id	sid	eid	entity	type
ejpam-3506	1	1	european journal	ORG
ejpam-3506	2	1	12	CARDINAL
ejpam-3506	2	2	3, 2019	DATE
ejpam-3506	2	3	1337-1349	DATE
ejpam-3506	2	4	new york	GPE
ejpam-3506	2	5	m. bent-usman1,∗	PERSON
ejpam-3506	2	6	rowena t. isla2	PERSON
ejpam-3506	2	7	sergio r. canoy	PERSON
ejpam-3506	2	8	jr.2 1	ORG
ejpam-3506	2	9	mindanao state university	ORG
ejpam-3506	2	10	9700	CARDINAL
ejpam-3506	2	11	marawi city	GPE
ejpam-3506	2	12	philippines 2 department	ORG
ejpam-3506	2	13	institute of science	ORG
ejpam-3506	2	14	mindanao state university-iligan institute of technology	ORG
ejpam-3506	2	15	9200	CARDINAL
ejpam-3506	2	16	iligan city	GPE
ejpam-3506	2	17	philippines	GPE
ejpam-3506	5	1	⊆	CARDINAL
ejpam-3506	5	2	|n(u)∩	NORP
ejpam-3506	5	3	〉	CARDINAL
ejpam-3506	8	1	cartesian	NORP
ejpam-3506	9	1	2010	DATE
ejpam-3506	9	2	05c69	CARDINAL
ejpam-3506	9	3	05c76	CARDINAL
ejpam-3506	9	4	1	CARDINAL
ejpam-3506	11	1	⊆	CARDINAL
ejpam-3506	11	2	∈	PERSON
ejpam-3506	11	3	∈	ORG
ejpam-3506	12	1	γ(g	NORP
ejpam-3506	12	2	γ(g	NORP
ejpam-3506	12	3	1	CARDINAL
ejpam-3506	13	1	ncd	ORG
ejpam-3506	13	2	〉	CARDINAL
ejpam-3506	13	3	n(s	GPE
ejpam-3506	14	1	ncd	ORG
ejpam-3506	16	1	w. bent-usman	PERSON
ejpam-3506	16	2	r. isla	PERSON
ejpam-3506	16	3	sergio.canoy@g.msuiit.edu.ph	ORG
ejpam-3506	16	4	s. canoy	PERSON
ejpam-3506	17	1	2019	DATE
ejpam-3506	18	1	w. bent-usman	PERSON
ejpam-3506	18	2	r. isla	PERSON
ejpam-3506	18	3	s. canoy / eur	PERSON
ejpam-3506	19	1	j. pure	PERSON
ejpam-3506	19	2	12	CARDINAL
ejpam-3506	19	3	3	CARDINAL
ejpam-3506	19	4	2019	CARDINAL
ejpam-3506	19	5	1337-1349 1338	DATE
ejpam-3506	20	1	ncd	ORG
ejpam-3506	21	1	hansberg	PERSON
ejpam-3506	21	2	henning	PERSON
ejpam-3506	22	1	3	CARDINAL
ejpam-3506	22	2	2011	DATE
ejpam-3506	23	1	1	CARDINAL
ejpam-3506	23	2	⊆	CARDINAL
ejpam-3506	25	1	maravilla	ORG
ejpam-3506	25	2	isla	PERSON
ejpam-3506	26	1	gomisong	PERSON
ejpam-3506	26	2	2	CARDINAL
ejpam-3506	28	1	⊆	CARDINAL
ejpam-3506	28	2	〉	CARDINAL
ejpam-3506	31	1	2	CARDINAL
ejpam-3506	31	2	1	CARDINAL
ejpam-3506	32	1	ncd	ORG
ejpam-3506	33	1	2	CARDINAL
ejpam-3506	34	1	2	CARDINAL
ejpam-3506	34	2	integer k	PERSON
ejpam-3506	34	3	1 ≤ γ(g	MONEY
ejpam-3506	39	1	1	CARDINAL
ejpam-3506	39	2	m. moreover	PERSON
ejpam-3506	40	1	4	CARDINAL
ejpam-3506	41	1	2	CARDINAL
ejpam-3506	42	1	1	CARDINAL
ejpam-3506	43	1	4	CARDINAL
ejpam-3506	44	1	n ≥ 2	ORG
ejpam-3506	44	2	k ≤ n.	ORG
ejpam-3506	44	3	≥ k. w. bent-usman	PERSON
ejpam-3506	44	4	r. isla	PERSON
ejpam-3506	44	5	s. canoy / eur	PERSON
ejpam-3506	45	1	j. pure	PERSON
ejpam-3506	45	2	12	CARDINAL
ejpam-3506	45	3	3	CARDINAL
ejpam-3506	45	4	2019	CARDINAL
ejpam-3506	45	5	1337-1349 1339	DATE
ejpam-3506	46	1	= k.	PERSON
ejpam-3506	47	1	k. proof	PERSON
ejpam-3506	49	1	≥ k.	PERSON
ejpam-3506	49	2	6=	CARDINAL
ejpam-3506	52	1	= k.	PERSON
ejpam-3506	52	2	= k.	PERSON
ejpam-3506	52	3	= k.	PERSON
ejpam-3506	53	1	k.	PERSON
ejpam-3506	56	1	kn	PERSON
ejpam-3506	56	2	〉	CARDINAL
ejpam-3506	56	3	kn−1	ORG
ejpam-3506	56	4	〉	CARDINAL
ejpam-3506	56	5	1	CARDINAL
ejpam-3506	56	6	kn	PERSON
ejpam-3506	58	1	3	CARDINAL
ejpam-3506	59	1	1	CARDINAL
ejpam-3506	59	2	1	CARDINAL
ejpam-3506	60	1	1	CARDINAL
ejpam-3506	60	2	1	CARDINAL
ejpam-3506	61	1	1	CARDINAL
ejpam-3506	61	2	1	CARDINAL
ejpam-3506	61	3	e. theorem 2	PERSON
ejpam-3506	62	1	1	CARDINAL
ejpam-3506	63	1	n ≡	ORG
ejpam-3506	63	2	4	CARDINAL
ejpam-3506	64	1	3	CARDINAL
ejpam-3506	65	1	1	CARDINAL
ejpam-3506	65	2	e. proof	PERSON
ejpam-3506	68	1	1	CARDINAL
ejpam-3506	68	2	2	DATE
ejpam-3506	68	3	3	CARDINAL
ejpam-3506	70	1	4l	CARDINAL
ejpam-3506	70	2	vi	PERSON
ejpam-3506	71	1	2a	CARDINAL
ejpam-3506	71	2	2a+ 1	DATE
ejpam-3506	74	1	〉	CARDINAL
ejpam-3506	75	1	4l + 2	DATE
ejpam-3506	76	1	〉	CARDINAL
ejpam-3506	77	1	4l + 3	DATE
ejpam-3506	77	2	〉	CARDINAL
ejpam-3506	77	3	dn2	NORP
ejpam-3506	77	4	ncd	ORG
ejpam-3506	77	5	2	CARDINAL
ejpam-3506	77	6	1	CARDINAL
ejpam-3506	77	7	e. � theorem	PERSON
ejpam-3506	77	8	4	CARDINAL
ejpam-3506	79	1	dn2	NORP
ejpam-3506	79	2	n ≡	ORG
ejpam-3506	79	3	1(mod 4	CARDINAL
ejpam-3506	79	4	1	CARDINAL
ejpam-3506	80	1	n ≡	ORG
ejpam-3506	81	1	w. bent-usman	PERSON
ejpam-3506	81	2	r. isla	PERSON
ejpam-3506	81	3	s. canoy / eur	PERSON
ejpam-3506	82	1	j. pure	PERSON
ejpam-3506	82	2	12	CARDINAL
ejpam-3506	82	3	3	CARDINAL
ejpam-3506	82	4	2019	CARDINAL
ejpam-3506	82	5	1337-1349 1340	DATE
ejpam-3506	85	1	1	CARDINAL
ejpam-3506	87	1	4l+	CARDINAL
ejpam-3506	87	2	0 ≤	MONEY
ejpam-3506	88	1	vi	PERSON
ejpam-3506	88	2	2j	CARDINAL
ejpam-3506	88	3	2j+ 1	DATE
ejpam-3506	88	4	1 ≤ j ≤ 2l− 1	QUANTITY
ejpam-3506	89	1	n ≡	ORG
ejpam-3506	89	2	1(mod 4	CARDINAL
ejpam-3506	89	3	n ≡	ORG
ejpam-3506	90	1	1fd	ORDINAL
ejpam-3506	91	1	〉	CARDINAL
ejpam-3506	91	2	n ≡	ORG
ejpam-3506	91	3	4	CARDINAL
ejpam-3506	92	1	n ≡	ORG
ejpam-3506	92	2	1(mod 4	CARDINAL
ejpam-3506	92	3	1	CARDINAL
ejpam-3506	93	1	n ≡	ORG
ejpam-3506	94	1	2	CARDINAL
ejpam-3506	94	2	2	CARDINAL
ejpam-3506	95	1	n ≡	ORG
ejpam-3506	96	1	n ≡ 0	ORG
ejpam-3506	96	2	≥ dn2 e.	PERSON
ejpam-3506	96	3	n ≡	ORG
ejpam-3506	96	4	4	CARDINAL
ejpam-3506	96	5	4	CARDINAL
ejpam-3506	96	6	two	CARDINAL
ejpam-3506	96	7	4	CARDINAL
ejpam-3506	96	8	one	CARDINAL
ejpam-3506	96	9	1(mod 4	CARDINAL
ejpam-3506	98	1	4	CARDINAL
ejpam-3506	100	1	5	CARDINAL
ejpam-3506	101	1	1	CARDINAL
ejpam-3506	105	1	1	CARDINAL
ejpam-3506	105	2	2	CARDINAL
ejpam-3506	105	3	1	CARDINAL
ejpam-3506	105	4	3	CARDINAL
ejpam-3506	106	1	〉	CARDINAL
ejpam-3506	106	2	〉	CARDINAL
ejpam-3506	107	1	〉	CARDINAL
ejpam-3506	107	2	1	CARDINAL
ejpam-3506	107	3	1	CARDINAL
ejpam-3506	109	1	1	CARDINAL
ejpam-3506	110	1	1	CARDINAL
ejpam-3506	110	2	3	CARDINAL
ejpam-3506	110	3	6	CARDINAL
ejpam-3506	111	1	2 ≤	QUANTITY
ejpam-3506	113	1	b. w. bent-usman	PERSON
ejpam-3506	113	2	r. isla	PERSON
ejpam-3506	113	3	s. canoy / eur	PERSON
ejpam-3506	114	1	j. pure	PERSON
ejpam-3506	114	2	12	CARDINAL
ejpam-3506	114	3	3	CARDINAL
ejpam-3506	114	4	2019	CARDINAL
ejpam-3506	114	5	1337-1349 1341	DATE
ejpam-3506	135	1	xa−1 xa	PERSON
ejpam-3506	135	2	•	CARDINAL
ejpam-3506	135	3	••	ORG
ejpam-3506	135	4	z3	DATE
ejpam-3506	135	5	z4 z5 za−1	FAC
ejpam-3506	135	6	1	CARDINAL
ejpam-3506	137	1	1	CARDINAL
ejpam-3506	138	1	1	CARDINAL
ejpam-3506	139	1	1	CARDINAL
ejpam-3506	139	2	2	DATE
ejpam-3506	141	1	2	CARDINAL
ejpam-3506	142	1	1	CARDINAL
ejpam-3506	143	1	1	CARDINAL
ejpam-3506	143	2	2	CARDINAL
ejpam-3506	144	1	xa−1	PERSON
ejpam-3506	144	2	w	GPE
ejpam-3506	145	1	1	CARDINAL
ejpam-3506	148	1	•	CARDINAL
ejpam-3506	148	2	z3	DATE
ejpam-3506	148	3	z4 z5 za−1	GPE
ejpam-3506	148	4	2	CARDINAL
ejpam-3506	149	1	2	CARDINAL
ejpam-3506	149	2	2	CARDINAL
ejpam-3506	149	3	2	CARDINAL
ejpam-3506	150	1	3	CARDINAL
ejpam-3506	166	1	........	PERSON
ejpam-3506	170	1	........	PERSON
ejpam-3506	191	1	...................................................................................................................................................	PERSON
ejpam-3506	206	1	xa−1 w	PERSON
ejpam-3506	206	2	•	CARDINAL
ejpam-3506	206	3	• •	DATE
ejpam-3506	206	4	y1 y2 y3 y4 y5 ya−1 z1 z2	PERSON
ejpam-3506	206	5	z3	DATE
ejpam-3506	206	6	z4 z5 za−1	GPE
ejpam-3506	206	7	3	CARDINAL
ejpam-3506	207	1	2	CARDINAL
ejpam-3506	207	2	w. bent-usman	PERSON
ejpam-3506	207	3	r. isla	PERSON
ejpam-3506	207	4	s. canoy / eur	PERSON
ejpam-3506	208	1	j. pure	PERSON
ejpam-3506	208	2	12	CARDINAL
ejpam-3506	208	3	3	CARDINAL
ejpam-3506	208	4	2019	CARDINAL
ejpam-3506	208	5	1337-1349 1342	DATE
ejpam-3506	208	6	xa−1	PERSON
ejpam-3506	209	1	⊆	CARDINAL
ejpam-3506	209	2	xa /∈ b.	PERSON
ejpam-3506	210	1	6=	CARDINAL
ejpam-3506	210	2	j = 1	WORK_OF_ART
ejpam-3506	210	3	2	CARDINAL
ejpam-3506	210	4	1fd	ORDINAL
ejpam-3506	211	1	xa ∈	PERSON
ejpam-3506	214	1	1	CARDINAL
ejpam-3506	214	2	2	DATE
ejpam-3506	215	1	1fd	ORDINAL
ejpam-3506	215	2	1	CARDINAL
ejpam-3506	215	3	2	DATE
ejpam-3506	216	1	xa	GPE
ejpam-3506	216	2	w	GPE
ejpam-3506	216	3	2	CARDINAL
ejpam-3506	221	1	2	CARDINAL
ejpam-3506	223	1	7	CARDINAL
ejpam-3506	224	1	4 ≤	QUANTITY
ejpam-3506	226	1	1	CARDINAL
ejpam-3506	227	1	4	CARDINAL
ejpam-3506	228	1	1	CARDINAL
ejpam-3506	228	2	2	DATE
ejpam-3506	257	1	xa−1 xa	PERSON
ejpam-3506	257	2	•	CARDINAL
ejpam-3506	257	3	••	ORG
ejpam-3506	257	4	z3	DATE
ejpam-3506	257	5	z4 z5 za−1	GPE
ejpam-3506	257	6	4	CARDINAL
ejpam-3506	258	1	2	CARDINAL
ejpam-3506	260	1	5	CARDINAL
ejpam-3506	261	1	xa−2	GPE
ejpam-3506	261	2	⊆	CARDINAL
ejpam-3506	261	3	a.	PERSON
ejpam-3506	261	4	xa−1 /∈ a.	PERSON
ejpam-3506	261	5	ya−2	GPE
ejpam-3506	261	6	za−2 ∈ a.	ORG
ejpam-3506	261	7	ya−1	PERSON
ejpam-3506	262	1	xa−1 ∈ a.	PERSON
ejpam-3506	262	2	xa /∈ a.	ORG
ejpam-3506	262	3	1	CARDINAL
ejpam-3506	262	4	h4	ORG
ejpam-3506	262	5	1	CARDINAL
ejpam-3506	263	1	ya−1 /∈	PERSON
ejpam-3506	264	1	xa−1	PERSON
ejpam-3506	265	1	〉	CARDINAL
ejpam-3506	265	2	∈	ORG
ejpam-3506	266	1	∈	ORG
ejpam-3506	266	2	⊆	CARDINAL
ejpam-3506	266	3	za−1	PERSON
ejpam-3506	266	4	∈	ORG
ejpam-3506	266	5	za−1	ORG
ejpam-3506	267	1	⊆	CARDINAL
ejpam-3506	267	2	{ya−1	PERSON
ejpam-3506	267	3	za−1	ORG
ejpam-3506	269	1	+ 1	DATE
ejpam-3506	270	1	∈	ORG
ejpam-3506	270	2	⊆	CARDINAL
ejpam-3506	270	3	∈	ORG
ejpam-3506	270	4	⊆	CARDINAL
ejpam-3506	273	1	w. bent-usman	PERSON
ejpam-3506	273	2	r. isla	PERSON
ejpam-3506	273	3	s. canoy / eur	PERSON
ejpam-3506	274	1	j. pure	PERSON
ejpam-3506	274	2	12	CARDINAL
ejpam-3506	274	3	3	CARDINAL
ejpam-3506	274	4	2019	CARDINAL
ejpam-3506	274	5	1337-1349 1343	DATE
ejpam-3506	373	1	xa−2	GPE
ejpam-3506	373	2	xa−1	GPE
ejpam-3506	375	1	ya−3	GPE
ejpam-3506	375	2	ya−1	GPE
ejpam-3506	375	3	5	CARDINAL
ejpam-3506	376	1	3	CARDINAL
ejpam-3506	378	1	8	CARDINAL
ejpam-3506	379	1	6	CARDINAL
ejpam-3506	379	2	1 ≤ k ≤ max{m	QUANTITY
ejpam-3506	380	1	⊆	CARDINAL
ejpam-3506	380	2	g+h	ORG
ejpam-3506	381	1	⊆	CARDINAL
ejpam-3506	381	2	= k	PERSON
ejpam-3506	381	3	⊆	CARDINAL
ejpam-3506	381	4	= k	PERSON
ejpam-3506	384	1	9	CARDINAL
ejpam-3506	385	1	6	CARDINAL
ejpam-3506	385	2	k ≤	ORG
ejpam-3506	386	1	⊆	CARDINAL
ejpam-3506	388	1	1	CARDINAL
ejpam-3506	388	2	⋃ v∈v	PERSON
ejpam-3506	388	3	sv	ORG
ejpam-3506	388	4	2	CARDINAL
ejpam-3506	390	1	⋃ v∈v	PERSON
ejpam-3506	390	2	sv	ORG
ejpam-3506	390	3	|sv| = k. theorem 10	PERSON
ejpam-3506	391	1	6	CARDINAL
ejpam-3506	392	1	⋃	PERSON
ejpam-3506	392	2	⊆	CARDINAL
ejpam-3506	393	1	w. bent-usman	PERSON
ejpam-3506	393	2	r. isla	PERSON
ejpam-3506	393	3	s. canoy / eur	PERSON
ejpam-3506	394	1	j. pure	PERSON
ejpam-3506	394	2	12	CARDINAL
ejpam-3506	394	3	3	CARDINAL
ejpam-3506	394	4	2019	CARDINAL
ejpam-3506	394	5	1337-1349 1344	DATE
ejpam-3506	394	6	tx	ORG
ejpam-3506	395	1	6=	CARDINAL
ejpam-3506	395	2	tx	ORG
ejpam-3506	395	3	∑ z∈ng(x)∩s |tz|	PERSON
ejpam-3506	396	1	k − r.	PERSON
ejpam-3506	396	2	∈	ORG
ejpam-3506	396	3	tx	ORG
ejpam-3506	396	4	tx	ORG
ejpam-3506	396	5	∑ v∈ng(y)∩s	PERSON
ejpam-3506	396	6	4	CARDINAL
ejpam-3506	397	1	6	CARDINAL
ejpam-3506	397	2	1 ≤ k ≤	QUANTITY
ejpam-3506	398	1	g�h. 3	ORG
ejpam-3506	398	2	two	CARDINAL
ejpam-3506	399	1	∈	ORG
ejpam-3506	400	1	11	CARDINAL
ejpam-3506	401	1	1 ≤ k ≤ max{m	QUANTITY
ejpam-3506	402	1	⊆	CARDINAL
ejpam-3506	402	2	g+h	ORG
ejpam-3506	403	1	g+h	ORG
ejpam-3506	405	1	⊆	CARDINAL
ejpam-3506	405	2	g+h	ORG
ejpam-3506	406	1	g+h	ORG
ejpam-3506	407	1	⊆	CARDINAL
ejpam-3506	407	2	g+h	ORG
ejpam-3506	408	1	at least one	CARDINAL
ejpam-3506	408	2	8	CARDINAL
ejpam-3506	409	1	〉	CARDINAL
ejpam-3506	412	1	⊆	CARDINAL
ejpam-3506	412	2	= k	PERSON
ejpam-3506	412	3	〉	CARDINAL
ejpam-3506	414	1	〉	CARDINAL
ejpam-3506	414	2	〉	CARDINAL
ejpam-3506	415	1	k−|sh	PERSON
ejpam-3506	417	1	〉	CARDINAL
ejpam-3506	417	2	+ h	GPE
ejpam-3506	418	1	k−m)fd	GPE
ejpam-3506	422	1	〉	CARDINAL
ejpam-3506	423	1	〉	CARDINAL
ejpam-3506	424	1	g+h	ORG
ejpam-3506	425	1	11	CARDINAL
ejpam-3506	426	1	5	CARDINAL
ejpam-3506	427	1	1 ≤ k ≤ max{m	QUANTITY
ejpam-3506	429	1	= k	PERSON
ejpam-3506	430	1	k. w. bent-usman	PERSON
ejpam-3506	430	2	r. isla	PERSON
ejpam-3506	430	3	s. canoy / eur	PERSON
ejpam-3506	431	1	j. pure	PERSON
ejpam-3506	431	2	12	CARDINAL
ejpam-3506	431	3	3	CARDINAL
ejpam-3506	431	4	2019	CARDINAL
ejpam-3506	431	5	1337-1349 1345 4	DATE
ejpam-3506	432	1	two	CARDINAL
ejpam-3506	432	2	g◦h	ORG
ejpam-3506	432	3	one	CARDINAL
ejpam-3506	432	4	∈	ORG
ejpam-3506	432	5	g ◦h	ORG
ejpam-3506	432	6	12	CARDINAL
ejpam-3506	433	1	2 ≤ k ≤	QUANTITY
ejpam-3506	434	1	⊆	CARDINAL
ejpam-3506	436	1	9	CARDINAL
ejpam-3506	437	1	⋃ v∈v	PERSON
ejpam-3506	437	2	sv	ORG
ejpam-3506	437	3	k ≥ 2	ORG
ejpam-3506	437	4	⋃ v∈v	PERSON
ejpam-3506	437	5	sv	ORG
ejpam-3506	437	6	|sv| =	PERSON
ejpam-3506	438	1	n(c	PERSON
ejpam-3506	438	2	〉	CARDINAL
ejpam-3506	440	1	1fd	ORDINAL
ejpam-3506	440	2	n(c	PERSON
ejpam-3506	440	3	〉	CARDINAL
ejpam-3506	442	1	⋃ v∈v	PERSON
ejpam-3506	442	2	sv	ORG
ejpam-3506	443	1	n(c	PERSON
ejpam-3506	443	2	〉	CARDINAL
ejpam-3506	443	3	〈 ⋃ v∈v	PERSON
ejpam-3506	445	1	n(c	PERSON
ejpam-3506	445	2	〉	CARDINAL
ejpam-3506	445	3	〈 ⋃ v∈v	PERSON
ejpam-3506	446	1	6	CARDINAL
ejpam-3506	447	1	1 ≤ k ≤ n.	QUANTITY
ejpam-3506	448	1	m. for k ≥ 2	WORK_OF_ART
ejpam-3506	448	2	m(1 + γ(k−1)fd(h	DATE
ejpam-3506	450	1	12	CARDINAL
ejpam-3506	451	1	5	CARDINAL
ejpam-3506	451	2	two	CARDINAL
ejpam-3506	451	3	g[h	ORG
ejpam-3506	451	4	v1)(u2	GPE
ejpam-3506	452	1	w. bent-usman	PERSON
ejpam-3506	452	2	r. isla	PERSON
ejpam-3506	452	3	s. canoy / eur	PERSON
ejpam-3506	453	1	j. pure	PERSON
ejpam-3506	453	2	12	CARDINAL
ejpam-3506	453	3	3	CARDINAL
ejpam-3506	453	4	2019	CARDINAL
ejpam-3506	453	5	1337-1349 1346	DATE
ejpam-3506	453	6	13	CARDINAL
ejpam-3506	454	1	1 ≤ k ≤ max{m	QUANTITY
ejpam-3506	455	1	⋃	PERSON
ejpam-3506	456	1	⊆	CARDINAL
ejpam-3506	456	2	tx	ORG
ejpam-3506	456	3	6=	CARDINAL
ejpam-3506	456	4	tx	ORG
ejpam-3506	456	5	∑ z∈ng(x)∩s |tz| = k−r	PERSON
ejpam-3506	456	6	∈	ORG
ejpam-3506	456	7	tx	ORG
ejpam-3506	456	8	6=	CARDINAL
ejpam-3506	457	1	tx	ORG
ejpam-3506	457	2	∑ v∈ng(y)∩s	PERSON
ejpam-3506	457	3	k. proof	PERSON
ejpam-3506	458	1	⋃	PERSON
ejpam-3506	458	2	g[h	ORG
ejpam-3506	459	1	10	CARDINAL
ejpam-3506	461	1	〉	CARDINAL
ejpam-3506	462	1	6=	CARDINAL
ejpam-3506	464	1	〉	CARDINAL
ejpam-3506	465	1	1	CARDINAL
ejpam-3506	467	1	∈	ORG
ejpam-3506	467	2	∈	ORG
ejpam-3506	468	1	⊆	CARDINAL
ejpam-3506	470	1	⊆	CARDINAL
ejpam-3506	471	1	∈	ORG
ejpam-3506	471	2	2	CARDINAL
ejpam-3506	472	1	6=	CARDINAL
ejpam-3506	473	1	∩ng[h](c	GPE
ejpam-3506	473	2	6=	CARDINAL
ejpam-3506	473	3	2	CARDINAL
ejpam-3506	473	4	3	DATE
ejpam-3506	473	5	1	CARDINAL
ejpam-3506	474	1	xj	GPE
ejpam-3506	474	2	aj	PERSON
ejpam-3506	474	3	2	CARDINAL
ejpam-3506	474	4	3	DATE
ejpam-3506	476	1	xk−1	ORG
ejpam-3506	477	1	〉	CARDINAL
ejpam-3506	478	1	g[h	ORG
ejpam-3506	479	1	13	CARDINAL
ejpam-3506	479	2	10	CARDINAL
ejpam-3506	481	1	1 ≤ k ≤ max{m	QUANTITY
ejpam-3506	482	1	⋃	PERSON
ejpam-3506	482	2	⊆	CARDINAL
ejpam-3506	483	1	g[h	ORG
ejpam-3506	485	1	6	CARDINAL
ejpam-3506	485	2	cartesian	NORP
ejpam-3506	485	3	cartesian	NORP
ejpam-3506	485	4	two	CARDINAL
ejpam-3506	485	5	v1)(u2	GPE
ejpam-3506	486	1	w. bent-usman	PERSON
ejpam-3506	486	2	r. isla	PERSON
ejpam-3506	486	3	s. canoy / eur	PERSON
ejpam-3506	487	1	j. pure	PERSON
ejpam-3506	487	2	12	CARDINAL
ejpam-3506	487	3	3	CARDINAL
ejpam-3506	487	4	2019	CARDINAL
ejpam-3506	487	5	1337-1349 1347	DATE
ejpam-3506	487	6	14	DATE
ejpam-3506	491	1	4	CARDINAL
ejpam-3506	491	2	〉	CARDINAL
ejpam-3506	492	1	1	CARDINAL
ejpam-3506	495	1	〉	CARDINAL
ejpam-3506	496	1	z}×v	ORG
ejpam-3506	496	2	⊆	CARDINAL
ejpam-3506	496	3	x}×v	ORG
ejpam-3506	497	1	〉	CARDINAL
ejpam-3506	497	2	2	CARDINAL
ejpam-3506	498	1	6=	CARDINAL
ejpam-3506	499	1	j ∈	PERSON
ejpam-3506	499	2	1	CARDINAL
ejpam-3506	499	3	2	DATE
ejpam-3506	500	1	xj ∈	PERSON
ejpam-3506	500	2	⊆	CARDINAL
ejpam-3506	501	1	xj ∈	PERSON
ejpam-3506	501	2	zj	ORG
ejpam-3506	501	3	∈	PRODUCT
ejpam-3506	502	1	⊆	CARDINAL
ejpam-3506	503	1	〉	CARDINAL
ejpam-3506	505	1	6=	CARDINAL
ejpam-3506	506	1	〉	CARDINAL
ejpam-3506	507	1	8	CARDINAL
ejpam-3506	508	1	6=	CARDINAL
ejpam-3506	509	1	g. (ii	PERSON
ejpam-3506	512	1	4	CARDINAL
ejpam-3506	513	1	c1	PERSON
ejpam-3506	513	2	∈	ORG
ejpam-3506	513	3	∈	ORG
ejpam-3506	515	1	a))∩c1|	CARDINAL
ejpam-3506	515	2	= k since c1	PERSON
ejpam-3506	517	1	14	DATE
ejpam-3506	520	1	9	CARDINAL
ejpam-3506	521	1	k ≤ min{m	ORG
ejpam-3506	524	1	1348	CARDINAL
ejpam-3506	524	2	4	CARDINAL
ejpam-3506	525	1	9	CARDINAL
ejpam-3506	527	1	6	CARDINAL
ejpam-3506	530	1	2	CARDINAL
ejpam-3506	530	2	1	CARDINAL
ejpam-3506	530	3	3	CARDINAL
ejpam-3506	530	4	1	CARDINAL
ejpam-3506	531	1	6	CARDINAL
ejpam-3506	531	2	2	DATE
ejpam-3506	531	3	3	CARDINAL
ejpam-3506	531	4	2	CARDINAL
ejpam-3506	531	5	c4)|	GPE
ejpam-3506	531	6	2	DATE
ejpam-3506	531	7	3	CARDINAL
ejpam-3506	531	8	3	CARDINAL
ejpam-3506	543	1	• •	CARDINAL
ejpam-3506	557	1	6	CARDINAL
ejpam-3506	558	1	the philippine commission	ORG
ejpam-3506	558	2	mindanao	LOC
ejpam-3506	558	3	university-iligan institute of technology	ORG
ejpam-3506	559	1	1	CARDINAL
ejpam-3506	559	2	s. arumugam	PERSON
ejpam-3506	559	3	c. sivagnanam	PERSON
ejpam-3506	561	1	73:55–64	CARDINAL
ejpam-3506	561	2	2010	DATE
ejpam-3506	562	1	2	CARDINAL
ejpam-3506	562	2	w. m. bent-usman d. p. gomisong	PERSON
ejpam-3506	562	3	r. t. isla	PERSON
ejpam-3506	563	1	cartesian	NORP
ejpam-3506	564	1	12:1341–1355	DATE
ejpam-3506	564	2	2018	DATE
ejpam-3506	565	1	3	CARDINAL
ejpam-3506	565	2	y. caro a. hansberg	PERSON
ejpam-3506	565	3	m. a. henning	PERSON
ejpam-3506	567	1	19:1–18,	DATE
ejpam-3506	567	2	2011	DATE
ejpam-3506	568	1	4	CARDINAL
ejpam-3506	568	2	e. maravilla r. isla	PERSON
ejpam-3506	568	3	s. r. canoy jr.	PERSON
ejpam-3506	569	1	93:4609–4620	CARDINAL
ejpam-3506	569	2	2014	DATE
ejpam-3506	570	1	1349	CARDINAL
ejpam-3506	571	1	5	CARDINAL
ejpam-3506	571	2	e. maravilla r. isla	PERSON
ejpam-3506	571	3	s. r. canoy jr.	PERSON
ejpam-3506	572	1	2685, 2014	DATE
ejpam-3506	573	1	6	CARDINAL
ejpam-3506	573	2	e. maravilla r. isla	PERSON
ejpam-3506	573	3	s. r. canoy jr.	PERSON
ejpam-3506	575	1	178:8863	CARDINAL
ejpam-3506	575	2	8874,	DATE
ejpam-3506	575	3	2014	DATE