id	sid	tid	token	lemma	pos
ejpam-5098	1	1	european	european	PROPN
ejpam-5098	1	2	journal	journal	PROPN
ejpam-5098	1	3	of	of	ADP
ejpam-5098	1	4	pure	pure	ADJ
ejpam-5098	1	5	and	and	CCONJ
ejpam-5098	1	6	applied	apply	VERB
ejpam-5098	1	7	mathematics	mathematic	NOUN
ejpam-5098	1	8	vol	vol	NOUN
ejpam-5098	1	9	.	.	PROPN
ejpam-5098	2	1	17	17	NUM
ejpam-5098	2	2	,	,	PUNCT
ejpam-5098	2	3	no	no	INTJ
ejpam-5098	2	4	.	.	NOUN
ejpam-5098	2	5	2	2	NUM
ejpam-5098	2	6	,	,	PUNCT
ejpam-5098	2	7	2024	2024	NUM
ejpam-5098	2	8	,	,	PUNCT
ejpam-5098	2	9	1094	1094	NUM
ejpam-5098	2	10	-	-	SYM
ejpam-5098	2	11	1112	1112	NUM
ejpam-5098	2	12	issn	issn	PROPN
ejpam-5098	2	13	1307	1307	NUM
ejpam-5098	2	14	-	-	SYM
ejpam-5098	2	15	5543	5543	NUM
ejpam-5098	2	16	–	–	PUNCT
ejpam-5098	2	17	ejpam.com	ejpam.com	X
ejpam-5098	2	18	published	publish	VERB
ejpam-5098	2	19	by	by	ADP
ejpam-5098	2	20	new	new	PROPN
ejpam-5098	2	21	york	york	PROPN
ejpam-5098	2	22	business	business	PROPN
ejpam-5098	2	23	global	global	PROPN
ejpam-5098	2	24	σ	σ	PROPN
ejpam-5098	2	25	-	-	PUNCT
ejpam-5098	2	26	prime	prime	ADJ
ejpam-5098	2	27	spectrum	spectrum	NOUN
ejpam-5098	2	28	of	of	ADP
ejpam-5098	2	29	almost	almost	ADV
ejpam-5098	2	30	distributive	distributive	ADJ
ejpam-5098	2	31	lattices	lattice	NOUN
ejpam-5098	2	32	rafi	rafi	PROPN
ejpam-5098	2	33	noorbhasha1	noorbhasha1	PROPN
ejpam-5098	2	34	,	,	PUNCT
ejpam-5098	2	35	ravikumar	ravikumar	PROPN
ejpam-5098	2	36	bandaru2	bandaru2	PROPN
ejpam-5098	2	37	,	,	PUNCT
ejpam-5098	2	38	aiyared	aiyare	VERB
ejpam-5098	2	39	iampan3,∗	iampan3,∗	ADJ
ejpam-5098	2	40	1	1	NUM
ejpam-5098	2	41	department	department	NOUN
ejpam-5098	2	42	of	of	ADP
ejpam-5098	2	43	mathematics	mathematic	NOUN
ejpam-5098	2	44	,	,	PUNCT
ejpam-5098	2	45	bapatla	bapatla	VERB
ejpam-5098	2	46	engineering	engineering	NOUN
ejpam-5098	2	47	college	college	NOUN
ejpam-5098	2	48	,	,	PUNCT
ejpam-5098	2	49	bapatla	bapatla	NOUN
ejpam-5098	2	50	,	,	PUNCT
ejpam-5098	2	51	andhra	andhra	PROPN
ejpam-5098	2	52	pradesh-522101	pradesh-522101	NOUN
ejpam-5098	2	53	,	,	PUNCT
ejpam-5098	2	54	india	india	PROPN
ejpam-5098	2	55	2	2	NUM
ejpam-5098	2	56	department	department	NOUN
ejpam-5098	2	57	of	of	ADP
ejpam-5098	2	58	mathematics	mathematic	NOUN
ejpam-5098	2	59	,	,	PUNCT
ejpam-5098	2	60	school	school	NOUN
ejpam-5098	2	61	of	of	ADP
ejpam-5098	2	62	advanced	advanced	ADJ
ejpam-5098	2	63	sciences	science	NOUN
ejpam-5098	2	64	,	,	PUNCT
ejpam-5098	2	65	vit	vit	PROPN
ejpam-5098	2	66	-	-	PUNCT
ejpam-5098	2	67	ap	ap	PROPN
ejpam-5098	2	68	university	university	PROPN
ejpam-5098	2	69	,	,	PUNCT
ejpam-5098	2	70	andhra	andhra	PROPN
ejpam-5098	2	71	pradesh-522237	pradesh-522237	NOUN
ejpam-5098	2	72	,	,	PUNCT
ejpam-5098	2	73	india	india	PROPN
ejpam-5098	2	74	3	3	NUM
ejpam-5098	2	75	department	department	NOUN
ejpam-5098	2	76	of	of	ADP
ejpam-5098	2	77	mathematics	mathematic	NOUN
ejpam-5098	2	78	,	,	PUNCT
ejpam-5098	2	79	school	school	NOUN
ejpam-5098	2	80	of	of	ADP
ejpam-5098	2	81	science	science	NOUN
ejpam-5098	2	82	,	,	PUNCT
ejpam-5098	2	83	university	university	NOUN
ejpam-5098	2	84	of	of	ADP
ejpam-5098	2	85	phayao	phayao	NOUN
ejpam-5098	2	86	,	,	PUNCT
ejpam-5098	2	87	mae	mae	PROPN
ejpam-5098	2	88	ka	ka	PROPN
ejpam-5098	2	89	,	,	PUNCT
ejpam-5098	2	90	mueang	mueang	PROPN
ejpam-5098	2	91	,	,	PUNCT
ejpam-5098	2	92	phayao	phayao	NOUN
ejpam-5098	2	93	56000	56000	NUM
ejpam-5098	2	94	,	,	PUNCT
ejpam-5098	2	95	thailand	thailand	PROPN
ejpam-5098	2	96	abstract	abstract	NOUN
ejpam-5098	2	97	.	.	PUNCT
ejpam-5098	3	1	for	for	ADP
ejpam-5098	3	2	each	each	DET
ejpam-5098	3	3	α	α	NOUN
ejpam-5098	3	4	-	-	NOUN
ejpam-5098	3	5	ideal	ideal	NOUN
ejpam-5098	3	6	of	of	ADP
ejpam-5098	3	7	an	an	DET
ejpam-5098	3	8	almost	almost	ADV
ejpam-5098	3	9	distributive	distributive	ADJ
ejpam-5098	3	10	lattice	lattice	NOUN
ejpam-5098	3	11	(	(	PUNCT
ejpam-5098	3	12	adl	adl	PROPN
ejpam-5098	3	13	)	)	PUNCT
ejpam-5098	3	14	to	to	PART
ejpam-5098	3	15	become	become	VERB
ejpam-5098	3	16	a	a	DET
ejpam-5098	3	17	σ	σ	NOUN
ejpam-5098	3	18	-	-	PUNCT
ejpam-5098	3	19	ideal	ideal	NOUN
ejpam-5098	3	20	,	,	PUNCT
ejpam-5098	3	21	a	a	DET
ejpam-5098	3	22	set	set	NOUN
ejpam-5098	3	23	of	of	ADP
ejpam-5098	3	24	equivalent	equivalent	ADJ
ejpam-5098	3	25	conditions	condition	NOUN
ejpam-5098	3	26	is	be	AUX
ejpam-5098	3	27	derived	derive	VERB
ejpam-5098	3	28	,	,	PUNCT
ejpam-5098	3	29	which	which	PRON
ejpam-5098	3	30	tends	tend	VERB
ejpam-5098	3	31	to	to	PART
ejpam-5098	3	32	result	result	VERB
ejpam-5098	3	33	in	in	ADP
ejpam-5098	3	34	a	a	DET
ejpam-5098	3	35	characterization	characterization	NOUN
ejpam-5098	3	36	of	of	ADP
ejpam-5098	3	37	generalized	generalized	ADJ
ejpam-5098	3	38	stone	stone	NOUN
ejpam-5098	3	39	adls	adls	PROPN
ejpam-5098	3	40	.	.	PUNCT
ejpam-5098	4	1	on	on	ADP
ejpam-5098	4	2	an	an	DET
ejpam-5098	4	3	adl	adl	NOUN
ejpam-5098	4	4	,	,	PUNCT
ejpam-5098	4	5	a	a	DET
ejpam-5098	4	6	one	one	NUM
ejpam-5098	4	7	-	-	PUNCT
ejpam-5098	4	8	to	to	ADP
ejpam-5098	4	9	-	-	PUNCT
ejpam-5098	4	10	one	one	NUM
ejpam-5098	4	11	correspondence	correspondence	NOUN
ejpam-5098	4	12	is	be	AUX
ejpam-5098	4	13	derived	derive	VERB
ejpam-5098	4	14	between	between	ADP
ejpam-5098	4	15	the	the	DET
ejpam-5098	4	16	set	set	NOUN
ejpam-5098	4	17	of	of	ADP
ejpam-5098	4	18	all	all	DET
ejpam-5098	4	19	prime	prime	ADJ
ejpam-5098	4	20	σ	σ	NOUN
ejpam-5098	4	21	-	-	PUNCT
ejpam-5098	4	22	ideals	ideal	NOUN
ejpam-5098	4	23	of	of	ADP
ejpam-5098	4	24	the	the	DET
ejpam-5098	4	25	adl	adl	PROPN
ejpam-5098	4	26	and	and	CCONJ
ejpam-5098	4	27	the	the	DET
ejpam-5098	4	28	set	set	NOUN
ejpam-5098	4	29	of	of	ADP
ejpam-5098	4	30	all	all	DET
ejpam-5098	4	31	prime	prime	ADJ
ejpam-5098	4	32	σ	σ	NOUN
ejpam-5098	4	33	-	-	PUNCT
ejpam-5098	4	34	ideals	ideal	NOUN
ejpam-5098	4	35	of	of	ADP
ejpam-5098	4	36	the	the	DET
ejpam-5098	4	37	quotient	quotient	NOUN
ejpam-5098	4	38	adl	adl	PROPN
ejpam-5098	4	39	.	.	PUNCT
ejpam-5098	5	1	finally	finally	ADV
ejpam-5098	5	2	,	,	PUNCT
ejpam-5098	5	3	proved	prove	VERB
ejpam-5098	5	4	some	some	DET
ejpam-5098	5	5	properties	property	NOUN
ejpam-5098	5	6	of	of	ADP
ejpam-5098	5	7	prime	prime	ADJ
ejpam-5098	5	8	σ	σ	NOUN
ejpam-5098	5	9	-	-	PUNCT
ejpam-5098	5	10	ideals	ideal	NOUN
ejpam-5098	5	11	of	of	ADP
ejpam-5098	5	12	a	a	DET
ejpam-5098	5	13	normal	normal	ADJ
ejpam-5098	5	14	adl	adl	NOUN
ejpam-5098	5	15	topologically	topologically	ADV
ejpam-5098	5	16	.	.	PUNCT
ejpam-5098	6	1	2020	2020	NUM
ejpam-5098	6	2	mathematics	mathematic	NOUN
ejpam-5098	6	3	subject	subject	NOUN
ejpam-5098	6	4	classifications	classification	NOUN
ejpam-5098	6	5	:	:	PUNCT
ejpam-5098	6	6	06d99	06d99	NUM
ejpam-5098	6	7	,	,	PUNCT
ejpam-5098	6	8	06d15	06d15	DET
ejpam-5098	6	9	key	key	ADJ
ejpam-5098	6	10	words	word	NOUN
ejpam-5098	6	11	and	and	CCONJ
ejpam-5098	6	12	phrases	phrase	NOUN
ejpam-5098	6	13	:	:	PUNCT
ejpam-5098	6	14	almost	almost	ADV
ejpam-5098	6	15	distributive	distributive	ADJ
ejpam-5098	6	16	lattice	lattice	NOUN
ejpam-5098	6	17	(	(	PUNCT
ejpam-5098	6	18	adl	adl	PROPN
ejpam-5098	6	19	)	)	PUNCT
ejpam-5098	6	20	,	,	PUNCT
ejpam-5098	6	21	generalized	generalize	VERB
ejpam-5098	6	22	stone	stone	NOUN
ejpam-5098	6	23	adl	adl	PROPN
ejpam-5098	6	24	,	,	PUNCT
ejpam-5098	6	25	complemented	complement	VERB
ejpam-5098	6	26	adl	adl	NOUN
ejpam-5098	6	27	,	,	PUNCT
ejpam-5098	6	28	relatively	relatively	ADV
ejpam-5098	6	29	complemented	complement	VERB
ejpam-5098	6	30	adl	adl	NOUN
ejpam-5098	6	31	,	,	PUNCT
ejpam-5098	6	32	normal	normal	ADJ
ejpam-5098	6	33	adl	adl	PROPN
ejpam-5098	6	34	,	,	PUNCT
ejpam-5098	6	35	minimal	minimal	ADJ
ejpam-5098	6	36	prime	prime	ADJ
ejpam-5098	6	37	ideal	ideal	NOUN
ejpam-5098	6	38	,	,	PUNCT
ejpam-5098	6	39	prime	prime	ADJ
ejpam-5098	6	40	σ	σ	PROPN
ejpam-5098	6	41	-	-	PUNCT
ejpam-5098	6	42	ideal	ideal	NOUN
ejpam-5098	6	43	,	,	PUNCT
ejpam-5098	6	44	prime	prime	ADJ
ejpam-5098	6	45	α	α	NOUN
ejpam-5098	6	46	-	-	PUNCT
ejpam-5098	6	47	ideal	ideal	ADJ
ejpam-5098	6	48	,	,	PUNCT
ejpam-5098	6	49	compact	compact	ADJ
ejpam-5098	6	50	space	space	NOUN
ejpam-5098	6	51	,	,	PUNCT
ejpam-5098	6	52	non	non	ADJ
ejpam-5098	6	53	-	-	ADJ
ejpam-5098	6	54	dense	dense	ADJ
ejpam-5098	6	55	.	.	PUNCT
ejpam-5098	7	1	1	1	X
ejpam-5098	7	2	.	.	X
ejpam-5098	7	3	introduction	introduction	NOUN
ejpam-5098	7	4	the	the	DET
ejpam-5098	7	5	concept	concept	NOUN
ejpam-5098	7	6	of	of	ADP
ejpam-5098	7	7	an	an	DET
ejpam-5098	7	8	almost	almost	ADV
ejpam-5098	7	9	distributive	distributive	ADJ
ejpam-5098	7	10	lattice	lattice	NOUN
ejpam-5098	7	11	(	(	PUNCT
ejpam-5098	7	12	adl	adl	PROPN
ejpam-5098	7	13	)	)	PUNCT
ejpam-5098	7	14	was	be	AUX
ejpam-5098	7	15	introduced	introduce	VERB
ejpam-5098	7	16	by	by	ADP
ejpam-5098	7	17	swamy	swamy	NOUN
ejpam-5098	7	18	and	and	CCONJ
ejpam-5098	7	19	rao	rao	PROPN
ejpam-5098	7	20	,	,	PUNCT
ejpam-5098	7	21	[	[	X
ejpam-5098	7	22	13	13	NUM
ejpam-5098	7	23	]	]	PUNCT
ejpam-5098	7	24	as	as	ADP
ejpam-5098	7	25	a	a	DET
ejpam-5098	7	26	common	common	ADJ
ejpam-5098	7	27	abstraction	abstraction	NOUN
ejpam-5098	7	28	of	of	ADP
ejpam-5098	7	29	many	many	ADJ
ejpam-5098	7	30	existing	exist	VERB
ejpam-5098	7	31	ring	ring	NOUN
ejpam-5098	7	32	theoretic	theoretic	NOUN
ejpam-5098	7	33	generalizations	generalization	NOUN
ejpam-5098	7	34	of	of	ADP
ejpam-5098	7	35	a	a	DET
ejpam-5098	7	36	boolean	boolean	ADJ
ejpam-5098	7	37	algebra	algebra	NOUN
ejpam-5098	7	38	on	on	ADP
ejpam-5098	7	39	one	one	NUM
ejpam-5098	7	40	hand	hand	NOUN
ejpam-5098	7	41	and	and	CCONJ
ejpam-5098	7	42	the	the	DET
ejpam-5098	7	43	class	class	NOUN
ejpam-5098	7	44	of	of	ADP
ejpam-5098	7	45	distributive	distributive	ADJ
ejpam-5098	7	46	lattices	lattice	NOUN
ejpam-5098	7	47	on	on	ADP
ejpam-5098	7	48	the	the	DET
ejpam-5098	7	49	other	other	ADJ
ejpam-5098	7	50	.	.	PUNCT
ejpam-5098	8	1	in	in	ADP
ejpam-5098	8	2	that	that	DET
ejpam-5098	8	3	paper	paper	NOUN
ejpam-5098	8	4	,	,	PUNCT
ejpam-5098	8	5	the	the	DET
ejpam-5098	8	6	concept	concept	NOUN
ejpam-5098	8	7	of	of	ADP
ejpam-5098	8	8	an	an	DET
ejpam-5098	8	9	ideal	ideal	NOUN
ejpam-5098	8	10	in	in	ADP
ejpam-5098	8	11	an	an	DET
ejpam-5098	8	12	adl	adl	NOUN
ejpam-5098	8	13	was	be	AUX
ejpam-5098	8	14	introduced	introduce	VERB
ejpam-5098	8	15	analogous	analogous	ADJ
ejpam-5098	8	16	to	to	ADP
ejpam-5098	8	17	that	that	PRON
ejpam-5098	8	18	in	in	ADP
ejpam-5098	8	19	a	a	DET
ejpam-5098	8	20	distributive	distributive	ADJ
ejpam-5098	8	21	lattice	lattice	NOUN
ejpam-5098	8	22	and	and	CCONJ
ejpam-5098	8	23	it	it	PRON
ejpam-5098	8	24	was	be	AUX
ejpam-5098	8	25	observed	observe	VERB
ejpam-5098	8	26	that	that	SCONJ
ejpam-5098	8	27	the	the	DET
ejpam-5098	8	28	set	set	NOUN
ejpam-5098	8	29	pi(r	pi(r	NOUN
ejpam-5098	8	30	)	)	PUNCT
ejpam-5098	8	31	of	of	ADP
ejpam-5098	8	32	all	all	DET
ejpam-5098	8	33	principal	principal	ADJ
ejpam-5098	8	34	ideals	ideal	NOUN
ejpam-5098	8	35	of	of	ADP
ejpam-5098	8	36	r	r	NOUN
ejpam-5098	8	37	forms	form	VERB
ejpam-5098	8	38	a	a	DET
ejpam-5098	8	39	distributive	distributive	ADJ
ejpam-5098	8	40	lattice	lattice	NOUN
ejpam-5098	8	41	.	.	PUNCT
ejpam-5098	9	1	also	also	ADV
ejpam-5098	9	2	,	,	PUNCT
ejpam-5098	9	3	the	the	DET
ejpam-5098	9	4	concepts	concept	NOUN
ejpam-5098	9	5	of	of	ADP
ejpam-5098	9	6	minimal	minimal	ADJ
ejpam-5098	9	7	prime	prime	ADJ
ejpam-5098	9	8	ideal	ideal	NOUN
ejpam-5098	9	9	belonging	belong	VERB
ejpam-5098	9	10	to	to	ADP
ejpam-5098	9	11	an	an	DET
ejpam-5098	9	12	ideal	ideal	NOUN
ejpam-5098	9	13	of	of	ADP
ejpam-5098	9	14	an	an	DET
ejpam-5098	9	15	adl	adl	NOUN
ejpam-5098	9	16	in	in	ADP
ejpam-5098	9	17	[	[	X
ejpam-5098	9	18	7	7	NUM
ejpam-5098	9	19	]	]	PUNCT
ejpam-5098	9	20	,	,	PUNCT
ejpam-5098	9	21	normal	normal	ADJ
ejpam-5098	9	22	adl	adl	NOUN
ejpam-5098	9	23	in	in	ADP
ejpam-5098	9	24	[	[	X
ejpam-5098	9	25	6	6	NUM
ejpam-5098	9	26	]	]	PUNCT
ejpam-5098	9	27	,	,	PUNCT
ejpam-5098	9	28	pseudo	pseudo	NOUN
ejpam-5098	9	29	-	-	ADJ
ejpam-5098	9	30	complemented	complement	VERB
ejpam-5098	9	31	adl	adl	NOUN
ejpam-5098	9	32	in	in	ADP
ejpam-5098	9	33	[	[	X
ejpam-5098	9	34	14	14	NUM
ejpam-5098	9	35	]	]	PUNCT
ejpam-5098	9	36	and	and	CCONJ
ejpam-5098	9	37	stone	stone	NOUN
ejpam-5098	9	38	adl	adl	PROPN
ejpam-5098	9	39	in	in	ADP
ejpam-5098	9	40	[	[	X
ejpam-5098	9	41	15	15	NUM
ejpam-5098	9	42	]	]	PUNCT
ejpam-5098	9	43	were	be	AUX
ejpam-5098	9	44	introduced	introduce	VERB
ejpam-5098	9	45	.	.	PUNCT
ejpam-5098	10	1	the	the	DET
ejpam-5098	10	2	notions	notion	NOUN
ejpam-5098	10	3	of	of	ADP
ejpam-5098	10	4	α	α	NOUN
ejpam-5098	10	5	-	-	PUNCT
ejpam-5098	10	6	ideals	ideal	NOUN
ejpam-5098	10	7	and	and	CCONJ
ejpam-5098	10	8	σ	σ	NOUN
ejpam-5098	10	9	-	-	PUNCT
ejpam-5098	10	10	ideals	ideal	NOUN
ejpam-5098	10	11	of	of	ADP
ejpam-5098	10	12	distributive	distributive	ADJ
ejpam-5098	10	13	lattices	lattice	NOUN
ejpam-5098	10	14	were	be	AUX
ejpam-5098	10	15	introduced	introduce	VERB
ejpam-5098	10	16	in	in	ADP
ejpam-5098	10	17	[	[	X
ejpam-5098	10	18	2	2	NUM
ejpam-5098	10	19	]	]	PUNCT
ejpam-5098	10	20	and	and	CCONJ
ejpam-5098	10	21	[	[	X
ejpam-5098	10	22	3	3	X
ejpam-5098	10	23	]	]	PUNCT
ejpam-5098	10	24	respectively	respectively	ADV
ejpam-5098	10	25	.	.	PUNCT
ejpam-5098	11	1	in	in	ADP
ejpam-5098	11	2	this	this	DET
ejpam-5098	11	3	paper	paper	NOUN
ejpam-5098	11	4	,	,	PUNCT
ejpam-5098	11	5	for	for	ADP
ejpam-5098	11	6	each	each	DET
ejpam-5098	11	7	α	α	NOUN
ejpam-5098	11	8	-	-	NOUN
ejpam-5098	11	9	ideal	ideal	NOUN
ejpam-5098	11	10	of	of	ADP
ejpam-5098	11	11	an	an	DET
ejpam-5098	11	12	adl	adl	NOUN
ejpam-5098	11	13	to	to	PART
ejpam-5098	11	14	become	become	VERB
ejpam-5098	11	15	a	a	DET
ejpam-5098	11	16	σ	σ	NOUN
ejpam-5098	11	17	-	-	PUNCT
ejpam-5098	11	18	ideal	ideal	NOUN
ejpam-5098	11	19	,	,	PUNCT
ejpam-5098	11	20	a	a	DET
ejpam-5098	11	21	set	set	NOUN
ejpam-5098	11	22	of	of	ADP
ejpam-5098	11	23	equivalent	equivalent	ADJ
ejpam-5098	11	24	conditions	condition	NOUN
ejpam-5098	11	25	is	be	AUX
ejpam-5098	11	26	derived	derive	VERB
ejpam-5098	11	27	,	,	PUNCT
ejpam-5098	11	28	which	which	PRON
ejpam-5098	11	29	tends	tend	VERB
ejpam-5098	11	30	to	to	PART
ejpam-5098	11	31	result	result	VERB
ejpam-5098	11	32	in	in	ADP
ejpam-5098	11	33	a	a	DET
ejpam-5098	11	34	characterization	characterization	NOUN
ejpam-5098	11	35	of	of	ADP
ejpam-5098	11	36	generalized	generalized	ADJ
ejpam-5098	11	37	stone	stone	NOUN
ejpam-5098	11	38	adls	adls	PROPN
ejpam-5098	11	39	.	.	PUNCT
ejpam-5098	12	1	some	some	DET
ejpam-5098	12	2	necessary	necessary	ADJ
ejpam-5098	12	3	and	and	CCONJ
ejpam-5098	12	4	sufficient	sufficient	ADJ
ejpam-5098	12	5	conditions	condition	NOUN
ejpam-5098	12	6	for	for	ADP
ejpam-5098	12	7	the	the	DET
ejpam-5098	12	8	maximal	maximal	ADJ
ejpam-5098	12	9	ideal	ideal	NOUN
ejpam-5098	12	10	∗corresponding	∗corresponde	VERB
ejpam-5098	12	11	author	author	NOUN
ejpam-5098	12	12	.	.	PUNCT
ejpam-5098	13	1	doi	doi	NOUN
ejpam-5098	13	2	:	:	PUNCT
ejpam-5098	13	3	https://doi.org/10.29020/nybg.ejpam.v17i2.5098	https://doi.org/10.29020/nybg.ejpam.v17i2.5098	PROPN
ejpam-5098	13	4	email	email	NOUN
ejpam-5098	13	5	addresses	address	VERB
ejpam-5098	13	6	:	:	PUNCT
ejpam-5098	13	7	rafimaths@gmail.com	rafimaths@gmail.com	X
ejpam-5098	13	8	(	(	PUNCT
ejpam-5098	13	9	r.	r.	PROPN
ejpam-5098	13	10	noorbhasha	noorbhasha	PROPN
ejpam-5098	13	11	)	)	PUNCT
ejpam-5098	13	12	,	,	PUNCT
ejpam-5098	13	13	ravimaths83@gmail.com	ravimaths83@gmail.com	PROPN
ejpam-5098	13	14	(	(	PUNCT
ejpam-5098	13	15	r.	r.	PROPN
ejpam-5098	13	16	bandaru	bandaru	PROPN
ejpam-5098	13	17	)	)	PUNCT
ejpam-5098	13	18	,	,	PUNCT
ejpam-5098	13	19	aiyared.ia@up.ac.th	aiyared.ia@up.ac.th	NOUN
ejpam-5098	13	20	(	(	PUNCT
ejpam-5098	13	21	a.	a.	NOUN
ejpam-5098	13	22	iampan	iampan	PROPN
ejpam-5098	13	23	)	)	PUNCT
ejpam-5098	13	24	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5098	13	25	1094	1094	NUM
ejpam-5098	14	1	©	©	ADP
ejpam-5098	14	2	2024	2024	NUM
ejpam-5098	14	3	ejpam	ejpam	NOUN
ejpam-5098	14	4	all	all	DET
ejpam-5098	14	5	rights	right	NOUN
ejpam-5098	14	6	reserved	reserve	VERB
ejpam-5098	14	7	.	.	PUNCT
ejpam-5098	15	1	r.	r.	PROPN
ejpam-5098	15	2	noorbhasha	noorbhasha	PROPN
ejpam-5098	15	3	,	,	PUNCT
ejpam-5098	15	4	r.	r.	PROPN
ejpam-5098	15	5	bandaru	bandaru	PROPN
ejpam-5098	15	6	,	,	PUNCT
ejpam-5098	15	7	a.	a.	NOUN
ejpam-5098	15	8	iampan	iampan	PROPN
ejpam-5098	15	9	/	/	SYM
ejpam-5098	15	10	eur	eur	PROPN
ejpam-5098	15	11	.	.	PUNCT
ejpam-5098	16	1	j.	j.	PROPN
ejpam-5098	16	2	pure	pure	PROPN
ejpam-5098	16	3	appl	appl	PROPN
ejpam-5098	16	4	.	.	PROPN
ejpam-5098	16	5	math	math	PROPN
ejpam-5098	16	6	,	,	PUNCT
ejpam-5098	16	7	17	17	NUM
ejpam-5098	16	8	(	(	PUNCT
ejpam-5098	16	9	2	2	NUM
ejpam-5098	16	10	)	)	PUNCT
ejpam-5098	16	11	(	(	PUNCT
ejpam-5098	16	12	2024	2024	NUM
ejpam-5098	16	13	)	)	PUNCT
ejpam-5098	16	14	,	,	PUNCT
ejpam-5098	16	15	1094	1094	NUM
ejpam-5098	16	16	-	-	SYM
ejpam-5098	16	17	1112	1112	NUM
ejpam-5098	16	18	1095	1095	NUM
ejpam-5098	16	19	of	of	ADP
ejpam-5098	16	20	an	an	DET
ejpam-5098	16	21	adl	adl	NOUN
ejpam-5098	16	22	are	be	AUX
ejpam-5098	16	23	derived	derive	VERB
ejpam-5098	16	24	.	.	PUNCT
ejpam-5098	17	1	for	for	ADP
ejpam-5098	17	2	each	each	DET
ejpam-5098	17	3	ideal	ideal	NOUN
ejpam-5098	17	4	of	of	ADP
ejpam-5098	17	5	an	an	DET
ejpam-5098	17	6	adl	adl	NOUN
ejpam-5098	17	7	that	that	PRON
ejpam-5098	17	8	becomes	become	VERB
ejpam-5098	17	9	a	a	DET
ejpam-5098	17	10	σ	σ	NOUN
ejpam-5098	17	11	-	-	PUNCT
ejpam-5098	17	12	ideal	ideal	NOUN
ejpam-5098	17	13	,	,	PUNCT
ejpam-5098	17	14	a	a	DET
ejpam-5098	17	15	set	set	NOUN
ejpam-5098	17	16	of	of	ADP
ejpam-5098	17	17	equivalent	equivalent	ADJ
ejpam-5098	17	18	conditions	condition	NOUN
ejpam-5098	17	19	is	be	AUX
ejpam-5098	17	20	derived	derive	VERB
ejpam-5098	17	21	,	,	PUNCT
ejpam-5098	17	22	resulting	result	VERB
ejpam-5098	17	23	in	in	ADP
ejpam-5098	17	24	a	a	DET
ejpam-5098	17	25	characterization	characterization	NOUN
ejpam-5098	17	26	of	of	ADP
ejpam-5098	17	27	relatively	relatively	ADV
ejpam-5098	17	28	complemented	complemented	ADJ
ejpam-5098	17	29	adls	adls	NOUN
ejpam-5098	17	30	.	.	PUNCT
ejpam-5098	18	1	on	on	ADP
ejpam-5098	18	2	an	an	DET
ejpam-5098	18	3	adl	adl	NOUN
ejpam-5098	18	4	,	,	PUNCT
ejpam-5098	18	5	a	a	DET
ejpam-5098	18	6	one	one	NUM
ejpam-5098	18	7	-	-	PUNCT
ejpam-5098	18	8	to	to	ADP
ejpam-5098	18	9	-	-	PUNCT
ejpam-5098	18	10	one	one	NUM
ejpam-5098	18	11	correspondence	correspondence	NOUN
ejpam-5098	18	12	is	be	AUX
ejpam-5098	18	13	derived	derive	VERB
ejpam-5098	18	14	between	between	ADP
ejpam-5098	18	15	the	the	DET
ejpam-5098	18	16	set	set	NOUN
ejpam-5098	18	17	of	of	ADP
ejpam-5098	18	18	all	all	DET
ejpam-5098	18	19	prime	prime	ADJ
ejpam-5098	18	20	σ	σ	NOUN
ejpam-5098	18	21	-	-	PUNCT
ejpam-5098	18	22	ideals	ideal	NOUN
ejpam-5098	18	23	of	of	ADP
ejpam-5098	18	24	the	the	DET
ejpam-5098	18	25	adl	adl	PROPN
ejpam-5098	18	26	and	and	CCONJ
ejpam-5098	18	27	the	the	DET
ejpam-5098	18	28	set	set	NOUN
ejpam-5098	18	29	of	of	ADP
ejpam-5098	18	30	all	all	DET
ejpam-5098	18	31	prime	prime	ADJ
ejpam-5098	18	32	σ	σ	NOUN
ejpam-5098	18	33	-	-	PUNCT
ejpam-5098	18	34	ideals	ideal	NOUN
ejpam-5098	18	35	of	of	ADP
ejpam-5098	18	36	the	the	DET
ejpam-5098	18	37	quotient	quotient	NOUN
ejpam-5098	18	38	adl	adl	PROPN
ejpam-5098	18	39	.	.	PUNCT
ejpam-5098	19	1	finally	finally	ADV
ejpam-5098	19	2	,	,	PUNCT
ejpam-5098	19	3	we	we	PRON
ejpam-5098	19	4	proved	prove	VERB
ejpam-5098	19	5	some	some	DET
ejpam-5098	19	6	properties	property	NOUN
ejpam-5098	19	7	of	of	ADP
ejpam-5098	19	8	the	the	DET
ejpam-5098	19	9	set	set	NOUN
ejpam-5098	19	10	of	of	ADP
ejpam-5098	19	11	all	all	DET
ejpam-5098	19	12	σ	σ	NOUN
ejpam-5098	19	13	-	-	PUNCT
ejpam-5098	19	14	ideals	ideal	NOUN
ejpam-5098	19	15	of	of	ADP
ejpam-5098	19	16	a	a	DET
ejpam-5098	19	17	normal	normal	ADJ
ejpam-5098	19	18	adl	adl	NOUN
ejpam-5098	19	19	topologically	topologically	ADV
ejpam-5098	19	20	.	.	PUNCT
ejpam-5098	20	1	throughout	throughout	ADP
ejpam-5098	20	2	this	this	DET
ejpam-5098	20	3	paper	paper	NOUN
ejpam-5098	20	4	,	,	PUNCT
ejpam-5098	20	5	r	r	NOUN
ejpam-5098	20	6	stands	stand	VERB
ejpam-5098	20	7	for	for	ADP
ejpam-5098	20	8	an	an	DET
ejpam-5098	20	9	adl	adl	NOUN
ejpam-5098	20	10	with	with	ADP
ejpam-5098	20	11	0	0	NUM
ejpam-5098	20	12	.	.	NOUN
ejpam-5098	20	13	2	2	NUM
ejpam-5098	20	14	.	.	X
ejpam-5098	20	15	preliminaries	preliminary	NOUN
ejpam-5098	20	16	this	this	DET
ejpam-5098	20	17	section	section	NOUN
ejpam-5098	20	18	contains	contain	VERB
ejpam-5098	20	19	definitions	definition	NOUN
ejpam-5098	20	20	and	and	CCONJ
ejpam-5098	20	21	important	important	ADJ
ejpam-5098	20	22	results	result	NOUN
ejpam-5098	20	23	from	from	ADP
ejpam-5098	20	24	[	[	X
ejpam-5098	20	25	5	5	NUM
ejpam-5098	20	26	]	]	PUNCT
ejpam-5098	20	27	and	and	CCONJ
ejpam-5098	20	28	[	[	X
ejpam-5098	20	29	13	13	NUM
ejpam-5098	20	30	]	]	PUNCT
ejpam-5098	20	31	,	,	PUNCT
ejpam-5098	20	32	which	which	PRON
ejpam-5098	20	33	will	will	AUX
ejpam-5098	20	34	be	be	AUX
ejpam-5098	20	35	required	require	VERB
ejpam-5098	20	36	in	in	ADP
ejpam-5098	20	37	the	the	DET
ejpam-5098	20	38	paper	paper	NOUN
ejpam-5098	20	39	’s	’s	PART
ejpam-5098	20	40	text	text	NOUN
ejpam-5098	20	41	.	.	PUNCT
ejpam-5098	21	1	definition	definition	NOUN
ejpam-5098	21	2	1	1	NUM
ejpam-5098	21	3	.	.	PUNCT
ejpam-5098	22	1	[	[	X
ejpam-5098	22	2	13	13	NUM
ejpam-5098	22	3	]	]	PUNCT
ejpam-5098	22	4	an	an	DET
ejpam-5098	22	5	algebra	algebra	NOUN
ejpam-5098	22	6	r	r	NOUN
ejpam-5098	22	7	=	=	PUNCT
ejpam-5098	22	8	(	(	PUNCT
ejpam-5098	22	9	r,∨,∧	r,∨,∧	NUM
ejpam-5098	22	10	,	,	PUNCT
ejpam-5098	22	11	0	0	NUM
ejpam-5098	22	12	)	)	PUNCT
ejpam-5098	22	13	of	of	ADP
ejpam-5098	22	14	type	type	NOUN
ejpam-5098	22	15	(	(	PUNCT
ejpam-5098	22	16	2	2	NUM
ejpam-5098	22	17	,	,	PUNCT
ejpam-5098	22	18	2	2	NUM
ejpam-5098	22	19	,	,	PUNCT
ejpam-5098	22	20	0	0	NUM
ejpam-5098	22	21	)	)	PUNCT
ejpam-5098	22	22	is	be	AUX
ejpam-5098	22	23	called	call	VERB
ejpam-5098	22	24	an	an	DET
ejpam-5098	22	25	almost	almost	ADV
ejpam-5098	22	26	distributive	distributive	ADJ
ejpam-5098	22	27	lattice	lattice	NOUN
ejpam-5098	22	28	(	(	PUNCT
ejpam-5098	22	29	abbreviated	abbreviate	VERB
ejpam-5098	22	30	as	as	ADP
ejpam-5098	22	31	adl	adl	PROPN
ejpam-5098	22	32	)	)	PUNCT
ejpam-5098	22	33	if	if	SCONJ
ejpam-5098	22	34	it	it	PRON
ejpam-5098	22	35	satisfies	satisfy	VERB
ejpam-5098	22	36	the	the	DET
ejpam-5098	22	37	following	follow	VERB
ejpam-5098	22	38	conditions	condition	NOUN
ejpam-5098	22	39	:	:	PUNCT
ejpam-5098	22	40	(	(	PUNCT
ejpam-5098	22	41	1	1	X
ejpam-5098	22	42	)	)	PUNCT
ejpam-5098	22	43	(	(	PUNCT
ejpam-5098	22	44	a	a	DET
ejpam-5098	22	45	∨	∨	NUM
ejpam-5098	22	46	b	b	NOUN
ejpam-5098	22	47	)	)	PUNCT
ejpam-5098	22	48	∧	∧	NOUN
ejpam-5098	22	49	c	c	NOUN
ejpam-5098	22	50	=	=	PUNCT
ejpam-5098	22	51	(	(	PUNCT
ejpam-5098	22	52	a	a	DET
ejpam-5098	22	53	∧	∧	PROPN
ejpam-5098	22	54	c	c	NOUN
ejpam-5098	22	55	)	)	PUNCT
ejpam-5098	22	56	∨	∨	PROPN
ejpam-5098	22	57	(	(	PUNCT
ejpam-5098	22	58	b	b	PROPN
ejpam-5098	22	59	∧	∧	PROPN
ejpam-5098	22	60	c	c	NOUN
ejpam-5098	22	61	)	)	PUNCT
ejpam-5098	22	62	(	(	PUNCT
ejpam-5098	22	63	2	2	X
ejpam-5098	22	64	)	)	PUNCT
ejpam-5098	22	65	a	a	DET
ejpam-5098	22	66	∧	∧	PROPN
ejpam-5098	22	67	(	(	PUNCT
ejpam-5098	22	68	b	b	PROPN
ejpam-5098	22	69	∨	∨	NUM
ejpam-5098	22	70	c	c	NOUN
ejpam-5098	22	71	)	)	PUNCT
ejpam-5098	22	72	=	=	NOUN
ejpam-5098	23	1	(	(	PUNCT
ejpam-5098	23	2	a	a	DET
ejpam-5098	23	3	∧	∧	PROPN
ejpam-5098	23	4	b	b	PROPN
ejpam-5098	23	5	)	)	PUNCT
ejpam-5098	23	6	∨	∨	NOUN
ejpam-5098	23	7	(	(	PUNCT
ejpam-5098	23	8	a	a	DET
ejpam-5098	23	9	∧	∧	PROPN
ejpam-5098	23	10	c	c	NOUN
ejpam-5098	23	11	)	)	PUNCT
ejpam-5098	23	12	(	(	PUNCT
ejpam-5098	23	13	3	3	X
ejpam-5098	23	14	)	)	PUNCT
ejpam-5098	23	15	(	(	PUNCT
ejpam-5098	23	16	a	a	DET
ejpam-5098	23	17	∨	∨	NUM
ejpam-5098	23	18	b	b	NOUN
ejpam-5098	23	19	)	)	PUNCT
ejpam-5098	23	20	∧	∧	PROPN
ejpam-5098	23	21	b	b	NOUN
ejpam-5098	23	22	=	=	SYM
ejpam-5098	23	23	b	b	PROPN
ejpam-5098	23	24	(	(	PUNCT
ejpam-5098	23	25	4	4	NUM
ejpam-5098	23	26	)	)	PUNCT
ejpam-5098	23	27	(	(	PUNCT
ejpam-5098	23	28	a	a	DET
ejpam-5098	23	29	∨	∨	NUM
ejpam-5098	23	30	b	b	NOUN
ejpam-5098	23	31	)	)	PUNCT
ejpam-5098	23	32	∧	∧	NOUN
ejpam-5098	23	33	a	a	PRON
ejpam-5098	23	34	=	=	X
ejpam-5098	23	35	a	a	X
ejpam-5098	23	36	(	(	PUNCT
ejpam-5098	23	37	5	5	NUM
ejpam-5098	23	38	)	)	PUNCT
ejpam-5098	23	39	a	a	DET
ejpam-5098	23	40	∨	∨	NOUN
ejpam-5098	23	41	(	(	PUNCT
ejpam-5098	23	42	a	a	DET
ejpam-5098	23	43	∧	∧	PROPN
ejpam-5098	23	44	b	b	NOUN
ejpam-5098	23	45	)	)	PUNCT
ejpam-5098	23	46	=	=	SYM
ejpam-5098	23	47	a	a	PRON
ejpam-5098	23	48	(	(	PUNCT
ejpam-5098	23	49	6	6	NUM
ejpam-5098	23	50	)	)	PUNCT
ejpam-5098	23	51	0	0	NUM
ejpam-5098	24	1	∧	∧	NOUN
ejpam-5098	24	2	a	a	DET
ejpam-5098	24	3	=	=	SYM
ejpam-5098	24	4	0	0	NUM
ejpam-5098	24	5	(	(	PUNCT
ejpam-5098	24	6	7	7	NUM
ejpam-5098	24	7	)	)	PUNCT
ejpam-5098	24	8	a	a	DET
ejpam-5098	24	9	∨	∨	NOUN
ejpam-5098	24	10	0	0	NUM
ejpam-5098	24	11	=	=	NOUN
ejpam-5098	24	12	a	a	PRON
ejpam-5098	24	13	for	for	ADP
ejpam-5098	24	14	all	all	DET
ejpam-5098	24	15	a	a	DET
ejpam-5098	24	16	,	,	PUNCT
ejpam-5098	24	17	b	b	NOUN
ejpam-5098	24	18	,	,	PUNCT
ejpam-5098	24	19	c	c	PROPN
ejpam-5098	24	20	∈	∈	PROPN
ejpam-5098	24	21	r.	r.	PROPN
ejpam-5098	24	22	theorem	theorem	NOUN
ejpam-5098	24	23	1	1	NUM
ejpam-5098	24	24	.	.	PUNCT
ejpam-5098	25	1	[	[	X
ejpam-5098	25	2	13	13	NUM
ejpam-5098	25	3	]	]	X
ejpam-5098	25	4	if	if	SCONJ
ejpam-5098	25	5	(	(	PUNCT
ejpam-5098	25	6	r,∨,∧	r,∨,∧	NUM
ejpam-5098	25	7	,	,	PUNCT
ejpam-5098	25	8	0	0	NUM
ejpam-5098	25	9	)	)	PUNCT
ejpam-5098	25	10	is	be	AUX
ejpam-5098	25	11	an	an	DET
ejpam-5098	25	12	adl	adl	NOUN
ejpam-5098	25	13	,	,	PUNCT
ejpam-5098	25	14	then	then	ADV
ejpam-5098	25	15	(	(	PUNCT
ejpam-5098	25	16	1	1	X
ejpam-5098	25	17	)	)	PUNCT
ejpam-5098	25	18	a	a	DET
ejpam-5098	25	19	∨	∨	NUM
ejpam-5098	25	20	b	b	X
ejpam-5098	25	21	=	=	NOUN
ejpam-5098	25	22	a⇔	a⇔	NOUN
ejpam-5098	25	23	a	a	DET
ejpam-5098	25	24	∧	∧	PROPN
ejpam-5098	25	25	b	b	PROPN
ejpam-5098	25	26	=	=	SYM
ejpam-5098	25	27	b	b	PROPN
ejpam-5098	25	28	(	(	PUNCT
ejpam-5098	25	29	2	2	NUM
ejpam-5098	25	30	)	)	PUNCT
ejpam-5098	25	31	a	a	DET
ejpam-5098	25	32	∨	∨	NUM
ejpam-5098	25	33	b	b	X
ejpam-5098	25	34	=	=	NOUN
ejpam-5098	25	35	b⇔	b⇔	PROPN
ejpam-5098	25	36	a	a	DET
ejpam-5098	25	37	∧	∧	PROPN
ejpam-5098	25	38	b	b	PROPN
ejpam-5098	25	39	=	=	SYM
ejpam-5098	25	40	a	a	X
ejpam-5098	25	41	(	(	PUNCT
ejpam-5098	25	42	3	3	NUM
ejpam-5098	25	43	)	)	PUNCT
ejpam-5098	25	44	∧	∧	NOUN
ejpam-5098	25	45	is	be	AUX
ejpam-5098	25	46	associative	associative	ADJ
ejpam-5098	25	47	in	in	ADP
ejpam-5098	25	48	r	r	NOUN
ejpam-5098	25	49	(	(	PUNCT
ejpam-5098	25	50	4	4	NUM
ejpam-5098	25	51	)	)	PUNCT
ejpam-5098	25	52	a	a	DET
ejpam-5098	25	53	∧	∧	PROPN
ejpam-5098	25	54	b	b	PROPN
ejpam-5098	25	55	∧	∧	PROPN
ejpam-5098	25	56	c	c	NOUN
ejpam-5098	25	57	=	=	SYM
ejpam-5098	25	58	b	b	PROPN
ejpam-5098	25	59	∧	∧	PROPN
ejpam-5098	25	60	a	a	DET
ejpam-5098	25	61	∧	∧	PROPN
ejpam-5098	25	62	c	c	X
ejpam-5098	25	63	(	(	PUNCT
ejpam-5098	25	64	5	5	NUM
ejpam-5098	25	65	)	)	PUNCT
ejpam-5098	25	66	(	(	PUNCT
ejpam-5098	25	67	a	a	DET
ejpam-5098	25	68	∨	∨	NUM
ejpam-5098	25	69	b	b	NOUN
ejpam-5098	25	70	)	)	PUNCT
ejpam-5098	25	71	∧	∧	NOUN
ejpam-5098	25	72	c	c	NOUN
ejpam-5098	25	73	=	=	SYM
ejpam-5098	25	74	(	(	PUNCT
ejpam-5098	25	75	b	b	PROPN
ejpam-5098	25	76	∨	∨	NUM
ejpam-5098	25	77	a	a	PRON
ejpam-5098	25	78	)	)	PUNCT
ejpam-5098	25	79	∧	∧	PROPN
ejpam-5098	25	80	c	c	PROPN
ejpam-5098	25	81	(	(	PUNCT
ejpam-5098	25	82	6	6	NUM
ejpam-5098	25	83	)	)	PUNCT
ejpam-5098	25	84	a	a	DET
ejpam-5098	25	85	∨	∨	NOUN
ejpam-5098	25	86	(	(	PUNCT
ejpam-5098	25	87	b	b	PROPN
ejpam-5098	25	88	∧	∧	PROPN
ejpam-5098	25	89	c	c	NOUN
ejpam-5098	25	90	)	)	PUNCT
ejpam-5098	26	1	=	=	NOUN
ejpam-5098	26	2	(	(	PUNCT
ejpam-5098	26	3	a	a	DET
ejpam-5098	26	4	∨	∨	NUM
ejpam-5098	26	5	b	b	NOUN
ejpam-5098	26	6	)	)	PUNCT
ejpam-5098	26	7	∧	∧	NOUN
ejpam-5098	26	8	(	(	PUNCT
ejpam-5098	26	9	a	a	DET
ejpam-5098	26	10	∨	∨	NUM
ejpam-5098	26	11	c	c	NOUN
ejpam-5098	26	12	)	)	PUNCT
ejpam-5098	26	13	(	(	PUNCT
ejpam-5098	26	14	7	7	X
ejpam-5098	26	15	)	)	PUNCT
ejpam-5098	26	16	a	a	DET
ejpam-5098	26	17	∧	∧	PROPN
ejpam-5098	26	18	(	(	PUNCT
ejpam-5098	26	19	a	a	DET
ejpam-5098	26	20	∨	∨	NUM
ejpam-5098	26	21	b	b	NOUN
ejpam-5098	26	22	)	)	PUNCT
ejpam-5098	26	23	=	=	SYM
ejpam-5098	26	24	a	a	PRON
ejpam-5098	26	25	,	,	PUNCT
ejpam-5098	26	26	(	(	PUNCT
ejpam-5098	26	27	a	a	DET
ejpam-5098	26	28	∧	∧	PROPN
ejpam-5098	26	29	b	b	PROPN
ejpam-5098	26	30	)	)	PUNCT
ejpam-5098	26	31	∨	∨	NUM
ejpam-5098	26	32	b	b	X
ejpam-5098	26	33	=	=	SYM
ejpam-5098	26	34	b	b	PROPN
ejpam-5098	26	35	and	and	CCONJ
ejpam-5098	26	36	a	a	DET
ejpam-5098	26	37	∨	∨	NOUN
ejpam-5098	26	38	(	(	PUNCT
ejpam-5098	26	39	b	b	PROPN
ejpam-5098	26	40	∧	∧	PROPN
ejpam-5098	26	41	a	a	NOUN
ejpam-5098	26	42	)	)	PUNCT
ejpam-5098	26	43	=	=	SYM
ejpam-5098	26	44	a	a	DET
ejpam-5098	26	45	(	(	PUNCT
ejpam-5098	26	46	8)	8)	NUM
ejpam-5098	26	47	a	a	DET
ejpam-5098	26	48	∧	∧	NOUN
ejpam-5098	26	49	a	a	DET
ejpam-5098	26	50	=	=	PUNCT
ejpam-5098	26	51	a	a	NOUN
ejpam-5098	26	52	and	and	CCONJ
ejpam-5098	26	53	a	a	DET
ejpam-5098	26	54	∨	∨	NOUN
ejpam-5098	26	55	a	a	DET
ejpam-5098	26	56	=	=	NOUN
ejpam-5098	26	57	a	a	NOUN
ejpam-5098	26	58	for	for	ADP
ejpam-5098	26	59	all	all	DET
ejpam-5098	26	60	a	a	DET
ejpam-5098	26	61	,	,	PUNCT
ejpam-5098	26	62	b	b	NOUN
ejpam-5098	26	63	,	,	PUNCT
ejpam-5098	26	64	c	c	PROPN
ejpam-5098	26	65	∈	∈	PROPN
ejpam-5098	26	66	r.	r.	PROPN
ejpam-5098	26	67	r.	r.	PROPN
ejpam-5098	26	68	noorbhasha	noorbhasha	PROPN
ejpam-5098	26	69	,	,	PUNCT
ejpam-5098	26	70	r.	r.	PROPN
ejpam-5098	26	71	bandaru	bandaru	PROPN
ejpam-5098	26	72	,	,	PUNCT
ejpam-5098	26	73	a.	a.	NOUN
ejpam-5098	26	74	iampan	iampan	PROPN
ejpam-5098	26	75	/	/	SYM
ejpam-5098	26	76	eur	eur	PROPN
ejpam-5098	26	77	.	.	PUNCT
ejpam-5098	27	1	j.	j.	PROPN
ejpam-5098	27	2	pure	pure	PROPN
ejpam-5098	27	3	appl	appl	PROPN
ejpam-5098	27	4	.	.	PROPN
ejpam-5098	27	5	math	math	PROPN
ejpam-5098	27	6	,	,	PUNCT
ejpam-5098	27	7	17	17	NUM
ejpam-5098	27	8	(	(	PUNCT
ejpam-5098	27	9	2	2	NUM
ejpam-5098	27	10	)	)	PUNCT
ejpam-5098	27	11	(	(	PUNCT
ejpam-5098	27	12	2024	2024	NUM
ejpam-5098	27	13	)	)	PUNCT
ejpam-5098	27	14	,	,	PUNCT
ejpam-5098	27	15	1094	1094	NUM
ejpam-5098	27	16	-	-	SYM
ejpam-5098	27	17	1112	1112	NUM
ejpam-5098	27	18	1096	1096	NUM
ejpam-5098	27	19	it	it	PRON
ejpam-5098	27	20	can	can	AUX
ejpam-5098	27	21	be	be	AUX
ejpam-5098	27	22	observed	observe	VERB
ejpam-5098	27	23	that	that	SCONJ
ejpam-5098	27	24	an	an	DET
ejpam-5098	27	25	adl	adl	NOUN
ejpam-5098	27	26	r	r	NOUN
ejpam-5098	27	27	satisfies	satisfie	NOUN
ejpam-5098	27	28	almost	almost	ADV
ejpam-5098	27	29	all	all	DET
ejpam-5098	27	30	the	the	DET
ejpam-5098	27	31	properties	property	NOUN
ejpam-5098	27	32	of	of	ADP
ejpam-5098	27	33	a	a	DET
ejpam-5098	27	34	distributive	distributive	ADJ
ejpam-5098	27	35	lattice	lattice	NOUN
ejpam-5098	27	36	except	except	SCONJ
ejpam-5098	27	37	the	the	DET
ejpam-5098	27	38	right	right	ADJ
ejpam-5098	27	39	distributivity	distributivity	NOUN
ejpam-5098	27	40	of	of	ADP
ejpam-5098	27	41	∨	∨	NUM
ejpam-5098	27	42	over	over	ADP
ejpam-5098	27	43	∧	∧	PROPN
ejpam-5098	27	44	,	,	PUNCT
ejpam-5098	27	45	commutativity	commutativity	NOUN
ejpam-5098	27	46	of	of	ADP
ejpam-5098	27	47	∨	∨	NOUN
ejpam-5098	27	48	,	,	PUNCT
ejpam-5098	27	49	commutativity	commutativity	NOUN
ejpam-5098	27	50	of	of	ADP
ejpam-5098	27	51	∧.	∧.	PROPN
ejpam-5098	27	52	any	any	DET
ejpam-5098	27	53	one	one	NUM
ejpam-5098	27	54	of	of	ADP
ejpam-5098	27	55	these	these	DET
ejpam-5098	27	56	properties	property	NOUN
ejpam-5098	27	57	makes	make	VERB
ejpam-5098	27	58	an	an	DET
ejpam-5098	27	59	adl	adl	NOUN
ejpam-5098	27	60	r	r	NOUN
ejpam-5098	27	61	a	a	DET
ejpam-5098	27	62	distributive	distributive	ADJ
ejpam-5098	27	63	lattice	lattice	NOUN
ejpam-5098	27	64	.	.	PUNCT
ejpam-5098	28	1	as	as	ADP
ejpam-5098	28	2	usual	usual	ADJ
ejpam-5098	28	3	,	,	PUNCT
ejpam-5098	28	4	an	an	DET
ejpam-5098	28	5	elementm	elementm	PROPN
ejpam-5098	28	6	∈	∈	PROPN
ejpam-5098	28	7	r	r	NOUN
ejpam-5098	28	8	is	be	AUX
ejpam-5098	28	9	called	call	VERB
ejpam-5098	28	10	maximal	maximal	ADJ
ejpam-5098	28	11	if	if	SCONJ
ejpam-5098	28	12	it	it	PRON
ejpam-5098	28	13	is	be	AUX
ejpam-5098	28	14	a	a	DET
ejpam-5098	28	15	maximal	maximal	ADJ
ejpam-5098	28	16	element	element	NOUN
ejpam-5098	28	17	in	in	ADP
ejpam-5098	28	18	the	the	DET
ejpam-5098	28	19	partially	partially	ADV
ejpam-5098	28	20	ordered	order	VERB
ejpam-5098	28	21	set	set	NOUN
ejpam-5098	28	22	(	(	PUNCT
ejpam-5098	28	23	r,≤	r,≤	PROPN
ejpam-5098	28	24	)	)	PUNCT
ejpam-5098	28	25	.	.	PUNCT
ejpam-5098	29	1	that	that	PRON
ejpam-5098	29	2	is	be	AUX
ejpam-5098	29	3	,	,	PUNCT
ejpam-5098	29	4	for	for	ADP
ejpam-5098	29	5	any	any	DET
ejpam-5098	29	6	a	a	DET
ejpam-5098	29	7	∈	∈	PROPN
ejpam-5098	29	8	r	r	NOUN
ejpam-5098	29	9	,	,	PUNCT
ejpam-5098	29	10	m	m	VERB
ejpam-5098	29	11	≤	≤	NOUN
ejpam-5098	29	12	a⇒	a⇒	PRON
ejpam-5098	30	1	m	m	NOUN
ejpam-5098	30	2	=	=	ADJ
ejpam-5098	30	3	a.	a.	NOUN
ejpam-5098	30	4	as	as	ADP
ejpam-5098	30	5	in	in	ADP
ejpam-5098	30	6	distributive	distributive	ADJ
ejpam-5098	30	7	lattices	lattice	NOUN
ejpam-5098	30	8	[	[	X
ejpam-5098	30	9	1	1	NUM
ejpam-5098	30	10	,	,	PUNCT
ejpam-5098	30	11	4	4	NUM
ejpam-5098	30	12	]	]	PUNCT
ejpam-5098	30	13	,	,	PUNCT
ejpam-5098	30	14	a	a	DET
ejpam-5098	30	15	non	non	ADJ
ejpam-5098	30	16	-	-	ADJ
ejpam-5098	30	17	empty	empty	ADJ
ejpam-5098	30	18	subset	subset	NOUN
ejpam-5098	30	19	i	i	PRON
ejpam-5098	30	20	of	of	ADP
ejpam-5098	30	21	an	an	DET
ejpam-5098	30	22	adl	adl	PROPN
ejpam-5098	30	23	r	r	NOUN
ejpam-5098	30	24	is	be	AUX
ejpam-5098	30	25	called	call	VERB
ejpam-5098	30	26	an	an	DET
ejpam-5098	30	27	ideal	ideal	NOUN
ejpam-5098	30	28	of	of	ADP
ejpam-5098	30	29	r	r	NOUN
ejpam-5098	30	30	if	if	SCONJ
ejpam-5098	30	31	a	a	DET
ejpam-5098	30	32	∨	∨	NUM
ejpam-5098	30	33	b	b	X
ejpam-5098	30	34	∈	∈	PROPN
ejpam-5098	30	35	i	i	PRON
ejpam-5098	30	36	and	and	CCONJ
ejpam-5098	30	37	a	a	DET
ejpam-5098	30	38	∧	∧	PROPN
ejpam-5098	30	39	x	x	SYM
ejpam-5098	30	40	∈	∈	PROPN
ejpam-5098	30	41	i	i	PRON
ejpam-5098	30	42	for	for	ADP
ejpam-5098	30	43	any	any	DET
ejpam-5098	30	44	a	a	NOUN
ejpam-5098	30	45	,	,	PUNCT
ejpam-5098	30	46	b	b	X
ejpam-5098	30	47	∈	∈	NOUN
ejpam-5098	31	1	i	i	PRON
ejpam-5098	31	2	and	and	CCONJ
ejpam-5098	31	3	x	x	PROPN
ejpam-5098	31	4	∈	∈	PROPN
ejpam-5098	31	5	r.	r.	PROPN
ejpam-5098	31	6	also	also	ADV
ejpam-5098	31	7	,	,	PUNCT
ejpam-5098	31	8	a	a	DET
ejpam-5098	31	9	non	non	ADJ
ejpam-5098	31	10	-	-	ADJ
ejpam-5098	31	11	empty	empty	ADJ
ejpam-5098	31	12	subset	subset	NOUN
ejpam-5098	31	13	f	f	NOUN
ejpam-5098	31	14	of	of	ADP
ejpam-5098	31	15	r	r	NOUN
ejpam-5098	31	16	is	be	AUX
ejpam-5098	31	17	said	say	VERB
ejpam-5098	31	18	to	to	PART
ejpam-5098	31	19	be	be	AUX
ejpam-5098	31	20	a	a	DET
ejpam-5098	31	21	filter	filter	NOUN
ejpam-5098	31	22	of	of	ADP
ejpam-5098	31	23	r	r	NOUN
ejpam-5098	31	24	if	if	SCONJ
ejpam-5098	31	25	a	a	DET
ejpam-5098	31	26	∧	∧	PROPN
ejpam-5098	31	27	b	b	PROPN
ejpam-5098	31	28	∈	∈	PROPN
ejpam-5098	31	29	f	f	PROPN
ejpam-5098	31	30	and	and	CCONJ
ejpam-5098	31	31	x	x	PROPN
ejpam-5098	31	32	∨	∨	NOUN
ejpam-5098	31	33	a	a	DET
ejpam-5098	31	34	∈	∈	PROPN
ejpam-5098	31	35	f	f	NOUN
ejpam-5098	31	36	for	for	ADP
ejpam-5098	31	37	a	a	DET
ejpam-5098	31	38	,	,	PUNCT
ejpam-5098	31	39	b	b	PROPN
ejpam-5098	31	40	∈	∈	PROPN
ejpam-5098	31	41	f	f	PROPN
ejpam-5098	31	42	and	and	CCONJ
ejpam-5098	31	43	x	x	PROPN
ejpam-5098	31	44	∈	∈	PROPN
ejpam-5098	31	45	r.	r.	NOUN
ejpam-5098	31	46	the	the	DET
ejpam-5098	31	47	set	set	PROPN
ejpam-5098	31	48	i(r	i(r	PROPN
ejpam-5098	31	49	)	)	PUNCT
ejpam-5098	31	50	of	of	ADP
ejpam-5098	31	51	all	all	DET
ejpam-5098	31	52	ideals	ideal	NOUN
ejpam-5098	31	53	of	of	ADP
ejpam-5098	31	54	r	r	NOUN
ejpam-5098	31	55	is	be	AUX
ejpam-5098	31	56	a	a	DET
ejpam-5098	31	57	bounded	bounded	ADJ
ejpam-5098	31	58	distributive	distributive	ADJ
ejpam-5098	31	59	lattice	lattice	NOUN
ejpam-5098	31	60	with	with	ADP
ejpam-5098	31	61	least	least	ADJ
ejpam-5098	31	62	element	element	ADJ
ejpam-5098	31	63	{	{	PUNCT
ejpam-5098	31	64	0	0	NUM
ejpam-5098	31	65	}	}	PUNCT
ejpam-5098	31	66	and	and	CCONJ
ejpam-5098	31	67	greatest	great	ADJ
ejpam-5098	31	68	element	element	NOUN
ejpam-5098	31	69	r	r	NOUN
ejpam-5098	31	70	under	under	ADP
ejpam-5098	31	71	set	set	NOUN
ejpam-5098	31	72	inclusion	inclusion	NOUN
ejpam-5098	31	73	in	in	ADP
ejpam-5098	31	74	which	which	PRON
ejpam-5098	31	75	,	,	PUNCT
ejpam-5098	31	76	for	for	ADP
ejpam-5098	31	77	any	any	DET
ejpam-5098	31	78	i	i	PROPN
ejpam-5098	31	79	,	,	PUNCT
ejpam-5098	31	80	j	j	PROPN
ejpam-5098	31	81	∈	∈	PROPN
ejpam-5098	31	82	i(r	i(r	PROPN
ejpam-5098	31	83	)	)	PUNCT
ejpam-5098	31	84	,	,	PUNCT
ejpam-5098	31	85	i	i	PROPN
ejpam-5098	31	86	∩	∩	VERB
ejpam-5098	31	87	j	j	PROPN
ejpam-5098	31	88	is	be	AUX
ejpam-5098	31	89	the	the	DET
ejpam-5098	31	90	infimum	infimum	NOUN
ejpam-5098	31	91	of	of	ADP
ejpam-5098	31	92	i	i	PRON
ejpam-5098	31	93	and	and	CCONJ
ejpam-5098	31	94	j	j	PROPN
ejpam-5098	31	95	while	while	SCONJ
ejpam-5098	31	96	the	the	DET
ejpam-5098	31	97	supremum	supremum	NOUN
ejpam-5098	31	98	is	be	AUX
ejpam-5098	31	99	given	give	VERB
ejpam-5098	31	100	by	by	ADP
ejpam-5098	31	101	i	i	PROPN
ejpam-5098	31	102	∨	∨	PROPN
ejpam-5098	31	103	j	j	PROPN
ejpam-5098	31	104	:	:	PUNCT
ejpam-5098	31	105	=	=	X
ejpam-5098	31	106	{	{	PUNCT
ejpam-5098	31	107	a	a	DET
ejpam-5098	31	108	∨	∨	NUM
ejpam-5098	31	109	b	b	NOUN
ejpam-5098	31	110	|	|	NOUN
ejpam-5098	31	111	a	a	DET
ejpam-5098	31	112	∈	∈	NOUN
ejpam-5098	31	113	i	i	NOUN
ejpam-5098	31	114	,	,	PUNCT
ejpam-5098	31	115	b	b	PROPN
ejpam-5098	31	116	∈	∈	PROPN
ejpam-5098	31	117	j	j	PROPN
ejpam-5098	31	118	}	}	PUNCT
ejpam-5098	31	119	.	.	PUNCT
ejpam-5098	32	1	a	a	DET
ejpam-5098	32	2	proper	proper	ADJ
ejpam-5098	32	3	ideal	ideal	NOUN
ejpam-5098	32	4	(	(	PUNCT
ejpam-5098	32	5	filter	filter	NOUN
ejpam-5098	32	6	)	)	PUNCT
ejpam-5098	32	7	p	p	NOUN
ejpam-5098	32	8	of	of	ADP
ejpam-5098	32	9	r	r	NOUN
ejpam-5098	32	10	is	be	AUX
ejpam-5098	32	11	called	call	VERB
ejpam-5098	32	12	a	a	DET
ejpam-5098	32	13	prime	prime	ADJ
ejpam-5098	32	14	ideal	ideal	NOUN
ejpam-5098	32	15	(	(	PUNCT
ejpam-5098	32	16	filter	filter	NOUN
ejpam-5098	32	17	)	)	PUNCT
ejpam-5098	32	18	if	if	SCONJ
ejpam-5098	32	19	for	for	ADP
ejpam-5098	32	20	any	any	DET
ejpam-5098	32	21	x	x	NOUN
ejpam-5098	32	22	,	,	PUNCT
ejpam-5098	32	23	y	y	PROPN
ejpam-5098	32	24	∈	∈	PROPN
ejpam-5098	32	25	r	r	NOUN
ejpam-5098	32	26	,	,	PUNCT
ejpam-5098	32	27	x	x	PUNCT
ejpam-5098	32	28	∧	∧	NOUN
ejpam-5098	32	29	y	y	PROPN
ejpam-5098	32	30	∈	∈	PROPN
ejpam-5098	32	31	p	p	X
ejpam-5098	32	32	(	(	PUNCT
ejpam-5098	32	33	x	x	PROPN
ejpam-5098	32	34	∨	∨	NUM
ejpam-5098	32	35	y	y	PROPN
ejpam-5098	32	36	∈	∈	PROPN
ejpam-5098	32	37	p	p	PROPN
ejpam-5098	32	38	)	)	PUNCT
ejpam-5098	32	39	⇒	⇒	NOUN
ejpam-5098	32	40	x	x	X
ejpam-5098	32	41	∈	∈	PROPN
ejpam-5098	32	42	p	p	NOUN
ejpam-5098	32	43	or	or	CCONJ
ejpam-5098	32	44	y	y	PROPN
ejpam-5098	32	45	∈	∈	PROPN
ejpam-5098	32	46	p	p	PROPN
ejpam-5098	32	47	.	.	PUNCT
ejpam-5098	33	1	a	a	DET
ejpam-5098	33	2	proper	proper	ADJ
ejpam-5098	33	3	ideal	ideal	NOUN
ejpam-5098	33	4	(	(	PUNCT
ejpam-5098	33	5	filter	filter	NOUN
ejpam-5098	33	6	)	)	PUNCT
ejpam-5098	33	7	m	m	NOUN
ejpam-5098	33	8	of	of	ADP
ejpam-5098	33	9	r	r	NOUN
ejpam-5098	33	10	is	be	AUX
ejpam-5098	33	11	said	say	VERB
ejpam-5098	33	12	to	to	PART
ejpam-5098	33	13	be	be	AUX
ejpam-5098	33	14	maximal	maximal	ADJ
ejpam-5098	33	15	if	if	SCONJ
ejpam-5098	33	16	it	it	PRON
ejpam-5098	33	17	is	be	AUX
ejpam-5098	33	18	not	not	PART
ejpam-5098	33	19	properly	properly	ADV
ejpam-5098	33	20	contained	contain	VERB
ejpam-5098	33	21	in	in	ADP
ejpam-5098	33	22	any	any	DET
ejpam-5098	33	23	proper	proper	ADJ
ejpam-5098	33	24	ideal	ideal	NOUN
ejpam-5098	33	25	(	(	PUNCT
ejpam-5098	33	26	filter	filter	NOUN
ejpam-5098	33	27	)	)	PUNCT
ejpam-5098	33	28	of	of	ADP
ejpam-5098	33	29	r.	r.	PROPN
ejpam-5098	33	30	it	it	PRON
ejpam-5098	33	31	can	can	AUX
ejpam-5098	33	32	be	be	AUX
ejpam-5098	33	33	observed	observe	VERB
ejpam-5098	33	34	that	that	SCONJ
ejpam-5098	33	35	every	every	DET
ejpam-5098	33	36	maximal	maximal	ADJ
ejpam-5098	33	37	ideal	ideal	NOUN
ejpam-5098	33	38	(	(	PUNCT
ejpam-5098	33	39	filter	filter	NOUN
ejpam-5098	33	40	)	)	PUNCT
ejpam-5098	33	41	of	of	ADP
ejpam-5098	33	42	r	r	NOUN
ejpam-5098	33	43	is	be	AUX
ejpam-5098	33	44	a	a	DET
ejpam-5098	33	45	prime	prime	ADJ
ejpam-5098	33	46	ideal	ideal	NOUN
ejpam-5098	33	47	(	(	PUNCT
ejpam-5098	33	48	filter	filter	NOUN
ejpam-5098	33	49	)	)	PUNCT
ejpam-5098	33	50	.	.	PUNCT
ejpam-5098	34	1	every	every	DET
ejpam-5098	34	2	proper	proper	ADJ
ejpam-5098	34	3	ideal	ideal	NOUN
ejpam-5098	34	4	(	(	PUNCT
ejpam-5098	34	5	filter	filter	NOUN
ejpam-5098	34	6	)	)	PUNCT
ejpam-5098	34	7	of	of	ADP
ejpam-5098	34	8	r	r	NOUN
ejpam-5098	34	9	is	be	AUX
ejpam-5098	34	10	contained	contain	VERB
ejpam-5098	34	11	in	in	ADP
ejpam-5098	34	12	a	a	DET
ejpam-5098	34	13	maximal	maximal	ADJ
ejpam-5098	34	14	ideal	ideal	NOUN
ejpam-5098	34	15	(	(	PUNCT
ejpam-5098	34	16	filter	filter	NOUN
ejpam-5098	34	17	)	)	PUNCT
ejpam-5098	34	18	.	.	PUNCT
ejpam-5098	35	1	for	for	ADP
ejpam-5098	35	2	any	any	DET
ejpam-5098	35	3	subset	subset	NOUN
ejpam-5098	35	4	s	s	NOUN
ejpam-5098	35	5	of	of	ADP
ejpam-5098	35	6	r	r	NOUN
ejpam-5098	35	7	the	the	DET
ejpam-5098	35	8	smallest	small	ADJ
ejpam-5098	35	9	ideal	ideal	NOUN
ejpam-5098	35	10	containing	contain	VERB
ejpam-5098	35	11	s	s	NOUN
ejpam-5098	35	12	is	be	AUX
ejpam-5098	35	13	given	give	VERB
ejpam-5098	35	14	by	by	ADP
ejpam-5098	35	15	(	(	PUNCT
ejpam-5098	35	16	s	s	X
ejpam-5098	35	17	]	]	X
ejpam-5098	35	18	:	:	PUNCT
ejpam-5098	35	19	=	=	SYM
ejpam-5098	35	20	{	{	PUNCT
ejpam-5098	35	21	(	(	PUNCT
ejpam-5098	35	22	n∨	n∨	PROPN
ejpam-5098	35	23	i=1	i=1	PROPN
ejpam-5098	35	24	si	si	PROPN
ejpam-5098	35	25	)	)	PUNCT
ejpam-5098	35	26	∧	∧	NOUN
ejpam-5098	35	27	x	x	INTJ
ejpam-5098	35	28	|	|	ADV
ejpam-5098	35	29	si	si	PROPN
ejpam-5098	35	30	∈	∈	PROPN
ejpam-5098	35	31	s	s	PROPN
ejpam-5098	35	32	,	,	PUNCT
ejpam-5098	35	33	x	x	SYM
ejpam-5098	35	34	∈	∈	NOUN
ejpam-5098	35	35	r	r	NOUN
ejpam-5098	35	36	and	and	CCONJ
ejpam-5098	35	37	n	n	CCONJ
ejpam-5098	35	38	∈	∈	PROPN
ejpam-5098	35	39	n	n	CCONJ
ejpam-5098	35	40	}	}	PUNCT
ejpam-5098	35	41	.	.	PUNCT
ejpam-5098	36	1	if	if	SCONJ
ejpam-5098	36	2	s	s	VERB
ejpam-5098	36	3	=	=	X
ejpam-5098	36	4	{	{	PUNCT
ejpam-5098	36	5	s	s	PROPN
ejpam-5098	36	6	}	}	PUNCT
ejpam-5098	36	7	,	,	PUNCT
ejpam-5098	36	8	we	we	PRON
ejpam-5098	36	9	write	write	VERB
ejpam-5098	36	10	(	(	PUNCT
ejpam-5098	36	11	s	s	X
ejpam-5098	36	12	]	]	X
ejpam-5098	36	13	instead	instead	ADV
ejpam-5098	36	14	of	of	ADP
ejpam-5098	36	15	(	(	PUNCT
ejpam-5098	36	16	s	s	X
ejpam-5098	36	17	]	]	X
ejpam-5098	36	18	and	and	CCONJ
ejpam-5098	36	19	such	such	DET
ejpam-5098	36	20	an	an	DET
ejpam-5098	36	21	ideal	ideal	NOUN
ejpam-5098	36	22	is	be	AUX
ejpam-5098	36	23	called	call	VERB
ejpam-5098	36	24	the	the	DET
ejpam-5098	36	25	principal	principal	ADJ
ejpam-5098	36	26	ideal	ideal	NOUN
ejpam-5098	36	27	of	of	ADP
ejpam-5098	36	28	r.	r.	PROPN
ejpam-5098	36	29	similarly	similarly	ADV
ejpam-5098	36	30	,	,	PUNCT
ejpam-5098	36	31	for	for	ADP
ejpam-5098	36	32	any	any	DET
ejpam-5098	36	33	s	s	NOUN
ejpam-5098	36	34	⊆	⊆	NUM
ejpam-5098	36	35	r	r	NOUN
ejpam-5098	36	36	,	,	PUNCT
ejpam-5098	36	37	[	[	X
ejpam-5098	36	38	s	s	X
ejpam-5098	36	39	)	)	PUNCT
ejpam-5098	36	40	:	:	PUNCT
ejpam-5098	36	41	=	=	SYM
ejpam-5098	36	42	{	{	PUNCT
ejpam-5098	36	43	x	x	PROPN
ejpam-5098	36	44	∨	∨	NOUN
ejpam-5098	36	45	(	(	PUNCT
ejpam-5098	36	46	n∧	n∧	NUM
ejpam-5098	36	47	i=1	i=1	PROPN
ejpam-5098	36	48	si	si	NOUN
ejpam-5098	36	49	)	)	PUNCT
ejpam-5098	37	1	|	|	ADV
ejpam-5098	37	2	si	si	PROPN
ejpam-5098	37	3	∈	∈	PROPN
ejpam-5098	37	4	s	s	PROPN
ejpam-5098	37	5	,	,	PUNCT
ejpam-5098	37	6	x	x	SYM
ejpam-5098	37	7	∈	∈	NOUN
ejpam-5098	37	8	r	r	NOUN
ejpam-5098	37	9	and	and	CCONJ
ejpam-5098	37	10	n	n	CCONJ
ejpam-5098	37	11	∈	∈	PROPN
ejpam-5098	37	12	n	n	CCONJ
ejpam-5098	37	13	}	}	PUNCT
ejpam-5098	37	14	.	.	PUNCT
ejpam-5098	38	1	if	if	SCONJ
ejpam-5098	38	2	s	s	VERB
ejpam-5098	38	3	=	=	X
ejpam-5098	38	4	{	{	PUNCT
ejpam-5098	38	5	s	s	PROPN
ejpam-5098	38	6	}	}	PUNCT
ejpam-5098	38	7	,	,	PUNCT
ejpam-5098	38	8	we	we	PRON
ejpam-5098	38	9	write	write	VERB
ejpam-5098	38	10	[	[	X
ejpam-5098	38	11	s	s	X
ejpam-5098	38	12	)	)	PUNCT
ejpam-5098	38	13	instead	instead	ADV
ejpam-5098	38	14	of	of	ADP
ejpam-5098	38	15	[	[	X
ejpam-5098	38	16	s	s	X
ejpam-5098	38	17	)	)	PUNCT
ejpam-5098	38	18	and	and	CCONJ
ejpam-5098	38	19	such	such	DET
ejpam-5098	38	20	a	a	DET
ejpam-5098	38	21	filter	filter	NOUN
ejpam-5098	38	22	is	be	AUX
ejpam-5098	38	23	called	call	VERB
ejpam-5098	38	24	the	the	DET
ejpam-5098	38	25	principal	principal	ADJ
ejpam-5098	38	26	filter	filter	NOUN
ejpam-5098	38	27	of	of	ADP
ejpam-5098	38	28	r.	r.	PROPN
ejpam-5098	38	29	for	for	ADP
ejpam-5098	38	30	any	any	DET
ejpam-5098	38	31	a	a	PRON
ejpam-5098	38	32	,	,	PUNCT
ejpam-5098	38	33	b	b	X
ejpam-5098	38	34	∈	∈	PROPN
ejpam-5098	38	35	r	r	NOUN
ejpam-5098	38	36	,	,	PUNCT
ejpam-5098	38	37	it	it	PRON
ejpam-5098	38	38	can	can	AUX
ejpam-5098	38	39	be	be	AUX
ejpam-5098	38	40	verified	verify	VERB
ejpam-5098	38	41	that	that	SCONJ
ejpam-5098	38	42	(	(	PUNCT
ejpam-5098	38	43	a]∨	a]∨	PROPN
ejpam-5098	38	44	(	(	PUNCT
ejpam-5098	38	45	b	b	NOUN
ejpam-5098	38	46	]	]	X
ejpam-5098	38	47	=	=	SYM
ejpam-5098	38	48	(	(	PUNCT
ejpam-5098	38	49	a∨	a∨	PROPN
ejpam-5098	38	50	b	b	PROPN
ejpam-5098	38	51	]	]	PUNCT
ejpam-5098	38	52	and	and	CCONJ
ejpam-5098	38	53	(	(	PUNCT
ejpam-5098	38	54	a]∧	a]∧	X
ejpam-5098	38	55	(	(	PUNCT
ejpam-5098	38	56	b	b	X
ejpam-5098	38	57	]	]	X
ejpam-5098	38	58	=	=	SYM
ejpam-5098	38	59	(	(	PUNCT
ejpam-5098	38	60	a∧	a∧	PROPN
ejpam-5098	38	61	b	b	PROPN
ejpam-5098	38	62	]	]	PUNCT
ejpam-5098	38	63	.	.	PUNCT
ejpam-5098	39	1	hence	hence	ADV
ejpam-5098	39	2	,	,	PUNCT
ejpam-5098	39	3	the	the	DET
ejpam-5098	39	4	set	set	NOUN
ejpam-5098	39	5	(	(	PUNCT
ejpam-5098	39	6	pi(r),∨,∩	pi(r),∨,∩	NOUN
ejpam-5098	39	7	)	)	PUNCT
ejpam-5098	39	8	of	of	ADP
ejpam-5098	39	9	all	all	DET
ejpam-5098	39	10	principal	principal	ADJ
ejpam-5098	39	11	ideals	ideal	NOUN
ejpam-5098	39	12	of	of	ADP
ejpam-5098	39	13	r	r	NOUN
ejpam-5098	39	14	is	be	AUX
ejpam-5098	39	15	a	a	DET
ejpam-5098	39	16	sublattice	sublattice	NOUN
ejpam-5098	39	17	of	of	ADP
ejpam-5098	39	18	the	the	DET
ejpam-5098	39	19	distributive	distributive	ADJ
ejpam-5098	39	20	lattice	lattice	NOUN
ejpam-5098	39	21	(	(	PUNCT
ejpam-5098	39	22	i(r),∨,∩	i(r),∨,∩	NOUN
ejpam-5098	39	23	)	)	PUNCT
ejpam-5098	39	24	of	of	ADP
ejpam-5098	39	25	all	all	DET
ejpam-5098	39	26	ideals	ideal	NOUN
ejpam-5098	39	27	of	of	ADP
ejpam-5098	39	28	r.	r.	PROPN
ejpam-5098	39	29	also	also	ADV
ejpam-5098	39	30	,	,	PUNCT
ejpam-5098	39	31	we	we	PRON
ejpam-5098	39	32	have	have	VERB
ejpam-5098	39	33	that	that	SCONJ
ejpam-5098	39	34	the	the	DET
ejpam-5098	39	35	set	set	NOUN
ejpam-5098	39	36	(	(	PUNCT
ejpam-5098	39	37	f(r),∨,∩	f(r),∨,∩	NOUN
ejpam-5098	39	38	)	)	PUNCT
ejpam-5098	39	39	of	of	ADP
ejpam-5098	39	40	all	all	DET
ejpam-5098	39	41	filters	filter	NOUN
ejpam-5098	39	42	of	of	ADP
ejpam-5098	39	43	r	r	NOUN
ejpam-5098	39	44	is	be	AUX
ejpam-5098	39	45	a	a	DET
ejpam-5098	39	46	bounded	bounded	ADJ
ejpam-5098	39	47	distributive	distributive	ADJ
ejpam-5098	39	48	lattice	lattice	NOUN
ejpam-5098	39	49	.	.	PUNCT
ejpam-5098	40	1	definition	definition	NOUN
ejpam-5098	40	2	2	2	NUM
ejpam-5098	40	3	.	.	PUNCT
ejpam-5098	41	1	[	[	X
ejpam-5098	41	2	9	9	NUM
ejpam-5098	41	3	]	]	X
ejpam-5098	41	4	an	an	DET
ejpam-5098	41	5	ideal	ideal	NOUN
ejpam-5098	41	6	i	i	PRON
ejpam-5098	41	7	of	of	ADP
ejpam-5098	41	8	an	an	DET
ejpam-5098	41	9	adl	adl	PROPN
ejpam-5098	41	10	r	r	NOUN
ejpam-5098	41	11	is	be	AUX
ejpam-5098	41	12	said	say	VERB
ejpam-5098	41	13	to	to	PART
ejpam-5098	41	14	be	be	AUX
ejpam-5098	41	15	dense	dense	ADJ
ejpam-5098	41	16	if	if	SCONJ
ejpam-5098	41	17	i∗	i∗	NOUN
ejpam-5098	41	18	=	=	SYM
ejpam-5098	41	19	(	(	PUNCT
ejpam-5098	41	20	0	0	NUM
ejpam-5098	41	21	]	]	PUNCT
ejpam-5098	41	22	.	.	PUNCT
ejpam-5098	42	1	otherwise	otherwise	ADV
ejpam-5098	42	2	,	,	PUNCT
ejpam-5098	42	3	i	i	PRON
ejpam-5098	42	4	is	be	AUX
ejpam-5098	42	5	called	call	VERB
ejpam-5098	42	6	a	a	DET
ejpam-5098	42	7	non	non	ADJ
ejpam-5098	42	8	-	-	ADJ
ejpam-5098	42	9	dense	dense	ADJ
ejpam-5098	42	10	ideal	ideal	NOUN
ejpam-5098	42	11	.	.	PUNCT
ejpam-5098	43	1	definition	definition	NOUN
ejpam-5098	43	2	3	3	NUM
ejpam-5098	43	3	.	.	PUNCT
ejpam-5098	44	1	[	[	X
ejpam-5098	44	2	10	10	NUM
ejpam-5098	44	3	]	]	PUNCT
ejpam-5098	44	4	for	for	ADP
ejpam-5098	44	5	any	any	DET
ejpam-5098	44	6	non	non	ADJ
ejpam-5098	44	7	-	-	ADJ
ejpam-5098	44	8	empty	empty	ADJ
ejpam-5098	44	9	subset	subset	VERB
ejpam-5098	44	10	a	a	PRON
ejpam-5098	44	11	of	of	ADP
ejpam-5098	44	12	an	an	DET
ejpam-5098	44	13	adl	adl	PROPN
ejpam-5098	44	14	r	r	NOUN
ejpam-5098	44	15	,	,	PUNCT
ejpam-5098	44	16	define	define	VERB
ejpam-5098	44	17	a∗	a∗	NOUN
ejpam-5098	44	18	=	=	SYM
ejpam-5098	44	19	{	{	PUNCT
ejpam-5098	44	20	x	x	SYM
ejpam-5098	44	21	∈	∈	NOUN
ejpam-5098	44	22	r	r	NOUN
ejpam-5098	44	23	|	|	ADV
ejpam-5098	44	24	a	a	DET
ejpam-5098	44	25	∧	∧	NOUN
ejpam-5098	44	26	x	x	PUNCT
ejpam-5098	44	27	=	=	SYM
ejpam-5098	44	28	0	0	NUM
ejpam-5098	44	29	for	for	ADP
ejpam-5098	44	30	all	all	DET
ejpam-5098	44	31	a	a	DET
ejpam-5098	44	32	∈	∈	PROPN
ejpam-5098	44	33	a	a	PRON
ejpam-5098	44	34	}	}	PUNCT
ejpam-5098	44	35	.	.	PUNCT
ejpam-5098	45	1	here	here	ADV
ejpam-5098	45	2	,	,	PUNCT
ejpam-5098	45	3	a∗	a∗	PROPN
ejpam-5098	45	4	is	be	AUX
ejpam-5098	45	5	called	call	VERB
ejpam-5098	45	6	the	the	DET
ejpam-5098	45	7	annihilator	annihilator	NOUN
ejpam-5098	45	8	of	of	ADP
ejpam-5098	45	9	a	a	PRON
ejpam-5098	45	10	in	in	ADP
ejpam-5098	45	11	r.	r.	NOUN
ejpam-5098	45	12	for	for	ADP
ejpam-5098	45	13	any	any	DET
ejpam-5098	45	14	a	a	DET
ejpam-5098	45	15	∈	∈	NOUN
ejpam-5098	45	16	r	r	NOUN
ejpam-5098	45	17	,	,	PUNCT
ejpam-5098	45	18	we	we	PRON
ejpam-5098	45	19	have	have	VERB
ejpam-5098	45	20	{	{	PUNCT
ejpam-5098	45	21	a}∗	a}∗	NOUN
ejpam-5098	45	22	=	=	SYM
ejpam-5098	45	23	(	(	PUNCT
ejpam-5098	45	24	a)∗.	a)∗.	PROPN
ejpam-5098	45	25	annihilators	annihilators	PROPN
ejpam-5098	45	26	have	have	VERB
ejpam-5098	45	27	many	many	ADJ
ejpam-5098	45	28	important	important	ADJ
ejpam-5098	45	29	properties	property	NOUN
ejpam-5098	45	30	.	.	PUNCT
ejpam-5098	46	1	we	we	PRON
ejpam-5098	46	2	give	give	VERB
ejpam-5098	46	3	some	some	PRON
ejpam-5098	46	4	of	of	ADP
ejpam-5098	46	5	them	they	PRON
ejpam-5098	46	6	in	in	ADP
ejpam-5098	46	7	the	the	DET
ejpam-5098	46	8	following	following	NOUN
ejpam-5098	46	9	:	:	PUNCT
ejpam-5098	46	10	theorem	theorem	NOUN
ejpam-5098	46	11	2	2	NUM
ejpam-5098	46	12	.	.	PUNCT
ejpam-5098	47	1	[	[	X
ejpam-5098	47	2	10	10	NUM
ejpam-5098	47	3	]	]	PUNCT
ejpam-5098	47	4	let	let	VERB
ejpam-5098	47	5	r	r	PRON
ejpam-5098	47	6	be	be	AUX
ejpam-5098	47	7	an	an	DET
ejpam-5098	47	8	adl	adl	NOUN
ejpam-5098	47	9	.	.	PUNCT
ejpam-5098	48	1	for	for	ADP
ejpam-5098	48	2	any	any	DET
ejpam-5098	48	3	x	x	NOUN
ejpam-5098	48	4	,	,	PUNCT
ejpam-5098	48	5	y	y	PROPN
ejpam-5098	48	6	∈	∈	PROPN
ejpam-5098	48	7	r	r	NOUN
ejpam-5098	48	8	,	,	PUNCT
ejpam-5098	48	9	we	we	PRON
ejpam-5098	48	10	have	have	VERB
ejpam-5098	48	11	:	:	PUNCT
ejpam-5098	48	12	(	(	PUNCT
ejpam-5098	48	13	1	1	X
ejpam-5098	48	14	)	)	PUNCT
ejpam-5098	48	15	x	x	PUNCT
ejpam-5098	48	16	≤	≤	NUM
ejpam-5098	48	17	y	y	PROPN
ejpam-5098	48	18	⇒	⇒	NOUN
ejpam-5098	48	19	(	(	PUNCT
ejpam-5098	48	20	y)∗	y)∗	NOUN
ejpam-5098	48	21	⊆	⊆	NUM
ejpam-5098	48	22	(	(	PUNCT
ejpam-5098	48	23	x)∗	x)∗	X
ejpam-5098	48	24	(	(	PUNCT
ejpam-5098	48	25	2	2	NUM
ejpam-5098	48	26	)	)	PUNCT
ejpam-5098	48	27	(	(	PUNCT
ejpam-5098	48	28	x	x	PUNCT
ejpam-5098	48	29	∧	∧	NOUN
ejpam-5098	48	30	y)∗	y)∗	NOUN
ejpam-5098	48	31	=	=	PUNCT
ejpam-5098	48	32	(	(	PUNCT
ejpam-5098	48	33	y	y	PROPN
ejpam-5098	48	34	∧	∧	PROPN
ejpam-5098	48	35	x)∗	x)∗	PROPN
ejpam-5098	49	1	(	(	PUNCT
ejpam-5098	49	2	3	3	NUM
ejpam-5098	49	3	)	)	PUNCT
ejpam-5098	49	4	(	(	PUNCT
ejpam-5098	50	1	x	x	X
ejpam-5098	50	2	∨	∨	NUM
ejpam-5098	50	3	y)∗	y)∗	NOUN
ejpam-5098	50	4	=	=	PUNCT
ejpam-5098	50	5	(	(	PUNCT
ejpam-5098	50	6	y	y	PROPN
ejpam-5098	50	7	∨	∨	PROPN
ejpam-5098	50	8	x)∗	x)∗	PROPN
ejpam-5098	50	9	(	(	PUNCT
ejpam-5098	50	10	4	4	NUM
ejpam-5098	50	11	)	)	PUNCT
ejpam-5098	50	12	(	(	PUNCT
ejpam-5098	50	13	x	x	X
ejpam-5098	50	14	∨	∨	NUM
ejpam-5098	50	15	y)∗	y)∗	NOUN
ejpam-5098	50	16	=	=	PUNCT
ejpam-5098	50	17	(	(	PUNCT
ejpam-5098	50	18	x)∗	x)∗	PROPN
ejpam-5098	50	19	∩	∩	NOUN
ejpam-5098	50	20	(	(	PUNCT
ejpam-5098	50	21	y)∗	y)∗	NOUN
ejpam-5098	50	22	(	(	PUNCT
ejpam-5098	50	23	5	5	NUM
ejpam-5098	50	24	)	)	PUNCT
ejpam-5098	50	25	(	(	PUNCT
ejpam-5098	50	26	x)∗	x)∗	PROPN
ejpam-5098	50	27	∨	∨	PROPN
ejpam-5098	50	28	(	(	PUNCT
ejpam-5098	50	29	y)∗	y)∗	NOUN
ejpam-5098	50	30	⊆	⊆	NUM
ejpam-5098	50	31	(	(	PUNCT
ejpam-5098	50	32	x	x	SYM
ejpam-5098	50	33	∧	∧	PROPN
ejpam-5098	50	34	y)∗	y)∗	PROPN
ejpam-5098	50	35	r.	r.	PROPN
ejpam-5098	50	36	noorbhasha	noorbhasha	PROPN
ejpam-5098	50	37	,	,	PUNCT
ejpam-5098	50	38	r.	r.	PROPN
ejpam-5098	50	39	bandaru	bandaru	PROPN
ejpam-5098	50	40	,	,	PUNCT
ejpam-5098	50	41	a.	a.	NOUN
ejpam-5098	50	42	iampan	iampan	PROPN
ejpam-5098	50	43	/	/	SYM
ejpam-5098	50	44	eur	eur	PROPN
ejpam-5098	50	45	.	.	PUNCT
ejpam-5098	51	1	j.	j.	PROPN
ejpam-5098	51	2	pure	pure	PROPN
ejpam-5098	51	3	appl	appl	PROPN
ejpam-5098	51	4	.	.	PROPN
ejpam-5098	51	5	math	math	PROPN
ejpam-5098	51	6	,	,	PUNCT
ejpam-5098	51	7	17	17	NUM
ejpam-5098	51	8	(	(	PUNCT
ejpam-5098	51	9	2	2	NUM
ejpam-5098	51	10	)	)	PUNCT
ejpam-5098	51	11	(	(	PUNCT
ejpam-5098	51	12	2024	2024	NUM
ejpam-5098	51	13	)	)	PUNCT
ejpam-5098	51	14	,	,	PUNCT
ejpam-5098	51	15	1094	1094	NUM
ejpam-5098	51	16	-	-	SYM
ejpam-5098	51	17	1112	1112	NUM
ejpam-5098	51	18	1097	1097	NUM
ejpam-5098	51	19	(	(	PUNCT
ejpam-5098	51	20	6	6	NUM
ejpam-5098	51	21	)	)	PUNCT
ejpam-5098	51	22	x	x	X
ejpam-5098	52	1	=	=	SYM
ejpam-5098	52	2	0	0	NUM
ejpam-5098	52	3	⇔	⇔	X
ejpam-5098	52	4	(	(	PUNCT
ejpam-5098	52	5	x)∗	x)∗	PROPN
ejpam-5098	52	6	=	=	SYM
ejpam-5098	52	7	r.	r.	PROPN
ejpam-5098	52	8	definition	definition	NOUN
ejpam-5098	52	9	4	4	NUM
ejpam-5098	52	10	.	.	PUNCT
ejpam-5098	53	1	[	[	X
ejpam-5098	53	2	7	7	X
ejpam-5098	53	3	]	]	X
ejpam-5098	53	4	a	a	DET
ejpam-5098	53	5	prime	prime	ADJ
ejpam-5098	53	6	ideal	ideal	NOUN
ejpam-5098	53	7	of	of	ADP
ejpam-5098	53	8	r	r	NOUN
ejpam-5098	53	9	is	be	AUX
ejpam-5098	53	10	called	call	VERB
ejpam-5098	53	11	a	a	DET
ejpam-5098	53	12	minimal	minimal	ADJ
ejpam-5098	53	13	prime	prime	ADJ
ejpam-5098	53	14	ideal	ideal	NOUN
ejpam-5098	53	15	if	if	SCONJ
ejpam-5098	53	16	it	it	PRON
ejpam-5098	53	17	is	be	AUX
ejpam-5098	53	18	a	a	DET
ejpam-5098	53	19	minimal	minimal	ADJ
ejpam-5098	53	20	element	element	NOUN
ejpam-5098	53	21	in	in	ADP
ejpam-5098	53	22	the	the	DET
ejpam-5098	53	23	set	set	NOUN
ejpam-5098	53	24	of	of	ADP
ejpam-5098	53	25	all	all	DET
ejpam-5098	53	26	prime	prime	ADJ
ejpam-5098	53	27	ideals	ideal	NOUN
ejpam-5098	53	28	of	of	ADP
ejpam-5098	53	29	r	r	NOUN
ejpam-5098	53	30	ordered	order	VERB
ejpam-5098	53	31	by	by	ADP
ejpam-5098	53	32	set	set	VERB
ejpam-5098	53	33	inclusion	inclusion	NOUN
ejpam-5098	53	34	.	.	PUNCT
ejpam-5098	54	1	theorem	theorem	VERB
ejpam-5098	54	2	3	3	NUM
ejpam-5098	54	3	.	.	PUNCT
ejpam-5098	55	1	[	[	X
ejpam-5098	55	2	7	7	X
ejpam-5098	55	3	]	]	PUNCT
ejpam-5098	55	4	let	let	VERB
ejpam-5098	55	5	r	r	PRON
ejpam-5098	55	6	be	be	AUX
ejpam-5098	55	7	an	an	DET
ejpam-5098	55	8	adl	adl	NOUN
ejpam-5098	55	9	.	.	PUNCT
ejpam-5098	56	1	then	then	ADV
ejpam-5098	56	2	a	a	DET
ejpam-5098	56	3	prime	prime	ADJ
ejpam-5098	56	4	ideal	ideal	NOUN
ejpam-5098	56	5	p	p	NOUN
ejpam-5098	56	6	is	be	AUX
ejpam-5098	56	7	minimal	minimal	ADJ
ejpam-5098	56	8	if	if	SCONJ
ejpam-5098	56	9	and	and	CCONJ
ejpam-5098	56	10	only	only	ADV
ejpam-5098	56	11	if	if	SCONJ
ejpam-5098	56	12	for	for	ADP
ejpam-5098	56	13	any	any	DET
ejpam-5098	56	14	x	x	SYM
ejpam-5098	56	15	∈	∈	PROPN
ejpam-5098	56	16	p	p	NOUN
ejpam-5098	56	17	,	,	PUNCT
ejpam-5098	56	18	there	there	PRON
ejpam-5098	56	19	exists	exist	VERB
ejpam-5098	56	20	an	an	DET
ejpam-5098	56	21	element	element	NOUN
ejpam-5098	56	22	y	y	PROPN
ejpam-5098	56	23	/∈	/∈	PUNCT
ejpam-5098	57	1	p	p	X
ejpam-5098	57	2	such	such	ADJ
ejpam-5098	57	3	that	that	SCONJ
ejpam-5098	57	4	x	x	PUNCT
ejpam-5098	57	5	∧	∧	NOUN
ejpam-5098	57	6	y	y	PROPN
ejpam-5098	57	7	=	=	SYM
ejpam-5098	57	8	0	0	PROPN
ejpam-5098	57	9	.	.	PUNCT
ejpam-5098	57	10	theorem	theorem	VERB
ejpam-5098	57	11	4	4	NUM
ejpam-5098	57	12	.	.	PUNCT
ejpam-5098	58	1	[	[	X
ejpam-5098	58	2	7	7	X
ejpam-5098	58	3	]	]	PUNCT
ejpam-5098	58	4	let	let	VERB
ejpam-5098	58	5	r	r	PRON
ejpam-5098	58	6	be	be	AUX
ejpam-5098	58	7	an	an	DET
ejpam-5098	58	8	adl	adl	NOUN
ejpam-5098	58	9	with	with	ADP
ejpam-5098	58	10	maximal	maximal	ADJ
ejpam-5098	58	11	elements	element	NOUN
ejpam-5098	58	12	.	.	PUNCT
ejpam-5098	59	1	then	then	ADV
ejpam-5098	59	2	p	p	NOUN
ejpam-5098	59	3	is	be	AUX
ejpam-5098	59	4	a	a	DET
ejpam-5098	59	5	prime	prime	ADJ
ejpam-5098	59	6	ideal	ideal	NOUN
ejpam-5098	59	7	of	of	ADP
ejpam-5098	59	8	r	r	NOUN
ejpam-5098	59	9	if	if	SCONJ
ejpam-5098	60	1	and	and	CCONJ
ejpam-5098	60	2	only	only	ADV
ejpam-5098	60	3	if	if	SCONJ
ejpam-5098	60	4	r	r	NOUN
ejpam-5098	60	5	\	\	PROPN
ejpam-5098	60	6	p	p	NOUN
ejpam-5098	60	7	is	be	AUX
ejpam-5098	60	8	a	a	DET
ejpam-5098	60	9	prime	prime	ADJ
ejpam-5098	60	10	filter	filter	NOUN
ejpam-5098	60	11	of	of	ADP
ejpam-5098	60	12	r.	r.	PROPN
ejpam-5098	60	13	definition	definition	NOUN
ejpam-5098	60	14	5	5	NUM
ejpam-5098	60	15	.	.	PUNCT
ejpam-5098	61	1	[	[	X
ejpam-5098	61	2	9	9	NUM
ejpam-5098	61	3	]	]	X
ejpam-5098	61	4	an	an	DET
ejpam-5098	61	5	ideal	ideal	NOUN
ejpam-5098	61	6	i	i	PRON
ejpam-5098	61	7	of	of	ADP
ejpam-5098	61	8	an	an	DET
ejpam-5098	61	9	adl	adl	PROPN
ejpam-5098	61	10	r	r	NOUN
ejpam-5098	61	11	is	be	AUX
ejpam-5098	61	12	said	say	VERB
ejpam-5098	61	13	to	to	PART
ejpam-5098	61	14	be	be	AUX
ejpam-5098	61	15	an	an	DET
ejpam-5098	61	16	α	α	NOUN
ejpam-5098	61	17	-	-	PUNCT
ejpam-5098	61	18	ideal	ideal	NOUN
ejpam-5098	61	19	if	if	SCONJ
ejpam-5098	61	20	(	(	PUNCT
ejpam-5098	61	21	a]∗∗	a]∗∗	NOUN
ejpam-5098	61	22	⊆	⊆	NUM
ejpam-5098	61	23	i	i	PRON
ejpam-5098	61	24	for	for	ADP
ejpam-5098	61	25	all	all	DET
ejpam-5098	61	26	a	a	DET
ejpam-5098	61	27	∈	∈	PROPN
ejpam-5098	61	28	i.	i.	NOUN
ejpam-5098	61	29	in	in	ADP
ejpam-5098	61	30	[	[	X
ejpam-5098	61	31	13	13	NUM
ejpam-5098	61	32	]	]	PUNCT
ejpam-5098	61	33	,	,	PUNCT
ejpam-5098	61	34	an	an	DET
ejpam-5098	61	35	adl	adl	NOUN
ejpam-5098	61	36	r	r	NOUN
ejpam-5098	61	37	is	be	AUX
ejpam-5098	61	38	said	say	VERB
ejpam-5098	61	39	to	to	PART
ejpam-5098	61	40	be	be	AUX
ejpam-5098	61	41	relatively	relatively	ADV
ejpam-5098	61	42	complemented	complement	VERB
ejpam-5098	61	43	if	if	SCONJ
ejpam-5098	61	44	for	for	ADP
ejpam-5098	61	45	any	any	DET
ejpam-5098	61	46	a	a	NOUN
ejpam-5098	61	47	,	,	PUNCT
ejpam-5098	61	48	b	b	X
ejpam-5098	61	49	∈	∈	NOUN
ejpam-5098	61	50	r	r	NOUN
ejpam-5098	61	51	with	with	ADP
ejpam-5098	61	52	a	a	DET
ejpam-5098	61	53	≤	≤	NUM
ejpam-5098	61	54	b	b	NOUN
ejpam-5098	61	55	,	,	PUNCT
ejpam-5098	61	56	the	the	DET
ejpam-5098	61	57	interval	interval	NOUN
ejpam-5098	61	58	[	[	X
ejpam-5098	61	59	a	a	X
ejpam-5098	61	60	,	,	PUNCT
ejpam-5098	61	61	b	b	X
ejpam-5098	61	62	]	]	X
ejpam-5098	61	63	is	be	AUX
ejpam-5098	61	64	a	a	DET
ejpam-5098	61	65	complemented	complemented	ADJ
ejpam-5098	61	66	lattice	lattice	NOUN
ejpam-5098	61	67	.	.	PUNCT
ejpam-5098	62	1	theorem	theorem	VERB
ejpam-5098	62	2	5	5	NUM
ejpam-5098	62	3	.	.	PUNCT
ejpam-5098	63	1	[	[	X
ejpam-5098	63	2	13	13	NUM
ejpam-5098	63	3	]	]	X
ejpam-5098	63	4	an	an	DET
ejpam-5098	63	5	adl	adl	PROPN
ejpam-5098	63	6	r	r	NOUN
ejpam-5098	63	7	is	be	AUX
ejpam-5098	63	8	relatively	relatively	ADV
ejpam-5098	63	9	complemented	complemented	ADJ
ejpam-5098	63	10	if	if	SCONJ
ejpam-5098	63	11	and	and	CCONJ
ejpam-5098	63	12	only	only	ADV
ejpam-5098	63	13	if	if	SCONJ
ejpam-5098	63	14	for	for	ADP
ejpam-5098	63	15	any	any	DET
ejpam-5098	63	16	a	a	NOUN
ejpam-5098	63	17	,	,	PUNCT
ejpam-5098	63	18	b	b	X
ejpam-5098	63	19	∈	∈	PROPN
ejpam-5098	63	20	r	r	NOUN
ejpam-5098	63	21	,	,	PUNCT
ejpam-5098	63	22	there	there	PRON
ejpam-5098	63	23	exists	exist	VERB
ejpam-5098	63	24	a	a	DET
ejpam-5098	63	25	unique	unique	ADJ
ejpam-5098	63	26	x	x	SYM
ejpam-5098	63	27	∈	∈	NOUN
ejpam-5098	63	28	r	r	NOUN
ejpam-5098	63	29	such	such	ADJ
ejpam-5098	63	30	that	that	SCONJ
ejpam-5098	63	31	a	a	DET
ejpam-5098	63	32	∨	∨	NOUN
ejpam-5098	63	33	x	x	X
ejpam-5098	63	34	=	=	PUNCT
ejpam-5098	63	35	a	a	DET
ejpam-5098	63	36	∨	∨	NUM
ejpam-5098	63	37	b	b	NOUN
ejpam-5098	63	38	and	and	CCONJ
ejpam-5098	63	39	a	a	DET
ejpam-5098	63	40	∧	∧	PROPN
ejpam-5098	63	41	b	b	PROPN
ejpam-5098	63	42	=	=	SYM
ejpam-5098	63	43	0	0	PROPN
ejpam-5098	63	44	.	.	PUNCT
ejpam-5098	64	1	definition	definition	NOUN
ejpam-5098	64	2	6	6	NUM
ejpam-5098	64	3	.	.	PUNCT
ejpam-5098	65	1	[	[	X
ejpam-5098	65	2	12	12	NUM
ejpam-5098	65	3	]	]	PUNCT
ejpam-5098	65	4	for	for	ADP
ejpam-5098	65	5	any	any	DET
ejpam-5098	65	6	adl	adl	NOUN
ejpam-5098	65	7	r	r	NOUN
ejpam-5098	65	8	with	with	ADP
ejpam-5098	65	9	maximal	maximal	ADJ
ejpam-5098	65	10	elements	element	NOUN
ejpam-5098	65	11	,	,	PUNCT
ejpam-5098	65	12	define	define	VERB
ejpam-5098	65	13	b	b	NOUN
ejpam-5098	65	14	=	=	PUNCT
ejpam-5098	65	15	{	{	PUNCT
ejpam-5098	65	16	a	a	DET
ejpam-5098	65	17	∈	∈	ADJ
ejpam-5098	65	18	l	l	NOUN
ejpam-5098	66	1	|	|	ADV
ejpam-5098	66	2	there	there	PRON
ejpam-5098	66	3	exists	exist	VERB
ejpam-5098	66	4	b	b	PROPN
ejpam-5098	66	5	∈	∈	NOUN
ejpam-5098	66	6	r	r	NOUN
ejpam-5098	66	7	such	such	DET
ejpam-5098	66	8	that	that	SCONJ
ejpam-5098	66	9	a	a	DET
ejpam-5098	66	10	∧	∧	PROPN
ejpam-5098	66	11	b	b	NOUN
ejpam-5098	66	12	=	=	SYM
ejpam-5098	66	13	0	0	PROPN
ejpam-5098	66	14	and	and	CCONJ
ejpam-5098	66	15	a	a	DET
ejpam-5098	66	16	∨	∨	NOUN
ejpam-5098	66	17	b	b	NOUN
ejpam-5098	66	18	is	be	AUX
ejpam-5098	66	19	maximal	maximal	ADJ
ejpam-5098	66	20	}	}	PUNCT
ejpam-5098	66	21	,	,	PUNCT
ejpam-5098	66	22	this	this	PRON
ejpam-5098	66	23	is	be	AUX
ejpam-5098	66	24	called	call	VERB
ejpam-5098	66	25	a	a	DET
ejpam-5098	66	26	birkhoff	birkhoff	NOUN
ejpam-5098	66	27	centre	centre	NOUN
ejpam-5098	66	28	of	of	ADP
ejpam-5098	66	29	an	an	DET
ejpam-5098	66	30	adl	adl	PROPN
ejpam-5098	66	31	r.	r.	PROPN
ejpam-5098	66	32	if	if	SCONJ
ejpam-5098	66	33	b	b	PROPN
ejpam-5098	66	34	=	=	SYM
ejpam-5098	66	35	r	r	NOUN
ejpam-5098	66	36	,	,	PUNCT
ejpam-5098	66	37	then	then	ADV
ejpam-5098	66	38	r	r	NOUN
ejpam-5098	66	39	is	be	AUX
ejpam-5098	66	40	called	call	VERB
ejpam-5098	66	41	a	a	DET
ejpam-5098	66	42	complemented	complement	VERB
ejpam-5098	66	43	adl	adl	PROPN
ejpam-5098	66	44	.	.	PUNCT
ejpam-5098	66	45	definition	definition	NOUN
ejpam-5098	66	46	7	7	NUM
ejpam-5098	66	47	.	.	PUNCT
ejpam-5098	67	1	[	[	X
ejpam-5098	67	2	14	14	NUM
ejpam-5098	67	3	]	]	X
ejpam-5098	67	4	let	let	VERB
ejpam-5098	67	5	(	(	PUNCT
ejpam-5098	67	6	r,∨,∧	r,∨,∧	NUM
ejpam-5098	67	7	,	,	PUNCT
ejpam-5098	67	8	0	0	NUM
ejpam-5098	67	9	)	)	PUNCT
ejpam-5098	67	10	be	be	AUX
ejpam-5098	67	11	an	an	DET
ejpam-5098	67	12	adl	adl	NOUN
ejpam-5098	67	13	.	.	PUNCT
ejpam-5098	68	1	then	then	ADV
ejpam-5098	68	2	a	a	DET
ejpam-5098	68	3	unary	unary	ADJ
ejpam-5098	68	4	operation	operation	NOUN
ejpam-5098	68	5	a	a	DET
ejpam-5098	68	6	−→	−→	NOUN
ejpam-5098	68	7	a∗	a∗	NOUN
ejpam-5098	68	8	on	on	ADP
ejpam-5098	68	9	r	r	NOUN
ejpam-5098	68	10	is	be	AUX
ejpam-5098	68	11	called	call	VERB
ejpam-5098	68	12	a	a	DET
ejpam-5098	68	13	pseudo	pseudo	NOUN
ejpam-5098	68	14	-	-	NOUN
ejpam-5098	68	15	complementation	complementation	NOUN
ejpam-5098	68	16	on	on	ADP
ejpam-5098	68	17	r	r	NOUN
ejpam-5098	68	18	if	if	SCONJ
ejpam-5098	68	19	for	for	ADP
ejpam-5098	68	20	any	any	DET
ejpam-5098	68	21	a	a	NOUN
ejpam-5098	68	22	,	,	PUNCT
ejpam-5098	68	23	b	b	X
ejpam-5098	68	24	∈	∈	PROPN
ejpam-5098	68	25	r	r	NOUN
ejpam-5098	68	26	,	,	PUNCT
ejpam-5098	68	27	it	it	PRON
ejpam-5098	68	28	satisfies	satisfy	VERB
ejpam-5098	68	29	the	the	DET
ejpam-5098	68	30	following	follow	VERB
ejpam-5098	68	31	conditions	condition	NOUN
ejpam-5098	68	32	:	:	PUNCT
ejpam-5098	68	33	(	(	PUNCT
ejpam-5098	68	34	1	1	X
ejpam-5098	68	35	)	)	PUNCT
ejpam-5098	68	36	a	a	DET
ejpam-5098	68	37	∧	∧	PROPN
ejpam-5098	68	38	b	b	NOUN
ejpam-5098	68	39	=	=	SYM
ejpam-5098	68	40	0	0	PROPN
ejpam-5098	68	41	⇒	⇒	NOUN
ejpam-5098	68	42	a∗	a∗	PROPN
ejpam-5098	68	43	∧	∧	PROPN
ejpam-5098	68	44	b	b	PROPN
ejpam-5098	68	45	=	=	SYM
ejpam-5098	68	46	b	b	PROPN
ejpam-5098	68	47	(	(	PUNCT
ejpam-5098	68	48	2	2	NUM
ejpam-5098	68	49	)	)	PUNCT
ejpam-5098	68	50	a	a	DET
ejpam-5098	68	51	∧	∧	NOUN
ejpam-5098	68	52	a∗	a∗	NOUN
ejpam-5098	68	53	=	=	SYM
ejpam-5098	68	54	0	0	NUM
ejpam-5098	68	55	(	(	PUNCT
ejpam-5098	68	56	3	3	NUM
ejpam-5098	68	57	)	)	PUNCT
ejpam-5098	68	58	(	(	PUNCT
ejpam-5098	68	59	a	a	DET
ejpam-5098	68	60	∨	∨	NOUN
ejpam-5098	68	61	b)∗	b)∗	NOUN
ejpam-5098	68	62	=	=	SYM
ejpam-5098	68	63	a∗	a∗	PROPN
ejpam-5098	68	64	∧	∧	PROPN
ejpam-5098	68	65	b∗.	b∗.	NOUN
ejpam-5098	68	66	then	then	ADV
ejpam-5098	68	67	(	(	PUNCT
ejpam-5098	68	68	r,∨,∧,∗	r,∨,∧,∗	PROPN
ejpam-5098	68	69	,	,	PUNCT
ejpam-5098	68	70	0	0	NUM
ejpam-5098	68	71	)	)	PUNCT
ejpam-5098	68	72	is	be	AUX
ejpam-5098	68	73	called	call	VERB
ejpam-5098	68	74	a	a	DET
ejpam-5098	68	75	pseudo	pseudo	NOUN
ejpam-5098	68	76	-	-	ADJ
ejpam-5098	68	77	complemented	complement	VERB
ejpam-5098	68	78	adl	adl	PROPN
ejpam-5098	68	79	.	.	PUNCT
ejpam-5098	69	1	definition	definition	NOUN
ejpam-5098	69	2	8	8	NUM
ejpam-5098	69	3	.	.	PUNCT
ejpam-5098	70	1	[	[	X
ejpam-5098	70	2	15	15	NUM
ejpam-5098	70	3	]	]	PUNCT
ejpam-5098	70	4	let	let	VERB
ejpam-5098	70	5	r	r	PRON
ejpam-5098	70	6	be	be	AUX
ejpam-5098	70	7	an	an	DET
ejpam-5098	70	8	adl	adl	NOUN
ejpam-5098	70	9	and	and	CCONJ
ejpam-5098	70	10	∗	∗	NOUN
ejpam-5098	70	11	a	a	DET
ejpam-5098	70	12	pseudo	pseudo	NOUN
ejpam-5098	70	13	-	-	NOUN
ejpam-5098	70	14	complementation	complementation	NOUN
ejpam-5098	70	15	on	on	ADP
ejpam-5098	70	16	r.	r.	PROPN
ejpam-5098	70	17	then	then	ADV
ejpam-5098	70	18	r	r	NOUN
ejpam-5098	70	19	is	be	AUX
ejpam-5098	70	20	called	call	VERB
ejpam-5098	70	21	a	a	DET
ejpam-5098	70	22	stone	stone	NOUN
ejpam-5098	70	23	adl	adl	NOUN
ejpam-5098	70	24	if	if	SCONJ
ejpam-5098	70	25	for	for	ADP
ejpam-5098	70	26	any	any	DET
ejpam-5098	70	27	x	x	SYM
ejpam-5098	70	28	∈	∈	PROPN
ejpam-5098	70	29	r	r	NOUN
ejpam-5098	70	30	,	,	PUNCT
ejpam-5098	70	31	x∗	x∗	X
ejpam-5098	70	32	∨	∨	NUM
ejpam-5098	70	33	x∗∗	x∗∗	PROPN
ejpam-5098	71	1	=	=	SYM
ejpam-5098	71	2	0∗.	0∗.	PROPN
ejpam-5098	71	3	lemma	lemma	PROPN
ejpam-5098	71	4	1	1	NUM
ejpam-5098	71	5	.	.	PUNCT
ejpam-5098	72	1	[	[	X
ejpam-5098	72	2	15	15	NUM
ejpam-5098	72	3	]	]	PUNCT
ejpam-5098	72	4	let	let	VERB
ejpam-5098	72	5	r	r	PRON
ejpam-5098	72	6	be	be	AUX
ejpam-5098	72	7	a	a	DET
ejpam-5098	72	8	stone	stone	NOUN
ejpam-5098	72	9	adl	adl	NOUN
ejpam-5098	72	10	and	and	CCONJ
ejpam-5098	72	11	a	a	DET
ejpam-5098	72	12	,	,	PUNCT
ejpam-5098	72	13	b	b	X
ejpam-5098	72	14	∈	∈	PROPN
ejpam-5098	72	15	r.	r.	NOUN
ejpam-5098	72	16	then	then	ADV
ejpam-5098	72	17	the	the	DET
ejpam-5098	72	18	following	follow	VERB
ejpam-5098	72	19	conditions	condition	NOUN
ejpam-5098	72	20	hold	hold	VERB
ejpam-5098	72	21	:	:	PUNCT
ejpam-5098	72	22	(	(	PUNCT
ejpam-5098	72	23	1	1	X
ejpam-5098	72	24	)	)	PUNCT
ejpam-5098	72	25	0∗	0∗	NUM
ejpam-5098	73	1	∧	∧	NOUN
ejpam-5098	73	2	a	a	PRON
ejpam-5098	73	3	=	=	PUNCT
ejpam-5098	73	4	a	a	NOUN
ejpam-5098	73	5	and	and	CCONJ
ejpam-5098	73	6	0∗	0∗	NUM
ejpam-5098	73	7	∨	∨	NOUN
ejpam-5098	73	8	a	a	PRON
ejpam-5098	73	9	=	=	NOUN
ejpam-5098	73	10	0∗	0∗	PUNCT
ejpam-5098	73	11	(	(	PUNCT
ejpam-5098	73	12	2	2	NUM
ejpam-5098	73	13	)	)	PUNCT
ejpam-5098	73	14	(	(	PUNCT
ejpam-5098	73	15	a	a	DET
ejpam-5098	73	16	∧	∧	PROPN
ejpam-5098	73	17	b)∗	b)∗	PROPN
ejpam-5098	73	18	=	=	SYM
ejpam-5098	73	19	a∗	a∗	PROPN
ejpam-5098	73	20	∨	∨	NUM
ejpam-5098	73	21	b∗.	b∗.	NOUN
ejpam-5098	73	22	definition	definition	NOUN
ejpam-5098	73	23	9	9	NUM
ejpam-5098	73	24	.	.	PUNCT
ejpam-5098	74	1	[	[	X
ejpam-5098	74	2	10	10	NUM
ejpam-5098	74	3	]	]	X
ejpam-5098	74	4	an	an	DET
ejpam-5098	74	5	adl	adl	PROPN
ejpam-5098	74	6	r	r	NOUN
ejpam-5098	74	7	is	be	AUX
ejpam-5098	74	8	said	say	VERB
ejpam-5098	74	9	to	to	PART
ejpam-5098	74	10	be	be	AUX
ejpam-5098	74	11	a	a	DET
ejpam-5098	74	12	generalized	generalized	ADJ
ejpam-5098	74	13	stone	stone	NOUN
ejpam-5098	74	14	adl	adl	NOUN
ejpam-5098	75	1	if	if	SCONJ
ejpam-5098	75	2	(	(	PUNCT
ejpam-5098	75	3	a]∗	a]∗	PROPN
ejpam-5098	75	4	∨	∨	PROPN
ejpam-5098	75	5	(	(	PUNCT
ejpam-5098	75	6	a]∗∗	a]∗∗	NOUN
ejpam-5098	75	7	=	=	SYM
ejpam-5098	75	8	r	r	NOUN
ejpam-5098	75	9	for	for	ADP
ejpam-5098	75	10	all	all	DET
ejpam-5098	75	11	a	a	DET
ejpam-5098	75	12	∈	∈	PROPN
ejpam-5098	75	13	r.	r.	PROPN
ejpam-5098	75	14	r.	r.	PROPN
ejpam-5098	75	15	noorbhasha	noorbhasha	PROPN
ejpam-5098	75	16	,	,	PUNCT
ejpam-5098	75	17	r.	r.	PROPN
ejpam-5098	75	18	bandaru	bandaru	PROPN
ejpam-5098	75	19	,	,	PUNCT
ejpam-5098	75	20	a.	a.	NOUN
ejpam-5098	75	21	iampan	iampan	PROPN
ejpam-5098	75	22	/	/	SYM
ejpam-5098	75	23	eur	eur	PROPN
ejpam-5098	75	24	.	.	PUNCT
ejpam-5098	76	1	j.	j.	PROPN
ejpam-5098	76	2	pure	pure	PROPN
ejpam-5098	76	3	appl	appl	PROPN
ejpam-5098	76	4	.	.	PROPN
ejpam-5098	76	5	math	math	PROPN
ejpam-5098	76	6	,	,	PUNCT
ejpam-5098	76	7	17	17	NUM
ejpam-5098	76	8	(	(	PUNCT
ejpam-5098	76	9	2	2	NUM
ejpam-5098	76	10	)	)	PUNCT
ejpam-5098	76	11	(	(	PUNCT
ejpam-5098	76	12	2024	2024	NUM
ejpam-5098	76	13	)	)	PUNCT
ejpam-5098	76	14	,	,	PUNCT
ejpam-5098	76	15	1094	1094	NUM
ejpam-5098	76	16	-	-	SYM
ejpam-5098	76	17	1112	1112	NUM
ejpam-5098	76	18	1098	1098	NUM
ejpam-5098	76	19	3	3	NUM
ejpam-5098	76	20	.	.	PUNCT
ejpam-5098	77	1	on	on	ADP
ejpam-5098	77	2	σ	σ	NOUN
ejpam-5098	77	3	-	-	PUNCT
ejpam-5098	77	4	ideals	ideal	NOUN
ejpam-5098	77	5	of	of	ADP
ejpam-5098	77	6	adls	adls	NOUN
ejpam-5098	77	7	for	for	ADP
ejpam-5098	77	8	each	each	DET
ejpam-5098	77	9	α	α	NOUN
ejpam-5098	77	10	-	-	NOUN
ejpam-5098	77	11	ideal	ideal	NOUN
ejpam-5098	77	12	of	of	ADP
ejpam-5098	77	13	an	an	DET
ejpam-5098	77	14	adl	adl	NOUN
ejpam-5098	77	15	to	to	PART
ejpam-5098	77	16	become	become	VERB
ejpam-5098	77	17	a	a	DET
ejpam-5098	77	18	σ	σ	NOUN
ejpam-5098	77	19	-	-	PUNCT
ejpam-5098	77	20	ideal	ideal	NOUN
ejpam-5098	77	21	,	,	PUNCT
ejpam-5098	77	22	a	a	DET
ejpam-5098	77	23	set	set	NOUN
ejpam-5098	77	24	of	of	ADP
ejpam-5098	77	25	equivalent	equivalent	ADJ
ejpam-5098	77	26	conditions	condition	NOUN
ejpam-5098	77	27	is	be	AUX
ejpam-5098	77	28	derived	derive	VERB
ejpam-5098	77	29	,	,	PUNCT
ejpam-5098	77	30	which	which	PRON
ejpam-5098	77	31	tends	tend	VERB
ejpam-5098	77	32	to	to	PART
ejpam-5098	77	33	result	result	VERB
ejpam-5098	77	34	in	in	ADP
ejpam-5098	77	35	a	a	DET
ejpam-5098	77	36	characterization	characterization	NOUN
ejpam-5098	77	37	of	of	ADP
ejpam-5098	77	38	generalized	generalized	ADJ
ejpam-5098	77	39	stone	stone	NOUN
ejpam-5098	77	40	adls	adls	PROPN
ejpam-5098	77	41	.	.	PUNCT
ejpam-5098	78	1	for	for	ADP
ejpam-5098	78	2	each	each	DET
ejpam-5098	78	3	ideal	ideal	NOUN
ejpam-5098	78	4	of	of	ADP
ejpam-5098	78	5	an	an	DET
ejpam-5098	78	6	adl	adl	NOUN
ejpam-5098	78	7	that	that	PRON
ejpam-5098	78	8	becomes	become	VERB
ejpam-5098	78	9	a	a	DET
ejpam-5098	78	10	σ	σ	NOUN
ejpam-5098	78	11	-	-	PUNCT
ejpam-5098	78	12	ideal	ideal	NOUN
ejpam-5098	78	13	,	,	PUNCT
ejpam-5098	78	14	a	a	DET
ejpam-5098	78	15	set	set	NOUN
ejpam-5098	78	16	of	of	ADP
ejpam-5098	78	17	equivalent	equivalent	ADJ
ejpam-5098	78	18	conditions	condition	NOUN
ejpam-5098	78	19	is	be	AUX
ejpam-5098	78	20	derived	derive	VERB
ejpam-5098	78	21	,	,	PUNCT
ejpam-5098	78	22	resulting	result	VERB
ejpam-5098	78	23	in	in	ADP
ejpam-5098	78	24	a	a	DET
ejpam-5098	78	25	characterization	characterization	NOUN
ejpam-5098	78	26	of	of	ADP
ejpam-5098	78	27	relatively	relatively	ADV
ejpam-5098	78	28	complemented	complemented	ADJ
ejpam-5098	78	29	adls	adls	NOUN
ejpam-5098	78	30	.	.	PUNCT
ejpam-5098	79	1	on	on	ADP
ejpam-5098	79	2	an	an	DET
ejpam-5098	79	3	adl	adl	NOUN
ejpam-5098	79	4	,	,	PUNCT
ejpam-5098	79	5	a	a	DET
ejpam-5098	79	6	one	one	NUM
ejpam-5098	79	7	-	-	PUNCT
ejpam-5098	79	8	to	to	ADP
ejpam-5098	79	9	-	-	PUNCT
ejpam-5098	79	10	one	one	NUM
ejpam-5098	79	11	correspondence	correspondence	NOUN
ejpam-5098	79	12	is	be	AUX
ejpam-5098	79	13	derived	derive	VERB
ejpam-5098	79	14	between	between	ADP
ejpam-5098	79	15	the	the	DET
ejpam-5098	79	16	set	set	NOUN
ejpam-5098	79	17	of	of	ADP
ejpam-5098	79	18	all	all	DET
ejpam-5098	79	19	prime	prime	ADJ
ejpam-5098	79	20	σ	σ	NOUN
ejpam-5098	79	21	-	-	PUNCT
ejpam-5098	79	22	ideals	ideal	NOUN
ejpam-5098	79	23	of	of	ADP
ejpam-5098	79	24	the	the	DET
ejpam-5098	79	25	adl	adl	PROPN
ejpam-5098	79	26	and	and	CCONJ
ejpam-5098	79	27	the	the	DET
ejpam-5098	79	28	set	set	NOUN
ejpam-5098	79	29	of	of	ADP
ejpam-5098	79	30	all	all	DET
ejpam-5098	79	31	prime	prime	ADJ
ejpam-5098	79	32	σ	σ	NOUN
ejpam-5098	79	33	-	-	PUNCT
ejpam-5098	79	34	ideals	ideal	NOUN
ejpam-5098	79	35	of	of	ADP
ejpam-5098	79	36	the	the	DET
ejpam-5098	79	37	quotient	quotient	NOUN
ejpam-5098	79	38	adl	adl	PROPN
ejpam-5098	79	39	.	.	PUNCT
ejpam-5098	80	1	the	the	DET
ejpam-5098	80	2	following	follow	VERB
ejpam-5098	80	3	definition	definition	NOUN
ejpam-5098	80	4	is	be	AUX
ejpam-5098	80	5	adopted	adopt	VERB
ejpam-5098	80	6	from	from	ADP
ejpam-5098	80	7	[	[	X
ejpam-5098	80	8	11	11	NUM
ejpam-5098	80	9	]	]	PUNCT
ejpam-5098	80	10	.	.	PUNCT
ejpam-5098	81	1	definition	definition	NOUN
ejpam-5098	81	2	10	10	NUM
ejpam-5098	81	3	.	.	PUNCT
ejpam-5098	82	1	an	an	DET
ejpam-5098	82	2	ideal	ideal	ADJ
ejpam-5098	82	3	k	k	PROPN
ejpam-5098	82	4	of	of	ADP
ejpam-5098	82	5	an	an	DET
ejpam-5098	82	6	adl	adl	PROPN
ejpam-5098	82	7	r	r	NOUN
ejpam-5098	82	8	is	be	AUX
ejpam-5098	82	9	said	say	VERB
ejpam-5098	82	10	to	to	PART
ejpam-5098	82	11	be	be	AUX
ejpam-5098	82	12	a	a	DET
ejpam-5098	82	13	σ	σ	NOUN
ejpam-5098	82	14	-	-	PUNCT
ejpam-5098	82	15	ideal	ideal	NOUN
ejpam-5098	82	16	of	of	ADP
ejpam-5098	82	17	r	r	NOUN
ejpam-5098	82	18	if	if	SCONJ
ejpam-5098	82	19	kσ	kσ	PROPN
ejpam-5098	82	20	=	=	SYM
ejpam-5098	82	21	k	k	PROPN
ejpam-5098	82	22	,	,	PUNCT
ejpam-5098	82	23	where	where	SCONJ
ejpam-5098	82	24	kσ	kσ	PROPN
ejpam-5098	82	25	=	=	PRON
ejpam-5098	82	26	{	{	PUNCT
ejpam-5098	82	27	x	x	SYM
ejpam-5098	82	28	∈	∈	NOUN
ejpam-5098	82	29	r	r	NOUN
ejpam-5098	82	30	|	|	NOUN
ejpam-5098	82	31	k	k	PROPN
ejpam-5098	82	32	∨	∨	X
ejpam-5098	83	1	(	(	PUNCT
ejpam-5098	83	2	x)∗	x)∗	PROPN
ejpam-5098	83	3	=	=	SYM
ejpam-5098	83	4	r	r	NOUN
ejpam-5098	83	5	}	}	PUNCT
ejpam-5098	83	6	.	.	PUNCT
ejpam-5098	84	1	proposition	proposition	NOUN
ejpam-5098	84	2	1	1	NUM
ejpam-5098	84	3	.	.	PUNCT
ejpam-5098	85	1	every	every	DET
ejpam-5098	85	2	prime	prime	PROPN
ejpam-5098	85	3	σ	σ	PROPN
ejpam-5098	85	4	-	-	PUNCT
ejpam-5098	85	5	ideal	ideal	NOUN
ejpam-5098	85	6	of	of	ADP
ejpam-5098	85	7	an	an	DET
ejpam-5098	85	8	adl	adl	NOUN
ejpam-5098	85	9	r	r	NOUN
ejpam-5098	85	10	with	with	ADP
ejpam-5098	85	11	maximal	maximal	ADJ
ejpam-5098	85	12	elements	element	NOUN
ejpam-5098	85	13	is	be	AUX
ejpam-5098	85	14	a	a	DET
ejpam-5098	85	15	minimal	minimal	ADJ
ejpam-5098	85	16	prime	prime	ADJ
ejpam-5098	85	17	ideal	ideal	NOUN
ejpam-5098	85	18	.	.	PUNCT
ejpam-5098	86	1	proof	proof	NOUN
ejpam-5098	86	2	.	.	PUNCT
ejpam-5098	87	1	let	let	VERB
ejpam-5098	87	2	m	m	PRON
ejpam-5098	87	3	be	be	AUX
ejpam-5098	87	4	any	any	DET
ejpam-5098	87	5	prime	prime	ADJ
ejpam-5098	87	6	σ	σ	NOUN
ejpam-5098	87	7	-	-	PUNCT
ejpam-5098	87	8	ideal	ideal	NOUN
ejpam-5098	87	9	of	of	ADP
ejpam-5098	87	10	an	an	DET
ejpam-5098	87	11	adl	adl	PROPN
ejpam-5098	87	12	r.	r.	PROPN
ejpam-5098	87	13	then	then	ADV
ejpam-5098	87	14	mσ	mσ	VERB
ejpam-5098	87	15	=	=	NOUN
ejpam-5098	87	16	m	m	VERB
ejpam-5098	87	17	.	.	PUNCT
ejpam-5098	88	1	let	let	VERB
ejpam-5098	88	2	t	t	PROPN
ejpam-5098	88	3	∈	∈	PROPN
ejpam-5098	89	1	m	m	PROPN
ejpam-5098	89	2	=	=	ADJ
ejpam-5098	89	3	mσ	mσ	PROPN
ejpam-5098	89	4	.	.	PUNCT
ejpam-5098	90	1	then	then	ADV
ejpam-5098	90	2	(	(	PUNCT
ejpam-5098	90	3	t)∗	t)∗	NOUN
ejpam-5098	90	4	∨m	∨m	NOUN
ejpam-5098	90	5	=	=	PUNCT
ejpam-5098	90	6	r.	r.	NOUN
ejpam-5098	90	7	that	that	PRON
ejpam-5098	90	8	implies	imply	VERB
ejpam-5098	90	9	x	x	PUNCT
ejpam-5098	90	10	∨	∨	NUM
ejpam-5098	90	11	y	y	PROPN
ejpam-5098	90	12	is	be	AUX
ejpam-5098	90	13	a	a	DET
ejpam-5098	90	14	maximal	maximal	ADJ
ejpam-5098	90	15	element	element	NOUN
ejpam-5098	90	16	of	of	ADP
ejpam-5098	90	17	r	r	NOUN
ejpam-5098	90	18	for	for	ADP
ejpam-5098	90	19	some	some	DET
ejpam-5098	90	20	x	x	SYM
ejpam-5098	90	21	∈	∈	PROPN
ejpam-5098	90	22	(	(	PUNCT
ejpam-5098	90	23	t)∗	t)∗	NOUN
ejpam-5098	90	24	and	and	CCONJ
ejpam-5098	90	25	y	y	PROPN
ejpam-5098	90	26	∈	∈	PROPN
ejpam-5098	90	27	m	m	VERB
ejpam-5098	90	28	.	.	PUNCT
ejpam-5098	91	1	since	since	SCONJ
ejpam-5098	91	2	m	m	PROPN
ejpam-5098	91	3	is	be	AUX
ejpam-5098	91	4	prime	prime	ADJ
ejpam-5098	91	5	,	,	PUNCT
ejpam-5098	91	6	we	we	PRON
ejpam-5098	91	7	get	get	VERB
ejpam-5098	91	8	x	x	PUNCT
ejpam-5098	91	9	/∈	/∈	PUNCT
ejpam-5098	92	1	m	m	VERB
ejpam-5098	92	2	.	.	PUNCT
ejpam-5098	93	1	hence	hence	ADV
ejpam-5098	93	2	for	for	ADP
ejpam-5098	93	3	any	any	DET
ejpam-5098	93	4	t	t	NOUN
ejpam-5098	93	5	∈	∈	PROPN
ejpam-5098	93	6	m	m	VERB
ejpam-5098	93	7	,	,	PUNCT
ejpam-5098	93	8	there	there	PRON
ejpam-5098	93	9	exists	exist	VERB
ejpam-5098	93	10	an	an	DET
ejpam-5098	93	11	element	element	NOUN
ejpam-5098	93	12	x	x	X
ejpam-5098	93	13	/∈m	/∈m	PUNCT
ejpam-5098	93	14	such	such	ADJ
ejpam-5098	93	15	that	that	SCONJ
ejpam-5098	93	16	x	x	SYM
ejpam-5098	93	17	∈	∈	PROPN
ejpam-5098	93	18	(	(	PUNCT
ejpam-5098	93	19	t)∗	t)∗	NOUN
ejpam-5098	93	20	(	(	PUNCT
ejpam-5098	93	21	i.e.	i.e.	X
ejpam-5098	93	22	,	,	PUNCT
ejpam-5098	93	23	x	x	PART
ejpam-5098	93	24	∧	∧	NOUN
ejpam-5098	93	25	t	t	NOUN
ejpam-5098	93	26	=	=	SYM
ejpam-5098	93	27	0	0	NUM
ejpam-5098	93	28	)	)	PUNCT
ejpam-5098	93	29	.	.	PUNCT
ejpam-5098	94	1	hence	hence	ADV
ejpam-5098	94	2	,	,	PUNCT
ejpam-5098	94	3	m	m	VERB
ejpam-5098	94	4	is	be	AUX
ejpam-5098	94	5	a	a	DET
ejpam-5098	94	6	minimal	minimal	ADJ
ejpam-5098	94	7	prime	prime	ADJ
ejpam-5098	94	8	ideal	ideal	NOUN
ejpam-5098	94	9	of	of	ADP
ejpam-5098	94	10	r.	r.	PROPN
ejpam-5098	94	11	in	in	ADP
ejpam-5098	94	12	general	general	ADJ
ejpam-5098	94	13	,	,	PUNCT
ejpam-5098	94	14	every	every	DET
ejpam-5098	94	15	minimal	minimal	ADJ
ejpam-5098	94	16	prime	prime	ADJ
ejpam-5098	94	17	ideal	ideal	NOUN
ejpam-5098	94	18	need	need	AUX
ejpam-5098	94	19	not	not	PART
ejpam-5098	94	20	be	be	AUX
ejpam-5098	94	21	a	a	DET
ejpam-5098	94	22	σ	σ	NOUN
ejpam-5098	94	23	-	-	PUNCT
ejpam-5098	94	24	ideal	ideal	NOUN
ejpam-5098	94	25	.	.	PUNCT
ejpam-5098	95	1	example	example	NOUN
ejpam-5098	96	1	1	1	NUM
ejpam-5098	96	2	.	.	X
ejpam-5098	96	3	consider	consider	VERB
ejpam-5098	96	4	a	a	DET
ejpam-5098	96	5	distributive	distributive	ADJ
ejpam-5098	96	6	lattice	lattice	NOUN
ejpam-5098	96	7	l	l	NOUN
ejpam-5098	96	8	=	=	PUNCT
ejpam-5098	96	9	{	{	PUNCT
ejpam-5098	96	10	0	0	NUM
ejpam-5098	96	11	,	,	PUNCT
ejpam-5098	96	12	a	a	DET
ejpam-5098	96	13	,	,	PUNCT
ejpam-5098	96	14	b	b	NOUN
ejpam-5098	96	15	,	,	PUNCT
ejpam-5098	96	16	c	c	NOUN
ejpam-5098	96	17	,	,	PUNCT
ejpam-5098	96	18	1	1	NUM
ejpam-5098	96	19	}	}	PUNCT
ejpam-5098	96	20	and	and	CCONJ
ejpam-5098	96	21	discrete	discrete	VERB
ejpam-5098	96	22	adl	adl	PROPN
ejpam-5098	96	23	d	d	NOUN
ejpam-5098	96	24	=	=	PUNCT
ejpam-5098	96	25	{	{	PUNCT
ejpam-5098	96	26	0′	0′	NUM
ejpam-5098	96	27	,	,	PUNCT
ejpam-5098	96	28	a′	a′	ADJ
ejpam-5098	96	29	}	}	PUNCT
ejpam-5098	96	30	.	.	PUNCT
ejpam-5098	97	1	�	�	PROPN
ejpam-5098	97	2	�	�	PROPN
ejpam-5098	97	3	�	�	PROPN
ejpam-5098	97	4	@	@	ADP
ejpam-5098	97	5	@	@	ADP
ejpam-5098	97	6	@	@	ADP
ejpam-5098	97	7	@	@	ADP
ejpam-5098	97	8	@	@	ADP
ejpam-5098	97	9	@	@	ADP
ejpam-5098	97	10	�	�	PROPN
ejpam-5098	97	11	�	�	PROPN
ejpam-5098	97	12	�	�	PROPN
ejpam-5098	98	1	d	d	PROPN
ejpam-5098	98	2	d	d	PROPN
ejpam-5098	98	3	d	d	PROPN
ejpam-5098	98	4	d	d	PROPN
ejpam-5098	98	5	d	d	PROPN
ejpam-5098	98	6	0	0	PUNCT
ejpam-5098	98	7	a	a	DET
ejpam-5098	98	8	b	b	NOUN
ejpam-5098	98	9	c	c	NOUN
ejpam-5098	98	10	1	1	NUM
ejpam-5098	98	11	clearly	clearly	ADV
ejpam-5098	98	12	,	,	PUNCT
ejpam-5098	98	13	r	r	NOUN
ejpam-5098	98	14	=	=	SYM
ejpam-5098	98	15	d×l	d×l	PROPN
ejpam-5098	98	16	=	=	SYM
ejpam-5098	98	17	{	{	PUNCT
ejpam-5098	98	18	(	(	PUNCT
ejpam-5098	98	19	0′	0′	NUM
ejpam-5098	98	20	,	,	PUNCT
ejpam-5098	98	21	0	0	NUM
ejpam-5098	98	22	)	)	PUNCT
ejpam-5098	98	23	,	,	PUNCT
ejpam-5098	98	24	(	(	PUNCT
ejpam-5098	98	25	0′	0′	NUM
ejpam-5098	98	26	,	,	PUNCT
ejpam-5098	98	27	a	a	PRON
ejpam-5098	98	28	)	)	PUNCT
ejpam-5098	98	29	,	,	PUNCT
ejpam-5098	98	30	(	(	PUNCT
ejpam-5098	98	31	0′	0′	NUM
ejpam-5098	98	32	,	,	PUNCT
ejpam-5098	98	33	b	b	NOUN
ejpam-5098	98	34	)	)	PUNCT
ejpam-5098	98	35	,	,	PUNCT
ejpam-5098	98	36	(	(	PUNCT
ejpam-5098	98	37	0′	0′	NUM
ejpam-5098	98	38	,	,	PUNCT
ejpam-5098	98	39	c	c	NOUN
ejpam-5098	98	40	)	)	PUNCT
ejpam-5098	98	41	,	,	PUNCT
ejpam-5098	98	42	(	(	PUNCT
ejpam-5098	98	43	0′	0′	NUM
ejpam-5098	98	44	,	,	PUNCT
ejpam-5098	98	45	1	1	NUM
ejpam-5098	98	46	)	)	PUNCT
ejpam-5098	98	47	,	,	PUNCT
ejpam-5098	98	48	(	(	PUNCT
ejpam-5098	98	49	a′	a′	PROPN
ejpam-5098	98	50	,	,	PUNCT
ejpam-5098	98	51	0	0	NUM
ejpam-5098	98	52	)	)	PUNCT
ejpam-5098	98	53	,	,	PUNCT
ejpam-5098	98	54	(	(	PUNCT
ejpam-5098	98	55	a′	a′	PROPN
ejpam-5098	98	56	,	,	PUNCT
ejpam-5098	98	57	a	a	PRON
ejpam-5098	98	58	)	)	PUNCT
ejpam-5098	98	59	,	,	PUNCT
ejpam-5098	98	60	(	(	PUNCT
ejpam-5098	98	61	a′	a′	PROPN
ejpam-5098	98	62	,	,	PUNCT
ejpam-5098	98	63	b	b	NOUN
ejpam-5098	98	64	)	)	PUNCT
ejpam-5098	98	65	,	,	PUNCT
ejpam-5098	98	66	(	(	PUNCT
ejpam-5098	98	67	a′	a′	PROPN
ejpam-5098	98	68	,	,	PUNCT
ejpam-5098	98	69	c	c	NOUN
ejpam-5098	98	70	)	)	PUNCT
ejpam-5098	98	71	,	,	PUNCT
ejpam-5098	98	72	(	(	PUNCT
ejpam-5098	98	73	a′	a′	PROPN
ejpam-5098	98	74	,	,	PUNCT
ejpam-5098	98	75	1	1	NUM
ejpam-5098	98	76	)	)	PUNCT
ejpam-5098	98	77	}	}	PUNCT
ejpam-5098	98	78	is	be	AUX
ejpam-5098	98	79	an	an	DET
ejpam-5098	98	80	adl	adl	NOUN
ejpam-5098	98	81	with	with	ADP
ejpam-5098	98	82	zero	zero	NUM
ejpam-5098	98	83	element	element	NOUN
ejpam-5098	98	84	(	(	PUNCT
ejpam-5098	98	85	0	0	NUM
ejpam-5098	98	86	,	,	PUNCT
ejpam-5098	98	87	0′	0′	NUM
ejpam-5098	98	88	)	)	PUNCT
ejpam-5098	98	89	.	.	PUNCT
ejpam-5098	99	1	consider	consider	VERB
ejpam-5098	99	2	the	the	DET
ejpam-5098	99	3	minimal	minimal	ADJ
ejpam-5098	99	4	prime	prime	ADJ
ejpam-5098	99	5	ideal	ideal	NOUN
ejpam-5098	99	6	p	p	NOUN
ejpam-5098	99	7	=	=	X
ejpam-5098	99	8	{	{	PUNCT
ejpam-5098	99	9	(	(	PUNCT
ejpam-5098	99	10	0′	0′	NUM
ejpam-5098	99	11	,	,	PUNCT
ejpam-5098	99	12	0	0	NUM
ejpam-5098	99	13	)	)	PUNCT
ejpam-5098	99	14	,	,	PUNCT
ejpam-5098	99	15	(	(	PUNCT
ejpam-5098	99	16	0′	0′	NUM
ejpam-5098	99	17	,	,	PUNCT
ejpam-5098	99	18	a	a	PRON
ejpam-5098	99	19	)	)	PUNCT
ejpam-5098	99	20	}	}	PUNCT
ejpam-5098	99	21	.	.	PUNCT
ejpam-5098	100	1	clearly	clearly	ADV
ejpam-5098	100	2	,	,	PUNCT
ejpam-5098	100	3	(	(	PUNCT
ejpam-5098	100	4	0′	0′	NUM
ejpam-5098	100	5	,	,	PUNCT
ejpam-5098	100	6	a)∗	a)∗	PROPN
ejpam-5098	100	7	∨	∨	NUM
ejpam-5098	100	8	p	p	X
ejpam-5098	100	9	̸=	̸=	PROPN
ejpam-5098	100	10	r.	r.	NOUN
ejpam-5098	100	11	that	that	PRON
ejpam-5098	100	12	implies	imply	VERB
ejpam-5098	100	13	(	(	PUNCT
ejpam-5098	100	14	0′	0′	NUM
ejpam-5098	100	15	,	,	PUNCT
ejpam-5098	100	16	a	a	PRON
ejpam-5098	100	17	)	)	PUNCT
ejpam-5098	100	18	/∈	/∈	PUNCT
ejpam-5098	101	1	p	p	PROPN
ejpam-5098	101	2	σ	σ	PROPN
ejpam-5098	101	3	.	.	PUNCT
ejpam-5098	102	1	hence	hence	ADV
ejpam-5098	102	2	,	,	PUNCT
ejpam-5098	102	3	p	p	PRON
ejpam-5098	102	4	is	be	AUX
ejpam-5098	102	5	not	not	PART
ejpam-5098	102	6	a	a	DET
ejpam-5098	102	7	σ	σ	NOUN
ejpam-5098	102	8	-	-	PUNCT
ejpam-5098	102	9	ideal	ideal	NOUN
ejpam-5098	102	10	of	of	ADP
ejpam-5098	102	11	r.	r.	PROPN
ejpam-5098	102	12	theorem	theorem	VERB
ejpam-5098	102	13	6	6	NUM
ejpam-5098	102	14	.	.	PUNCT
ejpam-5098	103	1	in	in	ADP
ejpam-5098	103	2	an	an	DET
ejpam-5098	103	3	adl	adl	NOUN
ejpam-5098	103	4	r	r	NOUN
ejpam-5098	103	5	with	with	ADP
ejpam-5098	103	6	maximal	maximal	ADJ
ejpam-5098	103	7	elements	element	NOUN
ejpam-5098	103	8	,	,	PUNCT
ejpam-5098	103	9	the	the	DET
ejpam-5098	103	10	following	follow	VERB
ejpam-5098	103	11	conditions	condition	NOUN
ejpam-5098	103	12	are	be	AUX
ejpam-5098	103	13	equivalent	equivalent	ADJ
ejpam-5098	103	14	:	:	PUNCT
ejpam-5098	103	15	(	(	PUNCT
ejpam-5098	103	16	1	1	X
ejpam-5098	103	17	)	)	PUNCT
ejpam-5098	103	18	r	r	NOUN
ejpam-5098	103	19	is	be	AUX
ejpam-5098	103	20	a	a	DET
ejpam-5098	103	21	generalized	generalized	ADJ
ejpam-5098	103	22	stone	stone	NOUN
ejpam-5098	103	23	adl	adl	NOUN
ejpam-5098	103	24	(	(	PUNCT
ejpam-5098	103	25	2	2	NUM
ejpam-5098	103	26	)	)	PUNCT
ejpam-5098	103	27	every	every	DET
ejpam-5098	103	28	α	α	NOUN
ejpam-5098	103	29	-	-	NOUN
ejpam-5098	103	30	ideal	ideal	NOUN
ejpam-5098	103	31	is	be	AUX
ejpam-5098	103	32	a	a	DET
ejpam-5098	103	33	σ	σ	NOUN
ejpam-5098	103	34	-	-	PUNCT
ejpam-5098	103	35	ideal	ideal	NOUN
ejpam-5098	103	36	(	(	PUNCT
ejpam-5098	103	37	3	3	NUM
ejpam-5098	103	38	)	)	PUNCT
ejpam-5098	103	39	every	every	DET
ejpam-5098	103	40	prime	prime	ADJ
ejpam-5098	103	41	α	α	NOUN
ejpam-5098	103	42	-	-	NOUN
ejpam-5098	103	43	ideal	ideal	NOUN
ejpam-5098	103	44	is	be	AUX
ejpam-5098	103	45	a	a	DET
ejpam-5098	103	46	σ	σ	NOUN
ejpam-5098	103	47	-	-	PUNCT
ejpam-5098	103	48	ideal	ideal	NOUN
ejpam-5098	103	49	(	(	PUNCT
ejpam-5098	103	50	4	4	NUM
ejpam-5098	103	51	)	)	PUNCT
ejpam-5098	103	52	every	every	DET
ejpam-5098	103	53	minimal	minimal	ADJ
ejpam-5098	103	54	prime	prime	ADJ
ejpam-5098	103	55	ideal	ideal	NOUN
ejpam-5098	103	56	is	be	AUX
ejpam-5098	103	57	a	a	DET
ejpam-5098	103	58	σ	σ	NOUN
ejpam-5098	103	59	-	-	PUNCT
ejpam-5098	103	60	ideal	ideal	NOUN
ejpam-5098	103	61	.	.	PUNCT
ejpam-5098	104	1	r.	r.	PROPN
ejpam-5098	104	2	noorbhasha	noorbhasha	PROPN
ejpam-5098	104	3	,	,	PUNCT
ejpam-5098	104	4	r.	r.	PROPN
ejpam-5098	104	5	bandaru	bandaru	PROPN
ejpam-5098	104	6	,	,	PUNCT
ejpam-5098	104	7	a.	a.	NOUN
ejpam-5098	104	8	iampan	iampan	PROPN
ejpam-5098	104	9	/	/	SYM
ejpam-5098	104	10	eur	eur	PROPN
ejpam-5098	104	11	.	.	PUNCT
ejpam-5098	105	1	j.	j.	PROPN
ejpam-5098	105	2	pure	pure	PROPN
ejpam-5098	105	3	appl	appl	PROPN
ejpam-5098	105	4	.	.	PROPN
ejpam-5098	105	5	math	math	PROPN
ejpam-5098	105	6	,	,	PUNCT
ejpam-5098	105	7	17	17	NUM
ejpam-5098	105	8	(	(	PUNCT
ejpam-5098	105	9	2	2	NUM
ejpam-5098	105	10	)	)	PUNCT
ejpam-5098	105	11	(	(	PUNCT
ejpam-5098	105	12	2024	2024	NUM
ejpam-5098	105	13	)	)	PUNCT
ejpam-5098	105	14	,	,	PUNCT
ejpam-5098	105	15	1094	1094	NUM
ejpam-5098	105	16	-	-	SYM
ejpam-5098	105	17	1112	1112	NUM
ejpam-5098	105	18	1099	1099	NUM
ejpam-5098	105	19	proof	proof	NOUN
ejpam-5098	105	20	.	.	PUNCT
ejpam-5098	106	1	(	(	PUNCT
ejpam-5098	106	2	1	1	X
ejpam-5098	106	3	)	)	PUNCT
ejpam-5098	106	4	⇒	⇒	NOUN
ejpam-5098	106	5	(	(	PUNCT
ejpam-5098	106	6	2	2	NUM
ejpam-5098	106	7	):	):	PUNCT
ejpam-5098	106	8	assume	assume	VERB
ejpam-5098	106	9	(	(	PUNCT
ejpam-5098	106	10	1	1	NUM
ejpam-5098	106	11	)	)	PUNCT
ejpam-5098	106	12	.	.	PUNCT
ejpam-5098	107	1	let	let	VERB
ejpam-5098	107	2	k	k	PRON
ejpam-5098	107	3	be	be	AUX
ejpam-5098	107	4	an	an	DET
ejpam-5098	107	5	α	α	NOUN
ejpam-5098	107	6	-	-	NOUN
ejpam-5098	107	7	ideal	ideal	NOUN
ejpam-5098	107	8	of	of	ADP
ejpam-5098	107	9	an	an	DET
ejpam-5098	107	10	adl	adl	PROPN
ejpam-5098	107	11	r.	r.	PROPN
ejpam-5098	107	12	then	then	ADV
ejpam-5098	107	13	(	(	PUNCT
ejpam-5098	107	14	a)∗∗	a)∗∗	X
ejpam-5098	107	15	⊆	⊆	NUM
ejpam-5098	107	16	k	k	NOUN
ejpam-5098	107	17	for	for	ADP
ejpam-5098	107	18	all	all	DET
ejpam-5098	107	19	a	a	DET
ejpam-5098	107	20	∈	∈	PROPN
ejpam-5098	107	21	k.	k.	NOUN
ejpam-5098	107	22	since	since	SCONJ
ejpam-5098	107	23	r	r	NOUN
ejpam-5098	107	24	is	be	AUX
ejpam-5098	107	25	a	a	DET
ejpam-5098	107	26	generalized	generalized	ADJ
ejpam-5098	107	27	stone	stone	NOUN
ejpam-5098	107	28	adl	adl	NOUN
ejpam-5098	107	29	,	,	PUNCT
ejpam-5098	107	30	we	we	PRON
ejpam-5098	107	31	have	have	VERB
ejpam-5098	107	32	(	(	PUNCT
ejpam-5098	107	33	a)∗	a)∗	PROPN
ejpam-5098	107	34	∨	∨	NUM
ejpam-5098	107	35	(	(	PUNCT
ejpam-5098	107	36	a)∗∗	a)∗∗	X
ejpam-5098	107	37	=	=	SYM
ejpam-5098	107	38	r	r	NOUN
ejpam-5098	107	39	and	and	CCONJ
ejpam-5098	107	40	hence	hence	ADV
ejpam-5098	107	41	(	(	PUNCT
ejpam-5098	107	42	a)∗	a)∗	PROPN
ejpam-5098	107	43	∨k	∨k	NOUN
ejpam-5098	108	1	=	=	SYM
ejpam-5098	108	2	r.	r.	NOUN
ejpam-5098	108	3	that	that	PRON
ejpam-5098	108	4	implies	imply	VERB
ejpam-5098	108	5	a	a	DET
ejpam-5098	108	6	∈	∈	NOUN
ejpam-5098	108	7	kσ	kσ	NOUN
ejpam-5098	108	8	for	for	ADP
ejpam-5098	108	9	all	all	DET
ejpam-5098	108	10	a	a	DET
ejpam-5098	108	11	∈	∈	PROPN
ejpam-5098	108	12	k.	k.	NOUN
ejpam-5098	108	13	therefore	therefore	ADV
ejpam-5098	108	14	,	,	PUNCT
ejpam-5098	108	15	k	k	PROPN
ejpam-5098	108	16	⊆	⊆	NUM
ejpam-5098	108	17	kσ	kσ	PROPN
ejpam-5098	108	18	.	.	PUNCT
ejpam-5098	108	19	clearly	clearly	ADV
ejpam-5098	108	20	,	,	PUNCT
ejpam-5098	108	21	we	we	PRON
ejpam-5098	108	22	have	have	VERB
ejpam-5098	108	23	kσ	kσ	PROPN
ejpam-5098	108	24	⊆	⊆	NUM
ejpam-5098	108	25	k.	k.	PROPN
ejpam-5098	108	26	thus	thus	ADV
ejpam-5098	108	27	,	,	PUNCT
ejpam-5098	108	28	kσ	kσ	PROPN
ejpam-5098	108	29	=	=	PROPN
ejpam-5098	108	30	k.	k.	PROPN
ejpam-5098	108	31	(	(	PUNCT
ejpam-5098	108	32	2	2	NUM
ejpam-5098	108	33	)	)	PUNCT
ejpam-5098	108	34	⇒	⇒	NOUN
ejpam-5098	108	35	(	(	PUNCT
ejpam-5098	108	36	3	3	NUM
ejpam-5098	108	37	):	):	PUNCT
ejpam-5098	108	38	it	it	PRON
ejpam-5098	108	39	is	be	AUX
ejpam-5098	108	40	obvious	obvious	ADJ
ejpam-5098	108	41	.	.	PUNCT
ejpam-5098	109	1	(	(	PUNCT
ejpam-5098	109	2	3	3	X
ejpam-5098	109	3	)	)	PUNCT
ejpam-5098	109	4	⇒	⇒	NOUN
ejpam-5098	109	5	(	(	PUNCT
ejpam-5098	109	6	4	4	NUM
ejpam-5098	109	7	):	):	PUNCT
ejpam-5098	109	8	assume	assume	VERB
ejpam-5098	109	9	(	(	PUNCT
ejpam-5098	109	10	3	3	NUM
ejpam-5098	109	11	)	)	PUNCT
ejpam-5098	109	12	.	.	PUNCT
ejpam-5098	110	1	clearly	clearly	ADV
ejpam-5098	110	2	,	,	PUNCT
ejpam-5098	110	3	we	we	PRON
ejpam-5098	110	4	have	have	VERB
ejpam-5098	110	5	that	that	SCONJ
ejpam-5098	110	6	every	every	DET
ejpam-5098	110	7	minimal	minimal	ADJ
ejpam-5098	110	8	prime	prime	ADJ
ejpam-5098	110	9	ideal	ideal	NOUN
ejpam-5098	110	10	is	be	AUX
ejpam-5098	110	11	a	a	DET
ejpam-5098	110	12	prime	prime	ADJ
ejpam-5098	110	13	α	α	NOUN
ejpam-5098	110	14	-	-	NOUN
ejpam-5098	110	15	ideal	ideal	NOUN
ejpam-5098	110	16	.	.	PUNCT
ejpam-5098	111	1	by	by	ADP
ejpam-5098	111	2	our	our	PRON
ejpam-5098	111	3	assumption	assumption	NOUN
ejpam-5098	111	4	,	,	PUNCT
ejpam-5098	111	5	we	we	PRON
ejpam-5098	111	6	get	get	VERB
ejpam-5098	111	7	that	that	SCONJ
ejpam-5098	111	8	every	every	DET
ejpam-5098	111	9	minimal	minimal	ADJ
ejpam-5098	111	10	prime	prime	ADJ
ejpam-5098	111	11	ideal	ideal	NOUN
ejpam-5098	111	12	is	be	AUX
ejpam-5098	111	13	a	a	DET
ejpam-5098	111	14	σ	σ	NOUN
ejpam-5098	111	15	-	-	PUNCT
ejpam-5098	111	16	ideal	ideal	NOUN
ejpam-5098	111	17	.	.	PUNCT
ejpam-5098	112	1	(	(	PUNCT
ejpam-5098	112	2	4	4	X
ejpam-5098	112	3	)	)	PUNCT
ejpam-5098	112	4	⇒	⇒	NOUN
ejpam-5098	112	5	(	(	PUNCT
ejpam-5098	112	6	1	1	NUM
ejpam-5098	112	7	):	):	PUNCT
ejpam-5098	112	8	assume	assume	VERB
ejpam-5098	112	9	(	(	PUNCT
ejpam-5098	112	10	4	4	NUM
ejpam-5098	112	11	)	)	PUNCT
ejpam-5098	112	12	.	.	PUNCT
ejpam-5098	113	1	let	let	VERB
ejpam-5098	113	2	a	a	DET
ejpam-5098	113	3	∈	∈	PROPN
ejpam-5098	113	4	r.	r.	NOUN
ejpam-5098	113	5	we	we	PRON
ejpam-5098	113	6	prove	prove	VERB
ejpam-5098	113	7	that	that	SCONJ
ejpam-5098	113	8	(	(	PUNCT
ejpam-5098	113	9	a)∗	a)∗	PROPN
ejpam-5098	113	10	∨	∨	NUM
ejpam-5098	113	11	(	(	PUNCT
ejpam-5098	113	12	a)∗∗	a)∗∗	X
ejpam-5098	113	13	=	=	SYM
ejpam-5098	113	14	r.	r.	PROPN
ejpam-5098	113	15	suppose	suppose	VERB
ejpam-5098	113	16	(	(	PUNCT
ejpam-5098	113	17	a)∗∨	a)∗∨	NOUN
ejpam-5098	113	18	(	(	PUNCT
ejpam-5098	113	19	a)∗∗	a)∗∗	NUM
ejpam-5098	113	20	̸=	̸=	PROPN
ejpam-5098	113	21	r.	r.	PROPN
ejpam-5098	113	22	then	then	ADV
ejpam-5098	113	23	there	there	PRON
ejpam-5098	113	24	exists	exist	VERB
ejpam-5098	113	25	a	a	DET
ejpam-5098	113	26	prime	prime	ADJ
ejpam-5098	113	27	filter	filter	NOUN
ejpam-5098	113	28	q	q	NOUN
ejpam-5098	113	29	such	such	ADJ
ejpam-5098	113	30	that	that	SCONJ
ejpam-5098	113	31	(	(	PUNCT
ejpam-5098	113	32	(	(	PUNCT
ejpam-5098	113	33	a)∗∨	a)∗∨	NOUN
ejpam-5098	113	34	(	(	PUNCT
ejpam-5098	113	35	a)∗∗)∩q	a)∗∗)∩q	NOUN
ejpam-5098	113	36	=	=	NOUN
ejpam-5098	113	37	∅.	∅.	NOUN
ejpam-5098	113	38	since	since	SCONJ
ejpam-5098	113	39	q	q	NOUN
ejpam-5098	113	40	is	be	AUX
ejpam-5098	113	41	a	a	DET
ejpam-5098	113	42	proper	proper	ADJ
ejpam-5098	113	43	filter	filter	NOUN
ejpam-5098	113	44	,	,	PUNCT
ejpam-5098	113	45	there	there	PRON
ejpam-5098	113	46	exists	exist	VERB
ejpam-5098	113	47	a	a	DET
ejpam-5098	113	48	maximal	maximal	ADJ
ejpam-5098	113	49	filter	filter	NOUN
ejpam-5098	113	50	n	n	PRON
ejpam-5098	113	51	such	such	ADJ
ejpam-5098	113	52	that	that	DET
ejpam-5098	113	53	q	q	PROPN
ejpam-5098	113	54	⊆	⊆	NUM
ejpam-5098	113	55	n	n	NOUN
ejpam-5098	113	56	.	.	PUNCT
ejpam-5098	114	1	clearly	clearly	ADV
ejpam-5098	114	2	,	,	PUNCT
ejpam-5098	114	3	r	r	NOUN
ejpam-5098	114	4	\n	\n	PROPN
ejpam-5098	114	5	is	be	AUX
ejpam-5098	114	6	a	a	DET
ejpam-5098	114	7	minimal	minimal	ADJ
ejpam-5098	114	8	prime	prime	ADJ
ejpam-5098	114	9	ideal	ideal	NOUN
ejpam-5098	114	10	.	.	PUNCT
ejpam-5098	115	1	by	by	ADP
ejpam-5098	115	2	our	our	PRON
ejpam-5098	115	3	assumption	assumption	NOUN
ejpam-5098	115	4	,	,	PUNCT
ejpam-5098	115	5	we	we	PRON
ejpam-5098	115	6	get	get	VERB
ejpam-5098	115	7	r	r	NOUN
ejpam-5098	115	8	\n	\n	NUM
ejpam-5098	115	9	is	be	AUX
ejpam-5098	115	10	a	a	DET
ejpam-5098	115	11	σ	σ	NOUN
ejpam-5098	115	12	-	-	PUNCT
ejpam-5098	115	13	ideal	ideal	NOUN
ejpam-5098	115	14	of	of	ADP
ejpam-5098	115	15	r.	r.	PROPN
ejpam-5098	115	16	if	if	SCONJ
ejpam-5098	115	17	a	a	DET
ejpam-5098	115	18	∈	∈	PROPN
ejpam-5098	115	19	n	n	X
ejpam-5098	115	20	,	,	PUNCT
ejpam-5098	115	21	there	there	PRON
ejpam-5098	115	22	exists	exist	VERB
ejpam-5098	115	23	an	an	DET
ejpam-5098	115	24	element	element	NOUN
ejpam-5098	115	25	b	b	PROPN
ejpam-5098	115	26	/∈	/∈	PUNCT
ejpam-5098	116	1	n	n	CCONJ
ejpam-5098	116	2	such	such	ADJ
ejpam-5098	116	3	that	that	SCONJ
ejpam-5098	116	4	a	a	DET
ejpam-5098	116	5	∨	∨	PROPN
ejpam-5098	116	6	b	b	PROPN
ejpam-5098	116	7	is	be	AUX
ejpam-5098	116	8	a	a	DET
ejpam-5098	116	9	maximal	maximal	ADJ
ejpam-5098	116	10	element	element	NOUN
ejpam-5098	116	11	of	of	ADP
ejpam-5098	116	12	r.	r.	PROPN
ejpam-5098	116	13	since	since	SCONJ
ejpam-5098	116	14	a	a	DET
ejpam-5098	116	15	∨	∨	NUM
ejpam-5098	116	16	b	b	X
ejpam-5098	116	17	∈	∈	PROPN
ejpam-5098	116	18	q	q	NOUN
ejpam-5098	116	19	⊆	⊆	NUM
ejpam-5098	116	20	n	n	NOUN
ejpam-5098	116	21	and	and	CCONJ
ejpam-5098	116	22	b	b	PROPN
ejpam-5098	116	23	/∈	/∈	PROPN
ejpam-5098	117	1	n	n	CCONJ
ejpam-5098	117	2	,	,	PUNCT
ejpam-5098	117	3	we	we	PRON
ejpam-5098	117	4	get	get	VERB
ejpam-5098	117	5	a	a	DET
ejpam-5098	117	6	∈	∈	NOUN
ejpam-5098	117	7	q.	q.	NOUN
ejpam-5098	117	8	since	since	SCONJ
ejpam-5098	117	9	a	a	DET
ejpam-5098	117	10	∈	∈	PROPN
ejpam-5098	117	11	(	(	PUNCT
ejpam-5098	117	12	a)∗∗	a)∗∗	X
ejpam-5098	117	13	⊆	⊆	NUM
ejpam-5098	117	14	(	(	PUNCT
ejpam-5098	117	15	a)∗	a)∗	PROPN
ejpam-5098	117	16	∨	∨	NUM
ejpam-5098	117	17	(	(	PUNCT
ejpam-5098	117	18	a)∗∗	a)∗∗	NUM
ejpam-5098	117	19	,	,	PUNCT
ejpam-5098	117	20	we	we	PRON
ejpam-5098	117	21	get	get	VERB
ejpam-5098	117	22	(	(	PUNCT
ejpam-5098	117	23	(	(	PUNCT
ejpam-5098	117	24	a)∗	a)∗	PROPN
ejpam-5098	117	25	∨	∨	NUM
ejpam-5098	117	26	(	(	PUNCT
ejpam-5098	117	27	a)∗∗	a)∗∗	NOUN
ejpam-5098	117	28	)	)	PUNCT
ejpam-5098	117	29	∩q	∩q	PROPN
ejpam-5098	118	1	̸=	̸=	NOUN
ejpam-5098	118	2	∅	∅	NOUN
ejpam-5098	118	3	,	,	PUNCT
ejpam-5098	118	4	it	it	PRON
ejpam-5098	118	5	gives	give	VERB
ejpam-5098	118	6	a	a	DET
ejpam-5098	118	7	contradiction	contradiction	NOUN
ejpam-5098	118	8	.	.	PUNCT
ejpam-5098	119	1	therefore	therefore	ADV
ejpam-5098	119	2	,	,	PUNCT
ejpam-5098	119	3	a	a	DET
ejpam-5098	119	4	/∈	/∈	NOUN
ejpam-5098	119	5	n	n	NOUN
ejpam-5098	119	6	and	and	CCONJ
ejpam-5098	119	7	hence	hence	ADV
ejpam-5098	119	8	a	a	DET
ejpam-5098	119	9	∈	∈	NOUN
ejpam-5098	119	10	r	r	NOUN
ejpam-5098	119	11	\	\	NOUN
ejpam-5098	119	12	n	n	NOUN
ejpam-5098	119	13	=	=	SYM
ejpam-5098	119	14	(	(	PUNCT
ejpam-5098	119	15	r	r	NOUN
ejpam-5098	119	16	\	\	NOUN
ejpam-5098	119	17	n)σ	n)σ	NOUN
ejpam-5098	119	18	.	.	PUNCT
ejpam-5098	120	1	that	that	PRON
ejpam-5098	120	2	implies	imply	VERB
ejpam-5098	120	3	(	(	PUNCT
ejpam-5098	120	4	a)∗	a)∗	PROPN
ejpam-5098	120	5	∨	∨	NUM
ejpam-5098	120	6	(	(	PUNCT
ejpam-5098	120	7	r	r	NOUN
ejpam-5098	120	8	\	\	PROPN
ejpam-5098	120	9	n	n	CCONJ
ejpam-5098	120	10	)	)	PUNCT
ejpam-5098	120	11	=	=	VERB
ejpam-5098	120	12	r.	r.	PROPN
ejpam-5098	120	13	so	so	SCONJ
ejpam-5098	120	14	that	that	SCONJ
ejpam-5098	120	15	there	there	PRON
ejpam-5098	120	16	exist	exist	VERB
ejpam-5098	120	17	x	x	SYM
ejpam-5098	120	18	∈	∈	PROPN
ejpam-5098	120	19	(	(	PUNCT
ejpam-5098	120	20	a)∗	a)∗	PROPN
ejpam-5098	120	21	and	and	CCONJ
ejpam-5098	120	22	y	y	PROPN
ejpam-5098	120	23	∈	∈	PROPN
ejpam-5098	120	24	r	r	NOUN
ejpam-5098	120	25	\	\	NOUN
ejpam-5098	120	26	n	n	CCONJ
ejpam-5098	120	27	such	such	ADJ
ejpam-5098	120	28	that	that	SCONJ
ejpam-5098	120	29	x	x	PROPN
ejpam-5098	120	30	∨	∨	NUM
ejpam-5098	120	31	y	y	PROPN
ejpam-5098	120	32	is	be	AUX
ejpam-5098	120	33	maximal	maximal	ADJ
ejpam-5098	120	34	.	.	PUNCT
ejpam-5098	121	1	since	since	SCONJ
ejpam-5098	121	2	y	y	PROPN
ejpam-5098	121	3	∈	∈	PROPN
ejpam-5098	121	4	r	r	NOUN
ejpam-5098	121	5	\	\	NOUN
ejpam-5098	121	6	n	n	X
ejpam-5098	121	7	,	,	PUNCT
ejpam-5098	121	8	we	we	PRON
ejpam-5098	121	9	get	get	VERB
ejpam-5098	121	10	y	y	PROPN
ejpam-5098	121	11	/∈	/∈	PUNCT
ejpam-5098	121	12	q.	q.	PROPN
ejpam-5098	121	13	since	since	SCONJ
ejpam-5098	121	14	x	x	PROPN
ejpam-5098	121	15	∨	∨	PROPN
ejpam-5098	121	16	y	y	PROPN
ejpam-5098	121	17	is	be	AUX
ejpam-5098	121	18	maximal	maximal	ADJ
ejpam-5098	121	19	,	,	PUNCT
ejpam-5098	121	20	we	we	PRON
ejpam-5098	121	21	get	get	VERB
ejpam-5098	121	22	x	x	X
ejpam-5098	121	23	∈	∈	PROPN
ejpam-5098	121	24	q.	q.	NOUN
ejpam-5098	121	25	since	since	SCONJ
ejpam-5098	121	26	x	x	PROPN
ejpam-5098	121	27	∈	∈	PROPN
ejpam-5098	121	28	(	(	PUNCT
ejpam-5098	121	29	a)∗	a)∗	PROPN
ejpam-5098	121	30	⊆	⊆	NUM
ejpam-5098	121	31	(	(	PUNCT
ejpam-5098	121	32	a)∗	a)∗	PROPN
ejpam-5098	121	33	∨	∨	NUM
ejpam-5098	121	34	(	(	PUNCT
ejpam-5098	121	35	a)∗∗	a)∗∗	NUM
ejpam-5098	121	36	,	,	PUNCT
ejpam-5098	121	37	we	we	PRON
ejpam-5098	121	38	get	get	VERB
ejpam-5098	121	39	(	(	PUNCT
ejpam-5098	121	40	(	(	PUNCT
ejpam-5098	121	41	a)∗	a)∗	PROPN
ejpam-5098	121	42	∨	∨	NUM
ejpam-5098	121	43	(	(	PUNCT
ejpam-5098	121	44	a)∗∗	a)∗∗	NOUN
ejpam-5098	121	45	)	)	PUNCT
ejpam-5098	121	46	∩	∩	NOUN
ejpam-5098	121	47	q	q	PROPN
ejpam-5098	121	48	̸=	̸=	PROPN
ejpam-5098	121	49	∅	∅	NOUN
ejpam-5098	121	50	,	,	PUNCT
ejpam-5098	121	51	which	which	PRON
ejpam-5098	121	52	is	be	AUX
ejpam-5098	121	53	a	a	DET
ejpam-5098	121	54	contradiction	contradiction	NOUN
ejpam-5098	121	55	.	.	PUNCT
ejpam-5098	122	1	hence	hence	ADV
ejpam-5098	122	2	,	,	PUNCT
ejpam-5098	122	3	(	(	PUNCT
ejpam-5098	122	4	a)∗	a)∗	PROPN
ejpam-5098	122	5	∨	∨	NUM
ejpam-5098	122	6	(	(	PUNCT
ejpam-5098	122	7	a)∗∗	a)∗∗	X
ejpam-5098	122	8	=	=	PUNCT
ejpam-5098	122	9	r.	r.	NOUN
ejpam-5098	122	10	we	we	PRON
ejpam-5098	122	11	denote	denote	VERB
ejpam-5098	122	12	specr	specr	NOUN
ejpam-5098	122	13	and	and	CCONJ
ejpam-5098	122	14	maxr	maxr	NOUN
ejpam-5098	122	15	as	as	ADP
ejpam-5098	122	16	the	the	DET
ejpam-5098	122	17	sets	set	NOUN
ejpam-5098	122	18	of	of	ADP
ejpam-5098	122	19	all	all	DET
ejpam-5098	122	20	prime	prime	ADJ
ejpam-5098	122	21	ideals	ideal	NOUN
ejpam-5098	122	22	and	and	CCONJ
ejpam-5098	122	23	maximal	maximal	ADJ
ejpam-5098	122	24	ideals	ideal	NOUN
ejpam-5098	122	25	of	of	ADP
ejpam-5098	122	26	r	r	NOUN
ejpam-5098	122	27	,	,	PUNCT
ejpam-5098	122	28	respectively	respectively	ADV
ejpam-5098	122	29	.	.	PUNCT
ejpam-5098	123	1	for	for	ADP
ejpam-5098	123	2	any	any	DET
ejpam-5098	123	3	n	n	PRON
ejpam-5098	123	4	∈	∈	PROPN
ejpam-5098	123	5	maxr	maxr	NOUN
ejpam-5098	123	6	,	,	PUNCT
ejpam-5098	123	7	define	define	VERB
ejpam-5098	123	8	no	no	PRON
ejpam-5098	123	9	=	=	PUNCT
ejpam-5098	123	10	{	{	PUNCT
ejpam-5098	123	11	a	a	PRON
ejpam-5098	123	12	∈	∈	PROPN
ejpam-5098	123	13	r	r	NOUN
ejpam-5098	123	14	|	|	NOUN
ejpam-5098	124	1	(	(	PUNCT
ejpam-5098	124	2	a)∗	a)∗	PROPN
ejpam-5098	124	3	⊈	⊈	PROPN
ejpam-5098	124	4	n	n	CCONJ
ejpam-5098	124	5	}	}	PUNCT
ejpam-5098	124	6	.	.	PUNCT
ejpam-5098	125	1	proposition	proposition	NOUN
ejpam-5098	125	2	2	2	NUM
ejpam-5098	125	3	.	.	X
ejpam-5098	126	1	for	for	ADP
ejpam-5098	126	2	any	any	DET
ejpam-5098	126	3	n	n	PRON
ejpam-5098	126	4	∈	∈	PROPN
ejpam-5098	126	5	maxr	maxr	NOUN
ejpam-5098	126	6	,	,	PUNCT
ejpam-5098	126	7	no	no	PRON
ejpam-5098	126	8	is	be	AUX
ejpam-5098	126	9	an	an	DET
ejpam-5098	126	10	ideal	ideal	NOUN
ejpam-5098	126	11	of	of	ADP
ejpam-5098	126	12	r	r	NOUN
ejpam-5098	126	13	contained	contain	VERB
ejpam-5098	126	14	in	in	ADP
ejpam-5098	126	15	n	n	PROPN
ejpam-5098	126	16	.	.	PUNCT
ejpam-5098	127	1	proof	proof	NOUN
ejpam-5098	127	2	.	.	PUNCT
ejpam-5098	128	1	clearly	clearly	ADV
ejpam-5098	128	2	,	,	PUNCT
ejpam-5098	128	3	(	(	PUNCT
ejpam-5098	128	4	0)∗	0)∗	PUNCT
ejpam-5098	128	5	=	=	PUNCT
ejpam-5098	128	6	r	r	NOUN
ejpam-5098	128	7	⊈	⊈	PROPN
ejpam-5098	128	8	n	n	CCONJ
ejpam-5098	128	9	and	and	CCONJ
ejpam-5098	128	10	hence	hence	ADV
ejpam-5098	128	11	no	no	PRON
ejpam-5098	128	12	is	be	AUX
ejpam-5098	128	13	non	non	ADJ
ejpam-5098	128	14	-	-	ADJ
ejpam-5098	128	15	empty	empty	ADJ
ejpam-5098	128	16	.	.	PUNCT
ejpam-5098	129	1	let	let	VERB
ejpam-5098	129	2	a	a	PRON
ejpam-5098	129	3	,	,	PUNCT
ejpam-5098	129	4	b	b	X
ejpam-5098	129	5	∈	∈	PROPN
ejpam-5098	130	1	no	no	INTJ
ejpam-5098	130	2	.	.	PUNCT
ejpam-5098	131	1	then	then	ADV
ejpam-5098	131	2	(	(	PUNCT
ejpam-5098	131	3	a)∗	a)∗	PROPN
ejpam-5098	131	4	⊈	⊈	PROPN
ejpam-5098	131	5	n	n	CCONJ
ejpam-5098	131	6	and	and	CCONJ
ejpam-5098	131	7	(	(	PUNCT
ejpam-5098	131	8	b)∗	b)∗	PROPN
ejpam-5098	131	9	⊈	⊈	PROPN
ejpam-5098	131	10	n	n	CCONJ
ejpam-5098	131	11	.	.	PUNCT
ejpam-5098	132	1	that	that	PRON
ejpam-5098	132	2	implies	imply	VERB
ejpam-5098	132	3	(	(	PUNCT
ejpam-5098	132	4	a	a	DET
ejpam-5098	132	5	∨	∨	NOUN
ejpam-5098	132	6	b)∗	b)∗	PROPN
ejpam-5098	132	7	⊆	⊆	NUM
ejpam-5098	132	8	(	(	PUNCT
ejpam-5098	132	9	a)∗	a)∗	PROPN
ejpam-5098	132	10	⊈	⊈	PROPN
ejpam-5098	132	11	n	n	CCONJ
ejpam-5098	132	12	.	.	PUNCT
ejpam-5098	133	1	hence	hence	ADV
ejpam-5098	133	2	,	,	PUNCT
ejpam-5098	133	3	a	a	DET
ejpam-5098	133	4	∨	∨	NUM
ejpam-5098	133	5	b	b	X
ejpam-5098	133	6	∈	∈	PROPN
ejpam-5098	133	7	no	no	INTJ
ejpam-5098	133	8	.	.	PUNCT
ejpam-5098	134	1	let	let	VERB
ejpam-5098	134	2	a	a	DET
ejpam-5098	134	3	∈	∈	ADJ
ejpam-5098	134	4	no	no	INTJ
ejpam-5098	134	5	.	.	PUNCT
ejpam-5098	135	1	then	then	ADV
ejpam-5098	135	2	(	(	PUNCT
ejpam-5098	135	3	a)∗	a)∗	PROPN
ejpam-5098	135	4	⊈	⊈	PROPN
ejpam-5098	135	5	n	n	CCONJ
ejpam-5098	135	6	.	.	PUNCT
ejpam-5098	136	1	let	let	VERB
ejpam-5098	136	2	r	r	NOUN
ejpam-5098	136	3	be	be	AUX
ejpam-5098	136	4	any	any	DET
ejpam-5098	136	5	element	element	NOUN
ejpam-5098	136	6	of	of	ADP
ejpam-5098	136	7	r.	r.	PROPN
ejpam-5098	136	8	since	since	SCONJ
ejpam-5098	136	9	r	r	PROPN
ejpam-5098	136	10	∧	∧	PROPN
ejpam-5098	136	11	a	a	DET
ejpam-5098	136	12	≤	≤	NOUN
ejpam-5098	136	13	a	a	X
ejpam-5098	136	14	,	,	PUNCT
ejpam-5098	136	15	we	we	PRON
ejpam-5098	136	16	get	get	VERB
ejpam-5098	136	17	(	(	PUNCT
ejpam-5098	136	18	a)∗	a)∗	PROPN
ejpam-5098	136	19	⊆	⊆	NUM
ejpam-5098	136	20	(	(	PUNCT
ejpam-5098	136	21	r∧	r∧	NOUN
ejpam-5098	136	22	a)∗	a)∗	PROPN
ejpam-5098	137	1	=	=	PRON
ejpam-5098	138	1	(	(	PUNCT
ejpam-5098	138	2	a∧	a∧	NOUN
ejpam-5098	138	3	r)∗.	r)∗.	PROPN
ejpam-5098	138	4	since	since	SCONJ
ejpam-5098	138	5	(	(	PUNCT
ejpam-5098	138	6	a)∗	a)∗	PROPN
ejpam-5098	138	7	⊈	⊈	PROPN
ejpam-5098	138	8	n	n	X
ejpam-5098	138	9	,	,	PUNCT
ejpam-5098	138	10	we	we	PRON
ejpam-5098	138	11	get	get	VERB
ejpam-5098	138	12	(	(	PUNCT
ejpam-5098	138	13	a∧	a∧	NOUN
ejpam-5098	138	14	r)∗	r)∗	PUNCT
ejpam-5098	138	15	⊈	⊈	PROPN
ejpam-5098	138	16	n	n	NOUN
ejpam-5098	138	17	.	.	PUNCT
ejpam-5098	139	1	that	that	PRON
ejpam-5098	139	2	implies	imply	VERB
ejpam-5098	139	3	a∧	a∧	NOUN
ejpam-5098	139	4	r	r	NOUN
ejpam-5098	139	5	∈	∈	PROPN
ejpam-5098	139	6	no	no	INTJ
ejpam-5098	139	7	.	.	PUNCT
ejpam-5098	140	1	therefore	therefore	ADV
ejpam-5098	140	2	,	,	PUNCT
ejpam-5098	140	3	no	no	PRON
ejpam-5098	140	4	is	be	AUX
ejpam-5098	140	5	an	an	DET
ejpam-5098	140	6	ideal	ideal	NOUN
ejpam-5098	140	7	of	of	ADP
ejpam-5098	140	8	r.	r.	PROPN
ejpam-5098	140	9	let	let	VERB
ejpam-5098	140	10	a	a	DET
ejpam-5098	140	11	∈	∈	ADJ
ejpam-5098	141	1	no	no	INTJ
ejpam-5098	141	2	.	.	PUNCT
ejpam-5098	142	1	then	then	ADV
ejpam-5098	142	2	(	(	PUNCT
ejpam-5098	142	3	a)∗	a)∗	PROPN
ejpam-5098	142	4	⊈	⊈	PROPN
ejpam-5098	142	5	n	n	CCONJ
ejpam-5098	142	6	.	.	PUNCT
ejpam-5098	143	1	there	there	PRON
ejpam-5098	143	2	exists	exist	VERB
ejpam-5098	143	3	t	t	PROPN
ejpam-5098	143	4	∈	∈	PROPN
ejpam-5098	143	5	(	(	PUNCT
ejpam-5098	143	6	a)∗	a)∗	PROPN
ejpam-5098	143	7	such	such	ADJ
ejpam-5098	143	8	that	that	DET
ejpam-5098	143	9	t	t	PROPN
ejpam-5098	143	10	/∈	/∈	PUNCT
ejpam-5098	144	1	n	n	INTJ
ejpam-5098	144	2	.	.	PUNCT
ejpam-5098	145	1	since	since	SCONJ
ejpam-5098	145	2	t	t	PROPN
ejpam-5098	145	3	∈	∈	PROPN
ejpam-5098	145	4	(	(	PUNCT
ejpam-5098	145	5	a)∗	a)∗	PROPN
ejpam-5098	145	6	,	,	PUNCT
ejpam-5098	145	7	we	we	PRON
ejpam-5098	145	8	have	have	VERB
ejpam-5098	145	9	a	a	DET
ejpam-5098	145	10	∧	∧	PROPN
ejpam-5098	145	11	t	t	NOUN
ejpam-5098	145	12	=	=	SYM
ejpam-5098	145	13	0	0	X
ejpam-5098	145	14	.	.	PUNCT
ejpam-5098	146	1	we	we	PRON
ejpam-5098	146	2	have	have	VERB
ejpam-5098	146	3	that	that	SCONJ
ejpam-5098	146	4	every	every	DET
ejpam-5098	146	5	maximal	maximal	ADJ
ejpam-5098	146	6	ideal	ideal	NOUN
ejpam-5098	146	7	is	be	AUX
ejpam-5098	146	8	prime	prime	ADJ
ejpam-5098	146	9	and	and	CCONJ
ejpam-5098	146	10	hence	hence	ADV
ejpam-5098	146	11	n	n	PRON
ejpam-5098	146	12	is	be	AUX
ejpam-5098	146	13	prime	prime	ADJ
ejpam-5098	146	14	.	.	PUNCT
ejpam-5098	147	1	since	since	SCONJ
ejpam-5098	147	2	a∧	a∧	NOUN
ejpam-5098	147	3	t	t	PROPN
ejpam-5098	147	4	=	=	SYM
ejpam-5098	147	5	0	0	PUNCT
ejpam-5098	147	6	∈	∈	PROPN
ejpam-5098	147	7	n	n	NOUN
ejpam-5098	147	8	and	and	CCONJ
ejpam-5098	147	9	t	t	PROPN
ejpam-5098	147	10	/∈	/∈	PUNCT
ejpam-5098	148	1	n	n	CCONJ
ejpam-5098	148	2	,	,	PUNCT
ejpam-5098	148	3	we	we	PRON
ejpam-5098	148	4	get	get	VERB
ejpam-5098	148	5	a	a	DET
ejpam-5098	148	6	∈	∈	NOUN
ejpam-5098	148	7	n	n	NOUN
ejpam-5098	148	8	.	.	PUNCT
ejpam-5098	149	1	therefore	therefore	ADV
ejpam-5098	149	2	,	,	PUNCT
ejpam-5098	149	3	no	no	DET
ejpam-5098	149	4	⊆	⊆	NUM
ejpam-5098	149	5	n	n	NOUN
ejpam-5098	149	6	.	.	PUNCT
ejpam-5098	150	1	for	for	ADP
ejpam-5098	150	2	any	any	DET
ejpam-5098	150	3	ideal	ideal	ADJ
ejpam-5098	150	4	k	k	PROPN
ejpam-5098	150	5	of	of	ADP
ejpam-5098	150	6	an	an	DET
ejpam-5098	150	7	adl	adl	PROPN
ejpam-5098	150	8	r	r	NOUN
ejpam-5098	150	9	,	,	PUNCT
ejpam-5098	150	10	define	define	VERB
ejpam-5098	150	11	w(k	w(k	NOUN
ejpam-5098	150	12	)	)	PUNCT
ejpam-5098	151	1	=	=	PRON
ejpam-5098	151	2	{	{	PUNCT
ejpam-5098	151	3	n	n	CCONJ
ejpam-5098	151	4	∈	∈	NOUN
ejpam-5098	151	5	maxr	maxr	NOUN
ejpam-5098	152	1	|	|	ADV
ejpam-5098	152	2	k	k	PROPN
ejpam-5098	152	3	⊆	⊆	NUM
ejpam-5098	152	4	n	n	CCONJ
ejpam-5098	152	5	}	}	PUNCT
ejpam-5098	152	6	.	.	PUNCT
ejpam-5098	153	1	proposition	proposition	NOUN
ejpam-5098	153	2	3	3	X
ejpam-5098	153	3	.	.	PUNCT
ejpam-5098	154	1	let	let	VERB
ejpam-5098	154	2	k	k	PRON
ejpam-5098	154	3	be	be	AUX
ejpam-5098	154	4	an	an	DET
ejpam-5098	154	5	ideal	ideal	NOUN
ejpam-5098	154	6	of	of	ADP
ejpam-5098	154	7	r.	r.	PROPN
ejpam-5098	154	8	then	then	ADV
ejpam-5098	154	9	we	we	PRON
ejpam-5098	154	10	have	have	VERB
ejpam-5098	154	11	the	the	DET
ejpam-5098	154	12	following	following	NOUN
ejpam-5098	154	13	:	:	PUNCT
ejpam-5098	154	14	(	(	PUNCT
ejpam-5098	154	15	1	1	X
ejpam-5098	154	16	)	)	PUNCT
ejpam-5098	154	17	kσ	kσ	PROPN
ejpam-5098	154	18	=	=	PUNCT
ejpam-5098	154	19	⋂	⋂	PROPN
ejpam-5098	154	20	n∈w(k	n∈w(k	PROPN
ejpam-5098	154	21	)	)	PUNCT
ejpam-5098	155	1	no	no	DET
ejpam-5098	155	2	(	(	PUNCT
ejpam-5098	155	3	2	2	NUM
ejpam-5098	155	4	)	)	PUNCT
ejpam-5098	155	5	if	if	SCONJ
ejpam-5098	155	6	k	k	PROPN
ejpam-5098	155	7	∈	∈	PROPN
ejpam-5098	155	8	specr	specr	PROPN
ejpam-5098	155	9	,	,	PUNCT
ejpam-5098	155	10	then	then	ADV
ejpam-5098	155	11	kσ	kσ	PROPN
ejpam-5098	155	12	⊆	⊆	NUM
ejpam-5098	155	13	ko	ko	PROPN
ejpam-5098	155	14	(	(	PUNCT
ejpam-5098	155	15	3	3	NUM
ejpam-5098	155	16	)	)	PUNCT
ejpam-5098	155	17	if	if	SCONJ
ejpam-5098	155	18	k	k	PROPN
ejpam-5098	155	19	∈	∈	PROPN
ejpam-5098	155	20	maxr	maxr	NOUN
ejpam-5098	155	21	,	,	PUNCT
ejpam-5098	155	22	then	then	ADV
ejpam-5098	155	23	kσ	kσ	PROPN
ejpam-5098	155	24	=	=	PUNCT
ejpam-5098	155	25	ko	ko	PROPN
ejpam-5098	155	26	.	.	PUNCT
ejpam-5098	156	1	proof	proof	NOUN
ejpam-5098	156	2	.	.	PUNCT
ejpam-5098	157	1	(	(	PUNCT
ejpam-5098	157	2	1	1	X
ejpam-5098	157	3	)	)	PUNCT
ejpam-5098	157	4	let	let	VERB
ejpam-5098	157	5	a	a	DET
ejpam-5098	157	6	∈	∈	NOUN
ejpam-5098	157	7	kσ	kσ	PROPN
ejpam-5098	157	8	.	.	PUNCT
ejpam-5098	158	1	then	then	ADV
ejpam-5098	158	2	(	(	PUNCT
ejpam-5098	158	3	a)∗	a)∗	PROPN
ejpam-5098	158	4	∨	∨	NUM
ejpam-5098	158	5	k	k	PROPN
ejpam-5098	158	6	=	=	X
ejpam-5098	158	7	r.	r.	PROPN
ejpam-5098	158	8	let	let	VERB
ejpam-5098	158	9	n	n	PRON
ejpam-5098	158	10	∈	∈	PROPN
ejpam-5098	158	11	w(k	w(k	PROPN
ejpam-5098	158	12	)	)	PUNCT
ejpam-5098	158	13	.	.	PUNCT
ejpam-5098	159	1	then	then	ADV
ejpam-5098	159	2	k	k	PROPN
ejpam-5098	159	3	⊆	⊆	NUM
ejpam-5098	159	4	n	n	NOUN
ejpam-5098	159	5	.	.	PUNCT
ejpam-5098	160	1	that	that	PRON
ejpam-5098	160	2	implies	imply	VERB
ejpam-5098	160	3	(	(	PUNCT
ejpam-5098	160	4	a)∗	a)∗	PROPN
ejpam-5098	160	5	∨	∨	NUM
ejpam-5098	160	6	n	n	NOUN
ejpam-5098	160	7	=	=	SYM
ejpam-5098	160	8	r	r	NOUN
ejpam-5098	160	9	and	and	CCONJ
ejpam-5098	160	10	hence	hence	ADV
ejpam-5098	160	11	(	(	PUNCT
ejpam-5098	160	12	a)∗	a)∗	PROPN
ejpam-5098	160	13	⊈	⊈	PROPN
ejpam-5098	160	14	n	n	CCONJ
ejpam-5098	160	15	.	.	PUNCT
ejpam-5098	161	1	that	that	PRON
ejpam-5098	161	2	implies	imply	VERB
ejpam-5098	161	3	a	a	DET
ejpam-5098	161	4	∈	∈	ADJ
ejpam-5098	161	5	no	no	NOUN
ejpam-5098	161	6	for	for	ADP
ejpam-5098	161	7	n	n	PRON
ejpam-5098	161	8	∈	∈	PROPN
ejpam-5098	161	9	maxr	maxr	NOUN
ejpam-5098	161	10	.	.	PUNCT
ejpam-5098	162	1	therefore	therefore	ADV
ejpam-5098	162	2	,	,	PUNCT
ejpam-5098	162	3	kσ	kσ	PROPN
ejpam-5098	162	4	⊆	⊆	NUM
ejpam-5098	162	5	⋂	⋂	PROPN
ejpam-5098	162	6	n∈w(k	n∈w(k	PROPN
ejpam-5098	162	7	)	)	PUNCT
ejpam-5098	162	8	no	no	INTJ
ejpam-5098	162	9	.	.	PUNCT
ejpam-5098	163	1	conversely	conversely	ADV
ejpam-5098	163	2	,	,	PUNCT
ejpam-5098	163	3	let	let	VERB
ejpam-5098	163	4	a	a	DET
ejpam-5098	163	5	∈	∈	NOUN
ejpam-5098	163	6	⋂	⋂	PROPN
ejpam-5098	163	7	n∈w(k	n∈w(k	PROPN
ejpam-5098	163	8	)	)	PUNCT
ejpam-5098	164	1	no	no	INTJ
ejpam-5098	164	2	.	.	PUNCT
ejpam-5098	165	1	then	then	ADV
ejpam-5098	165	2	a	a	DET
ejpam-5098	165	3	∈	∈	PROPN
ejpam-5098	165	4	o(n	o(n	PROPN
ejpam-5098	165	5	)	)	PUNCT
ejpam-5098	165	6	for	for	ADP
ejpam-5098	165	7	all	all	DET
ejpam-5098	165	8	r.	r.	PROPN
ejpam-5098	165	9	noorbhasha	noorbhasha	PROPN
ejpam-5098	165	10	,	,	PUNCT
ejpam-5098	165	11	r.	r.	PROPN
ejpam-5098	165	12	bandaru	bandaru	PROPN
ejpam-5098	165	13	,	,	PUNCT
ejpam-5098	165	14	a.	a.	NOUN
ejpam-5098	165	15	iampan	iampan	PROPN
ejpam-5098	165	16	/	/	SYM
ejpam-5098	165	17	eur	eur	PROPN
ejpam-5098	165	18	.	.	PUNCT
ejpam-5098	166	1	j.	j.	PROPN
ejpam-5098	166	2	pure	pure	PROPN
ejpam-5098	166	3	appl	appl	PROPN
ejpam-5098	166	4	.	.	PROPN
ejpam-5098	166	5	math	math	PROPN
ejpam-5098	166	6	,	,	PUNCT
ejpam-5098	166	7	17	17	NUM
ejpam-5098	166	8	(	(	PUNCT
ejpam-5098	166	9	2	2	NUM
ejpam-5098	166	10	)	)	PUNCT
ejpam-5098	166	11	(	(	PUNCT
ejpam-5098	166	12	2024	2024	NUM
ejpam-5098	166	13	)	)	PUNCT
ejpam-5098	166	14	,	,	PUNCT
ejpam-5098	166	15	1094	1094	NUM
ejpam-5098	166	16	-	-	SYM
ejpam-5098	166	17	1112	1112	NUM
ejpam-5098	166	18	1100	1100	NUM
ejpam-5098	166	19	n	n	PRON
ejpam-5098	166	20	∈	∈	PROPN
ejpam-5098	166	21	w(k	w(k	PROPN
ejpam-5098	166	22	)	)	PUNCT
ejpam-5098	166	23	.	.	PUNCT
ejpam-5098	167	1	that	that	PRON
ejpam-5098	167	2	implies	imply	VERB
ejpam-5098	167	3	(	(	PUNCT
ejpam-5098	167	4	a)∗	a)∗	PROPN
ejpam-5098	167	5	⊈	⊈	PROPN
ejpam-5098	167	6	n	n	PROPN
ejpam-5098	167	7	for	for	ADP
ejpam-5098	167	8	all	all	DET
ejpam-5098	167	9	n	n	PRON
ejpam-5098	167	10	∈	∈	PROPN
ejpam-5098	167	11	maxr	maxr	NOUN
ejpam-5098	167	12	,	,	PUNCT
ejpam-5098	167	13	k	k	PROPN
ejpam-5098	167	14	⊆	⊆	NUM
ejpam-5098	167	15	n	n	X
ejpam-5098	167	16	.	.	PUNCT
ejpam-5098	168	1	suppose	suppose	VERB
ejpam-5098	168	2	(	(	PUNCT
ejpam-5098	168	3	a)∗	a)∗	PROPN
ejpam-5098	168	4	∨k	∨k	ADJ
ejpam-5098	168	5	̸=	̸=	PROPN
ejpam-5098	168	6	r.	r.	PROPN
ejpam-5098	168	7	then	then	ADV
ejpam-5098	168	8	n	n	CCONJ
ejpam-5098	168	9	′	′	NOUN
ejpam-5098	168	10	∈	∈	PROPN
ejpam-5098	168	11	maxr	maxr	NOUN
ejpam-5098	168	12	such	such	ADJ
ejpam-5098	168	13	that	that	SCONJ
ejpam-5098	168	14	(	(	PUNCT
ejpam-5098	168	15	a)∗	a)∗	PROPN
ejpam-5098	168	16	∨	∨	NUM
ejpam-5098	168	17	k	k	PROPN
ejpam-5098	168	18	⊆	⊆	NUM
ejpam-5098	168	19	n	n	PRON
ejpam-5098	168	20	′.	′.	NOUN
ejpam-5098	168	21	that	that	PRON
ejpam-5098	168	22	implies	imply	VERB
ejpam-5098	168	23	(	(	PUNCT
ejpam-5098	168	24	a)∗	a)∗	PROPN
ejpam-5098	168	25	⊆	⊆	NUM
ejpam-5098	168	26	n	n	NUM
ejpam-5098	168	27	′	′	NOUN
ejpam-5098	168	28	and	and	CCONJ
ejpam-5098	168	29	k	k	PROPN
ejpam-5098	168	30	⊆	⊆	NUM
ejpam-5098	168	31	n	n	PRON
ejpam-5098	168	32	′	′	NOUN
ejpam-5098	168	33	,	,	PUNCT
ejpam-5098	168	34	which	which	PRON
ejpam-5098	168	35	is	be	AUX
ejpam-5098	168	36	a	a	DET
ejpam-5098	168	37	contradiction	contradiction	NOUN
ejpam-5098	168	38	.	.	PUNCT
ejpam-5098	169	1	that	that	PRON
ejpam-5098	169	2	implies	imply	VERB
ejpam-5098	169	3	(	(	PUNCT
ejpam-5098	169	4	a)∗	a)∗	PROPN
ejpam-5098	169	5	∨	∨	NUM
ejpam-5098	169	6	k	k	PROPN
ejpam-5098	169	7	=	=	SYM
ejpam-5098	169	8	r.	r.	PROPN
ejpam-5098	169	9	therefore	therefore	ADV
ejpam-5098	169	10	,	,	PUNCT
ejpam-5098	169	11	a	a	DET
ejpam-5098	169	12	∈	∈	NOUN
ejpam-5098	169	13	kσ	kσ	NOUN
ejpam-5098	169	14	and	and	CCONJ
ejpam-5098	169	15	hence⋂	hence⋂	PROPN
ejpam-5098	169	16	n∈w(k	n∈w(k	PROPN
ejpam-5098	169	17	)	)	PUNCT
ejpam-5098	170	1	no	no	DET
ejpam-5098	170	2	⊆	⊆	NUM
ejpam-5098	170	3	kσ	kσ	PROPN
ejpam-5098	170	4	.	.	PUNCT
ejpam-5098	171	1	thus	thus	ADV
ejpam-5098	171	2	,	,	PUNCT
ejpam-5098	171	3	kσ	kσ	PROPN
ejpam-5098	171	4	=	=	PUNCT
ejpam-5098	171	5	⋂	⋂	PROPN
ejpam-5098	171	6	n∈w(k	n∈w(k	PROPN
ejpam-5098	171	7	)	)	PUNCT
ejpam-5098	171	8	no	no	INTJ
ejpam-5098	171	9	.	.	PUNCT
ejpam-5098	172	1	(	(	PUNCT
ejpam-5098	172	2	2	2	X
ejpam-5098	172	3	)	)	PUNCT
ejpam-5098	172	4	let	let	VERB
ejpam-5098	172	5	k	k	PROPN
ejpam-5098	172	6	∈	∈	PROPN
ejpam-5098	172	7	specr	specr	NOUN
ejpam-5098	172	8	and	and	CCONJ
ejpam-5098	172	9	a	a	DET
ejpam-5098	172	10	∈	∈	PROPN
ejpam-5098	172	11	kσ	kσ	PROPN
ejpam-5098	172	12	.	.	PUNCT
ejpam-5098	173	1	then	then	ADV
ejpam-5098	173	2	(	(	PUNCT
ejpam-5098	173	3	a)∗	a)∗	PROPN
ejpam-5098	173	4	∨k	∨k	NOUN
ejpam-5098	173	5	=	=	SYM
ejpam-5098	173	6	r.	r.	NOUN
ejpam-5098	173	7	since	since	SCONJ
ejpam-5098	173	8	k	k	PROPN
ejpam-5098	173	9	is	be	AUX
ejpam-5098	173	10	a	a	DET
ejpam-5098	173	11	proper	proper	ADJ
ejpam-5098	173	12	ideal	ideal	NOUN
ejpam-5098	173	13	of	of	ADP
ejpam-5098	173	14	r	r	NOUN
ejpam-5098	173	15	,	,	PUNCT
ejpam-5098	173	16	we	we	PRON
ejpam-5098	173	17	get	get	VERB
ejpam-5098	173	18	(	(	PUNCT
ejpam-5098	173	19	a)∗	a)∗	PROPN
ejpam-5098	173	20	⊈	⊈	PROPN
ejpam-5098	174	1	k	k	NOUN
ejpam-5098	174	2	and	and	CCONJ
ejpam-5098	174	3	hence	hence	ADV
ejpam-5098	174	4	a	a	DET
ejpam-5098	174	5	∈	∈	PROPN
ejpam-5098	174	6	ko	ko	PROPN
ejpam-5098	174	7	.	.	PUNCT
ejpam-5098	175	1	therefore	therefore	ADV
ejpam-5098	175	2	,	,	PUNCT
ejpam-5098	175	3	kσ	kσ	PROPN
ejpam-5098	175	4	⊆	⊆	NUM
ejpam-5098	175	5	ko	ko	PROPN
ejpam-5098	175	6	.	.	PUNCT
ejpam-5098	176	1	(	(	PUNCT
ejpam-5098	176	2	3	3	X
ejpam-5098	176	3	)	)	PUNCT
ejpam-5098	176	4	let	let	VERB
ejpam-5098	176	5	k	k	PROPN
ejpam-5098	176	6	∈	∈	PROPN
ejpam-5098	176	7	maxr	maxr	NOUN
ejpam-5098	176	8	.	.	PUNCT
ejpam-5098	177	1	clearly	clearly	ADV
ejpam-5098	177	2	,	,	PUNCT
ejpam-5098	177	3	we	we	PRON
ejpam-5098	177	4	have	have	VERB
ejpam-5098	177	5	that	that	SCONJ
ejpam-5098	177	6	k	k	PROPN
ejpam-5098	177	7	is	be	AUX
ejpam-5098	177	8	a	a	DET
ejpam-5098	177	9	prime	prime	ADJ
ejpam-5098	177	10	ideal	ideal	NOUN
ejpam-5098	177	11	of	of	ADP
ejpam-5098	177	12	r.	r.	PROPN
ejpam-5098	177	13	by	by	ADP
ejpam-5098	177	14	(	(	PUNCT
ejpam-5098	177	15	2	2	NUM
ejpam-5098	177	16	)	)	PUNCT
ejpam-5098	177	17	,	,	PUNCT
ejpam-5098	177	18	we	we	PRON
ejpam-5098	177	19	get	get	VERB
ejpam-5098	177	20	kσ	kσ	PROPN
ejpam-5098	177	21	⊆	⊆	NUM
ejpam-5098	177	22	ko	ko	PROPN
ejpam-5098	177	23	.	.	PUNCT
ejpam-5098	178	1	let	let	VERB
ejpam-5098	178	2	a	a	DET
ejpam-5098	178	3	∈	∈	PROPN
ejpam-5098	178	4	ko	ko	PROPN
ejpam-5098	178	5	.	.	PUNCT
ejpam-5098	179	1	then	then	ADV
ejpam-5098	180	1	(	(	PUNCT
ejpam-5098	180	2	a)∗	a)∗	PROPN
ejpam-5098	180	3	⊈	⊈	PROPN
ejpam-5098	180	4	k.	k.	PROPN
ejpam-5098	180	5	that	that	PRON
ejpam-5098	180	6	implies	imply	VERB
ejpam-5098	180	7	(	(	PUNCT
ejpam-5098	180	8	a)∗	a)∗	PROPN
ejpam-5098	180	9	∨k	∨k	NOUN
ejpam-5098	180	10	=	=	SYM
ejpam-5098	180	11	r.	r.	PROPN
ejpam-5098	180	12	therefore	therefore	ADV
ejpam-5098	180	13	,	,	PUNCT
ejpam-5098	180	14	a	a	DET
ejpam-5098	180	15	∈	∈	NOUN
ejpam-5098	180	16	kσ	kσ	NOUN
ejpam-5098	180	17	and	and	CCONJ
ejpam-5098	180	18	hence	hence	ADV
ejpam-5098	180	19	ko	ko	PROPN
ejpam-5098	180	20	⊆	⊆	NUM
ejpam-5098	180	21	kσ	kσ	PROPN
ejpam-5098	180	22	.	.	PUNCT
ejpam-5098	181	1	thus	thus	ADV
ejpam-5098	181	2	,	,	PUNCT
ejpam-5098	181	3	kσ	kσ	PROPN
ejpam-5098	181	4	=	=	PUNCT
ejpam-5098	181	5	ko	ko	PROPN
ejpam-5098	181	6	.	.	PUNCT
ejpam-5098	181	7	theorem	theorem	VERB
ejpam-5098	181	8	7	7	NUM
ejpam-5098	181	9	.	.	PUNCT
ejpam-5098	182	1	in	in	ADP
ejpam-5098	182	2	an	an	DET
ejpam-5098	182	3	adl	adl	NOUN
ejpam-5098	182	4	r	r	NOUN
ejpam-5098	182	5	with	with	ADP
ejpam-5098	182	6	maximal	maximal	ADJ
ejpam-5098	182	7	elements	element	NOUN
ejpam-5098	182	8	,	,	PUNCT
ejpam-5098	182	9	the	the	DET
ejpam-5098	182	10	following	follow	VERB
ejpam-5098	182	11	are	be	AUX
ejpam-5098	182	12	equivalent	equivalent	ADJ
ejpam-5098	182	13	:	:	PUNCT
ejpam-5098	182	14	(	(	PUNCT
ejpam-5098	182	15	1	1	X
ejpam-5098	182	16	)	)	PUNCT
ejpam-5098	182	17	r	r	NOUN
ejpam-5098	182	18	is	be	AUX
ejpam-5098	182	19	complemented	complement	VERB
ejpam-5098	182	20	adl	adl	NOUN
ejpam-5098	182	21	(	(	PUNCT
ejpam-5098	182	22	2	2	NUM
ejpam-5098	182	23	)	)	PUNCT
ejpam-5098	182	24	for	for	ADP
ejpam-5098	182	25	any	any	DET
ejpam-5098	182	26	n	n	PRON
ejpam-5098	182	27	∈	∈	PROPN
ejpam-5098	182	28	maxr	maxr	NOUN
ejpam-5098	182	29	,	,	PUNCT
ejpam-5098	182	30	no	no	DET
ejpam-5098	182	31	∈	∈	PROPN
ejpam-5098	182	32	maxr	maxr	NOUN
ejpam-5098	182	33	(	(	PUNCT
ejpam-5098	182	34	3	3	NUM
ejpam-5098	182	35	)	)	PUNCT
ejpam-5098	182	36	for	for	ADP
ejpam-5098	182	37	any	any	DET
ejpam-5098	182	38	ideals	ideal	NOUN
ejpam-5098	182	39	n	n	CCONJ
ejpam-5098	182	40	,	,	PUNCT
ejpam-5098	182	41	n	n	PROPN
ejpam-5098	182	42	′	′	NOUN
ejpam-5098	182	43	of	of	ADP
ejpam-5098	182	44	r	r	NOUN
ejpam-5098	182	45	,	,	PUNCT
ejpam-5098	182	46	n	n	PRON
ejpam-5098	182	47	∨n	∨n	VERB
ejpam-5098	182	48	′	′	NOUN
ejpam-5098	183	1	=	=	NOUN
ejpam-5098	183	2	r⇒	r⇒	NOUN
ejpam-5098	183	3	nσ	nσ	NOUN
ejpam-5098	183	4	∨n	∨n	VERB
ejpam-5098	183	5	′σ	′σ	ADJ
ejpam-5098	183	6	=	=	SYM
ejpam-5098	183	7	r	r	NOUN
ejpam-5098	183	8	(	(	PUNCT
ejpam-5098	183	9	4	4	NUM
ejpam-5098	183	10	)	)	PUNCT
ejpam-5098	183	11	for	for	ADP
ejpam-5098	183	12	any	any	DET
ejpam-5098	183	13	ideals	ideal	NOUN
ejpam-5098	183	14	n	n	CCONJ
ejpam-5098	183	15	,	,	PUNCT
ejpam-5098	183	16	n	n	PROPN
ejpam-5098	183	17	′	′	NOUN
ejpam-5098	183	18	of	of	ADP
ejpam-5098	183	19	r	r	NOUN
ejpam-5098	183	20	,	,	PUNCT
ejpam-5098	183	21	n	n	PRON
ejpam-5098	183	22	∨n	∨n	VERB
ejpam-5098	183	23	′	′	NOUN
ejpam-5098	184	1	=	=	NOUN
ejpam-5098	184	2	r⇒	r⇒	NOUN
ejpam-5098	184	3	nσ	nσ	NOUN
ejpam-5098	184	4	∨n	∨n	VERB
ejpam-5098	184	5	′σ	′σ	ADJ
ejpam-5098	184	6	=	=	SYM
ejpam-5098	184	7	(	(	PUNCT
ejpam-5098	184	8	n	n	CCONJ
ejpam-5098	184	9	∨n	∨n	VERB
ejpam-5098	184	10	′)σ	′)σ	NOUN
ejpam-5098	184	11	(	(	PUNCT
ejpam-5098	184	12	5	5	NUM
ejpam-5098	184	13	)	)	PUNCT
ejpam-5098	184	14	for	for	ADP
ejpam-5098	184	15	any	any	DET
ejpam-5098	184	16	n	n	CCONJ
ejpam-5098	184	17	,	,	PUNCT
ejpam-5098	184	18	n	n	NOUN
ejpam-5098	184	19	′	′	NUM
ejpam-5098	184	20	∈	∈	PROPN
ejpam-5098	184	21	maxr	maxr	NOUN
ejpam-5098	184	22	with	with	ADP
ejpam-5098	184	23	n	n	CCONJ
ejpam-5098	184	24	̸=	̸=	PROPN
ejpam-5098	184	25	n	n	PROPN
ejpam-5098	184	26	′	′	NOUN
ejpam-5098	184	27	,	,	PUNCT
ejpam-5098	184	28	no	no	DET
ejpam-5098	184	29	∨n	∨n	VERB
ejpam-5098	184	30	′o	′o	NOUN
ejpam-5098	184	31	=	=	SYM
ejpam-5098	185	1	r	r	NOUN
ejpam-5098	185	2	(	(	PUNCT
ejpam-5098	185	3	6	6	NUM
ejpam-5098	185	4	)	)	PUNCT
ejpam-5098	185	5	for	for	ADP
ejpam-5098	185	6	any	any	DET
ejpam-5098	185	7	n	n	PRON
ejpam-5098	185	8	∈	∈	PROPN
ejpam-5098	185	9	maxr	maxr	NOUN
ejpam-5098	185	10	,	,	PUNCT
ejpam-5098	185	11	n	n	X
ejpam-5098	185	12	is	be	AUX
ejpam-5098	185	13	the	the	DET
ejpam-5098	185	14	unique	unique	ADJ
ejpam-5098	185	15	member	member	NOUN
ejpam-5098	185	16	of	of	ADP
ejpam-5098	185	17	maxr	maxr	NOUN
ejpam-5098	185	18	such	such	ADJ
ejpam-5098	185	19	that	that	SCONJ
ejpam-5098	185	20	no	no	DET
ejpam-5098	185	21	⊆	⊆	NUM
ejpam-5098	185	22	n	n	NOUN
ejpam-5098	185	23	.	.	PUNCT
ejpam-5098	186	1	proof	proof	NOUN
ejpam-5098	186	2	.	.	PUNCT
ejpam-5098	187	1	(	(	PUNCT
ejpam-5098	187	2	1	1	X
ejpam-5098	187	3	)	)	PUNCT
ejpam-5098	187	4	⇒	⇒	NOUN
ejpam-5098	187	5	(	(	PUNCT
ejpam-5098	187	6	2	2	NUM
ejpam-5098	187	7	):	):	PUNCT
ejpam-5098	187	8	assume	assume	VERB
ejpam-5098	187	9	(	(	PUNCT
ejpam-5098	187	10	1	1	NUM
ejpam-5098	187	11	)	)	PUNCT
ejpam-5098	187	12	.	.	PUNCT
ejpam-5098	188	1	let	let	VERB
ejpam-5098	188	2	n	n	PRON
ejpam-5098	188	3	∈	∈	PROPN
ejpam-5098	188	4	maxr	maxr	NOUN
ejpam-5098	188	5	and	and	CCONJ
ejpam-5098	188	6	a	a	DET
ejpam-5098	188	7	∈	∈	PROPN
ejpam-5098	188	8	n	n	ADV
ejpam-5098	188	9	.	.	PUNCT
ejpam-5098	189	1	by	by	ADP
ejpam-5098	189	2	our	our	PRON
ejpam-5098	189	3	assumption	assumption	NOUN
ejpam-5098	189	4	,	,	PUNCT
ejpam-5098	189	5	there	there	PRON
ejpam-5098	189	6	exists	exist	VERB
ejpam-5098	189	7	an	an	DET
ejpam-5098	189	8	element	element	NOUN
ejpam-5098	189	9	a′	a′	PROPN
ejpam-5098	189	10	∈	∈	PROPN
ejpam-5098	189	11	r	r	NOUN
ejpam-5098	189	12	such	such	ADJ
ejpam-5098	189	13	that	that	SCONJ
ejpam-5098	189	14	a	a	DET
ejpam-5098	189	15	∧	∧	NOUN
ejpam-5098	189	16	a′	a′	NOUN
ejpam-5098	189	17	=	=	SYM
ejpam-5098	189	18	0	0	NUM
ejpam-5098	189	19	and	and	CCONJ
ejpam-5098	189	20	a	a	DET
ejpam-5098	189	21	∨	∨	NOUN
ejpam-5098	189	22	a′	a′	NOUN
ejpam-5098	189	23	is	be	AUX
ejpam-5098	189	24	maximal	maximal	ADJ
ejpam-5098	189	25	.	.	PUNCT
ejpam-5098	190	1	clearly	clearly	ADV
ejpam-5098	190	2	,	,	PUNCT
ejpam-5098	190	3	we	we	PRON
ejpam-5098	190	4	get	get	VERB
ejpam-5098	190	5	that	that	PRON
ejpam-5098	190	6	a′	a′	PROPN
ejpam-5098	190	7	∈	∈	PROPN
ejpam-5098	190	8	(	(	PUNCT
ejpam-5098	190	9	a)∗	a)∗	PROPN
ejpam-5098	190	10	and	and	CCONJ
ejpam-5098	190	11	a′	a′	PROPN
ejpam-5098	190	12	/∈	/∈	PUNCT
ejpam-5098	191	1	n	n	PROPN
ejpam-5098	191	2	.	.	PUNCT
ejpam-5098	192	1	that	that	PRON
ejpam-5098	192	2	implies	imply	VERB
ejpam-5098	192	3	(	(	PUNCT
ejpam-5098	192	4	a)∗	a)∗	PROPN
ejpam-5098	192	5	⊈	⊈	PROPN
ejpam-5098	192	6	n	n	CCONJ
ejpam-5098	192	7	and	and	CCONJ
ejpam-5098	192	8	hence	hence	ADV
ejpam-5098	192	9	a	a	DET
ejpam-5098	192	10	∈	∈	ADJ
ejpam-5098	192	11	no	no	INTJ
ejpam-5098	192	12	.	.	PUNCT
ejpam-5098	193	1	therefore	therefore	ADV
ejpam-5098	193	2	,	,	PUNCT
ejpam-5098	193	3	n	n	PROPN
ejpam-5098	193	4	⊆	⊆	NUM
ejpam-5098	193	5	no	no	NOUN
ejpam-5098	193	6	.	.	PUNCT
ejpam-5098	194	1	since	since	SCONJ
ejpam-5098	194	2	n	n	PROPN
ejpam-5098	194	3	⊆	⊆	NUM
ejpam-5098	194	4	no	no	NOUN
ejpam-5098	194	5	,	,	PUNCT
ejpam-5098	194	6	we	we	PRON
ejpam-5098	194	7	get	get	VERB
ejpam-5098	194	8	n	n	PRON
ejpam-5098	194	9	=	=	ADJ
ejpam-5098	194	10	no	no	INTJ
ejpam-5098	194	11	.	.	PUNCT
ejpam-5098	195	1	since	since	SCONJ
ejpam-5098	195	2	n	n	PROPN
ejpam-5098	195	3	∈	∈	PROPN
ejpam-5098	195	4	maxr	maxr	NOUN
ejpam-5098	195	5	,	,	PUNCT
ejpam-5098	195	6	we	we	PRON
ejpam-5098	195	7	get	get	VERB
ejpam-5098	195	8	that	that	SCONJ
ejpam-5098	195	9	no	no	DET
ejpam-5098	195	10	∈	∈	PROPN
ejpam-5098	195	11	maxr	maxr	NOUN
ejpam-5098	195	12	.	.	PUNCT
ejpam-5098	196	1	(	(	PUNCT
ejpam-5098	196	2	2	2	X
ejpam-5098	196	3	)	)	PUNCT
ejpam-5098	196	4	⇒	⇒	NOUN
ejpam-5098	196	5	(	(	PUNCT
ejpam-5098	196	6	3	3	NUM
ejpam-5098	196	7	):	):	PUNCT
ejpam-5098	196	8	assume	assume	VERB
ejpam-5098	196	9	(	(	PUNCT
ejpam-5098	196	10	2	2	NUM
ejpam-5098	196	11	)	)	PUNCT
ejpam-5098	196	12	.	.	PUNCT
ejpam-5098	197	1	let	let	VERB
ejpam-5098	197	2	n	n	PRON
ejpam-5098	197	3	,	,	PUNCT
ejpam-5098	197	4	n	n	PRON
ejpam-5098	197	5	′	′	NUM
ejpam-5098	197	6	be	be	AUX
ejpam-5098	197	7	any	any	DET
ejpam-5098	197	8	two	two	NUM
ejpam-5098	197	9	ideals	ideal	NOUN
ejpam-5098	197	10	of	of	ADP
ejpam-5098	197	11	r	r	NOUN
ejpam-5098	197	12	such	such	ADJ
ejpam-5098	197	13	that	that	SCONJ
ejpam-5098	197	14	n	n	PROPN
ejpam-5098	197	15	∨	∨	NUM
ejpam-5098	197	16	n	n	NOUN
ejpam-5098	197	17	′	′	NOUN
ejpam-5098	197	18	=	=	PUNCT
ejpam-5098	197	19	r.	r.	PROPN
ejpam-5098	197	20	suppose	suppose	VERB
ejpam-5098	197	21	nσ	nσ	PRON
ejpam-5098	197	22	∨	∨	NOUN
ejpam-5098	197	23	n	n	CCONJ
ejpam-5098	197	24	′σ	′σ	PROPN
ejpam-5098	197	25	̸=	̸=	PROPN
ejpam-5098	197	26	r.	r.	NOUN
ejpam-5098	197	27	then	then	ADV
ejpam-5098	197	28	there	there	PRON
ejpam-5098	197	29	exists	exist	VERB
ejpam-5098	197	30	q	q	PROPN
ejpam-5098	197	31	∈	∈	PROPN
ejpam-5098	197	32	maxr	maxr	NOUN
ejpam-5098	197	33	such	such	ADJ
ejpam-5098	197	34	that	that	SCONJ
ejpam-5098	197	35	nσ	nσ	PROPN
ejpam-5098	197	36	∨	∨	NOUN
ejpam-5098	197	37	n	n	CCONJ
ejpam-5098	197	38	′σ	′σ	VERB
ejpam-5098	197	39	⊆	⊆	NUM
ejpam-5098	197	40	q.	q.	NOUN
ejpam-5098	197	41	that	that	PRON
ejpam-5098	197	42	impliesnσ	impliesnσ	VERB
ejpam-5098	197	43	⊆	⊆	NUM
ejpam-5098	197	44	q	q	NOUN
ejpam-5098	197	45	andn	andn	ADV
ejpam-5098	197	46	′σ	′σ	VERB
ejpam-5098	197	47	⊆	⊆	NUM
ejpam-5098	197	48	q.	q.	NOUN
ejpam-5098	197	49	by	by	ADP
ejpam-5098	197	50	proposition	proposition	NOUN
ejpam-5098	197	51	3	3	NUM
ejpam-5098	197	52	,	,	PUNCT
ejpam-5098	197	53	⋂	⋂	PROPN
ejpam-5098	197	54	ni∈w(n	ni∈w(n	NUM
ejpam-5098	197	55	)	)	PUNCT
ejpam-5098	198	1	no	no	DET
ejpam-5098	198	2	i	i	NOUN
ejpam-5098	198	3	⊆	⊆	NUM
ejpam-5098	198	4	q	q	NOUN
ejpam-5098	198	5	and	and	CCONJ
ejpam-5098	198	6	⋂	⋂	PROPN
ejpam-5098	198	7	n	n	PRON
ejpam-5098	198	8	′	′	NUM
ejpam-5098	198	9	i∈w(n	i∈w(n	NOUN
ejpam-5098	198	10	′	′	NOUN
ejpam-5098	198	11	)	)	PUNCT
ejpam-5098	199	1	n	n	NUM
ejpam-5098	199	2	′o	′o	NOUN
ejpam-5098	199	3	i	i	PROPN
ejpam-5098	199	4	⊆	⊆	NUM
ejpam-5098	199	5	q.	q.	NOUN
ejpam-5098	199	6	that	that	PRON
ejpam-5098	199	7	implies	imply	VERB
ejpam-5098	199	8	no	no	DET
ejpam-5098	199	9	i	i	NOUN
ejpam-5098	199	10	⊆	⊆	NUM
ejpam-5098	199	11	q	q	NOUN
ejpam-5098	199	12	and	and	CCONJ
ejpam-5098	199	13	n	n	NOUN
ejpam-5098	199	14	′o	′o	NOUN
ejpam-5098	199	15	i	i	PRON
ejpam-5098	199	16	⊆	⊆	NUM
ejpam-5098	199	17	q	q	NOUN
ejpam-5098	199	18	,	,	PUNCT
ejpam-5098	199	19	for	for	ADP
ejpam-5098	199	20	some	some	DET
ejpam-5098	199	21	ni	ni	PROPN
ejpam-5098	199	22	∈	∈	PROPN
ejpam-5098	199	23	w(n	w(n	PROPN
ejpam-5098	199	24	)	)	PUNCT
ejpam-5098	199	25	and	and	CCONJ
ejpam-5098	199	26	n	n	NUM
ejpam-5098	199	27	′	′	NUM
ejpam-5098	200	1	i	i	PRON
ejpam-5098	200	2	∈	∈	VERB
ejpam-5098	200	3	w(n	w(n	ADJ
ejpam-5098	200	4	′	′	NOUN
ejpam-5098	200	5	)	)	PUNCT
ejpam-5098	200	6	.	.	PUNCT
ejpam-5098	201	1	by	by	ADP
ejpam-5098	201	2	our	our	PRON
ejpam-5098	201	3	assumption	assumption	NOUN
ejpam-5098	201	4	,	,	PUNCT
ejpam-5098	201	5	we	we	PRON
ejpam-5098	201	6	get	get	VERB
ejpam-5098	201	7	that	that	PRON
ejpam-5098	201	8	ni	ni	PROPN
ejpam-5098	201	9	⊆	⊆	NUM
ejpam-5098	201	10	q	q	NOUN
ejpam-5098	201	11	and	and	CCONJ
ejpam-5098	201	12	n	n	NUM
ejpam-5098	201	13	′	′	NUM
ejpam-5098	202	1	i	i	PROPN
ejpam-5098	202	2	⊆	⊆	NUM
ejpam-5098	202	3	q.	q.	NOUN
ejpam-5098	202	4	since	since	SCONJ
ejpam-5098	202	5	n	n	PROPN
ejpam-5098	202	6	⊆	⊆	NUM
ejpam-5098	202	7	ni	ni	PROPN
ejpam-5098	202	8	,	,	PUNCT
ejpam-5098	202	9	n	n	CCONJ
ejpam-5098	202	10	′	′	NOUN
ejpam-5098	202	11	⊆	⊆	NUM
ejpam-5098	202	12	n	n	PRON
ejpam-5098	202	13	′	′	NUM
ejpam-5098	203	1	i	i	PRON
ejpam-5098	203	2	,	,	PUNCT
ejpam-5098	203	3	we	we	PRON
ejpam-5098	203	4	get	get	VERB
ejpam-5098	203	5	n	n	PRON
ejpam-5098	203	6	⊆	⊆	NUM
ejpam-5098	203	7	q	q	NOUN
ejpam-5098	204	1	and	and	CCONJ
ejpam-5098	204	2	n	n	NOUN
ejpam-5098	204	3	′	′	NUM
ejpam-5098	204	4	⊆	⊆	NUM
ejpam-5098	204	5	q.	q.	NOUN
ejpam-5098	204	6	that	that	PRON
ejpam-5098	204	7	implies	imply	VERB
ejpam-5098	204	8	l	l	NOUN
ejpam-5098	204	9	=	=	SYM
ejpam-5098	204	10	n	n	PRON
ejpam-5098	204	11	∨n	∨n	VERB
ejpam-5098	204	12	′	′	NUM
ejpam-5098	204	13	⊆	⊆	NUM
ejpam-5098	204	14	q	q	NOUN
ejpam-5098	204	15	and	and	CCONJ
ejpam-5098	204	16	hence	hence	ADV
ejpam-5098	204	17	q	q	NOUN
ejpam-5098	205	1	=	=	SYM
ejpam-5098	205	2	r	r	NOUN
ejpam-5098	205	3	,	,	PUNCT
ejpam-5098	205	4	we	we	PRON
ejpam-5098	205	5	get	get	VERB
ejpam-5098	205	6	a	a	DET
ejpam-5098	205	7	contradiction	contradiction	NOUN
ejpam-5098	205	8	.	.	PUNCT
ejpam-5098	206	1	thus	thus	ADV
ejpam-5098	206	2	,	,	PUNCT
ejpam-5098	206	3	nσ	nσ	PRON
ejpam-5098	206	4	∨n	∨n	VERB
ejpam-5098	206	5	′σ	′σ	PROPN
ejpam-5098	206	6	=	=	SYM
ejpam-5098	206	7	r.	r.	X
ejpam-5098	206	8	(	(	PUNCT
ejpam-5098	206	9	3	3	NUM
ejpam-5098	206	10	)	)	PUNCT
ejpam-5098	206	11	⇒	⇒	NOUN
ejpam-5098	206	12	(	(	PUNCT
ejpam-5098	206	13	4	4	NUM
ejpam-5098	206	14	):	):	PUNCT
ejpam-5098	206	15	assume	assume	VERB
ejpam-5098	206	16	(	(	PUNCT
ejpam-5098	206	17	3	3	NUM
ejpam-5098	206	18	)	)	PUNCT
ejpam-5098	206	19	.	.	PUNCT
ejpam-5098	207	1	let	let	VERB
ejpam-5098	207	2	n	n	PRON
ejpam-5098	207	3	,	,	PUNCT
ejpam-5098	207	4	n	n	PRON
ejpam-5098	207	5	′	′	NUM
ejpam-5098	207	6	be	be	AUX
ejpam-5098	207	7	two	two	NUM
ejpam-5098	207	8	ideals	ideal	NOUN
ejpam-5098	207	9	of	of	ADP
ejpam-5098	207	10	r.	r.	PROPN
ejpam-5098	207	11	then	then	ADV
ejpam-5098	207	12	nσ	nσ	PROPN
ejpam-5098	207	13	∨n	∨n	VERB
ejpam-5098	207	14	′σ	′σ	VERB
ejpam-5098	207	15	⊆	⊆	NUM
ejpam-5098	207	16	(	(	PUNCT
ejpam-5098	207	17	n	n	NUM
ejpam-5098	207	18	∨n	∨n	VERB
ejpam-5098	207	19	′)σ	′)σ	PROPN
ejpam-5098	207	20	.	.	PUNCT
ejpam-5098	208	1	let	let	VERB
ejpam-5098	208	2	a	a	DET
ejpam-5098	208	3	∈	∈	NOUN
ejpam-5098	208	4	(	(	PUNCT
ejpam-5098	208	5	n	n	X
ejpam-5098	208	6	∨n	∨n	VERB
ejpam-5098	208	7	′)σ	′)σ	PROPN
ejpam-5098	208	8	.	.	PUNCT
ejpam-5098	209	1	then	then	ADV
ejpam-5098	209	2	(	(	PUNCT
ejpam-5098	209	3	a)∗	a)∗	PROPN
ejpam-5098	209	4	∨	∨	PROPN
ejpam-5098	209	5	(	(	PUNCT
ejpam-5098	209	6	n	n	X
ejpam-5098	209	7	∨n	∨n	PROPN
ejpam-5098	209	8	′	′	NUM
ejpam-5098	209	9	)	)	PUNCT
ejpam-5098	209	10	=	=	VERB
ejpam-5098	210	1	r.	r.	NOUN
ejpam-5098	210	2	that	that	PRON
ejpam-5098	210	3	implies	imply	VERB
ejpam-5098	210	4	(	(	PUNCT
ejpam-5098	210	5	(	(	PUNCT
ejpam-5098	210	6	a∗)∨n)∨	a∗)∨n)∨	NOUN
ejpam-5098	210	7	(	(	PUNCT
ejpam-5098	210	8	(	(	PUNCT
ejpam-5098	210	9	a)∗	a)∗	PROPN
ejpam-5098	210	10	∨n	∨n	VERB
ejpam-5098	210	11	′	′	NUM
ejpam-5098	210	12	)	)	PUNCT
ejpam-5098	211	1	=	=	VERB
ejpam-5098	211	2	r.	r.	NOUN
ejpam-5098	211	3	by	by	ADP
ejpam-5098	211	4	our	our	PRON
ejpam-5098	211	5	assumption	assumption	NOUN
ejpam-5098	211	6	,	,	PUNCT
ejpam-5098	211	7	we	we	PRON
ejpam-5098	211	8	have	have	VERB
ejpam-5098	211	9	that	that	PRON
ejpam-5098	211	10	(	(	PUNCT
ejpam-5098	211	11	(	(	PUNCT
ejpam-5098	211	12	a∗	a∗	NOUN
ejpam-5098	211	13	)	)	PUNCT
ejpam-5098	211	14	∨	∨	NUM
ejpam-5098	211	15	n)σ	n)σ	ADJ
ejpam-5098	211	16	∨	∨	X
ejpam-5098	211	17	(	(	PUNCT
ejpam-5098	211	18	(	(	PUNCT
ejpam-5098	211	19	a)∗	a)∗	PROPN
ejpam-5098	211	20	∨	∨	NUM
ejpam-5098	211	21	n	n	CCONJ
ejpam-5098	211	22	′)σ	′)σ	PROPN
ejpam-5098	211	23	=	=	PUNCT
ejpam-5098	211	24	r.	r.	NOUN
ejpam-5098	211	25	that	that	PRON
ejpam-5098	211	26	implies	imply	VERB
ejpam-5098	211	27	a	a	DET
ejpam-5098	211	28	∈	∈	NOUN
ejpam-5098	211	29	(	(	PUNCT
ejpam-5098	211	30	(	(	PUNCT
ejpam-5098	211	31	a∗)∨n)σ	a∗)∨n)σ	PROPN
ejpam-5098	211	32	∨	∨	NUM
ejpam-5098	211	33	(	(	PUNCT
ejpam-5098	211	34	(	(	PUNCT
ejpam-5098	211	35	a)∗∨n	a)∗∨n	PROPN
ejpam-5098	211	36	′)σ	′)σ	PROPN
ejpam-5098	211	37	.	.	PUNCT
ejpam-5098	212	1	then	then	ADV
ejpam-5098	212	2	there	there	PRON
ejpam-5098	212	3	exist	exist	VERB
ejpam-5098	212	4	b	b	PROPN
ejpam-5098	212	5	∈	∈	PROPN
ejpam-5098	212	6	(	(	PUNCT
ejpam-5098	212	7	(	(	PUNCT
ejpam-5098	212	8	a∗)∨n)σ	a∗)∨n)σ	NOUN
ejpam-5098	212	9	and	and	CCONJ
ejpam-5098	212	10	c	c	NOUN
ejpam-5098	212	11	∈	∈	PROPN
ejpam-5098	212	12	(	(	PUNCT
ejpam-5098	212	13	(	(	PUNCT
ejpam-5098	212	14	a)∗∨n	a)∗∨n	PROPN
ejpam-5098	212	15	′)σ	′)σ	PROPN
ejpam-5098	212	16	such	such	ADJ
ejpam-5098	212	17	that	that	SCONJ
ejpam-5098	212	18	a	a	DET
ejpam-5098	212	19	=	=	PROPN
ejpam-5098	212	20	b∨c	b∨c	PROPN
ejpam-5098	212	21	.	.	PUNCT
ejpam-5098	213	1	since	since	SCONJ
ejpam-5098	213	2	b	b	PROPN
ejpam-5098	213	3	∈	∈	PROPN
ejpam-5098	213	4	(	(	PUNCT
ejpam-5098	213	5	(	(	PUNCT
ejpam-5098	213	6	a∗)∨n)σ	a∗)∨n)σ	NOUN
ejpam-5098	213	7	and	and	CCONJ
ejpam-5098	213	8	c	c	NOUN
ejpam-5098	213	9	∈	∈	PROPN
ejpam-5098	213	10	(	(	PUNCT
ejpam-5098	213	11	(	(	PUNCT
ejpam-5098	213	12	a)∗∨n	a)∗∨n	PROPN
ejpam-5098	213	13	′)σ	′)σ	PROPN
ejpam-5098	213	14	,	,	PUNCT
ejpam-5098	213	15	we	we	PRON
ejpam-5098	213	16	get	get	VERB
ejpam-5098	213	17	that	that	PRON
ejpam-5098	213	18	(	(	PUNCT
ejpam-5098	213	19	b)∗∨	b)∗∨	X
ejpam-5098	213	20	(	(	PUNCT
ejpam-5098	213	21	(	(	PUNCT
ejpam-5098	213	22	a∗)∨n	a∗)∨n	NOUN
ejpam-5098	213	23	)	)	PUNCT
ejpam-5098	213	24	=	=	SYM
ejpam-5098	213	25	r	r	NOUN
ejpam-5098	213	26	and	and	CCONJ
ejpam-5098	213	27	(	(	PUNCT
ejpam-5098	213	28	c)∗	c)∗	PROPN
ejpam-5098	213	29	∨	∨	PROPN
ejpam-5098	213	30	(	(	PUNCT
ejpam-5098	213	31	(	(	PUNCT
ejpam-5098	213	32	a)∗	a)∗	PROPN
ejpam-5098	213	33	∨n	∨n	VERB
ejpam-5098	213	34	′	′	NUM
ejpam-5098	213	35	)	)	PUNCT
ejpam-5098	214	1	=	=	VERB
ejpam-5098	214	2	r.	r.	NOUN
ejpam-5098	214	3	that	that	PRON
ejpam-5098	214	4	implies	imply	VERB
ejpam-5098	214	5	(	(	PUNCT
ejpam-5098	214	6	b	b	PROPN
ejpam-5098	214	7	∧	∧	PROPN
ejpam-5098	214	8	a)∗	a)∗	PROPN
ejpam-5098	214	9	∨n	∨n	VERB
ejpam-5098	214	10	=	=	SYM
ejpam-5098	214	11	r	r	NOUN
ejpam-5098	214	12	and	and	CCONJ
ejpam-5098	214	13	(	(	PUNCT
ejpam-5098	215	1	c	c	PROPN
ejpam-5098	215	2	∧	∧	PROPN
ejpam-5098	215	3	a)∗	a)∗	PROPN
ejpam-5098	215	4	∨n	∨n	VERB
ejpam-5098	215	5	′	′	NUM
ejpam-5098	216	1	=	=	SYM
ejpam-5098	216	2	r.	r.	PROPN
ejpam-5098	216	3	therefore	therefore	ADV
ejpam-5098	216	4	,	,	PUNCT
ejpam-5098	216	5	b∧	b∧	PROPN
ejpam-5098	216	6	a	a	DET
ejpam-5098	216	7	∈	∈	PROPN
ejpam-5098	216	8	nσ	nσ	NOUN
ejpam-5098	216	9	and	and	CCONJ
ejpam-5098	216	10	c∧	c∧	VERB
ejpam-5098	216	11	a	a	DET
ejpam-5098	216	12	∈	∈	PROPN
ejpam-5098	216	13	n	n	PRON
ejpam-5098	216	14	′σ	′σ	NOUN
ejpam-5098	216	15	.	.	PUNCT
ejpam-5098	217	1	hence	hence	ADV
ejpam-5098	217	2	,	,	PUNCT
ejpam-5098	217	3	a	a	DET
ejpam-5098	217	4	=	=	X
ejpam-5098	217	5	a∧	a∧	NOUN
ejpam-5098	217	6	a	a	X
ejpam-5098	217	7	=	=	X
ejpam-5098	217	8	(	(	PUNCT
ejpam-5098	217	9	b∨	b∨	PROPN
ejpam-5098	217	10	c)∧	c)∧	VERB
ejpam-5098	217	11	a	a	DET
ejpam-5098	217	12	=	=	X
ejpam-5098	217	13	(	(	PUNCT
ejpam-5098	217	14	b∧	b∧	PROPN
ejpam-5098	217	15	a)∨	a)∨	PROPN
ejpam-5098	217	16	(	(	PUNCT
ejpam-5098	217	17	c∧	c∧	VERB
ejpam-5098	217	18	a	a	PRON
ejpam-5098	217	19	)	)	PUNCT
ejpam-5098	217	20	∈	∈	PROPN
ejpam-5098	217	21	nσ	nσ	PROPN
ejpam-5098	217	22	∨n	∨n	VERB
ejpam-5098	217	23	′σ	′σ	VERB
ejpam-5098	217	24	.	.	PUNCT
ejpam-5098	218	1	thus	thus	ADV
ejpam-5098	218	2	,	,	PUNCT
ejpam-5098	218	3	(	(	PUNCT
ejpam-5098	218	4	n	n	CCONJ
ejpam-5098	218	5	∨n	∨n	VERB
ejpam-5098	218	6	′)σ	′)σ	PROPN
ejpam-5098	218	7	⊆	⊆	NUM
ejpam-5098	218	8	nσ	nσ	NOUN
ejpam-5098	218	9	∨n	∨n	VERB
ejpam-5098	218	10	′σ	′σ	VERB
ejpam-5098	218	11	.	.	PUNCT
ejpam-5098	219	1	r.	r.	PROPN
ejpam-5098	219	2	noorbhasha	noorbhasha	PROPN
ejpam-5098	219	3	,	,	PUNCT
ejpam-5098	219	4	r.	r.	PROPN
ejpam-5098	219	5	bandaru	bandaru	PROPN
ejpam-5098	219	6	,	,	PUNCT
ejpam-5098	219	7	a.	a.	NOUN
ejpam-5098	219	8	iampan	iampan	PROPN
ejpam-5098	219	9	/	/	SYM
ejpam-5098	219	10	eur	eur	PROPN
ejpam-5098	219	11	.	.	PUNCT
ejpam-5098	220	1	j.	j.	PROPN
ejpam-5098	220	2	pure	pure	PROPN
ejpam-5098	220	3	appl	appl	PROPN
ejpam-5098	220	4	.	.	PROPN
ejpam-5098	220	5	math	math	PROPN
ejpam-5098	220	6	,	,	PUNCT
ejpam-5098	220	7	17	17	NUM
ejpam-5098	220	8	(	(	PUNCT
ejpam-5098	220	9	2	2	NUM
ejpam-5098	220	10	)	)	PUNCT
ejpam-5098	220	11	(	(	PUNCT
ejpam-5098	220	12	2024	2024	NUM
ejpam-5098	220	13	)	)	PUNCT
ejpam-5098	220	14	,	,	PUNCT
ejpam-5098	220	15	1094	1094	NUM
ejpam-5098	220	16	-	-	SYM
ejpam-5098	220	17	1112	1112	NUM
ejpam-5098	220	18	1101	1101	NUM
ejpam-5098	220	19	(	(	PUNCT
ejpam-5098	220	20	4	4	NUM
ejpam-5098	220	21	)	)	PUNCT
ejpam-5098	220	22	⇒	⇒	NOUN
ejpam-5098	220	23	(	(	PUNCT
ejpam-5098	220	24	5	5	NUM
ejpam-5098	220	25	):	):	PUNCT
ejpam-5098	220	26	assume	assume	VERB
ejpam-5098	220	27	(	(	PUNCT
ejpam-5098	220	28	4	4	NUM
ejpam-5098	220	29	)	)	PUNCT
ejpam-5098	220	30	.	.	PUNCT
ejpam-5098	221	1	let	let	VERB
ejpam-5098	221	2	n	n	PRON
ejpam-5098	221	3	,	,	PUNCT
ejpam-5098	221	4	n	n	CCONJ
ejpam-5098	221	5	′	′	NUM
ejpam-5098	221	6	∈	∈	NOUN
ejpam-5098	221	7	maxr	maxr	NOUN
ejpam-5098	221	8	such	such	ADJ
ejpam-5098	221	9	that	that	SCONJ
ejpam-5098	221	10	n	n	CCONJ
ejpam-5098	221	11	̸=	̸=	PROPN
ejpam-5098	221	12	n	n	PRON
ejpam-5098	221	13	′.	′.	NOUN
ejpam-5098	221	14	then	then	ADV
ejpam-5098	221	15	there	there	PRON
ejpam-5098	221	16	exist	exist	VERB
ejpam-5098	221	17	a	a	DET
ejpam-5098	221	18	,	,	PUNCT
ejpam-5098	221	19	b	b	X
ejpam-5098	221	20	∈	∈	NOUN
ejpam-5098	221	21	r	r	NOUN
ejpam-5098	221	22	such	such	ADJ
ejpam-5098	221	23	that	that	SCONJ
ejpam-5098	221	24	a	a	DET
ejpam-5098	221	25	∈	∈	NOUN
ejpam-5098	221	26	n	n	NOUN
ejpam-5098	221	27	\n	\n	NOUN
ejpam-5098	221	28	′	′	NUM
ejpam-5098	222	1	and	and	CCONJ
ejpam-5098	222	2	b	b	X
ejpam-5098	222	3	∈	∈	PROPN
ejpam-5098	222	4	n	n	ADV
ejpam-5098	222	5	′	′	NOUN
ejpam-5098	222	6	\n	\n	PROPN
ejpam-5098	222	7	.	.	PUNCT
ejpam-5098	223	1	that	that	PRON
ejpam-5098	223	2	implies	imply	VERB
ejpam-5098	223	3	n	n	PROPN
ejpam-5098	223	4	′∨	′∨	PROPN
ejpam-5098	223	5	(	(	PUNCT
ejpam-5098	223	6	a	a	X
ejpam-5098	223	7	]	]	X
ejpam-5098	223	8	=	=	SYM
ejpam-5098	223	9	r	r	NOUN
ejpam-5098	223	10	and	and	CCONJ
ejpam-5098	223	11	n	n	CCONJ
ejpam-5098	223	12	∨	∨	NUM
ejpam-5098	223	13	(	(	PUNCT
ejpam-5098	223	14	b	b	NOUN
ejpam-5098	223	15	]	]	X
ejpam-5098	223	16	=	=	PUNCT
ejpam-5098	223	17	r.	r.	PROPN
ejpam-5098	223	18	so	so	SCONJ
ejpam-5098	223	19	that	that	SCONJ
ejpam-5098	223	20	n	n	NUM
ejpam-5098	223	21	′	′	NUM
ejpam-5098	223	22	∨n	∨n	PROPN
ejpam-5098	223	23	∨	∨	PROPN
ejpam-5098	223	24	(	(	PUNCT
ejpam-5098	223	25	a]∨	a]∨	PROPN
ejpam-5098	223	26	(	(	PUNCT
ejpam-5098	223	27	b	b	NOUN
ejpam-5098	223	28	]	]	X
ejpam-5098	223	29	=	=	SYM
ejpam-5098	223	30	r.	r.	PROPN
ejpam-5098	223	31	therefore	therefore	ADV
ejpam-5098	223	32	,	,	PUNCT
ejpam-5098	223	33	n	n	PROPN
ejpam-5098	223	34	′	′	NUM
ejpam-5098	223	35	∨n	∨n	PROPN
ejpam-5098	223	36	∨	∨	PROPN
ejpam-5098	223	37	(	(	PUNCT
ejpam-5098	223	38	a∨	a∨	PROPN
ejpam-5098	223	39	b	b	PROPN
ejpam-5098	223	40	]	]	X
ejpam-5098	223	41	=	=	X
ejpam-5098	223	42	r.	r.	NOUN
ejpam-5098	223	43	since	since	SCONJ
ejpam-5098	223	44	a∨	a∨	PROPN
ejpam-5098	223	45	b	b	PROPN
ejpam-5098	223	46	∈	∈	PROPN
ejpam-5098	223	47	n	n	PRON
ejpam-5098	223	48	∨n	∨n	NOUN
ejpam-5098	223	49	′	′	NOUN
ejpam-5098	223	50	,	,	PUNCT
ejpam-5098	223	51	we	we	PRON
ejpam-5098	223	52	get	get	VERB
ejpam-5098	223	53	n	n	PRON
ejpam-5098	223	54	∨n	∨n	VERB
ejpam-5098	223	55	′	′	NUM
ejpam-5098	224	1	=	=	PUNCT
ejpam-5098	224	2	r.	r.	NOUN
ejpam-5098	224	3	that	that	PRON
ejpam-5098	224	4	implies	imply	VERB
ejpam-5098	224	5	(	(	PUNCT
ejpam-5098	224	6	n	n	X
ejpam-5098	224	7	∨n	∨n	VERB
ejpam-5098	224	8	′)σ	′)σ	PROPN
ejpam-5098	224	9	=	=	PUNCT
ejpam-5098	224	10	r.	r.	NOUN
ejpam-5098	224	11	by	by	ADP
ejpam-5098	224	12	our	our	PRON
ejpam-5098	224	13	assumption	assumption	NOUN
ejpam-5098	224	14	,	,	PUNCT
ejpam-5098	224	15	we	we	PRON
ejpam-5098	224	16	get	get	VERB
ejpam-5098	224	17	nσ	nσ	NOUN
ejpam-5098	224	18	∨n	∨n	NOUN
ejpam-5098	224	19	′σ	′σ	PROPN
ejpam-5098	224	20	=	=	SYM
ejpam-5098	224	21	r.	r.	NOUN
ejpam-5098	224	22	since	since	SCONJ
ejpam-5098	224	23	nσ	nσ	PROPN
ejpam-5098	224	24	∨n	∨n	VERB
ejpam-5098	224	25	′σ	′σ	VERB
ejpam-5098	224	26	⊆	⊆	NUM
ejpam-5098	224	27	no	no	DET
ejpam-5098	224	28	∨n	∨n	NOUN
ejpam-5098	224	29	′o	′o	PROPN
ejpam-5098	224	30	,	,	PUNCT
ejpam-5098	224	31	we	we	PRON
ejpam-5098	224	32	get	get	VERB
ejpam-5098	224	33	no	no	DET
ejpam-5098	224	34	∨n	∨n	NOUN
ejpam-5098	224	35	′o	′o	X
ejpam-5098	224	36	=	=	SYM
ejpam-5098	224	37	r.	r.	X
ejpam-5098	224	38	(	(	PUNCT
ejpam-5098	224	39	5	5	NUM
ejpam-5098	224	40	)	)	PUNCT
ejpam-5098	224	41	⇒	⇒	NOUN
ejpam-5098	224	42	(	(	PUNCT
ejpam-5098	224	43	6	6	NUM
ejpam-5098	224	44	):	):	PUNCT
ejpam-5098	224	45	let	let	VERB
ejpam-5098	224	46	n	n	CCONJ
ejpam-5098	224	47	,	,	PUNCT
ejpam-5098	224	48	n	n	CCONJ
ejpam-5098	224	49	′	′	NUM
ejpam-5098	224	50	∈	∈	PROPN
ejpam-5098	224	51	maxr	maxr	NOUN
ejpam-5098	224	52	with	with	ADP
ejpam-5098	224	53	no	no	DET
ejpam-5098	224	54	⊆	⊆	NUM
ejpam-5098	224	55	n	n	NOUN
ejpam-5098	224	56	and	and	CCONJ
ejpam-5098	224	57	n	n	CCONJ
ejpam-5098	224	58	′o	′o	PROPN
ejpam-5098	225	1	⊆	⊆	NUM
ejpam-5098	225	2	n	n	NOUN
ejpam-5098	225	3	.	.	PUNCT
ejpam-5098	226	1	suppose	suppose	VERB
ejpam-5098	227	1	n	n	PRON
ejpam-5098	227	2	̸=	̸=	PROPN
ejpam-5098	227	3	n	n	PRON
ejpam-5098	227	4	′.	′.	NOUN
ejpam-5098	227	5	then	then	ADV
ejpam-5098	227	6	by	by	ADP
ejpam-5098	227	7	our	our	PRON
ejpam-5098	227	8	assumption	assumption	NOUN
ejpam-5098	227	9	,	,	PUNCT
ejpam-5098	227	10	we	we	PRON
ejpam-5098	227	11	have	have	VERB
ejpam-5098	227	12	that	that	SCONJ
ejpam-5098	227	13	no	no	DET
ejpam-5098	227	14	∨n	∨n	NOUN
ejpam-5098	227	15	′o	′o	X
ejpam-5098	227	16	=	=	PUNCT
ejpam-5098	227	17	r.	r.	NOUN
ejpam-5098	227	18	that	that	PRON
ejpam-5098	227	19	implies	imply	VERB
ejpam-5098	227	20	n	n	NOUN
ejpam-5098	227	21	=	=	SYM
ejpam-5098	227	22	r	r	NOUN
ejpam-5098	227	23	,	,	PUNCT
ejpam-5098	227	24	which	which	PRON
ejpam-5098	227	25	is	be	AUX
ejpam-5098	227	26	a	a	DET
ejpam-5098	227	27	contradiction	contradiction	NOUN
ejpam-5098	227	28	.	.	PUNCT
ejpam-5098	228	1	hence	hence	ADV
ejpam-5098	228	2	n	n	NOUN
ejpam-5098	228	3	=	=	SYM
ejpam-5098	228	4	n	n	PRON
ejpam-5098	228	5	′.	′.	NOUN
ejpam-5098	228	6	(	(	PUNCT
ejpam-5098	228	7	6	6	NUM
ejpam-5098	228	8	)	)	PUNCT
ejpam-5098	228	9	⇒	⇒	NOUN
ejpam-5098	228	10	(	(	PUNCT
ejpam-5098	228	11	1	1	NUM
ejpam-5098	228	12	):	):	PUNCT
ejpam-5098	228	13	let	let	VERB
ejpam-5098	228	14	a	a	DET
ejpam-5098	228	15	∈	∈	NOUN
ejpam-5098	228	16	r	r	NOUN
ejpam-5098	228	17	and	and	CCONJ
ejpam-5098	228	18	m	m	VERB
ejpam-5098	228	19	be	be	VERB
ejpam-5098	228	20	any	any	DET
ejpam-5098	228	21	maximal	maximal	ADJ
ejpam-5098	228	22	element	element	NOUN
ejpam-5098	228	23	of	of	ADP
ejpam-5098	228	24	r.	r.	PROPN
ejpam-5098	228	25	suppose	suppose	VERB
ejpam-5098	228	26	m	m	VERB
ejpam-5098	228	27	/∈	/∈	PUNCT
ejpam-5098	229	1	(	(	PUNCT
ejpam-5098	229	2	a	a	X
ejpam-5098	229	3	]	]	X
ejpam-5098	229	4	∨	∨	X
ejpam-5098	229	5	(	(	PUNCT
ejpam-5098	229	6	a)∗.	a)∗.	NOUN
ejpam-5098	229	7	then	then	ADV
ejpam-5098	229	8	(	(	PUNCT
ejpam-5098	229	9	a]∨	a]∨	PROPN
ejpam-5098	229	10	(	(	PUNCT
ejpam-5098	229	11	a)∗	a)∗	PROPN
ejpam-5098	229	12	⊆	⊆	NUM
ejpam-5098	229	13	n	n	NOUN
ejpam-5098	229	14	for	for	ADP
ejpam-5098	229	15	some	some	DET
ejpam-5098	229	16	maximal	maximal	ADJ
ejpam-5098	229	17	ideal	ideal	NOUN
ejpam-5098	229	18	n	n	PROPN
ejpam-5098	229	19	of	of	ADP
ejpam-5098	229	20	r.	r.	PROPN
ejpam-5098	229	21	that	that	PRON
ejpam-5098	229	22	implies	imply	VERB
ejpam-5098	229	23	(	(	PUNCT
ejpam-5098	229	24	a	a	X
ejpam-5098	229	25	]	]	X
ejpam-5098	229	26	⊆	⊆	NUM
ejpam-5098	229	27	n	n	NOUN
ejpam-5098	229	28	and	and	CCONJ
ejpam-5098	229	29	(	(	PUNCT
ejpam-5098	229	30	a)∗	a)∗	PROPN
ejpam-5098	229	31	⊆	⊆	NUM
ejpam-5098	229	32	n	n	NOUN
ejpam-5098	229	33	.	.	PUNCT
ejpam-5098	230	1	therefore	therefore	ADV
ejpam-5098	230	2	,	,	PUNCT
ejpam-5098	230	3	a	a	DET
ejpam-5098	230	4	∈	∈	PROPN
ejpam-5098	230	5	n	n	NOUN
ejpam-5098	230	6	and	and	CCONJ
ejpam-5098	230	7	a	a	DET
ejpam-5098	230	8	/∈	/∈	INTJ
ejpam-5098	230	9	no	no	INTJ
ejpam-5098	230	10	.	.	PUNCT
ejpam-5098	231	1	since	since	SCONJ
ejpam-5098	231	2	a	a	DET
ejpam-5098	231	3	/∈	/∈	INTJ
ejpam-5098	231	4	no	no	INTJ
ejpam-5098	231	5	,	,	PUNCT
ejpam-5098	231	6	there	there	PRON
ejpam-5098	231	7	is	be	VERB
ejpam-5098	231	8	a	a	DET
ejpam-5098	231	9	maximal	maximal	ADJ
ejpam-5098	231	10	ideal	ideal	ADJ
ejpam-5098	231	11	n1	n1	NOUN
ejpam-5098	231	12	of	of	ADP
ejpam-5098	231	13	r	r	NOUN
ejpam-5098	231	14	such	such	ADJ
ejpam-5098	231	15	that	that	SCONJ
ejpam-5098	231	16	a	a	DET
ejpam-5098	231	17	/∈	/∈	INTJ
ejpam-5098	231	18	n1	n1	NOUN
ejpam-5098	231	19	and	and	CCONJ
ejpam-5098	231	20	no	no	DET
ejpam-5098	231	21	⊆	⊆	NUM
ejpam-5098	231	22	n1	n1	NOUN
ejpam-5098	231	23	.	.	PUNCT
ejpam-5098	232	1	by	by	ADP
ejpam-5098	232	2	our	our	PRON
ejpam-5098	232	3	assumption	assumption	NOUN
ejpam-5098	232	4	,	,	PUNCT
ejpam-5098	232	5	we	we	PRON
ejpam-5098	232	6	get	get	VERB
ejpam-5098	232	7	that	that	DET
ejpam-5098	232	8	n	n	NOUN
ejpam-5098	232	9	=	=	SYM
ejpam-5098	232	10	n1	n1	PROPN
ejpam-5098	232	11	and	and	CCONJ
ejpam-5098	232	12	hence	hence	ADV
ejpam-5098	232	13	a	a	PRON
ejpam-5098	232	14	/∈	/∈	NOUN
ejpam-5098	232	15	n	n	NOUN
ejpam-5098	232	16	,	,	PUNCT
ejpam-5098	232	17	which	which	PRON
ejpam-5098	232	18	is	be	AUX
ejpam-5098	232	19	a	a	DET
ejpam-5098	232	20	contradiction	contradiction	NOUN
ejpam-5098	232	21	.	.	PUNCT
ejpam-5098	233	1	therefore	therefore	ADV
ejpam-5098	233	2	,	,	PUNCT
ejpam-5098	233	3	m	m	PROPN
ejpam-5098	233	4	∈	∈	ADJ
ejpam-5098	233	5	(	(	PUNCT
ejpam-5098	233	6	a]∨	a]∨	PROPN
ejpam-5098	233	7	(	(	PUNCT
ejpam-5098	233	8	a)∗.	a)∗.	NOUN
ejpam-5098	233	9	that	that	PRON
ejpam-5098	233	10	implies	imply	VERB
ejpam-5098	233	11	there	there	PRON
ejpam-5098	233	12	exists	exist	VERB
ejpam-5098	233	13	an	an	DET
ejpam-5098	233	14	element	element	NOUN
ejpam-5098	233	15	t	t	X
ejpam-5098	233	16	∈	∈	PROPN
ejpam-5098	233	17	(	(	PUNCT
ejpam-5098	233	18	a)∗	a)∗	PROPN
ejpam-5098	233	19	such	such	ADJ
ejpam-5098	233	20	that	that	SCONJ
ejpam-5098	233	21	a∨	a∨	PROPN
ejpam-5098	233	22	t	t	PROPN
ejpam-5098	233	23	=	=	PUNCT
ejpam-5098	233	24	m.	m.	NOUN
ejpam-5098	233	25	therefore	therefore	ADV
ejpam-5098	233	26	,	,	PUNCT
ejpam-5098	233	27	a∧	a∧	NOUN
ejpam-5098	233	28	t	t	PROPN
ejpam-5098	233	29	=	=	SYM
ejpam-5098	233	30	0	0	PUNCT
ejpam-5098	233	31	and	and	CCONJ
ejpam-5098	233	32	a∨	a∨	PROPN
ejpam-5098	233	33	t	t	PROPN
ejpam-5098	233	34	=	=	PUNCT
ejpam-5098	233	35	m.	m.	NOUN
ejpam-5098	233	36	thus	thus	ADV
ejpam-5098	233	37	,	,	PUNCT
ejpam-5098	233	38	l	l	NOUN
ejpam-5098	233	39	is	be	AUX
ejpam-5098	233	40	a	a	DET
ejpam-5098	233	41	complemented	complemented	ADJ
ejpam-5098	233	42	adl	adl	NOUN
ejpam-5098	233	43	.	.	PUNCT
ejpam-5098	234	1	in	in	ADP
ejpam-5098	234	2	general	general	ADJ
ejpam-5098	234	3	,	,	PUNCT
ejpam-5098	234	4	every	every	DET
ejpam-5098	234	5	maximal	maximal	ADJ
ejpam-5098	234	6	ideal	ideal	NOUN
ejpam-5098	234	7	of	of	ADP
ejpam-5098	234	8	r	r	NOUN
ejpam-5098	234	9	need	need	AUX
ejpam-5098	234	10	not	not	PART
ejpam-5098	234	11	be	be	AUX
ejpam-5098	234	12	a	a	DET
ejpam-5098	234	13	σ	σ	NOUN
ejpam-5098	234	14	-	-	PUNCT
ejpam-5098	234	15	ideal	ideal	NOUN
ejpam-5098	234	16	.	.	PUNCT
ejpam-5098	234	17	example	example	NOUN
ejpam-5098	235	1	2	2	NUM
ejpam-5098	235	2	.	.	PUNCT
ejpam-5098	235	3	let	let	VERB
ejpam-5098	235	4	r	r	NOUN
ejpam-5098	235	5	=	=	SYM
ejpam-5098	235	6	{	{	PUNCT
ejpam-5098	235	7	0	0	NUM
ejpam-5098	235	8	,	,	PUNCT
ejpam-5098	235	9	a	a	DET
ejpam-5098	235	10	,	,	PUNCT
ejpam-5098	235	11	b	b	NOUN
ejpam-5098	235	12	,	,	PUNCT
ejpam-5098	235	13	c	c	NOUN
ejpam-5098	235	14	}	}	PUNCT
ejpam-5098	235	15	.	.	PUNCT
ejpam-5098	236	1	define	define	VERB
ejpam-5098	236	2	two	two	NUM
ejpam-5098	236	3	binary	binary	ADJ
ejpam-5098	236	4	operations	operation	NOUN
ejpam-5098	236	5	∨	∨	NOUN
ejpam-5098	236	6	and	and	CCONJ
ejpam-5098	236	7	∧	∧	NOUN
ejpam-5098	236	8	on	on	ADP
ejpam-5098	236	9	r	r	NOUN
ejpam-5098	236	10	as	as	SCONJ
ejpam-5098	236	11	follows	follow	VERB
ejpam-5098	236	12	:	:	PUNCT
ejpam-5098	236	13	∨	∨	NUM
ejpam-5098	236	14	0	0	NUM
ejpam-5098	237	1	a	a	DET
ejpam-5098	237	2	b	b	NOUN
ejpam-5098	237	3	c	c	NOUN
ejpam-5098	237	4	0	0	NUM
ejpam-5098	237	5	0	0	NUM
ejpam-5098	237	6	a	a	DET
ejpam-5098	237	7	b	b	NOUN
ejpam-5098	237	8	c	c	ADP
ejpam-5098	237	9	a	a	DET
ejpam-5098	237	10	a	a	PRON
ejpam-5098	237	11	a	a	PRON
ejpam-5098	237	12	a	a	PRON
ejpam-5098	237	13	a	a	DET
ejpam-5098	237	14	b	b	PROPN
ejpam-5098	237	15	b	b	PROPN
ejpam-5098	237	16	b	b	PROPN
ejpam-5098	237	17	b	b	PROPN
ejpam-5098	237	18	b	b	PROPN
ejpam-5098	237	19	c	c	NOUN
ejpam-5098	237	20	c	c	PROPN
ejpam-5098	237	21	a	a	DET
ejpam-5098	237	22	b	b	NOUN
ejpam-5098	237	23	c	c	X
ejpam-5098	237	24	∧	∧	PROPN
ejpam-5098	237	25	0	0	NUM
ejpam-5098	237	26	a	a	DET
ejpam-5098	237	27	b	b	X
ejpam-5098	237	28	c	c	NOUN
ejpam-5098	237	29	0	0	NUM
ejpam-5098	237	30	0	0	NUM
ejpam-5098	237	31	0	0	NUM
ejpam-5098	237	32	0	0	NUM
ejpam-5098	237	33	0	0	NUM
ejpam-5098	237	34	a	a	DET
ejpam-5098	237	35	0	0	NUM
ejpam-5098	237	36	a	a	DET
ejpam-5098	237	37	b	b	NOUN
ejpam-5098	237	38	c	c	NOUN
ejpam-5098	237	39	b	b	PROPN
ejpam-5098	237	40	0	0	NUM
ejpam-5098	237	41	a	a	DET
ejpam-5098	237	42	b	b	NOUN
ejpam-5098	237	43	c	c	NOUN
ejpam-5098	237	44	c	c	NOUN
ejpam-5098	237	45	0	0	PUNCT
ejpam-5098	237	46	c	c	NOUN
ejpam-5098	237	47	c	c	NOUN
ejpam-5098	237	48	c	c	NOUN
ejpam-5098	237	49	consider	consider	VERB
ejpam-5098	237	50	the	the	DET
ejpam-5098	237	51	maximal	maximal	ADJ
ejpam-5098	237	52	ideal	ideal	NOUN
ejpam-5098	237	53	i	i	PRON
ejpam-5098	237	54	=	=	PUNCT
ejpam-5098	237	55	{	{	PUNCT
ejpam-5098	237	56	0	0	NUM
ejpam-5098	237	57	,	,	PUNCT
ejpam-5098	237	58	c	c	NOUN
ejpam-5098	237	59	}	}	PUNCT
ejpam-5098	237	60	.	.	PUNCT
ejpam-5098	238	1	now	now	ADV
ejpam-5098	238	2	,	,	PUNCT
ejpam-5098	238	3	(	(	PUNCT
ejpam-5098	238	4	c)∗	c)∗	PROPN
ejpam-5098	238	5	∨	∨	NOUN
ejpam-5098	238	6	i	i	NOUN
ejpam-5098	238	7	=	=	PUNCT
ejpam-5098	238	8	{	{	PUNCT
ejpam-5098	238	9	0	0	NUM
ejpam-5098	238	10	}	}	PUNCT
ejpam-5098	238	11	∨	∨	NUM
ejpam-5098	238	12	i	i	PRON
ejpam-5098	239	1	=	=	PROPN
ejpam-5098	240	1	i	i	PROPN
ejpam-5098	240	2	̸=	̸=	PROPN
ejpam-5098	240	3	r.	r.	PROPN
ejpam-5098	240	4	hence	hence	ADV
ejpam-5098	240	5	,	,	PUNCT
ejpam-5098	240	6	every	every	DET
ejpam-5098	240	7	maximal	maximal	ADJ
ejpam-5098	240	8	ideal	ideal	NOUN
ejpam-5098	240	9	of	of	ADP
ejpam-5098	240	10	r	r	NOUN
ejpam-5098	240	11	is	be	AUX
ejpam-5098	240	12	not	not	PART
ejpam-5098	240	13	a	a	DET
ejpam-5098	240	14	σ	σ	NOUN
ejpam-5098	240	15	-	-	PUNCT
ejpam-5098	240	16	ideal	ideal	NOUN
ejpam-5098	240	17	.	.	PUNCT
ejpam-5098	241	1	theorem	theorem	ADJ
ejpam-5098	241	2	8	8	NUM
ejpam-5098	241	3	.	.	PUNCT
ejpam-5098	242	1	in	in	ADP
ejpam-5098	242	2	an	an	DET
ejpam-5098	242	3	adl	adl	PROPN
ejpam-5098	242	4	r	r	NOUN
ejpam-5098	242	5	,	,	PUNCT
ejpam-5098	242	6	the	the	DET
ejpam-5098	242	7	following	follow	VERB
ejpam-5098	242	8	are	be	AUX
ejpam-5098	242	9	equivalent	equivalent	ADJ
ejpam-5098	242	10	:	:	PUNCT
ejpam-5098	242	11	(	(	PUNCT
ejpam-5098	242	12	1	1	X
ejpam-5098	242	13	)	)	PUNCT
ejpam-5098	242	14	r	r	NOUN
ejpam-5098	242	15	is	be	AUX
ejpam-5098	242	16	a	a	DET
ejpam-5098	242	17	complemented	complemented	ADJ
ejpam-5098	242	18	adl	adl	NOUN
ejpam-5098	242	19	(	(	PUNCT
ejpam-5098	242	20	2	2	NUM
ejpam-5098	242	21	)	)	PUNCT
ejpam-5098	242	22	every	every	DET
ejpam-5098	242	23	maximal	maximal	ADJ
ejpam-5098	242	24	ideal	ideal	NOUN
ejpam-5098	242	25	is	be	AUX
ejpam-5098	242	26	a	a	DET
ejpam-5098	242	27	σ	σ	NOUN
ejpam-5098	242	28	-	-	PUNCT
ejpam-5098	242	29	ideal	ideal	NOUN
ejpam-5098	242	30	(	(	PUNCT
ejpam-5098	242	31	3	3	NUM
ejpam-5098	242	32	)	)	PUNCT
ejpam-5098	242	33	every	every	DET
ejpam-5098	242	34	maximal	maximal	ADJ
ejpam-5098	242	35	ideal	ideal	NOUN
ejpam-5098	242	36	is	be	AUX
ejpam-5098	242	37	a	a	DET
ejpam-5098	242	38	minimal	minimal	ADJ
ejpam-5098	242	39	prime	prime	ADJ
ejpam-5098	242	40	ideal	ideal	NOUN
ejpam-5098	242	41	.	.	PUNCT
ejpam-5098	243	1	theorem	theorem	VERB
ejpam-5098	243	2	9	9	NUM
ejpam-5098	243	3	.	.	PUNCT
ejpam-5098	244	1	in	in	ADP
ejpam-5098	244	2	an	an	DET
ejpam-5098	244	3	adl	adl	PROPN
ejpam-5098	244	4	r	r	NOUN
ejpam-5098	244	5	,	,	PUNCT
ejpam-5098	244	6	the	the	DET
ejpam-5098	244	7	following	follow	VERB
ejpam-5098	244	8	conditions	condition	NOUN
ejpam-5098	244	9	are	be	AUX
ejpam-5098	244	10	equivalent	equivalent	ADJ
ejpam-5098	244	11	:	:	PUNCT
ejpam-5098	244	12	(	(	PUNCT
ejpam-5098	244	13	1	1	X
ejpam-5098	244	14	)	)	PUNCT
ejpam-5098	244	15	r	r	NOUN
ejpam-5098	244	16	is	be	AUX
ejpam-5098	244	17	relatively	relatively	ADV
ejpam-5098	244	18	complemented	complemented	ADJ
ejpam-5098	244	19	(	(	PUNCT
ejpam-5098	244	20	2	2	NUM
ejpam-5098	244	21	)	)	PUNCT
ejpam-5098	244	22	every	every	DET
ejpam-5098	244	23	principal	principal	ADJ
ejpam-5098	244	24	ideal	ideal	NOUN
ejpam-5098	244	25	is	be	AUX
ejpam-5098	244	26	a	a	DET
ejpam-5098	244	27	σ	σ	NOUN
ejpam-5098	244	28	-	-	PUNCT
ejpam-5098	244	29	ideal	ideal	NOUN
ejpam-5098	244	30	(	(	PUNCT
ejpam-5098	244	31	3	3	NUM
ejpam-5098	244	32	)	)	PUNCT
ejpam-5098	244	33	every	every	DET
ejpam-5098	244	34	ideal	ideal	NOUN
ejpam-5098	244	35	is	be	AUX
ejpam-5098	244	36	a	a	DET
ejpam-5098	244	37	σ	σ	NOUN
ejpam-5098	244	38	-	-	PUNCT
ejpam-5098	244	39	ideal	ideal	NOUN
ejpam-5098	244	40	(	(	PUNCT
ejpam-5098	244	41	4	4	NUM
ejpam-5098	244	42	)	)	PUNCT
ejpam-5098	244	43	every	every	DET
ejpam-5098	244	44	prime	prime	ADJ
ejpam-5098	244	45	ideal	ideal	NOUN
ejpam-5098	244	46	is	be	AUX
ejpam-5098	244	47	a	a	DET
ejpam-5098	244	48	σ	σ	NOUN
ejpam-5098	244	49	-	-	PUNCT
ejpam-5098	244	50	ideal	ideal	NOUN
ejpam-5098	244	51	(	(	PUNCT
ejpam-5098	244	52	5	5	NUM
ejpam-5098	244	53	)	)	PUNCT
ejpam-5098	244	54	every	every	DET
ejpam-5098	244	55	prime	prime	ADJ
ejpam-5098	244	56	ideal	ideal	NOUN
ejpam-5098	244	57	is	be	AUX
ejpam-5098	244	58	minimal	minimal	ADJ
ejpam-5098	244	59	.	.	PUNCT
ejpam-5098	245	1	r.	r.	PROPN
ejpam-5098	245	2	noorbhasha	noorbhasha	PROPN
ejpam-5098	245	3	,	,	PUNCT
ejpam-5098	245	4	r.	r.	PROPN
ejpam-5098	245	5	bandaru	bandaru	PROPN
ejpam-5098	245	6	,	,	PUNCT
ejpam-5098	245	7	a.	a.	NOUN
ejpam-5098	245	8	iampan	iampan	PROPN
ejpam-5098	245	9	/	/	SYM
ejpam-5098	245	10	eur	eur	PROPN
ejpam-5098	245	11	.	.	PUNCT
ejpam-5098	246	1	j.	j.	PROPN
ejpam-5098	246	2	pure	pure	PROPN
ejpam-5098	246	3	appl	appl	PROPN
ejpam-5098	246	4	.	.	PROPN
ejpam-5098	246	5	math	math	PROPN
ejpam-5098	246	6	,	,	PUNCT
ejpam-5098	246	7	17	17	NUM
ejpam-5098	246	8	(	(	PUNCT
ejpam-5098	246	9	2	2	NUM
ejpam-5098	246	10	)	)	PUNCT
ejpam-5098	246	11	(	(	PUNCT
ejpam-5098	246	12	2024	2024	NUM
ejpam-5098	246	13	)	)	PUNCT
ejpam-5098	246	14	,	,	PUNCT
ejpam-5098	246	15	1094	1094	NUM
ejpam-5098	246	16	-	-	SYM
ejpam-5098	246	17	1112	1112	NUM
ejpam-5098	246	18	1102	1102	NUM
ejpam-5098	246	19	proof	proof	NOUN
ejpam-5098	246	20	.	.	PUNCT
ejpam-5098	247	1	(	(	PUNCT
ejpam-5098	247	2	1	1	X
ejpam-5098	247	3	)	)	PUNCT
ejpam-5098	247	4	⇒	⇒	NOUN
ejpam-5098	247	5	(	(	PUNCT
ejpam-5098	247	6	2	2	NUM
ejpam-5098	247	7	):	):	PUNCT
ejpam-5098	247	8	assume	assume	VERB
ejpam-5098	247	9	(	(	PUNCT
ejpam-5098	247	10	1	1	NUM
ejpam-5098	247	11	)	)	PUNCT
ejpam-5098	247	12	.	.	PUNCT
ejpam-5098	248	1	clearly	clearly	ADV
ejpam-5098	248	2	,	,	PUNCT
ejpam-5098	248	3	we	we	PRON
ejpam-5098	248	4	have	have	VERB
ejpam-5098	248	5	that	that	PRON
ejpam-5098	248	6	(	(	PUNCT
ejpam-5098	248	7	a]σ	a]σ	VERB
ejpam-5098	248	8	⊆	⊆	NUM
ejpam-5098	248	9	(	(	PUNCT
ejpam-5098	248	10	a	a	X
ejpam-5098	248	11	]	]	X
ejpam-5098	248	12	for	for	ADP
ejpam-5098	248	13	all	all	DET
ejpam-5098	248	14	a	a	DET
ejpam-5098	248	15	∈	∈	PROPN
ejpam-5098	248	16	r.	r.	PROPN
ejpam-5098	248	17	suppose	suppose	VERB
ejpam-5098	248	18	(	(	PUNCT
ejpam-5098	248	19	a]σ	a]σ	VERB
ejpam-5098	248	20	⊈	⊈	PROPN
ejpam-5098	248	21	(	(	PUNCT
ejpam-5098	248	22	a	a	PRON
ejpam-5098	248	23	]	]	X
ejpam-5098	248	24	.	.	PUNCT
ejpam-5098	249	1	then	then	ADV
ejpam-5098	249	2	there	there	PRON
ejpam-5098	249	3	exists	exist	VERB
ejpam-5098	249	4	an	an	DET
ejpam-5098	249	5	element	element	NOUN
ejpam-5098	249	6	b	b	PROPN
ejpam-5098	249	7	∈	∈	PROPN
ejpam-5098	249	8	(	(	PUNCT
ejpam-5098	249	9	a	a	X
ejpam-5098	249	10	]	]	PUNCT
ejpam-5098	249	11	such	such	ADJ
ejpam-5098	249	12	that	that	DET
ejpam-5098	249	13	b	b	NOUN
ejpam-5098	249	14	/∈	/∈	PUNCT
ejpam-5098	249	15	(	(	PUNCT
ejpam-5098	249	16	a]σ	a]σ	ADJ
ejpam-5098	249	17	.	.	PUNCT
ejpam-5098	249	18	since	since	SCONJ
ejpam-5098	249	19	a	a	DET
ejpam-5098	249	20	,	,	PUNCT
ejpam-5098	249	21	b	b	X
ejpam-5098	249	22	∈	∈	PROPN
ejpam-5098	249	23	r	r	NOUN
ejpam-5098	249	24	and	and	CCONJ
ejpam-5098	249	25	r	r	NOUN
ejpam-5098	249	26	is	be	AUX
ejpam-5098	249	27	relatively	relatively	ADV
ejpam-5098	249	28	complemented	complemented	ADJ
ejpam-5098	249	29	,	,	PUNCT
ejpam-5098	249	30	there	there	PRON
ejpam-5098	249	31	exists	exist	VERB
ejpam-5098	249	32	an	an	DET
ejpam-5098	249	33	element	element	NOUN
ejpam-5098	249	34	x	x	SYM
ejpam-5098	249	35	∈	∈	NOUN
ejpam-5098	249	36	r	r	NOUN
ejpam-5098	249	37	such	such	ADJ
ejpam-5098	249	38	that	that	SCONJ
ejpam-5098	249	39	a	a	DET
ejpam-5098	249	40	∨	∨	NOUN
ejpam-5098	249	41	x	x	X
ejpam-5098	249	42	=	=	PUNCT
ejpam-5098	249	43	a	a	DET
ejpam-5098	249	44	∨	∨	NUM
ejpam-5098	249	45	b	b	NOUN
ejpam-5098	249	46	and	and	CCONJ
ejpam-5098	249	47	a	a	DET
ejpam-5098	249	48	∧	∧	NOUN
ejpam-5098	249	49	x	x	PUNCT
ejpam-5098	249	50	=	=	NOUN
ejpam-5098	249	51	0	0	NUM
ejpam-5098	249	52	.	.	PUNCT
ejpam-5098	250	1	since	since	SCONJ
ejpam-5098	250	2	a	a	DET
ejpam-5098	250	3	∧	∧	NOUN
ejpam-5098	250	4	x	x	PUNCT
ejpam-5098	250	5	=	=	SYM
ejpam-5098	250	6	0	0	NUM
ejpam-5098	250	7	,	,	PUNCT
ejpam-5098	250	8	we	we	PRON
ejpam-5098	250	9	get	get	VERB
ejpam-5098	250	10	that	that	SCONJ
ejpam-5098	250	11	a	a	DET
ejpam-5098	250	12	∈	∈	NOUN
ejpam-5098	250	13	(	(	PUNCT
ejpam-5098	250	14	x)∗	x)∗	PROPN
ejpam-5098	250	15	and	and	CCONJ
ejpam-5098	250	16	hence	hence	ADV
ejpam-5098	250	17	b	b	X
ejpam-5098	250	18	∈	∈	PROPN
ejpam-5098	250	19	(	(	PUNCT
ejpam-5098	250	20	x)∗.	x)∗.	PROPN
ejpam-5098	250	21	that	that	PRON
ejpam-5098	250	22	implies	imply	VERB
ejpam-5098	250	23	a	a	DET
ejpam-5098	250	24	∨	∨	NUM
ejpam-5098	250	25	b	b	X
ejpam-5098	250	26	∈	∈	PROPN
ejpam-5098	250	27	(	(	PUNCT
ejpam-5098	250	28	x)∗	x)∗	PROPN
ejpam-5098	250	29	,	,	PUNCT
ejpam-5098	250	30	which	which	PRON
ejpam-5098	250	31	gives	give	VERB
ejpam-5098	250	32	x	x	PUNCT
ejpam-5098	250	33	∨	∨	NUM
ejpam-5098	250	34	a	a	DET
ejpam-5098	250	35	∈	∈	NOUN
ejpam-5098	250	36	(	(	PUNCT
ejpam-5098	250	37	x)∗.	x)∗.	PROPN
ejpam-5098	250	38	so	so	SCONJ
ejpam-5098	250	39	that	that	SCONJ
ejpam-5098	250	40	x	x	SYM
ejpam-5098	250	41	∈	∈	PROPN
ejpam-5098	250	42	(	(	PUNCT
ejpam-5098	250	43	x)∗	x)∗	PROPN
ejpam-5098	250	44	and	and	CCONJ
ejpam-5098	250	45	hence	hence	ADV
ejpam-5098	250	46	x	x	X
ejpam-5098	250	47	=	=	SYM
ejpam-5098	250	48	0	0	NUM
ejpam-5098	250	49	,	,	PUNCT
ejpam-5098	250	50	which	which	PRON
ejpam-5098	250	51	is	be	AUX
ejpam-5098	250	52	a	a	DET
ejpam-5098	250	53	contraction	contraction	NOUN
ejpam-5098	250	54	.	.	PUNCT
ejpam-5098	251	1	hence	hence	ADV
ejpam-5098	251	2	,	,	PUNCT
ejpam-5098	251	3	(	(	PUNCT
ejpam-5098	251	4	a]σ	a]σ	VERB
ejpam-5098	251	5	⊆	⊆	NUM
ejpam-5098	251	6	(	(	PUNCT
ejpam-5098	251	7	a	a	PRON
ejpam-5098	251	8	]	]	X
ejpam-5098	251	9	.	.	PUNCT
ejpam-5098	252	1	thus	thus	ADV
ejpam-5098	252	2	,	,	PUNCT
ejpam-5098	252	3	every	every	DET
ejpam-5098	252	4	principal	principal	ADJ
ejpam-5098	252	5	ideal	ideal	NOUN
ejpam-5098	252	6	is	be	AUX
ejpam-5098	252	7	a	a	DET
ejpam-5098	252	8	σ	σ	NOUN
ejpam-5098	252	9	-	-	PUNCT
ejpam-5098	252	10	ideal	ideal	NOUN
ejpam-5098	252	11	.	.	PUNCT
ejpam-5098	253	1	(	(	PUNCT
ejpam-5098	253	2	2	2	X
ejpam-5098	253	3	)	)	PUNCT
ejpam-5098	253	4	⇒	⇒	NOUN
ejpam-5098	253	5	(	(	PUNCT
ejpam-5098	253	6	3	3	NUM
ejpam-5098	253	7	):	):	PUNCT
ejpam-5098	253	8	assume	assume	VERB
ejpam-5098	253	9	(	(	PUNCT
ejpam-5098	253	10	2	2	NUM
ejpam-5098	253	11	)	)	PUNCT
ejpam-5098	253	12	.	.	PUNCT
ejpam-5098	254	1	let	let	VERB
ejpam-5098	254	2	k	k	PRON
ejpam-5098	254	3	be	be	AUX
ejpam-5098	254	4	any	any	DET
ejpam-5098	254	5	ideal	ideal	NOUN
ejpam-5098	254	6	of	of	ADP
ejpam-5098	254	7	r	r	NOUN
ejpam-5098	254	8	and	and	CCONJ
ejpam-5098	254	9	a	a	DET
ejpam-5098	254	10	∈	∈	PROPN
ejpam-5098	254	11	k.	k.	NOUN
ejpam-5098	255	1	then	then	ADV
ejpam-5098	255	2	(	(	PUNCT
ejpam-5098	255	3	a	a	X
ejpam-5098	255	4	]	]	X
ejpam-5098	255	5	⊆	⊆	NUM
ejpam-5098	255	6	k.	k.	NOUN
ejpam-5098	255	7	that	that	PRON
ejpam-5098	255	8	implies	imply	VERB
ejpam-5098	255	9	(	(	PUNCT
ejpam-5098	255	10	a]σ	a]σ	VERB
ejpam-5098	255	11	⊆	⊆	NUM
ejpam-5098	255	12	kσ	kσ	PROPN
ejpam-5098	255	13	.	.	PROPN
ejpam-5098	255	14	by	by	ADP
ejpam-5098	255	15	our	our	PRON
ejpam-5098	255	16	assumption	assumption	NOUN
ejpam-5098	255	17	,	,	PUNCT
ejpam-5098	255	18	we	we	PRON
ejpam-5098	255	19	get	get	VERB
ejpam-5098	255	20	(	(	PUNCT
ejpam-5098	255	21	a	a	DET
ejpam-5098	255	22	]	]	X
ejpam-5098	255	23	⊆	⊆	NUM
ejpam-5098	255	24	kσ	kσ	PROPN
ejpam-5098	255	25	and	and	CCONJ
ejpam-5098	255	26	hence	hence	ADV
ejpam-5098	255	27	a	a	DET
ejpam-5098	255	28	∈	∈	PROPN
ejpam-5098	255	29	kσ	kσ	PROPN
ejpam-5098	255	30	.	.	PUNCT
ejpam-5098	256	1	therefore	therefore	ADV
ejpam-5098	256	2	,	,	PUNCT
ejpam-5098	256	3	k	k	PROPN
ejpam-5098	256	4	⊆	⊆	NUM
ejpam-5098	256	5	kσ	kσ	PROPN
ejpam-5098	256	6	.	.	PUNCT
ejpam-5098	256	7	since	since	SCONJ
ejpam-5098	256	8	kσ	kσ	PROPN
ejpam-5098	256	9	⊆	⊆	NUM
ejpam-5098	256	10	k	k	NOUN
ejpam-5098	256	11	,	,	PUNCT
ejpam-5098	256	12	we	we	PRON
ejpam-5098	256	13	get	get	VERB
ejpam-5098	256	14	kσ	kσ	PROPN
ejpam-5098	256	15	=	=	PROPN
ejpam-5098	256	16	k.	k.	PROPN
ejpam-5098	257	1	thus	thus	ADV
ejpam-5098	257	2	,	,	PUNCT
ejpam-5098	257	3	k	k	PROPN
ejpam-5098	257	4	is	be	AUX
ejpam-5098	257	5	a	a	DET
ejpam-5098	257	6	σ	σ	NOUN
ejpam-5098	257	7	-	-	PUNCT
ejpam-5098	257	8	ideal	ideal	NOUN
ejpam-5098	257	9	of	of	ADP
ejpam-5098	257	10	r.	r.	PROPN
ejpam-5098	257	11	(	(	PUNCT
ejpam-5098	257	12	3	3	NUM
ejpam-5098	257	13	)	)	PUNCT
ejpam-5098	257	14	⇒	⇒	NOUN
ejpam-5098	257	15	(	(	PUNCT
ejpam-5098	257	16	4	4	NUM
ejpam-5098	257	17	):	):	PUNCT
ejpam-5098	257	18	it	it	PRON
ejpam-5098	257	19	is	be	AUX
ejpam-5098	257	20	obvious	obvious	ADJ
ejpam-5098	257	21	.	.	PUNCT
ejpam-5098	258	1	(	(	PUNCT
ejpam-5098	258	2	4	4	X
ejpam-5098	258	3	)	)	PUNCT
ejpam-5098	258	4	⇒	⇒	NOUN
ejpam-5098	258	5	(	(	PUNCT
ejpam-5098	258	6	5	5	NUM
ejpam-5098	258	7	):	):	PUNCT
ejpam-5098	258	8	assume	assume	VERB
ejpam-5098	258	9	(	(	PUNCT
ejpam-5098	258	10	4	4	NUM
ejpam-5098	258	11	)	)	PUNCT
ejpam-5098	258	12	.	.	PUNCT
ejpam-5098	259	1	let	let	VERB
ejpam-5098	259	2	m	m	PRON
ejpam-5098	259	3	be	be	AUX
ejpam-5098	259	4	any	any	DET
ejpam-5098	259	5	prime	prime	ADJ
ejpam-5098	259	6	ideal	ideal	NOUN
ejpam-5098	259	7	of	of	ADP
ejpam-5098	259	8	r.	r.	PROPN
ejpam-5098	259	9	by	by	ADP
ejpam-5098	259	10	our	our	PRON
ejpam-5098	259	11	assumption	assumption	NOUN
ejpam-5098	259	12	,	,	PUNCT
ejpam-5098	259	13	we	we	PRON
ejpam-5098	259	14	have	have	VERB
ejpam-5098	259	15	that	that	PRON
ejpam-5098	259	16	σ	σ	NOUN
ejpam-5098	259	17	-	-	PUNCT
ejpam-5098	259	18	ideal	ideal	NOUN
ejpam-5098	259	19	of	of	ADP
ejpam-5098	259	20	r.	r.	PROPN
ejpam-5098	259	21	let	let	VERB
ejpam-5098	259	22	a	a	DET
ejpam-5098	259	23	∈	∈	NOUN
ejpam-5098	259	24	m	m	NOUN
ejpam-5098	259	25	.	.	PUNCT
ejpam-5098	260	1	then	then	ADV
ejpam-5098	260	2	a	a	DET
ejpam-5098	260	3	∈	∈	PROPN
ejpam-5098	260	4	mσ	mσ	NOUN
ejpam-5098	260	5	.	.	PUNCT
ejpam-5098	261	1	that	that	PRON
ejpam-5098	261	2	implies	imply	VERB
ejpam-5098	261	3	(	(	PUNCT
ejpam-5098	261	4	a)∗	a)∗	NOUN
ejpam-5098	261	5	∨m	∨m	NOUN
ejpam-5098	261	6	=	=	PUNCT
ejpam-5098	261	7	r.	r.	PROPN
ejpam-5098	261	8	there	there	PRON
ejpam-5098	261	9	exists	exist	VERB
ejpam-5098	261	10	s	s	PROPN
ejpam-5098	261	11	∈	∈	PROPN
ejpam-5098	261	12	(	(	PUNCT
ejpam-5098	261	13	a)∗	a)∗	PROPN
ejpam-5098	261	14	,	,	PUNCT
ejpam-5098	261	15	t	t	PROPN
ejpam-5098	261	16	∈	∈	PROPN
ejpam-5098	261	17	m	m	VERB
ejpam-5098	261	18	such	such	ADJ
ejpam-5098	261	19	that	that	PRON
ejpam-5098	261	20	s	s	PROPN
ejpam-5098	261	21	∨	∨	PROPN
ejpam-5098	261	22	t	t	PROPN
ejpam-5098	261	23	is	be	AUX
ejpam-5098	261	24	a	a	DET
ejpam-5098	261	25	maximal	maximal	ADJ
ejpam-5098	261	26	element	element	NOUN
ejpam-5098	261	27	of	of	ADP
ejpam-5098	261	28	r.	r.	PROPN
ejpam-5098	261	29	clearly	clearly	ADV
ejpam-5098	261	30	,	,	PUNCT
ejpam-5098	261	31	we	we	PRON
ejpam-5098	261	32	get	get	VERB
ejpam-5098	261	33	s	s	PRON
ejpam-5098	261	34	/∈	/∈	NOUN
ejpam-5098	262	1	m	m	VERB
ejpam-5098	262	2	and	and	CCONJ
ejpam-5098	262	3	s	s	VERB
ejpam-5098	262	4	∧	∧	NOUN
ejpam-5098	262	5	a	a	PRON
ejpam-5098	262	6	=	=	NOUN
ejpam-5098	262	7	0	0	NUM
ejpam-5098	262	8	.	.	PUNCT
ejpam-5098	263	1	therefore	therefore	ADV
ejpam-5098	263	2	,	,	PUNCT
ejpam-5098	263	3	for	for	ADP
ejpam-5098	263	4	any	any	DET
ejpam-5098	263	5	a	a	DET
ejpam-5098	263	6	∈	∈	NOUN
ejpam-5098	263	7	m	m	NOUN
ejpam-5098	263	8	,	,	PUNCT
ejpam-5098	263	9	there	there	PRON
ejpam-5098	263	10	exists	exist	VERB
ejpam-5098	263	11	an	an	DET
ejpam-5098	263	12	element	element	NOUN
ejpam-5098	263	13	s	s	PART
ejpam-5098	263	14	/∈	/∈	NOUN
ejpam-5098	263	15	m	m	VERB
ejpam-5098	263	16	such	such	ADJ
ejpam-5098	263	17	that	that	PRON
ejpam-5098	263	18	s	s	VERB
ejpam-5098	263	19	∧	∧	NOUN
ejpam-5098	263	20	a	a	DET
ejpam-5098	263	21	=	=	NOUN
ejpam-5098	263	22	0	0	NUM
ejpam-5098	263	23	.	.	PUNCT
ejpam-5098	264	1	thus	thus	ADV
ejpam-5098	264	2	,	,	PUNCT
ejpam-5098	264	3	m	m	VERB
ejpam-5098	264	4	is	be	AUX
ejpam-5098	264	5	a	a	DET
ejpam-5098	264	6	minimal	minimal	ADJ
ejpam-5098	264	7	prime	prime	ADJ
ejpam-5098	264	8	ideal	ideal	NOUN
ejpam-5098	264	9	.	.	PUNCT
ejpam-5098	265	1	(	(	PUNCT
ejpam-5098	265	2	5	5	X
ejpam-5098	265	3	)	)	PUNCT
ejpam-5098	265	4	⇒	⇒	NOUN
ejpam-5098	265	5	(	(	PUNCT
ejpam-5098	265	6	1	1	NUM
ejpam-5098	265	7	):	):	PUNCT
ejpam-5098	265	8	assume	assume	VERB
ejpam-5098	265	9	(	(	PUNCT
ejpam-5098	265	10	5	5	NUM
ejpam-5098	265	11	)	)	PUNCT
ejpam-5098	265	12	.	.	PUNCT
ejpam-5098	266	1	let	let	VERB
ejpam-5098	266	2	x	x	PRON
ejpam-5098	266	3	,	,	PUNCT
ejpam-5098	266	4	y	y	PROPN
ejpam-5098	266	5	∈	∈	PROPN
ejpam-5098	266	6	r	r	NOUN
ejpam-5098	267	1	such	such	ADJ
ejpam-5098	267	2	that	that	SCONJ
ejpam-5098	267	3	x	x	SYM
ejpam-5098	267	4	∈	∈	PROPN
ejpam-5098	268	1	[	[	X
ejpam-5098	268	2	0	0	NUM
ejpam-5098	268	3	,	,	PUNCT
ejpam-5098	268	4	y	y	NOUN
ejpam-5098	268	5	]	]	X
ejpam-5098	268	6	.	.	PUNCT
ejpam-5098	269	1	if	if	SCONJ
ejpam-5098	269	2	y	y	PROPN
ejpam-5098	269	3	/∈	/∈	PUNCT
ejpam-5098	269	4	(	(	PUNCT
ejpam-5098	269	5	x	x	X
ejpam-5098	269	6	]	]	X
ejpam-5098	269	7	∨	∨	NUM
ejpam-5098	269	8	(	(	PUNCT
ejpam-5098	269	9	x)∗.	x)∗.	PROPN
ejpam-5098	269	10	then	then	ADV
ejpam-5098	269	11	there	there	PRON
ejpam-5098	269	12	exists	exist	VERB
ejpam-5098	269	13	a	a	DET
ejpam-5098	269	14	prime	prime	ADJ
ejpam-5098	269	15	ideal	ideal	NOUN
ejpam-5098	269	16	q	q	NOUN
ejpam-5098	269	17	of	of	ADP
ejpam-5098	269	18	r	r	NOUN
ejpam-5098	269	19	such	such	ADJ
ejpam-5098	269	20	that	that	SCONJ
ejpam-5098	269	21	(	(	PUNCT
ejpam-5098	269	22	x	x	X
ejpam-5098	269	23	]	]	X
ejpam-5098	269	24	∨	∨	X
ejpam-5098	269	25	(	(	PUNCT
ejpam-5098	269	26	x)∗	x)∗	PROPN
ejpam-5098	269	27	⊆	⊆	NUM
ejpam-5098	269	28	q.	q.	NOUN
ejpam-5098	269	29	that	that	PRON
ejpam-5098	269	30	implies	imply	VERB
ejpam-5098	269	31	(	(	PUNCT
ejpam-5098	269	32	x)∗	x)∗	PROPN
ejpam-5098	269	33	⊆	⊆	NUM
ejpam-5098	269	34	q	q	NOUN
ejpam-5098	269	35	and	and	CCONJ
ejpam-5098	269	36	(	(	PUNCT
ejpam-5098	269	37	x	x	X
ejpam-5098	269	38	]	]	X
ejpam-5098	269	39	⊆	⊆	NUM
ejpam-5098	269	40	q.	q.	NOUN
ejpam-5098	269	41	so	so	SCONJ
ejpam-5098	269	42	that	that	SCONJ
ejpam-5098	269	43	(	(	PUNCT
ejpam-5098	269	44	x)∗	x)∗	PROPN
ejpam-5098	269	45	⊆	⊆	NUM
ejpam-5098	269	46	q	q	NOUN
ejpam-5098	269	47	and	and	CCONJ
ejpam-5098	269	48	x	x	SYM
ejpam-5098	269	49	∈	∈	NOUN
ejpam-5098	269	50	q.	q.	NOUN
ejpam-5098	269	51	by	by	ADP
ejpam-5098	269	52	our	our	PRON
ejpam-5098	269	53	assumption	assumption	NOUN
ejpam-5098	269	54	,	,	PUNCT
ejpam-5098	269	55	q	q	PUNCT
ejpam-5098	269	56	is	be	AUX
ejpam-5098	269	57	a	a	DET
ejpam-5098	269	58	minimal	minimal	ADJ
ejpam-5098	269	59	prime	prime	ADJ
ejpam-5098	269	60	ideal	ideal	NOUN
ejpam-5098	269	61	.	.	PUNCT
ejpam-5098	270	1	since	since	SCONJ
ejpam-5098	270	2	(	(	PUNCT
ejpam-5098	270	3	x)∗	x)∗	PROPN
ejpam-5098	270	4	⊆	⊆	NUM
ejpam-5098	270	5	q	q	NOUN
ejpam-5098	270	6	,	,	PUNCT
ejpam-5098	270	7	we	we	PRON
ejpam-5098	270	8	get	get	VERB
ejpam-5098	270	9	x	x	PUNCT
ejpam-5098	270	10	/∈	/∈	PUNCT
ejpam-5098	271	1	q	q	X
ejpam-5098	271	2	,	,	PUNCT
ejpam-5098	272	1	which	which	PRON
ejpam-5098	272	2	is	be	AUX
ejpam-5098	272	3	a	a	DET
ejpam-5098	272	4	contradiction	contradiction	NOUN
ejpam-5098	272	5	.	.	PUNCT
ejpam-5098	273	1	therefore	therefore	ADV
ejpam-5098	273	2	,	,	PUNCT
ejpam-5098	273	3	y	y	PROPN
ejpam-5098	273	4	∈	∈	PROPN
ejpam-5098	273	5	(	(	PUNCT
ejpam-5098	273	6	x)∗	x)∗	PROPN
ejpam-5098	273	7	∨	∨	PROPN
ejpam-5098	273	8	(	(	PUNCT
ejpam-5098	273	9	x	x	X
ejpam-5098	273	10	]	]	X
ejpam-5098	273	11	.	.	PUNCT
ejpam-5098	274	1	hence	hence	ADV
ejpam-5098	274	2	,	,	PUNCT
ejpam-5098	274	3	there	there	PRON
ejpam-5098	274	4	exist	exist	VERB
ejpam-5098	274	5	s	s	X
ejpam-5098	274	6	∈	∈	NOUN
ejpam-5098	274	7	(	(	PUNCT
ejpam-5098	274	8	x	x	X
ejpam-5098	274	9	]	]	X
ejpam-5098	274	10	,	,	PUNCT
ejpam-5098	274	11	t	t	PROPN
ejpam-5098	274	12	∈	∈	PROPN
ejpam-5098	274	13	(	(	PUNCT
ejpam-5098	274	14	x)∗	x)∗	X
ejpam-5098	274	15	such	such	ADJ
ejpam-5098	274	16	that	that	SCONJ
ejpam-5098	274	17	y	y	PROPN
ejpam-5098	274	18	=	=	SYM
ejpam-5098	274	19	t∨	t∨	PROPN
ejpam-5098	274	20	s.	s.	PROPN
ejpam-5098	274	21	since	since	SCONJ
ejpam-5098	274	22	t	t	PROPN
ejpam-5098	274	23	∈	∈	PROPN
ejpam-5098	274	24	(	(	PUNCT
ejpam-5098	274	25	x)∗	x)∗	PROPN
ejpam-5098	274	26	and	and	CCONJ
ejpam-5098	274	27	s	s	PROPN
ejpam-5098	274	28	∈	∈	PROPN
ejpam-5098	274	29	(	(	PUNCT
ejpam-5098	274	30	x	x	X
ejpam-5098	274	31	]	]	X
ejpam-5098	274	32	,	,	PUNCT
ejpam-5098	274	33	we	we	PRON
ejpam-5098	274	34	get	get	VERB
ejpam-5098	274	35	t∧x	t∧x	NOUN
ejpam-5098	274	36	=	=	SYM
ejpam-5098	274	37	0	0	NUM
ejpam-5098	275	1	and	and	CCONJ
ejpam-5098	275	2	x	x	PART
ejpam-5098	275	3	∧	∧	PROPN
ejpam-5098	275	4	s	s	PART
ejpam-5098	275	5	=	=	PUNCT
ejpam-5098	275	6	s.	s.	PROPN
ejpam-5098	275	7	now	now	ADV
ejpam-5098	275	8	,	,	PUNCT
ejpam-5098	275	9	x	x	X
ejpam-5098	275	10	=	=	PUNCT
ejpam-5098	275	11	x	x	SYM
ejpam-5098	275	12	∧	∧	NOUN
ejpam-5098	275	13	y	y	NOUN
ejpam-5098	275	14	=	=	PUNCT
ejpam-5098	275	15	x	x	SYM
ejpam-5098	275	16	∧	∧	PROPN
ejpam-5098	275	17	(	(	PUNCT
ejpam-5098	275	18	t	t	PROPN
ejpam-5098	275	19	∨	∨	NUM
ejpam-5098	275	20	s	s	PART
ejpam-5098	275	21	)	)	PUNCT
ejpam-5098	275	22	=	=	SYM
ejpam-5098	276	1	(	(	PUNCT
ejpam-5098	276	2	x	x	PUNCT
ejpam-5098	276	3	∧	∧	PROPN
ejpam-5098	276	4	t	t	PROPN
ejpam-5098	276	5	)	)	PUNCT
ejpam-5098	276	6	∨	∨	NOUN
ejpam-5098	276	7	(	(	PUNCT
ejpam-5098	276	8	x	x	PUNCT
ejpam-5098	276	9	∧	∧	NOUN
ejpam-5098	276	10	s	s	PART
ejpam-5098	276	11	)	)	PUNCT
ejpam-5098	276	12	=	=	PUNCT
ejpam-5098	277	1	x	x	PUNCT
ejpam-5098	277	2	∧	∧	PROPN
ejpam-5098	277	3	s	s	PART
ejpam-5098	277	4	=	=	PUNCT
ejpam-5098	277	5	s.	s.	PROPN
ejpam-5098	277	6	since	since	SCONJ
ejpam-5098	277	7	y	y	PROPN
ejpam-5098	277	8	=	=	SYM
ejpam-5098	277	9	t	t	PROPN
ejpam-5098	277	10	∨	∨	NUM
ejpam-5098	277	11	s	s	PART
ejpam-5098	277	12	=	=	X
ejpam-5098	277	13	t	t	PROPN
ejpam-5098	277	14	∨	∨	NUM
ejpam-5098	277	15	x	x	AUX
ejpam-5098	277	16	,	,	PUNCT
ejpam-5098	277	17	we	we	PRON
ejpam-5098	277	18	get	get	VERB
ejpam-5098	277	19	t	t	PRON
ejpam-5098	277	20	∈	∈	PROPN
ejpam-5098	278	1	[	[	X
ejpam-5098	278	2	0	0	NUM
ejpam-5098	278	3	,	,	PUNCT
ejpam-5098	278	4	y	y	PROPN
ejpam-5098	278	5	]	]	X
ejpam-5098	278	6	.	.	PUNCT
ejpam-5098	279	1	therefore	therefore	ADV
ejpam-5098	279	2	,	,	PUNCT
ejpam-5098	279	3	r	r	NOUN
ejpam-5098	279	4	is	be	AUX
ejpam-5098	279	5	relatively	relatively	ADV
ejpam-5098	279	6	complemented	complemented	ADJ
ejpam-5098	279	7	.	.	PUNCT
ejpam-5098	280	1	define	define	VERB
ejpam-5098	280	2	a	a	DET
ejpam-5098	280	3	binary	binary	PROPN
ejpam-5098	280	4	relation	relation	NOUN
ejpam-5098	280	5	ϕ	ϕ	PROPN
ejpam-5098	280	6	on	on	ADP
ejpam-5098	280	7	r	r	NOUN
ejpam-5098	280	8	as	as	ADP
ejpam-5098	280	9	(	(	PUNCT
ejpam-5098	280	10	a	a	PRON
ejpam-5098	280	11	,	,	PUNCT
ejpam-5098	280	12	b	b	NOUN
ejpam-5098	280	13	)	)	PUNCT
ejpam-5098	280	14	∈	∈	PROPN
ejpam-5098	280	15	ϕ	ϕ	NOUN
ejpam-5098	281	1	if	if	SCONJ
ejpam-5098	281	2	and	and	CCONJ
ejpam-5098	281	3	only	only	ADV
ejpam-5098	281	4	if	if	SCONJ
ejpam-5098	281	5	(	(	PUNCT
ejpam-5098	281	6	a)∗	a)∗	PROPN
ejpam-5098	281	7	=	=	PRON
ejpam-5098	281	8	(	(	PUNCT
ejpam-5098	281	9	b)∗	b)∗	PROPN
ejpam-5098	281	10	for	for	ADP
ejpam-5098	281	11	all	all	DET
ejpam-5098	281	12	a	a	PRON
ejpam-5098	281	13	,	,	PUNCT
ejpam-5098	281	14	b	b	PROPN
ejpam-5098	281	15	∈	∈	PROPN
ejpam-5098	281	16	r.	r.	PROPN
ejpam-5098	281	17	clearly	clearly	ADV
ejpam-5098	281	18	,	,	PUNCT
ejpam-5098	281	19	ϕ	ϕ	PROPN
ejpam-5098	281	20	is	be	AUX
ejpam-5098	281	21	a	a	DET
ejpam-5098	281	22	congruence	congruence	NOUN
ejpam-5098	281	23	relation	relation	NOUN
ejpam-5098	281	24	on	on	ADP
ejpam-5098	281	25	r	r	NOUN
ejpam-5098	281	26	with	with	ADP
ejpam-5098	281	27	kerϕ	kerϕ	NOUN
ejpam-5098	281	28	as	as	ADP
ejpam-5098	281	29	the	the	DET
ejpam-5098	281	30	smallest	small	ADJ
ejpam-5098	281	31	congruence	congruence	NOUN
ejpam-5098	281	32	class	class	NOUN
ejpam-5098	281	33	modulo	modulo	PROPN
ejpam-5098	281	34	ϕ.	ϕ.	PROPN
ejpam-5098	281	35	also	also	ADV
ejpam-5098	281	36	,	,	PUNCT
ejpam-5098	281	37	we	we	PRON
ejpam-5098	281	38	have	have	VERB
ejpam-5098	281	39	that	that	PRON
ejpam-5098	281	40	r/ϕ	r/ϕ	NOUN
ejpam-5098	281	41	is	be	AUX
ejpam-5098	281	42	a	a	DET
ejpam-5098	281	43	quotient	quotient	NOUN
ejpam-5098	281	44	adl	adl	NOUN
ejpam-5098	281	45	by	by	ADP
ejpam-5098	281	46	defining	define	VERB
ejpam-5098	281	47	[	[	X
ejpam-5098	281	48	a]ϕ	a]ϕ	NOUN
ejpam-5098	281	49	∧	∧	NOUN
ejpam-5098	282	1	[	[	X
ejpam-5098	282	2	b]ϕ	b]ϕ	NOUN
ejpam-5098	282	3	=	=	PUNCT
ejpam-5098	283	1	[	[	X
ejpam-5098	283	2	a	a	DET
ejpam-5098	283	3	∧	∧	PROPN
ejpam-5098	283	4	b]ϕ	b]ϕ	NOUN
ejpam-5098	283	5	and	and	CCONJ
ejpam-5098	283	6	[	[	X
ejpam-5098	283	7	a]ϕ	a]ϕ	NOUN
ejpam-5098	283	8	∨	∨	NOUN
ejpam-5098	283	9	[	[	X
ejpam-5098	283	10	b]ϕ	b]ϕ	NOUN
ejpam-5098	283	11	=	=	PUNCT
ejpam-5098	284	1	[	[	X
ejpam-5098	284	2	a	a	DET
ejpam-5098	284	3	∨	∨	NUM
ejpam-5098	284	4	b]ϕ	b]ϕ	NOUN
ejpam-5098	284	5	,	,	PUNCT
ejpam-5098	284	6	where	where	SCONJ
ejpam-5098	284	7	[	[	X
ejpam-5098	284	8	a]ϕ	a]ϕ	NOUN
ejpam-5098	284	9	is	be	AUX
ejpam-5098	284	10	the	the	DET
ejpam-5098	284	11	congruence	congruence	ADJ
ejpam-5098	284	12	class	class	NOUN
ejpam-5098	284	13	of	of	ADP
ejpam-5098	284	14	x	x	X
ejpam-5098	284	15	modulo	modulo	PROPN
ejpam-5098	284	16	ϕ.	ϕ.	PROPN
ejpam-5098	284	17	it	it	PRON
ejpam-5098	284	18	can	can	AUX
ejpam-5098	284	19	be	be	AUX
ejpam-5098	284	20	easily	easily	ADV
ejpam-5098	284	21	verified	verify	VERB
ejpam-5098	284	22	that	that	SCONJ
ejpam-5098	284	23	f	f	X
ejpam-5098	284	24	:	:	PUNCT
ejpam-5098	284	25	r→	r→	PROPN
ejpam-5098	284	26	r/ϕ	r/ϕ	PROPN
ejpam-5098	284	27	is	be	AUX
ejpam-5098	284	28	a	a	DET
ejpam-5098	284	29	homomorphism	homomorphism	NOUN
ejpam-5098	284	30	by	by	ADP
ejpam-5098	284	31	defining	define	VERB
ejpam-5098	284	32	f(a	f(a	NOUN
ejpam-5098	284	33	)	)	PUNCT
ejpam-5098	284	34	=	=	NOUN
ejpam-5098	285	1	[	[	X
ejpam-5098	285	2	a]ϕ.	a]ϕ.	X
ejpam-5098	285	3	lemma	lemma	PROPN
ejpam-5098	285	4	2	2	X
ejpam-5098	285	5	.	.	PUNCT
ejpam-5098	285	6	let	let	VERB
ejpam-5098	285	7	ϕ	ϕ	NOUN
ejpam-5098	285	8	be	be	AUX
ejpam-5098	285	9	a	a	DET
ejpam-5098	285	10	congruence	congruence	NOUN
ejpam-5098	285	11	relation	relation	NOUN
ejpam-5098	285	12	on	on	ADP
ejpam-5098	285	13	an	an	DET
ejpam-5098	285	14	adl	adl	NOUN
ejpam-5098	285	15	r	r	NOUN
ejpam-5098	285	16	with	with	ADP
ejpam-5098	285	17	a	a	DET
ejpam-5098	285	18	maximal	maximal	ADJ
ejpam-5098	285	19	element	element	NOUN
ejpam-5098	285	20	m	m	PROPN
ejpam-5098	285	21	and	and	CCONJ
ejpam-5098	285	22	a	a	DET
ejpam-5098	285	23	,	,	PUNCT
ejpam-5098	285	24	b	b	PROPN
ejpam-5098	285	25	∈	∈	PROPN
ejpam-5098	285	26	r.	r.	PROPN
ejpam-5098	285	27	then	then	ADV
ejpam-5098	285	28	(	(	PUNCT
ejpam-5098	285	29	1	1	X
ejpam-5098	285	30	)	)	PUNCT
ejpam-5098	285	31	a	a	DET
ejpam-5098	285	32	≤	≤	NUM
ejpam-5098	285	33	b⇒	b⇒	X
ejpam-5098	286	1	[	[	X
ejpam-5098	286	2	a]ϕ	a]ϕ	NOUN
ejpam-5098	286	3	⊆	⊆	NUM
ejpam-5098	286	4	[	[	X
ejpam-5098	286	5	b]ϕ	b]ϕ	X
ejpam-5098	286	6	(	(	PUNCT
ejpam-5098	286	7	2	2	NUM
ejpam-5098	286	8	)	)	PUNCT
ejpam-5098	287	1	[	[	X
ejpam-5098	287	2	a]ϕ	a]ϕ	NOUN
ejpam-5098	287	3	=	=	PUNCT
ejpam-5098	288	1	[	[	X
ejpam-5098	288	2	0]ϕ	0]ϕ	X
ejpam-5098	288	3	⇔	⇔	X
ejpam-5098	288	4	a	a	X
ejpam-5098	288	5	=	=	SYM
ejpam-5098	288	6	0	0	NUM
ejpam-5098	288	7	(	(	PUNCT
ejpam-5098	288	8	3	3	NUM
ejpam-5098	288	9	)	)	PUNCT
ejpam-5098	288	10	[	[	X
ejpam-5098	288	11	a]ϕ	a]ϕ	NOUN
ejpam-5098	288	12	=	=	PUNCT
ejpam-5098	288	13	[	[	X
ejpam-5098	288	14	m]ϕ	m]ϕ	X
ejpam-5098	288	15	⇔	⇔	NOUN
ejpam-5098	288	16	(	(	PUNCT
ejpam-5098	288	17	a)∗	a)∗	PROPN
ejpam-5098	288	18	=	=	PRON
ejpam-5098	288	19	{	{	PUNCT
ejpam-5098	288	20	0	0	NUM
ejpam-5098	288	21	}	}	PUNCT
ejpam-5098	288	22	.	.	PUNCT
ejpam-5098	289	1	for	for	ADP
ejpam-5098	289	2	any	any	DET
ejpam-5098	289	3	ideal	ideal	ADJ
ejpam-5098	289	4	k	k	PROPN
ejpam-5098	289	5	of	of	ADP
ejpam-5098	289	6	r	r	NOUN
ejpam-5098	289	7	,	,	PUNCT
ejpam-5098	289	8	define	define	VERB
ejpam-5098	289	9	k̃	k̃	PROPN
ejpam-5098	289	10	=	=	PUNCT
ejpam-5098	289	11	{	{	PUNCT
ejpam-5098	289	12	[	[	X
ejpam-5098	289	13	a]ϕ	a]ϕ	NOUN
ejpam-5098	289	14	∈	∈	NOUN
ejpam-5098	289	15	r/ϕ	r/ϕ	NOUN
ejpam-5098	289	16	|	|	ADV
ejpam-5098	289	17	a	a	DET
ejpam-5098	289	18	∈	∈	NOUN
ejpam-5098	289	19	k	k	NOUN
ejpam-5098	289	20	}	}	PUNCT
ejpam-5098	289	21	.	.	PUNCT
ejpam-5098	290	1	clearly	clearly	ADV
ejpam-5098	290	2	,	,	PUNCT
ejpam-5098	290	3	we	we	PRON
ejpam-5098	290	4	have	have	VERB
ejpam-5098	290	5	that	that	SCONJ
ejpam-5098	290	6	k	k	PROPN
ejpam-5098	290	7	⊆	⊆	NUM
ejpam-5098	290	8	k̃.	k̃.	PROPN
ejpam-5098	290	9	theorem	theorem	VERB
ejpam-5098	290	10	10	10	NUM
ejpam-5098	290	11	.	.	PUNCT
ejpam-5098	291	1	for	for	ADP
ejpam-5098	291	2	any	any	DET
ejpam-5098	291	3	α	α	NOUN
ejpam-5098	291	4	-	-	PUNCT
ejpam-5098	291	5	ideal	ideal	ADJ
ejpam-5098	291	6	k	k	PROPN
ejpam-5098	291	7	of	of	ADP
ejpam-5098	291	8	an	an	DET
ejpam-5098	291	9	adl	adl	PROPN
ejpam-5098	291	10	r	r	NOUN
ejpam-5098	291	11	,	,	PUNCT
ejpam-5098	291	12	we	we	PRON
ejpam-5098	291	13	have	have	VERB
ejpam-5098	291	14	the	the	DET
ejpam-5098	291	15	following	following	NOUN
ejpam-5098	291	16	:	:	PUNCT
ejpam-5098	291	17	(	(	PUNCT
ejpam-5098	291	18	1	1	X
ejpam-5098	291	19	)	)	PUNCT
ejpam-5098	292	1	[	[	X
ejpam-5098	292	2	x]ϕ	x]ϕ	NOUN
ejpam-5098	292	3	∈	∈	PROPN
ejpam-5098	292	4	k̃	k̃	PROPN
ejpam-5098	292	5	⇔	⇔	PROPN
ejpam-5098	292	6	x	x	SYM
ejpam-5098	292	7	∈	∈	PROPN
ejpam-5098	292	8	k	k	X
ejpam-5098	292	9	(	(	PUNCT
ejpam-5098	292	10	2	2	X
ejpam-5098	292	11	)	)	PUNCT
ejpam-5098	292	12	k̃	k̃	PROPN
ejpam-5098	292	13	is	be	AUX
ejpam-5098	292	14	an	an	DET
ejpam-5098	292	15	ideal	ideal	NOUN
ejpam-5098	292	16	of	of	ADP
ejpam-5098	292	17	r/ϕ	r/ϕ	NOUN
ejpam-5098	292	18	(	(	PUNCT
ejpam-5098	292	19	3	3	NUM
ejpam-5098	292	20	)	)	PUNCT
ejpam-5098	292	21	if	if	SCONJ
ejpam-5098	292	22	k̃	k̃	PROPN
ejpam-5098	292	23	is	be	AUX
ejpam-5098	292	24	a	a	DET
ejpam-5098	292	25	prime	prime	ADJ
ejpam-5098	292	26	ideal	ideal	NOUN
ejpam-5098	292	27	of	of	ADP
ejpam-5098	292	28	r	r	NOUN
ejpam-5098	292	29	,	,	PUNCT
ejpam-5098	292	30	then	then	ADV
ejpam-5098	292	31	k̃	k̃	PROPN
ejpam-5098	292	32	is	be	AUX
ejpam-5098	292	33	a	a	DET
ejpam-5098	292	34	prime	prime	ADJ
ejpam-5098	292	35	ideal	ideal	NOUN
ejpam-5098	292	36	of	of	ADP
ejpam-5098	292	37	r/ϕ.	r/ϕ.	PROPN
ejpam-5098	292	38	r.	r.	PROPN
ejpam-5098	292	39	noorbhasha	noorbhasha	PROPN
ejpam-5098	292	40	,	,	PUNCT
ejpam-5098	292	41	r.	r.	PROPN
ejpam-5098	292	42	bandaru	bandaru	PROPN
ejpam-5098	292	43	,	,	PUNCT
ejpam-5098	292	44	a.	a.	NOUN
ejpam-5098	292	45	iampan	iampan	PROPN
ejpam-5098	292	46	/	/	SYM
ejpam-5098	292	47	eur	eur	PROPN
ejpam-5098	292	48	.	.	PUNCT
ejpam-5098	293	1	j.	j.	PROPN
ejpam-5098	293	2	pure	pure	PROPN
ejpam-5098	293	3	appl	appl	PROPN
ejpam-5098	293	4	.	.	PROPN
ejpam-5098	293	5	math	math	PROPN
ejpam-5098	293	6	,	,	PUNCT
ejpam-5098	293	7	17	17	NUM
ejpam-5098	293	8	(	(	PUNCT
ejpam-5098	293	9	2	2	NUM
ejpam-5098	293	10	)	)	PUNCT
ejpam-5098	293	11	(	(	PUNCT
ejpam-5098	293	12	2024	2024	NUM
ejpam-5098	293	13	)	)	PUNCT
ejpam-5098	293	14	,	,	PUNCT
ejpam-5098	293	15	1094	1094	NUM
ejpam-5098	293	16	-	-	SYM
ejpam-5098	293	17	1112	1112	NUM
ejpam-5098	293	18	1103	1103	NUM
ejpam-5098	293	19	proof	proof	NOUN
ejpam-5098	293	20	.	.	PUNCT
ejpam-5098	294	1	(	(	PUNCT
ejpam-5098	294	2	1	1	X
ejpam-5098	294	3	)	)	PUNCT
ejpam-5098	294	4	assume	assume	VERB
ejpam-5098	294	5	that	that	SCONJ
ejpam-5098	294	6	[	[	X
ejpam-5098	294	7	x]ϕ	x]ϕ	PROPN
ejpam-5098	294	8	∈	∈	PROPN
ejpam-5098	294	9	k̃.	k̃.	PROPN
ejpam-5098	294	10	then	then	ADV
ejpam-5098	294	11	there	there	PRON
ejpam-5098	294	12	exists	exist	VERB
ejpam-5098	294	13	an	an	DET
ejpam-5098	294	14	element	element	NOUN
ejpam-5098	294	15	b	b	PROPN
ejpam-5098	294	16	∈	∈	PROPN
ejpam-5098	294	17	k	k	ADP
ejpam-5098	295	1	such	such	ADJ
ejpam-5098	295	2	that	that	SCONJ
ejpam-5098	296	1	[	[	X
ejpam-5098	296	2	a]ϕ	a]ϕ	NOUN
ejpam-5098	296	3	=	=	PUNCT
ejpam-5098	297	1	[	[	X
ejpam-5098	297	2	b]ϕ.	b]ϕ.	X
ejpam-5098	297	3	that	that	PRON
ejpam-5098	297	4	implies	imply	VERB
ejpam-5098	297	5	(	(	PUNCT
ejpam-5098	297	6	a	a	DET
ejpam-5098	297	7	,	,	PUNCT
ejpam-5098	297	8	b	b	NOUN
ejpam-5098	297	9	)	)	PUNCT
ejpam-5098	297	10	∈	∈	PROPN
ejpam-5098	297	11	ϕ	ϕ	NOUN
ejpam-5098	297	12	and	and	CCONJ
ejpam-5098	297	13	hence	hence	ADV
ejpam-5098	297	14	(	(	PUNCT
ejpam-5098	297	15	a)∗	a)∗	PROPN
ejpam-5098	297	16	=	=	PRON
ejpam-5098	297	17	(	(	PUNCT
ejpam-5098	297	18	b)∗.	b)∗.	PROPN
ejpam-5098	297	19	since	since	SCONJ
ejpam-5098	297	20	k	k	PROPN
ejpam-5098	297	21	is	be	AUX
ejpam-5098	297	22	an	an	DET
ejpam-5098	297	23	α	α	NOUN
ejpam-5098	297	24	-	-	NOUN
ejpam-5098	297	25	ideal	ideal	NOUN
ejpam-5098	297	26	of	of	ADP
ejpam-5098	297	27	r	r	NOUN
ejpam-5098	297	28	and	and	CCONJ
ejpam-5098	297	29	b	b	PROPN
ejpam-5098	297	30	∈	∈	PROPN
ejpam-5098	298	1	k	k	NOUN
ejpam-5098	298	2	,	,	PUNCT
ejpam-5098	298	3	we	we	PRON
ejpam-5098	298	4	get	get	VERB
ejpam-5098	298	5	that	that	SCONJ
ejpam-5098	298	6	a	a	DET
ejpam-5098	298	7	∈	∈	PROPN
ejpam-5098	298	8	k.	k.	PROPN
ejpam-5098	298	9	assume	assume	VERB
ejpam-5098	299	1	that	that	SCONJ
ejpam-5098	299	2	a	a	DET
ejpam-5098	299	3	∈	∈	PROPN
ejpam-5098	299	4	k.	k.	NOUN
ejpam-5098	300	1	then	then	ADV
ejpam-5098	300	2	[	[	X
ejpam-5098	300	3	a]ϕ	a]ϕ	NOUN
ejpam-5098	300	4	∈	∈	PROPN
ejpam-5098	300	5	k̃.	k̃.	NOUN
ejpam-5098	300	6	(	(	PUNCT
ejpam-5098	300	7	2	2	NUM
ejpam-5098	300	8	)	)	PUNCT
ejpam-5098	300	9	by	by	ADP
ejpam-5098	300	10	(	(	PUNCT
ejpam-5098	300	11	1	1	NUM
ejpam-5098	300	12	)	)	PUNCT
ejpam-5098	300	13	,	,	PUNCT
ejpam-5098	300	14	it	it	PRON
ejpam-5098	300	15	is	be	AUX
ejpam-5098	300	16	clear	clear	ADJ
ejpam-5098	300	17	.	.	PUNCT
ejpam-5098	301	1	(	(	PUNCT
ejpam-5098	301	2	3	3	X
ejpam-5098	301	3	)	)	PUNCT
ejpam-5098	301	4	by	by	ADP
ejpam-5098	301	5	(	(	PUNCT
ejpam-5098	301	6	1	1	NUM
ejpam-5098	301	7	)	)	PUNCT
ejpam-5098	301	8	and	and	CCONJ
ejpam-5098	301	9	(	(	PUNCT
ejpam-5098	301	10	2	2	NUM
ejpam-5098	301	11	)	)	PUNCT
ejpam-5098	301	12	,	,	PUNCT
ejpam-5098	301	13	it	it	PRON
ejpam-5098	301	14	can	can	AUX
ejpam-5098	301	15	be	be	AUX
ejpam-5098	301	16	verified	verify	VERB
ejpam-5098	301	17	easily	easily	ADV
ejpam-5098	301	18	.	.	PUNCT
ejpam-5098	302	1	from	from	ADP
ejpam-5098	302	2	the	the	DET
ejpam-5098	302	3	above	above	ADJ
ejpam-5098	302	4	result	result	NOUN
ejpam-5098	302	5	,	,	PUNCT
ejpam-5098	302	6	we	we	PRON
ejpam-5098	302	7	get	get	VERB
ejpam-5098	302	8	the	the	DET
ejpam-5098	302	9	following	follow	VERB
ejpam-5098	302	10	result	result	NOUN
ejpam-5098	302	11	:	:	PUNCT
ejpam-5098	302	12	corollary	corollary	ADJ
ejpam-5098	302	13	1	1	NUM
ejpam-5098	302	14	.	.	PUNCT
ejpam-5098	303	1	for	for	ADP
ejpam-5098	303	2	any	any	DET
ejpam-5098	303	3	α	α	NOUN
ejpam-5098	303	4	-	-	PUNCT
ejpam-5098	303	5	ideals	ideal	NOUN
ejpam-5098	303	6	k1,k2	k1,k2	PROPN
ejpam-5098	303	7	of	of	ADP
ejpam-5098	303	8	an	an	DET
ejpam-5098	303	9	adl	adl	PROPN
ejpam-5098	303	10	r	r	PROPN
ejpam-5098	303	11	,	,	PUNCT
ejpam-5098	303	12	k1	k1	NOUN
ejpam-5098	303	13	⊆	⊆	NUM
ejpam-5098	303	14	k2	k2	PROPN
ejpam-5098	303	15	⇔	⇔	PROPN
ejpam-5098	303	16	k̃1	k̃1	PROPN
ejpam-5098	303	17	⊆	⊆	NUM
ejpam-5098	303	18	k̃2	k̃2	PROPN
ejpam-5098	303	19	.	.	PROPN
ejpam-5098	304	1	for	for	ADP
ejpam-5098	304	2	any	any	DET
ejpam-5098	304	3	s	s	X
ejpam-5098	304	4	∈	∈	PROPN
ejpam-5098	304	5	r	r	NOUN
ejpam-5098	304	6	,	,	PUNCT
ejpam-5098	304	7	define	define	VERB
ejpam-5098	304	8	(	(	PUNCT
ejpam-5098	304	9	[	[	X
ejpam-5098	304	10	s]ϕ	s]ϕ	NOUN
ejpam-5098	304	11	)	)	PUNCT
ejpam-5098	304	12	◦	◦	NOUN
ejpam-5098	304	13	=	=	PUNCT
ejpam-5098	304	14	{	{	PUNCT
ejpam-5098	304	15	[	[	X
ejpam-5098	304	16	t]ϕ	t]ϕ	NOUN
ejpam-5098	304	17	∈	∈	NOUN
ejpam-5098	304	18	r/ϕ	r/ϕ	NOUN
ejpam-5098	305	1	|	|	ADV
ejpam-5098	306	1	[	[	X
ejpam-5098	306	2	s]ϕ	s]ϕ	NOUN
ejpam-5098	306	3	∧	∧	NOUN
ejpam-5098	307	1	[	[	X
ejpam-5098	307	2	t]ϕ	t]ϕ	NOUN
ejpam-5098	307	3	=	=	PUNCT
ejpam-5098	308	1	[	[	X
ejpam-5098	308	2	0]ϕ	0]ϕ	X
ejpam-5098	308	3	}	}	PUNCT
ejpam-5098	308	4	.	.	PUNCT
ejpam-5098	309	1	lemma	lemma	PROPN
ejpam-5098	309	2	3	3	X
ejpam-5098	309	3	.	.	X
ejpam-5098	310	1	for	for	ADP
ejpam-5098	310	2	any	any	DET
ejpam-5098	310	3	s	s	NOUN
ejpam-5098	310	4	,	,	PUNCT
ejpam-5098	310	5	t	t	PROPN
ejpam-5098	310	6	∈	∈	PROPN
ejpam-5098	310	7	r	r	NOUN
ejpam-5098	310	8	,	,	PUNCT
ejpam-5098	310	9	we	we	PRON
ejpam-5098	310	10	have	have	VERB
ejpam-5098	310	11	the	the	DET
ejpam-5098	310	12	following	following	NOUN
ejpam-5098	310	13	:	:	PUNCT
ejpam-5098	310	14	(	(	PUNCT
ejpam-5098	310	15	1	1	X
ejpam-5098	310	16	)	)	PUNCT
ejpam-5098	310	17	(	(	PUNCT
ejpam-5098	310	18	[	[	X
ejpam-5098	310	19	s]ϕ	s]ϕ	NOUN
ejpam-5098	310	20	)	)	PUNCT
ejpam-5098	310	21	◦	◦	NOUN
ejpam-5098	310	22	=	=	PUNCT
ejpam-5098	310	23	{	{	PUNCT
ejpam-5098	310	24	[	[	X
ejpam-5098	310	25	t]ϕ	t]ϕ	NOUN
ejpam-5098	310	26	∈	∈	NOUN
ejpam-5098	310	27	r/ϕ	r/ϕ	NOUN
ejpam-5098	310	28	|	|	NOUN
ejpam-5098	310	29	s	s	VERB
ejpam-5098	310	30	∧	∧	PROPN
ejpam-5098	310	31	t	t	NOUN
ejpam-5098	310	32	=	=	SYM
ejpam-5098	310	33	0	0	NUM
ejpam-5098	310	34	}	}	PUNCT
ejpam-5098	310	35	(	(	PUNCT
ejpam-5098	310	36	2	2	NUM
ejpam-5098	310	37	)	)	PUNCT
ejpam-5098	310	38	(	(	PUNCT
ejpam-5098	311	1	[	[	X
ejpam-5098	311	2	s]ϕ	s]ϕ	NOUN
ejpam-5098	311	3	)	)	PUNCT
ejpam-5098	311	4	◦	◦	NOUN
ejpam-5098	311	5	is	be	AUX
ejpam-5098	311	6	an	an	DET
ejpam-5098	311	7	ideal	ideal	NOUN
ejpam-5098	311	8	of	of	ADP
ejpam-5098	311	9	r/ϕ	r/ϕ	NOUN
ejpam-5098	311	10	(	(	PUNCT
ejpam-5098	311	11	3	3	NUM
ejpam-5098	311	12	)	)	PUNCT
ejpam-5098	311	13	(	(	PUNCT
ejpam-5098	311	14	[	[	X
ejpam-5098	311	15	0]ϕ	0]ϕ	NOUN
ejpam-5098	311	16	)	)	PUNCT
ejpam-5098	311	17	◦	◦	NOUN
ejpam-5098	311	18	=	=	SYM
ejpam-5098	311	19	r/ϕ	r/ϕ	NOUN
ejpam-5098	311	20	and	and	CCONJ
ejpam-5098	311	21	(	(	PUNCT
ejpam-5098	311	22	[	[	X
ejpam-5098	311	23	m]ϕ	m]ϕ	NOUN
ejpam-5098	311	24	)	)	PUNCT
ejpam-5098	311	25	◦	◦	NOUN
ejpam-5098	311	26	=	=	SYM
ejpam-5098	311	27	(	(	PUNCT
ejpam-5098	311	28	[	[	X
ejpam-5098	311	29	0]ϕ	0]ϕ	NOUN
ejpam-5098	311	30	)	)	PUNCT
ejpam-5098	311	31	◦	◦	NOUN
ejpam-5098	311	32	,	,	PUNCT
ejpam-5098	311	33	where	where	SCONJ
ejpam-5098	311	34	m	m	NOUN
ejpam-5098	311	35	is	be	AUX
ejpam-5098	311	36	any	any	DET
ejpam-5098	311	37	maximal	maximal	ADJ
ejpam-5098	311	38	element	element	NOUN
ejpam-5098	311	39	of	of	ADP
ejpam-5098	311	40	r	r	NOUN
ejpam-5098	311	41	(	(	PUNCT
ejpam-5098	311	42	4	4	NUM
ejpam-5098	311	43	)	)	PUNCT
ejpam-5098	311	44	t	t	NOUN
ejpam-5098	311	45	∈	∈	PROPN
ejpam-5098	311	46	(	(	PUNCT
ejpam-5098	311	47	s)∗	s)∗	ADJ
ejpam-5098	311	48	⇔	⇔	X
ejpam-5098	312	1	[	[	X
ejpam-5098	312	2	t]ϕ	t]ϕ	X
ejpam-5098	312	3	∈	∈	NOUN
ejpam-5098	312	4	(	(	PUNCT
ejpam-5098	312	5	[	[	X
ejpam-5098	312	6	s]ϕ	s]ϕ	NOUN
ejpam-5098	312	7	)	)	PUNCT
ejpam-5098	312	8	◦	◦	NOUN
ejpam-5098	312	9	(	(	PUNCT
ejpam-5098	312	10	5	5	NUM
ejpam-5098	312	11	)	)	PUNCT
ejpam-5098	312	12	(	(	PUNCT
ejpam-5098	312	13	s)∗	s)∗	ADJ
ejpam-5098	312	14	=	=	SYM
ejpam-5098	312	15	(	(	PUNCT
ejpam-5098	312	16	t)∗	t)∗	PROPN
ejpam-5098	312	17	⇔	⇔	X
ejpam-5098	312	18	(	(	PUNCT
ejpam-5098	312	19	[	[	X
ejpam-5098	312	20	s]ϕ	s]ϕ	NOUN
ejpam-5098	312	21	)	)	PUNCT
ejpam-5098	312	22	◦	◦	NOUN
ejpam-5098	312	23	=	=	SYM
ejpam-5098	312	24	(	(	PUNCT
ejpam-5098	312	25	[	[	X
ejpam-5098	312	26	t]ϕ	t]ϕ	NOUN
ejpam-5098	312	27	)	)	PUNCT
ejpam-5098	312	28	◦	◦	NOUN
ejpam-5098	312	29	(	(	PUNCT
ejpam-5098	312	30	6	6	NUM
ejpam-5098	312	31	)	)	PUNCT
ejpam-5098	313	1	[	[	X
ejpam-5098	313	2	s]ϕ	s]ϕ	NOUN
ejpam-5098	313	3	⊆	⊆	NUM
ejpam-5098	313	4	[	[	X
ejpam-5098	313	5	t]ϕ	t]ϕ	NOUN
ejpam-5098	313	6	⇒	⇒	NOUN
ejpam-5098	313	7	(	(	PUNCT
ejpam-5098	313	8	[	[	X
ejpam-5098	313	9	t]ϕ	t]ϕ	NOUN
ejpam-5098	313	10	)	)	PUNCT
ejpam-5098	313	11	◦	◦	NOUN
ejpam-5098	313	12	⊆	⊆	NUM
ejpam-5098	313	13	(	(	PUNCT
ejpam-5098	313	14	[	[	X
ejpam-5098	313	15	s]ϕ	s]ϕ	NOUN
ejpam-5098	313	16	)	)	PUNCT
ejpam-5098	313	17	◦	◦	NOUN
ejpam-5098	313	18	(	(	PUNCT
ejpam-5098	313	19	7	7	NUM
ejpam-5098	313	20	)	)	PUNCT
ejpam-5098	313	21	(	(	PUNCT
ejpam-5098	313	22	[	[	X
ejpam-5098	313	23	s]ϕ	s]ϕ	NOUN
ejpam-5098	313	24	)	)	PUNCT
ejpam-5098	313	25	◦	◦	NOUN
ejpam-5098	313	26	∩	∩	NOUN
ejpam-5098	313	27	(	(	PUNCT
ejpam-5098	313	28	[	[	X
ejpam-5098	313	29	t]ϕ	t]ϕ	NOUN
ejpam-5098	313	30	)	)	PUNCT
ejpam-5098	313	31	◦	◦	NOUN
ejpam-5098	313	32	=	=	SYM
ejpam-5098	313	33	(	(	PUNCT
ejpam-5098	313	34	[	[	X
ejpam-5098	313	35	s	s	X
ejpam-5098	313	36	∨	∨	NUM
ejpam-5098	313	37	t]ϕ)	t]ϕ)	PROPN
ejpam-5098	313	38	◦	◦	NOUN
ejpam-5098	313	39	.	.	PUNCT
ejpam-5098	314	1	definition	definition	NOUN
ejpam-5098	314	2	11	11	NUM
ejpam-5098	314	3	.	.	PUNCT
ejpam-5098	315	1	let	let	VERB
ejpam-5098	315	2	k	k	PRON
ejpam-5098	315	3	be	be	AUX
ejpam-5098	315	4	an	an	DET
ejpam-5098	315	5	ideal	ideal	NOUN
ejpam-5098	315	6	of	of	ADP
ejpam-5098	315	7	r/ϕ.	r/ϕ.	VERB
ejpam-5098	315	8	define	define	VERB
ejpam-5098	315	9	ρ(k	ρ(k	NOUN
ejpam-5098	315	10	)	)	PUNCT
ejpam-5098	316	1	=	=	PRON
ejpam-5098	316	2	{	{	PUNCT
ejpam-5098	316	3	[	[	X
ejpam-5098	316	4	a]ϕ	a]ϕ	NOUN
ejpam-5098	316	5	∈	∈	NOUN
ejpam-5098	316	6	r/ϕ	r/ϕ	NOUN
ejpam-5098	316	7	|	|	ADV
ejpam-5098	316	8	(	(	PUNCT
ejpam-5098	316	9	[	[	X
ejpam-5098	316	10	a]ϕ	a]ϕ	NOUN
ejpam-5098	316	11	)	)	PUNCT
ejpam-5098	316	12	◦	◦	NOUN
ejpam-5098	316	13	∨	∨	NOUN
ejpam-5098	316	14	k	k	X
ejpam-5098	316	15	=	=	SYM
ejpam-5098	316	16	r/ϕ	r/ϕ	NOUN
ejpam-5098	316	17	}	}	PUNCT
ejpam-5098	316	18	,	,	PUNCT
ejpam-5098	316	19	where	where	SCONJ
ejpam-5098	316	20	(	(	PUNCT
ejpam-5098	316	21	[	[	X
ejpam-5098	316	22	a]ϕ	a]ϕ	NOUN
ejpam-5098	316	23	)	)	PUNCT
ejpam-5098	316	24	◦	◦	NOUN
ejpam-5098	316	25	∨k	∨k	NOUN
ejpam-5098	316	26	is	be	AUX
ejpam-5098	316	27	the	the	DET
ejpam-5098	316	28	supremum	supremum	NOUN
ejpam-5098	316	29	of	of	ADP
ejpam-5098	316	30	(	(	PUNCT
ejpam-5098	316	31	[	[	X
ejpam-5098	316	32	a]ϕ	a]ϕ	NOUN
ejpam-5098	316	33	)	)	PUNCT
ejpam-5098	316	34	◦	◦	NOUN
ejpam-5098	316	35	and	and	CCONJ
ejpam-5098	316	36	k	k	PROPN
ejpam-5098	316	37	in	in	ADP
ejpam-5098	316	38	r/ϕ.	r/ϕ.	PROPN
ejpam-5098	316	39	theorem	theorem	VERB
ejpam-5098	316	40	11	11	NUM
ejpam-5098	316	41	.	.	PUNCT
ejpam-5098	317	1	let	let	VERB
ejpam-5098	317	2	k	k	PRON
ejpam-5098	317	3	be	be	AUX
ejpam-5098	317	4	an	an	DET
ejpam-5098	317	5	ideal	ideal	NOUN
ejpam-5098	317	6	of	of	ADP
ejpam-5098	317	7	r	r	NOUN
ejpam-5098	317	8	with	with	ADP
ejpam-5098	317	9	maximal	maximal	ADJ
ejpam-5098	317	10	elements	element	NOUN
ejpam-5098	317	11	and	and	CCONJ
ejpam-5098	317	12	ϕ	ϕ	NOUN
ejpam-5098	317	13	be	be	AUX
ejpam-5098	317	14	the	the	DET
ejpam-5098	317	15	congruence	congruence	NOUN
ejpam-5098	317	16	relation	relation	NOUN
ejpam-5098	317	17	on	on	ADP
ejpam-5098	317	18	r.	r.	PROPN
ejpam-5098	317	19	then	then	ADV
ejpam-5098	317	20	we	we	PRON
ejpam-5098	317	21	have	have	AUX
ejpam-5098	317	22	the	the	DET
ejpam-5098	317	23	following	following	NOUN
ejpam-5098	317	24	:	:	PUNCT
ejpam-5098	317	25	(	(	PUNCT
ejpam-5098	317	26	1	1	X
ejpam-5098	317	27	)	)	PUNCT
ejpam-5098	317	28	ρ(k	ρ(k	PROPN
ejpam-5098	317	29	)	)	PUNCT
ejpam-5098	317	30	is	be	AUX
ejpam-5098	317	31	an	an	DET
ejpam-5098	317	32	ideal	ideal	NOUN
ejpam-5098	317	33	of	of	ADP
ejpam-5098	317	34	r/ϕ	r/ϕ	NOUN
ejpam-5098	317	35	contained	contain	VERB
ejpam-5098	317	36	in	in	ADP
ejpam-5098	317	37	k	k	PROPN
ejpam-5098	317	38	(	(	PUNCT
ejpam-5098	317	39	2	2	NUM
ejpam-5098	317	40	)	)	PUNCT
ejpam-5098	317	41	if	if	SCONJ
ejpam-5098	317	42	k	k	PROPN
ejpam-5098	317	43	is	be	AUX
ejpam-5098	317	44	an	an	DET
ejpam-5098	317	45	α	α	NOUN
ejpam-5098	317	46	-	-	NOUN
ejpam-5098	317	47	ideal	ideal	NOUN
ejpam-5098	317	48	of	of	ADP
ejpam-5098	317	49	r	r	NOUN
ejpam-5098	317	50	,	,	PUNCT
ejpam-5098	317	51	then	then	ADV
ejpam-5098	317	52	so	so	ADV
ejpam-5098	317	53	is	be	AUX
ejpam-5098	317	54	k̃	k̃	PROPN
ejpam-5098	317	55	in	in	ADP
ejpam-5098	317	56	r/ϕ	r/ϕ	NOUN
ejpam-5098	317	57	(	(	PUNCT
ejpam-5098	317	58	3	3	NUM
ejpam-5098	317	59	)	)	PUNCT
ejpam-5098	317	60	if	if	SCONJ
ejpam-5098	317	61	k	k	PROPN
ejpam-5098	317	62	is	be	AUX
ejpam-5098	317	63	a	a	DET
ejpam-5098	317	64	σ	σ	NOUN
ejpam-5098	317	65	-	-	PUNCT
ejpam-5098	317	66	ideal	ideal	NOUN
ejpam-5098	317	67	of	of	ADP
ejpam-5098	317	68	r	r	NOUN
ejpam-5098	317	69	,	,	PUNCT
ejpam-5098	317	70	then	then	ADV
ejpam-5098	317	71	so	so	ADV
ejpam-5098	317	72	is	be	AUX
ejpam-5098	317	73	k̃	k̃	PROPN
ejpam-5098	317	74	in	in	ADP
ejpam-5098	317	75	r/ϕ.	r/ϕ.	NOUN
ejpam-5098	317	76	proof	proof	NOUN
ejpam-5098	317	77	.	.	PUNCT
ejpam-5098	318	1	(	(	PUNCT
ejpam-5098	318	2	1	1	X
ejpam-5098	318	3	)	)	PUNCT
ejpam-5098	318	4	clear	clear	ADJ
ejpam-5098	318	5	.	.	PUNCT
ejpam-5098	319	1	(	(	PUNCT
ejpam-5098	319	2	2	2	X
ejpam-5098	319	3	)	)	PUNCT
ejpam-5098	319	4	for	for	ADP
ejpam-5098	319	5	any	any	DET
ejpam-5098	319	6	α	α	NOUN
ejpam-5098	319	7	-	-	PUNCT
ejpam-5098	319	8	ideal	ideal	NOUN
ejpam-5098	319	9	k	k	NOUN
ejpam-5098	319	10	,	,	PUNCT
ejpam-5098	319	11	we	we	PRON
ejpam-5098	319	12	have	have	VERB
ejpam-5098	319	13	that	that	SCONJ
ejpam-5098	319	14	k̃	k̃	PROPN
ejpam-5098	319	15	is	be	AUX
ejpam-5098	319	16	an	an	DET
ejpam-5098	319	17	ideal	ideal	NOUN
ejpam-5098	319	18	of	of	ADP
ejpam-5098	319	19	r/ϕ.	r/ϕ.	PROPN
ejpam-5098	319	20	let	let	VERB
ejpam-5098	319	21	a	a	DET
ejpam-5098	319	22	,	,	PUNCT
ejpam-5098	319	23	b	b	X
ejpam-5098	319	24	∈	∈	NOUN
ejpam-5098	319	25	r	r	NOUN
ejpam-5098	320	1	such	such	ADJ
ejpam-5098	320	2	that	that	PRON
ejpam-5098	320	3	(	(	PUNCT
ejpam-5098	320	4	[	[	X
ejpam-5098	320	5	a]ϕ	a]ϕ	NOUN
ejpam-5098	320	6	)	)	PUNCT
ejpam-5098	320	7	◦	◦	NOUN
ejpam-5098	320	8	=	=	SYM
ejpam-5098	320	9	(	(	PUNCT
ejpam-5098	320	10	[	[	X
ejpam-5098	320	11	b]ϕ	b]ϕ	NOUN
ejpam-5098	320	12	)	)	PUNCT
ejpam-5098	320	13	◦	◦	NOUN
ejpam-5098	320	14	and	and	CCONJ
ejpam-5098	320	15	[	[	X
ejpam-5098	320	16	a]ϕ	a]ϕ	NOUN
ejpam-5098	320	17	∈	∈	PROPN
ejpam-5098	320	18	k̃.	k̃.	NOUN
ejpam-5098	320	19	then	then	ADV
ejpam-5098	320	20	we	we	PRON
ejpam-5098	320	21	have	have	VERB
ejpam-5098	320	22	that	that	PRON
ejpam-5098	320	23	(	(	PUNCT
ejpam-5098	320	24	a)∗	a)∗	PROPN
ejpam-5098	320	25	=	=	PRON
ejpam-5098	320	26	(	(	PUNCT
ejpam-5098	320	27	b)∗	b)∗	PROPN
ejpam-5098	320	28	and	and	CCONJ
ejpam-5098	320	29	a	a	DET
ejpam-5098	320	30	∈	∈	PROPN
ejpam-5098	320	31	k.	k.	NOUN
ejpam-5098	320	32	since	since	SCONJ
ejpam-5098	320	33	k	k	PROPN
ejpam-5098	320	34	is	be	AUX
ejpam-5098	320	35	an	an	DET
ejpam-5098	320	36	α	α	NOUN
ejpam-5098	320	37	-	-	PUNCT
ejpam-5098	320	38	ideal	ideal	ADJ
ejpam-5098	320	39	,	,	PUNCT
ejpam-5098	320	40	we	we	PRON
ejpam-5098	320	41	get	get	VERB
ejpam-5098	320	42	b	b	PRON
ejpam-5098	320	43	∈	∈	PROPN
ejpam-5098	320	44	k.	k.	NOUN
ejpam-5098	320	45	that	that	PRON
ejpam-5098	320	46	implies	imply	VERB
ejpam-5098	320	47	[	[	X
ejpam-5098	320	48	b]ϕ	b]ϕ	PROPN
ejpam-5098	320	49	∈	∈	PROPN
ejpam-5098	320	50	k̃.	k̃.	PROPN
ejpam-5098	320	51	therefore	therefore	ADV
ejpam-5098	320	52	,	,	PUNCT
ejpam-5098	320	53	k̃	k̃	PROPN
ejpam-5098	320	54	is	be	AUX
ejpam-5098	320	55	an	an	DET
ejpam-5098	320	56	α	α	NOUN
ejpam-5098	320	57	-	-	NOUN
ejpam-5098	320	58	ideal	ideal	NOUN
ejpam-5098	320	59	of	of	ADP
ejpam-5098	320	60	r/ϕ.	r/ϕ.	PROPN
ejpam-5098	320	61	(	(	PUNCT
ejpam-5098	320	62	3	3	X
ejpam-5098	320	63	)	)	PUNCT
ejpam-5098	320	64	let	let	VERB
ejpam-5098	320	65	k	k	X
ejpam-5098	320	66	be	be	AUX
ejpam-5098	320	67	a	a	DET
ejpam-5098	320	68	σ	σ	NOUN
ejpam-5098	320	69	-	-	PUNCT
ejpam-5098	320	70	ideal	ideal	NOUN
ejpam-5098	320	71	of	of	ADP
ejpam-5098	320	72	r.	r.	PROPN
ejpam-5098	320	73	clearly	clearly	ADV
ejpam-5098	320	74	,	,	PUNCT
ejpam-5098	320	75	k̃	k̃	PROPN
ejpam-5098	320	76	is	be	AUX
ejpam-5098	320	77	an	an	DET
ejpam-5098	320	78	ideal	ideal	NOUN
ejpam-5098	320	79	of	of	ADP
ejpam-5098	320	80	r/ϕ	r/ϕ	NOUN
ejpam-5098	320	81	and	and	CCONJ
ejpam-5098	320	82	ρ(k̃	ρ(k̃	NUM
ejpam-5098	320	83	)	)	PUNCT
ejpam-5098	320	84	⊆	⊆	NUM
ejpam-5098	320	85	k̃.	k̃.	PROPN
ejpam-5098	320	86	let	let	VERB
ejpam-5098	320	87	[	[	X
ejpam-5098	320	88	a]ϕ	a]ϕ	VERB
ejpam-5098	320	89	∈	∈	PROPN
ejpam-5098	320	90	k̃.	k̃.	PROPN
ejpam-5098	320	91	then	then	ADV
ejpam-5098	320	92	a	a	DET
ejpam-5098	320	93	∈	∈	PROPN
ejpam-5098	320	94	k.	k.	NOUN
ejpam-5098	320	95	since	since	SCONJ
ejpam-5098	320	96	k	k	PROPN
ejpam-5098	320	97	is	be	AUX
ejpam-5098	320	98	a	a	DET
ejpam-5098	320	99	σ	σ	NOUN
ejpam-5098	320	100	-	-	PUNCT
ejpam-5098	320	101	ideal	ideal	NOUN
ejpam-5098	320	102	of	of	ADP
ejpam-5098	320	103	r	r	NOUN
ejpam-5098	320	104	,	,	PUNCT
ejpam-5098	320	105	we	we	PRON
ejpam-5098	320	106	have	have	VERB
ejpam-5098	320	107	that	that	PRON
ejpam-5098	320	108	(	(	PUNCT
ejpam-5098	321	1	a)∗	a)∗	PROPN
ejpam-5098	321	2	∨	∨	NUM
ejpam-5098	321	3	k	k	PROPN
ejpam-5098	321	4	=	=	SYM
ejpam-5098	321	5	r.	r.	PROPN
ejpam-5098	321	6	then	then	ADV
ejpam-5098	321	7	there	there	PRON
ejpam-5098	321	8	exist	exist	VERB
ejpam-5098	321	9	s	s	X
ejpam-5098	321	10	∈	∈	NOUN
ejpam-5098	321	11	(	(	PUNCT
ejpam-5098	321	12	a)∗	a)∗	PROPN
ejpam-5098	321	13	,	,	PUNCT
ejpam-5098	321	14	t	t	PROPN
ejpam-5098	321	15	∈	∈	PROPN
ejpam-5098	321	16	k	k	X
ejpam-5098	321	17	such	such	ADJ
ejpam-5098	321	18	that	that	SCONJ
ejpam-5098	321	19	s∨	s∨	PROPN
ejpam-5098	321	20	t	t	PROPN
ejpam-5098	321	21	is	be	AUX
ejpam-5098	321	22	a	a	DET
ejpam-5098	321	23	maximal	maximal	ADJ
ejpam-5098	321	24	element	element	NOUN
ejpam-5098	321	25	of	of	ADP
ejpam-5098	321	26	r	r	NOUN
ejpam-5098	321	27	,	,	PUNCT
ejpam-5098	321	28	say	say	VERB
ejpam-5098	321	29	m.	m.	NOUN
ejpam-5098	321	30	that	that	PRON
ejpam-5098	321	31	implies	imply	VERB
ejpam-5098	321	32	[	[	X
ejpam-5098	321	33	m]ϕ	m]ϕ	X
ejpam-5098	321	34	=	=	NOUN
ejpam-5098	322	1	[	[	X
ejpam-5098	322	2	s	s	X
ejpam-5098	322	3	∨	∨	NUM
ejpam-5098	322	4	t]ϕ	t]ϕ	NOUN
ejpam-5098	322	5	=	=	PUNCT
ejpam-5098	323	1	[	[	X
ejpam-5098	323	2	s]ϕ	s]ϕ	NOUN
ejpam-5098	323	3	∨	∨	NOUN
ejpam-5098	324	1	[	[	X
ejpam-5098	324	2	t]ϕ	t]ϕ	NOUN
ejpam-5098	324	3	and	and	CCONJ
ejpam-5098	324	4	[	[	X
ejpam-5098	324	5	s]ϕ	s]ϕ	NOUN
ejpam-5098	324	6	∈	∈	PROPN
ejpam-5098	324	7	(	(	PUNCT
ejpam-5098	324	8	[	[	X
ejpam-5098	324	9	a]ϕ	a]ϕ	NOUN
ejpam-5098	324	10	)	)	PUNCT
ejpam-5098	324	11	◦	◦	NOUN
ejpam-5098	324	12	.	.	PUNCT
ejpam-5098	325	1	therefore	therefore	ADV
ejpam-5098	325	2	,	,	PUNCT
ejpam-5098	325	3	[	[	X
ejpam-5098	325	4	m]ϕ	m]ϕ	X
ejpam-5098	325	5	∈	∈	NOUN
ejpam-5098	325	6	(	(	PUNCT
ejpam-5098	325	7	[	[	X
ejpam-5098	325	8	a]ϕ	a]ϕ	NOUN
ejpam-5098	325	9	)	)	PUNCT
ejpam-5098	325	10	◦	◦	NOUN
ejpam-5098	325	11	∨	∨	NOUN
ejpam-5098	325	12	k̃	k̃	PROPN
ejpam-5098	325	13	and	and	CCONJ
ejpam-5098	325	14	hence	hence	ADV
ejpam-5098	325	15	(	(	PUNCT
ejpam-5098	325	16	[	[	X
ejpam-5098	325	17	a]ϕ	a]ϕ	NOUN
ejpam-5098	325	18	)	)	PUNCT
ejpam-5098	325	19	◦	◦	NOUN
ejpam-5098	325	20	∨	∨	NOUN
ejpam-5098	325	21	k̃	k̃	PROPN
ejpam-5098	325	22	=	=	PUNCT
ejpam-5098	325	23	r/ϕ.	r/ϕ.	PROPN
ejpam-5098	325	24	thus	thus	ADV
ejpam-5098	325	25	,	,	PUNCT
ejpam-5098	325	26	k̃	k̃	PROPN
ejpam-5098	325	27	is	be	AUX
ejpam-5098	325	28	a	a	DET
ejpam-5098	325	29	σ	σ	NOUN
ejpam-5098	325	30	-	-	PUNCT
ejpam-5098	325	31	ideal	ideal	NOUN
ejpam-5098	325	32	of	of	ADP
ejpam-5098	325	33	r/ϕ.	r/ϕ.	PROPN
ejpam-5098	325	34	we	we	PRON
ejpam-5098	325	35	denote	denote	VERB
ejpam-5098	325	36	specαr	specαr	ADJ
ejpam-5098	325	37	and	and	CCONJ
ejpam-5098	325	38	specαr/ϕ	specαr/ϕ	NOUN
ejpam-5098	325	39	as	as	ADP
ejpam-5098	325	40	the	the	DET
ejpam-5098	325	41	sets	set	NOUN
ejpam-5098	325	42	of	of	ADP
ejpam-5098	325	43	all	all	DET
ejpam-5098	325	44	prime	prime	ADJ
ejpam-5098	325	45	α	α	NOUN
ejpam-5098	325	46	-	-	NOUN
ejpam-5098	325	47	ideals	ideal	NOUN
ejpam-5098	325	48	of	of	ADP
ejpam-5098	325	49	r	r	NOUN
ejpam-5098	325	50	and	and	CCONJ
ejpam-5098	325	51	r/ϕ	r/ϕ	NOUN
ejpam-5098	325	52	,	,	PUNCT
ejpam-5098	325	53	respectively	respectively	ADV
ejpam-5098	325	54	.	.	PUNCT
ejpam-5098	326	1	r.	r.	PROPN
ejpam-5098	326	2	noorbhasha	noorbhasha	PROPN
ejpam-5098	326	3	,	,	PUNCT
ejpam-5098	326	4	r.	r.	PROPN
ejpam-5098	326	5	bandaru	bandaru	PROPN
ejpam-5098	326	6	,	,	PUNCT
ejpam-5098	326	7	a.	a.	NOUN
ejpam-5098	326	8	iampan	iampan	PROPN
ejpam-5098	326	9	/	/	SYM
ejpam-5098	326	10	eur	eur	PROPN
ejpam-5098	326	11	.	.	PUNCT
ejpam-5098	327	1	j.	j.	PROPN
ejpam-5098	327	2	pure	pure	PROPN
ejpam-5098	327	3	appl	appl	PROPN
ejpam-5098	327	4	.	.	PROPN
ejpam-5098	327	5	math	math	PROPN
ejpam-5098	327	6	,	,	PUNCT
ejpam-5098	327	7	17	17	NUM
ejpam-5098	327	8	(	(	PUNCT
ejpam-5098	327	9	2	2	NUM
ejpam-5098	327	10	)	)	PUNCT
ejpam-5098	327	11	(	(	PUNCT
ejpam-5098	327	12	2024	2024	NUM
ejpam-5098	327	13	)	)	PUNCT
ejpam-5098	327	14	,	,	PUNCT
ejpam-5098	327	15	1094	1094	NUM
ejpam-5098	327	16	-	-	SYM
ejpam-5098	327	17	1112	1112	NUM
ejpam-5098	327	18	1104	1104	NUM
ejpam-5098	327	19	theorem	theorem	NOUN
ejpam-5098	327	20	12	12	NUM
ejpam-5098	327	21	.	.	PUNCT
ejpam-5098	328	1	for	for	ADP
ejpam-5098	328	2	any	any	DET
ejpam-5098	328	3	congruence	congruence	PROPN
ejpam-5098	328	4	relation	relation	NOUN
ejpam-5098	328	5	ϕ	ϕ	PROPN
ejpam-5098	328	6	on	on	ADP
ejpam-5098	328	7	r	r	NOUN
ejpam-5098	328	8	,	,	PUNCT
ejpam-5098	328	9	the	the	DET
ejpam-5098	328	10	mapping	mapping	NOUN
ejpam-5098	328	11	is	be	AUX
ejpam-5098	328	12	an	an	DET
ejpam-5098	328	13	order	order	NOUN
ejpam-5098	328	14	isomorphism	isomorphism	NOUN
ejpam-5098	328	15	of	of	ADP
ejpam-5098	328	16	specαr	specαr	NOUN
ejpam-5098	328	17	onto	onto	ADP
ejpam-5098	328	18	specαr/ϕ.	specαr/ϕ.	ADJ
ejpam-5098	328	19	proof	proof	NOUN
ejpam-5098	328	20	.	.	PUNCT
ejpam-5098	329	1	let	let	VERB
ejpam-5098	329	2	m	m	PRON
ejpam-5098	329	3	∈	∈	PROPN
ejpam-5098	329	4	specαr	specαr	NOUN
ejpam-5098	329	5	.	.	PUNCT
ejpam-5098	330	1	then	then	ADV
ejpam-5098	330	2	clearly	clearly	ADV
ejpam-5098	330	3	,	,	PUNCT
ejpam-5098	330	4	we	we	PRON
ejpam-5098	330	5	have	have	VERB
ejpam-5098	330	6	that	that	DET
ejpam-5098	330	7	m̃	m̃	PROPN
ejpam-5098	330	8	is	be	AUX
ejpam-5098	330	9	an	an	DET
ejpam-5098	330	10	α	α	NOUN
ejpam-5098	330	11	-	-	NOUN
ejpam-5098	330	12	ideal	ideal	NOUN
ejpam-5098	330	13	of	of	ADP
ejpam-5098	330	14	r/ϕ.	r/ϕ.	PROPN
ejpam-5098	330	15	let	let	VERB
ejpam-5098	330	16	k	k	PROPN
ejpam-5098	330	17	∈	∈	PROPN
ejpam-5098	330	18	specαr/ϕ.	specαr/ϕ.	NOUN
ejpam-5098	330	19	take	take	VERB
ejpam-5098	330	20	h	h	NOUN
ejpam-5098	330	21	=	=	PUNCT
ejpam-5098	330	22	{	{	PUNCT
ejpam-5098	330	23	a	a	PRON
ejpam-5098	330	24	∈	∈	NOUN
ejpam-5098	330	25	r	r	NOUN
ejpam-5098	331	1	|	|	NOUN
ejpam-5098	332	1	[	[	X
ejpam-5098	332	2	a]ϕ	a]ϕ	NOUN
ejpam-5098	332	3	∈	∈	PROPN
ejpam-5098	332	4	k	k	NOUN
ejpam-5098	332	5	}	}	PUNCT
ejpam-5098	332	6	.	.	PUNCT
ejpam-5098	333	1	since	since	SCONJ
ejpam-5098	333	2	k	k	PROPN
ejpam-5098	333	3	is	be	AUX
ejpam-5098	333	4	an	an	DET
ejpam-5098	333	5	α	α	NOUN
ejpam-5098	333	6	-	-	NOUN
ejpam-5098	333	7	ideal	ideal	NOUN
ejpam-5098	333	8	of	of	ADP
ejpam-5098	333	9	r/ϕ	r/ϕ	NOUN
ejpam-5098	333	10	,	,	PUNCT
ejpam-5098	333	11	we	we	PRON
ejpam-5098	333	12	get	get	VERB
ejpam-5098	333	13	that	that	PRON
ejpam-5098	333	14	h	h	NOUN
ejpam-5098	333	15	is	be	AUX
ejpam-5098	333	16	an	an	DET
ejpam-5098	333	17	ideal	ideal	NOUN
ejpam-5098	333	18	of	of	ADP
ejpam-5098	333	19	r.	r.	PROPN
ejpam-5098	333	20	let	let	VERB
ejpam-5098	333	21	a	a	DET
ejpam-5098	333	22	,	,	PUNCT
ejpam-5098	333	23	b	b	X
ejpam-5098	333	24	∈	∈	NOUN
ejpam-5098	333	25	r	r	NOUN
ejpam-5098	333	26	with	with	ADP
ejpam-5098	333	27	(	(	PUNCT
ejpam-5098	333	28	a)∗	a)∗	PROPN
ejpam-5098	333	29	=	=	PRON
ejpam-5098	333	30	(	(	PUNCT
ejpam-5098	333	31	b)∗	b)∗	PROPN
ejpam-5098	333	32	and	and	CCONJ
ejpam-5098	333	33	a	a	DET
ejpam-5098	333	34	∈	∈	PROPN
ejpam-5098	333	35	h.	h.	NOUN
ejpam-5098	334	1	then	then	ADV
ejpam-5098	334	2	(	(	PUNCT
ejpam-5098	334	3	[	[	X
ejpam-5098	334	4	a]ϕ	a]ϕ	NOUN
ejpam-5098	334	5	)	)	PUNCT
ejpam-5098	334	6	◦	◦	NOUN
ejpam-5098	334	7	=	=	SYM
ejpam-5098	334	8	(	(	PUNCT
ejpam-5098	334	9	[	[	X
ejpam-5098	334	10	b]ϕ	b]ϕ	NOUN
ejpam-5098	334	11	)	)	PUNCT
ejpam-5098	334	12	◦	◦	NOUN
ejpam-5098	334	13	and	and	CCONJ
ejpam-5098	334	14	[	[	X
ejpam-5098	334	15	a]ϕ	a]ϕ	NOUN
ejpam-5098	334	16	∈	∈	PROPN
ejpam-5098	334	17	k.	k.	NOUN
ejpam-5098	334	18	since	since	SCONJ
ejpam-5098	334	19	k	k	PROPN
ejpam-5098	334	20	is	be	AUX
ejpam-5098	334	21	an	an	DET
ejpam-5098	334	22	α	α	NOUN
ejpam-5098	334	23	-	-	NOUN
ejpam-5098	334	24	ideal	ideal	NOUN
ejpam-5098	334	25	of	of	ADP
ejpam-5098	334	26	r/ϕ	r/ϕ	NOUN
ejpam-5098	334	27	,	,	PUNCT
ejpam-5098	334	28	we	we	PRON
ejpam-5098	334	29	get	get	VERB
ejpam-5098	334	30	[	[	PUNCT
ejpam-5098	334	31	b]ϕ	b]ϕ	VERB
ejpam-5098	334	32	∈	∈	PROPN
ejpam-5098	334	33	k.	k.	NOUN
ejpam-5098	335	1	that	that	PRON
ejpam-5098	335	2	implies	imply	VERB
ejpam-5098	335	3	b	b	PROPN
ejpam-5098	335	4	∈	∈	PROPN
ejpam-5098	335	5	h.	h.	NOUN
ejpam-5098	335	6	hence	hence	ADV
ejpam-5098	335	7	,	,	PUNCT
ejpam-5098	335	8	h	h	PROPN
ejpam-5098	335	9	is	be	AUX
ejpam-5098	335	10	an	an	DET
ejpam-5098	335	11	α	α	NOUN
ejpam-5098	335	12	-	-	NOUN
ejpam-5098	335	13	ideal	ideal	NOUN
ejpam-5098	335	14	of	of	ADP
ejpam-5098	335	15	r.	r.	PROPN
ejpam-5098	335	16	therefore	therefore	ADV
ejpam-5098	335	17	,	,	PUNCT
ejpam-5098	335	18	h̃	h̃	PROPN
ejpam-5098	335	19	=	=	SYM
ejpam-5098	335	20	k.	k.	PROPN
ejpam-5098	335	21	clearly	clearly	ADV
ejpam-5098	335	22	,	,	PUNCT
ejpam-5098	335	23	we	we	PRON
ejpam-5098	335	24	get	get	VERB
ejpam-5098	335	25	that	that	DET
ejpam-5098	335	26	h	h	PROPN
ejpam-5098	335	27	∈	∈	PROPN
ejpam-5098	335	28	specαr	specαr	NOUN
ejpam-5098	335	29	.	.	PUNCT
ejpam-5098	336	1	let	let	VERB
ejpam-5098	336	2	m	m	PRON
ejpam-5098	336	3	,	,	PUNCT
ejpam-5098	336	4	n	n	PROPN
ejpam-5098	336	5	∈	∈	PROPN
ejpam-5098	336	6	specαr	specαr	NOUN
ejpam-5098	336	7	with	with	ADP
ejpam-5098	336	8	m	m	PROPN
ejpam-5098	336	9	⊆	⊆	NUM
ejpam-5098	336	10	n	n	NOUN
ejpam-5098	336	11	.	.	PUNCT
ejpam-5098	337	1	then	then	ADV
ejpam-5098	337	2	m̃	m̃	PROPN
ejpam-5098	337	3	⊆	⊆	NUM
ejpam-5098	337	4	ñ	ñ	PROPN
ejpam-5098	337	5	.	.	PUNCT
ejpam-5098	338	1	hence	hence	ADV
ejpam-5098	338	2	,	,	PUNCT
ejpam-5098	338	3	the	the	DET
ejpam-5098	338	4	mapping	mapping	NOUN
ejpam-5098	338	5	is	be	AUX
ejpam-5098	338	6	an	an	DET
ejpam-5098	338	7	order	order	NOUN
ejpam-5098	338	8	isomorphism	isomorphism	NOUN
ejpam-5098	338	9	of	of	ADP
ejpam-5098	338	10	specαr	specαr	NOUN
ejpam-5098	338	11	onto	onto	ADP
ejpam-5098	338	12	specαr/ϕ.	specαr/ϕ.	PROPN
ejpam-5098	338	13	lemma	lemma	PROPN
ejpam-5098	338	14	4	4	X
ejpam-5098	338	15	.	.	PUNCT
ejpam-5098	339	1	in	in	ADP
ejpam-5098	339	2	an	an	DET
ejpam-5098	339	3	adl	adl	NOUN
ejpam-5098	339	4	r	r	NOUN
ejpam-5098	339	5	,	,	PUNCT
ejpam-5098	339	6	we	we	PRON
ejpam-5098	339	7	have	have	VERB
ejpam-5098	339	8	the	the	DET
ejpam-5098	339	9	following	following	NOUN
ejpam-5098	339	10	:	:	PUNCT
ejpam-5098	339	11	(	(	PUNCT
ejpam-5098	339	12	1	1	X
ejpam-5098	339	13	)	)	PUNCT
ejpam-5098	339	14	every	every	DET
ejpam-5098	339	15	proper	proper	ADJ
ejpam-5098	339	16	σ	σ	NOUN
ejpam-5098	339	17	-	-	PUNCT
ejpam-5098	339	18	ideal	ideal	NOUN
ejpam-5098	339	19	contains	contain	VERB
ejpam-5098	339	20	no	no	DET
ejpam-5098	339	21	dense	dense	ADJ
ejpam-5098	339	22	element	element	NOUN
ejpam-5098	339	23	(	(	PUNCT
ejpam-5098	339	24	2	2	NUM
ejpam-5098	339	25	)	)	PUNCT
ejpam-5098	339	26	every	every	DET
ejpam-5098	339	27	non	non	ADJ
ejpam-5098	339	28	-	-	ADJ
ejpam-5098	339	29	zero	zero	NUM
ejpam-5098	339	30	σ	σ	NOUN
ejpam-5098	339	31	-	-	PUNCT
ejpam-5098	339	32	ideal	ideal	NOUN
ejpam-5098	339	33	is	be	AUX
ejpam-5098	339	34	non	non	ADJ
ejpam-5098	339	35	-	-	ADJ
ejpam-5098	339	36	dense	dense	ADJ
ejpam-5098	339	37	(	(	PUNCT
ejpam-5098	339	38	3	3	NUM
ejpam-5098	339	39	)	)	PUNCT
ejpam-5098	339	40	every	every	DET
ejpam-5098	339	41	non	non	ADJ
ejpam-5098	339	42	-	-	ADJ
ejpam-5098	339	43	dense	dense	ADJ
ejpam-5098	339	44	prime	prime	ADJ
ejpam-5098	339	45	ideal	ideal	NOUN
ejpam-5098	339	46	is	be	AUX
ejpam-5098	339	47	an	an	DET
ejpam-5098	339	48	α	α	NOUN
ejpam-5098	339	49	-	-	PUNCT
ejpam-5098	339	50	ideal	ideal	ADJ
ejpam-5098	339	51	.	.	PUNCT
ejpam-5098	340	1	proof	proof	NOUN
ejpam-5098	340	2	.	.	PUNCT
ejpam-5098	341	1	(	(	PUNCT
ejpam-5098	341	2	1	1	X
ejpam-5098	341	3	)	)	PUNCT
ejpam-5098	341	4	let	let	VERB
ejpam-5098	341	5	k	k	PRON
ejpam-5098	341	6	be	be	AUX
ejpam-5098	341	7	any	any	DET
ejpam-5098	341	8	proper	proper	ADJ
ejpam-5098	341	9	σ	σ	NOUN
ejpam-5098	341	10	-	-	PUNCT
ejpam-5098	341	11	ideal	ideal	NOUN
ejpam-5098	341	12	of	of	ADP
ejpam-5098	341	13	r.	r.	PROPN
ejpam-5098	341	14	suppose	suppose	VERB
ejpam-5098	341	15	a	a	PRON
ejpam-5098	341	16	is	be	AUX
ejpam-5098	341	17	a	a	DET
ejpam-5098	341	18	dense	dense	ADJ
ejpam-5098	341	19	element	element	NOUN
ejpam-5098	341	20	of	of	ADP
ejpam-5098	341	21	r	r	NOUN
ejpam-5098	341	22	with	with	ADP
ejpam-5098	341	23	a	a	DET
ejpam-5098	341	24	∈	∈	PROPN
ejpam-5098	341	25	k.	k.	NOUN
ejpam-5098	342	1	then	then	ADV
ejpam-5098	342	2	(	(	PUNCT
ejpam-5098	342	3	a)∗	a)∗	PROPN
ejpam-5098	342	4	∨	∨	NUM
ejpam-5098	342	5	k	k	PROPN
ejpam-5098	342	6	=	=	PUNCT
ejpam-5098	342	7	r.	r.	PROPN
ejpam-5098	342	8	that	that	PRON
ejpam-5098	342	9	implies	imply	VERB
ejpam-5098	342	10	(	(	PUNCT
ejpam-5098	342	11	0	0	NUM
ejpam-5098	342	12	]	]	PUNCT
ejpam-5098	342	13	∨	∨	NUM
ejpam-5098	342	14	k	k	PROPN
ejpam-5098	342	15	=	=	SYM
ejpam-5098	342	16	r.	r.	PROPN
ejpam-5098	342	17	hence	hence	ADV
ejpam-5098	342	18	,	,	PUNCT
ejpam-5098	342	19	k	k	PROPN
ejpam-5098	342	20	=	=	SYM
ejpam-5098	342	21	r	r	NOUN
ejpam-5098	342	22	,	,	PUNCT
ejpam-5098	342	23	we	we	PRON
ejpam-5098	342	24	get	get	VERB
ejpam-5098	342	25	a	a	DET
ejpam-5098	342	26	contradiction	contradiction	NOUN
ejpam-5098	342	27	.	.	PUNCT
ejpam-5098	343	1	therefore	therefore	ADV
ejpam-5098	343	2	,	,	PUNCT
ejpam-5098	343	3	every	every	DET
ejpam-5098	343	4	proper	proper	ADJ
ejpam-5098	343	5	σ	σ	NOUN
ejpam-5098	343	6	-	-	PUNCT
ejpam-5098	343	7	ideal	ideal	NOUN
ejpam-5098	343	8	contains	contain	VERB
ejpam-5098	343	9	no	no	DET
ejpam-5098	343	10	dense	dense	ADJ
ejpam-5098	343	11	element	element	NOUN
ejpam-5098	343	12	.	.	PUNCT
ejpam-5098	344	1	(	(	PUNCT
ejpam-5098	344	2	2	2	X
ejpam-5098	344	3	)	)	PUNCT
ejpam-5098	344	4	let	let	VERB
ejpam-5098	344	5	k	k	PRON
ejpam-5098	344	6	be	be	AUX
ejpam-5098	344	7	a	a	DET
ejpam-5098	344	8	non	non	ADJ
ejpam-5098	344	9	-	-	ADJ
ejpam-5098	344	10	zero	zero	NUM
ejpam-5098	344	11	σ	σ	NOUN
ejpam-5098	344	12	-	-	PUNCT
ejpam-5098	344	13	ideal	ideal	NOUN
ejpam-5098	344	14	of	of	ADP
ejpam-5098	344	15	r.	r.	PROPN
ejpam-5098	344	16	then	then	ADV
ejpam-5098	344	17	there	there	PRON
ejpam-5098	344	18	exists	exist	VERB
ejpam-5098	344	19	a	a	DET
ejpam-5098	344	20	non	non	ADJ
ejpam-5098	344	21	-	-	ADJ
ejpam-5098	344	22	zero	zero	NUM
ejpam-5098	344	23	element	element	NOUN
ejpam-5098	344	24	a	a	DET
ejpam-5098	344	25	∈	∈	NOUN
ejpam-5098	344	26	r	r	NOUN
ejpam-5098	344	27	such	such	ADJ
ejpam-5098	344	28	that	that	SCONJ
ejpam-5098	344	29	a	a	DET
ejpam-5098	344	30	∈	∈	PROPN
ejpam-5098	344	31	k	k	X
ejpam-5098	345	1	=	=	PUNCT
ejpam-5098	345	2	kσ	kσ	PROPN
ejpam-5098	345	3	.	.	PROPN
ejpam-5098	346	1	that	that	PRON
ejpam-5098	346	2	implies	imply	VERB
ejpam-5098	346	3	(	(	PUNCT
ejpam-5098	346	4	a)∗	a)∗	PROPN
ejpam-5098	346	5	∨	∨	NUM
ejpam-5098	346	6	k	k	PROPN
ejpam-5098	346	7	=	=	SYM
ejpam-5098	346	8	r.	r.	PROPN
ejpam-5098	346	9	therefore	therefore	ADV
ejpam-5098	346	10	,	,	PUNCT
ejpam-5098	346	11	(	(	PUNCT
ejpam-5098	346	12	(	(	PUNCT
ejpam-5098	346	13	a)∗	a)∗	PROPN
ejpam-5098	346	14	∨	∨	NUM
ejpam-5098	346	15	k)∗	k)∗	PROPN
ejpam-5098	346	16	=	=	PUNCT
ejpam-5098	347	1	r∗	r∗	ADJ
ejpam-5098	347	2	and	and	CCONJ
ejpam-5098	347	3	hence	hence	ADV
ejpam-5098	347	4	(	(	PUNCT
ejpam-5098	347	5	a)∗∗	a)∗∗	PRON
ejpam-5098	347	6	∩	∩	NOUN
ejpam-5098	347	7	(	(	PUNCT
ejpam-5098	347	8	k)∗	k)∗	PROPN
ejpam-5098	347	9	=	=	SYM
ejpam-5098	347	10	{	{	PUNCT
ejpam-5098	347	11	0	0	NUM
ejpam-5098	347	12	}	}	PUNCT
ejpam-5098	347	13	.	.	PUNCT
ejpam-5098	348	1	suppose	suppose	VERB
ejpam-5098	348	2	k∗	k∗	PROPN
ejpam-5098	348	3	=	=	SYM
ejpam-5098	348	4	{	{	PUNCT
ejpam-5098	348	5	0	0	NUM
ejpam-5098	348	6	}	}	PUNCT
ejpam-5098	348	7	.	.	PUNCT
ejpam-5098	349	1	then	then	ADV
ejpam-5098	349	2	(	(	PUNCT
ejpam-5098	349	3	a)∗∗	a)∗∗	X
ejpam-5098	349	4	=	=	SYM
ejpam-5098	349	5	{	{	PUNCT
ejpam-5098	349	6	0	0	NUM
ejpam-5098	349	7	}	}	PUNCT
ejpam-5098	349	8	and	and	CCONJ
ejpam-5098	349	9	hence	hence	ADV
ejpam-5098	349	10	a	a	DET
ejpam-5098	349	11	=	=	SYM
ejpam-5098	349	12	0	0	NUM
ejpam-5098	349	13	,	,	PUNCT
ejpam-5098	349	14	we	we	PRON
ejpam-5098	349	15	get	get	VERB
ejpam-5098	349	16	a	a	DET
ejpam-5098	349	17	contradiction	contradiction	NOUN
ejpam-5098	349	18	.	.	PUNCT
ejpam-5098	350	1	therefore	therefore	ADV
ejpam-5098	350	2	,	,	PUNCT
ejpam-5098	350	3	k	k	PROPN
ejpam-5098	350	4	is	be	AUX
ejpam-5098	350	5	a	a	DET
ejpam-5098	350	6	non	non	ADJ
ejpam-5098	350	7	-	-	ADJ
ejpam-5098	350	8	dense	dense	ADJ
ejpam-5098	350	9	ideal	ideal	NOUN
ejpam-5098	350	10	of	of	ADP
ejpam-5098	350	11	r.	r.	PROPN
ejpam-5098	350	12	(	(	PUNCT
ejpam-5098	350	13	3	3	X
ejpam-5098	350	14	)	)	PUNCT
ejpam-5098	350	15	let	let	VERB
ejpam-5098	350	16	q	q	NOUN
ejpam-5098	350	17	be	be	AUX
ejpam-5098	350	18	any	any	DET
ejpam-5098	350	19	non	non	ADJ
ejpam-5098	350	20	-	-	ADJ
ejpam-5098	350	21	dense	dense	ADJ
ejpam-5098	350	22	prime	prime	ADJ
ejpam-5098	350	23	ideal	ideal	NOUN
ejpam-5098	350	24	of	of	ADP
ejpam-5098	350	25	r.	r.	PROPN
ejpam-5098	350	26	then	then	ADV
ejpam-5098	350	27	q	q	PROPN
ejpam-5098	351	1	=	=	PUNCT
ejpam-5098	351	2	(	(	PUNCT
ejpam-5098	351	3	a)∗	a)∗	PROPN
ejpam-5098	351	4	for	for	ADP
ejpam-5098	351	5	some	some	DET
ejpam-5098	351	6	non	non	ADJ
ejpam-5098	351	7	-	-	ADJ
ejpam-5098	351	8	zero	zero	NUM
ejpam-5098	351	9	element	element	NOUN
ejpam-5098	351	10	a	a	PRON
ejpam-5098	351	11	of	of	ADP
ejpam-5098	351	12	r.	r.	PROPN
ejpam-5098	351	13	let	let	VERB
ejpam-5098	351	14	b	b	PROPN
ejpam-5098	351	15	∈	∈	PROPN
ejpam-5098	351	16	q.	q.	NOUN
ejpam-5098	351	17	then	then	ADV
ejpam-5098	351	18	b	b	PROPN
ejpam-5098	351	19	∈	∈	PROPN
ejpam-5098	351	20	(	(	PUNCT
ejpam-5098	351	21	a)∗.	a)∗.	NOUN
ejpam-5098	351	22	that	that	PRON
ejpam-5098	351	23	implies	imply	VERB
ejpam-5098	351	24	(	(	PUNCT
ejpam-5098	351	25	b)∗∗	b)∗∗	NOUN
ejpam-5098	351	26	⊆	⊆	NUM
ejpam-5098	351	27	(	(	PUNCT
ejpam-5098	351	28	a)∗	a)∗	PROPN
ejpam-5098	351	29	=	=	SYM
ejpam-5098	351	30	q.	q.	PROPN
ejpam-5098	351	31	hence	hence	ADV
ejpam-5098	351	32	,	,	PUNCT
ejpam-5098	351	33	q	q	PROPN
ejpam-5098	351	34	is	be	AUX
ejpam-5098	351	35	an	an	DET
ejpam-5098	351	36	α	α	NOUN
ejpam-5098	351	37	-	-	NOUN
ejpam-5098	351	38	ideal	ideal	NOUN
ejpam-5098	351	39	of	of	ADP
ejpam-5098	351	40	r.	r.	PROPN
ejpam-5098	351	41	we	we	PRON
ejpam-5098	351	42	denote	denote	VERB
ejpam-5098	351	43	specσr	specσr	NOUN
ejpam-5098	351	44	and	and	CCONJ
ejpam-5098	351	45	specσr/ϕ	specσr/ϕ	PROPN
ejpam-5098	351	46	as	as	ADP
ejpam-5098	351	47	the	the	DET
ejpam-5098	351	48	sets	set	NOUN
ejpam-5098	351	49	of	of	ADP
ejpam-5098	351	50	all	all	DET
ejpam-5098	351	51	prime	prime	ADJ
ejpam-5098	351	52	σ	σ	NOUN
ejpam-5098	351	53	-	-	PUNCT
ejpam-5098	351	54	ideals	ideal	NOUN
ejpam-5098	351	55	of	of	ADP
ejpam-5098	351	56	r	r	NOUN
ejpam-5098	351	57	and	and	CCONJ
ejpam-5098	351	58	r/ϕ	r/ϕ	NOUN
ejpam-5098	351	59	,	,	PUNCT
ejpam-5098	351	60	respectively	respectively	ADV
ejpam-5098	351	61	.	.	PUNCT
ejpam-5098	352	1	theorem	theorem	VERB
ejpam-5098	352	2	13	13	NUM
ejpam-5098	352	3	.	.	PUNCT
ejpam-5098	353	1	for	for	ADP
ejpam-5098	353	2	any	any	DET
ejpam-5098	353	3	congruence	congruence	PROPN
ejpam-5098	353	4	relation	relation	NOUN
ejpam-5098	353	5	ϕ	ϕ	NOUN
ejpam-5098	353	6	on	on	ADP
ejpam-5098	353	7	r	r	NOUN
ejpam-5098	353	8	and	and	CCONJ
ejpam-5098	353	9	every	every	DET
ejpam-5098	353	10	member	member	NOUN
ejpam-5098	353	11	of	of	ADP
ejpam-5098	353	12	specr	specr	PROPN
ejpam-5098	353	13	is	be	AUX
ejpam-5098	353	14	nondense	nondense	NOUN
ejpam-5098	353	15	,	,	PUNCT
ejpam-5098	353	16	there	there	PRON
ejpam-5098	353	17	is	be	VERB
ejpam-5098	353	18	an	an	DET
ejpam-5098	353	19	order	order	NOUN
ejpam-5098	353	20	isomorphism	isomorphism	NOUN
ejpam-5098	353	21	between	between	ADP
ejpam-5098	353	22	specσr	specσr	PROPN
ejpam-5098	353	23	and	and	CCONJ
ejpam-5098	353	24	specσr/ϕ.	specσr/ϕ.	NOUN
ejpam-5098	353	25	proof	proof	NOUN
ejpam-5098	353	26	.	.	PUNCT
ejpam-5098	354	1	let	let	VERB
ejpam-5098	354	2	p	p	PRON
ejpam-5098	354	3	∈	∈	PROPN
ejpam-5098	354	4	specσr	specσr	NOUN
ejpam-5098	354	5	.	.	PUNCT
ejpam-5098	355	1	then	then	ADV
ejpam-5098	355	2	p̃	p̃	PROPN
ejpam-5098	355	3	∈	∈	PROPN
ejpam-5098	355	4	specσr/ϕ	specσr/ϕ	NOUN
ejpam-5098	355	5	and	and	CCONJ
ejpam-5098	355	6	hence	hence	ADV
ejpam-5098	355	7	p̃	p̃	PROPN
ejpam-5098	355	8	∈	∈	PROPN
ejpam-5098	355	9	specαr/ϕ.	specαr/ϕ.	NOUN
ejpam-5098	355	10	let	let	VERB
ejpam-5098	355	11	k	k	PROPN
ejpam-5098	355	12	∈	∈	PROPN
ejpam-5098	355	13	specσr/ϕ.	specσr/ϕ.	NOUN
ejpam-5098	355	14	then	then	ADV
ejpam-5098	355	15	we	we	PRON
ejpam-5098	355	16	have	have	VERB
ejpam-5098	355	17	that	that	PRON
ejpam-5098	355	18	k	k	PROPN
ejpam-5098	355	19	∈	∈	PROPN
ejpam-5098	355	20	specαr/ϕ.	specαr/ϕ.	NOUN
ejpam-5098	355	21	take	take	VERB
ejpam-5098	355	22	h	h	NOUN
ejpam-5098	355	23	=	=	PUNCT
ejpam-5098	355	24	{	{	PUNCT
ejpam-5098	355	25	a	a	PRON
ejpam-5098	355	26	∈	∈	NOUN
ejpam-5098	355	27	r	r	NOUN
ejpam-5098	356	1	|	|	NOUN
ejpam-5098	357	1	[	[	X
ejpam-5098	357	2	a]ϕ	a]ϕ	NOUN
ejpam-5098	357	3	∈	∈	PROPN
ejpam-5098	357	4	k	k	NOUN
ejpam-5098	357	5	}	}	PUNCT
ejpam-5098	357	6	.	.	PUNCT
ejpam-5098	358	1	since	since	SCONJ
ejpam-5098	358	2	k	k	PROPN
ejpam-5098	358	3	∈	∈	PROPN
ejpam-5098	358	4	specαr/ϕ	specαr/ϕ	NOUN
ejpam-5098	358	5	,	,	PUNCT
ejpam-5098	358	6	we	we	PRON
ejpam-5098	358	7	get	get	VERB
ejpam-5098	358	8	h	h	DET
ejpam-5098	358	9	∈	∈	PROPN
ejpam-5098	358	10	specαr	specαr	NOUN
ejpam-5098	358	11	.	.	PUNCT
ejpam-5098	359	1	hence	hence	ADV
ejpam-5098	359	2	,	,	PUNCT
ejpam-5098	359	3	h̃	h̃	PROPN
ejpam-5098	359	4	=	=	SYM
ejpam-5098	359	5	k.	k.	PROPN
ejpam-5098	359	6	let	let	VERB
ejpam-5098	359	7	a	a	DET
ejpam-5098	359	8	∈	∈	PROPN
ejpam-5098	359	9	h.	h.	NOUN
ejpam-5098	360	1	then	then	ADV
ejpam-5098	360	2	[	[	X
ejpam-5098	360	3	a]ϕ	a]ϕ	NOUN
ejpam-5098	360	4	∈	∈	PROPN
ejpam-5098	360	5	h̃	h̃	PROPN
ejpam-5098	360	6	=	=	SYM
ejpam-5098	360	7	k.	k.	PROPN
ejpam-5098	360	8	that	that	PRON
ejpam-5098	360	9	implies	imply	VERB
ejpam-5098	360	10	(	(	PUNCT
ejpam-5098	360	11	[	[	X
ejpam-5098	360	12	a]ϕ	a]ϕ	NOUN
ejpam-5098	360	13	)	)	PUNCT
ejpam-5098	360	14	◦	◦	VERB
ejpam-5098	360	15	∨k	∨k	ADJ
ejpam-5098	360	16	=	=	PUNCT
ejpam-5098	360	17	r/ϕ.	r/ϕ.	NOUN
ejpam-5098	360	18	we	we	PRON
ejpam-5098	360	19	prove	prove	VERB
ejpam-5098	360	20	that	that	SCONJ
ejpam-5098	360	21	(	(	PUNCT
ejpam-5098	360	22	a)∗	a)∗	PROPN
ejpam-5098	360	23	∨	∨	NUM
ejpam-5098	360	24	h	h	PROPN
ejpam-5098	360	25	=	=	SYM
ejpam-5098	360	26	r.	r.	PROPN
ejpam-5098	360	27	suppose	suppose	VERB
ejpam-5098	360	28	(	(	PUNCT
ejpam-5098	360	29	a)∗	a)∗	PROPN
ejpam-5098	360	30	∨	∨	NUM
ejpam-5098	360	31	h	h	NOUN
ejpam-5098	360	32	̸=	̸=	PROPN
ejpam-5098	360	33	r.	r.	PROPN
ejpam-5098	360	34	then	then	ADV
ejpam-5098	360	35	(	(	PUNCT
ejpam-5098	360	36	a)∗	a)∗	PROPN
ejpam-5098	360	37	∨	∨	NUM
ejpam-5098	360	38	h	h	NOUN
ejpam-5098	360	39	⊆	⊆	NUM
ejpam-5098	360	40	m	m	NOUN
ejpam-5098	360	41	for	for	ADP
ejpam-5098	360	42	some	some	DET
ejpam-5098	360	43	m	m	NOUN
ejpam-5098	360	44	∈	∈	PROPN
ejpam-5098	360	45	specr	specr	NOUN
ejpam-5098	360	46	.	.	PUNCT
ejpam-5098	361	1	that	that	PRON
ejpam-5098	361	2	implies	imply	VERB
ejpam-5098	361	3	(	(	PUNCT
ejpam-5098	361	4	a)∗	a)∗	PROPN
ejpam-5098	361	5	⊆	⊆	NUM
ejpam-5098	361	6	m	m	NOUN
ejpam-5098	361	7	and	and	CCONJ
ejpam-5098	361	8	h	h	NOUN
ejpam-5098	361	9	⊆	⊆	NUM
ejpam-5098	361	10	m	m	NOUN
ejpam-5098	361	11	.	.	PUNCT
ejpam-5098	362	1	by	by	ADP
ejpam-5098	362	2	our	our	PRON
ejpam-5098	362	3	assumption	assumption	NOUN
ejpam-5098	362	4	,	,	PUNCT
ejpam-5098	362	5	we	we	PRON
ejpam-5098	362	6	get	get	VERB
ejpam-5098	362	7	that	that	SCONJ
ejpam-5098	362	8	m	m	NOUN
ejpam-5098	362	9	is	be	AUX
ejpam-5098	362	10	non	non	ADJ
ejpam-5098	362	11	-	-	ADJ
ejpam-5098	362	12	dense	dense	ADJ
ejpam-5098	362	13	.	.	PUNCT
ejpam-5098	363	1	by	by	ADP
ejpam-5098	363	2	the	the	DET
ejpam-5098	363	3	above	above	ADJ
ejpam-5098	363	4	result	result	NOUN
ejpam-5098	363	5	,	,	PUNCT
ejpam-5098	363	6	we	we	PRON
ejpam-5098	363	7	have	have	VERB
ejpam-5098	363	8	that	that	PRON
ejpam-5098	363	9	p	p	NOUN
ejpam-5098	363	10	is	be	AUX
ejpam-5098	363	11	an	an	DET
ejpam-5098	363	12	α	α	NOUN
ejpam-5098	363	13	-	-	NOUN
ejpam-5098	363	14	ideal	ideal	NOUN
ejpam-5098	363	15	of	of	ADP
ejpam-5098	363	16	r.	r.	PROPN
ejpam-5098	363	17	hence	hence	ADV
ejpam-5098	363	18	,	,	PUNCT
ejpam-5098	363	19	p̃	p̃	PROPN
ejpam-5098	363	20	∈	∈	PROPN
ejpam-5098	363	21	specr/ϕ.	specr/ϕ.	VERB
ejpam-5098	363	22	since	since	SCONJ
ejpam-5098	363	23	(	(	PUNCT
ejpam-5098	363	24	a)∗	a)∗	PROPN
ejpam-5098	363	25	⊆m	⊆m	NOUN
ejpam-5098	363	26	and	and	CCONJ
ejpam-5098	363	27	h	h	NOUN
ejpam-5098	363	28	⊆m	⊆m	NOUN
ejpam-5098	363	29	,	,	PUNCT
ejpam-5098	363	30	we	we	PRON
ejpam-5098	363	31	get	get	AUX
ejpam-5098	363	32	(	(	PUNCT
ejpam-5098	363	33	[	[	X
ejpam-5098	363	34	a]ϕ	a]ϕ	NOUN
ejpam-5098	363	35	)	)	PUNCT
ejpam-5098	363	36	◦	◦	NOUN
ejpam-5098	363	37	⊆	⊆	NUM
ejpam-5098	363	38	m̃	m̃	PROPN
ejpam-5098	363	39	and	and	CCONJ
ejpam-5098	363	40	h̃	h̃	PROPN
ejpam-5098	363	41	⊆	⊆	NUM
ejpam-5098	363	42	m̃	m̃	PROPN
ejpam-5098	363	43	.	.	PUNCT
ejpam-5098	364	1	that	that	PRON
ejpam-5098	364	2	implies	imply	VERB
ejpam-5098	364	3	(	(	PUNCT
ejpam-5098	364	4	[	[	X
ejpam-5098	364	5	a]ϕ	a]ϕ	NOUN
ejpam-5098	364	6	)	)	PUNCT
ejpam-5098	364	7	◦	◦	NOUN
ejpam-5098	364	8	⊆	⊆	NUM
ejpam-5098	364	9	m̃	m̃	PROPN
ejpam-5098	364	10	and	and	CCONJ
ejpam-5098	364	11	k	k	PROPN
ejpam-5098	364	12	⊆	⊆	NUM
ejpam-5098	364	13	m̃	m̃	PROPN
ejpam-5098	364	14	.	.	PUNCT
ejpam-5098	365	1	so	so	ADV
ejpam-5098	365	2	that	that	SCONJ
ejpam-5098	365	3	r/ϕ	r/ϕ	NOUN
ejpam-5098	365	4	=	=	SYM
ejpam-5098	365	5	(	(	PUNCT
ejpam-5098	365	6	[	[	X
ejpam-5098	365	7	a]ϕ	a]ϕ	NOUN
ejpam-5098	365	8	)	)	PUNCT
ejpam-5098	365	9	◦	◦	VERB
ejpam-5098	365	10	∨k	∨k	ADJ
ejpam-5098	365	11	⊆	⊆	NUM
ejpam-5098	365	12	m̃	m̃	PROPN
ejpam-5098	365	13	,	,	PUNCT
ejpam-5098	365	14	which	which	PRON
ejpam-5098	365	15	gives	give	VERB
ejpam-5098	365	16	m̃	m̃	PROPN
ejpam-5098	365	17	=	=	SYM
ejpam-5098	365	18	r/ϕ	r/ϕ	NOUN
ejpam-5098	365	19	,	,	PUNCT
ejpam-5098	365	20	we	we	PRON
ejpam-5098	365	21	get	get	VERB
ejpam-5098	365	22	a	a	DET
ejpam-5098	365	23	contradiction	contradiction	NOUN
ejpam-5098	365	24	.	.	PUNCT
ejpam-5098	366	1	therefore	therefore	ADV
ejpam-5098	366	2	,	,	PUNCT
ejpam-5098	366	3	(	(	PUNCT
ejpam-5098	366	4	a)∗	a)∗	PROPN
ejpam-5098	366	5	∨	∨	NUM
ejpam-5098	366	6	h	h	NOUN
ejpam-5098	366	7	=	=	SYM
ejpam-5098	366	8	r	r	NOUN
ejpam-5098	366	9	and	and	CCONJ
ejpam-5098	366	10	hence	hence	ADV
ejpam-5098	366	11	a	a	DET
ejpam-5098	366	12	∈	∈	PROPN
ejpam-5098	366	13	hσ	hσ	NOUN
ejpam-5098	366	14	.	.	PUNCT
ejpam-5098	367	1	thus	thus	ADV
ejpam-5098	367	2	,	,	PUNCT
ejpam-5098	367	3	h	h	PROPN
ejpam-5098	367	4	⊆	⊆	NUM
ejpam-5098	367	5	hσ	hσ	NOUN
ejpam-5098	367	6	.	.	PUNCT
ejpam-5098	368	1	since	since	SCONJ
ejpam-5098	368	2	hσ	hσ	NOUN
ejpam-5098	368	3	⊆	⊆	NUM
ejpam-5098	368	4	h	h	NOUN
ejpam-5098	368	5	,	,	PUNCT
ejpam-5098	368	6	we	we	PRON
ejpam-5098	368	7	get	get	VERB
ejpam-5098	368	8	h	h	NOUN
ejpam-5098	368	9	is	be	AUX
ejpam-5098	368	10	a	a	DET
ejpam-5098	368	11	σ	σ	NOUN
ejpam-5098	368	12	-	-	PUNCT
ejpam-5098	368	13	ideal	ideal	NOUN
ejpam-5098	368	14	of	of	ADP
ejpam-5098	368	15	r.	r.	PROPN
ejpam-5098	368	16	r.	r.	PROPN
ejpam-5098	368	17	noorbhasha	noorbhasha	PROPN
ejpam-5098	368	18	,	,	PUNCT
ejpam-5098	368	19	r.	r.	PROPN
ejpam-5098	368	20	bandaru	bandaru	PROPN
ejpam-5098	368	21	,	,	PUNCT
ejpam-5098	368	22	a.	a.	NOUN
ejpam-5098	368	23	iampan	iampan	PROPN
ejpam-5098	368	24	/	/	SYM
ejpam-5098	368	25	eur	eur	PROPN
ejpam-5098	368	26	.	.	PUNCT
ejpam-5098	369	1	j.	j.	PROPN
ejpam-5098	369	2	pure	pure	PROPN
ejpam-5098	369	3	appl	appl	PROPN
ejpam-5098	369	4	.	.	PROPN
ejpam-5098	369	5	math	math	PROPN
ejpam-5098	369	6	,	,	PUNCT
ejpam-5098	369	7	17	17	NUM
ejpam-5098	369	8	(	(	PUNCT
ejpam-5098	369	9	2	2	NUM
ejpam-5098	369	10	)	)	PUNCT
ejpam-5098	369	11	(	(	PUNCT
ejpam-5098	369	12	2024	2024	NUM
ejpam-5098	369	13	)	)	PUNCT
ejpam-5098	369	14	,	,	PUNCT
ejpam-5098	369	15	1094	1094	NUM
ejpam-5098	369	16	-	-	SYM
ejpam-5098	369	17	1112	1112	NUM
ejpam-5098	369	18	1105	1105	NUM
ejpam-5098	369	19	corollary	corollary	NOUN
ejpam-5098	369	20	2	2	NUM
ejpam-5098	369	21	.	.	X
ejpam-5098	370	1	for	for	ADP
ejpam-5098	370	2	any	any	DET
ejpam-5098	370	3	congruence	congruence	PROPN
ejpam-5098	370	4	relation	relation	NOUN
ejpam-5098	370	5	ϕ	ϕ	PROPN
ejpam-5098	370	6	on	on	ADP
ejpam-5098	370	7	r	r	NOUN
ejpam-5098	370	8	,	,	PUNCT
ejpam-5098	370	9	we	we	PRON
ejpam-5098	370	10	have	have	VERB
ejpam-5098	370	11	the	the	DET
ejpam-5098	370	12	following	following	NOUN
ejpam-5098	370	13	:	:	PUNCT
ejpam-5098	370	14	(	(	PUNCT
ejpam-5098	370	15	1	1	X
ejpam-5098	370	16	)	)	PUNCT
ejpam-5098	370	17	there	there	PRON
ejpam-5098	370	18	is	be	VERB
ejpam-5098	370	19	one	one	NUM
ejpam-5098	370	20	-	-	PUNCT
ejpam-5098	370	21	to	to	ADP
ejpam-5098	370	22	-	-	PUNCT
ejpam-5098	370	23	one	one	NUM
ejpam-5098	370	24	correspondence	correspondence	NOUN
ejpam-5098	370	25	between	between	ADP
ejpam-5098	370	26	specαr	specαr	NOUN
ejpam-5098	370	27	and	and	CCONJ
ejpam-5098	370	28	specαr/ϕ	specαr/ϕ	PROPN
ejpam-5098	370	29	(	(	PUNCT
ejpam-5098	370	30	2	2	X
ejpam-5098	370	31	)	)	PUNCT
ejpam-5098	370	32	if	if	SCONJ
ejpam-5098	370	33	every	every	DET
ejpam-5098	370	34	member	member	NOUN
ejpam-5098	370	35	of	of	ADP
ejpam-5098	370	36	specr	specr	PROPN
ejpam-5098	370	37	is	be	AUX
ejpam-5098	370	38	non	non	ADJ
ejpam-5098	370	39	-	-	ADJ
ejpam-5098	370	40	dense	dense	ADJ
ejpam-5098	370	41	,	,	PUNCT
ejpam-5098	370	42	then	then	ADV
ejpam-5098	370	43	there	there	PRON
ejpam-5098	370	44	is	be	VERB
ejpam-5098	370	45	one	one	NUM
ejpam-5098	370	46	-	-	PUNCT
ejpam-5098	370	47	to	to	ADP
ejpam-5098	370	48	-	-	PUNCT
ejpam-5098	370	49	one	one	NUM
ejpam-5098	370	50	correspondence	correspondence	NOUN
ejpam-5098	370	51	between	between	ADP
ejpam-5098	370	52	specσr	specσr	PROPN
ejpam-5098	370	53	and	and	CCONJ
ejpam-5098	370	54	specσr/ϕ.	specσr/ϕ.	NOUN
ejpam-5098	370	55	theorem	theorem	VERB
ejpam-5098	370	56	14	14	NUM
ejpam-5098	370	57	.	.	PUNCT
ejpam-5098	371	1	for	for	ADP
ejpam-5098	371	2	any	any	DET
ejpam-5098	371	3	congruence	congruence	PROPN
ejpam-5098	371	4	relation	relation	NOUN
ejpam-5098	371	5	ϕ	ϕ	NOUN
ejpam-5098	371	6	on	on	ADP
ejpam-5098	371	7	r	r	NOUN
ejpam-5098	371	8	and	and	CCONJ
ejpam-5098	371	9	every	every	DET
ejpam-5098	371	10	member	member	NOUN
ejpam-5098	371	11	of	of	ADP
ejpam-5098	371	12	specr	specr	PROPN
ejpam-5098	371	13	is	be	AUX
ejpam-5098	371	14	nondense	nondense	NOUN
ejpam-5098	371	15	,	,	PUNCT
ejpam-5098	371	16	every	every	DET
ejpam-5098	371	17	α	α	NOUN
ejpam-5098	371	18	-	-	NOUN
ejpam-5098	371	19	ideal	ideal	NOUN
ejpam-5098	371	20	of	of	ADP
ejpam-5098	371	21	r	r	NOUN
ejpam-5098	371	22	is	be	AUX
ejpam-5098	371	23	a	a	DET
ejpam-5098	371	24	σ	σ	NOUN
ejpam-5098	371	25	-	-	PUNCT
ejpam-5098	371	26	ideal	ideal	NOUN
ejpam-5098	371	27	if	if	SCONJ
ejpam-5098	372	1	and	and	CCONJ
ejpam-5098	372	2	only	only	ADV
ejpam-5098	372	3	if	if	SCONJ
ejpam-5098	372	4	every	every	DET
ejpam-5098	372	5	α	α	NOUN
ejpam-5098	372	6	-	-	NOUN
ejpam-5098	372	7	ideal	ideal	NOUN
ejpam-5098	372	8	of	of	ADP
ejpam-5098	372	9	r/ϕ	r/ϕ	NOUN
ejpam-5098	372	10	is	be	AUX
ejpam-5098	372	11	a	a	DET
ejpam-5098	372	12	σ	σ	NOUN
ejpam-5098	372	13	-	-	PUNCT
ejpam-5098	372	14	ideal	ideal	NOUN
ejpam-5098	372	15	.	.	PUNCT
ejpam-5098	373	1	proof	proof	NOUN
ejpam-5098	373	2	.	.	PUNCT
ejpam-5098	374	1	assume	assume	VERB
ejpam-5098	374	2	that	that	SCONJ
ejpam-5098	374	3	every	every	DET
ejpam-5098	374	4	α	α	NOUN
ejpam-5098	374	5	-	-	NOUN
ejpam-5098	374	6	ideal	ideal	NOUN
ejpam-5098	374	7	of	of	ADP
ejpam-5098	374	8	r	r	NOUN
ejpam-5098	374	9	is	be	AUX
ejpam-5098	374	10	a	a	DET
ejpam-5098	374	11	σ	σ	NOUN
ejpam-5098	374	12	-	-	PUNCT
ejpam-5098	374	13	ideal	ideal	NOUN
ejpam-5098	374	14	.	.	PUNCT
ejpam-5098	375	1	let	let	VERB
ejpam-5098	375	2	k	k	PRON
ejpam-5098	375	3	be	be	AUX
ejpam-5098	375	4	an	an	DET
ejpam-5098	375	5	α	α	NOUN
ejpam-5098	375	6	-	-	NOUN
ejpam-5098	375	7	ideal	ideal	NOUN
ejpam-5098	375	8	of	of	ADP
ejpam-5098	375	9	r/ϕ.	r/ϕ.	PROPN
ejpam-5098	375	10	then	then	ADV
ejpam-5098	375	11	k	k	PROPN
ejpam-5098	375	12	=	=	SYM
ejpam-5098	375	13	h̃	h̃	PROPN
ejpam-5098	375	14	for	for	ADP
ejpam-5098	375	15	some	some	DET
ejpam-5098	375	16	α	α	NOUN
ejpam-5098	375	17	-	-	PUNCT
ejpam-5098	375	18	ideal	ideal	ADJ
ejpam-5098	375	19	h	h	NOUN
ejpam-5098	375	20	of	of	ADP
ejpam-5098	375	21	r.	r.	PROPN
ejpam-5098	375	22	by	by	ADP
ejpam-5098	375	23	our	our	PRON
ejpam-5098	375	24	assumption	assumption	NOUN
ejpam-5098	375	25	,	,	PUNCT
ejpam-5098	375	26	we	we	PRON
ejpam-5098	375	27	get	get	VERB
ejpam-5098	375	28	that	that	PRON
ejpam-5098	375	29	h	h	NOUN
ejpam-5098	375	30	is	be	AUX
ejpam-5098	375	31	a	a	DET
ejpam-5098	375	32	σ	σ	NOUN
ejpam-5098	375	33	-	-	PUNCT
ejpam-5098	375	34	ideal	ideal	NOUN
ejpam-5098	375	35	of	of	ADP
ejpam-5098	375	36	r.	r.	PROPN
ejpam-5098	375	37	hence	hence	ADV
ejpam-5098	375	38	,	,	PUNCT
ejpam-5098	375	39	h̃	h̃	PROPN
ejpam-5098	375	40	=	=	SYM
ejpam-5098	375	41	k	k	PROPN
ejpam-5098	375	42	is	be	AUX
ejpam-5098	375	43	a	a	DET
ejpam-5098	375	44	σ	σ	NOUN
ejpam-5098	375	45	-	-	PUNCT
ejpam-5098	375	46	ideal	ideal	NOUN
ejpam-5098	375	47	of	of	ADP
ejpam-5098	375	48	r/ϕ.	r/ϕ.	NOUN
ejpam-5098	375	49	conversely	conversely	ADV
ejpam-5098	375	50	,	,	PUNCT
ejpam-5098	375	51	assume	assume	VERB
ejpam-5098	375	52	that	that	SCONJ
ejpam-5098	375	53	every	every	DET
ejpam-5098	375	54	α	α	NOUN
ejpam-5098	375	55	-	-	NOUN
ejpam-5098	375	56	ideal	ideal	NOUN
ejpam-5098	375	57	of	of	ADP
ejpam-5098	375	58	r/ϕ	r/ϕ	NOUN
ejpam-5098	375	59	is	be	AUX
ejpam-5098	375	60	a	a	DET
ejpam-5098	375	61	σ	σ	NOUN
ejpam-5098	375	62	-	-	PUNCT
ejpam-5098	375	63	ideal	ideal	NOUN
ejpam-5098	375	64	.	.	PUNCT
ejpam-5098	376	1	let	let	VERB
ejpam-5098	376	2	k	k	PRON
ejpam-5098	376	3	be	be	AUX
ejpam-5098	376	4	an	an	DET
ejpam-5098	376	5	α	α	NOUN
ejpam-5098	376	6	-	-	NOUN
ejpam-5098	376	7	ideal	ideal	NOUN
ejpam-5098	376	8	of	of	ADP
ejpam-5098	376	9	r.	r.	PROPN
ejpam-5098	376	10	clearly	clearly	ADV
ejpam-5098	376	11	,	,	PUNCT
ejpam-5098	376	12	we	we	PRON
ejpam-5098	376	13	have	have	VERB
ejpam-5098	376	14	that	that	SCONJ
ejpam-5098	376	15	k̃	k̃	PROPN
ejpam-5098	376	16	is	be	AUX
ejpam-5098	376	17	an	an	DET
ejpam-5098	376	18	α	α	NOUN
ejpam-5098	376	19	-	-	NOUN
ejpam-5098	376	20	ideal	ideal	NOUN
ejpam-5098	376	21	of	of	ADP
ejpam-5098	376	22	r/ϕ.	r/ϕ.	PROPN
ejpam-5098	376	23	by	by	ADP
ejpam-5098	376	24	our	our	PRON
ejpam-5098	376	25	assumption	assumption	NOUN
ejpam-5098	376	26	,	,	PUNCT
ejpam-5098	376	27	we	we	PRON
ejpam-5098	376	28	get	get	VERB
ejpam-5098	376	29	k̃	k̃	PROPN
ejpam-5098	376	30	is	be	AUX
ejpam-5098	376	31	a	a	DET
ejpam-5098	376	32	σ	σ	NOUN
ejpam-5098	376	33	-	-	PUNCT
ejpam-5098	376	34	ideal	ideal	NOUN
ejpam-5098	376	35	of	of	ADP
ejpam-5098	376	36	r/ϕ.	r/ϕ.	PROPN
ejpam-5098	376	37	let	let	VERB
ejpam-5098	376	38	a	a	DET
ejpam-5098	376	39	∈	∈	PROPN
ejpam-5098	376	40	k.	k.	NOUN
ejpam-5098	377	1	then	then	ADV
ejpam-5098	377	2	[	[	X
ejpam-5098	377	3	a]ϕ	a]ϕ	NOUN
ejpam-5098	377	4	∈	∈	PROPN
ejpam-5098	377	5	k̃.	k̃.	NOUN
ejpam-5098	377	6	we	we	PRON
ejpam-5098	377	7	prove	prove	VERB
ejpam-5098	377	8	that	that	SCONJ
ejpam-5098	377	9	(	(	PUNCT
ejpam-5098	377	10	a)∗	a)∗	PROPN
ejpam-5098	377	11	∨k	∨k	NOUN
ejpam-5098	377	12	=	=	SYM
ejpam-5098	377	13	r.	r.	PROPN
ejpam-5098	377	14	if	if	SCONJ
ejpam-5098	377	15	(	(	PUNCT
ejpam-5098	377	16	a)∗	a)∗	PROPN
ejpam-5098	377	17	∨k	∨k	ADJ
ejpam-5098	377	18	̸=	̸=	PROPN
ejpam-5098	377	19	r	r	NOUN
ejpam-5098	377	20	,	,	PUNCT
ejpam-5098	377	21	then	then	ADV
ejpam-5098	377	22	(	(	PUNCT
ejpam-5098	377	23	a)∗	a)∗	PROPN
ejpam-5098	377	24	∨	∨	NUM
ejpam-5098	377	25	k	k	PROPN
ejpam-5098	377	26	⊆	⊆	NUM
ejpam-5098	377	27	m	m	NOUN
ejpam-5098	377	28	for	for	ADP
ejpam-5098	377	29	some	some	DET
ejpam-5098	377	30	m	m	NOUN
ejpam-5098	377	31	∈	∈	PROPN
ejpam-5098	377	32	specr	specr	NOUN
ejpam-5098	377	33	.	.	PUNCT
ejpam-5098	378	1	by	by	ADP
ejpam-5098	378	2	the	the	DET
ejpam-5098	378	3	hypothesis	hypothesis	NOUN
ejpam-5098	378	4	,	,	PUNCT
ejpam-5098	378	5	we	we	PRON
ejpam-5098	378	6	have	have	VERB
ejpam-5098	378	7	that	that	SCONJ
ejpam-5098	378	8	m	m	PROPN
ejpam-5098	378	9	is	be	AUX
ejpam-5098	378	10	non	non	ADJ
ejpam-5098	378	11	-	-	ADJ
ejpam-5098	378	12	dense	dense	ADJ
ejpam-5098	378	13	.	.	PUNCT
ejpam-5098	379	1	that	that	PRON
ejpam-5098	379	2	implies	imply	VERB
ejpam-5098	379	3	m	m	VERB
ejpam-5098	379	4	is	be	AUX
ejpam-5098	379	5	an	an	DET
ejpam-5098	379	6	α	α	NOUN
ejpam-5098	379	7	-	-	NOUN
ejpam-5098	379	8	ideal	ideal	NOUN
ejpam-5098	379	9	of	of	ADP
ejpam-5098	379	10	r.	r.	PROPN
ejpam-5098	379	11	clearly	clearly	ADV
ejpam-5098	379	12	,	,	PUNCT
ejpam-5098	379	13	we	we	PRON
ejpam-5098	379	14	get	get	VERB
ejpam-5098	379	15	m̃	m̃	PROPN
ejpam-5098	379	16	∈	∈	NOUN
ejpam-5098	379	17	specr/ϕ.	specr/ϕ.	VERB
ejpam-5098	379	18	since	since	SCONJ
ejpam-5098	379	19	(	(	PUNCT
ejpam-5098	379	20	a)∗	a)∗	PROPN
ejpam-5098	379	21	∨k	∨k	ADJ
ejpam-5098	379	22	⊆	⊆	NUM
ejpam-5098	379	23	m	m	NOUN
ejpam-5098	379	24	,	,	PUNCT
ejpam-5098	379	25	we	we	PRON
ejpam-5098	379	26	have	have	VERB
ejpam-5098	379	27	that	that	PRON
ejpam-5098	379	28	(	(	PUNCT
ejpam-5098	379	29	a)∗	a)∗	PROPN
ejpam-5098	379	30	⊆	⊆	NUM
ejpam-5098	379	31	m	m	NOUN
ejpam-5098	379	32	and	and	CCONJ
ejpam-5098	379	33	k	k	PROPN
ejpam-5098	379	34	⊆	⊆	NUM
ejpam-5098	379	35	m	m	NOUN
ejpam-5098	379	36	.	.	PUNCT
ejpam-5098	380	1	that	that	PRON
ejpam-5098	380	2	implies	imply	VERB
ejpam-5098	380	3	(	(	PUNCT
ejpam-5098	380	4	[	[	X
ejpam-5098	380	5	a]ϕ	a]ϕ	NOUN
ejpam-5098	380	6	)	)	PUNCT
ejpam-5098	380	7	◦	◦	NOUN
ejpam-5098	380	8	⊆	⊆	NUM
ejpam-5098	380	9	m̃	m̃	PROPN
ejpam-5098	380	10	and	and	CCONJ
ejpam-5098	380	11	k̃	k̃	PROPN
ejpam-5098	380	12	⊆	⊆	NUM
ejpam-5098	380	13	m̃	m̃	PROPN
ejpam-5098	380	14	.	.	PUNCT
ejpam-5098	381	1	therefore	therefore	ADV
ejpam-5098	381	2	,	,	PUNCT
ejpam-5098	381	3	r/ϕ	r/ϕ	NOUN
ejpam-5098	381	4	=	=	SYM
ejpam-5098	381	5	(	(	PUNCT
ejpam-5098	381	6	[	[	X
ejpam-5098	381	7	a]ϕ	a]ϕ	NOUN
ejpam-5098	381	8	)	)	PUNCT
ejpam-5098	381	9	◦	◦	NOUN
ejpam-5098	381	10	∨k̃	∨k̃	NOUN
ejpam-5098	381	11	⊆	⊆	NUM
ejpam-5098	381	12	m̃	m̃	PROPN
ejpam-5098	381	13	and	and	CCONJ
ejpam-5098	381	14	hence	hence	ADV
ejpam-5098	381	15	m̃	m̃	PROPN
ejpam-5098	381	16	=	=	SYM
ejpam-5098	381	17	r/ϕ	r/ϕ	PROPN
ejpam-5098	381	18	,	,	PUNCT
ejpam-5098	381	19	which	which	PRON
ejpam-5098	381	20	is	be	AUX
ejpam-5098	381	21	a	a	DET
ejpam-5098	381	22	contradiction	contradiction	NOUN
ejpam-5098	381	23	.	.	PUNCT
ejpam-5098	382	1	thus	thus	ADV
ejpam-5098	382	2	,	,	PUNCT
ejpam-5098	382	3	(	(	PUNCT
ejpam-5098	382	4	a)∗∨k	a)∗∨k	PROPN
ejpam-5098	382	5	=	=	SYM
ejpam-5098	382	6	r	r	PROPN
ejpam-5098	382	7	,	,	PUNCT
ejpam-5098	382	8	which	which	PRON
ejpam-5098	382	9	gives	give	VERB
ejpam-5098	382	10	k	k	PROPN
ejpam-5098	382	11	⊆	⊆	NUM
ejpam-5098	382	12	kσ	kσ	PROPN
ejpam-5098	382	13	.	.	PUNCT
ejpam-5098	383	1	since	since	SCONJ
ejpam-5098	383	2	kσ	kσ	PROPN
ejpam-5098	383	3	⊆	⊆	NUM
ejpam-5098	383	4	k	k	NOUN
ejpam-5098	383	5	,	,	PUNCT
ejpam-5098	383	6	we	we	PRON
ejpam-5098	383	7	get	get	VERB
ejpam-5098	383	8	kσ	kσ	PROPN
ejpam-5098	383	9	=	=	PUNCT
ejpam-5098	383	10	k.	k.	PROPN
ejpam-5098	383	11	hence	hence	ADV
ejpam-5098	383	12	,	,	PUNCT
ejpam-5098	383	13	k	k	PROPN
ejpam-5098	383	14	is	be	AUX
ejpam-5098	383	15	a	a	DET
ejpam-5098	383	16	σ	σ	NOUN
ejpam-5098	383	17	-	-	PUNCT
ejpam-5098	383	18	ideal	ideal	NOUN
ejpam-5098	383	19	of	of	ADP
ejpam-5098	383	20	r.	r.	PROPN
ejpam-5098	383	21	4	4	NUM
ejpam-5098	383	22	.	.	PUNCT
ejpam-5098	384	1	σ	σ	NOUN
ejpam-5098	384	2	-	-	PUNCT
ejpam-5098	384	3	prime	prime	ADJ
ejpam-5098	384	4	spectrum	spectrum	NOUN
ejpam-5098	384	5	of	of	ADP
ejpam-5098	384	6	normal	normal	ADJ
ejpam-5098	384	7	adls	adls	NOUN
ejpam-5098	384	8	in	in	ADP
ejpam-5098	384	9	this	this	DET
ejpam-5098	384	10	section	section	NOUN
ejpam-5098	384	11	,	,	PUNCT
ejpam-5098	384	12	we	we	PRON
ejpam-5098	384	13	derive	derive	VERB
ejpam-5098	384	14	the	the	DET
ejpam-5098	384	15	properties	property	NOUN
ejpam-5098	384	16	of	of	ADP
ejpam-5098	384	17	prime	prime	ADJ
ejpam-5098	384	18	σ	σ	NOUN
ejpam-5098	384	19	-	-	PUNCT
ejpam-5098	384	20	ideals	ideal	NOUN
ejpam-5098	384	21	of	of	ADP
ejpam-5098	384	22	a	a	DET
ejpam-5098	384	23	normal	normal	ADJ
ejpam-5098	384	24	adl	adl	PROPN
ejpam-5098	384	25	topologically	topologically	ADV
ejpam-5098	384	26	.	.	PUNCT
ejpam-5098	385	1	lemma	lemma	PROPN
ejpam-5098	385	2	5	5	NUM
ejpam-5098	385	3	.	.	PUNCT
ejpam-5098	386	1	a	a	DET
ejpam-5098	386	2	join	join	NOUN
ejpam-5098	386	3	of	of	ADP
ejpam-5098	386	4	two	two	NUM
ejpam-5098	386	5	σ	σ	NOUN
ejpam-5098	386	6	-	-	PUNCT
ejpam-5098	386	7	ideals	ideal	NOUN
ejpam-5098	386	8	of	of	ADP
ejpam-5098	386	9	an	an	DET
ejpam-5098	386	10	adl	adl	NOUN
ejpam-5098	386	11	r	r	NOUN
ejpam-5098	386	12	is	be	AUX
ejpam-5098	386	13	a	a	DET
ejpam-5098	386	14	σ	σ	NOUN
ejpam-5098	386	15	-	-	PUNCT
ejpam-5098	386	16	ideal	ideal	NOUN
ejpam-5098	386	17	of	of	ADP
ejpam-5098	386	18	r.	r.	PROPN
ejpam-5098	386	19	also	also	ADV
ejpam-5098	386	20	,	,	PUNCT
ejpam-5098	386	21	the	the	DET
ejpam-5098	386	22	intersection	intersection	NOUN
ejpam-5098	386	23	of	of	ADP
ejpam-5098	386	24	two	two	NUM
ejpam-5098	386	25	σ	σ	NOUN
ejpam-5098	386	26	-	-	PUNCT
ejpam-5098	386	27	ideals	ideal	NOUN
ejpam-5098	386	28	is	be	AUX
ejpam-5098	386	29	a	a	DET
ejpam-5098	386	30	σ	σ	NOUN
ejpam-5098	386	31	-	-	PUNCT
ejpam-5098	386	32	ideal	ideal	NOUN
ejpam-5098	386	33	of	of	ADP
ejpam-5098	386	34	r.	r.	PROPN
ejpam-5098	386	35	proof	proof	NOUN
ejpam-5098	386	36	.	.	PUNCT
ejpam-5098	387	1	let	let	VERB
ejpam-5098	387	2	i	i	PRON
ejpam-5098	387	3	,	,	PUNCT
ejpam-5098	387	4	j	j	PROPN
ejpam-5098	387	5	be	be	VERB
ejpam-5098	387	6	any	any	DET
ejpam-5098	387	7	two	two	NUM
ejpam-5098	387	8	σ	σ	NOUN
ejpam-5098	387	9	-	-	PUNCT
ejpam-5098	387	10	ideals	ideal	NOUN
ejpam-5098	387	11	of	of	ADP
ejpam-5098	387	12	r.	r.	PROPN
ejpam-5098	387	13	then	then	ADV
ejpam-5098	387	14	iσ	iσ	VERB
ejpam-5098	387	15	=	=	PUNCT
ejpam-5098	388	1	i	i	PROPN
ejpam-5098	388	2	and	and	CCONJ
ejpam-5098	388	3	jσ	jσ	PROPN
ejpam-5098	388	4	=	=	PROPN
ejpam-5098	388	5	j	j	PROPN
ejpam-5098	388	6	.	.	PUNCT
ejpam-5098	389	1	clearly	clearly	ADV
ejpam-5098	389	2	,	,	PUNCT
ejpam-5098	389	3	we	we	PRON
ejpam-5098	389	4	have	have	VERB
ejpam-5098	389	5	that	that	PRON
ejpam-5098	389	6	(	(	PUNCT
ejpam-5098	389	7	i∨j)σ	i∨j)σ	NOUN
ejpam-5098	389	8	⊆	⊆	NUM
ejpam-5098	389	9	i∨j	i∨j	ADP
ejpam-5098	389	10	.	.	PUNCT
ejpam-5098	390	1	and	and	CCONJ
ejpam-5098	390	2	also	also	ADV
ejpam-5098	390	3	,	,	PUNCT
ejpam-5098	390	4	we	we	PRON
ejpam-5098	390	5	have	have	VERB
ejpam-5098	390	6	that	that	PRON
ejpam-5098	390	7	iσ∨jσ	iσ∨jσ	ADP
ejpam-5098	390	8	⊆	⊆	NUM
ejpam-5098	390	9	(	(	PUNCT
ejpam-5098	390	10	i∨j)σ	i∨j)σ	NOUN
ejpam-5098	390	11	and	and	CCONJ
ejpam-5098	390	12	hence	hence	ADV
ejpam-5098	390	13	i∨j	i∨j	ADP
ejpam-5098	390	14	⊆	⊆	NUM
ejpam-5098	390	15	(	(	PUNCT
ejpam-5098	390	16	i∨j)σ	i∨j)σ	NOUN
ejpam-5098	390	17	.	.	PUNCT
ejpam-5098	391	1	therefore	therefore	ADV
ejpam-5098	391	2	,	,	PUNCT
ejpam-5098	391	3	(	(	PUNCT
ejpam-5098	391	4	i	i	NOUN
ejpam-5098	391	5	∨	∨	VERB
ejpam-5098	391	6	j)σ	j)σ	NOUN
ejpam-5098	391	7	=	=	SYM
ejpam-5098	392	1	i	i	PROPN
ejpam-5098	392	2	∨	∨	PROPN
ejpam-5098	392	3	j	j	PROPN
ejpam-5098	392	4	.	.	PUNCT
ejpam-5098	393	1	note	note	VERB
ejpam-5098	393	2	that	that	SCONJ
ejpam-5098	393	3	the	the	DET
ejpam-5098	393	4	set	set	NOUN
ejpam-5098	393	5	iσ(r	iσ(r	NOUN
ejpam-5098	393	6	)	)	PUNCT
ejpam-5098	393	7	of	of	ADP
ejpam-5098	393	8	all	all	DET
ejpam-5098	393	9	σ	σ	NOUN
ejpam-5098	393	10	-	-	PUNCT
ejpam-5098	393	11	ideals	ideal	NOUN
ejpam-5098	393	12	of	of	ADP
ejpam-5098	393	13	an	an	DET
ejpam-5098	393	14	adl	adl	NOUN
ejpam-5098	393	15	r	r	NOUN
ejpam-5098	393	16	is	be	AUX
ejpam-5098	393	17	a	a	DET
ejpam-5098	393	18	sublattice	sublattice	NOUN
ejpam-5098	393	19	of	of	ADP
ejpam-5098	393	20	all	all	DET
ejpam-5098	393	21	ideals	ideal	NOUN
ejpam-5098	393	22	of	of	ADP
ejpam-5098	393	23	r	r	NOUN
ejpam-5098	393	24	,	,	PUNCT
ejpam-5098	393	25	which	which	PRON
ejpam-5098	393	26	is	be	AUX
ejpam-5098	393	27	closed	close	VERB
ejpam-5098	393	28	under	under	ADP
ejpam-5098	393	29	arbitrary	arbitrary	ADJ
ejpam-5098	393	30	joins	join	NOUN
ejpam-5098	393	31	.	.	PUNCT
ejpam-5098	394	1	definition	definition	NOUN
ejpam-5098	394	2	12	12	NUM
ejpam-5098	394	3	.	.	PUNCT
ejpam-5098	395	1	[	[	X
ejpam-5098	395	2	6	6	NUM
ejpam-5098	395	3	]	]	PUNCT
ejpam-5098	395	4	an	an	DET
ejpam-5098	395	5	adl	adl	NOUN
ejpam-5098	395	6	r	r	NOUN
ejpam-5098	395	7	with	with	ADP
ejpam-5098	395	8	maximal	maximal	ADJ
ejpam-5098	395	9	elements	element	NOUN
ejpam-5098	395	10	is	be	AUX
ejpam-5098	395	11	said	say	VERB
ejpam-5098	395	12	to	to	PART
ejpam-5098	395	13	be	be	AUX
ejpam-5098	395	14	normal	normal	ADJ
ejpam-5098	395	15	if	if	SCONJ
ejpam-5098	395	16	for	for	ADP
ejpam-5098	395	17	any	any	DET
ejpam-5098	395	18	x	x	NOUN
ejpam-5098	395	19	,	,	PUNCT
ejpam-5098	395	20	y	y	PROPN
ejpam-5098	395	21	∈	∈	PROPN
ejpam-5098	395	22	l	l	NOUN
ejpam-5098	395	23	with	with	ADP
ejpam-5098	395	24	x	x	PROPN
ejpam-5098	395	25	∧	∧	NOUN
ejpam-5098	395	26	y	y	PROPN
ejpam-5098	395	27	=	=	SYM
ejpam-5098	395	28	0	0	PROPN
ejpam-5098	395	29	,	,	PUNCT
ejpam-5098	395	30	there	there	PRON
ejpam-5098	395	31	exist	exist	VERB
ejpam-5098	395	32	elements	element	NOUN
ejpam-5098	395	33	a	a	PRON
ejpam-5098	395	34	,	,	PUNCT
ejpam-5098	395	35	b	b	X
ejpam-5098	395	36	∈	∈	NOUN
ejpam-5098	395	37	r	r	NOUN
ejpam-5098	395	38	such	such	ADJ
ejpam-5098	395	39	that	that	SCONJ
ejpam-5098	395	40	x	x	SYM
ejpam-5098	395	41	∧	∧	NOUN
ejpam-5098	395	42	a	a	PRON
ejpam-5098	395	43	=	=	SYM
ejpam-5098	395	44	0	0	NUM
ejpam-5098	395	45	,	,	PUNCT
ejpam-5098	395	46	y	y	PROPN
ejpam-5098	395	47	∧	∧	PROPN
ejpam-5098	395	48	b	b	PROPN
ejpam-5098	395	49	=	=	SYM
ejpam-5098	395	50	0	0	PROPN
ejpam-5098	395	51	and	and	CCONJ
ejpam-5098	395	52	a	a	DET
ejpam-5098	395	53	∨	∨	NOUN
ejpam-5098	395	54	b	b	NOUN
ejpam-5098	395	55	is	be	AUX
ejpam-5098	395	56	maximal	maximal	ADJ
ejpam-5098	395	57	.	.	PUNCT
ejpam-5098	396	1	theorem	theorem	NOUN
ejpam-5098	396	2	15	15	NUM
ejpam-5098	396	3	.	.	PUNCT
ejpam-5098	397	1	let	let	VERB
ejpam-5098	397	2	r	r	PRON
ejpam-5098	397	3	be	be	AUX
ejpam-5098	397	4	an	an	DET
ejpam-5098	397	5	adl	adl	NOUN
ejpam-5098	397	6	with	with	ADP
ejpam-5098	397	7	maximal	maximal	ADJ
ejpam-5098	397	8	elements	element	NOUN
ejpam-5098	397	9	.	.	PUNCT
ejpam-5098	398	1	then	then	ADV
ejpam-5098	398	2	the	the	DET
ejpam-5098	398	3	following	follow	VERB
ejpam-5098	398	4	are	be	AUX
ejpam-5098	398	5	equivalent	equivalent	ADJ
ejpam-5098	398	6	:	:	PUNCT
ejpam-5098	398	7	(	(	PUNCT
ejpam-5098	398	8	1	1	X
ejpam-5098	398	9	)	)	PUNCT
ejpam-5098	398	10	r	r	NOUN
ejpam-5098	398	11	is	be	AUX
ejpam-5098	398	12	normal	normal	ADJ
ejpam-5098	398	13	(	(	PUNCT
ejpam-5098	398	14	2	2	NUM
ejpam-5098	398	15	)	)	PUNCT
ejpam-5098	398	16	every	every	DET
ejpam-5098	398	17	minimal	minimal	ADJ
ejpam-5098	398	18	prime	prime	ADJ
ejpam-5098	398	19	ideal	ideal	NOUN
ejpam-5098	398	20	is	be	AUX
ejpam-5098	398	21	a	a	DET
ejpam-5098	398	22	σ	σ	NOUN
ejpam-5098	398	23	-	-	PUNCT
ejpam-5098	398	24	ideal	ideal	NOUN
ejpam-5098	398	25	of	of	ADP
ejpam-5098	398	26	r	r	NOUN
ejpam-5098	398	27	(	(	PUNCT
ejpam-5098	398	28	3	3	NUM
ejpam-5098	398	29	)	)	PUNCT
ejpam-5098	398	30	every	every	DET
ejpam-5098	398	31	prime	prime	ADJ
ejpam-5098	398	32	ideal	ideal	NOUN
ejpam-5098	398	33	contains	contain	VERB
ejpam-5098	398	34	a	a	DET
ejpam-5098	398	35	unique	unique	ADJ
ejpam-5098	398	36	minimal	minimal	ADJ
ejpam-5098	398	37	prime	prime	ADJ
ejpam-5098	398	38	ideal	ideal	PROPN
ejpam-5098	398	39	r.	r.	PROPN
ejpam-5098	398	40	noorbhasha	noorbhasha	PROPN
ejpam-5098	398	41	,	,	PUNCT
ejpam-5098	398	42	r.	r.	PROPN
ejpam-5098	398	43	bandaru	bandaru	PROPN
ejpam-5098	398	44	,	,	PUNCT
ejpam-5098	398	45	a.	a.	NOUN
ejpam-5098	398	46	iampan	iampan	PROPN
ejpam-5098	398	47	/	/	SYM
ejpam-5098	398	48	eur	eur	PROPN
ejpam-5098	398	49	.	.	PUNCT
ejpam-5098	399	1	j.	j.	PROPN
ejpam-5098	399	2	pure	pure	PROPN
ejpam-5098	399	3	appl	appl	PROPN
ejpam-5098	399	4	.	.	PROPN
ejpam-5098	399	5	math	math	PROPN
ejpam-5098	399	6	,	,	PUNCT
ejpam-5098	399	7	17	17	NUM
ejpam-5098	399	8	(	(	PUNCT
ejpam-5098	399	9	2	2	NUM
ejpam-5098	399	10	)	)	PUNCT
ejpam-5098	399	11	(	(	PUNCT
ejpam-5098	399	12	2024	2024	NUM
ejpam-5098	399	13	)	)	PUNCT
ejpam-5098	399	14	,	,	PUNCT
ejpam-5098	399	15	1094	1094	NUM
ejpam-5098	399	16	-	-	SYM
ejpam-5098	399	17	1112	1112	NUM
ejpam-5098	399	18	1106	1106	NUM
ejpam-5098	399	19	(	(	PUNCT
ejpam-5098	399	20	4	4	NUM
ejpam-5098	399	21	)	)	PUNCT
ejpam-5098	399	22	for	for	ADP
ejpam-5098	399	23	every	every	DET
ejpam-5098	399	24	ideal	ideal	ADJ
ejpam-5098	399	25	q	q	NOUN
ejpam-5098	399	26	,	,	PUNCT
ejpam-5098	399	27	qo	qo	PROPN
ejpam-5098	399	28	is	be	AUX
ejpam-5098	399	29	a	a	DET
ejpam-5098	399	30	prime	prime	ADJ
ejpam-5098	399	31	ideal	ideal	NOUN
ejpam-5098	399	32	.	.	PUNCT
ejpam-5098	400	1	proof	proof	NOUN
ejpam-5098	400	2	.	.	PUNCT
ejpam-5098	401	1	(	(	PUNCT
ejpam-5098	401	2	1	1	X
ejpam-5098	401	3	)	)	PUNCT
ejpam-5098	401	4	⇒	⇒	NOUN
ejpam-5098	401	5	(	(	PUNCT
ejpam-5098	401	6	2	2	NUM
ejpam-5098	401	7	):	):	PUNCT
ejpam-5098	401	8	assume	assume	VERB
ejpam-5098	401	9	that	that	SCONJ
ejpam-5098	401	10	r	r	NOUN
ejpam-5098	401	11	is	be	AUX
ejpam-5098	401	12	normal	normal	ADJ
ejpam-5098	401	13	.	.	PUNCT
ejpam-5098	402	1	let	let	VERB
ejpam-5098	402	2	m	m	PRON
ejpam-5098	402	3	be	be	AUX
ejpam-5098	402	4	any	any	DET
ejpam-5098	402	5	minimal	minimal	ADJ
ejpam-5098	402	6	prime	prime	ADJ
ejpam-5098	402	7	ideal	ideal	NOUN
ejpam-5098	402	8	of	of	ADP
ejpam-5098	402	9	r.	r.	PROPN
ejpam-5098	402	10	we	we	PRON
ejpam-5098	402	11	prove	prove	VERB
ejpam-5098	402	12	that	that	SCONJ
ejpam-5098	402	13	m	m	PROPN
ejpam-5098	402	14	is	be	AUX
ejpam-5098	402	15	a	a	DET
ejpam-5098	402	16	σ	σ	NOUN
ejpam-5098	402	17	-	-	PUNCT
ejpam-5098	402	18	ideal	ideal	NOUN
ejpam-5098	402	19	of	of	ADP
ejpam-5098	402	20	r.	r.	PROPN
ejpam-5098	402	21	clearly	clearly	ADV
ejpam-5098	402	22	,	,	PUNCT
ejpam-5098	402	23	we	we	PRON
ejpam-5098	402	24	have	have	VERB
ejpam-5098	402	25	that	that	DET
ejpam-5098	402	26	mσ	mσ	PROPN
ejpam-5098	402	27	⊆m	⊆m	NOUN
ejpam-5098	402	28	.	.	PUNCT
ejpam-5098	403	1	let	let	VERB
ejpam-5098	403	2	x	x	PRON
ejpam-5098	403	3	∈m	∈m	VERB
ejpam-5098	403	4	.	.	PUNCT
ejpam-5098	404	1	since	since	SCONJ
ejpam-5098	404	2	m	m	PROPN
ejpam-5098	404	3	is	be	AUX
ejpam-5098	404	4	minimal	minimal	ADJ
ejpam-5098	404	5	,	,	PUNCT
ejpam-5098	404	6	there	there	PRON
ejpam-5098	404	7	exists	exist	VERB
ejpam-5098	404	8	an	an	DET
ejpam-5098	404	9	element	element	NOUN
ejpam-5098	404	10	y	y	NOUN
ejpam-5098	404	11	/∈	/∈	PUNCT
ejpam-5098	405	1	m	m	VERB
ejpam-5098	405	2	such	such	ADJ
ejpam-5098	405	3	that	that	SCONJ
ejpam-5098	405	4	x	x	PUNCT
ejpam-5098	405	5	∧	∧	NOUN
ejpam-5098	405	6	y	y	NOUN
ejpam-5098	405	7	=	=	NOUN
ejpam-5098	405	8	0	0	PROPN
ejpam-5098	405	9	.	.	PUNCT
ejpam-5098	406	1	since	since	SCONJ
ejpam-5098	406	2	r	r	NOUN
ejpam-5098	406	3	is	be	AUX
ejpam-5098	406	4	normal	normal	ADJ
ejpam-5098	406	5	,	,	PUNCT
ejpam-5098	406	6	there	there	PRON
ejpam-5098	406	7	exist	exist	VERB
ejpam-5098	406	8	elements	element	NOUN
ejpam-5098	406	9	a	a	PRON
ejpam-5098	406	10	,	,	PUNCT
ejpam-5098	406	11	b	b	X
ejpam-5098	406	12	∈	∈	NOUN
ejpam-5098	406	13	r	r	NOUN
ejpam-5098	406	14	such	such	ADJ
ejpam-5098	406	15	that	that	SCONJ
ejpam-5098	406	16	x	x	SYM
ejpam-5098	406	17	∧	∧	NOUN
ejpam-5098	406	18	a	a	PRON
ejpam-5098	406	19	=	=	SYM
ejpam-5098	406	20	0	0	NUM
ejpam-5098	406	21	,	,	PUNCT
ejpam-5098	406	22	y	y	PROPN
ejpam-5098	406	23	∧	∧	PROPN
ejpam-5098	406	24	b	b	PROPN
ejpam-5098	406	25	=	=	SYM
ejpam-5098	406	26	0	0	PROPN
ejpam-5098	406	27	and	and	CCONJ
ejpam-5098	406	28	a	a	DET
ejpam-5098	406	29	∨	∨	NOUN
ejpam-5098	406	30	b	b	NOUN
ejpam-5098	406	31	is	be	AUX
ejpam-5098	406	32	maximal	maximal	ADJ
ejpam-5098	406	33	.	.	PUNCT
ejpam-5098	407	1	that	that	PRON
ejpam-5098	407	2	implies	imply	VERB
ejpam-5098	407	3	a	a	DET
ejpam-5098	407	4	∈	∈	NOUN
ejpam-5098	407	5	(	(	PUNCT
ejpam-5098	407	6	x)∗	x)∗	PROPN
ejpam-5098	407	7	,	,	PUNCT
ejpam-5098	407	8	b	b	NOUN
ejpam-5098	407	9	∈m	∈m	NOUN
ejpam-5098	407	10	and	and	CCONJ
ejpam-5098	407	11	a	a	DET
ejpam-5098	407	12	∨	∨	NOUN
ejpam-5098	407	13	b	b	NOUN
ejpam-5098	407	14	is	be	AUX
ejpam-5098	407	15	maximal	maximal	ADJ
ejpam-5098	407	16	.	.	PUNCT
ejpam-5098	408	1	therefore	therefore	ADV
ejpam-5098	408	2	,	,	PUNCT
ejpam-5098	408	3	(	(	PUNCT
ejpam-5098	408	4	x)∗	x)∗	ADV
ejpam-5098	408	5	∨m	∨m	NOUN
ejpam-5098	408	6	=	=	SYM
ejpam-5098	408	7	r	r	NOUN
ejpam-5098	408	8	and	and	CCONJ
ejpam-5098	408	9	hence	hence	ADV
ejpam-5098	408	10	m	m	VERB
ejpam-5098	408	11	is	be	AUX
ejpam-5098	408	12	a	a	DET
ejpam-5098	408	13	σ	σ	NOUN
ejpam-5098	408	14	-	-	PUNCT
ejpam-5098	408	15	ideal	ideal	NOUN
ejpam-5098	408	16	of	of	ADP
ejpam-5098	408	17	r.	r.	PROPN
ejpam-5098	408	18	(	(	PUNCT
ejpam-5098	408	19	2	2	NUM
ejpam-5098	408	20	)	)	PUNCT
ejpam-5098	408	21	⇒	⇒	NOUN
ejpam-5098	408	22	(	(	PUNCT
ejpam-5098	408	23	3	3	NUM
ejpam-5098	408	24	):	):	PUNCT
ejpam-5098	408	25	assume	assume	VERB
ejpam-5098	408	26	(	(	PUNCT
ejpam-5098	408	27	2	2	NUM
ejpam-5098	408	28	)	)	PUNCT
ejpam-5098	408	29	.	.	PUNCT
ejpam-5098	409	1	let	let	VERB
ejpam-5098	409	2	p	p	PRON
ejpam-5098	409	3	be	be	AUX
ejpam-5098	409	4	any	any	DET
ejpam-5098	409	5	prime	prime	ADJ
ejpam-5098	409	6	ideal	ideal	NOUN
ejpam-5098	409	7	of	of	ADP
ejpam-5098	409	8	r	r	NOUN
ejpam-5098	409	9	and	and	CCONJ
ejpam-5098	409	10	m	m	PROPN
ejpam-5098	409	11	,	,	PUNCT
ejpam-5098	409	12	n	n	X
ejpam-5098	409	13	be	be	VERB
ejpam-5098	409	14	any	any	DET
ejpam-5098	409	15	two	two	NUM
ejpam-5098	409	16	minimal	minimal	ADJ
ejpam-5098	409	17	prime	prime	ADJ
ejpam-5098	409	18	ideals	ideal	NOUN
ejpam-5098	409	19	of	of	ADP
ejpam-5098	409	20	r	r	NOUN
ejpam-5098	409	21	such	such	ADJ
ejpam-5098	409	22	that	that	SCONJ
ejpam-5098	409	23	m	m	VERB
ejpam-5098	409	24	⊆	⊆	NUM
ejpam-5098	409	25	p	p	NOUN
ejpam-5098	409	26	and	and	CCONJ
ejpam-5098	409	27	n	n	CCONJ
ejpam-5098	409	28	⊆	⊆	NUM
ejpam-5098	409	29	p	p	NOUN
ejpam-5098	409	30	.	.	PUNCT
ejpam-5098	410	1	by	by	ADP
ejpam-5098	410	2	our	our	PRON
ejpam-5098	410	3	assumption	assumption	NOUN
ejpam-5098	410	4	,	,	PUNCT
ejpam-5098	410	5	we	we	PRON
ejpam-5098	410	6	have	have	VERB
ejpam-5098	410	7	that	that	PRON
ejpam-5098	410	8	m	m	VERB
ejpam-5098	410	9	and	and	CCONJ
ejpam-5098	410	10	n	n	PROPN
ejpam-5098	410	11	are	be	AUX
ejpam-5098	410	12	σ	σ	NOUN
ejpam-5098	410	13	-	-	PUNCT
ejpam-5098	410	14	ideals	ideal	NOUN
ejpam-5098	410	15	of	of	ADP
ejpam-5098	410	16	r.	r.	PROPN
ejpam-5098	410	17	that	that	PRON
ejpam-5098	410	18	implies	imply	VERB
ejpam-5098	410	19	mσ	mσ	NOUN
ejpam-5098	410	20	=	=	VERB
ejpam-5098	410	21	m	m	PROPN
ejpam-5098	410	22	and	and	CCONJ
ejpam-5098	410	23	nσ	nσ	PRON
ejpam-5098	410	24	=	=	SYM
ejpam-5098	410	25	n	n	PROPN
ejpam-5098	410	26	.	.	PUNCT
ejpam-5098	411	1	we	we	PRON
ejpam-5098	411	2	prove	prove	VERB
ejpam-5098	411	3	that	that	SCONJ
ejpam-5098	411	4	m	m	VERB
ejpam-5098	411	5	=	=	SYM
ejpam-5098	411	6	n	n	PROPN
ejpam-5098	411	7	.	.	PUNCT
ejpam-5098	412	1	suppose	suppose	VERB
ejpam-5098	412	2	m	m	PRON
ejpam-5098	412	3	̸=	̸=	PROPN
ejpam-5098	412	4	n	n	NOUN
ejpam-5098	412	5	.	.	PUNCT
ejpam-5098	413	1	then	then	ADV
ejpam-5098	413	2	choose	choose	VERB
ejpam-5098	413	3	an	an	DET
ejpam-5098	413	4	element	element	NOUN
ejpam-5098	413	5	x	x	PUNCT
ejpam-5098	413	6	∈m	∈m	NOUN
ejpam-5098	413	7	such	such	ADJ
ejpam-5098	413	8	that	that	PRON
ejpam-5098	413	9	x	x	SYM
ejpam-5098	413	10	/∈	/∈	PUNCT
ejpam-5098	413	11	n	n	INTJ
ejpam-5098	413	12	.	.	PUNCT
ejpam-5098	414	1	since	since	SCONJ
ejpam-5098	414	2	x	x	SYM
ejpam-5098	414	3	∈m	∈m	NOUN
ejpam-5098	414	4	,	,	PUNCT
ejpam-5098	414	5	we	we	PRON
ejpam-5098	414	6	get	get	VERB
ejpam-5098	414	7	that	that	PRON
ejpam-5098	414	8	x	x	SYM
ejpam-5098	414	9	∈	∈	NOUN
ejpam-5098	414	10	mσ	mσ	NOUN
ejpam-5098	414	11	and	and	CCONJ
ejpam-5098	414	12	hence	hence	ADV
ejpam-5098	414	13	(	(	PUNCT
ejpam-5098	414	14	x)∗	x)∗	ADV
ejpam-5098	414	15	∨m	∨m	PROPN
ejpam-5098	414	16	=	=	PUNCT
ejpam-5098	414	17	r.	r.	NOUN
ejpam-5098	414	18	that	that	PRON
ejpam-5098	414	19	implies	imply	VERB
ejpam-5098	414	20	a	a	DET
ejpam-5098	414	21	∨	∨	PROPN
ejpam-5098	414	22	b	b	NOUN
ejpam-5098	414	23	is	be	AUX
ejpam-5098	414	24	a	a	DET
ejpam-5098	414	25	maximal	maximal	ADJ
ejpam-5098	414	26	element	element	NOUN
ejpam-5098	414	27	of	of	ADP
ejpam-5098	414	28	r	r	NOUN
ejpam-5098	414	29	for	for	ADP
ejpam-5098	414	30	some	some	DET
ejpam-5098	414	31	a	a	DET
ejpam-5098	414	32	∈	∈	NOUN
ejpam-5098	414	33	(	(	PUNCT
ejpam-5098	414	34	x)∗	x)∗	PROPN
ejpam-5098	414	35	and	and	CCONJ
ejpam-5098	414	36	b	b	X
ejpam-5098	414	37	∈	∈	ADV
ejpam-5098	414	38	m	m	VERB
ejpam-5098	414	39	.	.	PUNCT
ejpam-5098	415	1	that	that	PRON
ejpam-5098	415	2	implies	imply	VERB
ejpam-5098	415	3	x	x	PUNCT
ejpam-5098	415	4	∧	∧	NOUN
ejpam-5098	415	5	a	a	PRON
ejpam-5098	415	6	=	=	SYM
ejpam-5098	415	7	0	0	NUM
ejpam-5098	415	8	,	,	PUNCT
ejpam-5098	415	9	b	b	X
ejpam-5098	415	10	∈	∈	NOUN
ejpam-5098	415	11	m	m	VERB
ejpam-5098	415	12	and	and	CCONJ
ejpam-5098	415	13	a	a	DET
ejpam-5098	415	14	∨	∨	NOUN
ejpam-5098	415	15	b	b	NOUN
ejpam-5098	415	16	is	be	AUX
ejpam-5098	415	17	maximal	maximal	ADJ
ejpam-5098	415	18	.	.	PUNCT
ejpam-5098	416	1	since	since	SCONJ
ejpam-5098	416	2	x	x	PROPN
ejpam-5098	416	3	/∈	/∈	PROPN
ejpam-5098	416	4	n	n	CCONJ
ejpam-5098	416	5	,	,	PUNCT
ejpam-5098	416	6	we	we	PRON
ejpam-5098	416	7	get	get	VERB
ejpam-5098	416	8	that	that	SCONJ
ejpam-5098	416	9	a	a	DET
ejpam-5098	416	10	∈	∈	NOUN
ejpam-5098	416	11	n	n	NOUN
ejpam-5098	416	12	.	.	PUNCT
ejpam-5098	417	1	since	since	SCONJ
ejpam-5098	417	2	m	m	PROPN
ejpam-5098	417	3	⊆	⊆	NUM
ejpam-5098	417	4	p	p	NOUN
ejpam-5098	417	5	and	and	CCONJ
ejpam-5098	417	6	n	n	CCONJ
ejpam-5098	417	7	⊆	⊆	NUM
ejpam-5098	417	8	p	p	NOUN
ejpam-5098	417	9	,	,	PUNCT
ejpam-5098	417	10	we	we	PRON
ejpam-5098	417	11	get	get	VERB
ejpam-5098	417	12	that	that	PRON
ejpam-5098	417	13	a	a	DET
ejpam-5098	417	14	,	,	PUNCT
ejpam-5098	417	15	b	b	PROPN
ejpam-5098	417	16	∈	∈	PROPN
ejpam-5098	417	17	p	p	NOUN
ejpam-5098	417	18	.	.	PUNCT
ejpam-5098	418	1	that	that	PRON
ejpam-5098	418	2	implies	imply	VERB
ejpam-5098	418	3	a∨	a∨	PROPN
ejpam-5098	418	4	b	b	PROPN
ejpam-5098	418	5	∈	∈	PROPN
ejpam-5098	418	6	p	p	NOUN
ejpam-5098	418	7	,	,	PUNCT
ejpam-5098	418	8	which	which	PRON
ejpam-5098	418	9	is	be	AUX
ejpam-5098	418	10	a	a	DET
ejpam-5098	418	11	contradiction	contradiction	NOUN
ejpam-5098	418	12	.	.	PUNCT
ejpam-5098	419	1	therefore	therefore	ADV
ejpam-5098	419	2	,	,	PUNCT
ejpam-5098	419	3	m	m	VERB
ejpam-5098	419	4	=	=	SYM
ejpam-5098	419	5	n	n	X
ejpam-5098	419	6	.	.	PUNCT
ejpam-5098	420	1	hence	hence	ADV
ejpam-5098	420	2	,	,	PUNCT
ejpam-5098	420	3	every	every	DET
ejpam-5098	420	4	prime	prime	ADJ
ejpam-5098	420	5	ideal	ideal	NOUN
ejpam-5098	420	6	contains	contain	VERB
ejpam-5098	420	7	a	a	DET
ejpam-5098	420	8	unique	unique	ADJ
ejpam-5098	420	9	minimal	minimal	ADJ
ejpam-5098	420	10	prime	prime	ADJ
ejpam-5098	420	11	ideal	ideal	NOUN
ejpam-5098	420	12	of	of	ADP
ejpam-5098	420	13	r.	r.	PROPN
ejpam-5098	420	14	(	(	PUNCT
ejpam-5098	420	15	3	3	NUM
ejpam-5098	420	16	)	)	PUNCT
ejpam-5098	420	17	⇒	⇒	NOUN
ejpam-5098	420	18	(	(	PUNCT
ejpam-5098	420	19	4	4	NUM
ejpam-5098	420	20	):	):	PUNCT
ejpam-5098	420	21	assume	assume	VERB
ejpam-5098	420	22	(	(	PUNCT
ejpam-5098	420	23	3	3	NUM
ejpam-5098	420	24	)	)	PUNCT
ejpam-5098	420	25	.	.	PUNCT
ejpam-5098	421	1	let	let	VERB
ejpam-5098	421	2	q	q	PRON
ejpam-5098	421	3	be	be	AUX
ejpam-5098	421	4	any	any	DET
ejpam-5098	421	5	prime	prime	ADJ
ejpam-5098	421	6	ideal	ideal	NOUN
ejpam-5098	421	7	of	of	ADP
ejpam-5098	421	8	r.	r.	PROPN
ejpam-5098	421	9	we	we	PRON
ejpam-5098	421	10	have	have	VERB
ejpam-5098	421	11	that	that	PRON
ejpam-5098	421	12	qo	qo	PROPN
ejpam-5098	421	13	is	be	AUX
ejpam-5098	421	14	the	the	DET
ejpam-5098	421	15	intersection	intersection	NOUN
ejpam-5098	421	16	of	of	ADP
ejpam-5098	421	17	all	all	DET
ejpam-5098	421	18	minimal	minimal	ADJ
ejpam-5098	421	19	prime	prime	ADJ
ejpam-5098	421	20	ideals	ideal	NOUN
ejpam-5098	421	21	contained	contain	VERB
ejpam-5098	421	22	in	in	ADP
ejpam-5098	421	23	q.	q.	NOUN
ejpam-5098	421	24	by	by	ADP
ejpam-5098	421	25	our	our	PRON
ejpam-5098	421	26	assumption	assumption	NOUN
ejpam-5098	421	27	,	,	PUNCT
ejpam-5098	421	28	we	we	PRON
ejpam-5098	421	29	get	get	VERB
ejpam-5098	421	30	that	that	PRON
ejpam-5098	421	31	qo	qo	PROPN
ejpam-5098	421	32	is	be	AUX
ejpam-5098	421	33	a	a	DET
ejpam-5098	421	34	minimal	minimal	ADJ
ejpam-5098	421	35	prime	prime	ADJ
ejpam-5098	421	36	ideal	ideal	NOUN
ejpam-5098	421	37	of	of	ADP
ejpam-5098	421	38	r	r	NOUN
ejpam-5098	421	39	contained	contain	VERB
ejpam-5098	421	40	in	in	ADP
ejpam-5098	421	41	q.	q.	PROPN
ejpam-5098	421	42	therefore	therefore	ADV
ejpam-5098	421	43	,	,	PUNCT
ejpam-5098	421	44	qo	qo	PROPN
ejpam-5098	421	45	is	be	AUX
ejpam-5098	421	46	prime	prime	ADJ
ejpam-5098	421	47	.	.	PUNCT
ejpam-5098	422	1	(	(	PUNCT
ejpam-5098	422	2	4	4	X
ejpam-5098	422	3	)	)	PUNCT
ejpam-5098	422	4	⇒	⇒	NOUN
ejpam-5098	422	5	(	(	PUNCT
ejpam-5098	422	6	1	1	NUM
ejpam-5098	422	7	):	):	PUNCT
ejpam-5098	422	8	assume	assume	VERB
ejpam-5098	422	9	(	(	PUNCT
ejpam-5098	422	10	4	4	NUM
ejpam-5098	422	11	)	)	PUNCT
ejpam-5098	422	12	.	.	PUNCT
ejpam-5098	423	1	let	let	VERB
ejpam-5098	423	2	a	a	DET
ejpam-5098	423	3	,	,	PUNCT
ejpam-5098	423	4	b	b	X
ejpam-5098	423	5	∈	∈	NOUN
ejpam-5098	423	6	r	r	NOUN
ejpam-5098	423	7	with	with	ADP
ejpam-5098	423	8	a	a	DET
ejpam-5098	423	9	∧	∧	PROPN
ejpam-5098	423	10	b	b	NOUN
ejpam-5098	423	11	=	=	SYM
ejpam-5098	423	12	0	0	PROPN
ejpam-5098	424	1	.	.	PUNCT
ejpam-5098	424	2	suppose	suppose	VERB
ejpam-5098	424	3	(	(	PUNCT
ejpam-5098	424	4	a)∗	a)∗	PROPN
ejpam-5098	424	5	∨	∨	PROPN
ejpam-5098	424	6	(	(	PUNCT
ejpam-5098	424	7	b)∗	b)∗	PROPN
ejpam-5098	424	8	̸=	̸=	PROPN
ejpam-5098	424	9	r.	r.	PROPN
ejpam-5098	424	10	then	then	ADV
ejpam-5098	424	11	there	there	PRON
ejpam-5098	424	12	exists	exist	VERB
ejpam-5098	424	13	n	n	PRON
ejpam-5098	424	14	∈	∈	NOUN
ejpam-5098	424	15	specr	specr	NOUN
ejpam-5098	424	16	such	such	ADJ
ejpam-5098	424	17	that	that	SCONJ
ejpam-5098	424	18	(	(	PUNCT
ejpam-5098	424	19	a)∗	a)∗	PROPN
ejpam-5098	424	20	∨	∨	PROPN
ejpam-5098	424	21	(	(	PUNCT
ejpam-5098	424	22	b)∗	b)∗	PROPN
ejpam-5098	424	23	⊆	⊆	NUM
ejpam-5098	424	24	n	n	NOUN
ejpam-5098	424	25	.	.	PUNCT
ejpam-5098	425	1	that	that	PRON
ejpam-5098	425	2	implies	imply	VERB
ejpam-5098	425	3	(	(	PUNCT
ejpam-5098	425	4	a)∗	a)∗	PROPN
ejpam-5098	425	5	⊆	⊆	NUM
ejpam-5098	425	6	n	n	PROPN
ejpam-5098	425	7	and	and	CCONJ
ejpam-5098	425	8	(	(	PUNCT
ejpam-5098	425	9	b)∗	b)∗	PROPN
ejpam-5098	425	10	⊆	⊆	NUM
ejpam-5098	425	11	n	n	CCONJ
ejpam-5098	425	12	,	,	PUNCT
ejpam-5098	425	13	which	which	PRON
ejpam-5098	425	14	give	give	VERB
ejpam-5098	425	15	that	that	DET
ejpam-5098	425	16	a	a	DET
ejpam-5098	425	17	/∈	/∈	INTJ
ejpam-5098	425	18	po	po	NOUN
ejpam-5098	425	19	and	and	CCONJ
ejpam-5098	425	20	b	b	PROPN
ejpam-5098	425	21	/∈	/∈	PUNCT
ejpam-5098	425	22	po	po	NOUN
ejpam-5098	425	23	,	,	PUNCT
ejpam-5098	425	24	which	which	PRON
ejpam-5098	425	25	is	be	AUX
ejpam-5098	425	26	a	a	DET
ejpam-5098	425	27	contradiction	contradiction	NOUN
ejpam-5098	425	28	to	to	ADP
ejpam-5098	425	29	po	po	PROPN
ejpam-5098	425	30	is	be	AUX
ejpam-5098	425	31	a	a	DET
ejpam-5098	425	32	prime	prime	ADJ
ejpam-5098	425	33	ideal	ideal	NOUN
ejpam-5098	425	34	.	.	PUNCT
ejpam-5098	426	1	therefore	therefore	ADV
ejpam-5098	426	2	,	,	PUNCT
ejpam-5098	426	3	(	(	PUNCT
ejpam-5098	426	4	a)∗	a)∗	PROPN
ejpam-5098	426	5	∨	∨	PROPN
ejpam-5098	426	6	(	(	PUNCT
ejpam-5098	426	7	b)∗	b)∗	PROPN
ejpam-5098	426	8	=	=	SYM
ejpam-5098	426	9	r.	r.	PROPN
ejpam-5098	426	10	hence	hence	ADV
ejpam-5098	426	11	,	,	PUNCT
ejpam-5098	426	12	there	there	PRON
ejpam-5098	426	13	exist	exist	VERB
ejpam-5098	426	14	elements	element	NOUN
ejpam-5098	426	15	s	s	PART
ejpam-5098	426	16	∈	∈	NOUN
ejpam-5098	426	17	(	(	PUNCT
ejpam-5098	426	18	a)∗	a)∗	PROPN
ejpam-5098	426	19	and	and	CCONJ
ejpam-5098	426	20	t	t	PROPN
ejpam-5098	426	21	∈	∈	PROPN
ejpam-5098	426	22	(	(	PUNCT
ejpam-5098	426	23	b)∗	b)∗	PROPN
ejpam-5098	426	24	such	such	ADJ
ejpam-5098	426	25	that	that	PRON
ejpam-5098	426	26	s	s	PART
ejpam-5098	426	27	∨	∨	PROPN
ejpam-5098	426	28	t	t	PROPN
ejpam-5098	426	29	is	be	AUX
ejpam-5098	426	30	maximal	maximal	ADJ
ejpam-5098	426	31	.	.	PUNCT
ejpam-5098	427	1	thus	thus	ADV
ejpam-5098	427	2	,	,	PUNCT
ejpam-5098	427	3	r	r	NOUN
ejpam-5098	427	4	is	be	AUX
ejpam-5098	427	5	normal	normal	ADJ
ejpam-5098	427	6	.	.	PUNCT
ejpam-5098	428	1	theorem	theorem	VERB
ejpam-5098	428	2	16	16	NUM
ejpam-5098	428	3	.	.	PUNCT
ejpam-5098	429	1	let	let	VERB
ejpam-5098	429	2	r	r	PRON
ejpam-5098	429	3	be	be	AUX
ejpam-5098	429	4	an	an	DET
ejpam-5098	429	5	adl	adl	NOUN
ejpam-5098	429	6	with	with	ADP
ejpam-5098	429	7	maximal	maximal	ADJ
ejpam-5098	429	8	elements	element	NOUN
ejpam-5098	429	9	.	.	PUNCT
ejpam-5098	430	1	then	then	ADV
ejpam-5098	430	2	r	r	NOUN
ejpam-5098	430	3	is	be	AUX
ejpam-5098	430	4	normal	normal	ADJ
ejpam-5098	430	5	if	if	SCONJ
ejpam-5098	430	6	and	and	CCONJ
ejpam-5098	430	7	only	only	ADV
ejpam-5098	430	8	if	if	SCONJ
ejpam-5098	430	9	for	for	ADP
ejpam-5098	430	10	every	every	DET
ejpam-5098	430	11	a	a	DET
ejpam-5098	430	12	∈	∈	PROPN
ejpam-5098	430	13	r	r	NOUN
ejpam-5098	430	14	,	,	PUNCT
ejpam-5098	430	15	(	(	PUNCT
ejpam-5098	430	16	a)∗	a)∗	PROPN
ejpam-5098	430	17	is	be	AUX
ejpam-5098	430	18	a	a	DET
ejpam-5098	430	19	σ	σ	NOUN
ejpam-5098	430	20	-	-	PUNCT
ejpam-5098	430	21	ideal	ideal	NOUN
ejpam-5098	430	22	of	of	ADP
ejpam-5098	430	23	r.	r.	PROPN
ejpam-5098	430	24	proof	proof	PROPN
ejpam-5098	430	25	.	.	PUNCT
ejpam-5098	431	1	assume	assume	VERB
ejpam-5098	431	2	that	that	SCONJ
ejpam-5098	431	3	r	r	NOUN
ejpam-5098	431	4	is	be	AUX
ejpam-5098	431	5	normal	normal	ADJ
ejpam-5098	431	6	.	.	PUNCT
ejpam-5098	432	1	let	let	VERB
ejpam-5098	432	2	a	a	DET
ejpam-5098	432	3	be	be	AUX
ejpam-5098	432	4	any	any	DET
ejpam-5098	432	5	element	element	NOUN
ejpam-5098	432	6	of	of	ADP
ejpam-5098	432	7	r.	r.	PROPN
ejpam-5098	432	8	we	we	PRON
ejpam-5098	432	9	prove	prove	VERB
ejpam-5098	432	10	that	that	SCONJ
ejpam-5098	432	11	(	(	PUNCT
ejpam-5098	432	12	a)∗	a)∗	PROPN
ejpam-5098	432	13	is	be	AUX
ejpam-5098	432	14	a	a	DET
ejpam-5098	432	15	σ	σ	NOUN
ejpam-5098	432	16	-	-	PUNCT
ejpam-5098	432	17	ideal	ideal	NOUN
ejpam-5098	432	18	of	of	ADP
ejpam-5098	432	19	r.	r.	PROPN
ejpam-5098	432	20	clearly	clearly	ADV
ejpam-5098	432	21	,	,	PUNCT
ejpam-5098	432	22	we	we	PRON
ejpam-5098	432	23	have	have	VERB
ejpam-5098	432	24	that	that	PRON
ejpam-5098	432	25	(	(	PUNCT
ejpam-5098	432	26	a)∗	a)∗	PROPN
ejpam-5098	432	27	σ	σ	PROPN
ejpam-5098	432	28	⊆	⊆	NUM
ejpam-5098	432	29	(	(	PUNCT
ejpam-5098	432	30	a)∗.	a)∗.	NOUN
ejpam-5098	432	31	let	let	VERB
ejpam-5098	432	32	b	b	PROPN
ejpam-5098	432	33	∈	∈	PROPN
ejpam-5098	432	34	(	(	PUNCT
ejpam-5098	432	35	a)∗.	a)∗.	NOUN
ejpam-5098	432	36	then	then	ADV
ejpam-5098	432	37	a	a	DET
ejpam-5098	432	38	∧	∧	PROPN
ejpam-5098	432	39	b	b	PROPN
ejpam-5098	432	40	=	=	NOUN
ejpam-5098	432	41	0	0	PROPN
ejpam-5098	432	42	.	.	PUNCT
ejpam-5098	433	1	since	since	SCONJ
ejpam-5098	433	2	r	r	NOUN
ejpam-5098	433	3	is	be	AUX
ejpam-5098	433	4	normal	normal	ADJ
ejpam-5098	433	5	,	,	PUNCT
ejpam-5098	433	6	we	we	PRON
ejpam-5098	433	7	have	have	VERB
ejpam-5098	433	8	that	that	PRON
ejpam-5098	433	9	(	(	PUNCT
ejpam-5098	433	10	a)∗	a)∗	PROPN
ejpam-5098	433	11	∨	∨	PROPN
ejpam-5098	433	12	(	(	PUNCT
ejpam-5098	433	13	b)∗	b)∗	PROPN
ejpam-5098	433	14	=	=	PUNCT
ejpam-5098	433	15	r.	r.	PROPN
ejpam-5098	433	16	that	that	PRON
ejpam-5098	433	17	implies	imply	VERB
ejpam-5098	433	18	b	b	PROPN
ejpam-5098	433	19	∈	∈	PROPN
ejpam-5098	433	20	(	(	PUNCT
ejpam-5098	433	21	a)∗	a)∗	PROPN
ejpam-5098	433	22	σ	σ	PROPN
ejpam-5098	433	23	and	and	CCONJ
ejpam-5098	433	24	hence	hence	ADV
ejpam-5098	433	25	(	(	PUNCT
ejpam-5098	433	26	a)∗	a)∗	PROPN
ejpam-5098	433	27	⊆	⊆	NUM
ejpam-5098	433	28	(	(	PUNCT
ejpam-5098	433	29	a)∗	a)∗	PROPN
ejpam-5098	433	30	σ	σ	PROPN
ejpam-5098	433	31	.	.	PUNCT
ejpam-5098	434	1	therefore	therefore	ADV
ejpam-5098	434	2	,	,	PUNCT
ejpam-5098	434	3	(	(	PUNCT
ejpam-5098	434	4	a)∗	a)∗	PROPN
ejpam-5098	434	5	σ	σ	PROPN
ejpam-5098	434	6	=	=	SYM
ejpam-5098	434	7	(	(	PUNCT
ejpam-5098	434	8	a)∗.	a)∗.	PROPN
ejpam-5098	434	9	thus	thus	ADV
ejpam-5098	434	10	,	,	PUNCT
ejpam-5098	434	11	(	(	PUNCT
ejpam-5098	434	12	a)∗	a)∗	PROPN
ejpam-5098	434	13	is	be	AUX
ejpam-5098	434	14	a	a	DET
ejpam-5098	434	15	σ	σ	NOUN
ejpam-5098	434	16	-	-	PUNCT
ejpam-5098	434	17	ideal	ideal	NOUN
ejpam-5098	434	18	of	of	ADP
ejpam-5098	434	19	r.	r.	PROPN
ejpam-5098	434	20	conversely	conversely	ADV
ejpam-5098	434	21	,	,	PUNCT
ejpam-5098	434	22	assume	assume	VERB
ejpam-5098	434	23	that	that	SCONJ
ejpam-5098	434	24	for	for	ADP
ejpam-5098	434	25	every	every	DET
ejpam-5098	434	26	a	a	DET
ejpam-5098	434	27	∈	∈	PROPN
ejpam-5098	434	28	r	r	NOUN
ejpam-5098	434	29	,	,	PUNCT
ejpam-5098	434	30	(	(	PUNCT
ejpam-5098	434	31	a)∗	a)∗	PROPN
ejpam-5098	434	32	is	be	AUX
ejpam-5098	434	33	a	a	DET
ejpam-5098	434	34	σ	σ	NOUN
ejpam-5098	434	35	-	-	PUNCT
ejpam-5098	434	36	ideal	ideal	NOUN
ejpam-5098	434	37	of	of	ADP
ejpam-5098	434	38	r.	r.	PROPN
ejpam-5098	434	39	we	we	PRON
ejpam-5098	434	40	prove	prove	VERB
ejpam-5098	434	41	that	that	SCONJ
ejpam-5098	434	42	r	r	NOUN
ejpam-5098	434	43	is	be	AUX
ejpam-5098	434	44	normal	normal	ADJ
ejpam-5098	434	45	.	.	PUNCT
ejpam-5098	435	1	it	it	PRON
ejpam-5098	435	2	is	be	AUX
ejpam-5098	435	3	enough	enough	ADJ
ejpam-5098	435	4	to	to	PART
ejpam-5098	435	5	prove	prove	VERB
ejpam-5098	435	6	that	that	SCONJ
ejpam-5098	435	7	every	every	DET
ejpam-5098	435	8	minimal	minimal	ADJ
ejpam-5098	435	9	prime	prime	ADJ
ejpam-5098	435	10	ideal	ideal	NOUN
ejpam-5098	435	11	of	of	ADP
ejpam-5098	435	12	r	r	NOUN
ejpam-5098	435	13	is	be	AUX
ejpam-5098	435	14	a	a	DET
ejpam-5098	435	15	σ	σ	NOUN
ejpam-5098	435	16	-	-	PUNCT
ejpam-5098	435	17	ideal	ideal	NOUN
ejpam-5098	435	18	of	of	ADP
ejpam-5098	435	19	r.	r.	PROPN
ejpam-5098	435	20	let	let	VERB
ejpam-5098	435	21	m	m	PRON
ejpam-5098	435	22	be	be	AUX
ejpam-5098	435	23	any	any	DET
ejpam-5098	435	24	minimal	minimal	ADJ
ejpam-5098	435	25	prime	prime	ADJ
ejpam-5098	435	26	ideal	ideal	NOUN
ejpam-5098	435	27	of	of	ADP
ejpam-5098	435	28	r	r	NOUN
ejpam-5098	435	29	with	with	ADP
ejpam-5098	435	30	a	a	DET
ejpam-5098	435	31	∈m	∈m	NOUN
ejpam-5098	435	32	.	.	PUNCT
ejpam-5098	436	1	then	then	ADV
ejpam-5098	436	2	there	there	PRON
ejpam-5098	436	3	exists	exist	VERB
ejpam-5098	436	4	an	an	DET
ejpam-5098	436	5	element	element	NOUN
ejpam-5098	436	6	b	b	NOUN
ejpam-5098	436	7	/∈m	/∈m	PUNCT
ejpam-5098	436	8	such	such	ADJ
ejpam-5098	436	9	that	that	SCONJ
ejpam-5098	436	10	a	a	DET
ejpam-5098	436	11	∧	∧	PROPN
ejpam-5098	436	12	b	b	PROPN
ejpam-5098	436	13	=	=	NOUN
ejpam-5098	436	14	0	0	PROPN
ejpam-5098	436	15	.	.	PUNCT
ejpam-5098	437	1	that	that	PRON
ejpam-5098	437	2	implies	imply	VERB
ejpam-5098	437	3	b	b	PROPN
ejpam-5098	437	4	∈	∈	PROPN
ejpam-5098	437	5	(	(	PUNCT
ejpam-5098	437	6	a)∗.	a)∗.	NOUN
ejpam-5098	437	7	by	by	ADP
ejpam-5098	437	8	our	our	PRON
ejpam-5098	437	9	assumption	assumption	NOUN
ejpam-5098	437	10	,	,	PUNCT
ejpam-5098	437	11	we	we	PRON
ejpam-5098	437	12	have	have	VERB
ejpam-5098	437	13	that	that	PRON
ejpam-5098	437	14	(	(	PUNCT
ejpam-5098	437	15	a)∗	a)∗	PROPN
ejpam-5098	437	16	is	be	AUX
ejpam-5098	437	17	a	a	DET
ejpam-5098	437	18	σ	σ	NOUN
ejpam-5098	437	19	-	-	PUNCT
ejpam-5098	437	20	ideal	ideal	NOUN
ejpam-5098	437	21	of	of	ADP
ejpam-5098	437	22	r	r	NOUN
ejpam-5098	437	23	and	and	CCONJ
ejpam-5098	437	24	hence	hence	ADV
ejpam-5098	437	25	b	b	X
ejpam-5098	437	26	∈	∈	PROPN
ejpam-5098	437	27	(	(	PUNCT
ejpam-5098	437	28	a)∗	a)∗	PROPN
ejpam-5098	437	29	σ	σ	PROPN
ejpam-5098	437	30	.	.	PUNCT
ejpam-5098	438	1	that	that	PRON
ejpam-5098	438	2	implies	imply	VERB
ejpam-5098	438	3	(	(	PUNCT
ejpam-5098	438	4	a)∗	a)∗	PROPN
ejpam-5098	438	5	∨	∨	PROPN
ejpam-5098	438	6	(	(	PUNCT
ejpam-5098	438	7	b)∗	b)∗	PROPN
ejpam-5098	438	8	=	=	PUNCT
ejpam-5098	438	9	r.	r.	PROPN
ejpam-5098	438	10	that	that	PRON
ejpam-5098	438	11	implies	imply	VERB
ejpam-5098	438	12	s	s	PROPN
ejpam-5098	438	13	∨	∨	PROPN
ejpam-5098	438	14	t	t	PROPN
ejpam-5098	438	15	is	be	AUX
ejpam-5098	438	16	a	a	DET
ejpam-5098	438	17	maximal	maximal	ADJ
ejpam-5098	438	18	element	element	NOUN
ejpam-5098	438	19	for	for	ADP
ejpam-5098	438	20	some	some	DET
ejpam-5098	438	21	s	s	X
ejpam-5098	438	22	∈	∈	NOUN
ejpam-5098	438	23	(	(	PUNCT
ejpam-5098	438	24	a)∗	a)∗	PROPN
ejpam-5098	438	25	and	and	CCONJ
ejpam-5098	438	26	t	t	PROPN
ejpam-5098	438	27	∈	∈	PROPN
ejpam-5098	438	28	(	(	PUNCT
ejpam-5098	438	29	b)∗	b)∗	PROPN
ejpam-5098	438	30	⊆	⊆	NUM
ejpam-5098	438	31	m	m	NOUN
ejpam-5098	438	32	.	.	PUNCT
ejpam-5098	439	1	that	that	PRON
ejpam-5098	439	2	implies	imply	VERB
ejpam-5098	439	3	(	(	PUNCT
ejpam-5098	439	4	a)∗	a)∗	NOUN
ejpam-5098	439	5	∨m	∨m	NOUN
ejpam-5098	439	6	=	=	SYM
ejpam-5098	439	7	r.	r.	PROPN
ejpam-5098	439	8	therefore	therefore	ADV
ejpam-5098	439	9	,	,	PUNCT
ejpam-5098	439	10	m	m	PROPN
ejpam-5098	439	11	is	be	AUX
ejpam-5098	439	12	a	a	DET
ejpam-5098	439	13	σ	σ	NOUN
ejpam-5098	439	14	-	-	PUNCT
ejpam-5098	439	15	ideal	ideal	NOUN
ejpam-5098	439	16	of	of	ADP
ejpam-5098	439	17	r.	r.	PROPN
ejpam-5098	439	18	definition	definition	NOUN
ejpam-5098	439	19	13	13	NUM
ejpam-5098	439	20	.	.	PUNCT
ejpam-5098	440	1	a	a	DET
ejpam-5098	440	2	σ	σ	NOUN
ejpam-5098	440	3	-	-	PUNCT
ejpam-5098	440	4	ideal	ideal	NOUN
ejpam-5098	440	5	i	i	PRON
ejpam-5098	440	6	of	of	ADP
ejpam-5098	440	7	an	an	DET
ejpam-5098	440	8	adl	adl	PROPN
ejpam-5098	440	9	r	r	NOUN
ejpam-5098	440	10	is	be	AUX
ejpam-5098	440	11	said	say	VERB
ejpam-5098	440	12	to	to	PART
ejpam-5098	440	13	be	be	AUX
ejpam-5098	440	14	a	a	DET
ejpam-5098	440	15	prime	prime	ADJ
ejpam-5098	440	16	σ	σ	NOUN
ejpam-5098	440	17	-	-	PUNCT
ejpam-5098	440	18	ideal	ideal	NOUN
ejpam-5098	440	19	if	if	SCONJ
ejpam-5098	440	20	for	for	ADP
ejpam-5098	440	21	any	any	DET
ejpam-5098	440	22	i1	i1	NOUN
ejpam-5098	440	23	,	,	PUNCT
ejpam-5098	440	24	i2	i2	PROPN
ejpam-5098	440	25	∈	∈	PROPN
ejpam-5098	440	26	iσ(r	iσ(r	NOUN
ejpam-5098	440	27	)	)	PUNCT
ejpam-5098	440	28	,	,	PUNCT
ejpam-5098	440	29	i1	i1	PROPN
ejpam-5098	440	30	∩	∩	PROPN
ejpam-5098	440	31	i2	i2	PROPN
ejpam-5098	440	32	⊆	⊆	NUM
ejpam-5098	440	33	i	i	PRON
ejpam-5098	440	34	⇒	⇒	VERB
ejpam-5098	440	35	i1	i1	PROPN
ejpam-5098	440	36	⊆	⊆	NUM
ejpam-5098	440	37	i	i	PROPN
ejpam-5098	440	38	or	or	CCONJ
ejpam-5098	440	39	i2	i2	PROPN
ejpam-5098	440	40	⊆	⊆	NUM
ejpam-5098	440	41	i.	i.	PROPN
ejpam-5098	440	42	r.	r.	PROPN
ejpam-5098	440	43	noorbhasha	noorbhasha	PROPN
ejpam-5098	440	44	,	,	PUNCT
ejpam-5098	440	45	r.	r.	PROPN
ejpam-5098	440	46	bandaru	bandaru	PROPN
ejpam-5098	440	47	,	,	PUNCT
ejpam-5098	440	48	a.	a.	NOUN
ejpam-5098	440	49	iampan	iampan	PROPN
ejpam-5098	440	50	/	/	SYM
ejpam-5098	440	51	eur	eur	PROPN
ejpam-5098	440	52	.	.	PUNCT
ejpam-5098	441	1	j.	j.	PROPN
ejpam-5098	441	2	pure	pure	PROPN
ejpam-5098	441	3	appl	appl	PROPN
ejpam-5098	441	4	.	.	PROPN
ejpam-5098	441	5	math	math	PROPN
ejpam-5098	441	6	,	,	PUNCT
ejpam-5098	441	7	17	17	NUM
ejpam-5098	441	8	(	(	PUNCT
ejpam-5098	441	9	2	2	NUM
ejpam-5098	441	10	)	)	PUNCT
ejpam-5098	441	11	(	(	PUNCT
ejpam-5098	441	12	2024	2024	NUM
ejpam-5098	441	13	)	)	PUNCT
ejpam-5098	441	14	,	,	PUNCT
ejpam-5098	441	15	1094	1094	NUM
ejpam-5098	441	16	-	-	SYM
ejpam-5098	441	17	1112	1112	NUM
ejpam-5098	441	18	1107	1107	NUM
ejpam-5098	441	19	let	let	VERB
ejpam-5098	441	20	specσ(r	specσ(r	NOUN
ejpam-5098	441	21	)	)	PUNCT
ejpam-5098	441	22	be	be	VERB
ejpam-5098	441	23	the	the	DET
ejpam-5098	441	24	set	set	NOUN
ejpam-5098	441	25	of	of	ADP
ejpam-5098	441	26	all	all	DET
ejpam-5098	441	27	prime	prime	ADJ
ejpam-5098	441	28	σ	σ	NOUN
ejpam-5098	441	29	-	-	PUNCT
ejpam-5098	441	30	ideals	ideal	NOUN
ejpam-5098	441	31	of	of	ADP
ejpam-5098	441	32	an	an	DET
ejpam-5098	441	33	adl	adl	PROPN
ejpam-5098	441	34	r.	r.	PROPN
ejpam-5098	441	35	for	for	ADP
ejpam-5098	441	36	any	any	DET
ejpam-5098	441	37	g	g	NOUN
ejpam-5098	441	38	⊆	⊆	NUM
ejpam-5098	441	39	r	r	NOUN
ejpam-5098	441	40	,	,	PUNCT
ejpam-5098	441	41	let	let	VERB
ejpam-5098	441	42	h(g	h(g	PRON
ejpam-5098	441	43	)	)	PUNCT
ejpam-5098	441	44	=	=	PRON
ejpam-5098	441	45	{	{	PUNCT
ejpam-5098	441	46	m	m	PROPN
ejpam-5098	441	47	∈	∈	PROPN
ejpam-5098	441	48	specσ(r	specσ(r	NOUN
ejpam-5098	441	49	)	)	PUNCT
ejpam-5098	441	50	|	|	ADV
ejpam-5098	441	51	g	g	NOUN
ejpam-5098	441	52	⊈m	⊈m	NUM
ejpam-5098	441	53	}	}	PUNCT
ejpam-5098	441	54	and	and	CCONJ
ejpam-5098	441	55	for	for	ADP
ejpam-5098	441	56	any	any	DET
ejpam-5098	441	57	a	a	DET
ejpam-5098	441	58	∈	∈	PROPN
ejpam-5098	441	59	r	r	NOUN
ejpam-5098	441	60	,	,	PUNCT
ejpam-5098	441	61	h(a	h(a	PROPN
ejpam-5098	441	62	)	)	PUNCT
ejpam-5098	442	1	=	=	SYM
ejpam-5098	442	2	h({a	h({a	PROPN
ejpam-5098	442	3	}	}	PUNCT
ejpam-5098	442	4	)	)	PUNCT
ejpam-5098	442	5	.	.	PUNCT
ejpam-5098	443	1	for	for	ADP
ejpam-5098	443	2	any	any	DET
ejpam-5098	443	3	two	two	NUM
ejpam-5098	443	4	subsets	subset	NOUN
ejpam-5098	443	5	g	g	NOUN
ejpam-5098	443	6	and	and	CCONJ
ejpam-5098	443	7	h	h	NOUN
ejpam-5098	443	8	of	of	ADP
ejpam-5098	443	9	r	r	NOUN
ejpam-5098	443	10	,	,	PUNCT
ejpam-5098	443	11	it	it	PRON
ejpam-5098	443	12	is	be	AUX
ejpam-5098	443	13	obvious	obvious	ADJ
ejpam-5098	443	14	that	that	SCONJ
ejpam-5098	443	15	g	g	PROPN
ejpam-5098	443	16	⊆	⊆	NUM
ejpam-5098	443	17	h	h	NOUN
ejpam-5098	443	18	implies	imply	VERB
ejpam-5098	443	19	h(g	h(g	NOUN
ejpam-5098	443	20	)	)	PUNCT
ejpam-5098	444	1	⊆	⊆	NUM
ejpam-5098	444	2	h(h	h(h	NOUN
ejpam-5098	444	3	)	)	PUNCT
ejpam-5098	444	4	.	.	PUNCT
ejpam-5098	445	1	the	the	DET
ejpam-5098	445	2	following	follow	VERB
ejpam-5098	445	3	observations	observation	NOUN
ejpam-5098	445	4	can	can	AUX
ejpam-5098	445	5	be	be	AUX
ejpam-5098	445	6	verified	verify	VERB
ejpam-5098	445	7	directly	directly	ADV
ejpam-5098	445	8	:	:	PUNCT
ejpam-5098	445	9	lemma	lemma	PROPN
ejpam-5098	445	10	6	6	NUM
ejpam-5098	445	11	.	.	PUNCT
ejpam-5098	446	1	for	for	ADP
ejpam-5098	446	2	any	any	DET
ejpam-5098	446	3	x	x	NOUN
ejpam-5098	446	4	,	,	PUNCT
ejpam-5098	446	5	y	y	PROPN
ejpam-5098	446	6	∈	∈	PROPN
ejpam-5098	446	7	r	r	NOUN
ejpam-5098	446	8	,	,	PUNCT
ejpam-5098	446	9	the	the	DET
ejpam-5098	446	10	following	follow	VERB
ejpam-5098	446	11	conditions	condition	NOUN
ejpam-5098	446	12	hold	hold	VERB
ejpam-5098	446	13	:	:	PUNCT
ejpam-5098	446	14	(	(	PUNCT
ejpam-5098	446	15	1	1	X
ejpam-5098	446	16	)	)	PUNCT
ejpam-5098	446	17	⋃	⋃	NOUN
ejpam-5098	446	18	a∈r	a∈r	NOUN
ejpam-5098	446	19	h(a	h(a	PROPN
ejpam-5098	446	20	)	)	PUNCT
ejpam-5098	447	1	=	=	SYM
ejpam-5098	447	2	specσ(r	specσ(r	PROPN
ejpam-5098	447	3	)	)	PUNCT
ejpam-5098	447	4	(	(	PUNCT
ejpam-5098	447	5	2	2	X
ejpam-5098	447	6	)	)	PUNCT
ejpam-5098	447	7	h(a	h(a	PROPN
ejpam-5098	447	8	)	)	PUNCT
ejpam-5098	447	9	∪	∪	ADP
ejpam-5098	447	10	h(b	h(b	PROPN
ejpam-5098	447	11	)	)	PUNCT
ejpam-5098	448	1	=	=	PUNCT
ejpam-5098	448	2	h(a	h(a	PROPN
ejpam-5098	448	3	∨	∨	NUM
ejpam-5098	448	4	b	b	NOUN
ejpam-5098	448	5	)	)	PUNCT
ejpam-5098	448	6	(	(	PUNCT
ejpam-5098	448	7	3	3	X
ejpam-5098	448	8	)	)	PUNCT
ejpam-5098	448	9	h(a	h(a	PROPN
ejpam-5098	448	10	)	)	PUNCT
ejpam-5098	448	11	∩	∩	NOUN
ejpam-5098	448	12	h(b	h(b	PROPN
ejpam-5098	448	13	)	)	PUNCT
ejpam-5098	449	1	=	=	PUNCT
ejpam-5098	449	2	h(a	h(a	PROPN
ejpam-5098	449	3	∧	∧	PROPN
ejpam-5098	449	4	b	b	PROPN
ejpam-5098	449	5	)	)	PUNCT
ejpam-5098	449	6	(	(	PUNCT
ejpam-5098	449	7	4	4	X
ejpam-5098	449	8	)	)	PUNCT
ejpam-5098	449	9	h(a	h(a	PROPN
ejpam-5098	449	10	)	)	PUNCT
ejpam-5098	450	1	=	=	NOUN
ejpam-5098	450	2	∅	∅	NOUN
ejpam-5098	450	3	⇔	⇔	X
ejpam-5098	450	4	a	a	X
ejpam-5098	450	5	=	=	SYM
ejpam-5098	450	6	0	0	NUM
ejpam-5098	450	7	(	(	PUNCT
ejpam-5098	450	8	5	5	NUM
ejpam-5098	450	9	)	)	PUNCT
ejpam-5098	450	10	h(a	h(a	PROPN
ejpam-5098	450	11	)	)	PUNCT
ejpam-5098	451	1	=	=	SYM
ejpam-5098	451	2	specσ(r	specσ(r	PROPN
ejpam-5098	451	3	)	)	PUNCT
ejpam-5098	451	4	⇔	⇔	PROPN
ejpam-5098	451	5	a	a	PROPN
ejpam-5098	451	6	is	be	AUX
ejpam-5098	451	7	a	a	DET
ejpam-5098	451	8	maximal	maximal	ADJ
ejpam-5098	451	9	element	element	NOUN
ejpam-5098	451	10	of	of	ADP
ejpam-5098	451	11	r.	r.	PROPN
ejpam-5098	451	12	from	from	ADP
ejpam-5098	451	13	the	the	DET
ejpam-5098	451	14	above	above	ADJ
ejpam-5098	451	15	lemma	lemma	PROPN
ejpam-5098	451	16	,	,	PUNCT
ejpam-5098	451	17	it	it	PRON
ejpam-5098	451	18	can	can	AUX
ejpam-5098	451	19	be	be	AUX
ejpam-5098	451	20	easily	easily	ADV
ejpam-5098	451	21	observed	observe	VERB
ejpam-5098	451	22	that	that	SCONJ
ejpam-5098	451	23	the	the	DET
ejpam-5098	451	24	collection	collection	NOUN
ejpam-5098	451	25	{	{	PUNCT
ejpam-5098	451	26	h(a	h(a	PROPN
ejpam-5098	451	27	)	)	PUNCT
ejpam-5098	451	28	|	|	ADV
ejpam-5098	451	29	a	a	DET
ejpam-5098	451	30	∈	∈	NOUN
ejpam-5098	451	31	r	r	NOUN
ejpam-5098	451	32	}	}	PUNCT
ejpam-5098	451	33	forms	form	VERB
ejpam-5098	451	34	a	a	DET
ejpam-5098	451	35	base	base	NOUN
ejpam-5098	451	36	for	for	ADP
ejpam-5098	451	37	a	a	DET
ejpam-5098	451	38	topology	topology	NOUN
ejpam-5098	451	39	on	on	ADP
ejpam-5098	451	40	specσ(r	specσ(r	PROPN
ejpam-5098	451	41	)	)	PUNCT
ejpam-5098	451	42	which	which	PRON
ejpam-5098	451	43	is	be	AUX
ejpam-5098	451	44	called	call	VERB
ejpam-5098	451	45	a	a	DET
ejpam-5098	451	46	hull	hull	NOUN
ejpam-5098	451	47	-	-	PUNCT
ejpam-5098	451	48	kernel	kernel	NOUN
ejpam-5098	451	49	topology	topology	NOUN
ejpam-5098	451	50	.	.	PUNCT
ejpam-5098	452	1	definition	definition	NOUN
ejpam-5098	452	2	14	14	NUM
ejpam-5098	452	3	.	.	PUNCT
ejpam-5098	453	1	for	for	ADP
ejpam-5098	453	2	any	any	DET
ejpam-5098	453	3	ideal	ideal	NOUN
ejpam-5098	453	4	i	i	PRON
ejpam-5098	453	5	of	of	ADP
ejpam-5098	453	6	an	an	DET
ejpam-5098	453	7	adl	adl	PROPN
ejpam-5098	453	8	r	r	NOUN
ejpam-5098	453	9	,	,	PUNCT
ejpam-5098	453	10	define	define	VERB
ejpam-5098	453	11	i	i	PRON
ejpam-5098	453	12	△	△	X
ejpam-5098	453	13	=	=	SYM
ejpam-5098	454	1	⋃	⋃	PROPN
ejpam-5098	454	2	{	{	PUNCT
ejpam-5098	454	3	j	j	PROPN
ejpam-5098	454	4	∈	∈	PROPN
ejpam-5098	454	5	iσ(r	iσ(r	NOUN
ejpam-5098	454	6	)	)	PUNCT
ejpam-5098	455	1	|	|	ADV
ejpam-5098	455	2	j	j	PROPN
ejpam-5098	455	3	⊆	⊆	NUM
ejpam-5098	455	4	i	i	PROPN
ejpam-5098	455	5	}	}	PUNCT
ejpam-5098	455	6	.	.	PUNCT
ejpam-5098	456	1	lemma	lemma	PROPN
ejpam-5098	456	2	7	7	X
ejpam-5098	456	3	.	.	PUNCT
ejpam-5098	457	1	let	let	VERB
ejpam-5098	457	2	i	i	PRON
ejpam-5098	457	3	be	be	AUX
ejpam-5098	457	4	any	any	DET
ejpam-5098	457	5	ideal	ideal	NOUN
ejpam-5098	457	6	of	of	ADP
ejpam-5098	457	7	r.	r.	PROPN
ejpam-5098	457	8	then	then	ADV
ejpam-5098	457	9	i	i	PRON
ejpam-5098	457	10	△	△	PROPN
ejpam-5098	457	11	is	be	AUX
ejpam-5098	457	12	the	the	DET
ejpam-5098	457	13	largest	large	ADJ
ejpam-5098	457	14	σ	σ	NOUN
ejpam-5098	457	15	-	-	PUNCT
ejpam-5098	457	16	ideal	ideal	NOUN
ejpam-5098	457	17	of	of	ADP
ejpam-5098	457	18	r	r	NOUN
ejpam-5098	457	19	contained	contain	VERB
ejpam-5098	457	20	in	in	ADP
ejpam-5098	457	21	i.	i.	NOUN
ejpam-5098	457	22	proof	proof	NOUN
ejpam-5098	457	23	.	.	PUNCT
ejpam-5098	458	1	clearly	clearly	ADV
ejpam-5098	458	2	,	,	PUNCT
ejpam-5098	458	3	we	we	PRON
ejpam-5098	458	4	have	have	VERB
ejpam-5098	458	5	that	that	PRON
ejpam-5098	458	6	{	{	PUNCT
ejpam-5098	458	7	0	0	NUM
ejpam-5098	458	8	}	}	PUNCT
ejpam-5098	458	9	is	be	AUX
ejpam-5098	458	10	a	a	DET
ejpam-5098	458	11	σ	σ	NOUN
ejpam-5098	458	12	-	-	PUNCT
ejpam-5098	458	13	ideal	ideal	NOUN
ejpam-5098	458	14	of	of	ADP
ejpam-5098	458	15	r	r	NOUN
ejpam-5098	458	16	and	and	CCONJ
ejpam-5098	458	17	{	{	PUNCT
ejpam-5098	458	18	0	0	NUM
ejpam-5098	458	19	}	}	SYM
ejpam-5098	458	20	⊆	⊆	NUM
ejpam-5098	458	21	i.	i.	NOUN
ejpam-5098	458	22	that	that	PRON
ejpam-5098	458	23	implies	imply	VERB
ejpam-5098	458	24	{	{	PUNCT
ejpam-5098	458	25	0	0	NUM
ejpam-5098	458	26	}	}	PUNCT
ejpam-5098	458	27	∈	∈	PROPN
ejpam-5098	458	28	i	i	PRON
ejpam-5098	458	29	△	△	PROPN
ejpam-5098	458	30	and	and	CCONJ
ejpam-5098	458	31	hence	hence	ADV
ejpam-5098	458	32	i	i	PRON
ejpam-5098	458	33	△	△	X
ejpam-5098	458	34	̸=	̸=	PROPN
ejpam-5098	458	35	∅.	∅.	ADV
ejpam-5098	458	36	let	let	VERB
ejpam-5098	458	37	x	x	PROPN
ejpam-5098	458	38	∈	∈	PROPN
ejpam-5098	458	39	i	i	PRON
ejpam-5098	458	40	△	△	PROPN
ejpam-5098	458	41	.	.	PUNCT
ejpam-5098	459	1	then	then	ADV
ejpam-5098	459	2	x	x	SYM
ejpam-5098	459	3	∈	∈	PROPN
ejpam-5098	459	4	⋃	⋃	PROPN
ejpam-5098	459	5	{	{	PUNCT
ejpam-5098	459	6	j	j	PROPN
ejpam-5098	459	7	∈	∈	PROPN
ejpam-5098	459	8	iσ(r	iσ(r	NOUN
ejpam-5098	459	9	)	)	PUNCT
ejpam-5098	459	10	|	|	ADV
ejpam-5098	459	11	j	j	PROPN
ejpam-5098	459	12	⊆	⊆	NUM
ejpam-5098	459	13	i	i	PROPN
ejpam-5098	459	14	}	}	PUNCT
ejpam-5098	459	15	.	.	PUNCT
ejpam-5098	460	1	then	then	ADV
ejpam-5098	460	2	there	there	PRON
ejpam-5098	460	3	exists	exist	VERB
ejpam-5098	460	4	j	j	PROPN
ejpam-5098	460	5	∈	∈	PROPN
ejpam-5098	460	6	iσ(r	iσ(r	NOUN
ejpam-5098	460	7	)	)	PUNCT
ejpam-5098	460	8	such	such	ADJ
ejpam-5098	460	9	that	that	SCONJ
ejpam-5098	460	10	j	j	PROPN
ejpam-5098	460	11	⊆	⊆	NUM
ejpam-5098	460	12	i	i	PROPN
ejpam-5098	460	13	and	and	CCONJ
ejpam-5098	460	14	x	x	PROPN
ejpam-5098	460	15	∈	∈	PROPN
ejpam-5098	460	16	j	j	PROPN
ejpam-5098	460	17	.	.	PUNCT
ejpam-5098	461	1	that	that	PRON
ejpam-5098	461	2	implies	imply	VERB
ejpam-5098	461	3	x	x	X
ejpam-5098	461	4	∈	∈	PROPN
ejpam-5098	461	5	i.	i.	NOUN
ejpam-5098	461	6	therefore	therefore	ADV
ejpam-5098	461	7	,	,	PUNCT
ejpam-5098	461	8	i	i	PROPN
ejpam-5098	461	9	△	△	PROPN
ejpam-5098	461	10	⊆	⊆	NUM
ejpam-5098	461	11	i.	i.	NOUN
ejpam-5098	461	12	let	let	VERB
ejpam-5098	461	13	x	x	PRON
ejpam-5098	461	14	,	,	PUNCT
ejpam-5098	461	15	y	y	PROPN
ejpam-5098	461	16	∈	∈	PROPN
ejpam-5098	461	17	i	i	PRON
ejpam-5098	461	18	△	△	PROPN
ejpam-5098	461	19	.	.	PUNCT
ejpam-5098	462	1	then	then	ADV
ejpam-5098	462	2	there	there	PRON
ejpam-5098	462	3	exist	exist	VERB
ejpam-5098	462	4	j	j	PROPN
ejpam-5098	462	5	,	,	PUNCT
ejpam-5098	462	6	k	k	PROPN
ejpam-5098	462	7	∈	∈	PROPN
ejpam-5098	462	8	iσ(r	iσ(r	NOUN
ejpam-5098	462	9	)	)	PUNCT
ejpam-5098	462	10	such	such	ADJ
ejpam-5098	462	11	that	that	SCONJ
ejpam-5098	462	12	x	x	SYM
ejpam-5098	462	13	∈	∈	PROPN
ejpam-5098	462	14	j	j	PROPN
ejpam-5098	462	15	,	,	PUNCT
ejpam-5098	462	16	j	j	PROPN
ejpam-5098	462	17	⊆	⊆	NUM
ejpam-5098	462	18	i	i	PROPN
ejpam-5098	462	19	and	and	CCONJ
ejpam-5098	462	20	y	y	PROPN
ejpam-5098	462	21	∈	∈	PROPN
ejpam-5098	462	22	k	k	PROPN
ejpam-5098	462	23	,	,	PUNCT
ejpam-5098	462	24	k	k	PROPN
ejpam-5098	462	25	⊆	⊆	NUM
ejpam-5098	462	26	i.	i.	NOUN
ejpam-5098	462	27	that	that	PRON
ejpam-5098	462	28	implies	imply	VERB
ejpam-5098	462	29	x∨	x∨	PROPN
ejpam-5098	462	30	y	y	PROPN
ejpam-5098	462	31	∈	∈	PROPN
ejpam-5098	462	32	j	j	PROPN
ejpam-5098	462	33	∨k	∨k	NOUN
ejpam-5098	462	34	⊆	⊆	NUM
ejpam-5098	462	35	i.	i.	NOUN
ejpam-5098	462	36	since	since	SCONJ
ejpam-5098	462	37	the	the	DET
ejpam-5098	462	38	join	join	NOUN
ejpam-5098	462	39	of	of	ADP
ejpam-5098	462	40	two	two	NUM
ejpam-5098	462	41	σ	σ	NOUN
ejpam-5098	462	42	-	-	PUNCT
ejpam-5098	462	43	ideals	ideal	NOUN
ejpam-5098	462	44	j	j	PROPN
ejpam-5098	462	45	and	and	CCONJ
ejpam-5098	462	46	k	k	PROPN
ejpam-5098	462	47	is	be	AUX
ejpam-5098	462	48	a	a	DET
ejpam-5098	462	49	σ	σ	NOUN
ejpam-5098	462	50	-	-	PUNCT
ejpam-5098	462	51	ideal	ideal	NOUN
ejpam-5098	462	52	of	of	ADP
ejpam-5098	462	53	r	r	NOUN
ejpam-5098	462	54	,	,	PUNCT
ejpam-5098	462	55	we	we	PRON
ejpam-5098	462	56	get	get	VERB
ejpam-5098	462	57	that	that	PRON
ejpam-5098	462	58	x	x	PROPN
ejpam-5098	462	59	∨	∨	NUM
ejpam-5098	462	60	y	y	PROPN
ejpam-5098	462	61	∈	∈	PROPN
ejpam-5098	462	62	i	i	PRON
ejpam-5098	462	63	△	△	PROPN
ejpam-5098	462	64	.	.	PUNCT
ejpam-5098	463	1	let	let	VERB
ejpam-5098	463	2	x	x	SYM
ejpam-5098	463	3	∈	∈	PROPN
ejpam-5098	463	4	i	i	PRON
ejpam-5098	463	5	△	△	PROPN
ejpam-5098	463	6	.	.	PUNCT
ejpam-5098	464	1	then	then	ADV
ejpam-5098	464	2	there	there	PRON
ejpam-5098	464	3	exists	exist	VERB
ejpam-5098	464	4	j	j	PROPN
ejpam-5098	464	5	∈	∈	PROPN
ejpam-5098	464	6	iσ(r	iσ(r	NOUN
ejpam-5098	464	7	)	)	PUNCT
ejpam-5098	464	8	such	such	ADJ
ejpam-5098	464	9	that	that	SCONJ
ejpam-5098	464	10	x	x	SYM
ejpam-5098	464	11	∈	∈	PROPN
ejpam-5098	464	12	j	j	PROPN
ejpam-5098	464	13	⊆	⊆	NUM
ejpam-5098	464	14	i.	i.	NOUN
ejpam-5098	464	15	let	let	VERB
ejpam-5098	464	16	r	r	NOUN
ejpam-5098	464	17	be	be	AUX
ejpam-5098	464	18	any	any	DET
ejpam-5098	464	19	element	element	NOUN
ejpam-5098	464	20	of	of	ADP
ejpam-5098	464	21	r.	r.	PROPN
ejpam-5098	464	22	then	then	ADV
ejpam-5098	464	23	x	x	PART
ejpam-5098	464	24	∧	∧	PROPN
ejpam-5098	464	25	r	r	NOUN
ejpam-5098	464	26	∈	∈	PROPN
ejpam-5098	464	27	j	j	PROPN
ejpam-5098	464	28	⊆	⊆	NUM
ejpam-5098	464	29	i.	i.	NOUN
ejpam-5098	464	30	that	that	PRON
ejpam-5098	464	31	implies	imply	VERB
ejpam-5098	464	32	x	x	PUNCT
ejpam-5098	464	33	∧	∧	NOUN
ejpam-5098	464	34	r	r	NOUN
ejpam-5098	464	35	∈	∈	PROPN
ejpam-5098	464	36	i	i	PRON
ejpam-5098	464	37	△	△	PROPN
ejpam-5098	464	38	.	.	PUNCT
ejpam-5098	465	1	hence	hence	ADV
ejpam-5098	465	2	,	,	PUNCT
ejpam-5098	465	3	i	i	PRON
ejpam-5098	465	4	△	△	PROPN
ejpam-5098	465	5	is	be	AUX
ejpam-5098	465	6	a	a	DET
ejpam-5098	465	7	σ	σ	NOUN
ejpam-5098	465	8	-	-	PUNCT
ejpam-5098	465	9	ideal	ideal	NOUN
ejpam-5098	465	10	of	of	ADP
ejpam-5098	465	11	r	r	NOUN
ejpam-5098	465	12	contained	contain	VERB
ejpam-5098	465	13	in	in	ADP
ejpam-5098	465	14	i.	i.	PROPN
ejpam-5098	465	15	clearly	clearly	ADV
ejpam-5098	465	16	,	,	PUNCT
ejpam-5098	465	17	i	i	PRON
ejpam-5098	465	18	△	△	PROPN
ejpam-5098	465	19	is	be	AUX
ejpam-5098	465	20	the	the	DET
ejpam-5098	465	21	largest	large	ADJ
ejpam-5098	465	22	σ	σ	NOUN
ejpam-5098	465	23	-	-	PUNCT
ejpam-5098	465	24	ideal	ideal	NOUN
ejpam-5098	465	25	of	of	ADP
ejpam-5098	465	26	r	r	NOUN
ejpam-5098	465	27	contained	contain	VERB
ejpam-5098	465	28	in	in	ADP
ejpam-5098	465	29	i.	i.	PROPN
ejpam-5098	465	30	lemma	lemma	PROPN
ejpam-5098	465	31	8	8	X
ejpam-5098	465	32	.	.	PUNCT
ejpam-5098	466	1	let	let	VERB
ejpam-5098	466	2	r	r	PRON
ejpam-5098	466	3	be	be	AUX
ejpam-5098	466	4	an	an	DET
ejpam-5098	466	5	adl	adl	NOUN
ejpam-5098	466	6	with	with	ADP
ejpam-5098	466	7	maximal	maximal	ADJ
ejpam-5098	466	8	elements	element	NOUN
ejpam-5098	466	9	.	.	PUNCT
ejpam-5098	467	1	then	then	ADV
ejpam-5098	467	2	for	for	ADP
ejpam-5098	467	3	any	any	DET
ejpam-5098	467	4	ideal	ideal	NOUN
ejpam-5098	467	5	k	k	PROPN
ejpam-5098	467	6	of	of	ADP
ejpam-5098	467	7	r	r	NOUN
ejpam-5098	467	8	,	,	PUNCT
ejpam-5098	467	9	we	we	PRON
ejpam-5098	467	10	have	have	VERB
ejpam-5098	467	11	k	k	PROPN
ejpam-5098	467	12	△	△	PROPN
ejpam-5098	467	13	⊆	⊆	NUM
ejpam-5098	467	14	kσ	kσ	PROPN
ejpam-5098	467	15	.	.	PROPN
ejpam-5098	467	16	proof	proof	NOUN
ejpam-5098	467	17	.	.	PUNCT
ejpam-5098	468	1	let	let	VERB
ejpam-5098	468	2	k	k	PRON
ejpam-5098	468	3	be	be	AUX
ejpam-5098	468	4	any	any	DET
ejpam-5098	468	5	ideal	ideal	NOUN
ejpam-5098	468	6	of	of	ADP
ejpam-5098	468	7	r	r	NOUN
ejpam-5098	468	8	with	with	ADP
ejpam-5098	468	9	a	a	DET
ejpam-5098	468	10	∈	∈	PROPN
ejpam-5098	468	11	k	k	NOUN
ejpam-5098	468	12	△	△	PROPN
ejpam-5098	468	13	.	.	PUNCT
ejpam-5098	469	1	then	then	ADV
ejpam-5098	469	2	there	there	PRON
ejpam-5098	469	3	exists	exist	VERB
ejpam-5098	469	4	h	h	PROPN
ejpam-5098	469	5	∈	∈	PROPN
ejpam-5098	469	6	iσ(r	iσ(r	NOUN
ejpam-5098	469	7	)	)	PUNCT
ejpam-5098	469	8	such	such	ADJ
ejpam-5098	469	9	that	that	SCONJ
ejpam-5098	469	10	a	a	DET
ejpam-5098	469	11	∈	∈	PROPN
ejpam-5098	469	12	h	h	NOUN
ejpam-5098	469	13	⊆	⊆	NUM
ejpam-5098	469	14	k.	k.	NOUN
ejpam-5098	469	15	since	since	SCONJ
ejpam-5098	469	16	h	h	PROPN
ejpam-5098	469	17	is	be	AUX
ejpam-5098	469	18	a	a	DET
ejpam-5098	469	19	σ	σ	NOUN
ejpam-5098	469	20	-	-	PUNCT
ejpam-5098	469	21	ideal	ideal	NOUN
ejpam-5098	469	22	of	of	ADP
ejpam-5098	469	23	r	r	NOUN
ejpam-5098	469	24	and	and	CCONJ
ejpam-5098	469	25	h	h	NOUN
ejpam-5098	469	26	⊆	⊆	NUM
ejpam-5098	469	27	k	k	NOUN
ejpam-5098	469	28	,	,	PUNCT
ejpam-5098	469	29	we	we	PRON
ejpam-5098	469	30	get	get	VERB
ejpam-5098	469	31	that	that	PRON
ejpam-5098	469	32	(	(	PUNCT
ejpam-5098	470	1	a)∗	a)∗	PROPN
ejpam-5098	470	2	∨	∨	NUM
ejpam-5098	470	3	k	k	PROPN
ejpam-5098	470	4	=	=	PUNCT
ejpam-5098	470	5	r.	r.	PROPN
ejpam-5098	470	6	that	that	PRON
ejpam-5098	470	7	implies	imply	VERB
ejpam-5098	470	8	a	a	DET
ejpam-5098	470	9	∈	∈	PROPN
ejpam-5098	470	10	kσ	kσ	PROPN
ejpam-5098	470	11	.	.	PUNCT
ejpam-5098	471	1	therefore	therefore	ADV
ejpam-5098	471	2	,	,	PUNCT
ejpam-5098	471	3	k	k	PROPN
ejpam-5098	471	4	△	△	PROPN
ejpam-5098	471	5	⊆	⊆	NUM
ejpam-5098	471	6	kσ	kσ	PROPN
ejpam-5098	471	7	.	.	PUNCT
ejpam-5098	471	8	lemma	lemma	PROPN
ejpam-5098	471	9	9	9	NUM
ejpam-5098	471	10	.	.	PUNCT
ejpam-5098	472	1	let	let	VERB
ejpam-5098	472	2	k	k	PRON
ejpam-5098	472	3	be	be	AUX
ejpam-5098	472	4	an	an	DET
ejpam-5098	472	5	ideal	ideal	NOUN
ejpam-5098	472	6	of	of	ADP
ejpam-5098	472	7	a	a	DET
ejpam-5098	472	8	normal	normal	ADJ
ejpam-5098	472	9	adl	adl	NOUN
ejpam-5098	472	10	r	r	NOUN
ejpam-5098	472	11	with	with	ADP
ejpam-5098	472	12	maximal	maximal	ADJ
ejpam-5098	472	13	elements	element	NOUN
ejpam-5098	472	14	.	.	PUNCT
ejpam-5098	473	1	then	then	ADV
ejpam-5098	473	2	k	k	X
ejpam-5098	473	3	△	△	X
ejpam-5098	473	4	=	=	SYM
ejpam-5098	473	5	kσ	kσ	PROPN
ejpam-5098	473	6	.	.	PROPN
ejpam-5098	474	1	moreover	moreover	ADV
ejpam-5098	474	2	,	,	PUNCT
ejpam-5098	474	3	kσ	kσ	PROPN
ejpam-5098	474	4	is	be	AUX
ejpam-5098	474	5	a	a	DET
ejpam-5098	474	6	σ	σ	NOUN
ejpam-5098	474	7	-	-	PUNCT
ejpam-5098	474	8	ideal	ideal	NOUN
ejpam-5098	474	9	of	of	ADP
ejpam-5098	474	10	r.	r.	PROPN
ejpam-5098	474	11	proof	proof	NOUN
ejpam-5098	474	12	.	.	PUNCT
ejpam-5098	475	1	clearly	clearly	ADV
ejpam-5098	475	2	,	,	PUNCT
ejpam-5098	475	3	we	we	PRON
ejpam-5098	475	4	have	have	VERB
ejpam-5098	475	5	that	that	PRON
ejpam-5098	475	6	k	k	PROPN
ejpam-5098	475	7	△	△	PROPN
ejpam-5098	475	8	⊆	⊆	NUM
ejpam-5098	475	9	kσ	kσ	PROPN
ejpam-5098	475	10	.	.	PROPN
ejpam-5098	475	11	let	let	VERB
ejpam-5098	475	12	a	a	DET
ejpam-5098	475	13	∈	∈	NOUN
ejpam-5098	475	14	kσ	kσ	PROPN
ejpam-5098	475	15	.	.	PUNCT
ejpam-5098	476	1	then	then	ADV
ejpam-5098	476	2	(	(	PUNCT
ejpam-5098	476	3	a)∗	a)∗	PROPN
ejpam-5098	476	4	∨	∨	NUM
ejpam-5098	476	5	k	k	PROPN
ejpam-5098	476	6	=	=	SYM
ejpam-5098	476	7	r.	r.	PROPN
ejpam-5098	476	8	then	then	ADV
ejpam-5098	476	9	there	there	PRON
ejpam-5098	476	10	exist	exist	VERB
ejpam-5098	476	11	elements	element	NOUN
ejpam-5098	476	12	s	s	PART
ejpam-5098	476	13	∈	∈	NOUN
ejpam-5098	476	14	(	(	PUNCT
ejpam-5098	476	15	a)∗	a)∗	PROPN
ejpam-5098	476	16	and	and	CCONJ
ejpam-5098	476	17	t	t	PROPN
ejpam-5098	476	18	∈	∈	PROPN
ejpam-5098	476	19	i	i	PRON
ejpam-5098	476	20	such	such	ADJ
ejpam-5098	476	21	that	that	SCONJ
ejpam-5098	476	22	a	a	DET
ejpam-5098	476	23	∨	∨	NUM
ejpam-5098	476	24	b	b	NOUN
ejpam-5098	476	25	is	be	AUX
ejpam-5098	476	26	maximal	maximal	ADJ
ejpam-5098	476	27	.	.	PUNCT
ejpam-5098	477	1	since	since	SCONJ
ejpam-5098	477	2	s	s	PROPN
ejpam-5098	477	3	∈	∈	PROPN
ejpam-5098	477	4	(	(	PUNCT
ejpam-5098	477	5	a)∗	a)∗	PROPN
ejpam-5098	477	6	,	,	PUNCT
ejpam-5098	477	7	we	we	PRON
ejpam-5098	477	8	have	have	VERB
ejpam-5098	477	9	that	that	SCONJ
ejpam-5098	477	10	a	a	DET
ejpam-5098	477	11	∧	∧	PROPN
ejpam-5098	477	12	s	s	PART
ejpam-5098	477	13	=	=	NOUN
ejpam-5098	477	14	0	0	PROPN
ejpam-5098	477	15	.	.	PUNCT
ejpam-5098	478	1	since	since	SCONJ
ejpam-5098	478	2	r	r	NOUN
ejpam-5098	478	3	is	be	AUX
ejpam-5098	478	4	normal	normal	ADJ
ejpam-5098	478	5	,	,	PUNCT
ejpam-5098	478	6	there	there	PRON
ejpam-5098	478	7	exist	exist	VERB
ejpam-5098	478	8	elements	element	NOUN
ejpam-5098	478	9	s1	s1	NOUN
ejpam-5098	478	10	,	,	PUNCT
ejpam-5098	478	11	t1	t1	NOUN
ejpam-5098	478	12	∈	∈	PROPN
ejpam-5098	478	13	r	r	NOUN
ejpam-5098	478	14	such	such	ADJ
ejpam-5098	478	15	that	that	DET
ejpam-5098	478	16	s	s	PART
ejpam-5098	478	17	∧	∧	NOUN
ejpam-5098	478	18	s1	s1	NOUN
ejpam-5098	478	19	=	=	SYM
ejpam-5098	478	20	0	0	NUM
ejpam-5098	478	21	,	,	PUNCT
ejpam-5098	478	22	a	a	DET
ejpam-5098	478	23	∧	∧	PROPN
ejpam-5098	478	24	t1	t1	NOUN
ejpam-5098	478	25	=	=	NOUN
ejpam-5098	478	26	0	0	NUM
ejpam-5098	478	27	and	and	CCONJ
ejpam-5098	478	28	s1	s1	PROPN
ejpam-5098	479	1	∨	∨	NUM
ejpam-5098	479	2	t1	t1	NOUN
ejpam-5098	479	3	is	be	AUX
ejpam-5098	479	4	maximal	maximal	ADJ
ejpam-5098	479	5	.	.	PUNCT
ejpam-5098	480	1	since	since	SCONJ
ejpam-5098	480	2	s	s	PROPN
ejpam-5098	480	3	∨	∨	PROPN
ejpam-5098	480	4	t	t	PROPN
ejpam-5098	480	5	is	be	AUX
ejpam-5098	480	6	maximal	maximal	ADJ
ejpam-5098	480	7	,	,	PUNCT
ejpam-5098	480	8	s	s	NOUN
ejpam-5098	480	9	∈	∈	PROPN
ejpam-5098	480	10	(	(	PUNCT
ejpam-5098	480	11	s1	s1	NOUN
ejpam-5098	480	12	)	)	PUNCT
ejpam-5098	480	13	∗	∗	NOUN
ejpam-5098	480	14	and	and	CCONJ
ejpam-5098	480	15	t	t	PROPN
ejpam-5098	480	16	∈	∈	PROPN
ejpam-5098	480	17	k	k	PROPN
ejpam-5098	480	18	,	,	PUNCT
ejpam-5098	480	19	r.	r.	PROPN
ejpam-5098	480	20	noorbhasha	noorbhasha	PROPN
ejpam-5098	480	21	,	,	PUNCT
ejpam-5098	480	22	r.	r.	PROPN
ejpam-5098	480	23	bandaru	bandaru	PROPN
ejpam-5098	480	24	,	,	PUNCT
ejpam-5098	480	25	a.	a.	NOUN
ejpam-5098	480	26	iampan	iampan	PROPN
ejpam-5098	480	27	/	/	SYM
ejpam-5098	480	28	eur	eur	PROPN
ejpam-5098	480	29	.	.	PUNCT
ejpam-5098	481	1	j.	j.	PROPN
ejpam-5098	481	2	pure	pure	PROPN
ejpam-5098	481	3	appl	appl	PROPN
ejpam-5098	481	4	.	.	PROPN
ejpam-5098	481	5	math	math	PROPN
ejpam-5098	481	6	,	,	PUNCT
ejpam-5098	481	7	17	17	NUM
ejpam-5098	481	8	(	(	PUNCT
ejpam-5098	481	9	2	2	NUM
ejpam-5098	481	10	)	)	PUNCT
ejpam-5098	481	11	(	(	PUNCT
ejpam-5098	481	12	2024	2024	NUM
ejpam-5098	481	13	)	)	PUNCT
ejpam-5098	481	14	,	,	PUNCT
ejpam-5098	481	15	1094	1094	NUM
ejpam-5098	481	16	-	-	SYM
ejpam-5098	481	17	1112	1112	NUM
ejpam-5098	481	18	1108	1108	NUM
ejpam-5098	481	19	we	we	PRON
ejpam-5098	481	20	get	get	VERB
ejpam-5098	481	21	that	that	PRON
ejpam-5098	481	22	(	(	PUNCT
ejpam-5098	481	23	s1	s1	NOUN
ejpam-5098	481	24	)	)	PUNCT
ejpam-5098	481	25	∗	∗	NOUN
ejpam-5098	481	26	∨k	∨k	NOUN
ejpam-5098	481	27	=	=	SYM
ejpam-5098	481	28	r.	r.	NOUN
ejpam-5098	481	29	that	that	PRON
ejpam-5098	481	30	implies	imply	VERB
ejpam-5098	481	31	s1	s1	PROPN
ejpam-5098	481	32	∈	∈	PROPN
ejpam-5098	481	33	kσ	kσ	PROPN
ejpam-5098	481	34	.	.	PUNCT
ejpam-5098	482	1	since	since	SCONJ
ejpam-5098	482	2	t1	t1	PROPN
ejpam-5098	482	3	∈	∈	PROPN
ejpam-5098	482	4	(	(	PUNCT
ejpam-5098	482	5	a)∗	a)∗	PROPN
ejpam-5098	482	6	,	,	PUNCT
ejpam-5098	482	7	s1	s1	PROPN
ejpam-5098	482	8	∨	∨	NUM
ejpam-5098	482	9	t1	t1	NOUN
ejpam-5098	482	10	is	be	AUX
ejpam-5098	482	11	maximal	maximal	ADJ
ejpam-5098	482	12	and	and	CCONJ
ejpam-5098	482	13	s1	s1	PROPN
ejpam-5098	482	14	∈	∈	PROPN
ejpam-5098	482	15	kσ	kσ	PROPN
ejpam-5098	482	16	,	,	PUNCT
ejpam-5098	482	17	we	we	PRON
ejpam-5098	482	18	get	get	VERB
ejpam-5098	482	19	that	that	PRON
ejpam-5098	482	20	(	(	PUNCT
ejpam-5098	482	21	a)∗∨kσ	a)∗∨kσ	PROPN
ejpam-5098	482	22	=	=	SYM
ejpam-5098	482	23	r.	r.	NOUN
ejpam-5098	482	24	that	that	PRON
ejpam-5098	482	25	implies	imply	VERB
ejpam-5098	482	26	a	a	DET
ejpam-5098	482	27	∈	∈	PROPN
ejpam-5098	482	28	(	(	PUNCT
ejpam-5098	482	29	kσ)σ	kσ)σ	PROPN
ejpam-5098	482	30	,	,	PUNCT
ejpam-5098	482	31	so	so	ADV
ejpam-5098	482	32	kσ	kσ	PROPN
ejpam-5098	482	33	⊆	⊆	NUM
ejpam-5098	482	34	(	(	PUNCT
ejpam-5098	482	35	kσ)σ	kσ)σ	PROPN
ejpam-5098	482	36	.	.	PUNCT
ejpam-5098	483	1	therefore	therefore	ADV
ejpam-5098	483	2	,	,	PUNCT
ejpam-5098	483	3	kσ	kσ	PROPN
ejpam-5098	483	4	is	be	AUX
ejpam-5098	483	5	a	a	DET
ejpam-5098	483	6	σ	σ	NOUN
ejpam-5098	483	7	-	-	PUNCT
ejpam-5098	483	8	ideal	ideal	NOUN
ejpam-5098	483	9	of	of	ADP
ejpam-5098	483	10	r	r	NOUN
ejpam-5098	483	11	contained	contain	VERB
ejpam-5098	483	12	in	in	ADP
ejpam-5098	483	13	k.	k.	PROPN
ejpam-5098	483	14	since	since	SCONJ
ejpam-5098	483	15	k	k	PROPN
ejpam-5098	483	16	△	△	PROPN
ejpam-5098	483	17	is	be	AUX
ejpam-5098	483	18	the	the	DET
ejpam-5098	483	19	largest	large	ADJ
ejpam-5098	483	20	σ	σ	NOUN
ejpam-5098	483	21	-	-	PUNCT
ejpam-5098	483	22	ideal	ideal	NOUN
ejpam-5098	483	23	of	of	ADP
ejpam-5098	483	24	r	r	NOUN
ejpam-5098	483	25	contained	contain	VERB
ejpam-5098	483	26	k	k	PROPN
ejpam-5098	483	27	,	,	PUNCT
ejpam-5098	483	28	we	we	PRON
ejpam-5098	483	29	get	get	VERB
ejpam-5098	483	30	that	that	DET
ejpam-5098	483	31	kσ	kσ	PROPN
ejpam-5098	483	32	⊆	⊆	NUM
ejpam-5098	483	33	k	k	X
ejpam-5098	483	34	△	△	X
ejpam-5098	483	35	.	.	PUNCT
ejpam-5098	484	1	hence	hence	ADV
ejpam-5098	484	2	,	,	PUNCT
ejpam-5098	484	3	k	k	X
ejpam-5098	484	4	△	△	X
ejpam-5098	484	5	=	=	SYM
ejpam-5098	484	6	σ(k	σ(k	PROPN
ejpam-5098	484	7	)	)	PUNCT
ejpam-5098	484	8	.	.	PUNCT
ejpam-5098	485	1	thus	thus	ADV
ejpam-5098	485	2	,	,	PUNCT
ejpam-5098	485	3	kσ	kσ	PROPN
ejpam-5098	485	4	is	be	AUX
ejpam-5098	485	5	a	a	DET
ejpam-5098	485	6	σ	σ	NOUN
ejpam-5098	485	7	-	-	PUNCT
ejpam-5098	485	8	ideal	ideal	NOUN
ejpam-5098	485	9	of	of	ADP
ejpam-5098	485	10	r.	r.	PROPN
ejpam-5098	485	11	theorem	theorem	PROPN
ejpam-5098	485	12	17	17	NUM
ejpam-5098	485	13	.	.	PUNCT
ejpam-5098	486	1	let	let	VERB
ejpam-5098	486	2	p	p	PRON
ejpam-5098	486	3	be	be	AUX
ejpam-5098	486	4	a	a	DET
ejpam-5098	486	5	proper	proper	ADJ
ejpam-5098	486	6	ideal	ideal	NOUN
ejpam-5098	486	7	of	of	ADP
ejpam-5098	486	8	an	an	DET
ejpam-5098	486	9	adl	adl	NOUN
ejpam-5098	486	10	r	r	NOUN
ejpam-5098	486	11	with	with	ADP
ejpam-5098	486	12	maximal	maximal	ADJ
ejpam-5098	486	13	elements	element	NOUN
ejpam-5098	486	14	.	.	PUNCT
ejpam-5098	487	1	then	then	ADV
ejpam-5098	487	2	σ	σ	PROPN
ejpam-5098	487	3	-	-	PUNCT
ejpam-5098	487	4	ideal	ideal	NOUN
ejpam-5098	487	5	p	p	NOUN
ejpam-5098	487	6	of	of	ADP
ejpam-5098	487	7	r	r	NOUN
ejpam-5098	487	8	,	,	PUNCT
ejpam-5098	487	9	p	p	NOUN
ejpam-5098	487	10	=	=	SYM
ejpam-5098	487	11	⋂	⋂	PROPN
ejpam-5098	487	12	p⊆n	p⊆n	PROPN
ejpam-5098	487	13	no	no	NOUN
ejpam-5098	487	14	,	,	PUNCT
ejpam-5098	487	15	where	where	SCONJ
ejpam-5098	487	16	n	n	PRON
ejpam-5098	487	17	runs	run	VERB
ejpam-5098	487	18	over	over	ADP
ejpam-5098	487	19	all	all	DET
ejpam-5098	487	20	maximal	maximal	ADJ
ejpam-5098	487	21	ideals	ideal	NOUN
ejpam-5098	487	22	of	of	ADP
ejpam-5098	487	23	r	r	NOUN
ejpam-5098	487	24	containing	contain	VERB
ejpam-5098	487	25	p	p	NOUN
ejpam-5098	487	26	.	.	PUNCT
ejpam-5098	488	1	proof	proof	NOUN
ejpam-5098	488	2	.	.	PUNCT
ejpam-5098	489	1	let	let	VERB
ejpam-5098	489	2	p	p	PRON
ejpam-5098	489	3	be	be	AUX
ejpam-5098	489	4	any	any	DET
ejpam-5098	489	5	proper	proper	ADJ
ejpam-5098	489	6	σ	σ	NOUN
ejpam-5098	489	7	-	-	PUNCT
ejpam-5098	489	8	ideal	ideal	NOUN
ejpam-5098	489	9	of	of	ADP
ejpam-5098	489	10	r	r	NOUN
ejpam-5098	489	11	and	and	CCONJ
ejpam-5098	489	12	n	n	CCONJ
ejpam-5098	489	13	be	be	VERB
ejpam-5098	489	14	a	a	DET
ejpam-5098	489	15	maximal	maximal	ADJ
ejpam-5098	489	16	ideal	ideal	NOUN
ejpam-5098	489	17	of	of	ADP
ejpam-5098	489	18	r	r	NOUN
ejpam-5098	489	19	with	with	ADP
ejpam-5098	489	20	p	p	PROPN
ejpam-5098	489	21	⊆	⊆	NUM
ejpam-5098	489	22	n	n	NOUN
ejpam-5098	489	23	.	.	PUNCT
ejpam-5098	490	1	let	let	VERB
ejpam-5098	490	2	a	a	DET
ejpam-5098	490	3	∈	∈	PROPN
ejpam-5098	490	4	p	p	NOUN
ejpam-5098	490	5	.	.	PUNCT
ejpam-5098	491	1	then	then	ADV
ejpam-5098	491	2	(	(	PUNCT
ejpam-5098	491	3	a)∗	a)∗	PROPN
ejpam-5098	491	4	∨	∨	NUM
ejpam-5098	491	5	p	p	X
ejpam-5098	491	6	=	=	PUNCT
ejpam-5098	491	7	r.	r.	NOUN
ejpam-5098	491	8	that	that	PRON
ejpam-5098	491	9	implies	imply	VERB
ejpam-5098	491	10	there	there	PRON
ejpam-5098	491	11	exist	exist	VERB
ejpam-5098	491	12	elements	element	NOUN
ejpam-5098	491	13	s	s	PART
ejpam-5098	491	14	∈	∈	NOUN
ejpam-5098	491	15	(	(	PUNCT
ejpam-5098	491	16	a)∗	a)∗	PROPN
ejpam-5098	491	17	and	and	CCONJ
ejpam-5098	491	18	t	t	PROPN
ejpam-5098	491	19	∈	∈	PROPN
ejpam-5098	491	20	p	p	NOUN
ejpam-5098	491	21	such	such	ADJ
ejpam-5098	491	22	that	that	DET
ejpam-5098	491	23	s	s	PROPN
ejpam-5098	491	24	∨	∨	PROPN
ejpam-5098	491	25	t	t	PROPN
ejpam-5098	491	26	is	be	AUX
ejpam-5098	491	27	maximal	maximal	ADJ
ejpam-5098	491	28	.	.	PUNCT
ejpam-5098	492	1	since	since	SCONJ
ejpam-5098	492	2	s	s	PROPN
ejpam-5098	492	3	∈	∈	PROPN
ejpam-5098	492	4	(	(	PUNCT
ejpam-5098	492	5	a)∗	a)∗	PROPN
ejpam-5098	492	6	and	and	CCONJ
ejpam-5098	492	7	t	t	PROPN
ejpam-5098	492	8	∈	∈	PROPN
ejpam-5098	492	9	p	p	NOUN
ejpam-5098	492	10	,	,	PUNCT
ejpam-5098	492	11	we	we	PRON
ejpam-5098	492	12	have	have	VERB
ejpam-5098	492	13	that	that	SCONJ
ejpam-5098	492	14	a	a	DET
ejpam-5098	492	15	∧	∧	PROPN
ejpam-5098	492	16	s	s	PART
ejpam-5098	492	17	=	=	NOUN
ejpam-5098	492	18	0	0	NUM
ejpam-5098	492	19	and	and	CCONJ
ejpam-5098	492	20	t	t	PROPN
ejpam-5098	492	21	∈	∈	PROPN
ejpam-5098	492	22	n	n	ADV
ejpam-5098	492	23	.	.	PUNCT
ejpam-5098	493	1	since	since	SCONJ
ejpam-5098	493	2	s	s	PROPN
ejpam-5098	493	3	∨	∨	PROPN
ejpam-5098	493	4	t	t	PROPN
ejpam-5098	493	5	is	be	AUX
ejpam-5098	493	6	maximal	maximal	ADJ
ejpam-5098	493	7	and	and	CCONJ
ejpam-5098	493	8	t	t	PROPN
ejpam-5098	493	9	∈	∈	PROPN
ejpam-5098	493	10	n	n	X
ejpam-5098	493	11	,	,	PUNCT
ejpam-5098	493	12	we	we	PRON
ejpam-5098	493	13	get	get	VERB
ejpam-5098	493	14	that	that	PRON
ejpam-5098	493	15	s	s	VERB
ejpam-5098	493	16	/∈	/∈	NOUN
ejpam-5098	493	17	n	n	INTJ
ejpam-5098	493	18	.	.	PUNCT
ejpam-5098	494	1	since	since	SCONJ
ejpam-5098	494	2	a	a	DET
ejpam-5098	494	3	∧	∧	PROPN
ejpam-5098	494	4	s	s	PART
ejpam-5098	494	5	=	=	NOUN
ejpam-5098	494	6	0	0	NUM
ejpam-5098	494	7	and	and	CCONJ
ejpam-5098	494	8	s	s	NOUN
ejpam-5098	494	9	/∈	/∈	PROPN
ejpam-5098	494	10	n	n	CCONJ
ejpam-5098	494	11	,	,	PUNCT
ejpam-5098	494	12	we	we	PRON
ejpam-5098	494	13	get	get	VERB
ejpam-5098	494	14	that	that	SCONJ
ejpam-5098	494	15	a	a	DET
ejpam-5098	494	16	∈	∈	ADJ
ejpam-5098	494	17	no	no	INTJ
ejpam-5098	494	18	.	.	PUNCT
ejpam-5098	495	1	therefore	therefore	ADV
ejpam-5098	495	2	,	,	PUNCT
ejpam-5098	495	3	p	p	PROPN
ejpam-5098	495	4	⊆	⊆	NUM
ejpam-5098	495	5	⋂	⋂	PROPN
ejpam-5098	495	6	p⊆n	p⊆n	PROPN
ejpam-5098	495	7	no	no	INTJ
ejpam-5098	495	8	.	.	PUNCT
ejpam-5098	496	1	let	let	VERB
ejpam-5098	496	2	a	a	DET
ejpam-5098	496	3	∈	∈	PROPN
ejpam-5098	496	4	⋂	⋂	PROPN
ejpam-5098	496	5	p⊆n	p⊆n	PROPN
ejpam-5098	496	6	no	no	PROPN
ejpam-5098	496	7	.	.	PUNCT
ejpam-5098	497	1	then	then	ADV
ejpam-5098	497	2	a	a	DET
ejpam-5098	497	3	∈	∈	NOUN
ejpam-5098	497	4	no	no	INTJ
ejpam-5098	497	5	for	for	ADP
ejpam-5098	497	6	all	all	DET
ejpam-5098	497	7	maximal	maximal	ADJ
ejpam-5098	497	8	ideal	ideal	NOUN
ejpam-5098	497	9	n	n	PROPN
ejpam-5098	497	10	of	of	ADP
ejpam-5098	497	11	r	r	NOUN
ejpam-5098	497	12	containing	contain	VERB
ejpam-5098	497	13	p	p	NOUN
ejpam-5098	497	14	.	.	PUNCT
ejpam-5098	498	1	now	now	ADV
ejpam-5098	498	2	,	,	PUNCT
ejpam-5098	498	3	we	we	PRON
ejpam-5098	498	4	prove	prove	VERB
ejpam-5098	498	5	that	that	SCONJ
ejpam-5098	498	6	a	a	DET
ejpam-5098	498	7	∈	∈	PROPN
ejpam-5098	498	8	p	p	NOUN
ejpam-5098	498	9	.	.	PUNCT
ejpam-5098	499	1	suppose	suppose	VERB
ejpam-5098	499	2	a	a	DET
ejpam-5098	499	3	/∈	/∈	INTJ
ejpam-5098	499	4	p	p	NOUN
ejpam-5098	499	5	.	.	PUNCT
ejpam-5098	500	1	then	then	ADV
ejpam-5098	500	2	there	there	PRON
ejpam-5098	500	3	exists	exist	VERB
ejpam-5098	500	4	a	a	DET
ejpam-5098	500	5	prime	prime	ADJ
ejpam-5098	500	6	ideal	ideal	NOUN
ejpam-5098	500	7	q	q	NOUN
ejpam-5098	500	8	of	of	ADP
ejpam-5098	500	9	r	r	NOUN
ejpam-5098	500	10	such	such	ADJ
ejpam-5098	500	11	that	that	SCONJ
ejpam-5098	500	12	a	a	DET
ejpam-5098	500	13	/∈	/∈	NOUN
ejpam-5098	500	14	q	q	NOUN
ejpam-5098	500	15	and	and	CCONJ
ejpam-5098	500	16	p	p	PRON
ejpam-5098	500	17	⊆	⊆	NUM
ejpam-5098	500	18	q.	q.	NOUN
ejpam-5098	500	19	we	we	PRON
ejpam-5098	500	20	know	know	VERB
ejpam-5098	500	21	that	that	SCONJ
ejpam-5098	500	22	a	a	DET
ejpam-5098	500	23	proper	proper	ADJ
ejpam-5098	500	24	ideal	ideal	NOUN
ejpam-5098	500	25	is	be	AUX
ejpam-5098	500	26	contained	contain	VERB
ejpam-5098	500	27	in	in	ADP
ejpam-5098	500	28	a	a	DET
ejpam-5098	500	29	maximal	maximal	ADJ
ejpam-5098	500	30	ideal	ideal	NOUN
ejpam-5098	500	31	.	.	PUNCT
ejpam-5098	501	1	so	so	ADV
ejpam-5098	501	2	that	that	DET
ejpam-5098	501	3	q	q	NOUN
ejpam-5098	501	4	⊆m	⊆m	NOUN
ejpam-5098	501	5	,	,	PUNCT
ejpam-5098	501	6	where	where	SCONJ
ejpam-5098	501	7	m	m	NOUN
ejpam-5098	501	8	is	be	AUX
ejpam-5098	501	9	a	a	DET
ejpam-5098	501	10	maximal	maximal	ADJ
ejpam-5098	501	11	ideal	ideal	NOUN
ejpam-5098	501	12	.	.	PUNCT
ejpam-5098	502	1	that	that	PRON
ejpam-5098	502	2	implies	imply	VERB
ejpam-5098	502	3	mo	mo	PROPN
ejpam-5098	502	4	⊆	⊆	NUM
ejpam-5098	502	5	qo	qo	PROPN
ejpam-5098	502	6	⊆	⊆	NUM
ejpam-5098	502	7	q.	q.	NOUN
ejpam-5098	502	8	that	that	PRON
ejpam-5098	502	9	implies	imply	VERB
ejpam-5098	502	10	mo	mo	PROPN
ejpam-5098	502	11	⊆	⊆	NUM
ejpam-5098	502	12	q.	q.	NOUN
ejpam-5098	502	13	since	since	SCONJ
ejpam-5098	502	14	a	a	DET
ejpam-5098	502	15	/∈	/∈	PUNCT
ejpam-5098	502	16	q	q	NOUN
ejpam-5098	502	17	,	,	PUNCT
ejpam-5098	502	18	we	we	PRON
ejpam-5098	502	19	get	get	VERB
ejpam-5098	502	20	that	that	DET
ejpam-5098	502	21	a	a	DET
ejpam-5098	502	22	/∈	/∈	SYM
ejpam-5098	502	23	mo	mo	NOUN
ejpam-5098	502	24	and	and	CCONJ
ejpam-5098	502	25	p	p	PRON
ejpam-5098	502	26	⊆	⊆	NUM
ejpam-5098	502	27	m	m	NOUN
ejpam-5098	502	28	,	,	PUNCT
ejpam-5098	502	29	which	which	PRON
ejpam-5098	502	30	is	be	AUX
ejpam-5098	502	31	a	a	DET
ejpam-5098	502	32	contradiction	contradiction	NOUN
ejpam-5098	502	33	.	.	PUNCT
ejpam-5098	503	1	therefore	therefore	ADV
ejpam-5098	503	2	,	,	PUNCT
ejpam-5098	503	3	a	a	DET
ejpam-5098	503	4	∈	∈	PROPN
ejpam-5098	503	5	p	p	NOUN
ejpam-5098	503	6	and	and	CCONJ
ejpam-5098	503	7	hence	hence	ADV
ejpam-5098	503	8	⋂	⋂	PROPN
ejpam-5098	503	9	p⊆n	p⊆n	PROPN
ejpam-5098	503	10	no	no	DET
ejpam-5098	503	11	⊆	⊆	NUM
ejpam-5098	503	12	p	p	NOUN
ejpam-5098	503	13	.	.	PUNCT
ejpam-5098	504	1	thus	thus	ADV
ejpam-5098	504	2	,	,	PUNCT
ejpam-5098	504	3	p	p	X
ejpam-5098	504	4	=	=	SYM
ejpam-5098	504	5	⋂	⋂	PROPN
ejpam-5098	504	6	p⊆n	p⊆n	PROPN
ejpam-5098	504	7	no	no	PROPN
ejpam-5098	504	8	.	.	PUNCT
ejpam-5098	504	9	theorem	theorem	NOUN
ejpam-5098	504	10	18	18	NUM
ejpam-5098	504	11	.	.	PUNCT
ejpam-5098	505	1	let	let	VERB
ejpam-5098	505	2	k	k	PRON
ejpam-5098	505	3	be	be	AUX
ejpam-5098	505	4	a	a	DET
ejpam-5098	505	5	σ	σ	NOUN
ejpam-5098	505	6	-	-	PUNCT
ejpam-5098	505	7	ideal	ideal	NOUN
ejpam-5098	505	8	of	of	ADP
ejpam-5098	505	9	a	a	DET
ejpam-5098	505	10	normal	normal	ADJ
ejpam-5098	505	11	adl	adl	NOUN
ejpam-5098	505	12	r	r	NOUN
ejpam-5098	505	13	with	with	ADP
ejpam-5098	505	14	maximal	maximal	ADJ
ejpam-5098	505	15	elements	element	NOUN
ejpam-5098	505	16	.	.	PUNCT
ejpam-5098	506	1	then	then	ADV
ejpam-5098	506	2	r	r	PROPN
ejpam-5098	506	3	θ(k	θ(k	PROPN
ejpam-5098	506	4	)	)	PUNCT
ejpam-5098	506	5	is	be	AUX
ejpam-5098	506	6	a	a	DET
ejpam-5098	506	7	normal	normal	ADJ
ejpam-5098	506	8	adl	adl	PROPN
ejpam-5098	506	9	.	.	PUNCT
ejpam-5098	507	1	definition	definition	NOUN
ejpam-5098	507	2	15	15	NUM
ejpam-5098	507	3	.	.	PUNCT
ejpam-5098	508	1	a	a	DET
ejpam-5098	508	2	σ	σ	NOUN
ejpam-5098	508	3	-	-	PUNCT
ejpam-5098	508	4	ideal	ideal	NOUN
ejpam-5098	508	5	m	m	NOUN
ejpam-5098	508	6	of	of	ADP
ejpam-5098	508	7	r	r	NOUN
ejpam-5098	508	8	is	be	AUX
ejpam-5098	508	9	said	say	VERB
ejpam-5098	508	10	to	to	PART
ejpam-5098	508	11	be	be	AUX
ejpam-5098	508	12	maximal	maximal	ADJ
ejpam-5098	508	13	if	if	SCONJ
ejpam-5098	508	14	it	it	PRON
ejpam-5098	508	15	is	be	AUX
ejpam-5098	508	16	maximal	maximal	ADJ
ejpam-5098	508	17	among	among	ADP
ejpam-5098	508	18	σ	σ	NOUN
ejpam-5098	508	19	-	-	PUNCT
ejpam-5098	508	20	ideals	ideal	NOUN
ejpam-5098	508	21	of	of	ADP
ejpam-5098	508	22	r.	r.	PROPN
ejpam-5098	508	23	lemma	lemma	PROPN
ejpam-5098	508	24	10	10	NUM
ejpam-5098	508	25	.	.	PUNCT
ejpam-5098	509	1	let	let	VERB
ejpam-5098	509	2	r	r	PRON
ejpam-5098	509	3	be	be	AUX
ejpam-5098	509	4	an	an	DET
ejpam-5098	509	5	adl	adl	NOUN
ejpam-5098	509	6	with	with	ADP
ejpam-5098	509	7	maximal	maximal	ADJ
ejpam-5098	509	8	elements	element	NOUN
ejpam-5098	509	9	.	.	PUNCT
ejpam-5098	510	1	then	then	ADV
ejpam-5098	510	2	(	(	PUNCT
ejpam-5098	510	3	1	1	X
ejpam-5098	510	4	)	)	PUNCT
ejpam-5098	510	5	every	every	DET
ejpam-5098	510	6	maximal	maximal	ADJ
ejpam-5098	510	7	σ	σ	NOUN
ejpam-5098	510	8	-	-	PUNCT
ejpam-5098	510	9	ideal	ideal	NOUN
ejpam-5098	510	10	is	be	AUX
ejpam-5098	510	11	a	a	DET
ejpam-5098	510	12	prime	prime	ADJ
ejpam-5098	510	13	σ	σ	NOUN
ejpam-5098	510	14	-	-	PUNCT
ejpam-5098	510	15	ideal	ideal	NOUN
ejpam-5098	510	16	(	(	PUNCT
ejpam-5098	510	17	2	2	NUM
ejpam-5098	510	18	)	)	PUNCT
ejpam-5098	510	19	every	every	DET
ejpam-5098	510	20	prime	prime	ADJ
ejpam-5098	510	21	σ	σ	PROPN
ejpam-5098	510	22	-	-	PUNCT
ejpam-5098	510	23	ideal	ideal	NOUN
ejpam-5098	510	24	is	be	AUX
ejpam-5098	510	25	contained	contain	VERB
ejpam-5098	510	26	in	in	ADP
ejpam-5098	510	27	a	a	DET
ejpam-5098	510	28	maximal	maximal	ADJ
ejpam-5098	510	29	σ	σ	NOUN
ejpam-5098	510	30	-	-	PUNCT
ejpam-5098	510	31	ideal	ideal	NOUN
ejpam-5098	510	32	.	.	PUNCT
ejpam-5098	511	1	proof	proof	NOUN
ejpam-5098	511	2	.	.	PUNCT
ejpam-5098	512	1	(	(	PUNCT
ejpam-5098	512	2	1	1	X
ejpam-5098	512	3	)	)	PUNCT
ejpam-5098	512	4	let	let	VERB
ejpam-5098	512	5	m	m	PRON
ejpam-5098	512	6	be	be	AUX
ejpam-5098	512	7	any	any	DET
ejpam-5098	512	8	maximal	maximal	ADJ
ejpam-5098	512	9	σ	σ	NOUN
ejpam-5098	512	10	-	-	PUNCT
ejpam-5098	512	11	ideal	ideal	NOUN
ejpam-5098	512	12	of	of	ADP
ejpam-5098	512	13	r.	r.	PROPN
ejpam-5098	512	14	let	let	VERB
ejpam-5098	512	15	i	i	PRON
ejpam-5098	512	16	,	,	PUNCT
ejpam-5098	512	17	j	j	PROPN
ejpam-5098	512	18	be	be	VERB
ejpam-5098	512	19	two	two	NUM
ejpam-5098	512	20	σ	σ	NOUN
ejpam-5098	512	21	-	-	PUNCT
ejpam-5098	512	22	ideals	ideal	NOUN
ejpam-5098	512	23	of	of	ADP
ejpam-5098	512	24	r	r	NOUN
ejpam-5098	512	25	such	such	ADJ
ejpam-5098	512	26	that	that	SCONJ
ejpam-5098	512	27	i	i	PROPN
ejpam-5098	512	28	∩	∩	VERB
ejpam-5098	512	29	j	j	PROPN
ejpam-5098	512	30	⊆	⊆	NUM
ejpam-5098	512	31	m	m	NOUN
ejpam-5098	512	32	.	.	PUNCT
ejpam-5098	513	1	now	now	ADV
ejpam-5098	513	2	,	,	PUNCT
ejpam-5098	513	3	we	we	PRON
ejpam-5098	513	4	prove	prove	VERB
ejpam-5098	513	5	that	that	SCONJ
ejpam-5098	513	6	i	i	PRON
ejpam-5098	513	7	⊆	⊆	NUM
ejpam-5098	513	8	m	m	VERB
ejpam-5098	513	9	or	or	CCONJ
ejpam-5098	513	10	j	j	PROPN
ejpam-5098	513	11	⊆	⊆	NUM
ejpam-5098	513	12	m	m	NOUN
ejpam-5098	513	13	.	.	PUNCT
ejpam-5098	514	1	suppose	suppose	VERB
ejpam-5098	514	2	i	i	PRON
ejpam-5098	514	3	⊈	⊈	PROPN
ejpam-5098	514	4	m	m	VERB
ejpam-5098	514	5	and	and	CCONJ
ejpam-5098	514	6	j	j	PROPN
ejpam-5098	514	7	⊈	⊈	PROPN
ejpam-5098	514	8	m	m	VERB
ejpam-5098	514	9	.	.	PUNCT
ejpam-5098	515	1	then	then	ADV
ejpam-5098	515	2	there	there	PRON
ejpam-5098	515	3	exist	exist	VERB
ejpam-5098	515	4	elements	element	NOUN
ejpam-5098	515	5	a	a	DET
ejpam-5098	515	6	∈	∈	NOUN
ejpam-5098	516	1	i	i	PRON
ejpam-5098	516	2	and	and	CCONJ
ejpam-5098	516	3	b	b	PROPN
ejpam-5098	516	4	∈	∈	PROPN
ejpam-5098	516	5	j	j	NOUN
ejpam-5098	516	6	such	such	ADJ
ejpam-5098	516	7	that	that	SCONJ
ejpam-5098	516	8	a	a	DET
ejpam-5098	516	9	,	,	PUNCT
ejpam-5098	516	10	b	b	NOUN
ejpam-5098	516	11	/∈m	/∈m	PUNCT
ejpam-5098	516	12	.	.	PUNCT
ejpam-5098	517	1	since	since	SCONJ
ejpam-5098	517	2	m	m	PROPN
ejpam-5098	517	3	is	be	AUX
ejpam-5098	517	4	σ	σ	PROPN
ejpam-5098	517	5	-	-	PUNCT
ejpam-5098	517	6	maximal	maximal	ADJ
ejpam-5098	517	7	,	,	PUNCT
ejpam-5098	517	8	we	we	PRON
ejpam-5098	517	9	have	have	VERB
ejpam-5098	517	10	m	m	PROPN
ejpam-5098	517	11	∨	∨	NUM
ejpam-5098	517	12	(	(	PUNCT
ejpam-5098	517	13	a	a	X
ejpam-5098	517	14	]	]	X
ejpam-5098	517	15	=	=	SYM
ejpam-5098	517	16	r	r	NOUN
ejpam-5098	517	17	and	and	CCONJ
ejpam-5098	517	18	m	m	PROPN
ejpam-5098	517	19	∨	∨	NOUN
ejpam-5098	517	20	(	(	PUNCT
ejpam-5098	517	21	b	b	X
ejpam-5098	517	22	]	]	X
ejpam-5098	517	23	=	=	PUNCT
ejpam-5098	517	24	r.	r.	NOUN
ejpam-5098	517	25	that	that	PRON
ejpam-5098	517	26	implies	imply	VERB
ejpam-5098	517	27	m	m	PRON
ejpam-5098	517	28	∨	∨	NUM
ejpam-5098	517	29	(	(	PUNCT
ejpam-5098	517	30	a∧	a∧	NOUN
ejpam-5098	517	31	b	b	NOUN
ejpam-5098	517	32	]	]	X
ejpam-5098	517	33	=	=	SYM
ejpam-5098	517	34	r.	r.	PROPN
ejpam-5098	517	35	therefore	therefore	ADV
ejpam-5098	517	36	,	,	PUNCT
ejpam-5098	517	37	a∧	a∧	NOUN
ejpam-5098	517	38	b	b	PROPN
ejpam-5098	517	39	/∈m	/∈m	PUNCT
ejpam-5098	517	40	.	.	PUNCT
ejpam-5098	518	1	since	since	SCONJ
ejpam-5098	518	2	i	i	PRON
ejpam-5098	518	3	,	,	PUNCT
ejpam-5098	518	4	j	j	PROPN
ejpam-5098	518	5	are	be	AUX
ejpam-5098	518	6	σ	σ	NOUN
ejpam-5098	518	7	-	-	PUNCT
ejpam-5098	518	8	ideals	ideal	NOUN
ejpam-5098	518	9	of	of	ADP
ejpam-5098	518	10	r	r	NOUN
ejpam-5098	518	11	and	and	CCONJ
ejpam-5098	518	12	a	a	DET
ejpam-5098	518	13	∈	∈	PROPN
ejpam-5098	519	1	i	i	NOUN
ejpam-5098	519	2	,	,	PUNCT
ejpam-5098	519	3	b	b	PROPN
ejpam-5098	519	4	∈	∈	PROPN
ejpam-5098	519	5	j	j	PROPN
ejpam-5098	519	6	,	,	PUNCT
ejpam-5098	519	7	we	we	PRON
ejpam-5098	519	8	get	get	VERB
ejpam-5098	519	9	that	that	PRON
ejpam-5098	519	10	(	(	PUNCT
ejpam-5098	519	11	a)∗	a)∗	PROPN
ejpam-5098	519	12	∨	∨	NUM
ejpam-5098	519	13	i	i	PRON
ejpam-5098	519	14	=	=	SYM
ejpam-5098	519	15	r	r	NOUN
ejpam-5098	519	16	and	and	CCONJ
ejpam-5098	519	17	(	(	PUNCT
ejpam-5098	519	18	b)∗	b)∗	PROPN
ejpam-5098	519	19	∨	∨	PROPN
ejpam-5098	519	20	j	j	PROPN
ejpam-5098	519	21	=	=	PUNCT
ejpam-5098	519	22	r.	r.	PROPN
ejpam-5098	519	23	that	that	PRON
ejpam-5098	519	24	implies	imply	VERB
ejpam-5098	519	25	(	(	PUNCT
ejpam-5098	519	26	a	a	DET
ejpam-5098	519	27	∧	∧	PROPN
ejpam-5098	519	28	b)∗	b)∗	PROPN
ejpam-5098	519	29	∨	∨	NUM
ejpam-5098	519	30	i	i	PROPN
ejpam-5098	519	31	∩	∩	PROPN
ejpam-5098	519	32	j	j	PROPN
ejpam-5098	519	33	=	=	SYM
ejpam-5098	519	34	r.	r.	PROPN
ejpam-5098	519	35	since	since	SCONJ
ejpam-5098	519	36	i	i	PROPN
ejpam-5098	519	37	∩	∩	PROPN
ejpam-5098	519	38	j	j	PROPN
ejpam-5098	519	39	⊆m	⊆m	NOUN
ejpam-5098	519	40	,	,	PUNCT
ejpam-5098	519	41	we	we	PRON
ejpam-5098	519	42	get	get	VERB
ejpam-5098	520	1	that	that	DET
ejpam-5098	520	2	(	(	PUNCT
ejpam-5098	520	3	a	a	DET
ejpam-5098	520	4	∧	∧	NOUN
ejpam-5098	520	5	b)∗	b)∗	PROPN
ejpam-5098	520	6	∨m	∨m	NOUN
ejpam-5098	520	7	=	=	PUNCT
ejpam-5098	520	8	r.	r.	NOUN
ejpam-5098	520	9	that	that	PRON
ejpam-5098	520	10	implies	imply	VERB
ejpam-5098	520	11	a	a	DET
ejpam-5098	520	12	∧	∧	PROPN
ejpam-5098	520	13	b	b	PROPN
ejpam-5098	520	14	∈	∈	PROPN
ejpam-5098	520	15	mσ	mσ	INTJ
ejpam-5098	521	1	=	=	NOUN
ejpam-5098	522	1	m	m	PROPN
ejpam-5098	522	2	and	and	CCONJ
ejpam-5098	522	3	hence	hence	ADV
ejpam-5098	522	4	a	a	DET
ejpam-5098	522	5	∧	∧	PROPN
ejpam-5098	522	6	b	b	PROPN
ejpam-5098	522	7	∈	∈	PROPN
ejpam-5098	522	8	m	m	NOUN
ejpam-5098	522	9	,	,	PUNCT
ejpam-5098	522	10	which	which	PRON
ejpam-5098	522	11	is	be	AUX
ejpam-5098	522	12	a	a	DET
ejpam-5098	522	13	contradiction	contradiction	NOUN
ejpam-5098	522	14	to	to	ADP
ejpam-5098	522	15	a	a	DET
ejpam-5098	522	16	∧	∧	PROPN
ejpam-5098	522	17	b	b	PROPN
ejpam-5098	522	18	/∈	/∈	PROPN
ejpam-5098	522	19	m	m	INTJ
ejpam-5098	522	20	.	.	PUNCT
ejpam-5098	523	1	therefore	therefore	ADV
ejpam-5098	523	2	,	,	PUNCT
ejpam-5098	523	3	i	i	PRON
ejpam-5098	523	4	⊆m	⊆m	VERB
ejpam-5098	523	5	or	or	CCONJ
ejpam-5098	523	6	j	j	PROPN
ejpam-5098	523	7	⊆m	⊆m	NOUN
ejpam-5098	523	8	.	.	PUNCT
ejpam-5098	524	1	thus	thus	ADV
ejpam-5098	524	2	,	,	PUNCT
ejpam-5098	524	3	m	m	VERB
ejpam-5098	524	4	is	be	AUX
ejpam-5098	524	5	a	a	DET
ejpam-5098	524	6	prime	prime	ADJ
ejpam-5098	524	7	σ	σ	NOUN
ejpam-5098	524	8	-	-	PUNCT
ejpam-5098	524	9	ideal	ideal	NOUN
ejpam-5098	524	10	of	of	ADP
ejpam-5098	524	11	r.	r.	PROPN
ejpam-5098	524	12	(	(	PUNCT
ejpam-5098	524	13	2	2	X
ejpam-5098	524	14	)	)	PUNCT
ejpam-5098	524	15	let	let	VERB
ejpam-5098	524	16	p	p	PRON
ejpam-5098	524	17	be	be	AUX
ejpam-5098	524	18	a	a	DET
ejpam-5098	524	19	prime	prime	ADJ
ejpam-5098	524	20	σ	σ	NOUN
ejpam-5098	524	21	-	-	PUNCT
ejpam-5098	524	22	ideal	ideal	NOUN
ejpam-5098	524	23	of	of	ADP
ejpam-5098	524	24	r.	r.	PROPN
ejpam-5098	524	25	consider	consider	VERB
ejpam-5098	524	26	f	f	PROPN
ejpam-5098	525	1	=	=	PRON
ejpam-5098	526	1	{	{	PUNCT
ejpam-5098	527	1	i	i	PRON
ejpam-5098	527	2	|	|	ADV
ejpam-5098	527	3	i	i	PRON
ejpam-5098	527	4	is	be	AUX
ejpam-5098	527	5	a	a	DET
ejpam-5098	527	6	σ	σ	NOUN
ejpam-5098	527	7	-	-	PUNCT
ejpam-5098	527	8	ideal	ideal	NOUN
ejpam-5098	527	9	of	of	ADP
ejpam-5098	527	10	r	r	NOUN
ejpam-5098	527	11	,	,	PUNCT
ejpam-5098	527	12	p	p	NOUN
ejpam-5098	527	13	⊆	⊆	NUM
ejpam-5098	527	14	i	i	PROPN
ejpam-5098	527	15	}	}	PUNCT
ejpam-5098	527	16	.	.	PUNCT
ejpam-5098	528	1	clearly	clearly	ADV
ejpam-5098	528	2	,	,	PUNCT
ejpam-5098	528	3	p	p	PROPN
ejpam-5098	528	4	∈	∈	PROPN
ejpam-5098	528	5	f.	f.	PROPN
ejpam-5098	528	6	let	let	VERB
ejpam-5098	528	7	{	{	PUNCT
ejpam-5098	528	8	iα}α∈∆	iα}α∈∆	NOUN
ejpam-5098	528	9	be	be	AUX
ejpam-5098	528	10	a	a	DET
ejpam-5098	528	11	chain	chain	NOUN
ejpam-5098	528	12	in	in	ADP
ejpam-5098	528	13	f.	f.	PROPN
ejpam-5098	528	14	clearly	clearly	ADV
ejpam-5098	528	15	,	,	PUNCT
ejpam-5098	528	16	we	we	PRON
ejpam-5098	528	17	have	have	VERB
ejpam-5098	528	18	that	that	PRON
ejpam-5098	528	19	⋃	⋃	NOUN
ejpam-5098	528	20	α∈∆	α∈∆	PRON
ejpam-5098	528	21	iα	iα	NOUN
ejpam-5098	528	22	is	be	AUX
ejpam-5098	528	23	a	a	DET
ejpam-5098	528	24	σ	σ	NOUN
ejpam-5098	528	25	-	-	PUNCT
ejpam-5098	528	26	ideal	ideal	NOUN
ejpam-5098	528	27	of	of	ADP
ejpam-5098	528	28	r	r	NOUN
ejpam-5098	528	29	and	and	CCONJ
ejpam-5098	528	30	it	it	PRON
ejpam-5098	528	31	is	be	AUX
ejpam-5098	528	32	an	an	DET
ejpam-5098	528	33	upper	upper	ADJ
ejpam-5098	528	34	bound	bind	VERB
ejpam-5098	528	35	for	for	ADP
ejpam-5098	528	36	{	{	PUNCT
ejpam-5098	528	37	iα	iα	INTJ
ejpam-5098	528	38	|	|	ADV
ejpam-5098	528	39	α	α	NOUN
ejpam-5098	528	40	∈	∈	NOUN
ejpam-5098	528	41	∆	∆	X
ejpam-5098	528	42	}	}	PUNCT
ejpam-5098	528	43	.	.	PUNCT
ejpam-5098	529	1	by	by	ADP
ejpam-5098	529	2	zorn	zorn	PROPN
ejpam-5098	529	3	’s	’s	PART
ejpam-5098	529	4	lemma	lemma	PROPN
ejpam-5098	529	5	,	,	PUNCT
ejpam-5098	529	6	f	f	PROPN
ejpam-5098	529	7	has	have	VERB
ejpam-5098	529	8	a	a	DET
ejpam-5098	529	9	maximal	maximal	ADJ
ejpam-5098	529	10	element	element	NOUN
ejpam-5098	529	11	;	;	PUNCT
ejpam-5098	529	12	let	let	VERB
ejpam-5098	529	13	it	it	PRON
ejpam-5098	529	14	be	be	AUX
ejpam-5098	529	15	m	m	PRON
ejpam-5098	529	16	.	.	PUNCT
ejpam-5098	530	1	therefore	therefore	ADV
ejpam-5098	530	2	,	,	PUNCT
ejpam-5098	530	3	prime	prime	ADJ
ejpam-5098	530	4	σ	σ	PROPN
ejpam-5098	530	5	-	-	PUNCT
ejpam-5098	530	6	ideal	ideal	NOUN
ejpam-5098	530	7	p	p	NOUN
ejpam-5098	530	8	is	be	AUX
ejpam-5098	530	9	contained	contain	VERB
ejpam-5098	530	10	in	in	ADP
ejpam-5098	530	11	a	a	DET
ejpam-5098	530	12	maximal	maximal	ADJ
ejpam-5098	530	13	σ	σ	NOUN
ejpam-5098	530	14	-	-	PUNCT
ejpam-5098	530	15	ideal	ideal	NOUN
ejpam-5098	530	16	m	m	PROPN
ejpam-5098	530	17	of	of	ADP
ejpam-5098	530	18	r.	r.	PROPN
ejpam-5098	530	19	r.	r.	PROPN
ejpam-5098	530	20	noorbhasha	noorbhasha	PROPN
ejpam-5098	530	21	,	,	PUNCT
ejpam-5098	530	22	r.	r.	PROPN
ejpam-5098	530	23	bandaru	bandaru	PROPN
ejpam-5098	530	24	,	,	PUNCT
ejpam-5098	530	25	a.	a.	NOUN
ejpam-5098	530	26	iampan	iampan	PROPN
ejpam-5098	530	27	/	/	SYM
ejpam-5098	530	28	eur	eur	PROPN
ejpam-5098	530	29	.	.	PUNCT
ejpam-5098	531	1	j.	j.	PROPN
ejpam-5098	531	2	pure	pure	PROPN
ejpam-5098	531	3	appl	appl	PROPN
ejpam-5098	531	4	.	.	PROPN
ejpam-5098	531	5	math	math	PROPN
ejpam-5098	531	6	,	,	PUNCT
ejpam-5098	531	7	17	17	NUM
ejpam-5098	531	8	(	(	PUNCT
ejpam-5098	531	9	2	2	NUM
ejpam-5098	531	10	)	)	PUNCT
ejpam-5098	531	11	(	(	PUNCT
ejpam-5098	531	12	2024	2024	NUM
ejpam-5098	531	13	)	)	PUNCT
ejpam-5098	531	14	,	,	PUNCT
ejpam-5098	531	15	1094	1094	NUM
ejpam-5098	531	16	-	-	SYM
ejpam-5098	531	17	1112	1112	NUM
ejpam-5098	531	18	1109	1109	NUM
ejpam-5098	531	19	theorem	theorem	NOUN
ejpam-5098	531	20	19	19	NUM
ejpam-5098	531	21	.	.	PUNCT
ejpam-5098	532	1	let	let	VERB
ejpam-5098	532	2	i	i	PRON
ejpam-5098	532	3	be	be	AUX
ejpam-5098	532	4	a	a	DET
ejpam-5098	532	5	σ	σ	NOUN
ejpam-5098	532	6	-	-	PUNCT
ejpam-5098	532	7	ideal	ideal	NOUN
ejpam-5098	532	8	of	of	ADP
ejpam-5098	532	9	r	r	NOUN
ejpam-5098	532	10	and	and	CCONJ
ejpam-5098	532	11	f	f	PROPN
ejpam-5098	532	12	a	a	DET
ejpam-5098	532	13	closed	close	VERB
ejpam-5098	532	14	under	under	ADP
ejpam-5098	532	15	∧	∧	NOUN
ejpam-5098	532	16	such	such	ADJ
ejpam-5098	532	17	that	that	SCONJ
ejpam-5098	532	18	i	i	PRON
ejpam-5098	532	19	∩f	∩f	NOUN
ejpam-5098	532	20	=	=	PUNCT
ejpam-5098	533	1	∅.	∅.	NOUN
ejpam-5098	533	2	then	then	ADV
ejpam-5098	533	3	there	there	PRON
ejpam-5098	533	4	exists	exist	VERB
ejpam-5098	533	5	a	a	DET
ejpam-5098	533	6	prime	prime	ADJ
ejpam-5098	533	7	σ	σ	NOUN
ejpam-5098	533	8	-	-	PUNCT
ejpam-5098	533	9	ideal	ideal	NOUN
ejpam-5098	533	10	p	p	NOUN
ejpam-5098	533	11	of	of	ADP
ejpam-5098	533	12	r	r	NOUN
ejpam-5098	533	13	such	such	ADJ
ejpam-5098	533	14	that	that	SCONJ
ejpam-5098	533	15	i	i	PRON
ejpam-5098	533	16	⊆	⊆	NUM
ejpam-5098	533	17	p	p	NOUN
ejpam-5098	533	18	and	and	CCONJ
ejpam-5098	533	19	f	f	PROPN
ejpam-5098	533	20	∩	∩	NOUN
ejpam-5098	533	21	p	p	X
ejpam-5098	533	22	=	=	PUNCT
ejpam-5098	533	23	∅.	∅.	NOUN
ejpam-5098	533	24	proof	proof	NOUN
ejpam-5098	533	25	.	.	PUNCT
ejpam-5098	534	1	consider	consider	VERB
ejpam-5098	534	2	f	f	NOUN
ejpam-5098	534	3	=	=	PRON
ejpam-5098	534	4	{	{	PUNCT
ejpam-5098	534	5	j	j	PROPN
ejpam-5098	534	6	|	|	ADV
ejpam-5098	534	7	j	j	PROPN
ejpam-5098	534	8	is	be	AUX
ejpam-5098	534	9	a	a	DET
ejpam-5098	534	10	σ	σ	NOUN
ejpam-5098	534	11	-	-	PUNCT
ejpam-5098	534	12	ideal	ideal	NOUN
ejpam-5098	534	13	of	of	ADP
ejpam-5098	534	14	r	r	NOUN
ejpam-5098	534	15	,	,	PUNCT
ejpam-5098	534	16	i	i	NOUN
ejpam-5098	534	17	⊆	⊆	NUM
ejpam-5098	534	18	p	p	NOUN
ejpam-5098	535	1	and	and	CCONJ
ejpam-5098	535	2	i	i	PROPN
ejpam-5098	535	3	∩	∩	ADJ
ejpam-5098	535	4	f	f	X
ejpam-5098	535	5	=	=	PUNCT
ejpam-5098	535	6	∅	∅	NOUN
ejpam-5098	535	7	}	}	PUNCT
ejpam-5098	535	8	.	.	PUNCT
ejpam-5098	536	1	clearly	clearly	ADV
ejpam-5098	536	2	,	,	PUNCT
ejpam-5098	536	3	i	i	PRON
ejpam-5098	536	4	∈	∈	PROPN
ejpam-5098	536	5	f.	f.	PROPN
ejpam-5098	536	6	let	let	VERB
ejpam-5098	536	7	{	{	PUNCT
ejpam-5098	536	8	jα}α∈∆	jα}α∈∆	NOUN
ejpam-5098	536	9	be	be	AUX
ejpam-5098	536	10	a	a	DET
ejpam-5098	536	11	chain	chain	NOUN
ejpam-5098	536	12	in	in	ADP
ejpam-5098	536	13	f.	f.	PROPN
ejpam-5098	536	14	clearly	clearly	ADV
ejpam-5098	536	15	,	,	PUNCT
ejpam-5098	536	16	we	we	PRON
ejpam-5098	536	17	have	have	VERB
ejpam-5098	536	18	that	that	PRON
ejpam-5098	536	19	⋃	⋃	NOUN
ejpam-5098	536	20	α∈∆	α∈∆	PRON
ejpam-5098	536	21	jα	jα	PROPN
ejpam-5098	536	22	is	be	AUX
ejpam-5098	536	23	a	a	DET
ejpam-5098	536	24	σ	σ	NOUN
ejpam-5098	536	25	-	-	PUNCT
ejpam-5098	536	26	ideal	ideal	NOUN
ejpam-5098	536	27	of	of	ADP
ejpam-5098	536	28	r	r	NOUN
ejpam-5098	536	29	and	and	CCONJ
ejpam-5098	536	30	it	it	PRON
ejpam-5098	536	31	is	be	AUX
ejpam-5098	536	32	an	an	DET
ejpam-5098	536	33	upper	upper	ADJ
ejpam-5098	536	34	bound	bind	VERB
ejpam-5098	536	35	for	for	ADP
ejpam-5098	536	36	{	{	PUNCT
ejpam-5098	536	37	jα	jα	NOUN
ejpam-5098	537	1	|	|	ADV
ejpam-5098	537	2	α	α	NOUN
ejpam-5098	537	3	∈	∈	NOUN
ejpam-5098	537	4	∆	∆	X
ejpam-5098	537	5	}	}	PUNCT
ejpam-5098	537	6	.	.	PUNCT
ejpam-5098	538	1	by	by	ADP
ejpam-5098	538	2	zorn	zorn	PROPN
ejpam-5098	538	3	’s	’s	PART
ejpam-5098	538	4	lemma	lemma	PROPN
ejpam-5098	538	5	,	,	PUNCT
ejpam-5098	538	6	f	f	PROPN
ejpam-5098	538	7	has	have	VERB
ejpam-5098	538	8	a	a	DET
ejpam-5098	538	9	maximal	maximal	ADJ
ejpam-5098	538	10	element	element	NOUN
ejpam-5098	538	11	;	;	PUNCT
ejpam-5098	538	12	let	let	VERB
ejpam-5098	538	13	it	it	PRON
ejpam-5098	538	14	be	be	AUX
ejpam-5098	538	15	m	m	PRON
ejpam-5098	538	16	.	.	PUNCT
ejpam-5098	539	1	let	let	VERB
ejpam-5098	539	2	i1	i1	PROPN
ejpam-5098	539	3	,	,	PUNCT
ejpam-5098	539	4	i2	i2	PROPN
ejpam-5098	539	5	be	be	VERB
ejpam-5098	539	6	any	any	DET
ejpam-5098	539	7	two	two	NUM
ejpam-5098	539	8	σ	σ	NOUN
ejpam-5098	539	9	-	-	PUNCT
ejpam-5098	539	10	ideals	ideal	NOUN
ejpam-5098	539	11	of	of	ADP
ejpam-5098	539	12	r	r	NOUN
ejpam-5098	539	13	such	such	ADJ
ejpam-5098	539	14	that	that	SCONJ
ejpam-5098	539	15	i1	i1	PROPN
ejpam-5098	539	16	∩	∩	PROPN
ejpam-5098	539	17	i2	i2	PROPN
ejpam-5098	539	18	⊆m	⊆m	NOUN
ejpam-5098	539	19	.	.	PUNCT
ejpam-5098	540	1	now	now	ADV
ejpam-5098	540	2	,	,	PUNCT
ejpam-5098	540	3	we	we	PRON
ejpam-5098	540	4	prove	prove	VERB
ejpam-5098	540	5	that	that	SCONJ
ejpam-5098	540	6	i1	i1	PROPN
ejpam-5098	540	7	⊆m	⊆m	NOUN
ejpam-5098	540	8	or	or	CCONJ
ejpam-5098	540	9	i2	i2	PROPN
ejpam-5098	540	10	⊆m	⊆m	NOUN
ejpam-5098	540	11	.	.	PUNCT
ejpam-5098	541	1	suppose	suppose	VERB
ejpam-5098	541	2	i1	i1	PROPN
ejpam-5098	541	3	⊈m	⊈m	PUNCT
ejpam-5098	541	4	and	and	CCONJ
ejpam-5098	541	5	i2	i2	PROPN
ejpam-5098	541	6	⊈m	⊈m	PUNCT
ejpam-5098	541	7	.	.	PUNCT
ejpam-5098	542	1	then	then	ADV
ejpam-5098	542	2	there	there	PRON
ejpam-5098	542	3	exist	exist	VERB
ejpam-5098	542	4	elements	element	NOUN
ejpam-5098	542	5	a	a	DET
ejpam-5098	542	6	∈	∈	PROPN
ejpam-5098	542	7	i1	i1	PROPN
ejpam-5098	542	8	and	and	CCONJ
ejpam-5098	542	9	b	b	PROPN
ejpam-5098	542	10	∈	∈	PROPN
ejpam-5098	542	11	i2	i2	PROPN
ejpam-5098	542	12	such	such	ADJ
ejpam-5098	542	13	that	that	SCONJ
ejpam-5098	542	14	a	a	DET
ejpam-5098	542	15	,	,	PUNCT
ejpam-5098	542	16	b	b	NOUN
ejpam-5098	542	17	/∈	/∈	INTJ
ejpam-5098	542	18	m	m	VERB
ejpam-5098	542	19	.	.	PUNCT
ejpam-5098	543	1	since	since	SCONJ
ejpam-5098	543	2	a	a	DET
ejpam-5098	543	3	∈	∈	PROPN
ejpam-5098	543	4	i1	i1	PROPN
ejpam-5098	543	5	and	and	CCONJ
ejpam-5098	543	6	b	b	PROPN
ejpam-5098	543	7	∈	∈	PROPN
ejpam-5098	543	8	i2	i2	PROPN
ejpam-5098	543	9	,	,	PUNCT
ejpam-5098	543	10	we	we	PRON
ejpam-5098	543	11	have	have	VERB
ejpam-5098	543	12	that	that	PRON
ejpam-5098	543	13	a	a	DET
ejpam-5098	543	14	∧	∧	PROPN
ejpam-5098	543	15	b	b	PROPN
ejpam-5098	543	16	∈	∈	PROPN
ejpam-5098	543	17	i1	i1	PROPN
ejpam-5098	543	18	∩	∩	PROPN
ejpam-5098	543	19	i2	i2	PROPN
ejpam-5098	543	20	⊆	⊆	NUM
ejpam-5098	543	21	m	m	NOUN
ejpam-5098	543	22	.	.	PUNCT
ejpam-5098	544	1	since	since	SCONJ
ejpam-5098	544	2	a	a	DET
ejpam-5098	544	3	/∈	/∈	NOUN
ejpam-5098	544	4	m	m	PROPN
ejpam-5098	544	5	and	and	CCONJ
ejpam-5098	544	6	b	b	PROPN
ejpam-5098	544	7	/∈	/∈	PUNCT
ejpam-5098	544	8	m	m	VERB
ejpam-5098	544	9	,	,	PUNCT
ejpam-5098	544	10	we	we	PRON
ejpam-5098	544	11	get	get	VERB
ejpam-5098	544	12	that	that	PRON
ejpam-5098	544	13	m	m	VERB
ejpam-5098	544	14	∨	∨	NOUN
ejpam-5098	544	15	(	(	PUNCT
ejpam-5098	544	16	a	a	X
ejpam-5098	544	17	]	]	X
ejpam-5098	544	18	⊆	⊆	NUM
ejpam-5098	544	19	m	m	NOUN
ejpam-5098	544	20	and	and	CCONJ
ejpam-5098	544	21	m	m	PROPN
ejpam-5098	544	22	∨	∨	NOUN
ejpam-5098	544	23	(	(	PUNCT
ejpam-5098	544	24	b	b	X
ejpam-5098	544	25	]	]	X
ejpam-5098	544	26	⊆	⊆	NUM
ejpam-5098	544	27	m	m	NOUN
ejpam-5098	544	28	.	.	PUNCT
ejpam-5098	545	1	that	that	PRON
ejpam-5098	545	2	implies	imply	VERB
ejpam-5098	545	3	m	m	PRON
ejpam-5098	545	4	∨	∨	NUM
ejpam-5098	545	5	(	(	PUNCT
ejpam-5098	545	6	a	a	DET
ejpam-5098	545	7	∧	∧	PROPN
ejpam-5098	545	8	b	b	PROPN
ejpam-5098	545	9	]	]	X
ejpam-5098	545	10	⊆	⊆	NUM
ejpam-5098	545	11	m	m	NOUN
ejpam-5098	545	12	.	.	PUNCT
ejpam-5098	546	1	by	by	ADP
ejpam-5098	546	2	maximality	maximality	NOUN
ejpam-5098	546	3	of	of	ADP
ejpam-5098	546	4	m	m	PROPN
ejpam-5098	546	5	in	in	ADP
ejpam-5098	546	6	f	f	PROPN
ejpam-5098	546	7	,	,	PUNCT
ejpam-5098	546	8	we	we	PRON
ejpam-5098	546	9	have	have	VERB
ejpam-5098	546	10	that	that	PRON
ejpam-5098	546	11	(	(	PUNCT
ejpam-5098	546	12	m	m	NOUN
ejpam-5098	546	13	∨	∨	NOUN
ejpam-5098	546	14	(	(	PUNCT
ejpam-5098	546	15	a	a	DET
ejpam-5098	546	16	]	]	X
ejpam-5098	546	17	)	)	PUNCT
ejpam-5098	546	18	∩	∩	NOUN
ejpam-5098	546	19	f	f	PROPN
ejpam-5098	546	20	̸=	̸=	PROPN
ejpam-5098	546	21	∅	∅	NOUN
ejpam-5098	546	22	and	and	CCONJ
ejpam-5098	546	23	(	(	PUNCT
ejpam-5098	546	24	m	m	PROPN
ejpam-5098	546	25	∨	∨	NOUN
ejpam-5098	546	26	(	(	PUNCT
ejpam-5098	546	27	b	b	NOUN
ejpam-5098	546	28	]	]	X
ejpam-5098	546	29	)	)	PUNCT
ejpam-5098	546	30	∩	∩	NOUN
ejpam-5098	546	31	f	f	PROPN
ejpam-5098	546	32	̸=	̸=	PROPN
ejpam-5098	546	33	∅.	∅.	ADV
ejpam-5098	546	34	then	then	ADV
ejpam-5098	546	35	choose	choose	VERB
ejpam-5098	546	36	elements	element	NOUN
ejpam-5098	546	37	x	x	SYM
ejpam-5098	547	1	∈	∈	PROPN
ejpam-5098	547	2	(	(	PUNCT
ejpam-5098	547	3	m	m	PROPN
ejpam-5098	547	4	∨	∨	NOUN
ejpam-5098	547	5	(	(	PUNCT
ejpam-5098	547	6	a	a	DET
ejpam-5098	547	7	]	]	X
ejpam-5098	547	8	)	)	PUNCT
ejpam-5098	547	9	∩	∩	PROPN
ejpam-5098	547	10	f	f	PROPN
ejpam-5098	547	11	and	and	CCONJ
ejpam-5098	547	12	y	y	PROPN
ejpam-5098	547	13	∈	∈	PROPN
ejpam-5098	547	14	(	(	PUNCT
ejpam-5098	547	15	m	m	NOUN
ejpam-5098	547	16	∨	∨	NOUN
ejpam-5098	547	17	(	(	PUNCT
ejpam-5098	547	18	b	b	NOUN
ejpam-5098	547	19	]	]	X
ejpam-5098	547	20	)	)	PUNCT
ejpam-5098	547	21	∩	∩	PROPN
ejpam-5098	547	22	f	f	X
ejpam-5098	547	23	.	.	PUNCT
ejpam-5098	548	1	since	since	SCONJ
ejpam-5098	548	2	f	f	PROPN
ejpam-5098	548	3	is	be	AUX
ejpam-5098	548	4	closed	close	VERB
ejpam-5098	548	5	under	under	ADP
ejpam-5098	548	6	∧	∧	PROPN
ejpam-5098	548	7	,	,	PUNCT
ejpam-5098	548	8	we	we	PRON
ejpam-5098	548	9	have	have	VERB
ejpam-5098	548	10	x	x	ADJ
ejpam-5098	548	11	∧	∧	PROPN
ejpam-5098	548	12	y	y	PROPN
ejpam-5098	548	13	∈	∈	PROPN
ejpam-5098	548	14	(	(	PUNCT
ejpam-5098	548	15	(	(	PUNCT
ejpam-5098	548	16	m	m	VERB
ejpam-5098	548	17	∨	∨	NOUN
ejpam-5098	548	18	(	(	PUNCT
ejpam-5098	548	19	a	a	DET
ejpam-5098	548	20	]	]	X
ejpam-5098	548	21	)	)	PUNCT
ejpam-5098	548	22	∩	∩	PROPN
ejpam-5098	548	23	f	f	PROPN
ejpam-5098	548	24	)	)	PUNCT
ejpam-5098	548	25	∩	∩	NOUN
ejpam-5098	548	26	(	(	PUNCT
ejpam-5098	548	27	(	(	PUNCT
ejpam-5098	548	28	m	m	VERB
ejpam-5098	548	29	∨	∨	NOUN
ejpam-5098	548	30	(	(	PUNCT
ejpam-5098	548	31	b	b	NOUN
ejpam-5098	548	32	]	]	X
ejpam-5098	548	33	)	)	PUNCT
ejpam-5098	548	34	∩	∩	PROPN
ejpam-5098	548	35	f	f	NOUN
ejpam-5098	548	36	)	)	PUNCT
ejpam-5098	548	37	=	=	SYM
ejpam-5098	548	38	(	(	PUNCT
ejpam-5098	548	39	m	m	VERB
ejpam-5098	548	40	∨	∨	NOUN
ejpam-5098	548	41	(	(	PUNCT
ejpam-5098	548	42	a	a	DET
ejpam-5098	548	43	∧	∧	PROPN
ejpam-5098	548	44	b	b	PROPN
ejpam-5098	548	45	]	]	X
ejpam-5098	548	46	)	)	PUNCT
ejpam-5098	548	47	∩	∩	PROPN
ejpam-5098	548	48	f	f	X
ejpam-5098	548	49	.	.	PUNCT
ejpam-5098	549	1	by	by	ADP
ejpam-5098	549	2	maximality	maximality	NOUN
ejpam-5098	549	3	of	of	ADP
ejpam-5098	549	4	m	m	PROPN
ejpam-5098	549	5	in	in	ADP
ejpam-5098	549	6	f	f	PROPN
ejpam-5098	549	7	,	,	PUNCT
ejpam-5098	549	8	we	we	PRON
ejpam-5098	549	9	have	have	VERB
ejpam-5098	549	10	that	that	PRON
ejpam-5098	549	11	a	a	DET
ejpam-5098	549	12	∧	∧	PROPN
ejpam-5098	549	13	b	b	PROPN
ejpam-5098	549	14	/∈	/∈	PROPN
ejpam-5098	549	15	m	m	VERB
ejpam-5098	549	16	,	,	PUNCT
ejpam-5098	549	17	which	which	PRON
ejpam-5098	549	18	is	be	AUX
ejpam-5098	549	19	a	a	DET
ejpam-5098	549	20	contradiction	contradiction	NOUN
ejpam-5098	549	21	to	to	ADP
ejpam-5098	549	22	a	a	DET
ejpam-5098	549	23	∧	∧	PROPN
ejpam-5098	549	24	b	b	PROPN
ejpam-5098	549	25	∈	∈	PROPN
ejpam-5098	549	26	m	m	VERB
ejpam-5098	549	27	.	.	PUNCT
ejpam-5098	550	1	therefore	therefore	ADV
ejpam-5098	550	2	,	,	PUNCT
ejpam-5098	550	3	i1	i1	PROPN
ejpam-5098	550	4	⊆	⊆	NUM
ejpam-5098	550	5	m	m	PROPN
ejpam-5098	550	6	or	or	CCONJ
ejpam-5098	550	7	i2	i2	PROPN
ejpam-5098	550	8	⊆m	⊆m	NOUN
ejpam-5098	550	9	.	.	PUNCT
ejpam-5098	551	1	hence	hence	ADV
ejpam-5098	551	2	,	,	PUNCT
ejpam-5098	551	3	m	m	VERB
ejpam-5098	551	4	is	be	AUX
ejpam-5098	551	5	a	a	DET
ejpam-5098	551	6	prime	prime	ADJ
ejpam-5098	551	7	σ	σ	NOUN
ejpam-5098	551	8	-	-	PUNCT
ejpam-5098	551	9	ideal	ideal	NOUN
ejpam-5098	551	10	of	of	ADP
ejpam-5098	551	11	r	r	NOUN
ejpam-5098	551	12	containing	contain	VERB
ejpam-5098	551	13	i	i	PRON
ejpam-5098	551	14	and	and	CCONJ
ejpam-5098	551	15	m	m	PROPN
ejpam-5098	551	16	∩	∩	ADJ
ejpam-5098	551	17	f	f	PROPN
ejpam-5098	551	18	=	=	PUNCT
ejpam-5098	551	19	∅.	∅.	NOUN
ejpam-5098	551	20	theorem	theorem	VERB
ejpam-5098	551	21	20	20	NUM
ejpam-5098	551	22	.	.	PUNCT
ejpam-5098	552	1	let	let	VERB
ejpam-5098	552	2	i	i	PRON
ejpam-5098	552	3	be	be	AUX
ejpam-5098	552	4	a	a	DET
ejpam-5098	552	5	proper	proper	ADJ
ejpam-5098	552	6	σ	σ	NOUN
ejpam-5098	552	7	-	-	PUNCT
ejpam-5098	552	8	ideal	ideal	NOUN
ejpam-5098	552	9	of	of	ADP
ejpam-5098	552	10	r.	r.	PROPN
ejpam-5098	553	1	then	then	ADV
ejpam-5098	553	2	i	i	PRON
ejpam-5098	553	3	=	=	SYM
ejpam-5098	553	4	⋂	⋂	PROPN
ejpam-5098	553	5	{	{	PUNCT
ejpam-5098	553	6	p	p	NOUN
ejpam-5098	553	7	|	|	NOUN
ejpam-5098	553	8	p	p	PROPN
ejpam-5098	553	9	∈	∈	PROPN
ejpam-5098	553	10	specσr	specσr	NOUN
ejpam-5098	553	11	and	and	CCONJ
ejpam-5098	553	12	i	i	PRON
ejpam-5098	553	13	⊆	⊆	NUM
ejpam-5098	553	14	p	p	X
ejpam-5098	553	15	}	}	PUNCT
ejpam-5098	553	16	.	.	PUNCT
ejpam-5098	554	1	proof	proof	NOUN
ejpam-5098	554	2	.	.	PUNCT
ejpam-5098	555	1	consider	consider	VERB
ejpam-5098	555	2	i	i	PRON
ejpam-5098	555	3	◦	◦	VERB
ejpam-5098	555	4	=	=	SYM
ejpam-5098	555	5	⋂	⋂	PROPN
ejpam-5098	555	6	{	{	PUNCT
ejpam-5098	555	7	p	p	NOUN
ejpam-5098	555	8	|	|	NOUN
ejpam-5098	555	9	p	p	PROPN
ejpam-5098	555	10	∈	∈	PROPN
ejpam-5098	555	11	specσr	specσr	NOUN
ejpam-5098	556	1	and	and	CCONJ
ejpam-5098	556	2	i	i	PRON
ejpam-5098	556	3	⊆	⊆	NUM
ejpam-5098	556	4	p	p	X
ejpam-5098	556	5	}	}	PUNCT
ejpam-5098	556	6	.	.	PUNCT
ejpam-5098	557	1	let	let	VERB
ejpam-5098	557	2	a	a	DET
ejpam-5098	557	3	/∈	/∈	NOUN
ejpam-5098	557	4	i.	i.	NOUN
ejpam-5098	557	5	then	then	ADV
ejpam-5098	557	6	there	there	PRON
ejpam-5098	557	7	exists	exist	VERB
ejpam-5098	557	8	a	a	DET
ejpam-5098	557	9	prime	prime	ADJ
ejpam-5098	557	10	σ	σ	NOUN
ejpam-5098	557	11	-	-	PUNCT
ejpam-5098	557	12	ideal	ideal	NOUN
ejpam-5098	557	13	of	of	ADP
ejpam-5098	557	14	r	r	NOUN
ejpam-5098	557	15	such	such	ADJ
ejpam-5098	557	16	that	that	SCONJ
ejpam-5098	557	17	a	a	DET
ejpam-5098	557	18	/∈	/∈	NOUN
ejpam-5098	557	19	p	p	NOUN
ejpam-5098	558	1	and	and	CCONJ
ejpam-5098	558	2	i	i	PRON
ejpam-5098	558	3	⊆	⊆	NUM
ejpam-5098	558	4	p	p	NOUN
ejpam-5098	558	5	.	.	PUNCT
ejpam-5098	559	1	that	that	PRON
ejpam-5098	559	2	implies	imply	VERB
ejpam-5098	559	3	a	a	DET
ejpam-5098	559	4	/∈	/∈	PUNCT
ejpam-5098	559	5	i	i	NOUN
ejpam-5098	559	6	◦	◦	NOUN
ejpam-5098	559	7	.	.	PUNCT
ejpam-5098	560	1	therefore	therefore	ADV
ejpam-5098	560	2	,	,	PUNCT
ejpam-5098	560	3	i	i	PRON
ejpam-5098	560	4	◦	◦	VERB
ejpam-5098	560	5	⊆	⊆	NUM
ejpam-5098	560	6	i.	i.	NOUN
ejpam-5098	560	7	let	let	VERB
ejpam-5098	560	8	x	x	X
ejpam-5098	560	9	∈	∈	PROPN
ejpam-5098	560	10	i.	i.	NOUN
ejpam-5098	560	11	then	then	ADV
ejpam-5098	560	12	(	(	PUNCT
ejpam-5098	560	13	x)∗	x)∗	PROPN
ejpam-5098	560	14	∨	∨	PROPN
ejpam-5098	560	15	i	i	PRON
ejpam-5098	560	16	=	=	PUNCT
ejpam-5098	560	17	r.	r.	NOUN
ejpam-5098	560	18	since	since	SCONJ
ejpam-5098	560	19	i	i	PRON
ejpam-5098	560	20	is	be	AUX
ejpam-5098	560	21	a	a	DET
ejpam-5098	560	22	proper	proper	ADJ
ejpam-5098	560	23	σ	σ	NOUN
ejpam-5098	560	24	-	-	PUNCT
ejpam-5098	560	25	ideal	ideal	NOUN
ejpam-5098	560	26	of	of	ADP
ejpam-5098	560	27	r	r	NOUN
ejpam-5098	560	28	,	,	PUNCT
ejpam-5098	560	29	choose	choose	VERB
ejpam-5098	560	30	a	a	DET
ejpam-5098	560	31	maximal	maximal	ADJ
ejpam-5098	560	32	σ	σ	NOUN
ejpam-5098	560	33	-	-	PUNCT
ejpam-5098	560	34	ideal	ideal	NOUN
ejpam-5098	560	35	m	m	VERB
ejpam-5098	560	36	such	such	ADJ
ejpam-5098	560	37	that	that	SCONJ
ejpam-5098	560	38	i	i	PRON
ejpam-5098	560	39	⊆	⊆	NUM
ejpam-5098	560	40	m	m	NOUN
ejpam-5098	560	41	.	.	PUNCT
ejpam-5098	561	1	that	that	PRON
ejpam-5098	561	2	implies	imply	VERB
ejpam-5098	561	3	(	(	PUNCT
ejpam-5098	561	4	x)∗	x)∗	INTJ
ejpam-5098	561	5	∨m	∨m	NOUN
ejpam-5098	561	6	=	=	SYM
ejpam-5098	561	7	r	r	NOUN
ejpam-5098	561	8	and	and	CCONJ
ejpam-5098	561	9	m	m	PROPN
ejpam-5098	561	10	is	be	AUX
ejpam-5098	561	11	a	a	DET
ejpam-5098	561	12	prime	prime	ADJ
ejpam-5098	561	13	σ	σ	NOUN
ejpam-5098	561	14	-	-	PUNCT
ejpam-5098	561	15	ideal	ideal	NOUN
ejpam-5098	561	16	of	of	ADP
ejpam-5098	561	17	r.	r.	PROPN
ejpam-5098	561	18	that	that	PRON
ejpam-5098	561	19	implies	imply	VERB
ejpam-5098	561	20	x	x	PUNCT
ejpam-5098	561	21	∈	∈	NOUN
ejpam-5098	561	22	mσ	mσ	NOUN
ejpam-5098	561	23	=	=	NOUN
ejpam-5098	561	24	m	m	PROPN
ejpam-5098	561	25	and	and	CCONJ
ejpam-5098	561	26	i	i	PRON
ejpam-5098	561	27	⊆	⊆	NUM
ejpam-5098	561	28	m	m	NOUN
ejpam-5098	561	29	.	.	PUNCT
ejpam-5098	562	1	that	that	PRON
ejpam-5098	562	2	implies	imply	VERB
ejpam-5098	562	3	a	a	DET
ejpam-5098	562	4	∈	∈	PROPN
ejpam-5098	562	5	i	i	NOUN
ejpam-5098	562	6	◦	◦	NOUN
ejpam-5098	562	7	.	.	PUNCT
ejpam-5098	563	1	therefore	therefore	ADV
ejpam-5098	563	2	,	,	PUNCT
ejpam-5098	563	3	i	i	PRON
ejpam-5098	563	4	⊆	⊆	NUM
ejpam-5098	563	5	i	i	NOUN
ejpam-5098	563	6	◦	◦	NOUN
ejpam-5098	563	7	and	and	CCONJ
ejpam-5098	563	8	hence	hence	ADV
ejpam-5098	563	9	i	i	PRON
ejpam-5098	563	10	=	=	PUNCT
ejpam-5098	563	11	i	i	PROPN
ejpam-5098	563	12	◦	◦	NOUN
ejpam-5098	563	13	.	.	PUNCT
ejpam-5098	564	1	theorem	theorem	VERB
ejpam-5098	564	2	21	21	NUM
ejpam-5098	564	3	.	.	PUNCT
ejpam-5098	565	1	in	in	ADP
ejpam-5098	565	2	an	an	DET
ejpam-5098	565	3	adl	adl	PROPN
ejpam-5098	565	4	r	r	NOUN
ejpam-5098	565	5	,	,	PUNCT
ejpam-5098	565	6	the	the	DET
ejpam-5098	565	7	mapping	mapping	NOUN
ejpam-5098	565	8	i	i	PRON
ejpam-5098	565	9	→	→	SYM
ejpam-5098	565	10	d(i	d(i	PROPN
ejpam-5098	565	11	)	)	PUNCT
ejpam-5098	565	12	from	from	ADP
ejpam-5098	565	13	the	the	DET
ejpam-5098	565	14	set	set	NOUN
ejpam-5098	565	15	iσ(r	iσ(r	NOUN
ejpam-5098	565	16	)	)	PUNCT
ejpam-5098	565	17	of	of	ADP
ejpam-5098	565	18	all	all	DET
ejpam-5098	565	19	σ	σ	NOUN
ejpam-5098	565	20	-	-	PUNCT
ejpam-5098	565	21	ideals	ideal	NOUN
ejpam-5098	565	22	to	to	ADP
ejpam-5098	565	23	the	the	DET
ejpam-5098	565	24	set	set	NOUN
ejpam-5098	565	25	{	{	PUNCT
ejpam-5098	565	26	k(i	k(i	PROPN
ejpam-5098	565	27	)	)	PUNCT
ejpam-5098	566	1	|	|	ADV
ejpam-5098	566	2	i	i	PRON
ejpam-5098	566	3	∈	∈	PROPN
ejpam-5098	566	4	iσ(r	iσ(r	NOUN
ejpam-5098	566	5	)	)	PUNCT
ejpam-5098	566	6	}	}	PUNCT
ejpam-5098	566	7	is	be	AUX
ejpam-5098	566	8	an	an	DET
ejpam-5098	566	9	isomorphism	isomorphism	NOUN
ejpam-5098	566	10	.	.	PUNCT
ejpam-5098	567	1	proof	proof	NOUN
ejpam-5098	567	2	.	.	PUNCT
ejpam-5098	568	1	let	let	VERB
ejpam-5098	569	1	i	i	PRON
ejpam-5098	569	2	,	,	PUNCT
ejpam-5098	569	3	j	j	PROPN
ejpam-5098	569	4	∈	∈	PROPN
ejpam-5098	569	5	iσ(r	iσ(r	NOUN
ejpam-5098	569	6	)	)	PUNCT
ejpam-5098	570	1	such	such	ADJ
ejpam-5098	570	2	that	that	SCONJ
ejpam-5098	570	3	i	i	PRON
ejpam-5098	570	4	=	=	SYM
ejpam-5098	570	5	j	j	PROPN
ejpam-5098	570	6	.	.	PUNCT
ejpam-5098	571	1	clearly	clearly	ADV
ejpam-5098	571	2	,	,	PUNCT
ejpam-5098	571	3	we	we	PRON
ejpam-5098	571	4	have	have	VERB
ejpam-5098	571	5	that	that	PRON
ejpam-5098	571	6	h(i	h(i	NOUN
ejpam-5098	571	7	)	)	PUNCT
ejpam-5098	571	8	=	=	PUNCT
ejpam-5098	571	9	h(j	h(j	PROPN
ejpam-5098	571	10	)	)	PUNCT
ejpam-5098	571	11	.	.	PUNCT
ejpam-5098	572	1	assume	assume	VERB
ejpam-5098	572	2	that	that	SCONJ
ejpam-5098	572	3	h(i	h(i	NOUN
ejpam-5098	572	4	)	)	PUNCT
ejpam-5098	572	5	=	=	PUNCT
ejpam-5098	572	6	h(j	h(j	PROPN
ejpam-5098	572	7	)	)	PUNCT
ejpam-5098	572	8	.	.	PUNCT
ejpam-5098	573	1	we	we	PRON
ejpam-5098	573	2	prove	prove	VERB
ejpam-5098	573	3	that	that	SCONJ
ejpam-5098	573	4	i	i	PRON
ejpam-5098	573	5	=	=	SYM
ejpam-5098	573	6	j	j	PROPN
ejpam-5098	573	7	.	.	PUNCT
ejpam-5098	574	1	let	let	VERB
ejpam-5098	574	2	x	x	SYM
ejpam-5098	574	3	∈	∈	PROPN
ejpam-5098	574	4	i.	i.	NOUN
ejpam-5098	574	5	suppose	suppose	VERB
ejpam-5098	574	6	that	that	SCONJ
ejpam-5098	574	7	x	x	PROPN
ejpam-5098	574	8	/∈	/∈	PROPN
ejpam-5098	575	1	j	j	PROPN
ejpam-5098	575	2	.	.	PUNCT
ejpam-5098	576	1	then	then	ADV
ejpam-5098	576	2	there	there	PRON
ejpam-5098	576	3	exists	exist	VERB
ejpam-5098	576	4	a	a	DET
ejpam-5098	576	5	prime	prime	ADJ
ejpam-5098	576	6	σ	σ	NOUN
ejpam-5098	576	7	-	-	PUNCT
ejpam-5098	576	8	ideal	ideal	NOUN
ejpam-5098	576	9	p	p	NOUN
ejpam-5098	576	10	of	of	ADP
ejpam-5098	576	11	r	r	NOUN
ejpam-5098	576	12	such	such	ADJ
ejpam-5098	576	13	that	that	PRON
ejpam-5098	576	14	x	x	PROPN
ejpam-5098	576	15	/∈	/∈	PUNCT
ejpam-5098	577	1	p	p	NOUN
ejpam-5098	577	2	and	and	CCONJ
ejpam-5098	577	3	j	j	PROPN
ejpam-5098	578	1	⊆	⊆	NUM
ejpam-5098	578	2	p	p	NOUN
ejpam-5098	578	3	.	.	PUNCT
ejpam-5098	579	1	since	since	SCONJ
ejpam-5098	579	2	j	j	PROPN
ejpam-5098	579	3	⊆	⊆	NUM
ejpam-5098	579	4	p	p	NOUN
ejpam-5098	579	5	,	,	PUNCT
ejpam-5098	579	6	we	we	PRON
ejpam-5098	579	7	have	have	VERB
ejpam-5098	579	8	that	that	DET
ejpam-5098	579	9	p	p	NOUN
ejpam-5098	579	10	/∈	/∈	PUNCT
ejpam-5098	579	11	h(j	h(j	PROPN
ejpam-5098	579	12	)	)	PUNCT
ejpam-5098	580	1	=	=	SYM
ejpam-5098	580	2	h(i	h(i	NOUN
ejpam-5098	580	3	)	)	PUNCT
ejpam-5098	580	4	.	.	PUNCT
ejpam-5098	581	1	that	that	PRON
ejpam-5098	581	2	implies	imply	VERB
ejpam-5098	581	3	i	i	PRON
ejpam-5098	581	4	⊆	⊆	NUM
ejpam-5098	581	5	p	p	NOUN
ejpam-5098	581	6	.	.	PUNCT
ejpam-5098	582	1	since	since	SCONJ
ejpam-5098	582	2	x	x	PROPN
ejpam-5098	582	3	/∈	/∈	PROPN
ejpam-5098	582	4	p	p	NOUN
ejpam-5098	582	5	,	,	PUNCT
ejpam-5098	582	6	we	we	PRON
ejpam-5098	582	7	get	get	VERB
ejpam-5098	582	8	that	that	PRON
ejpam-5098	582	9	x	x	PUNCT
ejpam-5098	582	10	/∈	/∈	PUNCT
ejpam-5098	583	1	i	i	PRON
ejpam-5098	583	2	,	,	PUNCT
ejpam-5098	583	3	which	which	PRON
ejpam-5098	583	4	is	be	AUX
ejpam-5098	583	5	a	a	DET
ejpam-5098	583	6	contradiction	contradiction	NOUN
ejpam-5098	583	7	.	.	PUNCT
ejpam-5098	584	1	therefore	therefore	ADV
ejpam-5098	584	2	,	,	PUNCT
ejpam-5098	584	3	x	x	PROPN
ejpam-5098	584	4	∈	∈	PROPN
ejpam-5098	584	5	j	j	NOUN
ejpam-5098	584	6	and	and	CCONJ
ejpam-5098	584	7	hence	hence	ADV
ejpam-5098	584	8	i	i	PRON
ejpam-5098	584	9	⊆	⊆	NUM
ejpam-5098	584	10	j	j	PROPN
ejpam-5098	584	11	.	.	PUNCT
ejpam-5098	585	1	similarly	similarly	ADV
ejpam-5098	585	2	,	,	PUNCT
ejpam-5098	585	3	we	we	PRON
ejpam-5098	585	4	get	get	VERB
ejpam-5098	585	5	that	that	PRON
ejpam-5098	585	6	j	j	PROPN
ejpam-5098	585	7	⊆	⊆	NUM
ejpam-5098	585	8	i.	i.	NOUN
ejpam-5098	585	9	therefore	therefore	ADV
ejpam-5098	585	10	,	,	PUNCT
ejpam-5098	585	11	i	i	PRON
ejpam-5098	585	12	=	=	SYM
ejpam-5098	585	13	j	j	PROPN
ejpam-5098	585	14	and	and	CCONJ
ejpam-5098	585	15	hence	hence	ADV
ejpam-5098	585	16	this	this	DET
ejpam-5098	585	17	map	map	NOUN
ejpam-5098	585	18	is	be	AUX
ejpam-5098	585	19	one	one	NUM
ejpam-5098	585	20	-	-	PUNCT
ejpam-5098	585	21	one	one	NUM
ejpam-5098	585	22	.	.	PUNCT
ejpam-5098	586	1	clearly	clearly	ADV
ejpam-5098	586	2	,	,	PUNCT
ejpam-5098	586	3	it	it	PRON
ejpam-5098	586	4	is	be	AUX
ejpam-5098	586	5	onto	onto	ADP
ejpam-5098	586	6	and	and	CCONJ
ejpam-5098	586	7	homomorphism	homomorphism	NOUN
ejpam-5098	586	8	.	.	PUNCT
ejpam-5098	587	1	thus	thus	ADV
ejpam-5098	587	2	,	,	PUNCT
ejpam-5098	587	3	it	it	PRON
ejpam-5098	587	4	is	be	AUX
ejpam-5098	587	5	an	an	DET
ejpam-5098	587	6	isomorphism	isomorphism	NOUN
ejpam-5098	587	7	.	.	PUNCT
ejpam-5098	588	1	theorem	theorem	PROPN
ejpam-5098	588	2	22	22	NUM
ejpam-5098	588	3	.	.	PUNCT
ejpam-5098	589	1	let	let	VERB
ejpam-5098	589	2	m	m	PRON
ejpam-5098	589	3	be	be	AUX
ejpam-5098	589	4	a	a	DET
ejpam-5098	589	5	prime	prime	ADJ
ejpam-5098	589	6	σ	σ	NOUN
ejpam-5098	589	7	-	-	PUNCT
ejpam-5098	589	8	ideal	ideal	NOUN
ejpam-5098	589	9	of	of	ADP
ejpam-5098	589	10	a	a	DET
ejpam-5098	589	11	normal	normal	ADJ
ejpam-5098	589	12	adl	adl	NOUN
ejpam-5098	589	13	r	r	NOUN
ejpam-5098	589	14	with	with	ADP
ejpam-5098	589	15	maximal	maximal	ADJ
ejpam-5098	589	16	elements	element	NOUN
ejpam-5098	589	17	.	.	PUNCT
ejpam-5098	590	1	then	then	ADV
ejpam-5098	590	2	m	m	PROPN
ejpam-5098	590	3	is	be	AUX
ejpam-5098	590	4	a	a	DET
ejpam-5098	590	5	maximal	maximal	ADJ
ejpam-5098	590	6	σ	σ	NOUN
ejpam-5098	590	7	-	-	PUNCT
ejpam-5098	590	8	ideal	ideal	NOUN
ejpam-5098	590	9	of	of	ADP
ejpam-5098	590	10	r.	r.	PROPN
ejpam-5098	590	11	proof	proof	NOUN
ejpam-5098	590	12	.	.	PUNCT
ejpam-5098	591	1	suppose	suppose	VERB
ejpam-5098	591	2	there	there	PRON
ejpam-5098	591	3	is	be	VERB
ejpam-5098	591	4	a	a	DET
ejpam-5098	591	5	proper	proper	ADJ
ejpam-5098	591	6	σ	σ	NOUN
ejpam-5098	591	7	-	-	PUNCT
ejpam-5098	591	8	ideal	ideal	NOUN
ejpam-5098	591	9	q	q	NOUN
ejpam-5098	591	10	such	such	ADJ
ejpam-5098	591	11	that	that	SCONJ
ejpam-5098	591	12	m	m	PROPN
ejpam-5098	591	13	⊆	⊆	NUM
ejpam-5098	591	14	q.	q.	NOUN
ejpam-5098	591	15	now	now	ADV
ejpam-5098	591	16	,	,	PUNCT
ejpam-5098	591	17	we	we	PRON
ejpam-5098	591	18	prove	prove	VERB
ejpam-5098	591	19	that	that	SCONJ
ejpam-5098	591	20	q	q	PROPN
ejpam-5098	591	21	⊆	⊆	NUM
ejpam-5098	591	22	m	m	NOUN
ejpam-5098	591	23	.	.	PUNCT
ejpam-5098	592	1	suppose	suppose	VERB
ejpam-5098	592	2	that	that	SCONJ
ejpam-5098	592	3	q	q	X
ejpam-5098	592	4	⊈	⊈	NUM
ejpam-5098	592	5	m	m	NOUN
ejpam-5098	592	6	.	.	PUNCT
ejpam-5098	593	1	then	then	ADV
ejpam-5098	593	2	,	,	PUNCT
ejpam-5098	593	3	there	there	PRON
ejpam-5098	593	4	exists	exist	VERB
ejpam-5098	593	5	an	an	DET
ejpam-5098	593	6	element	element	NOUN
ejpam-5098	593	7	x	x	SYM
ejpam-5098	593	8	∈	∈	PROPN
ejpam-5098	593	9	q	q	NOUN
ejpam-5098	593	10	such	such	ADJ
ejpam-5098	593	11	that	that	PRON
ejpam-5098	593	12	x	x	X
ejpam-5098	593	13	/∈	/∈	INTJ
ejpam-5098	594	1	m	m	VERB
ejpam-5098	594	2	.	.	PUNCT
ejpam-5098	595	1	since	since	SCONJ
ejpam-5098	595	2	x	x	PROPN
ejpam-5098	595	3	∈	∈	PROPN
ejpam-5098	595	4	q	q	NOUN
ejpam-5098	595	5	,	,	PUNCT
ejpam-5098	595	6	we	we	PRON
ejpam-5098	595	7	have	have	VERB
ejpam-5098	595	8	that	that	DET
ejpam-5098	595	9	(	(	PUNCT
ejpam-5098	595	10	x)∗∨q	x)∗∨q	PROPN
ejpam-5098	595	11	=	=	PROPN
ejpam-5098	595	12	r.	r.	NOUN
ejpam-5098	595	13	that	that	PRON
ejpam-5098	595	14	implies	imply	VERB
ejpam-5098	595	15	there	there	PRON
ejpam-5098	595	16	exist	exist	VERB
ejpam-5098	595	17	a	a	DET
ejpam-5098	595	18	∈	∈	NOUN
ejpam-5098	595	19	(	(	PUNCT
ejpam-5098	595	20	x)∗	x)∗	PROPN
ejpam-5098	595	21	and	and	CCONJ
ejpam-5098	595	22	b	b	X
ejpam-5098	595	23	∈	∈	PRON
ejpam-5098	595	24	q	q	NOUN
ejpam-5098	595	25	such	such	ADJ
ejpam-5098	595	26	that	that	SCONJ
ejpam-5098	595	27	a	a	DET
ejpam-5098	595	28	∨	∨	PROPN
ejpam-5098	595	29	b	b	PROPN
ejpam-5098	595	30	is	be	AUX
ejpam-5098	595	31	a	a	DET
ejpam-5098	595	32	maximal	maximal	ADJ
ejpam-5098	595	33	element	element	NOUN
ejpam-5098	595	34	of	of	ADP
ejpam-5098	595	35	r.	r.	PROPN
ejpam-5098	595	36	since	since	SCONJ
ejpam-5098	595	37	a	a	DET
ejpam-5098	595	38	∈	∈	PROPN
ejpam-5098	595	39	(	(	PUNCT
ejpam-5098	595	40	x)∗	x)∗	PROPN
ejpam-5098	595	41	,	,	PUNCT
ejpam-5098	595	42	we	we	PRON
ejpam-5098	595	43	have	have	AUX
ejpam-5098	595	44	that	that	PRON
ejpam-5098	595	45	a	a	DET
ejpam-5098	595	46	∧	∧	PROPN
ejpam-5098	595	47	x	x	PUNCT
ejpam-5098	596	1	=	=	NOUN
ejpam-5098	596	2	0	0	X
ejpam-5098	596	3	.	.	PUNCT
ejpam-5098	597	1	since	since	SCONJ
ejpam-5098	597	2	r	r	NOUN
ejpam-5098	597	3	is	be	AUX
ejpam-5098	597	4	normal	normal	ADJ
ejpam-5098	597	5	,	,	PUNCT
ejpam-5098	597	6	there	there	PRON
ejpam-5098	597	7	exist	exist	VERB
ejpam-5098	597	8	elements	element	NOUN
ejpam-5098	597	9	c	c	NOUN
ejpam-5098	597	10	,	,	PUNCT
ejpam-5098	597	11	d	d	PROPN
ejpam-5098	597	12	∈	∈	NOUN
ejpam-5098	597	13	r	r	NOUN
ejpam-5098	597	14	such	such	DET
ejpam-5098	597	15	that	that	DET
ejpam-5098	597	16	a∧	a∧	NOUN
ejpam-5098	597	17	c	c	NOUN
ejpam-5098	597	18	=	=	SYM
ejpam-5098	597	19	0	0	PROPN
ejpam-5098	597	20	,	,	PUNCT
ejpam-5098	597	21	x∧	x∧	PROPN
ejpam-5098	598	1	d	d	NOUN
ejpam-5098	598	2	=	=	SYM
ejpam-5098	598	3	0	0	PROPN
ejpam-5098	598	4	and	and	CCONJ
ejpam-5098	598	5	c∨	c∨	PROPN
ejpam-5098	598	6	d	d	PROPN
ejpam-5098	598	7	is	be	AUX
ejpam-5098	598	8	a	a	DET
ejpam-5098	598	9	maximal	maximal	ADJ
ejpam-5098	598	10	r.	r.	NOUN
ejpam-5098	598	11	noorbhasha	noorbhasha	PROPN
ejpam-5098	598	12	,	,	PUNCT
ejpam-5098	598	13	r.	r.	PROPN
ejpam-5098	598	14	bandaru	bandaru	PROPN
ejpam-5098	598	15	,	,	PUNCT
ejpam-5098	598	16	a.	a.	NOUN
ejpam-5098	598	17	iampan	iampan	PROPN
ejpam-5098	598	18	/	/	SYM
ejpam-5098	598	19	eur	eur	PROPN
ejpam-5098	598	20	.	.	PUNCT
ejpam-5098	599	1	j.	j.	PROPN
ejpam-5098	599	2	pure	pure	PROPN
ejpam-5098	599	3	appl	appl	PROPN
ejpam-5098	599	4	.	.	PROPN
ejpam-5098	599	5	math	math	PROPN
ejpam-5098	599	6	,	,	PUNCT
ejpam-5098	599	7	17	17	NUM
ejpam-5098	599	8	(	(	PUNCT
ejpam-5098	599	9	2	2	NUM
ejpam-5098	599	10	)	)	PUNCT
ejpam-5098	599	11	(	(	PUNCT
ejpam-5098	599	12	2024	2024	NUM
ejpam-5098	599	13	)	)	PUNCT
ejpam-5098	599	14	,	,	PUNCT
ejpam-5098	599	15	1094	1094	NUM
ejpam-5098	599	16	-	-	SYM
ejpam-5098	599	17	1112	1112	NUM
ejpam-5098	599	18	1110	1110	NUM
ejpam-5098	599	19	element	element	NOUN
ejpam-5098	599	20	.	.	PUNCT
ejpam-5098	600	1	since	since	SCONJ
ejpam-5098	600	2	c	c	PROPN
ejpam-5098	600	3	∨	∨	PROPN
ejpam-5098	600	4	d	d	PROPN
ejpam-5098	600	5	is	be	AUX
ejpam-5098	600	6	maximal	maximal	ADJ
ejpam-5098	600	7	,	,	PUNCT
ejpam-5098	600	8	we	we	PRON
ejpam-5098	600	9	have	have	VERB
ejpam-5098	600	10	that	that	PRON
ejpam-5098	600	11	(	(	PUNCT
ejpam-5098	600	12	c)∗	c)∗	PROPN
ejpam-5098	600	13	∩	∩	NOUN
ejpam-5098	600	14	(	(	PUNCT
ejpam-5098	600	15	d)∗	d)∗	PROPN
ejpam-5098	600	16	=	=	SYM
ejpam-5098	600	17	(	(	PUNCT
ejpam-5098	600	18	0	0	NUM
ejpam-5098	600	19	]	]	X
ejpam-5098	600	20	⊆	⊆	NUM
ejpam-5098	600	21	m	m	NOUN
ejpam-5098	600	22	.	.	PUNCT
ejpam-5098	601	1	since	since	SCONJ
ejpam-5098	601	2	a	a	DET
ejpam-5098	601	3	∈	∈	NOUN
ejpam-5098	601	4	(	(	PUNCT
ejpam-5098	601	5	c)∗	c)∗	PROPN
ejpam-5098	601	6	and	and	CCONJ
ejpam-5098	601	7	a	a	DET
ejpam-5098	601	8	/∈	/∈	NOUN
ejpam-5098	601	9	q	q	NOUN
ejpam-5098	601	10	,	,	PUNCT
ejpam-5098	601	11	we	we	PRON
ejpam-5098	601	12	have	have	VERB
ejpam-5098	601	13	that	that	PRON
ejpam-5098	601	14	(	(	PUNCT
ejpam-5098	601	15	c)∗	c)∗	VERB
ejpam-5098	601	16	⊈	⊈	PROPN
ejpam-5098	601	17	q	q	X
ejpam-5098	601	18	and	and	CCONJ
ejpam-5098	601	19	hence	hence	ADV
ejpam-5098	601	20	(	(	PUNCT
ejpam-5098	601	21	c)∗	c)∗	VERB
ejpam-5098	601	22	⊈	⊈	PROPN
ejpam-5098	601	23	m	m	VERB
ejpam-5098	601	24	.	.	PUNCT
ejpam-5098	602	1	since	since	SCONJ
ejpam-5098	602	2	m	m	PROPN
ejpam-5098	602	3	is	be	AUX
ejpam-5098	602	4	a	a	DET
ejpam-5098	602	5	prime	prime	ADJ
ejpam-5098	602	6	σ	σ	NOUN
ejpam-5098	602	7	-	-	PUNCT
ejpam-5098	602	8	ideal	ideal	NOUN
ejpam-5098	602	9	of	of	ADP
ejpam-5098	602	10	r	r	NOUN
ejpam-5098	602	11	,	,	PUNCT
ejpam-5098	602	12	(	(	PUNCT
ejpam-5098	602	13	c)∗	c)∗	PROPN
ejpam-5098	602	14	∩	∩	NOUN
ejpam-5098	602	15	(	(	PUNCT
ejpam-5098	602	16	d)∗	d)∗	PROPN
ejpam-5098	602	17	⊆	⊆	NUM
ejpam-5098	602	18	m	m	PRON
ejpam-5098	602	19	and	and	CCONJ
ejpam-5098	602	20	(	(	PUNCT
ejpam-5098	602	21	c)∗	c)∗	VERB
ejpam-5098	602	22	⊈	⊈	PROPN
ejpam-5098	602	23	m	m	VERB
ejpam-5098	602	24	,	,	PUNCT
ejpam-5098	602	25	we	we	PRON
ejpam-5098	602	26	get	get	VERB
ejpam-5098	602	27	that	that	PRON
ejpam-5098	602	28	(	(	PUNCT
ejpam-5098	602	29	d)∗	d)∗	PROPN
ejpam-5098	602	30	⊆	⊆	NUM
ejpam-5098	602	31	m	m	NOUN
ejpam-5098	602	32	.	.	PUNCT
ejpam-5098	603	1	since	since	SCONJ
ejpam-5098	603	2	d	d	PROPN
ejpam-5098	603	3	∧	∧	PROPN
ejpam-5098	603	4	x	x	PUNCT
ejpam-5098	603	5	=	=	SYM
ejpam-5098	603	6	0	0	NUM
ejpam-5098	603	7	,	,	PUNCT
ejpam-5098	603	8	we	we	PRON
ejpam-5098	603	9	have	have	VERB
ejpam-5098	603	10	that	that	PRON
ejpam-5098	603	11	x	x	SYM
ejpam-5098	603	12	∈	∈	PROPN
ejpam-5098	603	13	(	(	PUNCT
ejpam-5098	603	14	d)∗	d)∗	PROPN
ejpam-5098	603	15	⊆	⊆	NUM
ejpam-5098	603	16	m	m	NOUN
ejpam-5098	603	17	.	.	PUNCT
ejpam-5098	604	1	that	that	PRON
ejpam-5098	604	2	implies	imply	VERB
ejpam-5098	604	3	x	x	PUNCT
ejpam-5098	604	4	∈	∈	NOUN
ejpam-5098	604	5	m	m	PROPN
ejpam-5098	604	6	,	,	PUNCT
ejpam-5098	604	7	which	which	PRON
ejpam-5098	604	8	contradicts	contradict	VERB
ejpam-5098	604	9	x	x	X
ejpam-5098	604	10	/∈	/∈	INTJ
ejpam-5098	604	11	m	m	VERB
ejpam-5098	604	12	.	.	PUNCT
ejpam-5098	605	1	therefore	therefore	ADV
ejpam-5098	605	2	,	,	PUNCT
ejpam-5098	605	3	q	q	PROPN
ejpam-5098	605	4	⊆	⊆	NUM
ejpam-5098	605	5	m	m	NOUN
ejpam-5098	605	6	and	and	CCONJ
ejpam-5098	605	7	hence	hence	ADV
ejpam-5098	605	8	m	m	VERB
ejpam-5098	605	9	=	=	SYM
ejpam-5098	605	10	q.	q.	PROPN
ejpam-5098	605	11	thus	thus	ADV
ejpam-5098	605	12	,	,	PUNCT
ejpam-5098	605	13	m	m	VERB
ejpam-5098	605	14	is	be	AUX
ejpam-5098	605	15	a	a	DET
ejpam-5098	605	16	maximal	maximal	ADJ
ejpam-5098	605	17	σ	σ	NOUN
ejpam-5098	605	18	-	-	PUNCT
ejpam-5098	605	19	ideal	ideal	NOUN
ejpam-5098	605	20	of	of	ADP
ejpam-5098	605	21	r.	r.	PROPN
ejpam-5098	605	22	theorem	theorem	VERB
ejpam-5098	605	23	23	23	NUM
ejpam-5098	605	24	.	.	PUNCT
ejpam-5098	606	1	let	let	VERB
ejpam-5098	606	2	r	r	PRON
ejpam-5098	606	3	be	be	AUX
ejpam-5098	606	4	a	a	DET
ejpam-5098	606	5	normal	normal	ADJ
ejpam-5098	606	6	adl	adl	NOUN
ejpam-5098	606	7	with	with	ADP
ejpam-5098	606	8	maximal	maximal	ADJ
ejpam-5098	606	9	elements	element	NOUN
ejpam-5098	606	10	and	and	CCONJ
ejpam-5098	606	11	p	p	X
ejpam-5098	606	12	a	a	DET
ejpam-5098	606	13	minimal	minimal	ADJ
ejpam-5098	606	14	prime	prime	ADJ
ejpam-5098	606	15	ideal	ideal	NOUN
ejpam-5098	606	16	of	of	ADP
ejpam-5098	606	17	r.	r.	PROPN
ejpam-5098	606	18	for	for	ADP
ejpam-5098	606	19	every	every	DET
ejpam-5098	606	20	maximal	maximal	ADJ
ejpam-5098	606	21	ideal	ideal	NOUN
ejpam-5098	606	22	m	m	AUX
ejpam-5098	606	23	containing	contain	VERB
ejpam-5098	606	24	p	p	NOUN
ejpam-5098	606	25	,	,	PUNCT
ejpam-5098	606	26	p	p	X
ejpam-5098	606	27	=	=	PUNCT
ejpam-5098	606	28	mo	mo	PROPN
ejpam-5098	606	29	and	and	CCONJ
ejpam-5098	606	30	p	p	NOUN
ejpam-5098	606	31	is	be	AUX
ejpam-5098	606	32	a	a	DET
ejpam-5098	606	33	prime	prime	ADJ
ejpam-5098	606	34	σ	σ	NOUN
ejpam-5098	606	35	-	-	PUNCT
ejpam-5098	606	36	ideal	ideal	NOUN
ejpam-5098	606	37	of	of	ADP
ejpam-5098	606	38	r.	r.	PROPN
ejpam-5098	606	39	proof	proof	NOUN
ejpam-5098	606	40	.	.	PUNCT
ejpam-5098	607	1	let	let	VERB
ejpam-5098	607	2	p	p	PRON
ejpam-5098	607	3	be	be	AUX
ejpam-5098	607	4	a	a	DET
ejpam-5098	607	5	minimal	minimal	ADJ
ejpam-5098	607	6	prime	prime	ADJ
ejpam-5098	607	7	ideal	ideal	NOUN
ejpam-5098	607	8	of	of	ADP
ejpam-5098	607	9	r	r	NOUN
ejpam-5098	607	10	and	and	CCONJ
ejpam-5098	607	11	m	m	VERB
ejpam-5098	607	12	be	be	AUX
ejpam-5098	607	13	a	a	DET
ejpam-5098	607	14	maximal	maximal	ADJ
ejpam-5098	607	15	ideal	ideal	NOUN
ejpam-5098	607	16	with	with	ADP
ejpam-5098	607	17	p	p	NOUN
ejpam-5098	607	18	⊆m	⊆m	NOUN
ejpam-5098	607	19	.	.	PUNCT
ejpam-5098	608	1	we	we	PRON
ejpam-5098	608	2	prove	prove	VERB
ejpam-5098	608	3	that	that	SCONJ
ejpam-5098	608	4	mo	mo	PROPN
ejpam-5098	608	5	=	=	PROPN
ejpam-5098	608	6	p	p	PROPN
ejpam-5098	608	7	.	.	PUNCT
ejpam-5098	609	1	let	let	VERB
ejpam-5098	609	2	x	x	PUNCT
ejpam-5098	609	3	∈	∈	PROPN
ejpam-5098	609	4	p	p	NOUN
ejpam-5098	609	5	.	.	PUNCT
ejpam-5098	610	1	then	then	ADV
ejpam-5098	610	2	there	there	PRON
ejpam-5098	610	3	exists	exist	VERB
ejpam-5098	610	4	an	an	DET
ejpam-5098	610	5	element	element	NOUN
ejpam-5098	610	6	y	y	PROPN
ejpam-5098	610	7	/∈	/∈	PUNCT
ejpam-5098	611	1	p	p	X
ejpam-5098	611	2	such	such	ADJ
ejpam-5098	611	3	that	that	SCONJ
ejpam-5098	611	4	x	x	PUNCT
ejpam-5098	611	5	∧	∧	NOUN
ejpam-5098	611	6	y	y	NOUN
ejpam-5098	611	7	=	=	NOUN
ejpam-5098	611	8	0	0	PROPN
ejpam-5098	611	9	.	.	PUNCT
ejpam-5098	612	1	since	since	SCONJ
ejpam-5098	612	2	r	r	NOUN
ejpam-5098	612	3	is	be	AUX
ejpam-5098	612	4	normal	normal	ADJ
ejpam-5098	612	5	,	,	PUNCT
ejpam-5098	612	6	there	there	PRON
ejpam-5098	612	7	exist	exist	VERB
ejpam-5098	612	8	elements	element	NOUN
ejpam-5098	612	9	a	a	PRON
ejpam-5098	612	10	,	,	PUNCT
ejpam-5098	612	11	b	b	X
ejpam-5098	612	12	∈	∈	NOUN
ejpam-5098	612	13	r	r	NOUN
ejpam-5098	612	14	such	such	ADJ
ejpam-5098	612	15	that	that	SCONJ
ejpam-5098	612	16	x	x	SYM
ejpam-5098	612	17	∧	∧	NOUN
ejpam-5098	612	18	a	a	PRON
ejpam-5098	612	19	=	=	SYM
ejpam-5098	612	20	0	0	NUM
ejpam-5098	612	21	,	,	PUNCT
ejpam-5098	612	22	y	y	PROPN
ejpam-5098	612	23	∧	∧	PROPN
ejpam-5098	612	24	b	b	PROPN
ejpam-5098	612	25	=	=	SYM
ejpam-5098	612	26	0	0	PROPN
ejpam-5098	612	27	and	and	CCONJ
ejpam-5098	612	28	a	a	DET
ejpam-5098	612	29	∨	∨	NOUN
ejpam-5098	612	30	b	b	NOUN
ejpam-5098	612	31	is	be	AUX
ejpam-5098	612	32	maximal	maximal	ADJ
ejpam-5098	612	33	.	.	PUNCT
ejpam-5098	613	1	since	since	SCONJ
ejpam-5098	613	2	y	y	PROPN
ejpam-5098	613	3	/∈	/∈	PROPN
ejpam-5098	613	4	p	p	NOUN
ejpam-5098	613	5	,	,	PUNCT
ejpam-5098	613	6	we	we	PRON
ejpam-5098	613	7	get	get	VERB
ejpam-5098	613	8	that	that	DET
ejpam-5098	613	9	b	b	NOUN
ejpam-5098	613	10	∈	∈	PROPN
ejpam-5098	613	11	p	p	NOUN
ejpam-5098	613	12	.	.	PUNCT
ejpam-5098	614	1	since	since	SCONJ
ejpam-5098	614	2	p	p	PRON
ejpam-5098	614	3	⊆m	⊆m	NOUN
ejpam-5098	614	4	,	,	PUNCT
ejpam-5098	614	5	we	we	PRON
ejpam-5098	614	6	get	get	VERB
ejpam-5098	614	7	that	that	PRON
ejpam-5098	614	8	a	a	DET
ejpam-5098	614	9	/∈m	/∈m	PUNCT
ejpam-5098	614	10	.	.	PUNCT
ejpam-5098	615	1	since	since	SCONJ
ejpam-5098	615	2	a	a	DET
ejpam-5098	615	3	/∈	/∈	NOUN
ejpam-5098	615	4	m	m	VERB
ejpam-5098	615	5	and	and	CCONJ
ejpam-5098	615	6	x	x	PUNCT
ejpam-5098	615	7	∧	∧	NOUN
ejpam-5098	615	8	a	a	DET
ejpam-5098	615	9	=	=	SYM
ejpam-5098	615	10	0	0	NUM
ejpam-5098	615	11	,	,	PUNCT
ejpam-5098	615	12	we	we	PRON
ejpam-5098	615	13	get	get	VERB
ejpam-5098	615	14	that	that	PRON
ejpam-5098	615	15	x	x	PROPN
ejpam-5098	615	16	∈	∈	PROPN
ejpam-5098	615	17	mo	mo	PROPN
ejpam-5098	615	18	.	.	PROPN
ejpam-5098	616	1	therefore	therefore	ADV
ejpam-5098	616	2	,	,	PUNCT
ejpam-5098	616	3	p	p	PROPN
ejpam-5098	616	4	⊆	⊆	NUM
ejpam-5098	616	5	mo	mo	PROPN
ejpam-5098	616	6	.	.	PROPN
ejpam-5098	616	7	let	let	VERB
ejpam-5098	616	8	x	x	SYM
ejpam-5098	616	9	∈	∈	PROPN
ejpam-5098	616	10	mo	mo	PROPN
ejpam-5098	616	11	.	.	PROPN
ejpam-5098	617	1	then	then	ADV
ejpam-5098	617	2	there	there	PRON
ejpam-5098	617	3	exists	exist	VERB
ejpam-5098	617	4	an	an	DET
ejpam-5098	617	5	element	element	NOUN
ejpam-5098	617	6	y	y	NOUN
ejpam-5098	617	7	/∈	/∈	PUNCT
ejpam-5098	618	1	m	m	VERB
ejpam-5098	618	2	such	such	ADJ
ejpam-5098	618	3	that	that	SCONJ
ejpam-5098	618	4	x	x	PUNCT
ejpam-5098	618	5	∧	∧	NOUN
ejpam-5098	618	6	y	y	NOUN
ejpam-5098	618	7	=	=	NOUN
ejpam-5098	618	8	0	0	PROPN
ejpam-5098	618	9	.	.	PUNCT
ejpam-5098	619	1	since	since	SCONJ
ejpam-5098	619	2	p	p	PROPN
ejpam-5098	619	3	⊆	⊆	NUM
ejpam-5098	619	4	m	m	NOUN
ejpam-5098	619	5	,	,	PUNCT
ejpam-5098	619	6	we	we	PRON
ejpam-5098	619	7	get	get	VERB
ejpam-5098	619	8	that	that	PRON
ejpam-5098	619	9	y	y	PROPN
ejpam-5098	619	10	/∈	/∈	PUNCT
ejpam-5098	620	1	p	p	X
ejpam-5098	620	2	.	.	PUNCT
ejpam-5098	621	1	since	since	SCONJ
ejpam-5098	621	2	x	x	PROPN
ejpam-5098	621	3	∧	∧	NOUN
ejpam-5098	621	4	y	y	PROPN
ejpam-5098	621	5	=	=	SYM
ejpam-5098	621	6	0	0	PROPN
ejpam-5098	621	7	,	,	PUNCT
ejpam-5098	621	8	we	we	PRON
ejpam-5098	621	9	get	get	VERB
ejpam-5098	621	10	that	that	PRON
ejpam-5098	621	11	x	x	PROPN
ejpam-5098	621	12	∈	∈	PROPN
ejpam-5098	621	13	p	p	NOUN
ejpam-5098	621	14	.	.	PUNCT
ejpam-5098	622	1	therefore	therefore	ADV
ejpam-5098	622	2	,	,	PUNCT
ejpam-5098	622	3	mo	mo	PROPN
ejpam-5098	622	4	⊆	⊆	NUM
ejpam-5098	622	5	p	p	NOUN
ejpam-5098	622	6	.	.	PUNCT
ejpam-5098	623	1	hence	hence	ADV
ejpam-5098	623	2	,	,	PUNCT
ejpam-5098	623	3	p	p	PROPN
ejpam-5098	623	4	=	=	PROPN
ejpam-5098	623	5	mo	mo	PROPN
ejpam-5098	623	6	.	.	PROPN
ejpam-5098	624	1	we	we	PRON
ejpam-5098	624	2	prove	prove	VERB
ejpam-5098	624	3	that	that	SCONJ
ejpam-5098	624	4	p	p	NOUN
ejpam-5098	624	5	is	be	AUX
ejpam-5098	624	6	a	a	DET
ejpam-5098	624	7	prime	prime	ADJ
ejpam-5098	624	8	σ	σ	NOUN
ejpam-5098	624	9	-	-	PUNCT
ejpam-5098	624	10	ideal	ideal	NOUN
ejpam-5098	624	11	of	of	ADP
ejpam-5098	624	12	r.	r.	PROPN
ejpam-5098	624	13	it	it	PRON
ejpam-5098	624	14	is	be	AUX
ejpam-5098	624	15	enough	enough	ADJ
ejpam-5098	624	16	to	to	PART
ejpam-5098	624	17	prove	prove	VERB
ejpam-5098	624	18	that	that	SCONJ
ejpam-5098	624	19	p	p	NOUN
ejpam-5098	624	20	is	be	AUX
ejpam-5098	624	21	a	a	DET
ejpam-5098	624	22	maximal	maximal	ADJ
ejpam-5098	624	23	σ	σ	NOUN
ejpam-5098	624	24	-	-	PUNCT
ejpam-5098	624	25	ideal	ideal	NOUN
ejpam-5098	624	26	of	of	ADP
ejpam-5098	624	27	r.	r.	PROPN
ejpam-5098	624	28	suppose	suppose	VERB
ejpam-5098	624	29	q	q	X
ejpam-5098	624	30	is	be	AUX
ejpam-5098	624	31	a	a	DET
ejpam-5098	624	32	proper	proper	ADJ
ejpam-5098	624	33	σ	σ	NOUN
ejpam-5098	624	34	-	-	PUNCT
ejpam-5098	624	35	ideal	ideal	NOUN
ejpam-5098	624	36	of	of	ADP
ejpam-5098	624	37	r	r	NOUN
ejpam-5098	624	38	such	such	ADJ
ejpam-5098	624	39	that	that	SCONJ
ejpam-5098	624	40	p	p	PROPN
ejpam-5098	624	41	⊆	⊆	NUM
ejpam-5098	624	42	q.	q.	NOUN
ejpam-5098	624	43	let	let	VERB
ejpam-5098	624	44	x	x	SYM
ejpam-5098	624	45	∈	∈	PROPN
ejpam-5098	624	46	q.	q.	NOUN
ejpam-5098	624	47	then	then	ADV
ejpam-5098	624	48	(	(	PUNCT
ejpam-5098	624	49	x)∗	x)∗	PROPN
ejpam-5098	624	50	∨q	∨q	PROPN
ejpam-5098	624	51	=	=	PROPN
ejpam-5098	624	52	r.	r.	PROPN
ejpam-5098	624	53	that	that	PRON
ejpam-5098	624	54	implies	imply	VERB
ejpam-5098	624	55	there	there	PRON
ejpam-5098	624	56	exist	exist	VERB
ejpam-5098	624	57	elements	element	NOUN
ejpam-5098	624	58	a	a	DET
ejpam-5098	624	59	∈	∈	NOUN
ejpam-5098	624	60	(	(	PUNCT
ejpam-5098	624	61	x)∗	x)∗	PROPN
ejpam-5098	624	62	and	and	CCONJ
ejpam-5098	624	63	b	b	X
ejpam-5098	624	64	∈	∈	PRON
ejpam-5098	624	65	q	q	NOUN
ejpam-5098	624	66	such	such	ADJ
ejpam-5098	624	67	that	that	SCONJ
ejpam-5098	624	68	a∨b	a∨b	NOUN
ejpam-5098	624	69	is	be	AUX
ejpam-5098	624	70	a	a	DET
ejpam-5098	624	71	maximal	maximal	ADJ
ejpam-5098	624	72	element	element	NOUN
ejpam-5098	624	73	.	.	PUNCT
ejpam-5098	625	1	that	that	PRON
ejpam-5098	625	2	implies	imply	VERB
ejpam-5098	625	3	a	a	DET
ejpam-5098	625	4	/∈	/∈	ADJ
ejpam-5098	625	5	q	q	NOUN
ejpam-5098	625	6	and	and	CCONJ
ejpam-5098	625	7	hence	hence	ADV
ejpam-5098	625	8	a	a	PRON
ejpam-5098	625	9	/∈	/∈	INTJ
ejpam-5098	625	10	p	p	NOUN
ejpam-5098	625	11	.	.	PUNCT
ejpam-5098	626	1	since	since	SCONJ
ejpam-5098	626	2	a	a	DET
ejpam-5098	626	3	∧	∧	NOUN
ejpam-5098	626	4	x	x	PUNCT
ejpam-5098	626	5	=	=	SYM
ejpam-5098	626	6	0	0	NUM
ejpam-5098	626	7	,	,	PUNCT
ejpam-5098	626	8	we	we	PRON
ejpam-5098	626	9	get	get	VERB
ejpam-5098	626	10	that	that	PRON
ejpam-5098	626	11	x	x	PROPN
ejpam-5098	626	12	∈	∈	PROPN
ejpam-5098	626	13	p	p	NOUN
ejpam-5098	626	14	.	.	PUNCT
ejpam-5098	627	1	that	that	PRON
ejpam-5098	627	2	implies	imply	VERB
ejpam-5098	627	3	q	q	NOUN
ejpam-5098	627	4	⊆	⊆	NUM
ejpam-5098	627	5	p	p	NOUN
ejpam-5098	627	6	.	.	PUNCT
ejpam-5098	628	1	therefore	therefore	ADV
ejpam-5098	628	2	,	,	PUNCT
ejpam-5098	628	3	q	q	X
ejpam-5098	628	4	=	=	PUNCT
ejpam-5098	628	5	p	p	NOUN
ejpam-5098	628	6	and	and	CCONJ
ejpam-5098	628	7	hence	hence	ADV
ejpam-5098	628	8	p	p	PRON
ejpam-5098	628	9	is	be	AUX
ejpam-5098	628	10	a	a	DET
ejpam-5098	628	11	maximal	maximal	ADJ
ejpam-5098	628	12	σ	σ	NOUN
ejpam-5098	628	13	-	-	PUNCT
ejpam-5098	628	14	ideal	ideal	NOUN
ejpam-5098	628	15	of	of	ADP
ejpam-5098	628	16	r.	r.	PROPN
ejpam-5098	628	17	thus	thus	ADV
ejpam-5098	628	18	,	,	PUNCT
ejpam-5098	628	19	p	p	PROPN
ejpam-5098	628	20	is	be	AUX
ejpam-5098	628	21	a	a	DET
ejpam-5098	628	22	prime	prime	ADJ
ejpam-5098	628	23	σ	σ	NOUN
ejpam-5098	628	24	-	-	PUNCT
ejpam-5098	628	25	ideal	ideal	NOUN
ejpam-5098	628	26	of	of	ADP
ejpam-5098	628	27	r.	r.	PROPN
ejpam-5098	628	28	theorem	theorem	PROPN
ejpam-5098	628	29	24	24	NUM
ejpam-5098	628	30	.	.	PUNCT
ejpam-5098	629	1	let	let	VERB
ejpam-5098	629	2	r	r	PRON
ejpam-5098	629	3	be	be	AUX
ejpam-5098	629	4	a	a	DET
ejpam-5098	629	5	normal	normal	ADJ
ejpam-5098	629	6	adl	adl	NOUN
ejpam-5098	629	7	and	and	CCONJ
ejpam-5098	629	8	minr	minr	NOUN
ejpam-5098	629	9	be	be	AUX
ejpam-5098	629	10	the	the	DET
ejpam-5098	629	11	set	set	NOUN
ejpam-5098	629	12	of	of	ADP
ejpam-5098	629	13	all	all	DET
ejpam-5098	629	14	minimal	minimal	ADJ
ejpam-5098	629	15	prime	prime	ADJ
ejpam-5098	629	16	ideals	ideal	NOUN
ejpam-5098	629	17	of	of	ADP
ejpam-5098	629	18	r.	r.	PROPN
ejpam-5098	629	19	then	then	ADV
ejpam-5098	629	20	there	there	PRON
ejpam-5098	629	21	is	be	VERB
ejpam-5098	629	22	a	a	DET
ejpam-5098	629	23	continuous	continuous	ADJ
ejpam-5098	629	24	bijection	bijection	NOUN
ejpam-5098	629	25	ψ	ψ	NOUN
ejpam-5098	629	26	:	:	PUNCT
ejpam-5098	629	27	minr→	minr→	PROPN
ejpam-5098	629	28	specσr	specσr	NOUN
ejpam-5098	629	29	defined	define	VERB
ejpam-5098	629	30	by	by	ADP
ejpam-5098	629	31	ψ(p	ψ(p	NOUN
ejpam-5098	629	32	)	)	PUNCT
ejpam-5098	630	1	=	=	SYM
ejpam-5098	630	2	p	p	NOUN
ejpam-5098	630	3	.	.	PUNCT
ejpam-5098	631	1	proof	proof	NOUN
ejpam-5098	631	2	.	.	PUNCT
ejpam-5098	632	1	clearly	clearly	ADV
ejpam-5098	632	2	,	,	PUNCT
ejpam-5098	632	3	ψ	ψ	X
ejpam-5098	632	4	is	be	AUX
ejpam-5098	632	5	a	a	DET
ejpam-5098	632	6	bijection	bijection	NOUN
ejpam-5098	632	7	.	.	PUNCT
ejpam-5098	633	1	now	now	ADV
ejpam-5098	633	2	,	,	PUNCT
ejpam-5098	633	3	we	we	PRON
ejpam-5098	633	4	prove	prove	VERB
ejpam-5098	633	5	that	that	SCONJ
ejpam-5098	633	6	ψ	ψ	NOUN
ejpam-5098	633	7	is	be	AUX
ejpam-5098	633	8	continuous	continuous	ADJ
ejpam-5098	633	9	.	.	PUNCT
ejpam-5098	634	1	let	let	VERB
ejpam-5098	634	2	h(a	h(a	PROPN
ejpam-5098	634	3	)	)	PUNCT
ejpam-5098	634	4	be	be	AUX
ejpam-5098	634	5	open	open	ADJ
ejpam-5098	634	6	in	in	ADP
ejpam-5098	634	7	specσ(r	specσ(r	NOUN
ejpam-5098	634	8	)	)	PUNCT
ejpam-5098	634	9	.	.	PUNCT
ejpam-5098	635	1	we	we	PRON
ejpam-5098	635	2	prove	prove	VERB
ejpam-5098	635	3	that	that	SCONJ
ejpam-5098	635	4	ψ−1(h(a	ψ−1(h(a	NOUN
ejpam-5098	635	5	)	)	PUNCT
ejpam-5098	635	6	)	)	PUNCT
ejpam-5098	635	7	is	be	AUX
ejpam-5098	635	8	open	open	ADJ
ejpam-5098	635	9	in	in	ADP
ejpam-5098	635	10	minr	minr	PROPN
ejpam-5098	635	11	.	.	PUNCT
ejpam-5098	636	1	now	now	ADV
ejpam-5098	636	2	,	,	PUNCT
ejpam-5098	636	3	ψ−1(h(a	ψ−1(h(a	PROPN
ejpam-5098	636	4	)	)	PUNCT
ejpam-5098	636	5	)	)	PUNCT
ejpam-5098	637	1	=	=	PRON
ejpam-5098	637	2	{	{	PUNCT
ejpam-5098	637	3	m	m	PROPN
ejpam-5098	637	4	∈	∈	PROPN
ejpam-5098	637	5	minr	minr	NOUN
ejpam-5098	637	6	|	|	ADP
ejpam-5098	637	7	ψ(m	ψ(m	NOUN
ejpam-5098	637	8	)	)	PUNCT
ejpam-5098	638	1	∈	∈	PROPN
ejpam-5098	638	2	h(a	h(a	PROPN
ejpam-5098	638	3	)	)	PUNCT
ejpam-5098	638	4	}	}	PUNCT
ejpam-5098	639	1	=	=	SYM
ejpam-5098	639	2	{	{	PUNCT
ejpam-5098	639	3	m	m	PROPN
ejpam-5098	639	4	∈	∈	PROPN
ejpam-5098	639	5	minr	minr	NOUN
ejpam-5098	640	1	|	|	ADP
ejpam-5098	640	2	m	m	PROPN
ejpam-5098	640	3	∈	∈	PROPN
ejpam-5098	640	4	h(a	h(a	PROPN
ejpam-5098	640	5	)	)	PUNCT
ejpam-5098	640	6	}	}	PUNCT
ejpam-5098	641	1	=	=	SYM
ejpam-5098	641	2	{	{	PUNCT
ejpam-5098	641	3	m	m	PROPN
ejpam-5098	641	4	∈	∈	PROPN
ejpam-5098	641	5	minr	minr	NOUN
ejpam-5098	641	6	|	|	ADV
ejpam-5098	641	7	a	a	DET
ejpam-5098	641	8	/∈	/∈	NOUN
ejpam-5098	641	9	m	m	VERB
ejpam-5098	641	10	}	}	PUNCT
ejpam-5098	641	11	=	=	SYM
ejpam-5098	641	12	h(a	h(a	PROPN
ejpam-5098	641	13	)	)	PUNCT
ejpam-5098	641	14	∩	∩	ADJ
ejpam-5098	641	15	minr	minr	PROPN
ejpam-5098	641	16	.	.	PUNCT
ejpam-5098	642	1	therefore	therefore	ADV
ejpam-5098	642	2	,	,	PUNCT
ejpam-5098	642	3	ψ−1(h(a	ψ−1(h(a	PROPN
ejpam-5098	642	4	)	)	PUNCT
ejpam-5098	642	5	)	)	PUNCT
ejpam-5098	643	1	is	be	AUX
ejpam-5098	643	2	open	open	ADJ
ejpam-5098	643	3	in	in	ADP
ejpam-5098	643	4	minr	minr	PROPN
ejpam-5098	643	5	.	.	PUNCT
ejpam-5098	644	1	hence	hence	ADV
ejpam-5098	644	2	,	,	PUNCT
ejpam-5098	644	3	ψ	ψ	NOUN
ejpam-5098	644	4	is	be	AUX
ejpam-5098	644	5	continuous	continuous	ADJ
ejpam-5098	644	6	.	.	PUNCT
ejpam-5098	645	1	theorem	theorem	ADJ
ejpam-5098	645	2	25	25	NUM
ejpam-5098	645	3	.	.	PUNCT
ejpam-5098	646	1	an	an	DET
ejpam-5098	646	2	adl	adl	PROPN
ejpam-5098	646	3	r	r	NOUN
ejpam-5098	646	4	is	be	AUX
ejpam-5098	646	5	stone	stone	NOUN
ejpam-5098	646	6	if	if	SCONJ
ejpam-5098	646	7	and	and	CCONJ
ejpam-5098	646	8	only	only	ADV
ejpam-5098	646	9	if	if	SCONJ
ejpam-5098	646	10	r	r	NOUN
ejpam-5098	646	11	is	be	AUX
ejpam-5098	646	12	normal	normal	ADJ
ejpam-5098	646	13	and	and	CCONJ
ejpam-5098	646	14	minr	minr	NOUN
ejpam-5098	646	15	is	be	AUX
ejpam-5098	646	16	compact	compact	ADJ
ejpam-5098	646	17	.	.	PUNCT
ejpam-5098	647	1	theorem	theorem	NOUN
ejpam-5098	647	2	26	26	NUM
ejpam-5098	647	3	.	.	PUNCT
ejpam-5098	648	1	let	let	VERB
ejpam-5098	648	2	r	r	PRON
ejpam-5098	648	3	be	be	AUX
ejpam-5098	648	4	a	a	DET
ejpam-5098	648	5	normal	normal	ADJ
ejpam-5098	648	6	adl	adl	NOUN
ejpam-5098	648	7	.	.	PUNCT
ejpam-5098	649	1	then	then	ADV
ejpam-5098	649	2	ψ	ψ	X
ejpam-5098	649	3	:	:	PUNCT
ejpam-5098	649	4	minr	minr	PROPN
ejpam-5098	649	5	→	→	SYM
ejpam-5098	649	6	specσr	specσr	PROPN
ejpam-5098	649	7	defined	define	VERB
ejpam-5098	649	8	above	above	ADV
ejpam-5098	649	9	is	be	AUX
ejpam-5098	649	10	a	a	DET
ejpam-5098	649	11	homeomorphism	homeomorphism	PROPN
ejpam-5098	649	12	if	if	SCONJ
ejpam-5098	650	1	and	and	CCONJ
ejpam-5098	650	2	only	only	ADV
ejpam-5098	650	3	if	if	SCONJ
ejpam-5098	650	4	r	r	NOUN
ejpam-5098	650	5	is	be	AUX
ejpam-5098	650	6	a	a	DET
ejpam-5098	650	7	stone	stone	NOUN
ejpam-5098	650	8	adl	adl	PROPN
ejpam-5098	650	9	.	.	PUNCT
ejpam-5098	651	1	definition	definition	NOUN
ejpam-5098	651	2	16	16	NUM
ejpam-5098	651	3	.	.	PUNCT
ejpam-5098	652	1	[	[	X
ejpam-5098	652	2	8	8	NUM
ejpam-5098	652	3	]	]	PUNCT
ejpam-5098	652	4	let	let	VERB
ejpam-5098	652	5	r	r	PRON
ejpam-5098	652	6	be	be	AUX
ejpam-5098	652	7	an	an	DET
ejpam-5098	652	8	adl	adl	NOUN
ejpam-5098	652	9	with	with	ADP
ejpam-5098	652	10	maximal	maximal	ADJ
ejpam-5098	652	11	elements	element	NOUN
ejpam-5098	652	12	and	and	CCONJ
ejpam-5098	652	13	b	b	ADP
ejpam-5098	652	14	a	a	DET
ejpam-5098	652	15	birkhoff	birkhoff	NOUN
ejpam-5098	652	16	center	center	NOUN
ejpam-5098	652	17	of	of	ADP
ejpam-5098	652	18	r.	r.	PROPN
ejpam-5098	652	19	an	an	DET
ejpam-5098	652	20	ideal	ideal	NOUN
ejpam-5098	652	21	i	i	PRON
ejpam-5098	652	22	of	of	ADP
ejpam-5098	652	23	r	r	NOUN
ejpam-5098	652	24	is	be	AUX
ejpam-5098	652	25	said	say	VERB
ejpam-5098	652	26	to	to	PART
ejpam-5098	652	27	be	be	AUX
ejpam-5098	652	28	a	a	DET
ejpam-5098	652	29	b	b	NOUN
ejpam-5098	652	30	-	-	PUNCT
ejpam-5098	652	31	ideal	ideal	NOUN
ejpam-5098	652	32	of	of	ADP
ejpam-5098	652	33	r	r	NOUN
ejpam-5098	652	34	if	if	SCONJ
ejpam-5098	652	35	for	for	ADP
ejpam-5098	652	36	any	any	DET
ejpam-5098	652	37	x	x	SYM
ejpam-5098	652	38	∈	∈	PROPN
ejpam-5098	652	39	i	i	PRON
ejpam-5098	652	40	there	there	PRON
ejpam-5098	652	41	exists	exist	VERB
ejpam-5098	652	42	an	an	DET
ejpam-5098	652	43	element	element	NOUN
ejpam-5098	652	44	e	e	NOUN
ejpam-5098	652	45	∈	∈	PROPN
ejpam-5098	652	46	i	i	PRON
ejpam-5098	652	47	∩b	∩b	VERB
ejpam-5098	652	48	such	such	ADJ
ejpam-5098	652	49	that	that	SCONJ
ejpam-5098	652	50	e	e	X
ejpam-5098	652	51	∧	∧	NOUN
ejpam-5098	652	52	x	x	X
ejpam-5098	652	53	=	=	PUNCT
ejpam-5098	652	54	x.	x.	NOUN
ejpam-5098	652	55	lemma	lemma	PROPN
ejpam-5098	652	56	11	11	NUM
ejpam-5098	652	57	.	.	PUNCT
ejpam-5098	653	1	let	let	VERB
ejpam-5098	653	2	r	r	PRON
ejpam-5098	653	3	be	be	AUX
ejpam-5098	653	4	an	an	DET
ejpam-5098	653	5	adl	adl	NOUN
ejpam-5098	653	6	with	with	ADP
ejpam-5098	653	7	maximal	maximal	ADJ
ejpam-5098	653	8	elements	element	NOUN
ejpam-5098	653	9	.	.	PUNCT
ejpam-5098	654	1	then	then	ADV
ejpam-5098	654	2	every	every	DET
ejpam-5098	654	3	b	b	NOUN
ejpam-5098	654	4	-	-	PUNCT
ejpam-5098	654	5	ideal	ideal	NOUN
ejpam-5098	654	6	of	of	ADP
ejpam-5098	654	7	r	r	NOUN
ejpam-5098	654	8	is	be	AUX
ejpam-5098	654	9	a	a	DET
ejpam-5098	654	10	σ	σ	NOUN
ejpam-5098	654	11	-	-	PUNCT
ejpam-5098	654	12	ideal	ideal	NOUN
ejpam-5098	654	13	of	of	ADP
ejpam-5098	654	14	r.	r.	PROPN
ejpam-5098	654	15	r.	r.	PROPN
ejpam-5098	654	16	noorbhasha	noorbhasha	PROPN
ejpam-5098	654	17	,	,	PUNCT
ejpam-5098	654	18	r.	r.	PROPN
ejpam-5098	654	19	bandaru	bandaru	PROPN
ejpam-5098	654	20	,	,	PUNCT
ejpam-5098	654	21	a.	a.	NOUN
ejpam-5098	654	22	iampan	iampan	PROPN
ejpam-5098	654	23	/	/	SYM
ejpam-5098	654	24	eur	eur	PROPN
ejpam-5098	654	25	.	.	PUNCT
ejpam-5098	655	1	j.	j.	PROPN
ejpam-5098	655	2	pure	pure	PROPN
ejpam-5098	655	3	appl	appl	PROPN
ejpam-5098	655	4	.	.	PROPN
ejpam-5098	655	5	math	math	PROPN
ejpam-5098	655	6	,	,	PUNCT
ejpam-5098	655	7	17	17	NUM
ejpam-5098	655	8	(	(	PUNCT
ejpam-5098	655	9	2	2	NUM
ejpam-5098	655	10	)	)	PUNCT
ejpam-5098	655	11	(	(	PUNCT
ejpam-5098	655	12	2024	2024	NUM
ejpam-5098	655	13	)	)	PUNCT
ejpam-5098	655	14	,	,	PUNCT
ejpam-5098	655	15	1094	1094	NUM
ejpam-5098	655	16	-	-	SYM
ejpam-5098	655	17	1112	1112	NUM
ejpam-5098	655	18	1111	1111	NUM
ejpam-5098	655	19	proof	proof	NOUN
ejpam-5098	655	20	.	.	PUNCT
ejpam-5098	656	1	let	let	VERB
ejpam-5098	656	2	i	i	PRON
ejpam-5098	656	3	be	be	AUX
ejpam-5098	656	4	any	any	DET
ejpam-5098	656	5	b	b	NOUN
ejpam-5098	656	6	-	-	PUNCT
ejpam-5098	656	7	ideal	ideal	NOUN
ejpam-5098	656	8	of	of	ADP
ejpam-5098	656	9	r.	r.	PROPN
ejpam-5098	656	10	we	we	PRON
ejpam-5098	656	11	prove	prove	VERB
ejpam-5098	656	12	that	that	SCONJ
ejpam-5098	656	13	i	i	PRON
ejpam-5098	656	14	is	be	AUX
ejpam-5098	656	15	a	a	DET
ejpam-5098	656	16	σ	σ	NOUN
ejpam-5098	656	17	-	-	PUNCT
ejpam-5098	656	18	ideal	ideal	NOUN
ejpam-5098	656	19	of	of	ADP
ejpam-5098	656	20	r.	r.	PROPN
ejpam-5098	656	21	clearly	clearly	ADV
ejpam-5098	656	22	,	,	PUNCT
ejpam-5098	656	23	we	we	PRON
ejpam-5098	656	24	have	have	VERB
ejpam-5098	656	25	that	that	PRON
ejpam-5098	656	26	iσ	iσ	VERB
ejpam-5098	656	27	⊆	⊆	NUM
ejpam-5098	656	28	i.	i.	NOUN
ejpam-5098	656	29	let	let	VERB
ejpam-5098	656	30	x	x	X
ejpam-5098	656	31	∈	∈	PROPN
ejpam-5098	656	32	i.	i.	NOUN
ejpam-5098	656	33	since	since	SCONJ
ejpam-5098	656	34	i	i	PRON
ejpam-5098	656	35	is	be	AUX
ejpam-5098	656	36	a	a	DET
ejpam-5098	656	37	b	b	NOUN
ejpam-5098	656	38	-	-	PUNCT
ejpam-5098	656	39	ideal	ideal	NOUN
ejpam-5098	656	40	of	of	ADP
ejpam-5098	656	41	r	r	NOUN
ejpam-5098	656	42	,	,	PUNCT
ejpam-5098	656	43	there	there	PRON
ejpam-5098	656	44	exists	exist	VERB
ejpam-5098	656	45	an	an	DET
ejpam-5098	656	46	element	element	NOUN
ejpam-5098	656	47	b	b	PROPN
ejpam-5098	656	48	∈	∈	PROPN
ejpam-5098	656	49	i	i	PRON
ejpam-5098	656	50	∩	∩	NOUN
ejpam-5098	657	1	b	b	X
ejpam-5098	657	2	such	such	ADJ
ejpam-5098	657	3	that	that	DET
ejpam-5098	657	4	b∧x	b∧x	NOUN
ejpam-5098	657	5	=	=	PUNCT
ejpam-5098	657	6	x.	x.	NOUN
ejpam-5098	657	7	since	since	SCONJ
ejpam-5098	657	8	b	b	PROPN
ejpam-5098	657	9	is	be	AUX
ejpam-5098	657	10	a	a	DET
ejpam-5098	657	11	birkhoff	birkhoff	NOUN
ejpam-5098	657	12	center	center	NOUN
ejpam-5098	657	13	and	and	CCONJ
ejpam-5098	657	14	b	b	X
ejpam-5098	657	15	∈	∈	PROPN
ejpam-5098	657	16	b	b	NOUN
ejpam-5098	657	17	,	,	PUNCT
ejpam-5098	657	18	there	there	PRON
ejpam-5098	657	19	exists	exist	VERB
ejpam-5098	657	20	an	an	DET
ejpam-5098	657	21	element	element	NOUN
ejpam-5098	657	22	c	c	PROPN
ejpam-5098	657	23	∈	∈	NOUN
ejpam-5098	657	24	r	r	NOUN
ejpam-5098	657	25	such	such	ADJ
ejpam-5098	657	26	that	that	DET
ejpam-5098	657	27	b∧c	b∧c	NOUN
ejpam-5098	658	1	=	=	SYM
ejpam-5098	658	2	0	0	PROPN
ejpam-5098	658	3	and	and	CCONJ
ejpam-5098	658	4	b∨c	b∨c	PROPN
ejpam-5098	658	5	is	be	AUX
ejpam-5098	658	6	maximal	maximal	ADJ
ejpam-5098	658	7	.	.	PUNCT
ejpam-5098	659	1	now	now	ADV
ejpam-5098	659	2	,	,	PUNCT
ejpam-5098	659	3	x∧c	x∧c	X
ejpam-5098	659	4	=	=	SYM
ejpam-5098	659	5	b∧x∧c	b∧x∧c	NOUN
ejpam-5098	659	6	=	=	SYM
ejpam-5098	659	7	0	0	NUM
ejpam-5098	659	8	.	.	PUNCT
ejpam-5098	660	1	that	that	PRON
ejpam-5098	660	2	implies	imply	VERB
ejpam-5098	660	3	c	c	PROPN
ejpam-5098	660	4	∈	∈	PROPN
ejpam-5098	660	5	(	(	PUNCT
ejpam-5098	660	6	x)∗.	x)∗.	PROPN
ejpam-5098	660	7	since	since	SCONJ
ejpam-5098	660	8	b	b	PROPN
ejpam-5098	660	9	∈	∈	PROPN
ejpam-5098	660	10	i	i	PRON
ejpam-5098	660	11	and	and	CCONJ
ejpam-5098	660	12	c	c	PROPN
ejpam-5098	660	13	∨	∨	PROPN
ejpam-5098	660	14	b	b	PROPN
ejpam-5098	660	15	is	be	AUX
ejpam-5098	660	16	maximal	maximal	ADJ
ejpam-5098	660	17	,	,	PUNCT
ejpam-5098	660	18	we	we	PRON
ejpam-5098	660	19	get	get	VERB
ejpam-5098	660	20	that	that	DET
ejpam-5098	660	21	(	(	PUNCT
ejpam-5098	660	22	x)∗	x)∗	PROPN
ejpam-5098	660	23	∨	∨	NOUN
ejpam-5098	660	24	i	i	PRON
ejpam-5098	660	25	=	=	PUNCT
ejpam-5098	660	26	r.	r.	NOUN
ejpam-5098	660	27	that	that	PRON
ejpam-5098	660	28	implies	imply	VERB
ejpam-5098	660	29	x	x	X
ejpam-5098	660	30	∈	∈	NOUN
ejpam-5098	660	31	iσ	iσ	VERB
ejpam-5098	660	32	.	.	PUNCT
ejpam-5098	661	1	therefore	therefore	ADV
ejpam-5098	661	2	,	,	PUNCT
ejpam-5098	661	3	i	i	PRON
ejpam-5098	661	4	⊆	⊆	NUM
ejpam-5098	661	5	iσ	iσ	ADV
ejpam-5098	661	6	and	and	CCONJ
ejpam-5098	661	7	hence	hence	ADV
ejpam-5098	661	8	iσ	iσ	VERB
ejpam-5098	661	9	=	=	PUNCT
ejpam-5098	661	10	i.	i.	PROPN
ejpam-5098	661	11	thus	thus	ADV
ejpam-5098	661	12	,	,	PUNCT
ejpam-5098	661	13	i	i	PRON
ejpam-5098	661	14	is	be	AUX
ejpam-5098	661	15	a	a	DET
ejpam-5098	661	16	σ	σ	NOUN
ejpam-5098	661	17	-	-	PUNCT
ejpam-5098	661	18	ideal	ideal	NOUN
ejpam-5098	661	19	of	of	ADP
ejpam-5098	661	20	r.	r.	PROPN
ejpam-5098	661	21	definition	definition	NOUN
ejpam-5098	661	22	17	17	NUM
ejpam-5098	661	23	.	.	PUNCT
ejpam-5098	662	1	an	an	DET
ejpam-5098	662	2	adl	adl	PROPN
ejpam-5098	662	3	r	r	NOUN
ejpam-5098	662	4	is	be	AUX
ejpam-5098	662	5	said	say	VERB
ejpam-5098	662	6	to	to	PART
ejpam-5098	662	7	be	be	AUX
ejpam-5098	662	8	strongly	strongly	ADV
ejpam-5098	662	9	normal	normal	ADJ
ejpam-5098	662	10	if	if	SCONJ
ejpam-5098	662	11	for	for	ADP
ejpam-5098	662	12	any	any	DET
ejpam-5098	662	13	a	a	DET
ejpam-5098	662	14	∈	∈	PROPN
ejpam-5098	662	15	r	r	NOUN
ejpam-5098	662	16	,	,	PUNCT
ejpam-5098	662	17	(	(	PUNCT
ejpam-5098	662	18	a)∗	a)∗	PROPN
ejpam-5098	662	19	is	be	AUX
ejpam-5098	662	20	a	a	DET
ejpam-5098	662	21	b	b	NOUN
ejpam-5098	662	22	-	-	PUNCT
ejpam-5098	662	23	ideal	ideal	NOUN
ejpam-5098	662	24	of	of	ADP
ejpam-5098	662	25	r.	r.	PROPN
ejpam-5098	662	26	theorem	theorem	VERB
ejpam-5098	662	27	27	27	NUM
ejpam-5098	662	28	.	.	PUNCT
ejpam-5098	663	1	an	an	DET
ejpam-5098	663	2	adl	adl	PROPN
ejpam-5098	663	3	r	r	NOUN
ejpam-5098	663	4	with	with	ADP
ejpam-5098	663	5	maximal	maximal	ADJ
ejpam-5098	663	6	elements	element	NOUN
ejpam-5098	663	7	is	be	AUX
ejpam-5098	663	8	strongly	strongly	ADV
ejpam-5098	663	9	normal	normal	ADJ
ejpam-5098	663	10	if	if	SCONJ
ejpam-5098	663	11	and	and	CCONJ
ejpam-5098	663	12	only	only	ADV
ejpam-5098	663	13	if	if	SCONJ
ejpam-5098	663	14	every	every	DET
ejpam-5098	663	15	σ	σ	NOUN
ejpam-5098	663	16	-	-	PUNCT
ejpam-5098	663	17	ideal	ideal	NOUN
ejpam-5098	663	18	of	of	ADP
ejpam-5098	663	19	r	r	NOUN
ejpam-5098	663	20	is	be	AUX
ejpam-5098	663	21	a	a	DET
ejpam-5098	663	22	b	b	NOUN
ejpam-5098	663	23	-	-	PUNCT
ejpam-5098	663	24	ideal	ideal	NOUN
ejpam-5098	663	25	.	.	PUNCT
ejpam-5098	664	1	proof	proof	NOUN
ejpam-5098	664	2	.	.	PUNCT
ejpam-5098	665	1	assume	assume	VERB
ejpam-5098	665	2	that	that	SCONJ
ejpam-5098	665	3	r	r	NOUN
ejpam-5098	665	4	is	be	AUX
ejpam-5098	665	5	strongly	strongly	ADV
ejpam-5098	665	6	normal	normal	ADJ
ejpam-5098	665	7	.	.	PUNCT
ejpam-5098	666	1	let	let	VERB
ejpam-5098	666	2	i	i	PRON
ejpam-5098	666	3	be	be	AUX
ejpam-5098	666	4	any	any	DET
ejpam-5098	666	5	σ	σ	NOUN
ejpam-5098	666	6	-	-	PUNCT
ejpam-5098	666	7	ideal	ideal	NOUN
ejpam-5098	666	8	of	of	ADP
ejpam-5098	666	9	r.	r.	PROPN
ejpam-5098	666	10	let	let	VERB
ejpam-5098	666	11	x	x	X
ejpam-5098	666	12	∈	∈	PROPN
ejpam-5098	666	13	i.	i.	NOUN
ejpam-5098	667	1	then	then	ADV
ejpam-5098	667	2	i	i	PRON
ejpam-5098	667	3	∨	∨	X
ejpam-5098	667	4	(	(	PUNCT
ejpam-5098	667	5	x)∗	x)∗	PROPN
ejpam-5098	667	6	=	=	NOUN
ejpam-5098	667	7	r.	r.	NOUN
ejpam-5098	667	8	that	that	PRON
ejpam-5098	667	9	implies	imply	VERB
ejpam-5098	667	10	there	there	PRON
ejpam-5098	667	11	exist	exist	VERB
ejpam-5098	667	12	elements	element	NOUN
ejpam-5098	667	13	i	i	PRON
ejpam-5098	667	14	∈	∈	VERB
ejpam-5098	668	1	i	i	PRON
ejpam-5098	668	2	and	and	CCONJ
ejpam-5098	668	3	y	y	PROPN
ejpam-5098	668	4	∈	∈	PROPN
ejpam-5098	668	5	(	(	PUNCT
ejpam-5098	668	6	x)∗	x)∗	INTJ
ejpam-5098	668	7	such	such	ADJ
ejpam-5098	668	8	that	that	SCONJ
ejpam-5098	668	9	i	i	PRON
ejpam-5098	668	10	∨	∨	NOUN
ejpam-5098	668	11	y	y	PROPN
ejpam-5098	668	12	is	be	AUX
ejpam-5098	668	13	maximal	maximal	ADJ
ejpam-5098	668	14	.	.	PUNCT
ejpam-5098	669	1	that	that	PRON
ejpam-5098	669	2	implies	imply	VERB
ejpam-5098	669	3	y	y	PROPN
ejpam-5098	669	4	∧	∧	PROPN
ejpam-5098	669	5	x	x	PUNCT
ejpam-5098	670	1	=	=	SYM
ejpam-5098	670	2	0	0	PUNCT
ejpam-5098	671	1	and	and	CCONJ
ejpam-5098	671	2	i	i	PRON
ejpam-5098	671	3	∨	∨	PROPN
ejpam-5098	671	4	y	y	PROPN
ejpam-5098	671	5	is	be	AUX
ejpam-5098	671	6	maximal	maximal	ADJ
ejpam-5098	671	7	.	.	PUNCT
ejpam-5098	672	1	by	by	ADP
ejpam-5098	672	2	our	our	PRON
ejpam-5098	672	3	assumption	assumption	NOUN
ejpam-5098	672	4	,	,	PUNCT
ejpam-5098	672	5	we	we	PRON
ejpam-5098	672	6	have	have	VERB
ejpam-5098	672	7	that	that	PRON
ejpam-5098	672	8	(	(	PUNCT
ejpam-5098	672	9	x)∗	x)∗	PROPN
ejpam-5098	672	10	is	be	AUX
ejpam-5098	672	11	a	a	DET
ejpam-5098	672	12	b	b	NOUN
ejpam-5098	672	13	-	-	PUNCT
ejpam-5098	672	14	ideal	ideal	NOUN
ejpam-5098	672	15	of	of	ADP
ejpam-5098	672	16	r.	r.	PROPN
ejpam-5098	672	17	since	since	SCONJ
ejpam-5098	672	18	y	y	PROPN
ejpam-5098	672	19	∈	∈	PROPN
ejpam-5098	672	20	(	(	PUNCT
ejpam-5098	672	21	x)∗	x)∗	PROPN
ejpam-5098	672	22	,	,	PUNCT
ejpam-5098	672	23	there	there	PRON
ejpam-5098	672	24	exists	exist	VERB
ejpam-5098	672	25	an	an	DET
ejpam-5098	672	26	element	element	NOUN
ejpam-5098	672	27	b	b	PROPN
ejpam-5098	672	28	∈	∈	PROPN
ejpam-5098	672	29	(	(	PUNCT
ejpam-5098	672	30	x)∗	x)∗	PROPN
ejpam-5098	672	31	∩	∩	PROPN
ejpam-5098	672	32	b	b	X
ejpam-5098	672	33	such	such	ADJ
ejpam-5098	672	34	that	that	DET
ejpam-5098	672	35	b	b	PROPN
ejpam-5098	672	36	∧	∧	NOUN
ejpam-5098	672	37	y	y	PROPN
ejpam-5098	672	38	=	=	SYM
ejpam-5098	672	39	y.	y.	PROPN
ejpam-5098	672	40	since	since	SCONJ
ejpam-5098	672	41	b	b	PROPN
ejpam-5098	672	42	∈	∈	PROPN
ejpam-5098	672	43	b	b	NOUN
ejpam-5098	672	44	,	,	PUNCT
ejpam-5098	672	45	there	there	PRON
ejpam-5098	672	46	exists	exist	VERB
ejpam-5098	672	47	an	an	DET
ejpam-5098	672	48	element	element	NOUN
ejpam-5098	672	49	c	c	PROPN
ejpam-5098	672	50	∈	∈	NOUN
ejpam-5098	672	51	r	r	NOUN
ejpam-5098	672	52	such	such	DET
ejpam-5098	672	53	that	that	DET
ejpam-5098	672	54	b	b	PROPN
ejpam-5098	672	55	∧	∧	NOUN
ejpam-5098	672	56	c	c	NOUN
ejpam-5098	672	57	=	=	SYM
ejpam-5098	672	58	0	0	NUM
ejpam-5098	672	59	and	and	CCONJ
ejpam-5098	672	60	b	b	NUM
ejpam-5098	672	61	∨	∨	NOUN
ejpam-5098	672	62	c	c	PROPN
ejpam-5098	672	63	is	be	AUX
ejpam-5098	672	64	maximal	maximal	ADJ
ejpam-5098	672	65	.	.	PUNCT
ejpam-5098	673	1	since	since	SCONJ
ejpam-5098	673	2	b	b	PROPN
ejpam-5098	673	3	∧	∧	PROPN
ejpam-5098	673	4	c	c	NOUN
ejpam-5098	673	5	=	=	SYM
ejpam-5098	673	6	0	0	NUM
ejpam-5098	673	7	,	,	PUNCT
ejpam-5098	673	8	we	we	PRON
ejpam-5098	673	9	have	have	VERB
ejpam-5098	673	10	that	that	PRON
ejpam-5098	673	11	y	y	PROPN
ejpam-5098	673	12	∧	∧	PROPN
ejpam-5098	673	13	c	c	PROPN
ejpam-5098	673	14	=	=	SYM
ejpam-5098	673	15	0	0	X
ejpam-5098	673	16	.	.	PUNCT
ejpam-5098	674	1	now	now	ADV
ejpam-5098	674	2	,	,	PUNCT
ejpam-5098	674	3	i	i	PRON
ejpam-5098	674	4	∧	∧	NOUN
ejpam-5098	674	5	c	c	NOUN
ejpam-5098	674	6	=	=	PUNCT
ejpam-5098	674	7	(	(	PUNCT
ejpam-5098	674	8	i	i	PRON
ejpam-5098	674	9	∧	∧	PROPN
ejpam-5098	674	10	c	c	NOUN
ejpam-5098	674	11	)	)	PUNCT
ejpam-5098	674	12	∨	∨	NOUN
ejpam-5098	674	13	0	0	NUM
ejpam-5098	675	1	=	=	SYM
ejpam-5098	675	2	(	(	PUNCT
ejpam-5098	675	3	i	i	PRON
ejpam-5098	675	4	∧	∧	PROPN
ejpam-5098	675	5	c	c	NOUN
ejpam-5098	675	6	)	)	PUNCT
ejpam-5098	675	7	∨	∨	PROPN
ejpam-5098	675	8	(	(	PUNCT
ejpam-5098	675	9	y	y	PROPN
ejpam-5098	675	10	∧	∧	PROPN
ejpam-5098	675	11	c	c	NOUN
ejpam-5098	675	12	)	)	PUNCT
ejpam-5098	675	13	=	=	PUNCT
ejpam-5098	675	14	(	(	PUNCT
ejpam-5098	675	15	i	i	PROPN
ejpam-5098	675	16	∨	∨	PROPN
ejpam-5098	675	17	y	y	NOUN
ejpam-5098	675	18	)	)	PUNCT
ejpam-5098	675	19	∧	∧	PROPN
ejpam-5098	675	20	c	c	NOUN
ejpam-5098	675	21	=	=	SYM
ejpam-5098	675	22	c	c	NOUN
ejpam-5098	675	23	,	,	PUNCT
ejpam-5098	675	24	since	since	SCONJ
ejpam-5098	675	25	i	i	PRON
ejpam-5098	675	26	∨	∨	NOUN
ejpam-5098	675	27	y	y	PROPN
ejpam-5098	675	28	is	be	AUX
ejpam-5098	675	29	maximal	maximal	ADJ
ejpam-5098	675	30	.	.	PUNCT
ejpam-5098	676	1	since	since	SCONJ
ejpam-5098	676	2	i	i	PRON
ejpam-5098	676	3	∈	∈	PROPN
ejpam-5098	676	4	i	i	PRON
ejpam-5098	676	5	,	,	PUNCT
ejpam-5098	676	6	we	we	PRON
ejpam-5098	676	7	get	get	VERB
ejpam-5098	676	8	that	that	DET
ejpam-5098	676	9	c	c	PROPN
ejpam-5098	676	10	∈	∈	PROPN
ejpam-5098	676	11	i.	i.	NOUN
ejpam-5098	676	12	now	now	ADV
ejpam-5098	676	13	,	,	PUNCT
ejpam-5098	676	14	c	c	PROPN
ejpam-5098	676	15	∧	∧	NOUN
ejpam-5098	676	16	x	x	PUNCT
ejpam-5098	677	1	=	=	SYM
ejpam-5098	677	2	0	0	NUM
ejpam-5098	677	3	∨	∨	NOUN
ejpam-5098	677	4	(	(	PUNCT
ejpam-5098	677	5	c	c	PROPN
ejpam-5098	677	6	∧	∧	PROPN
ejpam-5098	677	7	x	x	NOUN
ejpam-5098	677	8	)	)	PUNCT
ejpam-5098	677	9	=	=	SYM
ejpam-5098	677	10	(	(	PUNCT
ejpam-5098	677	11	b	b	X
ejpam-5098	677	12	∧	∧	PROPN
ejpam-5098	677	13	x	x	NOUN
ejpam-5098	677	14	)	)	PUNCT
ejpam-5098	677	15	∨	∨	NUM
ejpam-5098	677	16	(	(	PUNCT
ejpam-5098	677	17	c	c	PROPN
ejpam-5098	677	18	∧	∧	PROPN
ejpam-5098	677	19	x	x	NOUN
ejpam-5098	677	20	)	)	PUNCT
ejpam-5098	677	21	=	=	SYM
ejpam-5098	677	22	(	(	PUNCT
ejpam-5098	677	23	b	b	PROPN
ejpam-5098	677	24	∨	∨	NUM
ejpam-5098	677	25	c	c	NOUN
ejpam-5098	677	26	)	)	PUNCT
ejpam-5098	677	27	∧	∧	NOUN
ejpam-5098	677	28	x	x	X
ejpam-5098	677	29	=	=	SYM
ejpam-5098	677	30	x	x	NOUN
ejpam-5098	677	31	,	,	PUNCT
ejpam-5098	677	32	since	since	SCONJ
ejpam-5098	677	33	b	b	NOUN
ejpam-5098	677	34	∨	∨	PROPN
ejpam-5098	677	35	c	c	PROPN
ejpam-5098	677	36	is	be	AUX
ejpam-5098	677	37	maximal	maximal	ADJ
ejpam-5098	677	38	.	.	PUNCT
ejpam-5098	678	1	therefore	therefore	ADV
ejpam-5098	678	2	,	,	PUNCT
ejpam-5098	678	3	i	i	PRON
ejpam-5098	678	4	is	be	AUX
ejpam-5098	678	5	a	a	DET
ejpam-5098	678	6	b	b	NOUN
ejpam-5098	678	7	-	-	PUNCT
ejpam-5098	678	8	ideal	ideal	NOUN
ejpam-5098	678	9	of	of	ADP
ejpam-5098	678	10	r.	r.	PROPN
ejpam-5098	678	11	conversely	conversely	ADV
ejpam-5098	678	12	,	,	PUNCT
ejpam-5098	678	13	assume	assume	VERB
ejpam-5098	678	14	that	that	SCONJ
ejpam-5098	678	15	every	every	DET
ejpam-5098	678	16	σ	σ	NOUN
ejpam-5098	678	17	-	-	PUNCT
ejpam-5098	678	18	ideal	ideal	NOUN
ejpam-5098	678	19	of	of	ADP
ejpam-5098	678	20	r	r	NOUN
ejpam-5098	678	21	is	be	AUX
ejpam-5098	678	22	a	a	DET
ejpam-5098	678	23	b	b	NOUN
ejpam-5098	678	24	-	-	PUNCT
ejpam-5098	678	25	ideal	ideal	NOUN
ejpam-5098	678	26	.	.	PUNCT
ejpam-5098	679	1	we	we	PRON
ejpam-5098	679	2	prove	prove	VERB
ejpam-5098	679	3	that	that	SCONJ
ejpam-5098	679	4	r	r	NOUN
ejpam-5098	679	5	is	be	AUX
ejpam-5098	679	6	strongly	strongly	ADV
ejpam-5098	679	7	normal	normal	ADJ
ejpam-5098	679	8	.	.	PUNCT
ejpam-5098	680	1	let	let	VERB
ejpam-5098	680	2	x	x	SYM
ejpam-5098	680	3	∈	∈	PROPN
ejpam-5098	680	4	r.	r.	PROPN
ejpam-5098	680	5	we	we	PRON
ejpam-5098	680	6	prove	prove	VERB
ejpam-5098	680	7	that	that	SCONJ
ejpam-5098	680	8	(	(	PUNCT
ejpam-5098	680	9	x)∗	x)∗	PROPN
ejpam-5098	680	10	is	be	AUX
ejpam-5098	680	11	a	a	DET
ejpam-5098	680	12	b	b	NOUN
ejpam-5098	680	13	-	-	PUNCT
ejpam-5098	680	14	ideal	ideal	NOUN
ejpam-5098	680	15	of	of	ADP
ejpam-5098	680	16	r.	r.	PROPN
ejpam-5098	680	17	let	let	VERB
ejpam-5098	680	18	y	y	PROPN
ejpam-5098	680	19	∈	∈	PROPN
ejpam-5098	680	20	(	(	PUNCT
ejpam-5098	680	21	x)∗.	x)∗.	PROPN
ejpam-5098	680	22	then	then	ADV
ejpam-5098	680	23	x	x	PART
ejpam-5098	680	24	∧	∧	NOUN
ejpam-5098	680	25	y	y	PROPN
ejpam-5098	680	26	=	=	NOUN
ejpam-5098	680	27	0	0	PROPN
ejpam-5098	680	28	.	.	PUNCT
ejpam-5098	681	1	since	since	SCONJ
ejpam-5098	681	2	l	l	NOUN
ejpam-5098	681	3	is	be	AUX
ejpam-5098	681	4	normal	normal	ADJ
ejpam-5098	681	5	,	,	PUNCT
ejpam-5098	681	6	there	there	PRON
ejpam-5098	681	7	exist	exist	VERB
ejpam-5098	681	8	elements	element	NOUN
ejpam-5098	681	9	a	a	PRON
ejpam-5098	681	10	,	,	PUNCT
ejpam-5098	681	11	b	b	X
ejpam-5098	681	12	∈	∈	NOUN
ejpam-5098	681	13	r	r	NOUN
ejpam-5098	681	14	such	such	ADJ
ejpam-5098	681	15	that	that	SCONJ
ejpam-5098	681	16	x	x	SYM
ejpam-5098	681	17	∧	∧	NOUN
ejpam-5098	681	18	a	a	PRON
ejpam-5098	681	19	=	=	SYM
ejpam-5098	681	20	0	0	NUM
ejpam-5098	681	21	,	,	PUNCT
ejpam-5098	681	22	y	y	PROPN
ejpam-5098	681	23	∧	∧	PROPN
ejpam-5098	681	24	b	b	PROPN
ejpam-5098	681	25	=	=	SYM
ejpam-5098	681	26	0	0	PROPN
ejpam-5098	681	27	and	and	CCONJ
ejpam-5098	681	28	a	a	DET
ejpam-5098	681	29	∨	∨	NOUN
ejpam-5098	681	30	b	b	NOUN
ejpam-5098	681	31	is	be	AUX
ejpam-5098	681	32	maximal	maximal	ADJ
ejpam-5098	681	33	.	.	PUNCT
ejpam-5098	682	1	that	that	PRON
ejpam-5098	682	2	implies	imply	VERB
ejpam-5098	682	3	(	(	PUNCT
ejpam-5098	682	4	y)∗∨(x)∗	y)∗∨(x)∗	NOUN
ejpam-5098	682	5	=	=	PROPN
ejpam-5098	682	6	r.	r.	PROPN
ejpam-5098	682	7	that	that	PRON
ejpam-5098	682	8	implies	imply	VERB
ejpam-5098	682	9	(	(	PUNCT
ejpam-5098	682	10	x)∗	x)∗	PROPN
ejpam-5098	682	11	⊆	⊆	NUM
ejpam-5098	682	12	(	(	PUNCT
ejpam-5098	682	13	x)∗	x)∗	PROPN
ejpam-5098	682	14	σ	σ	PROPN
ejpam-5098	682	15	and	and	CCONJ
ejpam-5098	682	16	hence	hence	ADV
ejpam-5098	682	17	(	(	PUNCT
ejpam-5098	682	18	x)∗	x)∗	PROPN
ejpam-5098	682	19	σ	σ	PROPN
ejpam-5098	682	20	=	=	PUNCT
ejpam-5098	683	1	(	(	PUNCT
ejpam-5098	683	2	x)∗.	x)∗.	PROPN
ejpam-5098	683	3	therefore	therefore	ADV
ejpam-5098	683	4	,	,	PUNCT
ejpam-5098	683	5	(	(	PUNCT
ejpam-5098	683	6	x)∗	x)∗	PROPN
ejpam-5098	683	7	is	be	AUX
ejpam-5098	683	8	a	a	DET
ejpam-5098	683	9	σ	σ	NOUN
ejpam-5098	683	10	-	-	PUNCT
ejpam-5098	683	11	ideal	ideal	NOUN
ejpam-5098	683	12	of	of	ADP
ejpam-5098	683	13	r.	r.	PROPN
ejpam-5098	683	14	by	by	ADP
ejpam-5098	683	15	our	our	PRON
ejpam-5098	683	16	assumption	assumption	NOUN
ejpam-5098	683	17	,	,	PUNCT
ejpam-5098	683	18	we	we	PRON
ejpam-5098	683	19	get	get	VERB
ejpam-5098	683	20	that	that	PRON
ejpam-5098	683	21	(	(	PUNCT
ejpam-5098	683	22	x)∗	x)∗	PROPN
ejpam-5098	683	23	is	be	AUX
ejpam-5098	683	24	a	a	DET
ejpam-5098	683	25	b	b	NOUN
ejpam-5098	683	26	-	-	PUNCT
ejpam-5098	683	27	ideal	ideal	NOUN
ejpam-5098	683	28	of	of	ADP
ejpam-5098	683	29	r.	r.	PROPN
ejpam-5098	683	30	theorem	theorem	PROPN
ejpam-5098	683	31	28	28	NUM
ejpam-5098	683	32	.	.	PUNCT
ejpam-5098	684	1	for	for	ADP
ejpam-5098	684	2	any	any	DET
ejpam-5098	684	3	σ	σ	NOUN
ejpam-5098	684	4	-	-	PUNCT
ejpam-5098	684	5	ideal	ideal	NOUN
ejpam-5098	684	6	i	i	PRON
ejpam-5098	684	7	of	of	ADP
ejpam-5098	684	8	an	an	DET
ejpam-5098	684	9	adl	adl	NOUN
ejpam-5098	684	10	r	r	NOUN
ejpam-5098	684	11	with	with	ADP
ejpam-5098	684	12	maximal	maximal	ADJ
ejpam-5098	684	13	elements	element	NOUN
ejpam-5098	684	14	,	,	PUNCT
ejpam-5098	684	15	h(i	h(i	PROPN
ejpam-5098	684	16	)	)	PUNCT
ejpam-5098	684	17	is	be	AUX
ejpam-5098	684	18	clopen	clopen	ADJ
ejpam-5098	684	19	in	in	ADP
ejpam-5098	684	20	specσ(r	specσ(r	NOUN
ejpam-5098	684	21	)	)	PUNCT
ejpam-5098	684	22	if	if	SCONJ
ejpam-5098	684	23	and	and	CCONJ
ejpam-5098	684	24	only	only	ADV
ejpam-5098	684	25	if	if	SCONJ
ejpam-5098	684	26	there	there	PRON
ejpam-5098	684	27	exists	exist	VERB
ejpam-5098	684	28	an	an	DET
ejpam-5098	684	29	element	element	NOUN
ejpam-5098	684	30	e	e	NOUN
ejpam-5098	684	31	∈	∈	PROPN
ejpam-5098	684	32	b	b	PROPN
ejpam-5098	684	33	such	such	ADJ
ejpam-5098	684	34	that	that	SCONJ
ejpam-5098	684	35	i	i	PRON
ejpam-5098	684	36	=	=	PUNCT
ejpam-5098	684	37	(	(	PUNCT
ejpam-5098	684	38	e	e	NOUN
ejpam-5098	684	39	]	]	PUNCT
ejpam-5098	684	40	.	.	PUNCT
ejpam-5098	685	1	proof	proof	NOUN
ejpam-5098	685	2	.	.	PUNCT
ejpam-5098	686	1	let	let	VERB
ejpam-5098	686	2	i	i	PRON
ejpam-5098	686	3	be	be	AUX
ejpam-5098	686	4	any	any	DET
ejpam-5098	686	5	σ	σ	NOUN
ejpam-5098	686	6	-	-	PUNCT
ejpam-5098	686	7	ideal	ideal	NOUN
ejpam-5098	686	8	of	of	ADP
ejpam-5098	686	9	r.	r.	PROPN
ejpam-5098	686	10	assume	assume	VERB
ejpam-5098	686	11	that	that	SCONJ
ejpam-5098	686	12	h(i	h(i	NOUN
ejpam-5098	686	13	)	)	PUNCT
ejpam-5098	686	14	is	be	AUX
ejpam-5098	686	15	clopen	clopen	ADJ
ejpam-5098	686	16	.	.	PUNCT
ejpam-5098	687	1	then	then	ADV
ejpam-5098	687	2	specσr	specσr	PROPN
ejpam-5098	687	3	\	\	PROPN
ejpam-5098	687	4	h(i	h(i	PROPN
ejpam-5098	687	5	)	)	PUNCT
ejpam-5098	687	6	=	=	PUNCT
ejpam-5098	687	7	h(j	h(j	PROPN
ejpam-5098	687	8	)	)	PUNCT
ejpam-5098	687	9	.	.	PUNCT
ejpam-5098	688	1	that	that	PRON
ejpam-5098	688	2	implies	imply	VERB
ejpam-5098	688	3	h(i	h(i	NOUN
ejpam-5098	688	4	)	)	PUNCT
ejpam-5098	688	5	∪	∪	ADJ
ejpam-5098	688	6	h(j	h(j	NOUN
ejpam-5098	688	7	)	)	PUNCT
ejpam-5098	689	1	=	=	SYM
ejpam-5098	689	2	specσr	specσr	NOUN
ejpam-5098	689	3	and	and	CCONJ
ejpam-5098	689	4	h(i	h(i	NOUN
ejpam-5098	689	5	)	)	PUNCT
ejpam-5098	689	6	∩	∩	NOUN
ejpam-5098	689	7	h(j	h(j	VERB
ejpam-5098	689	8	)	)	PUNCT
ejpam-5098	690	1	=	=	PUNCT
ejpam-5098	690	2	∅.	∅.	NOUN
ejpam-5098	690	3	that	that	PRON
ejpam-5098	690	4	implies	imply	VERB
ejpam-5098	690	5	h(i	h(i	PROPN
ejpam-5098	690	6	∨	∨	NUM
ejpam-5098	690	7	j	j	NOUN
ejpam-5098	690	8	)	)	PUNCT
ejpam-5098	690	9	=	=	SYM
ejpam-5098	690	10	specσr	specσr	NOUN
ejpam-5098	690	11	and	and	CCONJ
ejpam-5098	690	12	h(i	h(i	PROPN
ejpam-5098	690	13	∩	∩	PROPN
ejpam-5098	690	14	j	j	PROPN
ejpam-5098	690	15	)	)	PUNCT
ejpam-5098	690	16	=	=	PUNCT
ejpam-5098	690	17	∅.	∅.	NOUN
ejpam-5098	690	18	that	that	PRON
ejpam-5098	690	19	implies	imply	VERB
ejpam-5098	690	20	i	i	PRON
ejpam-5098	690	21	∨	∨	NUM
ejpam-5098	690	22	j	j	PROPN
ejpam-5098	690	23	=	=	SYM
ejpam-5098	690	24	r	r	NOUN
ejpam-5098	690	25	and	and	CCONJ
ejpam-5098	690	26	i	i	PROPN
ejpam-5098	690	27	∩	∩	PROPN
ejpam-5098	690	28	j	j	PROPN
ejpam-5098	690	29	=	=	PUNCT
ejpam-5098	690	30	{	{	PUNCT
ejpam-5098	690	31	0	0	NUM
ejpam-5098	690	32	}	}	PUNCT
ejpam-5098	690	33	.	.	PUNCT
ejpam-5098	691	1	let	let	VERB
ejpam-5098	691	2	x	x	SYM
ejpam-5098	691	3	∈	∈	PROPN
ejpam-5098	691	4	i.	i.	NOUN
ejpam-5098	691	5	then	then	ADV
ejpam-5098	691	6	x	x	SYM
ejpam-5098	691	7	∈	∈	NOUN
ejpam-5098	691	8	iσ	iσ	VERB
ejpam-5098	691	9	.	.	PUNCT
ejpam-5098	692	1	that	that	PRON
ejpam-5098	692	2	implies	imply	VERB
ejpam-5098	692	3	(	(	PUNCT
ejpam-5098	692	4	x)∗	x)∗	PROPN
ejpam-5098	692	5	∨	∨	NOUN
ejpam-5098	692	6	i	i	PRON
ejpam-5098	692	7	=	=	PUNCT
ejpam-5098	692	8	r.	r.	NOUN
ejpam-5098	692	9	that	that	PRON
ejpam-5098	692	10	implies	imply	VERB
ejpam-5098	692	11	there	there	PRON
ejpam-5098	692	12	exist	exist	VERB
ejpam-5098	692	13	elements	element	NOUN
ejpam-5098	692	14	y	y	PROPN
ejpam-5098	692	15	∈	∈	PROPN
ejpam-5098	692	16	(	(	PUNCT
ejpam-5098	693	1	x)∗	x)∗	PROPN
ejpam-5098	693	2	and	and	CCONJ
ejpam-5098	693	3	i	i	PRON
ejpam-5098	693	4	∈	∈	PROPN
ejpam-5098	693	5	i	i	PRON
ejpam-5098	693	6	such	such	ADJ
ejpam-5098	693	7	that	that	SCONJ
ejpam-5098	693	8	y	y	PROPN
ejpam-5098	693	9	∨	∨	NOUN
ejpam-5098	693	10	i	i	PRON
ejpam-5098	693	11	is	be	AUX
ejpam-5098	693	12	maximal	maximal	ADJ
ejpam-5098	693	13	.	.	PUNCT
ejpam-5098	694	1	since	since	SCONJ
ejpam-5098	694	2	i	i	PRON
ejpam-5098	694	3	∨	∨	PROPN
ejpam-5098	694	4	j	j	PROPN
ejpam-5098	694	5	=	=	SYM
ejpam-5098	694	6	r	r	NOUN
ejpam-5098	694	7	and	and	CCONJ
ejpam-5098	694	8	i	i	PROPN
ejpam-5098	694	9	∩	∩	PROPN
ejpam-5098	694	10	j	j	PROPN
ejpam-5098	694	11	=	=	PUNCT
ejpam-5098	694	12	{	{	PUNCT
ejpam-5098	694	13	0	0	NUM
ejpam-5098	694	14	}	}	PUNCT
ejpam-5098	694	15	,	,	PUNCT
ejpam-5098	694	16	we	we	PRON
ejpam-5098	694	17	have	have	VERB
ejpam-5098	694	18	that	that	SCONJ
ejpam-5098	694	19	i	i	PRON
ejpam-5098	694	20	∨	∨	PROPN
ejpam-5098	694	21	j	j	PROPN
ejpam-5098	694	22	is	be	AUX
ejpam-5098	694	23	maximal	maximal	ADJ
ejpam-5098	694	24	and	and	CCONJ
ejpam-5098	694	25	i	i	PRON
ejpam-5098	695	1	∧	∧	PROPN
ejpam-5098	695	2	j	j	PROPN
ejpam-5098	695	3	=	=	PUNCT
ejpam-5098	695	4	0	0	PROPN
ejpam-5098	695	5	for	for	ADP
ejpam-5098	695	6	all	all	DET
ejpam-5098	695	7	j	j	PROPN
ejpam-5098	695	8	∈	∈	PROPN
ejpam-5098	695	9	j	j	PROPN
ejpam-5098	695	10	.	.	PUNCT
ejpam-5098	696	1	that	that	PRON
ejpam-5098	696	2	implies	imply	VERB
ejpam-5098	696	3	i	i	PRON
ejpam-5098	696	4	∈	∈	PROPN
ejpam-5098	696	5	b.	b.	PROPN
ejpam-5098	697	1	now	now	ADV
ejpam-5098	697	2	,	,	PUNCT
ejpam-5098	697	3	we	we	PRON
ejpam-5098	697	4	prove	prove	VERB
ejpam-5098	697	5	that	that	SCONJ
ejpam-5098	697	6	i	i	PRON
ejpam-5098	697	7	⊆	⊆	NUM
ejpam-5098	697	8	(	(	PUNCT
ejpam-5098	697	9	i	i	PRON
ejpam-5098	697	10	]	]	PUNCT
ejpam-5098	697	11	.	.	PUNCT
ejpam-5098	698	1	let	let	VERB
ejpam-5098	698	2	x	x	SYM
ejpam-5098	698	3	∈	∈	PROPN
ejpam-5098	698	4	i.	i.	NOUN
ejpam-5098	698	5	now	now	ADV
ejpam-5098	698	6	,	,	PUNCT
ejpam-5098	698	7	x	x	PUNCT
ejpam-5098	698	8	=	=	PUNCT
ejpam-5098	698	9	(	(	PUNCT
ejpam-5098	698	10	i	i	PROPN
ejpam-5098	698	11	∨	∨	PROPN
ejpam-5098	698	12	j	j	PROPN
ejpam-5098	698	13	)	)	PUNCT
ejpam-5098	698	14	∧	∧	NOUN
ejpam-5098	698	15	x	x	X
ejpam-5098	698	16	=	=	PUNCT
ejpam-5098	698	17	(	(	PUNCT
ejpam-5098	698	18	i	i	PRON
ejpam-5098	698	19	∧	∧	PROPN
ejpam-5098	698	20	x	x	X
ejpam-5098	698	21	)	)	PUNCT
ejpam-5098	698	22	∨	∨	PROPN
ejpam-5098	698	23	(	(	PUNCT
ejpam-5098	698	24	j	j	PROPN
ejpam-5098	698	25	∧	∧	PROPN
ejpam-5098	698	26	x	x	NOUN
ejpam-5098	698	27	)	)	PUNCT
ejpam-5098	698	28	=	=	SYM
ejpam-5098	699	1	(	(	PUNCT
ejpam-5098	699	2	i	i	PRON
ejpam-5098	699	3	∧	∧	PROPN
ejpam-5098	699	4	x	x	X
ejpam-5098	699	5	)	)	PUNCT
ejpam-5098	699	6	∨	∨	NOUN
ejpam-5098	699	7	0	0	NUM
ejpam-5098	700	1	=	=	SYM
ejpam-5098	700	2	i	i	PRON
ejpam-5098	700	3	∧	∧	PROPN
ejpam-5098	700	4	x.	x.	NOUN
ejpam-5098	700	5	that	that	PRON
ejpam-5098	700	6	implies	imply	VERB
ejpam-5098	700	7	x	x	X
ejpam-5098	700	8	∈	∈	PROPN
ejpam-5098	700	9	(	(	PUNCT
ejpam-5098	700	10	i	i	NOUN
ejpam-5098	700	11	]	]	PUNCT
ejpam-5098	700	12	and	and	CCONJ
ejpam-5098	700	13	hence	hence	ADV
ejpam-5098	700	14	i	i	PRON
ejpam-5098	700	15	⊆	⊆	NUM
ejpam-5098	700	16	(	(	PUNCT
ejpam-5098	700	17	i	i	PRON
ejpam-5098	700	18	]	]	X
ejpam-5098	700	19	.	.	PUNCT
ejpam-5098	701	1	therefore	therefore	ADV
ejpam-5098	701	2	,	,	PUNCT
ejpam-5098	701	3	i	i	PRON
ejpam-5098	701	4	=	=	PUNCT
ejpam-5098	701	5	(	(	PUNCT
ejpam-5098	701	6	i	i	X
ejpam-5098	701	7	]	]	X
ejpam-5098	701	8	.	.	PUNCT
ejpam-5098	702	1	conversely	conversely	ADV
ejpam-5098	702	2	,	,	PUNCT
ejpam-5098	702	3	assume	assume	VERB
ejpam-5098	702	4	that	that	SCONJ
ejpam-5098	702	5	there	there	PRON
ejpam-5098	702	6	is	be	VERB
ejpam-5098	702	7	an	an	DET
ejpam-5098	702	8	element	element	NOUN
ejpam-5098	702	9	e	e	NOUN
ejpam-5098	702	10	∈	∈	PROPN
ejpam-5098	702	11	b	b	PROPN
ejpam-5098	702	12	such	such	ADJ
ejpam-5098	702	13	that	that	SCONJ
ejpam-5098	702	14	i	i	PRON
ejpam-5098	702	15	=	=	PUNCT
ejpam-5098	702	16	(	(	PUNCT
ejpam-5098	702	17	e	e	NOUN
ejpam-5098	702	18	]	]	X
ejpam-5098	702	19	.	.	PUNCT
ejpam-5098	703	1	since	since	SCONJ
ejpam-5098	703	2	e	e	PROPN
ejpam-5098	703	3	∈	∈	PROPN
ejpam-5098	703	4	b	b	PROPN
ejpam-5098	703	5	,	,	PUNCT
ejpam-5098	703	6	there	there	PRON
ejpam-5098	703	7	exists	exist	VERB
ejpam-5098	703	8	an	an	DET
ejpam-5098	703	9	element	element	NOUN
ejpam-5098	703	10	f	f	PROPN
ejpam-5098	703	11	∈	∈	PROPN
ejpam-5098	703	12	r	r	NOUN
ejpam-5098	703	13	such	such	ADJ
ejpam-5098	703	14	that	that	SCONJ
ejpam-5098	703	15	e	e	PROPN
ejpam-5098	703	16	∧	∧	PROPN
ejpam-5098	703	17	f	f	PROPN
ejpam-5098	703	18	=	=	SYM
ejpam-5098	703	19	0	0	PROPN
ejpam-5098	703	20	and	and	CCONJ
ejpam-5098	703	21	e	e	PROPN
ejpam-5098	703	22	∨	∨	PROPN
ejpam-5098	703	23	f	f	PROPN
ejpam-5098	703	24	is	be	AUX
ejpam-5098	703	25	maximal	maximal	ADJ
ejpam-5098	703	26	.	.	PUNCT
ejpam-5098	704	1	now	now	ADV
ejpam-5098	704	2	,	,	PUNCT
ejpam-5098	704	3	h(e	h(e	PROPN
ejpam-5098	704	4	)	)	PUNCT
ejpam-5098	704	5	∩	∩	NOUN
ejpam-5098	704	6	h(f	h(f	X
ejpam-5098	704	7	)	)	PUNCT
ejpam-5098	705	1	=	=	SYM
ejpam-5098	705	2	h(e	h(e	PROPN
ejpam-5098	705	3	∧	∧	PROPN
ejpam-5098	705	4	f	f	NOUN
ejpam-5098	705	5	)	)	PUNCT
ejpam-5098	705	6	=	=	SYM
ejpam-5098	705	7	h(0	h(0	PROPN
ejpam-5098	705	8	)	)	PUNCT
ejpam-5098	705	9	=	=	SYM
ejpam-5098	705	10	∅	∅	NOUN
ejpam-5098	705	11	and	and	CCONJ
ejpam-5098	705	12	h(e	h(e	PROPN
ejpam-5098	705	13	)	)	PUNCT
ejpam-5098	705	14	∪	∪	ADP
ejpam-5098	705	15	h(f	h(f	NOUN
ejpam-5098	705	16	)	)	PUNCT
ejpam-5098	705	17	=	=	SYM
ejpam-5098	705	18	h(e	h(e	PROPN
ejpam-5098	705	19	∨	∨	NUM
ejpam-5098	705	20	f	f	NOUN
ejpam-5098	705	21	)	)	PUNCT
ejpam-5098	705	22	=	=	SYM
ejpam-5098	705	23	specσ(r	specσ(r	PROPN
ejpam-5098	705	24	)	)	PUNCT
ejpam-5098	705	25	,	,	PUNCT
ejpam-5098	705	26	since	since	SCONJ
ejpam-5098	705	27	e	e	PROPN
ejpam-5098	705	28	∨	∨	PROPN
ejpam-5098	705	29	f	f	PROPN
ejpam-5098	705	30	is	be	AUX
ejpam-5098	705	31	maximal	maximal	ADJ
ejpam-5098	705	32	.	.	PUNCT
ejpam-5098	706	1	therefore	therefore	ADV
ejpam-5098	706	2	,	,	PUNCT
ejpam-5098	706	3	h(i	h(i	PROPN
ejpam-5098	706	4	)	)	PUNCT
ejpam-5098	706	5	is	be	AUX
ejpam-5098	706	6	clopen	clopen	ADJ
ejpam-5098	706	7	.	.	PUNCT
ejpam-5098	707	1	references	reference	NOUN
ejpam-5098	707	2	1112	1112	NUM
ejpam-5098	707	3	acknowledgements	acknowledgement	NOUN
ejpam-5098	707	4	this	this	DET
ejpam-5098	707	5	research	research	NOUN
ejpam-5098	707	6	was	be	AUX
ejpam-5098	707	7	supported	support	VERB
ejpam-5098	707	8	by	by	ADP
ejpam-5098	707	9	university	university	NOUN
ejpam-5098	707	10	of	of	ADP
ejpam-5098	707	11	phayao	phayao	NOUN
ejpam-5098	707	12	and	and	CCONJ
ejpam-5098	707	13	thailand	thailand	PROPN
ejpam-5098	707	14	science	science	PROPN
ejpam-5098	707	15	research	research	PROPN
ejpam-5098	707	16	and	and	CCONJ
ejpam-5098	707	17	innovation	innovation	NOUN
ejpam-5098	707	18	fund	fund	NOUN
ejpam-5098	707	19	(	(	PUNCT
ejpam-5098	707	20	fundamental	fundamental	ADJ
ejpam-5098	707	21	fund	fund	NOUN
ejpam-5098	707	22	2024	2024	NUM
ejpam-5098	707	23	)	)	PUNCT
ejpam-5098	707	24	.	.	PUNCT
ejpam-5098	708	1	references	reference	NOUN
ejpam-5098	708	2	[	[	X
ejpam-5098	708	3	1	1	NUM
ejpam-5098	708	4	]	]	PUNCT
ejpam-5098	708	5	g.	g.	NOUN
ejpam-5098	708	6	birkhoff	birkhoff	PROPN
ejpam-5098	708	7	.	.	PUNCT
ejpam-5098	709	1	lattice	lattice	PROPN
ejpam-5098	709	2	theory	theory	NOUN
ejpam-5098	709	3	.	.	PUNCT
ejpam-5098	710	1	american	american	PROPN
ejpam-5098	710	2	mathematical	mathematical	PROPN
ejpam-5098	710	3	society	society	NOUN
ejpam-5098	710	4	colloquium	colloquium	NOUN
ejpam-5098	710	5	publications	publication	NOUN
ejpam-5098	710	6	xxv	xxv	PROPN
ejpam-5098	710	7	,	,	PUNCT
ejpam-5098	710	8	providence	providence	NOUN
ejpam-5098	710	9	,	,	PUNCT
ejpam-5098	710	10	u.s.a	u.s.a	PROPN
ejpam-5098	710	11	.	.	PROPN
ejpam-5098	710	12	,	,	PUNCT
ejpam-5098	710	13	1967	1967	NUM
ejpam-5098	710	14	.	.	PUNCT
ejpam-5098	711	1	[	[	X
ejpam-5098	711	2	2	2	NUM
ejpam-5098	711	3	]	]	X
ejpam-5098	711	4	w.h	w.h	PROPN
ejpam-5098	711	5	.	.	PROPN
ejpam-5098	711	6	cornish	cornish	PROPN
ejpam-5098	711	7	.	.	PUNCT
ejpam-5098	711	8	annulets	annulet	NOUN
ejpam-5098	711	9	and	and	CCONJ
ejpam-5098	711	10	α	α	NOUN
ejpam-5098	711	11	-	-	NOUN
ejpam-5098	711	12	ideals	ideal	NOUN
ejpam-5098	711	13	in	in	ADP
ejpam-5098	711	14	distributive	distributive	ADJ
ejpam-5098	711	15	lattices	lattice	NOUN
ejpam-5098	711	16	.	.	PUNCT
ejpam-5098	712	1	journal	journal	NOUN
ejpam-5098	712	2	of	of	ADP
ejpam-5098	712	3	the	the	DET
ejpam-5098	712	4	australian	australian	ADJ
ejpam-5098	712	5	mathematical	mathematical	ADJ
ejpam-5098	712	6	society	society	NOUN
ejpam-5098	712	7	,	,	PUNCT
ejpam-5098	712	8	15:70–77	15:70–77	NUM
ejpam-5098	712	9	,	,	PUNCT
ejpam-5098	712	10	1973	1973	NUM
ejpam-5098	712	11	.	.	PUNCT
ejpam-5098	713	1	[	[	X
ejpam-5098	713	2	3	3	X
ejpam-5098	713	3	]	]	X
ejpam-5098	713	4	w.h	w.h	PROPN
ejpam-5098	713	5	.	.	PROPN
ejpam-5098	713	6	cornish	cornish	PROPN
ejpam-5098	713	7	.	.	PUNCT
ejpam-5098	714	1	o	o	NOUN
ejpam-5098	714	2	-	-	NOUN
ejpam-5098	714	3	ideals	ideal	NOUN
ejpam-5098	714	4	,	,	PUNCT
ejpam-5098	714	5	congruences	congruence	NOUN
ejpam-5098	714	6	,	,	PUNCT
ejpam-5098	714	7	sheaf	sheaf	NOUN
ejpam-5098	714	8	representation	representation	NOUN
ejpam-5098	714	9	of	of	ADP
ejpam-5098	714	10	distributive	distributive	ADJ
ejpam-5098	714	11	lattices	lattice	NOUN
ejpam-5098	714	12	.	.	PUNCT
ejpam-5098	715	1	revue	revue	NOUN
ejpam-5098	715	2	roumaine	roumaine	NOUN
ejpam-5098	715	3	de	de	PROPN
ejpam-5098	715	4	mathématiques	mathématiques	PROPN
ejpam-5098	715	5	pures	pure	NOUN
ejpam-5098	715	6	et	et	NOUN
ejpam-5098	715	7	appliquées	appliquée	NOUN
ejpam-5098	715	8	,	,	PUNCT
ejpam-5098	715	9	22:1059–1067	22:1059–1067	NUM
ejpam-5098	715	10	,	,	PUNCT
ejpam-5098	715	11	1977	1977	NUM
ejpam-5098	715	12	.	.	PUNCT
ejpam-5098	716	1	[	[	X
ejpam-5098	716	2	4	4	X
ejpam-5098	716	3	]	]	X
ejpam-5098	716	4	g.	g.	PROPN
ejpam-5098	716	5	gratzer	gratzer	PROPN
ejpam-5098	716	6	.	.	PUNCT
ejpam-5098	717	1	general	general	PROPN
ejpam-5098	717	2	lattice	lattice	PROPN
ejpam-5098	717	3	theory	theory	NOUN
ejpam-5098	717	4	.	.	PUNCT
ejpam-5098	718	1	academic	academic	ADJ
ejpam-5098	718	2	press	press	NOUN
ejpam-5098	718	3	,	,	PUNCT
ejpam-5098	718	4	new	new	PROPN
ejpam-5098	718	5	york	york	PROPN
ejpam-5098	718	6	,	,	PUNCT
ejpam-5098	718	7	san	san	PROPN
ejpam-5098	718	8	francisco	francisco	PROPN
ejpam-5098	718	9	,	,	PUNCT
ejpam-5098	718	10	1978	1978	NUM
ejpam-5098	718	11	.	.	PUNCT
ejpam-5098	719	1	[	[	X
ejpam-5098	719	2	5	5	X
ejpam-5098	719	3	]	]	X
ejpam-5098	719	4	g.c	g.c	PROPN
ejpam-5098	719	5	.	.	PROPN
ejpam-5098	719	6	rao	rao	PROPN
ejpam-5098	719	7	.	.	PUNCT
ejpam-5098	720	1	almost	almost	ADV
ejpam-5098	720	2	distributive	distributive	ADJ
ejpam-5098	720	3	lattices	lattice	NOUN
ejpam-5098	720	4	.	.	PUNCT
ejpam-5098	721	1	doctoral	doctoral	ADJ
ejpam-5098	721	2	thesis	thesis	NOUN
ejpam-5098	721	3	,	,	PUNCT
ejpam-5098	721	4	department	department	NOUN
ejpam-5098	721	5	of	of	ADP
ejpam-5098	721	6	mathematics	mathematics	PROPN
ejpam-5098	721	7	,	,	PUNCT
ejpam-5098	721	8	andhra	andhra	PROPN
ejpam-5098	721	9	university	university	PROPN
ejpam-5098	721	10	,	,	PUNCT
ejpam-5098	721	11	visakhapatnam	visakhapatnam	PROPN
ejpam-5098	721	12	,	,	PUNCT
ejpam-5098	721	13	1980	1980	NUM
ejpam-5098	721	14	.	.	PUNCT
ejpam-5098	722	1	[	[	X
ejpam-5098	722	2	6	6	NUM
ejpam-5098	722	3	]	]	X
ejpam-5098	722	4	g.c	g.c	PROPN
ejpam-5098	722	5	.	.	PROPN
ejpam-5098	722	6	rao	rao	PROPN
ejpam-5098	722	7	and	and	CCONJ
ejpam-5098	722	8	s.	s.	PROPN
ejpam-5098	722	9	ravi	ravi	PROPN
ejpam-5098	722	10	kumar	kumar	PROPN
ejpam-5098	722	11	.	.	PUNCT
ejpam-5098	723	1	normal	normal	ADJ
ejpam-5098	723	2	almost	almost	ADV
ejpam-5098	723	3	distributive	distributive	ADJ
ejpam-5098	723	4	lattices	lattice	NOUN
ejpam-5098	723	5	.	.	PUNCT
ejpam-5098	724	1	southeast	southeast	ADJ
ejpam-5098	724	2	asian	asian	ADJ
ejpam-5098	724	3	bulletin	bulletin	NOUN
ejpam-5098	724	4	of	of	ADP
ejpam-5098	724	5	mathematics	mathematic	NOUN
ejpam-5098	724	6	,	,	PUNCT
ejpam-5098	724	7	32:831–841	32:831–841	PROPN
ejpam-5098	724	8	,	,	PUNCT
ejpam-5098	724	9	2008	2008	NUM
ejpam-5098	724	10	.	.	PUNCT
ejpam-5098	725	1	[	[	X
ejpam-5098	725	2	7	7	X
ejpam-5098	725	3	]	]	X
ejpam-5098	725	4	g.c	g.c	PROPN
ejpam-5098	725	5	.	.	PROPN
ejpam-5098	725	6	rao	rao	PROPN
ejpam-5098	725	7	and	and	CCONJ
ejpam-5098	725	8	s.	s.	PROPN
ejpam-5098	725	9	ravi	ravi	PROPN
ejpam-5098	725	10	kumar	kumar	PROPN
ejpam-5098	725	11	.	.	PROPN
ejpam-5098	725	12	minimal	minimal	ADJ
ejpam-5098	725	13	prime	prime	ADJ
ejpam-5098	725	14	ideals	ideal	NOUN
ejpam-5098	725	15	in	in	ADP
ejpam-5098	725	16	almost	almost	ADV
ejpam-5098	725	17	distributive	distributive	ADJ
ejpam-5098	725	18	lattices	lattice	NOUN
ejpam-5098	725	19	.	.	PUNCT
ejpam-5098	726	1	international	international	ADJ
ejpam-5098	726	2	journal	journal	PROPN
ejpam-5098	726	3	of	of	ADP
ejpam-5098	726	4	contemporary	contemporary	PROPN
ejpam-5098	726	5	mathematical	mathematical	PROPN
ejpam-5098	726	6	sciences	sciences	PROPN
ejpam-5098	726	7	,	,	PUNCT
ejpam-5098	726	8	4(9	4(9	NUM
ejpam-5098	726	9	-	-	SYM
ejpam-5098	726	10	12):475–484	12):475–484	PROPN
ejpam-5098	726	11	,	,	PUNCT
ejpam-5098	726	12	2009	2009	NUM
ejpam-5098	726	13	.	.	PUNCT
ejpam-5098	727	1	[	[	X
ejpam-5098	727	2	8	8	NUM
ejpam-5098	727	3	]	]	X
ejpam-5098	727	4	g.c	g.c	PROPN
ejpam-5098	727	5	.	.	PROPN
ejpam-5098	727	6	rao	rao	PROPN
ejpam-5098	727	7	,	,	PUNCT
ejpam-5098	727	8	n.	n.	PROPN
ejpam-5098	727	9	rafi	rafi	PROPN
ejpam-5098	727	10	,	,	PUNCT
ejpam-5098	727	11	and	and	CCONJ
ejpam-5098	727	12	r.	r.	PROPN
ejpam-5098	727	13	bandaru	bandaru	PROPN
ejpam-5098	727	14	.	.	PUNCT
ejpam-5098	728	1	s	s	NOUN
ejpam-5098	728	2	-	-	PUNCT
ejpam-5098	728	3	ideals	ideal	NOUN
ejpam-5098	728	4	in	in	ADP
ejpam-5098	728	5	almost	almost	ADV
ejpam-5098	728	6	distributive	distributive	ADJ
ejpam-5098	728	7	lattices	lattice	NOUN
ejpam-5098	728	8	.	.	PUNCT
ejpam-5098	729	1	southeast	southeast	ADJ
ejpam-5098	729	2	asian	asian	ADJ
ejpam-5098	729	3	bulletin	bulletin	NOUN
ejpam-5098	729	4	of	of	ADP
ejpam-5098	729	5	mathematics	mathematic	NOUN
ejpam-5098	729	6	,	,	PUNCT
ejpam-5098	729	7	35:825–836	35:825–836	NUM
ejpam-5098	729	8	,	,	PUNCT
ejpam-5098	729	9	2011	2011	NUM
ejpam-5098	729	10	.	.	PUNCT
ejpam-5098	730	1	[	[	X
ejpam-5098	730	2	9	9	NUM
ejpam-5098	730	3	]	]	X
ejpam-5098	730	4	g.c	g.c	PROPN
ejpam-5098	730	5	.	.	PROPN
ejpam-5098	730	6	rao	rao	PROPN
ejpam-5098	730	7	and	and	CCONJ
ejpam-5098	730	8	m.	m.	PROPN
ejpam-5098	730	9	sambasiva	sambasiva	PROPN
ejpam-5098	730	10	rao	rao	PROPN
ejpam-5098	730	11	.	.	PUNCT
ejpam-5098	731	1	α	α	X
ejpam-5098	731	2	-	-	PUNCT
ejpam-5098	731	3	ideals	ideal	NOUN
ejpam-5098	731	4	and	and	CCONJ
ejpam-5098	731	5	prime	prime	ADJ
ejpam-5098	731	6	ideals	ideal	NOUN
ejpam-5098	731	7	in	in	ADP
ejpam-5098	731	8	almost	almost	ADV
ejpam-5098	731	9	distributive	distributive	ADJ
ejpam-5098	731	10	lattices	lattice	NOUN
ejpam-5098	731	11	.	.	PUNCT
ejpam-5098	732	1	international	international	ADJ
ejpam-5098	732	2	journal	journal	NOUN
ejpam-5098	732	3	of	of	ADP
ejpam-5098	732	4	algebra	algebra	PROPN
ejpam-5098	732	5	,	,	PUNCT
ejpam-5098	732	6	3(5):221–229	3(5):221–229	NUM
ejpam-5098	732	7	,	,	PUNCT
ejpam-5098	732	8	2009	2009	NUM
ejpam-5098	732	9	.	.	PUNCT
ejpam-5098	733	1	[	[	X
ejpam-5098	733	2	10	10	NUM
ejpam-5098	733	3	]	]	X
ejpam-5098	733	4	g.c	g.c	PROPN
ejpam-5098	733	5	.	.	PROPN
ejpam-5098	733	6	rao	rao	PROPN
ejpam-5098	733	7	and	and	CCONJ
ejpam-5098	733	8	m.	m.	PROPN
ejpam-5098	733	9	sambasiva	sambasiva	PROPN
ejpam-5098	733	10	rao	rao	PROPN
ejpam-5098	733	11	.	.	PUNCT
ejpam-5098	734	1	annulets	annulet	NOUN
ejpam-5098	734	2	in	in	ADP
ejpam-5098	734	3	almost	almost	ADV
ejpam-5098	734	4	distributive	distributive	ADJ
ejpam-5098	734	5	lattices	lattice	NOUN
ejpam-5098	734	6	.	.	PUNCT
ejpam-5098	735	1	european	european	ADJ
ejpam-5098	735	2	journal	journal	PROPN
ejpam-5098	735	3	of	of	ADP
ejpam-5098	735	4	pure	pure	ADJ
ejpam-5098	735	5	and	and	CCONJ
ejpam-5098	735	6	applied	applied	ADJ
ejpam-5098	735	7	mathematics	mathematic	NOUN
ejpam-5098	735	8	,	,	PUNCT
ejpam-5098	735	9	2(1):58–72	2(1):58–72	NUM
ejpam-5098	735	10	,	,	PUNCT
ejpam-5098	735	11	2009	2009	NUM
ejpam-5098	735	12	.	.	PUNCT
ejpam-5098	736	1	[	[	X
ejpam-5098	736	2	11	11	NUM
ejpam-5098	736	3	]	]	X
ejpam-5098	736	4	g.c	g.c	PROPN
ejpam-5098	736	5	.	.	PROPN
ejpam-5098	736	6	rao	rao	PROPN
ejpam-5098	736	7	and	and	CCONJ
ejpam-5098	736	8	m.	m.	PROPN
ejpam-5098	736	9	sambasiva	sambasiva	PROPN
ejpam-5098	736	10	rao	rao	PROPN
ejpam-5098	736	11	.	.	PUNCT
ejpam-5098	737	1	σ	σ	NOUN
ejpam-5098	737	2	-	-	PUNCT
ejpam-5098	737	3	ideals	ideal	NOUN
ejpam-5098	737	4	of	of	ADP
ejpam-5098	737	5	almost	almost	ADV
ejpam-5098	737	6	distributive	distributive	ADJ
ejpam-5098	737	7	lattices	lattice	NOUN
ejpam-5098	737	8	.	.	PUNCT
ejpam-5098	738	1	asianeuropean	asianeuropean	PROPN
ejpam-5098	738	2	journal	journal	PROPN
ejpam-5098	738	3	of	of	ADP
ejpam-5098	738	4	mathematics	mathematic	NOUN
ejpam-5098	738	5	,	,	PUNCT
ejpam-5098	738	6	5(4):1250057	5(4):1250057	NUM
ejpam-5098	738	7	,	,	PUNCT
ejpam-5098	738	8	2012	2012	NUM
ejpam-5098	738	9	.	.	PUNCT
ejpam-5098	739	1	[	[	X
ejpam-5098	739	2	12	12	NUM
ejpam-5098	739	3	]	]	X
ejpam-5098	739	4	u.m	u.m	PROPN
ejpam-5098	739	5	.	.	PROPN
ejpam-5098	739	6	swamy	swamy	PROPN
ejpam-5098	739	7	and	and	CCONJ
ejpam-5098	739	8	s.	s.	PROPN
ejpam-5098	739	9	ramesh	ramesh	PROPN
ejpam-5098	739	10	.	.	PUNCT
ejpam-5098	740	1	birkhoff	birkhoff	PROPN
ejpam-5098	740	2	centre	centre	PROPN
ejpam-5098	740	3	of	of	ADP
ejpam-5098	740	4	an	an	DET
ejpam-5098	740	5	almost	almost	ADV
ejpam-5098	740	6	distributive	distributive	ADJ
ejpam-5098	740	7	lattice	lattice	NOUN
ejpam-5098	740	8	.	.	PUNCT
ejpam-5098	741	1	international	international	ADJ
ejpam-5098	741	2	journal	journal	PROPN
ejpam-5098	741	3	of	of	ADP
ejpam-5098	741	4	algebra	algebra	PROPN
ejpam-5098	741	5	,	,	PUNCT
ejpam-5098	741	6	3(11):539–546	3(11):539–546	NUM
ejpam-5098	741	7	,	,	PUNCT
ejpam-5098	741	8	2009	2009	NUM
ejpam-5098	741	9	.	.	PUNCT
ejpam-5098	742	1	[	[	X
ejpam-5098	742	2	13	13	NUM
ejpam-5098	742	3	]	]	X
ejpam-5098	742	4	u.m	u.m	PROPN
ejpam-5098	742	5	.	.	PROPN
ejpam-5098	742	6	swamy	swamy	PROPN
ejpam-5098	742	7	and	and	CCONJ
ejpam-5098	742	8	g.c	g.c	PROPN
ejpam-5098	742	9	.	.	PROPN
ejpam-5098	742	10	rao	rao	PROPN
ejpam-5098	742	11	.	.	PUNCT
ejpam-5098	743	1	almost	almost	ADV
ejpam-5098	743	2	distributive	distributive	ADJ
ejpam-5098	743	3	lattices	lattice	NOUN
ejpam-5098	743	4	.	.	PUNCT
ejpam-5098	744	1	journal	journal	NOUN
ejpam-5098	744	2	of	of	ADP
ejpam-5098	744	3	the	the	DET
ejpam-5098	744	4	australian	australian	ADJ
ejpam-5098	744	5	mathematical	mathematical	ADJ
ejpam-5098	744	6	society	society	NOUN
ejpam-5098	744	7	(	(	PUNCT
ejpam-5098	744	8	series	series	PROPN
ejpam-5098	744	9	a	a	PROPN
ejpam-5098	744	10	)	)	PUNCT
ejpam-5098	744	11	,	,	PUNCT
ejpam-5098	744	12	31:77–91	31:77–91	NUM
ejpam-5098	744	13	,	,	PUNCT
ejpam-5098	744	14	1981	1981	NUM
ejpam-5098	744	15	.	.	PUNCT
ejpam-5098	745	1	[	[	X
ejpam-5098	745	2	14	14	NUM
ejpam-5098	745	3	]	]	X
ejpam-5098	745	4	u.m	u.m	PROPN
ejpam-5098	745	5	.	.	PROPN
ejpam-5098	745	6	swamy	swamy	PROPN
ejpam-5098	745	7	,	,	PUNCT
ejpam-5098	745	8	g.c	g.c	PROPN
ejpam-5098	745	9	.	.	PROPN
ejpam-5098	745	10	rao	rao	PROPN
ejpam-5098	745	11	,	,	PUNCT
ejpam-5098	745	12	and	and	CCONJ
ejpam-5098	745	13	g.	g.	PROPN
ejpam-5098	745	14	nanaji	nanaji	PROPN
ejpam-5098	745	15	rao	rao	PROPN
ejpam-5098	745	16	.	.	PUNCT
ejpam-5098	746	1	pseudo	pseudo	NOUN
ejpam-5098	746	2	-	-	NOUN
ejpam-5098	746	3	complementation	complementation	NOUN
ejpam-5098	746	4	on	on	ADP
ejpam-5098	746	5	almost	almost	ADV
ejpam-5098	746	6	distributive	distributive	ADJ
ejpam-5098	746	7	lattices	lattice	NOUN
ejpam-5098	746	8	.	.	PUNCT
ejpam-5098	747	1	southeast	southeast	ADJ
ejpam-5098	747	2	asian	asian	ADJ
ejpam-5098	747	3	bulletin	bulletin	NOUN
ejpam-5098	747	4	of	of	ADP
ejpam-5098	747	5	mathematics	mathematic	NOUN
ejpam-5098	747	6	,	,	PUNCT
ejpam-5098	747	7	24:95–104	24:95–104	NUM
ejpam-5098	747	8	,	,	PUNCT
ejpam-5098	747	9	2000	2000	NUM
ejpam-5098	747	10	.	.	PUNCT
ejpam-5098	748	1	[	[	X
ejpam-5098	748	2	15	15	NUM
ejpam-5098	748	3	]	]	X
ejpam-5098	748	4	u.m	u.m	PROPN
ejpam-5098	748	5	.	.	PROPN
ejpam-5098	748	6	swamy	swamy	PROPN
ejpam-5098	748	7	,	,	PUNCT
ejpam-5098	748	8	g.c	g.c	PROPN
ejpam-5098	748	9	.	.	PROPN
ejpam-5098	748	10	rao	rao	PROPN
ejpam-5098	748	11	,	,	PUNCT
ejpam-5098	748	12	and	and	CCONJ
ejpam-5098	748	13	g.	g.	PROPN
ejpam-5098	748	14	nanaji	nanaji	PROPN
ejpam-5098	748	15	rao	rao	PROPN
ejpam-5098	748	16	.	.	PUNCT
ejpam-5098	749	1	stone	stone	NOUN
ejpam-5098	749	2	almost	almost	ADV
ejpam-5098	749	3	distributive	distributive	ADJ
ejpam-5098	749	4	lattices	lattice	NOUN
ejpam-5098	749	5	.	.	PUNCT
ejpam-5098	750	1	southeast	southeast	ADJ
ejpam-5098	750	2	asian	asian	ADJ
ejpam-5098	750	3	bulletin	bulletin	NOUN
ejpam-5098	750	4	of	of	ADP
ejpam-5098	750	5	mathematics	mathematic	NOUN
ejpam-5098	750	6	,	,	PUNCT
ejpam-5098	750	7	24:513–526	24:513–526	PROPN
ejpam-5098	750	8	,	,	PUNCT
ejpam-5098	750	9	2000	2000	NUM
ejpam-5098	750	10	.	.	PUNCT