496	222	11	b	b	NOUN
ejpam-6496	222	12	]	]	PUNCT
ejpam-6496	222	13	such	such	ADJ
ejpam-6496	222	14	that	that	SCONJ
ejpam-6496	222	15	f	f	PROPN
ejpam-6496	222	16	=	=	PUNCT
ejpam-6496	222	17	l∑	l∑	PROPN
ejpam-6496	222	18	j=1	j=1	PROPN
ejpam-6496	222	19	kjφjj	kjφjj	ADV
ejpam-6496	222	20	and	and	CCONJ
ejpam-6496	222	21	let	let	VERB
ejpam-6496	222	22	ε	ε	PROPN
ejpam-6496	222	23	>	>	X
ejpam-6496	222	24	0	0	PUNCT
ejpam-6496	222	25	be	be	AUX
ejpam-6496	222	26	given	give	VERB
ejpam-6496	222	27	.	.	PUNCT
ejpam-6496	223	1	define	define	VERB
ejpam-6496	223	2	a	a	DET
ejpam-6496	223	3	gauge	gauge	NOUN
ejpam-6496	223	4	δ	δ	NOUN
ejpam-6496	223	5	:	:	PUNCT
ejpam-6496	223	6	[	[	PUNCT
ejpam-6496	223	7	a	a	X
ejpam-6496	223	8	,	,	PUNCT
ejpam-6496	223	9	b	b	NOUN
ejpam-6496	223	10	]	]	PUNCT
ejpam-6496	223	11	→	→	SYM
ejpam-6496	223	12	r+	r+	NOUN
ejpam-6496	223	13	by	by	ADP
ejpam-6496	223	14	δ(x	δ(x	NOUN
ejpam-6496	223	15	)	)	PUNCT
ejpam-6496	223	16	=	=	SYM
ejpam-6496	224	1	ε	ε	PROPN
ejpam-6496	224	2	k+1	k+1	X
ejpam-6496	224	3	,	,	PUNCT
ejpam-6496	224	4	where	where	SCONJ
ejpam-6496	224	5	k	k	PROPN
ejpam-6496	224	6	=	=	SYM
ejpam-6496	224	7	max{|kj1	max{|kj1	PROPN
ejpam-6496	224	8	−kj2	−kj2	PROPN
ejpam-6496	224	9	|	|	NOUN
ejpam-6496	224	10	:	:	PUNCT
ejpam-6496	224	11	j1	j1	PROPN
ejpam-6496	224	12	,	,	PUNCT
ejpam-6496	224	13	j2	j2	NOUN
ejpam-6496	224	14	=	=	SYM
ejpam-6496	224	15	1	1	NUM
ejpam-6496	224	16	,	,	PUNCT
ejpam-6496	224	17	.	.	PUNCT
ejpam-6496	224	18	.	.	PUNCT
ejpam-6496	224	19	.	.	PUNCT
ejpam-6496	224	20	,	,	PUNCT
ejpam-6496	224	21	l	l	NOUN
ejpam-6496	224	22	}	}	PUNCT
ejpam-6496	224	23	.	.	PUNCT
ejpam-6496	225	1	let	let	VERB
ejpam-6496	225	2	p	p	NOUN
ejpam-6496	225	3	:	:	PUNCT
ejpam-6496	225	4	=	=	SYM
ejpam-6496	225	5	{	{	PUNCT
ejpam-6496	225	6	(	(	PUNCT
ejpam-6496	225	7	ti	ti	NOUN
ejpam-6496	225	8	,	,	PUNCT
ejpam-6496	225	9	[	[	X
ejpam-6496	225	10	xi−1	xi−1	PROPN
ejpam-6496	225	11	,	,	PUNCT
ejpam-6496	225	12	xi	xi	X
ejpam-6496	225	13	]	]	PUNCT
ejpam-6496	225	14	)	)	PUNCT
ejpam-6496	225	15	:	:	PUNCT
ejpam-6496	226	1	i	i	PRON
ejpam-6496	226	2	=	=	NOUN
ejpam-6496	226	3	1	1	NUM
ejpam-6496	226	4	,	,	PUNCT
ejpam-6496	226	5	2	2	NUM
ejpam-6496	226	6	,	,	PUNCT
ejpam-6496	226	7	.	.	PUNCT
ejpam-6496	226	8	.	.	PUNCT
ejpam-6496	226	9	.	.	PUNCT
ejpam-6496	227	1	,	,	PUNCT
ejpam-6496	227	2	n	n	CCONJ
ejpam-6496	227	3	}	}	PUNCT
ejpam-6496	227	4	be	be	AUX
ejpam-6496	227	5	a	a	DET
ejpam-6496	227	6	δ	δ	NOUN
ejpam-6496	227	7	-	-	PUNCT
ejpam-6496	227	8	fine	fine	ADJ
ejpam-6496	227	9	tagged	tag	VERB
ejpam-6496	227	10	partition	partition	NOUN
ejpam-6496	227	11	of	of	ADP
ejpam-6496	227	12	[	[	PUNCT
ejpam-6496	227	13	a	a	PROPN
ejpam-6496	227	14	,	,	PUNCT
ejpam-6496	227	15	b	b	NOUN
ejpam-6496	227	16	]	]	X
ejpam-6496	227	17	.	.	PUNCT
ejpam-6496	228	1	for	for	ADP
ejpam-6496	228	2	each	each	DET
ejpam-6496	228	3	i	i	NOUN
ejpam-6496	228	4	=	=	NOUN
ejpam-6496	228	5	1	1	NUM
ejpam-6496	228	6	,	,	PUNCT
ejpam-6496	228	7	.	.	PUNCT
ejpam-6496	228	8	.	.	PUNCT
ejpam-6496	228	9	.	.	PUNCT
ejpam-6496	229	1	,	,	PUNCT
ejpam-6496	229	2	n	n	CCONJ
ejpam-6496	229	3	,	,	PUNCT
ejpam-6496	229	4	we	we	PRON
ejpam-6496	229	5	separate	separate	VERB
ejpam-6496	229	6	into	into	ADP
ejpam-6496	229	7	two	two	NUM
ejpam-6496	229	8	cases	case	NOUN
ejpam-6496	229	9	.	.	PUNCT
ejpam-6496	230	1	case	case	NOUN
ejpam-6496	230	2	1	1	NUM
ejpam-6496	230	3	:	:	PUNCT
ejpam-6496	231	1	[	[	X
ejpam-6496	231	2	xi−1	xi−1	PROPN
ejpam-6496	231	3	,	,	PUNCT
ejpam-6496	231	4	xi	xi	X
ejpam-6496	231	5	]	]	PUNCT
ejpam-6496	231	6	⊆	⊆	NUM
ejpam-6496	231	7	]	]	PUNCT
ejpam-6496	231	8	cj−1	cj−1	NOUN
ejpam-6496	231	9	,	,	PUNCT
ejpam-6496	231	10	cj	cj	NOUN
ejpam-6496	231	11	[	[	PUNCT
ejpam-6496	231	12	for	for	ADP
ejpam-6496	231	13	some	some	DET
ejpam-6496	231	14	j	j	NOUN
ejpam-6496	231	15	=	=	SYM
ejpam-6496	231	16	1	1	NUM
ejpam-6496	231	17	,	,	PUNCT
ejpam-6496	231	18	2	2	NUM
ejpam-6496	231	19	,	,	PUNCT
ejpam-6496	231	20	.	.	PUNCT
ejpam-6496	231	21	.	.	PUNCT
ejpam-6496	232	1	.	.	PUNCT
ejpam-6496	233	1	,	,	PUNCT
ejpam-6496	233	2	l.	l.	PROPN
ejpam-6496	233	3	since	since	SCONJ
ejpam-6496	233	4	|f(t)−f(s)|	|f(t)−f(s)|	PROPN
ejpam-6496	233	5	=	=	PROPN
ejpam-6496	233	6	|kj−kj	|kj−kj	VERB
ejpam-6496	233	7	|	|	ADV
ejpam-6496	233	8	=	=	SYM
ejpam-6496	233	9	0	0	NUM
ejpam-6496	233	10	for	for	ADP
ejpam-6496	233	11	all	all	DET
ejpam-6496	233	12	s	s	PROPN
ejpam-6496	233	13	,	,	PUNCT
ejpam-6496	233	14	t	t	PROPN
ejpam-6496	233	15	∈	∈	PROPN
ejpam-6496	234	1	[	[	X
ejpam-6496	234	2	xi−1	xi−1	PROPN
ejpam-6496	234	3	,	,	PUNCT
ejpam-6496	234	4	xi	xi	X
ejpam-6496	234	5	]	]	PUNCT
ejpam-6496	234	6	,	,	PUNCT
ejpam-6496	234	7	then	then	ADV
ejpam-6496	234	8	ω(f	ω(f	ADJ
ejpam-6496	234	9	,	,	PUNCT
ejpam-6496	234	10	[	[	X
ejpam-6496	234	11	xi−1	xi−1	PROPN
ejpam-6496	234	12	,	,	PUNCT
ejpam-6496	234	13	xi	xi	X
ejpam-6496	234	14	]	]	PUNCT
ejpam-6496	234	15	)	)	PUNCT
ejpam-6496	234	16	=	=	SYM
ejpam-6496	234	17	0	0	X
ejpam-6496	234	18	.	.	PUNCT
ejpam-6496	234	19	case	case	NOUN
ejpam-6496	234	20	2	2	NUM
ejpam-6496	234	21	:	:	PUNCT
ejpam-6496	234	22	[	[	X
ejpam-6496	234	23	xi−1	xi−1	PROPN
ejpam-6496	234	24	,	,	PUNCT
ejpam-6496	234	25	xi	xi	X
ejpam-6496	234	26	]	]	PUNCT
ejpam-6496	234	27	̸⊆	̸⊆	PUNCT
ejpam-6496	234	28	]	]	PUNCT
ejpam-6496	234	29	cj−1	cj−1	NOUN
ejpam-6496	234	30	,	,	PUNCT
ejpam-6496	234	31	cj	cj	NOUN
ejpam-6496	234	32	[	[	PUNCT
ejpam-6496	234	33	for	for	ADP
ejpam-6496	234	34	all	all	DET
ejpam-6496	234	35	j	j	NOUN
ejpam-6496	234	36	=	=	SYM
ejpam-6496	234	37	1	1	NUM
ejpam-6496	234	38	,	,	PUNCT
ejpam-6496	234	39	2	2	NUM
ejpam-6496	234	40	,	,	PUNCT
ejpam-6496	234	41	.	.	PUNCT
ejpam-6496	234	42	.	.	PUNCT
ejpam-6496	235	1	.	.	PUNCT
ejpam-6496	236	1	,	,	PUNCT
ejpam-6496	236	2	l.	l.	PROPN
ejpam-6496	236	3	since	since	SCONJ
ejpam-6496	236	4	|f(t	|f(t	PROPN
ejpam-6496	236	5	)	)	PUNCT
ejpam-6496	236	6	−	−	NUM
ejpam-6496	236	7	f(s)|	f(s)|	PROPN
ejpam-6496	236	8	≤	≤	PROPN
ejpam-6496	236	9	k	k	NOUN
ejpam-6496	236	10	for	for	ADP
ejpam-6496	236	11	all	all	DET
ejpam-6496	236	12	s	s	PROPN
ejpam-6496	236	13	,	,	PUNCT
ejpam-6496	236	14	t	t	PROPN
ejpam-6496	236	15	∈	∈	PROPN
ejpam-6496	237	1	[	[	X
ejpam-6496	237	2	xi−1	xi−1	PROPN
ejpam-6496	237	3	,	,	PUNCT
ejpam-6496	237	4	xi	xi	X
ejpam-6496	237	5	]	]	PUNCT
ejpam-6496	237	6	,	,	PUNCT
ejpam-6496	237	7	then	then	ADV
ejpam-6496	237	8	ω(f	ω(f	ADJ
ejpam-6496	237	9	,	,	PUNCT
ejpam-6496	237	10	[	[	X
ejpam-6496	237	11	xi−1	xi−1	PROPN
ejpam-6496	237	12	,	,	PUNCT
ejpam-6496	237	13	xi	xi	X
ejpam-6496	237	14	]	]	PUNCT
ejpam-6496	237	15	)	)	PUNCT
ejpam-6496	237	16	≤	≤	PROPN
ejpam-6496	237	17	k.	k.	PUNCT
ejpam-6496	238	1	furthermore	furthermore	ADV
ejpam-6496	238	2	,	,	PUNCT
ejpam-6496	238	3	since	since	SCONJ
ejpam-6496	238	4	[	[	X
ejpam-6496	238	5	xi−1	xi−1	PROPN
ejpam-6496	238	6	,	,	PUNCT
ejpam-6496	238	7	xi	xi	X
ejpam-6496	238	8	]	]	PUNCT
ejpam-6496	239	1	⊂	⊂	X
ejpam-6496	239	2	]	]	PUNCT
ejpam-6496	239	3	ti	ti	X
ejpam-6496	239	4	−	−	PROPN
ejpam-6496	239	5	ε	ε	PROPN
ejpam-6496	239	6	k+1	k+1	X
ejpam-6496	239	7	,	,	PUNCT
ejpam-6496	239	8	ti	ti	X
ejpam-6496	239	9	+	+	CCONJ
ejpam-6496	239	10	ε	ε	PROPN
ejpam-6496	239	11	k+1	k+1	X
ejpam-6496	239	12	[	[	PUNCT
ejpam-6496	239	13	,	,	PUNCT
ejpam-6496	239	14	so	so	ADV
ejpam-6496	239	15	xi	xi	INTJ
ejpam-6496	239	16	−	−	PROPN
ejpam-6496	240	1	xi−1	xi−1	PROPN
ejpam-6496	240	2	<	<	X
ejpam-6496	240	3	2ε	2ε	X
ejpam-6496	240	4	k+1	k+1	X
ejpam-6496	240	5	for	for	ADP
ejpam-6496	240	6	all	all	DET
ejpam-6496	240	7	i	i	PRON
ejpam-6496	240	8	=	=	NOUN
ejpam-6496	240	9	1	1	NUM
ejpam-6496	240	10	,	,	PUNCT
ejpam-6496	240	11	.	.	PUNCT
ejpam-6496	240	12	.	.	PUNCT
ejpam-6496	240	13	.	.	PUNCT
ejpam-6496	241	1	,	,	PUNCT
ejpam-6496	241	2	n.	n.	PROPN
ejpam-6496	241	3	let	let	VERB
ejpam-6496	241	4	i	i	PRON
ejpam-6496	241	5	:	:	PUNCT
ejpam-6496	241	6	=	=	SYM
ejpam-6496	241	7	{	{	PUNCT
ejpam-6496	241	8	i	i	NOUN
ejpam-6496	241	9	:	:	PUNCT
ejpam-6496	242	1	[	[	X
ejpam-6496	242	2	xi−1	xi−1	PROPN
ejpam-6496	242	3	,	,	PUNCT
ejpam-6496	242	4	xi	xi	X
ejpam-6496	242	5	]	]	PUNCT
ejpam-6496	242	6	̸⊆	̸⊆	PUNCT
ejpam-6496	242	7	]	]	PUNCT
ejpam-6496	242	8	cj−1	cj−1	NOUN
ejpam-6496	242	9	,	,	PUNCT
ejpam-6496	242	10	cj	cj	NOUN
ejpam-6496	242	11	[	[	PUNCT
ejpam-6496	242	12	for	for	ADP
ejpam-6496	242	13	all	all	DET
ejpam-6496	242	14	j	j	NOUN
ejpam-6496	242	15	=	=	SYM
ejpam-6496	242	16	1	1	NUM
ejpam-6496	242	17	,	,	PUNCT
ejpam-6496	242	18	2	2	NUM
ejpam-6496	242	19	,	,	PUNCT
ejpam-6496	242	20	.	.	PUNCT
ejpam-6496	242	21	.	.	PUNCT
ejpam-6496	242	22	.	.	PUNCT
ejpam-6496	243	1	,	,	PUNCT
ejpam-6496	243	2	l	l	NOUN
ejpam-6496	243	3	}	}	PUNCT
ejpam-6496	243	4	.	.	PUNCT
ejpam-6496	244	1	it	it	PRON
ejpam-6496	244	2	follows	follow	VERB
ejpam-6496	244	3	from	from	ADP
ejpam-6496	244	4	lemma	lemma	PROPN
ejpam-6496	244	5	3	3	NUM
ejpam-6496	244	6	that	that	PRON
ejpam-6496	244	7	|i|	|i|	VERB
ejpam-6496	244	8	≤	≤	NOUN
ejpam-6496	244	9	2l	2l	NUM
ejpam-6496	244	10	.	.	PUNCT
ejpam-6496	245	1	consequently	consequently	ADV
ejpam-6496	245	2	,	,	PUNCT
ejpam-6496	245	3	n∑	n∑	PROPN
ejpam-6496	245	4	i=1	i=1	PROPN
ejpam-6496	246	1	ω(f	ω(f	ADJ
ejpam-6496	246	2	,	,	PUNCT
ejpam-6496	246	3	[	[	X
ejpam-6496	246	4	xi−1	xi−1	PROPN
ejpam-6496	246	5	,	,	PUNCT
ejpam-6496	246	6	xi	xi	X
ejpam-6496	246	7	]	]	X
ejpam-6496	246	8	)	)	PUNCT
ejpam-6496	246	9	(	(	PUNCT
ejpam-6496	246	10	xi	xi	X
ejpam-6496	246	11	−	−	PROPN
ejpam-6496	246	12	xi−1	xi−1	PROPN
ejpam-6496	246	13	)	)	PUNCT
ejpam-6496	246	14	=	=	PUNCT
ejpam-6496	247	1	∑	∑	PROPN
ejpam-6496	247	2	i∈i	i∈i	ADJ
ejpam-6496	247	3	ω(f	ω(f	PROPN
ejpam-6496	247	4	,	,	PUNCT
ejpam-6496	247	5	[	[	X
ejpam-6496	247	6	xi−1	xi−1	PROPN
ejpam-6496	247	7	,	,	PUNCT
ejpam-6496	247	8	xi	xi	X
ejpam-6496	247	9	]	]	X
ejpam-6496	247	10	)	)	PUNCT
ejpam-6496	247	11	(	(	PUNCT
ejpam-6496	247	12	xi	xi	X
ejpam-6496	247	13	−	−	PROPN
ejpam-6496	247	14	xi−1	xi−1	PROPN
ejpam-6496	247	15	)	)	PUNCT
ejpam-6496	247	16	<	<	X
ejpam-6496	247	17	∑	∑	PROPN
ejpam-6496	247	18	i∈i	i∈i	ADJ
ejpam-6496	247	19	k	k	PROPN
ejpam-6496	247	20	(	(	PUNCT
ejpam-6496	247	21	2ε	2ε	NOUN
ejpam-6496	247	22	k	k	PROPN
ejpam-6496	248	1	+	+	PROPN
ejpam-6496	248	2	1	1	X
ejpam-6496	248	3	)	)	PUNCT
ejpam-6496	248	4	<	<	X
ejpam-6496	248	5	2ε|i|	2ε|i|	NUM
ejpam-6496	248	6	≤	≤	X
ejpam-6496	248	7	4lε	4lε	NOUN
ejpam-6496	248	8	.	.	PUNCT
ejpam-6496	249	1	therefore	therefore	ADV
ejpam-6496	249	2	,	,	PUNCT
ejpam-6496	249	3	f	f	PROPN
ejpam-6496	249	4	is	be	AUX
ejpam-6496	249	5	riemann	riemann	PROPN
ejpam-6496	249	6	integrable	integrable	ADJ
ejpam-6496	249	7	on	on	ADP
ejpam-6496	249	8	[	[	PUNCT
ejpam-6496	249	9	a	a	PROPN
ejpam-6496	249	10	,	,	PUNCT
ejpam-6496	249	11	b	b	NOUN
ejpam-6496	249	12	]	]	PUNCT
ejpam-6496	249	13	.	.	PUNCT
ejpam-6496	250	1	lemma	lemma	PROPN
ejpam-6496	250	2	4	4	X
ejpam-6496	250	3	.	.	PUNCT
ejpam-6496	251	1	let	let	VERB
ejpam-6496	251	2	f	f	NOUN
ejpam-6496	251	3	:	:	PUNCT
ejpam-6496	251	4	[	[	PUNCT
ejpam-6496	251	5	a	a	X
ejpam-6496	251	6	,	,	PUNCT
ejpam-6496	251	7	b	b	NOUN
ejpam-6496	251	8	]	]	PUNCT
ejpam-6496	251	9	→	→	PUNCT
ejpam-6496	251	10	r	r	NOUN
ejpam-6496	251	11	be	be	AUX
ejpam-6496	251	12	a	a	DET
ejpam-6496	251	13	function	function	NOUN
ejpam-6496	251	14	,	,	PUNCT
ejpam-6496	251	15	and	and	CCONJ
ejpam-6496	251	16	let	let	VERB
ejpam-6496	251	17	{	{	PUNCT
ejpam-6496	251	18	[	[	PUNCT
ejpam-6496	251	19	yj−1	yj−1	PROPN
ejpam-6496	251	20	,	,	PUNCT
ejpam-6496	251	21	yj	yj	PROPN
ejpam-6496	251	22	]	]	PUNCT
ejpam-6496	251	23	:	:	PUNCT
ejpam-6496	252	1	j	j	X
ejpam-6496	252	2	=	=	SYM
ejpam-6496	252	3	1	1	NUM
ejpam-6496	252	4	,	,	PUNCT
ejpam-6496	252	5	.	.	PUNCT
ejpam-6496	252	6	.	.	PUNCT
ejpam-6496	253	1	.	.	PUNCT
ejpam-6496	254	1	,	,	PUNCT
ejpam-6496	254	2	m	m	AUX
ejpam-6496	254	3	}	}	PUNCT
ejpam-6496	254	4	be	be	AUX
ejpam-6496	254	5	a	a	DET
ejpam-6496	254	6	partition	partition	NOUN
ejpam-6496	254	7	of	of	ADP
ejpam-6496	254	8	[	[	PUNCT
ejpam-6496	254	9	a	a	PROPN
ejpam-6496	254	10	,	,	PUNCT
ejpam-6496	254	11	b	b	NOUN
ejpam-6496	254	12	]	]	X
ejpam-6496	254	13	.	.	PUNCT
ejpam-6496	255	1	if	if	SCONJ
ejpam-6496	255	2	m∑	m∑	CCONJ
ejpam-6496	255	3	j=1	j=1	PROPN
ejpam-6496	255	4	ω(f	ω(f	ADJ
ejpam-6496	255	5	,	,	PUNCT
ejpam-6496	255	6	[	[	PUNCT
ejpam-6496	255	7	yj−1	yj−1	PROPN
ejpam-6496	255	8	,	,	PUNCT
ejpam-6496	255	9	yj	yj	PROPN
ejpam-6496	255	10	]	]	X
ejpam-6496	255	11	)	)	PUNCT
ejpam-6496	255	12	(	(	PUNCT
ejpam-6496	255	13	yj−yj−1	yj−yj−1	NOUN
ejpam-6496	255	14	)	)	PUNCT
ejpam-6496	255	15	<	<	X
ejpam-6496	255	16	1	1	NUM
ejpam-6496	255	17	,	,	PUNCT
ejpam-6496	255	18	then	then	ADV
ejpam-6496	255	19	ω(f	ω(f	ADJ
ejpam-6496	255	20	,	,	PUNCT
ejpam-6496	255	21	[	[	PUNCT
ejpam-6496	255	22	yj−1	yj−1	PROPN
ejpam-6496	255	23	,	,	PUNCT
ejpam-6496	255	24	yj	yj	PROPN
ejpam-6496	255	25	]	]	X
ejpam-6496	255	26	)	)	PUNCT
ejpam-6496	255	27	,	,	PUNCT
ejpam-6496	255	28	j	j	PROPN
ejpam-6496	255	29	=	=	SYM
ejpam-6496	255	30	1	1	NUM
ejpam-6496	255	31	,	,	PUNCT
ejpam-6496	255	32	.	.	PUNCT
ejpam-6496	255	33	.	.	PUNCT
ejpam-6496	256	1	.	.	PUNCT
ejpam-6496	257	1	,	,	PUNCT
ejpam-6496	257	2	m	m	PROPN
ejpam-6496	257	3	,	,	PUNCT
ejpam-6496	257	4	are	be	AUX
ejpam-6496	257	5	all	all	PRON
ejpam-6496	257	6	nonnegative	nonnegative	ADJ
ejpam-6496	257	7	real	real	ADJ
ejpam-6496	257	8	numbers	number	NOUN
ejpam-6496	257	9	.	.	PUNCT
ejpam-6496	258	1	proof	proof	NOUN
ejpam-6496	258	2	.	.	PUNCT
ejpam-6496	259	1	let	let	VERB
ejpam-6496	259	2	m	m	PRON
ejpam-6496	259	3	:	:	PUNCT
ejpam-6496	259	4	=	=	SYM
ejpam-6496	259	5	min{yj	min{yj	PROPN
ejpam-6496	259	6	−	−	PROPN
ejpam-6496	259	7	yj−1	yj−1	NOUN
ejpam-6496	259	8	:	:	PUNCT
ejpam-6496	259	9	j	j	PROPN
ejpam-6496	259	10	=	=	SYM
ejpam-6496	259	11	1	1	NUM
ejpam-6496	259	12	,	,	PUNCT
ejpam-6496	259	13	.	.	PUNCT
ejpam-6496	259	14	.	.	PUNCT
ejpam-6496	259	15	.	.	PUNCT
ejpam-6496	260	1	,	,	PUNCT
ejpam-6496	260	2	m	m	VERB
ejpam-6496	260	3	}	}	PUNCT
ejpam-6496	260	4	.	.	PUNCT
ejpam-6496	261	1	let	let	VERB
ejpam-6496	261	2	us	we	PRON
ejpam-6496	261	3	consider	consider	VERB
ejpam-6496	261	4	,	,	PUNCT
ejpam-6496	261	5	1	1	NUM
ejpam-6496	261	6	m	m	NOUN
ejpam-6496	261	7	>	>	X
ejpam-6496	261	8	1	1	NUM
ejpam-6496	261	9	m	m	NOUN
ejpam-6496	261	10	m∑	m∑	NOUN
ejpam-6496	261	11	j=1	j=1	NOUN
ejpam-6496	261	12	ω(f	ω(f	ADV
ejpam-6496	261	13	,	,	PUNCT
ejpam-6496	261	14	[	[	PUNCT
ejpam-6496	261	15	yj−1	yj−1	PROPN
ejpam-6496	261	16	,	,	PUNCT
ejpam-6496	261	17	yj	yj	PROPN
ejpam-6496	261	18	]	]	X
ejpam-6496	261	19	)	)	PUNCT
ejpam-6496	261	20	(	(	PUNCT
ejpam-6496	261	21	yj	yj	PROPN
ejpam-6496	261	22	−	−	PROPN
ejpam-6496	261	23	yj−1	yj−1	PROPN
ejpam-6496	261	24	)	)	PUNCT
ejpam-6496	261	25	=	=	PUNCT
ejpam-6496	262	1	m∑	m∑	ADV
ejpam-6496	262	2	j=1	j=1	PROPN
ejpam-6496	262	3	ω(f	ω(f	ADJ
ejpam-6496	262	4	,	,	PUNCT
ejpam-6496	262	5	[	[	PUNCT
ejpam-6496	262	6	yj−1	yj−1	PROPN
ejpam-6496	262	7	,	,	PUNCT
ejpam-6496	262	8	yj	yj	PROPN
ejpam-6496	262	9	]	]	X
ejpam-6496	262	10	)	)	PUNCT
ejpam-6496	262	11	(	(	PUNCT
ejpam-6496	262	12	yj	yj	PROPN
ejpam-6496	262	13	−	−	PROPN
ejpam-6496	262	14	yj−1	yj−1	PROPN
ejpam-6496	262	15	)	)	PUNCT
ejpam-6496	262	16	m	m	VERB
ejpam-6496	262	17	≥	≥	NOUN
ejpam-6496	262	18	m∑	m∑	VERB
ejpam-6496	262	19	j=1	j=1	PROPN
ejpam-6496	262	20	ω(f	ω(f	PROPN
ejpam-6496	262	21	,	,	PUNCT
ejpam-6496	262	22	[	[	PUNCT
ejpam-6496	262	23	yj−1	yj−1	PROPN
ejpam-6496	262	24	,	,	PUNCT
ejpam-6496	262	25	yj	yj	PROPN
ejpam-6496	262	26	]	]	X
ejpam-6496	262	27	)	)	PUNCT
ejpam-6496	262	28	.	.	PUNCT
ejpam-6496	263	1	since	since	SCONJ
ejpam-6496	263	2	ω(f	ω(f	ADJ
ejpam-6496	263	3	,	,	PUNCT
ejpam-6496	263	4	[	[	PUNCT
ejpam-6496	263	5	yj−1	yj−1	PROPN
ejpam-6496	263	6	,	,	PUNCT
ejpam-6496	263	7	yj	yj	PROPN
ejpam-6496	263	8	]	]	PUNCT
ejpam-6496	263	9	)	)	PUNCT
ejpam-6496	263	10	≥	≥	NOUN
ejpam-6496	263	11	0	0	NUM
ejpam-6496	263	12	for	for	ADP
ejpam-6496	263	13	all	all	DET
ejpam-6496	263	14	j	j	NOUN
ejpam-6496	263	15	=	=	SYM
ejpam-6496	263	16	1	1	NUM
ejpam-6496	263	17	,	,	PUNCT
ejpam-6496	263	18	.	.	PUNCT
ejpam-6496	263	19	.	.	PUNCT
ejpam-6496	263	20	.	.	PUNCT
ejpam-6496	264	1	,	,	PUNCT
ejpam-6496	264	2	m	m	PROPN
ejpam-6496	264	3	,	,	PUNCT
ejpam-6496	264	4	then	then	ADV
ejpam-6496	264	5	each	each	DET
ejpam-6496	264	6	ω(f	ω(f	ADJ
ejpam-6496	264	7	,	,	PUNCT
ejpam-6496	264	8	[	[	PUNCT
ejpam-6496	264	9	yj−1	yj−1	PROPN
ejpam-6496	264	10	,	,	PUNCT
ejpam-6496	264	11	yj	yj	PROPN
ejpam-6496	264	12	]	]	X
ejpam-6496	264	13	)	)	PUNCT
ejpam-6496	264	14	<	<	X
ejpam-6496	264	15	1	1	NUM
ejpam-6496	264	16	m	m	NOUN
ejpam-6496	264	17	.	.	PUNCT
ejpam-6496	265	1	therefore	therefore	ADV
ejpam-6496	265	2	,	,	PUNCT
ejpam-6496	265	3	we	we	PRON
ejpam-6496	265	4	conclude	conclude	VERB
ejpam-6496	265	5	that	that	SCONJ
ejpam-6496	265	6	,	,	PUNCT
ejpam-6496	265	7	for	for	ADP
ejpam-6496	265	8	each	each	DET
ejpam-6496	265	9	j	j	PROPN
ejpam-6496	265	10	=	=	SYM
ejpam-6496	265	11	1	1	NUM
ejpam-6496	265	12	,	,	PUNCT
ejpam-6496	265	13	.	.	PUNCT
ejpam-6496	265	14	.	.	PUNCT
ejpam-6496	265	15	.	.	PUNCT
ejpam-6496	266	1	,	,	PUNCT
ejpam-6496	266	2	m	m	X
ejpam-6496	266	3	,	,	PUNCT
ejpam-6496	266	4	there	there	PRON
ejpam-6496	266	5	exists	exist	VERB
ejpam-6496	266	6	ωj	ωj	ADP
ejpam-6496	266	7	∈	∈	NOUN
ejpam-6496	266	8	r	r	NOUN
ejpam-6496	266	9	with	with	ADP
ejpam-6496	266	10	0	0	NUM
ejpam-6496	266	11	≤	≤	NOUN
ejpam-6496	266	12	ωj	ωj	ADP
ejpam-6496	266	13	<	<	X
ejpam-6496	266	14	1	1	NUM
ejpam-6496	266	15	m	m	NOUN
ejpam-6496	266	16	such	such	ADJ
ejpam-6496	266	17	that	that	SCONJ
ejpam-6496	266	18	ω(f	ω(f	ADJ
ejpam-6496	266	19	,	,	PUNCT
ejpam-6496	266	20	[	[	PUNCT
ejpam-6496	266	21	yj−1	yj−1	PROPN
ejpam-6496	266	22	,	,	PUNCT
ejpam-6496	266	23	yj	yj	PROPN
ejpam-6496	266	24	]	]	X
ejpam-6496	266	25	)	)	PUNCT
ejpam-6496	266	26	=	=	SYM
ejpam-6496	266	27	ωj	ωj	PROPN
ejpam-6496	266	28	.	.	PUNCT
ejpam-6496	267	1	s.	s.	PROPN
ejpam-6496	267	2	prongjit	prongjit	ADV
ejpam-6496	267	3	,	,	PUNCT
ejpam-6496	267	4	p.	p.	PROPN
ejpam-6496	267	5	prasertsang	prasertsang	PROPN
ejpam-6496	267	6	/	/	SYM
ejpam-6496	267	7	eur	eur	PROPN
ejpam-6496	267	8	.	.	PUNCT
ejpam-6496	268	1	j.	j.	PROPN
ejpam-6496	268	2	pure	pure	PROPN
ejpam-6496	268	3	appl	appl	PROPN
ejpam-6496	268	4	.	.	PROPN
ejpam-6496	268	5	math	math	PROPN
ejpam-6496	268	6	,	,	PUNCT
ejpam-6496	268	7	18	18	NUM
ejpam-6496	268	8	(	(	PUNCT
ejpam-6496	268	9	3	3	NUM
ejpam-6496	268	10	)	)	PUNCT
ejpam-6496	268	11	(	(	PUNCT
ejpam-6496	268	12	2025	2025	NUM
ejpam-6496	268	13	)	)	PUNCT
ejpam-6496	268	14	,	,	PUNCT
ejpam-6496	268	15	6496	6496	NUM
ejpam-6496	268	16	9	9	NUM
ejpam-6496	268	17	of	of	ADP
ejpam-6496	268	18	10	10	NUM
ejpam-6496	268	19	remark	remark	NOUN
ejpam-6496	268	20	1	1	NUM
ejpam-6496	268	21	.	.	PUNCT
ejpam-6496	269	1	lemma	lemma	PROPN
ejpam-6496	269	2	4	4	NUM
ejpam-6496	269	3	also	also	ADV
ejpam-6496	269	4	holds	hold	VERB
ejpam-6496	269	5	for	for	ADP
ejpam-6496	269	6	the	the	DET
ejpam-6496	269	7	condition	condition	NOUN
ejpam-6496	269	8	m∑	m∑	VERB
ejpam-6496	269	9	j=1	j=1	PROPN
ejpam-6496	269	10	ω(f	ω(f	PROPN
ejpam-6496	269	11	,	,	PUNCT
ejpam-6496	269	12	[	[	PUNCT
ejpam-6496	269	13	yj−1	yj−1	PROPN
ejpam-6496	269	14	,	,	PUNCT
ejpam-6496	269	15	yj	yj	PROPN
ejpam-6496	269	16	]	]	X
ejpam-6496	269	17	)	)	PUNCT
ejpam-6496	269	18	(	(	PUNCT
ejpam-6496	269	19	yj	yj	PROPN
ejpam-6496	269	20	−	−	PROPN
ejpam-6496	269	21	yj−1	yj−1	PROPN
ejpam-6496	269	22	)	)	PUNCT
ejpam-6496	269	23	<	<	X
ejpam-6496	269	24	m	m	X
ejpam-6496	269	25	,	,	PUNCT
ejpam-6496	269	26	where	where	SCONJ
ejpam-6496	269	27	m	m	NOUN
ejpam-6496	269	28	is	be	AUX
ejpam-6496	269	29	any	any	DET
ejpam-6496	269	30	positive	positive	ADJ
ejpam-6496	269	31	real	real	ADJ
ejpam-6496	269	32	number	number	NOUN
ejpam-6496	269	33	.	.	PUNCT
ejpam-6496	270	1	we	we	PRON
ejpam-6496	270	2	prove	prove	VERB
ejpam-6496	270	3	only	only	ADV
ejpam-6496	270	4	a	a	DET
ejpam-6496	270	5	special	special	ADJ
ejpam-6496	270	6	case	case	NOUN
ejpam-6496	270	7	m	m	NOUN
ejpam-6496	270	8	=	=	NOUN
ejpam-6496	270	9	1	1	NUM
ejpam-6496	270	10	for	for	ADP
ejpam-6496	270	11	using	use	VERB
ejpam-6496	270	12	in	in	ADP
ejpam-6496	270	13	the	the	DET
ejpam-6496	270	14	next	next	ADJ
ejpam-6496	270	15	theorem	theorem	PROPN
ejpam-6496	270	16	.	.	PUNCT
ejpam-6496	270	17	theorem	theorem	VERB
ejpam-6496	270	18	9	9	NUM
ejpam-6496	270	19	.	.	PUNCT
ejpam-6496	271	1	if	if	SCONJ
ejpam-6496	271	2	f	f	PROPN
ejpam-6496	271	3	is	be	AUX
ejpam-6496	271	4	riemann	riemann	PROPN
ejpam-6496	271	5	integrable	integrable	ADJ
ejpam-6496	271	6	on	on	ADP
ejpam-6496	271	7	[	[	PUNCT
ejpam-6496	271	8	a	a	PROPN
ejpam-6496	271	9	,	,	PUNCT
ejpam-6496	271	10	b	b	NOUN
ejpam-6496	271	11	]	]	X
ejpam-6496	271	12	,	,	PUNCT
ejpam-6496	271	13	then	then	ADV
ejpam-6496	271	14	f	f	PROPN
ejpam-6496	271	15	is	be	AUX
ejpam-6496	271	16	bounded	bound	VERB
ejpam-6496	271	17	on	on	ADP
ejpam-6496	271	18	[	[	PUNCT
ejpam-6496	271	19	a	a	PROPN
ejpam-6496	271	20	,	,	PUNCT
ejpam-6496	271	21	b	b	NOUN
ejpam-6496	271	22	]	]	PUNCT
ejpam-6496	271	23	.	.	PUNCT
ejpam-6496	272	1	proof	proof	NOUN
ejpam-6496	272	2	.	.	PUNCT
ejpam-6496	273	1	by	by	ADP
ejpam-6496	273	2	assumption	assumption	NOUN
ejpam-6496	273	3	,	,	PUNCT
ejpam-6496	273	4	there	there	PRON
ejpam-6496	273	5	exists	exist	VERB
ejpam-6496	273	6	a	a	DET
ejpam-6496	273	7	partition	partition	NOUN
ejpam-6496	273	8	{	{	PUNCT
ejpam-6496	273	9	[	[	PUNCT
ejpam-6496	273	10	yj−1	yj−1	PROPN
ejpam-6496	273	11	,	,	PUNCT
ejpam-6496	273	12	yj	yj	PROPN
ejpam-6496	273	13	]	]	PUNCT
ejpam-6496	273	14	:	:	PUNCT
ejpam-6496	273	15	j	j	X
ejpam-6496	273	16	=	=	SYM
ejpam-6496	273	17	1	1	NUM
ejpam-6496	273	18	,	,	PUNCT
ejpam-6496	273	19	.	.	PUNCT
ejpam-6496	273	20	.	.	PUNCT
ejpam-6496	274	1	.	.	PUNCT
ejpam-6496	275	1	,	,	PUNCT
ejpam-6496	275	2	m	m	VERB
ejpam-6496	275	3	}	}	PUNCT
ejpam-6496	275	4	of	of	ADP
ejpam-6496	275	5	[	[	PUNCT
ejpam-6496	275	6	a	a	PROPN
ejpam-6496	275	7	,	,	PUNCT
ejpam-6496	275	8	b	b	NOUN
ejpam-6496	275	9	]	]	PUNCT
ejpam-6496	275	10	such	such	ADJ
ejpam-6496	275	11	that	that	SCONJ
ejpam-6496	275	12	m∑	m∑	VERB
ejpam-6496	275	13	j=1	j=1	PROPN
ejpam-6496	275	14	ω(f	ω(f	PROPN
ejpam-6496	275	15	,	,	PUNCT
ejpam-6496	275	16	[	[	PUNCT
ejpam-6496	275	17	yj−1	yj−1	PROPN
ejpam-6496	275	18	,	,	PUNCT
ejpam-6496	275	19	yj	yj	PROPN
ejpam-6496	275	20	]	]	X
ejpam-6496	275	21	)	)	PUNCT
ejpam-6496	275	22	(	(	PUNCT
ejpam-6496	275	23	yj	yj	PROPN
ejpam-6496	275	24	−	−	PROPN
ejpam-6496	275	25	yj−1	yj−1	PROPN
ejpam-6496	275	26	)	)	PUNCT
ejpam-6496	275	27	<	<	X
ejpam-6496	275	28	1	1	X
ejpam-6496	275	29	.	.	PUNCT
ejpam-6496	276	1	it	it	PRON
ejpam-6496	276	2	follows	follow	VERB
ejpam-6496	276	3	from	from	ADP
ejpam-6496	276	4	lemma	lemma	PROPN
ejpam-6496	276	5	4	4	NUM
ejpam-6496	276	6	that	that	PRON
ejpam-6496	276	7	,	,	PUNCT
ejpam-6496	276	8	for	for	ADP
ejpam-6496	276	9	each	each	DET
ejpam-6496	276	10	j	j	PROPN
ejpam-6496	276	11	=	=	SYM
ejpam-6496	276	12	1	1	NUM
ejpam-6496	276	13	,	,	PUNCT
ejpam-6496	276	14	.	.	PUNCT
ejpam-6496	276	15	.	.	PUNCT
ejpam-6496	277	1	.	.	PUNCT
ejpam-6496	278	1	,	,	PUNCT
ejpam-6496	278	2	m	m	PROPN
ejpam-6496	278	3	,	,	PUNCT
ejpam-6496	278	4	ω(f	ω(f	ADJ
ejpam-6496	278	5	,	,	PUNCT
ejpam-6496	278	6	[	[	PUNCT
ejpam-6496	278	7	yj−1	yj−1	PROPN
ejpam-6496	278	8	,	,	PUNCT
ejpam-6496	278	9	yj	yj	PROPN
ejpam-6496	278	10	]	]	PUNCT
ejpam-6496	278	11	)	)	PUNCT
ejpam-6496	278	12	is	be	AUX
ejpam-6496	278	13	a	a	DET
ejpam-6496	278	14	nonnegative	nonnegative	ADJ
ejpam-6496	278	15	real	real	ADJ
ejpam-6496	278	16	number	number	NOUN
ejpam-6496	278	17	.	.	PUNCT
ejpam-6496	279	1	we	we	PRON
ejpam-6496	279	2	denote	denote	VERB
ejpam-6496	279	3	ωj	ωj	ADP
ejpam-6496	279	4	:	:	PUNCT
ejpam-6496	279	5	=	=	PUNCT
ejpam-6496	279	6	ω(f	ω(f	X
ejpam-6496	279	7	,	,	PUNCT
ejpam-6496	279	8	[	[	PUNCT
ejpam-6496	279	9	yj−1	yj−1	PROPN
ejpam-6496	279	10	,	,	PUNCT
ejpam-6496	279	11	yj	yj	PROPN
ejpam-6496	279	12	]	]	X
ejpam-6496	279	13	)	)	PUNCT
ejpam-6496	279	14	for	for	ADP
ejpam-6496	279	15	convenience	convenience	NOUN
ejpam-6496	279	16	.	.	PUNCT
ejpam-6496	280	1	to	to	PART
ejpam-6496	280	2	show	show	VERB
ejpam-6496	280	3	that	that	SCONJ
ejpam-6496	280	4	f	f	PROPN
ejpam-6496	280	5	is	be	AUX
ejpam-6496	280	6	bounded	bound	VERB
ejpam-6496	280	7	on	on	ADP
ejpam-6496	280	8	[	[	PUNCT
ejpam-6496	280	9	y0	y0	NOUN
ejpam-6496	280	10	,	,	PUNCT
ejpam-6496	280	11	y1	y1	NOUN
ejpam-6496	280	12	]	]	PUNCT
ejpam-6496	280	13	,	,	PUNCT
ejpam-6496	280	14	let	let	VERB
ejpam-6496	280	15	c	c	X
ejpam-6496	280	16	,	,	PUNCT
ejpam-6496	280	17	d	d	PROPN
ejpam-6496	280	18	∈	∈	PROPN
ejpam-6496	280	19	]	]	PUNCT
ejpam-6496	280	20	y0	y0	NOUN
ejpam-6496	280	21	,	,	PUNCT
ejpam-6496	280	22	y1	y1	NOUN
ejpam-6496	280	23	[	[	PUNCT
ejpam-6496	280	24	with	with	ADP
ejpam-6496	280	25	c	c	PROPN
ejpam-6496	280	26	<	<	X
ejpam-6496	280	27	d.	d.	PROPN
ejpam-6496	280	28	define	define	VERB
ejpam-6496	280	29	a	a	DET
ejpam-6496	280	30	gauge	gauge	NOUN
ejpam-6496	280	31	δ	δ	NOUN
ejpam-6496	280	32	on	on	ADP
ejpam-6496	280	33	[	[	PUNCT
ejpam-6496	280	34	c	c	X
ejpam-6496	280	35	,	,	PUNCT
ejpam-6496	280	36	d	d	X
ejpam-6496	280	37	]	]	PUNCT
ejpam-6496	280	38	as	as	SCONJ
ejpam-6496	280	39	follows	follow	VERB
ejpam-6496	280	40	:	:	PUNCT
ejpam-6496	280	41	let	let	VERB
ejpam-6496	280	42	x	x	X
ejpam-6496	280	43	∈	∈	PROPN
ejpam-6496	280	44	[	[	PUNCT
ejpam-6496	280	45	c	c	X
ejpam-6496	280	46	,	,	PUNCT
ejpam-6496	280	47	d	d	X
ejpam-6496	280	48	]	]	PUNCT
ejpam-6496	280	49	be	be	AUX
ejpam-6496	280	50	fixed	fix	VERB
ejpam-6496	280	51	,	,	PUNCT
ejpam-6496	280	52	then	then	ADV
ejpam-6496	280	53	there	there	PRON
ejpam-6496	280	54	exists	exist	VERB
ejpam-6496	280	55	a	a	DET
ejpam-6496	280	56	number	number	NOUN
ejpam-6496	280	57	δx	δx	NOUN
ejpam-6496	280	58	>	>	X
ejpam-6496	280	59	0	0	NUM
ejpam-6496	280	60	such	such	ADJ
ejpam-6496	280	61	that	that	SCONJ
ejpam-6496	280	62	]	]	X
ejpam-6496	280	63	x−	x−	PROPN
ejpam-6496	280	64	δx	δx	PROPN
ejpam-6496	280	65	,	,	PUNCT
ejpam-6496	280	66	x+	x+	ADJ
ejpam-6496	280	67	δx	δx	PROPN
ejpam-6496	280	68	[	[	PUNCT
ejpam-6496	280	69	⊂	⊂	X
ejpam-6496	280	70	[	[	PUNCT
ejpam-6496	280	71	y0	y0	NOUN
ejpam-6496	280	72	,	,	PUNCT
ejpam-6496	280	73	y1	y1	NOUN
ejpam-6496	280	74	]	]	PUNCT
ejpam-6496	280	75	.	.	PUNCT
ejpam-6496	281	1	we	we	PRON
ejpam-6496	281	2	define	define	VERB
ejpam-6496	281	3	δ(x	δ(x	PROPN
ejpam-6496	281	4	)	)	PUNCT
ejpam-6496	281	5	:	:	PUNCT
ejpam-6496	281	6	=	=	NUM
ejpam-6496	281	7	δx	δx	NOUN
ejpam-6496	281	8	,	,	PUNCT
ejpam-6496	281	9	and	and	CCONJ
ejpam-6496	281	10	let	let	VERB
ejpam-6496	281	11	p	p	NOUN
ejpam-6496	281	12	:	:	PUNCT
ejpam-6496	281	13	=	=	SYM
ejpam-6496	281	14	{	{	PUNCT
ejpam-6496	281	15	(	(	PUNCT
ejpam-6496	281	16	ti	ti	NOUN
ejpam-6496	281	17	,	,	PUNCT
ejpam-6496	281	18	[	[	X
ejpam-6496	281	19	xi−1	xi−1	PROPN
ejpam-6496	281	20	,	,	PUNCT
ejpam-6496	281	21	xi	xi	X
ejpam-6496	281	22	]	]	PUNCT
ejpam-6496	281	23	)	)	PUNCT
ejpam-6496	281	24	:	:	PUNCT
ejpam-6496	282	1	i	i	PRON
ejpam-6496	282	2	=	=	NOUN
ejpam-6496	282	3	1	1	NUM
ejpam-6496	282	4	,	,	PUNCT
ejpam-6496	282	5	2	2	NUM
ejpam-6496	282	6	,	,	PUNCT
ejpam-6496	282	7	.	.	PUNCT
ejpam-6496	282	8	.	.	PUNCT
ejpam-6496	282	9	.	.	PUNCT
ejpam-6496	283	1	,	,	PUNCT
ejpam-6496	283	2	n	n	CCONJ
ejpam-6496	283	3	}	}	PUNCT
ejpam-6496	283	4	be	be	AUX
ejpam-6496	283	5	a	a	DET
ejpam-6496	283	6	δ	δ	NOUN
ejpam-6496	283	7	-	-	PUNCT
ejpam-6496	283	8	fine	fine	ADJ
ejpam-6496	283	9	tagged	tag	VERB
ejpam-6496	283	10	partition	partition	NOUN
ejpam-6496	283	11	of	of	ADP
ejpam-6496	283	12	[	[	PUNCT
ejpam-6496	283	13	c	c	X
ejpam-6496	283	14	,	,	PUNCT
ejpam-6496	283	15	d	d	NOUN
ejpam-6496	283	16	]	]	PUNCT
ejpam-6496	283	17	.	.	PUNCT
ejpam-6496	284	1	set	set	VERB
ejpam-6496	284	2	m	m	VERB
ejpam-6496	284	3	:	:	PUNCT
ejpam-6496	284	4	=	=	SYM
ejpam-6496	284	5	max{|f(ti)|	max{|f(ti)|	NOUN
ejpam-6496	284	6	:	:	PUNCT
ejpam-6496	284	7	i	i	NOUN
ejpam-6496	284	8	=	=	NOUN
ejpam-6496	284	9	1	1	NUM
ejpam-6496	284	10	,	,	PUNCT
ejpam-6496	284	11	.	.	PUNCT
ejpam-6496	284	12	.	.	PUNCT
ejpam-6496	285	1	.	.	PUNCT
ejpam-6496	285	2	,	,	PUNCT
ejpam-6496	285	3	n	n	CCONJ
ejpam-6496	285	4	}	}	PUNCT
ejpam-6496	285	5	.	.	PUNCT
ejpam-6496	286	1	let	let	VERB
ejpam-6496	286	2	x	x	PRON
ejpam-6496	286	3	be	be	AUX
ejpam-6496	286	4	any	any	DET
ejpam-6496	286	5	element	element	NOUN
ejpam-6496	286	6	of	of	ADP
ejpam-6496	286	7	[	[	PUNCT
ejpam-6496	286	8	c	c	X
ejpam-6496	286	9	,	,	PUNCT
ejpam-6496	286	10	d	d	X
ejpam-6496	286	11	]	]	X
ejpam-6496	286	12	,	,	PUNCT
ejpam-6496	286	13	then	then	ADV
ejpam-6496	286	14	x	x	X
ejpam-6496	286	15	∈	∈	PROPN
ejpam-6496	286	16	[	[	X
ejpam-6496	286	17	xi−1	xi−1	PROPN
ejpam-6496	286	18	,	,	PUNCT
ejpam-6496	286	19	xi	xi	X
ejpam-6496	286	20	]	]	PUNCT
ejpam-6496	286	21	for	for	ADP
ejpam-6496	286	22	some	some	DET
ejpam-6496	286	23	i	i	NOUN
ejpam-6496	286	24	=	=	NOUN
ejpam-6496	286	25	1	1	NUM
ejpam-6496	286	26	,	,	PUNCT
ejpam-6496	286	27	2	2	NUM
ejpam-6496	286	28	,	,	PUNCT
ejpam-6496	286	29	.	.	PUNCT
ejpam-6496	286	30	.	.	PUNCT
ejpam-6496	286	31	.	.	PUNCT
ejpam-6496	287	1	,	,	PUNCT
ejpam-6496	287	2	n	n	CCONJ
ejpam-6496	287	3	,	,	PUNCT
ejpam-6496	287	4	and	and	CCONJ
ejpam-6496	287	5	hence	hence	ADV
ejpam-6496	287	6	|f(x)|	|f(x)|	PROPN
ejpam-6496	287	7	≤	≤	NUM
ejpam-6496	287	8	|f(x)−	|f(x)−	NOUN
ejpam-6496	287	9	f(ti)|+	f(ti)|+	PROPN
ejpam-6496	287	10	|f(ti)|	|f(ti)|	ADP
ejpam-6496	287	11	≤	≤	NOUN
ejpam-6496	287	12	ωj	ωj	ADP
ejpam-6496	287	13	+	+	ADJ
ejpam-6496	287	14	m.	m.	NOUN
ejpam-6496	287	15	for	for	ADP
ejpam-6496	287	16	this	this	DET
ejpam-6496	287	17	reason	reason	NOUN
ejpam-6496	287	18	,	,	PUNCT
ejpam-6496	287	19	f	f	PROPN
ejpam-6496	287	20	is	be	AUX
ejpam-6496	287	21	bounded	bound	VERB
ejpam-6496	287	22	on	on	ADP
ejpam-6496	287	23	[	[	PUNCT
ejpam-6496	287	24	c	c	X
ejpam-6496	287	25	,	,	PUNCT
ejpam-6496	287	26	d	d	NOUN
ejpam-6496	287	27	]	]	X
ejpam-6496	287	28	.	.	PUNCT
ejpam-6496	288	1	since	since	SCONJ
ejpam-6496	288	2	c	c	PROPN
ejpam-6496	288	3	and	and	CCONJ
ejpam-6496	288	4	d	d	PROPN
ejpam-6496	288	5	are	be	AUX
ejpam-6496	288	6	arbitrary	arbitrary	ADJ
ejpam-6496	288	7	elements	element	NOUN
ejpam-6496	288	8	in	in	ADP
ejpam-6496	288	9	]	]	X
ejpam-6496	288	10	y0	y0	NOUN
ejpam-6496	288	11	,	,	PUNCT
ejpam-6496	288	12	y1	y1	INTJ
ejpam-6496	288	13	[	[	PUNCT
ejpam-6496	288	14	,	,	PUNCT
ejpam-6496	288	15	so	so	CCONJ
ejpam-6496	288	16	f	f	PROPN
ejpam-6496	288	17	is	be	AUX
ejpam-6496	288	18	bounded	bound	VERB
ejpam-6496	288	19	on	on	ADP
ejpam-6496	288	20	]	]	PUNCT
ejpam-6496	288	21	y0	y0	NOUN
ejpam-6496	288	22	,	,	PUNCT
ejpam-6496	288	23	y1	y1	INTJ
ejpam-6496	288	24	[	[	PUNCT
ejpam-6496	288	25	.	.	PUNCT
ejpam-6496	289	1	we	we	PRON
ejpam-6496	289	2	assume	assume	VERB
ejpam-6496	289	3	further	far	ADV
ejpam-6496	289	4	that	that	SCONJ
ejpam-6496	289	5	f	f	PROPN
ejpam-6496	289	6	is	be	AUX
ejpam-6496	289	7	bounded	bound	VERB
ejpam-6496	289	8	on	on	ADP
ejpam-6496	289	9	]	]	PUNCT
ejpam-6496	289	10	y0	y0	NOUN
ejpam-6496	289	11	,	,	PUNCT
ejpam-6496	289	12	y1	y1	NOUN
ejpam-6496	289	13	[	[	PUNCT
ejpam-6496	289	14	by	by	ADP
ejpam-6496	289	15	a	a	DET
ejpam-6496	289	16	positive	positive	ADJ
ejpam-6496	289	17	real	real	ADJ
ejpam-6496	289	18	number	number	NOUN
ejpam-6496	289	19	m̂	m̂	NUM
ejpam-6496	289	20	,	,	PUNCT
ejpam-6496	289	21	and	and	CCONJ
ejpam-6496	289	22	let	let	VERB
ejpam-6496	289	23	m1	m1	PROPN
ejpam-6496	289	24	:	:	PUNCT
ejpam-6496	289	25	=	=	SYM
ejpam-6496	289	26	max{m̂	max{m̂	PROPN
ejpam-6496	289	27	,	,	PUNCT
ejpam-6496	289	28	|f(y0)|	|f(y0)|	PROPN
ejpam-6496	289	29	,	,	PUNCT
ejpam-6496	289	30	|f(y1)|	|f(y1)|	NOUN
ejpam-6496	289	31	}	}	PUNCT
ejpam-6496	289	32	.	.	PUNCT
ejpam-6496	290	1	clearly	clearly	ADV
ejpam-6496	290	2	,	,	PUNCT
ejpam-6496	290	3	f	f	PROPN
ejpam-6496	290	4	is	be	AUX
ejpam-6496	290	5	bounded	bound	VERB
ejpam-6496	290	6	on	on	ADP
ejpam-6496	290	7	[	[	PUNCT
ejpam-6496	290	8	y0	y0	NOUN
ejpam-6496	290	9	,	,	PUNCT
ejpam-6496	290	10	y1	y1	NOUN
ejpam-6496	290	11	]	]	PUNCT
ejpam-6496	290	12	by	by	ADP
ejpam-6496	290	13	m1	m1	PROPN
ejpam-6496	290	14	as	as	SCONJ
ejpam-6496	290	15	desired	desire	VERB
ejpam-6496	290	16	.	.	PUNCT
ejpam-6496	291	1	moreover	moreover	ADV
ejpam-6496	291	2	,	,	PUNCT
ejpam-6496	291	3	for	for	ADP
ejpam-6496	291	4	each	each	DET
ejpam-6496	291	5	j	j	PROPN
ejpam-6496	291	6	=	=	SYM
ejpam-6496	291	7	2	2	NUM
ejpam-6496	291	8	,	,	PUNCT
ejpam-6496	291	9	.	.	PUNCT
ejpam-6496	291	10	.	.	PUNCT
ejpam-6496	291	11	.	.	PUNCT
ejpam-6496	292	1	,	,	PUNCT
ejpam-6496	292	2	m	m	X
ejpam-6496	292	3	,	,	PUNCT
ejpam-6496	292	4	we	we	PRON
ejpam-6496	292	5	can	can	AUX
ejpam-6496	292	6	prove	prove	VERB
ejpam-6496	292	7	in	in	ADP
ejpam-6496	292	8	the	the	DET
ejpam-6496	292	9	same	same	ADJ
ejpam-6496	292	10	way	way	NOUN
ejpam-6496	292	11	that	that	PRON
ejpam-6496	292	12	f	f	PROPN
ejpam-6496	292	13	is	be	AUX
ejpam-6496	292	14	bounded	bound	VERB
ejpam-6496	292	15	on	on	ADP
ejpam-6496	292	16	[	[	PUNCT
ejpam-6496	292	17	yj−1	yj−1	PROPN
ejpam-6496	292	18	,	,	PUNCT
ejpam-6496	292	19	yj	yj	PROPN
ejpam-6496	292	20	]	]	PUNCT
ejpam-6496	292	21	by	by	ADP
ejpam-6496	292	22	some	some	DET
ejpam-6496	292	23	positive	positive	ADJ
ejpam-6496	292	24	real	real	ADJ
ejpam-6496	292	25	number	number	NOUN
ejpam-6496	292	26	mj	mj	PROPN
ejpam-6496	292	27	.	.	PUNCT
ejpam-6496	293	1	finally	finally	ADV
ejpam-6496	293	2	,	,	PUNCT
ejpam-6496	293	3	the	the	DET
ejpam-6496	293	4	conclusion	conclusion	NOUN
ejpam-6496	293	5	of	of	ADP
ejpam-6496	293	6	this	this	DET
ejpam-6496	293	7	theorem	theorem	NOUN
ejpam-6496	293	8	is	be	AUX
ejpam-6496	293	9	attained	attain	VERB
ejpam-6496	293	10	by	by	ADP
ejpam-6496	293	11	letting	let	VERB
ejpam-6496	293	12	m	m	PRON
ejpam-6496	293	13	:	:	PUNCT
ejpam-6496	293	14	=	=	SYM
ejpam-6496	293	15	max{mj	max{mj	X
ejpam-6496	293	16	:	:	PUNCT
ejpam-6496	293	17	j	j	PROPN
ejpam-6496	293	18	=	=	SYM
ejpam-6496	293	19	1	1	NUM
ejpam-6496	293	20	,	,	PUNCT
ejpam-6496	293	21	.	.	PUNCT
ejpam-6496	293	22	.	.	PUNCT
ejpam-6496	293	23	.	.	PUNCT
ejpam-6496	294	1	,	,	PUNCT
ejpam-6496	294	2	m	m	VERB
ejpam-6496	294	3	}	}	PUNCT
ejpam-6496	294	4	,	,	PUNCT
ejpam-6496	294	5	so	so	SCONJ
ejpam-6496	294	6	that	that	SCONJ
ejpam-6496	294	7	|f(y)|	|f(y)|	PROPN
ejpam-6496	294	8	≤	≤	PROPN
ejpam-6496	294	9	m	m	VERB
ejpam-6496	294	10	for	for	ADP
ejpam-6496	294	11	all	all	DET
ejpam-6496	294	12	y	y	PROPN
ejpam-6496	294	13	∈	∈	PROPN
ejpam-6496	294	14	[	[	PUNCT
ejpam-6496	294	15	a	a	PROPN
ejpam-6496	294	16	,	,	PUNCT
ejpam-6496	294	17	b	b	NOUN
ejpam-6496	294	18	]	]	X
ejpam-6496	294	19	.	.	PUNCT
ejpam-6496	295	1	4	4	X
ejpam-6496	295	2	.	.	X
ejpam-6496	295	3	conclusions	conclusion	NOUN
ejpam-6496	295	4	the	the	DET
ejpam-6496	295	5	concept	concept	NOUN
ejpam-6496	295	6	of	of	ADP
ejpam-6496	295	7	δ	δ	PROPN
ejpam-6496	295	8	-	-	PUNCT
ejpam-6496	295	9	fine	fine	ADJ
ejpam-6496	295	10	tagged	tag	VERB
ejpam-6496	295	11	partitions	partition	NOUN
ejpam-6496	295	12	allows	allow	VERB
ejpam-6496	295	13	us	we	PRON
ejpam-6496	295	14	to	to	PART
ejpam-6496	295	15	perceive	perceive	VERB
ejpam-6496	295	16	knowledge	knowledge	NOUN
ejpam-6496	295	17	of	of	ADP
ejpam-6496	295	18	elementary	elementary	ADJ
ejpam-6496	295	19	real	real	ADJ
ejpam-6496	295	20	analysis	analysis	NOUN
ejpam-6496	295	21	in	in	ADP
ejpam-6496	295	22	a	a	DET
ejpam-6496	295	23	different	different	ADJ
ejpam-6496	295	24	facet	facet	NOUN
ejpam-6496	295	25	.	.	PUNCT
ejpam-6496	296	1	this	this	DET
ejpam-6496	296	2	research	research	NOUN
ejpam-6496	296	3	has	have	AUX
ejpam-6496	296	4	shown	show	VERB
ejpam-6496	296	5	a	a	DET
ejpam-6496	296	6	little	little	ADJ
ejpam-6496	296	7	new	new	ADJ
ejpam-6496	296	8	property	property	NOUN
ejpam-6496	296	9	of	of	ADP
ejpam-6496	296	10	differentiable	differentiable	ADJ
ejpam-6496	296	11	functions	function	NOUN
ejpam-6496	296	12	in	in	ADP
ejpam-6496	296	13	lemma	lemma	PROPN
ejpam-6496	296	14	2	2	NUM
ejpam-6496	296	15	.	.	PUNCT
ejpam-6496	296	16	by	by	ADP
ejpam-6496	296	17	this	this	DET
ejpam-6496	296	18	property	property	NOUN
ejpam-6496	296	19	,	,	PUNCT
ejpam-6496	296	20	we	we	PRON
ejpam-6496	296	21	can	can	AUX
ejpam-6496	296	22	define	define	VERB
ejpam-6496	296	23	the	the	DET
ejpam-6496	296	24	gauge	gauge	NOUN
ejpam-6496	296	25	δ	δ	PROPN
ejpam-6496	296	26	to	to	PART
ejpam-6496	296	27	reach	reach	VERB
ejpam-6496	296	28	the	the	DET
ejpam-6496	296	29	conclusion	conclusion	NOUN
ejpam-6496	296	30	of	of	ADP
ejpam-6496	296	31	theorem	theorem	ADJ
ejpam-6496	296	32	3	3	NUM
ejpam-6496	296	33	more	more	ADV
ejpam-6496	296	34	elegant	elegant	ADJ
ejpam-6496	296	35	than	than	ADP
ejpam-6496	296	36	the	the	DET
ejpam-6496	296	37	author	author	NOUN
ejpam-6496	296	38	’s	’s	PART
ejpam-6496	296	39	previous	previous	ADJ
ejpam-6496	296	40	some	some	DET
ejpam-6496	296	41	work	work	NOUN
ejpam-6496	296	42	which	which	PRON
ejpam-6496	296	43	similar	similar	ADJ
ejpam-6496	296	44	to	to	ADP
ejpam-6496	296	45	this	this	PRON
ejpam-6496	296	46	(	(	PUNCT
ejpam-6496	296	47	see	see	VERB
ejpam-6496	296	48	[	[	X
ejpam-6496	296	49	6	6	NUM
ejpam-6496	296	50	]	]	NUM
ejpam-6496	296	51	)	)	PUNCT
ejpam-6496	296	52	.	.	PUNCT
ejpam-6496	297	1	the	the	DET
ejpam-6496	297	2	proofs	proof	NOUN
ejpam-6496	297	3	of	of	ADP
ejpam-6496	297	4	mean	mean	ADJ
ejpam-6496	297	5	value	value	NOUN
ejpam-6496	297	6	theorem	theorem	NOUN
ejpam-6496	297	7	and	and	CCONJ
ejpam-6496	297	8	cauchy	cauchy	PROPN
ejpam-6496	297	9	mean	mean	NOUN
ejpam-6496	297	10	value	value	NOUN
ejpam-6496	297	11	theorem	theorem	NOUN
ejpam-6496	297	12	have	have	AUX
ejpam-6496	297	13	been	be	AUX
ejpam-6496	297	14	adjusted	adjust	VERB
ejpam-6496	297	15	a	a	DET
ejpam-6496	297	16	little	little	ADJ
ejpam-6496	297	17	bit	bit	NOUN
ejpam-6496	297	18	through	through	ADP
ejpam-6496	297	19	theorem	theorem	NOUN
ejpam-6496	297	20	4	4	NUM
ejpam-6496	297	21	to	to	PART
ejpam-6496	297	22	give	give	VERB
ejpam-6496	297	23	another	another	DET
ejpam-6496	297	24	views	view	NOUN
ejpam-6496	297	25	of	of	ADP
ejpam-6496	297	26	these	these	DET
ejpam-6496	297	27	two	two	NUM
ejpam-6496	297	28	well	well	ADV
ejpam-6496	297	29	known	know	VERB
ejpam-6496	297	30	theorems	theorem	NOUN
ejpam-6496	297	31	.	.	PUNCT
ejpam-6496	298	1	in	in	ADP
ejpam-6496	298	2	the	the	DET
ejpam-6496	298	3	part	part	NOUN
ejpam-6496	298	4	of	of	ADP
ejpam-6496	298	5	riemann	riemann	PROPN
ejpam-6496	298	6	integrable	integrable	PROPN
ejpam-6496	298	7	function	function	PROPN
ejpam-6496	298	8	,	,	PUNCT
ejpam-6496	298	9	we	we	PRON
ejpam-6496	298	10	have	have	AUX
ejpam-6496	298	11	presented	present	VERB
ejpam-6496	298	12	new	new	ADJ
ejpam-6496	298	13	proofs	proof	NOUN
ejpam-6496	298	14	of	of	ADP
ejpam-6496	298	15	three	three	NUM
ejpam-6496	298	16	basic	basic	ADJ
ejpam-6496	298	17	theorems	theorem	NOUN
ejpam-6496	298	18	by	by	ADP
ejpam-6496	298	19	applying	apply	VERB
ejpam-6496	298	20	this	this	DET
ejpam-6496	298	21	concept	concept	NOUN
ejpam-6496	298	22	.	.	PUNCT
ejpam-6496	299	1	sincerely	sincerely	ADV
ejpam-6496	299	2	,	,	PUNCT
ejpam-6496	299	3	we	we	PRON
ejpam-6496	299	4	are	be	AUX
ejpam-6496	299	5	awe	awe	ADJ
ejpam-6496	299	6	in	in	ADP
ejpam-6496	299	7	gordon	gordon	PROPN
ejpam-6496	299	8	’	'	PUNCT
ejpam-6496	299	9	work	work	NOUN
ejpam-6496	299	10	for	for	ADP
ejpam-6496	299	11	his	his	PRON
ejpam-6496	299	12	version	version	NOUN
ejpam-6496	299	13	of	of	ADP
ejpam-6496	299	14	criterion	criterion	NOUN
ejpam-6496	299	15	for	for	ADP
ejpam-6496	299	16	riemann	riemann	PROPN
ejpam-6496	299	17	integrability	integrability	NOUN
ejpam-6496	299	18	that	that	PRON
ejpam-6496	299	19	is	be	AUX
ejpam-6496	299	20	suitable	suitable	ADJ
ejpam-6496	299	21	to	to	ADP
ejpam-6496	299	22	δ	δ	PROPN
ejpam-6496	299	23	-	-	PUNCT
ejpam-6496	299	24	fine	fine	ADJ
ejpam-6496	299	25	tagged	tag	VERB
ejpam-6496	299	26	partitions	partition	NOUN
ejpam-6496	299	27	technique	technique	NOUN
ejpam-6496	299	28	resulting	result	VERB
ejpam-6496	299	29	in	in	ADP
ejpam-6496	299	30	simplicity	simplicity	NOUN
ejpam-6496	299	31	and	and	CCONJ
ejpam-6496	299	32	unity	unity	NOUN
ejpam-6496	299	33	of	of	ADP
ejpam-6496	299	34	these	these	DET
ejpam-6496	299	35	three	three	NUM
ejpam-6496	299	36	proofs	proof	NOUN
ejpam-6496	299	37	,	,	PUNCT
ejpam-6496	299	38	at	at	ADP
ejpam-6496	299	39	least	least	ADJ
ejpam-6496	299	40	in	in	ADP
ejpam-6496	299	41	our	our	PRON
ejpam-6496	299	42	opinion	opinion	NOUN
ejpam-6496	299	43	.	.	PUNCT
ejpam-6496	300	1	there	there	PRON
ejpam-6496	300	2	are	be	VERB
ejpam-6496	300	3	many	many	ADJ
ejpam-6496	300	4	results	result	NOUN
ejpam-6496	300	5	in	in	ADP
ejpam-6496	300	6	real	real	ADJ
ejpam-6496	300	7	analysis	analysis	NOUN
ejpam-6496	300	8	that	that	PRON
ejpam-6496	300	9	have	have	AUX
ejpam-6496	300	10	not	not	PART
ejpam-6496	300	11	yet	yet	ADV
ejpam-6496	300	12	been	be	AUX
ejpam-6496	300	13	studied	study	VERB
ejpam-6496	300	14	via	via	ADP
ejpam-6496	300	15	δ	δ	PROPN
ejpam-6496	300	16	-	-	PUNCT
ejpam-6496	300	17	fine	fine	ADJ
ejpam-6496	300	18	tagged	tag	VERB
ejpam-6496	300	19	partitions	partition	NOUN
ejpam-6496	300	20	.	.	PUNCT
ejpam-6496	301	1	in	in	ADP
ejpam-6496	301	2	the	the	DET
ejpam-6496	301	3	next	next	ADJ
ejpam-6496	301	4	project	project	NOUN
ejpam-6496	301	5	,	,	PUNCT
ejpam-6496	301	6	we	we	PRON
ejpam-6496	301	7	would	would	AUX
ejpam-6496	301	8	extend	extend	VERB
ejpam-6496	301	9	our	our	PRON
ejpam-6496	301	10	research	research	NOUN
ejpam-6496	301	11	to	to	ADP
ejpam-6496	301	12	some	some	DET
ejpam-6496	301	13	other	other	ADJ
ejpam-6496	301	14	interesting	interesting	ADJ
ejpam-6496	301	15	theorems	theorem	NOUN
ejpam-6496	301	16	or	or	CCONJ
ejpam-6496	301	17	properties	property	NOUN
ejpam-6496	301	18	through	through	ADP
ejpam-6496	301	19	this	this	DET
ejpam-6496	301	20	idea	idea	NOUN
ejpam-6496	301	21	.	.	PUNCT
ejpam-6496	302	1	s.	s.	PROPN
ejpam-6496	302	2	prongjit	prongjit	ADV
ejpam-6496	302	3	,	,	PUNCT
ejpam-6496	302	4	p.	p.	PROPN
ejpam-6496	302	5	prasertsang	prasertsang	PROPN
ejpam-6496	302	6	/	/	SYM
ejpam-6496	302	7	eur	eur	PROPN
ejpam-6496	302	8	.	.	PUNCT
ejpam-6496	303	1	j.	j.	PROPN
ejpam-6496	303	2	pure	pure	PROPN
ejpam-6496	303	3	appl	appl	PROPN
ejpam-6496	303	4	.	.	PROPN
ejpam-6496	303	5	math	math	PROPN
ejpam-6496	303	6	,	,	PUNCT
ejpam-6496	303	7	18	18	NUM
ejpam-6496	303	8	(	(	PUNCT
ejpam-6496	303	9	3	3	NUM
ejpam-6496	303	10	)	)	PUNCT
ejpam-6496	303	11	(	(	PUNCT
ejpam-6496	303	12	2025	2025	NUM
ejpam-6496	303	13	)	)	PUNCT
ejpam-6496	303	14	,	,	PUNCT
ejpam-6496	303	15	6496	6496	NUM
ejpam-6496	303	16	10	10	NUM
ejpam-6496	303	17	of	of	ADP
ejpam-6496	303	18	10	10	NUM
ejpam-6496	303	19	acknowledgements	acknowledgement	NOUN
ejpam-6496	303	20	we	we	PRON
ejpam-6496	303	21	would	would	AUX
ejpam-6496	303	22	like	like	VERB
ejpam-6496	303	23	to	to	PART
ejpam-6496	303	24	express	express	VERB
ejpam-6496	303	25	our	our	PRON
ejpam-6496	303	26	sincere	sincere	ADJ
ejpam-6496	303	27	appreciation	appreciation	NOUN
ejpam-6496	303	28	to	to	ADP
ejpam-6496	303	29	research	research	NOUN
ejpam-6496	303	30	and	and	CCONJ
ejpam-6496	303	31	academic	academic	ADJ
ejpam-6496	303	32	service	service	NOUN
ejpam-6496	303	33	division	division	NOUN
ejpam-6496	303	34	and	and	CCONJ
ejpam-6496	303	35	faculty	faculty	NOUN
ejpam-6496	303	36	of	of	ADP
ejpam-6496	303	37	sciences	science	NOUN
ejpam-6496	303	38	and	and	CCONJ
ejpam-6496	303	39	engineering	engineering	NOUN
ejpam-6496	303	40	,	,	PUNCT
ejpam-6496	303	41	kasetsart	kasetsart	PROPN
ejpam-6496	303	42	university	university	PROPN
ejpam-6496	303	43	,	,	PUNCT
ejpam-6496	303	44	chalermprakiat	chalermprakiat	PROPN
ejpam-6496	303	45	sakon	sakon	PROPN
ejpam-6496	303	46	nakhon	nakhon	PROPN
ejpam-6496	303	47	province	province	PROPN
ejpam-6496	303	48	campus	campus	PROPN
ejpam-6496	303	49	,	,	PUNCT
ejpam-6496	303	50	sakon	sakon	PROPN
ejpam-6496	303	51	nakhon	nakhon	PROPN
ejpam-6496	303	52	,	,	PUNCT
ejpam-6496	303	53	thailand	thailand	PROPN
ejpam-6496	303	54	for	for	ADP
ejpam-6496	303	55	their	their	PRON
ejpam-6496	303	56	generous	generous	ADJ
ejpam-6496	303	57	financial	financial	ADJ
ejpam-6496	303	58	support	support	NOUN
ejpam-6496	303	59	for	for	ADP
ejpam-6496	303	60	this	this	DET
ejpam-6496	303	61	research	research	NOUN
ejpam-6496	303	62	.	.	PUNCT
ejpam-6496	304	1	additionally	additionally	ADV
ejpam-6496	304	2	,	,	PUNCT
ejpam-6496	304	3	the	the	DET
ejpam-6496	304	4	authors	author	NOUN
ejpam-6496	304	5	wish	wish	VERB
ejpam-6496	304	6	to	to	PART
ejpam-6496	304	7	thank	thank	VERB
ejpam-6496	304	8	the	the	DET
ejpam-6496	304	9	reviewers	reviewer	NOUN
ejpam-6496	304	10	of	of	ADP
ejpam-6496	304	11	european	european	ADJ
ejpam-6496	304	12	journal	journal	PROPN
ejpam-6496	304	13	of	of	ADP
ejpam-6496	304	14	pure	pure	ADJ
ejpam-6496	304	15	and	and	CCONJ
ejpam-6496	304	16	applied	applied	ADJ
ejpam-6496	304	17	mathematics	mathematic	NOUN
ejpam-6496	304	18	,	,	PUNCT
ejpam-6496	304	19	for	for	ADP
ejpam-6496	304	20	making	make	VERB
ejpam-6496	304	21	our	our	PRON
ejpam-6496	304	22	journal	journal	NOUN
ejpam-6496	304	23	successful	successful	ADJ
ejpam-6496	304	24	.	.	PUNCT
ejpam-6496	305	1	references	reference	NOUN
ejpam-6496	305	2	[	[	X
ejpam-6496	305	3	1	1	NUM
ejpam-6496	305	4	]	]	PUNCT
ejpam-6496	305	5	m.	m.	NOUN
ejpam-6496	305	6	w.	w.	PROPN
ejpam-6496	305	7	botsko	botsko	PROPN
ejpam-6496	305	8	.	.	PUNCT
ejpam-6496	306	1	a	a	DET
ejpam-6496	306	2	unified	unified	ADJ
ejpam-6496	306	3	treatment	treatment	NOUN
ejpam-6496	306	4	of	of	ADP
ejpam-6496	306	5	various	various	ADJ
ejpam-6496	306	6	theorems	theorem	NOUN
ejpam-6496	306	7	in	in	ADP
ejpam-6496	306	8	elementary	elementary	ADJ
ejpam-6496	306	9	analysis	analysis	NOUN
ejpam-6496	306	10	.	.	PUNCT
ejpam-6496	307	1	the	the	DET
ejpam-6496	307	2	american	american	PROPN
ejpam-6496	307	3	mathematical	mathematical	PROPN
ejpam-6496	307	4	monthly	monthly	ADV
ejpam-6496	307	5	,	,	PUNCT
ejpam-6496	307	6	94:450–452	94:450–452	PROPN
ejpam-6496	307	7	,	,	PUNCT
ejpam-6496	307	8	1987	1987	NUM
ejpam-6496	307	9	.	.	PUNCT
ejpam-6496	308	1	[	[	X
ejpam-6496	308	2	2	2	NUM
ejpam-6496	308	3	]	]	PUNCT
ejpam-6496	308	4	m.	m.	NOUN
ejpam-6496	308	5	w.	w.	PROPN
ejpam-6496	308	6	botsko	botsko	PROPN
ejpam-6496	308	7	.	.	PUNCT
ejpam-6496	309	1	the	the	DET
ejpam-6496	309	2	use	use	NOUN
ejpam-6496	309	3	of	of	ADP
ejpam-6496	309	4	full	full	ADJ
ejpam-6496	309	5	covers	cover	NOUN
ejpam-6496	309	6	in	in	ADP
ejpam-6496	309	7	real	real	ADJ
ejpam-6496	309	8	analysis	analysis	NOUN
ejpam-6496	309	9	.	.	PUNCT
ejpam-6496	310	1	the	the	DET
ejpam-6496	310	2	american	american	PROPN
ejpam-6496	310	3	mathematical	mathematical	PROPN
ejpam-6496	310	4	monthly	monthly	ADV
ejpam-6496	310	5	,	,	PUNCT
ejpam-6496	310	6	96:328–333	96:328–333	NUM
ejpam-6496	310	7	,	,	PUNCT
ejpam-6496	310	8	1989	1989	NUM
ejpam-6496	310	9	.	.	PUNCT
ejpam-6496	311	1	[	[	X
ejpam-6496	311	2	3	3	X
ejpam-6496	311	3	]	]	PUNCT
ejpam-6496	311	4	k.	k.	PROPN
ejpam-6496	311	5	klaimon	klaimon	PROPN
ejpam-6496	311	6	.	.	PUNCT
ejpam-6496	312	1	applications	application	NOUN
ejpam-6496	312	2	of	of	ADP
ejpam-6496	312	3	full	full	ADJ
ejpam-6496	312	4	covering	covering	NOUN
ejpam-6496	312	5	.	.	PUNCT
ejpam-6496	313	1	the	the	DET
ejpam-6496	313	2	pi	pi	PROPN
ejpam-6496	313	3	mu	mu	PROPN
ejpam-6496	313	4	epsilon	epsilon	PROPN
ejpam-6496	313	5	journal	journal	PROPN
ejpam-6496	313	6	,	,	PUNCT
ejpam-6496	313	7	9:156–161	9:156–161	NUM
ejpam-6496	313	8	,	,	PUNCT
ejpam-6496	313	9	1990	1990	NUM
ejpam-6496	313	10	.	.	PUNCT
ejpam-6496	314	1	[	[	X
ejpam-6496	314	2	4	4	X
ejpam-6496	314	3	]	]	PUNCT
ejpam-6496	314	4	k.	k.	PROPN
ejpam-6496	314	5	zangara	zangara	PROPN
ejpam-6496	314	6	and	and	CCONJ
ejpam-6496	314	7	j.	j.	PROPN
ejpam-6496	314	8	marafino	marafino	PROPN
ejpam-6496	314	9	.	.	PUNCT
ejpam-6496	315	1	applications	application	NOUN
ejpam-6496	315	2	of	of	ADP
ejpam-6496	315	3	full	full	ADJ
ejpam-6496	315	4	covers	cover	NOUN
ejpam-6496	315	5	in	in	ADP
ejpam-6496	315	6	real	real	ADJ
ejpam-6496	315	7	analysis	analysis	NOUN
ejpam-6496	315	8	.	.	PUNCT
ejpam-6496	316	1	involve	involve	NOUN
ejpam-6496	316	2	,	,	PUNCT
ejpam-6496	316	3	2:297–304	2:297–304	NUM
ejpam-6496	316	4	,	,	PUNCT
ejpam-6496	316	5	2009	2009	NUM
ejpam-6496	316	6	.	.	PUNCT
ejpam-6496	317	1	[	[	X
ejpam-6496	317	2	5	5	X
ejpam-6496	317	3	]	]	PUNCT
ejpam-6496	317	4	r.	r.	PROPN
ejpam-6496	317	5	a.	a.	PROPN
ejpam-6496	317	6	gordon	gordon	PROPN
ejpam-6496	317	7	.	.	PUNCT
ejpam-6496	318	1	the	the	DET
ejpam-6496	318	2	use	use	NOUN
ejpam-6496	318	3	of	of	ADP
ejpam-6496	318	4	tagged	tag	VERB
ejpam-6496	318	5	partitions	partition	NOUN
ejpam-6496	318	6	in	in	ADP
ejpam-6496	318	7	elementary	elementary	ADJ
ejpam-6496	318	8	real	real	ADJ
ejpam-6496	318	9	analysis	analysis	NOUN
ejpam-6496	318	10	.	.	PUNCT
ejpam-6496	319	1	the	the	DET
ejpam-6496	319	2	american	american	PROPN
ejpam-6496	319	3	mathematical	mathematical	PROPN
ejpam-6496	319	4	monthly	monthly	ADV
ejpam-6496	319	5	,	,	PUNCT
ejpam-6496	319	6	105(1):107–117	105(1):107–117	NUM
ejpam-6496	319	7	,	,	PUNCT
ejpam-6496	319	8	1998	1998	NUM
ejpam-6496	319	9	.	.	PUNCT
ejpam-6496	320	1	[	[	X
ejpam-6496	320	2	6	6	X
ejpam-6496	320	3	]	]	PUNCT
ejpam-6496	320	4	s.	s.	PROPN
ejpam-6496	320	5	prongjit	prongjit	PROPN
ejpam-6496	320	6	and	and	CCONJ
ejpam-6496	320	7	w.	w.	PROPN
ejpam-6496	320	8	sodsiri	sodsiri	PROPN
ejpam-6496	320	9	.	.	PUNCT
ejpam-6496	321	1	applications	application	NOUN
ejpam-6496	321	2	of	of	ADP
ejpam-6496	321	3	δ	δ	NOUN
ejpam-6496	321	4	-	-	PUNCT
ejpam-6496	321	5	fine	fine	ADJ
ejpam-6496	321	6	tagged	tag	VERB
ejpam-6496	321	7	partitions	partition	NOUN
ejpam-6496	321	8	in	in	ADP
ejpam-6496	321	9	real	real	ADJ
ejpam-6496	321	10	analysis	analysis	NOUN
ejpam-6496	321	11	.	.	PUNCT
ejpam-6496	322	1	far	far	PROPN
ejpam-6496	322	2	east	east	PROPN
ejpam-6496	322	3	journal	journal	PROPN
ejpam-6496	322	4	of	of	ADP
ejpam-6496	322	5	mathematical	mathematical	ADJ
ejpam-6496	322	6	sciences	sciences	PROPN
ejpam-6496	322	7	(	(	PUNCT
ejpam-6496	322	8	fjms	fjms	NOUN
ejpam-6496	322	9	)	)	PUNCT
ejpam-6496	322	10	,	,	PUNCT
ejpam-6496	322	11	91(1):97–109	91(1):97–109	NUM
ejpam-6496	322	12	,	,	PUNCT
ejpam-6496	322	13	2014	2014	NUM
ejpam-6496	322	14	.	.	PUNCT
ejpam-6496	323	1	[	[	X
ejpam-6496	323	2	7	7	X
ejpam-6496	323	3	]	]	X
ejpam-6496	323	4	s.	s.	PROPN
ejpam-6496	323	5	prongjit	prongjit	PROPN
ejpam-6496	323	6	and	and	CCONJ
ejpam-6496	323	7	w.	w.	PROPN
ejpam-6496	323	8	sodsiri	sodsiri	PROPN
ejpam-6496	323	9	.	.	PUNCT
ejpam-6496	324	1	more	more	ADJ
ejpam-6496	324	2	applications	application	NOUN
ejpam-6496	324	3	of	of	ADP
ejpam-6496	324	4	δ	δ	NOUN
ejpam-6496	324	5	-	-	PUNCT
ejpam-6496	324	6	fine	fine	ADJ
ejpam-6496	324	7	tagged	tag	VERB
ejpam-6496	324	8	partitions	partition	NOUN
ejpam-6496	324	9	in	in	ADP
ejpam-6496	324	10	real	real	ADJ
ejpam-6496	324	11	analysis	analysis	NOUN
ejpam-6496	324	12	.	.	PUNCT
ejpam-6496	325	1	far	far	PROPN
ejpam-6496	325	2	east	east	PROPN
ejpam-6496	325	3	journal	journal	PROPN
ejpam-6496	325	4	of	of	ADP
ejpam-6496	325	5	mathematical	mathematical	ADJ
ejpam-6496	325	6	sciences	sciences	PROPN
ejpam-6496	325	7	(	(	PUNCT
ejpam-6496	325	8	fjms	fjms	NOUN
ejpam-6496	325	9	)	)	PUNCT
ejpam-6496	325	10	,	,	PUNCT
ejpam-6496	325	11	91(2):233–245	91(2):233–245	PROPN
ejpam-6496	325	12	,	,	PUNCT
ejpam-6496	325	13	2014	2014	NUM
ejpam-6496	325	14	.	.	PUNCT
ejpam-6496	326	1	[	[	X
ejpam-6496	326	2	8	8	NUM
ejpam-6496	326	3	]	]	PUNCT
ejpam-6496	326	4	p.	p.	NOUN
ejpam-6496	326	5	zheng	zheng	PROPN
ejpam-6496	326	6	and	and	CCONJ
ejpam-6496	326	7	x.	x.	PROPN
ejpam-6496	326	8	shi	shi	PROPN
ejpam-6496	326	9	.	.	PUNCT
ejpam-6496	327	1	the	the	DET
ejpam-6496	327	2	use	use	NOUN
ejpam-6496	327	3	of	of	ADP
ejpam-6496	327	4	the	the	DET
ejpam-6496	327	5	dyadic	dyadic	ADJ
ejpam-6496	327	6	partition	partition	NOUN
ejpam-6496	327	7	in	in	ADP
ejpam-6496	327	8	real	real	ADJ
ejpam-6496	327	9	analysis	analysis	NOUN
ejpam-6496	327	10	.	.	PUNCT
ejpam-6496	328	1	journal	journal	NOUN
ejpam-6496	328	2	of	of	ADP
ejpam-6496	328	3	computational	computational	ADJ
ejpam-6496	328	4	and	and	CCONJ
ejpam-6496	328	5	applied	applied	ADJ
ejpam-6496	328	6	mathematics	mathematic	NOUN
ejpam-6496	328	7	,	,	PUNCT
ejpam-6496	328	8	329:344–352	329:344–352	NUM
ejpam-6496	328	9	,	,	PUNCT
ejpam-6496	328	10	2018	2018	NUM
ejpam-6496	328	11	.	.	PUNCT
ejpam-6496	329	1	[	[	X
ejpam-6496	329	2	9	9	NUM
ejpam-6496	329	3	]	]	PUNCT
ejpam-6496	329	4	r.	r.	PROPN
ejpam-6496	329	5	g.	g.	PROPN
ejpam-6496	329	6	bartle	bartle	PROPN
ejpam-6496	329	7	and	and	CCONJ
ejpam-6496	329	8	d.	d.	PROPN
ejpam-6496	329	9	r.	r.	PROPN
ejpam-6496	329	10	sherbert	sherbert	PROPN
ejpam-6496	329	11	.	.	PUNCT
ejpam-6496	330	1	introduction	introduction	NOUN
ejpam-6496	330	2	to	to	ADP
ejpam-6496	330	3	real	real	ADJ
ejpam-6496	330	4	analysis	analysis	NOUN
ejpam-6496	330	5	.	.	PUNCT
ejpam-6496	331	1	john	john	PROPN
ejpam-6496	331	2	wiley	wiley	PROPN
ejpam-6496	331	3	&	&	CCONJ
ejpam-6496	331	4	sons	son	NOUN
ejpam-6496	331	5	,	,	PUNCT
ejpam-6496	331	6	new	new	PROPN
ejpam-6496	331	7	york	york	PROPN
ejpam-6496	331	8	,	,	PUNCT
ejpam-6496	331	9	2011	2011	NUM
ejpam-6496	331	10	.	.	PUNCT