id	sid	tid	token	lemma	pos
ejpam-4548	1	1	european	european	PROPN
ejpam-4548	1	2	journal	journal	PROPN
ejpam-4548	1	3	of	of	ADP
ejpam-4548	1	4	pure	pure	ADJ
ejpam-4548	1	5	and	and	CCONJ
ejpam-4548	1	6	applied	apply	VERB
ejpam-4548	1	7	mathematics	mathematic	NOUN
ejpam-4548	1	8	vol	vol	NOUN
ejpam-4548	1	9	.	.	PROPN
ejpam-4548	2	1	15	15	NUM
ejpam-4548	2	2	,	,	PUNCT
ejpam-4548	2	3	no	no	INTJ
ejpam-4548	2	4	.	.	NOUN
ejpam-4548	2	5	4	4	NUM
ejpam-4548	2	6	,	,	PUNCT
ejpam-4548	2	7	2022	2022	NUM
ejpam-4548	2	8	,	,	PUNCT
ejpam-4548	2	9	2086	2086	NUM
ejpam-4548	2	10	-	-	SYM
ejpam-4548	2	11	2115	2115	NUM
ejpam-4548	2	12	issn	issn	VERB
ejpam-4548	2	13	1307	1307	NUM
ejpam-4548	2	14	-	-	SYM
ejpam-4548	2	15	5543	5543	NUM
ejpam-4548	2	16	–	–	PUNCT
ejpam-4548	2	17	ejpam.com	ejpam.com	X
ejpam-4548	2	18	published	publish	VERB
ejpam-4548	2	19	by	by	ADP
ejpam-4548	2	20	new	new	PROPN
ejpam-4548	2	21	york	york	PROPN
ejpam-4548	2	22	business	business	PROPN
ejpam-4548	2	23	global	global	ADJ
ejpam-4548	2	24	subnormality	subnormality	NOUN
ejpam-4548	2	25	and	and	CCONJ
ejpam-4548	2	26	theory	theory	NOUN
ejpam-4548	2	27	of	of	ADP
ejpam-4548	2	28	l	l	NOUN
ejpam-4548	2	29	-	-	PUNCT
ejpam-4548	2	30	subgroups	subgroup	NOUN
ejpam-4548	2	31	naseem	naseem	PROPN
ejpam-4548	2	32	ajmal1	ajmal1	NOUN
ejpam-4548	2	33	,	,	PUNCT
ejpam-4548	2	34	ifffat	ifffat	NOUN
ejpam-4548	2	35	jahan2,∗	jahan2,∗	PROPN
ejpam-4548	2	36	,	,	PUNCT
ejpam-4548	2	37	bijan	bijan	PROPN
ejpam-4548	2	38	davvaz3	davvaz3	ADJ
ejpam-4548	2	39	1	1	NUM
ejpam-4548	2	40	department	department	NOUN
ejpam-4548	2	41	of	of	ADP
ejpam-4548	2	42	mathematics	mathematic	NOUN
ejpam-4548	2	43	,	,	PUNCT
ejpam-4548	2	44	zakir	zakir	PROPN
ejpam-4548	2	45	husain	husain	PROPN
ejpam-4548	2	46	delhi	delhi	PROPN
ejpam-4548	2	47	college	college	PROPN
ejpam-4548	2	48	,	,	PUNCT
ejpam-4548	2	49	university	university	PROPN
ejpam-4548	2	50	of	of	ADP
ejpam-4548	2	51	delhi	delhi	PROPN
ejpam-4548	2	52	,	,	PUNCT
ejpam-4548	2	53	delhi	delhi	PROPN
ejpam-4548	2	54	,	,	PUNCT
ejpam-4548	2	55	india	india	PROPN
ejpam-4548	2	56	2	2	NUM
ejpam-4548	2	57	department	department	NOUN
ejpam-4548	2	58	of	of	ADP
ejpam-4548	2	59	mathematics	mathematics	PROPN
ejpam-4548	2	60	,	,	PUNCT
ejpam-4548	2	61	ramjas	ramjas	PROPN
ejpam-4548	2	62	college	college	PROPN
ejpam-4548	2	63	,	,	PUNCT
ejpam-4548	2	64	university	university	NOUN
ejpam-4548	2	65	of	of	ADP
ejpam-4548	2	66	delhi	delhi	PROPN
ejpam-4548	2	67	,	,	PUNCT
ejpam-4548	2	68	delhi	delhi	PROPN
ejpam-4548	2	69	,	,	PUNCT
ejpam-4548	2	70	india	india	PROPN
ejpam-4548	2	71	3	3	NUM
ejpam-4548	2	72	department	department	NOUN
ejpam-4548	2	73	of	of	ADP
ejpam-4548	2	74	mathematics	mathematics	PROPN
ejpam-4548	2	75	,	,	PUNCT
ejpam-4548	2	76	yazd	yazd	PROPN
ejpam-4548	2	77	university	university	PROPN
ejpam-4548	2	78	,	,	PUNCT
ejpam-4548	2	79	yazd	yazd	PROPN
ejpam-4548	2	80	,	,	PUNCT
ejpam-4548	2	81	iran	iran	PROPN
ejpam-4548	2	82	abstract	abstract	ADJ
ejpam-4548	2	83	.	.	PUNCT
ejpam-4548	3	1	the	the	DET
ejpam-4548	3	2	main	main	ADJ
ejpam-4548	3	3	focus	focus	NOUN
ejpam-4548	3	4	in	in	ADP
ejpam-4548	3	5	this	this	DET
ejpam-4548	3	6	work	work	NOUN
ejpam-4548	3	7	is	be	AUX
ejpam-4548	3	8	to	to	PART
ejpam-4548	3	9	establish	establish	VERB
ejpam-4548	3	10	that	that	SCONJ
ejpam-4548	3	11	l	l	NOUN
ejpam-4548	3	12	-	-	NOUN
ejpam-4548	3	13	group	group	NOUN
ejpam-4548	3	14	theory	theory	NOUN
ejpam-4548	3	15	,	,	PUNCT
ejpam-4548	3	16	which	which	PRON
ejpam-4548	3	17	uses	use	VERB
ejpam-4548	3	18	the	the	DET
ejpam-4548	3	19	language	language	NOUN
ejpam-4548	3	20	of	of	ADP
ejpam-4548	3	21	functions	function	NOUN
ejpam-4548	3	22	instead	instead	ADV
ejpam-4548	3	23	of	of	ADP
ejpam-4548	3	24	formal	formal	ADJ
ejpam-4548	3	25	set	set	VERB
ejpam-4548	3	26	theoretic	theoretic	NOUN
ejpam-4548	3	27	language	language	NOUN
ejpam-4548	3	28	,	,	PUNCT
ejpam-4548	3	29	is	be	AUX
ejpam-4548	3	30	capable	capable	ADJ
ejpam-4548	3	31	of	of	ADP
ejpam-4548	3	32	capturing	capture	VERB
ejpam-4548	3	33	most	most	ADJ
ejpam-4548	3	34	of	of	ADP
ejpam-4548	3	35	the	the	DET
ejpam-4548	3	36	refined	refined	ADJ
ejpam-4548	3	37	ideas	idea	NOUN
ejpam-4548	3	38	and	and	CCONJ
ejpam-4548	3	39	concepts	concept	NOUN
ejpam-4548	3	40	of	of	ADP
ejpam-4548	3	41	classical	classical	ADJ
ejpam-4548	3	42	group	group	NOUN
ejpam-4548	3	43	theory	theory	NOUN
ejpam-4548	3	44	.	.	PUNCT
ejpam-4548	4	1	we	we	PRON
ejpam-4548	4	2	demonstrate	demonstrate	VERB
ejpam-4548	4	3	this	this	PRON
ejpam-4548	4	4	by	by	ADP
ejpam-4548	4	5	extending	extend	VERB
ejpam-4548	4	6	the	the	DET
ejpam-4548	4	7	notion	notion	NOUN
ejpam-4548	4	8	of	of	ADP
ejpam-4548	4	9	subnormality	subnormality	NOUN
ejpam-4548	4	10	to	to	ADP
ejpam-4548	4	11	the	the	DET
ejpam-4548	4	12	l	l	NOUN
ejpam-4548	4	13	-	-	ADJ
ejpam-4548	4	14	setting	set	VERB
ejpam-4548	4	15	and	and	CCONJ
ejpam-4548	4	16	investigating	investigate	VERB
ejpam-4548	4	17	its	its	PRON
ejpam-4548	4	18	properties	property	NOUN
ejpam-4548	4	19	.	.	PUNCT
ejpam-4548	5	1	we	we	PRON
ejpam-4548	5	2	develop	develop	VERB
ejpam-4548	5	3	a	a	DET
ejpam-4548	5	4	mechanism	mechanism	NOUN
ejpam-4548	5	5	to	to	PART
ejpam-4548	5	6	tackle	tackle	VERB
ejpam-4548	5	7	the	the	DET
ejpam-4548	5	8	join	join	NOUN
ejpam-4548	5	9	problem	problem	NOUN
ejpam-4548	5	10	of	of	ADP
ejpam-4548	5	11	subnormal	subnormal	ADJ
ejpam-4548	5	12	l	l	NOUN
ejpam-4548	5	13	-	-	NOUN
ejpam-4548	5	14	subgroups	subgroup	NOUN
ejpam-4548	5	15	.	.	PUNCT
ejpam-4548	6	1	the	the	DET
ejpam-4548	6	2	conjugate	conjugate	ADJ
ejpam-4548	6	3	l	l	NOUN
ejpam-4548	6	4	-	-	NOUN
ejpam-4548	6	5	subgroup	subgroup	NOUN
ejpam-4548	6	6	as	as	SCONJ
ejpam-4548	6	7	is	be	AUX
ejpam-4548	6	8	defined	define	VERB
ejpam-4548	6	9	in	in	ADP
ejpam-4548	6	10	our	our	PRON
ejpam-4548	6	11	previous	previous	ADJ
ejpam-4548	6	12	paper	paper	NOUN
ejpam-4548	6	13	[	[	X
ejpam-4548	6	14	4	4	X
ejpam-4548	6	15	]	]	PUNCT
ejpam-4548	6	16	has	have	AUX
ejpam-4548	6	17	been	be	AUX
ejpam-4548	6	18	used	use	VERB
ejpam-4548	6	19	to	to	PART
ejpam-4548	6	20	formulate	formulate	VERB
ejpam-4548	6	21	the	the	DET
ejpam-4548	6	22	concept	concept	NOUN
ejpam-4548	6	23	of	of	ADP
ejpam-4548	6	24	normal	normal	ADJ
ejpam-4548	6	25	closure	closure	NOUN
ejpam-4548	6	26	and	and	CCONJ
ejpam-4548	6	27	normal	normal	ADJ
ejpam-4548	6	28	closure	closure	NOUN
ejpam-4548	6	29	series	series	NOUN
ejpam-4548	6	30	of	of	ADP
ejpam-4548	6	31	an	an	DET
ejpam-4548	6	32	l	l	NOUN
ejpam-4548	6	33	-	-	NOUN
ejpam-4548	6	34	subgroup	subgroup	NOUN
ejpam-4548	6	35	which	which	PRON
ejpam-4548	6	36	,	,	PUNCT
ejpam-4548	6	37	in	in	ADP
ejpam-4548	6	38	turn	turn	NOUN
ejpam-4548	6	39	,	,	PUNCT
ejpam-4548	6	40	is	be	AUX
ejpam-4548	6	41	used	use	VERB
ejpam-4548	6	42	to	to	PART
ejpam-4548	6	43	define	define	VERB
ejpam-4548	6	44	subnormal	subnormal	ADJ
ejpam-4548	6	45	l	l	NOUN
ejpam-4548	6	46	-	-	NOUN
ejpam-4548	6	47	subgroups	subgroup	NOUN
ejpam-4548	6	48	.	.	PUNCT
ejpam-4548	7	1	further	far	ADV
ejpam-4548	7	2	,	,	PUNCT
ejpam-4548	7	3	the	the	DET
ejpam-4548	7	4	concept	concept	NOUN
ejpam-4548	7	5	of	of	ADP
ejpam-4548	7	6	subnormal	subnormal	ADJ
ejpam-4548	7	7	series	series	PROPN
ejpam-4548	7	8	has	have	AUX
ejpam-4548	7	9	been	be	AUX
ejpam-4548	7	10	introduced	introduce	VERB
ejpam-4548	7	11	in	in	ADP
ejpam-4548	7	12	l	l	NOUN
ejpam-4548	7	13	-	-	PUNCT
ejpam-4548	7	14	setting	set	VERB
ejpam-4548	7	15	and	and	CCONJ
ejpam-4548	7	16	utilized	utilize	VERB
ejpam-4548	7	17	to	to	PART
ejpam-4548	7	18	establish	establish	VERB
ejpam-4548	7	19	the	the	DET
ejpam-4548	7	20	subnormality	subnormality	NOUN
ejpam-4548	7	21	of	of	ADP
ejpam-4548	7	22	l	l	NOUN
ejpam-4548	7	23	-	-	PUNCT
ejpam-4548	7	24	subgroups	subgroup	NOUN
ejpam-4548	7	25	.	.	PUNCT
ejpam-4548	8	1	also	also	ADV
ejpam-4548	8	2	,	,	PUNCT
ejpam-4548	8	3	several	several	ADJ
ejpam-4548	8	4	results	result	NOUN
ejpam-4548	8	5	pertaining	pertain	VERB
ejpam-4548	8	6	to	to	ADP
ejpam-4548	8	7	the	the	DET
ejpam-4548	8	8	notion	notion	NOUN
ejpam-4548	8	9	of	of	ADP
ejpam-4548	8	10	subnormality	subnormality	NOUN
ejpam-4548	8	11	have	have	AUX
ejpam-4548	8	12	been	be	AUX
ejpam-4548	8	13	established	establish	VERB
ejpam-4548	8	14	.	.	PUNCT
ejpam-4548	9	1	lastly	lastly	ADV
ejpam-4548	9	2	,	,	PUNCT
ejpam-4548	9	3	the	the	DET
ejpam-4548	9	4	level	level	NOUN
ejpam-4548	9	5	subset	subset	NOUN
ejpam-4548	9	6	characterization	characterization	NOUN
ejpam-4548	9	7	of	of	ADP
ejpam-4548	9	8	a	a	DET
ejpam-4548	9	9	subnormal	subnormal	ADJ
ejpam-4548	9	10	l	l	NOUN
ejpam-4548	9	11	-	-	NOUN
ejpam-4548	9	12	subgroup	subgroup	NOUN
ejpam-4548	9	13	is	be	AUX
ejpam-4548	9	14	provided	provide	VERB
ejpam-4548	9	15	after	after	ADP
ejpam-4548	9	16	developing	develop	VERB
ejpam-4548	9	17	a	a	DET
ejpam-4548	9	18	necessary	necessary	ADJ
ejpam-4548	9	19	mechanism	mechanism	NOUN
ejpam-4548	9	20	.	.	PUNCT
ejpam-4548	10	1	finally	finally	ADV
ejpam-4548	10	2	,	,	PUNCT
ejpam-4548	10	3	we	we	PRON
ejpam-4548	10	4	establish	establish	VERB
ejpam-4548	10	5	that	that	SCONJ
ejpam-4548	10	6	every	every	DET
ejpam-4548	10	7	subgroup	subgroup	NOUN
ejpam-4548	10	8	of	of	ADP
ejpam-4548	10	9	a	a	DET
ejpam-4548	10	10	nilpotent	nilpotent	ADJ
ejpam-4548	10	11	lgroup	lgroup	NOUN
ejpam-4548	10	12	is	be	AUX
ejpam-4548	10	13	subnormal	subnormal	ADJ
ejpam-4548	10	14	.	.	PUNCT
ejpam-4548	11	1	in	in	ADP
ejpam-4548	11	2	fact	fact	NOUN
ejpam-4548	11	3	,	,	PUNCT
ejpam-4548	11	4	it	it	PRON
ejpam-4548	11	5	has	have	AUX
ejpam-4548	11	6	been	be	AUX
ejpam-4548	11	7	exhibited	exhibit	VERB
ejpam-4548	11	8	through	through	ADP
ejpam-4548	11	9	this	this	DET
ejpam-4548	11	10	work	work	NOUN
ejpam-4548	11	11	that	that	SCONJ
ejpam-4548	11	12	l	l	NOUN
ejpam-4548	11	13	-	-	NOUN
ejpam-4548	11	14	group	group	NOUN
ejpam-4548	11	15	theory	theory	NOUN
ejpam-4548	11	16	presents	present	VERB
ejpam-4548	11	17	a	a	DET
ejpam-4548	11	18	modernized	modernized	ADJ
ejpam-4548	11	19	approach	approach	NOUN
ejpam-4548	11	20	to	to	PART
ejpam-4548	11	21	study	study	VERB
ejpam-4548	11	22	classical	classical	ADJ
ejpam-4548	11	23	group	group	NOUN
ejpam-4548	11	24	theory	theory	NOUN
ejpam-4548	11	25	.	.	PUNCT
ejpam-4548	12	1	2020	2020	NUM
ejpam-4548	12	2	mathematics	mathematic	NOUN
ejpam-4548	12	3	subject	subject	NOUN
ejpam-4548	12	4	classifications	classification	NOUN
ejpam-4548	12	5	:	:	PUNCT
ejpam-4548	12	6	11e57	11e57	NUM
ejpam-4548	12	7	,	,	PUNCT
ejpam-4548	12	8	20n25	20n25	NUM
ejpam-4548	12	9	,	,	PUNCT
ejpam-4548	12	10	20a10	20a10	NUM
ejpam-4548	12	11	key	key	ADJ
ejpam-4548	12	12	words	word	NOUN
ejpam-4548	12	13	and	and	CCONJ
ejpam-4548	12	14	phrases	phrase	NOUN
ejpam-4548	12	15	:	:	PUNCT
ejpam-4548	12	16	l	l	NOUN
ejpam-4548	12	17	-	-	NOUN
ejpam-4548	12	18	subgroup	subgroup	NOUN
ejpam-4548	12	19	,	,	PUNCT
ejpam-4548	12	20	normal	normal	ADJ
ejpam-4548	12	21	l	l	NOUN
ejpam-4548	12	22	-	-	NOUN
ejpam-4548	12	23	subgroup	subgroup	NOUN
ejpam-4548	12	24	,	,	PUNCT
ejpam-4548	12	25	nilpotent	nilpotent	ADJ
ejpam-4548	12	26	l	l	NOUN
ejpam-4548	12	27	-	-	NOUN
ejpam-4548	12	28	subgroup	subgroup	ADJ
ejpam-4548	12	29	,	,	PUNCT
ejpam-4548	12	30	normal	normal	ADJ
ejpam-4548	12	31	closure	closure	NOUN
ejpam-4548	12	32	,	,	PUNCT
ejpam-4548	12	33	subnormal	subnormal	ADJ
ejpam-4548	12	34	subgroup	subgroup	PROPN
ejpam-4548	12	35	,	,	PUNCT
ejpam-4548	12	36	subnormal	subnormal	PROPN
ejpam-4548	12	37	series	series	PROPN
ejpam-4548	12	38	1	1	NUM
ejpam-4548	12	39	.	.	PUNCT
ejpam-4548	12	40	introduction	introduction	NOUN
ejpam-4548	12	41	the	the	DET
ejpam-4548	12	42	notion	notion	NOUN
ejpam-4548	12	43	of	of	ADP
ejpam-4548	12	44	subnormality	subnormality	NOUN
ejpam-4548	12	45	in	in	ADP
ejpam-4548	12	46	classical	classical	ADJ
ejpam-4548	12	47	group	group	NOUN
ejpam-4548	12	48	theory	theory	NOUN
ejpam-4548	12	49	is	be	AUX
ejpam-4548	12	50	an	an	DET
ejpam-4548	12	51	important	important	ADJ
ejpam-4548	12	52	generalization	generalization	NOUN
ejpam-4548	12	53	of	of	ADP
ejpam-4548	12	54	the	the	DET
ejpam-4548	12	55	notion	notion	NOUN
ejpam-4548	12	56	of	of	ADP
ejpam-4548	12	57	normality	normality	NOUN
ejpam-4548	12	58	.	.	PUNCT
ejpam-4548	13	1	this	this	DET
ejpam-4548	13	2	notion	notion	NOUN
ejpam-4548	13	3	is	be	AUX
ejpam-4548	13	4	introduced	introduce	VERB
ejpam-4548	13	5	to	to	PART
ejpam-4548	13	6	repair	repair	VERB
ejpam-4548	13	7	the	the	DET
ejpam-4548	13	8	deficiency	deficiency	NOUN
ejpam-4548	13	9	of	of	ADP
ejpam-4548	13	10	the	the	DET
ejpam-4548	13	11	notion	notion	NOUN
ejpam-4548	13	12	of	of	ADP
ejpam-4548	13	13	normality	normality	NOUN
ejpam-4548	13	14	that	that	SCONJ
ejpam-4548	13	15	it	it	PRON
ejpam-4548	13	16	fails	fail	VERB
ejpam-4548	13	17	to	to	PART
ejpam-4548	13	18	be	be	AUX
ejpam-4548	13	19	a	a	DET
ejpam-4548	13	20	transitive	transitive	ADJ
ejpam-4548	13	21	relation	relation	NOUN
ejpam-4548	13	22	.	.	PUNCT
ejpam-4548	14	1	in	in	ADP
ejpam-4548	14	2	fact	fact	NOUN
ejpam-4548	14	3	,	,	PUNCT
ejpam-4548	14	4	the	the	DET
ejpam-4548	14	5	relation	relation	NOUN
ejpam-4548	14	6	of	of	ADP
ejpam-4548	14	7	subnormality	subnormality	NOUN
ejpam-4548	14	8	can	can	AUX
ejpam-4548	14	9	be	be	AUX
ejpam-4548	14	10	defined	define	VERB
ejpam-4548	14	11	as	as	ADP
ejpam-4548	14	12	the	the	DET
ejpam-4548	14	13	transitive	transitive	ADJ
ejpam-4548	14	14	closure	closure	NOUN
ejpam-4548	14	15	of	of	ADP
ejpam-4548	14	16	the	the	DET
ejpam-4548	14	17	relation	relation	NOUN
ejpam-4548	14	18	of	of	ADP
ejpam-4548	14	19	normality	normality	NOUN
ejpam-4548	14	20	.	.	PUNCT
ejpam-4548	15	1	the	the	DET
ejpam-4548	15	2	class	class	NOUN
ejpam-4548	15	3	of	of	ADP
ejpam-4548	15	4	subnormal	subnormal	ADJ
ejpam-4548	15	5	subgroups	subgroup	NOUN
ejpam-4548	15	6	not	not	PART
ejpam-4548	15	7	only	only	ADV
ejpam-4548	15	8	properly	properly	ADV
ejpam-4548	15	9	contains	contain	VERB
ejpam-4548	15	10	the	the	DET
ejpam-4548	15	11	class	class	NOUN
ejpam-4548	15	12	of	of	ADP
ejpam-4548	15	13	normal	normal	ADJ
ejpam-4548	15	14	subgroups	subgroup	NOUN
ejpam-4548	15	15	but	but	CCONJ
ejpam-4548	15	16	in	in	ADP
ejpam-4548	15	17	the	the	DET
ejpam-4548	15	18	class	class	NOUN
ejpam-4548	15	19	of	of	ADP
ejpam-4548	15	20	finitely	finitely	ADV
ejpam-4548	15	21	generated	generate	VERB
ejpam-4548	15	22	groups	group	NOUN
ejpam-4548	15	23	this	this	DET
ejpam-4548	15	24	class	class	NOUN
ejpam-4548	15	25	coincides	coincide	VERB
ejpam-4548	15	26	with	with	ADP
ejpam-4548	15	27	the	the	DET
ejpam-4548	15	28	important	important	ADJ
ejpam-4548	15	29	subclass	subclass	NOUN
ejpam-4548	15	30	of	of	ADP
ejpam-4548	15	31	nilpotent	nilpotent	ADJ
ejpam-4548	15	32	subgroups	subgroup	NOUN
ejpam-4548	15	33	.	.	PUNCT
ejpam-4548	16	1	besides	besides	SCONJ
ejpam-4548	16	2	,	,	PUNCT
ejpam-4548	16	3	this	this	DET
ejpam-4548	16	4	notion	notion	NOUN
ejpam-4548	16	5	has	have	VERB
ejpam-4548	16	6	many	many	ADJ
ejpam-4548	16	7	more	more	ADV
ejpam-4548	16	8	pleasing	pleasing	ADJ
ejpam-4548	16	9	properties	property	NOUN
ejpam-4548	16	10	which	which	PRON
ejpam-4548	16	11	have	have	AUX
ejpam-4548	16	12	been	be	AUX
ejpam-4548	16	13	the	the	DET
ejpam-4548	16	14	focus	focus	NOUN
ejpam-4548	16	15	of	of	ADP
ejpam-4548	16	16	attention	attention	NOUN
ejpam-4548	16	17	of	of	ADP
ejpam-4548	16	18	algebraist	algebraist	NOUN
ejpam-4548	16	19	for	for	ADP
ejpam-4548	16	20	a	a	DET
ejpam-4548	16	21	long	long	ADJ
ejpam-4548	16	22	period	period	NOUN
ejpam-4548	16	23	of	of	ADP
ejpam-4548	16	24	time	time	NOUN
ejpam-4548	16	25	.	.	PUNCT
ejpam-4548	17	1	the	the	DET
ejpam-4548	17	2	work	work	NOUN
ejpam-4548	17	3	of	of	ADP
ejpam-4548	17	4	wielandt	wielandt	NOUN
ejpam-4548	17	5	in	in	ADP
ejpam-4548	17	6	the	the	DET
ejpam-4548	17	7	∗corresponding	∗corresponding	NOUN
ejpam-4548	17	8	author	author	NOUN
ejpam-4548	17	9	.	.	PUNCT
ejpam-4548	18	1	doi	doi	NOUN
ejpam-4548	18	2	:	:	PUNCT
ejpam-4548	18	3	https://doi.org/10.29020/nybg.ejpam.v15i4.4548	https://doi.org/10.29020/nybg.ejpam.v15i4.4548	NOUN
ejpam-4548	18	4	email	email	NOUN
ejpam-4548	18	5	addresses	address	NOUN
ejpam-4548	18	6	:	:	PUNCT
ejpam-4548	18	7	ij.umar@yahoo.com	ij.umar@yahoo.com	PROPN
ejpam-4548	18	8	(	(	PUNCT
ejpam-4548	18	9	i.	i.	PROPN
ejpam-4548	18	10	jahan	jahan	PROPN
ejpam-4548	18	11	)	)	PUNCT
ejpam-4548	18	12	,	,	PUNCT
ejpam-4548	18	13	nasajmal@yahoo.com	nasajmal@yahoo.com	X
ejpam-4548	18	14	(	(	PUNCT
ejpam-4548	18	15	n.	n.	NOUN
ejpam-4548	18	16	ajmal	ajmal	PROPN
ejpam-4548	18	17	)	)	PUNCT
ejpam-4548	18	18	,	,	PUNCT
ejpam-4548	18	19	davvaz@yazd.ac.ir	davvaz@yazd.ac.ir	PROPN
ejpam-4548	18	20	(	(	PUNCT
ejpam-4548	18	21	b.	b.	PROPN
ejpam-4548	18	22	davvaz	davvaz	PROPN
ejpam-4548	18	23	)	)	PUNCT
ejpam-4548	18	24	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4548	18	25	2086	2086	NUM
ejpam-4548	19	1	©	©	ADP
ejpam-4548	19	2	2022	2022	NUM
ejpam-4548	19	3	ejpam	ejpam	VERB
ejpam-4548	19	4	all	all	DET
ejpam-4548	19	5	rights	right	NOUN
ejpam-4548	19	6	reserved	reserve	VERB
ejpam-4548	19	7	.	.	PUNCT
ejpam-4548	20	1	n.	n.	PROPN
ejpam-4548	20	2	ajmal	ajmal	PROPN
ejpam-4548	20	3	,	,	PUNCT
ejpam-4548	20	4	i.	i.	PROPN
ejpam-4548	20	5	jahan	jahan	PROPN
ejpam-4548	20	6	.	.	PROPN
ejpam-4548	20	7	,	,	PUNCT
ejpam-4548	20	8	b.	b.	PROPN
ejpam-4548	20	9	davvaz	davvaz	PROPN
ejpam-4548	20	10	/	/	SYM
ejpam-4548	20	11	eur	eur	PROPN
ejpam-4548	20	12	.	.	PUNCT
ejpam-4548	21	1	j.	j.	PROPN
ejpam-4548	21	2	pure	pure	PROPN
ejpam-4548	21	3	appl	appl	PROPN
ejpam-4548	21	4	.	.	PROPN
ejpam-4548	21	5	math	math	PROPN
ejpam-4548	21	6	,	,	PUNCT
ejpam-4548	21	7	15	15	NUM
ejpam-4548	21	8	(	(	PUNCT
ejpam-4548	21	9	4	4	NUM
ejpam-4548	21	10	)	)	PUNCT
ejpam-4548	21	11	(	(	PUNCT
ejpam-4548	21	12	2022	2022	NUM
ejpam-4548	21	13	)	)	PUNCT
ejpam-4548	21	14	,	,	PUNCT
ejpam-4548	21	15	2086	2086	NUM
ejpam-4548	21	16	-	-	SYM
ejpam-4548	21	17	2115	2115	NUM
ejpam-4548	21	18	2087	2087	NUM
ejpam-4548	21	19	year	year	NOUN
ejpam-4548	21	20	1939	1939	NUM
ejpam-4548	21	21	on	on	ADP
ejpam-4548	21	22	the	the	DET
ejpam-4548	21	23	join	join	NOUN
ejpam-4548	21	24	of	of	ADP
ejpam-4548	21	25	two	two	NUM
ejpam-4548	21	26	subnormal	subnormal	ADJ
ejpam-4548	21	27	subgroups	subgroup	NOUN
ejpam-4548	21	28	in	in	ADP
ejpam-4548	21	29	a	a	DET
ejpam-4548	21	30	finite	finite	ADJ
ejpam-4548	21	31	group	group	NOUN
ejpam-4548	21	32	provided	provide	VERB
ejpam-4548	21	33	an	an	DET
ejpam-4548	21	34	impetus	impetus	NOUN
ejpam-4548	21	35	for	for	ADP
ejpam-4548	21	36	the	the	DET
ejpam-4548	21	37	development	development	NOUN
ejpam-4548	21	38	of	of	ADP
ejpam-4548	21	39	group	group	NOUN
ejpam-4548	21	40	theory	theory	NOUN
ejpam-4548	21	41	.	.	PUNCT
ejpam-4548	22	1	the	the	DET
ejpam-4548	22	2	question	question	NOUN
ejpam-4548	22	3	of	of	ADP
ejpam-4548	22	4	finding	find	VERB
ejpam-4548	22	5	necessary	necessary	ADJ
ejpam-4548	22	6	and	and	CCONJ
ejpam-4548	22	7	sufficient	sufficient	ADJ
ejpam-4548	22	8	conditions	condition	NOUN
ejpam-4548	22	9	for	for	ADP
ejpam-4548	22	10	the	the	DET
ejpam-4548	22	11	join	join	NOUN
ejpam-4548	22	12	of	of	ADP
ejpam-4548	22	13	subnormal	subnormal	ADJ
ejpam-4548	22	14	subgroups	subgroup	NOUN
ejpam-4548	22	15	to	to	PART
ejpam-4548	22	16	be	be	AUX
ejpam-4548	22	17	subnormal	subnormal	ADJ
ejpam-4548	22	18	is	be	AUX
ejpam-4548	22	19	one	one	NUM
ejpam-4548	22	20	of	of	ADP
ejpam-4548	22	21	the	the	DET
ejpam-4548	22	22	most	most	ADV
ejpam-4548	22	23	important	important	ADJ
ejpam-4548	22	24	problems	problem	NOUN
ejpam-4548	22	25	in	in	ADP
ejpam-4548	22	26	the	the	DET
ejpam-4548	22	27	area	area	NOUN
ejpam-4548	22	28	of	of	ADP
ejpam-4548	22	29	group	group	NOUN
ejpam-4548	22	30	theory	theory	NOUN
ejpam-4548	22	31	even	even	ADV
ejpam-4548	22	32	today	today	NOUN
ejpam-4548	22	33	.	.	PUNCT
ejpam-4548	23	1	here	here	ADV
ejpam-4548	23	2	in	in	ADP
ejpam-4548	23	3	the	the	DET
ejpam-4548	23	4	part	part	NOUN
ejpam-4548	23	5	i	i	PRON
ejpam-4548	23	6	of	of	ADP
ejpam-4548	23	7	this	this	DET
ejpam-4548	23	8	work	work	NOUN
ejpam-4548	23	9	,	,	PUNCT
ejpam-4548	23	10	we	we	PRON
ejpam-4548	23	11	introduce	introduce	VERB
ejpam-4548	23	12	and	and	CCONJ
ejpam-4548	23	13	study	study	VERB
ejpam-4548	23	14	the	the	DET
ejpam-4548	23	15	notion	notion	NOUN
ejpam-4548	23	16	of	of	ADP
ejpam-4548	23	17	subnormality	subnormality	NOUN
ejpam-4548	23	18	in	in	ADP
ejpam-4548	23	19	l	l	NOUN
ejpam-4548	23	20	-	-	NOUN
ejpam-4548	23	21	group	group	NOUN
ejpam-4548	23	22	theory	theory	NOUN
ejpam-4548	23	23	with	with	ADP
ejpam-4548	23	24	a	a	DET
ejpam-4548	23	25	special	special	ADJ
ejpam-4548	23	26	attention	attention	NOUN
ejpam-4548	23	27	of	of	ADP
ejpam-4548	23	28	approaching	approach	VERB
ejpam-4548	23	29	the	the	DET
ejpam-4548	23	30	join	join	NOUN
ejpam-4548	23	31	problem	problem	NOUN
ejpam-4548	23	32	of	of	ADP
ejpam-4548	23	33	subnormal	subnormal	ADJ
ejpam-4548	23	34	l	l	NOUN
ejpam-4548	23	35	-	-	NOUN
ejpam-4548	23	36	subgroups	subgroup	NOUN
ejpam-4548	23	37	.	.	PUNCT
ejpam-4548	24	1	this	this	DET
ejpam-4548	24	2	study	study	NOUN
ejpam-4548	24	3	is	be	AUX
ejpam-4548	24	4	carried	carry	VERB
ejpam-4548	24	5	out	out	ADP
ejpam-4548	24	6	in	in	ADP
ejpam-4548	24	7	detail	detail	NOUN
ejpam-4548	24	8	in	in	ADP
ejpam-4548	24	9	part	part	NOUN
ejpam-4548	24	10	ii	ii	NOUN
ejpam-4548	24	11	of	of	ADP
ejpam-4548	24	12	this	this	DET
ejpam-4548	24	13	work	work	NOUN
ejpam-4548	24	14	and	and	CCONJ
ejpam-4548	24	15	we	we	PRON
ejpam-4548	24	16	establish	establish	VERB
ejpam-4548	24	17	necessary	necessary	ADJ
ejpam-4548	24	18	and	and	CCONJ
ejpam-4548	24	19	sufficient	sufficient	ADJ
ejpam-4548	24	20	conditions	condition	NOUN
ejpam-4548	24	21	for	for	ADP
ejpam-4548	24	22	the	the	DET
ejpam-4548	24	23	subnormality	subnormality	NOUN
ejpam-4548	24	24	of	of	ADP
ejpam-4548	24	25	the	the	DET
ejpam-4548	24	26	join	join	NOUN
ejpam-4548	24	27	of	of	ADP
ejpam-4548	24	28	two	two	NUM
ejpam-4548	24	29	subnormal	subnormal	ADJ
ejpam-4548	24	30	l	l	NOUN
ejpam-4548	24	31	-	-	NOUN
ejpam-4548	24	32	subgroups	subgroup	NOUN
ejpam-4548	24	33	of	of	ADP
ejpam-4548	24	34	an	an	DET
ejpam-4548	24	35	arbitrary	arbitrary	ADJ
ejpam-4548	24	36	group	group	NOUN
ejpam-4548	24	37	.	.	PUNCT
ejpam-4548	25	1	in	in	ADP
ejpam-4548	25	2	the	the	DET
ejpam-4548	25	3	end	end	NOUN
ejpam-4548	25	4	of	of	ADP
ejpam-4548	25	5	this	this	DET
ejpam-4548	25	6	paper	paper	NOUN
ejpam-4548	25	7	,	,	PUNCT
ejpam-4548	25	8	we	we	PRON
ejpam-4548	25	9	present	present	VERB
ejpam-4548	25	10	the	the	DET
ejpam-4548	25	11	main	main	ADJ
ejpam-4548	25	12	result	result	NOUN
ejpam-4548	25	13	of	of	ADP
ejpam-4548	25	14	part	part	PROPN
ejpam-4548	25	15	ii	ii	PROPN
ejpam-4548	25	16	without	without	ADP
ejpam-4548	25	17	proof	proof	NOUN
ejpam-4548	25	18	.	.	PUNCT
ejpam-4548	26	1	in	in	ADP
ejpam-4548	26	2	this	this	DET
ejpam-4548	26	3	paper	paper	NOUN
ejpam-4548	26	4	,	,	PUNCT
ejpam-4548	26	5	we	we	PRON
ejpam-4548	26	6	present	present	VERB
ejpam-4548	26	7	a	a	DET
ejpam-4548	26	8	systematic	systematic	ADJ
ejpam-4548	26	9	and	and	CCONJ
ejpam-4548	26	10	successful	successful	ADJ
ejpam-4548	26	11	development	development	NOUN
ejpam-4548	26	12	of	of	ADP
ejpam-4548	26	13	l	l	NOUN
ejpam-4548	26	14	-	-	NOUN
ejpam-4548	26	15	group	group	NOUN
ejpam-4548	26	16	theory	theory	NOUN
ejpam-4548	26	17	which	which	PRON
ejpam-4548	26	18	has	have	AUX
ejpam-4548	26	19	been	be	AUX
ejpam-4548	26	20	made	make	VERB
ejpam-4548	26	21	in	in	ADP
ejpam-4548	26	22	papers	paper	NOUN
ejpam-4548	26	23	[	[	X
ejpam-4548	26	24	2–7	2–7	NOUN
ejpam-4548	26	25	,	,	PUNCT
ejpam-4548	26	26	10	10	NUM
ejpam-4548	26	27	]	]	PUNCT
ejpam-4548	26	28	.	.	PUNCT
ejpam-4548	27	1	the	the	DET
ejpam-4548	27	2	important	important	ADJ
ejpam-4548	27	3	concepts	concept	NOUN
ejpam-4548	27	4	of	of	ADP
ejpam-4548	27	5	nilpoent	nilpoent	ADJ
ejpam-4548	27	6	l	l	NOUN
ejpam-4548	27	7	-	-	NOUN
ejpam-4548	27	8	subgroups	subgroup	NOUN
ejpam-4548	27	9	,	,	PUNCT
ejpam-4548	27	10	solvable	solvable	ADJ
ejpam-4548	27	11	l	l	NOUN
ejpam-4548	27	12	-	-	NOUN
ejpam-4548	27	13	subgroups	subgroup	NOUN
ejpam-4548	27	14	have	have	AUX
ejpam-4548	27	15	been	be	AUX
ejpam-4548	27	16	introduced	introduce	VERB
ejpam-4548	27	17	and	and	CCONJ
ejpam-4548	27	18	studied	study	VERB
ejpam-4548	27	19	and	and	CCONJ
ejpam-4548	27	20	their	their	PRON
ejpam-4548	27	21	inter	inter	NOUN
ejpam-4548	27	22	-	-	NOUN
ejpam-4548	27	23	relationship	relationship	NOUN
ejpam-4548	27	24	is	be	AUX
ejpam-4548	27	25	established	establish	VERB
ejpam-4548	27	26	in	in	ADP
ejpam-4548	27	27	[	[	X
ejpam-4548	27	28	3	3	NUM
ejpam-4548	27	29	,	,	PUNCT
ejpam-4548	27	30	6	6	NUM
ejpam-4548	27	31	]	]	PUNCT
ejpam-4548	27	32	.	.	PUNCT
ejpam-4548	28	1	moreover	moreover	ADV
ejpam-4548	28	2	,	,	PUNCT
ejpam-4548	28	3	the	the	DET
ejpam-4548	28	4	concept	concept	NOUN
ejpam-4548	28	5	of	of	ADP
ejpam-4548	28	6	normalizer	normalizer	NOUN
ejpam-4548	28	7	of	of	ADP
ejpam-4548	28	8	an	an	DET
ejpam-4548	28	9	l	l	NOUN
ejpam-4548	28	10	-	-	NOUN
ejpam-4548	28	11	subgroup	subgroup	NOUN
ejpam-4548	28	12	in	in	ADP
ejpam-4548	28	13	an	an	DET
ejpam-4548	28	14	l	l	NOUN
ejpam-4548	28	15	-	-	NOUN
ejpam-4548	28	16	group	group	NOUN
ejpam-4548	28	17	is	be	AUX
ejpam-4548	28	18	formulated	formulate	VERB
ejpam-4548	28	19	in	in	ADP
ejpam-4548	28	20	[	[	X
ejpam-4548	28	21	2	2	NUM
ejpam-4548	28	22	]	]	PUNCT
ejpam-4548	28	23	.	.	PUNCT
ejpam-4548	29	1	it	it	PRON
ejpam-4548	29	2	has	have	AUX
ejpam-4548	29	3	been	be	AUX
ejpam-4548	29	4	proved	prove	VERB
ejpam-4548	29	5	in	in	ADP
ejpam-4548	29	6	[	[	X
ejpam-4548	29	7	10	10	NUM
ejpam-4548	29	8	]	]	PUNCT
ejpam-4548	29	9	that	that	SCONJ
ejpam-4548	29	10	a	a	DET
ejpam-4548	29	11	nilpotent	nilpotent	ADJ
ejpam-4548	29	12	l	l	NOUN
ejpam-4548	29	13	-	-	NOUN
ejpam-4548	29	14	subgroup	subgroup	NOUN
ejpam-4548	29	15	of	of	ADP
ejpam-4548	29	16	an	an	DET
ejpam-4548	29	17	l	l	NOUN
ejpam-4548	29	18	-	-	NOUN
ejpam-4548	29	19	group	group	NOUN
ejpam-4548	29	20	satisfies	satisfy	VERB
ejpam-4548	29	21	the	the	DET
ejpam-4548	29	22	normalizer	normalizer	PROPN
ejpam-4548	29	23	condition	condition	NOUN
ejpam-4548	29	24	.	.	PUNCT
ejpam-4548	30	1	as	as	ADP
ejpam-4548	30	2	an	an	DET
ejpam-4548	30	3	application	application	NOUN
ejpam-4548	30	4	and	and	CCONJ
ejpam-4548	30	5	motivation	motivation	NOUN
ejpam-4548	30	6	,	,	PUNCT
ejpam-4548	30	7	here	here	ADV
ejpam-4548	30	8	we	we	PRON
ejpam-4548	30	9	mention	mention	VERB
ejpam-4548	30	10	that	that	SCONJ
ejpam-4548	30	11	if	if	SCONJ
ejpam-4548	30	12	we	we	PRON
ejpam-4548	30	13	replace	replace	VERB
ejpam-4548	30	14	the	the	DET
ejpam-4548	30	15	lattice	lattice	PROPN
ejpam-4548	30	16	l	l	PROPN
ejpam-4548	30	17	,	,	PUNCT
ejpam-4548	30	18	in	in	ADP
ejpam-4548	30	19	our	our	PRON
ejpam-4548	30	20	work	work	NOUN
ejpam-4548	30	21	by	by	ADP
ejpam-4548	30	22	the	the	DET
ejpam-4548	30	23	closed	closed	ADJ
ejpam-4548	30	24	unit	unit	NOUN
ejpam-4548	30	25	interval	interval	NOUN
ejpam-4548	30	26	[	[	X
ejpam-4548	30	27	0	0	NUM
ejpam-4548	30	28	,	,	PUNCT
ejpam-4548	30	29	1	1	NUM
ejpam-4548	30	30	]	]	PUNCT
ejpam-4548	30	31	,	,	PUNCT
ejpam-4548	30	32	then	then	ADV
ejpam-4548	30	33	we	we	PRON
ejpam-4548	30	34	retrieve	retrieve	VERB
ejpam-4548	30	35	the	the	DET
ejpam-4548	30	36	corresponding	corresponding	ADJ
ejpam-4548	30	37	version	version	NOUN
ejpam-4548	30	38	of	of	ADP
ejpam-4548	30	39	fuzzy	fuzzy	ADJ
ejpam-4548	30	40	group	group	NOUN
ejpam-4548	30	41	theory	theory	NOUN
ejpam-4548	30	42	.	.	PUNCT
ejpam-4548	31	1	moreover	moreover	ADV
ejpam-4548	31	2	,	,	PUNCT
ejpam-4548	31	3	as	as	ADP
ejpam-4548	31	4	an	an	DET
ejpam-4548	31	5	application	application	NOUN
ejpam-4548	31	6	of	of	ADP
ejpam-4548	31	7	this	this	DET
ejpam-4548	31	8	theory	theory	NOUN
ejpam-4548	31	9	we	we	PRON
ejpam-4548	31	10	also	also	ADV
ejpam-4548	31	11	mention	mention	VERB
ejpam-4548	31	12	that	that	SCONJ
ejpam-4548	31	13	if	if	SCONJ
ejpam-4548	31	14	we	we	PRON
ejpam-4548	31	15	replace	replace	VERB
ejpam-4548	31	16	the	the	DET
ejpam-4548	31	17	lattice	lattice	NOUN
ejpam-4548	31	18	l	l	NOUN
ejpam-4548	31	19	by	by	ADP
ejpam-4548	31	20	the	the	DET
ejpam-4548	31	21	two	two	NUM
ejpam-4548	31	22	elements	element	NOUN
ejpam-4548	31	23	set	set	VERB
ejpam-4548	31	24	{	{	PUNCT
ejpam-4548	31	25	0	0	NUM
ejpam-4548	31	26	,	,	PUNCT
ejpam-4548	31	27	1	1	NUM
ejpam-4548	31	28	}	}	PUNCT
ejpam-4548	31	29	,	,	PUNCT
ejpam-4548	31	30	then	then	ADV
ejpam-4548	31	31	the	the	DET
ejpam-4548	31	32	results	result	NOUN
ejpam-4548	31	33	of	of	ADP
ejpam-4548	31	34	classical	classical	ADJ
ejpam-4548	31	35	group	group	NOUN
ejpam-4548	31	36	theory	theory	NOUN
ejpam-4548	31	37	follow	follow	VERB
ejpam-4548	31	38	as	as	ADP
ejpam-4548	31	39	simple	simple	ADJ
ejpam-4548	31	40	corollaries	corollary	NOUN
ejpam-4548	31	41	of	of	ADP
ejpam-4548	31	42	the	the	DET
ejpam-4548	31	43	corresponding	corresponding	ADJ
ejpam-4548	31	44	results	result	NOUN
ejpam-4548	31	45	of	of	ADP
ejpam-4548	31	46	l	l	NOUN
ejpam-4548	31	47	-	-	NOUN
ejpam-4548	31	48	group	group	NOUN
ejpam-4548	31	49	theory	theory	NOUN
ejpam-4548	31	50	.	.	PUNCT
ejpam-4548	32	1	moreover	moreover	ADV
ejpam-4548	32	2	,	,	PUNCT
ejpam-4548	32	3	this	this	DET
ejpam-4548	32	4	development	development	NOUN
ejpam-4548	32	5	of	of	ADP
ejpam-4548	32	6	l	l	PROPN
ejpam-4548	32	7	-	-	NOUN
ejpam-4548	32	8	group	group	NOUN
ejpam-4548	32	9	theory	theory	NOUN
ejpam-4548	32	10	is	be	AUX
ejpam-4548	32	11	beyond	beyond	ADP
ejpam-4548	32	12	the	the	DET
ejpam-4548	32	13	purview	purview	NOUN
ejpam-4548	32	14	of	of	ADP
ejpam-4548	32	15	metatheorem	metatheorem	PROPN
ejpam-4548	32	16	,	,	PUNCT
ejpam-4548	32	17	contrary	contrary	ADJ
ejpam-4548	32	18	to	to	ADP
ejpam-4548	32	19	the	the	DET
ejpam-4548	32	20	development	development	NOUN
ejpam-4548	32	21	of	of	ADP
ejpam-4548	32	22	fuzzy	fuzzy	ADJ
ejpam-4548	32	23	group	group	NOUN
ejpam-4548	32	24	theory	theory	NOUN
ejpam-4548	32	25	.	.	PUNCT
ejpam-4548	33	1	the	the	DET
ejpam-4548	33	2	development	development	NOUN
ejpam-4548	33	3	of	of	ADP
ejpam-4548	33	4	fuzzy	fuzzy	ADJ
ejpam-4548	33	5	group	group	NOUN
ejpam-4548	33	6	theory	theory	NOUN
ejpam-4548	33	7	could	could	AUX
ejpam-4548	33	8	not	not	PART
ejpam-4548	33	9	be	be	AUX
ejpam-4548	33	10	sustained	sustain	VERB
ejpam-4548	33	11	due	due	ADP
ejpam-4548	33	12	to	to	ADP
ejpam-4548	33	13	some	some	DET
ejpam-4548	33	14	inherent	inherent	ADJ
ejpam-4548	33	15	problems	problem	NOUN
ejpam-4548	33	16	and	and	CCONJ
ejpam-4548	33	17	the	the	DET
ejpam-4548	33	18	emergence	emergence	NOUN
ejpam-4548	33	19	of	of	ADP
ejpam-4548	33	20	metatheorem	metatheorem	VERB
ejpam-4548	33	21	formulated	formulate	VERB
ejpam-4548	33	22	by	by	ADP
ejpam-4548	33	23	head	head	NOUN
ejpam-4548	33	24	[	[	X
ejpam-4548	33	25	9	9	NUM
ejpam-4548	33	26	]	]	PUNCT
ejpam-4548	33	27	.	.	PUNCT
ejpam-4548	34	1	the	the	DET
ejpam-4548	34	2	hurdles	hurdle	NOUN
ejpam-4548	34	3	faced	face	VERB
ejpam-4548	34	4	by	by	ADP
ejpam-4548	34	5	the	the	DET
ejpam-4548	34	6	researchers	researcher	NOUN
ejpam-4548	34	7	in	in	ADP
ejpam-4548	34	8	this	this	DET
ejpam-4548	34	9	development	development	NOUN
ejpam-4548	34	10	are	be	AUX
ejpam-4548	34	11	removed	remove	VERB
ejpam-4548	34	12	in	in	ADP
ejpam-4548	34	13	l	l	NOUN
ejpam-4548	34	14	-	-	NOUN
ejpam-4548	34	15	group	group	NOUN
ejpam-4548	34	16	theory	theory	NOUN
ejpam-4548	34	17	by	by	ADP
ejpam-4548	34	18	considering	consider	VERB
ejpam-4548	34	19	the	the	DET
ejpam-4548	34	20	parent	parent	NOUN
ejpam-4548	34	21	structure	structure	NOUN
ejpam-4548	34	22	as	as	ADP
ejpam-4548	34	23	an	an	DET
ejpam-4548	34	24	l	l	NOUN
ejpam-4548	34	25	-	-	NOUN
ejpam-4548	34	26	group	group	NOUN
ejpam-4548	34	27	rather	rather	ADV
ejpam-4548	34	28	than	than	ADP
ejpam-4548	34	29	an	an	DET
ejpam-4548	34	30	ordinary	ordinary	ADJ
ejpam-4548	34	31	group	group	NOUN
ejpam-4548	34	32	.	.	PUNCT
ejpam-4548	35	1	many	many	ADJ
ejpam-4548	35	2	higher	high	ADJ
ejpam-4548	35	3	concepts	concept	NOUN
ejpam-4548	35	4	of	of	ADP
ejpam-4548	35	5	classical	classical	ADJ
ejpam-4548	35	6	group	group	NOUN
ejpam-4548	35	7	theory	theory	NOUN
ejpam-4548	35	8	such	such	ADJ
ejpam-4548	35	9	as	as	ADP
ejpam-4548	35	10	supersolvability	supersolvability	NOUN
ejpam-4548	35	11	,	,	PUNCT
ejpam-4548	35	12	nilpotency	nilpotency	NOUN
ejpam-4548	35	13	and	and	CCONJ
ejpam-4548	35	14	subnormality	subnormality	NOUN
ejpam-4548	35	15	could	could	AUX
ejpam-4548	35	16	not	not	PART
ejpam-4548	35	17	be	be	AUX
ejpam-4548	35	18	studied	study	VERB
ejpam-4548	35	19	in	in	ADP
ejpam-4548	35	20	the	the	DET
ejpam-4548	35	21	framework	framework	NOUN
ejpam-4548	35	22	of	of	ADP
ejpam-4548	35	23	fuzzy	fuzzy	ADJ
ejpam-4548	35	24	group	group	NOUN
ejpam-4548	35	25	theory	theory	NOUN
ejpam-4548	35	26	.	.	PUNCT
ejpam-4548	36	1	the	the	DET
ejpam-4548	36	2	above	above	ADV
ejpam-4548	36	3	mentioned	mention	VERB
ejpam-4548	36	4	modification	modification	NOUN
ejpam-4548	36	5	allowed	allow	VERB
ejpam-4548	36	6	the	the	DET
ejpam-4548	36	7	formation	formation	NOUN
ejpam-4548	36	8	of	of	ADP
ejpam-4548	36	9	normalizer	normalizer	NOUN
ejpam-4548	36	10	of	of	ADP
ejpam-4548	36	11	an	an	DET
ejpam-4548	36	12	l	l	NOUN
ejpam-4548	36	13	-	-	NOUN
ejpam-4548	36	14	subgroup	subgroup	NOUN
ejpam-4548	36	15	of	of	ADP
ejpam-4548	36	16	an	an	DET
ejpam-4548	36	17	l	l	NOUN
ejpam-4548	36	18	-	-	NOUN
ejpam-4548	36	19	group	group	NOUN
ejpam-4548	36	20	similar	similar	ADJ
ejpam-4548	36	21	to	to	ADP
ejpam-4548	36	22	that	that	PRON
ejpam-4548	36	23	of	of	ADP
ejpam-4548	36	24	classical	classical	ADJ
ejpam-4548	36	25	group	group	NOUN
ejpam-4548	36	26	theory.the	theory.the	DET
ejpam-4548	36	27	question	question	NOUN
ejpam-4548	36	28	of	of	ADP
ejpam-4548	36	29	formulation	formulation	NOUN
ejpam-4548	36	30	of	of	ADP
ejpam-4548	36	31	the	the	DET
ejpam-4548	36	32	notion	notion	NOUN
ejpam-4548	36	33	of	of	ADP
ejpam-4548	36	34	subnormality	subnormality	NOUN
ejpam-4548	36	35	in	in	ADP
ejpam-4548	36	36	lgroup	lgroup	ADJ
ejpam-4548	36	37	theory	theory	NOUN
ejpam-4548	36	38	leads	lead	VERB
ejpam-4548	36	39	to	to	ADP
ejpam-4548	36	40	the	the	DET
ejpam-4548	36	41	question	question	NOUN
ejpam-4548	36	42	of	of	ADP
ejpam-4548	36	43	formation	formation	NOUN
ejpam-4548	36	44	of	of	ADP
ejpam-4548	36	45	normal	normal	ADJ
ejpam-4548	36	46	closure	closure	NOUN
ejpam-4548	36	47	of	of	ADP
ejpam-4548	36	48	an	an	DET
ejpam-4548	36	49	l	l	NOUN
ejpam-4548	36	50	-	-	NOUN
ejpam-4548	36	51	subgroup	subgroup	NOUN
ejpam-4548	36	52	in	in	ADP
ejpam-4548	36	53	an	an	DET
ejpam-4548	36	54	l	l	NOUN
ejpam-4548	36	55	-	-	NOUN
ejpam-4548	36	56	group	group	NOUN
ejpam-4548	36	57	.	.	PUNCT
ejpam-4548	37	1	the	the	DET
ejpam-4548	37	2	concept	concept	NOUN
ejpam-4548	37	3	of	of	ADP
ejpam-4548	37	4	normal	normal	ADJ
ejpam-4548	37	5	closure	closure	NOUN
ejpam-4548	37	6	of	of	ADP
ejpam-4548	37	7	an	an	DET
ejpam-4548	37	8	l	l	NOUN
ejpam-4548	37	9	-	-	NOUN
ejpam-4548	37	10	subgroup	subgroup	NOUN
ejpam-4548	37	11	in	in	ADP
ejpam-4548	37	12	an	an	DET
ejpam-4548	37	13	l	l	NOUN
ejpam-4548	37	14	-	-	NOUN
ejpam-4548	37	15	group	group	NOUN
ejpam-4548	37	16	should	should	AUX
ejpam-4548	37	17	be	be	AUX
ejpam-4548	37	18	defined	define	VERB
ejpam-4548	37	19	in	in	ADP
ejpam-4548	37	20	such	such	DET
ejpam-4548	37	21	a	a	DET
ejpam-4548	37	22	manner	manner	NOUN
ejpam-4548	37	23	that	that	SCONJ
ejpam-4548	37	24	the	the	DET
ejpam-4548	37	25	normal	normal	ADJ
ejpam-4548	37	26	closure	closure	NOUN
ejpam-4548	37	27	series	series	NOUN
ejpam-4548	37	28	arising	arise	VERB
ejpam-4548	37	29	by	by	ADP
ejpam-4548	37	30	this	this	DET
ejpam-4548	37	31	normal	normal	ADJ
ejpam-4548	37	32	closure	closure	NOUN
ejpam-4548	37	33	should	should	AUX
ejpam-4548	37	34	be	be	AUX
ejpam-4548	37	35	compatible	compatible	ADJ
ejpam-4548	37	36	with	with	ADP
ejpam-4548	37	37	a	a	DET
ejpam-4548	37	38	subnormal	subnormal	ADJ
ejpam-4548	37	39	series	series	NOUN
ejpam-4548	37	40	arising	arise	VERB
ejpam-4548	37	41	by	by	ADP
ejpam-4548	37	42	the	the	DET
ejpam-4548	37	43	given	give	VERB
ejpam-4548	37	44	l	l	NOUN
ejpam-4548	37	45	-	-	NOUN
ejpam-4548	37	46	subgroup	subgroup	NOUN
ejpam-4548	37	47	likewise	likewise	ADV
ejpam-4548	37	48	its	its	PRON
ejpam-4548	37	49	classical	classical	ADJ
ejpam-4548	37	50	counterparts	counterpart	NOUN
ejpam-4548	37	51	.	.	PUNCT
ejpam-4548	38	1	for	for	ADP
ejpam-4548	38	2	this	this	DET
ejpam-4548	38	3	purpose	purpose	NOUN
ejpam-4548	38	4	,	,	PUNCT
ejpam-4548	38	5	we	we	PRON
ejpam-4548	38	6	define	define	VERB
ejpam-4548	38	7	the	the	DET
ejpam-4548	38	8	conjugate	conjugate	NOUN
ejpam-4548	38	9	of	of	ADP
ejpam-4548	38	10	an	an	DET
ejpam-4548	38	11	l	l	NOUN
ejpam-4548	38	12	-	-	NOUN
ejpam-4548	38	13	subset	subset	NOUN
ejpam-4548	38	14	by	by	ADP
ejpam-4548	38	15	an	an	DET
ejpam-4548	38	16	l	l	NOUN
ejpam-4548	38	17	-	-	NOUN
ejpam-4548	38	18	subset	subset	NOUN
ejpam-4548	38	19	of	of	ADP
ejpam-4548	38	20	an	an	DET
ejpam-4548	38	21	l	l	NOUN
ejpam-4548	38	22	-	-	NOUN
ejpam-4548	38	23	group	group	NOUN
ejpam-4548	38	24	instead	instead	ADV
ejpam-4548	38	25	of	of	ADP
ejpam-4548	38	26	a	a	DET
ejpam-4548	38	27	crisp	crisp	ADJ
ejpam-4548	38	28	point	point	NOUN
ejpam-4548	38	29	.	.	PUNCT
ejpam-4548	39	1	then	then	ADV
ejpam-4548	39	2	,	,	PUNCT
ejpam-4548	39	3	the	the	DET
ejpam-4548	39	4	l	l	PROPN
ejpam-4548	39	5	-	-	NOUN
ejpam-4548	39	6	subgroup	subgroup	NOUN
ejpam-4548	39	7	generated	generate	VERB
ejpam-4548	39	8	by	by	ADP
ejpam-4548	39	9	the	the	DET
ejpam-4548	39	10	conjugate	conjugate	NOUN
ejpam-4548	39	11	of	of	ADP
ejpam-4548	39	12	an	an	DET
ejpam-4548	39	13	l	l	NOUN
ejpam-4548	39	14	-	-	NOUN
ejpam-4548	39	15	subgroup	subgroup	NOUN
ejpam-4548	39	16	by	by	ADP
ejpam-4548	39	17	the	the	PRON
ejpam-4548	39	18	given	give	VERB
ejpam-4548	39	19	parent	parent	NOUN
ejpam-4548	39	20	l	l	NOUN
ejpam-4548	39	21	-	-	NOUN
ejpam-4548	39	22	group	group	NOUN
ejpam-4548	39	23	is	be	AUX
ejpam-4548	39	24	called	call	VERB
ejpam-4548	39	25	the	the	DET
ejpam-4548	39	26	normal	normal	ADJ
ejpam-4548	39	27	closure	closure	NOUN
ejpam-4548	39	28	of	of	ADP
ejpam-4548	39	29	the	the	DET
ejpam-4548	39	30	given	give	VERB
ejpam-4548	39	31	l	l	NOUN
ejpam-4548	39	32	-	-	NOUN
ejpam-4548	39	33	subgroup	subgroup	NOUN
ejpam-4548	39	34	.	.	PUNCT
ejpam-4548	40	1	this	this	DET
ejpam-4548	40	2	notion	notion	NOUN
ejpam-4548	40	3	satisfies	satisfy	VERB
ejpam-4548	40	4	most	most	ADJ
ejpam-4548	40	5	of	of	ADP
ejpam-4548	40	6	the	the	DET
ejpam-4548	40	7	desirable	desirable	ADJ
ejpam-4548	40	8	properties	property	NOUN
ejpam-4548	40	9	related	relate	VERB
ejpam-4548	40	10	to	to	ADP
ejpam-4548	40	11	this	this	DET
ejpam-4548	40	12	concept	concept	NOUN
ejpam-4548	40	13	.	.	PUNCT
ejpam-4548	41	1	for	for	ADP
ejpam-4548	41	2	example	example	NOUN
ejpam-4548	41	3	,	,	PUNCT
ejpam-4548	41	4	the	the	DET
ejpam-4548	41	5	compatibility	compatibility	NOUN
ejpam-4548	41	6	of	of	ADP
ejpam-4548	41	7	this	this	DET
ejpam-4548	41	8	notion	notion	NOUN
ejpam-4548	41	9	with	with	ADP
ejpam-4548	41	10	that	that	PRON
ejpam-4548	41	11	of	of	ADP
ejpam-4548	41	12	commutator	commutator	NOUN
ejpam-4548	41	13	l	l	PROPN
ejpam-4548	41	14	-	-	NOUN
ejpam-4548	41	15	subgroup	subgroup	NOUN
ejpam-4548	41	16	of	of	ADP
ejpam-4548	41	17	an	an	DET
ejpam-4548	41	18	l	l	NOUN
ejpam-4548	41	19	-	-	NOUN
ejpam-4548	41	20	group	group	NOUN
ejpam-4548	41	21	exists	exist	VERB
ejpam-4548	41	22	like	like	ADP
ejpam-4548	41	23	its	its	PRON
ejpam-4548	41	24	counterpart	counterpart	NOUN
ejpam-4548	41	25	in	in	ADP
ejpam-4548	41	26	classical	classical	ADJ
ejpam-4548	41	27	group	group	NOUN
ejpam-4548	41	28	theory	theory	NOUN
ejpam-4548	41	29	.	.	PUNCT
ejpam-4548	42	1	then	then	ADV
ejpam-4548	42	2	,	,	PUNCT
ejpam-4548	42	3	this	this	DET
ejpam-4548	42	4	notion	notion	NOUN
ejpam-4548	42	5	is	be	AUX
ejpam-4548	42	6	utilized	utilize	VERB
ejpam-4548	42	7	n.	n.	NOUN
ejpam-4548	42	8	ajmal	ajmal	PROPN
ejpam-4548	42	9	,	,	PUNCT
ejpam-4548	42	10	i.	i.	PROPN
ejpam-4548	42	11	jahan	jahan	PROPN
ejpam-4548	42	12	.	.	PROPN
ejpam-4548	42	13	,	,	PUNCT
ejpam-4548	42	14	b.	b.	PROPN
ejpam-4548	42	15	davvaz	davvaz	PROPN
ejpam-4548	42	16	/	/	SYM
ejpam-4548	42	17	eur	eur	PROPN
ejpam-4548	42	18	.	.	PUNCT
ejpam-4548	43	1	j.	j.	PROPN
ejpam-4548	43	2	pure	pure	PROPN
ejpam-4548	43	3	appl	appl	PROPN
ejpam-4548	43	4	.	.	PROPN
ejpam-4548	43	5	math	math	PROPN
ejpam-4548	43	6	,	,	PUNCT
ejpam-4548	43	7	15	15	NUM
ejpam-4548	43	8	(	(	PUNCT
ejpam-4548	43	9	4	4	NUM
ejpam-4548	43	10	)	)	PUNCT
ejpam-4548	43	11	(	(	PUNCT
ejpam-4548	43	12	2022	2022	NUM
ejpam-4548	43	13	)	)	PUNCT
ejpam-4548	43	14	,	,	PUNCT
ejpam-4548	43	15	2086	2086	NUM
ejpam-4548	43	16	-	-	SYM
ejpam-4548	43	17	2115	2115	NUM
ejpam-4548	43	18	2088	2088	NUM
ejpam-4548	43	19	in	in	ADP
ejpam-4548	43	20	the	the	DET
ejpam-4548	43	21	formation	formation	NOUN
ejpam-4548	43	22	of	of	ADP
ejpam-4548	43	23	normal	normal	ADJ
ejpam-4548	43	24	closure	closure	NOUN
ejpam-4548	43	25	series	series	NOUN
ejpam-4548	43	26	of	of	ADP
ejpam-4548	43	27	an	an	DET
ejpam-4548	43	28	l	l	NOUN
ejpam-4548	43	29	-	-	NOUN
ejpam-4548	43	30	subgroup	subgroup	NOUN
ejpam-4548	43	31	of	of	ADP
ejpam-4548	43	32	an	an	DET
ejpam-4548	43	33	l	l	NOUN
ejpam-4548	43	34	-	-	NOUN
ejpam-4548	43	35	group	group	NOUN
ejpam-4548	43	36	which	which	PRON
ejpam-4548	43	37	leads	lead	VERB
ejpam-4548	43	38	to	to	ADP
ejpam-4548	43	39	the	the	DET
ejpam-4548	43	40	definition	definition	NOUN
ejpam-4548	43	41	of	of	ADP
ejpam-4548	43	42	subnormal	subnormal	ADJ
ejpam-4548	43	43	l	l	PROPN
ejpam-4548	43	44	-	-	NOUN
ejpam-4548	43	45	subgroup	subgroup	NOUN
ejpam-4548	43	46	.	.	PUNCT
ejpam-4548	44	1	the	the	DET
ejpam-4548	44	2	notion	notion	NOUN
ejpam-4548	44	3	of	of	ADP
ejpam-4548	44	4	subnormal	subnormal	ADJ
ejpam-4548	44	5	series	series	NOUN
ejpam-4548	44	6	for	for	ADP
ejpam-4548	44	7	an	an	DET
ejpam-4548	44	8	lsubgroup	lsubgroup	NOUN
ejpam-4548	44	9	is	be	AUX
ejpam-4548	44	10	introduced	introduce	VERB
ejpam-4548	44	11	which	which	PRON
ejpam-4548	44	12	exihibits	exihibit	VERB
ejpam-4548	44	13	the	the	DET
ejpam-4548	44	14	desired	desire	VERB
ejpam-4548	44	15	relationship	relationship	NOUN
ejpam-4548	44	16	with	with	ADP
ejpam-4548	44	17	the	the	DET
ejpam-4548	44	18	normal	normal	ADJ
ejpam-4548	44	19	closure	closure	NOUN
ejpam-4548	44	20	series	series	NOUN
ejpam-4548	44	21	.	.	PUNCT
ejpam-4548	45	1	in	in	ADP
ejpam-4548	45	2	the	the	DET
ejpam-4548	45	3	end	end	NOUN
ejpam-4548	45	4	,	,	PUNCT
ejpam-4548	45	5	we	we	PRON
ejpam-4548	45	6	establish	establish	VERB
ejpam-4548	45	7	the	the	DET
ejpam-4548	45	8	inter	inter	NOUN
ejpam-4548	45	9	-	-	NOUN
ejpam-4548	45	10	connection	connection	NOUN
ejpam-4548	45	11	of	of	ADP
ejpam-4548	45	12	subnormality	subnormality	NOUN
ejpam-4548	45	13	of	of	ADP
ejpam-4548	45	14	an	an	DET
ejpam-4548	45	15	l	l	NOUN
ejpam-4548	45	16	-	-	NOUN
ejpam-4548	45	17	subgroup	subgroup	NOUN
ejpam-4548	45	18	with	with	ADP
ejpam-4548	45	19	the	the	DET
ejpam-4548	45	20	subnormality	subnormality	NOUN
ejpam-4548	45	21	in	in	ADP
ejpam-4548	45	22	the	the	DET
ejpam-4548	45	23	usual	usual	ADJ
ejpam-4548	45	24	group	group	NOUN
ejpam-4548	45	25	theoretic	theoretic	ADJ
ejpam-4548	45	26	sense	sense	NOUN
ejpam-4548	45	27	of	of	ADP
ejpam-4548	45	28	the	the	DET
ejpam-4548	45	29	level	level	NOUN
ejpam-4548	45	30	subsets	subset	NOUN
ejpam-4548	45	31	of	of	ADP
ejpam-4548	45	32	the	the	DET
ejpam-4548	45	33	given	give	VERB
ejpam-4548	45	34	l	l	NOUN
ejpam-4548	45	35	-	-	NOUN
ejpam-4548	45	36	subgroup	subgroup	NOUN
ejpam-4548	45	37	by	by	ADP
ejpam-4548	45	38	providing	provide	VERB
ejpam-4548	45	39	a	a	DET
ejpam-4548	45	40	result	result	NOUN
ejpam-4548	45	41	which	which	PRON
ejpam-4548	45	42	is	be	AUX
ejpam-4548	45	43	called	call	VERB
ejpam-4548	45	44	the	the	DET
ejpam-4548	45	45	level	level	NOUN
ejpam-4548	45	46	subset	subset	NOUN
ejpam-4548	45	47	characterization	characterization	NOUN
ejpam-4548	45	48	.	.	PUNCT
ejpam-4548	46	1	finally	finally	ADV
ejpam-4548	46	2	,	,	PUNCT
ejpam-4548	46	3	by	by	ADP
ejpam-4548	46	4	an	an	DET
ejpam-4548	46	5	application	application	NOUN
ejpam-4548	46	6	of	of	ADP
ejpam-4548	46	7	this	this	DET
ejpam-4548	46	8	result	result	NOUN
ejpam-4548	46	9	we	we	PRON
ejpam-4548	46	10	establish	establish	VERB
ejpam-4548	46	11	that	that	SCONJ
ejpam-4548	46	12	every	every	DET
ejpam-4548	46	13	subgroup	subgroup	NOUN
ejpam-4548	46	14	of	of	ADP
ejpam-4548	46	15	a	a	DET
ejpam-4548	46	16	nilpotent	nilpotent	ADJ
ejpam-4548	46	17	l	l	NOUN
ejpam-4548	46	18	-	-	NOUN
ejpam-4548	46	19	subgroup	subgroup	NOUN
ejpam-4548	46	20	is	be	AUX
ejpam-4548	46	21	subnormal	subnormal	ADJ
ejpam-4548	46	22	.	.	PUNCT
ejpam-4548	47	1	although	although	SCONJ
ejpam-4548	47	2	so	so	ADV
ejpam-4548	47	3	much	much	ADJ
ejpam-4548	47	4	has	have	AUX
ejpam-4548	47	5	been	be	AUX
ejpam-4548	47	6	said	say	VERB
ejpam-4548	47	7	and	and	CCONJ
ejpam-4548	47	8	done	do	VERB
ejpam-4548	47	9	in	in	ADP
ejpam-4548	47	10	the	the	DET
ejpam-4548	47	11	area	area	NOUN
ejpam-4548	47	12	of	of	ADP
ejpam-4548	47	13	l	l	PROPN
ejpam-4548	47	14	-	-	NOUN
ejpam-4548	47	15	group	group	NOUN
ejpam-4548	47	16	theory	theory	NOUN
ejpam-4548	47	17	,	,	PUNCT
ejpam-4548	47	18	but	but	CCONJ
ejpam-4548	47	19	still	still	ADV
ejpam-4548	47	20	researchers	researcher	NOUN
ejpam-4548	47	21	from	from	ADP
ejpam-4548	47	22	all	all	ADV
ejpam-4548	47	23	over	over	ADP
ejpam-4548	47	24	the	the	DET
ejpam-4548	47	25	world	world	NOUN
ejpam-4548	47	26	are	be	AUX
ejpam-4548	47	27	involved	involve	VERB
ejpam-4548	47	28	in	in	ADP
ejpam-4548	47	29	investigating	investigate	VERB
ejpam-4548	47	30	various	various	ADJ
ejpam-4548	47	31	types	type	NOUN
ejpam-4548	47	32	of	of	ADP
ejpam-4548	47	33	structures	structure	NOUN
ejpam-4548	47	34	in	in	ADP
ejpam-4548	47	35	the	the	DET
ejpam-4548	47	36	fuzzy	fuzzy	ADJ
ejpam-4548	47	37	environments	environment	NOUN
ejpam-4548	47	38	.	.	PUNCT
ejpam-4548	48	1	a	a	DET
ejpam-4548	48	2	significant	significant	ADJ
ejpam-4548	48	3	attempt	attempt	NOUN
ejpam-4548	48	4	made	make	VERB
ejpam-4548	48	5	in	in	ADP
ejpam-4548	48	6	this	this	DET
ejpam-4548	48	7	direction	direction	NOUN
ejpam-4548	48	8	is	be	AUX
ejpam-4548	48	9	due	due	ADJ
ejpam-4548	48	10	to	to	ADP
ejpam-4548	48	11	a.	a.	NOUN
ejpam-4548	48	12	razzaque	razzaque	NOUN
ejpam-4548	48	13	and	and	CCONJ
ejpam-4548	48	14	a.	a.	NOUN
ejpam-4548	48	15	razaq	razaq	NOUN
ejpam-4548	48	16	[	[	X
ejpam-4548	48	17	12	12	NUM
ejpam-4548	48	18	]	]	PUNCT
ejpam-4548	48	19	wherein	wherein	SCONJ
ejpam-4548	48	20	they	they	PRON
ejpam-4548	48	21	have	have	AUX
ejpam-4548	48	22	defined	define	VERB
ejpam-4548	48	23	and	and	CCONJ
ejpam-4548	48	24	studied	study	VERB
ejpam-4548	48	25	a	a	DET
ejpam-4548	48	26	class	class	NOUN
ejpam-4548	48	27	of	of	ADP
ejpam-4548	48	28	fuzzy	fuzzy	ADJ
ejpam-4548	48	29	structure	structure	NOUN
ejpam-4548	48	30	which	which	PRON
ejpam-4548	48	31	properly	properly	ADV
ejpam-4548	48	32	contains	contain	VERB
ejpam-4548	48	33	the	the	DET
ejpam-4548	48	34	classes	class	NOUN
ejpam-4548	48	35	of	of	ADP
ejpam-4548	48	36	intuitionistic	intuitionistic	ADJ
ejpam-4548	48	37	fuzzy	fuzzy	ADJ
ejpam-4548	48	38	subgroups	subgroup	NOUN
ejpam-4548	48	39	and	and	CCONJ
ejpam-4548	48	40	pythagorean	pythagorean	PROPN
ejpam-4548	48	41	fuzzy	fuzzy	ADJ
ejpam-4548	48	42	subgroups	subgroup	NOUN
ejpam-4548	48	43	as	as	ADP
ejpam-4548	48	44	its	its	PRON
ejpam-4548	48	45	subclasses	subclass	NOUN
ejpam-4548	48	46	.	.	PUNCT
ejpam-4548	49	1	2	2	X
ejpam-4548	49	2	.	.	NUM
ejpam-4548	49	3	preliminaries	preliminary	NOUN
ejpam-4548	49	4	throughout	throughout	ADP
ejpam-4548	49	5	this	this	DET
ejpam-4548	49	6	paper	paper	NOUN
ejpam-4548	49	7	l	l	NOUN
ejpam-4548	50	1	=	=	PUNCT
ejpam-4548	50	2	⟨l,≤,∨,∧⟩	⟨l,≤,∨,∧⟩	NOUN
ejpam-4548	50	3	denotes	denote	VERB
ejpam-4548	50	4	a	a	DET
ejpam-4548	50	5	complete	complete	ADJ
ejpam-4548	50	6	and	and	CCONJ
ejpam-4548	50	7	completely	completely	ADV
ejpam-4548	50	8	distributive	distributive	ADJ
ejpam-4548	50	9	lattice	lattice	NOUN
ejpam-4548	50	10	where	where	SCONJ
ejpam-4548	50	11	‘	'	PUNCT
ejpam-4548	50	12	≤	≤	NUM
ejpam-4548	50	13	’	'	PUNCT
ejpam-4548	50	14	denotes	denote	VERB
ejpam-4548	50	15	the	the	DET
ejpam-4548	50	16	partial	partial	ADJ
ejpam-4548	50	17	ordering	ordering	NOUN
ejpam-4548	50	18	of	of	ADP
ejpam-4548	50	19	l	l	NOUN
ejpam-4548	50	20	,	,	PUNCT
ejpam-4548	50	21	the	the	DET
ejpam-4548	50	22	join	join	NOUN
ejpam-4548	50	23	(	(	PUNCT
ejpam-4548	50	24	sup	sup	NOUN
ejpam-4548	50	25	)	)	PUNCT
ejpam-4548	50	26	and	and	CCONJ
ejpam-4548	50	27	the	the	DET
ejpam-4548	50	28	meet	meet	NOUN
ejpam-4548	50	29	(	(	PUNCT
ejpam-4548	50	30	inf	inf	NOUN
ejpam-4548	50	31	)	)	PUNCT
ejpam-4548	50	32	of	of	ADP
ejpam-4548	50	33	the	the	DET
ejpam-4548	50	34	elements	element	NOUN
ejpam-4548	50	35	of	of	ADP
ejpam-4548	50	36	l	l	NOUN
ejpam-4548	50	37	are	be	AUX
ejpam-4548	50	38	denoted	denote	VERB
ejpam-4548	50	39	by	by	ADP
ejpam-4548	50	40	‘	'	PUNCT
ejpam-4548	50	41	∨	∨	NOUN
ejpam-4548	50	42	’	'	PUNCT
ejpam-4548	50	43	and	and	CCONJ
ejpam-4548	50	44	‘	'	PUNCT
ejpam-4548	50	45	∧	∧	NOUN
ejpam-4548	50	46	’	'	PUNCT
ejpam-4548	50	47	respectively	respectively	ADV
ejpam-4548	50	48	.	.	PUNCT
ejpam-4548	51	1	also	also	ADV
ejpam-4548	51	2	,	,	PUNCT
ejpam-4548	51	3	we	we	PRON
ejpam-4548	51	4	write	write	VERB
ejpam-4548	51	5	1	1	NUM
ejpam-4548	51	6	and	and	CCONJ
ejpam-4548	51	7	0	0	NUM
ejpam-4548	51	8	for	for	ADP
ejpam-4548	51	9	maximal	maximal	ADJ
ejpam-4548	51	10	and	and	CCONJ
ejpam-4548	51	11	minimal	minimal	ADJ
ejpam-4548	51	12	elements	element	NOUN
ejpam-4548	51	13	of	of	ADP
ejpam-4548	51	14	l	l	NOUN
ejpam-4548	51	15	respectively	respectively	ADV
ejpam-4548	51	16	.	.	PUNCT
ejpam-4548	52	1	the	the	DET
ejpam-4548	52	2	definition	definition	NOUN
ejpam-4548	52	3	of	of	ADP
ejpam-4548	52	4	a	a	DET
ejpam-4548	52	5	completely	completely	ADV
ejpam-4548	52	6	distributive	distributive	ADJ
ejpam-4548	52	7	lattice	lattice	NOUN
ejpam-4548	52	8	is	be	AUX
ejpam-4548	52	9	well	well	ADV
ejpam-4548	52	10	known	know	VERB
ejpam-4548	52	11	in	in	ADP
ejpam-4548	52	12	the	the	DET
ejpam-4548	52	13	literature	literature	NOUN
ejpam-4548	52	14	and	and	CCONJ
ejpam-4548	52	15	can	can	AUX
ejpam-4548	52	16	be	be	AUX
ejpam-4548	52	17	found	find	VERB
ejpam-4548	52	18	in	in	ADP
ejpam-4548	52	19	any	any	DET
ejpam-4548	52	20	standard	standard	ADJ
ejpam-4548	52	21	text	text	NOUN
ejpam-4548	52	22	on	on	ADP
ejpam-4548	52	23	the	the	DET
ejpam-4548	52	24	subject	subject	NOUN
ejpam-4548	52	25	.	.	PUNCT
ejpam-4548	53	1	if	if	SCONJ
ejpam-4548	53	2	{	{	PUNCT
ejpam-4548	53	3	ji	ji	NOUN
ejpam-4548	53	4	:	:	PUNCT
ejpam-4548	53	5	i	i	PRON
ejpam-4548	53	6	∈	∈	VERB
ejpam-4548	53	7	i	i	PRON
ejpam-4548	53	8	}	}	PUNCT
ejpam-4548	53	9	is	be	AUX
ejpam-4548	53	10	any	any	DET
ejpam-4548	53	11	family	family	NOUN
ejpam-4548	53	12	of	of	ADP
ejpam-4548	53	13	subsets	subset	NOUN
ejpam-4548	53	14	of	of	ADP
ejpam-4548	53	15	a	a	DET
ejpam-4548	53	16	complete	complete	ADJ
ejpam-4548	53	17	lattice	lattice	NOUN
ejpam-4548	53	18	l	l	NOUN
ejpam-4548	53	19	,	,	PUNCT
ejpam-4548	53	20	let	let	VERB
ejpam-4548	53	21	f	f	PRON
ejpam-4548	53	22	denote	denote	VERB
ejpam-4548	53	23	the	the	DET
ejpam-4548	53	24	set	set	NOUN
ejpam-4548	53	25	of	of	ADP
ejpam-4548	53	26	choice	choice	NOUN
ejpam-4548	53	27	functions	function	NOUN
ejpam-4548	53	28	for	for	ADP
ejpam-4548	53	29	ji	ji	PROPN
ejpam-4548	53	30	,	,	PUNCT
ejpam-4548	53	31	i.e.	i.e.	X
ejpam-4548	53	32	functions	function	NOUN
ejpam-4548	54	1	f	f	NOUN
ejpam-4548	54	2	:	:	PUNCT
ejpam-4548	54	3	i	i	PROPN
ejpam-4548	54	4	→	→	SYM
ejpam-4548	54	5	∏	∏	PROPN
ejpam-4548	54	6	i∈i	i∈i	NOUN
ejpam-4548	54	7	ji	ji	PROPN
ejpam-4548	54	8	such	such	ADJ
ejpam-4548	54	9	that	that	SCONJ
ejpam-4548	54	10	f(i	f(i	PROPN
ejpam-4548	54	11	)	)	PUNCT
ejpam-4548	54	12	∈	∈	PROPN
ejpam-4548	54	13	ji	ji	PROPN
ejpam-4548	54	14	for	for	ADP
ejpam-4548	54	15	each	each	DET
ejpam-4548	54	16	i	i	PRON
ejpam-4548	54	17	∈	∈	PROPN
ejpam-4548	54	18	i.	i.	NOUN
ejpam-4548	54	19	then	then	ADV
ejpam-4548	54	20	,	,	PUNCT
ejpam-4548	54	21	we	we	PRON
ejpam-4548	54	22	say	say	VERB
ejpam-4548	54	23	that	that	SCONJ
ejpam-4548	54	24	l	l	NOUN
ejpam-4548	54	25	is	be	AUX
ejpam-4548	54	26	a	a	DET
ejpam-4548	54	27	completely	completely	ADV
ejpam-4548	54	28	distributive	distributive	ADJ
ejpam-4548	54	29	lattice	lattice	NOUN
ejpam-4548	54	30	,	,	PUNCT
ejpam-4548	54	31	if∧{∨	if∧{∨	PROPN
ejpam-4548	54	32	i∈i	i∈i	ADJ
ejpam-4548	54	33	ji	ji	NOUN
ejpam-4548	54	34	}	}	PUNCT
ejpam-4548	54	35	=	=	PUNCT
ejpam-4548	54	36	∨	∨	NUM
ejpam-4548	54	37	f∈f	f∈f	NOUN
ejpam-4548	54	38	{	{	PUNCT
ejpam-4548	54	39	∧	∧	PROPN
ejpam-4548	54	40	i∈i	i∈i	ADJ
ejpam-4548	54	41	f(i	f(i	PROPN
ejpam-4548	54	42	)	)	PUNCT
ejpam-4548	54	43	}	}	PUNCT
ejpam-4548	54	44	.	.	PUNCT
ejpam-4548	55	1	the	the	DET
ejpam-4548	55	2	above	above	ADJ
ejpam-4548	55	3	law	law	NOUN
ejpam-4548	55	4	is	be	AUX
ejpam-4548	55	5	known	know	VERB
ejpam-4548	55	6	as	as	ADP
ejpam-4548	55	7	the	the	DET
ejpam-4548	55	8	complete	complete	ADJ
ejpam-4548	55	9	distributive	distributive	ADJ
ejpam-4548	55	10	law	law	NOUN
ejpam-4548	55	11	.	.	PUNCT
ejpam-4548	56	1	moreover	moreover	ADV
ejpam-4548	56	2	,	,	PUNCT
ejpam-4548	56	3	a	a	DET
ejpam-4548	56	4	lattice	lattice	NOUN
ejpam-4548	56	5	l	l	NOUN
ejpam-4548	56	6	is	be	AUX
ejpam-4548	56	7	said	say	VERB
ejpam-4548	56	8	to	to	PART
ejpam-4548	56	9	be	be	AUX
ejpam-4548	56	10	infinitely	infinitely	ADV
ejpam-4548	56	11	meet	meet	VERB
ejpam-4548	56	12	distributive	distributive	ADJ
ejpam-4548	56	13	if	if	SCONJ
ejpam-4548	56	14	for	for	ADP
ejpam-4548	56	15	every	every	DET
ejpam-4548	56	16	subset	subset	NOUN
ejpam-4548	56	17	{	{	PUNCT
ejpam-4548	56	18	bβ	bβ	NOUN
ejpam-4548	56	19	:	:	PUNCT
ejpam-4548	56	20	β	β	X
ejpam-4548	56	21	∈	∈	PROPN
ejpam-4548	56	22	b	b	X
ejpam-4548	56	23	}	}	PUNCT
ejpam-4548	56	24	of	of	ADP
ejpam-4548	56	25	l	l	NOUN
ejpam-4548	56	26	,	,	PUNCT
ejpam-4548	56	27	we	we	PRON
ejpam-4548	56	28	have	have	VERB
ejpam-4548	56	29	:	:	PUNCT
ejpam-4548	56	30	a	a	DET
ejpam-4548	56	31	∧	∧	PROPN
ejpam-4548	56	32			PUNCT
ejpam-4548	56	33	∨	∨	NOUN
ejpam-4548	56	34	β∈b	β∈b	NOUN
ejpam-4548	56	35	bβ	bβ	NOUN
ejpam-4548	56	36			NOUN
ejpam-4548	56	37	=	=	PUNCT
ejpam-4548	56	38	∨	∨	NUM
ejpam-4548	56	39	β∈b	β∈b	NOUN
ejpam-4548	56	40	{	{	PUNCT
ejpam-4548	56	41	a	a	DET
ejpam-4548	56	42	∧	∧	PROPN
ejpam-4548	56	43	bβ	bβ	NOUN
ejpam-4548	56	44	}	}	PUNCT
ejpam-4548	56	45	,	,	PUNCT
ejpam-4548	56	46	provided	provide	VERB
ejpam-4548	56	47	l	l	NOUN
ejpam-4548	56	48	is	be	AUX
ejpam-4548	56	49	join	join	VERB
ejpam-4548	56	50	complete	complete	ADJ
ejpam-4548	56	51	.	.	PUNCT
ejpam-4548	57	1	the	the	DET
ejpam-4548	57	2	above	above	ADJ
ejpam-4548	57	3	law	law	NOUN
ejpam-4548	57	4	is	be	AUX
ejpam-4548	57	5	known	know	VERB
ejpam-4548	57	6	as	as	ADP
ejpam-4548	57	7	the	the	DET
ejpam-4548	57	8	infinitely	infinitely	ADV
ejpam-4548	57	9	meet	meet	VERB
ejpam-4548	57	10	distributive	distributive	ADJ
ejpam-4548	57	11	law	law	NOUN
ejpam-4548	57	12	.	.	PUNCT
ejpam-4548	58	1	the	the	DET
ejpam-4548	58	2	definition	definition	NOUN
ejpam-4548	58	3	of	of	ADP
ejpam-4548	58	4	infinitely	infinitely	ADV
ejpam-4548	58	5	join	join	VERB
ejpam-4548	58	6	distributive	distributive	ADJ
ejpam-4548	58	7	lattice	lattice	NOUN
ejpam-4548	58	8	is	be	AUX
ejpam-4548	58	9	dual	dual	ADJ
ejpam-4548	58	10	to	to	ADP
ejpam-4548	58	11	the	the	DET
ejpam-4548	58	12	above	above	ADJ
ejpam-4548	58	13	definition	definition	NOUN
ejpam-4548	58	14	.	.	PUNCT
ejpam-4548	59	1	a	a	DET
ejpam-4548	59	2	complete	complete	ADJ
ejpam-4548	59	3	lattice	lattice	NOUN
ejpam-4548	59	4	which	which	PRON
ejpam-4548	59	5	satisfies	satisfy	VERB
ejpam-4548	59	6	infinitely	infinitely	ADV
ejpam-4548	59	7	meet	meet	VERB
ejpam-4548	59	8	distributive	distributive	ADJ
ejpam-4548	59	9	law	law	NOUN
ejpam-4548	59	10	is	be	AUX
ejpam-4548	59	11	known	know	VERB
ejpam-4548	59	12	as	as	ADP
ejpam-4548	59	13	a	a	DET
ejpam-4548	59	14	complete	complete	ADJ
ejpam-4548	59	15	heyting	heyting	NOUN
ejpam-4548	59	16	algebra	algebra	NOUN
ejpam-4548	59	17	or	or	CCONJ
ejpam-4548	59	18	a	a	DET
ejpam-4548	59	19	frame.therefore	frame.therefore	NOUN
ejpam-4548	59	20	,	,	PUNCT
ejpam-4548	59	21	a	a	DET
ejpam-4548	59	22	completely	completely	ADV
ejpam-4548	59	23	distributive	distributive	ADJ
ejpam-4548	59	24	lattice	lattice	NOUN
ejpam-4548	59	25	is	be	AUX
ejpam-4548	59	26	always	always	ADV
ejpam-4548	59	27	a	a	DET
ejpam-4548	59	28	complete	complete	ADJ
ejpam-4548	59	29	heyting	heyting	NOUN
ejpam-4548	59	30	algebra	algebra	NOUN
ejpam-4548	59	31	.	.	PUNCT
ejpam-4548	60	1	further	far	ADV
ejpam-4548	60	2	,	,	PUNCT
ejpam-4548	60	3	we	we	PRON
ejpam-4548	60	4	introduce	introduce	VERB
ejpam-4548	60	5	some	some	DET
ejpam-4548	60	6	basic	basic	ADJ
ejpam-4548	60	7	definitions	definition	NOUN
ejpam-4548	60	8	and	and	CCONJ
ejpam-4548	60	9	results	result	NOUN
ejpam-4548	60	10	which	which	PRON
ejpam-4548	60	11	are	be	AUX
ejpam-4548	60	12	used	use	VERB
ejpam-4548	60	13	in	in	ADP
ejpam-4548	60	14	the	the	DET
ejpam-4548	60	15	sequel	sequel	NOUN
ejpam-4548	60	16	.	.	PUNCT
ejpam-4548	61	1	for	for	ADP
ejpam-4548	61	2	details	detail	NOUN
ejpam-4548	61	3	we	we	PRON
ejpam-4548	61	4	refer	refer	VERB
ejpam-4548	61	5	to	to	ADP
ejpam-4548	61	6	[	[	X
ejpam-4548	61	7	1	1	NUM
ejpam-4548	61	8	,	,	PUNCT
ejpam-4548	61	9	3	3	NUM
ejpam-4548	61	10	,	,	PUNCT
ejpam-4548	61	11	8	8	NUM
ejpam-4548	61	12	,	,	PUNCT
ejpam-4548	61	13	14	14	NUM
ejpam-4548	61	14	?	?	PUNCT
ejpam-4548	61	15	,	,	PUNCT
ejpam-4548	61	16	15	15	NUM
ejpam-4548	61	17	]	]	PUNCT
ejpam-4548	61	18	.	.	PUNCT
ejpam-4548	62	1	an	an	DET
ejpam-4548	62	2	l	l	NOUN
ejpam-4548	62	3	-	-	NOUN
ejpam-4548	62	4	subset	subset	NOUN
ejpam-4548	62	5	of	of	ADP
ejpam-4548	62	6	x	x	PUNCT
ejpam-4548	62	7	is	be	AUX
ejpam-4548	62	8	a	a	DET
ejpam-4548	62	9	function	function	NOUN
ejpam-4548	62	10	from	from	ADP
ejpam-4548	62	11	x	x	PUNCT
ejpam-4548	62	12	into	into	ADP
ejpam-4548	62	13	l.	l.	NOUN
ejpam-4548	62	14	the	the	DET
ejpam-4548	62	15	set	set	NOUN
ejpam-4548	62	16	of	of	ADP
ejpam-4548	62	17	l	l	NOUN
ejpam-4548	62	18	-	-	NOUN
ejpam-4548	62	19	subsets	subset	NOUN
ejpam-4548	62	20	of	of	ADP
ejpam-4548	62	21	x	x	SYM
ejpam-4548	62	22	is	be	AUX
ejpam-4548	62	23	called	call	VERB
ejpam-4548	62	24	the	the	DET
ejpam-4548	62	25	l	l	NOUN
ejpam-4548	62	26	-	-	NOUN
ejpam-4548	62	27	power	power	NOUN
ejpam-4548	62	28	set	set	NOUN
ejpam-4548	62	29	of	of	ADP
ejpam-4548	62	30	x	x	PUNCT
ejpam-4548	62	31	and	and	CCONJ
ejpam-4548	62	32	is	be	AUX
ejpam-4548	62	33	denoted	denote	VERB
ejpam-4548	62	34	by	by	ADP
ejpam-4548	62	35	lx	lx	X
ejpam-4548	62	36	.	.	PUNCT
ejpam-4548	63	1	for	for	ADP
ejpam-4548	63	2	µ	µ	NOUN
ejpam-4548	63	3	∈	∈	NOUN
ejpam-4548	63	4	lx	lx	NOUN
ejpam-4548	63	5	,	,	PUNCT
ejpam-4548	63	6	the	the	DET
ejpam-4548	63	7	tip	tip	NOUN
ejpam-4548	63	8	of	of	ADP
ejpam-4548	63	9	µ	µ	PROPN
ejpam-4548	63	10	is	be	AUX
ejpam-4548	63	11	defined	define	VERB
ejpam-4548	63	12	as	as	ADP
ejpam-4548	63	13	∨	∨	NUM
ejpam-4548	63	14	x∈x	x∈x	NOUN
ejpam-4548	63	15	{	{	PUNCT
ejpam-4548	63	16	µ(x	µ(x	NUM
ejpam-4548	63	17	)	)	PUNCT
ejpam-4548	63	18	}	}	PUNCT
ejpam-4548	63	19	.	.	PUNCT
ejpam-4548	64	1	n.	n.	PROPN
ejpam-4548	64	2	ajmal	ajmal	PROPN
ejpam-4548	64	3	,	,	PUNCT
ejpam-4548	64	4	i.	i.	PROPN
ejpam-4548	64	5	jahan	jahan	PROPN
ejpam-4548	64	6	.	.	PROPN
ejpam-4548	64	7	,	,	PUNCT
ejpam-4548	64	8	b.	b.	PROPN
ejpam-4548	64	9	davvaz	davvaz	PROPN
ejpam-4548	64	10	/	/	SYM
ejpam-4548	64	11	eur	eur	PROPN
ejpam-4548	64	12	.	.	PUNCT
ejpam-4548	65	1	j.	j.	PROPN
ejpam-4548	65	2	pure	pure	PROPN
ejpam-4548	65	3	appl	appl	PROPN
ejpam-4548	65	4	.	.	PROPN
ejpam-4548	65	5	math	math	PROPN
ejpam-4548	65	6	,	,	PUNCT
ejpam-4548	65	7	15	15	NUM
ejpam-4548	65	8	(	(	PUNCT
ejpam-4548	65	9	4	4	NUM
ejpam-4548	65	10	)	)	PUNCT
ejpam-4548	65	11	(	(	PUNCT
ejpam-4548	65	12	2022	2022	NUM
ejpam-4548	65	13	)	)	PUNCT
ejpam-4548	65	14	,	,	PUNCT
ejpam-4548	65	15	2086	2086	NUM
ejpam-4548	65	16	-	-	SYM
ejpam-4548	65	17	2115	2115	NUM
ejpam-4548	65	18	2089	2089	NUM
ejpam-4548	65	19	if	if	SCONJ
ejpam-4548	65	20	µ	µ	NUM
ejpam-4548	65	21	,	,	PUNCT
ejpam-4548	65	22	ν	ν	X
ejpam-4548	65	23	∈	∈	NOUN
ejpam-4548	65	24	lx	lx	NOUN
ejpam-4548	66	1	,	,	PUNCT
ejpam-4548	66	2	then	then	ADV
ejpam-4548	66	3	we	we	PRON
ejpam-4548	66	4	say	say	VERB
ejpam-4548	66	5	that	that	SCONJ
ejpam-4548	66	6	µ	µ	NOUN
ejpam-4548	66	7	is	be	AUX
ejpam-4548	66	8	contained	contain	VERB
ejpam-4548	66	9	in	in	ADP
ejpam-4548	66	10	ν	ν	NOUN
ejpam-4548	66	11	if	if	SCONJ
ejpam-4548	66	12	µ(x	µ(x	NOUN
ejpam-4548	66	13	)	)	PUNCT
ejpam-4548	66	14	≤	≤	NOUN
ejpam-4548	66	15	ν(x	ν(x	PROPN
ejpam-4548	66	16	)	)	PUNCT
ejpam-4548	66	17	for	for	ADP
ejpam-4548	66	18	every	every	DET
ejpam-4548	66	19	x	x	SYM
ejpam-4548	66	20	∈	∈	PROPN
ejpam-4548	66	21	x	x	PUNCT
ejpam-4548	66	22	and	and	CCONJ
ejpam-4548	66	23	is	be	AUX
ejpam-4548	66	24	denoted	denote	VERB
ejpam-4548	66	25	by	by	ADP
ejpam-4548	66	26	µ	µ	PRON
ejpam-4548	66	27	⊆	⊆	NUM
ejpam-4548	66	28	ν	ν	NOUN
ejpam-4548	66	29	.	.	PROPN
ejpam-4548	66	30	for	for	ADP
ejpam-4548	66	31	a	a	DET
ejpam-4548	66	32	family	family	NOUN
ejpam-4548	66	33	{	{	PUNCT
ejpam-4548	66	34	µi	µi	INTJ
ejpam-4548	66	35	:	:	PUNCT
ejpam-4548	66	36	i	i	PRON
ejpam-4548	66	37	∈	∈	VERB
ejpam-4548	66	38	i	i	PRON
ejpam-4548	66	39	}	}	PUNCT
ejpam-4548	66	40	of	of	ADP
ejpam-4548	66	41	l	l	NOUN
ejpam-4548	66	42	-	-	NOUN
ejpam-4548	66	43	subsets	subset	NOUN
ejpam-4548	66	44	of	of	ADP
ejpam-4548	66	45	x	x	NOUN
ejpam-4548	66	46	,	,	PUNCT
ejpam-4548	66	47	where	where	SCONJ
ejpam-4548	66	48	i	i	PRON
ejpam-4548	66	49	is	be	AUX
ejpam-4548	66	50	a	a	DET
ejpam-4548	66	51	non	non	ADJ
ejpam-4548	66	52	-	-	ADJ
ejpam-4548	66	53	empty	empty	ADJ
ejpam-4548	66	54	index	index	NOUN
ejpam-4548	66	55	set	set	NOUN
ejpam-4548	66	56	,	,	PUNCT
ejpam-4548	66	57	the	the	DET
ejpam-4548	66	58	union	union	NOUN
ejpam-4548	66	59	⋃	⋃	PUNCT
ejpam-4548	66	60	i∈i	i∈i	ADJ
ejpam-4548	66	61	µi	µi	PROPN
ejpam-4548	66	62	and	and	CCONJ
ejpam-4548	66	63	the	the	DET
ejpam-4548	66	64	intersection	intersection	NOUN
ejpam-4548	66	65	⋂	⋂	PROPN
ejpam-4548	66	66	i∈i	i∈i	ADV
ejpam-4548	66	67	µi	µi	ADP
ejpam-4548	66	68	of	of	ADP
ejpam-4548	66	69	{	{	PUNCT
ejpam-4548	66	70	µi	µi	INTJ
ejpam-4548	66	71	:	:	PUNCT
ejpam-4548	66	72	i	i	PRON
ejpam-4548	66	73	∈	∈	PROPN
ejpam-4548	67	1	i	i	PRON
ejpam-4548	67	2	}	}	PUNCT
ejpam-4548	67	3	are	be	AUX
ejpam-4548	67	4	respectively	respectively	ADV
ejpam-4548	67	5	defined	define	VERB
ejpam-4548	67	6	by	by	ADP
ejpam-4548	67	7	:	:	PUNCT
ejpam-4548	67	8	(	(	PUNCT
ejpam-4548	67	9	⋃	⋃	ADP
ejpam-4548	67	10	i∈i	i∈i	ADJ
ejpam-4548	67	11	µi	µi	PROPN
ejpam-4548	67	12	)	)	PUNCT
ejpam-4548	67	13	(	(	PUNCT
ejpam-4548	67	14	x	x	X
ejpam-4548	67	15	)	)	PUNCT
ejpam-4548	67	16	=	=	PUNCT
ejpam-4548	68	1	∨	∨	NOUN
ejpam-4548	68	2	i∈i	i∈i	ADJ
ejpam-4548	68	3	{	{	PUNCT
ejpam-4548	68	4	µi(x	µi(x	NUM
ejpam-4548	68	5	)	)	PUNCT
ejpam-4548	68	6	}	}	PUNCT
ejpam-4548	68	7	and	and	CCONJ
ejpam-4548	68	8	(	(	PUNCT
ejpam-4548	68	9	⋂	⋂	PROPN
ejpam-4548	68	10	i∈i	i∈i	ADJ
ejpam-4548	68	11	µi	µi	PROPN
ejpam-4548	68	12	)	)	PUNCT
ejpam-4548	68	13	(	(	PUNCT
ejpam-4548	68	14	x	x	X
ejpam-4548	68	15	)	)	PUNCT
ejpam-4548	68	16	=	=	SYM
ejpam-4548	68	17	∧	∧	PROPN
ejpam-4548	68	18	i∈i	i∈i	ADJ
ejpam-4548	68	19	{	{	PUNCT
ejpam-4548	68	20	µi(x	µi(x	NUM
ejpam-4548	68	21	)	)	PUNCT
ejpam-4548	68	22	}	}	PUNCT
ejpam-4548	68	23	,	,	PUNCT
ejpam-4548	68	24	for	for	ADP
ejpam-4548	68	25	each	each	DET
ejpam-4548	68	26	x	x	SYM
ejpam-4548	68	27	∈	∈	PROPN
ejpam-4548	68	28	x.	x.	NOUN
ejpam-4548	68	29	let	let	VERB
ejpam-4548	68	30	f	f	PRON
ejpam-4548	68	31	be	be	AUX
ejpam-4548	68	32	a	a	DET
ejpam-4548	68	33	mapping	mapping	NOUN
ejpam-4548	68	34	from	from	ADP
ejpam-4548	68	35	a	a	DET
ejpam-4548	68	36	set	set	NOUN
ejpam-4548	68	37	x	x	PUNCT
ejpam-4548	68	38	to	to	ADP
ejpam-4548	68	39	a	a	DET
ejpam-4548	68	40	set	set	NOUN
ejpam-4548	68	41	y	y	PROPN
ejpam-4548	68	42	.	.	PUNCT
ejpam-4548	69	1	if	if	SCONJ
ejpam-4548	69	2	µ	µ	X
ejpam-4548	69	3	∈	∈	NOUN
ejpam-4548	69	4	lx	lx	NOUN
ejpam-4548	69	5	and	and	CCONJ
ejpam-4548	69	6	ν	ν	X
ejpam-4548	69	7	∈	∈	NOUN
ejpam-4548	69	8	ly	ly	X
ejpam-4548	69	9	,	,	PUNCT
ejpam-4548	69	10	then	then	ADV
ejpam-4548	69	11	the	the	DET
ejpam-4548	69	12	image	image	NOUN
ejpam-4548	69	13	f(µ	f(µ	PROPN
ejpam-4548	69	14	)	)	PUNCT
ejpam-4548	69	15	of	of	ADP
ejpam-4548	69	16	µ	µ	NUM
ejpam-4548	69	17	under	under	ADP
ejpam-4548	69	18	f	f	PROPN
ejpam-4548	69	19	and	and	CCONJ
ejpam-4548	69	20	the	the	DET
ejpam-4548	69	21	preimage	preimage	PROPN
ejpam-4548	69	22	f−1(ν	f−1(ν	PROPN
ejpam-4548	69	23	)	)	PUNCT
ejpam-4548	69	24	of	of	ADP
ejpam-4548	69	25	ν	ν	PROPN
ejpam-4548	69	26	under	under	ADP
ejpam-4548	69	27	f	f	PROPN
ejpam-4548	69	28	are	be	AUX
ejpam-4548	69	29	l	l	NOUN
ejpam-4548	69	30	-	-	NOUN
ejpam-4548	69	31	subsets	subset	NOUN
ejpam-4548	69	32	of	of	ADP
ejpam-4548	69	33	y	y	PROPN
ejpam-4548	69	34	and	and	CCONJ
ejpam-4548	69	35	x	x	X
ejpam-4548	69	36	respectively	respectively	ADV
ejpam-4548	69	37	,	,	PUNCT
ejpam-4548	69	38	defined	define	VERB
ejpam-4548	69	39	by	by	ADP
ejpam-4548	69	40	f(µ)(y	f(µ)(y	NUM
ejpam-4548	69	41	)	)	PUNCT
ejpam-4548	69	42	=	=	SYM
ejpam-4548	69	43	∨	∨	PROPN
ejpam-4548	69	44	x∈f−1(y	x∈f−1(y	PROPN
ejpam-4548	69	45	)	)	PUNCT
ejpam-4548	69	46	{	{	PUNCT
ejpam-4548	69	47	µ(x	µ(x	NOUN
ejpam-4548	69	48	)	)	PUNCT
ejpam-4548	69	49	}	}	PUNCT
ejpam-4548	69	50	and	and	CCONJ
ejpam-4548	69	51	f−1(ν)(x	f−1(ν)(x	PROPN
ejpam-4548	69	52	)	)	PUNCT
ejpam-4548	69	53	=	=	SYM
ejpam-4548	69	54	ν(f(x	ν(f(x	PROPN
ejpam-4548	69	55	)	)	PUNCT
ejpam-4548	69	56	)	)	PUNCT
ejpam-4548	69	57	.	.	PUNCT
ejpam-4548	70	1	here	here	ADV
ejpam-4548	70	2	we	we	PRON
ejpam-4548	70	3	point	point	VERB
ejpam-4548	70	4	out	out	ADP
ejpam-4548	70	5	that	that	SCONJ
ejpam-4548	70	6	in	in	ADP
ejpam-4548	70	7	the	the	DET
ejpam-4548	70	8	above	above	ADJ
ejpam-4548	70	9	definition	definition	NOUN
ejpam-4548	70	10	if	if	SCONJ
ejpam-4548	70	11	f−1(y	f−1(y	PROPN
ejpam-4548	70	12	)	)	PUNCT
ejpam-4548	70	13	=	=	SYM
ejpam-4548	70	14	ϕ	ϕ	NOUN
ejpam-4548	70	15	,	,	PUNCT
ejpam-4548	70	16	then	then	ADV
ejpam-4548	70	17	f(µ)(y	f(µ)(y	NUM
ejpam-4548	70	18	)	)	PUNCT
ejpam-4548	70	19	,	,	PUNCT
ejpam-4548	70	20	being	be	AUX
ejpam-4548	70	21	the	the	DET
ejpam-4548	70	22	least	least	ADJ
ejpam-4548	70	23	upper	upper	ADJ
ejpam-4548	70	24	bound	bind	VERB
ejpam-4548	70	25	of	of	ADP
ejpam-4548	70	26	the	the	DET
ejpam-4548	70	27	empty	empty	ADJ
ejpam-4548	70	28	set	set	NOUN
ejpam-4548	70	29	,	,	PUNCT
ejpam-4548	70	30	is	be	AUX
ejpam-4548	70	31	zero	zero	NUM
ejpam-4548	70	32	.	.	PUNCT
ejpam-4548	71	1	the	the	DET
ejpam-4548	71	2	set	set	ADJ
ejpam-4548	71	3	product	product	NOUN
ejpam-4548	71	4	µ	µ	PRON
ejpam-4548	71	5	◦	◦	NOUN
ejpam-4548	71	6	ν	ν	NOUN
ejpam-4548	71	7	of	of	ADP
ejpam-4548	71	8	µ	µ	NOUN
ejpam-4548	71	9	,	,	PUNCT
ejpam-4548	71	10	ν	ν	PROPN
ejpam-4548	71	11	∈	∈	PROPN
ejpam-4548	71	12	ls	ls	ADJ
ejpam-4548	71	13	where	where	SCONJ
ejpam-4548	71	14	s	s	NOUN
ejpam-4548	71	15	is	be	AUX
ejpam-4548	71	16	a	a	DET
ejpam-4548	71	17	groupoid	groupoid	NOUN
ejpam-4548	71	18	,	,	PUNCT
ejpam-4548	71	19	is	be	AUX
ejpam-4548	71	20	an	an	DET
ejpam-4548	71	21	l	l	NOUN
ejpam-4548	71	22	-	-	NOUN
ejpam-4548	71	23	subset	subset	NOUN
ejpam-4548	71	24	of	of	ADP
ejpam-4548	71	25	s	s	PRON
ejpam-4548	71	26	defined	define	VERB
ejpam-4548	71	27	by	by	ADP
ejpam-4548	71	28	µ	µ	NUM
ejpam-4548	71	29	◦	◦	NOUN
ejpam-4548	71	30	ν(x	ν(x	PROPN
ejpam-4548	71	31	)	)	PUNCT
ejpam-4548	71	32	=	=	PUNCT
ejpam-4548	72	1	∨	∨	NUM
ejpam-4548	72	2	x	x	X
ejpam-4548	72	3	=	=	PROPN
ejpam-4548	72	4	yz	yz	X
ejpam-4548	72	5	{	{	PUNCT
ejpam-4548	72	6	µ(y	µ(y	PROPN
ejpam-4548	72	7	)	)	PUNCT
ejpam-4548	72	8	∧	∧	PROPN
ejpam-4548	72	9	ν(z	ν(z	NOUN
ejpam-4548	72	10	)	)	PUNCT
ejpam-4548	72	11	}	}	PUNCT
ejpam-4548	72	12	.	.	PUNCT
ejpam-4548	73	1	if	if	SCONJ
ejpam-4548	73	2	µ	µ	X
ejpam-4548	73	3	∈	∈	NOUN
ejpam-4548	73	4	lx	lx	NOUN
ejpam-4548	73	5	and	and	CCONJ
ejpam-4548	73	6	a	a	DET
ejpam-4548	73	7	∈	∈	PROPN
ejpam-4548	73	8	l	l	NOUN
ejpam-4548	73	9	,	,	PUNCT
ejpam-4548	73	10	then	then	ADV
ejpam-4548	73	11	the	the	DET
ejpam-4548	73	12	notions	notion	NOUN
ejpam-4548	73	13	of	of	ADP
ejpam-4548	73	14	level	level	NOUN
ejpam-4548	73	15	subset	subset	VERB
ejpam-4548	73	16	µa	µa	NOUN
ejpam-4548	73	17	and	and	CCONJ
ejpam-4548	73	18	strong	strong	ADJ
ejpam-4548	73	19	level	level	NOUN
ejpam-4548	73	20	subset	subset	VERB
ejpam-4548	73	21	µ	µ	VERB
ejpam-4548	73	22	>	>	X
ejpam-4548	73	23	a	a	PRON
ejpam-4548	73	24	of	of	ADP
ejpam-4548	73	25	µ	µ	NOUN
ejpam-4548	73	26	are	be	AUX
ejpam-4548	73	27	,	,	PUNCT
ejpam-4548	73	28	respectively	respectively	ADV
ejpam-4548	73	29	,	,	PUNCT
ejpam-4548	73	30	defined	define	VERB
ejpam-4548	73	31	by	by	ADP
ejpam-4548	73	32	:	:	PUNCT
ejpam-4548	73	33	µa	µa	NOUN
ejpam-4548	73	34	=	=	PUNCT
ejpam-4548	73	35	{	{	PUNCT
ejpam-4548	73	36	x	x	SYM
ejpam-4548	73	37	∈	∈	PROPN
ejpam-4548	73	38	x	x	X
ejpam-4548	73	39	:	:	PUNCT
ejpam-4548	73	40	µ(x	µ(x	NUM
ejpam-4548	73	41	)	)	PUNCT
ejpam-4548	73	42	≥	≥	NOUN
ejpam-4548	73	43	a	a	NOUN
ejpam-4548	73	44	}	}	PUNCT
ejpam-4548	73	45	and	and	CCONJ
ejpam-4548	73	46	µ	µ	X
ejpam-4548	73	47	>	>	X
ejpam-4548	73	48	a	a	DET
ejpam-4548	73	49	=	=	X
ejpam-4548	73	50	{	{	PUNCT
ejpam-4548	73	51	x	x	SYM
ejpam-4548	73	52	∈	∈	PROPN
ejpam-4548	73	53	x	x	X
ejpam-4548	73	54	:	:	PUNCT
ejpam-4548	73	55	µ(x	µ(x	X
ejpam-4548	73	56	)	)	PUNCT
ejpam-4548	73	57	>	>	X
ejpam-4548	73	58	a	a	X
ejpam-4548	73	59	}	}	PUNCT
ejpam-4548	73	60	.	.	PUNCT
ejpam-4548	74	1	clearly	clearly	ADV
ejpam-4548	74	2	,	,	PUNCT
ejpam-4548	74	3	if	if	SCONJ
ejpam-4548	74	4	a	a	DET
ejpam-4548	74	5	≤	≤	NUM
ejpam-4548	74	6	b	b	NOUN
ejpam-4548	74	7	for	for	ADP
ejpam-4548	74	8	a	a	DET
ejpam-4548	74	9	,	,	PUNCT
ejpam-4548	74	10	b	b	PROPN
ejpam-4548	74	11	∈	∈	PROPN
ejpam-4548	74	12	l	l	NOUN
ejpam-4548	74	13	,	,	PUNCT
ejpam-4548	74	14	then	then	ADV
ejpam-4548	74	15	µb	µb	VERB
ejpam-4548	74	16	⊆	⊆	NUM
ejpam-4548	74	17	µa	µa	NOUN
ejpam-4548	74	18	and	and	CCONJ
ejpam-4548	74	19	µ	µ	PRON
ejpam-4548	74	20	>	>	X
ejpam-4548	74	21	b	b	PROPN
ejpam-4548	74	22	⊆	⊆	NUM
ejpam-4548	74	23	µ	µ	X
ejpam-4548	74	24	>	>	X
ejpam-4548	74	25	a	a	PRON
ejpam-4548	74	26	.	.	PUNCT
ejpam-4548	75	1	proposition	proposition	NOUN
ejpam-4548	75	2	1	1	NUM
ejpam-4548	75	3	.	.	PUNCT
ejpam-4548	76	1	[	[	X
ejpam-4548	76	2	15	15	NUM
ejpam-4548	76	3	]	]	X
ejpam-4548	76	4	let	let	VERB
ejpam-4548	76	5	µ	µ	NUM
ejpam-4548	76	6	,	,	PUNCT
ejpam-4548	76	7	ν	ν	X
ejpam-4548	76	8	∈	∈	NOUN
ejpam-4548	76	9	lx	lx	NOUN
ejpam-4548	76	10	and	and	CCONJ
ejpam-4548	76	11	µ	µ	PRON
ejpam-4548	76	12	⊆	⊆	NUM
ejpam-4548	76	13	ν	ν	NOUN
ejpam-4548	76	14	.	.	PUNCT
ejpam-4548	77	1	then	then	ADV
ejpam-4548	77	2	,	,	PUNCT
ejpam-4548	77	3	(	(	PUNCT
ejpam-4548	77	4	i	i	NOUN
ejpam-4548	77	5	)	)	PUNCT
ejpam-4548	77	6	if	if	SCONJ
ejpam-4548	77	7	µ	µ	PRON
ejpam-4548	77	8	⊆	⊆	NUM
ejpam-4548	77	9	ν	ν	NOUN
ejpam-4548	77	10	,	,	PUNCT
ejpam-4548	77	11	then	then	ADV
ejpam-4548	77	12	µa	µa	NOUN
ejpam-4548	77	13	⊆	⊆	NUM
ejpam-4548	77	14	νa	νa	NOUN
ejpam-4548	77	15	for	for	ADP
ejpam-4548	77	16	each	each	DET
ejpam-4548	77	17	a	a	DET
ejpam-4548	77	18	∈	∈	PROPN
ejpam-4548	77	19	l	l	NOUN
ejpam-4548	77	20	,	,	PUNCT
ejpam-4548	77	21	(	(	PUNCT
ejpam-4548	77	22	ii	ii	NOUN
ejpam-4548	77	23	)	)	PUNCT
ejpam-4548	77	24	if	if	SCONJ
ejpam-4548	77	25	µa	µa	NOUN
ejpam-4548	77	26	⊆	⊆	NUM
ejpam-4548	77	27	νa	νa	NOUN
ejpam-4548	77	28	for	for	ADP
ejpam-4548	77	29	each	each	DET
ejpam-4548	77	30	a	a	DET
ejpam-4548	77	31	∈	∈	PROPN
ejpam-4548	77	32	imµ	imµ	NOUN
ejpam-4548	77	33	,	,	PUNCT
ejpam-4548	77	34	then	then	ADV
ejpam-4548	77	35	µ	µ	PROPN
ejpam-4548	77	36	⊆	⊆	NUM
ejpam-4548	77	37	ν	ν	NOUN
ejpam-4548	77	38	,	,	PUNCT
ejpam-4548	77	39	(	(	PUNCT
ejpam-4548	77	40	iii	iii	X
ejpam-4548	77	41	)	)	PUNCT
ejpam-4548	77	42	if	if	SCONJ
ejpam-4548	77	43	µ	µ	PRON
ejpam-4548	77	44	⊆	⊆	NUM
ejpam-4548	77	45	ν	ν	NOUN
ejpam-4548	77	46	,	,	PUNCT
ejpam-4548	77	47	then	then	ADV
ejpam-4548	77	48	µ	µ	X
ejpam-4548	77	49	>	>	X
ejpam-4548	77	50	a	a	DET
ejpam-4548	77	51	⊆	⊆	NUM
ejpam-4548	77	52	ν	ν	NOUN
ejpam-4548	77	53	>	>	X
ejpam-4548	77	54	a	a	PRON
ejpam-4548	77	55	for	for	ADP
ejpam-4548	77	56	each	each	DET
ejpam-4548	77	57	a	a	DET
ejpam-4548	77	58	∈	∈	PROPN
ejpam-4548	77	59	l	l	NOUN
ejpam-4548	77	60	provided	provide	VERB
ejpam-4548	77	61	l	l	PROPN
ejpam-4548	77	62	is	be	AUX
ejpam-4548	77	63	a	a	DET
ejpam-4548	77	64	chain	chain	NOUN
ejpam-4548	77	65	,	,	PUNCT
ejpam-4548	77	66	(	(	PUNCT
ejpam-4548	77	67	iv	iv	X
ejpam-4548	77	68	)	)	PUNCT
ejpam-4548	77	69	if	if	SCONJ
ejpam-4548	77	70	µ	µ	X
ejpam-4548	77	71	>	>	X
ejpam-4548	77	72	a	a	DET
ejpam-4548	77	73	⊆	⊆	NUM
ejpam-4548	77	74	ν	ν	NOUN
ejpam-4548	77	75	>	>	X
ejpam-4548	77	76	a	a	PRON
ejpam-4548	77	77	for	for	ADP
ejpam-4548	77	78	each	each	DET
ejpam-4548	77	79	a	a	DET
ejpam-4548	77	80	∈	∈	PROPN
ejpam-4548	77	81	imν	imν	NOUN
ejpam-4548	77	82	,	,	PUNCT
ejpam-4548	77	83	then	then	ADV
ejpam-4548	77	84	µ	µ	NUM
ejpam-4548	77	85	⊆	⊆	NUM
ejpam-4548	77	86	ν	ν	NOUN
ejpam-4548	77	87	provided	provide	VERB
ejpam-4548	77	88	l	l	NOUN
ejpam-4548	77	89	is	be	AUX
ejpam-4548	77	90	a	a	DET
ejpam-4548	77	91	chain	chain	NOUN
ejpam-4548	77	92	.	.	PUNCT
ejpam-4548	78	1	throughout	throughout	ADP
ejpam-4548	78	2	this	this	DET
ejpam-4548	78	3	paper	paper	NOUN
ejpam-4548	78	4	g	g	PROPN
ejpam-4548	78	5	denotes	denote	VERB
ejpam-4548	78	6	an	an	DET
ejpam-4548	78	7	ordinary	ordinary	ADJ
ejpam-4548	78	8	group	group	NOUN
ejpam-4548	78	9	with	with	ADP
ejpam-4548	78	10	the	the	DET
ejpam-4548	78	11	identity	identity	NOUN
ejpam-4548	78	12	element	element	NOUN
ejpam-4548	78	13	‘	'	PUNCT
ejpam-4548	78	14	e	e	NOUN
ejpam-4548	78	15	’	'	PUNCT
ejpam-4548	78	16	and	and	CCONJ
ejpam-4548	78	17	i	i	PRON
ejpam-4548	78	18	denotes	denote	VERB
ejpam-4548	78	19	a	a	DET
ejpam-4548	78	20	non	non	ADJ
ejpam-4548	78	21	-	-	ADJ
ejpam-4548	78	22	empty	empty	ADJ
ejpam-4548	78	23	index	index	NOUN
ejpam-4548	78	24	set	set	NOUN
ejpam-4548	78	25	.	.	PUNCT
ejpam-4548	79	1	for	for	ADP
ejpam-4548	79	2	a	a	DET
ejpam-4548	79	3	⊆	⊆	NUM
ejpam-4548	79	4	x	x	NOUN
ejpam-4548	79	5	,	,	PUNCT
ejpam-4548	79	6	1a	1a	PROPN
ejpam-4548	79	7	denotes	denote	VERB
ejpam-4548	79	8	the	the	DET
ejpam-4548	79	9	characteristic	characteristic	ADJ
ejpam-4548	79	10	function	function	NOUN
ejpam-4548	79	11	of	of	ADP
ejpam-4548	79	12	a	a	DET
ejpam-4548	79	13	in	in	ADP
ejpam-4548	79	14	x.	x.	NOUN
ejpam-4548	79	15	definition	definition	NOUN
ejpam-4548	79	16	1	1	NUM
ejpam-4548	79	17	.	.	PUNCT
ejpam-4548	80	1	[	[	X
ejpam-4548	80	2	15	15	NUM
ejpam-4548	80	3	]	]	PUNCT
ejpam-4548	80	4	let	let	VERB
ejpam-4548	80	5	µ	µ	PRON
ejpam-4548	80	6	∈	∈	PROPN
ejpam-4548	80	7	lg	lg	NOUN
ejpam-4548	80	8	.	.	PROPN
ejpam-4548	81	1	then	then	ADV
ejpam-4548	81	2	,	,	PUNCT
ejpam-4548	81	3	µ	µ	X
ejpam-4548	81	4	is	be	AUX
ejpam-4548	81	5	called	call	VERB
ejpam-4548	81	6	an	an	DET
ejpam-4548	81	7	l	l	NOUN
ejpam-4548	81	8	-	-	NOUN
ejpam-4548	81	9	subgroup	subgroup	NOUN
ejpam-4548	81	10	of	of	ADP
ejpam-4548	81	11	g	g	PROPN
ejpam-4548	81	12	if	if	SCONJ
ejpam-4548	81	13	for	for	ADP
ejpam-4548	81	14	each	each	DET
ejpam-4548	81	15	x	x	NOUN
ejpam-4548	81	16	,	,	PUNCT
ejpam-4548	81	17	y	y	PROPN
ejpam-4548	81	18	∈	∈	PROPN
ejpam-4548	81	19	g	g	PROPN
ejpam-4548	81	20	(	(	PUNCT
ejpam-4548	81	21	i	i	PROPN
ejpam-4548	81	22	)	)	PUNCT
ejpam-4548	81	23	µ(xy	µ(xy	PROPN
ejpam-4548	81	24	)	)	PUNCT
ejpam-4548	81	25	≥	≥	NOUN
ejpam-4548	81	26	µ(x	µ(x	VERB
ejpam-4548	81	27	)	)	PUNCT
ejpam-4548	81	28	∧	∧	PROPN
ejpam-4548	81	29	µ(y	µ(y	PROPN
ejpam-4548	81	30	)	)	PUNCT
ejpam-4548	81	31	,	,	PUNCT
ejpam-4548	81	32	(	(	PUNCT
ejpam-4548	81	33	ii	ii	NOUN
ejpam-4548	81	34	)	)	PUNCT
ejpam-4548	81	35	µ(x−1	µ(x−1	NOUN
ejpam-4548	81	36	)	)	PUNCT
ejpam-4548	81	37	=	=	SYM
ejpam-4548	81	38	µ(x	µ(x	NUM
ejpam-4548	81	39	)	)	PUNCT
ejpam-4548	81	40	.	.	PUNCT
ejpam-4548	82	1	n.	n.	PROPN
ejpam-4548	82	2	ajmal	ajmal	PROPN
ejpam-4548	82	3	,	,	PUNCT
ejpam-4548	82	4	i.	i.	PROPN
ejpam-4548	82	5	jahan	jahan	PROPN
ejpam-4548	82	6	.	.	PROPN
ejpam-4548	82	7	,	,	PUNCT
ejpam-4548	82	8	b.	b.	PROPN
ejpam-4548	82	9	davvaz	davvaz	PROPN
ejpam-4548	82	10	/	/	SYM
ejpam-4548	82	11	eur	eur	PROPN
ejpam-4548	82	12	.	.	PUNCT
ejpam-4548	83	1	j.	j.	PROPN
ejpam-4548	83	2	pure	pure	PROPN
ejpam-4548	83	3	appl	appl	PROPN
ejpam-4548	83	4	.	.	PROPN
ejpam-4548	83	5	math	math	PROPN
ejpam-4548	83	6	,	,	PUNCT
ejpam-4548	83	7	15	15	NUM
ejpam-4548	83	8	(	(	PUNCT
ejpam-4548	83	9	4	4	NUM
ejpam-4548	83	10	)	)	PUNCT
ejpam-4548	83	11	(	(	PUNCT
ejpam-4548	83	12	2022	2022	NUM
ejpam-4548	83	13	)	)	PUNCT
ejpam-4548	83	14	,	,	PUNCT
ejpam-4548	83	15	2086	2086	NUM
ejpam-4548	83	16	-	-	SYM
ejpam-4548	83	17	2115	2115	NUM
ejpam-4548	83	18	2090	2090	NUM
ejpam-4548	83	19	the	the	DET
ejpam-4548	83	20	set	set	NOUN
ejpam-4548	83	21	of	of	ADP
ejpam-4548	83	22	l	l	NOUN
ejpam-4548	83	23	-	-	NOUN
ejpam-4548	83	24	subgroups	subgroup	NOUN
ejpam-4548	83	25	of	of	ADP
ejpam-4548	83	26	g	g	PROPN
ejpam-4548	83	27	is	be	AUX
ejpam-4548	83	28	denoted	denote	VERB
ejpam-4548	83	29	by	by	ADP
ejpam-4548	83	30	l(g	l(g	NOUN
ejpam-4548	83	31	)	)	PUNCT
ejpam-4548	83	32	.	.	PUNCT
ejpam-4548	84	1	clearly	clearly	ADV
ejpam-4548	84	2	,	,	PUNCT
ejpam-4548	84	3	the	the	DET
ejpam-4548	84	4	tip	tip	NOUN
ejpam-4548	84	5	of	of	ADP
ejpam-4548	84	6	an	an	DET
ejpam-4548	84	7	l	l	NOUN
ejpam-4548	84	8	-	-	NOUN
ejpam-4548	84	9	subgroup	subgroup	NOUN
ejpam-4548	84	10	is	be	AUX
ejpam-4548	84	11	attained	attain	VERB
ejpam-4548	84	12	at	at	ADP
ejpam-4548	84	13	the	the	DET
ejpam-4548	84	14	identity	identity	NOUN
ejpam-4548	84	15	element	element	NOUN
ejpam-4548	84	16	‘	'	PUNCT
ejpam-4548	84	17	e	e	NOUN
ejpam-4548	84	18	’	'	PUNCT
ejpam-4548	84	19	of	of	ADP
ejpam-4548	84	20	g.	g.	PROPN
ejpam-4548	84	21	theorem	theorem	VERB
ejpam-4548	84	22	1	1	X
ejpam-4548	84	23	.	.	PUNCT
ejpam-4548	85	1	let	let	VERB
ejpam-4548	85	2	µ	µ	PRON
ejpam-4548	85	3	∈	∈	NOUN
ejpam-4548	85	4	lg	lg	NOUN
ejpam-4548	85	5	with	with	ADP
ejpam-4548	85	6	tip	tip	PROPN
ejpam-4548	85	7	a0	a0	PROPN
ejpam-4548	85	8	.	.	PUNCT
ejpam-4548	86	1	then	then	ADV
ejpam-4548	86	2	,	,	PUNCT
ejpam-4548	86	3	(	(	PUNCT
ejpam-4548	86	4	i	i	NOUN
ejpam-4548	86	5	)	)	PUNCT
ejpam-4548	86	6	µ	µ	PRON
ejpam-4548	86	7	∈	∈	PROPN
ejpam-4548	86	8	l(g	l(g	NOUN
ejpam-4548	86	9	)	)	PUNCT
ejpam-4548	86	10	if	if	SCONJ
ejpam-4548	86	11	and	and	CCONJ
ejpam-4548	86	12	only	only	ADV
ejpam-4548	86	13	if	if	SCONJ
ejpam-4548	86	14	µa	µa	NOUN
ejpam-4548	86	15	is	be	AUX
ejpam-4548	86	16	a	a	DET
ejpam-4548	86	17	subgroup	subgroup	NOUN
ejpam-4548	86	18	of	of	ADP
ejpam-4548	86	19	g	g	PROPN
ejpam-4548	86	20	for	for	ADP
ejpam-4548	86	21	each	each	DET
ejpam-4548	86	22	a	a	DET
ejpam-4548	86	23	≤	≤	NUM
ejpam-4548	86	24	a0	a0	NOUN
ejpam-4548	86	25	,	,	PUNCT
ejpam-4548	86	26	(	(	PUNCT
ejpam-4548	86	27	ii	ii	NOUN
ejpam-4548	86	28	)	)	PUNCT
ejpam-4548	86	29	µ	µ	PRON
ejpam-4548	86	30	∈	∈	PROPN
ejpam-4548	86	31	l(g	l(g	NOUN
ejpam-4548	86	32	)	)	PUNCT
ejpam-4548	86	33	if	if	SCONJ
ejpam-4548	86	34	and	and	CCONJ
ejpam-4548	86	35	only	only	ADV
ejpam-4548	86	36	if	if	SCONJ
ejpam-4548	86	37	µ	µ	X
ejpam-4548	86	38	>	>	X
ejpam-4548	86	39	a	a	PRON
ejpam-4548	86	40	is	be	AUX
ejpam-4548	86	41	a	a	DET
ejpam-4548	86	42	subgroup	subgroup	NOUN
ejpam-4548	86	43	of	of	ADP
ejpam-4548	86	44	g	g	PROPN
ejpam-4548	86	45	for	for	ADP
ejpam-4548	86	46	each	each	DET
ejpam-4548	86	47	a	a	DET
ejpam-4548	86	48	<	<	X
ejpam-4548	86	49	a0	a0	NOUN
ejpam-4548	86	50	.	.	PUNCT
ejpam-4548	87	1	it	it	PRON
ejpam-4548	87	2	is	be	AUX
ejpam-4548	87	3	well	well	ADV
ejpam-4548	87	4	known	know	VERB
ejpam-4548	87	5	in	in	ADP
ejpam-4548	87	6	the	the	DET
ejpam-4548	87	7	literature	literature	NOUN
ejpam-4548	87	8	that	that	PRON
ejpam-4548	87	9	the	the	DET
ejpam-4548	87	10	intersection	intersection	NOUN
ejpam-4548	87	11	of	of	ADP
ejpam-4548	87	12	an	an	DET
ejpam-4548	87	13	arbitrary	arbitrary	ADJ
ejpam-4548	87	14	family	family	NOUN
ejpam-4548	87	15	of	of	ADP
ejpam-4548	87	16	lsubgroups	lsubgroup	NOUN
ejpam-4548	87	17	of	of	ADP
ejpam-4548	87	18	a	a	DET
ejpam-4548	87	19	group	group	NOUN
ejpam-4548	87	20	is	be	AUX
ejpam-4548	87	21	an	an	DET
ejpam-4548	87	22	l	l	NOUN
ejpam-4548	87	23	-	-	NOUN
ejpam-4548	87	24	subgroup	subgroup	NOUN
ejpam-4548	87	25	.	.	PUNCT
ejpam-4548	88	1	definition	definition	NOUN
ejpam-4548	88	2	2	2	NUM
ejpam-4548	88	3	.	.	PUNCT
ejpam-4548	89	1	let	let	VERB
ejpam-4548	89	2	µ	µ	PRON
ejpam-4548	89	3	∈	∈	PROPN
ejpam-4548	89	4	lg	lg	NOUN
ejpam-4548	89	5	.	.	PROPN
ejpam-4548	90	1	then	then	ADV
ejpam-4548	90	2	,	,	PUNCT
ejpam-4548	90	3	the	the	DET
ejpam-4548	90	4	l	l	NOUN
ejpam-4548	90	5	-	-	NOUN
ejpam-4548	90	6	subgroup	subgroup	NOUN
ejpam-4548	90	7	of	of	ADP
ejpam-4548	90	8	g	g	PROPN
ejpam-4548	90	9	generated	generate	VERB
ejpam-4548	90	10	by	by	ADP
ejpam-4548	90	11	µ	µ	PROPN
ejpam-4548	90	12	is	be	AUX
ejpam-4548	90	13	defined	define	VERB
ejpam-4548	90	14	as	as	ADP
ejpam-4548	90	15	the	the	DET
ejpam-4548	90	16	smallest	small	ADJ
ejpam-4548	90	17	l	l	NOUN
ejpam-4548	90	18	-	-	NOUN
ejpam-4548	90	19	subgroup	subgroup	NOUN
ejpam-4548	90	20	of	of	ADP
ejpam-4548	90	21	g	g	PROPN
ejpam-4548	90	22	which	which	PRON
ejpam-4548	90	23	contains	contain	VERB
ejpam-4548	90	24	µ.	µ.	NOUN
ejpam-4548	90	25	it	it	PRON
ejpam-4548	90	26	is	be	AUX
ejpam-4548	90	27	denoted	denote	VERB
ejpam-4548	90	28	by	by	ADP
ejpam-4548	90	29	⟨µ⟩.	⟨µ⟩.	PROPN
ejpam-4548	90	30	that	that	PRON
ejpam-4548	90	31	is	be	AUX
ejpam-4548	91	1	⟨µ⟩	⟨µ⟩	PROPN
ejpam-4548	91	2	=	=	SYM
ejpam-4548	91	3	⋂	⋂	PROPN
ejpam-4548	91	4	i∈i	i∈i	ADJ
ejpam-4548	91	5	{	{	PUNCT
ejpam-4548	91	6	µi	µi	PROPN
ejpam-4548	91	7	∈	∈	PROPN
ejpam-4548	91	8	l(g	l(g	NOUN
ejpam-4548	91	9	)	)	PUNCT
ejpam-4548	91	10	:	:	PUNCT
ejpam-4548	91	11	µ	µ	PROPN
ejpam-4548	91	12	⊆	⊆	NUM
ejpam-4548	91	13	µi	µi	PROPN
ejpam-4548	91	14	}	}	PUNCT
ejpam-4548	91	15	.	.	PUNCT
ejpam-4548	92	1	definition	definition	NOUN
ejpam-4548	92	2	3	3	X
ejpam-4548	92	3	.	.	PUNCT
ejpam-4548	93	1	let	let	VERB
ejpam-4548	93	2	µ	µ	PRON
ejpam-4548	93	3	∈	∈	PROPN
ejpam-4548	93	4	l(g	l(g	NOUN
ejpam-4548	93	5	)	)	PUNCT
ejpam-4548	93	6	.	.	PUNCT
ejpam-4548	94	1	then	then	ADV
ejpam-4548	94	2	,	,	PUNCT
ejpam-4548	94	3	µ	µ	X
ejpam-4548	94	4	is	be	AUX
ejpam-4548	94	5	called	call	VERB
ejpam-4548	94	6	an	an	DET
ejpam-4548	94	7	l	l	NOUN
ejpam-4548	94	8	-	-	ADJ
ejpam-4548	94	9	normal	normal	ADJ
ejpam-4548	94	10	subgroup	subgroup	NOUN
ejpam-4548	94	11	of	of	ADP
ejpam-4548	94	12	g	g	PROPN
ejpam-4548	94	13	if	if	SCONJ
ejpam-4548	94	14	for	for	ADP
ejpam-4548	94	15	all	all	DET
ejpam-4548	94	16	x	x	NOUN
ejpam-4548	94	17	,	,	PUNCT
ejpam-4548	94	18	y	y	PROPN
ejpam-4548	94	19	∈	∈	PROPN
ejpam-4548	94	20	g	g	PROPN
ejpam-4548	94	21	,	,	PUNCT
ejpam-4548	94	22	µ(xy	µ(xy	PROPN
ejpam-4548	94	23	)	)	PUNCT
ejpam-4548	94	24	=	=	PUNCT
ejpam-4548	94	25	µ(yx	µ(yx	NOUN
ejpam-4548	94	26	)	)	PUNCT
ejpam-4548	94	27	.	.	PUNCT
ejpam-4548	95	1	definition	definition	NOUN
ejpam-4548	95	2	4	4	NUM
ejpam-4548	95	3	.	.	PUNCT
ejpam-4548	96	1	if	if	SCONJ
ejpam-4548	96	2	µ	µ	NUM
ejpam-4548	96	3	,	,	PUNCT
ejpam-4548	96	4	η	η	PROPN
ejpam-4548	96	5	∈	∈	PROPN
ejpam-4548	96	6	lx	lx	NOUN
ejpam-4548	96	7	and	and	CCONJ
ejpam-4548	96	8	η	η	PROPN
ejpam-4548	96	9	⊆	⊆	PROPN
ejpam-4548	96	10	µ	µ	NUM
ejpam-4548	96	11	,	,	PUNCT
ejpam-4548	96	12	then	then	ADV
ejpam-4548	96	13	we	we	PRON
ejpam-4548	96	14	say	say	VERB
ejpam-4548	96	15	that	that	SCONJ
ejpam-4548	96	16	η	η	PROPN
ejpam-4548	96	17	is	be	AUX
ejpam-4548	96	18	an	an	DET
ejpam-4548	96	19	l	l	NOUN
ejpam-4548	96	20	-	-	NOUN
ejpam-4548	96	21	subset	subset	NOUN
ejpam-4548	96	22	of	of	ADP
ejpam-4548	96	23	µ.	µ.	PROPN
ejpam-4548	96	24	the	the	DET
ejpam-4548	96	25	set	set	NOUN
ejpam-4548	96	26	of	of	ADP
ejpam-4548	96	27	l	l	NOUN
ejpam-4548	96	28	-	-	NOUN
ejpam-4548	96	29	subsets	subset	NOUN
ejpam-4548	96	30	of	of	ADP
ejpam-4548	96	31	µ	µ	NOUN
ejpam-4548	96	32	is	be	AUX
ejpam-4548	96	33	denoted	denote	VERB
ejpam-4548	96	34	by	by	ADP
ejpam-4548	96	35	lµ.	lµ.	ADJ
ejpam-4548	96	36	definition	definition	NOUN
ejpam-4548	96	37	5	5	NUM
ejpam-4548	96	38	.	.	PUNCT
ejpam-4548	97	1	if	if	SCONJ
ejpam-4548	97	2	µ	µ	NUM
ejpam-4548	97	3	,	,	PUNCT
ejpam-4548	97	4	η	η	PROPN
ejpam-4548	97	5	∈	∈	PROPN
ejpam-4548	97	6	l(g	l(g	PROPN
ejpam-4548	97	7	)	)	PUNCT
ejpam-4548	97	8	and	and	CCONJ
ejpam-4548	97	9	η	η	PROPN
ejpam-4548	97	10	⊆	⊆	PROPN
ejpam-4548	97	11	µ	µ	NUM
ejpam-4548	97	12	,	,	PUNCT
ejpam-4548	97	13	then	then	ADV
ejpam-4548	97	14	we	we	PRON
ejpam-4548	97	15	say	say	VERB
ejpam-4548	97	16	that	that	SCONJ
ejpam-4548	97	17	η	η	PROPN
ejpam-4548	97	18	is	be	AUX
ejpam-4548	97	19	an	an	DET
ejpam-4548	97	20	l	l	NOUN
ejpam-4548	97	21	-	-	NOUN
ejpam-4548	97	22	subgroup	subgroup	NOUN
ejpam-4548	97	23	of	of	ADP
ejpam-4548	97	24	µ.the	µ.the	DET
ejpam-4548	97	25	set	set	NOUN
ejpam-4548	97	26	of	of	ADP
ejpam-4548	97	27	l	l	NOUN
ejpam-4548	97	28	-	-	NOUN
ejpam-4548	97	29	subgroups	subgroup	NOUN
ejpam-4548	97	30	of	of	ADP
ejpam-4548	97	31	µ	µ	NOUN
ejpam-4548	97	32	is	be	AUX
ejpam-4548	97	33	denoted	denote	VERB
ejpam-4548	97	34	by	by	ADP
ejpam-4548	97	35	l(µ	l(µ	PROPN
ejpam-4548	97	36	)	)	PUNCT
ejpam-4548	97	37	.	.	PUNCT
ejpam-4548	98	1	hence	hence	ADV
ejpam-4548	98	2	onwards	onward	VERB
ejpam-4548	98	3	µ	µ	DET
ejpam-4548	98	4	denotes	denote	NOUN
ejpam-4548	98	5	an	an	DET
ejpam-4548	98	6	l	l	NOUN
ejpam-4548	98	7	-	-	NOUN
ejpam-4548	98	8	subgroup	subgroup	NOUN
ejpam-4548	98	9	of	of	ADP
ejpam-4548	98	10	g	g	PROPN
ejpam-4548	99	1	and	and	CCONJ
ejpam-4548	99	2	we	we	PRON
ejpam-4548	99	3	shall	shall	AUX
ejpam-4548	99	4	call	call	VERB
ejpam-4548	99	5	the	the	DET
ejpam-4548	99	6	parent	parent	NOUN
ejpam-4548	99	7	l	l	PROPN
ejpam-4548	99	8	-	-	NOUN
ejpam-4548	99	9	subgroup	subgroup	NOUN
ejpam-4548	99	10	µ	µ	X
ejpam-4548	99	11	simply	simply	ADV
ejpam-4548	99	12	an	an	DET
ejpam-4548	99	13	l	l	NOUN
ejpam-4548	99	14	-	-	NOUN
ejpam-4548	99	15	group	group	NOUN
ejpam-4548	99	16	.	.	PUNCT
ejpam-4548	100	1	theorem	theorem	NOUN
ejpam-4548	100	2	2	2	NUM
ejpam-4548	100	3	.	.	PUNCT
ejpam-4548	101	1	let	let	VERB
ejpam-4548	101	2	η	η	PROPN
ejpam-4548	101	3	∈	∈	PROPN
ejpam-4548	101	4	lµ	lµ	PROPN
ejpam-4548	101	5	with	with	ADP
ejpam-4548	101	6	tip	tip	PROPN
ejpam-4548	101	7	a0	a0	PROPN
ejpam-4548	101	8	.	.	PUNCT
ejpam-4548	102	1	then	then	ADV
ejpam-4548	102	2	,	,	PUNCT
ejpam-4548	102	3	(	(	PUNCT
ejpam-4548	102	4	i	i	NOUN
ejpam-4548	102	5	)	)	PUNCT
ejpam-4548	102	6	η	η	PROPN
ejpam-4548	102	7	∈	∈	PROPN
ejpam-4548	102	8	l(µ	l(µ	PROPN
ejpam-4548	102	9	)	)	PUNCT
ejpam-4548	102	10	if	if	SCONJ
ejpam-4548	102	11	and	and	CCONJ
ejpam-4548	102	12	only	only	ADV
ejpam-4548	102	13	if	if	SCONJ
ejpam-4548	102	14	ηa	ηa	PROPN
ejpam-4548	102	15	is	be	AUX
ejpam-4548	102	16	a	a	DET
ejpam-4548	102	17	subgroup	subgroup	NOUN
ejpam-4548	102	18	of	of	ADP
ejpam-4548	102	19	µa	µa	NOUN
ejpam-4548	102	20	for	for	ADP
ejpam-4548	102	21	each	each	DET
ejpam-4548	102	22	a	a	DET
ejpam-4548	102	23	≤	≤	NUM
ejpam-4548	102	24	a0	a0	NOUN
ejpam-4548	102	25	,	,	PUNCT
ejpam-4548	102	26	(	(	PUNCT
ejpam-4548	102	27	ii	ii	NOUN
ejpam-4548	102	28	)	)	PUNCT
ejpam-4548	102	29	η	η	PROPN
ejpam-4548	102	30	∈	∈	PROPN
ejpam-4548	102	31	l(µ	l(µ	PROPN
ejpam-4548	102	32	)	)	PUNCT
ejpam-4548	103	1	if	if	SCONJ
ejpam-4548	103	2	and	and	CCONJ
ejpam-4548	103	3	only	only	ADV
ejpam-4548	103	4	if	if	SCONJ
ejpam-4548	103	5	η	η	NOUN
ejpam-4548	103	6	>	>	X
ejpam-4548	103	7	a	a	PRON
ejpam-4548	103	8	is	be	AUX
ejpam-4548	103	9	a	a	DET
ejpam-4548	103	10	subgroup	subgroup	NOUN
ejpam-4548	103	11	of	of	ADP
ejpam-4548	103	12	µ	µ	PROPN
ejpam-4548	103	13	>	>	X
ejpam-4548	103	14	a	a	PRON
ejpam-4548	103	15	for	for	ADP
ejpam-4548	103	16	each	each	PRON
ejpam-4548	103	17	a	a	DET
ejpam-4548	103	18	<	<	X
ejpam-4548	103	19	a0	a0	NOUN
ejpam-4548	103	20	provided	provide	VERB
ejpam-4548	103	21	l	l	PROPN
ejpam-4548	103	22	is	be	AUX
ejpam-4548	103	23	a	a	DET
ejpam-4548	103	24	chain	chain	NOUN
ejpam-4548	103	25	.	.	PUNCT
ejpam-4548	104	1	below	below	ADP
ejpam-4548	104	2	we	we	PRON
ejpam-4548	104	3	extend	extend	VERB
ejpam-4548	104	4	a	a	DET
ejpam-4548	104	5	result	result	NOUN
ejpam-4548	104	6	from	from	ADP
ejpam-4548	104	7	[	[	X
ejpam-4548	104	8	1	1	NUM
ejpam-4548	104	9	]	]	PUNCT
ejpam-4548	104	10	to	to	ADP
ejpam-4548	104	11	l	l	NOUN
ejpam-4548	104	12	-	-	ADJ
ejpam-4548	104	13	setting	setting	NOUN
ejpam-4548	104	14	:	:	PUNCT
ejpam-4548	104	15	proposition	proposition	NOUN
ejpam-4548	104	16	2	2	NUM
ejpam-4548	104	17	.	.	PUNCT
ejpam-4548	105	1	[	[	X
ejpam-4548	105	2	?	?	PUNCT
ejpam-4548	105	3	]	]	PUNCT
ejpam-4548	105	4	let	let	VERB
ejpam-4548	105	5	η	η	PROPN
ejpam-4548	105	6	,	,	PUNCT
ejpam-4548	105	7	θ	θ	PROPN
ejpam-4548	105	8	∈	∈	PROPN
ejpam-4548	105	9	l(µ	l(µ	PROPN
ejpam-4548	105	10	)	)	PUNCT
ejpam-4548	105	11	.	.	PUNCT
ejpam-4548	106	1	then	then	ADV
ejpam-4548	106	2	,	,	PUNCT
ejpam-4548	106	3	η	η	PROPN
ejpam-4548	106	4	◦	◦	NOUN
ejpam-4548	106	5	θ	θ	X
ejpam-4548	106	6	∈	∈	PROPN
ejpam-4548	106	7	l(µ	l(µ	PROPN
ejpam-4548	106	8	)	)	PUNCT
ejpam-4548	107	1	if	if	SCONJ
ejpam-4548	107	2	and	and	CCONJ
ejpam-4548	107	3	only	only	ADV
ejpam-4548	107	4	if	if	SCONJ
ejpam-4548	107	5	η	η	X
ejpam-4548	107	6	◦	◦	NOUN
ejpam-4548	107	7	θ	θ	NOUN
ejpam-4548	107	8	=	=	SYM
ejpam-4548	107	9	θ	θ	PROPN
ejpam-4548	107	10	◦	◦	NOUN
ejpam-4548	107	11	η	η	PROPN
ejpam-4548	107	12	.	.	PUNCT
ejpam-4548	107	13	proposition	proposition	NOUN
ejpam-4548	107	14	3	3	NUM
ejpam-4548	107	15	.	.	PUNCT
ejpam-4548	108	1	let	let	VERB
ejpam-4548	108	2	η	η	PROPN
ejpam-4548	108	3	,	,	PUNCT
ejpam-4548	108	4	θ	θ	PROPN
ejpam-4548	108	5	∈	∈	PROPN
ejpam-4548	108	6	l(µ	l(µ	PROPN
ejpam-4548	108	7	)	)	PUNCT
ejpam-4548	108	8	.	.	PUNCT
ejpam-4548	109	1	then	then	ADV
ejpam-4548	109	2	,	,	PUNCT
ejpam-4548	109	3	η	η	PROPN
ejpam-4548	109	4	and	and	CCONJ
ejpam-4548	109	5	θ	θ	PROPN
ejpam-4548	109	6	⊆	⊆	NUM
ejpam-4548	109	7	η	η	PROPN
ejpam-4548	109	8	◦	◦	NOUN
ejpam-4548	109	9	θ	θ	PROPN
ejpam-4548	109	10	if	if	SCONJ
ejpam-4548	109	11	and	and	CCONJ
ejpam-4548	109	12	only	only	ADV
ejpam-4548	109	13	if	if	SCONJ
ejpam-4548	109	14	η(e	η(e	NOUN
ejpam-4548	109	15	)	)	PUNCT
ejpam-4548	109	16	=	=	SYM
ejpam-4548	109	17	θ(e	θ(e	NUM
ejpam-4548	109	18	)	)	PUNCT
ejpam-4548	109	19	.	.	PUNCT
ejpam-4548	110	1	if	if	SCONJ
ejpam-4548	110	2	η	η	PROPN
ejpam-4548	110	3	∈	∈	PROPN
ejpam-4548	110	4	lµ	lµ	PROPN
ejpam-4548	110	5	and	and	CCONJ
ejpam-4548	110	6	⟨η⟩µ	⟨η⟩µ	PROPN
ejpam-4548	110	7	denotes	denote	VERB
ejpam-4548	110	8	the	the	DET
ejpam-4548	110	9	l	l	NOUN
ejpam-4548	110	10	-	-	NOUN
ejpam-4548	110	11	subgroup	subgroup	NOUN
ejpam-4548	110	12	of	of	ADP
ejpam-4548	110	13	µ	µ	PROPN
ejpam-4548	110	14	generated	generate	VERB
ejpam-4548	110	15	by	by	ADP
ejpam-4548	110	16	η	η	PROPN
ejpam-4548	110	17	,	,	PUNCT
ejpam-4548	110	18	then	then	ADV
ejpam-4548	110	19	it	it	PRON
ejpam-4548	110	20	can	can	AUX
ejpam-4548	110	21	be	be	AUX
ejpam-4548	110	22	easily	easily	ADV
ejpam-4548	110	23	verified	verify	VERB
ejpam-4548	110	24	that	that	SCONJ
ejpam-4548	110	25	⟨η⟩µ	⟨η⟩µ	PROPN
ejpam-4548	110	26	=	=	SYM
ejpam-4548	110	27	⟨η⟩.	⟨η⟩.	PROPN
ejpam-4548	110	28	we	we	PRON
ejpam-4548	110	29	recall	recall	VERB
ejpam-4548	110	30	the	the	DET
ejpam-4548	110	31	definition	definition	NOUN
ejpam-4548	110	32	of	of	ADP
ejpam-4548	110	33	a	a	DET
ejpam-4548	110	34	normal	normal	ADJ
ejpam-4548	110	35	l	l	NOUN
ejpam-4548	110	36	-	-	NOUN
ejpam-4548	110	37	subgroup	subgroup	NOUN
ejpam-4548	110	38	of	of	ADP
ejpam-4548	110	39	an	an	DET
ejpam-4548	110	40	l	l	NOUN
ejpam-4548	110	41	-	-	NOUN
ejpam-4548	110	42	group	group	NOUN
ejpam-4548	110	43	.	.	PUNCT
ejpam-4548	111	1	n.	n.	PROPN
ejpam-4548	111	2	ajmal	ajmal	PROPN
ejpam-4548	111	3	,	,	PUNCT
ejpam-4548	111	4	i.	i.	PROPN
ejpam-4548	111	5	jahan	jahan	PROPN
ejpam-4548	111	6	.	.	PROPN
ejpam-4548	111	7	,	,	PUNCT
ejpam-4548	111	8	b.	b.	PROPN
ejpam-4548	111	9	davvaz	davvaz	PROPN
ejpam-4548	111	10	/	/	SYM
ejpam-4548	111	11	eur	eur	PROPN
ejpam-4548	111	12	.	.	PUNCT
ejpam-4548	112	1	j.	j.	PROPN
ejpam-4548	112	2	pure	pure	PROPN
ejpam-4548	112	3	appl	appl	PROPN
ejpam-4548	112	4	.	.	PROPN
ejpam-4548	112	5	math	math	PROPN
ejpam-4548	112	6	,	,	PUNCT
ejpam-4548	112	7	15	15	NUM
ejpam-4548	112	8	(	(	PUNCT
ejpam-4548	112	9	4	4	NUM
ejpam-4548	112	10	)	)	PUNCT
ejpam-4548	112	11	(	(	PUNCT
ejpam-4548	112	12	2022	2022	NUM
ejpam-4548	112	13	)	)	PUNCT
ejpam-4548	112	14	,	,	PUNCT
ejpam-4548	112	15	2086	2086	NUM
ejpam-4548	112	16	-	-	SYM
ejpam-4548	112	17	2115	2115	NUM
ejpam-4548	112	18	2091	2091	NUM
ejpam-4548	112	19	definition	definition	NOUN
ejpam-4548	112	20	6	6	NUM
ejpam-4548	112	21	.	.	PUNCT
ejpam-4548	113	1	[	[	X
ejpam-4548	113	2	3	3	X
ejpam-4548	113	3	]	]	X
ejpam-4548	113	4	let	let	VERB
ejpam-4548	113	5	η	η	PROPN
ejpam-4548	113	6	∈	∈	PROPN
ejpam-4548	113	7	l(µ	l(µ	PROPN
ejpam-4548	113	8	)	)	PUNCT
ejpam-4548	113	9	.	.	PUNCT
ejpam-4548	114	1	then	then	ADV
ejpam-4548	114	2	,	,	PUNCT
ejpam-4548	114	3	η	η	PROPN
ejpam-4548	114	4	is	be	AUX
ejpam-4548	114	5	said	say	VERB
ejpam-4548	114	6	to	to	PART
ejpam-4548	114	7	be	be	AUX
ejpam-4548	114	8	a	a	DET
ejpam-4548	114	9	normal	normal	ADJ
ejpam-4548	114	10	l	l	NOUN
ejpam-4548	114	11	-	-	NOUN
ejpam-4548	114	12	subgroup	subgroup	NOUN
ejpam-4548	114	13	of	of	ADP
ejpam-4548	114	14	µ	µ	NOUN
ejpam-4548	114	15	,	,	PUNCT
ejpam-4548	114	16	if	if	SCONJ
ejpam-4548	114	17	for	for	ADP
ejpam-4548	114	18	all	all	DET
ejpam-4548	114	19	x	x	NOUN
ejpam-4548	114	20	,	,	PUNCT
ejpam-4548	114	21	y	y	PROPN
ejpam-4548	114	22	∈	∈	PROPN
ejpam-4548	114	23	g	g	ADP
ejpam-4548	114	24	η(yxy−1	η(yxy−1	NOUN
ejpam-4548	114	25	)	)	PUNCT
ejpam-4548	114	26	≥	≥	NOUN
ejpam-4548	114	27	η(x	η(x	NOUN
ejpam-4548	114	28	)	)	PUNCT
ejpam-4548	114	29	∧	∧	PROPN
ejpam-4548	114	30	µ(y	µ(y	PROPN
ejpam-4548	114	31	)	)	PUNCT
ejpam-4548	114	32	.	.	PUNCT
ejpam-4548	115	1	the	the	DET
ejpam-4548	115	2	set	set	NOUN
ejpam-4548	115	3	of	of	ADP
ejpam-4548	115	4	normal	normal	ADJ
ejpam-4548	115	5	l	l	NOUN
ejpam-4548	115	6	-	-	NOUN
ejpam-4548	115	7	subgroup	subgroup	NOUN
ejpam-4548	115	8	of	of	ADP
ejpam-4548	115	9	µ	µ	PROPN
ejpam-4548	115	10	is	be	AUX
ejpam-4548	115	11	denoted	denote	VERB
ejpam-4548	115	12	by	by	ADP
ejpam-4548	115	13	nl(µ	nl(µ	NOUN
ejpam-4548	115	14	)	)	PUNCT
ejpam-4548	115	15	.	.	PUNCT
ejpam-4548	116	1	theorem	theorem	NOUN
ejpam-4548	116	2	3	3	X
ejpam-4548	116	3	.	.	PUNCT
ejpam-4548	117	1	let	let	VERB
ejpam-4548	117	2	η	η	PROPN
ejpam-4548	117	3	∈	∈	PROPN
ejpam-4548	117	4	l(µ	l(µ	PROPN
ejpam-4548	117	5	)	)	PUNCT
ejpam-4548	117	6	.	.	PUNCT
ejpam-4548	118	1	then	then	ADV
ejpam-4548	118	2	,	,	PUNCT
ejpam-4548	118	3	η	η	PROPN
ejpam-4548	118	4	∈	∈	PROPN
ejpam-4548	118	5	nl(µ	nl(µ	NOUN
ejpam-4548	118	6	)	)	PUNCT
ejpam-4548	118	7	if	if	SCONJ
ejpam-4548	118	8	and	and	CCONJ
ejpam-4548	118	9	only	only	ADV
ejpam-4548	118	10	if	if	SCONJ
ejpam-4548	118	11	ηa	ηa	PRON
ejpam-4548	118	12	is	be	AUX
ejpam-4548	118	13	a	a	DET
ejpam-4548	118	14	normal	normal	ADJ
ejpam-4548	118	15	subgroup	subgroup	NOUN
ejpam-4548	118	16	of	of	ADP
ejpam-4548	118	17	µa	µa	NOUN
ejpam-4548	118	18	for	for	ADP
ejpam-4548	118	19	each	each	DET
ejpam-4548	118	20	a	a	DET
ejpam-4548	118	21	≤	≤	NUM
ejpam-4548	118	22	η(e	η(e	PROPN
ejpam-4548	118	23	)	)	PUNCT
ejpam-4548	118	24	.	.	PUNCT
ejpam-4548	119	1	theorem	theorem	ADJ
ejpam-4548	119	2	4	4	NUM
ejpam-4548	119	3	.	.	PUNCT
ejpam-4548	120	1	let	let	VERB
ejpam-4548	120	2	f	f	NOUN
ejpam-4548	120	3	:	:	PUNCT
ejpam-4548	120	4	g	g	PROPN
ejpam-4548	120	5	→	→	SYM
ejpam-4548	120	6	k	k	X
ejpam-4548	120	7	be	be	AUX
ejpam-4548	120	8	a	a	DET
ejpam-4548	120	9	group	group	NOUN
ejpam-4548	120	10	homomorphism	homomorphism	NOUN
ejpam-4548	120	11	.	.	PUNCT
ejpam-4548	121	1	if	if	SCONJ
ejpam-4548	121	2	µ	µ	PRON
ejpam-4548	121	3	∈	∈	PROPN
ejpam-4548	121	4	l(g	l(g	NOUN
ejpam-4548	121	5	)	)	PUNCT
ejpam-4548	121	6	and	and	CCONJ
ejpam-4548	121	7	ν	ν	PROPN
ejpam-4548	121	8	∈	∈	PROPN
ejpam-4548	121	9	l(k	l(k	PROPN
ejpam-4548	121	10	)	)	PUNCT
ejpam-4548	121	11	,	,	PUNCT
ejpam-4548	121	12	then	then	ADV
ejpam-4548	121	13	(	(	PUNCT
ejpam-4548	121	14	i	i	NOUN
ejpam-4548	121	15	)	)	PUNCT
ejpam-4548	121	16	f(η	f(η	PROPN
ejpam-4548	121	17	)	)	PUNCT
ejpam-4548	121	18	is	be	AUX
ejpam-4548	121	19	an	an	DET
ejpam-4548	121	20	l	l	NOUN
ejpam-4548	121	21	-	-	NOUN
ejpam-4548	121	22	subgroup	subgroup	NOUN
ejpam-4548	121	23	of	of	ADP
ejpam-4548	121	24	f(µ	f(µ	PROPN
ejpam-4548	121	25	)	)	PUNCT
ejpam-4548	121	26	for	for	ADP
ejpam-4548	121	27	each	each	DET
ejpam-4548	121	28	η	η	PROPN
ejpam-4548	121	29	∈	∈	PROPN
ejpam-4548	121	30	l(µ	l(µ	PROPN
ejpam-4548	121	31	)	)	PUNCT
ejpam-4548	121	32	,	,	PUNCT
ejpam-4548	121	33	(	(	PUNCT
ejpam-4548	121	34	ii	ii	NOUN
ejpam-4548	121	35	)	)	PUNCT
ejpam-4548	121	36	f−1(θ	f−1(θ	PROPN
ejpam-4548	121	37	)	)	PUNCT
ejpam-4548	121	38	is	be	AUX
ejpam-4548	121	39	an	an	DET
ejpam-4548	121	40	l	l	NOUN
ejpam-4548	121	41	-	-	NOUN
ejpam-4548	121	42	subgroup	subgroup	NOUN
ejpam-4548	121	43	of	of	ADP
ejpam-4548	121	44	f−1(ν	f−1(ν	PROPN
ejpam-4548	121	45	)	)	PUNCT
ejpam-4548	121	46	for	for	ADP
ejpam-4548	121	47	each	each	DET
ejpam-4548	121	48	θ	θ	PROPN
ejpam-4548	121	49	∈	∈	PROPN
ejpam-4548	121	50	l(ν	l(ν	PROPN
ejpam-4548	121	51	)	)	PUNCT
ejpam-4548	121	52	.	.	PUNCT
ejpam-4548	122	1	theorem	theorem	ADJ
ejpam-4548	122	2	5	5	NUM
ejpam-4548	122	3	.	.	PUNCT
ejpam-4548	123	1	[	[	X
ejpam-4548	123	2	3	3	X
ejpam-4548	123	3	]	]	X
ejpam-4548	123	4	let	let	VERB
ejpam-4548	123	5	η	η	PROPN
ejpam-4548	123	6	,	,	PUNCT
ejpam-4548	123	7	θ	θ	PROPN
ejpam-4548	123	8	∈	∈	PROPN
ejpam-4548	123	9	l(µ	l(µ	PROPN
ejpam-4548	123	10	)	)	PUNCT
ejpam-4548	123	11	.	.	PUNCT
ejpam-4548	124	1	then	then	ADV
ejpam-4548	124	2	,	,	PUNCT
ejpam-4548	124	3	(	(	PUNCT
ejpam-4548	124	4	i	i	NOUN
ejpam-4548	124	5	)	)	PUNCT
ejpam-4548	124	6	η	η	PROPN
ejpam-4548	124	7	◦	◦	NOUN
ejpam-4548	124	8	θ	θ	X
ejpam-4548	124	9	∈	∈	PROPN
ejpam-4548	124	10	l(µ	l(µ	PROPN
ejpam-4548	124	11	)	)	PUNCT
ejpam-4548	124	12	if	if	SCONJ
ejpam-4548	124	13	η	η	PROPN
ejpam-4548	124	14	or	or	CCONJ
ejpam-4548	124	15	θ	θ	PROPN
ejpam-4548	124	16	∈	∈	PROPN
ejpam-4548	124	17	nl(µ	nl(µ	NOUN
ejpam-4548	124	18	)	)	PUNCT
ejpam-4548	124	19	,	,	PUNCT
ejpam-4548	124	20	(	(	PUNCT
ejpam-4548	124	21	ii	ii	NOUN
ejpam-4548	124	22	)	)	PUNCT
ejpam-4548	124	23	η	η	PROPN
ejpam-4548	124	24	◦	◦	NOUN
ejpam-4548	124	25	θ	θ	X
ejpam-4548	124	26	∈	∈	NOUN
ejpam-4548	124	27	nl(µ	nl(µ	NOUN
ejpam-4548	124	28	)	)	PUNCT
ejpam-4548	124	29	if	if	SCONJ
ejpam-4548	124	30	η	η	PROPN
ejpam-4548	124	31	and	and	CCONJ
ejpam-4548	124	32	θ	θ	PROPN
ejpam-4548	124	33	∈	∈	PROPN
ejpam-4548	124	34	nl(µ	nl(µ	NOUN
ejpam-4548	124	35	)	)	PUNCT
ejpam-4548	124	36	.	.	PUNCT
ejpam-4548	125	1	theorem	theorem	VERB
ejpam-4548	125	2	6	6	NUM
ejpam-4548	125	3	.	.	PUNCT
ejpam-4548	126	1	[	[	X
ejpam-4548	126	2	?	?	PUNCT
ejpam-4548	126	3	]	]	PUNCT
ejpam-4548	126	4	let	let	VERB
ejpam-4548	126	5	η	η	PROPN
ejpam-4548	126	6	∈	∈	PROPN
ejpam-4548	126	7	l	l	PROPN
ejpam-4548	126	8	µ	µ	X
ejpam-4548	126	9	.	.	PUNCT
ejpam-4548	127	1	let	let	VERB
ejpam-4548	127	2	a0	a0	PROPN
ejpam-4548	127	3	=	=	SYM
ejpam-4548	127	4	∨	∨	PROPN
ejpam-4548	127	5	x∈g	x∈g	PROPN
ejpam-4548	127	6	{	{	PUNCT
ejpam-4548	127	7	η	η	PROPN
ejpam-4548	127	8	(	(	PUNCT
ejpam-4548	127	9	x	x	NOUN
ejpam-4548	127	10	)	)	PUNCT
ejpam-4548	127	11	}	}	PUNCT
ejpam-4548	127	12	and	and	CCONJ
ejpam-4548	127	13	define	define	VERB
ejpam-4548	127	14	an	an	DET
ejpam-4548	127	15	l	l	NOUN
ejpam-4548	127	16	-	-	NOUN
ejpam-4548	127	17	subset	subset	NOUN
ejpam-4548	127	18	η̂	η̂	PUNCT
ejpam-4548	127	19	of	of	ADP
ejpam-4548	127	20	g	g	NOUN
ejpam-4548	127	21	by	by	ADP
ejpam-4548	127	22	η̂	η̂	PROPN
ejpam-4548	127	23	(	(	PUNCT
ejpam-4548	127	24	x	x	X
ejpam-4548	127	25	)	)	PUNCT
ejpam-4548	127	26	=	=	SYM
ejpam-4548	128	1	∨	∨	X
ejpam-4548	128	2	a≤a0	a≤a0	X
ejpam-4548	128	3	{	{	PUNCT
ejpam-4548	128	4	a	a	X
ejpam-4548	128	5	:	:	PUNCT
ejpam-4548	128	6	x	x	X
ejpam-4548	128	7	∈	∈	NOUN
ejpam-4548	128	8	⟨ηa⟩	⟨ηa⟩	NOUN
ejpam-4548	128	9	}	}	PUNCT
ejpam-4548	128	10	.	.	PUNCT
ejpam-4548	129	1	then	then	ADV
ejpam-4548	129	2	,	,	PUNCT
ejpam-4548	129	3	η̂	η̂	PROPN
ejpam-4548	129	4	∈	∈	PROPN
ejpam-4548	129	5	l(µ	l(µ	PROPN
ejpam-4548	129	6	)	)	PUNCT
ejpam-4548	129	7	and	and	CCONJ
ejpam-4548	129	8	η̂	η̂	NUM
ejpam-4548	129	9	=	=	SYM
ejpam-4548	129	10	⟨η⟩.	⟨η⟩.	PROPN
ejpam-4548	129	11	corollary	corollary	NOUN
ejpam-4548	129	12	1	1	NUM
ejpam-4548	129	13	.	.	PUNCT
ejpam-4548	130	1	let	let	VERB
ejpam-4548	130	2	η	η	PROPN
ejpam-4548	130	3	∈	∈	PROPN
ejpam-4548	130	4	l	l	PROPN
ejpam-4548	130	5	µ	µ	X
ejpam-4548	130	6	.	.	PUNCT
ejpam-4548	131	1	then	then	ADV
ejpam-4548	131	2	,	,	PUNCT
ejpam-4548	131	3	η̂(e	η̂(e	PROPN
ejpam-4548	131	4	)	)	PUNCT
ejpam-4548	131	5	=	=	PUNCT
ejpam-4548	132	1	∨	∨	NUM
ejpam-4548	132	2	x∈g	x∈g	PROPN
ejpam-4548	132	3	{	{	PUNCT
ejpam-4548	132	4	η(x	η(x	PROPN
ejpam-4548	132	5	)	)	PUNCT
ejpam-4548	132	6	}	}	PUNCT
ejpam-4548	132	7	.	.	PUNCT
ejpam-4548	133	1	definition	definition	NOUN
ejpam-4548	133	2	7	7	NUM
ejpam-4548	133	3	.	.	PUNCT
ejpam-4548	134	1	[	[	X
ejpam-4548	134	2	15	15	NUM
ejpam-4548	134	3	]	]	X
ejpam-4548	134	4	let	let	VERB
ejpam-4548	134	5	η	η	PROPN
ejpam-4548	134	6	∈	∈	PROPN
ejpam-4548	134	7	lµ.	lµ.	NOUN
ejpam-4548	134	8	then	then	ADV
ejpam-4548	134	9	,	,	PUNCT
ejpam-4548	134	10	η	η	PROPN
ejpam-4548	134	11	is	be	AUX
ejpam-4548	134	12	said	say	VERB
ejpam-4548	134	13	to	to	PART
ejpam-4548	134	14	have	have	VERB
ejpam-4548	134	15	sup	sup	NOUN
ejpam-4548	134	16	-	-	PUNCT
ejpam-4548	134	17	property	property	NOUN
ejpam-4548	134	18	if	if	SCONJ
ejpam-4548	134	19	for	for	ADP
ejpam-4548	134	20	every	every	DET
ejpam-4548	134	21	non	non	ADJ
ejpam-4548	134	22	-	-	ADJ
ejpam-4548	134	23	empty	empty	ADJ
ejpam-4548	134	24	subset	subset	NOUN
ejpam-4548	134	25	a	a	PRON
ejpam-4548	134	26	of	of	ADP
ejpam-4548	134	27	g	g	NOUN
ejpam-4548	134	28	,	,	PUNCT
ejpam-4548	134	29	there	there	PRON
ejpam-4548	134	30	exists	exist	VERB
ejpam-4548	134	31	a0	a0	PROPN
ejpam-4548	134	32	∈	∈	PROPN
ejpam-4548	134	33	a	a	DET
ejpam-4548	134	34	such	such	ADJ
ejpam-4548	134	35	that	that	SCONJ
ejpam-4548	134	36	∨	∨	NUM
ejpam-4548	134	37	a∈a	a∈a	ADJ
ejpam-4548	134	38	{	{	PUNCT
ejpam-4548	134	39	η(a	η(a	NOUN
ejpam-4548	134	40	)	)	PUNCT
ejpam-4548	134	41	}	}	PUNCT
ejpam-4548	134	42	=	=	SYM
ejpam-4548	134	43	η(a0	η(a0	NOUN
ejpam-4548	134	44	)	)	PUNCT
ejpam-4548	134	45	.	.	PUNCT
ejpam-4548	135	1	theorem	theorem	VERB
ejpam-4548	135	2	7	7	NUM
ejpam-4548	135	3	.	.	PUNCT
ejpam-4548	136	1	[	[	X
ejpam-4548	136	2	?	?	PUNCT
ejpam-4548	136	3	]	]	PUNCT
ejpam-4548	136	4	let	let	VERB
ejpam-4548	136	5	η	η	PROPN
ejpam-4548	136	6	∈	∈	PROPN
ejpam-4548	136	7	lµ	lµ	PROPN
ejpam-4548	136	8	and	and	CCONJ
ejpam-4548	136	9	possesses	possess	VERB
ejpam-4548	136	10	sup	sup	ADJ
ejpam-4548	136	11	-	-	PUNCT
ejpam-4548	136	12	property	property	NOUN
ejpam-4548	136	13	.	.	PUNCT
ejpam-4548	137	1	then	then	ADV
ejpam-4548	137	2	,	,	PUNCT
ejpam-4548	137	3	define	define	VERB
ejpam-4548	137	4	an	an	DET
ejpam-4548	137	5	l	l	NOUN
ejpam-4548	137	6	-	-	NOUN
ejpam-4548	137	7	subset	subset	NOUN
ejpam-4548	137	8	η̂	η̂	PUNCT
ejpam-4548	137	9	of	of	ADP
ejpam-4548	137	10	g	g	NOUN
ejpam-4548	137	11	by	by	ADP
ejpam-4548	137	12	η̂	η̂	PROPN
ejpam-4548	137	13	(	(	PUNCT
ejpam-4548	137	14	x	x	NOUN
ejpam-4548	137	15	)	)	PUNCT
ejpam-4548	137	16	=	=	SYM
ejpam-4548	137	17	∨	∨	X
ejpam-4548	137	18	a∈imη	a∈imη	PROPN
ejpam-4548	137	19	{	{	PUNCT
ejpam-4548	137	20	a	a	X
ejpam-4548	137	21	:	:	PUNCT
ejpam-4548	137	22	x	x	SYM
ejpam-4548	137	23	∈	∈	NOUN
ejpam-4548	137	24	⟨ηa⟩	⟨ηa⟩	NOUN
ejpam-4548	137	25	}	}	PUNCT
ejpam-4548	137	26	.	.	PUNCT
ejpam-4548	138	1	then	then	ADV
ejpam-4548	138	2	,	,	PUNCT
ejpam-4548	138	3	η̂	η̂	PROPN
ejpam-4548	138	4	∈	∈	PROPN
ejpam-4548	138	5	l(µ	l(µ	PROPN
ejpam-4548	138	6	)	)	PUNCT
ejpam-4548	138	7	and	and	CCONJ
ejpam-4548	138	8	η̂	η̂	NUM
ejpam-4548	138	9	=	=	SYM
ejpam-4548	138	10	⟨η⟩.	⟨η⟩.	PROPN
ejpam-4548	138	11	moreover	moreover	ADV
ejpam-4548	138	12	,	,	PUNCT
ejpam-4548	138	13	η̂	η̂	NUM
ejpam-4548	138	14	possesses	possess	VERB
ejpam-4548	138	15	sup	sup	ADJ
ejpam-4548	138	16	-	-	PUNCT
ejpam-4548	138	17	property	property	NOUN
ejpam-4548	138	18	and	and	CCONJ
ejpam-4548	138	19	i	i	PRON
ejpam-4548	138	20	m	m	VERB
ejpam-4548	138	21	η̂	η̂	NUM
ejpam-4548	138	22	⊆	⊆	NUM
ejpam-4548	138	23	i	i	PROPN
ejpam-4548	138	24	m	m	PROPN
ejpam-4548	138	25	η	η	PROPN
ejpam-4548	138	26	.	.	PROPN
ejpam-4548	138	27	recall	recall	VERB
ejpam-4548	138	28	the	the	DET
ejpam-4548	138	29	following	following	NOUN
ejpam-4548	138	30	from	from	ADP
ejpam-4548	138	31	[	[	X
ejpam-4548	138	32	3	3	NUM
ejpam-4548	138	33	]	]	PUNCT
ejpam-4548	138	34	:	:	PUNCT
ejpam-4548	138	35	definition	definition	NOUN
ejpam-4548	138	36	8	8	NUM
ejpam-4548	138	37	.	.	PUNCT
ejpam-4548	139	1	let	let	VERB
ejpam-4548	139	2	η	η	PROPN
ejpam-4548	139	3	,	,	PUNCT
ejpam-4548	139	4	θ	θ	PROPN
ejpam-4548	139	5	∈	∈	PROPN
ejpam-4548	139	6	lµ.	lµ.	NOUN
ejpam-4548	139	7	then	then	ADV
ejpam-4548	139	8	,	,	PUNCT
ejpam-4548	139	9	the	the	DET
ejpam-4548	139	10	commutator	commutator	NOUN
ejpam-4548	139	11	of	of	ADP
ejpam-4548	139	12	η	η	PROPN
ejpam-4548	139	13	and	and	CCONJ
ejpam-4548	139	14	θ	θ	PROPN
ejpam-4548	139	15	is	be	AUX
ejpam-4548	139	16	an	an	DET
ejpam-4548	139	17	l	l	NOUN
ejpam-4548	139	18	-	-	NOUN
ejpam-4548	139	19	subset	subset	NOUN
ejpam-4548	139	20	(	(	PUNCT
ejpam-4548	139	21	η	η	PROPN
ejpam-4548	139	22	,	,	PUNCT
ejpam-4548	139	23	θ	θ	NOUN
ejpam-4548	139	24	)	)	PUNCT
ejpam-4548	139	25	of	of	ADP
ejpam-4548	139	26	g	g	PROPN
ejpam-4548	139	27	defined	define	VERB
ejpam-4548	139	28	as	as	SCONJ
ejpam-4548	139	29	follows	follow	VERB
ejpam-4548	139	30	:	:	PUNCT
ejpam-4548	139	31	(	(	PUNCT
ejpam-4548	139	32	η	η	PROPN
ejpam-4548	139	33	,	,	PUNCT
ejpam-4548	139	34	θ)(x	θ)(x	PROPN
ejpam-4548	139	35	)	)	PUNCT
ejpam-4548	140	1	=	=	PRON
ejpam-4548	140	2	{	{	PUNCT
ejpam-4548	140	3	∨{η(y	∨{η(y	NOUN
ejpam-4548	140	4	)	)	PUNCT
ejpam-4548	140	5	∧	∧	PROPN
ejpam-4548	140	6	θ(z	θ(z	NOUN
ejpam-4548	140	7	)	)	PUNCT
ejpam-4548	140	8	}	}	PUNCT
ejpam-4548	140	9	if	if	SCONJ
ejpam-4548	140	10	x	x	X
ejpam-4548	140	11	=	=	PUNCT
ejpam-4548	141	1	[	[	X
ejpam-4548	141	2	y	y	PROPN
ejpam-4548	141	3	,	,	PUNCT
ejpam-4548	141	4	z	z	X
ejpam-4548	141	5	]	]	X
ejpam-4548	141	6	for	for	ADP
ejpam-4548	141	7	some	some	DET
ejpam-4548	141	8	y	y	PROPN
ejpam-4548	141	9	,	,	PUNCT
ejpam-4548	141	10	z	z	PROPN
ejpam-4548	141	11	∈	∈	PROPN
ejpam-4548	141	12	g	g	PROPN
ejpam-4548	141	13	,	,	PUNCT
ejpam-4548	141	14	inf	inf	PROPN
ejpam-4548	141	15	η	η	PROPN
ejpam-4548	141	16	∧	∧	PROPN
ejpam-4548	141	17	inf	inf	PROPN
ejpam-4548	141	18	θ	θ	NOUN
ejpam-4548	141	19	if	if	SCONJ
ejpam-4548	141	20	x	x	X
ejpam-4548	141	21	̸=	̸=	PROPN
ejpam-4548	141	22	[	[	X
ejpam-4548	141	23	y	y	PROPN
ejpam-4548	141	24	,	,	PUNCT
ejpam-4548	141	25	z	z	X
ejpam-4548	141	26	]	]	X
ejpam-4548	141	27	for	for	ADP
ejpam-4548	141	28	any	any	DET
ejpam-4548	141	29	y	y	NOUN
ejpam-4548	141	30	,	,	PUNCT
ejpam-4548	141	31	z	z	PROPN
ejpam-4548	141	32	∈	∈	PROPN
ejpam-4548	141	33	g.	g.	NOUN
ejpam-4548	141	34	clearly	clearly	ADV
ejpam-4548	141	35	,	,	PUNCT
ejpam-4548	141	36	(	(	PUNCT
ejpam-4548	141	37	η	η	PROPN
ejpam-4548	141	38	,	,	PUNCT
ejpam-4548	141	39	θ	θ	NOUN
ejpam-4548	141	40	)	)	PUNCT
ejpam-4548	141	41	⊆	⊆	NUM
ejpam-4548	141	42	µ.	µ.	NOUN
ejpam-4548	141	43	the	the	DET
ejpam-4548	141	44	commutator	commutator	NOUN
ejpam-4548	141	45	l	l	PROPN
ejpam-4548	141	46	-	-	PROPN
ejpam-4548	141	47	subgroup	subgroup	NOUN
ejpam-4548	141	48	of	of	ADP
ejpam-4548	141	49	η	η	PROPN
ejpam-4548	141	50	and	and	CCONJ
ejpam-4548	141	51	θ	θ	PROPN
ejpam-4548	141	52	is	be	AUX
ejpam-4548	141	53	defined	define	VERB
ejpam-4548	141	54	as	as	ADP
ejpam-4548	141	55	⟨(η	⟨(η	ADJ
ejpam-4548	141	56	,	,	PUNCT
ejpam-4548	141	57	θ)⟩	θ)⟩	ADJ
ejpam-4548	141	58	and	and	CCONJ
ejpam-4548	141	59	we	we	PRON
ejpam-4548	141	60	write	write	VERB
ejpam-4548	141	61	⟨(η	⟨(η	ADJ
ejpam-4548	141	62	,	,	PUNCT
ejpam-4548	141	63	θ)⟩	θ)⟩	ADJ
ejpam-4548	141	64	=	=	SYM
ejpam-4548	141	65	[	[	X
ejpam-4548	141	66	η	η	X
ejpam-4548	141	67	,	,	PUNCT
ejpam-4548	141	68	θ	θ	NOUN
ejpam-4548	141	69	]	]	PUNCT
ejpam-4548	141	70	.	.	PUNCT
ejpam-4548	142	1	note	note	VERB
ejpam-4548	142	2	that	that	SCONJ
ejpam-4548	142	3	inf(η	inf(η	PROPN
ejpam-4548	142	4	,	,	PUNCT
ejpam-4548	142	5	θ	θ	PROPN
ejpam-4548	142	6	)	)	PUNCT
ejpam-4548	142	7	=	=	SYM
ejpam-4548	142	8	inf	inf	PROPN
ejpam-4548	142	9	η	η	PROPN
ejpam-4548	142	10	∧	∧	PROPN
ejpam-4548	142	11	inf	inf	PROPN
ejpam-4548	142	12	θ	θ	PROPN
ejpam-4548	142	13	and	and	CCONJ
ejpam-4548	142	14	[	[	X
ejpam-4548	142	15	η	η	PROPN
ejpam-4548	142	16	,	,	PUNCT
ejpam-4548	142	17	θ	θ	X
ejpam-4548	142	18	]	]	X
ejpam-4548	142	19	∈	∈	PROPN
ejpam-4548	142	20	l(µ	l(µ	PROPN
ejpam-4548	142	21	)	)	PUNCT
ejpam-4548	142	22	.	.	PUNCT
ejpam-4548	143	1	n.	n.	PROPN
ejpam-4548	143	2	ajmal	ajmal	PROPN
ejpam-4548	143	3	,	,	PUNCT
ejpam-4548	143	4	i.	i.	PROPN
ejpam-4548	143	5	jahan	jahan	PROPN
ejpam-4548	143	6	.	.	PROPN
ejpam-4548	143	7	,	,	PUNCT
ejpam-4548	143	8	b.	b.	PROPN
ejpam-4548	143	9	davvaz	davvaz	PROPN
ejpam-4548	143	10	/	/	SYM
ejpam-4548	143	11	eur	eur	PROPN
ejpam-4548	143	12	.	.	PUNCT
ejpam-4548	144	1	j.	j.	PROPN
ejpam-4548	144	2	pure	pure	PROPN
ejpam-4548	144	3	appl	appl	PROPN
ejpam-4548	144	4	.	.	PROPN
ejpam-4548	144	5	math	math	PROPN
ejpam-4548	144	6	,	,	PUNCT
ejpam-4548	144	7	15	15	NUM
ejpam-4548	144	8	(	(	PUNCT
ejpam-4548	144	9	4	4	NUM
ejpam-4548	144	10	)	)	PUNCT
ejpam-4548	144	11	(	(	PUNCT
ejpam-4548	144	12	2022	2022	NUM
ejpam-4548	144	13	)	)	PUNCT
ejpam-4548	144	14	,	,	PUNCT
ejpam-4548	144	15	2086	2086	NUM
ejpam-4548	144	16	-	-	SYM
ejpam-4548	144	17	2115	2115	NUM
ejpam-4548	144	18	2092	2092	NUM
ejpam-4548	144	19	here	here	ADV
ejpam-4548	144	20	recall	recall	VERB
ejpam-4548	144	21	that	that	SCONJ
ejpam-4548	144	22	if	if	SCONJ
ejpam-4548	144	23	η	η	PROPN
ejpam-4548	144	24	is	be	AUX
ejpam-4548	144	25	non	non	ADJ
ejpam-4548	144	26	-	-	ADJ
ejpam-4548	144	27	constant	constant	ADJ
ejpam-4548	144	28	and	and	CCONJ
ejpam-4548	144	29	η	η	PROPN
ejpam-4548	144	30	̸=	̸=	PROPN
ejpam-4548	144	31	µ	µ	NUM
ejpam-4548	144	32	,	,	PUNCT
ejpam-4548	144	33	then	then	ADV
ejpam-4548	144	34	η	η	PROPN
ejpam-4548	144	35	is	be	AUX
ejpam-4548	144	36	said	say	VERB
ejpam-4548	144	37	to	to	PART
ejpam-4548	144	38	be	be	AUX
ejpam-4548	144	39	a	a	DET
ejpam-4548	144	40	proper	proper	ADJ
ejpam-4548	144	41	l	l	NOUN
ejpam-4548	144	42	-	-	NOUN
ejpam-4548	144	43	subgroup	subgroup	NOUN
ejpam-4548	144	44	of	of	ADP
ejpam-4548	144	45	µ.	µ.	PROPN
ejpam-4548	144	46	also	also	ADV
ejpam-4548	144	47	,	,	PUNCT
ejpam-4548	144	48	η	η	PROPN
ejpam-4548	144	49	is	be	AUX
ejpam-4548	144	50	said	say	VERB
ejpam-4548	144	51	to	to	PART
ejpam-4548	144	52	be	be	AUX
ejpam-4548	144	53	a	a	DET
ejpam-4548	144	54	trivial	trivial	ADJ
ejpam-4548	144	55	l	l	NOUN
ejpam-4548	144	56	-	-	NOUN
ejpam-4548	144	57	subgroup	subgroup	NOUN
ejpam-4548	144	58	of	of	ADP
ejpam-4548	144	59	µ	µ	PROPN
ejpam-4548	144	60	if	if	SCONJ
ejpam-4548	144	61	its	its	PRON
ejpam-4548	144	62	chain	chain	NOUN
ejpam-4548	144	63	of	of	ADP
ejpam-4548	144	64	level	level	NOUN
ejpam-4548	144	65	subgroups	subgroup	NOUN
ejpam-4548	144	66	contains	contain	VERB
ejpam-4548	144	67	only	only	ADV
ejpam-4548	144	68	{	{	PUNCT
ejpam-4548	144	69	e	e	NOUN
ejpam-4548	144	70	}	}	PUNCT
ejpam-4548	144	71	and	and	CCONJ
ejpam-4548	144	72	g.	g.	PROPN
ejpam-4548	144	73	clearly	clearly	ADV
ejpam-4548	144	74	,	,	PUNCT
ejpam-4548	144	75	η	η	PROPN
ejpam-4548	144	76	is	be	AUX
ejpam-4548	144	77	a	a	DET
ejpam-4548	144	78	proper	proper	ADJ
ejpam-4548	144	79	l	l	NOUN
ejpam-4548	144	80	-	-	NOUN
ejpam-4548	144	81	subgroup	subgroup	NOUN
ejpam-4548	144	82	of	of	ADP
ejpam-4548	144	83	µ	µ	NOUN
ejpam-4548	144	84	if	if	NOUN
ejpam-4548	145	1	and	and	CCONJ
ejpam-4548	145	2	only	only	ADV
ejpam-4548	145	3	if	if	SCONJ
ejpam-4548	145	4	η	η	PROPN
ejpam-4548	145	5	has	have	VERB
ejpam-4548	145	6	distinct	distinct	ADJ
ejpam-4548	145	7	tip	tip	NOUN
ejpam-4548	145	8	and	and	CCONJ
ejpam-4548	145	9	tail	tail	NOUN
ejpam-4548	145	10	,	,	PUNCT
ejpam-4548	145	11	and	and	CCONJ
ejpam-4548	145	12	η	η	PROPN
ejpam-4548	145	13	̸=	̸=	PROPN
ejpam-4548	145	14	µ.	µ.	NOUN
ejpam-4548	145	15	definition	definition	NOUN
ejpam-4548	145	16	9	9	NUM
ejpam-4548	145	17	.	.	PUNCT
ejpam-4548	146	1	let	let	VERB
ejpam-4548	146	2	η	η	PROPN
ejpam-4548	146	3	be	be	AUX
ejpam-4548	146	4	a	a	DET
ejpam-4548	146	5	proper	proper	ADJ
ejpam-4548	146	6	l	l	NOUN
ejpam-4548	146	7	-	-	NOUN
ejpam-4548	146	8	subgroup	subgroup	NOUN
ejpam-4548	146	9	of	of	ADP
ejpam-4548	146	10	µ.	µ.	PROPN
ejpam-4548	146	11	then	then	ADV
ejpam-4548	146	12	,	,	PUNCT
ejpam-4548	146	13	we	we	PRON
ejpam-4548	146	14	define	define	VERB
ejpam-4548	146	15	an	an	DET
ejpam-4548	146	16	l	l	NOUN
ejpam-4548	146	17	-	-	NOUN
ejpam-4548	146	18	subgroup	subgroup	NOUN
ejpam-4548	146	19	of	of	ADP
ejpam-4548	146	20	µ	µ	PROPN
ejpam-4548	146	21	contained	contain	VERB
ejpam-4548	146	22	in	in	ADP
ejpam-4548	146	23	η	η	PROPN
ejpam-4548	146	24	,	,	PUNCT
ejpam-4548	146	25	denoted	denote	VERB
ejpam-4548	146	26	by	by	ADP
ejpam-4548	146	27	ηa0t0	ηa0t0	PROPN
ejpam-4548	146	28	,	,	PUNCT
ejpam-4548	146	29	as	as	SCONJ
ejpam-4548	146	30	follows	follow	VERB
ejpam-4548	146	31	:	:	PUNCT
ejpam-4548	146	32	ηa0t0	ηa0t0	PROPN
ejpam-4548	146	33	(	(	PUNCT
ejpam-4548	146	34	y	y	NOUN
ejpam-4548	146	35	)	)	PUNCT
ejpam-4548	146	36	=	=	PRON
ejpam-4548	146	37	{	{	PUNCT
ejpam-4548	146	38	a0	a0	NOUN
ejpam-4548	146	39	,	,	PUNCT
ejpam-4548	146	40	if	if	SCONJ
ejpam-4548	146	41	y	y	PROPN
ejpam-4548	146	42	=	=	SYM
ejpam-4548	146	43	e	e	PROPN
ejpam-4548	146	44	,	,	PUNCT
ejpam-4548	146	45	t0	t0	PROPN
ejpam-4548	146	46	,	,	PUNCT
ejpam-4548	146	47	if	if	SCONJ
ejpam-4548	146	48	y	y	PROPN
ejpam-4548	146	49	̸=	̸=	PROPN
ejpam-4548	146	50	e	e	NOUN
ejpam-4548	146	51	,	,	PUNCT
ejpam-4548	146	52	where	where	SCONJ
ejpam-4548	146	53	a0	a0	PROPN
ejpam-4548	146	54	=	=	SYM
ejpam-4548	146	55	η(e	η(e	PROPN
ejpam-4548	146	56	)	)	PUNCT
ejpam-4548	146	57	and	and	CCONJ
ejpam-4548	146	58	t0	t0	PROPN
ejpam-4548	146	59	=	=	PROPN
ejpam-4548	146	60	inf	inf	PROPN
ejpam-4548	146	61	η	η	PROPN
ejpam-4548	146	62	.	.	PROPN
ejpam-4548	146	63	here	here	ADV
ejpam-4548	146	64	ηa0t0	ηa0t0	PROPN
ejpam-4548	146	65	,	,	PUNCT
ejpam-4548	146	66	a	a	DET
ejpam-4548	146	67	trivial	trivial	ADJ
ejpam-4548	146	68	l	l	NOUN
ejpam-4548	146	69	-	-	NOUN
ejpam-4548	146	70	subgroup	subgroup	NOUN
ejpam-4548	146	71	of	of	ADP
ejpam-4548	146	72	µ	µ	NUM
ejpam-4548	146	73	,	,	PUNCT
ejpam-4548	146	74	is	be	AUX
ejpam-4548	146	75	called	call	VERB
ejpam-4548	146	76	the	the	DET
ejpam-4548	146	77	trivial	trivial	ADJ
ejpam-4548	146	78	l	l	NOUN
ejpam-4548	146	79	-	-	NOUN
ejpam-4548	146	80	subgroup	subgroup	NOUN
ejpam-4548	146	81	of	of	ADP
ejpam-4548	146	82	η	η	PROPN
ejpam-4548	146	83	.	.	PROPN
ejpam-4548	146	84	3	3	NUM
ejpam-4548	146	85	.	.	PUNCT
ejpam-4548	146	86	subnormal	subnormal	ADJ
ejpam-4548	146	87	l	l	PROPN
ejpam-4548	146	88	-	-	NOUN
ejpam-4548	146	89	subgroup	subgroup	NOUN
ejpam-4548	146	90	in	in	ADP
ejpam-4548	146	91	order	order	NOUN
ejpam-4548	146	92	to	to	PART
ejpam-4548	146	93	introduce	introduce	VERB
ejpam-4548	146	94	the	the	DET
ejpam-4548	146	95	concept	concept	NOUN
ejpam-4548	146	96	of	of	ADP
ejpam-4548	146	97	subnormality	subnormality	NOUN
ejpam-4548	146	98	in	in	ADP
ejpam-4548	146	99	the	the	DET
ejpam-4548	146	100	l	l	NOUN
ejpam-4548	146	101	-	-	NOUN
ejpam-4548	146	102	setting	setting	NOUN
ejpam-4548	146	103	,	,	PUNCT
ejpam-4548	146	104	we	we	PRON
ejpam-4548	146	105	start	start	VERB
ejpam-4548	146	106	with	with	ADP
ejpam-4548	146	107	the	the	DET
ejpam-4548	146	108	definition	definition	NOUN
ejpam-4548	146	109	of	of	ADP
ejpam-4548	146	110	conjugate	conjugate	NOUN
ejpam-4548	146	111	of	of	ADP
ejpam-4548	146	112	an	an	DET
ejpam-4548	146	113	l	l	NOUN
ejpam-4548	146	114	-	-	NOUN
ejpam-4548	146	115	subset	subset	NOUN
ejpam-4548	146	116	by	by	ADP
ejpam-4548	146	117	an	an	DET
ejpam-4548	146	118	l	l	NOUN
ejpam-4548	146	119	-	-	NOUN
ejpam-4548	146	120	subset	subset	VERB
ejpam-4548	146	121	like	like	ADP
ejpam-4548	146	122	its	its	PRON
ejpam-4548	146	123	counterpart	counterpart	NOUN
ejpam-4548	146	124	in	in	ADP
ejpam-4548	146	125	classical	classical	ADJ
ejpam-4548	146	126	group	group	NOUN
ejpam-4548	146	127	theory.then	theory.then	NOUN
ejpam-4548	146	128	,	,	PUNCT
ejpam-4548	146	129	the	the	DET
ejpam-4548	146	130	notion	notion	NOUN
ejpam-4548	146	131	of	of	ADP
ejpam-4548	146	132	normal	normal	ADJ
ejpam-4548	146	133	closure	closure	NOUN
ejpam-4548	146	134	of	of	ADP
ejpam-4548	146	135	an	an	DET
ejpam-4548	146	136	l	l	NOUN
ejpam-4548	146	137	-	-	NOUN
ejpam-4548	146	138	subgroup	subgroup	NOUN
ejpam-4548	146	139	is	be	AUX
ejpam-4548	146	140	defined	define	VERB
ejpam-4548	146	141	and	and	CCONJ
ejpam-4548	146	142	utilized	utilize	VERB
ejpam-4548	146	143	to	to	PART
ejpam-4548	146	144	formulate	formulate	VERB
ejpam-4548	146	145	the	the	DET
ejpam-4548	146	146	concept	concept	NOUN
ejpam-4548	146	147	of	of	ADP
ejpam-4548	146	148	subnormal	subnormal	ADJ
ejpam-4548	146	149	l	l	NOUN
ejpam-4548	146	150	-	-	NOUN
ejpam-4548	146	151	subgroups	subgroup	NOUN
ejpam-4548	146	152	.	.	PUNCT
ejpam-4548	147	1	definition	definition	NOUN
ejpam-4548	147	2	10	10	NUM
ejpam-4548	147	3	.	.	PUNCT
ejpam-4548	148	1	[	[	X
ejpam-4548	148	2	4	4	X
ejpam-4548	148	3	]	]	PUNCT
ejpam-4548	148	4	let	let	AUX
ejpam-4548	148	5	η	η	PROPN
ejpam-4548	148	6	,	,	PUNCT
ejpam-4548	148	7	θ	θ	PROPN
ejpam-4548	148	8	∈	∈	PROPN
ejpam-4548	148	9	lµ.	lµ.	NOUN
ejpam-4548	148	10	define	define	VERB
ejpam-4548	148	11	an	an	DET
ejpam-4548	148	12	l	l	NOUN
ejpam-4548	148	13	-	-	NOUN
ejpam-4548	148	14	subset	subset	ADJ
ejpam-4548	148	15	θηθ−1	θηθ−1	PROPN
ejpam-4548	148	16	of	of	ADP
ejpam-4548	148	17	g	g	PROPN
ejpam-4548	148	18	as	as	SCONJ
ejpam-4548	148	19	follows	follow	VERB
ejpam-4548	148	20	:	:	PUNCT
ejpam-4548	148	21	θηθ−1(x	θηθ−1(x	NOUN
ejpam-4548	148	22	)	)	PUNCT
ejpam-4548	148	23	=	=	PUNCT
ejpam-4548	149	1	∨	∨	NUM
ejpam-4548	149	2	x	x	X
ejpam-4548	149	3	=	=	PROPN
ejpam-4548	149	4	zyz−1	zyz−1	PROPN
ejpam-4548	149	5	{	{	PUNCT
ejpam-4548	149	6	η(y	η(y	PROPN
ejpam-4548	149	7	)	)	PUNCT
ejpam-4548	149	8	∧	∧	PROPN
ejpam-4548	149	9	θ(z	θ(z	NOUN
ejpam-4548	149	10	)	)	PUNCT
ejpam-4548	149	11	}	}	PUNCT
ejpam-4548	149	12	for	for	ADP
ejpam-4548	149	13	each	each	DET
ejpam-4548	149	14	x	x	SYM
ejpam-4548	149	15	∈	∈	PROPN
ejpam-4548	149	16	g.	g.	NOUN
ejpam-4548	149	17	we	we	PRON
ejpam-4548	149	18	call	call	VERB
ejpam-4548	149	19	θηθ−1	θηθ−1	PROPN
ejpam-4548	149	20	the	the	DET
ejpam-4548	149	21	conjugate	conjugate	NOUN
ejpam-4548	149	22	of	of	ADP
ejpam-4548	149	23	η	η	PROPN
ejpam-4548	149	24	by	by	ADP
ejpam-4548	149	25	θ	θ	PROPN
ejpam-4548	149	26	.	.	PUNCT
ejpam-4548	150	1	clearly	clearly	ADV
ejpam-4548	150	2	,	,	PUNCT
ejpam-4548	150	3	θηθ−1	θηθ−1	PROPN
ejpam-4548	150	4	⊆	⊆	NUM
ejpam-4548	150	5	µ.	µ.	NOUN
ejpam-4548	150	6	hence	hence	ADV
ejpam-4548	150	7	the	the	DET
ejpam-4548	150	8	l	l	PROPN
ejpam-4548	150	9	-	-	PROPN
ejpam-4548	150	10	subgroup	subgroup	NOUN
ejpam-4548	150	11	⟨θηθ−1⟩	⟨θηθ−1⟩	PROPN
ejpam-4548	150	12	∈	∈	PROPN
ejpam-4548	150	13	l(µ	l(µ	PROPN
ejpam-4548	150	14	)	)	PUNCT
ejpam-4548	150	15	and	and	CCONJ
ejpam-4548	150	16	is	be	AUX
ejpam-4548	150	17	denoted	denote	VERB
ejpam-4548	150	18	by	by	ADP
ejpam-4548	150	19	ηθ	ηθ	NOUN
ejpam-4548	150	20	.	.	PUNCT
ejpam-4548	151	1	the	the	DET
ejpam-4548	151	2	following	follow	VERB
ejpam-4548	151	3	lemma	lemma	PROPN
ejpam-4548	151	4	is	be	AUX
ejpam-4548	151	5	instrumental	instrumental	ADJ
ejpam-4548	151	6	in	in	ADP
ejpam-4548	151	7	the	the	DET
ejpam-4548	151	8	development	development	NOUN
ejpam-4548	151	9	of	of	ADP
ejpam-4548	151	10	this	this	DET
ejpam-4548	151	11	paper	paper	NOUN
ejpam-4548	151	12	:	:	PUNCT
ejpam-4548	151	13	lemma	lemma	PROPN
ejpam-4548	151	14	1	1	X
ejpam-4548	151	15	.	.	PUNCT
ejpam-4548	152	1	let	let	VERB
ejpam-4548	152	2	η	η	PROPN
ejpam-4548	152	3	,	,	PUNCT
ejpam-4548	152	4	θ	θ	PROPN
ejpam-4548	152	5	∈	∈	PROPN
ejpam-4548	152	6	l(µ	l(µ	PROPN
ejpam-4548	152	7	)	)	PUNCT
ejpam-4548	152	8	.	.	PUNCT
ejpam-4548	153	1	then	then	ADV
ejpam-4548	153	2	,	,	PUNCT
ejpam-4548	153	3	(	(	PUNCT
ejpam-4548	153	4	i	i	NOUN
ejpam-4548	153	5	)	)	PUNCT
ejpam-4548	153	6	η	η	PROPN
ejpam-4548	153	7	⊆	⊆	X
ejpam-4548	153	8	θηθ−1	θηθ−1	PROPN
ejpam-4548	153	9	provided	provide	VERB
ejpam-4548	153	10	η(e	η(e	PROPN
ejpam-4548	153	11	)	)	PUNCT
ejpam-4548	153	12	⩽	⩽	PROPN
ejpam-4548	153	13	θ(e	θ(e	PROPN
ejpam-4548	153	14	)	)	PUNCT
ejpam-4548	153	15	,	,	PUNCT
ejpam-4548	153	16	(	(	PUNCT
ejpam-4548	153	17	ii	ii	NOUN
ejpam-4548	153	18	)	)	PUNCT
ejpam-4548	153	19	∨	∨	NUM
ejpam-4548	153	20	x∈g	x∈g	PROPN
ejpam-4548	153	21	{	{	PUNCT
ejpam-4548	153	22	θηθ−1(x	θηθ−1(x	NOUN
ejpam-4548	153	23	)	)	PUNCT
ejpam-4548	153	24	}	}	PUNCT
ejpam-4548	153	25	=	=	SYM
ejpam-4548	153	26	θηθ−1(e	θηθ−1(e	X
ejpam-4548	153	27	)	)	PUNCT
ejpam-4548	153	28	=	=	PUNCT
ejpam-4548	153	29	η(e	η(e	X
ejpam-4548	153	30	)	)	PUNCT
ejpam-4548	153	31	∧	∧	PROPN
ejpam-4548	153	32	θ(e	θ(e	PROPN
ejpam-4548	153	33	)	)	PUNCT
ejpam-4548	153	34	,	,	PUNCT
ejpam-4548	153	35	(	(	PUNCT
ejpam-4548	153	36	iii	iii	NOUN
ejpam-4548	153	37	)	)	PUNCT
ejpam-4548	153	38	θηθ−1(x	θηθ−1(x	NOUN
ejpam-4548	153	39	)	)	PUNCT
ejpam-4548	153	40	=	=	SYM
ejpam-4548	153	41	θηθ−1(x−1	θηθ−1(x−1	NOUN
ejpam-4548	153	42	)	)	PUNCT
ejpam-4548	153	43	for	for	ADP
ejpam-4548	153	44	each	each	DET
ejpam-4548	153	45	x	x	SYM
ejpam-4548	153	46	∈	∈	PROPN
ejpam-4548	153	47	g	g	PROPN
ejpam-4548	153	48	,	,	PUNCT
ejpam-4548	153	49	(	(	PUNCT
ejpam-4548	153	50	iv	iv	X
ejpam-4548	153	51	)	)	PUNCT
ejpam-4548	153	52	θηθ−1(gxg−1	θηθ−1(gxg−1	ADJ
ejpam-4548	153	53	)	)	PUNCT
ejpam-4548	153	54	≥	≥	NUM
ejpam-4548	153	55	θηθ−1(x	θηθ−1(x	NOUN
ejpam-4548	153	56	)	)	PUNCT
ejpam-4548	153	57	∧	∧	NOUN
ejpam-4548	153	58	θ(g	θ(g	PROPN
ejpam-4548	153	59	)	)	PUNCT
ejpam-4548	153	60	for	for	ADP
ejpam-4548	153	61	each	each	DET
ejpam-4548	153	62	x	x	NOUN
ejpam-4548	153	63	,	,	PUNCT
ejpam-4548	153	64	g	g	PROPN
ejpam-4548	153	65	∈	∈	PROPN
ejpam-4548	153	66	g.	g.	NOUN
ejpam-4548	153	67	proof	proof	NOUN
ejpam-4548	153	68	.	.	PUNCT
ejpam-4548	154	1	the	the	DET
ejpam-4548	154	2	proofs	proof	NOUN
ejpam-4548	154	3	of	of	ADP
ejpam-4548	154	4	(	(	PUNCT
ejpam-4548	154	5	i)-(iii	i)-(iii	NOUN
ejpam-4548	154	6	)	)	PUNCT
ejpam-4548	154	7	are	be	AUX
ejpam-4548	154	8	obvious	obvious	ADJ
ejpam-4548	154	9	.	.	PUNCT
ejpam-4548	155	1	we	we	PRON
ejpam-4548	155	2	only	only	ADV
ejpam-4548	155	3	prove	prove	VERB
ejpam-4548	155	4	(	(	PUNCT
ejpam-4548	155	5	iv	iv	NUM
ejpam-4548	155	6	)	)	PUNCT
ejpam-4548	155	7	.	.	PUNCT
ejpam-4548	156	1	so	so	ADV
ejpam-4548	156	2	,	,	PUNCT
ejpam-4548	156	3	let	let	VERB
ejpam-4548	156	4	x	x	PRON
ejpam-4548	156	5	,	,	PUNCT
ejpam-4548	156	6	g	g	PROPN
ejpam-4548	156	7	∈	∈	PROPN
ejpam-4548	156	8	g.	g.	NOUN
ejpam-4548	156	9	then	then	ADV
ejpam-4548	156	10	,	,	PUNCT
ejpam-4548	156	11	θηθ−1(x	θηθ−1(x	NOUN
ejpam-4548	156	12	)	)	PUNCT
ejpam-4548	156	13	∧	∧	NOUN
ejpam-4548	156	14	θ(g	θ(g	PROPN
ejpam-4548	156	15	)	)	PUNCT
ejpam-4548	156	16	=	=	SYM
ejpam-4548	156	17	{	{	PUNCT
ejpam-4548	156	18	∨	∨	NOUN
ejpam-4548	157	1	x	x	X
ejpam-4548	157	2	=	=	PROPN
ejpam-4548	157	3	zyz−1	zyz−1	PROPN
ejpam-4548	157	4	{	{	PUNCT
ejpam-4548	157	5	η(y	η(y	PROPN
ejpam-4548	157	6	)	)	PUNCT
ejpam-4548	157	7	∧	∧	PROPN
ejpam-4548	157	8	θ(z	θ(z	NOUN
ejpam-4548	157	9	)	)	PUNCT
ejpam-4548	157	10	}	}	PUNCT
ejpam-4548	157	11	}	}	PUNCT
ejpam-4548	157	12	∧	∧	PROPN
ejpam-4548	157	13	θ(g	θ(g	NOUN
ejpam-4548	157	14	)	)	PUNCT
ejpam-4548	157	15	n.	n.	NOUN
ejpam-4548	157	16	ajmal	ajmal	PROPN
ejpam-4548	157	17	,	,	PUNCT
ejpam-4548	157	18	i.	i.	PROPN
ejpam-4548	157	19	jahan	jahan	PROPN
ejpam-4548	157	20	.	.	PROPN
ejpam-4548	157	21	,	,	PUNCT
ejpam-4548	157	22	b.	b.	PROPN
ejpam-4548	157	23	davvaz	davvaz	PROPN
ejpam-4548	157	24	/	/	SYM
ejpam-4548	157	25	eur	eur	PROPN
ejpam-4548	157	26	.	.	PUNCT
ejpam-4548	158	1	j.	j.	PROPN
ejpam-4548	158	2	pure	pure	PROPN
ejpam-4548	158	3	appl	appl	PROPN
ejpam-4548	158	4	.	.	PROPN
ejpam-4548	158	5	math	math	PROPN
ejpam-4548	158	6	,	,	PUNCT
ejpam-4548	158	7	15	15	NUM
ejpam-4548	158	8	(	(	PUNCT
ejpam-4548	158	9	4	4	NUM
ejpam-4548	158	10	)	)	PUNCT
ejpam-4548	158	11	(	(	PUNCT
ejpam-4548	158	12	2022	2022	NUM
ejpam-4548	158	13	)	)	PUNCT
ejpam-4548	158	14	,	,	PUNCT
ejpam-4548	158	15	2086	2086	NUM
ejpam-4548	158	16	-	-	SYM
ejpam-4548	158	17	2115	2115	NUM
ejpam-4548	158	18	2093	2093	NUM
ejpam-4548	158	19	=	=	SYM
ejpam-4548	159	1	∨	∨	NUM
ejpam-4548	159	2	x	x	X
ejpam-4548	159	3	=	=	PROPN
ejpam-4548	159	4	zyz−1	zyz−1	PROPN
ejpam-4548	159	5	{	{	PUNCT
ejpam-4548	159	6	η(y	η(y	PROPN
ejpam-4548	159	7	)	)	PUNCT
ejpam-4548	159	8	∧	∧	PROPN
ejpam-4548	159	9	θ(z	θ(z	NOUN
ejpam-4548	159	10	)	)	PUNCT
ejpam-4548	159	11	∧	∧	PROPN
ejpam-4548	159	12	θ(g	θ(g	NUM
ejpam-4548	159	13	)	)	PUNCT
ejpam-4548	159	14	}	}	PUNCT
ejpam-4548	159	15	(	(	PUNCT
ejpam-4548	159	16	as	as	SCONJ
ejpam-4548	159	17	l	l	NOUN
ejpam-4548	159	18	is	be	AUX
ejpam-4548	159	19	a	a	DET
ejpam-4548	159	20	completely	completely	ADV
ejpam-4548	159	21	distributive	distributive	ADJ
ejpam-4548	159	22	lattice	lattice	NOUN
ejpam-4548	159	23	)	)	PUNCT
ejpam-4548	159	24	≤	≤	NOUN
ejpam-4548	159	25	∨	∨	NUM
ejpam-4548	159	26	x	x	X
ejpam-4548	159	27	=	=	NOUN
ejpam-4548	159	28	zyz−1	zyz−1	NOUN
ejpam-4548	159	29	gxg−1=(gz)y(gz)−1	gxg−1=(gz)y(gz)−1	NOUN
ejpam-4548	159	30	{	{	PUNCT
ejpam-4548	159	31	η(y	η(y	NOUN
ejpam-4548	159	32	)	)	PUNCT
ejpam-4548	159	33	∧	∧	PROPN
ejpam-4548	159	34	θ(gz	θ(gz	PROPN
ejpam-4548	159	35	)	)	PUNCT
ejpam-4548	159	36	}	}	PUNCT
ejpam-4548	159	37	=	=	SYM
ejpam-4548	159	38	θηθ−1(gxg−1	θηθ−1(gxg−1	ADJ
ejpam-4548	159	39	)	)	PUNCT
ejpam-4548	159	40	.	.	PUNCT
ejpam-4548	160	1	corollary	corollary	ADJ
ejpam-4548	160	2	2	2	NUM
ejpam-4548	160	3	.	.	PUNCT
ejpam-4548	160	4	let	let	VERB
ejpam-4548	160	5	η	η	PROPN
ejpam-4548	160	6	∈	∈	PROPN
ejpam-4548	160	7	l(µ	l(µ	PROPN
ejpam-4548	160	8	)	)	PUNCT
ejpam-4548	160	9	.	.	PUNCT
ejpam-4548	161	1	then	then	ADV
ejpam-4548	161	2	,	,	PUNCT
ejpam-4548	161	3	η	η	PROPN
ejpam-4548	161	4	⊆	⊆	NUM
ejpam-4548	161	5	µηµ−1	µηµ−1	PROPN
ejpam-4548	161	6	⊆	⊆	NUM
ejpam-4548	161	7	µ.	µ.	NOUN
ejpam-4548	161	8	our	our	PRON
ejpam-4548	161	9	next	next	ADJ
ejpam-4548	161	10	definition	definition	NOUN
ejpam-4548	161	11	leads	lead	VERB
ejpam-4548	161	12	to	to	ADP
ejpam-4548	161	13	the	the	DET
ejpam-4548	161	14	notion	notion	NOUN
ejpam-4548	161	15	of	of	ADP
ejpam-4548	161	16	normal	normal	ADJ
ejpam-4548	161	17	closure	closure	NOUN
ejpam-4548	161	18	of	of	ADP
ejpam-4548	161	19	an	an	DET
ejpam-4548	161	20	l	l	NOUN
ejpam-4548	161	21	-	-	NOUN
ejpam-4548	161	22	subgroup	subgroup	NOUN
ejpam-4548	161	23	of	of	ADP
ejpam-4548	161	24	an	an	DET
ejpam-4548	161	25	lgroup	lgroup	NOUN
ejpam-4548	161	26	which	which	PRON
ejpam-4548	161	27	in	in	ADP
ejpam-4548	161	28	its	its	PRON
ejpam-4548	161	29	essence	essence	NOUN
ejpam-4548	161	30	is	be	AUX
ejpam-4548	161	31	parallel	parallel	ADJ
ejpam-4548	161	32	to	to	ADP
ejpam-4548	161	33	classical	classical	ADJ
ejpam-4548	161	34	group	group	NOUN
ejpam-4548	161	35	theory	theory	NOUN
ejpam-4548	161	36	.	.	PUNCT
ejpam-4548	162	1	definition	definition	NOUN
ejpam-4548	162	2	11	11	NUM
ejpam-4548	162	3	.	.	PUNCT
ejpam-4548	163	1	let	let	VERB
ejpam-4548	163	2	η	η	PROPN
ejpam-4548	163	3	∈	∈	PROPN
ejpam-4548	163	4	l(µ	l(µ	PROPN
ejpam-4548	163	5	)	)	PUNCT
ejpam-4548	163	6	.	.	PUNCT
ejpam-4548	164	1	then	then	ADV
ejpam-4548	164	2	,	,	PUNCT
ejpam-4548	164	3	the	the	DET
ejpam-4548	164	4	normal	normal	ADJ
ejpam-4548	164	5	closure	closure	NOUN
ejpam-4548	164	6	of	of	ADP
ejpam-4548	164	7	η	η	PROPN
ejpam-4548	164	8	in	in	ADP
ejpam-4548	164	9	µ	µ	PROPN
ejpam-4548	164	10	is	be	AUX
ejpam-4548	164	11	defined	define	VERB
ejpam-4548	164	12	as	as	ADP
ejpam-4548	164	13	the	the	DET
ejpam-4548	164	14	l	l	NOUN
ejpam-4548	164	15	-	-	NOUN
ejpam-4548	164	16	subgroup	subgroup	NOUN
ejpam-4548	164	17	of	of	ADP
ejpam-4548	164	18	µ	µ	PROPN
ejpam-4548	164	19	generated	generate	VERB
ejpam-4548	164	20	by	by	ADP
ejpam-4548	164	21	the	the	DET
ejpam-4548	164	22	conjugate	conjugate	NOUN
ejpam-4548	164	23	‘	'	PUNCT
ejpam-4548	164	24	µηµ−1	µηµ−1	PROPN
ejpam-4548	164	25	’	'	PUNCT
ejpam-4548	164	26	.	.	PUNCT
ejpam-4548	165	1	it	it	PRON
ejpam-4548	165	2	is	be	AUX
ejpam-4548	165	3	denoted	denote	VERB
ejpam-4548	165	4	by	by	ADP
ejpam-4548	165	5	ηµ.	ηµ.	NOUN
ejpam-4548	165	6	thus	thus	ADV
ejpam-4548	165	7	,	,	PUNCT
ejpam-4548	165	8	ηµ	ηµ	ADP
ejpam-4548	165	9	=	=	PROPN
ejpam-4548	165	10	⟨µηµ−1⟩.	⟨µηµ−1⟩.	PROPN
ejpam-4548	165	11	note	note	NOUN
ejpam-4548	165	12	that	that	SCONJ
ejpam-4548	165	13	by	by	ADP
ejpam-4548	165	14	corollary	corollary	ADJ
ejpam-4548	165	15	2	2	NUM
ejpam-4548	165	16	,	,	PUNCT
ejpam-4548	165	17	the	the	DET
ejpam-4548	165	18	normal	normal	ADJ
ejpam-4548	165	19	closure	closure	NOUN
ejpam-4548	165	20	ηµ	ηµ	VERB
ejpam-4548	165	21	is	be	AUX
ejpam-4548	165	22	an	an	DET
ejpam-4548	165	23	l	l	NOUN
ejpam-4548	165	24	-	-	NOUN
ejpam-4548	165	25	subgroup	subgroup	NOUN
ejpam-4548	165	26	of	of	ADP
ejpam-4548	165	27	µ	µ	X
ejpam-4548	165	28	containing	contain	VERB
ejpam-4548	165	29	η	η	PROPN
ejpam-4548	165	30	.	.	PROPN
ejpam-4548	165	31	in	in	ADP
ejpam-4548	165	32	classical	classical	ADJ
ejpam-4548	165	33	group	group	NOUN
ejpam-4548	165	34	theory	theory	NOUN
ejpam-4548	165	35	,	,	PUNCT
ejpam-4548	165	36	the	the	DET
ejpam-4548	165	37	normal	normal	ADJ
ejpam-4548	165	38	closure	closure	NOUN
ejpam-4548	165	39	of	of	ADP
ejpam-4548	165	40	a	a	DET
ejpam-4548	165	41	subgroup	subgroup	NOUN
ejpam-4548	165	42	of	of	ADP
ejpam-4548	165	43	a	a	DET
ejpam-4548	165	44	group	group	NOUN
ejpam-4548	165	45	g	g	NOUN
ejpam-4548	165	46	is	be	AUX
ejpam-4548	165	47	the	the	DET
ejpam-4548	165	48	smallest	small	ADJ
ejpam-4548	165	49	normal	normal	ADJ
ejpam-4548	165	50	subgroup	subgroup	NOUN
ejpam-4548	165	51	of	of	ADP
ejpam-4548	165	52	g	g	PROPN
ejpam-4548	165	53	which	which	PRON
ejpam-4548	165	54	contains	contain	VERB
ejpam-4548	165	55	the	the	DET
ejpam-4548	165	56	given	give	VERB
ejpam-4548	165	57	subgroup	subgroup	NOUN
ejpam-4548	165	58	.	.	PUNCT
ejpam-4548	166	1	for	for	ADP
ejpam-4548	166	2	the	the	DET
ejpam-4548	166	3	sake	sake	NOUN
ejpam-4548	166	4	of	of	ADP
ejpam-4548	166	5	completeness	completeness	NOUN
ejpam-4548	166	6	,	,	PUNCT
ejpam-4548	166	7	recall	recall	VERB
ejpam-4548	166	8	the	the	DET
ejpam-4548	166	9	following	following	NOUN
ejpam-4548	166	10	from	from	ADP
ejpam-4548	166	11	[	[	X
ejpam-4548	166	12	2	2	NUM
ejpam-4548	166	13	]	]	PUNCT
ejpam-4548	166	14	:	:	PUNCT
ejpam-4548	166	15	theorem	theorem	NOUN
ejpam-4548	166	16	8	8	NUM
ejpam-4548	166	17	.	.	PUNCT
ejpam-4548	167	1	let	let	VERB
ejpam-4548	167	2	η	η	PROPN
ejpam-4548	167	3	∈	∈	PROPN
ejpam-4548	167	4	l(µ	l(µ	PROPN
ejpam-4548	167	5	)	)	PUNCT
ejpam-4548	167	6	.	.	PUNCT
ejpam-4548	168	1	then	then	ADV
ejpam-4548	168	2	,	,	PUNCT
ejpam-4548	168	3	the	the	DET
ejpam-4548	168	4	normal	normal	ADJ
ejpam-4548	168	5	closure	closure	NOUN
ejpam-4548	168	6	ηµ	ηµ	VERB
ejpam-4548	168	7	is	be	AUX
ejpam-4548	168	8	the	the	DET
ejpam-4548	168	9	least	least	ADJ
ejpam-4548	168	10	normal	normal	ADJ
ejpam-4548	168	11	l	l	NOUN
ejpam-4548	168	12	-	-	NOUN
ejpam-4548	168	13	subgroup	subgroup	NOUN
ejpam-4548	168	14	of	of	ADP
ejpam-4548	168	15	µ	µ	X
ejpam-4548	168	16	containing	contain	VERB
ejpam-4548	168	17	η	η	PROPN
ejpam-4548	168	18	.	.	PROPN
ejpam-4548	168	19	proof	proof	NOUN
ejpam-4548	168	20	.	.	PUNCT
ejpam-4548	169	1	as	as	SCONJ
ejpam-4548	169	2	ηµ	ηµ	VERB
ejpam-4548	169	3	is	be	AUX
ejpam-4548	169	4	an	an	DET
ejpam-4548	169	5	l	l	NOUN
ejpam-4548	169	6	-	-	NOUN
ejpam-4548	169	7	subgroup	subgroup	NOUN
ejpam-4548	169	8	of	of	ADP
ejpam-4548	169	9	µ	µ	X
ejpam-4548	169	10	containing	contain	VERB
ejpam-4548	169	11	η	η	PROPN
ejpam-4548	169	12	,	,	PUNCT
ejpam-4548	169	13	we	we	PRON
ejpam-4548	169	14	first	first	ADV
ejpam-4548	169	15	establish	establish	VERB
ejpam-4548	169	16	that	that	SCONJ
ejpam-4548	169	17	ηµ	ηµ	VERB
ejpam-4548	169	18	∈	∈	PROPN
ejpam-4548	169	19	nl(µ	nl(µ	NOUN
ejpam-4548	169	20	)	)	PUNCT
ejpam-4548	169	21	.	.	PUNCT
ejpam-4548	170	1	for	for	ADP
ejpam-4548	170	2	this	this	DET
ejpam-4548	170	3	purpose	purpose	NOUN
ejpam-4548	170	4	,	,	PUNCT
ejpam-4548	170	5	let	let	VERB
ejpam-4548	170	6	x	x	PRON
ejpam-4548	170	7	,	,	PUNCT
ejpam-4548	170	8	g	g	PROPN
ejpam-4548	170	9	∈	∈	PROPN
ejpam-4548	170	10	g.	g.	PROPN
ejpam-4548	170	11	note	note	VERB
ejpam-4548	170	12	that	that	SCONJ
ejpam-4548	170	13	,	,	PUNCT
ejpam-4548	170	14	by	by	ADP
ejpam-4548	170	15	lemma	lemma	PROPN
ejpam-4548	170	16	1,∨	1,∨	PROPN
ejpam-4548	170	17	y∈g	y∈g	PROPN
ejpam-4548	170	18	µηµ−1(y	µηµ−1(y	X
ejpam-4548	170	19	)	)	PUNCT
ejpam-4548	171	1	=	=	SYM
ejpam-4548	171	2	η(e	η(e	PROPN
ejpam-4548	171	3	)	)	PUNCT
ejpam-4548	171	4	.	.	PUNCT
ejpam-4548	172	1	hence	hence	ADV
ejpam-4548	172	2	in	in	ADP
ejpam-4548	172	3	view	view	NOUN
ejpam-4548	172	4	of	of	ADP
ejpam-4548	172	5	theorem	theorem	ADJ
ejpam-4548	172	6	6	6	NUM
ejpam-4548	172	7	,	,	PUNCT
ejpam-4548	172	8	ηµ(x	ηµ(x	NOUN
ejpam-4548	172	9	)	)	PUNCT
ejpam-4548	172	10	∧	∧	NOUN
ejpam-4548	172	11	µ(g	µ(g	PROPN
ejpam-4548	172	12	)	)	PUNCT
ejpam-4548	172	13	=	=	SYM
ejpam-4548	172	14	{	{	PUNCT
ejpam-4548	172	15	∨	∨	NOUN
ejpam-4548	172	16	a≤η(e	a≤η(e	NOUN
ejpam-4548	172	17	)	)	PUNCT
ejpam-4548	172	18	{	{	PUNCT
ejpam-4548	172	19	a	a	X
ejpam-4548	172	20	:	:	PUNCT
ejpam-4548	172	21	x	x	SYM
ejpam-4548	172	22	∈	∈	PROPN
ejpam-4548	172	23	⟨(µηµ−1)a⟩	⟨(µηµ−1)a⟩	NOUN
ejpam-4548	172	24	}	}	PUNCT
ejpam-4548	172	25	}	}	PUNCT
ejpam-4548	172	26	∧	∧	PROPN
ejpam-4548	172	27	µ(g	µ(g	PROPN
ejpam-4548	172	28	)	)	PUNCT
ejpam-4548	172	29	=	=	PUNCT
ejpam-4548	172	30	∨	∨	NOUN
ejpam-4548	172	31	a≤η(e	a≤η(e	NOUN
ejpam-4548	172	32	)	)	PUNCT
ejpam-4548	172	33	{	{	PUNCT
ejpam-4548	172	34	a	a	DET
ejpam-4548	172	35	∧	∧	PROPN
ejpam-4548	172	36	µ(g	µ(g	PROPN
ejpam-4548	172	37	)	)	PUNCT
ejpam-4548	172	38	:	:	PUNCT
ejpam-4548	172	39	x	x	X
ejpam-4548	172	40	∈	∈	NOUN
ejpam-4548	172	41	⟨(µηµ−1)a⟩	⟨(µηµ−1)a⟩	NOUN
ejpam-4548	172	42	}	}	PUNCT
ejpam-4548	172	43	.	.	PUNCT
ejpam-4548	173	1	(	(	PUNCT
ejpam-4548	173	2	as	as	SCONJ
ejpam-4548	173	3	l	l	NOUN
ejpam-4548	173	4	is	be	AUX
ejpam-4548	173	5	a	a	DET
ejpam-4548	173	6	completely	completely	ADV
ejpam-4548	173	7	distributive	distributive	ADJ
ejpam-4548	173	8	lattice	lattice	NOUN
ejpam-4548	173	9	)	)	PUNCT
ejpam-4548	173	10	next	next	ADV
ejpam-4548	173	11	,	,	PUNCT
ejpam-4548	173	12	claim	claim	VERB
ejpam-4548	173	13	that	that	SCONJ
ejpam-4548	173	14	if	if	SCONJ
ejpam-4548	173	15	x	x	PROPN
ejpam-4548	173	16	∈	∈	PROPN
ejpam-4548	173	17	⟨(µηµ−1)a⟩	⟨(µηµ−1)a⟩	NOUN
ejpam-4548	173	18	,	,	PUNCT
ejpam-4548	173	19	then	then	ADV
ejpam-4548	173	20	ηµ(gxg−1	ηµ(gxg−1	PROPN
ejpam-4548	173	21	)	)	PUNCT
ejpam-4548	173	22	≥	≥	NOUN
ejpam-4548	173	23	a	a	DET
ejpam-4548	173	24	∧	∧	PROPN
ejpam-4548	173	25	µ(g	µ(g	PROPN
ejpam-4548	173	26	)	)	PUNCT
ejpam-4548	173	27	.	.	PUNCT
ejpam-4548	174	1	n.	n.	PROPN
ejpam-4548	174	2	ajmal	ajmal	PROPN
ejpam-4548	174	3	,	,	PUNCT
ejpam-4548	174	4	i.	i.	PROPN
ejpam-4548	174	5	jahan	jahan	PROPN
ejpam-4548	174	6	.	.	PROPN
ejpam-4548	174	7	,	,	PUNCT
ejpam-4548	174	8	b.	b.	PROPN
ejpam-4548	174	9	davvaz	davvaz	PROPN
ejpam-4548	174	10	/	/	SYM
ejpam-4548	174	11	eur	eur	PROPN
ejpam-4548	174	12	.	.	PUNCT
ejpam-4548	175	1	j.	j.	PROPN
ejpam-4548	175	2	pure	pure	PROPN
ejpam-4548	175	3	appl	appl	PROPN
ejpam-4548	175	4	.	.	PROPN
ejpam-4548	175	5	math	math	PROPN
ejpam-4548	175	6	,	,	PUNCT
ejpam-4548	175	7	15	15	NUM
ejpam-4548	175	8	(	(	PUNCT
ejpam-4548	175	9	4	4	NUM
ejpam-4548	175	10	)	)	PUNCT
ejpam-4548	175	11	(	(	PUNCT
ejpam-4548	175	12	2022	2022	NUM
ejpam-4548	175	13	)	)	PUNCT
ejpam-4548	175	14	,	,	PUNCT
ejpam-4548	175	15	2086	2086	NUM
ejpam-4548	175	16	-	-	SYM
ejpam-4548	175	17	2115	2115	NUM
ejpam-4548	175	18	2094	2094	NUM
ejpam-4548	175	19	in	in	ADP
ejpam-4548	175	20	order	order	NOUN
ejpam-4548	175	21	to	to	PART
ejpam-4548	175	22	prove	prove	VERB
ejpam-4548	175	23	the	the	DET
ejpam-4548	175	24	claim	claim	NOUN
ejpam-4548	175	25	,	,	PUNCT
ejpam-4548	175	26	let	let	VERB
ejpam-4548	175	27	x	x	SYM
ejpam-4548	175	28	∈	∈	PROPN
ejpam-4548	175	29	⟨(µηµ−1)a⟩.	⟨(µηµ−1)a⟩.	X
ejpam-4548	175	30	then	then	ADV
ejpam-4548	175	31	,	,	PUNCT
ejpam-4548	175	32	x	x	PUNCT
ejpam-4548	175	33	=	=	SYM
ejpam-4548	176	1	x1	x1	NUM
ejpam-4548	176	2	x2	x2	PROPN
ejpam-4548	176	3	·	·	PUNCT
ejpam-4548	176	4	·	·	PUNCT
ejpam-4548	176	5	·	·	PUNCT
ejpam-4548	177	1	xn	xn	X
ejpam-4548	177	2	,	,	PUNCT
ejpam-4548	177	3	where	where	SCONJ
ejpam-4548	177	4	either	either	CCONJ
ejpam-4548	177	5	xi	xi	X
ejpam-4548	177	6	or	or	CCONJ
ejpam-4548	177	7	x−1	x−1	PROPN
ejpam-4548	177	8	i	i	PRON
ejpam-4548	177	9	∈	∈	PROPN
ejpam-4548	177	10	(	(	PUNCT
ejpam-4548	177	11	µηµ−1)a	µηµ−1)a	NUM
ejpam-4548	177	12	for	for	ADP
ejpam-4548	177	13	each	each	DET
ejpam-4548	177	14	i.	i.	NOUN
ejpam-4548	177	15	thus	thus	ADV
ejpam-4548	177	16	,	,	PUNCT
ejpam-4548	177	17	gxg−1	gxg−1	PROPN
ejpam-4548	177	18	=	=	PUNCT
ejpam-4548	177	19	(	(	PUNCT
ejpam-4548	177	20	gx1	gx1	PROPN
ejpam-4548	177	21	g	g	PROPN
ejpam-4548	177	22	−1)(gx2	−1)(gx2	PROPN
ejpam-4548	177	23	g	g	PROPN
ejpam-4548	177	24	−1	−1	NOUN
ejpam-4548	177	25	)	)	PUNCT
ejpam-4548	177	26	·	·	PUNCT
ejpam-4548	177	27	·	·	PUNCT
ejpam-4548	177	28	·	·	PUNCT
ejpam-4548	177	29	(	(	PUNCT
ejpam-4548	177	30	gxng−1	gxng−1	PROPN
ejpam-4548	177	31	)	)	PUNCT
ejpam-4548	177	32	.	.	PUNCT
ejpam-4548	178	1	by	by	ADP
ejpam-4548	178	2	lemma	lemma	PROPN
ejpam-4548	178	3	1	1	NUM
ejpam-4548	178	4	,	,	PUNCT
ejpam-4548	178	5	for	for	ADP
ejpam-4548	178	6	each	each	DET
ejpam-4548	178	7	i	i	PRON
ejpam-4548	178	8	µηµ−1(gxig	µηµ−1(gxig	PROPN
ejpam-4548	178	9	−1	−1	NOUN
ejpam-4548	178	10	)	)	PUNCT
ejpam-4548	178	11	≥	≥	NOUN
ejpam-4548	178	12	µηµ−1(xi	µηµ−1(xi	NOUN
ejpam-4548	178	13	)	)	PUNCT
ejpam-4548	178	14	∧	∧	PROPN
ejpam-4548	178	15	µ(g	µ(g	PROPN
ejpam-4548	178	16	)	)	PUNCT
ejpam-4548	178	17	.	.	PUNCT
ejpam-4548	179	1	again	again	ADV
ejpam-4548	179	2	by	by	ADP
ejpam-4548	179	3	lemma	lemma	PROPN
ejpam-4548	179	4	1	1	NUM
ejpam-4548	179	5	,	,	PUNCT
ejpam-4548	179	6	for	for	SCONJ
ejpam-4548	179	7	each	each	DET
ejpam-4548	179	8	i	i	PRON
ejpam-4548	179	9	µηµ−1(xi	µηµ−1(xi	PROPN
ejpam-4548	179	10	)	)	PUNCT
ejpam-4548	180	1	=	=	PUNCT
ejpam-4548	181	1	µηµ−1(x−1	µηµ−1(x−1	NOUN
ejpam-4548	181	2	i	i	NOUN
ejpam-4548	181	3	)	)	PUNCT
ejpam-4548	181	4	.	.	PUNCT
ejpam-4548	182	1	as	as	ADP
ejpam-4548	182	2	xi	xi	NUM
ejpam-4548	182	3	or	or	CCONJ
ejpam-4548	182	4	x	x	ADP
ejpam-4548	182	5	−1	−1	NOUN
ejpam-4548	182	6	i	i	PRON
ejpam-4548	182	7	∈	∈	PROPN
ejpam-4548	182	8	(	(	PUNCT
ejpam-4548	182	9	µηµ−1)a	µηµ−1)a	NUM
ejpam-4548	182	10	,	,	PUNCT
ejpam-4548	182	11	it	it	PRON
ejpam-4548	182	12	follows	follow	VERB
ejpam-4548	182	13	that	that	SCONJ
ejpam-4548	182	14	µηµ−1(gxig	µηµ−1(gxig	PROPN
ejpam-4548	182	15	−1	−1	NOUN
ejpam-4548	182	16	)	)	PUNCT
ejpam-4548	182	17	≥	≥	NOUN
ejpam-4548	182	18	a	a	DET
ejpam-4548	182	19	∧	∧	PROPN
ejpam-4548	182	20	µ(g	µ(g	PROPN
ejpam-4548	182	21	)	)	PUNCT
ejpam-4548	182	22	for	for	ADP
ejpam-4548	182	23	each	each	DET
ejpam-4548	182	24	i.	i.	NOUN
ejpam-4548	182	25	hence	hence	ADV
ejpam-4548	182	26	,	,	PUNCT
ejpam-4548	182	27	n∧	n∧	PROPN
ejpam-4548	182	28	i=1	i=1	PROPN
ejpam-4548	182	29	{	{	PUNCT
ejpam-4548	182	30	µηµ−1(gxig	µηµ−1(gxig	PROPN
ejpam-4548	182	31	−1	−1	NOUN
ejpam-4548	182	32	)	)	PUNCT
ejpam-4548	182	33	}	}	PUNCT
ejpam-4548	182	34	≥	≥	VERB
ejpam-4548	182	35	a	a	DET
ejpam-4548	182	36	∧	∧	PROPN
ejpam-4548	182	37	µ(g	µ(g	PROPN
ejpam-4548	182	38	)	)	PUNCT
ejpam-4548	182	39	.	.	PUNCT
ejpam-4548	183	1	therefore	therefore	ADV
ejpam-4548	183	2	,	,	PUNCT
ejpam-4548	183	3	ηµ(gxg−1	ηµ(gxg−1	PROPN
ejpam-4548	183	4	)	)	PUNCT
ejpam-4548	183	5	≥	≥	PROPN
ejpam-4548	183	6	n∧	n∧	NUM
ejpam-4548	183	7	i=0	i=0	PROPN
ejpam-4548	183	8	{	{	PUNCT
ejpam-4548	183	9	ηµ(gxig−1	ηµ(gxig−1	NOUN
ejpam-4548	183	10	)	)	PUNCT
ejpam-4548	183	11	}	}	PUNCT
ejpam-4548	183	12	≥	≥	PROPN
ejpam-4548	183	13	n∧	n∧	NUM
ejpam-4548	183	14	i=1	i=1	PROPN
ejpam-4548	183	15	{	{	PUNCT
ejpam-4548	183	16	µηµ−1(gxig	µηµ−1(gxig	PROPN
ejpam-4548	183	17	−1	−1	NOUN
ejpam-4548	183	18	)	)	PUNCT
ejpam-4548	183	19	}	}	PUNCT
ejpam-4548	183	20	≥	≥	VERB
ejpam-4548	183	21	a	a	DET
ejpam-4548	183	22	∧	∧	PROPN
ejpam-4548	183	23	µ(g	µ(g	PROPN
ejpam-4548	183	24	)	)	PUNCT
ejpam-4548	183	25	.	.	PUNCT
ejpam-4548	184	1	this	this	PRON
ejpam-4548	184	2	establishes	establish	VERB
ejpam-4548	184	3	the	the	DET
ejpam-4548	184	4	claim	claim	NOUN
ejpam-4548	184	5	.	.	PUNCT
ejpam-4548	185	1	consequently	consequently	ADV
ejpam-4548	185	2	,	,	PUNCT
ejpam-4548	185	3	ηµ(x	ηµ(x	NOUN
ejpam-4548	185	4	)	)	PUNCT
ejpam-4548	185	5	∧	∧	NOUN
ejpam-4548	185	6	µ(g	µ(g	PROPN
ejpam-4548	185	7	)	)	PUNCT
ejpam-4548	185	8	=	=	PUNCT
ejpam-4548	185	9	∨	∨	NOUN
ejpam-4548	185	10	a≤η(e	a≤η(e	NOUN
ejpam-4548	185	11	)	)	PUNCT
ejpam-4548	185	12	{	{	PUNCT
ejpam-4548	185	13	a	a	DET
ejpam-4548	185	14	∧	∧	PROPN
ejpam-4548	185	15	µ(g	µ(g	PROPN
ejpam-4548	185	16	)	)	PUNCT
ejpam-4548	185	17	:	:	PUNCT
ejpam-4548	186	1	x	x	X
ejpam-4548	186	2	∈	∈	PROPN
ejpam-4548	186	3	⟨(µηµ−1)a⟩	⟨(µηµ−1)a⟩	NOUN
ejpam-4548	186	4	}	}	PUNCT
ejpam-4548	186	5	≤	≤	ADJ
ejpam-4548	186	6	∨	∨	NUM
ejpam-4548	186	7	a≤η(e	a≤η(e	NOUN
ejpam-4548	186	8	)	)	PUNCT
ejpam-4548	186	9	{	{	PUNCT
ejpam-4548	186	10	ηµ(gxg−1	ηµ(gxg−1	PROPN
ejpam-4548	186	11	)	)	PUNCT
ejpam-4548	186	12	:	:	PUNCT
ejpam-4548	186	13	x	x	X
ejpam-4548	186	14	∈	∈	NOUN
ejpam-4548	186	15	⟨(µηµ−1)a⟩	⟨(µηµ−1)a⟩	NOUN
ejpam-4548	186	16	}	}	PUNCT
ejpam-4548	186	17	=	=	SYM
ejpam-4548	186	18	ηµ(gxg−1	ηµ(gxg−1	PROPN
ejpam-4548	186	19	)	)	PUNCT
ejpam-4548	186	20	.	.	PUNCT
ejpam-4548	187	1	lastly	lastly	ADV
ejpam-4548	187	2	to	to	PART
ejpam-4548	187	3	prove	prove	VERB
ejpam-4548	187	4	that	that	SCONJ
ejpam-4548	187	5	ηµ	ηµ	VERB
ejpam-4548	187	6	is	be	AUX
ejpam-4548	187	7	the	the	DET
ejpam-4548	187	8	least	least	ADJ
ejpam-4548	187	9	normal	normal	ADJ
ejpam-4548	187	10	l	l	NOUN
ejpam-4548	187	11	-	-	NOUN
ejpam-4548	187	12	subgroup	subgroup	NOUN
ejpam-4548	187	13	of	of	ADP
ejpam-4548	187	14	µ	µ	X
ejpam-4548	187	15	containing	contain	VERB
ejpam-4548	187	16	η	η	PROPN
ejpam-4548	187	17	,	,	PUNCT
ejpam-4548	187	18	let	let	VERB
ejpam-4548	187	19	λ	λ	PRON
ejpam-4548	187	20	be	be	AUX
ejpam-4548	187	21	a	a	DET
ejpam-4548	187	22	normal	normal	ADJ
ejpam-4548	187	23	l	l	NOUN
ejpam-4548	187	24	-	-	NOUN
ejpam-4548	187	25	subgroup	subgroup	NOUN
ejpam-4548	187	26	of	of	ADP
ejpam-4548	187	27	µ	µ	X
ejpam-4548	187	28	containing	contain	VERB
ejpam-4548	187	29	η	η	PROPN
ejpam-4548	187	30	and	and	CCONJ
ejpam-4548	187	31	x	x	PROPN
ejpam-4548	187	32	∈	∈	PROPN
ejpam-4548	187	33	g.	g.	NOUN
ejpam-4548	187	34	then	then	ADV
ejpam-4548	187	35	,	,	PUNCT
ejpam-4548	187	36	µηµ−1(x	µηµ−1(x	NOUN
ejpam-4548	187	37	)	)	PUNCT
ejpam-4548	187	38	=	=	PUNCT
ejpam-4548	188	1	∨	∨	NUM
ejpam-4548	188	2	x	x	X
ejpam-4548	188	3	=	=	PROPN
ejpam-4548	188	4	zyz−1	zyz−1	PROPN
ejpam-4548	188	5	{	{	PUNCT
ejpam-4548	188	6	η(y	η(y	PROPN
ejpam-4548	188	7	)	)	PUNCT
ejpam-4548	188	8	∧	∧	PROPN
ejpam-4548	188	9	µ(z	µ(z	PROPN
ejpam-4548	188	10	)	)	PUNCT
ejpam-4548	188	11	}	}	PUNCT
ejpam-4548	188	12	≤	≤	NOUN
ejpam-4548	188	13	∨	∨	NUM
ejpam-4548	188	14	x	x	X
ejpam-4548	188	15	=	=	PROPN
ejpam-4548	188	16	zyz−1	zyz−1	PROPN
ejpam-4548	188	17	{	{	PUNCT
ejpam-4548	188	18	λ(y	λ(y	PROPN
ejpam-4548	188	19	)	)	PUNCT
ejpam-4548	188	20	∧	∧	PROPN
ejpam-4548	188	21	µ(z	µ(z	PROPN
ejpam-4548	188	22	)	)	PUNCT
ejpam-4548	188	23	}	}	PUNCT
ejpam-4548	188	24	(	(	PUNCT
ejpam-4548	188	25	as	as	ADP
ejpam-4548	188	26	η	η	PROPN
ejpam-4548	188	27	⊆	⊆	NUM
ejpam-4548	188	28	λ	λ	PROPN
ejpam-4548	188	29	)	)	PUNCT
ejpam-4548	188	30	n.	n.	NOUN
ejpam-4548	188	31	ajmal	ajmal	PROPN
ejpam-4548	188	32	,	,	PUNCT
ejpam-4548	188	33	i.	i.	PROPN
ejpam-4548	188	34	jahan	jahan	PROPN
ejpam-4548	188	35	.	.	PROPN
ejpam-4548	188	36	,	,	PUNCT
ejpam-4548	188	37	b.	b.	PROPN
ejpam-4548	188	38	davvaz	davvaz	PROPN
ejpam-4548	188	39	/	/	SYM
ejpam-4548	188	40	eur	eur	PROPN
ejpam-4548	188	41	.	.	PUNCT
ejpam-4548	189	1	j.	j.	PROPN
ejpam-4548	189	2	pure	pure	PROPN
ejpam-4548	189	3	appl	appl	PROPN
ejpam-4548	189	4	.	.	PROPN
ejpam-4548	189	5	math	math	PROPN
ejpam-4548	189	6	,	,	PUNCT
ejpam-4548	189	7	15	15	NUM
ejpam-4548	189	8	(	(	PUNCT
ejpam-4548	189	9	4	4	NUM
ejpam-4548	189	10	)	)	PUNCT
ejpam-4548	189	11	(	(	PUNCT
ejpam-4548	189	12	2022	2022	NUM
ejpam-4548	189	13	)	)	PUNCT
ejpam-4548	189	14	,	,	PUNCT
ejpam-4548	189	15	2086	2086	NUM
ejpam-4548	189	16	-	-	SYM
ejpam-4548	189	17	2115	2115	NUM
ejpam-4548	189	18	2095	2095	NUM
ejpam-4548	189	19	≤	≤	NOUN
ejpam-4548	189	20	∨	∨	NUM
ejpam-4548	189	21	x	x	X
ejpam-4548	189	22	=	=	PROPN
ejpam-4548	189	23	zyz−1	zyz−1	PROPN
ejpam-4548	189	24	{	{	PUNCT
ejpam-4548	189	25	λ(zyz−1	λ(zyz−1	NOUN
ejpam-4548	189	26	)	)	PUNCT
ejpam-4548	189	27	}	}	PUNCT
ejpam-4548	189	28	(	(	PUNCT
ejpam-4548	189	29	as	as	ADP
ejpam-4548	189	30	η	η	PROPN
ejpam-4548	189	31	∈	∈	PROPN
ejpam-4548	189	32	nl(µ	nl(µ	NOUN
ejpam-4548	189	33	)	)	PUNCT
ejpam-4548	189	34	)	)	PUNCT
ejpam-4548	190	1	=	=	PUNCT
ejpam-4548	190	2	λ(x	λ(x	X
ejpam-4548	190	3	)	)	PUNCT
ejpam-4548	190	4	.	.	PUNCT
ejpam-4548	191	1	thus	thus	ADV
ejpam-4548	191	2	µηµ−1	µηµ−1	PROPN
ejpam-4548	191	3	⊆	⊆	NUM
ejpam-4548	191	4	λ	λ	NOUN
ejpam-4548	191	5	and	and	CCONJ
ejpam-4548	191	6	as	as	SCONJ
ejpam-4548	191	7	λ	λ	PROPN
ejpam-4548	191	8	is	be	AUX
ejpam-4548	191	9	an	an	DET
ejpam-4548	191	10	l	l	NOUN
ejpam-4548	191	11	-	-	NOUN
ejpam-4548	191	12	subgroup	subgroup	NOUN
ejpam-4548	191	13	of	of	ADP
ejpam-4548	191	14	µ	µ	NUM
ejpam-4548	191	15	,	,	PUNCT
ejpam-4548	191	16	we	we	PRON
ejpam-4548	191	17	have	have	AUX
ejpam-4548	191	18	ηµ	ηµ	VERB
ejpam-4548	191	19	=	=	PUNCT
ejpam-4548	191	20	⟨µηµ−1⟩	⟨µηµ−1⟩	PROPN
ejpam-4548	191	21	⊆	⊆	NUM
ejpam-4548	191	22	λ	λ	PROPN
ejpam-4548	191	23	.	.	PUNCT
ejpam-4548	192	1	this	this	PRON
ejpam-4548	192	2	proves	prove	VERB
ejpam-4548	192	3	our	our	PRON
ejpam-4548	192	4	result	result	NOUN
ejpam-4548	192	5	.	.	PUNCT
ejpam-4548	193	1	corollary	corollary	ADJ
ejpam-4548	193	2	3	3	X
ejpam-4548	193	3	.	.	PUNCT
ejpam-4548	194	1	let	let	VERB
ejpam-4548	194	2	η	η	PROPN
ejpam-4548	194	3	∈	∈	PROPN
ejpam-4548	194	4	l(µ	l(µ	PROPN
ejpam-4548	194	5	)	)	PUNCT
ejpam-4548	194	6	.	.	PUNCT
ejpam-4548	195	1	then	then	ADV
ejpam-4548	195	2	,	,	PUNCT
ejpam-4548	195	3	η	η	PROPN
ejpam-4548	195	4	∈	∈	PROPN
ejpam-4548	195	5	nl(µ	nl(µ	NOUN
ejpam-4548	195	6	)	)	PUNCT
ejpam-4548	195	7	if	if	SCONJ
ejpam-4548	195	8	and	and	CCONJ
ejpam-4548	195	9	only	only	ADV
ejpam-4548	195	10	if	if	SCONJ
ejpam-4548	195	11	ηµ	ηµ	ADP
ejpam-4548	195	12	=	=	SYM
ejpam-4548	195	13	η	η	PROPN
ejpam-4548	195	14	.	.	PROPN
ejpam-4548	195	15	moreover	moreover	ADV
ejpam-4548	195	16	,	,	PUNCT
ejpam-4548	195	17	proposition	proposition	NOUN
ejpam-4548	195	18	4	4	NUM
ejpam-4548	195	19	.	.	PUNCT
ejpam-4548	195	20	let	let	VERB
ejpam-4548	195	21	η	η	PROPN
ejpam-4548	195	22	∈	∈	PROPN
ejpam-4548	195	23	l(µ	l(µ	PROPN
ejpam-4548	195	24	)	)	PUNCT
ejpam-4548	195	25	.	.	PUNCT
ejpam-4548	196	1	then	then	ADV
ejpam-4548	196	2	,	,	PUNCT
ejpam-4548	196	3	ηµ(e	ηµ(e	X
ejpam-4548	196	4	)	)	PUNCT
ejpam-4548	196	5	=	=	SYM
ejpam-4548	196	6	η(e	η(e	PROPN
ejpam-4548	196	7	)	)	PUNCT
ejpam-4548	196	8	.	.	PUNCT
ejpam-4548	197	1	next	next	ADV
ejpam-4548	197	2	,	,	PUNCT
ejpam-4548	197	3	we	we	PRON
ejpam-4548	197	4	provide	provide	VERB
ejpam-4548	197	5	an	an	DET
ejpam-4548	197	6	example	example	NOUN
ejpam-4548	197	7	of	of	ADP
ejpam-4548	197	8	the	the	DET
ejpam-4548	197	9	‘	'	PUNCT
ejpam-4548	197	10	normal	normal	ADJ
ejpam-4548	197	11	closure	closure	NOUN
ejpam-4548	197	12	’	'	PUNCT
ejpam-4548	197	13	of	of	ADP
ejpam-4548	197	14	an	an	DET
ejpam-4548	197	15	l	l	NOUN
ejpam-4548	197	16	-	-	NOUN
ejpam-4548	197	17	subgroup	subgroup	NOUN
ejpam-4548	197	18	of	of	ADP
ejpam-4548	197	19	an	an	DET
ejpam-4548	197	20	l	l	NOUN
ejpam-4548	197	21	-	-	NOUN
ejpam-4548	197	22	group	group	NOUN
ejpam-4548	197	23	:	:	PUNCT
ejpam-4548	197	24	example	example	NOUN
ejpam-4548	197	25	1	1	X
ejpam-4548	197	26	.	.	PUNCT
ejpam-4548	197	27	let	let	VERB
ejpam-4548	197	28	d8	d8	ADV
ejpam-4548	197	29	=	=	PRON
ejpam-4548	197	30	{	{	PUNCT
ejpam-4548	197	31	⟨x	⟨x	NUM
ejpam-4548	197	32	,	,	PUNCT
ejpam-4548	197	33	y⟩	y⟩	NOUN
ejpam-4548	197	34	:	:	PUNCT
ejpam-4548	198	1	x2	x2	NOUN
ejpam-4548	198	2	=	=	PUNCT
ejpam-4548	198	3	e	e	X
ejpam-4548	198	4	=	=	PUNCT
ejpam-4548	198	5	y8;xy	y8;xy	NOUN
ejpam-4548	198	6	=	=	SYM
ejpam-4548	198	7	y−1x	y−1x	NOUN
ejpam-4548	198	8	}	}	PUNCT
ejpam-4548	198	9	be	be	VERB
ejpam-4548	198	10	the	the	DET
ejpam-4548	198	11	dihedral	dihedral	ADJ
ejpam-4548	198	12	group	group	NOUN
ejpam-4548	198	13	of	of	ADP
ejpam-4548	198	14	degree	degree	NOUN
ejpam-4548	198	15	8	8	NUM
ejpam-4548	198	16	.	.	PUNCT
ejpam-4548	199	1	if	if	SCONJ
ejpam-4548	199	2	d4	d4	PROPN
ejpam-4548	199	3	=	=	SYM
ejpam-4548	199	4	{	{	PUNCT
ejpam-4548	199	5	⟨x	⟨x	NUM
ejpam-4548	199	6	,	,	PUNCT
ejpam-4548	199	7	y2⟩	y2⟩	VERB
ejpam-4548	199	8	:	:	PUNCT
ejpam-4548	200	1	x2	x2	X
ejpam-4548	200	2	=	=	PUNCT
ejpam-4548	200	3	e	e	X
ejpam-4548	200	4	=	=	SYM
ejpam-4548	200	5	(	(	PUNCT
ejpam-4548	200	6	y2)4;xy−2	y2)4;xy−2	NOUN
ejpam-4548	200	7	=	=	SYM
ejpam-4548	200	8	y−2x	y−2x	NOUN
ejpam-4548	200	9	}	}	PUNCT
ejpam-4548	200	10	,	,	PUNCT
ejpam-4548	200	11	is	be	AUX
ejpam-4548	200	12	a	a	DET
ejpam-4548	200	13	dihedral	dihedral	ADJ
ejpam-4548	200	14	subgroup	subgroup	NOUN
ejpam-4548	200	15	of	of	ADP
ejpam-4548	200	16	d8	d8	PROPN
ejpam-4548	200	17	,	,	PUNCT
ejpam-4548	200	18	then	then	ADV
ejpam-4548	200	19	define	define	VERB
ejpam-4548	200	20	the	the	DET
ejpam-4548	200	21	following	follow	VERB
ejpam-4548	200	22	l	l	NOUN
ejpam-4548	200	23	-	-	NOUN
ejpam-4548	200	24	subsets	subset	NOUN
ejpam-4548	200	25	of	of	ADP
ejpam-4548	200	26	d8	d8	PROPN
ejpam-4548	200	27	:	:	PUNCT
ejpam-4548	200	28	µ(z	µ(z	PROPN
ejpam-4548	200	29	)	)	PUNCT
ejpam-4548	200	30	=	=	SYM
ejpam-4548	200	31			NUM
ejpam-4548	200	32	1	1	NUM
ejpam-4548	200	33	2	2	NUM
ejpam-4548	200	34	if	if	SCONJ
ejpam-4548	200	35	z	z	PROPN
ejpam-4548	200	36	∈	∈	PROPN
ejpam-4548	200	37	d4	d4	PROPN
ejpam-4548	200	38	,	,	PUNCT
ejpam-4548	200	39	1	1	NUM
ejpam-4548	200	40	4	4	NUM
ejpam-4548	200	41	if	if	SCONJ
ejpam-4548	200	42	z	z	NOUN
ejpam-4548	200	43	∈	∈	PROPN
ejpam-4548	200	44	d8	d8	NOUN
ejpam-4548	200	45	∼	∼	NOUN
ejpam-4548	200	46	d4	d4	PROPN
ejpam-4548	200	47	;	;	PUNCT
ejpam-4548	200	48	η(z	η(z	PROPN
ejpam-4548	200	49	)	)	PUNCT
ejpam-4548	200	50	=	=	PUNCT
ejpam-4548	200	51			NUM
ejpam-4548	200	52	1	1	NUM
ejpam-4548	200	53	3	3	NUM
ejpam-4548	200	54	if	if	SCONJ
ejpam-4548	200	55	z	z	PROPN
ejpam-4548	200	56	∈	∈	PROPN
ejpam-4548	200	57	⟨x⟩	⟨x⟩	PROPN
ejpam-4548	200	58	,	,	PUNCT
ejpam-4548	200	59	1	1	NUM
ejpam-4548	200	60	6	6	NUM
ejpam-4548	200	61	if	if	SCONJ
ejpam-4548	200	62	z	z	NOUN
ejpam-4548	200	63	∈	∈	PROPN
ejpam-4548	200	64	d4	d4	VERB
ejpam-4548	200	65	∼	∼	NOUN
ejpam-4548	200	66	⟨x⟩	⟨x⟩	PROPN
ejpam-4548	200	67	,	,	PUNCT
ejpam-4548	200	68	1	1	NUM
ejpam-4548	200	69	9	9	NUM
ejpam-4548	200	70	if	if	SCONJ
ejpam-4548	200	71	z	z	NOUN
ejpam-4548	200	72	∈	∈	PROPN
ejpam-4548	200	73	d8	d8	VERB
ejpam-4548	200	74	∼	∼	NOUN
ejpam-4548	200	75	d4	d4	PROPN
ejpam-4548	200	76	.	.	PUNCT
ejpam-4548	201	1	here	here	ADV
ejpam-4548	201	2	a	a	DET
ejpam-4548	201	3	∼	∼	NOUN
ejpam-4548	201	4	b	b	NOUN
ejpam-4548	201	5	means	mean	VERB
ejpam-4548	201	6	usual	usual	ADJ
ejpam-4548	201	7	set	set	NOUN
ejpam-4548	201	8	difference	difference	NOUN
ejpam-4548	201	9	and	and	CCONJ
ejpam-4548	201	10	⟨x⟩	⟨x⟩	PROPN
ejpam-4548	201	11	denotes	denote	NOUN
ejpam-4548	201	12	subgroup	subgroup	ADV
ejpam-4548	201	13	of	of	ADP
ejpam-4548	201	14	d8	d8	PROPN
ejpam-4548	201	15	generated	generate	VERB
ejpam-4548	201	16	by	by	ADP
ejpam-4548	201	17	‘	'	PUNCT
ejpam-4548	201	18	x	x	X
ejpam-4548	201	19	’	'	PUNCT
ejpam-4548	201	20	.	.	PUNCT
ejpam-4548	202	1	clearly	clearly	ADV
ejpam-4548	202	2	η	η	PROPN
ejpam-4548	202	3	⊆	⊆	PROPN
ejpam-4548	202	4	µ	µ	NUM
ejpam-4548	202	5	,	,	PUNCT
ejpam-4548	202	6	η	η	PROPN
ejpam-4548	202	7	̸=	̸=	PROPN
ejpam-4548	202	8	µ	µ	X
ejpam-4548	202	9	and	and	CCONJ
ejpam-4548	202	10	η	η	PROPN
ejpam-4548	202	11	,	,	PUNCT
ejpam-4548	202	12	µ	µ	PRON
ejpam-4548	202	13	∈	∈	PROPN
ejpam-4548	202	14	l(g	l(g	NOUN
ejpam-4548	202	15	)	)	PUNCT
ejpam-4548	202	16	.	.	PUNCT
ejpam-4548	203	1	it	it	PRON
ejpam-4548	203	2	can	can	AUX
ejpam-4548	203	3	be	be	AUX
ejpam-4548	203	4	verified	verify	VERB
ejpam-4548	203	5	easily	easily	ADV
ejpam-4548	203	6	that	that	SCONJ
ejpam-4548	203	7	the	the	DET
ejpam-4548	203	8	conjugate	conjugate	NOUN
ejpam-4548	203	9	‘	'	PUNCT
ejpam-4548	203	10	µηµ−1	µηµ−1	PROPN
ejpam-4548	203	11	’	'	PUNCT
ejpam-4548	203	12	is	be	AUX
ejpam-4548	203	13	defined	define	VERB
ejpam-4548	203	14	by	by	ADP
ejpam-4548	203	15	the	the	DET
ejpam-4548	203	16	following	follow	VERB
ejpam-4548	203	17	level	level	NOUN
ejpam-4548	203	18	subsets	subset	NOUN
ejpam-4548	203	19	:	:	PUNCT
ejpam-4548	203	20	(	(	PUNCT
ejpam-4548	203	21	µηµ−1	µηµ−1	PROPN
ejpam-4548	203	22	)	)	PUNCT
ejpam-4548	203	23	1	1	NUM
ejpam-4548	203	24	3	3	NUM
ejpam-4548	203	25	=	=	SYM
ejpam-4548	203	26	{	{	PUNCT
ejpam-4548	203	27	e	e	NOUN
ejpam-4548	203	28	,	,	PUNCT
ejpam-4548	203	29	x	x	X
ejpam-4548	203	30	,	,	PUNCT
ejpam-4548	203	31	xy4	xy4	PROPN
ejpam-4548	203	32	}	}	PUNCT
ejpam-4548	203	33	,	,	PUNCT
ejpam-4548	203	34	(	(	PUNCT
ejpam-4548	203	35	µηµ−1	µηµ−1	PROPN
ejpam-4548	203	36	)	)	PUNCT
ejpam-4548	203	37	1	1	NUM
ejpam-4548	203	38	4	4	NUM
ejpam-4548	203	39	=	=	SYM
ejpam-4548	203	40	{	{	PUNCT
ejpam-4548	203	41	e	e	NOUN
ejpam-4548	203	42	,	,	PUNCT
ejpam-4548	203	43	x	x	X
ejpam-4548	203	44	,	,	PUNCT
ejpam-4548	203	45	xy2	xy2	PROPN
ejpam-4548	203	46	,	,	PUNCT
ejpam-4548	203	47	xy4	xy4	PROPN
ejpam-4548	203	48	,	,	PUNCT
ejpam-4548	203	49	xy6	xy6	PROPN
ejpam-4548	203	50	}	}	PUNCT
ejpam-4548	203	51	,	,	PUNCT
ejpam-4548	203	52	(	(	PUNCT
ejpam-4548	203	53	µηµ−1	µηµ−1	PROPN
ejpam-4548	203	54	)	)	PUNCT
ejpam-4548	203	55	1	1	NUM
ejpam-4548	203	56	6	6	NUM
ejpam-4548	203	57	=	=	SYM
ejpam-4548	203	58	{	{	PUNCT
ejpam-4548	203	59	e	e	NOUN
ejpam-4548	203	60	,	,	PUNCT
ejpam-4548	203	61	x	x	X
ejpam-4548	203	62	,	,	PUNCT
ejpam-4548	203	63	xy2	xy2	PROPN
ejpam-4548	203	64	,	,	PUNCT
ejpam-4548	203	65	xy4	xy4	PROPN
ejpam-4548	203	66	,	,	PUNCT
ejpam-4548	203	67	xy6	xy6	PROPN
ejpam-4548	203	68	,	,	PUNCT
ejpam-4548	203	69	y2	y2	PROPN
ejpam-4548	203	70	,	,	PUNCT
ejpam-4548	203	71	y4	y4	PROPN
ejpam-4548	203	72	,	,	PUNCT
ejpam-4548	203	73	y6}and	y6}and	X
ejpam-4548	203	74	(	(	PUNCT
ejpam-4548	203	75	µηµ−1	µηµ−1	PROPN
ejpam-4548	203	76	)	)	PUNCT
ejpam-4548	203	77	1	1	NUM
ejpam-4548	203	78	9	9	NUM
ejpam-4548	203	79	=	=	SYM
ejpam-4548	203	80	d8	d8	PROPN
ejpam-4548	203	81	.	.	PUNCT
ejpam-4548	204	1	consequently	consequently	ADV
ejpam-4548	204	2	,	,	PUNCT
ejpam-4548	204	3	if	if	SCONJ
ejpam-4548	204	4	k4	k4	ADJ
ejpam-4548	204	5	=	=	SYM
ejpam-4548	204	6	{	{	PUNCT
ejpam-4548	204	7	e	e	NOUN
ejpam-4548	204	8	,	,	PUNCT
ejpam-4548	204	9	x	x	X
ejpam-4548	204	10	,	,	PUNCT
ejpam-4548	204	11	y4	y4	PROPN
ejpam-4548	204	12	,	,	PUNCT
ejpam-4548	204	13	xy4	xy4	PROPN
ejpam-4548	204	14	}	}	PUNCT
ejpam-4548	204	15	is	be	AUX
ejpam-4548	204	16	a	a	DET
ejpam-4548	204	17	klein-4	klein-4	PROPN
ejpam-4548	204	18	subgroup	subgroup	NOUN
ejpam-4548	204	19	of	of	ADP
ejpam-4548	204	20	d8	d8	PROPN
ejpam-4548	204	21	,	,	PUNCT
ejpam-4548	204	22	then	then	ADV
ejpam-4548	204	23	by	by	ADP
ejpam-4548	204	24	applying	apply	VERB
ejpam-4548	204	25	the	the	DET
ejpam-4548	204	26	construction	construction	NOUN
ejpam-4548	204	27	of	of	ADP
ejpam-4548	204	28	theorem	theorem	NOUN
ejpam-4548	204	29	6	6	NUM
ejpam-4548	204	30	on	on	ADP
ejpam-4548	204	31	the	the	DET
ejpam-4548	204	32	l	l	NOUN
ejpam-4548	204	33	-	-	NOUN
ejpam-4548	204	34	subset	subset	VERB
ejpam-4548	204	35	‘	'	PUNCT
ejpam-4548	204	36	η	η	NOUN
ejpam-4548	204	37	’	'	PUNCT
ejpam-4548	204	38	of	of	ADP
ejpam-4548	204	39	µ	µ	PRON
ejpam-4548	204	40	and	and	CCONJ
ejpam-4548	204	41	after	after	ADP
ejpam-4548	204	42	doing	do	VERB
ejpam-4548	204	43	necessary	necessary	ADJ
ejpam-4548	204	44	calculations	calculation	NOUN
ejpam-4548	204	45	,	,	PUNCT
ejpam-4548	204	46	n.	n.	PROPN
ejpam-4548	204	47	ajmal	ajmal	PROPN
ejpam-4548	204	48	,	,	PUNCT
ejpam-4548	204	49	i.	i.	PROPN
ejpam-4548	204	50	jahan	jahan	PROPN
ejpam-4548	204	51	.	.	PROPN
ejpam-4548	204	52	,	,	PUNCT
ejpam-4548	204	53	b.	b.	PROPN
ejpam-4548	204	54	davvaz	davvaz	PROPN
ejpam-4548	204	55	/	/	SYM
ejpam-4548	204	56	eur	eur	PROPN
ejpam-4548	204	57	.	.	PUNCT
ejpam-4548	205	1	j.	j.	PROPN
ejpam-4548	205	2	pure	pure	PROPN
ejpam-4548	205	3	appl	appl	PROPN
ejpam-4548	205	4	.	.	PROPN
ejpam-4548	205	5	math	math	PROPN
ejpam-4548	205	6	,	,	PUNCT
ejpam-4548	205	7	15	15	NUM
ejpam-4548	205	8	(	(	PUNCT
ejpam-4548	205	9	4	4	NUM
ejpam-4548	205	10	)	)	PUNCT
ejpam-4548	205	11	(	(	PUNCT
ejpam-4548	205	12	2022	2022	NUM
ejpam-4548	205	13	)	)	PUNCT
ejpam-4548	205	14	,	,	PUNCT
ejpam-4548	205	15	2086	2086	NUM
ejpam-4548	205	16	-	-	SYM
ejpam-4548	205	17	2115	2115	NUM
ejpam-4548	205	18	2096	2096	NUM
ejpam-4548	205	19	we	we	PRON
ejpam-4548	205	20	get	get	VERB
ejpam-4548	205	21	l	l	NOUN
ejpam-4548	205	22	-	-	NOUN
ejpam-4548	205	23	subgroup	subgroup	NOUN
ejpam-4548	205	24	‘	'	PUNCT
ejpam-4548	205	25	ηµ	ηµ	VERB
ejpam-4548	205	26	=	=	SYM
ejpam-4548	205	27	⟨µηµ−1⟩	⟨µηµ−1⟩	PROPN
ejpam-4548	205	28	’	'	PUNCT
ejpam-4548	205	29	as	as	SCONJ
ejpam-4548	205	30	follows	follow	VERB
ejpam-4548	205	31	:	:	PUNCT
ejpam-4548	205	32	ηµ(z	ηµ(z	NOUN
ejpam-4548	205	33	)	)	PUNCT
ejpam-4548	205	34	=	=	SYM
ejpam-4548	205	35			NUM
ejpam-4548	206	1	1	1	NUM
ejpam-4548	206	2	3	3	NUM
ejpam-4548	206	3	if	if	SCONJ
ejpam-4548	206	4	z	z	PROPN
ejpam-4548	206	5	∈	∈	PROPN
ejpam-4548	206	6	k4	k4	NOUN
ejpam-4548	206	7	,	,	PUNCT
ejpam-4548	206	8	1	1	NUM
ejpam-4548	206	9	4	4	NUM
ejpam-4548	206	10	if	if	SCONJ
ejpam-4548	206	11	z	z	NOUN
ejpam-4548	206	12	∈	∈	PROPN
ejpam-4548	206	13	d4	d4	PROPN
ejpam-4548	206	14	∼	∼	NOUN
ejpam-4548	206	15	k4	k4	NOUN
ejpam-4548	206	16	,	,	PUNCT
ejpam-4548	206	17	1	1	NUM
ejpam-4548	206	18	9	9	NUM
ejpam-4548	206	19	if	if	SCONJ
ejpam-4548	206	20	z	z	NOUN
ejpam-4548	206	21	∈	∈	PROPN
ejpam-4548	206	22	d8	d8	VERB
ejpam-4548	206	23	∼	∼	NOUN
ejpam-4548	206	24	d4	d4	PROPN
ejpam-4548	206	25	.	.	PUNCT
ejpam-4548	207	1	in	in	ADP
ejpam-4548	207	2	the	the	DET
ejpam-4548	207	3	following	following	NOUN
ejpam-4548	207	4	,	,	PUNCT
ejpam-4548	207	5	we	we	PRON
ejpam-4548	207	6	discuss	discuss	VERB
ejpam-4548	207	7	some	some	DET
ejpam-4548	207	8	properties	property	NOUN
ejpam-4548	207	9	of	of	ADP
ejpam-4548	207	10	conjugate	conjugate	NOUN
ejpam-4548	207	11	of	of	ADP
ejpam-4548	207	12	l	l	NOUN
ejpam-4548	207	13	-	-	NOUN
ejpam-4548	207	14	subsets	subset	NOUN
ejpam-4548	207	15	where	where	SCONJ
ejpam-4548	207	16	the	the	DET
ejpam-4548	207	17	lsubsets	lsubset	NOUN
ejpam-4548	207	18	in	in	ADP
ejpam-4548	207	19	question	question	NOUN
ejpam-4548	207	20	are	be	AUX
ejpam-4548	207	21	l	l	NOUN
ejpam-4548	207	22	-	-	NOUN
ejpam-4548	207	23	subgroups	subgroup	NOUN
ejpam-4548	207	24	of	of	ADP
ejpam-4548	207	25	a	a	DET
ejpam-4548	207	26	given	give	VERB
ejpam-4548	207	27	parent	parent	NOUN
ejpam-4548	207	28	l	l	NOUN
ejpam-4548	207	29	-	-	NOUN
ejpam-4548	207	30	group	group	NOUN
ejpam-4548	207	31	.	.	PUNCT
ejpam-4548	208	1	the	the	DET
ejpam-4548	208	2	significance	significance	NOUN
ejpam-4548	208	3	of	of	ADP
ejpam-4548	208	4	such	such	ADJ
ejpam-4548	208	5	properties	property	NOUN
ejpam-4548	208	6	have	have	AUX
ejpam-4548	208	7	already	already	ADV
ejpam-4548	208	8	been	be	AUX
ejpam-4548	208	9	shown	show	VERB
ejpam-4548	208	10	in	in	ADP
ejpam-4548	208	11	classical	classical	ADJ
ejpam-4548	208	12	group	group	NOUN
ejpam-4548	208	13	theory	theory	NOUN
ejpam-4548	208	14	for	for	ADP
ejpam-4548	208	15	establishing	establish	VERB
ejpam-4548	208	16	certain	certain	ADJ
ejpam-4548	208	17	properties	property	NOUN
ejpam-4548	208	18	of	of	ADP
ejpam-4548	208	19	ith	ith	PROPN
ejpam-4548	208	20	normal	normal	ADJ
ejpam-4548	208	21	closure	closure	NOUN
ejpam-4548	208	22	of	of	ADP
ejpam-4548	208	23	a	a	DET
ejpam-4548	208	24	subgroup	subgroup	NOUN
ejpam-4548	208	25	of	of	ADP
ejpam-4548	208	26	a	a	DET
ejpam-4548	208	27	group	group	NOUN
ejpam-4548	208	28	.	.	PUNCT
ejpam-4548	209	1	theorem	theorem	VERB
ejpam-4548	209	2	9	9	NUM
ejpam-4548	209	3	.	.	PUNCT
ejpam-4548	210	1	let	let	VERB
ejpam-4548	210	2	η	η	PROPN
ejpam-4548	210	3	,	,	PUNCT
ejpam-4548	210	4	θ	θ	PROPN
ejpam-4548	210	5	,	,	PUNCT
ejpam-4548	210	6	γ	γ	PROPN
ejpam-4548	210	7	∈	∈	PROPN
ejpam-4548	210	8	l(µ	l(µ	PROPN
ejpam-4548	210	9	)	)	PUNCT
ejpam-4548	210	10	.	.	PUNCT
ejpam-4548	211	1	let	let	VERB
ejpam-4548	211	2	θ	θ	PROPN
ejpam-4548	211	3	⊆	⊆	NUM
ejpam-4548	211	4	γ	γ	NOUN
ejpam-4548	211	5	and	and	CCONJ
ejpam-4548	211	6	η(e	η(e	PROPN
ejpam-4548	211	7	)	)	PUNCT
ejpam-4548	212	1	=	=	SYM
ejpam-4548	212	2	θ(e	θ(e	NUM
ejpam-4548	212	3	)	)	PUNCT
ejpam-4548	212	4	.	.	PUNCT
ejpam-4548	213	1	then	then	ADV
ejpam-4548	213	2	,	,	PUNCT
ejpam-4548	213	3	(	(	PUNCT
ejpam-4548	213	4	i	i	NOUN
ejpam-4548	213	5	)	)	PUNCT
ejpam-4548	213	6	(	(	PUNCT
ejpam-4548	213	7	ηγ)θ	ηγ)θ	PROPN
ejpam-4548	213	8	=	=	SYM
ejpam-4548	213	9	ηγ	ηγ	PROPN
ejpam-4548	213	10	,	,	PUNCT
ejpam-4548	213	11	(	(	PUNCT
ejpam-4548	213	12	ii	ii	NOUN
ejpam-4548	213	13	)	)	PUNCT
ejpam-4548	213	14	(	(	PUNCT
ejpam-4548	213	15	ηθ	ηθ	X
ejpam-4548	213	16	)	)	PUNCT
ejpam-4548	213	17	γ	γ	NOUN
ejpam-4548	213	18	=	=	SYM
ejpam-4548	213	19	ηγ	ηγ	PROPN
ejpam-4548	213	20	,	,	PUNCT
ejpam-4548	213	21	(	(	PUNCT
ejpam-4548	213	22	iii	iii	NOUN
ejpam-4548	213	23	)	)	PUNCT
ejpam-4548	213	24	ηθ	ηθ	PROPN
ejpam-4548	213	25	◦	◦	NOUN
ejpam-4548	213	26	η	η	NOUN
ejpam-4548	213	27	=	=	SYM
ejpam-4548	213	28	ηθ	ηθ	NOUN
ejpam-4548	213	29	.	.	NOUN
ejpam-4548	213	30	proof	proof	NOUN
ejpam-4548	213	31	.	.	PUNCT
ejpam-4548	214	1	(	(	PUNCT
ejpam-4548	214	2	i	i	NOUN
ejpam-4548	214	3	)	)	PUNCT
ejpam-4548	214	4	in	in	ADP
ejpam-4548	214	5	view	view	NOUN
ejpam-4548	214	6	of	of	ADP
ejpam-4548	214	7	lemma	lemma	PROPN
ejpam-4548	214	8	1	1	NUM
ejpam-4548	214	9	and	and	CCONJ
ejpam-4548	214	10	corollary	corollary	ADJ
ejpam-4548	214	11	1	1	NUM
ejpam-4548	214	12	and	and	CCONJ
ejpam-4548	214	13	using	use	VERB
ejpam-4548	214	14	the	the	DET
ejpam-4548	214	15	fact	fact	NOUN
ejpam-4548	214	16	θ	θ	PROPN
ejpam-4548	214	17	⊆	⊆	NUM
ejpam-4548	214	18	γ	γ	X
ejpam-4548	214	19	,	,	PUNCT
ejpam-4548	214	20	we	we	PRON
ejpam-4548	214	21	have	have	VERB
ejpam-4548	214	22	ηγ(e	ηγ(e	NOUN
ejpam-4548	214	23	)	)	PUNCT
ejpam-4548	215	1	=	=	SYM
ejpam-4548	215	2	η(e	η(e	X
ejpam-4548	215	3	)	)	PUNCT
ejpam-4548	215	4	∧	∧	NOUN
ejpam-4548	215	5	γ(e	γ(e	NOUN
ejpam-4548	215	6	)	)	PUNCT
ejpam-4548	215	7	=	=	SYM
ejpam-4548	215	8	θ(e	θ(e	NUM
ejpam-4548	215	9	)	)	PUNCT
ejpam-4548	215	10	.	.	PUNCT
ejpam-4548	216	1	hence	hence	ADV
ejpam-4548	216	2	by	by	ADP
ejpam-4548	216	3	lemma	lemma	PROPN
ejpam-4548	216	4	1	1	NUM
ejpam-4548	216	5	,	,	PUNCT
ejpam-4548	216	6	ηγ	ηγ	PROPN
ejpam-4548	216	7	⊆	⊆	NUM
ejpam-4548	216	8	(	(	PUNCT
ejpam-4548	216	9	ηγ)θ	ηγ)θ	PROPN
ejpam-4548	216	10	.	.	PUNCT
ejpam-4548	217	1	in	in	ADP
ejpam-4548	217	2	order	order	NOUN
ejpam-4548	217	3	to	to	PART
ejpam-4548	217	4	prove	prove	VERB
ejpam-4548	217	5	the	the	DET
ejpam-4548	217	6	reverse	reverse	ADJ
ejpam-4548	217	7	inclusion	inclusion	NOUN
ejpam-4548	217	8	,	,	PUNCT
ejpam-4548	217	9	firstly	firstly	ADV
ejpam-4548	217	10	we	we	PRON
ejpam-4548	217	11	shall	shall	AUX
ejpam-4548	217	12	show	show	VERB
ejpam-4548	217	13	that	that	SCONJ
ejpam-4548	217	14	θηγθ−1	θηγθ−1	PROPN
ejpam-4548	217	15	⊆	⊆	NUM
ejpam-4548	217	16	ηγ	ηγ	NOUN
ejpam-4548	217	17	.	.	PUNCT
ejpam-4548	218	1	so	so	ADV
ejpam-4548	218	2	let	let	VERB
ejpam-4548	218	3	x	x	PUNCT
ejpam-4548	218	4	∈	∈	PROPN
ejpam-4548	218	5	g.	g.	NOUN
ejpam-4548	218	6	then	then	ADV
ejpam-4548	218	7	,	,	PUNCT
ejpam-4548	218	8	θηγθ−1(x	θηγθ−1(x	NOUN
ejpam-4548	218	9	)	)	PUNCT
ejpam-4548	218	10	=	=	PUNCT
ejpam-4548	219	1	∨	∨	NUM
ejpam-4548	219	2	x	x	X
ejpam-4548	219	3	=	=	NOUN
ejpam-4548	219	4	yiziy	yiziy	NOUN
ejpam-4548	219	5	−1	−1	NOUN
ejpam-4548	219	6	i	i	PRON
ejpam-4548	219	7	{	{	PUNCT
ejpam-4548	219	8	ηγ(zi	ηγ(zi	X
ejpam-4548	219	9	)	)	PUNCT
ejpam-4548	219	10	∧	∧	PROPN
ejpam-4548	219	11	θ(yi	θ(yi	NOUN
ejpam-4548	219	12	)	)	PUNCT
ejpam-4548	219	13	}	}	PUNCT
ejpam-4548	219	14	.	.	PUNCT
ejpam-4548	220	1	here	here	ADV
ejpam-4548	220	2	,	,	PUNCT
ejpam-4548	220	3	in	in	ADP
ejpam-4548	220	4	view	view	NOUN
ejpam-4548	220	5	of	of	ADP
ejpam-4548	220	6	lemma	lemma	PROPN
ejpam-4548	220	7	1	1	NUM
ejpam-4548	220	8	and	and	CCONJ
ejpam-4548	220	9	corollary	corollary	ADJ
ejpam-4548	220	10	1	1	NUM
ejpam-4548	220	11	,	,	PUNCT
ejpam-4548	220	12	we	we	PRON
ejpam-4548	220	13	observe	observe	VERB
ejpam-4548	220	14	that∨	that∨	PROPN
ejpam-4548	220	15	x∈g	x∈g	PROPN
ejpam-4548	220	16	{	{	PUNCT
ejpam-4548	220	17	γηγ−1(x	γηγ−1(x	NOUN
ejpam-4548	220	18	)	)	PUNCT
ejpam-4548	220	19	}	}	PUNCT
ejpam-4548	220	20	=	=	SYM
ejpam-4548	220	21	ηγ(e	ηγ(e	NOUN
ejpam-4548	220	22	)	)	PUNCT
ejpam-4548	220	23	=	=	SYM
ejpam-4548	221	1	θ(e	θ(e	NUM
ejpam-4548	221	2	)	)	PUNCT
ejpam-4548	221	3	=	=	PUNCT
ejpam-4548	221	4	η(e	η(e	PROPN
ejpam-4548	221	5	)	)	PUNCT
ejpam-4548	221	6	.	.	PUNCT
ejpam-4548	222	1	thus	thus	ADV
ejpam-4548	222	2	γηγ−1(e	γηγ−1(e	PUNCT
ejpam-4548	222	3	)	)	PUNCT
ejpam-4548	222	4	=	=	SYM
ejpam-4548	222	5	η(e	η(e	PROPN
ejpam-4548	222	6	)	)	PUNCT
ejpam-4548	222	7	.	.	PUNCT
ejpam-4548	223	1	therefore	therefore	ADV
ejpam-4548	223	2	,	,	PUNCT
ejpam-4548	223	3	in	in	ADP
ejpam-4548	223	4	view	view	NOUN
ejpam-4548	223	5	of	of	ADP
ejpam-4548	223	6	theorem	theorem	NOUN
ejpam-4548	223	7	6	6	NUM
ejpam-4548	223	8	,	,	PUNCT
ejpam-4548	223	9	as	as	SCONJ
ejpam-4548	223	10	ηγ	ηγ	PROPN
ejpam-4548	223	11	=	=	SYM
ejpam-4548	223	12	⟨γηγ−1⟩	⟨γηγ−1⟩	PROPN
ejpam-4548	223	13	,	,	PUNCT
ejpam-4548	223	14	we	we	PRON
ejpam-4548	223	15	have	have	AUX
ejpam-4548	223	16	ηγ(zi	ηγ(zi	NOUN
ejpam-4548	223	17	)	)	PUNCT
ejpam-4548	223	18	=	=	PUNCT
ejpam-4548	223	19	∨	∨	NUM
ejpam-4548	223	20	aij≤η(e	aij≤η(e	NOUN
ejpam-4548	223	21	)	)	PUNCT
ejpam-4548	223	22	{	{	PUNCT
ejpam-4548	223	23	aij	aij	PROPN
ejpam-4548	223	24	:	:	PUNCT
ejpam-4548	223	25	zi	zi	PROPN
ejpam-4548	223	26	∈	∈	PROPN
ejpam-4548	223	27	⟨(γηγ−1)aij	⟨(γηγ−1)aij	PROPN
ejpam-4548	223	28	⟩	⟩	NOUN
ejpam-4548	223	29	}	}	PUNCT
ejpam-4548	223	30	.	.	PUNCT
ejpam-4548	224	1	therefore	therefore	ADV
ejpam-4548	224	2	,	,	PUNCT
ejpam-4548	224	3	θηγθ−1(x	θηγθ−1(x	NOUN
ejpam-4548	224	4	)	)	PUNCT
ejpam-4548	224	5	=	=	PUNCT
ejpam-4548	225	1	∨	∨	NUM
ejpam-4548	225	2	x	x	X
ejpam-4548	225	3	=	=	NOUN
ejpam-4548	225	4	yiziy	yiziy	NOUN
ejpam-4548	225	5	−1	−1	NOUN
ejpam-4548	225	6	i	i	PRON
ejpam-4548	225	7	{	{	PUNCT
ejpam-4548	225	8	ηγ(zi	ηγ(zi	X
ejpam-4548	225	9	)	)	PUNCT
ejpam-4548	225	10	∧	∧	PROPN
ejpam-4548	225	11	θ(yi	θ(yi	NOUN
ejpam-4548	225	12	)	)	PUNCT
ejpam-4548	225	13	}	}	PUNCT
ejpam-4548	225	14	n.	n.	PROPN
ejpam-4548	225	15	ajmal	ajmal	PROPN
ejpam-4548	225	16	,	,	PUNCT
ejpam-4548	225	17	i.	i.	PROPN
ejpam-4548	225	18	jahan	jahan	PROPN
ejpam-4548	225	19	.	.	PROPN
ejpam-4548	225	20	,	,	PUNCT
ejpam-4548	225	21	b.	b.	PROPN
ejpam-4548	225	22	davvaz	davvaz	PROPN
ejpam-4548	225	23	/	/	SYM
ejpam-4548	225	24	eur	eur	PROPN
ejpam-4548	225	25	.	.	PUNCT
ejpam-4548	226	1	j.	j.	PROPN
ejpam-4548	226	2	pure	pure	PROPN
ejpam-4548	226	3	appl	appl	PROPN
ejpam-4548	226	4	.	.	PROPN
ejpam-4548	226	5	math	math	PROPN
ejpam-4548	226	6	,	,	PUNCT
ejpam-4548	226	7	15	15	NUM
ejpam-4548	226	8	(	(	PUNCT
ejpam-4548	226	9	4	4	NUM
ejpam-4548	226	10	)	)	PUNCT
ejpam-4548	226	11	(	(	PUNCT
ejpam-4548	226	12	2022	2022	NUM
ejpam-4548	226	13	)	)	PUNCT
ejpam-4548	226	14	,	,	PUNCT
ejpam-4548	226	15	2086	2086	NUM
ejpam-4548	226	16	-	-	SYM
ejpam-4548	226	17	2115	2115	NUM
ejpam-4548	226	18	2097	2097	NUM
ejpam-4548	226	19	=	=	SYM
ejpam-4548	227	1	∨	∨	NUM
ejpam-4548	227	2	x	x	X
ejpam-4548	227	3	=	=	NOUN
ejpam-4548	227	4	yiziy	yiziy	NOUN
ejpam-4548	227	5	−1	−1	NOUN
ejpam-4548	227	6	i	i	PRON
ejpam-4548	227	7	{	{	PUNCT
ejpam-4548	227	8	∨	∨	PROPN
ejpam-4548	227	9	aij≤η(e	aij≤η(e	X
ejpam-4548	227	10	)	)	PUNCT
ejpam-4548	227	11	{	{	PUNCT
ejpam-4548	227	12	aij	aij	PROPN
ejpam-4548	227	13	:	:	PUNCT
ejpam-4548	227	14	zi	zi	NOUN
ejpam-4548	227	15	∈	∈	PROPN
ejpam-4548	227	16	⟨	⟨	VERB
ejpam-4548	227	17	(	(	PUNCT
ejpam-4548	227	18	γηγ−1	γηγ−1	PROPN
ejpam-4548	227	19	)	)	PUNCT
ejpam-4548	227	20	aij	aij	PROPN
ejpam-4548	227	21	⟩	⟩	PROPN
ejpam-4548	227	22	}	}	PUNCT
ejpam-4548	227	23	∧	∧	PROPN
ejpam-4548	227	24	θ(yi	θ(yi	NOUN
ejpam-4548	227	25	)	)	PUNCT
ejpam-4548	227	26	}	}	PUNCT
ejpam-4548	227	27	=	=	PUNCT
ejpam-4548	228	1	∨	∨	NUM
ejpam-4548	228	2	x	x	X
ejpam-4548	228	3	=	=	NOUN
ejpam-4548	228	4	yiziy	yiziy	NOUN
ejpam-4548	228	5	−1	−1	NOUN
ejpam-4548	228	6	i	i	PRON
ejpam-4548	228	7	{	{	PUNCT
ejpam-4548	228	8	∨	∨	PROPN
ejpam-4548	228	9	aij≤η(e	aij≤η(e	X
ejpam-4548	228	10	)	)	PUNCT
ejpam-4548	228	11	{	{	PUNCT
ejpam-4548	228	12	aij	aij	PROPN
ejpam-4548	228	13	∧	∧	PROPN
ejpam-4548	228	14	θ(yi	θ(yi	PROPN
ejpam-4548	228	15	)	)	PUNCT
ejpam-4548	228	16	:	:	PUNCT
ejpam-4548	228	17	zi	zi	NOUN
ejpam-4548	228	18	∈	∈	PROPN
ejpam-4548	228	19	⟨	⟨	VERB
ejpam-4548	228	20	(	(	PUNCT
ejpam-4548	228	21	γηγ−1	γηγ−1	PROPN
ejpam-4548	228	22	)	)	PUNCT
ejpam-4548	228	23	aij	aij	PROPN
ejpam-4548	228	24	⟩	⟩	PROPN
ejpam-4548	228	25	}	}	PUNCT
ejpam-4548	228	26	}	}	PUNCT
ejpam-4548	228	27	(	(	PUNCT
ejpam-4548	228	28	as	as	SCONJ
ejpam-4548	228	29	l	l	NOUN
ejpam-4548	228	30	is	be	AUX
ejpam-4548	228	31	a	a	DET
ejpam-4548	228	32	completely	completely	ADV
ejpam-4548	228	33	distributive	distributive	ADJ
ejpam-4548	228	34	lattice	lattice	NOUN
ejpam-4548	228	35	)	)	PUNCT
ejpam-4548	228	36	=	=	PUNCT
ejpam-4548	229	1	∨	∨	NUM
ejpam-4548	229	2	x	x	X
ejpam-4548	229	3	=	=	NOUN
ejpam-4548	229	4	yiziy	yiziy	NOUN
ejpam-4548	229	5	−1	−1	NOUN
ejpam-4548	230	1	i	i	PRON
ejpam-4548	230	2	ai	ai	VERB
ejpam-4548	230	3	j	j	PROPN
ejpam-4548	230	4	≤η(e	≤η(e	PROPN
ejpam-4548	230	5	)	)	PUNCT
ejpam-4548	230	6	{	{	PUNCT
ejpam-4548	230	7	aij	aij	PROPN
ejpam-4548	230	8	∧	∧	PROPN
ejpam-4548	230	9	θ(yi	θ(yi	PROPN
ejpam-4548	230	10	)	)	PUNCT
ejpam-4548	230	11	:	:	PUNCT
ejpam-4548	230	12	zi	zi	NOUN
ejpam-4548	230	13	∈	∈	PROPN
ejpam-4548	230	14	⟨	⟨	VERB
ejpam-4548	230	15	(	(	PUNCT
ejpam-4548	230	16	γηγ−1	γηγ−1	PROPN
ejpam-4548	230	17	)	)	PUNCT
ejpam-4548	230	18	aij	aij	PROPN
ejpam-4548	230	19	⟩	⟩	PROPN
ejpam-4548	230	20	}	}	PUNCT
ejpam-4548	230	21	≤	≤	NOUN
ejpam-4548	230	22	∨	∨	NUM
ejpam-4548	230	23	x	x	X
ejpam-4548	230	24	=	=	NOUN
ejpam-4548	230	25	yiziy	yiziy	NOUN
ejpam-4548	231	1	−1	−1	NOUN
ejpam-4548	232	1	i	i	PRON
ejpam-4548	232	2	ai	ai	VERB
ejpam-4548	232	3	j	j	PROPN
ejpam-4548	232	4	≤η(e	≤η(e	PROPN
ejpam-4548	232	5	)	)	PUNCT
ejpam-4548	232	6	{	{	PUNCT
ejpam-4548	232	7	aij	aij	PROPN
ejpam-4548	232	8	∧	∧	PROPN
ejpam-4548	232	9	γ(yi	γ(yi	PROPN
ejpam-4548	232	10	)	)	PUNCT
ejpam-4548	232	11	:	:	PUNCT
ejpam-4548	232	12	zi	zi	NOUN
ejpam-4548	232	13	∈	∈	PROPN
ejpam-4548	232	14	⟨	⟨	VERB
ejpam-4548	232	15	(	(	PUNCT
ejpam-4548	232	16	γηγ−1	γηγ−1	PROPN
ejpam-4548	232	17	)	)	PUNCT
ejpam-4548	232	18	aij	aij	PROPN
ejpam-4548	232	19	⟩	⟩	PROPN
ejpam-4548	232	20	}	}	PUNCT
ejpam-4548	232	21	.	.	PUNCT
ejpam-4548	233	1	(	(	PUNCT
ejpam-4548	233	2	as	as	ADP
ejpam-4548	233	3	θ	θ	PROPN
ejpam-4548	233	4	⊆	⊆	NUM
ejpam-4548	233	5	γ	γ	NOUN
ejpam-4548	233	6	)	)	PUNCT
ejpam-4548	233	7	now	now	ADV
ejpam-4548	233	8	,	,	PUNCT
ejpam-4548	233	9	let	let	VERB
ejpam-4548	233	10	yi	yi	PRON
ejpam-4548	233	11	∈	∈	PROPN
ejpam-4548	233	12	g	g	PROPN
ejpam-4548	233	13	be	be	AUX
ejpam-4548	233	14	such	such	ADJ
ejpam-4548	233	15	that	that	SCONJ
ejpam-4548	233	16	x	x	SYM
ejpam-4548	233	17	=	=	PUNCT
ejpam-4548	233	18	yiziy	yiziy	NOUN
ejpam-4548	233	19	−1	−1	NOUN
ejpam-4548	233	20	i	i	PROPN
ejpam-4548	233	21	for	for	ADP
ejpam-4548	233	22	some	some	DET
ejpam-4548	233	23	zi	zi	NOUN
ejpam-4548	233	24	∈	∈	PROPN
ejpam-4548	233	25	⟨	⟨	VERB
ejpam-4548	233	26	(	(	PUNCT
ejpam-4548	233	27	γηγ−1	γηγ−1	PROPN
ejpam-4548	233	28	)	)	PUNCT
ejpam-4548	233	29	aij	aij	PROPN
ejpam-4548	233	30	⟩	⟩	PROPN
ejpam-4548	233	31	and	and	CCONJ
ejpam-4548	233	32	aij	aij	PROPN
ejpam-4548	233	33	≤	≤	PROPN
ejpam-4548	233	34	η(e	η(e	PROPN
ejpam-4548	233	35	)	)	PUNCT
ejpam-4548	233	36	.	.	PUNCT
ejpam-4548	234	1	we	we	PRON
ejpam-4548	234	2	write	write	VERB
ejpam-4548	234	3	caij	caij	PROPN
ejpam-4548	234	4	,	,	PUNCT
ejpam-4548	234	5	yi	yi	X
ejpam-4548	234	6	=	=	SYM
ejpam-4548	234	7	aij	aij	PROPN
ejpam-4548	234	8	∧	∧	PROPN
ejpam-4548	234	9	γ(yi	γ(yi	PROPN
ejpam-4548	234	10	)	)	PUNCT
ejpam-4548	234	11	.	.	PUNCT
ejpam-4548	235	1	then	then	ADV
ejpam-4548	235	2	,	,	PUNCT
ejpam-4548	235	3	caij	caij	PROPN
ejpam-4548	235	4	,	,	PUNCT
ejpam-4548	235	5	yi	yi	PROPN
ejpam-4548	235	6	≤	≤	NUM
ejpam-4548	235	7	η(e	η(e	PROPN
ejpam-4548	235	8	)	)	PUNCT
ejpam-4548	235	9	.	.	PUNCT
ejpam-4548	236	1	we	we	PRON
ejpam-4548	236	2	shall	shall	AUX
ejpam-4548	236	3	establish	establish	VERB
ejpam-4548	236	4	that	that	DET
ejpam-4548	236	5	yiziy	yiziy	ADJ
ejpam-4548	236	6	−1	−1	NOUN
ejpam-4548	236	7	i	i	PRON
ejpam-4548	236	8	∈	∈	VERB
ejpam-4548	236	9	⟨	⟨	VERB
ejpam-4548	236	10	(	(	PUNCT
ejpam-4548	236	11	γηγ−1	γηγ−1	PROPN
ejpam-4548	236	12	)	)	PUNCT
ejpam-4548	237	1	c	c	AUX
ejpam-4548	237	2	ai	ai	VERB
ejpam-4548	237	3	j	j	PROPN
ejpam-4548	237	4	,	,	PUNCT
ejpam-4548	237	5	yi	yi	PROPN
ejpam-4548	237	6	⟩.	⟩.	PROPN
ejpam-4548	237	7	as	as	SCONJ
ejpam-4548	237	8	zi	zi	NOUN
ejpam-4548	237	9	∈	∈	PROPN
ejpam-4548	237	10	⟨	⟨	VERB
ejpam-4548	237	11	(	(	PUNCT
ejpam-4548	237	12	γηγ−1	γηγ−1	PROPN
ejpam-4548	237	13	)	)	PUNCT
ejpam-4548	237	14	aij	aij	PROPN
ejpam-4548	237	15	⟩	⟩	PROPN
ejpam-4548	237	16	,	,	PUNCT
ejpam-4548	237	17	we	we	PRON
ejpam-4548	237	18	have	have	VERB
ejpam-4548	237	19	,	,	PUNCT
ejpam-4548	237	20	zi	zi	X
ejpam-4548	237	21	=	=	PUNCT
ejpam-4548	238	1	zi1z	zi1z	NOUN
ejpam-4548	238	2	i	i	PRON
ejpam-4548	238	3	2	2	NUM
ejpam-4548	238	4	...	...	PUNCT
ejpam-4548	239	1	z	z	NOUN
ejpam-4548	240	1	i	i	PRON
ejpam-4548	241	1	n	n	PROPN
ejpam-4548	242	1	where	where	SCONJ
ejpam-4548	242	2	either	either	CCONJ
ejpam-4548	242	3	zik	zik	PROPN
ejpam-4548	242	4	or	or	CCONJ
ejpam-4548	242	5	(	(	PUNCT
ejpam-4548	242	6	zik	zik	PROPN
ejpam-4548	242	7	)	)	PUNCT
ejpam-4548	242	8	−1	−1	NOUN
ejpam-4548	242	9	∈	∈	PROPN
ejpam-4548	242	10	(	(	PUNCT
ejpam-4548	242	11	γηγ−1)aij	γηγ−1)aij	X
ejpam-4548	242	12	for	for	ADP
ejpam-4548	242	13	k	k	PROPN
ejpam-4548	242	14	=	=	SYM
ejpam-4548	242	15	1	1	NUM
ejpam-4548	242	16	,	,	PUNCT
ejpam-4548	242	17	2	2	NUM
ejpam-4548	242	18	,	,	PUNCT
ejpam-4548	242	19	...	...	PUNCT
ejpam-4548	242	20	,	,	PUNCT
ejpam-4548	242	21	n.	n.	PROPN
ejpam-4548	242	22	thus	thus	ADV
ejpam-4548	242	23	,	,	PUNCT
ejpam-4548	242	24	yiziy	yiziy	ADJ
ejpam-4548	242	25	−1	−1	NOUN
ejpam-4548	243	1	i	i	NOUN
ejpam-4548	243	2	=	=	PUNCT
ejpam-4548	243	3	(	(	PUNCT
ejpam-4548	243	4	yiz	yiz	INTJ
ejpam-4548	243	5	i	i	PRON
ejpam-4548	243	6	1y	1y	VERB
ejpam-4548	243	7	−1	−1	VERB
ejpam-4548	243	8	i	i	NOUN
ejpam-4548	243	9	)	)	PUNCT
ejpam-4548	244	1	(	(	PUNCT
ejpam-4548	244	2	yiz	yiz	INTJ
ejpam-4548	244	3	i	i	PRON
ejpam-4548	244	4	2y	2y	NUM
ejpam-4548	244	5	−1	−1	NOUN
ejpam-4548	244	6	i	i	NOUN
ejpam-4548	244	7	)	)	PUNCT
ejpam-4548	244	8	...	...	PUNCT
ejpam-4548	245	1	(	(	PUNCT
ejpam-4548	245	2	yiz	yiz	INTJ
ejpam-4548	245	3	i	i	PRON
ejpam-4548	245	4	ny	ny	PROPN
ejpam-4548	245	5	−1	−1	NOUN
ejpam-4548	245	6	i	i	PROPN
ejpam-4548	245	7	)	)	PUNCT
ejpam-4548	245	8	.	.	PUNCT
ejpam-4548	246	1	by	by	ADP
ejpam-4548	246	2	lemma	lemma	PROPN
ejpam-4548	246	3	1	1	NUM
ejpam-4548	246	4	,	,	PUNCT
ejpam-4548	246	5	for	for	ADP
ejpam-4548	246	6	each	each	DET
ejpam-4548	246	7	k	k	PROPN
ejpam-4548	246	8	γηγ−1(yiz	γηγ−1(yiz	PROPN
ejpam-4548	247	1	i	i	PRON
ejpam-4548	247	2	ky	ky	VERB
ejpam-4548	247	3	−1	−1	NOUN
ejpam-4548	247	4	i	i	PROPN
ejpam-4548	247	5	)	)	PUNCT
ejpam-4548	247	6	≥	≥	NOUN
ejpam-4548	247	7	γηγ−1(zik	γηγ−1(zik	NOUN
ejpam-4548	247	8	)	)	PUNCT
ejpam-4548	247	9	∧	∧	PROPN
ejpam-4548	247	10	γ(yi	γ(yi	PROPN
ejpam-4548	247	11	)	)	PUNCT
ejpam-4548	247	12	.	.	PUNCT
ejpam-4548	248	1	moreover	moreover	ADV
ejpam-4548	248	2	,	,	PUNCT
ejpam-4548	248	3	again	again	ADV
ejpam-4548	248	4	by	by	ADP
ejpam-4548	248	5	lemma	lemma	PROPN
ejpam-4548	248	6	1	1	NUM
ejpam-4548	248	7	,	,	PUNCT
ejpam-4548	248	8	γηγ−1(zik	γηγ−1(zik	NOUN
ejpam-4548	248	9	)	)	PUNCT
ejpam-4548	248	10	=	=	SYM
ejpam-4548	248	11	γηγ−1((zik	γηγ−1((zik	PROPN
ejpam-4548	248	12	)	)	PUNCT
ejpam-4548	248	13	−1	−1	NOUN
ejpam-4548	248	14	)	)	PUNCT
ejpam-4548	248	15	and	and	CCONJ
ejpam-4548	248	16	as	as	ADP
ejpam-4548	248	17	zik	zik	PROPN
ejpam-4548	248	18	or	or	CCONJ
ejpam-4548	248	19	(	(	PUNCT
ejpam-4548	248	20	z	z	NOUN
ejpam-4548	248	21	i	i	NOUN
ejpam-4548	248	22	k	k	NOUN
ejpam-4548	248	23	)	)	PUNCT
ejpam-4548	248	24	−1	−1	NOUN
ejpam-4548	248	25	∈	∈	NOUN
ejpam-4548	248	26	(	(	PUNCT
ejpam-4548	248	27	γηγ−1)aij	γηγ−1)aij	X
ejpam-4548	248	28	,	,	PUNCT
ejpam-4548	248	29	it	it	PRON
ejpam-4548	248	30	follows	follow	VERB
ejpam-4548	248	31	that	that	SCONJ
ejpam-4548	248	32	γηγ−1(yiz	γηγ−1(yiz	PROPN
ejpam-4548	249	1	i	i	PRON
ejpam-4548	249	2	ky	ky	VERB
ejpam-4548	249	3	−1	−1	NOUN
ejpam-4548	249	4	i	i	PROPN
ejpam-4548	249	5	)	)	PUNCT
ejpam-4548	249	6	≥	≥	PROPN
ejpam-4548	249	7	aij	aij	PROPN
ejpam-4548	249	8	∧	∧	PROPN
ejpam-4548	249	9	γ(yi	γ(yi	PROPN
ejpam-4548	249	10	)	)	PUNCT
ejpam-4548	250	1	=	=	VERB
ejpam-4548	250	2	caij	caij	NOUN
ejpam-4548	250	3	,	,	PUNCT
ejpam-4548	250	4	yi	yi	PROPN
ejpam-4548	250	5	.	.	PUNCT
ejpam-4548	251	1	hence	hence	ADV
ejpam-4548	251	2	yiz	yiz	INTJ
ejpam-4548	252	1	i	i	PRON
ejpam-4548	252	2	ky	ky	VERB
ejpam-4548	252	3	−1	−1	NOUN
ejpam-4548	253	1	i	i	PRON
ejpam-4548	253	2	∈	∈	PROPN
ejpam-4548	253	3	(	(	PUNCT
ejpam-4548	253	4	γηγ−1)c	γηγ−1)c	NOUN
ejpam-4548	253	5	ai	ai	VERB
ejpam-4548	253	6	j	j	PROPN
ejpam-4548	253	7	,	,	PUNCT
ejpam-4548	253	8	yi	yi	PROPN
ejpam-4548	253	9	.	.	PUNCT
ejpam-4548	254	1	again	again	ADV
ejpam-4548	254	2	,	,	PUNCT
ejpam-4548	254	3	in	in	ADP
ejpam-4548	254	4	view	view	NOUN
ejpam-4548	254	5	of	of	ADP
ejpam-4548	254	6	lemma	lemma	PROPN
ejpam-4548	254	7	1	1	NUM
ejpam-4548	254	8	,	,	PUNCT
ejpam-4548	254	9	γηγ−1(yi(z	γηγ−1(yi(z	PROPN
ejpam-4548	254	10	i	i	NOUN
ejpam-4548	254	11	k	k	PROPN
ejpam-4548	254	12	)	)	PUNCT
ejpam-4548	254	13	−1y−1	−1y−1	PROPN
ejpam-4548	254	14	i	i	PROPN
ejpam-4548	254	15	)	)	PUNCT
ejpam-4548	255	1	=	=	SYM
ejpam-4548	255	2	γηγ−1(yiz	γηγ−1(yiz	NOUN
ejpam-4548	256	1	i	i	PRON
ejpam-4548	256	2	ky	ky	VERB
ejpam-4548	256	3	−1	−1	NOUN
ejpam-4548	256	4	i	i	PROPN
ejpam-4548	256	5	)	)	PUNCT
ejpam-4548	256	6	.	.	PUNCT
ejpam-4548	257	1	then	then	ADV
ejpam-4548	257	2	,	,	PUNCT
ejpam-4548	257	3	using	use	VERB
ejpam-4548	257	4	the	the	DET
ejpam-4548	257	5	above	above	ADJ
ejpam-4548	257	6	arguments	argument	NOUN
ejpam-4548	257	7	,	,	PUNCT
ejpam-4548	257	8	we	we	PRON
ejpam-4548	257	9	have	have	VERB
ejpam-4548	257	10	yi(z	yi(z	PRON
ejpam-4548	257	11	i	i	PROPN
ejpam-4548	257	12	k	k	PROPN
ejpam-4548	257	13	)	)	PUNCT
ejpam-4548	258	1	−1y−1	−1y−1	PROPN
ejpam-4548	258	2	i	i	PRON
ejpam-4548	258	3	∈	∈	PROPN
ejpam-4548	258	4	(	(	PUNCT
ejpam-4548	258	5	γηγ−1)c	γηγ−1)c	NOUN
ejpam-4548	258	6	ai	ai	VERB
ejpam-4548	258	7	j	j	PROPN
ejpam-4548	258	8	,	,	PUNCT
ejpam-4548	258	9	yi	yi	PROPN
ejpam-4548	258	10	.	.	PUNCT
ejpam-4548	259	1	consequently	consequently	ADV
ejpam-4548	259	2	,	,	PUNCT
ejpam-4548	259	3	yi(z	yi(z	PUNCT
ejpam-4548	259	4	i	i	PROPN
ejpam-4548	259	5	k	k	PROPN
ejpam-4548	259	6	)	)	PUNCT
ejpam-4548	259	7	−1y−1	−1y−1	PROPN
ejpam-4548	259	8	i	i	PROPN
ejpam-4548	259	9	and	and	CCONJ
ejpam-4548	259	10	yiz	yiz	NOUN
ejpam-4548	260	1	i	i	PRON
ejpam-4548	260	2	ky	ky	VERB
ejpam-4548	260	3	−1	−1	NOUN
ejpam-4548	261	1	i	i	PRON
ejpam-4548	261	2	∈	∈	PROPN
ejpam-4548	261	3	(	(	PUNCT
ejpam-4548	261	4	γηγ−1)c	γηγ−1)c	NOUN
ejpam-4548	261	5	ai	ai	VERB
ejpam-4548	261	6	j	j	PROPN
ejpam-4548	261	7	,	,	PUNCT
ejpam-4548	261	8	yi	yi	PROPN
ejpam-4548	261	9	.	.	PUNCT
ejpam-4548	262	1	this	this	PRON
ejpam-4548	262	2	implies	imply	VERB
ejpam-4548	262	3	yiziy	yiziy	ADJ
ejpam-4548	262	4	−1	−1	NOUN
ejpam-4548	263	1	i	i	PRON
ejpam-4548	263	2	∈	∈	PROPN
ejpam-4548	264	1	⟨(γηγ−1)c	⟨(γηγ−1)c	NOUN
ejpam-4548	264	2	ai	ai	VERB
ejpam-4548	264	3	j	j	PROPN
ejpam-4548	264	4	,	,	PUNCT
ejpam-4548	264	5	yi	yi	PROPN
ejpam-4548	264	6	⟩.	⟩.	PROPN
ejpam-4548	264	7	n.	n.	PROPN
ejpam-4548	264	8	ajmal	ajmal	PROPN
ejpam-4548	264	9	,	,	PUNCT
ejpam-4548	264	10	i.	i.	PROPN
ejpam-4548	264	11	jahan	jahan	PROPN
ejpam-4548	264	12	.	.	PROPN
ejpam-4548	264	13	,	,	PUNCT
ejpam-4548	264	14	b.	b.	PROPN
ejpam-4548	264	15	davvaz	davvaz	PROPN
ejpam-4548	264	16	/	/	SYM
ejpam-4548	264	17	eur	eur	PROPN
ejpam-4548	264	18	.	.	PUNCT
ejpam-4548	265	1	j.	j.	PROPN
ejpam-4548	265	2	pure	pure	PROPN
ejpam-4548	265	3	appl	appl	PROPN
ejpam-4548	265	4	.	.	PROPN
ejpam-4548	265	5	math	math	PROPN
ejpam-4548	265	6	,	,	PUNCT
ejpam-4548	265	7	15	15	NUM
ejpam-4548	265	8	(	(	PUNCT
ejpam-4548	265	9	4	4	NUM
ejpam-4548	265	10	)	)	PUNCT
ejpam-4548	265	11	(	(	PUNCT
ejpam-4548	265	12	2022	2022	NUM
ejpam-4548	265	13	)	)	PUNCT
ejpam-4548	265	14	,	,	PUNCT
ejpam-4548	265	15	2086	2086	NUM
ejpam-4548	265	16	-	-	SYM
ejpam-4548	265	17	2115	2115	NUM
ejpam-4548	265	18	2098	2098	NUM
ejpam-4548	265	19	thus	thus	ADV
ejpam-4548	265	20	our	our	PRON
ejpam-4548	265	21	claim	claim	NOUN
ejpam-4548	265	22	is	be	AUX
ejpam-4548	265	23	established	establish	VERB
ejpam-4548	265	24	.	.	PUNCT
ejpam-4548	266	1	therefore	therefore	ADV
ejpam-4548	266	2	,	,	PUNCT
ejpam-4548	266	3	we	we	PRON
ejpam-4548	266	4	have	have	VERB
ejpam-4548	266	5	θηγθ−1(x	θηγθ−1(x	NOUN
ejpam-4548	266	6	)	)	PUNCT
ejpam-4548	266	7	≤	≤	NOUN
ejpam-4548	266	8	∨	∨	NUM
ejpam-4548	266	9	x	x	X
ejpam-4548	266	10	=	=	NOUN
ejpam-4548	266	11	yiziy	yiziy	NOUN
ejpam-4548	267	1	−1	−1	NOUN
ejpam-4548	268	1	i	i	PRON
ejpam-4548	268	2	ai	ai	VERB
ejpam-4548	268	3	j	j	PROPN
ejpam-4548	268	4	≤η(e	≤η(e	PROPN
ejpam-4548	268	5	)	)	PUNCT
ejpam-4548	268	6	{	{	PUNCT
ejpam-4548	268	7	aij	aij	PROPN
ejpam-4548	268	8	∧	∧	PROPN
ejpam-4548	268	9	γ(yi	γ(yi	PROPN
ejpam-4548	268	10	)	)	PUNCT
ejpam-4548	268	11	:	:	PUNCT
ejpam-4548	268	12	zi	zi	NOUN
ejpam-4548	268	13	∈	∈	PROPN
ejpam-4548	268	14	⟨	⟨	VERB
ejpam-4548	268	15	(	(	PUNCT
ejpam-4548	268	16	γηγ−1	γηγ−1	PROPN
ejpam-4548	268	17	)	)	PUNCT
ejpam-4548	268	18	aij	aij	PROPN
ejpam-4548	268	19	⟩	⟩	PROPN
ejpam-4548	268	20	}	}	PUNCT
ejpam-4548	268	21	≤	≤	NOUN
ejpam-4548	268	22	∨	∨	NUM
ejpam-4548	268	23	x	x	X
ejpam-4548	268	24	=	=	NOUN
ejpam-4548	268	25	yiziy	yiziy	ADJ
ejpam-4548	268	26	−1	−1	NOUN
ejpam-4548	269	1	i	i	PROPN
ejpam-4548	269	2	)	)	PUNCT
ejpam-4548	270	1	c	c	AUX
ejpam-4548	270	2	ai	ai	VERB
ejpam-4548	270	3	j	j	PROPN
ejpam-4548	270	4	,	,	PUNCT
ejpam-4548	270	5	yi	yi	PROPN
ejpam-4548	270	6	≤η(e	≤η(e	PROPN
ejpam-4548	270	7	)	)	PUNCT
ejpam-4548	270	8	{	{	PUNCT
ejpam-4548	270	9	caij	caij	NOUN
ejpam-4548	270	10	,	,	PUNCT
ejpam-4548	270	11	yi	yi	PROPN
ejpam-4548	270	12	:	:	PUNCT
ejpam-4548	271	1	yiziy	yiziy	ADJ
ejpam-4548	271	2	−1	−1	NOUN
ejpam-4548	272	1	i	i	PRON
ejpam-4548	272	2	∈	∈	PROPN
ejpam-4548	273	1	⟨(γηγ−1)c	⟨(γηγ−1)c	NOUN
ejpam-4548	273	2	ai	ai	VERB
ejpam-4548	273	3	j	j	PROPN
ejpam-4548	273	4	,	,	PUNCT
ejpam-4548	273	5	yi	yi	PROPN
ejpam-4548	273	6	⟩	⟩	PROPN
ejpam-4548	273	7	}	}	PUNCT
ejpam-4548	273	8	≤	≤	NOUN
ejpam-4548	273	9	∨	∨	NUM
ejpam-4548	273	10	x	x	X
ejpam-4548	273	11	=	=	NOUN
ejpam-4548	273	12	yiziy	yiziy	NOUN
ejpam-4548	273	13	−1	−1	NOUN
ejpam-4548	274	1	i	i	PRON
ejpam-4548	274	2	b≤η(e	b≤η(e	VERB
ejpam-4548	274	3	)	)	PUNCT
ejpam-4548	274	4	{	{	PUNCT
ejpam-4548	274	5	b	b	NOUN
ejpam-4548	274	6	:	:	PUNCT
ejpam-4548	274	7	yiziy−1	yiziy−1	PROPN
ejpam-4548	274	8	i	i	PROPN
ejpam-4548	274	9	∈	∈	PROPN
ejpam-4548	274	10	⟨(γηγ−1)b⟩	⟨(γηγ−1)b⟩	NOUN
ejpam-4548	274	11	}	}	PUNCT
ejpam-4548	274	12	≤	≤	NOUN
ejpam-4548	274	13	∨	∨	NUM
ejpam-4548	274	14	d≤η(e	d≤η(e	NUM
ejpam-4548	274	15	)	)	PUNCT
ejpam-4548	274	16	{	{	PUNCT
ejpam-4548	274	17	d	d	NOUN
ejpam-4548	274	18	:	:	PUNCT
ejpam-4548	274	19	x	x	SYM
ejpam-4548	274	20	∈	∈	PROPN
ejpam-4548	274	21	⟨(γηγ−1)d⟩	⟨(γηγ−1)d⟩	NOUN
ejpam-4548	274	22	}	}	PUNCT
ejpam-4548	274	23	=	=	NOUN
ejpam-4548	274	24	⟨γηγ−1⟩(x	⟨γηγ−1⟩(x	X
ejpam-4548	274	25	)	)	PUNCT
ejpam-4548	274	26	(	(	PUNCT
ejpam-4548	274	27	by	by	ADP
ejpam-4548	274	28	theorem	theorem	NOUN
ejpam-4548	274	29	6	6	NUM
ejpam-4548	274	30	)	)	PUNCT
ejpam-4548	274	31	=	=	NOUN
ejpam-4548	274	32	ηγ(x	ηγ(x	NUM
ejpam-4548	274	33	)	)	PUNCT
ejpam-4548	274	34	.	.	PUNCT
ejpam-4548	275	1	consequently	consequently	ADV
ejpam-4548	275	2	,	,	PUNCT
ejpam-4548	275	3	θηγθ−1	θηγθ−1	PROPN
ejpam-4548	275	4	⊆	⊆	NUM
ejpam-4548	275	5	ηγ	ηγ	NOUN
ejpam-4548	275	6	.	.	PUNCT
ejpam-4548	276	1	as	as	ADP
ejpam-4548	276	2	ηγ	ηγ	PROPN
ejpam-4548	276	3	∈	∈	PROPN
ejpam-4548	276	4	l(µ	l(µ	PROPN
ejpam-4548	276	5	)	)	PUNCT
ejpam-4548	276	6	,	,	PUNCT
ejpam-4548	276	7	it	it	PRON
ejpam-4548	276	8	follows	follow	VERB
ejpam-4548	276	9	that	that	PRON
ejpam-4548	276	10	⟨θηγθ−1⟩	⟨θηγθ−1⟩	PROPN
ejpam-4548	276	11	=	=	SYM
ejpam-4548	276	12	(	(	PUNCT
ejpam-4548	276	13	ηγ)θ	ηγ)θ	PROPN
ejpam-4548	276	14	⊆	⊆	NUM
ejpam-4548	276	15	ηγ	ηγ	PROPN
ejpam-4548	276	16	.	.	PUNCT
ejpam-4548	277	1	this	this	PRON
ejpam-4548	277	2	establishes	establish	VERB
ejpam-4548	277	3	the	the	DET
ejpam-4548	277	4	desired	desire	VERB
ejpam-4548	277	5	equality	equality	NOUN
ejpam-4548	277	6	.	.	PUNCT
ejpam-4548	278	1	(	(	PUNCT
ejpam-4548	278	2	ii	ii	NOUN
ejpam-4548	278	3	)	)	PUNCT
ejpam-4548	278	4	as	as	ADP
ejpam-4548	278	5	θ	θ	PROPN
ejpam-4548	278	6	⊆	⊆	NUM
ejpam-4548	278	7	γ	γ	X
ejpam-4548	278	8	,	,	PUNCT
ejpam-4548	278	9	by	by	ADP
ejpam-4548	278	10	the	the	DET
ejpam-4548	278	11	definition	definition	NOUN
ejpam-4548	278	12	of	of	ADP
ejpam-4548	278	13	a	a	DET
ejpam-4548	278	14	conjugate	conjugate	ADJ
ejpam-4548	278	15	l	l	NOUN
ejpam-4548	278	16	-	-	NOUN
ejpam-4548	278	17	subgroup	subgroup	NOUN
ejpam-4548	278	18	,	,	PUNCT
ejpam-4548	278	19	we	we	PRON
ejpam-4548	278	20	have	have	VERB
ejpam-4548	278	21	ηθ	ηθ	NOUN
ejpam-4548	278	22	⊆	⊆	NUM
ejpam-4548	278	23	ηγ	ηγ	ADV
ejpam-4548	278	24	and	and	CCONJ
ejpam-4548	278	25	hence	hence	ADV
ejpam-4548	278	26	(	(	PUNCT
ejpam-4548	278	27	ηθ)γ	ηθ)γ	PROPN
ejpam-4548	278	28	⊆	⊆	NUM
ejpam-4548	278	29	(	(	PUNCT
ejpam-4548	278	30	ηγ)γ	ηγ)γ	PROPN
ejpam-4548	278	31	.	.	PUNCT
ejpam-4548	279	1	by	by	ADP
ejpam-4548	279	2	part(i	part(i	PROPN
ejpam-4548	279	3	)	)	PUNCT
ejpam-4548	279	4	,	,	PUNCT
ejpam-4548	279	5	(	(	PUNCT
ejpam-4548	279	6	ηγ)γ	ηγ)γ	PROPN
ejpam-4548	279	7	=	=	SYM
ejpam-4548	279	8	ηγ	ηγ	PROPN
ejpam-4548	279	9	.	.	PUNCT
ejpam-4548	280	1	thus	thus	ADV
ejpam-4548	280	2	,	,	PUNCT
ejpam-4548	280	3	(	(	PUNCT
ejpam-4548	280	4	ηθ)γ	ηθ)γ	PROPN
ejpam-4548	280	5	⊆	⊆	NUM
ejpam-4548	280	6	ηγ	ηγ	ADV
ejpam-4548	280	7	.	.	PUNCT
ejpam-4548	281	1	in	in	ADP
ejpam-4548	281	2	order	order	NOUN
ejpam-4548	281	3	to	to	PART
ejpam-4548	281	4	prove	prove	VERB
ejpam-4548	281	5	the	the	DET
ejpam-4548	281	6	reverse	reverse	ADJ
ejpam-4548	281	7	inclusion	inclusion	NOUN
ejpam-4548	281	8	,	,	PUNCT
ejpam-4548	281	9	note	note	VERB
ejpam-4548	281	10	that	that	SCONJ
ejpam-4548	281	11	η(e	η(e	PROPN
ejpam-4548	281	12	)	)	PUNCT
ejpam-4548	281	13	=	=	SYM
ejpam-4548	281	14	θ(e	θ(e	NUM
ejpam-4548	281	15	)	)	PUNCT
ejpam-4548	281	16	so	so	SCONJ
ejpam-4548	281	17	that	that	SCONJ
ejpam-4548	281	18	,	,	PUNCT
ejpam-4548	281	19	by	by	ADP
ejpam-4548	281	20	lemma	lemma	PROPN
ejpam-4548	281	21	1	1	NUM
ejpam-4548	281	22	,	,	PUNCT
ejpam-4548	281	23	η	η	PROPN
ejpam-4548	281	24	⊆	⊆	NUM
ejpam-4548	281	25	ηθ	ηθ	NOUN
ejpam-4548	281	26	.	.	PUNCT
ejpam-4548	281	27	therefore	therefore	ADV
ejpam-4548	281	28	,	,	PUNCT
ejpam-4548	281	29	we	we	PRON
ejpam-4548	281	30	have	have	VERB
ejpam-4548	281	31	ηγ	ηγ	ADV
ejpam-4548	281	32	⊆	⊆	NUM
ejpam-4548	281	33	(	(	PUNCT
ejpam-4548	281	34	ηθ)γ	ηθ)γ	PROPN
ejpam-4548	281	35	.	.	PUNCT
ejpam-4548	282	1	this	this	PRON
ejpam-4548	282	2	establishes	establish	VERB
ejpam-4548	282	3	the	the	DET
ejpam-4548	282	4	desired	desire	VERB
ejpam-4548	282	5	equality	equality	NOUN
ejpam-4548	282	6	.	.	PUNCT
ejpam-4548	283	1	(	(	PUNCT
ejpam-4548	283	2	iii	iii	NOUN
ejpam-4548	283	3	)	)	PUNCT
ejpam-4548	283	4	as	as	ADP
ejpam-4548	283	5	η(e	η(e	NOUN
ejpam-4548	283	6	)	)	PUNCT
ejpam-4548	283	7	=	=	SYM
ejpam-4548	283	8	θ(e	θ(e	NUM
ejpam-4548	283	9	)	)	PUNCT
ejpam-4548	283	10	,	,	PUNCT
ejpam-4548	283	11	by	by	ADP
ejpam-4548	283	12	proposition	proposition	NOUN
ejpam-4548	283	13	3	3	NUM
ejpam-4548	283	14	,	,	PUNCT
ejpam-4548	283	15	we	we	PRON
ejpam-4548	283	16	have	have	VERB
ejpam-4548	283	17	θ	θ	PROPN
ejpam-4548	283	18	⊆	⊆	NUM
ejpam-4548	283	19	θ	θ	PROPN
ejpam-4548	283	20	◦	◦	PROPN
ejpam-4548	283	21	η	η	PROPN
ejpam-4548	283	22	.	.	PUNCT
ejpam-4548	284	1	thus	thus	ADV
ejpam-4548	284	2	by	by	ADP
ejpam-4548	284	3	the	the	DET
ejpam-4548	284	4	definition	definition	NOUN
ejpam-4548	284	5	of	of	ADP
ejpam-4548	284	6	a	a	DET
ejpam-4548	284	7	conjugate	conjugate	ADJ
ejpam-4548	284	8	l	l	NOUN
ejpam-4548	284	9	-	-	NOUN
ejpam-4548	284	10	subset	subset	NOUN
ejpam-4548	284	11	,	,	PUNCT
ejpam-4548	284	12	we	we	PRON
ejpam-4548	284	13	get	get	VERB
ejpam-4548	284	14	ηθ	ηθ	PRON
ejpam-4548	284	15	⊆	⊆	NUM
ejpam-4548	284	16	ηθ	ηθ	SYM
ejpam-4548	284	17	◦	◦	NOUN
ejpam-4548	284	18	η	η	PROPN
ejpam-4548	284	19	.	.	PROPN
ejpam-4548	284	20	now	now	ADV
ejpam-4548	284	21	to	to	PART
ejpam-4548	284	22	prove	prove	VERB
ejpam-4548	284	23	the	the	DET
ejpam-4548	284	24	reverse	reverse	ADJ
ejpam-4548	284	25	inclusion	inclusion	NOUN
ejpam-4548	284	26	,	,	PUNCT
ejpam-4548	284	27	we	we	PRON
ejpam-4548	284	28	take	take	VERB
ejpam-4548	284	29	λ	λ	X
ejpam-4548	284	30	=	=	SYM
ejpam-4548	284	31	θ	θ	PROPN
ejpam-4548	284	32	◦	◦	PROPN
ejpam-4548	284	33	η	η	PROPN
ejpam-4548	284	34	.	.	PROPN
ejpam-4548	285	1	firstly	firstly	ADV
ejpam-4548	285	2	,	,	PUNCT
ejpam-4548	285	3	we	we	PRON
ejpam-4548	285	4	shall	shall	AUX
ejpam-4548	285	5	show	show	VERB
ejpam-4548	285	6	that	that	SCONJ
ejpam-4548	285	7	ληλ−1	ληλ−1	PROPN
ejpam-4548	285	8	⊆	⊆	NUM
ejpam-4548	285	9	ηθ	ηθ	NOUN
ejpam-4548	285	10	.	.	PUNCT
ejpam-4548	286	1	so	so	ADV
ejpam-4548	286	2	let	let	VERB
ejpam-4548	286	3	x	x	PUNCT
ejpam-4548	286	4	∈	∈	PROPN
ejpam-4548	286	5	g	g	PROPN
ejpam-4548	286	6	and	and	CCONJ
ejpam-4548	286	7	consider	consider	VERB
ejpam-4548	286	8	ληλ−1(x	ληλ−1(x	NOUN
ejpam-4548	286	9	)	)	PUNCT
ejpam-4548	286	10	=	=	PUNCT
ejpam-4548	287	1	∨	∨	NUM
ejpam-4548	287	2	x	x	X
ejpam-4548	287	3	=	=	NOUN
ejpam-4548	287	4	yiziy	yiziy	NOUN
ejpam-4548	287	5	−1	−1	NOUN
ejpam-4548	287	6	i	i	PRON
ejpam-4548	287	7	{	{	PUNCT
ejpam-4548	287	8	η(zi	η(zi	PROPN
ejpam-4548	287	9	)	)	PUNCT
ejpam-4548	287	10	∧	∧	PROPN
ejpam-4548	287	11	λ(yi	λ(yi	NOUN
ejpam-4548	287	12	)	)	PUNCT
ejpam-4548	287	13	}	}	PUNCT
ejpam-4548	287	14	=	=	PUNCT
ejpam-4548	287	15	∨	∨	NUM
ejpam-4548	287	16	x	x	X
ejpam-4548	287	17	=	=	NOUN
ejpam-4548	287	18	yiziy	yiziy	NOUN
ejpam-4548	287	19	−1	−1	NOUN
ejpam-4548	288	1	i	i	PRON
ejpam-4548	288	2	{	{	PUNCT
ejpam-4548	288	3	η(zi	η(zi	PROPN
ejpam-4548	288	4	)	)	PUNCT
ejpam-4548	288	5	∧	∧	NOUN
ejpam-4548	288	6	{	{	PUNCT
ejpam-4548	288	7	∨	∨	NUM
ejpam-4548	288	8	y	y	PROPN
ejpam-4548	288	9	=	=	PROPN
ejpam-4548	288	10	ui	ui	PROPN
ejpam-4548	288	11	jv	jv	NOUN
ejpam-4548	288	12	i	i	PRON
ejpam-4548	288	13	j	j	PROPN
ejpam-4548	288	14	{	{	PUNCT
ejpam-4548	288	15	θ(uij	θ(uij	PROPN
ejpam-4548	288	16	)	)	PUNCT
ejpam-4548	288	17	∧	∧	PROPN
ejpam-4548	288	18	η(vij	η(vij	NOUN
ejpam-4548	288	19	)	)	PUNCT
ejpam-4548	288	20	}	}	PUNCT
ejpam-4548	288	21	}	}	PUNCT
ejpam-4548	288	22	}	}	PUNCT
ejpam-4548	288	23	=	=	SYM
ejpam-4548	288	24	∨	∨	NUM
ejpam-4548	288	25	x	x	X
ejpam-4548	288	26	=	=	NOUN
ejpam-4548	288	27	yiziy	yiziy	NOUN
ejpam-4548	288	28	−1	−1	NOUN
ejpam-4548	288	29	i	i	PRON
ejpam-4548	288	30	{	{	PUNCT
ejpam-4548	288	31	∨	∨	NUM
ejpam-4548	288	32	yi	yi	NOUN
ejpam-4548	288	33	=	=	NOUN
ejpam-4548	288	34	ui	ui	NOUN
ejpam-4548	288	35	jv	jv	NOUN
ejpam-4548	289	1	i	i	PRON
ejpam-4548	289	2	j	j	PROPN
ejpam-4548	289	3	{	{	PUNCT
ejpam-4548	289	4	η(zi	η(zi	PROPN
ejpam-4548	289	5	)	)	PUNCT
ejpam-4548	289	6	∧	∧	NOUN
ejpam-4548	289	7	{	{	PUNCT
ejpam-4548	289	8	θ(uij	θ(uij	PROPN
ejpam-4548	289	9	)	)	PUNCT
ejpam-4548	289	10	∧	∧	PROPN
ejpam-4548	289	11	η(vij	η(vij	NOUN
ejpam-4548	289	12	)	)	PUNCT
ejpam-4548	289	13	}	}	PUNCT
ejpam-4548	289	14	}	}	PUNCT
ejpam-4548	289	15	}	}	PUNCT
ejpam-4548	289	16	(	(	PUNCT
ejpam-4548	289	17	as	as	SCONJ
ejpam-4548	289	18	l	l	NOUN
ejpam-4548	289	19	is	be	AUX
ejpam-4548	289	20	a	a	DET
ejpam-4548	289	21	completely	completely	ADV
ejpam-4548	289	22	distributive	distributive	ADJ
ejpam-4548	289	23	lattice	lattice	NOUN
ejpam-4548	289	24	)	)	PUNCT
ejpam-4548	289	25	n.	n.	PROPN
ejpam-4548	289	26	ajmal	ajmal	PROPN
ejpam-4548	289	27	,	,	PUNCT
ejpam-4548	289	28	i.	i.	PROPN
ejpam-4548	289	29	jahan	jahan	PROPN
ejpam-4548	289	30	.	.	PROPN
ejpam-4548	289	31	,	,	PUNCT
ejpam-4548	289	32	b.	b.	PROPN
ejpam-4548	289	33	davvaz	davvaz	PROPN
ejpam-4548	289	34	/	/	SYM
ejpam-4548	289	35	eur	eur	PROPN
ejpam-4548	289	36	.	.	PUNCT
ejpam-4548	290	1	j.	j.	PROPN
ejpam-4548	290	2	pure	pure	PROPN
ejpam-4548	290	3	appl	appl	PROPN
ejpam-4548	290	4	.	.	PROPN
ejpam-4548	290	5	math	math	PROPN
ejpam-4548	290	6	,	,	PUNCT
ejpam-4548	290	7	15	15	NUM
ejpam-4548	290	8	(	(	PUNCT
ejpam-4548	290	9	4	4	NUM
ejpam-4548	290	10	)	)	PUNCT
ejpam-4548	290	11	(	(	PUNCT
ejpam-4548	290	12	2022	2022	NUM
ejpam-4548	290	13	)	)	PUNCT
ejpam-4548	290	14	,	,	PUNCT
ejpam-4548	290	15	2086	2086	NUM
ejpam-4548	290	16	-	-	SYM
ejpam-4548	290	17	2115	2115	NUM
ejpam-4548	290	18	2099	2099	NUM
ejpam-4548	290	19	=	=	SYM
ejpam-4548	290	20	∨	∨	NUM
ejpam-4548	290	21	x	x	X
ejpam-4548	290	22	=	=	NOUN
ejpam-4548	290	23	yiziy	yiziy	NOUN
ejpam-4548	290	24	−1	−1	NOUN
ejpam-4548	291	1	i	i	PRON
ejpam-4548	291	2	yi	yi	PUNCT
ejpam-4548	292	1	=	=	PROPN
ejpam-4548	292	2	ui	ui	PROPN
ejpam-4548	292	3	j	j	PROPN
ejpam-4548	292	4	vi	vi	PROPN
ejpam-4548	292	5	j	j	PROPN
ejpam-4548	292	6	{	{	PUNCT
ejpam-4548	292	7	η(zi	η(zi	PROPN
ejpam-4548	292	8	)	)	PUNCT
ejpam-4548	292	9	∧	∧	PROPN
ejpam-4548	292	10	θ(uij	θ(uij	NOUN
ejpam-4548	292	11	)	)	PUNCT
ejpam-4548	292	12	∧	∧	PROPN
ejpam-4548	292	13	η(vij	η(vij	NOUN
ejpam-4548	292	14	)	)	PUNCT
ejpam-4548	292	15	}	}	PUNCT
ejpam-4548	292	16	≤	≤	NOUN
ejpam-4548	292	17	∨	∨	NUM
ejpam-4548	292	18	x	x	X
ejpam-4548	292	19	=	=	NOUN
ejpam-4548	292	20	ui	ui	NOUN
ejpam-4548	292	21	j(v	j(v	NOUN
ejpam-4548	293	1	i	i	PRON
ejpam-4548	293	2	jzi(v	jzi(v	VERB
ejpam-4548	293	3	i	i	PRON
ejpam-4548	293	4	j	j	PROPN
ejpam-4548	293	5	)	)	PUNCT
ejpam-4548	293	6	−1)(ui	−1)(ui	PROPN
ejpam-4548	293	7	j	j	X
ejpam-4548	293	8	)	)	PUNCT
ejpam-4548	293	9	−1	−1	NOUN
ejpam-4548	293	10	{	{	PUNCT
ejpam-4548	293	11	η(vijzi(vij)−1	η(vijzi(vij)−1	NOUN
ejpam-4548	293	12	)	)	PUNCT
ejpam-4548	293	13	∧	∧	PROPN
ejpam-4548	293	14	θ(uij	θ(uij	NOUN
ejpam-4548	293	15	)	)	PUNCT
ejpam-4548	293	16	}	}	PUNCT
ejpam-4548	293	17	≤	≤	NUM
ejpam-4548	293	18	θηθ−1(x	θηθ−1(x	NOUN
ejpam-4548	293	19	)	)	PUNCT
ejpam-4548	293	20	≤	≤	NOUN
ejpam-4548	293	21	ηθ(x	ηθ(x	NUM
ejpam-4548	293	22	)	)	PUNCT
ejpam-4548	293	23	.	.	PUNCT
ejpam-4548	294	1	this	this	PRON
ejpam-4548	294	2	establishes	establish	VERB
ejpam-4548	294	3	the	the	DET
ejpam-4548	294	4	claim	claim	NOUN
ejpam-4548	294	5	.	.	PUNCT
ejpam-4548	295	1	as	as	ADP
ejpam-4548	295	2	ηθ	ηθ	PRON
ejpam-4548	295	3	∈	∈	PROPN
ejpam-4548	295	4	l(µ	l(µ	PROPN
ejpam-4548	295	5	)	)	PUNCT
ejpam-4548	295	6	,	,	PUNCT
ejpam-4548	295	7	it	it	PRON
ejpam-4548	295	8	follows	follow	VERB
ejpam-4548	295	9	that	that	SCONJ
ejpam-4548	295	10	ηθ	ηθ	NOUN
ejpam-4548	295	11	◦	◦	NOUN
ejpam-4548	295	12	η	η	NOUN
ejpam-4548	295	13	=	=	PROPN
ejpam-4548	295	14	⟨ληλ−1⟩	⟨ληλ−1⟩	PROPN
ejpam-4548	295	15	⊆	⊆	NUM
ejpam-4548	295	16	ηθ	ηθ	NOUN
ejpam-4548	295	17	.	.	PUNCT
ejpam-4548	296	1	this	this	PRON
ejpam-4548	296	2	establishes	establish	VERB
ejpam-4548	296	3	the	the	DET
ejpam-4548	296	4	desired	desire	VERB
ejpam-4548	296	5	equality	equality	NOUN
ejpam-4548	296	6	.	.	PUNCT
ejpam-4548	297	1	theorem	theorem	NOUN
ejpam-4548	297	2	10	10	NUM
ejpam-4548	297	3	.	.	PUNCT
ejpam-4548	298	1	let	let	VERB
ejpam-4548	298	2	η	η	PROPN
ejpam-4548	298	3	,	,	PUNCT
ejpam-4548	298	4	θ	θ	PROPN
ejpam-4548	298	5	,	,	PUNCT
ejpam-4548	298	6	γ	γ	PROPN
ejpam-4548	298	7	∈	∈	PROPN
ejpam-4548	298	8	l(µ	l(µ	PROPN
ejpam-4548	298	9	)	)	PUNCT
ejpam-4548	298	10	be	be	AUX
ejpam-4548	298	11	such	such	ADJ
ejpam-4548	298	12	that	that	SCONJ
ejpam-4548	298	13	γ(e	γ(e	NOUN
ejpam-4548	298	14	)	)	PUNCT
ejpam-4548	299	1	=	=	SYM
ejpam-4548	299	2	η(e	η(e	PROPN
ejpam-4548	299	3	)	)	PUNCT
ejpam-4548	299	4	=	=	SYM
ejpam-4548	299	5	θ(e	θ(e	NUM
ejpam-4548	299	6	)	)	PUNCT
ejpam-4548	299	7	.	.	PUNCT
ejpam-4548	300	1	then	then	ADV
ejpam-4548	300	2	,	,	PUNCT
ejpam-4548	300	3	(	(	PUNCT
ejpam-4548	300	4	ηθ)γ	ηθ)γ	PROPN
ejpam-4548	300	5	=	=	SYM
ejpam-4548	300	6	ηγ	ηγ	PROPN
ejpam-4548	300	7	◦	◦	NOUN
ejpam-4548	300	8	θ	θ	NOUN
ejpam-4548	300	9	.	.	PUNCT
ejpam-4548	300	10	proof	proof	NOUN
ejpam-4548	300	11	.	.	PUNCT
ejpam-4548	301	1	as	as	ADP
ejpam-4548	301	2	γ(e	γ(e	NOUN
ejpam-4548	301	3	)	)	PUNCT
ejpam-4548	301	4	=	=	SYM
ejpam-4548	301	5	θ(e	θ(e	NUM
ejpam-4548	301	6	)	)	PUNCT
ejpam-4548	301	7	,	,	PUNCT
ejpam-4548	301	8	by	by	ADP
ejpam-4548	301	9	proposition	proposition	NOUN
ejpam-4548	301	10	3	3	NUM
ejpam-4548	301	11	,	,	PUNCT
ejpam-4548	301	12	γ	γ	X
ejpam-4548	301	13	⊆	⊆	NUM
ejpam-4548	301	14	γ	γ	PROPN
ejpam-4548	301	15	◦	◦	NOUN
ejpam-4548	301	16	θ	θ	PROPN
ejpam-4548	301	17	.	.	PUNCT
ejpam-4548	301	18	hence	hence	ADV
ejpam-4548	301	19	,	,	PUNCT
ejpam-4548	301	20	(	(	PUNCT
ejpam-4548	301	21	ηθ)γ	ηθ)γ	PROPN
ejpam-4548	301	22	⊆	⊆	NUM
ejpam-4548	301	23	(	(	PUNCT
ejpam-4548	301	24	ηθ)γ	ηθ)γ	NOUN
ejpam-4548	301	25	◦	◦	NOUN
ejpam-4548	301	26	θ	θ	NOUN
ejpam-4548	301	27	.	.	PUNCT
ejpam-4548	301	28	by	by	ADP
ejpam-4548	301	29	theorem	theorem	NOUN
ejpam-4548	301	30	9	9	NUM
ejpam-4548	301	31	,	,	PUNCT
ejpam-4548	301	32	we	we	PRON
ejpam-4548	301	33	have	have	VERB
ejpam-4548	301	34	(	(	PUNCT
ejpam-4548	301	35	ηθ)γ	ηθ)γ	NOUN
ejpam-4548	301	36	◦	◦	NOUN
ejpam-4548	301	37	θ	θ	NOUN
ejpam-4548	301	38	=	=	SYM
ejpam-4548	301	39	ηγ	ηγ	PROPN
ejpam-4548	301	40	◦	◦	NOUN
ejpam-4548	301	41	θ	θ	NOUN
ejpam-4548	301	42	.	.	PUNCT
ejpam-4548	302	1	therefore	therefore	ADV
ejpam-4548	302	2	,	,	PUNCT
ejpam-4548	302	3	(	(	PUNCT
ejpam-4548	302	4	ηθ)γ	ηθ)γ	PROPN
ejpam-4548	302	5	⊆	⊆	NUM
ejpam-4548	302	6	ηγ	ηγ	PROPN
ejpam-4548	302	7	◦	◦	NOUN
ejpam-4548	302	8	θ	θ	NOUN
ejpam-4548	302	9	.	.	PROPN
ejpam-4548	302	10	for	for	ADP
ejpam-4548	302	11	the	the	DET
ejpam-4548	302	12	reverse	reverse	ADJ
ejpam-4548	302	13	inclusion	inclusion	NOUN
ejpam-4548	302	14	,	,	PUNCT
ejpam-4548	302	15	we	we	PRON
ejpam-4548	302	16	take	take	VERB
ejpam-4548	302	17	λ	λ	X
ejpam-4548	302	18	=	=	PUNCT
ejpam-4548	302	19	γ	γ	X
ejpam-4548	302	20	◦	◦	NOUN
ejpam-4548	302	21	θ	θ	PROPN
ejpam-4548	302	22	.	.	PUNCT
ejpam-4548	303	1	we	we	PRON
ejpam-4548	303	2	shall	shall	AUX
ejpam-4548	303	3	establish	establish	VERB
ejpam-4548	303	4	that	that	SCONJ
ejpam-4548	303	5	ληλ−1	ληλ−1	PROPN
ejpam-4548	303	6	⊆	⊆	NUM
ejpam-4548	303	7	γηθγ−1	γηθγ−1	NOUN
ejpam-4548	303	8	.	.	PUNCT
ejpam-4548	304	1	so	so	ADV
ejpam-4548	304	2	let	let	VERB
ejpam-4548	304	3	x	x	PUNCT
ejpam-4548	304	4	∈	∈	PROPN
ejpam-4548	304	5	g	g	PROPN
ejpam-4548	304	6	and	and	CCONJ
ejpam-4548	304	7	consider	consider	VERB
ejpam-4548	304	8	ληλ−1(x	ληλ−1(x	NOUN
ejpam-4548	304	9	)	)	PUNCT
ejpam-4548	304	10	=	=	PUNCT
ejpam-4548	305	1	∨	∨	NUM
ejpam-4548	305	2	x	x	X
ejpam-4548	305	3	=	=	NOUN
ejpam-4548	305	4	yiziy	yiziy	NOUN
ejpam-4548	305	5	−1	−1	NOUN
ejpam-4548	305	6	i	i	PRON
ejpam-4548	305	7	{	{	PUNCT
ejpam-4548	305	8	η(zi	η(zi	PROPN
ejpam-4548	305	9	)	)	PUNCT
ejpam-4548	305	10	∧	∧	PROPN
ejpam-4548	305	11	λ(yi	λ(yi	NOUN
ejpam-4548	305	12	)	)	PUNCT
ejpam-4548	305	13	}	}	PUNCT
ejpam-4548	305	14	=	=	PUNCT
ejpam-4548	305	15	∨	∨	NUM
ejpam-4548	305	16	x	x	X
ejpam-4548	305	17	=	=	NOUN
ejpam-4548	305	18	yiziy	yiziy	NOUN
ejpam-4548	305	19	−1	−1	NOUN
ejpam-4548	305	20	i	i	PRON
ejpam-4548	305	21	{	{	PUNCT
ejpam-4548	305	22	η(zi	η(zi	PROPN
ejpam-4548	305	23	)	)	PUNCT
ejpam-4548	305	24	∧	∧	PROPN
ejpam-4548	305	25	{	{	PUNCT
ejpam-4548	305	26	∨	∨	NUM
ejpam-4548	305	27	yi	yi	NOUN
ejpam-4548	305	28	=	=	NOUN
ejpam-4548	305	29	ui	ui	NOUN
ejpam-4548	305	30	jv	jv	NOUN
ejpam-4548	305	31	i	i	PRON
ejpam-4548	305	32	j	j	PROPN
ejpam-4548	305	33	{	{	PUNCT
ejpam-4548	305	34	γ(uij	γ(uij	PROPN
ejpam-4548	305	35	)	)	PUNCT
ejpam-4548	305	36	∧	∧	PROPN
ejpam-4548	305	37	θ(vij	θ(vij	PROPN
ejpam-4548	305	38	)	)	PUNCT
ejpam-4548	305	39	}	}	PUNCT
ejpam-4548	305	40	}	}	PUNCT
ejpam-4548	305	41	}	}	PUNCT
ejpam-4548	305	42	=	=	SYM
ejpam-4548	306	1	∨	∨	NUM
ejpam-4548	306	2	x	x	X
ejpam-4548	306	3	=	=	NOUN
ejpam-4548	306	4	yiziy	yiziy	NOUN
ejpam-4548	306	5	−1	−1	NOUN
ejpam-4548	306	6	i	i	PRON
ejpam-4548	306	7	{	{	PUNCT
ejpam-4548	306	8	∨	∨	NUM
ejpam-4548	306	9	yi	yi	NOUN
ejpam-4548	306	10	=	=	NOUN
ejpam-4548	306	11	ui	ui	NOUN
ejpam-4548	306	12	jv	jv	NOUN
ejpam-4548	306	13	i	i	PRON
ejpam-4548	306	14	j	j	PROPN
ejpam-4548	306	15	{	{	PUNCT
ejpam-4548	306	16	η(zi	η(zi	PROPN
ejpam-4548	306	17	)	)	PUNCT
ejpam-4548	306	18	∧	∧	PROPN
ejpam-4548	306	19	{	{	PUNCT
ejpam-4548	306	20	γ(uij	γ(uij	PROPN
ejpam-4548	306	21	)	)	PUNCT
ejpam-4548	306	22	∧	∧	PROPN
ejpam-4548	306	23	θ(vij	θ(vij	PROPN
ejpam-4548	306	24	)	)	PUNCT
ejpam-4548	306	25	}	}	PUNCT
ejpam-4548	306	26	}	}	PUNCT
ejpam-4548	306	27	}	}	PUNCT
ejpam-4548	306	28	(	(	PUNCT
ejpam-4548	306	29	as	as	SCONJ
ejpam-4548	306	30	l	l	NOUN
ejpam-4548	306	31	is	be	AUX
ejpam-4548	306	32	a	a	DET
ejpam-4548	306	33	completely	completely	ADV
ejpam-4548	306	34	distributive	distributive	ADJ
ejpam-4548	306	35	lattice	lattice	NOUN
ejpam-4548	306	36	)	)	PUNCT
ejpam-4548	306	37	=	=	PUNCT
ejpam-4548	307	1	∨	∨	NUM
ejpam-4548	307	2	x	x	X
ejpam-4548	307	3	=	=	NOUN
ejpam-4548	307	4	yiziy	yiziy	NOUN
ejpam-4548	307	5	−1	−1	NOUN
ejpam-4548	307	6	i	i	PRON
ejpam-4548	307	7	yi	yi	PUNCT
ejpam-4548	308	1	=	=	PROPN
ejpam-4548	308	2	ui	ui	PROPN
ejpam-4548	308	3	j	j	PROPN
ejpam-4548	308	4	vi	vi	PROPN
ejpam-4548	308	5	j	j	PROPN
ejpam-4548	308	6	{	{	PUNCT
ejpam-4548	308	7	η(zi	η(zi	PROPN
ejpam-4548	308	8	)	)	PUNCT
ejpam-4548	308	9	∧	∧	PROPN
ejpam-4548	308	10	γ(uij	γ(uij	PROPN
ejpam-4548	308	11	)	)	PUNCT
ejpam-4548	308	12	∧	∧	PROPN
ejpam-4548	308	13	θ(vij	θ(vij	PROPN
ejpam-4548	308	14	)	)	PUNCT
ejpam-4548	308	15	}	}	PUNCT
ejpam-4548	308	16	≤	≤	NOUN
ejpam-4548	308	17	∨	∨	NUM
ejpam-4548	308	18	x	x	X
ejpam-4548	308	19	=	=	NOUN
ejpam-4548	308	20	ui	ui	NOUN
ejpam-4548	308	21	j(v	j(v	NOUN
ejpam-4548	309	1	i	i	PRON
ejpam-4548	309	2	jzi(v	jzi(v	VERB
ejpam-4548	309	3	i	i	PRON
ejpam-4548	309	4	j	j	PROPN
ejpam-4548	309	5	)	)	PUNCT
ejpam-4548	309	6	−1)(ui	−1)(ui	PROPN
ejpam-4548	309	7	j	j	X
ejpam-4548	309	8	)	)	PUNCT
ejpam-4548	309	9	−1	−1	NOUN
ejpam-4548	309	10	{	{	PUNCT
ejpam-4548	309	11	θηθ−1(vijzi(v	θηθ−1(vijzi(v	NOUN
ejpam-4548	309	12	i	i	PROPN
ejpam-4548	309	13	j	j	NOUN
ejpam-4548	309	14	)	)	PUNCT
ejpam-4548	309	15	−1	−1	NOUN
ejpam-4548	309	16	)	)	PUNCT
ejpam-4548	309	17	∧	∧	PROPN
ejpam-4548	309	18	γ(uij	γ(uij	PROPN
ejpam-4548	309	19	)	)	PUNCT
ejpam-4548	309	20	}	}	PUNCT
ejpam-4548	309	21	(	(	PUNCT
ejpam-4548	309	22	by	by	ADP
ejpam-4548	309	23	the	the	DET
ejpam-4548	309	24	definition	definition	NOUN
ejpam-4548	309	25	of	of	ADP
ejpam-4548	309	26	a	a	DET
ejpam-4548	309	27	conjugate	conjugate	ADJ
ejpam-4548	309	28	l	l	NOUN
ejpam-4548	309	29	-	-	NOUN
ejpam-4548	309	30	subset	subset	NOUN
ejpam-4548	309	31	)	)	PUNCT
ejpam-4548	309	32	n.	n.	PROPN
ejpam-4548	309	33	ajmal	ajmal	PROPN
ejpam-4548	309	34	,	,	PUNCT
ejpam-4548	309	35	i.	i.	PROPN
ejpam-4548	309	36	jahan	jahan	PROPN
ejpam-4548	309	37	.	.	PROPN
ejpam-4548	309	38	,	,	PUNCT
ejpam-4548	309	39	b.	b.	PROPN
ejpam-4548	309	40	davvaz	davvaz	PROPN
ejpam-4548	309	41	/	/	SYM
ejpam-4548	309	42	eur	eur	PROPN
ejpam-4548	309	43	.	.	PUNCT
ejpam-4548	310	1	j.	j.	PROPN
ejpam-4548	310	2	pure	pure	PROPN
ejpam-4548	310	3	appl	appl	PROPN
ejpam-4548	310	4	.	.	PROPN
ejpam-4548	310	5	math	math	PROPN
ejpam-4548	310	6	,	,	PUNCT
ejpam-4548	310	7	15	15	NUM
ejpam-4548	310	8	(	(	PUNCT
ejpam-4548	310	9	4	4	NUM
ejpam-4548	310	10	)	)	PUNCT
ejpam-4548	310	11	(	(	PUNCT
ejpam-4548	310	12	2022	2022	NUM
ejpam-4548	310	13	)	)	PUNCT
ejpam-4548	310	14	,	,	PUNCT
ejpam-4548	310	15	2086	2086	NUM
ejpam-4548	310	16	-	-	SYM
ejpam-4548	310	17	2115	2115	NUM
ejpam-4548	310	18	2100	2100	NUM
ejpam-4548	310	19	≤	≤	NOUN
ejpam-4548	310	20	∨	∨	NUM
ejpam-4548	310	21	x	x	X
ejpam-4548	310	22	=	=	NOUN
ejpam-4548	310	23	ui	ui	NOUN
ejpam-4548	310	24	j(v	j(v	NOUN
ejpam-4548	311	1	i	i	PRON
ejpam-4548	311	2	jzi(v	jzi(v	VERB
ejpam-4548	311	3	i	i	PRON
ejpam-4548	311	4	j	j	PROPN
ejpam-4548	311	5	)	)	PUNCT
ejpam-4548	311	6	−1)(ui	−1)(ui	PROPN
ejpam-4548	311	7	j	j	X
ejpam-4548	311	8	)	)	PUNCT
ejpam-4548	311	9	−1	−1	NOUN
ejpam-4548	311	10	{	{	PUNCT
ejpam-4548	311	11	ηθ(vijzi(vij)−1	ηθ(vijzi(vij)−1	PROPN
ejpam-4548	311	12	)	)	PUNCT
ejpam-4548	311	13	∧	∧	PROPN
ejpam-4548	311	14	γ(uij	γ(uij	PROPN
ejpam-4548	311	15	)	)	PUNCT
ejpam-4548	311	16	}	}	PUNCT
ejpam-4548	311	17	≤	≤	NUM
ejpam-4548	311	18	γηθγ−1(x	γηθγ−1(x	NOUN
ejpam-4548	311	19	)	)	PUNCT
ejpam-4548	311	20	.	.	PUNCT
ejpam-4548	312	1	this	this	PRON
ejpam-4548	312	2	establishes	establish	VERB
ejpam-4548	312	3	the	the	DET
ejpam-4548	312	4	claim	claim	NOUN
ejpam-4548	312	5	and	and	CCONJ
ejpam-4548	312	6	hence	hence	ADV
ejpam-4548	312	7	ηγ	ηγ	ADP
ejpam-4548	312	8	◦	◦	NOUN
ejpam-4548	312	9	θ	θ	NOUN
ejpam-4548	312	10	=	=	SYM
ejpam-4548	312	11	⟨ληλ−1⟩	⟨ληλ−1⟩	PROPN
ejpam-4548	312	12	⊆	⊆	NUM
ejpam-4548	312	13	⟨γηθγ−1⟩	⟨γηθγ−1⟩	X
ejpam-4548	312	14	=	=	SYM
ejpam-4548	312	15	(	(	PUNCT
ejpam-4548	312	16	ηθ)γ	ηθ)γ	PROPN
ejpam-4548	312	17	.	.	PUNCT
ejpam-4548	313	1	this	this	PRON
ejpam-4548	313	2	proves	prove	VERB
ejpam-4548	313	3	the	the	DET
ejpam-4548	313	4	result	result	NOUN
ejpam-4548	313	5	.	.	PUNCT
ejpam-4548	314	1	theorem	theorem	ADJ
ejpam-4548	314	2	11	11	NUM
ejpam-4548	314	3	.	.	PUNCT
ejpam-4548	315	1	let	let	VERB
ejpam-4548	315	2	η	η	PROPN
ejpam-4548	315	3	,	,	PUNCT
ejpam-4548	315	4	θ	θ	PROPN
ejpam-4548	315	5	∈	∈	PROPN
ejpam-4548	315	6	l(µ	l(µ	PROPN
ejpam-4548	315	7	)	)	PUNCT
ejpam-4548	315	8	and	and	CCONJ
ejpam-4548	315	9	ηθ	ηθ	NOUN
ejpam-4548	315	10	=	=	PROPN
ejpam-4548	315	11	η	η	PROPN
ejpam-4548	315	12	.	.	PROPN
ejpam-4548	315	13	then	then	ADV
ejpam-4548	315	14	,	,	PUNCT
ejpam-4548	315	15	η	η	PROPN
ejpam-4548	315	16	◦	◦	NOUN
ejpam-4548	315	17	θ	θ	X
ejpam-4548	315	18	∈	∈	PROPN
ejpam-4548	315	19	l(µ	l(µ	PROPN
ejpam-4548	315	20	)	)	PUNCT
ejpam-4548	315	21	.	.	PUNCT
ejpam-4548	316	1	proof	proof	NOUN
ejpam-4548	316	2	.	.	PUNCT
ejpam-4548	317	1	firstly	firstly	ADV
ejpam-4548	317	2	we	we	PRON
ejpam-4548	317	3	establish	establish	VERB
ejpam-4548	317	4	that	that	SCONJ
ejpam-4548	317	5	ηθ	ηθ	ADP
ejpam-4548	317	6	◦	◦	NOUN
ejpam-4548	317	7	θ	θ	NOUN
ejpam-4548	317	8	=	=	SYM
ejpam-4548	317	9	θ	θ	X
ejpam-4548	317	10	◦	◦	NOUN
ejpam-4548	317	11	ηθ	ηθ	NOUN
ejpam-4548	317	12	.	.	PROPN
ejpam-4548	317	13	note	note	VERB
ejpam-4548	317	14	that	that	SCONJ
ejpam-4548	317	15	,	,	PUNCT
ejpam-4548	317	16	by	by	ADP
ejpam-4548	317	17	lemma	lemma	PROPN
ejpam-4548	317	18	1	1	NUM
ejpam-4548	317	19	,	,	PUNCT
ejpam-4548	317	20	we	we	PRON
ejpam-4548	317	21	have	have	VERB
ejpam-4548	317	22	∨	∨	NUM
ejpam-4548	317	23	x∈g	x∈g	PROPN
ejpam-4548	317	24	θηθ−1(x	θηθ−1(x	NOUN
ejpam-4548	317	25	)	)	PUNCT
ejpam-4548	317	26	=	=	SYM
ejpam-4548	317	27	η(e)∧θ(e	η(e)∧θ(e	NUM
ejpam-4548	317	28	)	)	PUNCT
ejpam-4548	317	29	.	.	PUNCT
ejpam-4548	318	1	let	let	VERB
ejpam-4548	318	2	x	x	SYM
ejpam-4548	318	3	∈	∈	PROPN
ejpam-4548	318	4	g	g	NOUN
ejpam-4548	318	5	and	and	CCONJ
ejpam-4548	318	6	a0	a0	PROPN
ejpam-4548	318	7	=	=	SYM
ejpam-4548	318	8	η(e)∧θ(e	η(e)∧θ(e	NUM
ejpam-4548	318	9	)	)	PUNCT
ejpam-4548	318	10	.	.	PUNCT
ejpam-4548	319	1	then	then	ADV
ejpam-4548	319	2	,	,	PUNCT
ejpam-4548	319	3	ηθ	ηθ	ADP
ejpam-4548	319	4	◦	◦	NOUN
ejpam-4548	319	5	θ(x	θ(x	PROPN
ejpam-4548	319	6	)	)	PUNCT
ejpam-4548	319	7	=	=	PUNCT
ejpam-4548	320	1	∨	∨	NUM
ejpam-4548	320	2	x	x	X
ejpam-4548	320	3	=	=	NOUN
ejpam-4548	320	4	uivi	uivi	NOUN
ejpam-4548	320	5	{	{	PUNCT
ejpam-4548	320	6	ηθ(ui	ηθ(ui	NOUN
ejpam-4548	320	7	)	)	PUNCT
ejpam-4548	320	8	∧	∧	PROPN
ejpam-4548	320	9	θ(vi	θ(vi	NUM
ejpam-4548	320	10	)	)	PUNCT
ejpam-4548	320	11	}	}	PUNCT
ejpam-4548	320	12	=	=	PUNCT
ejpam-4548	321	1	∨	∨	NUM
ejpam-4548	321	2	x	x	X
ejpam-4548	321	3	=	=	NOUN
ejpam-4548	321	4	uivi	uivi	NOUN
ejpam-4548	321	5	{	{	PUNCT
ejpam-4548	321	6	{	{	PUNCT
ejpam-4548	321	7	∨	∨	PROPN
ejpam-4548	321	8	ai	ai	PROPN
ejpam-4548	321	9	j	j	PROPN
ejpam-4548	321	10	≤a0	≤a0	X
ejpam-4548	321	11	{	{	PUNCT
ejpam-4548	321	12	aij	aij	PROPN
ejpam-4548	321	13	:	:	PUNCT
ejpam-4548	321	14	ui	ui	PROPN
ejpam-4548	321	15	∈	∈	PROPN
ejpam-4548	321	16	⟨	⟨	VERB
ejpam-4548	321	17	(	(	PUNCT
ejpam-4548	321	18	θηθ−1	θηθ−1	PROPN
ejpam-4548	321	19	)	)	PUNCT
ejpam-4548	321	20	aij	aij	PROPN
ejpam-4548	321	21	⟩	⟩	PROPN
ejpam-4548	321	22	}	}	PUNCT
ejpam-4548	321	23	}	}	PUNCT
ejpam-4548	321	24	∧	∧	PROPN
ejpam-4548	321	25	θ(vi	θ(vi	NUM
ejpam-4548	321	26	)	)	PUNCT
ejpam-4548	321	27	}	}	PUNCT
ejpam-4548	321	28	=	=	PUNCT
ejpam-4548	322	1	∨	∨	NUM
ejpam-4548	322	2	x	x	X
ejpam-4548	322	3	=	=	NOUN
ejpam-4548	322	4	uivi	uivi	NOUN
ejpam-4548	322	5	{	{	PUNCT
ejpam-4548	322	6	{	{	PUNCT
ejpam-4548	322	7	∨	∨	NUM
ejpam-4548	322	8	aij≤a0	aij≤a0	ADJ
ejpam-4548	322	9	{	{	PUNCT
ejpam-4548	322	10	aij	aij	PROPN
ejpam-4548	322	11	∧	∧	PROPN
ejpam-4548	322	12	θ(vi	θ(vi	NUM
ejpam-4548	322	13	)	)	PUNCT
ejpam-4548	322	14	:	:	PUNCT
ejpam-4548	322	15	ui	ui	PROPN
ejpam-4548	322	16	∈	∈	PROPN
ejpam-4548	322	17	⟨	⟨	VERB
ejpam-4548	322	18	(	(	PUNCT
ejpam-4548	322	19	θηθ−1	θηθ−1	PROPN
ejpam-4548	322	20	)	)	PUNCT
ejpam-4548	322	21	aij	aij	PROPN
ejpam-4548	322	22	⟩	⟩	PROPN
ejpam-4548	322	23	}	}	PUNCT
ejpam-4548	322	24	}	}	PUNCT
ejpam-4548	322	25	∧	∧	PROPN
ejpam-4548	322	26	θ(vi	θ(vi	NOUN
ejpam-4548	322	27	)	)	PUNCT
ejpam-4548	322	28	}	}	PUNCT
ejpam-4548	322	29	.	.	PUNCT
ejpam-4548	323	1	(	(	PUNCT
ejpam-4548	323	2	as	as	SCONJ
ejpam-4548	323	3	l	l	NOUN
ejpam-4548	323	4	is	be	AUX
ejpam-4548	323	5	a	a	DET
ejpam-4548	323	6	completely	completely	ADV
ejpam-4548	323	7	distributive	distributive	ADJ
ejpam-4548	323	8	lattice	lattice	NOUN
ejpam-4548	323	9	)	)	PUNCT
ejpam-4548	323	10	let	let	VERB
ejpam-4548	323	11	aij	aij	PROPN
ejpam-4548	323	12	≤	≤	PROPN
ejpam-4548	323	13	a0	a0	NOUN
ejpam-4548	323	14	and	and	CCONJ
ejpam-4548	323	15	ui	ui	PROPN
ejpam-4548	323	16	∈	∈	PROPN
ejpam-4548	323	17	⟨(θηθ−1)aij	⟨(θηθ−1)aij	PROPN
ejpam-4548	323	18	⟩.	⟩.	PROPN
ejpam-4548	323	19	it	it	PRON
ejpam-4548	323	20	can	can	AUX
ejpam-4548	323	21	be	be	AUX
ejpam-4548	323	22	verified	verify	VERB
ejpam-4548	323	23	,	,	PUNCT
ejpam-4548	323	24	as	as	ADP
ejpam-4548	323	25	in	in	ADP
ejpam-4548	323	26	theorem	theorem	NOUN
ejpam-4548	323	27	9	9	NUM
ejpam-4548	323	28	,	,	PUNCT
ejpam-4548	324	1	that	that	SCONJ
ejpam-4548	324	2	v−1	v−1	PROPN
ejpam-4548	324	3	i	i	PRON
ejpam-4548	324	4	uivi	uivi	VERB
ejpam-4548	324	5	∈	∈	PROPN
ejpam-4548	324	6	⟨(θηθ−1)aij∧θ(vi	⟨(θηθ−1)aij∧θ(vi	X
ejpam-4548	324	7	)	)	PUNCT
ejpam-4548	324	8	⟩.	⟩.	PROPN
ejpam-4548	324	9	thus	thus	ADV
ejpam-4548	324	10	,	,	PUNCT
ejpam-4548	324	11	ηθ	ηθ	ADP
ejpam-4548	324	12	◦	◦	NOUN
ejpam-4548	324	13	θ(x	θ(x	PROPN
ejpam-4548	324	14	)	)	PUNCT
ejpam-4548	324	15	=	=	PUNCT
ejpam-4548	325	1	∨	∨	NUM
ejpam-4548	325	2	x	x	X
ejpam-4548	325	3	=	=	NOUN
ejpam-4548	325	4	uivi	uivi	NOUN
ejpam-4548	325	5	{	{	PUNCT
ejpam-4548	325	6	{	{	PUNCT
ejpam-4548	325	7	∨	∨	NUM
ejpam-4548	325	8	aij≤a0	aij≤a0	ADJ
ejpam-4548	325	9	{	{	PUNCT
ejpam-4548	325	10	aij	aij	PROPN
ejpam-4548	325	11	∧	∧	PROPN
ejpam-4548	325	12	θ(vi	θ(vi	NUM
ejpam-4548	325	13	)	)	PUNCT
ejpam-4548	325	14	:	:	PUNCT
ejpam-4548	325	15	ui	ui	PROPN
ejpam-4548	325	16	∈	∈	PROPN
ejpam-4548	325	17	⟨	⟨	VERB
ejpam-4548	325	18	(	(	PUNCT
ejpam-4548	325	19	θηθ−1	θηθ−1	PROPN
ejpam-4548	325	20	)	)	PUNCT
ejpam-4548	325	21	aij	aij	PROPN
ejpam-4548	325	22	⟩	⟩	PROPN
ejpam-4548	325	23	}	}	PUNCT
ejpam-4548	325	24	}	}	PUNCT
ejpam-4548	325	25	∧	∧	PROPN
ejpam-4548	325	26	θ(vi	θ(vi	NOUN
ejpam-4548	325	27	)	)	PUNCT
ejpam-4548	325	28	}	}	PUNCT
ejpam-4548	325	29	≤	≤	NOUN
ejpam-4548	325	30	∨	∨	NUM
ejpam-4548	325	31	x	x	X
ejpam-4548	325	32	=	=	NOUN
ejpam-4548	325	33	uivi	uivi	NOUN
ejpam-4548	325	34	{	{	PUNCT
ejpam-4548	325	35	{	{	PUNCT
ejpam-4548	325	36	∨	∨	NUM
ejpam-4548	325	37	c≤a0	c≤a0	PROPN
ejpam-4548	325	38	{	{	PUNCT
ejpam-4548	325	39	c	c	NOUN
ejpam-4548	325	40	:	:	PUNCT
ejpam-4548	326	1	v−1	v−1	NOUN
ejpam-4548	326	2	i	i	PRON
ejpam-4548	326	3	uivi	uivi	NOUN
ejpam-4548	326	4	∈	∈	NOUN
ejpam-4548	326	5	⟨	⟨	VERB
ejpam-4548	326	6	(	(	PUNCT
ejpam-4548	326	7	θηθ−1	θηθ−1	PROPN
ejpam-4548	326	8	)	)	PUNCT
ejpam-4548	326	9	c	c	PROPN
ejpam-4548	326	10	⟩	⟩	NOUN
ejpam-4548	326	11	}	}	PUNCT
ejpam-4548	326	12	∧	∧	PROPN
ejpam-4548	326	13	θ(vi	θ(vi	NUM
ejpam-4548	326	14	)	)	PUNCT
ejpam-4548	326	15	}	}	PUNCT
ejpam-4548	326	16	=	=	PUNCT
ejpam-4548	326	17	∨	∨	NOUN
ejpam-4548	326	18	x	x	X
ejpam-4548	326	19	=	=	NOUN
ejpam-4548	326	20	vi(v	vi(v	ADJ
ejpam-4548	326	21	−1	−1	NOUN
ejpam-4548	326	22	i	i	PRON
ejpam-4548	326	23	uivi	uivi	VERB
ejpam-4548	326	24	)	)	PUNCT
ejpam-4548	326	25	{	{	PUNCT
ejpam-4548	326	26	ηθ(v−1	ηθ(v−1	PROPN
ejpam-4548	326	27	i	i	PRON
ejpam-4548	326	28	uivi	uivi	VERB
ejpam-4548	326	29	)	)	PUNCT
ejpam-4548	326	30	∧	∧	PROPN
ejpam-4548	326	31	θ(vi	θ(vi	NUM
ejpam-4548	326	32	)	)	PUNCT
ejpam-4548	326	33	}	}	PUNCT
ejpam-4548	326	34	(	(	PUNCT
ejpam-4548	326	35	by	by	ADP
ejpam-4548	326	36	theorem	theorem	NOUN
ejpam-4548	326	37	6	6	NUM
ejpam-4548	326	38	)	)	PUNCT
ejpam-4548	326	39	≤	≤	NUM
ejpam-4548	326	40	θ	θ	PROPN
ejpam-4548	326	41	◦	◦	NOUN
ejpam-4548	326	42	ηθ(x	ηθ(x	NUM
ejpam-4548	326	43	)	)	PUNCT
ejpam-4548	326	44	.	.	PUNCT
ejpam-4548	327	1	n.	n.	PROPN
ejpam-4548	327	2	ajmal	ajmal	PROPN
ejpam-4548	327	3	,	,	PUNCT
ejpam-4548	327	4	i.	i.	PROPN
ejpam-4548	327	5	jahan	jahan	PROPN
ejpam-4548	327	6	.	.	PROPN
ejpam-4548	327	7	,	,	PUNCT
ejpam-4548	327	8	b.	b.	PROPN
ejpam-4548	327	9	davvaz	davvaz	PROPN
ejpam-4548	327	10	/	/	SYM
ejpam-4548	327	11	eur	eur	PROPN
ejpam-4548	327	12	.	.	PUNCT
ejpam-4548	328	1	j.	j.	PROPN
ejpam-4548	328	2	pure	pure	PROPN
ejpam-4548	328	3	appl	appl	PROPN
ejpam-4548	328	4	.	.	PROPN
ejpam-4548	328	5	math	math	PROPN
ejpam-4548	328	6	,	,	PUNCT
ejpam-4548	328	7	15	15	NUM
ejpam-4548	328	8	(	(	PUNCT
ejpam-4548	328	9	4	4	NUM
ejpam-4548	328	10	)	)	PUNCT
ejpam-4548	328	11	(	(	PUNCT
ejpam-4548	328	12	2022	2022	NUM
ejpam-4548	328	13	)	)	PUNCT
ejpam-4548	328	14	,	,	PUNCT
ejpam-4548	328	15	2086	2086	NUM
ejpam-4548	328	16	-	-	SYM
ejpam-4548	328	17	2115	2115	NUM
ejpam-4548	328	18	2101	2101	NUM
ejpam-4548	328	19	similarly	similarly	ADV
ejpam-4548	328	20	,	,	PUNCT
ejpam-4548	328	21	we	we	PRON
ejpam-4548	328	22	can	can	AUX
ejpam-4548	328	23	prove	prove	VERB
ejpam-4548	328	24	that	that	SCONJ
ejpam-4548	328	25	θ	θ	PROPN
ejpam-4548	328	26	◦	◦	NOUN
ejpam-4548	328	27	ηθ	ηθ	ADP
ejpam-4548	328	28	⊆	⊆	NUM
ejpam-4548	328	29	ηθ	ηθ	ADP
ejpam-4548	328	30	◦	◦	PROPN
ejpam-4548	328	31	θ	θ	PROPN
ejpam-4548	328	32	.	.	PUNCT
ejpam-4548	329	1	this	this	PRON
ejpam-4548	329	2	implies	imply	VERB
ejpam-4548	329	3	θ	θ	PROPN
ejpam-4548	329	4	◦	◦	NOUN
ejpam-4548	329	5	ηθ	ηθ	ADP
ejpam-4548	329	6	=	=	NOUN
ejpam-4548	329	7	ηθ	ηθ	PROPN
ejpam-4548	329	8	◦	◦	PROPN
ejpam-4548	329	9	θ	θ	PROPN
ejpam-4548	329	10	.	.	PUNCT
ejpam-4548	330	1	as	as	ADP
ejpam-4548	330	2	ηθ	ηθ	PROPN
ejpam-4548	330	3	=	=	SYM
ejpam-4548	330	4	η	η	PROPN
ejpam-4548	330	5	,	,	PUNCT
ejpam-4548	330	6	we	we	PRON
ejpam-4548	330	7	have	have	VERB
ejpam-4548	330	8	η	η	NOUN
ejpam-4548	330	9	◦	◦	NOUN
ejpam-4548	330	10	θ	θ	NOUN
ejpam-4548	330	11	=	=	SYM
ejpam-4548	330	12	θ	θ	PROPN
ejpam-4548	330	13	◦	◦	PROPN
ejpam-4548	330	14	η	η	PROPN
ejpam-4548	330	15	.	.	PROPN
ejpam-4548	330	16	therefore	therefore	ADV
ejpam-4548	330	17	,	,	PUNCT
ejpam-4548	330	18	by	by	ADP
ejpam-4548	330	19	proposition	proposition	NOUN
ejpam-4548	330	20	2	2	NUM
ejpam-4548	330	21	,	,	PUNCT
ejpam-4548	330	22	η	η	PROPN
ejpam-4548	330	23	◦	◦	NOUN
ejpam-4548	330	24	θ	θ	X
ejpam-4548	330	25	∈	∈	PROPN
ejpam-4548	330	26	l(µ	l(µ	PROPN
ejpam-4548	330	27	)	)	PUNCT
ejpam-4548	330	28	.	.	PUNCT
ejpam-4548	331	1	in	in	ADP
ejpam-4548	331	2	order	order	NOUN
ejpam-4548	331	3	to	to	PART
ejpam-4548	331	4	introduce	introduce	VERB
ejpam-4548	331	5	the	the	DET
ejpam-4548	331	6	notion	notion	NOUN
ejpam-4548	331	7	of	of	ADP
ejpam-4548	331	8	subnormality	subnormality	NOUN
ejpam-4548	331	9	of	of	ADP
ejpam-4548	331	10	an	an	DET
ejpam-4548	331	11	l	l	NOUN
ejpam-4548	331	12	-	-	NOUN
ejpam-4548	331	13	subgroup	subgroup	NOUN
ejpam-4548	331	14	of	of	ADP
ejpam-4548	331	15	an	an	DET
ejpam-4548	331	16	l	l	NOUN
ejpam-4548	331	17	-	-	NOUN
ejpam-4548	331	18	group	group	NOUN
ejpam-4548	331	19	,	,	PUNCT
ejpam-4548	331	20	firstly	firstly	ADV
ejpam-4548	331	21	we	we	PRON
ejpam-4548	331	22	define	define	VERB
ejpam-4548	331	23	a	a	DET
ejpam-4548	331	24	descending	descend	VERB
ejpam-4548	331	25	series	series	NOUN
ejpam-4548	331	26	.	.	PUNCT
ejpam-4548	332	1	for	for	ADP
ejpam-4548	332	2	η	η	PROPN
ejpam-4548	332	3	∈	∈	PROPN
ejpam-4548	332	4	l(µ	l(µ	PROPN
ejpam-4548	332	5	)	)	PUNCT
ejpam-4548	332	6	,	,	PUNCT
ejpam-4548	332	7	define	define	VERB
ejpam-4548	332	8	a	a	DET
ejpam-4548	332	9	series	series	NOUN
ejpam-4548	332	10	of	of	ADP
ejpam-4548	332	11	l	l	NOUN
ejpam-4548	332	12	-	-	NOUN
ejpam-4548	332	13	subgroups	subgroup	NOUN
ejpam-4548	332	14	of	of	ADP
ejpam-4548	332	15	µ	µ	PRON
ejpam-4548	332	16	inductively	inductively	ADV
ejpam-4548	332	17	as	as	SCONJ
ejpam-4548	332	18	follows	follow	VERB
ejpam-4548	332	19	:	:	PUNCT
ejpam-4548	332	20	η0	η0	PROPN
ejpam-4548	332	21	=	=	SYM
ejpam-4548	332	22	µ	µ	NUM
ejpam-4548	332	23	,	,	PUNCT
ejpam-4548	332	24	η1	η1	NOUN
ejpam-4548	332	25	=	=	SYM
ejpam-4548	332	26	ηµ	ηµ	ADJ
ejpam-4548	332	27	,	,	PUNCT
ejpam-4548	332	28	η2	η2	ADJ
ejpam-4548	332	29	=	=	PUNCT
ejpam-4548	332	30	ηη1	ηη1	NOUN
ejpam-4548	332	31	,	,	PUNCT
ejpam-4548	332	32	.	.	PUNCT
ejpam-4548	332	33	.	.	PUNCT
ejpam-4548	333	1	.	.	PUNCT
ejpam-4548	334	1	,	,	PUNCT
ejpam-4548	334	2	ηi	ηi	X
ejpam-4548	334	3	=	=	PUNCT
ejpam-4548	334	4	ηηi−1	ηηi−1	PROPN
ejpam-4548	334	5	.	.	PUNCT
ejpam-4548	334	6	.	.	PUNCT
ejpam-4548	334	7	.	.	PUNCT
ejpam-4548	335	1	by	by	ADP
ejpam-4548	335	2	theorem	theorem	ADJ
ejpam-4548	335	3	8	8	NUM
ejpam-4548	335	4	,	,	PUNCT
ejpam-4548	335	5	η1	η1	NOUN
ejpam-4548	335	6	is	be	AUX
ejpam-4548	335	7	the	the	DET
ejpam-4548	335	8	smallest	small	ADJ
ejpam-4548	335	9	normal	normal	ADJ
ejpam-4548	335	10	l	l	NOUN
ejpam-4548	335	11	-	-	NOUN
ejpam-4548	335	12	subgroup	subgroup	NOUN
ejpam-4548	335	13	of	of	ADP
ejpam-4548	335	14	µ	µ	X
ejpam-4548	335	15	containing	contain	VERB
ejpam-4548	335	16	η	η	PROPN
ejpam-4548	335	17	and	and	CCONJ
ejpam-4548	335	18	η2	η2	PROPN
ejpam-4548	335	19	is	be	AUX
ejpam-4548	335	20	the	the	DET
ejpam-4548	335	21	smallest	small	ADJ
ejpam-4548	335	22	normal	normal	ADJ
ejpam-4548	335	23	l	l	NOUN
ejpam-4548	335	24	-	-	NOUN
ejpam-4548	335	25	subgroup	subgroup	NOUN
ejpam-4548	335	26	of	of	ADP
ejpam-4548	335	27	η1	η1	NOUN
ejpam-4548	335	28	containing	contain	VERB
ejpam-4548	335	29	η	η	PROPN
ejpam-4548	335	30	and	and	CCONJ
ejpam-4548	335	31	so	so	ADV
ejpam-4548	335	32	on	on	ADV
ejpam-4548	335	33	.	.	PUNCT
ejpam-4548	336	1	thus	thus	ADV
ejpam-4548	336	2	,	,	PUNCT
ejpam-4548	336	3	we	we	PRON
ejpam-4548	336	4	have	have	VERB
ejpam-4548	336	5	η	η	PROPN
ejpam-4548	336	6	⊆	⊆	NUM
ejpam-4548	336	7	·	·	PUNCT
ejpam-4548	336	8	·	·	PUNCT
ejpam-4548	337	1	·	·	PUNCT
ejpam-4548	337	2	◁	◁	X
ejpam-4548	337	3	ηi+1	ηi+1	NUM
ejpam-4548	337	4	◁	◁	PUNCT
ejpam-4548	337	5	ηi	ηi	INTJ
ejpam-4548	337	6	◁	◁	X
ejpam-4548	337	7	·	·	PUNCT
ejpam-4548	337	8	·	·	PUNCT
ejpam-4548	337	9	·	·	PUNCT
ejpam-4548	337	10	◁	◁	X
ejpam-4548	337	11	η1	η1	NOUN
ejpam-4548	337	12	◁	◁	NOUN
ejpam-4548	337	13	η0	η0	NOUN
ejpam-4548	337	14	=	=	SYM
ejpam-4548	337	15	µ	µ	X
ejpam-4548	337	16	.	.	PUNCT
ejpam-4548	338	1	this	this	DET
ejpam-4548	338	2	inductively	inductively	ADV
ejpam-4548	338	3	defined	define	VERB
ejpam-4548	338	4	series	series	NOUN
ejpam-4548	338	5	is	be	AUX
ejpam-4548	338	6	known	know	VERB
ejpam-4548	338	7	as	as	ADP
ejpam-4548	338	8	the	the	DET
ejpam-4548	338	9	normal	normal	ADJ
ejpam-4548	338	10	closure	closure	NOUN
ejpam-4548	338	11	series	series	NOUN
ejpam-4548	338	12	of	of	ADP
ejpam-4548	338	13	η	η	PROPN
ejpam-4548	338	14	in	in	ADP
ejpam-4548	338	15	µ	µ	PROPN
ejpam-4548	338	16	and	and	CCONJ
ejpam-4548	338	17	we	we	PRON
ejpam-4548	338	18	call	call	VERB
ejpam-4548	338	19	ηi	ηi	ADP
ejpam-4548	338	20	the	the	DET
ejpam-4548	338	21	ith	ith	PROPN
ejpam-4548	338	22	normal	normal	ADJ
ejpam-4548	338	23	closure	closure	NOUN
ejpam-4548	338	24	of	of	ADP
ejpam-4548	338	25	η	η	PROPN
ejpam-4548	338	26	in	in	ADP
ejpam-4548	338	27	µ.	µ.	NOUN
ejpam-4548	338	28	in	in	ADP
ejpam-4548	338	29	view	view	NOUN
ejpam-4548	338	30	of	of	ADP
ejpam-4548	338	31	proposition	proposition	NOUN
ejpam-4548	338	32	4	4	NUM
ejpam-4548	338	33	and	and	CCONJ
ejpam-4548	338	34	using	use	VERB
ejpam-4548	338	35	the	the	DET
ejpam-4548	338	36	principle	principle	NOUN
ejpam-4548	338	37	of	of	ADP
ejpam-4548	338	38	mathematical	mathematical	ADJ
ejpam-4548	338	39	induction	induction	NOUN
ejpam-4548	338	40	,	,	PUNCT
ejpam-4548	338	41	we	we	PRON
ejpam-4548	338	42	have	have	VERB
ejpam-4548	338	43	:	:	PUNCT
ejpam-4548	338	44	proposition	proposition	NOUN
ejpam-4548	338	45	5	5	NUM
ejpam-4548	338	46	.	.	PUNCT
ejpam-4548	339	1	let	let	VERB
ejpam-4548	339	2	η	η	PROPN
ejpam-4548	339	3	∈	∈	PROPN
ejpam-4548	339	4	l(µ	l(µ	PROPN
ejpam-4548	339	5	)	)	PUNCT
ejpam-4548	339	6	.	.	PUNCT
ejpam-4548	340	1	then	then	ADV
ejpam-4548	340	2	,	,	PUNCT
ejpam-4548	340	3	ηi(e	ηi(e	NOUN
ejpam-4548	340	4	)	)	PUNCT
ejpam-4548	340	5	=	=	SYM
ejpam-4548	341	1	η(e	η(e	PROPN
ejpam-4548	341	2	)	)	PUNCT
ejpam-4548	341	3	for	for	ADP
ejpam-4548	341	4	each	each	DET
ejpam-4548	341	5	i.	i.	NOUN
ejpam-4548	341	6	theorem	theorem	PROPN
ejpam-4548	341	7	12	12	NUM
ejpam-4548	341	8	.	.	PUNCT
ejpam-4548	342	1	let	let	VERB
ejpam-4548	342	2	η	η	PROPN
ejpam-4548	342	3	,	,	PUNCT
ejpam-4548	342	4	θ	θ	PROPN
ejpam-4548	342	5	∈	∈	PROPN
ejpam-4548	342	6	l(µ	l(µ	PROPN
ejpam-4548	342	7	)	)	PUNCT
ejpam-4548	342	8	such	such	ADJ
ejpam-4548	342	9	that	that	SCONJ
ejpam-4548	342	10	η(e	η(e	PROPN
ejpam-4548	342	11	)	)	PUNCT
ejpam-4548	342	12	=	=	SYM
ejpam-4548	342	13	θ(e	θ(e	NUM
ejpam-4548	342	14	)	)	PUNCT
ejpam-4548	342	15	.	.	PUNCT
ejpam-4548	343	1	let	let	VERB
ejpam-4548	343	2	ηi	ηi	PRON
ejpam-4548	343	3	be	be	AUX
ejpam-4548	343	4	the	the	DET
ejpam-4548	343	5	ith	ith	PROPN
ejpam-4548	343	6	normal	normal	ADJ
ejpam-4548	343	7	closure	closure	NOUN
ejpam-4548	343	8	of	of	ADP
ejpam-4548	343	9	η	η	PROPN
ejpam-4548	343	10	in	in	ADP
ejpam-4548	343	11	µ	µ	NOUN
ejpam-4548	343	12	and	and	CCONJ
ejpam-4548	343	13	ηθi	ηθi	NOUN
ejpam-4548	343	14	=	=	SYM
ejpam-4548	343	15	ηi	ηi	PROPN
ejpam-4548	343	16	.	.	PUNCT
ejpam-4548	344	1	then	then	ADV
ejpam-4548	344	2	,	,	PUNCT
ejpam-4548	344	3	ηi+1	ηi+1	PROPN
ejpam-4548	344	4	∈	∈	PROPN
ejpam-4548	344	5	nl(ηi	nl(ηi	X
ejpam-4548	344	6	◦	◦	NOUN
ejpam-4548	344	7	θ	θ	NOUN
ejpam-4548	344	8	)	)	PUNCT
ejpam-4548	344	9	.	.	PUNCT
ejpam-4548	345	1	proof	proof	NOUN
ejpam-4548	345	2	.	.	PUNCT
ejpam-4548	346	1	in	in	ADP
ejpam-4548	346	2	order	order	NOUN
ejpam-4548	346	3	to	to	PART
ejpam-4548	346	4	establish	establish	VERB
ejpam-4548	346	5	the	the	DET
ejpam-4548	346	6	result	result	NOUN
ejpam-4548	346	7	,	,	PUNCT
ejpam-4548	346	8	firstly	firstly	ADV
ejpam-4548	346	9	we	we	PRON
ejpam-4548	346	10	show	show	VERB
ejpam-4548	346	11	that	that	SCONJ
ejpam-4548	346	12	ηi+1	ηi+1	PROPN
ejpam-4548	346	13	⊆	⊆	NUM
ejpam-4548	346	14	ηi	ηi	PROPN
ejpam-4548	346	15	◦	◦	PROPN
ejpam-4548	346	16	θ	θ	PROPN
ejpam-4548	346	17	.	.	PUNCT
ejpam-4548	346	18	by	by	ADP
ejpam-4548	346	19	proposition	proposition	NOUN
ejpam-4548	346	20	5	5	NUM
ejpam-4548	346	21	,	,	PUNCT
ejpam-4548	346	22	for	for	ADP
ejpam-4548	346	23	each	each	DET
ejpam-4548	346	24	i	i	PRON
ejpam-4548	346	25	,	,	PUNCT
ejpam-4548	346	26	we	we	PRON
ejpam-4548	346	27	have	have	VERB
ejpam-4548	346	28	ηi(e	ηi(e	NOUN
ejpam-4548	346	29	)	)	PUNCT
ejpam-4548	346	30	=	=	SYM
ejpam-4548	346	31	η(e	η(e	PROPN
ejpam-4548	346	32	)	)	PUNCT
ejpam-4548	346	33	.	.	PUNCT
ejpam-4548	347	1	and	and	CCONJ
ejpam-4548	347	2	since	since	SCONJ
ejpam-4548	347	3	η(e	η(e	PROPN
ejpam-4548	347	4	)	)	PUNCT
ejpam-4548	347	5	=	=	SYM
ejpam-4548	347	6	θ(e	θ(e	NUM
ejpam-4548	347	7	)	)	PUNCT
ejpam-4548	347	8	,	,	PUNCT
ejpam-4548	347	9	we	we	PRON
ejpam-4548	347	10	have	have	VERB
ejpam-4548	347	11	ηi(e	ηi(e	NOUN
ejpam-4548	347	12	)	)	PUNCT
ejpam-4548	348	1	=	=	SYM
ejpam-4548	348	2	θ(e	θ(e	NUM
ejpam-4548	348	3	)	)	PUNCT
ejpam-4548	348	4	for	for	ADP
ejpam-4548	348	5	each	each	DET
ejpam-4548	348	6	i.	i.	NOUN
ejpam-4548	348	7	thus	thus	ADV
ejpam-4548	348	8	in	in	ADP
ejpam-4548	348	9	view	view	NOUN
ejpam-4548	348	10	of	of	ADP
ejpam-4548	348	11	proposition	proposition	NOUN
ejpam-4548	348	12	3	3	NUM
ejpam-4548	348	13	,	,	PUNCT
ejpam-4548	348	14	ηi	ηi	VERB
ejpam-4548	348	15	⊆	⊆	NUM
ejpam-4548	348	16	ηi	ηi	NOUN
ejpam-4548	348	17	◦	◦	NOUN
ejpam-4548	348	18	θ	θ	PROPN
ejpam-4548	348	19	and	and	CCONJ
ejpam-4548	348	20	hence	hence	ADV
ejpam-4548	348	21	by	by	ADP
ejpam-4548	348	22	the	the	DET
ejpam-4548	348	23	definition	definition	NOUN
ejpam-4548	348	24	of	of	ADP
ejpam-4548	348	25	ith	ith	PROPN
ejpam-4548	348	26	normal	normal	ADJ
ejpam-4548	348	27	closure	closure	NOUN
ejpam-4548	348	28	of	of	ADP
ejpam-4548	348	29	η	η	PROPN
ejpam-4548	348	30	in	in	ADP
ejpam-4548	348	31	µ	µ	NUM
ejpam-4548	348	32	,	,	PUNCT
ejpam-4548	348	33	we	we	PRON
ejpam-4548	348	34	obtain	obtain	VERB
ejpam-4548	348	35	ηi+1	ηi+1	NUM
ejpam-4548	348	36	⊆	⊆	NUM
ejpam-4548	348	37	ηi	ηi	NOUN
ejpam-4548	348	38	⊆	⊆	NUM
ejpam-4548	348	39	ηi	ηi	PROPN
ejpam-4548	348	40	◦	◦	PROPN
ejpam-4548	348	41	θ	θ	PROPN
ejpam-4548	348	42	.	.	PUNCT
ejpam-4548	349	1	moreover	moreover	ADV
ejpam-4548	349	2	as	as	ADP
ejpam-4548	349	3	ηθi	ηθi	NOUN
ejpam-4548	349	4	=	=	SYM
ejpam-4548	349	5	ηi	ηi	NOUN
ejpam-4548	349	6	,	,	PUNCT
ejpam-4548	349	7	by	by	ADP
ejpam-4548	349	8	theorem	theorem	NOUN
ejpam-4548	349	9	11	11	NUM
ejpam-4548	349	10	,	,	PUNCT
ejpam-4548	349	11	ηi	ηi	NOUN
ejpam-4548	349	12	◦	◦	NOUN
ejpam-4548	349	13	θ	θ	X
ejpam-4548	349	14	∈	∈	PROPN
ejpam-4548	349	15	l(µ	l(µ	PROPN
ejpam-4548	349	16	)	)	PUNCT
ejpam-4548	349	17	.	.	PUNCT
ejpam-4548	350	1	thus	thus	ADV
ejpam-4548	350	2	,	,	PUNCT
ejpam-4548	350	3	ηi+1	ηi+1	PROPN
ejpam-4548	350	4	∈	∈	PROPN
ejpam-4548	350	5	l(ηi	l(ηi	PUNCT
ejpam-4548	350	6	◦	◦	NOUN
ejpam-4548	350	7	θ	θ	NOUN
ejpam-4548	350	8	)	)	PUNCT
ejpam-4548	350	9	.	.	PUNCT
ejpam-4548	351	1	here	here	ADV
ejpam-4548	351	2	note	note	VERB
ejpam-4548	351	3	that	that	SCONJ
ejpam-4548	351	4	θηi+1θ	θηi+1θ	NOUN
ejpam-4548	351	5	−1(e	−1(e	NOUN
ejpam-4548	351	6	)	)	PUNCT
ejpam-4548	351	7	=	=	PUNCT
ejpam-4548	351	8	ηi+1(e	ηi+1(e	ADJ
ejpam-4548	351	9	)	)	PUNCT
ejpam-4548	351	10	∧	∧	PROPN
ejpam-4548	351	11	θ(e	θ(e	NUM
ejpam-4548	351	12	)	)	PUNCT
ejpam-4548	351	13	(	(	PUNCT
ejpam-4548	351	14	by	by	ADP
ejpam-4548	351	15	lemma	lemma	PROPN
ejpam-4548	351	16	1	1	NUM
ejpam-4548	351	17	)	)	PUNCT
ejpam-4548	351	18	=	=	SYM
ejpam-4548	351	19	η(e	η(e	X
ejpam-4548	351	20	)	)	PUNCT
ejpam-4548	351	21	∧	∧	PROPN
ejpam-4548	351	22	θ(e	θ(e	NUM
ejpam-4548	351	23	)	)	PUNCT
ejpam-4548	351	24	(	(	PUNCT
ejpam-4548	351	25	as	as	ADP
ejpam-4548	351	26	ηi(e	ηi(e	NOUN
ejpam-4548	351	27	)	)	PUNCT
ejpam-4548	351	28	=	=	SYM
ejpam-4548	351	29	η(e	η(e	PROPN
ejpam-4548	351	30	)	)	PUNCT
ejpam-4548	351	31	)	)	PUNCT
ejpam-4548	352	1	=	=	SYM
ejpam-4548	352	2	η(e	η(e	PROPN
ejpam-4548	352	3	)	)	PUNCT
ejpam-4548	352	4	(	(	PUNCT
ejpam-4548	352	5	as	as	ADP
ejpam-4548	352	6	η(e	η(e	NOUN
ejpam-4548	352	7	)	)	PUNCT
ejpam-4548	352	8	=	=	SYM
ejpam-4548	352	9	θ(e	θ(e	NUM
ejpam-4548	352	10	)	)	PUNCT
ejpam-4548	352	11	)	)	PUNCT
ejpam-4548	353	1	=	=	PUNCT
ejpam-4548	353	2	ηi+1(e	ηi+1(e	ADJ
ejpam-4548	353	3	)	)	PUNCT
ejpam-4548	353	4	.	.	PUNCT
ejpam-4548	354	1	so	so	ADV
ejpam-4548	354	2	by	by	ADP
ejpam-4548	354	3	lemma	lemma	PROPN
ejpam-4548	354	4	1	1	NUM
ejpam-4548	354	5	,	,	PUNCT
ejpam-4548	354	6	ηi+1	ηi+1	NUM
ejpam-4548	354	7	⊆	⊆	NUM
ejpam-4548	354	8	θηi+1θ	θηi+1θ	NOUN
ejpam-4548	354	9	−1	−1	NOUN
ejpam-4548	354	10	.	.	PUNCT
ejpam-4548	355	1	now	now	ADV
ejpam-4548	355	2	to	to	PART
ejpam-4548	355	3	prove	prove	VERB
ejpam-4548	355	4	that	that	SCONJ
ejpam-4548	355	5	ηi+1	ηi+1	PROPN
ejpam-4548	355	6	∈	∈	PROPN
ejpam-4548	355	7	nl(ηi	nl(ηi	X
ejpam-4548	355	8	◦	◦	NOUN
ejpam-4548	355	9	θ	θ	PROPN
ejpam-4548	355	10	)	)	PUNCT
ejpam-4548	355	11	,	,	PUNCT
ejpam-4548	355	12	let	let	VERB
ejpam-4548	355	13	x	x	PRON
ejpam-4548	355	14	,	,	PUNCT
ejpam-4548	355	15	g	g	PROPN
ejpam-4548	355	16	∈	∈	PROPN
ejpam-4548	355	17	g	g	PROPN
ejpam-4548	355	18	and	and	CCONJ
ejpam-4548	355	19	consider	consider	VERB
ejpam-4548	355	20	ηi+1(x	ηi+1(x	NOUN
ejpam-4548	355	21	)	)	PUNCT
ejpam-4548	355	22	∧	∧	NOUN
ejpam-4548	355	23	ηi	ηi	NOUN
ejpam-4548	355	24	◦	◦	NOUN
ejpam-4548	355	25	θ(g	θ(g	NUM
ejpam-4548	355	26	)	)	PUNCT
ejpam-4548	355	27	≤	≤	NOUN
ejpam-4548	355	28	θηi+1θ	θηi+1θ	NOUN
ejpam-4548	355	29	−1(x	−1(x	NOUN
ejpam-4548	355	30	)	)	PUNCT
ejpam-4548	355	31	∧	∧	PROPN
ejpam-4548	355	32	ηi	ηi	NOUN
ejpam-4548	355	33	◦	◦	NOUN
ejpam-4548	355	34	θ(g	θ(g	NUM
ejpam-4548	355	35	)	)	PUNCT
ejpam-4548	356	1	=	=	SYM
ejpam-4548	356	2			PROPN
ejpam-4548	356	3	∨	∨	NOUN
ejpam-4548	356	4	x	x	X
ejpam-4548	356	5	=	=	NOUN
ejpam-4548	356	6	yjzjy	yjzjy	NUM
ejpam-4548	356	7	−1	−1	NOUN
ejpam-4548	356	8	j	j	PROPN
ejpam-4548	356	9	{	{	PUNCT
ejpam-4548	356	10	ηi+1(zj	ηi+1(zj	PROPN
ejpam-4548	356	11	)	)	PUNCT
ejpam-4548	356	12	∧	∧	PROPN
ejpam-4548	356	13	θ(yj	θ(yj	NOUN
ejpam-4548	356	14	)	)	PUNCT
ejpam-4548	356	15	}	}	PUNCT
ejpam-4548	356	16			ADP
ejpam-4548	356	17	∧	∧	PROPN
ejpam-4548	356	18	{	{	PUNCT
ejpam-4548	356	19	∨	∨	ADV
ejpam-4548	356	20	g	g	NOUN
ejpam-4548	356	21	=	=	SYM
ejpam-4548	356	22	ukvk	ukvk	X
ejpam-4548	356	23	{	{	PUNCT
ejpam-4548	356	24	ηi(uk	ηi(uk	PROPN
ejpam-4548	356	25	)	)	PUNCT
ejpam-4548	356	26	∧	∧	PROPN
ejpam-4548	356	27	θ(vk	θ(vk	PROPN
ejpam-4548	356	28	)	)	PUNCT
ejpam-4548	356	29	}	}	PUNCT
ejpam-4548	356	30	}	}	PUNCT
ejpam-4548	356	31	n.	n.	PROPN
ejpam-4548	356	32	ajmal	ajmal	PROPN
ejpam-4548	356	33	,	,	PUNCT
ejpam-4548	356	34	i.	i.	PROPN
ejpam-4548	356	35	jahan	jahan	PROPN
ejpam-4548	356	36	.	.	PROPN
ejpam-4548	356	37	,	,	PUNCT
ejpam-4548	356	38	b.	b.	PROPN
ejpam-4548	356	39	davvaz	davvaz	PROPN
ejpam-4548	356	40	/	/	SYM
ejpam-4548	356	41	eur	eur	PROPN
ejpam-4548	356	42	.	.	PUNCT
ejpam-4548	357	1	j.	j.	PROPN
ejpam-4548	357	2	pure	pure	PROPN
ejpam-4548	357	3	appl	appl	PROPN
ejpam-4548	357	4	.	.	PROPN
ejpam-4548	357	5	math	math	PROPN
ejpam-4548	357	6	,	,	PUNCT
ejpam-4548	357	7	15	15	NUM
ejpam-4548	357	8	(	(	PUNCT
ejpam-4548	357	9	4	4	NUM
ejpam-4548	357	10	)	)	PUNCT
ejpam-4548	357	11	(	(	PUNCT
ejpam-4548	357	12	2022	2022	NUM
ejpam-4548	357	13	)	)	PUNCT
ejpam-4548	357	14	,	,	PUNCT
ejpam-4548	357	15	2086	2086	NUM
ejpam-4548	357	16	-	-	SYM
ejpam-4548	357	17	2115	2115	NUM
ejpam-4548	357	18	2102	2102	NUM
ejpam-4548	357	19	=	=	SYM
ejpam-4548	357	20	∨	∨	NOUN
ejpam-4548	357	21	x	x	X
ejpam-4548	357	22	=	=	NOUN
ejpam-4548	357	23	yjzjy	yjzjy	NUM
ejpam-4548	357	24	−1	−1	NOUN
ejpam-4548	357	25	j	j	PROPN
ejpam-4548	357	26	{	{	PUNCT
ejpam-4548	357	27	{	{	PUNCT
ejpam-4548	357	28	ηi+1(zj	ηi+1(zj	ADV
ejpam-4548	357	29	)	)	PUNCT
ejpam-4548	357	30	∧	∧	PROPN
ejpam-4548	357	31	θ(yj	θ(yj	NOUN
ejpam-4548	357	32	)	)	PUNCT
ejpam-4548	357	33	}	}	PUNCT
ejpam-4548	357	34	∧	∧	PROPN
ejpam-4548	357	35	{	{	PUNCT
ejpam-4548	357	36	∨	∨	ADV
ejpam-4548	357	37	g	g	NOUN
ejpam-4548	357	38	=	=	SYM
ejpam-4548	357	39	ukvk	ukvk	X
ejpam-4548	357	40	{	{	PUNCT
ejpam-4548	357	41	ηi(uk	ηi(uk	PROPN
ejpam-4548	357	42	)	)	PUNCT
ejpam-4548	357	43	∧	∧	PROPN
ejpam-4548	357	44	θ(vk	θ(vk	PROPN
ejpam-4548	357	45	)	)	PUNCT
ejpam-4548	357	46	}	}	PUNCT
ejpam-4548	357	47	}	}	PUNCT
ejpam-4548	357	48	}	}	PUNCT
ejpam-4548	357	49	(	(	PUNCT
ejpam-4548	357	50	as	as	SCONJ
ejpam-4548	357	51	l	l	NOUN
ejpam-4548	357	52	is	be	AUX
ejpam-4548	357	53	a	a	DET
ejpam-4548	357	54	completely	completely	ADV
ejpam-4548	357	55	distributive	distributive	ADJ
ejpam-4548	357	56	lattice	lattice	NOUN
ejpam-4548	357	57	)	)	PUNCT
ejpam-4548	357	58	=	=	PUNCT
ejpam-4548	358	1	∨	∨	NUM
ejpam-4548	358	2	x	x	X
ejpam-4548	358	3	=	=	NOUN
ejpam-4548	358	4	yjzjy	yjzjy	NUM
ejpam-4548	358	5	−1	−1	NOUN
ejpam-4548	358	6	j	j	PROPN
ejpam-4548	358	7	{	{	PUNCT
ejpam-4548	358	8	∨	∨	ADV
ejpam-4548	358	9	g	g	NOUN
ejpam-4548	358	10	=	=	SYM
ejpam-4548	358	11	ukvk	ukvk	ADJ
ejpam-4548	358	12	{	{	PUNCT
ejpam-4548	358	13	ηi+1(zj	ηi+1(zj	ADJ
ejpam-4548	358	14	)	)	PUNCT
ejpam-4548	358	15	∧	∧	PROPN
ejpam-4548	358	16	θ(yj	θ(yj	NOUN
ejpam-4548	358	17	)	)	PUNCT
ejpam-4548	358	18	}	}	PUNCT
ejpam-4548	358	19	∧	∧	PROPN
ejpam-4548	358	20	{	{	PUNCT
ejpam-4548	358	21	ηi(uk	ηi(uk	PROPN
ejpam-4548	358	22	)	)	PUNCT
ejpam-4548	358	23	∧	∧	PROPN
ejpam-4548	358	24	θ(vk	θ(vk	PROPN
ejpam-4548	358	25	)	)	PUNCT
ejpam-4548	358	26	}	}	PUNCT
ejpam-4548	358	27	}	}	PUNCT
ejpam-4548	358	28	(	(	PUNCT
ejpam-4548	358	29	as	as	SCONJ
ejpam-4548	358	30	l	l	NOUN
ejpam-4548	358	31	is	be	AUX
ejpam-4548	358	32	a	a	DET
ejpam-4548	358	33	completely	completely	ADV
ejpam-4548	358	34	distributive	distributive	ADJ
ejpam-4548	358	35	lattice	lattice	NOUN
ejpam-4548	358	36	)	)	PUNCT
ejpam-4548	358	37	=	=	PUNCT
ejpam-4548	359	1	∨	∨	NUM
ejpam-4548	359	2	x	x	X
ejpam-4548	359	3	=	=	NOUN
ejpam-4548	359	4	yjzjy	yjzjy	NUM
ejpam-4548	359	5	−1	−1	NOUN
ejpam-4548	359	6	j	j	PROPN
ejpam-4548	359	7	g	g	PROPN
ejpam-4548	359	8	=	=	SYM
ejpam-4548	359	9	ukvk	ukvk	X
ejpam-4548	359	10	{	{	PUNCT
ejpam-4548	359	11	ηi+1(zj	ηi+1(zj	ADJ
ejpam-4548	359	12	)	)	PUNCT
ejpam-4548	359	13	∧	∧	PROPN
ejpam-4548	359	14	θ(yj	θ(yj	NOUN
ejpam-4548	359	15	)	)	PUNCT
ejpam-4548	359	16	∧	∧	PROPN
ejpam-4548	359	17	ηi(uk	ηi(uk	PROPN
ejpam-4548	359	18	)	)	PUNCT
ejpam-4548	359	19	∧	∧	PROPN
ejpam-4548	359	20	θ(vk	θ(vk	PROPN
ejpam-4548	359	21	)	)	PUNCT
ejpam-4548	359	22	}	}	PUNCT
ejpam-4548	359	23	}	}	PUNCT
ejpam-4548	359	24	≤	≤	NOUN
ejpam-4548	359	25	∨	∨	NUM
ejpam-4548	359	26	x	x	X
ejpam-4548	359	27	=	=	NOUN
ejpam-4548	359	28	yjzjy	yjzjy	NUM
ejpam-4548	359	29	−1	−1	NOUN
ejpam-4548	359	30	j	j	PROPN
ejpam-4548	359	31	g	g	PROPN
ejpam-4548	359	32	=	=	SYM
ejpam-4548	359	33	ukvk	ukvk	X
ejpam-4548	359	34	{	{	PUNCT
ejpam-4548	359	35	ηi+1(zj	ηi+1(zj	ADJ
ejpam-4548	359	36	)	)	PUNCT
ejpam-4548	359	37	∧	∧	NOUN
ejpam-4548	359	38	θ(vkyj	θ(vkyj	NOUN
ejpam-4548	359	39	)	)	PUNCT
ejpam-4548	359	40	∧	∧	PROPN
ejpam-4548	359	41	ηi(uk	ηi(uk	PROPN
ejpam-4548	359	42	)	)	PUNCT
ejpam-4548	359	43	}	}	PUNCT
ejpam-4548	359	44	(	(	PUNCT
ejpam-4548	359	45	as	as	ADP
ejpam-4548	359	46	θ	θ	PROPN
ejpam-4548	359	47	∈	∈	PROPN
ejpam-4548	359	48	l(µ	l(µ	PROPN
ejpam-4548	359	49	)	)	PUNCT
ejpam-4548	359	50	)	)	PUNCT
ejpam-4548	360	1	=	=	PUNCT
ejpam-4548	360	2	∨	∨	NUM
ejpam-4548	360	3	gxg−1	gxg−1	PROPN
ejpam-4548	360	4	=	=	ADJ
ejpam-4548	360	5	ukvkyjzjy	ukvkyjzjy	X
ejpam-4548	360	6	−1	−1	NOUN
ejpam-4548	360	7	j	j	NOUN
ejpam-4548	361	1	v−1	v−1	NOUN
ejpam-4548	361	2	k	k	PROPN
ejpam-4548	362	1	u−1	u−1	PROPN
ejpam-4548	362	2	k	k	PROPN
ejpam-4548	362	3	{	{	PUNCT
ejpam-4548	362	4	ηi+1(zj	ηi+1(zj	ADJ
ejpam-4548	362	5	)	)	PUNCT
ejpam-4548	362	6	∧	∧	NOUN
ejpam-4548	362	7	θ(vkyj	θ(vkyj	NOUN
ejpam-4548	362	8	)	)	PUNCT
ejpam-4548	362	9	∧	∧	PROPN
ejpam-4548	362	10	ηi(uk	ηi(uk	PROPN
ejpam-4548	362	11	)	)	PUNCT
ejpam-4548	362	12	}	}	PUNCT
ejpam-4548	362	13	≤	≤	NOUN
ejpam-4548	362	14	∨	∨	NUM
ejpam-4548	363	1	gxg−1	gxg−1	PROPN
ejpam-4548	363	2	=	=	ADJ
ejpam-4548	363	3	uk(vkyjzjy	uk(vkyjzjy	X
ejpam-4548	363	4	−1	−1	NOUN
ejpam-4548	363	5	j	j	NOUN
ejpam-4548	363	6	v−1	v−1	PROPN
ejpam-4548	363	7	k	k	PROPN
ejpam-4548	363	8	)	)	PUNCT
ejpam-4548	364	1	u−1	u−1	PROPN
ejpam-4548	364	2	k	k	PROPN
ejpam-4548	364	3	{	{	PUNCT
ejpam-4548	364	4	θηi+1θ	θηi+1θ	NOUN
ejpam-4548	364	5	−1(vkyjzjy	−1(vkyjzjy	PROPN
ejpam-4548	364	6	−1	−1	NOUN
ejpam-4548	365	1	j	j	NOUN
ejpam-4548	365	2	v−1	v−1	PROPN
ejpam-4548	365	3	k	k	PROPN
ejpam-4548	365	4	)	)	PUNCT
ejpam-4548	365	5	∧	∧	PROPN
ejpam-4548	365	6	ηi(uk	ηi(uk	PROPN
ejpam-4548	365	7	)	)	PUNCT
ejpam-4548	365	8	}	}	PUNCT
ejpam-4548	365	9	≤	≤	NOUN
ejpam-4548	365	10	∨	∨	NUM
ejpam-4548	365	11	gxg−1	gxg−1	PROPN
ejpam-4548	365	12	=	=	ADJ
ejpam-4548	365	13	uk(vkyjzjy	uk(vkyjzjy	X
ejpam-4548	365	14	−1	−1	NOUN
ejpam-4548	365	15	j	j	NOUN
ejpam-4548	366	1	v−1	v−1	PROPN
ejpam-4548	366	2	k	k	PROPN
ejpam-4548	366	3	)	)	PUNCT
ejpam-4548	367	1	u−1	u−1	PROPN
ejpam-4548	367	2	k	k	PROPN
ejpam-4548	367	3	{	{	PUNCT
ejpam-4548	367	4	ηθi+1(vkyjzjy	ηθi+1(vkyjzjy	VERB
ejpam-4548	367	5	−1	−1	NOUN
ejpam-4548	367	6	j	j	PROPN
ejpam-4548	367	7	v−1	v−1	PROPN
ejpam-4548	367	8	k	k	PROPN
ejpam-4548	367	9	)	)	PUNCT
ejpam-4548	367	10	∧	∧	PROPN
ejpam-4548	367	11	ηi(uk	ηi(uk	PROPN
ejpam-4548	367	12	)	)	PUNCT
ejpam-4548	367	13	}	}	PUNCT
ejpam-4548	368	1	=	=	PUNCT
ejpam-4548	368	2	∨	∨	ADV
ejpam-4548	368	3	gxg−1	gxg−1	PROPN
ejpam-4548	368	4	=	=	ADJ
ejpam-4548	368	5	uk(vkyjzjy	uk(vkyjzjy	X
ejpam-4548	368	6	−1	−1	NOUN
ejpam-4548	368	7	j	j	NOUN
ejpam-4548	369	1	v−1	v−1	PROPN
ejpam-4548	369	2	k	k	PROPN
ejpam-4548	369	3	)	)	PUNCT
ejpam-4548	370	1	u−1	u−1	PROPN
ejpam-4548	370	2	k	k	PROPN
ejpam-4548	370	3	{	{	PUNCT
ejpam-4548	370	4	ηi+1(vkyjzjv	ηi+1(vkyjzjv	X
ejpam-4548	370	5	−1	−1	NOUN
ejpam-4548	370	6	k	k	PROPN
ejpam-4548	370	7	y−1	y−1	PROPN
ejpam-4548	370	8	j	j	PROPN
ejpam-4548	370	9	)	)	PUNCT
ejpam-4548	370	10	∧	∧	PROPN
ejpam-4548	370	11	ηi(uk	ηi(uk	PROPN
ejpam-4548	370	12	)	)	PUNCT
ejpam-4548	370	13	}	}	PUNCT
ejpam-4548	370	14	(	(	PUNCT
ejpam-4548	370	15	as	as	ADP
ejpam-4548	370	16	ηθi+1	ηθi+1	NOUN
ejpam-4548	370	17	=	=	SYM
ejpam-4548	370	18	ηi+1	ηi+1	NOUN
ejpam-4548	370	19	)	)	PUNCT
ejpam-4548	370	20	≤	≤	NOUN
ejpam-4548	370	21	∨	∨	NUM
ejpam-4548	370	22	gxg−1	gxg−1	PROPN
ejpam-4548	370	23	=	=	ADJ
ejpam-4548	370	24	uk(vkyjzjy	uk(vkyjzjy	X
ejpam-4548	370	25	−1	−1	NOUN
ejpam-4548	370	26	j	j	NOUN
ejpam-4548	371	1	v−1	v−1	PROPN
ejpam-4548	371	2	k	k	PROPN
ejpam-4548	371	3	)	)	PUNCT
ejpam-4548	372	1	u−1	u−1	PROPN
ejpam-4548	372	2	k	k	PROPN
ejpam-4548	372	3	{	{	PUNCT
ejpam-4548	372	4	ηi+1(ukyjvkzjv	ηi+1(ukyjvkzjv	NOUN
ejpam-4548	372	5	−1	−1	NOUN
ejpam-4548	372	6	k	k	PROPN
ejpam-4548	372	7	y−1	y−1	PROPN
ejpam-4548	372	8	j	j	PROPN
ejpam-4548	373	1	u−1	u−1	PROPN
ejpam-4548	373	2	k	k	PROPN
ejpam-4548	373	3	)	)	PUNCT
ejpam-4548	373	4	}	}	PUNCT
ejpam-4548	373	5	(	(	PUNCT
ejpam-4548	373	6	as	as	ADP
ejpam-4548	373	7	ηi+1	ηi+1	NUM
ejpam-4548	373	8	∈	∈	PROPN
ejpam-4548	373	9	nl(ηi	nl(ηi	NOUN
ejpam-4548	373	10	)	)	PUNCT
ejpam-4548	373	11	)	)	PUNCT
ejpam-4548	374	1	=	=	PRON
ejpam-4548	374	2	ηi+1(gxg	ηi+1(gxg	NOUN
ejpam-4548	374	3	−1	−1	NOUN
ejpam-4548	374	4	)	)	PUNCT
ejpam-4548	374	5	.	.	PUNCT
ejpam-4548	375	1	now	now	ADV
ejpam-4548	375	2	,	,	PUNCT
ejpam-4548	375	3	define	define	VERB
ejpam-4548	375	4	the	the	DET
ejpam-4548	375	5	notion	notion	NOUN
ejpam-4548	375	6	of	of	ADP
ejpam-4548	375	7	subnormal	subnormal	ADJ
ejpam-4548	375	8	l	l	PROPN
ejpam-4548	375	9	-	-	NOUN
ejpam-4548	375	10	subgroup	subgroup	NOUN
ejpam-4548	375	11	of	of	ADP
ejpam-4548	375	12	an	an	DET
ejpam-4548	375	13	l	l	NOUN
ejpam-4548	375	14	-	-	NOUN
ejpam-4548	375	15	group	group	NOUN
ejpam-4548	375	16	as	as	SCONJ
ejpam-4548	375	17	follows	follow	VERB
ejpam-4548	375	18	:	:	PUNCT
ejpam-4548	375	19	definition	definition	NOUN
ejpam-4548	375	20	12	12	NUM
ejpam-4548	375	21	.	.	PUNCT
ejpam-4548	376	1	let	let	VERB
ejpam-4548	376	2	η	η	PROPN
ejpam-4548	376	3	∈	∈	PROPN
ejpam-4548	376	4	l(µ	l(µ	PROPN
ejpam-4548	376	5	)	)	PUNCT
ejpam-4548	376	6	and	and	CCONJ
ejpam-4548	376	7	ηi	ηi	PROPN
ejpam-4548	376	8	be	be	AUX
ejpam-4548	376	9	the	the	DET
ejpam-4548	376	10	ith	ith	PROPN
ejpam-4548	376	11	normal	normal	ADJ
ejpam-4548	376	12	closure	closure	NOUN
ejpam-4548	376	13	of	of	ADP
ejpam-4548	376	14	η	η	PROPN
ejpam-4548	376	15	in	in	ADP
ejpam-4548	376	16	µ.	µ.	NOUN
ejpam-4548	376	17	if	if	SCONJ
ejpam-4548	376	18	there	there	PRON
ejpam-4548	376	19	exists	exist	VERB
ejpam-4548	376	20	a	a	DET
ejpam-4548	376	21	non	non	X
ejpam-4548	376	22	negative	negative	ADJ
ejpam-4548	376	23	integer	integer	NOUN
ejpam-4548	376	24	m	m	VERB
ejpam-4548	376	25	such	such	ADJ
ejpam-4548	376	26	that	that	DET
ejpam-4548	376	27	ηm	ηm	PROPN
ejpam-4548	376	28	=	=	SYM
ejpam-4548	376	29	η	η	PROPN
ejpam-4548	376	30	◁	◁	PROPN
ejpam-4548	377	1	ηm−1	ηm−1	NOUN
ejpam-4548	377	2	◁	◁	X
ejpam-4548	377	3	·	·	PUNCT
ejpam-4548	377	4	·	·	PUNCT
ejpam-4548	377	5	·	·	PUNCT
ejpam-4548	377	6	◁	◁	NOUN
ejpam-4548	377	7	η0	η0	NOUN
ejpam-4548	377	8	=	=	SYM
ejpam-4548	377	9	µ	µ	NUM
ejpam-4548	377	10	,	,	PUNCT
ejpam-4548	377	11	then	then	ADV
ejpam-4548	377	12	η	η	PROPN
ejpam-4548	377	13	is	be	AUX
ejpam-4548	377	14	known	know	VERB
ejpam-4548	377	15	as	as	ADP
ejpam-4548	377	16	a	a	DET
ejpam-4548	377	17	subnormal	subnormal	ADJ
ejpam-4548	377	18	l	l	NOUN
ejpam-4548	377	19	-	-	NOUN
ejpam-4548	377	20	subgroup	subgroup	NOUN
ejpam-4548	377	21	of	of	ADP
ejpam-4548	377	22	µ	µ	NOUN
ejpam-4548	377	23	with	with	ADP
ejpam-4548	377	24	defect	defect	ADJ
ejpam-4548	377	25	m.	m.	NOUN
ejpam-4548	377	26	we	we	PRON
ejpam-4548	377	27	shall	shall	AUX
ejpam-4548	377	28	denote	denote	VERB
ejpam-4548	377	29	a	a	DET
ejpam-4548	377	30	subnormal	subnormal	ADJ
ejpam-4548	377	31	l	l	PROPN
ejpam-4548	377	32	-	-	PROPN
ejpam-4548	377	33	subgroup	subgroup	PROPN
ejpam-4548	377	34	η	η	PROPN
ejpam-4548	377	35	of	of	ADP
ejpam-4548	377	36	µ	µ	PROPN
ejpam-4548	377	37	with	with	ADP
ejpam-4548	377	38	defect	defect	NOUN
ejpam-4548	377	39	m	m	VERB
ejpam-4548	377	40	by	by	ADP
ejpam-4548	377	41	η	η	PROPN
ejpam-4548	377	42	m	m	PROPN
ejpam-4548	377	43	◁	◁	NOUN
ejpam-4548	377	44	µ.	µ.	NOUN
ejpam-4548	377	45	if	if	SCONJ
ejpam-4548	377	46	η	η	PROPN
ejpam-4548	377	47	is	be	AUX
ejpam-4548	377	48	a	a	DET
ejpam-4548	377	49	subnormal	subnormal	ADJ
ejpam-4548	377	50	l	l	NOUN
ejpam-4548	377	51	-	-	NOUN
ejpam-4548	377	52	subgroup	subgroup	NOUN
ejpam-4548	377	53	of	of	ADP
ejpam-4548	377	54	µ	µ	NUM
ejpam-4548	377	55	,	,	PUNCT
ejpam-4548	377	56	then	then	ADV
ejpam-4548	377	57	we	we	PRON
ejpam-4548	377	58	shall	shall	AUX
ejpam-4548	377	59	also	also	ADV
ejpam-4548	377	60	write	write	VERB
ejpam-4548	377	61	’	'	PUNCT
ejpam-4548	377	62	η	η	PROPN
ejpam-4548	377	63	is	be	AUX
ejpam-4548	377	64	subnormal	subnormal	ADJ
ejpam-4548	377	65	in	in	ADP
ejpam-4548	377	66	µ.	µ.	NOUN
ejpam-4548	377	67	’	'	PUNCT
ejpam-4548	377	68	remark	remark	NOUN
ejpam-4548	377	69	1	1	NUM
ejpam-4548	377	70	.	.	PUNCT
ejpam-4548	378	1	obviously	obviously	ADV
ejpam-4548	378	2	m	m	VERB
ejpam-4548	378	3	equals	equal	VERB
ejpam-4548	378	4	0	0	PUNCT
ejpam-4548	379	1	if	if	SCONJ
ejpam-4548	379	2	η	η	PROPN
ejpam-4548	379	3	=	=	PROPN
ejpam-4548	379	4	µ	µ	PROPN
ejpam-4548	379	5	and	and	CCONJ
ejpam-4548	379	6	m	m	VERB
ejpam-4548	379	7	=	=	ADJ
ejpam-4548	379	8	1	1	NUM
ejpam-4548	379	9	if	if	SCONJ
ejpam-4548	379	10	η	η	PROPN
ejpam-4548	379	11	∈	∈	PROPN
ejpam-4548	379	12	nl(µ	nl(µ	NOUN
ejpam-4548	379	13	)	)	PUNCT
ejpam-4548	379	14	and	and	CCONJ
ejpam-4548	379	15	η	η	PROPN
ejpam-4548	379	16	̸=	̸=	PROPN
ejpam-4548	379	17	µ.	µ.	NOUN
ejpam-4548	379	18	the	the	DET
ejpam-4548	379	19	following	follow	VERB
ejpam-4548	379	20	example	example	NOUN
ejpam-4548	379	21	illustrates	illustrate	VERB
ejpam-4548	379	22	the	the	DET
ejpam-4548	379	23	notion	notion	NOUN
ejpam-4548	379	24	of	of	ADP
ejpam-4548	379	25	subnormal	subnormal	ADJ
ejpam-4548	379	26	l	l	PROPN
ejpam-4548	379	27	-	-	NOUN
ejpam-4548	379	28	subgroup	subgroup	NOUN
ejpam-4548	379	29	of	of	ADP
ejpam-4548	379	30	an	an	DET
ejpam-4548	379	31	l	l	NOUN
ejpam-4548	379	32	-	-	NOUN
ejpam-4548	379	33	group	group	NOUN
ejpam-4548	379	34	:	:	PUNCT
ejpam-4548	379	35	n.	n.	PROPN
ejpam-4548	379	36	ajmal	ajmal	PROPN
ejpam-4548	379	37	,	,	PUNCT
ejpam-4548	379	38	i.	i.	PROPN
ejpam-4548	379	39	jahan	jahan	PROPN
ejpam-4548	379	40	.	.	PROPN
ejpam-4548	379	41	,	,	PUNCT
ejpam-4548	379	42	b.	b.	PROPN
ejpam-4548	379	43	davvaz	davvaz	PROPN
ejpam-4548	379	44	/	/	SYM
ejpam-4548	379	45	eur	eur	PROPN
ejpam-4548	379	46	.	.	PUNCT
ejpam-4548	380	1	j.	j.	PROPN
ejpam-4548	380	2	pure	pure	PROPN
ejpam-4548	380	3	appl	appl	PROPN
ejpam-4548	380	4	.	.	PROPN
ejpam-4548	380	5	math	math	PROPN
ejpam-4548	380	6	,	,	PUNCT
ejpam-4548	380	7	15	15	NUM
ejpam-4548	380	8	(	(	PUNCT
ejpam-4548	380	9	4	4	NUM
ejpam-4548	380	10	)	)	PUNCT
ejpam-4548	380	11	(	(	PUNCT
ejpam-4548	380	12	2022	2022	NUM
ejpam-4548	380	13	)	)	PUNCT
ejpam-4548	380	14	,	,	PUNCT
ejpam-4548	380	15	2086	2086	NUM
ejpam-4548	380	16	-	-	SYM
ejpam-4548	380	17	2115	2115	NUM
ejpam-4548	380	18	2103	2103	NUM
ejpam-4548	380	19	example	example	NOUN
ejpam-4548	380	20	2	2	NUM
ejpam-4548	380	21	.	.	X
ejpam-4548	380	22	recall	recall	VERB
ejpam-4548	380	23	that	that	SCONJ
ejpam-4548	380	24	in	in	ADP
ejpam-4548	380	25	example	example	NOUN
ejpam-4548	380	26	1	1	NUM
ejpam-4548	380	27	,	,	PUNCT
ejpam-4548	380	28	the	the	DET
ejpam-4548	380	29	normal	normal	ADJ
ejpam-4548	380	30	closure	closure	NOUN
ejpam-4548	380	31	‘	'	PUNCT
ejpam-4548	380	32	ηµ	ηµ	VERB
ejpam-4548	380	33	’	'	PUNCT
ejpam-4548	380	34	of	of	ADP
ejpam-4548	380	35	η	η	PROPN
ejpam-4548	380	36	in	in	ADP
ejpam-4548	380	37	µ	µ	PROPN
ejpam-4548	380	38	is	be	AUX
ejpam-4548	380	39	an	an	DET
ejpam-4548	380	40	l	l	NOUN
ejpam-4548	380	41	-	-	NOUN
ejpam-4548	380	42	subgroup	subgroup	NOUN
ejpam-4548	380	43	of	of	ADP
ejpam-4548	380	44	µ	µ	PROPN
ejpam-4548	380	45	defined	define	VERB
ejpam-4548	380	46	as	as	SCONJ
ejpam-4548	380	47	follows	follow	VERB
ejpam-4548	380	48	:	:	PUNCT
ejpam-4548	380	49	η1(z	η1(z	NUM
ejpam-4548	380	50	)	)	PUNCT
ejpam-4548	380	51	=	=	NOUN
ejpam-4548	380	52	ηµ(z	ηµ(z	NOUN
ejpam-4548	380	53	)	)	PUNCT
ejpam-4548	380	54	=	=	SYM
ejpam-4548	380	55			NUM
ejpam-4548	380	56	1	1	NUM
ejpam-4548	380	57	3	3	NUM
ejpam-4548	380	58	if	if	SCONJ
ejpam-4548	380	59	z	z	PROPN
ejpam-4548	380	60	∈	∈	PROPN
ejpam-4548	380	61	k4	k4	NOUN
ejpam-4548	380	62	,	,	PUNCT
ejpam-4548	380	63	1	1	NUM
ejpam-4548	380	64	4	4	NUM
ejpam-4548	380	65	if	if	SCONJ
ejpam-4548	380	66	z	z	NOUN
ejpam-4548	380	67	∈	∈	PROPN
ejpam-4548	380	68	d4	d4	PROPN
ejpam-4548	380	69	∼	∼	NOUN
ejpam-4548	380	70	k4	k4	NOUN
ejpam-4548	380	71	,	,	PUNCT
ejpam-4548	380	72	1	1	NUM
ejpam-4548	380	73	9	9	NUM
ejpam-4548	380	74	if	if	SCONJ
ejpam-4548	380	75	z	z	NOUN
ejpam-4548	380	76	∈	∈	PROPN
ejpam-4548	380	77	d8	d8	VERB
ejpam-4548	380	78	∼	∼	NOUN
ejpam-4548	380	79	d4	d4	PROPN
ejpam-4548	380	80	.	.	PUNCT
ejpam-4548	381	1	similarly	similarly	ADV
ejpam-4548	381	2	,	,	PUNCT
ejpam-4548	381	3	one	one	PRON
ejpam-4548	381	4	can	can	AUX
ejpam-4548	381	5	verify	verify	VERB
ejpam-4548	381	6	that	that	SCONJ
ejpam-4548	381	7	the	the	DET
ejpam-4548	381	8	conjugate	conjugate	NOUN
ejpam-4548	381	9	‘	'	PUNCT
ejpam-4548	381	10	η1η(η1	η1η(η1	NOUN
ejpam-4548	381	11	)	)	PUNCT
ejpam-4548	381	12	−1	−1	NOUN
ejpam-4548	381	13	’	'	PUNCT
ejpam-4548	381	14	is	be	AUX
ejpam-4548	381	15	defined	define	VERB
ejpam-4548	381	16	by	by	ADP
ejpam-4548	381	17	the	the	DET
ejpam-4548	381	18	following	follow	VERB
ejpam-4548	381	19	level	level	NOUN
ejpam-4548	381	20	subsets	subset	NOUN
ejpam-4548	381	21	:	:	PUNCT
ejpam-4548	381	22	(	(	PUNCT
ejpam-4548	381	23	η1η(η1	η1η(η1	NOUN
ejpam-4548	381	24	)	)	PUNCT
ejpam-4548	381	25	−1	−1	NOUN
ejpam-4548	381	26	)	)	PUNCT
ejpam-4548	381	27	1	1	NUM
ejpam-4548	381	28	3	3	NUM
ejpam-4548	381	29	=	=	SYM
ejpam-4548	381	30	{	{	PUNCT
ejpam-4548	381	31	e	e	NOUN
ejpam-4548	381	32	,	,	PUNCT
ejpam-4548	381	33	x	x	X
ejpam-4548	381	34	,	,	PUNCT
ejpam-4548	381	35	xy4	xy4	PROPN
ejpam-4548	381	36	}	}	PUNCT
ejpam-4548	381	37	,	,	PUNCT
ejpam-4548	381	38	(	(	PUNCT
ejpam-4548	381	39	η1η(η1	η1η(η1	NOUN
ejpam-4548	381	40	)	)	PUNCT
ejpam-4548	381	41	−1	−1	NOUN
ejpam-4548	381	42	)	)	PUNCT
ejpam-4548	381	43	1	1	NUM
ejpam-4548	381	44	6	6	NUM
ejpam-4548	381	45	=	=	SYM
ejpam-4548	381	46	{	{	PUNCT
ejpam-4548	381	47	e	e	NOUN
ejpam-4548	381	48	,	,	PUNCT
ejpam-4548	381	49	x	x	X
ejpam-4548	381	50	,	,	PUNCT
ejpam-4548	381	51	xy2	xy2	PROPN
ejpam-4548	381	52	,	,	PUNCT
ejpam-4548	381	53	xy4	xy4	PROPN
ejpam-4548	381	54	,	,	PUNCT
ejpam-4548	381	55	xy6	xy6	PROPN
ejpam-4548	381	56	,	,	PUNCT
ejpam-4548	381	57	y2	y2	PROPN
ejpam-4548	381	58	,	,	PUNCT
ejpam-4548	381	59	y4	y4	PROPN
ejpam-4548	381	60	,	,	PUNCT
ejpam-4548	381	61	y6	y6	PROPN
ejpam-4548	381	62	}	}	PUNCT
ejpam-4548	381	63	,	,	PUNCT
ejpam-4548	381	64	(	(	PUNCT
ejpam-4548	381	65	η1η(η1	η1η(η1	NOUN
ejpam-4548	381	66	)	)	PUNCT
ejpam-4548	381	67	−1	−1	NOUN
ejpam-4548	381	68	)	)	PUNCT
ejpam-4548	381	69	1	1	NUM
ejpam-4548	381	70	9	9	NUM
ejpam-4548	381	71	=	=	SYM
ejpam-4548	381	72	d8	d8	NOUN
ejpam-4548	381	73	.	.	PUNCT
ejpam-4548	382	1	so	so	ADV
ejpam-4548	382	2	in	in	ADP
ejpam-4548	382	3	view	view	NOUN
ejpam-4548	382	4	of	of	ADP
ejpam-4548	382	5	theorem	theorem	NOUN
ejpam-4548	382	6	6	6	NUM
ejpam-4548	382	7	,	,	PUNCT
ejpam-4548	382	8	the	the	DET
ejpam-4548	382	9	normal	normal	ADJ
ejpam-4548	382	10	closure	closure	NOUN
ejpam-4548	382	11	of	of	ADP
ejpam-4548	382	12	η	η	PROPN
ejpam-4548	382	13	in	in	ADP
ejpam-4548	382	14	η1	η1	PROPN
ejpam-4548	382	15	has	have	VERB
ejpam-4548	382	16	the	the	DET
ejpam-4548	382	17	following	follow	VERB
ejpam-4548	382	18	definition	definition	NOUN
ejpam-4548	382	19	:	:	PUNCT
ejpam-4548	382	20	η2(z	η2(z	X
ejpam-4548	382	21	)	)	PUNCT
ejpam-4548	382	22	=	=	SYM
ejpam-4548	382	23	⟨η1η(η1)−1⟩(z	⟨η1η(η1)−1⟩(z	NOUN
ejpam-4548	382	24	)	)	PUNCT
ejpam-4548	382	25	=	=	SYM
ejpam-4548	382	26			NUM
ejpam-4548	382	27	1	1	NUM
ejpam-4548	382	28	3	3	NUM
ejpam-4548	382	29	if	if	SCONJ
ejpam-4548	382	30	z	z	PROPN
ejpam-4548	382	31	∈	∈	PROPN
ejpam-4548	382	32	k4	k4	NOUN
ejpam-4548	382	33	,	,	PUNCT
ejpam-4548	382	34	1	1	NUM
ejpam-4548	382	35	6	6	NUM
ejpam-4548	382	36	if	if	SCONJ
ejpam-4548	382	37	z	z	NOUN
ejpam-4548	382	38	∈	∈	PROPN
ejpam-4548	382	39	d4	d4	PROPN
ejpam-4548	382	40	∼	∼	NOUN
ejpam-4548	382	41	k4	k4	NOUN
ejpam-4548	382	42	,	,	PUNCT
ejpam-4548	382	43	1	1	NUM
ejpam-4548	382	44	9	9	NUM
ejpam-4548	382	45	if	if	SCONJ
ejpam-4548	382	46	z	z	NOUN
ejpam-4548	382	47	∈	∈	PROPN
ejpam-4548	382	48	d8	d8	VERB
ejpam-4548	382	49	∼	∼	NOUN
ejpam-4548	382	50	d4	d4	NOUN
ejpam-4548	382	51	.	.	PUNCT
ejpam-4548	383	1	moreover	moreover	ADV
ejpam-4548	383	2	,	,	PUNCT
ejpam-4548	383	3	one	one	PRON
ejpam-4548	383	4	can	can	AUX
ejpam-4548	383	5	verify	verify	VERB
ejpam-4548	383	6	that	that	SCONJ
ejpam-4548	383	7	the	the	DET
ejpam-4548	383	8	conjugate	conjugate	ADJ
ejpam-4548	383	9	‘	'	PUNCT
ejpam-4548	383	10	η2η(η2	η2η(η2	ADJ
ejpam-4548	383	11	)	)	PUNCT
ejpam-4548	383	12	−1	−1	NOUN
ejpam-4548	383	13	’	'	PUNCT
ejpam-4548	383	14	is	be	AUX
ejpam-4548	383	15	defined	define	VERB
ejpam-4548	383	16	by	by	ADP
ejpam-4548	383	17	the	the	DET
ejpam-4548	383	18	following	follow	VERB
ejpam-4548	383	19	level	level	NOUN
ejpam-4548	383	20	subsets	subset	NOUN
ejpam-4548	383	21	:	:	PUNCT
ejpam-4548	383	22	(	(	PUNCT
ejpam-4548	383	23	η2η(η2	η2η(η2	ADJ
ejpam-4548	383	24	)	)	PUNCT
ejpam-4548	383	25	−1	−1	NOUN
ejpam-4548	383	26	)	)	PUNCT
ejpam-4548	383	27	1	1	NUM
ejpam-4548	383	28	3	3	NUM
ejpam-4548	383	29	=	=	SYM
ejpam-4548	383	30	⟨x⟩	⟨x⟩	PROPN
ejpam-4548	383	31	,	,	PUNCT
ejpam-4548	383	32	(	(	PUNCT
ejpam-4548	383	33	η2η(η2	η2η(η2	ADJ
ejpam-4548	383	34	)	)	PUNCT
ejpam-4548	383	35	−1	−1	NOUN
ejpam-4548	383	36	)	)	PUNCT
ejpam-4548	383	37	1	1	NUM
ejpam-4548	383	38	6	6	NUM
ejpam-4548	383	39	=	=	SYM
ejpam-4548	383	40	d4	d4	PROPN
ejpam-4548	383	41	,	,	PUNCT
ejpam-4548	383	42	and	and	CCONJ
ejpam-4548	383	43	(	(	PUNCT
ejpam-4548	383	44	η2η(η2	η2η(η2	ADJ
ejpam-4548	383	45	)	)	PUNCT
ejpam-4548	383	46	−1	−1	NOUN
ejpam-4548	383	47	)	)	PUNCT
ejpam-4548	383	48	1	1	NUM
ejpam-4548	383	49	9	9	NUM
ejpam-4548	383	50	=	=	SYM
ejpam-4548	383	51	d8	d8	PROPN
ejpam-4548	383	52	.	.	PUNCT
ejpam-4548	384	1	as	as	ADP
ejpam-4548	384	2	η2η(η2	η2η(η2	ADJ
ejpam-4548	384	3	)	)	PUNCT
ejpam-4548	384	4	−1	−1	NOUN
ejpam-4548	384	5	∈	∈	PROPN
ejpam-4548	384	6	l(µ	l(µ	PROPN
ejpam-4548	384	7	)	)	PUNCT
ejpam-4548	384	8	,	,	PUNCT
ejpam-4548	384	9	we	we	PRON
ejpam-4548	384	10	have	have	VERB
ejpam-4548	384	11	η3	η3	NOUN
ejpam-4548	384	12	=	=	SYM
ejpam-4548	384	13	ηη2	ηη2	NOUN
ejpam-4548	384	14	=	=	PUNCT
ejpam-4548	384	15	η2η(η2	η2η(η2	ADJ
ejpam-4548	384	16	)	)	PUNCT
ejpam-4548	384	17	−1	−1	NOUN
ejpam-4548	384	18	=	=	SYM
ejpam-4548	384	19	η	η	PROPN
ejpam-4548	384	20	.	.	PROPN
ejpam-4548	384	21	consequently	consequently	ADV
ejpam-4548	384	22	,	,	PUNCT
ejpam-4548	384	23	we	we	PRON
ejpam-4548	384	24	have	have	VERB
ejpam-4548	384	25	η3	η3	NOUN
ejpam-4548	384	26	=	=	SYM
ejpam-4548	384	27	η	η	PROPN
ejpam-4548	384	28	◁	◁	PRON
ejpam-4548	384	29	η2	η2	PUNCT
ejpam-4548	384	30	◁	◁	NOUN
ejpam-4548	384	31	η1	η1	NOUN
ejpam-4548	384	32	◁	◁	NOUN
ejpam-4548	384	33	η0	η0	NOUN
ejpam-4548	384	34	=	=	PUNCT
ejpam-4548	384	35	µ.	µ.	NOUN
ejpam-4548	384	36	hence	hence	ADV
ejpam-4548	384	37	η	η	PROPN
ejpam-4548	384	38	is	be	AUX
ejpam-4548	384	39	a	a	DET
ejpam-4548	384	40	subnormal	subnormal	ADJ
ejpam-4548	384	41	l	l	NOUN
ejpam-4548	384	42	-	-	NOUN
ejpam-4548	384	43	subgroup	subgroup	NOUN
ejpam-4548	384	44	of	of	ADP
ejpam-4548	384	45	µ	µ	NOUN
ejpam-4548	384	46	with	with	ADP
ejpam-4548	384	47	defect	defect	NOUN
ejpam-4548	384	48	3	3	NUM
ejpam-4548	384	49	.	.	PUNCT
ejpam-4548	385	1	next	next	ADV
ejpam-4548	385	2	,	,	PUNCT
ejpam-4548	385	3	we	we	PRON
ejpam-4548	385	4	provide	provide	VERB
ejpam-4548	385	5	the	the	DET
ejpam-4548	385	6	definition	definition	NOUN
ejpam-4548	385	7	of	of	ADP
ejpam-4548	385	8	a	a	DET
ejpam-4548	385	9	subnormal	subnormal	ADJ
ejpam-4548	385	10	seies	seie	NOUN
ejpam-4548	385	11	for	for	ADP
ejpam-4548	385	12	an	an	DET
ejpam-4548	385	13	l	l	NOUN
ejpam-4548	385	14	-	-	NOUN
ejpam-4548	385	15	subgroup	subgroup	NOUN
ejpam-4548	385	16	:	:	PUNCT
ejpam-4548	385	17	definition	definition	NOUN
ejpam-4548	385	18	13	13	NUM
ejpam-4548	385	19	.	.	PUNCT
ejpam-4548	386	1	let	let	VERB
ejpam-4548	386	2	η	η	PROPN
ejpam-4548	386	3	∈	∈	PROPN
ejpam-4548	386	4	l(µ	l(µ	PROPN
ejpam-4548	386	5	)	)	PUNCT
ejpam-4548	386	6	.	.	PUNCT
ejpam-4548	387	1	a	a	DET
ejpam-4548	387	2	finite	finite	PROPN
ejpam-4548	387	3	series	series	NOUN
ejpam-4548	387	4	θ0	θ0	PROPN
ejpam-4548	387	5	=	=	PROPN
ejpam-4548	387	6	µ	µ	PROPN
ejpam-4548	387	7	,	,	PUNCT
ejpam-4548	387	8	θ1	θ1	NOUN
ejpam-4548	387	9	,	,	PUNCT
ejpam-4548	387	10	θ2	θ2	PROPN
ejpam-4548	387	11	,	,	PUNCT
ejpam-4548	387	12	...	...	PUNCT
ejpam-4548	387	13	,	,	PUNCT
ejpam-4548	387	14	θm	θm	PROPN
ejpam-4548	387	15	=	=	SYM
ejpam-4548	387	16	η	η	PROPN
ejpam-4548	387	17	of	of	ADP
ejpam-4548	387	18	l	l	PROPN
ejpam-4548	387	19	-	-	NOUN
ejpam-4548	387	20	subgroups	subgroup	NOUN
ejpam-4548	387	21	of	of	ADP
ejpam-4548	387	22	µ	µ	PRON
ejpam-4548	387	23	such	such	ADJ
ejpam-4548	387	24	that	that	SCONJ
ejpam-4548	387	25	η	η	PROPN
ejpam-4548	387	26	=	=	PROPN
ejpam-4548	387	27	θm	θm	PROPN
ejpam-4548	387	28	◁	◁	PROPN
ejpam-4548	388	1	θm−1	θm−1	PROPN
ejpam-4548	388	2	◁	◁	X
ejpam-4548	388	3	...	...	PUNCT
ejpam-4548	389	1	◁	◁	X
ejpam-4548	389	2	θ0	θ0	PROPN
ejpam-4548	389	3	=	=	SYM
ejpam-4548	389	4	µ	µ	X
ejpam-4548	389	5	is	be	AUX
ejpam-4548	389	6	said	say	VERB
ejpam-4548	389	7	to	to	PART
ejpam-4548	389	8	be	be	AUX
ejpam-4548	389	9	a	a	DET
ejpam-4548	389	10	subnormal	subnormal	ADJ
ejpam-4548	389	11	series	series	NOUN
ejpam-4548	389	12	of	of	ADP
ejpam-4548	389	13	η	η	PROPN
ejpam-4548	389	14	.	.	PROPN
ejpam-4548	389	15	n.	n.	PROPN
ejpam-4548	389	16	ajmal	ajmal	PROPN
ejpam-4548	389	17	,	,	PUNCT
ejpam-4548	389	18	i.	i.	PROPN
ejpam-4548	389	19	jahan	jahan	PROPN
ejpam-4548	389	20	.	.	PROPN
ejpam-4548	389	21	,	,	PUNCT
ejpam-4548	389	22	b.	b.	PROPN
ejpam-4548	389	23	davvaz	davvaz	PROPN
ejpam-4548	389	24	/	/	SYM
ejpam-4548	389	25	eur	eur	PROPN
ejpam-4548	389	26	.	.	PUNCT
ejpam-4548	390	1	j.	j.	PROPN
ejpam-4548	390	2	pure	pure	PROPN
ejpam-4548	390	3	appl	appl	PROPN
ejpam-4548	390	4	.	.	PROPN
ejpam-4548	390	5	math	math	PROPN
ejpam-4548	390	6	,	,	PUNCT
ejpam-4548	390	7	15	15	NUM
ejpam-4548	390	8	(	(	PUNCT
ejpam-4548	390	9	4	4	NUM
ejpam-4548	390	10	)	)	PUNCT
ejpam-4548	390	11	(	(	PUNCT
ejpam-4548	390	12	2022	2022	NUM
ejpam-4548	390	13	)	)	PUNCT
ejpam-4548	390	14	,	,	PUNCT
ejpam-4548	390	15	2086	2086	NUM
ejpam-4548	390	16	-	-	SYM
ejpam-4548	390	17	2115	2115	NUM
ejpam-4548	390	18	2104	2104	NUM
ejpam-4548	391	1	we	we	PRON
ejpam-4548	391	2	shall	shall	AUX
ejpam-4548	391	3	describe	describe	VERB
ejpam-4548	391	4	the	the	DET
ejpam-4548	391	5	notion	notion	NOUN
ejpam-4548	391	6	of	of	ADP
ejpam-4548	391	7	a	a	DET
ejpam-4548	391	8	subnormal	subnormal	ADJ
ejpam-4548	391	9	l	l	NOUN
ejpam-4548	391	10	-	-	NOUN
ejpam-4548	391	11	subgroup	subgroup	NOUN
ejpam-4548	391	12	through	through	ADP
ejpam-4548	391	13	the	the	DET
ejpam-4548	391	14	notion	notion	NOUN
ejpam-4548	391	15	of	of	ADP
ejpam-4548	391	16	above	above	ADV
ejpam-4548	391	17	defined	define	VERB
ejpam-4548	391	18	subnormal	subnormal	ADJ
ejpam-4548	391	19	series	series	NOUN
ejpam-4548	391	20	like	like	ADP
ejpam-4548	391	21	their	their	PRON
ejpam-4548	391	22	classical	classical	ADJ
ejpam-4548	391	23	counterparts	counterpart	NOUN
ejpam-4548	391	24	.	.	PUNCT
ejpam-4548	392	1	the	the	DET
ejpam-4548	392	2	following	follow	VERB
ejpam-4548	392	3	lemma	lemma	PROPN
ejpam-4548	392	4	,	,	PUNCT
ejpam-4548	392	5	is	be	AUX
ejpam-4548	392	6	a	a	DET
ejpam-4548	392	7	step	step	NOUN
ejpam-4548	392	8	in	in	ADP
ejpam-4548	392	9	this	this	DET
ejpam-4548	392	10	direction	direction	NOUN
ejpam-4548	392	11	:	:	PUNCT
ejpam-4548	392	12	lemma	lemma	PROPN
ejpam-4548	392	13	2	2	X
ejpam-4548	392	14	.	.	PUNCT
ejpam-4548	392	15	let	let	VERB
ejpam-4548	392	16	η	η	PROPN
ejpam-4548	392	17	∈	∈	PROPN
ejpam-4548	392	18	l(µ	l(µ	PROPN
ejpam-4548	392	19	)	)	PUNCT
ejpam-4548	392	20	and	and	CCONJ
ejpam-4548	392	21	η	η	PROPN
ejpam-4548	392	22	⊆	⊆	NUM
ejpam-4548	392	23	·	·	PUNCT
ejpam-4548	392	24	·	·	PUNCT
ejpam-4548	393	1	·	·	PUNCT
ejpam-4548	393	2	◁	◁	X
ejpam-4548	393	3	ηi+1	ηi+1	NUM
ejpam-4548	393	4	◁	◁	PUNCT
ejpam-4548	393	5	ηi	ηi	INTJ
ejpam-4548	393	6	◁	◁	X
ejpam-4548	393	7	·	·	PUNCT
ejpam-4548	393	8	·	·	PUNCT
ejpam-4548	393	9	·	·	PUNCT
ejpam-4548	393	10	◁	◁	X
ejpam-4548	393	11	η1	η1	NOUN
ejpam-4548	393	12	◁	◁	NOUN
ejpam-4548	393	13	η0	η0	NOUN
ejpam-4548	393	14	=	=	PUNCT
ejpam-4548	393	15	µ	µ	X
ejpam-4548	393	16	be	be	AUX
ejpam-4548	393	17	the	the	DET
ejpam-4548	393	18	normal	normal	ADJ
ejpam-4548	393	19	closure	closure	NOUN
ejpam-4548	393	20	series	series	NOUN
ejpam-4548	393	21	of	of	ADP
ejpam-4548	393	22	η	η	PROPN
ejpam-4548	393	23	.	.	PROPN
ejpam-4548	393	24	if	if	SCONJ
ejpam-4548	393	25	there	there	PRON
ejpam-4548	393	26	exists	exist	VERB
ejpam-4548	393	27	a	a	DET
ejpam-4548	393	28	descending	descend	VERB
ejpam-4548	393	29	series	series	NOUN
ejpam-4548	393	30	γ0	γ0	PROPN
ejpam-4548	393	31	=	=	SYM
ejpam-4548	393	32	µ	µ	NOUN
ejpam-4548	393	33	,	,	PUNCT
ejpam-4548	393	34	γ1	γ1	NOUN
ejpam-4548	393	35	,	,	PUNCT
ejpam-4548	393	36	...	...	PUNCT
ejpam-4548	393	37	,	,	PUNCT
ejpam-4548	393	38	γi	γi	INTJ
ejpam-4548	393	39	...	...	PUNCT
ejpam-4548	393	40	of	of	ADP
ejpam-4548	393	41	l	l	NOUN
ejpam-4548	393	42	-	-	NOUN
ejpam-4548	393	43	subgroups	subgroup	NOUN
ejpam-4548	393	44	of	of	ADP
ejpam-4548	393	45	µ	µ	PRON
ejpam-4548	393	46	such	such	ADJ
ejpam-4548	393	47	that	that	SCONJ
ejpam-4548	393	48	η	η	PROPN
ejpam-4548	393	49	⊆	⊆	NUM
ejpam-4548	393	50	...	...	PUNCT
ejpam-4548	393	51	◁	◁	X
ejpam-4548	393	52	γi+1	γi+1	ADP
ejpam-4548	393	53	◁	◁	X
ejpam-4548	393	54	...	...	PUNCT
ejpam-4548	393	55	γ1	γ1	PROPN
ejpam-4548	394	1	◁	◁	X
ejpam-4548	394	2	γ0	γ0	PROPN
ejpam-4548	394	3	=	=	SYM
ejpam-4548	394	4	µ	µ	NOUN
ejpam-4548	394	5	,	,	PUNCT
ejpam-4548	394	6	then	then	ADV
ejpam-4548	394	7	ηi	ηi	VERB
ejpam-4548	394	8	⊆	⊆	NUM
ejpam-4548	394	9	γi	γi	NOUN
ejpam-4548	394	10	.	.	PUNCT
ejpam-4548	395	1	proof	proof	NOUN
ejpam-4548	395	2	.	.	PUNCT
ejpam-4548	396	1	we	we	PRON
ejpam-4548	396	2	shall	shall	AUX
ejpam-4548	396	3	prove	prove	VERB
ejpam-4548	396	4	the	the	DET
ejpam-4548	396	5	result	result	NOUN
ejpam-4548	396	6	by	by	ADP
ejpam-4548	396	7	induction	induction	NOUN
ejpam-4548	396	8	on	on	ADP
ejpam-4548	396	9	i.	i.	NOUN
ejpam-4548	396	10	for	for	ADP
ejpam-4548	396	11	i	i	PROPN
ejpam-4548	396	12	=	=	SYM
ejpam-4548	396	13	1	1	NUM
ejpam-4548	396	14	,	,	PUNCT
ejpam-4548	396	15	γ1	γ1	NOUN
ejpam-4548	396	16	is	be	AUX
ejpam-4548	396	17	a	a	DET
ejpam-4548	396	18	normal	normal	ADJ
ejpam-4548	396	19	lsubgroup	lsubgroup	NOUN
ejpam-4548	396	20	of	of	ADP
ejpam-4548	396	21	µ	µ	X
ejpam-4548	396	22	containing	contain	VERB
ejpam-4548	396	23	η	η	PROPN
ejpam-4548	396	24	.	.	PROPN
ejpam-4548	396	25	but	but	CCONJ
ejpam-4548	396	26	by	by	ADP
ejpam-4548	396	27	theorem	theorem	ADJ
ejpam-4548	396	28	8	8	NUM
ejpam-4548	396	29	,	,	PUNCT
ejpam-4548	396	30	η1	η1	NOUN
ejpam-4548	396	31	=	=	SYM
ejpam-4548	396	32	ηµ	ηµ	VERB
ejpam-4548	396	33	is	be	AUX
ejpam-4548	396	34	the	the	DET
ejpam-4548	396	35	smallest	small	ADJ
ejpam-4548	396	36	normal	normal	ADJ
ejpam-4548	396	37	subgroup	subgroup	NOUN
ejpam-4548	396	38	of	of	ADP
ejpam-4548	396	39	µ	µ	X
ejpam-4548	396	40	containing	contain	VERB
ejpam-4548	396	41	η	η	PROPN
ejpam-4548	396	42	.	.	PROPN
ejpam-4548	396	43	hence	hence	ADV
ejpam-4548	396	44	η1	η1	VERB
ejpam-4548	396	45	⊆	⊆	NUM
ejpam-4548	396	46	γ1	γ1	NOUN
ejpam-4548	396	47	.	.	PUNCT
ejpam-4548	397	1	now	now	ADV
ejpam-4548	397	2	,	,	PUNCT
ejpam-4548	397	3	we	we	PRON
ejpam-4548	397	4	set	set	VERB
ejpam-4548	397	5	the	the	DET
ejpam-4548	397	6	induction	induction	NOUN
ejpam-4548	397	7	hypothesis	hypothesis	NOUN
ejpam-4548	397	8	as	as	ADP
ejpam-4548	397	9	ηk	ηk	PROPN
ejpam-4548	397	10	⊆	⊆	NUM
ejpam-4548	397	11	γk	γk	NOUN
ejpam-4548	397	12	.	.	PUNCT
ejpam-4548	398	1	thus	thus	ADV
ejpam-4548	398	2	,	,	PUNCT
ejpam-4548	398	3	in	in	ADP
ejpam-4548	398	4	view	view	NOUN
ejpam-4548	398	5	of	of	ADP
ejpam-4548	398	6	the	the	DET
ejpam-4548	398	7	definition	definition	NOUN
ejpam-4548	398	8	of	of	ADP
ejpam-4548	398	9	a	a	DET
ejpam-4548	398	10	conjugate	conjugate	ADJ
ejpam-4548	398	11	l	l	NOUN
ejpam-4548	398	12	-	-	NOUN
ejpam-4548	398	13	subset	subset	NOUN
ejpam-4548	398	14	,	,	PUNCT
ejpam-4548	398	15	we	we	PRON
ejpam-4548	398	16	have	have	VERB
ejpam-4548	398	17	ηk+1	ηk+1	VERB
ejpam-4548	398	18	=	=	SYM
ejpam-4548	398	19	ηηk	ηηk	NOUN
ejpam-4548	398	20	⊆	⊆	NUM
ejpam-4548	398	21	ηγk	ηγk	NOUN
ejpam-4548	398	22	.	.	PUNCT
ejpam-4548	399	1	(	(	PUNCT
ejpam-4548	399	2	1	1	X
ejpam-4548	399	3	)	)	PUNCT
ejpam-4548	399	4	note	note	NOUN
ejpam-4548	399	5	that	that	SCONJ
ejpam-4548	399	6	as	as	ADP
ejpam-4548	399	7	η	η	PROPN
ejpam-4548	399	8	⊆	⊆	NUM
ejpam-4548	399	9	γk	γk	NOUN
ejpam-4548	399	10	,	,	PUNCT
ejpam-4548	399	11	by	by	ADP
ejpam-4548	399	12	theorem	theorem	NOUN
ejpam-4548	399	13	8	8	NUM
ejpam-4548	399	14	,	,	PUNCT
ejpam-4548	399	15	ηγk	ηγk	NOUN
ejpam-4548	399	16	is	be	AUX
ejpam-4548	399	17	the	the	DET
ejpam-4548	399	18	smallest	small	ADJ
ejpam-4548	399	19	normal	normal	ADJ
ejpam-4548	399	20	l	l	NOUN
ejpam-4548	399	21	-	-	NOUN
ejpam-4548	399	22	subgroup	subgroup	NOUN
ejpam-4548	399	23	of	of	ADP
ejpam-4548	399	24	γk	γk	PROPN
ejpam-4548	399	25	containing	contain	VERB
ejpam-4548	399	26	η	η	PROPN
ejpam-4548	399	27	.	.	PROPN
ejpam-4548	399	28	moreover	moreover	ADV
ejpam-4548	399	29	,	,	PUNCT
ejpam-4548	399	30	γk+1	γk+1	X
ejpam-4548	399	31	is	be	AUX
ejpam-4548	399	32	also	also	ADV
ejpam-4548	399	33	a	a	DET
ejpam-4548	399	34	normal	normal	ADJ
ejpam-4548	399	35	l	l	NOUN
ejpam-4548	399	36	-	-	NOUN
ejpam-4548	399	37	subgroup	subgroup	NOUN
ejpam-4548	399	38	of	of	ADP
ejpam-4548	399	39	γk	γk	PROPN
ejpam-4548	399	40	containing	contain	VERB
ejpam-4548	399	41	η	η	PROPN
ejpam-4548	399	42	.	.	PROPN
ejpam-4548	400	1	this	this	PRON
ejpam-4548	400	2	implies	imply	VERB
ejpam-4548	400	3	ηγk	ηγk	NOUN
ejpam-4548	400	4	⊆	⊆	NUM
ejpam-4548	400	5	γk+1	γk+1	NOUN
ejpam-4548	400	6	.	.	PUNCT
ejpam-4548	401	1	therefore	therefore	ADV
ejpam-4548	401	2	,	,	PUNCT
ejpam-4548	401	3	by	by	ADP
ejpam-4548	401	4	using	use	VERB
ejpam-4548	401	5	(	(	PUNCT
ejpam-4548	401	6	1	1	NUM
ejpam-4548	401	7	)	)	PUNCT
ejpam-4548	401	8	,	,	PUNCT
ejpam-4548	401	9	we	we	PRON
ejpam-4548	401	10	have	have	VERB
ejpam-4548	401	11	ηk+1	ηk+1	VERB
ejpam-4548	401	12	=	=	SYM
ejpam-4548	401	13	ηηk	ηηk	NOUN
ejpam-4548	401	14	⊆	⊆	NUM
ejpam-4548	401	15	ηγk	ηγk	NOUN
ejpam-4548	401	16	⊆	⊆	NUM
ejpam-4548	401	17	γk+1	γk+1	NOUN
ejpam-4548	401	18	.	.	PUNCT
ejpam-4548	402	1	this	this	PRON
ejpam-4548	402	2	proves	prove	VERB
ejpam-4548	402	3	the	the	DET
ejpam-4548	402	4	result	result	NOUN
ejpam-4548	402	5	completely	completely	ADV
ejpam-4548	402	6	.	.	PUNCT
ejpam-4548	403	1	the	the	DET
ejpam-4548	403	2	following	follow	VERB
ejpam-4548	403	3	result	result	VERB
ejpam-4548	403	4	inter	inter	NOUN
ejpam-4548	403	5	-	-	PUNCT
ejpam-4548	403	6	connects	connect	VERB
ejpam-4548	403	7	the	the	DET
ejpam-4548	403	8	notions	notion	NOUN
ejpam-4548	403	9	of	of	ADP
ejpam-4548	403	10	subnormality	subnormality	NOUN
ejpam-4548	403	11	and	and	CCONJ
ejpam-4548	403	12	subnormal	subnormal	ADJ
ejpam-4548	403	13	l	l	PROPN
ejpam-4548	403	14	-	-	NOUN
ejpam-4548	403	15	series	series	NOUN
ejpam-4548	403	16	of	of	ADP
ejpam-4548	403	17	an	an	DET
ejpam-4548	403	18	l	l	NOUN
ejpam-4548	403	19	-	-	NOUN
ejpam-4548	403	20	subgroup	subgroup	NOUN
ejpam-4548	403	21	:	:	PUNCT
ejpam-4548	403	22	theorem	theorem	NOUN
ejpam-4548	403	23	13	13	NUM
ejpam-4548	403	24	.	.	PUNCT
ejpam-4548	404	1	let	let	VERB
ejpam-4548	404	2	η	η	PROPN
ejpam-4548	404	3	∈	∈	PROPN
ejpam-4548	404	4	l(µ	l(µ	PROPN
ejpam-4548	404	5	)	)	PUNCT
ejpam-4548	404	6	.	.	PUNCT
ejpam-4548	405	1	then	then	ADV
ejpam-4548	405	2	,	,	PUNCT
ejpam-4548	405	3	η	η	PROPN
ejpam-4548	405	4	is	be	AUX
ejpam-4548	405	5	a	a	DET
ejpam-4548	405	6	subnormal	subnormal	ADJ
ejpam-4548	405	7	l	l	NOUN
ejpam-4548	405	8	-	-	NOUN
ejpam-4548	405	9	subgroup	subgroup	NOUN
ejpam-4548	405	10	of	of	ADP
ejpam-4548	405	11	µ	µ	NOUN
ejpam-4548	405	12	having	have	VERB
ejpam-4548	405	13	defect	defect	VERB
ejpam-4548	405	14	m	m	NOUN
ejpam-4548	405	15	if	if	SCONJ
ejpam-4548	406	1	and	and	CCONJ
ejpam-4548	406	2	only	only	ADV
ejpam-4548	406	3	if	if	SCONJ
ejpam-4548	406	4	η	η	PROPN
ejpam-4548	406	5	has	have	VERB
ejpam-4548	406	6	a	a	DET
ejpam-4548	406	7	subnormal	subnormal	ADJ
ejpam-4548	406	8	series	series	NOUN
ejpam-4548	406	9	η	η	PROPN
ejpam-4548	406	10	=	=	PROPN
ejpam-4548	406	11	γm	γm	PROPN
ejpam-4548	406	12	◁	◁	X
ejpam-4548	406	13	...	...	PUNCT
ejpam-4548	407	1	◁	◁	X
ejpam-4548	407	2	γi+1	γi+1	ADP
ejpam-4548	407	3	◁	◁	X
ejpam-4548	407	4	...	...	PUNCT
ejpam-4548	407	5	γ1	γ1	PROPN
ejpam-4548	407	6	◁	◁	X
ejpam-4548	407	7	γ0	γ0	PROPN
ejpam-4548	407	8	=	=	SYM
ejpam-4548	407	9	µ	µ	NOUN
ejpam-4548	407	10	,	,	PUNCT
ejpam-4548	407	11	of	of	ADP
ejpam-4548	407	12	length	length	NOUN
ejpam-4548	407	13	m	m	PROPN
ejpam-4548	407	14	and	and	CCONJ
ejpam-4548	407	15	m	m	PROPN
ejpam-4548	407	16	is	be	AUX
ejpam-4548	407	17	the	the	DET
ejpam-4548	407	18	smallest	small	ADJ
ejpam-4548	407	19	length	length	NOUN
ejpam-4548	407	20	of	of	ADP
ejpam-4548	407	21	such	such	DET
ejpam-4548	407	22	a	a	DET
ejpam-4548	407	23	subnormal	subnormal	ADJ
ejpam-4548	407	24	series	series	NOUN
ejpam-4548	407	25	.	.	PUNCT
ejpam-4548	408	1	proof	proof	NOUN
ejpam-4548	408	2	.	.	PUNCT
ejpam-4548	409	1	if	if	SCONJ
ejpam-4548	409	2	η	η	PROPN
ejpam-4548	409	3	is	be	AUX
ejpam-4548	409	4	a	a	DET
ejpam-4548	409	5	subnormal	subnormal	ADJ
ejpam-4548	409	6	l	l	NOUN
ejpam-4548	409	7	-	-	NOUN
ejpam-4548	409	8	subgroup	subgroup	NOUN
ejpam-4548	409	9	of	of	ADP
ejpam-4548	409	10	µ	µ	NOUN
ejpam-4548	409	11	,	,	PUNCT
ejpam-4548	409	12	then	then	ADV
ejpam-4548	409	13	the	the	DET
ejpam-4548	409	14	normal	normal	ADJ
ejpam-4548	409	15	closure	closure	NOUN
ejpam-4548	409	16	series	series	NOUN
ejpam-4548	409	17	is	be	AUX
ejpam-4548	409	18	the	the	DET
ejpam-4548	409	19	required	require	VERB
ejpam-4548	409	20	subnormal	subnormal	ADJ
ejpam-4548	409	21	series	series	NOUN
ejpam-4548	409	22	.	.	PUNCT
ejpam-4548	410	1	conversely	conversely	ADV
ejpam-4548	410	2	,	,	PUNCT
ejpam-4548	410	3	suppose	suppose	VERB
ejpam-4548	410	4	that	that	SCONJ
ejpam-4548	410	5	η	η	PROPN
ejpam-4548	410	6	has	have	VERB
ejpam-4548	410	7	a	a	DET
ejpam-4548	410	8	subnormal	subnormal	ADJ
ejpam-4548	410	9	series	series	NOUN
ejpam-4548	410	10	say	say	VERB
ejpam-4548	410	11	η	η	PROPN
ejpam-4548	410	12	=	=	PROPN
ejpam-4548	410	13	θm	θm	PROPN
ejpam-4548	410	14	◁	◁	PROPN
ejpam-4548	410	15	θm−1	θm−1	PROPN
ejpam-4548	410	16	◁	◁	X
ejpam-4548	410	17	...	...	PUNCT
ejpam-4548	411	1	◁	◁	X
ejpam-4548	411	2	θ0	θ0	PROPN
ejpam-4548	411	3	=	=	SYM
ejpam-4548	411	4	µ.	µ.	PROPN
ejpam-4548	411	5	n.	n.	PROPN
ejpam-4548	411	6	ajmal	ajmal	PROPN
ejpam-4548	411	7	,	,	PUNCT
ejpam-4548	411	8	i.	i.	PROPN
ejpam-4548	411	9	jahan	jahan	PROPN
ejpam-4548	411	10	.	.	PROPN
ejpam-4548	411	11	,	,	PUNCT
ejpam-4548	411	12	b.	b.	PROPN
ejpam-4548	411	13	davvaz	davvaz	PROPN
ejpam-4548	411	14	/	/	SYM
ejpam-4548	411	15	eur	eur	PROPN
ejpam-4548	411	16	.	.	PUNCT
ejpam-4548	412	1	j.	j.	PROPN
ejpam-4548	412	2	pure	pure	PROPN
ejpam-4548	412	3	appl	appl	PROPN
ejpam-4548	412	4	.	.	PROPN
ejpam-4548	412	5	math	math	PROPN
ejpam-4548	412	6	,	,	PUNCT
ejpam-4548	412	7	15	15	NUM
ejpam-4548	412	8	(	(	PUNCT
ejpam-4548	412	9	4	4	NUM
ejpam-4548	412	10	)	)	PUNCT
ejpam-4548	412	11	(	(	PUNCT
ejpam-4548	412	12	2022	2022	NUM
ejpam-4548	412	13	)	)	PUNCT
ejpam-4548	412	14	,	,	PUNCT
ejpam-4548	412	15	2086	2086	NUM
ejpam-4548	412	16	-	-	SYM
ejpam-4548	412	17	2115	2115	NUM
ejpam-4548	412	18	2105	2105	NUM
ejpam-4548	412	19	then	then	ADV
ejpam-4548	412	20	,	,	PUNCT
ejpam-4548	412	21	by	by	ADP
ejpam-4548	412	22	lemma	lemma	PROPN
ejpam-4548	412	23	2	2	NUM
ejpam-4548	412	24	,	,	PUNCT
ejpam-4548	412	25	ηm	ηm	NOUN
ejpam-4548	412	26	⊆	⊆	NUM
ejpam-4548	412	27	θm	θm	NOUN
ejpam-4548	412	28	.	.	PUNCT
ejpam-4548	413	1	as	as	ADP
ejpam-4548	413	2	η	η	PROPN
ejpam-4548	413	3	=	=	SYM
ejpam-4548	413	4	θm	θm	PROPN
ejpam-4548	413	5	,	,	PUNCT
ejpam-4548	413	6	we	we	PRON
ejpam-4548	413	7	have	have	VERB
ejpam-4548	413	8	η	η	PROPN
ejpam-4548	413	9	⊆	⊆	NUM
ejpam-4548	413	10	ηm	ηm	NOUN
ejpam-4548	413	11	⊆	⊆	NUM
ejpam-4548	413	12	θm	θm	PROPN
ejpam-4548	413	13	=	=	SYM
ejpam-4548	413	14	η	η	PROPN
ejpam-4548	413	15	.	.	PROPN
ejpam-4548	413	16	hence	hence	ADV
ejpam-4548	413	17	,	,	PUNCT
ejpam-4548	413	18	ηm	ηm	PROPN
ejpam-4548	413	19	=	=	SYM
ejpam-4548	413	20	η	η	PROPN
ejpam-4548	413	21	.	.	PROPN
ejpam-4548	414	1	this	this	PRON
ejpam-4548	414	2	proves	prove	VERB
ejpam-4548	414	3	that	that	SCONJ
ejpam-4548	414	4	η	η	PROPN
ejpam-4548	414	5	is	be	AUX
ejpam-4548	414	6	a	a	DET
ejpam-4548	414	7	subnormal	subnormal	ADJ
ejpam-4548	414	8	l	l	NOUN
ejpam-4548	414	9	-	-	NOUN
ejpam-4548	414	10	subgroup	subgroup	NOUN
ejpam-4548	414	11	of	of	ADP
ejpam-4548	414	12	µ.	µ.	PROPN
ejpam-4548	414	13	the	the	DET
ejpam-4548	414	14	following	follow	VERB
ejpam-4548	414	15	results	result	NOUN
ejpam-4548	414	16	are	be	AUX
ejpam-4548	414	17	established	establish	VERB
ejpam-4548	414	18	by	by	ADP
ejpam-4548	414	19	using	use	VERB
ejpam-4548	414	20	the	the	DET
ejpam-4548	414	21	above	above	ADJ
ejpam-4548	414	22	theorem	theorem	NOUN
ejpam-4548	414	23	:	:	PUNCT
ejpam-4548	414	24	theorem	theorem	NOUN
ejpam-4548	414	25	14	14	NUM
ejpam-4548	414	26	.	.	PUNCT
ejpam-4548	415	1	let	let	VERB
ejpam-4548	415	2	η	η	X
ejpam-4548	415	3	be	be	AUX
ejpam-4548	415	4	a	a	DET
ejpam-4548	415	5	subnormal	subnormal	ADJ
ejpam-4548	415	6	l	l	NOUN
ejpam-4548	415	7	-	-	NOUN
ejpam-4548	415	8	subgroup	subgroup	NOUN
ejpam-4548	415	9	of	of	ADP
ejpam-4548	415	10	µ	µ	NOUN
ejpam-4548	415	11	with	with	ADP
ejpam-4548	415	12	defect	defect	NOUN
ejpam-4548	415	13	m.	m.	NOUN
ejpam-4548	415	14	(	(	PUNCT
ejpam-4548	415	15	i	i	NOUN
ejpam-4548	415	16	)	)	PUNCT
ejpam-4548	415	17	let	let	VERB
ejpam-4548	415	18	θ	θ	PROPN
ejpam-4548	415	19	∈	∈	PROPN
ejpam-4548	415	20	l(µ	l(µ	PROPN
ejpam-4548	415	21	)	)	PUNCT
ejpam-4548	415	22	.	.	PUNCT
ejpam-4548	416	1	then	then	ADV
ejpam-4548	416	2	,	,	PUNCT
ejpam-4548	416	3	η	η	PROPN
ejpam-4548	416	4	∩	∩	PROPN
ejpam-4548	416	5	θ	θ	PROPN
ejpam-4548	416	6	is	be	AUX
ejpam-4548	416	7	a	a	DET
ejpam-4548	416	8	subnormal	subnormal	ADJ
ejpam-4548	416	9	l	l	NOUN
ejpam-4548	416	10	-	-	NOUN
ejpam-4548	416	11	subgroup	subgroup	NOUN
ejpam-4548	416	12	of	of	ADP
ejpam-4548	416	13	θ	θ	PROPN
ejpam-4548	416	14	with	with	ADP
ejpam-4548	416	15	defect	defect	NOUN
ejpam-4548	416	16	c	c	NOUN
ejpam-4548	416	17	where	where	SCONJ
ejpam-4548	416	18	c	c	NOUN
ejpam-4548	416	19	≤	≤	X
ejpam-4548	416	20	m.	m.	NOUN
ejpam-4548	416	21	in	in	ADP
ejpam-4548	416	22	particular	particular	ADJ
ejpam-4548	416	23	,	,	PUNCT
ejpam-4548	416	24	η	η	PROPN
ejpam-4548	416	25	is	be	AUX
ejpam-4548	416	26	a	a	DET
ejpam-4548	416	27	subnormal	subnormal	ADJ
ejpam-4548	416	28	l	l	NOUN
ejpam-4548	416	29	-	-	NOUN
ejpam-4548	416	30	subgroup	subgroup	NOUN
ejpam-4548	416	31	of	of	ADP
ejpam-4548	416	32	λ	λ	PROPN
ejpam-4548	416	33	where	where	SCONJ
ejpam-4548	416	34	λ	λ	X
ejpam-4548	416	35	∈	∈	PROPN
ejpam-4548	416	36	l(µ	l(µ	PROPN
ejpam-4548	416	37	)	)	PUNCT
ejpam-4548	416	38	such	such	ADJ
ejpam-4548	416	39	that	that	SCONJ
ejpam-4548	416	40	η	η	PROPN
ejpam-4548	416	41	⊆	⊆	NUM
ejpam-4548	416	42	λ	λ	PROPN
ejpam-4548	416	43	with	with	ADP
ejpam-4548	416	44	defect	defect	NOUN
ejpam-4548	416	45	c	c	NOUN
ejpam-4548	416	46	where	where	SCONJ
ejpam-4548	416	47	c	c	NOUN
ejpam-4548	416	48	≤	≤	NOUN
ejpam-4548	416	49	m	m	ADP
ejpam-4548	416	50	,	,	PUNCT
ejpam-4548	416	51	(	(	PUNCT
ejpam-4548	416	52	ii	ii	NOUN
ejpam-4548	416	53	)	)	PUNCT
ejpam-4548	416	54	let	let	VERB
ejpam-4548	416	55	θ	θ	PROPN
ejpam-4548	416	56	∈	∈	PROPN
ejpam-4548	416	57	nl(µ	nl(µ	NOUN
ejpam-4548	416	58	)	)	PUNCT
ejpam-4548	416	59	.	.	PUNCT
ejpam-4548	417	1	then	then	ADV
ejpam-4548	417	2	,	,	PUNCT
ejpam-4548	417	3	η	η	PROPN
ejpam-4548	417	4	◦	◦	PROPN
ejpam-4548	417	5	θ	θ	PROPN
ejpam-4548	417	6	is	be	AUX
ejpam-4548	417	7	a	a	DET
ejpam-4548	417	8	subnormal	subnormal	ADJ
ejpam-4548	417	9	l	l	NOUN
ejpam-4548	417	10	-	-	NOUN
ejpam-4548	417	11	subgroup	subgroup	NOUN
ejpam-4548	417	12	of	of	ADP
ejpam-4548	417	13	µ	µ	NOUN
ejpam-4548	417	14	with	with	ADP
ejpam-4548	417	15	defect	defect	NOUN
ejpam-4548	417	16	c	c	NOUN
ejpam-4548	417	17	where	where	SCONJ
ejpam-4548	417	18	c	c	NOUN
ejpam-4548	417	19	≤	≤	NUM
ejpam-4548	417	20	m.	m.	NOUN
ejpam-4548	417	21	it	it	PRON
ejpam-4548	417	22	can	can	AUX
ejpam-4548	417	23	be	be	AUX
ejpam-4548	417	24	seen	see	VERB
ejpam-4548	417	25	easily	easily	ADV
ejpam-4548	417	26	that	that	SCONJ
ejpam-4548	417	27	the	the	DET
ejpam-4548	417	28	intersection	intersection	NOUN
ejpam-4548	417	29	of	of	ADP
ejpam-4548	417	30	any	any	DET
ejpam-4548	417	31	finite	finite	NOUN
ejpam-4548	417	32	set	set	NOUN
ejpam-4548	417	33	of	of	ADP
ejpam-4548	417	34	subnormal	subnormal	ADJ
ejpam-4548	417	35	l	l	NOUN
ejpam-4548	417	36	-	-	NOUN
ejpam-4548	417	37	subgroups	subgroup	NOUN
ejpam-4548	417	38	is	be	AUX
ejpam-4548	417	39	again	again	ADV
ejpam-4548	417	40	subnormal	subnormal	ADJ
ejpam-4548	417	41	.	.	PUNCT
ejpam-4548	418	1	more	more	ADV
ejpam-4548	418	2	generally	generally	ADV
ejpam-4548	418	3	:	:	PUNCT
ejpam-4548	418	4	theorem	theorem	VERB
ejpam-4548	418	5	15	15	NUM
ejpam-4548	418	6	.	.	PUNCT
ejpam-4548	419	1	let	let	VERB
ejpam-4548	419	2	{	{	PUNCT
ejpam-4548	419	3	θi	θi	X
ejpam-4548	419	4	:	:	PUNCT
ejpam-4548	419	5	i	i	PRON
ejpam-4548	419	6	∈	∈	PROPN
ejpam-4548	420	1	i	i	PRON
ejpam-4548	420	2	}	}	PUNCT
ejpam-4548	420	3	be	be	VERB
ejpam-4548	420	4	a	a	DET
ejpam-4548	420	5	family	family	NOUN
ejpam-4548	420	6	of	of	ADP
ejpam-4548	420	7	subnormal	subnormal	ADJ
ejpam-4548	420	8	l	l	NOUN
ejpam-4548	420	9	-	-	NOUN
ejpam-4548	420	10	subgroups	subgroup	NOUN
ejpam-4548	420	11	such	such	ADJ
ejpam-4548	420	12	that	that	DET
ejpam-4548	420	13	defect	defect	NOUN
ejpam-4548	420	14	of	of	ADP
ejpam-4548	420	15	θi	θi	NOUN
ejpam-4548	420	16	is	be	AUX
ejpam-4548	420	17	mi	mi	PROPN
ejpam-4548	420	18	where	where	SCONJ
ejpam-4548	420	19	mi	mi	PROPN
ejpam-4548	420	20	≤	≤	PROPN
ejpam-4548	420	21	m.	m.	NOUN
ejpam-4548	420	22	then	then	ADV
ejpam-4548	420	23	,	,	PUNCT
ejpam-4548	420	24	⋂	⋂	PROPN
ejpam-4548	420	25	i∈i	i∈i	ADJ
ejpam-4548	420	26	θi	θi	PROPN
ejpam-4548	420	27	is	be	AUX
ejpam-4548	420	28	a	a	DET
ejpam-4548	420	29	subnormal	subnormal	ADJ
ejpam-4548	420	30	l	l	NOUN
ejpam-4548	420	31	-	-	NOUN
ejpam-4548	420	32	subgroup	subgroup	NOUN
ejpam-4548	420	33	of	of	ADP
ejpam-4548	420	34	µ	µ	NOUN
ejpam-4548	420	35	with	with	ADP
ejpam-4548	420	36	defect	defect	NOUN
ejpam-4548	420	37	c	c	NOUN
ejpam-4548	420	38	where	where	SCONJ
ejpam-4548	420	39	c	c	NOUN
ejpam-4548	420	40	≤	≤	X
ejpam-4548	420	41	m.	m.	NOUN
ejpam-4548	420	42	next	next	ADJ
ejpam-4548	420	43	result	result	NOUN
ejpam-4548	420	44	determines	determine	VERB
ejpam-4548	420	45	the	the	DET
ejpam-4548	420	46	transitivity	transitivity	NOUN
ejpam-4548	420	47	of	of	ADP
ejpam-4548	420	48	the	the	DET
ejpam-4548	420	49	notion	notion	NOUN
ejpam-4548	420	50	of	of	ADP
ejpam-4548	420	51	subnormality	subnormality	NOUN
ejpam-4548	420	52	.	.	PUNCT
ejpam-4548	421	1	theorem	theorem	VERB
ejpam-4548	421	2	16	16	NUM
ejpam-4548	421	3	.	.	PUNCT
ejpam-4548	422	1	let	let	VERB
ejpam-4548	422	2	η	η	PROPN
ejpam-4548	422	3	,	,	PUNCT
ejpam-4548	422	4	θ	θ	PROPN
ejpam-4548	422	5	∈	∈	PROPN
ejpam-4548	422	6	l(µ	l(µ	PROPN
ejpam-4548	422	7	)	)	PUNCT
ejpam-4548	422	8	such	such	ADJ
ejpam-4548	422	9	that	that	SCONJ
ejpam-4548	422	10	η	η	PROPN
ejpam-4548	422	11	is	be	AUX
ejpam-4548	422	12	a	a	DET
ejpam-4548	422	13	subnormal	subnormal	ADJ
ejpam-4548	422	14	l	l	NOUN
ejpam-4548	422	15	-	-	NOUN
ejpam-4548	422	16	subgroup	subgroup	NOUN
ejpam-4548	422	17	of	of	ADP
ejpam-4548	422	18	θ	θ	PROPN
ejpam-4548	422	19	with	with	ADP
ejpam-4548	422	20	defect	defect	NOUN
ejpam-4548	422	21	m	m	NOUN
ejpam-4548	422	22	and	and	CCONJ
ejpam-4548	422	23	θ	θ	PROPN
ejpam-4548	422	24	is	be	AUX
ejpam-4548	422	25	a	a	DET
ejpam-4548	422	26	subnormal	subnormal	ADJ
ejpam-4548	422	27	l	l	NOUN
ejpam-4548	422	28	-	-	NOUN
ejpam-4548	422	29	subgroup	subgroup	NOUN
ejpam-4548	422	30	of	of	ADP
ejpam-4548	422	31	µ	µ	NOUN
ejpam-4548	422	32	with	with	ADP
ejpam-4548	422	33	defect	defect	NOUN
ejpam-4548	422	34	n.	n.	NOUN
ejpam-4548	422	35	then	then	ADV
ejpam-4548	422	36	,	,	PUNCT
ejpam-4548	422	37	η	η	PROPN
ejpam-4548	422	38	is	be	AUX
ejpam-4548	422	39	a	a	DET
ejpam-4548	422	40	subnormal	subnormal	ADJ
ejpam-4548	422	41	l	l	NOUN
ejpam-4548	422	42	-	-	NOUN
ejpam-4548	422	43	subgroup	subgroup	NOUN
ejpam-4548	422	44	of	of	ADP
ejpam-4548	422	45	µ	µ	NOUN
ejpam-4548	422	46	with	with	ADP
ejpam-4548	422	47	defect	defect	NOUN
ejpam-4548	422	48	m+n	m+n	PROPN
ejpam-4548	422	49	.	.	PUNCT
ejpam-4548	423	1	the	the	DET
ejpam-4548	423	2	following	follow	VERB
ejpam-4548	423	3	theorem	theorem	NOUN
ejpam-4548	423	4	establishes	establish	VERB
ejpam-4548	423	5	that	that	SCONJ
ejpam-4548	423	6	the	the	DET
ejpam-4548	423	7	subnormality	subnormality	NOUN
ejpam-4548	423	8	in	in	ADP
ejpam-4548	423	9	l	l	NOUN
ejpam-4548	423	10	-	-	PUNCT
ejpam-4548	423	11	setting	setting	NOUN
ejpam-4548	423	12	is	be	AUX
ejpam-4548	423	13	also	also	ADV
ejpam-4548	423	14	preserved	preserve	VERB
ejpam-4548	423	15	under	under	ADP
ejpam-4548	423	16	the	the	DET
ejpam-4548	423	17	action	action	NOUN
ejpam-4548	423	18	of	of	ADP
ejpam-4548	423	19	a	a	DET
ejpam-4548	423	20	homomorphism	homomorphism	NOUN
ejpam-4548	423	21	and	and	CCONJ
ejpam-4548	423	22	its	its	PRON
ejpam-4548	423	23	inverse	inverse	NOUN
ejpam-4548	423	24	image	image	NOUN
ejpam-4548	423	25	:	:	PUNCT
ejpam-4548	423	26	theorem	theorem	VERB
ejpam-4548	423	27	17	17	NUM
ejpam-4548	423	28	.	.	PUNCT
ejpam-4548	424	1	let	let	VERB
ejpam-4548	424	2	η	η	PROPN
ejpam-4548	424	3	∈	∈	PROPN
ejpam-4548	424	4	l(µ	l(µ	PROPN
ejpam-4548	424	5	)	)	PUNCT
ejpam-4548	424	6	and	and	CCONJ
ejpam-4548	424	7	f	f	X
ejpam-4548	424	8	:	:	PUNCT
ejpam-4548	424	9	g	g	PROPN
ejpam-4548	424	10	→	→	SYM
ejpam-4548	424	11	k	k	X
ejpam-4548	424	12	be	be	AUX
ejpam-4548	424	13	a	a	DET
ejpam-4548	424	14	group	group	NOUN
ejpam-4548	424	15	homomorphism	homomorphism	NOUN
ejpam-4548	424	16	.	.	PUNCT
ejpam-4548	425	1	then	then	ADV
ejpam-4548	425	2	,	,	PUNCT
ejpam-4548	425	3	(	(	PUNCT
ejpam-4548	425	4	i	i	NOUN
ejpam-4548	425	5	)	)	PUNCT
ejpam-4548	425	6	if	if	SCONJ
ejpam-4548	425	7	η	η	PROPN
ejpam-4548	425	8	is	be	AUX
ejpam-4548	425	9	a	a	DET
ejpam-4548	425	10	subnormal	subnormal	ADJ
ejpam-4548	425	11	l	l	NOUN
ejpam-4548	425	12	-	-	NOUN
ejpam-4548	425	13	subgroup	subgroup	NOUN
ejpam-4548	425	14	of	of	ADP
ejpam-4548	425	15	µ	µ	NOUN
ejpam-4548	425	16	with	with	ADP
ejpam-4548	425	17	defect	defect	NOUN
ejpam-4548	425	18	n	n	CCONJ
ejpam-4548	425	19	,	,	PUNCT
ejpam-4548	425	20	then	then	ADV
ejpam-4548	425	21	f(η	f(η	PROPN
ejpam-4548	425	22	)	)	PUNCT
ejpam-4548	425	23	is	be	AUX
ejpam-4548	425	24	a	a	DET
ejpam-4548	425	25	subnormal	subnormal	ADJ
ejpam-4548	425	26	l	l	NOUN
ejpam-4548	425	27	-	-	NOUN
ejpam-4548	425	28	subgroup	subgroup	NOUN
ejpam-4548	425	29	of	of	ADP
ejpam-4548	425	30	f(µ	f(µ	PROPN
ejpam-4548	425	31	)	)	PUNCT
ejpam-4548	425	32	with	with	ADP
ejpam-4548	425	33	defect	defect	NOUN
ejpam-4548	425	34	m	m	VERB
ejpam-4548	425	35	where	where	SCONJ
ejpam-4548	425	36	m	m	VERB
ejpam-4548	425	37	≤	≤	NOUN
ejpam-4548	425	38	n	n	CCONJ
ejpam-4548	425	39	,	,	PUNCT
ejpam-4548	425	40	(	(	PUNCT
ejpam-4548	425	41	ii	ii	NOUN
ejpam-4548	425	42	)	)	PUNCT
ejpam-4548	425	43	if	if	SCONJ
ejpam-4548	425	44	η	η	PROPN
ejpam-4548	425	45	is	be	AUX
ejpam-4548	425	46	a	a	DET
ejpam-4548	425	47	subnormal	subnormal	ADJ
ejpam-4548	425	48	l	l	NOUN
ejpam-4548	425	49	-	-	NOUN
ejpam-4548	425	50	subgroup	subgroup	NOUN
ejpam-4548	425	51	of	of	ADP
ejpam-4548	425	52	µ	µ	NOUN
ejpam-4548	425	53	with	with	ADP
ejpam-4548	425	54	defect	defect	NOUN
ejpam-4548	425	55	n	n	CCONJ
ejpam-4548	425	56	,	,	PUNCT
ejpam-4548	425	57	then	then	ADV
ejpam-4548	425	58	f−1(η	f−1(η	PROPN
ejpam-4548	425	59	)	)	PUNCT
ejpam-4548	425	60	is	be	AUX
ejpam-4548	425	61	a	a	DET
ejpam-4548	425	62	subnormal	subnormal	ADJ
ejpam-4548	425	63	l	l	NOUN
ejpam-4548	425	64	-	-	NOUN
ejpam-4548	425	65	subgroup	subgroup	NOUN
ejpam-4548	425	66	of	of	ADP
ejpam-4548	425	67	f−1(µ	f−1(µ	PROPN
ejpam-4548	425	68	)	)	PUNCT
ejpam-4548	425	69	with	with	ADP
ejpam-4548	425	70	defect	defect	NOUN
ejpam-4548	425	71	m	m	VERB
ejpam-4548	425	72	where	where	SCONJ
ejpam-4548	425	73	m	m	VERB
ejpam-4548	425	74	≤	≤	NOUN
ejpam-4548	425	75	n	n	CCONJ
ejpam-4548	425	76	,	,	PUNCT
ejpam-4548	425	77	provided	provide	VERB
ejpam-4548	425	78	that	that	SCONJ
ejpam-4548	425	79	the	the	DET
ejpam-4548	425	80	group	group	NOUN
ejpam-4548	425	81	homomoirphism	homomoirphism	PROPN
ejpam-4548	425	82	f	f	PROPN
ejpam-4548	425	83	is	be	AUX
ejpam-4548	425	84	onto	onto	ADP
ejpam-4548	425	85	.	.	PROPN
ejpam-4548	426	1	4	4	NUM
ejpam-4548	426	2	.	.	X
ejpam-4548	426	3	subnormal	subnormal	ADJ
ejpam-4548	426	4	l	l	NOUN
ejpam-4548	426	5	-	-	PUNCT
ejpam-4548	426	6	subgroups	subgroup	NOUN
ejpam-4548	426	7	and	and	CCONJ
ejpam-4548	426	8	nilpotency	nilpotency	NOUN
ejpam-4548	426	9	in	in	ADP
ejpam-4548	426	10	this	this	DET
ejpam-4548	426	11	section	section	NOUN
ejpam-4548	426	12	,	,	PUNCT
ejpam-4548	426	13	we	we	PRON
ejpam-4548	426	14	characterize	characterize	VERB
ejpam-4548	426	15	subnormal	subnormal	ADJ
ejpam-4548	426	16	l	l	NOUN
ejpam-4548	426	17	-	-	NOUN
ejpam-4548	426	18	subgroups	subgroup	NOUN
ejpam-4548	426	19	by	by	ADP
ejpam-4548	426	20	the	the	DET
ejpam-4548	426	21	usual	usual	ADJ
ejpam-4548	426	22	group	group	NOUN
ejpam-4548	426	23	theoretic	theoretic	ADJ
ejpam-4548	426	24	subnormality	subnormality	NOUN
ejpam-4548	426	25	of	of	ADP
ejpam-4548	426	26	the	the	DET
ejpam-4548	426	27	level	level	NOUN
ejpam-4548	426	28	subsets	subset	NOUN
ejpam-4548	426	29	of	of	ADP
ejpam-4548	426	30	the	the	DET
ejpam-4548	426	31	given	give	VERB
ejpam-4548	426	32	l	l	NOUN
ejpam-4548	426	33	-	-	NOUN
ejpam-4548	426	34	subgroups	subgroup	NOUN
ejpam-4548	426	35	.	.	PUNCT
ejpam-4548	427	1	we	we	PRON
ejpam-4548	427	2	shall	shall	AUX
ejpam-4548	427	3	refer	refer	VERB
ejpam-4548	427	4	this	this	PRON
ejpam-4548	427	5	as	as	ADP
ejpam-4548	427	6	a	a	DET
ejpam-4548	427	7	level	level	NOUN
ejpam-4548	427	8	subset	subset	NOUN
ejpam-4548	427	9	characterization	characterization	NOUN
ejpam-4548	427	10	of	of	ADP
ejpam-4548	427	11	subnormality	subnormality	NOUN
ejpam-4548	427	12	.	.	PUNCT
ejpam-4548	428	1	then	then	ADV
ejpam-4548	428	2	,	,	PUNCT
ejpam-4548	428	3	this	this	DET
ejpam-4548	428	4	characterization	characterization	NOUN
ejpam-4548	428	5	is	be	AUX
ejpam-4548	428	6	used	use	VERB
ejpam-4548	428	7	to	to	PART
ejpam-4548	428	8	establish	establish	VERB
ejpam-4548	428	9	that	that	SCONJ
ejpam-4548	428	10	when	when	SCONJ
ejpam-4548	428	11	the	the	DET
ejpam-4548	428	12	lattice	lattice	PROPN
ejpam-4548	428	13	l	l	PROPN
ejpam-4548	428	14	is	be	AUX
ejpam-4548	428	15	an	an	DET
ejpam-4548	428	16	upper	upper	ADJ
ejpam-4548	428	17	well	well	NOUN
ejpam-4548	428	18	ordered	order	VERB
ejpam-4548	428	19	chain	chain	NOUN
ejpam-4548	428	20	,	,	PUNCT
ejpam-4548	428	21	then	then	ADV
ejpam-4548	428	22	every	every	DET
ejpam-4548	428	23	l	l	NOUN
ejpam-4548	428	24	-	-	NOUN
ejpam-4548	428	25	subgroup	subgroup	NOUN
ejpam-4548	428	26	of	of	ADP
ejpam-4548	428	27	a	a	DET
ejpam-4548	428	28	nilpotent	nilpotent	ADJ
ejpam-4548	428	29	l	l	NOUN
ejpam-4548	428	30	-	-	NOUN
ejpam-4548	428	31	group	group	NOUN
ejpam-4548	428	32	is	be	AUX
ejpam-4548	428	33	subnormal	subnormal	ADJ
ejpam-4548	428	34	.	.	PUNCT
ejpam-4548	429	1	for	for	ADP
ejpam-4548	429	2	this	this	DET
ejpam-4548	429	3	purpose	purpose	NOUN
ejpam-4548	429	4	,	,	PUNCT
ejpam-4548	429	5	we	we	PRON
ejpam-4548	429	6	need	need	VERB
ejpam-4548	429	7	to	to	PART
ejpam-4548	429	8	develop	develop	VERB
ejpam-4548	429	9	a	a	DET
ejpam-4548	429	10	nececcary	nececcary	ADJ
ejpam-4548	429	11	mechanism	mechanism	NOUN
ejpam-4548	429	12	.	.	PUNCT
ejpam-4548	430	1	so	so	ADV
ejpam-4548	430	2	we	we	PRON
ejpam-4548	430	3	start	start	VERB
ejpam-4548	430	4	with	with	ADP
ejpam-4548	430	5	a	a	DET
ejpam-4548	430	6	characterization	characterization	NOUN
ejpam-4548	430	7	of	of	ADP
ejpam-4548	430	8	the	the	DET
ejpam-4548	430	9	notion	notion	NOUN
ejpam-4548	430	10	of	of	ADP
ejpam-4548	430	11	sup	sup	NOUN
ejpam-4548	430	12	-	-	PUNCT
ejpam-4548	430	13	property	property	NOUN
ejpam-4548	430	14	which	which	PRON
ejpam-4548	430	15	lends	lend	VERB
ejpam-4548	430	16	itself	itself	PRON
ejpam-4548	430	17	more	more	ADV
ejpam-4548	430	18	easily	easily	ADV
ejpam-4548	430	19	for	for	ADP
ejpam-4548	430	20	applications	application	NOUN
ejpam-4548	430	21	.	.	PUNCT
ejpam-4548	431	1	n.	n.	PROPN
ejpam-4548	431	2	ajmal	ajmal	PROPN
ejpam-4548	431	3	,	,	PUNCT
ejpam-4548	431	4	i.	i.	PROPN
ejpam-4548	431	5	jahan	jahan	PROPN
ejpam-4548	431	6	.	.	PROPN
ejpam-4548	431	7	,	,	PUNCT
ejpam-4548	431	8	b.	b.	PROPN
ejpam-4548	431	9	davvaz	davvaz	PROPN
ejpam-4548	431	10	/	/	SYM
ejpam-4548	431	11	eur	eur	PROPN
ejpam-4548	431	12	.	.	PUNCT
ejpam-4548	432	1	j.	j.	PROPN
ejpam-4548	432	2	pure	pure	PROPN
ejpam-4548	432	3	appl	appl	PROPN
ejpam-4548	432	4	.	.	PROPN
ejpam-4548	432	5	math	math	PROPN
ejpam-4548	432	6	,	,	PUNCT
ejpam-4548	432	7	15	15	NUM
ejpam-4548	432	8	(	(	PUNCT
ejpam-4548	432	9	4	4	NUM
ejpam-4548	432	10	)	)	PUNCT
ejpam-4548	432	11	(	(	PUNCT
ejpam-4548	432	12	2022	2022	NUM
ejpam-4548	432	13	)	)	PUNCT
ejpam-4548	432	14	,	,	PUNCT
ejpam-4548	432	15	2086	2086	NUM
ejpam-4548	432	16	-	-	SYM
ejpam-4548	432	17	2115	2115	NUM
ejpam-4548	432	18	2106	2106	NUM
ejpam-4548	432	19	definition	definition	NOUN
ejpam-4548	432	20	14	14	NUM
ejpam-4548	432	21	.	.	PUNCT
ejpam-4548	433	1	a	a	DET
ejpam-4548	433	2	non	non	ADJ
ejpam-4548	433	3	-	-	ADJ
ejpam-4548	433	4	empty	empty	ADJ
ejpam-4548	433	5	subset	subset	NOUN
ejpam-4548	433	6	x	x	PUNCT
ejpam-4548	433	7	of	of	ADP
ejpam-4548	433	8	a	a	DET
ejpam-4548	433	9	lattice	lattice	NOUN
ejpam-4548	433	10	l	l	NOUN
ejpam-4548	433	11	is	be	AUX
ejpam-4548	433	12	said	say	VERB
ejpam-4548	433	13	to	to	PART
ejpam-4548	433	14	be	be	AUX
ejpam-4548	433	15	a	a	DET
ejpam-4548	433	16	supstar	supstar	NOUN
ejpam-4548	433	17	subset	subset	NOUN
ejpam-4548	433	18	of	of	ADP
ejpam-4548	433	19	l	l	NOUN
ejpam-4548	433	20	if	if	SCONJ
ejpam-4548	433	21	every	every	DET
ejpam-4548	433	22	non	non	ADJ
ejpam-4548	433	23	-	-	ADJ
ejpam-4548	433	24	empty	empty	ADJ
ejpam-4548	433	25	subset	subset	NOUN
ejpam-4548	433	26	a	a	PRON
ejpam-4548	433	27	of	of	ADP
ejpam-4548	433	28	x	x	PUNCT
ejpam-4548	433	29	contains	contain	VERB
ejpam-4548	433	30	its	its	PRON
ejpam-4548	433	31	supremum	supremum	NOUN
ejpam-4548	433	32	.	.	PUNCT
ejpam-4548	434	1	that	that	PRON
ejpam-4548	434	2	is	be	AUX
ejpam-4548	434	3	,	,	PUNCT
ejpam-4548	434	4	if	if	SCONJ
ejpam-4548	434	5	supa	supa	NOUN
ejpam-4548	434	6	=	=	SYM
ejpam-4548	434	7	a0	a0	PROPN
ejpam-4548	434	8	,	,	PUNCT
ejpam-4548	434	9	then	then	ADV
ejpam-4548	434	10	a0	a0	PROPN
ejpam-4548	434	11	∈	∈	PROPN
ejpam-4548	434	12	a.	a.	NOUN
ejpam-4548	434	13	clearly	clearly	ADV
ejpam-4548	434	14	,	,	PUNCT
ejpam-4548	434	15	a	a	DET
ejpam-4548	434	16	subset	subset	NOUN
ejpam-4548	434	17	of	of	ADP
ejpam-4548	434	18	a	a	DET
ejpam-4548	434	19	supstar	supstar	NOUN
ejpam-4548	434	20	subset	subset	NOUN
ejpam-4548	434	21	is	be	AUX
ejpam-4548	434	22	again	again	ADV
ejpam-4548	434	23	supstar	supstar	ADJ
ejpam-4548	434	24	.	.	PUNCT
ejpam-4548	435	1	proposition	proposition	NOUN
ejpam-4548	435	2	6	6	NUM
ejpam-4548	435	3	.	.	PUNCT
ejpam-4548	436	1	let	let	VERB
ejpam-4548	436	2	η	η	PROPN
ejpam-4548	436	3	∈	∈	PROPN
ejpam-4548	436	4	lµ.	lµ.	NOUN
ejpam-4548	436	5	then	then	ADV
ejpam-4548	436	6	,	,	PUNCT
ejpam-4548	436	7	η	η	PROPN
ejpam-4548	436	8	has	have	VERB
ejpam-4548	436	9	sup	sup	ADJ
ejpam-4548	436	10	-	-	PUNCT
ejpam-4548	436	11	property	property	NOUN
ejpam-4548	436	12	if	if	NOUN
ejpam-4548	436	13	and	and	CCONJ
ejpam-4548	436	14	only	only	ADV
ejpam-4548	436	15	if	if	SCONJ
ejpam-4548	436	16	imη	imη	ADJ
ejpam-4548	436	17	is	be	AUX
ejpam-4548	436	18	a	a	DET
ejpam-4548	436	19	supstar	supstar	NOUN
ejpam-4548	436	20	subset	subset	NOUN
ejpam-4548	436	21	of	of	ADP
ejpam-4548	436	22	l.	l.	PROPN
ejpam-4548	436	23	the	the	DET
ejpam-4548	436	24	above	above	ADJ
ejpam-4548	436	25	characterization	characterization	NOUN
ejpam-4548	436	26	also	also	ADV
ejpam-4548	436	27	allows	allow	VERB
ejpam-4548	436	28	a	a	DET
ejpam-4548	436	29	generalization	generalization	NOUN
ejpam-4548	436	30	of	of	ADP
ejpam-4548	436	31	the	the	DET
ejpam-4548	436	32	concept	concept	NOUN
ejpam-4548	436	33	of	of	ADP
ejpam-4548	436	34	sup	sup	NOUN
ejpam-4548	436	35	-	-	PUNCT
ejpam-4548	436	36	property	property	NOUN
ejpam-4548	436	37	to	to	ADP
ejpam-4548	436	38	an	an	DET
ejpam-4548	436	39	arbitrary	arbitrary	ADJ
ejpam-4548	436	40	family	family	NOUN
ejpam-4548	436	41	of	of	ADP
ejpam-4548	436	42	l	l	NOUN
ejpam-4548	436	43	-	-	NOUN
ejpam-4548	436	44	subsets	subset	NOUN
ejpam-4548	436	45	and	and	CCONJ
ejpam-4548	436	46	hence	hence	ADV
ejpam-4548	436	47	widens	widen	VERB
ejpam-4548	436	48	the	the	DET
ejpam-4548	436	49	scope	scope	NOUN
ejpam-4548	436	50	of	of	ADP
ejpam-4548	436	51	its	its	PRON
ejpam-4548	436	52	applications	application	NOUN
ejpam-4548	436	53	.	.	PUNCT
ejpam-4548	437	1	definition	definition	NOUN
ejpam-4548	437	2	15	15	NUM
ejpam-4548	437	3	.	.	PUNCT
ejpam-4548	438	1	let	let	VERB
ejpam-4548	438	2	{	{	PUNCT
ejpam-4548	438	3	ηi}i∈i	ηi}i∈i	ADP
ejpam-4548	438	4	⊆	⊆	NUM
ejpam-4548	438	5	lµ.	lµ.	NOUN
ejpam-4548	438	6	then	then	ADV
ejpam-4548	438	7	,	,	PUNCT
ejpam-4548	438	8	{	{	PUNCT
ejpam-4548	438	9	ηi}i∈i	ηi}i∈i	ADP
ejpam-4548	438	10	is	be	AUX
ejpam-4548	438	11	said	say	VERB
ejpam-4548	438	12	to	to	PART
ejpam-4548	438	13	be	be	AUX
ejpam-4548	438	14	a	a	DET
ejpam-4548	438	15	supstar	supstar	ADJ
ejpam-4548	438	16	family	family	NOUN
ejpam-4548	438	17	if	if	SCONJ
ejpam-4548	438	18	∪	∪	ADJ
ejpam-4548	438	19	i∈i	i∈i	ADJ
ejpam-4548	438	20	imηi	imηi	NOUN
ejpam-4548	438	21	is	be	AUX
ejpam-4548	438	22	a	a	DET
ejpam-4548	438	23	supstar	supstar	NOUN
ejpam-4548	438	24	subset	subset	NOUN
ejpam-4548	438	25	of	of	ADP
ejpam-4548	438	26	l.	l.	PROPN
ejpam-4548	438	27	as	as	ADP
ejpam-4548	438	28	a	a	DET
ejpam-4548	438	29	particular	particular	ADJ
ejpam-4548	438	30	case	case	NOUN
ejpam-4548	438	31	,	,	PUNCT
ejpam-4548	438	32	we	we	PRON
ejpam-4548	438	33	say	say	VERB
ejpam-4548	438	34	that	that	SCONJ
ejpam-4548	438	35	two	two	NUM
ejpam-4548	438	36	l	l	NOUN
ejpam-4548	438	37	-	-	PUNCT
ejpam-4548	438	38	subsets	subsets	NOUN
ejpam-4548	438	39	θ	θ	PROPN
ejpam-4548	438	40	and	and	CCONJ
ejpam-4548	438	41	η	η	PROPN
ejpam-4548	438	42	are	be	AUX
ejpam-4548	438	43	jointly	jointly	ADV
ejpam-4548	438	44	supstar	supstar	ADJ
ejpam-4548	438	45	if	if	SCONJ
ejpam-4548	438	46	imθ	imθ	NOUN
ejpam-4548	438	47	∪	∪	ADJ
ejpam-4548	438	48	imη	imη	NOUN
ejpam-4548	438	49	is	be	AUX
ejpam-4548	438	50	a	a	DET
ejpam-4548	438	51	supstar	supstar	NOUN
ejpam-4548	438	52	subset	subset	NOUN
ejpam-4548	438	53	of	of	ADP
ejpam-4548	438	54	l.	l.	PROPN
ejpam-4548	438	55	in	in	ADP
ejpam-4548	438	56	view	view	NOUN
ejpam-4548	438	57	of	of	ADP
ejpam-4548	438	58	the	the	DET
ejpam-4548	438	59	above	above	ADJ
ejpam-4548	438	60	definition	definition	NOUN
ejpam-4548	438	61	and	and	CCONJ
ejpam-4548	438	62	proposition	proposition	NOUN
ejpam-4548	438	63	6	6	NUM
ejpam-4548	438	64	,	,	PUNCT
ejpam-4548	438	65	we	we	PRON
ejpam-4548	438	66	have	have	VERB
ejpam-4548	438	67	the	the	DET
ejpam-4548	438	68	following	following	NOUN
ejpam-4548	438	69	:	:	PUNCT
ejpam-4548	438	70	proposition	proposition	NOUN
ejpam-4548	438	71	7	7	NUM
ejpam-4548	438	72	.	.	PUNCT
ejpam-4548	439	1	each	each	DET
ejpam-4548	439	2	member	member	NOUN
ejpam-4548	439	3	of	of	ADP
ejpam-4548	439	4	a	a	DET
ejpam-4548	439	5	supstar	supstar	ADJ
ejpam-4548	439	6	family	family	NOUN
ejpam-4548	439	7	of	of	ADP
ejpam-4548	439	8	l	l	NOUN
ejpam-4548	439	9	-	-	PUNCT
ejpam-4548	439	10	subsets	subset	NOUN
ejpam-4548	439	11	satisfies	satisfy	VERB
ejpam-4548	439	12	sup	sup	NOUN
ejpam-4548	439	13	-	-	PUNCT
ejpam-4548	439	14	property	property	NOUN
ejpam-4548	439	15	.	.	PUNCT
ejpam-4548	440	1	lemma	lemma	PROPN
ejpam-4548	440	2	3	3	X
ejpam-4548	440	3	.	.	PUNCT
ejpam-4548	440	4	let	let	VERB
ejpam-4548	440	5	η	η	PROPN
ejpam-4548	440	6	∈	∈	PROPN
ejpam-4548	440	7	lµ	lµ	PROPN
ejpam-4548	440	8	and	and	CCONJ
ejpam-4548	440	9	η	η	PROPN
ejpam-4548	440	10	has	have	VERB
ejpam-4548	440	11	sup	sup	ADJ
ejpam-4548	440	12	-	-	PUNCT
ejpam-4548	440	13	property	property	NOUN
ejpam-4548	440	14	.	.	PUNCT
ejpam-4548	441	1	if	if	SCONJ
ejpam-4548	441	2	a0	a0	PROPN
ejpam-4548	441	3	=	=	PUNCT
ejpam-4548	441	4	∨	∨	PROPN
ejpam-4548	441	5	x∈g	x∈g	PROPN
ejpam-4548	441	6	{	{	PUNCT
ejpam-4548	441	7	η(x	η(x	PROPN
ejpam-4548	441	8	)	)	PUNCT
ejpam-4548	441	9	}	}	PUNCT
ejpam-4548	441	10	,	,	PUNCT
ejpam-4548	441	11	then	then	ADV
ejpam-4548	441	12	⟨ηb⟩	⟨ηb⟩	PROPN
ejpam-4548	441	13	=	=	PUNCT
ejpam-4548	441	14	⟨η⟩b	⟨η⟩b	PROPN
ejpam-4548	441	15	for	for	ADP
ejpam-4548	441	16	each	each	DET
ejpam-4548	441	17	b	b	PROPN
ejpam-4548	441	18	≤	≤	NUM
ejpam-4548	441	19	a0	a0	NOUN
ejpam-4548	441	20	.	.	PUNCT
ejpam-4548	442	1	lemma	lemma	PROPN
ejpam-4548	442	2	4	4	X
ejpam-4548	442	3	.	.	PUNCT
ejpam-4548	442	4	let	let	VERB
ejpam-4548	442	5	η	η	PROPN
ejpam-4548	442	6	∈	∈	PROPN
ejpam-4548	442	7	l(µ	l(µ	PROPN
ejpam-4548	442	8	)	)	PUNCT
ejpam-4548	442	9	be	be	AUX
ejpam-4548	442	10	such	such	ADJ
ejpam-4548	442	11	that	that	SCONJ
ejpam-4548	442	12	µ	µ	NOUN
ejpam-4548	442	13	and	and	CCONJ
ejpam-4548	442	14	η	η	PROPN
ejpam-4548	442	15	are	be	AUX
ejpam-4548	442	16	jointly	jointly	ADV
ejpam-4548	442	17	supstar	supstar	ADJ
ejpam-4548	442	18	..	..	PUNCT
ejpam-4548	442	19	then	then	ADV
ejpam-4548	442	20	,	,	PUNCT
ejpam-4548	442	21	imη	imη	VERB
ejpam-4548	442	22	∪	∪	X
ejpam-4548	442	23	imµ	imµ	PROPN
ejpam-4548	442	24	is	be	AUX
ejpam-4548	442	25	a	a	DET
ejpam-4548	442	26	chain	chain	NOUN
ejpam-4548	442	27	.	.	PUNCT
ejpam-4548	443	1	next	next	ADV
ejpam-4548	443	2	,	,	PUNCT
ejpam-4548	443	3	we	we	PRON
ejpam-4548	443	4	prove	prove	VERB
ejpam-4548	443	5	:	:	PUNCT
ejpam-4548	443	6	lemma	lemma	PROPN
ejpam-4548	443	7	5	5	X
ejpam-4548	443	8	.	.	PUNCT
ejpam-4548	443	9	let	let	VERB
ejpam-4548	443	10	η	η	PROPN
ejpam-4548	443	11	∈	∈	PROPN
ejpam-4548	443	12	l(µ	l(µ	PROPN
ejpam-4548	443	13	)	)	PUNCT
ejpam-4548	443	14	be	be	AUX
ejpam-4548	443	15	such	such	ADJ
ejpam-4548	443	16	that	that	SCONJ
ejpam-4548	443	17	µ	µ	NOUN
ejpam-4548	443	18	and	and	CCONJ
ejpam-4548	443	19	η	η	PROPN
ejpam-4548	443	20	are	be	AUX
ejpam-4548	443	21	jointly	jointly	ADV
ejpam-4548	443	22	supstar	supstar	ADJ
ejpam-4548	443	23	.	.	PUNCT
ejpam-4548	444	1	then	then	ADV
ejpam-4548	444	2	,	,	PUNCT
ejpam-4548	444	3	i	i	PRON
ejpam-4548	444	4	m	m	AUX
ejpam-4548	444	5	ηµ	ηµ	VERB
ejpam-4548	444	6	⊆	⊆	NUM
ejpam-4548	444	7	i	i	PRON
ejpam-4548	444	8	m	m	PROPN
ejpam-4548	444	9	(	(	PUNCT
ejpam-4548	444	10	µηµ−1	µηµ−1	PROPN
ejpam-4548	444	11	)	)	PUNCT
ejpam-4548	444	12	⊆	⊆	NUM
ejpam-4548	444	13	imµ	imµ	PROPN
ejpam-4548	444	14	∪	∪	X
ejpam-4548	444	15	imη	imη	X
ejpam-4548	444	16	.	.	PUNCT
ejpam-4548	445	1	proof	proof	NOUN
ejpam-4548	445	2	.	.	PUNCT
ejpam-4548	446	1	let	let	VERB
ejpam-4548	446	2	a	a	DET
ejpam-4548	446	3	∈	∈	PROPN
ejpam-4548	446	4	im(µηµ−1	im(µηµ−1	NOUN
ejpam-4548	446	5	)	)	PUNCT
ejpam-4548	446	6	.	.	PUNCT
ejpam-4548	447	1	so	so	ADV
ejpam-4548	447	2	,	,	PUNCT
ejpam-4548	447	3	there	there	PRON
ejpam-4548	447	4	exists	exist	VERB
ejpam-4548	447	5	x	x	X
ejpam-4548	447	6	∈	∈	PROPN
ejpam-4548	447	7	g	g	NOUN
ejpam-4548	447	8	such	such	DET
ejpam-4548	447	9	that	that	SCONJ
ejpam-4548	447	10	a	a	DET
ejpam-4548	447	11	=	=	X
ejpam-4548	447	12	(	(	PUNCT
ejpam-4548	447	13	µηµ−1)(x	µηµ−1)(x	NOUN
ejpam-4548	447	14	)	)	PUNCT
ejpam-4548	447	15	.	.	PUNCT
ejpam-4548	448	1	now	now	ADV
ejpam-4548	448	2	,	,	PUNCT
ejpam-4548	448	3	define	define	VERB
ejpam-4548	448	4	the	the	DET
ejpam-4548	448	5	following	follow	VERB
ejpam-4548	448	6	subset	subset	NOUN
ejpam-4548	448	7	of	of	ADP
ejpam-4548	448	8	g×g	g×g	PROPN
ejpam-4548	448	9	:	:	PUNCT
ejpam-4548	448	10	c(x	c(x	NOUN
ejpam-4548	448	11	)	)	PUNCT
ejpam-4548	448	12	=	=	PRON
ejpam-4548	448	13	{	{	PUNCT
ejpam-4548	448	14	(	(	PUNCT
ejpam-4548	448	15	v	v	NOUN
ejpam-4548	448	16	,	,	PUNCT
ejpam-4548	448	17	u	u	NOUN
ejpam-4548	448	18	)	)	PUNCT
ejpam-4548	448	19	∈	∈	PROPN
ejpam-4548	449	1	g×g	g×g	PROPN
ejpam-4548	449	2	:	:	PUNCT
ejpam-4548	450	1	x	x	X
ejpam-4548	450	2	=	=	SYM
ejpam-4548	450	3	uvu−1	uvu−1	NOUN
ejpam-4548	450	4	}	}	PUNCT
ejpam-4548	450	5	.	.	PUNCT
ejpam-4548	451	1	thus	thus	ADV
ejpam-4548	451	2	,	,	PUNCT
ejpam-4548	451	3	a	a	DET
ejpam-4548	451	4	=	=	X
ejpam-4548	451	5	(	(	PUNCT
ejpam-4548	451	6	µηµ−1)(x	µηµ−1)(x	NOUN
ejpam-4548	451	7	)	)	PUNCT
ejpam-4548	451	8	=	=	SYM
ejpam-4548	451	9	∨	∨	X
ejpam-4548	451	10	(	(	PUNCT
ejpam-4548	451	11	y	y	NOUN
ejpam-4548	451	12	,	,	PUNCT
ejpam-4548	451	13	z)∈c(x	z)∈c(x	NUM
ejpam-4548	451	14	)	)	PUNCT
ejpam-4548	451	15	{	{	PUNCT
ejpam-4548	451	16	η(y	η(y	NOUN
ejpam-4548	451	17	)	)	PUNCT
ejpam-4548	451	18	∧	∧	PROPN
ejpam-4548	451	19	µ(z	µ(z	PROPN
ejpam-4548	451	20	)	)	PUNCT
ejpam-4548	451	21	}	}	PUNCT
ejpam-4548	451	22	.	.	PUNCT
ejpam-4548	452	1	(	(	PUNCT
ejpam-4548	452	2	1	1	X
ejpam-4548	452	3	)	)	PUNCT
ejpam-4548	452	4	now	now	ADV
ejpam-4548	452	5	,	,	PUNCT
ejpam-4548	452	6	for	for	ADP
ejpam-4548	452	7	any	any	DET
ejpam-4548	452	8	(	(	PUNCT
ejpam-4548	452	9	y	y	PROPN
ejpam-4548	452	10	,	,	PUNCT
ejpam-4548	452	11	z	z	NOUN
ejpam-4548	452	12	)	)	PUNCT
ejpam-4548	452	13	∈	∈	PROPN
ejpam-4548	452	14	c(x	c(x	NOUN
ejpam-4548	452	15	)	)	PUNCT
ejpam-4548	452	16	,	,	PUNCT
ejpam-4548	452	17	{	{	PUNCT
ejpam-4548	452	18	η(y	η(y	NOUN
ejpam-4548	452	19	)	)	PUNCT
ejpam-4548	452	20	,	,	PUNCT
ejpam-4548	452	21	µ(z	µ(z	PROPN
ejpam-4548	452	22	)	)	PUNCT
ejpam-4548	452	23	}	}	PUNCT
ejpam-4548	452	24	⊆	⊆	NUM
ejpam-4548	452	25	imη	imη	X
ejpam-4548	452	26	∪	∪	NOUN
ejpam-4548	452	27	imµ.	imµ.	NOUN
ejpam-4548	452	28	as	as	ADP
ejpam-4548	452	29	µ	µ	NOUN
ejpam-4548	452	30	and	and	CCONJ
ejpam-4548	452	31	η	η	PROPN
ejpam-4548	452	32	are	be	AUX
ejpam-4548	452	33	jointly	jointly	ADV
ejpam-4548	452	34	supstar	supstar	ADJ
ejpam-4548	452	35	,	,	PUNCT
ejpam-4548	452	36	by	by	ADP
ejpam-4548	452	37	lemma	lemma	PROPN
ejpam-4548	452	38	4	4	NUM
ejpam-4548	452	39	,	,	PUNCT
ejpam-4548	452	40	imη	imη	VERB
ejpam-4548	452	41	∪	∪	X
ejpam-4548	452	42	imη	imη	NOUN
ejpam-4548	452	43	is	be	AUX
ejpam-4548	452	44	a	a	DET
ejpam-4548	452	45	chain	chain	NOUN
ejpam-4548	452	46	.	.	PUNCT
ejpam-4548	453	1	hence	hence	ADV
ejpam-4548	453	2	η(y	η(y	NOUN
ejpam-4548	453	3	)	)	PUNCT
ejpam-4548	453	4	∧	∧	PROPN
ejpam-4548	453	5	µ(z	µ(z	PROPN
ejpam-4548	453	6	)	)	PUNCT
ejpam-4548	453	7	=	=	SYM
ejpam-4548	453	8	η(y	η(y	NOUN
ejpam-4548	453	9	)	)	PUNCT
ejpam-4548	453	10	or	or	CCONJ
ejpam-4548	453	11	µ(z	µ(z	PROPN
ejpam-4548	453	12	)	)	PUNCT
ejpam-4548	453	13	which	which	PRON
ejpam-4548	453	14	is	be	AUX
ejpam-4548	453	15	an	an	DET
ejpam-4548	453	16	element	element	NOUN
ejpam-4548	453	17	of	of	ADP
ejpam-4548	453	18	imη	imη	X
ejpam-4548	453	19	∪	∪	X
ejpam-4548	453	20	imµ.	imµ.	NOUN
ejpam-4548	453	21	as	as	ADP
ejpam-4548	453	22	imη	imη	X
ejpam-4548	453	23	∪	∪	X
ejpam-4548	453	24	imµ	imµ	PROPN
ejpam-4548	453	25	is	be	AUX
ejpam-4548	453	26	supstar	supstar	ADJ
ejpam-4548	453	27	,	,	PUNCT
ejpam-4548	453	28	the	the	DET
ejpam-4548	453	29	subset	subset	NOUN
ejpam-4548	453	30	n.	n.	PROPN
ejpam-4548	453	31	ajmal	ajmal	PROPN
ejpam-4548	453	32	,	,	PUNCT
ejpam-4548	453	33	i.	i.	PROPN
ejpam-4548	453	34	jahan	jahan	PROPN
ejpam-4548	453	35	.	.	PROPN
ejpam-4548	453	36	,	,	PUNCT
ejpam-4548	453	37	b.	b.	PROPN
ejpam-4548	453	38	davvaz	davvaz	PROPN
ejpam-4548	453	39	/	/	SYM
ejpam-4548	453	40	eur	eur	PROPN
ejpam-4548	453	41	.	.	PUNCT
ejpam-4548	454	1	j.	j.	PROPN
ejpam-4548	454	2	pure	pure	PROPN
ejpam-4548	454	3	appl	appl	PROPN
ejpam-4548	454	4	.	.	PROPN
ejpam-4548	454	5	math	math	PROPN
ejpam-4548	454	6	,	,	PUNCT
ejpam-4548	454	7	15	15	NUM
ejpam-4548	454	8	(	(	PUNCT
ejpam-4548	454	9	4	4	NUM
ejpam-4548	454	10	)	)	PUNCT
ejpam-4548	454	11	(	(	PUNCT
ejpam-4548	454	12	2022	2022	NUM
ejpam-4548	454	13	)	)	PUNCT
ejpam-4548	454	14	,	,	PUNCT
ejpam-4548	454	15	2086	2086	NUM
ejpam-4548	454	16	-	-	SYM
ejpam-4548	454	17	2115	2115	NUM
ejpam-4548	454	18	2107	2107	NUM
ejpam-4548	454	19	{	{	PUNCT
ejpam-4548	454	20	η(y	η(y	NOUN
ejpam-4548	454	21	)	)	PUNCT
ejpam-4548	454	22	∧	∧	PROPN
ejpam-4548	454	23	µ(z	µ(z	PROPN
ejpam-4548	454	24	)	)	PUNCT
ejpam-4548	454	25	:	:	PUNCT
ejpam-4548	454	26	(	(	PUNCT
ejpam-4548	454	27	y	y	PROPN
ejpam-4548	454	28	,	,	PUNCT
ejpam-4548	454	29	z	z	NOUN
ejpam-4548	454	30	)	)	PUNCT
ejpam-4548	454	31	∈	∈	PROPN
ejpam-4548	454	32	c(x	c(x	NOUN
ejpam-4548	454	33	)	)	PUNCT
ejpam-4548	454	34	}	}	PUNCT
ejpam-4548	455	1	⊆	⊆	NUM
ejpam-4548	455	2	imη	imη	X
ejpam-4548	455	3	∪	∪	X
ejpam-4548	455	4	imµ	imµ	PROPN
ejpam-4548	455	5	contains	contain	VERB
ejpam-4548	455	6	its	its	PRON
ejpam-4548	455	7	supremum	supremum	ADJ
ejpam-4548	455	8	a	a	PRON
ejpam-4548	455	9	and	and	CCONJ
ejpam-4548	455	10	hence	hence	ADV
ejpam-4548	455	11	a	a	DET
ejpam-4548	455	12	=	=	PUNCT
ejpam-4548	455	13	(	(	PUNCT
ejpam-4548	455	14	µηµ−1)(x	µηµ−1)(x	NOUN
ejpam-4548	455	15	)	)	PUNCT
ejpam-4548	455	16	∈	∈	PROPN
ejpam-4548	455	17	imµ	imµ	VERB
ejpam-4548	455	18	∪	∪	X
ejpam-4548	455	19	imη	imη	X
ejpam-4548	455	20	.	.	PUNCT
ejpam-4548	456	1	therefore	therefore	ADV
ejpam-4548	456	2	,	,	PUNCT
ejpam-4548	456	3	i	i	PRON
ejpam-4548	456	4	m	m	VERB
ejpam-4548	456	5	(	(	PUNCT
ejpam-4548	456	6	µηµ−1	µηµ−1	PROPN
ejpam-4548	456	7	)	)	PUNCT
ejpam-4548	456	8	⊆	⊆	NUM
ejpam-4548	456	9	imµ	imµ	PROPN
ejpam-4548	456	10	∪	∪	X
ejpam-4548	456	11	imη	imη	X
ejpam-4548	456	12	.	.	PUNCT
ejpam-4548	456	13	further	far	ADV
ejpam-4548	456	14	,	,	PUNCT
ejpam-4548	456	15	as	as	ADP
ejpam-4548	456	16	a	a	DET
ejpam-4548	456	17	subset	subset	NOUN
ejpam-4548	456	18	of	of	ADP
ejpam-4548	456	19	a	a	DET
ejpam-4548	456	20	supstar	supstar	NOUN
ejpam-4548	456	21	subset	subset	NOUN
ejpam-4548	456	22	is	be	AUX
ejpam-4548	456	23	again	again	ADV
ejpam-4548	456	24	supstar	supstar	ADJ
ejpam-4548	456	25	,	,	PUNCT
ejpam-4548	456	26	it	it	PRON
ejpam-4548	456	27	follows	follow	VERB
ejpam-4548	456	28	that	that	DET
ejpam-4548	456	29	im(µηµ−1	im(µηµ−1	NOUN
ejpam-4548	456	30	)	)	PUNCT
ejpam-4548	456	31	is	be	AUX
ejpam-4548	456	32	also	also	ADV
ejpam-4548	456	33	supstar	supstar	ADJ
ejpam-4548	456	34	.	.	PUNCT
ejpam-4548	457	1	hence	hence	ADV
ejpam-4548	457	2	by	by	ADP
ejpam-4548	457	3	proposition	proposition	NOUN
ejpam-4548	457	4	6	6	NUM
ejpam-4548	457	5	,	,	PUNCT
ejpam-4548	457	6	µηµ−1	µηµ−1	AUX
ejpam-4548	457	7	possesses	possess	VERB
ejpam-4548	457	8	sup	sup	ADJ
ejpam-4548	457	9	-	-	PUNCT
ejpam-4548	457	10	property	property	NOUN
ejpam-4548	457	11	.	.	PUNCT
ejpam-4548	458	1	hence	hence	ADV
ejpam-4548	458	2	in	in	ADP
ejpam-4548	458	3	view	view	NOUN
ejpam-4548	458	4	of	of	ADP
ejpam-4548	458	5	theorem	theorem	NOUN
ejpam-4548	458	6	7	7	NUM
ejpam-4548	458	7	,	,	PUNCT
ejpam-4548	458	8	it	it	PRON
ejpam-4548	458	9	follows	follow	VERB
ejpam-4548	458	10	that	that	SCONJ
ejpam-4548	458	11	imηµ	imηµ	NOUN
ejpam-4548	458	12	=	=	PUNCT
ejpam-4548	458	13	im⟨µηµ−1⟩	im⟨µηµ−1⟩	PROPN
ejpam-4548	458	14	⊆	⊆	NUM
ejpam-4548	458	15	i	i	PRON
ejpam-4548	458	16	m	m	VERB
ejpam-4548	458	17	(	(	PUNCT
ejpam-4548	458	18	µηµ−1	µηµ−1	PROPN
ejpam-4548	458	19	)	)	PUNCT
ejpam-4548	459	1	⊆	⊆	NUM
ejpam-4548	459	2	imµ	imµ	NOUN
ejpam-4548	459	3	∪	∪	X
ejpam-4548	459	4	imη	imη	X
ejpam-4548	459	5	.	.	PUNCT
ejpam-4548	460	1	more	more	ADV
ejpam-4548	460	2	generally	generally	ADV
ejpam-4548	460	3	we	we	PRON
ejpam-4548	460	4	have	have	VERB
ejpam-4548	460	5	:	:	PUNCT
ejpam-4548	460	6	lemma	lemma	PROPN
ejpam-4548	460	7	6	6	X
ejpam-4548	460	8	.	.	PUNCT
ejpam-4548	461	1	let	let	VERB
ejpam-4548	461	2	η	η	PROPN
ejpam-4548	461	3	∈	∈	PROPN
ejpam-4548	461	4	l(µ	l(µ	PROPN
ejpam-4548	461	5	)	)	PUNCT
ejpam-4548	461	6	be	be	AUX
ejpam-4548	461	7	such	such	ADJ
ejpam-4548	461	8	that	that	SCONJ
ejpam-4548	461	9	µ	µ	NOUN
ejpam-4548	461	10	and	and	CCONJ
ejpam-4548	461	11	η	η	PROPN
ejpam-4548	461	12	are	be	AUX
ejpam-4548	461	13	jointly	jointly	ADV
ejpam-4548	461	14	supstar	supstar	ADJ
ejpam-4548	461	15	.	.	PUNCT
ejpam-4548	462	1	then	then	ADV
ejpam-4548	462	2	,	,	PUNCT
ejpam-4548	462	3	for	for	SCONJ
ejpam-4548	462	4	each	each	DET
ejpam-4548	462	5	i	i	PRON
ejpam-4548	462	6	i	i	PRON
ejpam-4548	462	7	m	m	VERB
ejpam-4548	462	8	ηi+1	ηi+1	NUM
ejpam-4548	462	9	⊆	⊆	NUM
ejpam-4548	463	1	i	i	PRON
ejpam-4548	463	2	m	m	VERB
ejpam-4548	463	3	(	(	PUNCT
ejpam-4548	463	4	ηiηη	ηiηη	NOUN
ejpam-4548	463	5	−1	−1	NOUN
ejpam-4548	463	6	i	i	PROPN
ejpam-4548	463	7	)	)	PUNCT
ejpam-4548	464	1	⊆	⊆	NUM
ejpam-4548	464	2	imµ	imµ	NOUN
ejpam-4548	464	3	∪	∪	X
ejpam-4548	464	4	imη	imη	NOUN
ejpam-4548	464	5	,	,	PUNCT
ejpam-4548	464	6	where	where	SCONJ
ejpam-4548	464	7	ηi	ηi	PROPN
ejpam-4548	464	8	is	be	AUX
ejpam-4548	464	9	the	the	DET
ejpam-4548	464	10	ith	ith	PROPN
ejpam-4548	464	11	normal	normal	ADJ
ejpam-4548	464	12	closure	closure	NOUN
ejpam-4548	464	13	of	of	ADP
ejpam-4548	464	14	η	η	PROPN
ejpam-4548	464	15	in	in	ADP
ejpam-4548	464	16	µ.	µ.	PROPN
ejpam-4548	464	17	corollary	corollary	ADJ
ejpam-4548	464	18	4	4	NUM
ejpam-4548	464	19	.	.	PUNCT
ejpam-4548	464	20	let	let	VERB
ejpam-4548	464	21	η	η	PROPN
ejpam-4548	464	22	∈	∈	PROPN
ejpam-4548	464	23	l(µ	l(µ	PROPN
ejpam-4548	464	24	)	)	PUNCT
ejpam-4548	464	25	be	be	AUX
ejpam-4548	464	26	such	such	ADJ
ejpam-4548	464	27	that	that	SCONJ
ejpam-4548	464	28	µ	µ	NOUN
ejpam-4548	464	29	and	and	CCONJ
ejpam-4548	464	30	η	η	PROPN
ejpam-4548	464	31	are	be	AUX
ejpam-4548	464	32	jointly	jointly	ADV
ejpam-4548	464	33	supstar	supstar	ADJ
ejpam-4548	464	34	.	.	PUNCT
ejpam-4548	465	1	then	then	ADV
ejpam-4548	465	2	for	for	ADP
ejpam-4548	465	3	each	each	DET
ejpam-4548	465	4	i	i	PROPN
ejpam-4548	465	5	,	,	PUNCT
ejpam-4548	465	6	η	η	PROPN
ejpam-4548	465	7	and	and	CCONJ
ejpam-4548	465	8	ηi	ηi	NOUN
ejpam-4548	465	9	are	be	AUX
ejpam-4548	465	10	jointly	jointly	ADV
ejpam-4548	465	11	supstar	supstar	ADJ
ejpam-4548	465	12	.	.	PUNCT
ejpam-4548	466	1	remark	remark	PROPN
ejpam-4548	466	2	2	2	NUM
ejpam-4548	466	3	.	.	PUNCT
ejpam-4548	466	4	note	note	VERB
ejpam-4548	466	5	that	that	SCONJ
ejpam-4548	466	6	if	if	SCONJ
ejpam-4548	466	7	η	η	PROPN
ejpam-4548	466	8	∈	∈	PROPN
ejpam-4548	466	9	l(µ	l(µ	PROPN
ejpam-4548	466	10	)	)	PUNCT
ejpam-4548	466	11	and	and	CCONJ
ejpam-4548	466	12	µ	µ	NOUN
ejpam-4548	466	13	and	and	CCONJ
ejpam-4548	466	14	η	η	PROPN
ejpam-4548	466	15	be	be	VERB
ejpam-4548	466	16	jointly	jointly	ADV
ejpam-4548	466	17	supstar	supstar	ADJ
ejpam-4548	466	18	,	,	PUNCT
ejpam-4548	466	19	then	then	ADV
ejpam-4548	466	20	the	the	DET
ejpam-4548	466	21	l	l	NOUN
ejpam-4548	466	22	-	-	ADJ
ejpam-4548	466	23	subset	subset	ADJ
ejpam-4548	466	24	ηiηη	ηiηη	NOUN
ejpam-4548	466	25	−1	−1	NOUN
ejpam-4548	466	26	i	i	PRON
ejpam-4548	466	27	possesses	possess	VERB
ejpam-4548	466	28	sup	sup	ADJ
ejpam-4548	466	29	-	-	PUNCT
ejpam-4548	466	30	property	property	NOUN
ejpam-4548	466	31	for	for	ADP
ejpam-4548	466	32	each	each	DET
ejpam-4548	466	33	i.	i.	NOUN
ejpam-4548	466	34	below	below	ADV
ejpam-4548	466	35	,	,	PUNCT
ejpam-4548	466	36	we	we	PRON
ejpam-4548	466	37	discuss	discuss	VERB
ejpam-4548	466	38	the	the	DET
ejpam-4548	466	39	level	level	NOUN
ejpam-4548	466	40	subset	subset	NOUN
ejpam-4548	466	41	of	of	ADP
ejpam-4548	466	42	the	the	DET
ejpam-4548	466	43	normal	normal	ADJ
ejpam-4548	466	44	closure	closure	NOUN
ejpam-4548	466	45	of	of	ADP
ejpam-4548	466	46	η	η	PROPN
ejpam-4548	466	47	in	in	ADP
ejpam-4548	466	48	µ.	µ.	PROPN
ejpam-4548	466	49	lemma	lemma	PROPN
ejpam-4548	466	50	7	7	X
ejpam-4548	466	51	.	.	PUNCT
ejpam-4548	467	1	let	let	VERB
ejpam-4548	467	2	η	η	PROPN
ejpam-4548	467	3	∈	∈	PROPN
ejpam-4548	467	4	l(µ	l(µ	PROPN
ejpam-4548	467	5	)	)	PUNCT
ejpam-4548	467	6	.	.	PUNCT
ejpam-4548	468	1	then	then	ADV
ejpam-4548	468	2	,	,	PUNCT
ejpam-4548	468	3	(	(	PUNCT
ejpam-4548	468	4	i	i	NOUN
ejpam-4548	468	5	)	)	PUNCT
ejpam-4548	468	6	(	(	PUNCT
ejpam-4548	468	7	ηµ)a	ηµ)a	PROPN
ejpam-4548	468	8	=	=	X
ejpam-4548	468	9	(	(	PUNCT
ejpam-4548	468	10	ηa	ηa	NOUN
ejpam-4548	468	11	)	)	PUNCT
ejpam-4548	468	12	µa	µa	NOUN
ejpam-4548	468	13	for	for	ADP
ejpam-4548	468	14	each	each	DET
ejpam-4548	468	15	a	a	DET
ejpam-4548	468	16	≤	≤	NUM
ejpam-4548	468	17	η(e	η(e	PROPN
ejpam-4548	468	18	)	)	PUNCT
ejpam-4548	468	19	provided	provide	VERB
ejpam-4548	468	20	µ	µ	PROPN
ejpam-4548	468	21	and	and	CCONJ
ejpam-4548	468	22	η	η	PROPN
ejpam-4548	468	23	are	be	AUX
ejpam-4548	468	24	supstar	supstar	ADJ
ejpam-4548	468	25	,	,	PUNCT
ejpam-4548	468	26	(	(	PUNCT
ejpam-4548	468	27	ii	ii	NOUN
ejpam-4548	468	28	)	)	PUNCT
ejpam-4548	468	29	(	(	PUNCT
ejpam-4548	468	30	ηµ)>a	ηµ)>a	NOUN
ejpam-4548	468	31	=	=	SYM
ejpam-4548	468	32	(	(	PUNCT
ejpam-4548	468	33	η	η	X
ejpam-4548	468	34	>	>	X
ejpam-4548	468	35	a	a	PRON
ejpam-4548	468	36	)	)	PUNCT
ejpam-4548	468	37	µ	µ	X
ejpam-4548	468	38	>	>	X
ejpam-4548	468	39	a	a	DET
ejpam-4548	468	40	provided	provide	VERB
ejpam-4548	468	41	l	l	NOUN
ejpam-4548	468	42	is	be	AUX
ejpam-4548	468	43	a	a	DET
ejpam-4548	468	44	chain	chain	NOUN
ejpam-4548	468	45	.	.	PUNCT
ejpam-4548	469	1	proof	proof	NOUN
ejpam-4548	469	2	.	.	PUNCT
ejpam-4548	470	1	(	(	PUNCT
ejpam-4548	470	2	i	i	NOUN
ejpam-4548	470	3	)	)	PUNCT
ejpam-4548	470	4	let	let	VERB
ejpam-4548	470	5	a	a	DET
ejpam-4548	470	6	≤	≤	NUM
ejpam-4548	470	7	η(e	η(e	PROPN
ejpam-4548	470	8	)	)	PUNCT
ejpam-4548	470	9	.	.	PUNCT
ejpam-4548	471	1	we	we	PRON
ejpam-4548	471	2	show	show	VERB
ejpam-4548	471	3	that	that	SCONJ
ejpam-4548	471	4	(	(	PUNCT
ejpam-4548	471	5	µηµ−1)a	µηµ−1)a	NUM
ejpam-4548	471	6	⊆	⊆	NUM
ejpam-4548	471	7	µaηa(µa	µaηa(µa	NOUN
ejpam-4548	471	8	)	)	PUNCT
ejpam-4548	471	9	−1	−1	NOUN
ejpam-4548	471	10	.	.	PUNCT
ejpam-4548	472	1	(	(	PUNCT
ejpam-4548	472	2	1	1	X
ejpam-4548	472	3	)	)	PUNCT
ejpam-4548	472	4	so	so	ADV
ejpam-4548	472	5	,	,	PUNCT
ejpam-4548	472	6	let	let	VERB
ejpam-4548	472	7	x	x	X
ejpam-4548	472	8	∈	∈	PROPN
ejpam-4548	472	9	(	(	PUNCT
ejpam-4548	472	10	µηµ−1)a	µηµ−1)a	NUM
ejpam-4548	472	11	.	.	PUNCT
ejpam-4548	473	1	then	then	ADV
ejpam-4548	473	2	,	,	PUNCT
ejpam-4548	473	3	µηµ−1(x	µηµ−1(x	NOUN
ejpam-4548	473	4	)	)	PUNCT
ejpam-4548	473	5	=	=	PUNCT
ejpam-4548	474	1	∨	∨	NUM
ejpam-4548	474	2	x	x	X
ejpam-4548	474	3	=	=	PROPN
ejpam-4548	474	4	zyz−1	zyz−1	PROPN
ejpam-4548	474	5	{	{	PUNCT
ejpam-4548	474	6	η(y	η(y	PROPN
ejpam-4548	474	7	)	)	PUNCT
ejpam-4548	474	8	∧	∧	PROPN
ejpam-4548	474	9	µ(z	µ(z	PROPN
ejpam-4548	474	10	)	)	PUNCT
ejpam-4548	474	11	}	}	PUNCT
ejpam-4548	474	12	≥	≥	NUM
ejpam-4548	474	13	a.	a.	NOUN
ejpam-4548	474	14	as	as	ADP
ejpam-4548	474	15	µ	µ	PROPN
ejpam-4548	474	16	and	and	CCONJ
ejpam-4548	474	17	η	η	PROPN
ejpam-4548	474	18	are	be	AUX
ejpam-4548	474	19	jointly	jointly	ADV
ejpam-4548	474	20	supstar	supstar	ADJ
ejpam-4548	474	21	,	,	PUNCT
ejpam-4548	474	22	it	it	PRON
ejpam-4548	474	23	can	can	AUX
ejpam-4548	474	24	be	be	AUX
ejpam-4548	474	25	verified	verify	VERB
ejpam-4548	474	26	,	,	PUNCT
ejpam-4548	474	27	as	as	ADP
ejpam-4548	474	28	in	in	ADP
ejpam-4548	474	29	lemma	lemma	PROPN
ejpam-4548	474	30	5	5	NUM
ejpam-4548	474	31	,	,	PUNCT
ejpam-4548	474	32	that	that	SCONJ
ejpam-4548	474	33	there	there	PRON
ejpam-4548	474	34	exists	exist	VERB
ejpam-4548	474	35	y0	y0	PROPN
ejpam-4548	474	36	and	and	CCONJ
ejpam-4548	474	37	z0	z0	PROPN
ejpam-4548	474	38	∈	∈	PROPN
ejpam-4548	474	39	g	g	NOUN
ejpam-4548	474	40	such	such	ADJ
ejpam-4548	474	41	that	that	SCONJ
ejpam-4548	474	42	x	x	X
ejpam-4548	474	43	=	=	PUNCT
ejpam-4548	474	44	z0y0z	z0y0z	NUM
ejpam-4548	474	45	−1	−1	NOUN
ejpam-4548	474	46	0	0	PUNCT
ejpam-4548	475	1	and∨	and∨	PROPN
ejpam-4548	475	2	x	x	PROPN
ejpam-4548	475	3	=	=	PROPN
ejpam-4548	475	4	zyz−1	zyz−1	PROPN
ejpam-4548	475	5	{	{	PUNCT
ejpam-4548	475	6	η(y	η(y	PROPN
ejpam-4548	475	7	)	)	PUNCT
ejpam-4548	475	8	∧	∧	PROPN
ejpam-4548	475	9	µ(z	µ(z	PROPN
ejpam-4548	475	10	)	)	PUNCT
ejpam-4548	475	11	}	}	PUNCT
ejpam-4548	475	12	=	=	SYM
ejpam-4548	475	13	η(y0	η(y0	ADJ
ejpam-4548	475	14	)	)	PUNCT
ejpam-4548	475	15	∧	∧	PROPN
ejpam-4548	475	16	µ(z0	µ(z0	ADV
ejpam-4548	475	17	)	)	PUNCT
ejpam-4548	475	18	.	.	PUNCT
ejpam-4548	476	1	n.	n.	PROPN
ejpam-4548	476	2	ajmal	ajmal	PROPN
ejpam-4548	476	3	,	,	PUNCT
ejpam-4548	476	4	i.	i.	PROPN
ejpam-4548	476	5	jahan	jahan	PROPN
ejpam-4548	476	6	.	.	PROPN
ejpam-4548	476	7	,	,	PUNCT
ejpam-4548	476	8	b.	b.	PROPN
ejpam-4548	476	9	davvaz	davvaz	PROPN
ejpam-4548	476	10	/	/	SYM
ejpam-4548	476	11	eur	eur	PROPN
ejpam-4548	476	12	.	.	PUNCT
ejpam-4548	477	1	j.	j.	PROPN
ejpam-4548	477	2	pure	pure	PROPN
ejpam-4548	477	3	appl	appl	PROPN
ejpam-4548	477	4	.	.	PROPN
ejpam-4548	477	5	math	math	PROPN
ejpam-4548	477	6	,	,	PUNCT
ejpam-4548	477	7	15	15	NUM
ejpam-4548	477	8	(	(	PUNCT
ejpam-4548	477	9	4	4	NUM
ejpam-4548	477	10	)	)	PUNCT
ejpam-4548	477	11	(	(	PUNCT
ejpam-4548	477	12	2022	2022	NUM
ejpam-4548	477	13	)	)	PUNCT
ejpam-4548	477	14	,	,	PUNCT
ejpam-4548	477	15	2086	2086	NUM
ejpam-4548	477	16	-	-	SYM
ejpam-4548	477	17	2115	2115	NUM
ejpam-4548	477	18	2108	2108	NUM
ejpam-4548	477	19	this	this	PRON
ejpam-4548	477	20	implies	imply	VERB
ejpam-4548	477	21	µηµ−1(x	µηµ−1(x	NOUN
ejpam-4548	477	22	)	)	PUNCT
ejpam-4548	477	23	=	=	SYM
ejpam-4548	477	24	η(y0	η(y0	NOUN
ejpam-4548	477	25	)	)	PUNCT
ejpam-4548	477	26	∧	∧	PROPN
ejpam-4548	477	27	µ(z0	µ(z0	ADV
ejpam-4548	477	28	)	)	PUNCT
ejpam-4548	478	1	≥	≥	PROPN
ejpam-4548	478	2	a.	a.	NOUN
ejpam-4548	478	3	therefore	therefore	ADV
ejpam-4548	478	4	,	,	PUNCT
ejpam-4548	478	5	η(y0	η(y0	NOUN
ejpam-4548	478	6	)	)	PUNCT
ejpam-4548	478	7	and	and	CCONJ
ejpam-4548	478	8	µ(z0	µ(z0	ADV
ejpam-4548	478	9	)	)	PUNCT
ejpam-4548	478	10	≥	≥	PROPN
ejpam-4548	478	11	a.	a.	NOUN
ejpam-4548	479	1	so	so	ADV
ejpam-4548	479	2	,	,	PUNCT
ejpam-4548	479	3	we	we	PRON
ejpam-4548	479	4	have	have	VERB
ejpam-4548	479	5	y0	y0	NOUN
ejpam-4548	479	6	∈	∈	NOUN
ejpam-4548	479	7	ηa	ηa	NOUN
ejpam-4548	479	8	and	and	CCONJ
ejpam-4548	479	9	z0	z0	PROPN
ejpam-4548	479	10	∈	∈	PROPN
ejpam-4548	479	11	µa	µa	PROPN
ejpam-4548	479	12	.	.	PUNCT
ejpam-4548	480	1	hence	hence	ADV
ejpam-4548	480	2	,	,	PUNCT
ejpam-4548	480	3	x	x	PUNCT
ejpam-4548	480	4	=	=	PUNCT
ejpam-4548	480	5	z0y0z	z0y0z	NUM
ejpam-4548	480	6	−1	−1	NOUN
ejpam-4548	480	7	0	0	NUM
ejpam-4548	480	8	∈	∈	PROPN
ejpam-4548	480	9	µaηa(µa	µaηa(µa	NOUN
ejpam-4548	480	10	)	)	PUNCT
ejpam-4548	480	11	−1	−1	NOUN
ejpam-4548	480	12	.	.	PUNCT
ejpam-4548	481	1	this	this	PRON
ejpam-4548	481	2	establishes	establish	VERB
ejpam-4548	481	3	(	(	PUNCT
ejpam-4548	481	4	1	1	NUM
ejpam-4548	481	5	)	)	PUNCT
ejpam-4548	481	6	.	.	PUNCT
ejpam-4548	482	1	in	in	ADP
ejpam-4548	482	2	order	order	NOUN
ejpam-4548	482	3	to	to	PART
ejpam-4548	482	4	prove	prove	VERB
ejpam-4548	482	5	the	the	DET
ejpam-4548	482	6	reverse	reverse	ADJ
ejpam-4548	482	7	inclusion	inclusion	NOUN
ejpam-4548	482	8	,	,	PUNCT
ejpam-4548	482	9	let	let	VERB
ejpam-4548	482	10	x	x	PUNCT
ejpam-4548	482	11	∈	∈	PROPN
ejpam-4548	482	12	µaηa(µa	µaηa(µa	NOUN
ejpam-4548	482	13	)	)	PUNCT
ejpam-4548	482	14	−1	−1	NOUN
ejpam-4548	482	15	.	.	PUNCT
ejpam-4548	483	1	then	then	ADV
ejpam-4548	483	2	,	,	PUNCT
ejpam-4548	483	3	there	there	PRON
ejpam-4548	483	4	exist	exist	VERB
ejpam-4548	483	5	y0	y0	PROPN
ejpam-4548	483	6	∈	∈	NOUN
ejpam-4548	483	7	ηa	ηa	NOUN
ejpam-4548	483	8	and	and	CCONJ
ejpam-4548	483	9	z0	z0	PROPN
ejpam-4548	483	10	∈	∈	PROPN
ejpam-4548	483	11	µa	µa	ADP
ejpam-4548	483	12	such	such	ADJ
ejpam-4548	483	13	that	that	SCONJ
ejpam-4548	483	14	x	x	X
ejpam-4548	483	15	=	=	PUNCT
ejpam-4548	483	16	z0y0(z0	z0y0(z0	X
ejpam-4548	483	17	)	)	PUNCT
ejpam-4548	483	18	−1	−1	NOUN
ejpam-4548	483	19	.	.	PUNCT
ejpam-4548	484	1	therefore	therefore	ADV
ejpam-4548	484	2	,	,	PUNCT
ejpam-4548	484	3	µηµ−1(x	µηµ−1(x	NOUN
ejpam-4548	484	4	)	)	PUNCT
ejpam-4548	484	5	=	=	PUNCT
ejpam-4548	485	1	∨	∨	NUM
ejpam-4548	485	2	x	x	X
ejpam-4548	485	3	=	=	PROPN
ejpam-4548	485	4	zyz−1	zyz−1	PROPN
ejpam-4548	485	5	{	{	PUNCT
ejpam-4548	485	6	η(y	η(y	PROPN
ejpam-4548	485	7	)	)	PUNCT
ejpam-4548	485	8	∧	∧	PROPN
ejpam-4548	485	9	µ(z	µ(z	PROPN
ejpam-4548	485	10	)	)	PUNCT
ejpam-4548	485	11	}	}	PUNCT
ejpam-4548	485	12	≥	≥	NOUN
ejpam-4548	485	13	η(y0	η(y0	NOUN
ejpam-4548	485	14	)	)	PUNCT
ejpam-4548	485	15	∧	∧	NOUN
ejpam-4548	485	16	µ(z0	µ(z0	ADV
ejpam-4548	485	17	)	)	PUNCT
ejpam-4548	485	18	≥	≥	NOUN
ejpam-4548	485	19	a.	a.	NOUN
ejpam-4548	485	20	hence	hence	ADV
ejpam-4548	485	21	x	x	X
ejpam-4548	485	22	∈	∈	PROPN
ejpam-4548	485	23	(	(	PUNCT
ejpam-4548	485	24	µηµ−1)a	µηµ−1)a	NUM
ejpam-4548	485	25	.	.	PUNCT
ejpam-4548	486	1	thus	thus	ADV
ejpam-4548	486	2	,	,	PUNCT
ejpam-4548	486	3	µaηa(µa	µaηa(µa	ADJ
ejpam-4548	486	4	)	)	PUNCT
ejpam-4548	486	5	−1	−1	NOUN
ejpam-4548	486	6	⊆	⊆	NUM
ejpam-4548	486	7	(	(	PUNCT
ejpam-4548	486	8	µηµ−1)a	µηµ−1)a	NUM
ejpam-4548	486	9	.	.	PUNCT
ejpam-4548	487	1	therefore	therefore	ADV
ejpam-4548	487	2	,	,	PUNCT
ejpam-4548	487	3	µaηa(µa	µaηa(µa	ADJ
ejpam-4548	487	4	)	)	PUNCT
ejpam-4548	487	5	−1	−1	NOUN
ejpam-4548	487	6	=	=	PUNCT
ejpam-4548	487	7	(	(	PUNCT
ejpam-4548	487	8	µηµ−1)a	µηµ−1)a	NUM
ejpam-4548	487	9	.	.	PUNCT
ejpam-4548	488	1	further	far	ADV
ejpam-4548	488	2	,	,	PUNCT
ejpam-4548	488	3	as	as	SCONJ
ejpam-4548	488	4	µ	µ	NOUN
ejpam-4548	488	5	and	and	CCONJ
ejpam-4548	488	6	η	η	PROPN
ejpam-4548	488	7	are	be	AUX
ejpam-4548	488	8	jointly	jointly	ADV
ejpam-4548	488	9	supstar	supstar	ADJ
ejpam-4548	488	10	,	,	PUNCT
ejpam-4548	488	11	by	by	ADP
ejpam-4548	488	12	remark	remark	NOUN
ejpam-4548	488	13	2	2	NUM
ejpam-4548	488	14	the	the	DET
ejpam-4548	488	15	l	l	NOUN
ejpam-4548	488	16	-	-	NOUN
ejpam-4548	488	17	subset	subset	NOUN
ejpam-4548	488	18	µηµ−1	µηµ−1	AUX
ejpam-4548	488	19	possesses	possess	VERB
ejpam-4548	488	20	supproperty	supproperty	NOUN
ejpam-4548	488	21	.	.	PUNCT
ejpam-4548	489	1	hence	hence	ADV
ejpam-4548	489	2	,	,	PUNCT
ejpam-4548	489	3	by	by	ADP
ejpam-4548	489	4	lemma	lemma	PROPN
ejpam-4548	489	5	3	3	NUM
ejpam-4548	489	6	,	,	PUNCT
ejpam-4548	489	7	we	we	PRON
ejpam-4548	489	8	have	have	VERB
ejpam-4548	489	9	(	(	PUNCT
ejpam-4548	489	10	ηµ)a	ηµ)a	PROPN
ejpam-4548	489	11	=	=	SYM
ejpam-4548	489	12	⟨µηµ−1⟩a	⟨µηµ−1⟩a	X
ejpam-4548	489	13	=	=	SYM
ejpam-4548	489	14	⟨(µηµ−1)a⟩	⟨(µηµ−1)a⟩	X
ejpam-4548	489	15	=	=	SYM
ejpam-4548	489	16	⟨µaηa(µa	⟨µaηa(µa	NOUN
ejpam-4548	489	17	)	)	PUNCT
ejpam-4548	489	18	−1⟩	−1⟩	PROPN
ejpam-4548	490	1	=	=	PUNCT
ejpam-4548	490	2	(	(	PUNCT
ejpam-4548	490	3	ηa	ηa	NOUN
ejpam-4548	490	4	)	)	PUNCT
ejpam-4548	490	5	µa	µa	NOUN
ejpam-4548	490	6	.	.	PUNCT
ejpam-4548	491	1	this	this	PRON
ejpam-4548	491	2	proves	prove	VERB
ejpam-4548	491	3	(	(	PUNCT
ejpam-4548	491	4	i	i	NOUN
ejpam-4548	491	5	)	)	PUNCT
ejpam-4548	491	6	.	.	PUNCT
ejpam-4548	492	1	(	(	PUNCT
ejpam-4548	492	2	ii	ii	X
ejpam-4548	492	3	)	)	PUNCT
ejpam-4548	492	4	it	it	PRON
ejpam-4548	492	5	can	can	AUX
ejpam-4548	492	6	be	be	AUX
ejpam-4548	492	7	prove	prove	VERB
ejpam-4548	492	8	by	by	ADP
ejpam-4548	492	9	using	use	VERB
ejpam-4548	492	10	the	the	DET
ejpam-4548	492	11	arguments	argument	NOUN
ejpam-4548	492	12	of	of	ADP
ejpam-4548	492	13	part(i	part(i	NOUN
ejpam-4548	492	14	)	)	PUNCT
ejpam-4548	492	15	.	.	PUNCT
ejpam-4548	493	1	let	let	VERB
ejpam-4548	493	2	ηi	ηi	PRON
ejpam-4548	493	3	be	be	AUX
ejpam-4548	493	4	the	the	DET
ejpam-4548	493	5	ith	ith	PROPN
ejpam-4548	493	6	normal	normal	ADJ
ejpam-4548	493	7	closure	closure	NOUN
ejpam-4548	493	8	of	of	ADP
ejpam-4548	493	9	η	η	PROPN
ejpam-4548	493	10	in	in	ADP
ejpam-4548	493	11	µ.	µ.	PROPN
ejpam-4548	493	12	then	then	ADV
ejpam-4548	493	13	,	,	PUNCT
ejpam-4548	493	14	more	more	ADV
ejpam-4548	493	15	generally	generally	ADV
ejpam-4548	493	16	we	we	PRON
ejpam-4548	493	17	have	have	VERB
ejpam-4548	493	18	:	:	PUNCT
ejpam-4548	493	19	lemma	lemma	PROPN
ejpam-4548	493	20	8	8	NUM
ejpam-4548	493	21	.	.	PUNCT
ejpam-4548	494	1	let	let	VERB
ejpam-4548	494	2	η	η	PROPN
ejpam-4548	494	3	∈	∈	PROPN
ejpam-4548	494	4	l(µ	l(µ	PROPN
ejpam-4548	494	5	)	)	PUNCT
ejpam-4548	494	6	.	.	PUNCT
ejpam-4548	495	1	then	then	ADV
ejpam-4548	495	2	,	,	PUNCT
ejpam-4548	495	3	(	(	PUNCT
ejpam-4548	495	4	i	i	NOUN
ejpam-4548	495	5	)	)	PUNCT
ejpam-4548	495	6	(	(	PUNCT
ejpam-4548	495	7	ηi)a	ηi)a	PROPN
ejpam-4548	495	8	=	=	SYM
ejpam-4548	495	9	(	(	PUNCT
ejpam-4548	495	10	ηa)i	ηa)i	PROPN
ejpam-4548	495	11	for	for	ADP
ejpam-4548	495	12	each	each	DET
ejpam-4548	495	13	a	a	DET
ejpam-4548	495	14	≤	≤	NUM
ejpam-4548	495	15	η(e	η(e	PROPN
ejpam-4548	495	16	)	)	PUNCT
ejpam-4548	495	17	provided	provide	VERB
ejpam-4548	495	18	µ	µ	PROPN
ejpam-4548	495	19	and	and	CCONJ
ejpam-4548	495	20	η	η	PROPN
ejpam-4548	495	21	are	be	AUX
ejpam-4548	495	22	supstar	supstar	ADJ
ejpam-4548	495	23	,	,	PUNCT
ejpam-4548	495	24	(	(	PUNCT
ejpam-4548	495	25	ii	ii	NOUN
ejpam-4548	495	26	)	)	PUNCT
ejpam-4548	495	27	(	(	PUNCT
ejpam-4548	495	28	ηi	ηi	PROPN
ejpam-4548	495	29	)	)	PUNCT
ejpam-4548	495	30	>	>	X
ejpam-4548	496	1	a	a	PRON
ejpam-4548	496	2	=	=	SYM
ejpam-4548	496	3	(	(	PUNCT
ejpam-4548	496	4	η	η	X
ejpam-4548	496	5	>	>	X
ejpam-4548	496	6	a	a	PRON
ejpam-4548	496	7	)	)	PUNCT
ejpam-4548	496	8	i	i	PRON
ejpam-4548	496	9	provided	provide	VERB
ejpam-4548	496	10	l	l	NOUN
ejpam-4548	496	11	is	be	AUX
ejpam-4548	496	12	a	a	DET
ejpam-4548	496	13	chain	chain	NOUN
ejpam-4548	496	14	.	.	PUNCT
ejpam-4548	497	1	theorem	theorem	NOUN
ejpam-4548	497	2	18	18	NUM
ejpam-4548	497	3	.	.	PUNCT
ejpam-4548	498	1	let	let	VERB
ejpam-4548	498	2	η	η	PROPN
ejpam-4548	498	3	∈	∈	PROPN
ejpam-4548	498	4	l(µ	l(µ	PROPN
ejpam-4548	498	5	)	)	PUNCT
ejpam-4548	498	6	be	be	AUX
ejpam-4548	498	7	such	such	ADJ
ejpam-4548	498	8	that	that	SCONJ
ejpam-4548	498	9	η	η	PROPN
ejpam-4548	498	10	and	and	CCONJ
ejpam-4548	498	11	µ	µ	PROPN
ejpam-4548	498	12	are	be	AUX
ejpam-4548	498	13	jointly	jointly	ADV
ejpam-4548	498	14	supstar	supstar	ADJ
ejpam-4548	498	15	.	.	PUNCT
ejpam-4548	499	1	then	then	ADV
ejpam-4548	499	2	,	,	PUNCT
ejpam-4548	499	3	η	η	PROPN
ejpam-4548	499	4	is	be	AUX
ejpam-4548	499	5	subnormal	subnormal	ADJ
ejpam-4548	499	6	having	have	VERB
ejpam-4548	499	7	defect	defect	NOUN
ejpam-4548	499	8	at	at	ADP
ejpam-4548	499	9	most	most	ADJ
ejpam-4548	499	10	n	n	NOUN
ejpam-4548	499	11	if	if	ADV
ejpam-4548	500	1	and	and	CCONJ
ejpam-4548	500	2	only	only	ADV
ejpam-4548	500	3	if	if	SCONJ
ejpam-4548	500	4	each	each	DET
ejpam-4548	500	5	level	level	NOUN
ejpam-4548	500	6	subset	subset	VERB
ejpam-4548	500	7	ηa	ηa	NOUN
ejpam-4548	500	8	is	be	AUX
ejpam-4548	500	9	subnormal	subnormal	ADJ
ejpam-4548	500	10	having	having	AUX
ejpam-4548	500	11	defect	defect	VERB
ejpam-4548	500	12	atmost	atmost	NOUN
ejpam-4548	500	13	n	n	CCONJ
ejpam-4548	500	14	where	where	SCONJ
ejpam-4548	500	15	a	a	DET
ejpam-4548	500	16	≤	≤	NUM
ejpam-4548	500	17	η(e	η(e	PROPN
ejpam-4548	500	18	)	)	PUNCT
ejpam-4548	500	19	.	.	PUNCT
ejpam-4548	501	1	proof	proof	NOUN
ejpam-4548	501	2	.	.	PUNCT
ejpam-4548	502	1	(	(	PUNCT
ejpam-4548	502	2	condition	condition	NOUN
ejpam-4548	502	3	is	be	AUX
ejpam-4548	502	4	necessary	necessary	ADJ
ejpam-4548	502	5	.	.	PUNCT
ejpam-4548	502	6	)	)	PUNCT
ejpam-4548	503	1	since	since	SCONJ
ejpam-4548	503	2	η	η	PROPN
ejpam-4548	503	3	is	be	AUX
ejpam-4548	503	4	subnormal	subnormal	ADJ
ejpam-4548	503	5	in	in	ADP
ejpam-4548	503	6	µ	µ	AUX
ejpam-4548	503	7	having	having	AUX
ejpam-4548	503	8	defect	defect	VERB
ejpam-4548	503	9	atmost	atmost	PROPN
ejpam-4548	503	10	n	n	CCONJ
ejpam-4548	503	11	,	,	PUNCT
ejpam-4548	503	12	by	by	ADP
ejpam-4548	503	13	the	the	DET
ejpam-4548	503	14	definition	definition	NOUN
ejpam-4548	503	15	,	,	PUNCT
ejpam-4548	503	16	there	there	PRON
ejpam-4548	503	17	exists	exist	VERB
ejpam-4548	503	18	a	a	DET
ejpam-4548	503	19	positive	positive	ADJ
ejpam-4548	503	20	integer	integer	NOUN
ejpam-4548	503	21	m	m	VERB
ejpam-4548	503	22	≤	≤	NOUN
ejpam-4548	503	23	n	n	CCONJ
ejpam-4548	503	24	such	such	ADJ
ejpam-4548	503	25	that	that	SCONJ
ejpam-4548	503	26	µ	µ	ADJ
ejpam-4548	503	27	=	=	SYM
ejpam-4548	503	28	η0	η0	NOUN
ejpam-4548	503	29	⊇	⊇	NOUN
ejpam-4548	503	30	η1	η1	PROPN
ejpam-4548	503	31	⊇	⊇	X
ejpam-4548	503	32	·	·	PUNCT
ejpam-4548	503	33	·	·	PUNCT
ejpam-4548	503	34	·	·	PUNCT
ejpam-4548	503	35	⊇	⊇	X
ejpam-4548	503	36	ηm	ηm	PROPN
ejpam-4548	503	37	=	=	SYM
ejpam-4548	503	38	η	η	PROPN
ejpam-4548	503	39	,	,	PUNCT
ejpam-4548	503	40	(	(	PUNCT
ejpam-4548	503	41	1	1	X
ejpam-4548	503	42	)	)	PUNCT
ejpam-4548	503	43	where	where	SCONJ
ejpam-4548	503	44	ηi	ηi	PROPN
ejpam-4548	503	45	is	be	AUX
ejpam-4548	503	46	the	the	DET
ejpam-4548	503	47	ith	ith	PROPN
ejpam-4548	503	48	normal	normal	ADJ
ejpam-4548	503	49	closure	closure	NOUN
ejpam-4548	503	50	of	of	ADP
ejpam-4548	503	51	η	η	PROPN
ejpam-4548	503	52	in	in	ADP
ejpam-4548	503	53	µ.	µ.	NOUN
ejpam-4548	503	54	let	let	VERB
ejpam-4548	503	55	a	a	DET
ejpam-4548	503	56	≤	≤	NUM
ejpam-4548	503	57	η(e	η(e	PROPN
ejpam-4548	503	58	)	)	PUNCT
ejpam-4548	503	59	.	.	PUNCT
ejpam-4548	504	1	as	as	ADP
ejpam-4548	504	2	η	η	PROPN
ejpam-4548	504	3	∈	∈	PROPN
ejpam-4548	504	4	l(µ	l(µ	PROPN
ejpam-4548	504	5	)	)	PUNCT
ejpam-4548	504	6	,	,	PUNCT
ejpam-4548	504	7	by	by	ADP
ejpam-4548	504	8	theorem	theorem	NOUN
ejpam-4548	504	9	2	2	NUM
ejpam-4548	504	10	,	,	PUNCT
ejpam-4548	504	11	ηa	ηa	PROPN
ejpam-4548	504	12	is	be	AUX
ejpam-4548	504	13	a	a	DET
ejpam-4548	504	14	subgroup	subgroup	NOUN
ejpam-4548	504	15	of	of	ADP
ejpam-4548	504	16	µa	µa	PROPN
ejpam-4548	504	17	.	.	PUNCT
ejpam-4548	505	1	we	we	PRON
ejpam-4548	505	2	show	show	VERB
ejpam-4548	505	3	that	that	SCONJ
ejpam-4548	505	4	ηa	ηa	PROPN
ejpam-4548	505	5	is	be	AUX
ejpam-4548	505	6	subnormal	subnormal	ADJ
ejpam-4548	505	7	in	in	ADP
ejpam-4548	505	8	µa	µa	PROPN
ejpam-4548	505	9	.	.	PUNCT
ejpam-4548	505	10	note	note	VERB
ejpam-4548	505	11	that	that	SCONJ
ejpam-4548	505	12	,	,	PUNCT
ejpam-4548	505	13	in	in	ADP
ejpam-4548	505	14	view	view	NOUN
ejpam-4548	505	15	of	of	ADP
ejpam-4548	505	16	(	(	PUNCT
ejpam-4548	505	17	1	1	X
ejpam-4548	505	18	)	)	PUNCT
ejpam-4548	505	19	we	we	PRON
ejpam-4548	505	20	have	have	VERB
ejpam-4548	505	21	µa	µa	NOUN
ejpam-4548	505	22	=	=	PUNCT
ejpam-4548	505	23	(	(	PUNCT
ejpam-4548	505	24	η0)a	η0)a	PROPN
ejpam-4548	505	25	⊇	⊇	NOUN
ejpam-4548	505	26	(	(	PUNCT
ejpam-4548	505	27	η1)a	η1)a	PROPN
ejpam-4548	505	28	⊇	⊇	PROPN
ejpam-4548	505	29	·	·	PUNCT
ejpam-4548	505	30	·	·	PUNCT
ejpam-4548	505	31	·	·	PUNCT
ejpam-4548	505	32	⊇	⊇	X
ejpam-4548	505	33	(	(	PUNCT
ejpam-4548	505	34	ηm)a	ηm)a	NOUN
ejpam-4548	505	35	=	=	SYM
ejpam-4548	505	36	ηa	ηa	PROPN
ejpam-4548	505	37	.	.	PUNCT
ejpam-4548	506	1	further	far	ADV
ejpam-4548	506	2	,	,	PUNCT
ejpam-4548	506	3	as	as	SCONJ
ejpam-4548	506	4	η	η	PROPN
ejpam-4548	506	5	and	and	CCONJ
ejpam-4548	506	6	µ	µ	PROPN
ejpam-4548	506	7	are	be	AUX
ejpam-4548	506	8	jointly	jointly	ADV
ejpam-4548	506	9	supstar	supstar	ADJ
ejpam-4548	506	10	,	,	PUNCT
ejpam-4548	506	11	by	by	ADP
ejpam-4548	506	12	lemma	lemma	PROPN
ejpam-4548	506	13	8	8	NUM
ejpam-4548	506	14	n.	n.	NOUN
ejpam-4548	506	15	ajmal	ajmal	PROPN
ejpam-4548	506	16	,	,	PUNCT
ejpam-4548	506	17	i.	i.	PROPN
ejpam-4548	506	18	jahan	jahan	PROPN
ejpam-4548	506	19	.	.	PROPN
ejpam-4548	506	20	,	,	PUNCT
ejpam-4548	506	21	b.	b.	PROPN
ejpam-4548	506	22	davvaz	davvaz	PROPN
ejpam-4548	506	23	/	/	SYM
ejpam-4548	506	24	eur	eur	PROPN
ejpam-4548	506	25	.	.	PUNCT
ejpam-4548	507	1	j.	j.	PROPN
ejpam-4548	507	2	pure	pure	PROPN
ejpam-4548	507	3	appl	appl	PROPN
ejpam-4548	507	4	.	.	PROPN
ejpam-4548	507	5	math	math	PROPN
ejpam-4548	507	6	,	,	PUNCT
ejpam-4548	507	7	15	15	NUM
ejpam-4548	507	8	(	(	PUNCT
ejpam-4548	507	9	4	4	NUM
ejpam-4548	507	10	)	)	PUNCT
ejpam-4548	507	11	(	(	PUNCT
ejpam-4548	507	12	2022	2022	NUM
ejpam-4548	507	13	)	)	PUNCT
ejpam-4548	507	14	,	,	PUNCT
ejpam-4548	507	15	2086	2086	NUM
ejpam-4548	507	16	-	-	SYM
ejpam-4548	507	17	2115	2115	NUM
ejpam-4548	507	18	2109	2109	NUM
ejpam-4548	507	19	(	(	PUNCT
ejpam-4548	507	20	ηa)i	ηa)i	NOUN
ejpam-4548	507	21	=	=	SYM
ejpam-4548	507	22	(	(	PUNCT
ejpam-4548	507	23	ηi)a	ηi)a	PROPN
ejpam-4548	507	24	for	for	ADP
ejpam-4548	507	25	all	all	DET
ejpam-4548	507	26	i.	i.	NOUN
ejpam-4548	507	27	thus	thus	ADV
ejpam-4548	507	28	,	,	PUNCT
ejpam-4548	507	29	we	we	PRON
ejpam-4548	507	30	have	have	VERB
ejpam-4548	507	31	µa	µa	NOUN
ejpam-4548	507	32	=	=	SYM
ejpam-4548	507	33	(	(	PUNCT
ejpam-4548	507	34	ηa)0	ηa)0	PROPN
ejpam-4548	507	35	⊇	⊇	NOUN
ejpam-4548	507	36	(	(	PUNCT
ejpam-4548	507	37	ηa)1	ηa)1	PROPN
ejpam-4548	507	38	⊇	⊇	PROPN
ejpam-4548	507	39	·	·	PUNCT
ejpam-4548	507	40	·	·	PUNCT
ejpam-4548	507	41	·	·	PUNCT
ejpam-4548	507	42	⊇	⊇	X
ejpam-4548	507	43	(	(	PUNCT
ejpam-4548	507	44	ηa)m	ηa)m	NOUN
ejpam-4548	507	45	=	=	SYM
ejpam-4548	507	46	ηa	ηa	PROPN
ejpam-4548	507	47	.	.	PUNCT
ejpam-4548	508	1	consequently	consequently	ADV
ejpam-4548	508	2	,	,	PUNCT
ejpam-4548	508	3	the	the	DET
ejpam-4548	508	4	normal	normal	ADJ
ejpam-4548	508	5	closure	closure	NOUN
ejpam-4548	508	6	series	series	NOUN
ejpam-4548	508	7	of	of	ADP
ejpam-4548	508	8	ηa	ηa	PRON
ejpam-4548	508	9	in	in	ADP
ejpam-4548	508	10	µa	µa	NOUN
ejpam-4548	508	11	has	have	VERB
ejpam-4548	508	12	the	the	DET
ejpam-4548	508	13	length	length	NOUN
ejpam-4548	508	14	atmost	atmost	PROPN
ejpam-4548	508	15	m.	m.	NOUN
ejpam-4548	508	16	as	as	ADP
ejpam-4548	508	17	m	m	PROPN
ejpam-4548	508	18	≤	≤	NOUN
ejpam-4548	508	19	n	n	CCONJ
ejpam-4548	508	20	,	,	PUNCT
ejpam-4548	508	21	ηa	ηa	PROPN
ejpam-4548	508	22	is	be	AUX
ejpam-4548	508	23	subnormal	subnormal	ADJ
ejpam-4548	508	24	in	in	ADP
ejpam-4548	508	25	µa	µa	NOUN
ejpam-4548	508	26	having	having	AUX
ejpam-4548	508	27	defect	defect	VERB
ejpam-4548	508	28	atmost	atmost	PROPN
ejpam-4548	508	29	n.	n.	NOUN
ejpam-4548	508	30	(	(	PUNCT
ejpam-4548	508	31	condition	condition	NOUN
ejpam-4548	508	32	is	be	AUX
ejpam-4548	508	33	sufficient	sufficient	ADJ
ejpam-4548	508	34	.	.	PUNCT
ejpam-4548	508	35	)	)	PUNCT
ejpam-4548	509	1	in	in	ADP
ejpam-4548	509	2	order	order	NOUN
ejpam-4548	509	3	to	to	PART
ejpam-4548	509	4	show	show	VERB
ejpam-4548	509	5	that	that	SCONJ
ejpam-4548	509	6	η	η	PROPN
ejpam-4548	509	7	is	be	AUX
ejpam-4548	509	8	subnormal	subnormal	ADJ
ejpam-4548	509	9	having	having	AUX
ejpam-4548	509	10	defect	defect	VERB
ejpam-4548	509	11	atmost	atmost	PROPN
ejpam-4548	509	12	n	n	CCONJ
ejpam-4548	509	13	,	,	PUNCT
ejpam-4548	509	14	we	we	PRON
ejpam-4548	509	15	construct	construct	VERB
ejpam-4548	509	16	the	the	DET
ejpam-4548	509	17	normal	normal	ADJ
ejpam-4548	509	18	closure	closure	NOUN
ejpam-4548	509	19	series	series	NOUN
ejpam-4548	509	20	of	of	ADP
ejpam-4548	509	21	η	η	PROPN
ejpam-4548	509	22	in	in	ADP
ejpam-4548	509	23	µ	µ	NOUN
ejpam-4548	509	24	and	and	CCONJ
ejpam-4548	509	25	show	show	VERB
ejpam-4548	509	26	that	that	SCONJ
ejpam-4548	509	27	its	its	PRON
ejpam-4548	509	28	length	length	NOUN
ejpam-4548	509	29	is	be	AUX
ejpam-4548	509	30	atmost	atmost	PROPN
ejpam-4548	509	31	n.	n.	PROPN
ejpam-4548	509	32	now	now	ADV
ejpam-4548	509	33	,	,	PUNCT
ejpam-4548	509	34	by	by	ADP
ejpam-4548	509	35	the	the	DET
ejpam-4548	509	36	hypothesis	hypothesis	NOUN
ejpam-4548	509	37	,	,	PUNCT
ejpam-4548	509	38	each	each	DET
ejpam-4548	509	39	non	non	ADJ
ejpam-4548	509	40	-	-	ADJ
ejpam-4548	509	41	empty	empty	ADJ
ejpam-4548	509	42	level	level	NOUN
ejpam-4548	509	43	subgroup	subgroup	NOUN
ejpam-4548	509	44	of	of	ADP
ejpam-4548	509	45	η	η	PROPN
ejpam-4548	509	46	is	be	AUX
ejpam-4548	509	47	subnormal	subnormal	ADJ
ejpam-4548	509	48	having	having	AUX
ejpam-4548	509	49	defect	defect	VERB
ejpam-4548	509	50	atmost	atmost	PROPN
ejpam-4548	509	51	n.	n.	PROPN
ejpam-4548	509	52	let	let	VERB
ejpam-4548	509	53	maj	maj	PROPN
ejpam-4548	509	54	be	be	AUX
ejpam-4548	509	55	the	the	DET
ejpam-4548	509	56	defect	defect	NOUN
ejpam-4548	509	57	of	of	ADP
ejpam-4548	509	58	the	the	DET
ejpam-4548	509	59	level	level	NOUN
ejpam-4548	509	60	subgroup	subgroup	NOUN
ejpam-4548	509	61	ηaj	ηaj	NOUN
ejpam-4548	509	62	in	in	ADP
ejpam-4548	509	63	µaj	µaj	NOUN
ejpam-4548	509	64	for	for	ADP
ejpam-4548	509	65	any	any	DET
ejpam-4548	509	66	aj	aj	PROPN
ejpam-4548	509	67	≤	≤	PROPN
ejpam-4548	509	68	η(e	η(e	PROPN
ejpam-4548	509	69	)	)	PUNCT
ejpam-4548	509	70	.	.	PUNCT
ejpam-4548	510	1	then	then	ADV
ejpam-4548	510	2	,	,	PUNCT
ejpam-4548	510	3	the	the	DET
ejpam-4548	510	4	normal	normal	ADJ
ejpam-4548	510	5	closure	closure	NOUN
ejpam-4548	510	6	series	series	NOUN
ejpam-4548	510	7	of	of	ADP
ejpam-4548	510	8	ηaj	ηaj	PROPN
ejpam-4548	510	9	in	in	ADP
ejpam-4548	510	10	µaj	µaj	NOUN
ejpam-4548	510	11	is	be	AUX
ejpam-4548	510	12	given	give	VERB
ejpam-4548	510	13	by	by	ADP
ejpam-4548	510	14	µaj	µaj	NOUN
ejpam-4548	510	15	=	=	SYM
ejpam-4548	510	16	(	(	PUNCT
ejpam-4548	510	17	ηaj	ηaj	PROPN
ejpam-4548	510	18	)	)	PUNCT
ejpam-4548	510	19	0	0	PROPN
ejpam-4548	510	20	⊇	⊇	NOUN
ejpam-4548	510	21	(	(	PUNCT
ejpam-4548	510	22	ηaj	ηaj	PROPN
ejpam-4548	510	23	)	)	PUNCT
ejpam-4548	510	24	1	1	NUM
ejpam-4548	510	25	⊇	⊇	NOUN
ejpam-4548	510	26	...	...	PUNCT
ejpam-4548	510	27	⊇	⊇	X
ejpam-4548	510	28	(	(	PUNCT
ejpam-4548	510	29	ηaj	ηaj	PROPN
ejpam-4548	510	30	)	)	PUNCT
ejpam-4548	510	31	maj	maj	PROPN
ejpam-4548	511	1	=	=	PROPN
ejpam-4548	511	2	ηaj	ηaj	PROPN
ejpam-4548	511	3	for	for	ADP
ejpam-4548	511	4	all	all	DET
ejpam-4548	511	5	aj	aj	PROPN
ejpam-4548	511	6	≤	≤	PROPN
ejpam-4548	511	7	η(e	η(e	PROPN
ejpam-4548	511	8	)	)	PUNCT
ejpam-4548	511	9	,	,	PUNCT
ejpam-4548	511	10	(	(	PUNCT
ejpam-4548	511	11	2	2	X
ejpam-4548	511	12	)	)	PUNCT
ejpam-4548	511	13	where	where	SCONJ
ejpam-4548	511	14	(	(	PUNCT
ejpam-4548	511	15	ηaj	ηaj	PROPN
ejpam-4548	511	16	)	)	PUNCT
ejpam-4548	511	17	i	i	PRON
ejpam-4548	511	18	is	be	AUX
ejpam-4548	511	19	the	the	DET
ejpam-4548	511	20	ith	ith	PROPN
ejpam-4548	511	21	normal	normal	ADJ
ejpam-4548	511	22	closure	closure	NOUN
ejpam-4548	511	23	of	of	ADP
ejpam-4548	511	24	ηaj	ηaj	NOUN
ejpam-4548	511	25	in	in	ADP
ejpam-4548	511	26	µaj	µaj	NOUN
ejpam-4548	511	27	.	.	PUNCT
ejpam-4548	512	1	next	next	ADV
ejpam-4548	512	2	,	,	PUNCT
ejpam-4548	512	3	let	let	VERB
ejpam-4548	512	4	c	c	NOUN
ejpam-4548	512	5	=	=	SYM
ejpam-4548	512	6	imη	imη	VERB
ejpam-4548	512	7	∪	∪	ADV
ejpam-4548	512	8	{	{	PUNCT
ejpam-4548	512	9	a	a	DET
ejpam-4548	512	10	∈	∈	PROPN
ejpam-4548	512	11	imµ	imµ	NOUN
ejpam-4548	512	12	:	:	PUNCT
ejpam-4548	512	13	a	a	DET
ejpam-4548	512	14	≤	≤	NUM
ejpam-4548	512	15	η(e	η(e	PROPN
ejpam-4548	512	16	)	)	PUNCT
ejpam-4548	512	17	}	}	PUNCT
ejpam-4548	512	18	.	.	PUNCT
ejpam-4548	513	1	then	then	ADV
ejpam-4548	513	2	,	,	PUNCT
ejpam-4548	513	3	by	by	ADP
ejpam-4548	513	4	lemma	lemma	PROPN
ejpam-4548	513	5	4	4	NUM
ejpam-4548	513	6	,	,	PUNCT
ejpam-4548	513	7	c	c	PROPN
ejpam-4548	513	8	is	be	AUX
ejpam-4548	513	9	a	a	DET
ejpam-4548	513	10	chain	chain	NOUN
ejpam-4548	513	11	.	.	PUNCT
ejpam-4548	514	1	let	let	VERB
ejpam-4548	514	2	s	s	PRON
ejpam-4548	514	3	=	=	X
ejpam-4548	514	4	{	{	PUNCT
ejpam-4548	514	5	maj	maj	PROPN
ejpam-4548	514	6	≤	≤	PROPN
ejpam-4548	514	7	n	n	PROPN
ejpam-4548	514	8	:	:	PUNCT
ejpam-4548	514	9	aj	aj	PROPN
ejpam-4548	514	10	∈	∈	PROPN
ejpam-4548	514	11	c	c	AUX
ejpam-4548	514	12	}	}	PUNCT
ejpam-4548	514	13	.	.	PUNCT
ejpam-4548	515	1	then	then	ADV
ejpam-4548	515	2	,	,	PUNCT
ejpam-4548	515	3	sups	sup	NOUN
ejpam-4548	515	4	≤	≤	PROPN
ejpam-4548	515	5	n.	n.	NOUN
ejpam-4548	515	6	we	we	PRON
ejpam-4548	515	7	write	write	VERB
ejpam-4548	515	8	m	m	NOUN
ejpam-4548	515	9	=	=	NOUN
ejpam-4548	515	10	sups	sup	NOUN
ejpam-4548	515	11	≤	≤	X
ejpam-4548	515	12	n.	n.	NOUN
ejpam-4548	515	13	now	now	ADV
ejpam-4548	515	14	,	,	PUNCT
ejpam-4548	515	15	we	we	PRON
ejpam-4548	515	16	shall	shall	AUX
ejpam-4548	515	17	construct	construct	VERB
ejpam-4548	515	18	the	the	DET
ejpam-4548	515	19	normal	normal	ADJ
ejpam-4548	515	20	closure	closure	NOUN
ejpam-4548	515	21	series	series	NOUN
ejpam-4548	515	22	of	of	ADP
ejpam-4548	515	23	η	η	PROPN
ejpam-4548	515	24	in	in	ADP
ejpam-4548	515	25	µ	µ	NOUN
ejpam-4548	515	26	having	have	VERB
ejpam-4548	515	27	the	the	DET
ejpam-4548	515	28	length	length	NOUN
ejpam-4548	515	29	m.	m.	NOUN
ejpam-4548	515	30	in	in	ADP
ejpam-4548	515	31	view	view	NOUN
ejpam-4548	515	32	of	of	ADP
ejpam-4548	515	33	(	(	PUNCT
ejpam-4548	515	34	2	2	NUM
ejpam-4548	515	35	)	)	PUNCT
ejpam-4548	515	36	,	,	PUNCT
ejpam-4548	515	37	we	we	PRON
ejpam-4548	515	38	have	have	VERB
ejpam-4548	515	39	µaj	µaj	NOUN
ejpam-4548	515	40	=	=	SYM
ejpam-4548	515	41	(	(	PUNCT
ejpam-4548	515	42	ηaj	ηaj	PROPN
ejpam-4548	515	43	)	)	PUNCT
ejpam-4548	515	44	0	0	PROPN
ejpam-4548	516	1	⊇	⊇	NOUN
ejpam-4548	516	2	(	(	PUNCT
ejpam-4548	516	3	ηaj	ηaj	PROPN
ejpam-4548	516	4	)	)	PUNCT
ejpam-4548	516	5	1	1	NUM
ejpam-4548	516	6	⊇	⊇	NOUN
ejpam-4548	516	7	...	...	PUNCT
ejpam-4548	516	8	⊇	⊇	X
ejpam-4548	516	9	(	(	PUNCT
ejpam-4548	516	10	ηaj	ηaj	PROPN
ejpam-4548	516	11	)	)	PUNCT
ejpam-4548	516	12	maj	maj	PROPN
ejpam-4548	517	1	=	=	PROPN
ejpam-4548	517	2	ηaj	ηaj	PROPN
ejpam-4548	517	3	for	for	ADP
ejpam-4548	517	4	all	all	DET
ejpam-4548	517	5	aj	aj	PROPN
ejpam-4548	517	6	∈	∈	PROPN
ejpam-4548	517	7	c.	c.	NOUN
ejpam-4548	517	8	further	far	ADV
ejpam-4548	517	9	,	,	PUNCT
ejpam-4548	517	10	as	as	SCONJ
ejpam-4548	517	11	maj	maj	PROPN
ejpam-4548	517	12	≤	≤	X
ejpam-4548	517	13	m	m	VERB
ejpam-4548	517	14	for	for	ADP
ejpam-4548	517	15	each	each	DET
ejpam-4548	517	16	aj	aj	PROPN
ejpam-4548	517	17	∈	∈	PROPN
ejpam-4548	517	18	c	c	AUX
ejpam-4548	517	19	,	,	PUNCT
ejpam-4548	517	20	we	we	PRON
ejpam-4548	517	21	insert	insert	VERB
ejpam-4548	517	22	m	m	VERB
ejpam-4548	517	23	−maj	−maj	NOUN
ejpam-4548	517	24	times	time	NOUN
ejpam-4548	517	25	the	the	DET
ejpam-4548	517	26	level	level	NOUN
ejpam-4548	517	27	subgroup	subgroup	NOUN
ejpam-4548	517	28	ηaj	ηaj	NOUN
ejpam-4548	517	29	in	in	ADP
ejpam-4548	517	30	the	the	DET
ejpam-4548	517	31	normal	normal	ADJ
ejpam-4548	517	32	closure	closure	NOUN
ejpam-4548	517	33	series	series	NOUN
ejpam-4548	517	34	of	of	ADP
ejpam-4548	517	35	ηaj	ηaj	PROPN
ejpam-4548	517	36	in	in	ADP
ejpam-4548	517	37	µaj	µaj	NOUN
ejpam-4548	517	38	.	.	PUNCT
ejpam-4548	518	1	consequently	consequently	ADV
ejpam-4548	518	2	,	,	PUNCT
ejpam-4548	518	3	the	the	DET
ejpam-4548	518	4	normal	normal	ADJ
ejpam-4548	518	5	closure	closure	NOUN
ejpam-4548	518	6	series	series	NOUN
ejpam-4548	518	7	of	of	ADP
ejpam-4548	518	8	ηaj	ηaj	PROPN
ejpam-4548	518	9	in	in	ADP
ejpam-4548	518	10	µaj	µaj	NOUN
ejpam-4548	518	11	aquire	aquire	VERB
ejpam-4548	518	12	the	the	DET
ejpam-4548	518	13	length	length	NOUN
ejpam-4548	518	14	m.	m.	NOUN
ejpam-4548	518	15	obviously	obviously	ADV
ejpam-4548	518	16	,	,	PUNCT
ejpam-4548	518	17	for	for	ADP
ejpam-4548	518	18	all	all	DET
ejpam-4548	518	19	l	l	NOUN
ejpam-4548	518	20	=	=	SYM
ejpam-4548	518	21	1	1	NUM
ejpam-4548	518	22	,	,	PUNCT
ejpam-4548	518	23	2	2	NUM
ejpam-4548	518	24	,	,	PUNCT
ejpam-4548	518	25	...	...	PUNCT
ejpam-4548	518	26	,	,	PUNCT
ejpam-4548	518	27	m−maj	m−maj	X
ejpam-4548	518	28	(	(	PUNCT
ejpam-4548	518	29	ηaj	ηaj	NOUN
ejpam-4548	518	30	)	)	PUNCT
ejpam-4548	518	31	maj+l	maj+l	PROPN
ejpam-4548	518	32	=	=	SYM
ejpam-4548	518	33	ηaj	ηaj	NOUN
ejpam-4548	518	34	.	.	PUNCT
ejpam-4548	519	1	thus	thus	ADV
ejpam-4548	519	2	µaj	µaj	NOUN
ejpam-4548	519	3	=	=	SYM
ejpam-4548	519	4	(	(	PUNCT
ejpam-4548	519	5	ηaj	ηaj	PROPN
ejpam-4548	519	6	)	)	PUNCT
ejpam-4548	519	7	0	0	PROPN
ejpam-4548	520	1	⊇	⊇	NOUN
ejpam-4548	520	2	(	(	PUNCT
ejpam-4548	520	3	ηaj	ηaj	PROPN
ejpam-4548	520	4	)	)	PUNCT
ejpam-4548	520	5	1	1	NUM
ejpam-4548	520	6	⊇	⊇	NOUN
ejpam-4548	520	7	...	...	PUNCT
ejpam-4548	520	8	⊇	⊇	X
ejpam-4548	520	9	(	(	PUNCT
ejpam-4548	520	10	ηaj	ηaj	PROPN
ejpam-4548	520	11	)	)	PUNCT
ejpam-4548	520	12	maj	maj	PROPN
ejpam-4548	521	1	=	=	PROPN
ejpam-4548	521	2	ηaj	ηaj	PROPN
ejpam-4548	521	3	for	for	ADP
ejpam-4548	521	4	all	all	DET
ejpam-4548	521	5	aj	aj	PROPN
ejpam-4548	521	6	∈	∈	PROPN
ejpam-4548	521	7	c.	c.	NOUN
ejpam-4548	521	8	(	(	PUNCT
ejpam-4548	521	9	3	3	NUM
ejpam-4548	521	10	)	)	PUNCT
ejpam-4548	521	11	in	in	ADP
ejpam-4548	521	12	view	view	NOUN
ejpam-4548	521	13	of	of	ADP
ejpam-4548	521	14	lemma	lemma	PROPN
ejpam-4548	521	15	8	8	NUM
ejpam-4548	521	16	,	,	PUNCT
ejpam-4548	521	17	we	we	PRON
ejpam-4548	521	18	have	have	VERB
ejpam-4548	521	19	µaj	µaj	NOUN
ejpam-4548	521	20	=	=	SYM
ejpam-4548	521	21	(	(	PUNCT
ejpam-4548	521	22	η0)aj	η0)aj	PROPN
ejpam-4548	521	23	⊇	⊇	PROPN
ejpam-4548	521	24	(	(	PUNCT
ejpam-4548	521	25	η1)aj	η1)aj	PROPN
ejpam-4548	521	26	⊇	⊇	NOUN
ejpam-4548	521	27	...	...	PUNCT
ejpam-4548	522	1	⊇	⊇	NOUN
ejpam-4548	522	2	(	(	PUNCT
ejpam-4548	522	3	ηm)aj	ηm)aj	PROPN
ejpam-4548	522	4	=	=	PUNCT
ejpam-4548	522	5	ηaj	ηaj	PROPN
ejpam-4548	522	6	for	for	ADP
ejpam-4548	522	7	all	all	DET
ejpam-4548	522	8	aj	aj	PROPN
ejpam-4548	522	9	∈	∈	PROPN
ejpam-4548	522	10	c.	c.	NOUN
ejpam-4548	522	11	let	let	VERB
ejpam-4548	522	12	aj	aj	PROPN
ejpam-4548	522	13	,	,	PUNCT
ejpam-4548	522	14	ak	ak	PROPN
ejpam-4548	522	15	∈	∈	PROPN
ejpam-4548	522	16	c	c	PROPN
ejpam-4548	522	17	and	and	CCONJ
ejpam-4548	522	18	aj	aj	PROPN
ejpam-4548	522	19	≥	≥	PROPN
ejpam-4548	522	20	ak	ak	PROPN
ejpam-4548	522	21	.	.	PROPN
ejpam-4548	522	22	hence	hence	ADV
ejpam-4548	522	23	for	for	ADP
ejpam-4548	522	24	each	each	DET
ejpam-4548	522	25	fixed	fix	VERB
ejpam-4548	522	26	i	i	PRON
ejpam-4548	522	27	,	,	PUNCT
ejpam-4548	522	28	it	it	PRON
ejpam-4548	522	29	follows	follow	VERB
ejpam-4548	522	30	that	that	SCONJ
ejpam-4548	522	31	(	(	PUNCT
ejpam-4548	522	32	ηaj	ηaj	PROPN
ejpam-4548	522	33	)	)	PUNCT
ejpam-4548	522	34	i	i	PRON
ejpam-4548	522	35	=	=	PUNCT
ejpam-4548	522	36	(	(	PUNCT
ejpam-4548	522	37	ηaj	ηaj	PROPN
ejpam-4548	522	38	)	)	PUNCT
ejpam-4548	522	39	(	(	PUNCT
ejpam-4548	522	40	ηaj	ηaj	PROPN
ejpam-4548	522	41	)	)	PUNCT
ejpam-4548	522	42	i−1	i−1	PROPN
ejpam-4548	522	43	(	(	PUNCT
ejpam-4548	522	44	by	by	ADP
ejpam-4548	522	45	the	the	DET
ejpam-4548	522	46	definition	definition	NOUN
ejpam-4548	522	47	of	of	ADP
ejpam-4548	522	48	classical	classical	ADJ
ejpam-4548	522	49	ith	ith	PROPN
ejpam-4548	522	50	normal	normal	ADJ
ejpam-4548	522	51	closure	closure	NOUN
ejpam-4548	522	52	)	)	PUNCT
ejpam-4548	523	1	=	=	SYM
ejpam-4548	523	2	⟨(ηaj	⟨(ηaj	NOUN
ejpam-4548	523	3	)	)	PUNCT
ejpam-4548	524	1	i−1ηaj	i−1ηaj	NOUN
ejpam-4548	524	2	(	(	PUNCT
ejpam-4548	524	3	(	(	PUNCT
ejpam-4548	524	4	ηaj	ηaj	PROPN
ejpam-4548	524	5	)	)	PUNCT
ejpam-4548	524	6	i−1	i−1	PROPN
ejpam-4548	524	7	)	)	PUNCT
ejpam-4548	524	8	−1⟩	−1⟩	PROPN
ejpam-4548	524	9	⊆	⊆	NUM
ejpam-4548	524	10	⟨(ηak)i−1ηak((ηak)i−1	⟨(ηak)i−1ηak((ηak)i−1	PROPN
ejpam-4548	524	11	)	)	PUNCT
ejpam-4548	524	12	−1⟩	−1⟩	PROPN
ejpam-4548	524	13	=	=	SYM
ejpam-4548	524	14	(	(	PUNCT
ejpam-4548	524	15	ηak)i	ηak)i	PROPN
ejpam-4548	524	16	.	.	PUNCT
ejpam-4548	525	1	(	(	PUNCT
ejpam-4548	525	2	4	4	X
ejpam-4548	525	3	)	)	PUNCT
ejpam-4548	525	4	n.	n.	NOUN
ejpam-4548	525	5	ajmal	ajmal	PROPN
ejpam-4548	525	6	,	,	PUNCT
ejpam-4548	525	7	i.	i.	PROPN
ejpam-4548	525	8	jahan	jahan	PROPN
ejpam-4548	525	9	.	.	PROPN
ejpam-4548	525	10	,	,	PUNCT
ejpam-4548	525	11	b.	b.	PROPN
ejpam-4548	525	12	davvaz	davvaz	PROPN
ejpam-4548	525	13	/	/	SYM
ejpam-4548	525	14	eur	eur	PROPN
ejpam-4548	525	15	.	.	PUNCT
ejpam-4548	526	1	j.	j.	PROPN
ejpam-4548	526	2	pure	pure	PROPN
ejpam-4548	526	3	appl	appl	PROPN
ejpam-4548	526	4	.	.	PROPN
ejpam-4548	526	5	math	math	PROPN
ejpam-4548	526	6	,	,	PUNCT
ejpam-4548	526	7	15	15	NUM
ejpam-4548	526	8	(	(	PUNCT
ejpam-4548	526	9	4	4	NUM
ejpam-4548	526	10	)	)	PUNCT
ejpam-4548	526	11	(	(	PUNCT
ejpam-4548	526	12	2022	2022	NUM
ejpam-4548	526	13	)	)	PUNCT
ejpam-4548	526	14	,	,	PUNCT
ejpam-4548	526	15	2086	2086	NUM
ejpam-4548	526	16	-	-	SYM
ejpam-4548	526	17	2115	2115	NUM
ejpam-4548	526	18	2110	2110	NUM
ejpam-4548	526	19	thus	thus	ADV
ejpam-4548	526	20	for	for	ADP
ejpam-4548	526	21	a	a	DET
ejpam-4548	526	22	fixed	fix	VERB
ejpam-4548	526	23	i	i	PRON
ejpam-4548	526	24	the	the	DET
ejpam-4548	526	25	ith	ith	PROPN
ejpam-4548	526	26	normal	normal	ADJ
ejpam-4548	526	27	closures	closure	NOUN
ejpam-4548	526	28	of	of	ADP
ejpam-4548	526	29	the	the	DET
ejpam-4548	526	30	level	level	NOUN
ejpam-4548	526	31	subsets	subset	NOUN
ejpam-4548	526	32	ηaj	ηaj	NOUN
ejpam-4548	526	33	for	for	ADP
ejpam-4548	526	34	each	each	DET
ejpam-4548	526	35	aj	aj	PROPN
ejpam-4548	526	36	∈	∈	PROPN
ejpam-4548	526	37	c	c	PROPN
ejpam-4548	526	38	constitutes	constitute	VERB
ejpam-4548	526	39	a	a	DET
ejpam-4548	526	40	chain	chain	NOUN
ejpam-4548	526	41	.	.	PUNCT
ejpam-4548	527	1	in	in	ADP
ejpam-4548	527	2	view	view	NOUN
ejpam-4548	527	3	of	of	ADP
ejpam-4548	527	4	the	the	DET
ejpam-4548	527	5	fact	fact	NOUN
ejpam-4548	527	6	that	that	SCONJ
ejpam-4548	527	7	the	the	DET
ejpam-4548	527	8	level	level	NOUN
ejpam-4548	527	9	subsets	subset	NOUN
ejpam-4548	527	10	ηaj	ηaj	PROPN
ejpam-4548	527	11	are	be	AUX
ejpam-4548	527	12	ordinary	ordinary	ADJ
ejpam-4548	527	13	subgroups	subgroup	NOUN
ejpam-4548	527	14	,	,	PUNCT
ejpam-4548	527	15	η	η	PROPN
ejpam-4548	527	16	ηaj	ηaj	PROPN
ejpam-4548	527	17	aj	aj	PROPN
ejpam-4548	527	18	=	=	PROPN
ejpam-4548	527	19	ηaj	ηaj	PROPN
ejpam-4548	527	20	,	,	PUNCT
ejpam-4548	527	21	insertion	insertion	NOUN
ejpam-4548	527	22	of	of	ADP
ejpam-4548	527	23	the	the	DET
ejpam-4548	527	24	level	level	NOUN
ejpam-4548	527	25	subgroup	subgroup	NOUN
ejpam-4548	527	26	ηaj	ηaj	NOUN
ejpam-4548	527	27	in	in	ADP
ejpam-4548	527	28	the	the	DET
ejpam-4548	527	29	chain	chain	NOUN
ejpam-4548	527	30	of	of	ADP
ejpam-4548	527	31	its	its	PRON
ejpam-4548	527	32	ith	ith	PROPN
ejpam-4548	527	33	normal	normal	ADJ
ejpam-4548	527	34	closures	closure	NOUN
ejpam-4548	527	35	does	do	AUX
ejpam-4548	527	36	not	not	PART
ejpam-4548	527	37	affect	affect	VERB
ejpam-4548	527	38	the	the	DET
ejpam-4548	527	39	above	above	ADJ
ejpam-4548	527	40	containment	containment	NOUN
ejpam-4548	527	41	.	.	PUNCT
ejpam-4548	528	1	in	in	ADP
ejpam-4548	528	2	view	view	NOUN
ejpam-4548	528	3	of	of	ADP
ejpam-4548	528	4	lemma	lemma	PROPN
ejpam-4548	528	5	8	8	NUM
ejpam-4548	528	6	and	and	CCONJ
ejpam-4548	528	7	by	by	ADP
ejpam-4548	528	8	(	(	PUNCT
ejpam-4548	528	9	4	4	NUM
ejpam-4548	528	10	)	)	PUNCT
ejpam-4548	528	11	,	,	PUNCT
ejpam-4548	528	12	we	we	PRON
ejpam-4548	528	13	obtain	obtain	VERB
ejpam-4548	528	14	(	(	PUNCT
ejpam-4548	528	15	ηi)aj	ηi)aj	NOUN
ejpam-4548	528	16	⊆	⊆	NUM
ejpam-4548	528	17	(	(	PUNCT
ejpam-4548	528	18	ηi)ak	ηi)ak	X
ejpam-4548	528	19	.	.	PUNCT
ejpam-4548	529	1	in	in	ADP
ejpam-4548	529	2	view	view	NOUN
ejpam-4548	529	3	of	of	ADP
ejpam-4548	529	4	(	(	PUNCT
ejpam-4548	529	5	3	3	NUM
ejpam-4548	529	6	)	)	PUNCT
ejpam-4548	529	7	,	,	PUNCT
ejpam-4548	529	8	for	for	ADP
ejpam-4548	529	9	each	each	DET
ejpam-4548	529	10	i	i	PRON
ejpam-4548	529	11	,	,	PUNCT
ejpam-4548	529	12	we	we	PRON
ejpam-4548	529	13	have	have	VERB
ejpam-4548	529	14	(	(	PUNCT
ejpam-4548	529	15	ηi−1)aj	ηi−1)aj	PROPN
ejpam-4548	529	16	⊇	⊇	PROPN
ejpam-4548	529	17	(	(	PUNCT
ejpam-4548	529	18	ηi)aj	ηi)aj	X
ejpam-4548	529	19	for	for	ADP
ejpam-4548	529	20	each	each	DET
ejpam-4548	529	21	aj	aj	PROPN
ejpam-4548	529	22	∈	∈	PROPN
ejpam-4548	529	23	c.	c.	NOUN
ejpam-4548	529	24	now	now	ADV
ejpam-4548	529	25	we	we	PRON
ejpam-4548	529	26	claim	claim	VERB
ejpam-4548	529	27	that	that	SCONJ
ejpam-4548	529	28	(	(	PUNCT
ejpam-4548	529	29	ηi−1)aj	ηi−1)aj	PROPN
ejpam-4548	529	30	⊇	⊇	PROPN
ejpam-4548	529	31	(	(	PUNCT
ejpam-4548	529	32	ηi)aj	ηi)aj	X
ejpam-4548	529	33	for	for	ADP
ejpam-4548	529	34	each	each	DET
ejpam-4548	529	35	aj	aj	PROPN
ejpam-4548	529	36	∈	∈	PROPN
ejpam-4548	529	37	i	i	PRON
ejpam-4548	529	38	m	m	VERB
ejpam-4548	529	39	µ	µ	PRON
ejpam-4548	529	40	∪	∪	X
ejpam-4548	529	41	i	i	PROPN
ejpam-4548	529	42	m	m	PROPN
ejpam-4548	529	43	η	η	PROPN
ejpam-4548	529	44	.	.	PROPN
ejpam-4548	529	45	here	here	ADV
ejpam-4548	529	46	observe	observe	VERB
ejpam-4548	529	47	that	that	SCONJ
ejpam-4548	529	48	,	,	PUNCT
ejpam-4548	529	49	in	in	ADP
ejpam-4548	529	50	view	view	NOUN
ejpam-4548	529	51	of	of	ADP
ejpam-4548	529	52	proposition	proposition	NOUN
ejpam-4548	529	53	5	5	NUM
ejpam-4548	529	54	,	,	PUNCT
ejpam-4548	529	55	the	the	DET
ejpam-4548	529	56	tip	tip	NOUN
ejpam-4548	529	57	of	of	ADP
ejpam-4548	529	58	ηi	ηi	PROPN
ejpam-4548	529	59	,	,	PUNCT
ejpam-4548	529	60	for	for	ADP
ejpam-4548	529	61	each	each	DET
ejpam-4548	529	62	i	i	PRON
ejpam-4548	529	63	,	,	PUNCT
ejpam-4548	529	64	is	be	AUX
ejpam-4548	529	65	η(e	η(e	NOUN
ejpam-4548	529	66	)	)	PUNCT
ejpam-4548	529	67	.	.	PUNCT
ejpam-4548	530	1	hence	hence	ADV
ejpam-4548	530	2	(	(	PUNCT
ejpam-4548	530	3	ηi)aj	ηi)aj	X
ejpam-4548	530	4	=	=	SYM
ejpam-4548	530	5	ϕ	ϕ	PROPN
ejpam-4548	530	6	for	for	ADP
ejpam-4548	530	7	all	all	DET
ejpam-4548	530	8	aj	aj	PROPN
ejpam-4548	530	9	∈	∈	PROPN
ejpam-4548	530	10	{	{	PUNCT
ejpam-4548	530	11	a	a	DET
ejpam-4548	530	12	∈	∈	PROPN
ejpam-4548	531	1	i	i	PRON
ejpam-4548	531	2	m	m	VERB
ejpam-4548	531	3	µ	µ	VERB
ejpam-4548	531	4	:	:	PUNCT
ejpam-4548	531	5	a	a	DET
ejpam-4548	531	6	≰	≰	PROPN
ejpam-4548	531	7	η(e	η(e	PROPN
ejpam-4548	531	8	)	)	PUNCT
ejpam-4548	531	9	}	}	PUNCT
ejpam-4548	531	10	.	.	PUNCT
ejpam-4548	532	1	this	this	PRON
ejpam-4548	532	2	establishes	establish	VERB
ejpam-4548	532	3	the	the	DET
ejpam-4548	532	4	claim	claim	NOUN
ejpam-4548	532	5	.	.	PUNCT
ejpam-4548	533	1	further	far	ADV
ejpam-4548	533	2	,	,	PUNCT
ejpam-4548	533	3	by	by	ADP
ejpam-4548	533	4	lemma	lemma	PROPN
ejpam-4548	533	5	6	6	NUM
ejpam-4548	533	6	,	,	PUNCT
ejpam-4548	533	7	for	for	AUX
ejpam-4548	533	8	each	each	DET
ejpam-4548	533	9	i	i	PRON
ejpam-4548	533	10	i	i	PRON
ejpam-4548	533	11	m	m	VERB
ejpam-4548	533	12	ηi	ηi	VERB
ejpam-4548	533	13	⊆	⊆	NUM
ejpam-4548	533	14	i	i	PROPN
ejpam-4548	533	15	m	m	PROPN
ejpam-4548	533	16	η	η	PROPN
ejpam-4548	533	17	∪	∪	X
ejpam-4548	533	18	i	i	PRON
ejpam-4548	533	19	m	m	VERB
ejpam-4548	533	20	µ.	µ.	NOUN
ejpam-4548	533	21	thus	thus	ADV
ejpam-4548	533	22	,	,	PUNCT
ejpam-4548	533	23	in	in	ADP
ejpam-4548	533	24	view	view	NOUN
ejpam-4548	533	25	of	of	ADP
ejpam-4548	533	26	proposition	proposition	NOUN
ejpam-4548	533	27	1	1	NUM
ejpam-4548	533	28	,	,	PUNCT
ejpam-4548	533	29	we	we	PRON
ejpam-4548	533	30	have	have	VERB
ejpam-4548	533	31	ηi−1	ηi−1	PROPN
ejpam-4548	533	32	⊇	⊇	PROPN
ejpam-4548	533	33	ηi	ηi	PROPN
ejpam-4548	533	34	for	for	ADP
ejpam-4548	533	35	each	each	DET
ejpam-4548	533	36	i.	i.	NOUN
ejpam-4548	533	37	consequently	consequently	ADV
ejpam-4548	533	38	,	,	PUNCT
ejpam-4548	533	39	µ	µ	NOUN
ejpam-4548	533	40	=	=	SYM
ejpam-4548	533	41	η0	η0	NOUN
ejpam-4548	533	42	⊇	⊇	NOUN
ejpam-4548	533	43	η1	η1	PROPN
ejpam-4548	533	44	⊇	⊇	X
ejpam-4548	533	45	·	·	PUNCT
ejpam-4548	533	46	·	·	PUNCT
ejpam-4548	533	47	·	·	PUNCT
ejpam-4548	533	48	⊇	⊇	X
ejpam-4548	533	49	ηm	ηm	PROPN
ejpam-4548	533	50	=	=	SYM
ejpam-4548	533	51	η	η	PROPN
ejpam-4548	533	52	.	.	PROPN
ejpam-4548	533	53	then	then	ADV
ejpam-4548	533	54	,	,	PUNCT
ejpam-4548	533	55	the	the	DET
ejpam-4548	533	56	normal	normal	ADJ
ejpam-4548	533	57	closure	closure	NOUN
ejpam-4548	533	58	series	series	NOUN
ejpam-4548	533	59	of	of	ADP
ejpam-4548	533	60	η	η	PROPN
ejpam-4548	533	61	in	in	ADP
ejpam-4548	533	62	µ	µ	PROPN
ejpam-4548	533	63	has	have	VERB
ejpam-4548	533	64	the	the	DET
ejpam-4548	533	65	length	length	NOUN
ejpam-4548	533	66	atmost	atmost	PROPN
ejpam-4548	533	67	m	m	PROPN
ejpam-4548	533	68	≤	≤	PROPN
ejpam-4548	533	69	n.	n.	NOUN
ejpam-4548	533	70	therefore	therefore	ADV
ejpam-4548	533	71	,	,	PUNCT
ejpam-4548	533	72	η	η	PROPN
ejpam-4548	533	73	is	be	AUX
ejpam-4548	533	74	a	a	DET
ejpam-4548	533	75	subnormal	subnormal	ADJ
ejpam-4548	533	76	l	l	NOUN
ejpam-4548	533	77	-	-	NOUN
ejpam-4548	533	78	subgroup	subgroup	NOUN
ejpam-4548	533	79	of	of	ADP
ejpam-4548	533	80	µ	µ	NOUN
ejpam-4548	533	81	having	having	AUX
ejpam-4548	533	82	defect	defect	VERB
ejpam-4548	533	83	atmost	atmost	PROPN
ejpam-4548	533	84	n.	n.	PROPN
ejpam-4548	533	85	corollary	corollary	PROPN
ejpam-4548	533	86	5	5	NUM
ejpam-4548	533	87	.	.	PUNCT
ejpam-4548	534	1	let	let	VERB
ejpam-4548	534	2	g	g	PRON
ejpam-4548	534	3	be	be	AUX
ejpam-4548	534	4	a	a	DET
ejpam-4548	534	5	group	group	NOUN
ejpam-4548	534	6	and	and	CCONJ
ejpam-4548	534	7	h	h	NOUN
ejpam-4548	534	8	be	be	AUX
ejpam-4548	534	9	its	its	PRON
ejpam-4548	534	10	subgroup	subgroup	NOUN
ejpam-4548	534	11	.	.	PUNCT
ejpam-4548	535	1	then	then	ADV
ejpam-4548	535	2	,	,	PUNCT
ejpam-4548	535	3	h	h	NOUN
ejpam-4548	535	4	is	be	AUX
ejpam-4548	535	5	a	a	DET
ejpam-4548	535	6	subnormal	subnormal	ADJ
ejpam-4548	535	7	subgroup	subgroup	NOUN
ejpam-4548	535	8	of	of	ADP
ejpam-4548	535	9	g	g	PROPN
ejpam-4548	535	10	if	if	SCONJ
ejpam-4548	536	1	and	and	CCONJ
ejpam-4548	536	2	only	only	ADV
ejpam-4548	536	3	if	if	SCONJ
ejpam-4548	536	4	1h	1h	NUM
ejpam-4548	536	5	is	be	AUX
ejpam-4548	536	6	a	a	DET
ejpam-4548	536	7	subnormal	subnormal	ADJ
ejpam-4548	536	8	l	l	NOUN
ejpam-4548	536	9	-	-	NOUN
ejpam-4548	536	10	subgroup	subgroup	NOUN
ejpam-4548	536	11	of	of	ADP
ejpam-4548	536	12	1	1	NUM
ejpam-4548	536	13	g.	g.	NOUN
ejpam-4548	536	14	the	the	DET
ejpam-4548	536	15	strong	strong	ADJ
ejpam-4548	536	16	level	level	NOUN
ejpam-4548	536	17	subset	subset	NOUN
ejpam-4548	536	18	characterization	characterization	NOUN
ejpam-4548	536	19	of	of	ADP
ejpam-4548	536	20	subnormal	subnormal	ADJ
ejpam-4548	536	21	l	l	PROPN
ejpam-4548	536	22	-	-	NOUN
ejpam-4548	536	23	subgroup	subgroup	NOUN
ejpam-4548	536	24	can	can	AUX
ejpam-4548	536	25	be	be	AUX
ejpam-4548	536	26	obtained	obtain	VERB
ejpam-4548	536	27	easily	easily	ADV
ejpam-4548	536	28	by	by	ADP
ejpam-4548	536	29	using	use	VERB
ejpam-4548	536	30	the	the	DET
ejpam-4548	536	31	arguments	argument	NOUN
ejpam-4548	536	32	of	of	ADP
ejpam-4548	536	33	theorem	theorem	ADJ
ejpam-4548	536	34	18	18	NUM
ejpam-4548	536	35	.	.	PUNCT
ejpam-4548	537	1	so	so	ADV
ejpam-4548	537	2	,	,	PUNCT
ejpam-4548	537	3	we	we	PRON
ejpam-4548	537	4	state	state	VERB
ejpam-4548	537	5	it	it	PRON
ejpam-4548	537	6	without	without	ADP
ejpam-4548	537	7	proof	proof	NOUN
ejpam-4548	537	8	:	:	PUNCT
ejpam-4548	537	9	theorem	theorem	NOUN
ejpam-4548	537	10	19	19	NUM
ejpam-4548	537	11	.	.	PUNCT
ejpam-4548	538	1	let	let	VERB
ejpam-4548	538	2	l	l	NOUN
ejpam-4548	538	3	be	be	AUX
ejpam-4548	538	4	chain	chain	NOUN
ejpam-4548	538	5	and	and	CCONJ
ejpam-4548	538	6	η	η	PROPN
ejpam-4548	538	7	∈	∈	PROPN
ejpam-4548	538	8	l(µ	l(µ	PROPN
ejpam-4548	538	9	)	)	PUNCT
ejpam-4548	538	10	.	.	PUNCT
ejpam-4548	539	1	then	then	ADV
ejpam-4548	539	2	,	,	PUNCT
ejpam-4548	539	3	η	η	PROPN
ejpam-4548	539	4	is	be	AUX
ejpam-4548	539	5	subnormal	subnormal	ADJ
ejpam-4548	539	6	having	have	VERB
ejpam-4548	539	7	defect	defect	NOUN
ejpam-4548	539	8	at	at	ADP
ejpam-4548	539	9	most	most	ADJ
ejpam-4548	539	10	n	n	NOUN
ejpam-4548	539	11	if	if	ADV
ejpam-4548	540	1	and	and	CCONJ
ejpam-4548	540	2	only	only	ADV
ejpam-4548	540	3	if	if	SCONJ
ejpam-4548	540	4	each	each	DET
ejpam-4548	540	5	strong	strong	ADJ
ejpam-4548	540	6	level	level	NOUN
ejpam-4548	540	7	subset	subset	VERB
ejpam-4548	540	8	η	η	PROPN
ejpam-4548	540	9	>	>	X
ejpam-4548	540	10	a	a	PRON
ejpam-4548	540	11	is	be	AUX
ejpam-4548	540	12	subnormal	subnormal	ADJ
ejpam-4548	540	13	having	having	AUX
ejpam-4548	540	14	defect	defect	VERB
ejpam-4548	540	15	atmost	atmost	NOUN
ejpam-4548	540	16	n	n	CCONJ
ejpam-4548	540	17	where	where	SCONJ
ejpam-4548	540	18	a	a	DET
ejpam-4548	540	19	<	<	X
ejpam-4548	540	20	η(e	η(e	NOUN
ejpam-4548	540	21	)	)	PUNCT
ejpam-4548	540	22	.	.	PUNCT
ejpam-4548	541	1	recall	recall	VERB
ejpam-4548	541	2	the	the	DET
ejpam-4548	541	3	following	following	ADJ
ejpam-4548	541	4	result	result	NOUN
ejpam-4548	541	5	from	from	ADP
ejpam-4548	541	6	classical	classical	ADJ
ejpam-4548	541	7	group	group	NOUN
ejpam-4548	541	8	theory	theory	NOUN
ejpam-4548	541	9	[	[	X
ejpam-4548	541	10	11	11	NUM
ejpam-4548	541	11	,	,	PUNCT
ejpam-4548	541	12	13	13	NUM
ejpam-4548	541	13	]	]	NUM
ejpam-4548	541	14	:	:	PUNCT
ejpam-4548	541	15	n.	n.	PROPN
ejpam-4548	541	16	ajmal	ajmal	PROPN
ejpam-4548	541	17	,	,	PUNCT
ejpam-4548	541	18	i.	i.	PROPN
ejpam-4548	541	19	jahan	jahan	PROPN
ejpam-4548	541	20	.	.	PROPN
ejpam-4548	541	21	,	,	PUNCT
ejpam-4548	541	22	b.	b.	PROPN
ejpam-4548	541	23	davvaz	davvaz	PROPN
ejpam-4548	541	24	/	/	SYM
ejpam-4548	541	25	eur	eur	PROPN
ejpam-4548	541	26	.	.	PUNCT
ejpam-4548	542	1	j.	j.	PROPN
ejpam-4548	542	2	pure	pure	PROPN
ejpam-4548	542	3	appl	appl	PROPN
ejpam-4548	542	4	.	.	PROPN
ejpam-4548	542	5	math	math	PROPN
ejpam-4548	542	6	,	,	PUNCT
ejpam-4548	542	7	15	15	NUM
ejpam-4548	542	8	(	(	PUNCT
ejpam-4548	542	9	4	4	NUM
ejpam-4548	542	10	)	)	PUNCT
ejpam-4548	542	11	(	(	PUNCT
ejpam-4548	542	12	2022	2022	NUM
ejpam-4548	542	13	)	)	PUNCT
ejpam-4548	542	14	,	,	PUNCT
ejpam-4548	542	15	2086	2086	NUM
ejpam-4548	542	16	-	-	SYM
ejpam-4548	542	17	2115	2115	NUM
ejpam-4548	542	18	2111	2111	NUM
ejpam-4548	542	19	theorem	theorem	VERB
ejpam-4548	542	20	20	20	NUM
ejpam-4548	542	21	.	.	PUNCT
ejpam-4548	543	1	every	every	DET
ejpam-4548	543	2	subgroup	subgroup	NOUN
ejpam-4548	543	3	of	of	ADP
ejpam-4548	543	4	a	a	DET
ejpam-4548	543	5	nilpotent	nilpotent	ADJ
ejpam-4548	543	6	group	group	NOUN
ejpam-4548	543	7	is	be	AUX
ejpam-4548	543	8	subnormal	subnormal	ADJ
ejpam-4548	543	9	.	.	PUNCT
ejpam-4548	544	1	now	now	ADV
ejpam-4548	544	2	,	,	PUNCT
ejpam-4548	544	3	we	we	PRON
ejpam-4548	544	4	shall	shall	AUX
ejpam-4548	544	5	deal	deal	VERB
ejpam-4548	544	6	with	with	ADP
ejpam-4548	544	7	this	this	DET
ejpam-4548	544	8	result	result	NOUN
ejpam-4548	544	9	in	in	ADP
ejpam-4548	544	10	l	l	NOUN
ejpam-4548	544	11	-	-	NOUN
ejpam-4548	544	12	group	group	NOUN
ejpam-4548	544	13	theory	theory	NOUN
ejpam-4548	544	14	provided	provide	VERB
ejpam-4548	544	15	l	l	PROPN
ejpam-4548	544	16	is	be	AUX
ejpam-4548	544	17	an	an	DET
ejpam-4548	544	18	upper	upper	ADJ
ejpam-4548	544	19	well	well	NOUN
ejpam-4548	544	20	ordered	order	VERB
ejpam-4548	544	21	chain	chain	NOUN
ejpam-4548	544	22	.	.	PUNCT
ejpam-4548	545	1	a	a	DET
ejpam-4548	545	2	chain	chain	NOUN
ejpam-4548	545	3	is	be	AUX
ejpam-4548	545	4	said	say	VERB
ejpam-4548	545	5	to	to	PART
ejpam-4548	545	6	be	be	AUX
ejpam-4548	545	7	upper	upper	ADJ
ejpam-4548	545	8	well	well	ADV
ejpam-4548	545	9	ordered	order	VERB
ejpam-4548	545	10	if	if	SCONJ
ejpam-4548	545	11	every	every	DET
ejpam-4548	545	12	non	non	ADJ
ejpam-4548	545	13	-	-	ADJ
ejpam-4548	545	14	empty	empty	ADJ
ejpam-4548	545	15	subset	subset	NOUN
ejpam-4548	545	16	of	of	ADP
ejpam-4548	545	17	the	the	DET
ejpam-4548	545	18	given	give	VERB
ejpam-4548	545	19	chain	chain	NOUN
ejpam-4548	545	20	has	have	VERB
ejpam-4548	545	21	a	a	DET
ejpam-4548	545	22	greatest	great	ADJ
ejpam-4548	545	23	element	element	NOUN
ejpam-4548	545	24	.	.	PUNCT
ejpam-4548	546	1	clearly	clearly	ADV
ejpam-4548	546	2	,	,	PUNCT
ejpam-4548	546	3	every	every	DET
ejpam-4548	546	4	subset	subset	NOUN
ejpam-4548	546	5	of	of	ADP
ejpam-4548	546	6	an	an	DET
ejpam-4548	546	7	upper	upper	ADJ
ejpam-4548	546	8	well	well	NOUN
ejpam-4548	546	9	ordred	ordre	VERB
ejpam-4548	546	10	chain	chain	NOUN
ejpam-4548	546	11	is	be	AUX
ejpam-4548	546	12	a	a	DET
ejpam-4548	546	13	supstar	supstar	NOUN
ejpam-4548	546	14	subset	subset	NOUN
ejpam-4548	546	15	.	.	PUNCT
ejpam-4548	547	1	consequently	consequently	ADV
ejpam-4548	547	2	,	,	PUNCT
ejpam-4548	547	3	each	each	DET
ejpam-4548	547	4	l	l	NOUN
ejpam-4548	547	5	-	-	NOUN
ejpam-4548	547	6	subset	subset	NOUN
ejpam-4548	547	7	for	for	ADP
ejpam-4548	547	8	an	an	DET
ejpam-4548	547	9	upper	upper	ADJ
ejpam-4548	547	10	well	well	NOUN
ejpam-4548	547	11	ordered	order	VERB
ejpam-4548	547	12	chain	chain	NOUN
ejpam-4548	547	13	l	l	NOUN
ejpam-4548	547	14	satisfies	satisfie	NOUN
ejpam-4548	547	15	sup	sup	NOUN
ejpam-4548	547	16	-	-	PUNCT
ejpam-4548	547	17	property	property	NOUN
ejpam-4548	547	18	.	.	PUNCT
ejpam-4548	548	1	proposition	proposition	NOUN
ejpam-4548	548	2	8	8	NUM
ejpam-4548	548	3	.	.	PUNCT
ejpam-4548	549	1	let	let	VERB
ejpam-4548	549	2	l	l	NOUN
ejpam-4548	549	3	be	be	AUX
ejpam-4548	549	4	an	an	DET
ejpam-4548	549	5	upper	upper	ADJ
ejpam-4548	549	6	well	well	NOUN
ejpam-4548	549	7	ordered	order	VERB
ejpam-4548	549	8	chain	chain	NOUN
ejpam-4548	549	9	and	and	CCONJ
ejpam-4548	549	10	η	η	PROPN
ejpam-4548	549	11	,	,	PUNCT
ejpam-4548	549	12	θ	θ	PROPN
ejpam-4548	549	13	∈	∈	PROPN
ejpam-4548	549	14	lµ.	lµ.	NOUN
ejpam-4548	549	15	then	then	ADV
ejpam-4548	549	16	,	,	PUNCT
ejpam-4548	549	17	η	η	PROPN
ejpam-4548	549	18	and	and	CCONJ
ejpam-4548	549	19	θ	θ	PROPN
ejpam-4548	549	20	are	be	AUX
ejpam-4548	549	21	jointly	jointly	ADV
ejpam-4548	549	22	supstar	supstar	ADJ
ejpam-4548	549	23	if	if	SCONJ
ejpam-4548	549	24	and	and	CCONJ
ejpam-4548	549	25	only	only	ADV
ejpam-4548	549	26	if	if	SCONJ
ejpam-4548	549	27	η	η	PROPN
ejpam-4548	549	28	and	and	CCONJ
ejpam-4548	549	29	θ	θ	PROPN
ejpam-4548	549	30	possess	possess	VERB
ejpam-4548	549	31	sup	sup	NOUN
ejpam-4548	549	32	-	-	PUNCT
ejpam-4548	549	33	property	property	NOUN
ejpam-4548	549	34	.	.	PUNCT
ejpam-4548	550	1	below	below	ADV
ejpam-4548	550	2	,	,	PUNCT
ejpam-4548	550	3	we	we	PRON
ejpam-4548	550	4	recall	recall	VERB
ejpam-4548	550	5	the	the	DET
ejpam-4548	550	6	definition	definition	NOUN
ejpam-4548	550	7	of	of	ADP
ejpam-4548	550	8	descending	descend	VERB
ejpam-4548	550	9	central	central	ADJ
ejpam-4548	550	10	chain	chain	NOUN
ejpam-4548	550	11	of	of	ADP
ejpam-4548	550	12	an	an	DET
ejpam-4548	550	13	l	l	NOUN
ejpam-4548	550	14	-	-	PROPN
ejpam-4548	550	15	subgroup	subgroup	PROPN
ejpam-4548	550	16	η	η	PROPN
ejpam-4548	550	17	of	of	ADP
ejpam-4548	550	18	µ	µ	NUM
ejpam-4548	550	19	from	from	ADP
ejpam-4548	550	20	[	[	X
ejpam-4548	550	21	3	3	NUM
ejpam-4548	550	22	]	]	PUNCT
ejpam-4548	550	23	:	:	PUNCT
ejpam-4548	550	24	take	take	VERB
ejpam-4548	550	25	γ0(η	γ0(η	PRON
ejpam-4548	550	26	)	)	PUNCT
ejpam-4548	550	27	=	=	SYM
ejpam-4548	550	28	η	η	PROPN
ejpam-4548	550	29	,	,	PUNCT
ejpam-4548	550	30	γ1(η	γ1(η	PROPN
ejpam-4548	550	31	)	)	PUNCT
ejpam-4548	550	32	=	=	PUNCT
ejpam-4548	551	1	[	[	X
ejpam-4548	551	2	γ0(η	γ0(η	X
ejpam-4548	551	3	)	)	PUNCT
ejpam-4548	551	4	,	,	PUNCT
ejpam-4548	551	5	η	η	PROPN
ejpam-4548	551	6	]	]	X
ejpam-4548	551	7	.	.	PUNCT
ejpam-4548	552	1	and	and	CCONJ
ejpam-4548	552	2	in	in	ADP
ejpam-4548	552	3	general	general	ADJ
ejpam-4548	552	4	,	,	PUNCT
ejpam-4548	552	5	for	for	ADP
ejpam-4548	552	6	each	each	DET
ejpam-4548	552	7	i	i	PRON
ejpam-4548	552	8	,	,	PUNCT
ejpam-4548	552	9	we	we	PRON
ejpam-4548	552	10	define	define	VERB
ejpam-4548	552	11	γi(η	γi(η	PUNCT
ejpam-4548	552	12	)	)	PUNCT
ejpam-4548	552	13	=	=	PUNCT
ejpam-4548	553	1	[	[	X
ejpam-4548	553	2	γi−1(η	γi−1(η	NOUN
ejpam-4548	553	3	)	)	PUNCT
ejpam-4548	553	4	,	,	PUNCT
ejpam-4548	553	5	η	η	PROPN
ejpam-4548	553	6	]	]	X
ejpam-4548	553	7	.	.	PUNCT
ejpam-4548	554	1	proposition	proposition	NOUN
ejpam-4548	554	2	9	9	NUM
ejpam-4548	554	3	.	.	PUNCT
ejpam-4548	554	4	let	let	VERB
ejpam-4548	554	5	η	η	PROPN
ejpam-4548	554	6	∈	∈	PROPN
ejpam-4548	554	7	l(µ	l(µ	PROPN
ejpam-4548	554	8	)	)	PUNCT
ejpam-4548	554	9	.	.	PUNCT
ejpam-4548	555	1	then	then	ADV
ejpam-4548	555	2	for	for	ADP
ejpam-4548	555	3	each	each	DET
ejpam-4548	555	4	i	i	PROPN
ejpam-4548	555	5	,	,	PUNCT
ejpam-4548	555	6	γi(η	γi(η	NUM
ejpam-4548	555	7	)	)	PUNCT
ejpam-4548	555	8	⊆	⊆	NUM
ejpam-4548	555	9	γi−1(η	γi−1(η	NUM
ejpam-4548	555	10	)	)	PUNCT
ejpam-4548	555	11	.	.	PUNCT
ejpam-4548	556	1	definition	definition	NOUN
ejpam-4548	556	2	16	16	NUM
ejpam-4548	556	3	.	.	PUNCT
ejpam-4548	557	1	let	let	VERB
ejpam-4548	557	2	η	η	PROPN
ejpam-4548	557	3	∈	∈	PROPN
ejpam-4548	557	4	l(µ	l(µ	PROPN
ejpam-4548	557	5	)	)	PUNCT
ejpam-4548	557	6	.	.	PUNCT
ejpam-4548	558	1	then	then	ADV
ejpam-4548	558	2	,	,	PUNCT
ejpam-4548	558	3	the	the	DET
ejpam-4548	558	4	chain	chain	NOUN
ejpam-4548	558	5	η	η	PROPN
ejpam-4548	558	6	=	=	PROPN
ejpam-4548	558	7	γ0(η	γ0(η	PROPN
ejpam-4548	558	8	)	)	PUNCT
ejpam-4548	558	9	⊇	⊇	PROPN
ejpam-4548	558	10	γ1(η	γ1(η	PROPN
ejpam-4548	558	11	)	)	PUNCT
ejpam-4548	558	12	⊇	⊇	X
ejpam-4548	558	13	·	·	PUNCT
ejpam-4548	558	14	·	·	PUNCT
ejpam-4548	558	15	·	·	PUNCT
ejpam-4548	558	16	⊇	⊇	NOUN
ejpam-4548	558	17	γi(η	γi(η	NUM
ejpam-4548	558	18	)	)	PUNCT
ejpam-4548	558	19	⊇	⊇	X
ejpam-4548	558	20	·	·	PUNCT
ejpam-4548	558	21	·	·	PUNCT
ejpam-4548	558	22	·	·	PUNCT
ejpam-4548	558	23	of	of	ADP
ejpam-4548	558	24	l	l	NOUN
ejpam-4548	558	25	-	-	NOUN
ejpam-4548	558	26	subgroups	subgroup	NOUN
ejpam-4548	558	27	of	of	ADP
ejpam-4548	558	28	µ	µ	NOUN
ejpam-4548	558	29	is	be	AUX
ejpam-4548	558	30	called	call	VERB
ejpam-4548	558	31	the	the	DET
ejpam-4548	558	32	descending	descend	VERB
ejpam-4548	558	33	central	central	ADJ
ejpam-4548	558	34	chain	chain	NOUN
ejpam-4548	558	35	of	of	ADP
ejpam-4548	558	36	η	η	PROPN
ejpam-4548	558	37	.	.	PROPN
ejpam-4548	558	38	definition	definition	NOUN
ejpam-4548	558	39	17	17	NUM
ejpam-4548	558	40	.	.	PUNCT
ejpam-4548	559	1	let	let	VERB
ejpam-4548	559	2	η	η	PROPN
ejpam-4548	559	3	∈	∈	PROPN
ejpam-4548	559	4	l(µ	l(µ	PROPN
ejpam-4548	559	5	)	)	PUNCT
ejpam-4548	559	6	with	with	ADP
ejpam-4548	559	7	tip	tip	NOUN
ejpam-4548	559	8	a0	a0	PROPN
ejpam-4548	559	9	and	and	CCONJ
ejpam-4548	559	10	tail	tail	NOUN
ejpam-4548	559	11	t0	t0	PROPN
ejpam-4548	559	12	and	and	CCONJ
ejpam-4548	559	13	a0	a0	PROPN
ejpam-4548	559	14	̸=	̸=	PROPN
ejpam-4548	559	15	t0	t0	PROPN
ejpam-4548	559	16	.	.	PUNCT
ejpam-4548	560	1	if	if	SCONJ
ejpam-4548	560	2	the	the	DET
ejpam-4548	560	3	descending	descend	VERB
ejpam-4548	560	4	central	central	ADJ
ejpam-4548	560	5	chain	chain	NOUN
ejpam-4548	560	6	η	η	PROPN
ejpam-4548	560	7	=	=	SYM
ejpam-4548	560	8	γ0(η	γ0(η	PROPN
ejpam-4548	560	9	)	)	PUNCT
ejpam-4548	560	10	⊇	⊇	PROPN
ejpam-4548	560	11	γ1(η	γ1(η	PROPN
ejpam-4548	560	12	)	)	PUNCT
ejpam-4548	560	13	⊇	⊇	X
ejpam-4548	560	14	·	·	PUNCT
ejpam-4548	560	15	·	·	PUNCT
ejpam-4548	560	16	·	·	PUNCT
ejpam-4548	560	17	⊇	⊇	NOUN
ejpam-4548	560	18	γi(η	γi(η	NUM
ejpam-4548	560	19	)	)	PUNCT
ejpam-4548	560	20	⊇	⊇	X
ejpam-4548	560	21	·	·	PUNCT
ejpam-4548	560	22	·	·	PUNCT
ejpam-4548	560	23	·	·	PUNCT
ejpam-4548	560	24	terminates	terminate	VERB
ejpam-4548	560	25	finitely	finitely	ADV
ejpam-4548	560	26	to	to	ADP
ejpam-4548	560	27	the	the	DET
ejpam-4548	560	28	trivial	trivial	ADJ
ejpam-4548	560	29	l	l	NOUN
ejpam-4548	560	30	-	-	NOUN
ejpam-4548	560	31	subgroup	subgroup	NOUN
ejpam-4548	560	32	ηa0t0	ηa0t0	PROPN
ejpam-4548	560	33	,	,	PUNCT
ejpam-4548	560	34	then	then	ADV
ejpam-4548	560	35	η	η	PROPN
ejpam-4548	560	36	is	be	AUX
ejpam-4548	560	37	known	know	VERB
ejpam-4548	560	38	as	as	ADP
ejpam-4548	560	39	a	a	DET
ejpam-4548	560	40	nilpotent	nilpotent	ADJ
ejpam-4548	560	41	l	l	NOUN
ejpam-4548	560	42	-	-	NOUN
ejpam-4548	560	43	subgroup	subgroup	NOUN
ejpam-4548	560	44	of	of	ADP
ejpam-4548	560	45	µ.	µ.	PROPN
ejpam-4548	560	46	more	more	ADV
ejpam-4548	560	47	precisely	precisely	ADV
ejpam-4548	560	48	,	,	PUNCT
ejpam-4548	560	49	η	η	PROPN
ejpam-4548	560	50	is	be	AUX
ejpam-4548	560	51	said	say	VERB
ejpam-4548	560	52	to	to	PART
ejpam-4548	560	53	be	be	AUX
ejpam-4548	560	54	nilpotent	nilpotent	ADJ
ejpam-4548	560	55	of	of	ADP
ejpam-4548	560	56	class	class	NOUN
ejpam-4548	560	57	c	c	NOUN
ejpam-4548	560	58	if	if	SCONJ
ejpam-4548	560	59	c	c	PROPN
ejpam-4548	560	60	is	be	AUX
ejpam-4548	560	61	the	the	DET
ejpam-4548	560	62	least	least	ADJ
ejpam-4548	560	63	non	non	ADJ
ejpam-4548	560	64	-	-	ADJ
ejpam-4548	560	65	negative	negative	ADJ
ejpam-4548	560	66	integer	integer	NOUN
ejpam-4548	560	67	such	such	ADJ
ejpam-4548	560	68	that	that	PRON
ejpam-4548	560	69	γc(η	γc(η	PUNCT
ejpam-4548	560	70	)	)	PUNCT
ejpam-4548	560	71	=	=	SYM
ejpam-4548	561	1	ηa0t0	ηa0t0	PROPN
ejpam-4548	561	2	.	.	PUNCT
ejpam-4548	562	1	in	in	ADP
ejpam-4548	562	2	this	this	DET
ejpam-4548	562	3	case	case	NOUN
ejpam-4548	562	4	,	,	PUNCT
ejpam-4548	562	5	the	the	DET
ejpam-4548	562	6	series	series	NOUN
ejpam-4548	562	7	η	η	PROPN
ejpam-4548	562	8	=	=	PROPN
ejpam-4548	562	9	γ0(η	γ0(η	PROPN
ejpam-4548	562	10	)	)	PUNCT
ejpam-4548	562	11	⊇	⊇	PROPN
ejpam-4548	562	12	γ1(η	γ1(η	PROPN
ejpam-4548	562	13	)	)	PUNCT
ejpam-4548	562	14	⊇	⊇	X
ejpam-4548	562	15	·	·	PUNCT
ejpam-4548	562	16	·	·	PUNCT
ejpam-4548	562	17	·	·	PUNCT
ejpam-4548	562	18	⊇	⊇	NOUN
ejpam-4548	562	19	γc(η	γc(η	PUNCT
ejpam-4548	562	20	)	)	PUNCT
ejpam-4548	562	21	=	=	SYM
ejpam-4548	562	22	ηa0t0	ηa0t0	PROPN
ejpam-4548	562	23	,	,	PUNCT
ejpam-4548	562	24	is	be	AUX
ejpam-4548	562	25	called	call	VERB
ejpam-4548	562	26	the	the	DET
ejpam-4548	562	27	descending	descend	VERB
ejpam-4548	562	28	central	central	ADJ
ejpam-4548	562	29	series	series	NOUN
ejpam-4548	562	30	of	of	ADP
ejpam-4548	562	31	η	η	PROPN
ejpam-4548	562	32	.	.	PROPN
ejpam-4548	562	33	if	if	SCONJ
ejpam-4548	562	34	it	it	PRON
ejpam-4548	562	35	is	be	AUX
ejpam-4548	562	36	a	a	DET
ejpam-4548	562	37	nilpotent	nilpotent	ADJ
ejpam-4548	562	38	l	l	NOUN
ejpam-4548	562	39	-	-	NOUN
ejpam-4548	562	40	subgroup	subgroup	NOUN
ejpam-4548	562	41	of	of	ADP
ejpam-4548	562	42	µ	µ	NUM
ejpam-4548	562	43	,	,	PUNCT
ejpam-4548	562	44	then	then	ADV
ejpam-4548	562	45	we	we	PRON
ejpam-4548	562	46	simply	simply	ADV
ejpam-4548	562	47	write	write	VERB
ejpam-4548	562	48	η	η	PROPN
ejpam-4548	562	49	is	be	AUX
ejpam-4548	562	50	nilpotent	nilpotent	ADJ
ejpam-4548	562	51	.	.	PUNCT
ejpam-4548	563	1	theorem	theorem	NOUN
ejpam-4548	563	2	21	21	NUM
ejpam-4548	563	3	.	.	PUNCT
ejpam-4548	564	1	let	let	VERB
ejpam-4548	564	2	η	η	PROPN
ejpam-4548	564	3	∈	∈	PROPN
ejpam-4548	564	4	l(µ	l(µ	PROPN
ejpam-4548	564	5	)	)	PUNCT
ejpam-4548	564	6	and	and	CCONJ
ejpam-4548	564	7	possesses	possess	VERB
ejpam-4548	564	8	sup	sup	ADJ
ejpam-4548	564	9	-	-	PUNCT
ejpam-4548	564	10	property	property	NOUN
ejpam-4548	564	11	.	.	PUNCT
ejpam-4548	565	1	then	then	ADV
ejpam-4548	565	2	,	,	PUNCT
ejpam-4548	565	3	η	η	PROPN
ejpam-4548	565	4	is	be	AUX
ejpam-4548	565	5	a	a	DET
ejpam-4548	565	6	nilpotent	nilpotent	ADJ
ejpam-4548	565	7	l	l	NOUN
ejpam-4548	565	8	-	-	NOUN
ejpam-4548	565	9	subgroup	subgroup	NOUN
ejpam-4548	565	10	of	of	ADP
ejpam-4548	565	11	µ	µ	NOUN
ejpam-4548	565	12	of	of	ADP
ejpam-4548	565	13	nilpotent	nilpotent	ADJ
ejpam-4548	565	14	length	length	NOUN
ejpam-4548	565	15	at	at	ADP
ejpam-4548	565	16	most	most	ADV
ejpam-4548	565	17	n	n	NOUN
ejpam-4548	565	18	if	if	ADV
ejpam-4548	566	1	and	and	CCONJ
ejpam-4548	566	2	only	only	ADV
ejpam-4548	566	3	if	if	SCONJ
ejpam-4548	566	4	each	each	DET
ejpam-4548	566	5	level	level	NOUN
ejpam-4548	566	6	subgroup	subgroup	NOUN
ejpam-4548	566	7	ηa	ηa	PROPN
ejpam-4548	566	8	is	be	AUX
ejpam-4548	566	9	a	a	DET
ejpam-4548	566	10	nilpotent	nilpotent	ADJ
ejpam-4548	566	11	subgroup	subgroup	NOUN
ejpam-4548	566	12	of	of	ADP
ejpam-4548	566	13	µa	µa	NOUN
ejpam-4548	566	14	of	of	ADP
ejpam-4548	566	15	nilpotent	nilpotent	ADJ
ejpam-4548	566	16	length	length	NOUN
ejpam-4548	566	17	at	at	ADP
ejpam-4548	566	18	most	most	ADJ
ejpam-4548	566	19	n	n	ADJ
ejpam-4548	566	20	for	for	ADP
ejpam-4548	566	21	each	each	PRON
ejpam-4548	566	22	a	a	DET
ejpam-4548	566	23	̸≤	̸≤	NOUN
ejpam-4548	566	24	inf	inf	PROPN
ejpam-4548	566	25	η	η	PROPN
ejpam-4548	566	26	and	and	CCONJ
ejpam-4548	566	27	a	a	DET
ejpam-4548	566	28	≤	≤	NUM
ejpam-4548	566	29	η(e	η(e	PROPN
ejpam-4548	566	30	)	)	PUNCT
ejpam-4548	566	31	.	.	PUNCT
ejpam-4548	567	1	theorem	theorem	NOUN
ejpam-4548	567	2	22	22	NUM
ejpam-4548	567	3	.	.	PUNCT
ejpam-4548	568	1	let	let	VERB
ejpam-4548	568	2	η	η	PROPN
ejpam-4548	568	3	∈	∈	PROPN
ejpam-4548	568	4	l(µ	l(µ	PROPN
ejpam-4548	568	5	)	)	PUNCT
ejpam-4548	568	6	and	and	CCONJ
ejpam-4548	568	7	l	l	NOUN
ejpam-4548	568	8	be	be	AUX
ejpam-4548	568	9	a	a	DET
ejpam-4548	568	10	chain	chain	NOUN
ejpam-4548	568	11	.	.	PUNCT
ejpam-4548	569	1	then	then	ADV
ejpam-4548	569	2	,	,	PUNCT
ejpam-4548	569	3	η	η	PROPN
ejpam-4548	569	4	is	be	AUX
ejpam-4548	569	5	a	a	DET
ejpam-4548	569	6	nilpotent	nilpotent	ADJ
ejpam-4548	569	7	l	l	NOUN
ejpam-4548	569	8	-	-	NOUN
ejpam-4548	569	9	subgroup	subgroup	NOUN
ejpam-4548	569	10	of	of	ADP
ejpam-4548	569	11	µ	µ	NOUN
ejpam-4548	569	12	of	of	ADP
ejpam-4548	569	13	nilpotent	nilpotent	ADJ
ejpam-4548	569	14	length	length	NOUN
ejpam-4548	569	15	at	at	ADP
ejpam-4548	569	16	most	most	ADV
ejpam-4548	569	17	n	n	NOUN
ejpam-4548	569	18	if	if	ADV
ejpam-4548	570	1	and	and	CCONJ
ejpam-4548	570	2	only	only	ADV
ejpam-4548	570	3	if	if	SCONJ
ejpam-4548	570	4	η	η	NOUN
ejpam-4548	570	5	>	>	X
ejpam-4548	570	6	a	a	PRON
ejpam-4548	570	7	is	be	AUX
ejpam-4548	570	8	a	a	DET
ejpam-4548	570	9	nilpotent	nilpotent	ADJ
ejpam-4548	570	10	subgroup	subgroup	NOUN
ejpam-4548	570	11	of	of	ADP
ejpam-4548	570	12	µ	µ	PROPN
ejpam-4548	570	13	>	>	X
ejpam-4548	570	14	a	a	PRON
ejpam-4548	570	15	of	of	ADP
ejpam-4548	570	16	nilpotent	nilpotent	ADJ
ejpam-4548	570	17	length	length	NOUN
ejpam-4548	570	18	at	at	ADP
ejpam-4548	570	19	most	most	ADV
ejpam-4548	570	20	n	n	CCONJ
ejpam-4548	570	21	,	,	PUNCT
ejpam-4548	570	22	where	where	SCONJ
ejpam-4548	570	23	inf	inf	PROPN
ejpam-4548	570	24	η	η	PROPN
ejpam-4548	570	25	≤	≤	PROPN
ejpam-4548	570	26	a	a	DET
ejpam-4548	570	27	<	<	X
ejpam-4548	570	28	η(e	η(e	NOUN
ejpam-4548	570	29	)	)	PUNCT
ejpam-4548	570	30	.	.	PUNCT
ejpam-4548	571	1	the	the	DET
ejpam-4548	571	2	notion	notion	NOUN
ejpam-4548	571	3	of	of	ADP
ejpam-4548	571	4	nilpotency	nilpotency	NOUN
ejpam-4548	571	5	of	of	ADP
ejpam-4548	571	6	l	l	PROPN
ejpam-4548	571	7	-	-	NOUN
ejpam-4548	571	8	subgroup	subgroup	NOUN
ejpam-4548	571	9	is	be	AUX
ejpam-4548	571	10	exihibited	exihibite	VERB
ejpam-4548	571	11	in	in	ADP
ejpam-4548	571	12	the	the	DET
ejpam-4548	571	13	following	following	NOUN
ejpam-4548	571	14	:	:	PUNCT
ejpam-4548	571	15	n.	n.	PROPN
ejpam-4548	571	16	ajmal	ajmal	PROPN
ejpam-4548	571	17	,	,	PUNCT
ejpam-4548	571	18	i.	i.	PROPN
ejpam-4548	571	19	jahan	jahan	PROPN
ejpam-4548	571	20	.	.	PROPN
ejpam-4548	571	21	,	,	PUNCT
ejpam-4548	571	22	b.	b.	PROPN
ejpam-4548	571	23	davvaz	davvaz	PROPN
ejpam-4548	571	24	/	/	SYM
ejpam-4548	571	25	eur	eur	PROPN
ejpam-4548	571	26	.	.	PUNCT
ejpam-4548	572	1	j.	j.	PROPN
ejpam-4548	572	2	pure	pure	PROPN
ejpam-4548	572	3	appl	appl	PROPN
ejpam-4548	572	4	.	.	PROPN
ejpam-4548	572	5	math	math	PROPN
ejpam-4548	572	6	,	,	PUNCT
ejpam-4548	572	7	15	15	NUM
ejpam-4548	572	8	(	(	PUNCT
ejpam-4548	572	9	4	4	NUM
ejpam-4548	572	10	)	)	PUNCT
ejpam-4548	572	11	(	(	PUNCT
ejpam-4548	572	12	2022	2022	NUM
ejpam-4548	572	13	)	)	PUNCT
ejpam-4548	572	14	,	,	PUNCT
ejpam-4548	572	15	2086	2086	NUM
ejpam-4548	572	16	-	-	SYM
ejpam-4548	572	17	2115	2115	NUM
ejpam-4548	572	18	2112	2112	NUM
ejpam-4548	572	19	example	example	NOUN
ejpam-4548	572	20	3	3	X
ejpam-4548	572	21	.	.	PUNCT
ejpam-4548	573	1	let	let	VERB
ejpam-4548	573	2	g	g	PROPN
ejpam-4548	573	3	=	=	PROPN
ejpam-4548	573	4	q8	q8	PROPN
ejpam-4548	573	5	and	and	CCONJ
ejpam-4548	573	6	the	the	DET
ejpam-4548	573	7	evaluation	evaluation	NOUN
ejpam-4548	573	8	lattice	lattice	NOUN
ejpam-4548	573	9	be	be	AUX
ejpam-4548	573	10	given	give	VERB
ejpam-4548	573	11	by	by	ADP
ejpam-4548	573	12	the	the	DET
ejpam-4548	573	13	diagram	diagram	NOUN
ejpam-4548	573	14	:	:	PUNCT
ejpam-4548	574	1	u	u	NOUN
ejpam-4548	574	2	f	f	PROPN
ejpam-4548	574	3	ba	ba	PROPN
ejpam-4548	574	4	d	d	NOUN
ejpam-4548	574	5	fig	fig	NOUN
ejpam-4548	574	6	.	.	PUNCT
ejpam-4548	575	1	1	1	NUM
ejpam-4548	575	2	consider	consider	VERB
ejpam-4548	575	3	the	the	DET
ejpam-4548	575	4	parent	parent	NOUN
ejpam-4548	575	5	l	l	NOUN
ejpam-4548	575	6	-	-	NOUN
ejpam-4548	575	7	subgroup	subgroup	NOUN
ejpam-4548	575	8	of	of	ADP
ejpam-4548	575	9	g	g	NOUN
ejpam-4548	575	10	given	give	VERB
ejpam-4548	575	11	by	by	ADP
ejpam-4548	575	12	:	:	PUNCT
ejpam-4548	575	13	µ(x	µ(x	X
ejpam-4548	575	14	)	)	PUNCT
ejpam-4548	575	15	=	=	PRON
ejpam-4548	575	16	{	{	PUNCT
ejpam-4548	575	17	u	u	NOUN
ejpam-4548	575	18	if	if	SCONJ
ejpam-4548	575	19	x	x	PROPN
ejpam-4548	575	20	∈	∈	PROPN
ejpam-4548	575	21	c	c	NOUN
ejpam-4548	575	22	,	,	PUNCT
ejpam-4548	575	23	d	d	X
ejpam-4548	575	24	if	if	SCONJ
ejpam-4548	575	25	x	x	PROPN
ejpam-4548	575	26	∈	∈	PROPN
ejpam-4548	575	27	g	g	NOUN
ejpam-4548	575	28	\	\	PROPN
ejpam-4548	575	29	c.	c.	NOUN
ejpam-4548	575	30	now	now	ADV
ejpam-4548	575	31	define	define	VERB
ejpam-4548	575	32	l	l	NOUN
ejpam-4548	575	33	-	-	PUNCT
ejpam-4548	575	34	subset	subset	VERB
ejpam-4548	575	35	η	η	PROPN
ejpam-4548	575	36	of	of	ADP
ejpam-4548	575	37	µ	µ	NOUN
ejpam-4548	575	38	as	as	SCONJ
ejpam-4548	575	39	given	give	VERB
ejpam-4548	575	40	below	below	ADV
ejpam-4548	575	41	:	:	PUNCT
ejpam-4548	575	42	η(x	η(x	X
ejpam-4548	575	43	)	)	PUNCT
ejpam-4548	575	44	=	=	SYM
ejpam-4548	575	45			PUNCT
ejpam-4548	575	46	u	u	NOUN
ejpam-4548	575	47	if	if	SCONJ
ejpam-4548	575	48	x	x	PROPN
ejpam-4548	575	49	∈	∈	PROPN
ejpam-4548	575	50	c	c	NOUN
ejpam-4548	575	51	,	,	PUNCT
ejpam-4548	575	52	d	d	X
ejpam-4548	576	1	if	if	SCONJ
ejpam-4548	576	2	x	x	X
ejpam-4548	576	3	∈	∈	PROPN
ejpam-4548	576	4	h1	h1	PROPN
ejpam-4548	576	5	\	\	PROPN
ejpam-4548	576	6	c	c	PROPN
ejpam-4548	576	7	,	,	PUNCT
ejpam-4548	576	8	a	a	DET
ejpam-4548	576	9	if	if	SCONJ
ejpam-4548	576	10	x	x	SYM
ejpam-4548	576	11	∈	∈	PROPN
ejpam-4548	576	12	h2	h2	NOUN
ejpam-4548	576	13	\	\	PROPN
ejpam-4548	576	14	c	c	PROPN
ejpam-4548	576	15	,	,	PUNCT
ejpam-4548	576	16	b	b	NOUN
ejpam-4548	576	17	if	if	SCONJ
ejpam-4548	576	18	x	x	PROPN
ejpam-4548	576	19	∈	∈	NOUN
ejpam-4548	576	20	h3	h3	NOUN
ejpam-4548	576	21	\	\	NOUN
ejpam-4548	576	22	c	c	NOUN
ejpam-4548	576	23	;	;	PUNCT
ejpam-4548	576	24	where	where	SCONJ
ejpam-4548	576	25	c	c	NOUN
ejpam-4548	576	26	=	=	PRON
ejpam-4548	576	27	{	{	PUNCT
ejpam-4548	576	28	±1	±1	NOUN
ejpam-4548	576	29	}	}	PUNCT
ejpam-4548	576	30	,	,	PUNCT
ejpam-4548	576	31	h1	h1	PROPN
ejpam-4548	576	32	=	=	SYM
ejpam-4548	576	33	{	{	PUNCT
ejpam-4548	576	34	±1,±i	±1,±i	ADV
ejpam-4548	576	35	}	}	PUNCT
ejpam-4548	576	36	,	,	PUNCT
ejpam-4548	576	37	h2	h2	NOUN
ejpam-4548	576	38	=	=	PUNCT
ejpam-4548	576	39	{	{	PUNCT
ejpam-4548	576	40	±1,±j	±1,±j	PROPN
ejpam-4548	576	41	}	}	PUNCT
ejpam-4548	576	42	,	,	PUNCT
ejpam-4548	576	43	h3	h3	NOUN
ejpam-4548	576	44	=	=	SYM
ejpam-4548	576	45	{	{	PUNCT
ejpam-4548	576	46	±1,±k	±1,±k	NOUN
ejpam-4548	576	47	}	}	PUNCT
ejpam-4548	576	48	.	.	PUNCT
ejpam-4548	577	1	since	since	SCONJ
ejpam-4548	577	2	the	the	DET
ejpam-4548	577	3	level	level	NOUN
ejpam-4548	577	4	subsets	subset	NOUN
ejpam-4548	577	5	of	of	ADP
ejpam-4548	577	6	η	η	PROPN
ejpam-4548	577	7	are	be	AUX
ejpam-4548	577	8	normal	normal	ADJ
ejpam-4548	577	9	subgroups	subgroup	NOUN
ejpam-4548	577	10	of	of	ADP
ejpam-4548	577	11	g	g	PROPN
ejpam-4548	577	12	,	,	PUNCT
ejpam-4548	577	13	η	η	PROPN
ejpam-4548	577	14	is	be	AUX
ejpam-4548	577	15	a	a	DET
ejpam-4548	577	16	normal	normal	ADJ
ejpam-4548	577	17	l	l	NOUN
ejpam-4548	577	18	-	-	NOUN
ejpam-4548	577	19	subgroup	subgroup	NOUN
ejpam-4548	577	20	of	of	ADP
ejpam-4548	577	21	g	g	PROPN
ejpam-4548	577	22	and	and	CCONJ
ejpam-4548	577	23	hence	hence	ADV
ejpam-4548	577	24	of	of	ADP
ejpam-4548	577	25	µ.	µ.	NOUN
ejpam-4548	577	26	now	now	ADV
ejpam-4548	577	27	that	that	SCONJ
ejpam-4548	577	28	η	η	PROPN
ejpam-4548	577	29	is	be	AUX
ejpam-4548	577	30	a	a	DET
ejpam-4548	577	31	nilpotent	nilpotent	ADJ
ejpam-4548	577	32	l	l	NOUN
ejpam-4548	577	33	-	-	NOUN
ejpam-4548	577	34	subgroup	subgroup	NOUN
ejpam-4548	577	35	of	of	ADP
ejpam-4548	577	36	µ	µ	NOUN
ejpam-4548	577	37	in	in	ADP
ejpam-4548	577	38	view	view	NOUN
ejpam-4548	577	39	of	of	ADP
ejpam-4548	577	40	definition	definition	NOUN
ejpam-4548	577	41	17	17	NUM
ejpam-4548	577	42	,	,	PUNCT
ejpam-4548	577	43	we	we	PRON
ejpam-4548	577	44	demonstrate	demonstrate	VERB
ejpam-4548	577	45	this	this	PRON
ejpam-4548	577	46	as	as	SCONJ
ejpam-4548	577	47	follows	follow	VERB
ejpam-4548	577	48	:	:	PUNCT
ejpam-4548	577	49	note	note	VERB
ejpam-4548	577	50	that	that	SCONJ
ejpam-4548	577	51	g′	g′	NOUN
ejpam-4548	577	52	=	=	PUNCT
ejpam-4548	577	53	{	{	PUNCT
ejpam-4548	577	54	1,−1	1,−1	NUM
ejpam-4548	577	55	}	}	PUNCT
ejpam-4548	577	56	.	.	PUNCT
ejpam-4548	578	1	in	in	ADP
ejpam-4548	578	2	order	order	NOUN
ejpam-4548	578	3	to	to	PART
ejpam-4548	578	4	obtain	obtain	VERB
ejpam-4548	578	5	the	the	DET
ejpam-4548	578	6	members	member	NOUN
ejpam-4548	578	7	of	of	ADP
ejpam-4548	578	8	descending	descend	VERB
ejpam-4548	578	9	central	central	ADJ
ejpam-4548	578	10	series	series	NOUN
ejpam-4548	578	11	of	of	ADP
ejpam-4548	578	12	η	η	PROPN
ejpam-4548	578	13	,	,	PUNCT
ejpam-4548	578	14	we	we	PRON
ejpam-4548	578	15	set	set	VERB
ejpam-4548	578	16	γ0(η	γ0(η	PRON
ejpam-4548	578	17	)	)	PUNCT
ejpam-4548	578	18	=	=	SYM
ejpam-4548	578	19	η	η	PROPN
ejpam-4548	578	20	and	and	CCONJ
ejpam-4548	578	21	consider	consider	VERB
ejpam-4548	578	22	the	the	DET
ejpam-4548	578	23	commutator	commutator	NOUN
ejpam-4548	578	24	(	(	PUNCT
ejpam-4548	578	25	η	η	PROPN
ejpam-4548	578	26	,	,	PUNCT
ejpam-4548	578	27	η	η	PROPN
ejpam-4548	578	28	)	)	PUNCT
ejpam-4548	578	29	(	(	PUNCT
ejpam-4548	578	30	η	η	PROPN
ejpam-4548	578	31	,	,	PUNCT
ejpam-4548	578	32	η	η	NOUN
ejpam-4548	578	33	)	)	PUNCT
ejpam-4548	578	34	(	(	PUNCT
ejpam-4548	578	35	x	x	X
ejpam-4548	578	36	)	)	PUNCT
ejpam-4548	578	37	=	=	PUNCT
ejpam-4548	579	1			PUNCT
ejpam-4548	579	2	u	u	NOUN
ejpam-4548	579	3	if	if	SCONJ
ejpam-4548	579	4	x	x	SYM
ejpam-4548	579	5	=	=	SYM
ejpam-4548	579	6	1	1	NUM
ejpam-4548	579	7	,	,	PUNCT
ejpam-4548	579	8	d	d	NOUN
ejpam-4548	579	9	if	if	SCONJ
ejpam-4548	579	10	x	x	SYM
ejpam-4548	579	11	∈	∈	PROPN
ejpam-4548	579	12	c	c	NOUN
ejpam-4548	579	13	\	\	X
ejpam-4548	579	14	{	{	PUNCT
ejpam-4548	579	15	1	1	NUM
ejpam-4548	579	16	}	}	PUNCT
ejpam-4548	579	17	,	,	PUNCT
ejpam-4548	579	18	f	f	PROPN
ejpam-4548	579	19	if	if	SCONJ
ejpam-4548	579	20	x	x	PROPN
ejpam-4548	579	21	∈	∈	PROPN
ejpam-4548	579	22	g	g	NOUN
ejpam-4548	579	23	\	\	PROPN
ejpam-4548	579	24	c.	c.	PROPN
ejpam-4548	579	25	n.	n.	PROPN
ejpam-4548	579	26	ajmal	ajmal	PROPN
ejpam-4548	579	27	,	,	PUNCT
ejpam-4548	579	28	i.	i.	PROPN
ejpam-4548	579	29	jahan	jahan	PROPN
ejpam-4548	579	30	.	.	PROPN
ejpam-4548	579	31	,	,	PUNCT
ejpam-4548	579	32	b.	b.	PROPN
ejpam-4548	579	33	davvaz	davvaz	PROPN
ejpam-4548	579	34	/	/	SYM
ejpam-4548	579	35	eur	eur	PROPN
ejpam-4548	579	36	.	.	PUNCT
ejpam-4548	580	1	j.	j.	PROPN
ejpam-4548	580	2	pure	pure	PROPN
ejpam-4548	580	3	appl	appl	PROPN
ejpam-4548	580	4	.	.	PROPN
ejpam-4548	580	5	math	math	PROPN
ejpam-4548	580	6	,	,	PUNCT
ejpam-4548	580	7	15	15	NUM
ejpam-4548	580	8	(	(	PUNCT
ejpam-4548	580	9	4	4	NUM
ejpam-4548	580	10	)	)	PUNCT
ejpam-4548	580	11	(	(	PUNCT
ejpam-4548	580	12	2022	2022	NUM
ejpam-4548	580	13	)	)	PUNCT
ejpam-4548	580	14	,	,	PUNCT
ejpam-4548	580	15	2086	2086	NUM
ejpam-4548	580	16	-	-	SYM
ejpam-4548	580	17	2115	2115	NUM
ejpam-4548	580	18	2113	2113	NUM
ejpam-4548	580	19	as	as	SCONJ
ejpam-4548	580	20	the	the	DET
ejpam-4548	580	21	level	level	NOUN
ejpam-4548	580	22	subsets	subset	NOUN
ejpam-4548	580	23	of	of	ADP
ejpam-4548	580	24	(	(	PUNCT
ejpam-4548	580	25	η	η	PROPN
ejpam-4548	580	26	,	,	PUNCT
ejpam-4548	580	27	η	η	NOUN
ejpam-4548	580	28	)	)	PUNCT
ejpam-4548	580	29	are	be	AUX
ejpam-4548	580	30	subgroups	subgroup	NOUN
ejpam-4548	580	31	of	of	ADP
ejpam-4548	580	32	g	g	NOUN
ejpam-4548	580	33	,	,	PUNCT
ejpam-4548	580	34	γ1(η	γ1(η	PROPN
ejpam-4548	580	35	)	)	PUNCT
ejpam-4548	580	36	=	=	PUNCT
ejpam-4548	581	1	[	[	X
ejpam-4548	581	2	η	η	PROPN
ejpam-4548	581	3	,	,	PUNCT
ejpam-4548	581	4	η	η	PROPN
ejpam-4548	581	5	]	]	X
ejpam-4548	581	6	=	=	SYM
ejpam-4548	581	7	(	(	PUNCT
ejpam-4548	581	8	η	η	PROPN
ejpam-4548	581	9	,	,	PUNCT
ejpam-4548	581	10	η	η	NOUN
ejpam-4548	581	11	)	)	PUNCT
ejpam-4548	581	12	.	.	PUNCT
ejpam-4548	582	1	next	next	ADV
ejpam-4548	582	2	,	,	PUNCT
ejpam-4548	582	3	we	we	PRON
ejpam-4548	582	4	calculate	calculate	VERB
ejpam-4548	582	5	the	the	DET
ejpam-4548	582	6	commutator	commutator	NOUN
ejpam-4548	582	7	:	:	PUNCT
ejpam-4548	582	8	(	(	PUNCT
ejpam-4548	582	9	(	(	PUNCT
ejpam-4548	582	10	η	η	PROPN
ejpam-4548	582	11	,	,	PUNCT
ejpam-4548	582	12	η	η	NOUN
ejpam-4548	582	13	)	)	PUNCT
ejpam-4548	582	14	,	,	PUNCT
ejpam-4548	582	15	η	η	PROPN
ejpam-4548	582	16	)	)	PUNCT
ejpam-4548	582	17	(	(	PUNCT
ejpam-4548	582	18	x	x	X
ejpam-4548	582	19	)	)	PUNCT
ejpam-4548	582	20	=	=	SYM
ejpam-4548	582	21	{	{	PUNCT
ejpam-4548	582	22	u	u	NOUN
ejpam-4548	582	23	if	if	SCONJ
ejpam-4548	582	24	x	x	PROPN
ejpam-4548	582	25	=	=	SYM
ejpam-4548	582	26	1	1	NUM
ejpam-4548	582	27	,	,	PUNCT
ejpam-4548	582	28	f	f	PROPN
ejpam-4548	582	29	if	if	SCONJ
ejpam-4548	582	30	x	x	PROPN
ejpam-4548	582	31	∈	∈	PROPN
ejpam-4548	582	32	g	g	NOUN
ejpam-4548	582	33	\	\	PROPN
ejpam-4548	582	34	{	{	PUNCT
ejpam-4548	582	35	1	1	NUM
ejpam-4548	582	36	}	}	PUNCT
ejpam-4548	582	37	.	.	PUNCT
ejpam-4548	583	1	again	again	ADV
ejpam-4548	583	2	,	,	PUNCT
ejpam-4548	583	3	by	by	ADP
ejpam-4548	583	4	the	the	DET
ejpam-4548	583	5	reasons	reason	NOUN
ejpam-4548	583	6	as	as	SCONJ
ejpam-4548	583	7	given	give	VERB
ejpam-4548	583	8	above	above	ADP
ejpam-4548	583	9	γ2(η	γ2(η	NOUN
ejpam-4548	583	10	)	)	PUNCT
ejpam-4548	583	11	=	=	NOUN
ejpam-4548	584	1	[	[	X
ejpam-4548	584	2	[	[	X
ejpam-4548	584	3	η	η	PROPN
ejpam-4548	584	4	,	,	PUNCT
ejpam-4548	584	5	η	η	PROPN
ejpam-4548	584	6	]	]	X
ejpam-4548	584	7	,	,	PUNCT
ejpam-4548	584	8	η	η	PROPN
ejpam-4548	584	9	]	]	X
ejpam-4548	584	10	=	=	SYM
ejpam-4548	584	11	(	(	PUNCT
ejpam-4548	584	12	(	(	PUNCT
ejpam-4548	584	13	η	η	PROPN
ejpam-4548	584	14	,	,	PUNCT
ejpam-4548	584	15	η	η	NOUN
ejpam-4548	584	16	)	)	PUNCT
ejpam-4548	584	17	,	,	PUNCT
ejpam-4548	584	18	η	η	PROPN
ejpam-4548	584	19	)	)	PUNCT
ejpam-4548	584	20	.	.	PUNCT
ejpam-4548	585	1	observe	observe	VERB
ejpam-4548	585	2	that	that	SCONJ
ejpam-4548	585	3	γ2(η	γ2(η	NOUN
ejpam-4548	585	4	)	)	PUNCT
ejpam-4548	585	5	is	be	AUX
ejpam-4548	585	6	not	not	PART
ejpam-4548	585	7	only	only	ADV
ejpam-4548	585	8	an	an	DET
ejpam-4548	585	9	l	l	NOUN
ejpam-4548	585	10	-	-	NOUN
ejpam-4548	585	11	subgroup	subgroup	NOUN
ejpam-4548	585	12	,	,	PUNCT
ejpam-4548	585	13	it	it	PRON
ejpam-4548	585	14	is	be	AUX
ejpam-4548	585	15	the	the	DET
ejpam-4548	585	16	trivial	trivial	ADJ
ejpam-4548	585	17	l	l	NOUN
ejpam-4548	585	18	-	-	NOUN
ejpam-4548	585	19	subgroup	subgroup	NOUN
ejpam-4548	585	20	of	of	ADP
ejpam-4548	585	21	η	η	PROPN
ejpam-4548	585	22	and	and	CCONJ
ejpam-4548	585	23	so	so	ADV
ejpam-4548	585	24	the	the	DET
ejpam-4548	585	25	descending	descend	VERB
ejpam-4548	585	26	central	central	ADJ
ejpam-4548	585	27	series	series	NOUN
ejpam-4548	585	28	terminates	terminate	VERB
ejpam-4548	585	29	at	at	ADP
ejpam-4548	585	30	γ2(η	γ2(η	NOUN
ejpam-4548	585	31	)	)	PUNCT
ejpam-4548	585	32	,	,	PUNCT
ejpam-4548	585	33	i.	i.	PROPN
ejpam-4548	585	34	e.	e.	PROPN
ejpam-4548	585	35	η	η	PROPN
ejpam-4548	585	36	=	=	PROPN
ejpam-4548	585	37	γ0(η	γ0(η	PROPN
ejpam-4548	585	38	)	)	PUNCT
ejpam-4548	585	39	⊇	⊇	PROPN
ejpam-4548	585	40	γ1(η	γ1(η	PROPN
ejpam-4548	585	41	)	)	PUNCT
ejpam-4548	585	42	⊇	⊇	NOUN
ejpam-4548	585	43	γ2(η	γ2(η	NOUN
ejpam-4548	585	44	)	)	PUNCT
ejpam-4548	585	45	=	=	PRON
ejpam-4548	585	46	ηuf	ηuf	VERB
ejpam-4548	585	47	.	.	PUNCT
ejpam-4548	586	1	consequently	consequently	ADV
ejpam-4548	586	2	,	,	PUNCT
ejpam-4548	586	3	η	η	PROPN
ejpam-4548	586	4	is	be	AUX
ejpam-4548	586	5	a	a	DET
ejpam-4548	586	6	nilpotent	nilpotent	ADJ
ejpam-4548	586	7	l	l	NOUN
ejpam-4548	586	8	-	-	NOUN
ejpam-4548	586	9	subgroup	subgroup	NOUN
ejpam-4548	586	10	of	of	ADP
ejpam-4548	586	11	µ	µ	NOUN
ejpam-4548	586	12	having	have	VERB
ejpam-4548	586	13	nilpotent	nilpotent	ADJ
ejpam-4548	586	14	length	length	NOUN
ejpam-4548	586	15	2	2	NUM
ejpam-4548	586	16	.	.	PUNCT
ejpam-4548	586	17	theorem	theorem	NOUN
ejpam-4548	586	18	23	23	NUM
ejpam-4548	586	19	.	.	PUNCT
ejpam-4548	587	1	let	let	VERB
ejpam-4548	587	2	l	l	NOUN
ejpam-4548	587	3	be	be	AUX
ejpam-4548	587	4	an	an	DET
ejpam-4548	587	5	upper	upper	ADJ
ejpam-4548	587	6	well	well	NOUN
ejpam-4548	587	7	ordered	order	VERB
ejpam-4548	587	8	chain	chain	NOUN
ejpam-4548	587	9	.	.	PUNCT
ejpam-4548	588	1	let	let	VERB
ejpam-4548	588	2	η	η	PROPN
ejpam-4548	588	3	∈	∈	PROPN
ejpam-4548	588	4	l(µ	l(µ	PROPN
ejpam-4548	588	5	)	)	PUNCT
ejpam-4548	588	6	and	and	CCONJ
ejpam-4548	588	7	η	η	PROPN
ejpam-4548	588	8	be	be	AUX
ejpam-4548	588	9	nilpotent	nilpotent	ADJ
ejpam-4548	588	10	having	have	VERB
ejpam-4548	588	11	the	the	DET
ejpam-4548	588	12	tip	tip	NOUN
ejpam-4548	588	13	a0	a0	NOUN
ejpam-4548	588	14	and	and	CCONJ
ejpam-4548	588	15	the	the	DET
ejpam-4548	588	16	tail	tail	NOUN
ejpam-4548	588	17	t0	t0	NOUN
ejpam-4548	588	18	.	.	PUNCT
ejpam-4548	589	1	if	if	SCONJ
ejpam-4548	589	2	θ	θ	PROPN
ejpam-4548	589	3	∈	∈	PROPN
ejpam-4548	589	4	l(η	l(η	PROPN
ejpam-4548	589	5	)	)	PUNCT
ejpam-4548	589	6	having	have	VERB
ejpam-4548	589	7	the	the	DET
ejpam-4548	589	8	tail	tail	NOUN
ejpam-4548	589	9	t0	t0	NOUN
ejpam-4548	589	10	,	,	PUNCT
ejpam-4548	589	11	then	then	ADV
ejpam-4548	589	12	θ	θ	PROPN
ejpam-4548	589	13	is	be	AUX
ejpam-4548	589	14	a	a	DET
ejpam-4548	589	15	subnormal	subnormal	ADJ
ejpam-4548	589	16	l	l	NOUN
ejpam-4548	589	17	-	-	NOUN
ejpam-4548	589	18	subgroup	subgroup	NOUN
ejpam-4548	589	19	of	of	ADP
ejpam-4548	589	20	η	η	PROPN
ejpam-4548	589	21	.	.	PROPN
ejpam-4548	589	22	proof	proof	NOUN
ejpam-4548	589	23	.	.	PUNCT
ejpam-4548	590	1	let	let	VERB
ejpam-4548	590	2	θ	θ	PROPN
ejpam-4548	590	3	∈	∈	PROPN
ejpam-4548	590	4	l(η	l(η	PROPN
ejpam-4548	590	5	)	)	PUNCT
ejpam-4548	590	6	having	have	VERB
ejpam-4548	590	7	the	the	DET
ejpam-4548	590	8	tail	tail	NOUN
ejpam-4548	590	9	t0	t0	NOUN
ejpam-4548	590	10	.	.	PUNCT
ejpam-4548	591	1	we	we	PRON
ejpam-4548	591	2	shall	shall	AUX
ejpam-4548	591	3	show	show	VERB
ejpam-4548	591	4	that	that	SCONJ
ejpam-4548	591	5	the	the	DET
ejpam-4548	591	6	level	level	NOUN
ejpam-4548	591	7	subgroup	subgroup	NOUN
ejpam-4548	591	8	θa	θa	NOUN
ejpam-4548	591	9	is	be	AUX
ejpam-4548	591	10	a	a	DET
ejpam-4548	591	11	subnormal	subnormal	ADJ
ejpam-4548	591	12	lsubgroup	lsubgroup	NOUN
ejpam-4548	591	13	of	of	ADP
ejpam-4548	591	14	ηa	ηa	PRON
ejpam-4548	591	15	for	for	ADP
ejpam-4548	591	16	each	each	DET
ejpam-4548	591	17	a	a	DET
ejpam-4548	591	18	≤	≤	NUM
ejpam-4548	591	19	θ(e	θ(e	NUM
ejpam-4548	591	20	)	)	PUNCT
ejpam-4548	591	21	.	.	PUNCT
ejpam-4548	592	1	if	if	SCONJ
ejpam-4548	592	2	a	a	DET
ejpam-4548	592	3	≤	≤	NUM
ejpam-4548	592	4	t0	t0	PROPN
ejpam-4548	592	5	,	,	PUNCT
ejpam-4548	592	6	then	then	ADV
ejpam-4548	592	7	obviously	obviously	ADV
ejpam-4548	592	8	θa	θa	PRON
ejpam-4548	592	9	is	be	AUX
ejpam-4548	592	10	a	a	DET
ejpam-4548	592	11	subnormal	subnormal	ADJ
ejpam-4548	592	12	subgroup	subgroup	NOUN
ejpam-4548	592	13	of	of	ADP
ejpam-4548	592	14	ηa	ηa	PROPN
ejpam-4548	592	15	.	.	PUNCT
ejpam-4548	593	1	if	if	SCONJ
ejpam-4548	593	2	so	so	ADV
ejpam-4548	593	3	,	,	PUNCT
ejpam-4548	593	4	let	let	VERB
ejpam-4548	593	5	t0	t0	PROPN
ejpam-4548	593	6	<	<	X
ejpam-4548	593	7	a	a	DET
ejpam-4548	593	8	≤	≤	PROPN
ejpam-4548	593	9	θ(e	θ(e	NUM
ejpam-4548	593	10	)	)	PUNCT
ejpam-4548	593	11	.	.	PUNCT
ejpam-4548	594	1	as	as	SCONJ
ejpam-4548	594	2	l	l	NOUN
ejpam-4548	594	3	is	be	AUX
ejpam-4548	594	4	upper	upper	ADJ
ejpam-4548	594	5	well	well	ADV
ejpam-4548	594	6	ordered	order	VERB
ejpam-4548	594	7	,	,	PUNCT
ejpam-4548	594	8	it	it	PRON
ejpam-4548	594	9	follows	follow	VERB
ejpam-4548	594	10	the	the	DET
ejpam-4548	594	11	l	l	NOUN
ejpam-4548	594	12	-	-	PUNCT
ejpam-4548	594	13	subset	subset	VERB
ejpam-4548	594	14	η	η	PROPN
ejpam-4548	594	15	possesses	possesse	NOUN
ejpam-4548	594	16	sup	sup	ADJ
ejpam-4548	594	17	-	-	PUNCT
ejpam-4548	594	18	property	property	NOUN
ejpam-4548	594	19	.	.	PUNCT
ejpam-4548	595	1	now	now	ADV
ejpam-4548	595	2	,	,	PUNCT
ejpam-4548	595	3	as	as	SCONJ
ejpam-4548	595	4	t0	t0	PROPN
ejpam-4548	595	5	<	<	X
ejpam-4548	595	6	a	a	DET
ejpam-4548	595	7	≤	≤	PROPN
ejpam-4548	595	8	θ(e	θ(e	NUM
ejpam-4548	595	9	)	)	PUNCT
ejpam-4548	595	10	≤	≤	NUM
ejpam-4548	595	11	η(e	η(e	PROPN
ejpam-4548	595	12	)	)	PUNCT
ejpam-4548	595	13	and	and	CCONJ
ejpam-4548	595	14	η	η	PROPN
ejpam-4548	595	15	is	be	AUX
ejpam-4548	595	16	nilpotent	nilpotent	ADJ
ejpam-4548	595	17	,	,	PUNCT
ejpam-4548	595	18	by	by	ADP
ejpam-4548	595	19	theorem	theorem	NOUN
ejpam-4548	595	20	21	21	NUM
ejpam-4548	595	21	,	,	PUNCT
ejpam-4548	595	22	the	the	DET
ejpam-4548	595	23	level	level	NOUN
ejpam-4548	595	24	subgroup	subgroup	NOUN
ejpam-4548	595	25	ηa	ηa	PROPN
ejpam-4548	595	26	is	be	AUX
ejpam-4548	595	27	nilpotent	nilpotent	ADJ
ejpam-4548	595	28	.	.	PUNCT
ejpam-4548	596	1	thus	thus	ADV
ejpam-4548	596	2	θa	θa	PRON
ejpam-4548	596	3	is	be	AUX
ejpam-4548	596	4	a	a	DET
ejpam-4548	596	5	subgroup	subgroup	NOUN
ejpam-4548	596	6	of	of	ADP
ejpam-4548	596	7	the	the	DET
ejpam-4548	596	8	nilpotent	nilpotent	ADJ
ejpam-4548	596	9	subgroup	subgroup	NOUN
ejpam-4548	596	10	ηa	ηa	PROPN
ejpam-4548	596	11	,	,	PUNCT
ejpam-4548	596	12	hence	hence	ADV
ejpam-4548	596	13	by	by	ADP
ejpam-4548	596	14	theorem	theorem	NOUN
ejpam-4548	596	15	20	20	NUM
ejpam-4548	596	16	,	,	PUNCT
ejpam-4548	596	17	θa	θa	NUM
ejpam-4548	596	18	is	be	AUX
ejpam-4548	596	19	a	a	DET
ejpam-4548	596	20	subnormal	subnormal	ADJ
ejpam-4548	596	21	subgroup	subgroup	NOUN
ejpam-4548	596	22	of	of	ADP
ejpam-4548	596	23	ηa	ηa	PROPN
ejpam-4548	596	24	.	.	PUNCT
ejpam-4548	597	1	again	again	ADV
ejpam-4548	597	2	as	as	SCONJ
ejpam-4548	597	3	l	l	NOUN
ejpam-4548	597	4	is	be	AUX
ejpam-4548	597	5	an	an	DET
ejpam-4548	597	6	upper	upper	ADJ
ejpam-4548	597	7	well	well	NOUN
ejpam-4548	597	8	ordered	order	VERB
ejpam-4548	597	9	chain	chain	NOUN
ejpam-4548	597	10	,	,	PUNCT
ejpam-4548	597	11	by	by	ADP
ejpam-4548	597	12	using	use	VERB
ejpam-4548	597	13	the	the	DET
ejpam-4548	597	14	arguments	argument	NOUN
ejpam-4548	597	15	as	as	ADP
ejpam-4548	597	16	above	above	ADV
ejpam-4548	597	17	,	,	PUNCT
ejpam-4548	597	18	it	it	PRON
ejpam-4548	597	19	follows	follow	VERB
ejpam-4548	597	20	that	that	SCONJ
ejpam-4548	597	21	both	both	CCONJ
ejpam-4548	597	22	the	the	DET
ejpam-4548	597	23	l	l	NOUN
ejpam-4548	597	24	-	-	PUNCT
ejpam-4548	597	25	subgroups	subgroup	NOUN
ejpam-4548	597	26	η	η	PROPN
ejpam-4548	597	27	and	and	CCONJ
ejpam-4548	597	28	θ	θ	PROPN
ejpam-4548	597	29	possess	possess	VERB
ejpam-4548	597	30	sup	sup	NOUN
ejpam-4548	597	31	-	-	PUNCT
ejpam-4548	597	32	property	property	NOUN
ejpam-4548	597	33	.	.	PUNCT
ejpam-4548	598	1	therefore	therefore	ADV
ejpam-4548	598	2	,	,	PUNCT
ejpam-4548	598	3	by	by	ADP
ejpam-4548	598	4	proposition	proposition	NOUN
ejpam-4548	598	5	8	8	NUM
ejpam-4548	598	6	,	,	PUNCT
ejpam-4548	598	7	η	η	PROPN
ejpam-4548	598	8	and	and	CCONJ
ejpam-4548	598	9	θ	θ	PROPN
ejpam-4548	598	10	are	be	AUX
ejpam-4548	598	11	joinly	joinly	ADV
ejpam-4548	598	12	supstar	supstar	ADJ
ejpam-4548	598	13	.	.	PUNCT
ejpam-4548	599	1	consequently	consequently	ADV
ejpam-4548	599	2	,	,	PUNCT
ejpam-4548	599	3	by	by	ADP
ejpam-4548	599	4	theorem	theorem	NOUN
ejpam-4548	599	5	18	18	NUM
ejpam-4548	599	6	,	,	PUNCT
ejpam-4548	599	7	θ	θ	PROPN
ejpam-4548	599	8	is	be	AUX
ejpam-4548	599	9	a	a	DET
ejpam-4548	599	10	subnormal	subnormal	ADJ
ejpam-4548	599	11	l	l	NOUN
ejpam-4548	599	12	-	-	NOUN
ejpam-4548	599	13	subgroup	subgroup	NOUN
ejpam-4548	599	14	of	of	ADP
ejpam-4548	599	15	η	η	PROPN
ejpam-4548	599	16	.	.	PROPN
ejpam-4548	599	17	in	in	ADP
ejpam-4548	599	18	the	the	DET
ejpam-4548	599	19	part	part	NOUN
ejpam-4548	599	20	ii	ii	NOUN
ejpam-4548	599	21	of	of	ADP
ejpam-4548	599	22	this	this	DET
ejpam-4548	599	23	paper	paper	NOUN
ejpam-4548	599	24	,	,	PUNCT
ejpam-4548	599	25	we	we	PRON
ejpam-4548	599	26	develop	develop	VERB
ejpam-4548	599	27	a	a	DET
ejpam-4548	599	28	mechanism	mechanism	NOUN
ejpam-4548	599	29	in	in	ADP
ejpam-4548	599	30	order	order	NOUN
ejpam-4548	599	31	to	to	PART
ejpam-4548	599	32	tackle	tackle	VERB
ejpam-4548	599	33	the	the	DET
ejpam-4548	599	34	join	join	NOUN
ejpam-4548	599	35	problem	problem	NOUN
ejpam-4548	599	36	for	for	ADP
ejpam-4548	599	37	subnormal	subnormal	ADJ
ejpam-4548	599	38	l	l	PROPN
ejpam-4548	599	39	-	-	NOUN
ejpam-4548	599	40	subgroup	subgroup	NOUN
ejpam-4548	599	41	.	.	PUNCT
ejpam-4548	600	1	theorem	theorem	PROPN
ejpam-4548	600	2	24	24	NUM
ejpam-4548	600	3	.	.	PUNCT
ejpam-4548	601	1	let	let	VERB
ejpam-4548	601	2	η	η	PROPN
ejpam-4548	601	3	and	and	CCONJ
ejpam-4548	601	4	θ	θ	PROPN
ejpam-4548	601	5	be	be	VERB
ejpam-4548	601	6	subnormal	subnormal	ADJ
ejpam-4548	601	7	l	l	NOUN
ejpam-4548	601	8	-	-	NOUN
ejpam-4548	601	9	subgroups	subgroup	NOUN
ejpam-4548	601	10	of	of	ADP
ejpam-4548	601	11	µ	µ	NUM
ejpam-4548	601	12	.	.	PUNCT
ejpam-4548	602	1	let	let	VERB
ejpam-4548	602	2	η(e	η(e	NOUN
ejpam-4548	602	3	)	)	PUNCT
ejpam-4548	603	1	=	=	SYM
ejpam-4548	603	2	θ(e	θ(e	NUM
ejpam-4548	603	3	)	)	PUNCT
ejpam-4548	603	4	and	and	CCONJ
ejpam-4548	603	5	η	η	PROPN
ejpam-4548	603	6	◦	◦	NOUN
ejpam-4548	603	7	θ	θ	X
ejpam-4548	603	8	∈	∈	PROPN
ejpam-4548	603	9	l(µ	l(µ	PROPN
ejpam-4548	603	10	)	)	PUNCT
ejpam-4548	603	11	.	.	PUNCT
ejpam-4548	604	1	then	then	ADV
ejpam-4548	604	2	,	,	PUNCT
ejpam-4548	604	3	the	the	DET
ejpam-4548	604	4	following	follow	VERB
ejpam-4548	604	5	are	be	AUX
ejpam-4548	604	6	equivalent	equivalent	ADJ
ejpam-4548	604	7	:	:	PUNCT
ejpam-4548	604	8	(	(	PUNCT
ejpam-4548	604	9	i	i	NOUN
ejpam-4548	604	10	)	)	PUNCT
ejpam-4548	604	11	η	η	PROPN
ejpam-4548	604	12	◦	◦	PROPN
ejpam-4548	604	13	θ	θ	PROPN
ejpam-4548	604	14	is	be	AUX
ejpam-4548	604	15	subnormal	subnormal	ADJ
ejpam-4548	604	16	in	in	ADP
ejpam-4548	604	17	µ	µ	NUM
ejpam-4548	604	18	,	,	PUNCT
ejpam-4548	604	19	(	(	PUNCT
ejpam-4548	604	20	ii	ii	NOUN
ejpam-4548	604	21	)	)	PUNCT
ejpam-4548	604	22	ηθ	ηθ	NOUN
ejpam-4548	604	23	is	be	AUX
ejpam-4548	604	24	subnormal	subnormal	ADJ
ejpam-4548	604	25	in	in	ADP
ejpam-4548	604	26	µ	µ	NUM
ejpam-4548	604	27	,	,	PUNCT
ejpam-4548	604	28	(	(	PUNCT
ejpam-4548	604	29	iii	iii	NOUN
ejpam-4548	604	30	)	)	PUNCT
ejpam-4548	605	1	[	[	X
ejpam-4548	605	2	η	η	X
ejpam-4548	605	3	,	,	PUNCT
ejpam-4548	605	4	θ	θ	PROPN
ejpam-4548	605	5	]	]	PUNCT
ejpam-4548	605	6	is	be	AUX
ejpam-4548	605	7	subnormal	subnormal	ADJ
ejpam-4548	605	8	in	in	ADP
ejpam-4548	605	9	µ	µ	NUM
ejpam-4548	605	10	,	,	PUNCT
ejpam-4548	605	11	references	reference	NOUN
ejpam-4548	605	12	2114	2114	NUM
ejpam-4548	605	13	5	5	NUM
ejpam-4548	605	14	.	.	PUNCT
ejpam-4548	606	1	conclusion	conclusion	VERB
ejpam-4548	606	2	the	the	DET
ejpam-4548	606	3	theory	theory	NOUN
ejpam-4548	606	4	of	of	ADP
ejpam-4548	606	5	l	l	NOUN
ejpam-4548	606	6	-	-	NOUN
ejpam-4548	606	7	groups	group	NOUN
ejpam-4548	606	8	is	be	AUX
ejpam-4548	606	9	a	a	DET
ejpam-4548	606	10	very	very	ADV
ejpam-4548	606	11	rich	rich	ADJ
ejpam-4548	606	12	generalization	generalization	NOUN
ejpam-4548	606	13	of	of	ADP
ejpam-4548	606	14	classical	classical	ADJ
ejpam-4548	606	15	group	group	NOUN
ejpam-4548	606	16	theory	theory	NOUN
ejpam-4548	606	17	.	.	PUNCT
ejpam-4548	607	1	here	here	ADV
ejpam-4548	607	2	we	we	PRON
ejpam-4548	607	3	study	study	VERB
ejpam-4548	607	4	the	the	DET
ejpam-4548	607	5	group	group	NOUN
ejpam-4548	607	6	theoretic	theoretic	NOUN
ejpam-4548	607	7	properties	property	NOUN
ejpam-4548	607	8	of	of	ADP
ejpam-4548	607	9	posets	poset	NOUN
ejpam-4548	607	10	of	of	ADP
ejpam-4548	607	11	subgroups	subgroup	NOUN
ejpam-4548	607	12	of	of	ADP
ejpam-4548	607	13	a	a	DET
ejpam-4548	607	14	group	group	NOUN
ejpam-4548	607	15	or	or	CCONJ
ejpam-4548	607	16	in	in	ADP
ejpam-4548	607	17	particular	particular	ADJ
ejpam-4548	607	18	chains	chain	NOUN
ejpam-4548	607	19	of	of	ADP
ejpam-4548	607	20	subgroups	subgroup	NOUN
ejpam-4548	607	21	of	of	ADP
ejpam-4548	607	22	a	a	DET
ejpam-4548	607	23	group	group	NOUN
ejpam-4548	607	24	rather	rather	ADV
ejpam-4548	607	25	than	than	ADP
ejpam-4548	607	26	properties	property	NOUN
ejpam-4548	607	27	of	of	ADP
ejpam-4548	607	28	a	a	DET
ejpam-4548	607	29	single	single	ADJ
ejpam-4548	607	30	subgroup	subgroup	NOUN
ejpam-4548	607	31	.	.	PUNCT
ejpam-4548	608	1	as	as	ADP
ejpam-4548	608	2	an	an	DET
ejpam-4548	608	3	application	application	NOUN
ejpam-4548	608	4	of	of	ADP
ejpam-4548	608	5	this	this	DET
ejpam-4548	608	6	work	work	NOUN
ejpam-4548	608	7	and	and	CCONJ
ejpam-4548	608	8	the	the	DET
ejpam-4548	608	9	motivation	motivation	NOUN
ejpam-4548	608	10	for	for	ADP
ejpam-4548	608	11	the	the	DET
ejpam-4548	608	12	development	development	NOUN
ejpam-4548	608	13	of	of	ADP
ejpam-4548	608	14	l	l	NOUN
ejpam-4548	608	15	-	-	NOUN
ejpam-4548	608	16	group	group	NOUN
ejpam-4548	608	17	theory	theory	NOUN
ejpam-4548	608	18	,	,	PUNCT
ejpam-4548	608	19	we	we	PRON
ejpam-4548	608	20	mention	mention	VERB
ejpam-4548	608	21	that	that	SCONJ
ejpam-4548	608	22	if	if	SCONJ
ejpam-4548	608	23	we	we	PRON
ejpam-4548	608	24	replace	replace	VERB
ejpam-4548	608	25	the	the	DET
ejpam-4548	608	26	lattice	lattice	PROPN
ejpam-4548	608	27	l	l	PROPN
ejpam-4548	608	28	,	,	PUNCT
ejpam-4548	608	29	in	in	ADP
ejpam-4548	608	30	our	our	PRON
ejpam-4548	608	31	work	work	NOUN
ejpam-4548	608	32	by	by	ADP
ejpam-4548	608	33	the	the	DET
ejpam-4548	608	34	closed	closed	ADJ
ejpam-4548	608	35	unit	unit	NOUN
ejpam-4548	608	36	interval	interval	NOUN
ejpam-4548	608	37	[	[	X
ejpam-4548	608	38	0	0	NUM
ejpam-4548	608	39	,	,	PUNCT
ejpam-4548	608	40	1	1	NUM
ejpam-4548	608	41	]	]	PUNCT
ejpam-4548	608	42	,	,	PUNCT
ejpam-4548	608	43	then	then	ADV
ejpam-4548	608	44	we	we	PRON
ejpam-4548	608	45	retrieve	retrieve	VERB
ejpam-4548	608	46	the	the	DET
ejpam-4548	608	47	corresponding	corresponding	ADJ
ejpam-4548	608	48	version	version	NOUN
ejpam-4548	608	49	of	of	ADP
ejpam-4548	608	50	fuzzy	fuzzy	ADJ
ejpam-4548	608	51	group	group	NOUN
ejpam-4548	608	52	theory	theory	NOUN
ejpam-4548	608	53	.	.	PUNCT
ejpam-4548	609	1	moreover	moreover	ADV
ejpam-4548	609	2	,	,	PUNCT
ejpam-4548	609	3	as	as	ADP
ejpam-4548	609	4	an	an	DET
ejpam-4548	609	5	application	application	NOUN
ejpam-4548	609	6	of	of	ADP
ejpam-4548	609	7	this	this	DET
ejpam-4548	609	8	theory	theory	NOUN
ejpam-4548	609	9	we	we	PRON
ejpam-4548	609	10	also	also	ADV
ejpam-4548	609	11	mention	mention	VERB
ejpam-4548	609	12	that	that	SCONJ
ejpam-4548	609	13	if	if	SCONJ
ejpam-4548	609	14	we	we	PRON
ejpam-4548	609	15	replace	replace	VERB
ejpam-4548	609	16	the	the	DET
ejpam-4548	609	17	lattice	lattice	NOUN
ejpam-4548	609	18	l	l	NOUN
ejpam-4548	609	19	by	by	ADP
ejpam-4548	609	20	the	the	DET
ejpam-4548	609	21	two	two	NUM
ejpam-4548	609	22	elements	element	NOUN
ejpam-4548	609	23	set	set	VERB
ejpam-4548	609	24	{	{	PUNCT
ejpam-4548	609	25	0	0	NUM
ejpam-4548	609	26	,	,	PUNCT
ejpam-4548	609	27	1	1	NUM
ejpam-4548	609	28	}	}	PUNCT
ejpam-4548	609	29	,	,	PUNCT
ejpam-4548	609	30	then	then	ADV
ejpam-4548	609	31	the	the	DET
ejpam-4548	609	32	results	result	NOUN
ejpam-4548	609	33	of	of	ADP
ejpam-4548	609	34	classical	classical	ADJ
ejpam-4548	609	35	group	group	NOUN
ejpam-4548	609	36	theory	theory	NOUN
ejpam-4548	609	37	follow	follow	VERB
ejpam-4548	609	38	as	as	ADP
ejpam-4548	609	39	simple	simple	ADJ
ejpam-4548	609	40	corollaries	corollary	NOUN
ejpam-4548	609	41	of	of	ADP
ejpam-4548	609	42	the	the	DET
ejpam-4548	609	43	corresponding	corresponding	ADJ
ejpam-4548	609	44	results	result	NOUN
ejpam-4548	609	45	of	of	ADP
ejpam-4548	609	46	l	l	NOUN
ejpam-4548	609	47	group	group	NOUN
ejpam-4548	609	48	theory	theory	NOUN
ejpam-4548	609	49	.	.	PUNCT
ejpam-4548	610	1	this	this	DET
ejpam-4548	610	2	way	way	NOUN
ejpam-4548	610	3	,	,	PUNCT
ejpam-4548	610	4	l	l	NOUN
ejpam-4548	610	5	-	-	NOUN
ejpam-4548	610	6	group	group	NOUN
ejpam-4548	610	7	theory	theory	NOUN
ejpam-4548	610	8	provides	provide	VERB
ejpam-4548	610	9	us	we	PRON
ejpam-4548	610	10	a	a	DET
ejpam-4548	610	11	new	new	ADJ
ejpam-4548	610	12	language	language	NOUN
ejpam-4548	610	13	and	and	CCONJ
ejpam-4548	610	14	a	a	DET
ejpam-4548	610	15	new	new	ADJ
ejpam-4548	610	16	tool	tool	NOUN
ejpam-4548	610	17	for	for	ADP
ejpam-4548	610	18	the	the	DET
ejpam-4548	610	19	study	study	NOUN
ejpam-4548	610	20	of	of	ADP
ejpam-4548	610	21	the	the	DET
ejpam-4548	610	22	classical	classical	ADJ
ejpam-4548	610	23	group	group	NOUN
ejpam-4548	610	24	theory	theory	NOUN
ejpam-4548	610	25	.	.	PUNCT
ejpam-4548	611	1	the	the	DET
ejpam-4548	611	2	classical	classical	ADJ
ejpam-4548	611	3	group	group	NOUN
ejpam-4548	611	4	theory	theory	NOUN
ejpam-4548	611	5	has	have	AUX
ejpam-4548	611	6	been	be	AUX
ejpam-4548	611	7	founded	found	VERB
ejpam-4548	611	8	on	on	ADP
ejpam-4548	611	9	abstract	abstract	ADJ
ejpam-4548	611	10	sets	set	NOUN
ejpam-4548	611	11	and	and	CCONJ
ejpam-4548	611	12	therefore	therefore	ADV
ejpam-4548	611	13	the	the	DET
ejpam-4548	611	14	language	language	NOUN
ejpam-4548	611	15	used	use	VERB
ejpam-4548	611	16	for	for	ADP
ejpam-4548	611	17	its	its	PRON
ejpam-4548	611	18	development	development	NOUN
ejpam-4548	611	19	is	be	AUX
ejpam-4548	611	20	formal	formal	ADJ
ejpam-4548	611	21	set	set	NOUN
ejpam-4548	611	22	theory	theory	NOUN
ejpam-4548	611	23	.	.	PUNCT
ejpam-4548	612	1	on	on	ADP
ejpam-4548	612	2	the	the	DET
ejpam-4548	612	3	other	other	ADJ
ejpam-4548	612	4	hand	hand	NOUN
ejpam-4548	612	5	,	,	PUNCT
ejpam-4548	612	6	l	l	NOUN
ejpam-4548	612	7	-	-	NOUN
ejpam-4548	612	8	group	group	NOUN
ejpam-4548	612	9	theory	theory	NOUN
ejpam-4548	612	10	expresses	express	VERB
ejpam-4548	612	11	itself	itself	PRON
ejpam-4548	612	12	through	through	ADP
ejpam-4548	612	13	the	the	DET
ejpam-4548	612	14	language	language	NOUN
ejpam-4548	612	15	of	of	ADP
ejpam-4548	612	16	functions	function	NOUN
ejpam-4548	612	17	.	.	PUNCT
ejpam-4548	613	1	the	the	DET
ejpam-4548	613	2	functions	function	NOUN
ejpam-4548	613	3	which	which	PRON
ejpam-4548	613	4	are	be	AUX
ejpam-4548	613	5	lattice	lattice	NOUN
ejpam-4548	613	6	valued	value	VERB
ejpam-4548	613	7	.	.	PUNCT
ejpam-4548	614	1	therefore	therefore	ADV
ejpam-4548	614	2	the	the	DET
ejpam-4548	614	3	approach	approach	NOUN
ejpam-4548	614	4	adopted	adopt	VERB
ejpam-4548	614	5	in	in	ADP
ejpam-4548	614	6	the	the	DET
ejpam-4548	614	7	studies	study	NOUN
ejpam-4548	614	8	of	of	ADP
ejpam-4548	614	9	l	l	NOUN
ejpam-4548	614	10	-	-	NOUN
ejpam-4548	614	11	group	group	NOUN
ejpam-4548	614	12	theory	theory	NOUN
ejpam-4548	614	13	can	can	AUX
ejpam-4548	614	14	be	be	AUX
ejpam-4548	614	15	looked	look	VERB
ejpam-4548	614	16	upon	upon	SCONJ
ejpam-4548	614	17	as	as	ADP
ejpam-4548	614	18	a	a	DET
ejpam-4548	614	19	modernization	modernization	NOUN
ejpam-4548	614	20	of	of	ADP
ejpam-4548	614	21	the	the	DET
ejpam-4548	614	22	approach	approach	NOUN
ejpam-4548	614	23	of	of	ADP
ejpam-4548	614	24	classical	classical	ADJ
ejpam-4548	614	25	group	group	NOUN
ejpam-4548	614	26	theory	theory	NOUN
ejpam-4548	614	27	.	.	PUNCT
ejpam-4548	615	1	acknowledgements	acknowledgement	NOUN
ejpam-4548	615	2	this	this	DET
ejpam-4548	615	3	work	work	NOUN
ejpam-4548	615	4	is	be	AUX
ejpam-4548	615	5	carried	carry	VERB
ejpam-4548	615	6	out	out	ADP
ejpam-4548	615	7	under	under	ADP
ejpam-4548	615	8	the	the	DET
ejpam-4548	615	9	support	support	NOUN
ejpam-4548	615	10	of	of	ADP
ejpam-4548	615	11	emeritus	emeritus	ADJ
ejpam-4548	615	12	fellowship	fellowship	NOUN
ejpam-4548	615	13	,	,	PUNCT
ejpam-4548	615	14	2015	2015	NUM
ejpam-4548	615	15	-	-	SYM
ejpam-4548	615	16	2017	2017	NUM
ejpam-4548	615	17	,	,	PUNCT
ejpam-4548	615	18	ugc	ugc	PROPN
ejpam-4548	615	19	,	,	PUNCT
ejpam-4548	615	20	india	india	PROPN
ejpam-4548	615	21	.	.	PUNCT
ejpam-4548	616	1	references	reference	NOUN
ejpam-4548	617	1	[	[	X
ejpam-4548	617	2	1	1	NUM
ejpam-4548	617	3	]	]	PUNCT
ejpam-4548	617	4	n	n	PRON
ejpam-4548	617	5	ajmal	ajmal	ADJ
ejpam-4548	617	6	.	.	PUNCT
ejpam-4548	618	1	set	set	VERB
ejpam-4548	618	2	product	product	NOUN
ejpam-4548	618	3	and	and	CCONJ
ejpam-4548	618	4	fuzzy	fuzzy	ADJ
ejpam-4548	618	5	subgroups	subgroup	NOUN
ejpam-4548	618	6	.	.	PUNCT
ejpam-4548	619	1	in	in	ADP
ejpam-4548	619	2	r	r	PROPN
ejpam-4548	619	3	lowen	lowen	PROPN
ejpam-4548	619	4	and	and	CCONJ
ejpam-4548	619	5	m	m	PROPN
ejpam-4548	619	6	roubens	rouben	NOUN
ejpam-4548	619	7	,	,	PUNCT
ejpam-4548	619	8	editors	editor	NOUN
ejpam-4548	619	9	,	,	PUNCT
ejpam-4548	619	10	proc	proc	NOUN
ejpam-4548	619	11	.	.	PUNCT
ejpam-4548	620	1	of	of	ADP
ejpam-4548	620	2	the	the	DET
ejpam-4548	620	3	fourth	fourth	PROPN
ejpam-4548	620	4	ifsa	ifsa	PROPN
ejpam-4548	620	5	world	world	PROPN
ejpam-4548	620	6	congress	congress	PROPN
ejpam-4548	620	7	.	.	PUNCT
ejpam-4548	621	1	,	,	PUNCT
ejpam-4548	621	2	pages	page	NOUN
ejpam-4548	621	3	3–7	3–7	NUM
ejpam-4548	621	4	,	,	PUNCT
ejpam-4548	621	5	belgium	belgium	NOUN
ejpam-4548	621	6	,	,	PUNCT
ejpam-4548	621	7	1991	1991	NUM
ejpam-4548	621	8	.	.	PUNCT
ejpam-4548	622	1	university	university	NOUN
ejpam-4548	622	2	of	of	ADP
ejpam-4548	622	3	antwerp	antwerp	PROPN
ejpam-4548	622	4	,	,	PUNCT
ejpam-4548	622	5	antwerp	antwerp	PROPN
ejpam-4548	622	6	.	.	PUNCT
ejpam-4548	623	1	[	[	X
ejpam-4548	623	2	2	2	NUM
ejpam-4548	623	3	]	]	PUNCT
ejpam-4548	623	4	n	n	X
ejpam-4548	623	5	ajmal	ajmal	ADJ
ejpam-4548	623	6	and	and	CCONJ
ejpam-4548	623	7	i	i	PROPN
ejpam-4548	623	8	jahan	jahan	PROPN
ejpam-4548	623	9	.	.	PUNCT
ejpam-4548	624	1	an	an	DET
ejpam-4548	624	2	l	l	NOUN
ejpam-4548	624	3	-	-	PUNCT
ejpam-4548	624	4	point	point	NOUN
ejpam-4548	624	5	characterization	characterization	NOUN
ejpam-4548	624	6	of	of	ADP
ejpam-4548	624	7	normality	normality	NOUN
ejpam-4548	624	8	and	and	CCONJ
ejpam-4548	624	9	normalizer	normalizer	NOUN
ejpam-4548	624	10	of	of	ADP
ejpam-4548	624	11	an	an	DET
ejpam-4548	624	12	l	l	NOUN
ejpam-4548	624	13	-	-	NOUN
ejpam-4548	624	14	subgroup	subgroup	NOUN
ejpam-4548	624	15	of	of	ADP
ejpam-4548	624	16	an	an	DET
ejpam-4548	624	17	l	l	NOUN
ejpam-4548	624	18	-	-	NOUN
ejpam-4548	624	19	group	group	NOUN
ejpam-4548	624	20	.	.	PUNCT
ejpam-4548	625	1	fuzzy	fuzzy	ADJ
ejpam-4548	625	2	information	information	NOUN
ejpam-4548	625	3	and	and	CCONJ
ejpam-4548	625	4	engineering	engineering	NOUN
ejpam-4548	625	5	,	,	PUNCT
ejpam-4548	625	6	6(1):147–166	6(1):147–166	NUM
ejpam-4548	625	7	,	,	PUNCT
ejpam-4548	625	8	2014	2014	NUM
ejpam-4548	625	9	.	.	PUNCT
ejpam-4548	626	1	[	[	X
ejpam-4548	626	2	3	3	X
ejpam-4548	626	3	]	]	PUNCT
ejpam-4548	626	4	n	n	X
ejpam-4548	626	5	ajmal	ajmal	ADJ
ejpam-4548	626	6	and	and	CCONJ
ejpam-4548	626	7	i	i	PROPN
ejpam-4548	626	8	jahan	jahan	PROPN
ejpam-4548	626	9	.	.	PUNCT
ejpam-4548	627	1	nilpotency	nilpotency	NOUN
ejpam-4548	627	2	and	and	CCONJ
ejpam-4548	627	3	theory	theory	NOUN
ejpam-4548	627	4	of	of	ADP
ejpam-4548	627	5	l	l	NOUN
ejpam-4548	627	6	subgroups	subgroup	NOUN
ejpam-4548	627	7	of	of	ADP
ejpam-4548	627	8	an	an	DET
ejpam-4548	627	9	l	l	NOUN
ejpam-4548	627	10	-	-	NOUN
ejpam-4548	627	11	group	group	NOUN
ejpam-4548	627	12	.	.	PUNCT
ejpam-4548	628	1	fuzzy	fuzzy	ADJ
ejpam-4548	628	2	information	information	NOUN
ejpam-4548	628	3	and	and	CCONJ
ejpam-4548	628	4	engineering	engineering	NOUN
ejpam-4548	628	5	,	,	PUNCT
ejpam-4548	628	6	6(1):1–17	6(1):1–17	PROPN
ejpam-4548	628	7	,	,	PUNCT
ejpam-4548	628	8	2014	2014	NUM
ejpam-4548	628	9	.	.	PUNCT
ejpam-4548	629	1	[	[	X
ejpam-4548	629	2	4	4	X
ejpam-4548	629	3	]	]	PUNCT
ejpam-4548	629	4	n	n	X
ejpam-4548	629	5	ajmal	ajmal	ADJ
ejpam-4548	629	6	and	and	CCONJ
ejpam-4548	629	7	i	i	PROPN
ejpam-4548	629	8	jahan	jahan	PROPN
ejpam-4548	629	9	.	.	PUNCT
ejpam-4548	630	1	normal	normal	ADJ
ejpam-4548	630	2	closure	closure	NOUN
ejpam-4548	630	3	of	of	ADP
ejpam-4548	630	4	an	an	DET
ejpam-4548	630	5	l	l	NOUN
ejpam-4548	630	6	-	-	NOUN
ejpam-4548	630	7	subgroup	subgroup	NOUN
ejpam-4548	630	8	of	of	ADP
ejpam-4548	630	9	an	an	DET
ejpam-4548	630	10	l	l	NOUN
ejpam-4548	630	11	-	-	NOUN
ejpam-4548	630	12	group	group	NOUN
ejpam-4548	630	13	.	.	PUNCT
ejpam-4548	631	1	the	the	DET
ejpam-4548	631	2	journal	journal	NOUN
ejpam-4548	631	3	of	of	ADP
ejpam-4548	631	4	fuzzy	fuzzy	ADJ
ejpam-4548	631	5	mathematics	mathematic	NOUN
ejpam-4548	631	6	,	,	PUNCT
ejpam-4548	631	7	22(1):115–126	22(1):115–126	PROPN
ejpam-4548	631	8	,	,	PUNCT
ejpam-4548	631	9	2014	2014	NUM
ejpam-4548	631	10	.	.	PUNCT
ejpam-4548	632	1	[	[	X
ejpam-4548	632	2	5	5	NUM
ejpam-4548	632	3	]	]	PUNCT
ejpam-4548	632	4	n	n	X
ejpam-4548	632	5	ajmal	ajmal	ADJ
ejpam-4548	632	6	and	and	CCONJ
ejpam-4548	632	7	i	i	PROPN
ejpam-4548	632	8	jahan	jahan	PROPN
ejpam-4548	632	9	.	.	PUNCT
ejpam-4548	633	1	generated	generate	VERB
ejpam-4548	633	2	l	l	NOUN
ejpam-4548	633	3	-	-	NOUN
ejpam-4548	633	4	subgroup	subgroup	NOUN
ejpam-4548	633	5	of	of	ADP
ejpam-4548	633	6	an	an	DET
ejpam-4548	633	7	l	l	NOUN
ejpam-4548	633	8	-	-	NOUN
ejpam-4548	633	9	group	group	NOUN
ejpam-4548	633	10	.	.	PUNCT
ejpam-4548	634	1	iranian	iranian	PROPN
ejpam-4548	634	2	journal	journal	PROPN
ejpam-4548	634	3	of	of	ADP
ejpam-4548	634	4	fuzzy	fuzzy	ADJ
ejpam-4548	634	5	systems	system	NOUN
ejpam-4548	634	6	,	,	PUNCT
ejpam-4548	634	7	12(1):129–136	12(1):129–136	NUM
ejpam-4548	634	8	,	,	PUNCT
ejpam-4548	634	9	2015	2015	NUM
ejpam-4548	634	10	.	.	PUNCT
ejpam-4548	635	1	[	[	X
ejpam-4548	635	2	6	6	NUM
ejpam-4548	635	3	]	]	PUNCT
ejpam-4548	635	4	n	n	X
ejpam-4548	635	5	ajmal	ajmal	ADJ
ejpam-4548	635	6	and	and	CCONJ
ejpam-4548	635	7	i	i	PROPN
ejpam-4548	635	8	jahan	jahan	PROPN
ejpam-4548	635	9	.	.	PUNCT
ejpam-4548	636	1	solvable	solvable	ADJ
ejpam-4548	636	2	l	l	NOUN
ejpam-4548	636	3	-	-	NOUN
ejpam-4548	636	4	subgroup	subgroup	NOUN
ejpam-4548	636	5	of	of	ADP
ejpam-4548	636	6	an	an	DET
ejpam-4548	636	7	l	l	NOUN
ejpam-4548	636	8	-	-	NOUN
ejpam-4548	636	9	group	group	NOUN
ejpam-4548	636	10	.	.	PUNCT
ejpam-4548	637	1	iranian	iranian	PROPN
ejpam-4548	637	2	journal	journal	PROPN
ejpam-4548	637	3	of	of	ADP
ejpam-4548	637	4	fuzzy	fuzzy	ADJ
ejpam-4548	637	5	systems	system	NOUN
ejpam-4548	637	6	,	,	PUNCT
ejpam-4548	637	7	12(3):151–161	12(3):151–161	NOUN
ejpam-4548	637	8	,	,	PUNCT
ejpam-4548	637	9	2015	2015	NUM
ejpam-4548	637	10	.	.	PUNCT
ejpam-4548	638	1	[	[	X
ejpam-4548	638	2	7	7	X
ejpam-4548	638	3	]	]	X
ejpam-4548	638	4	n	n	X
ejpam-4548	638	5	ajmal	ajmal	ADJ
ejpam-4548	638	6	,	,	PUNCT
ejpam-4548	638	7	i	i	PROPN
ejpam-4548	638	8	jahan	jahan	PROPN
ejpam-4548	638	9	,	,	PUNCT
ejpam-4548	638	10	and	and	CCONJ
ejpam-4548	638	11	b	b	DET
ejpam-4548	638	12	davvaz	davvaz	NOUN
ejpam-4548	638	13	.	.	PUNCT
ejpam-4548	639	1	nilpotent	nilpotent	PROPN
ejpam-4548	639	2	l	l	NOUN
ejpam-4548	639	3	-	-	PUNCT
ejpam-4548	639	4	subgroups	subgroup	NOUN
ejpam-4548	639	5	and	and	CCONJ
ejpam-4548	639	6	the	the	DET
ejpam-4548	639	7	set	set	ADJ
ejpam-4548	639	8	product	product	NOUN
ejpam-4548	639	9	of	of	ADP
ejpam-4548	639	10	l	l	NOUN
ejpam-4548	639	11	-	-	NOUN
ejpam-4548	639	12	subsets	subset	NOUN
ejpam-4548	639	13	.	.	PUNCT
ejpam-4548	640	1	european	european	ADJ
ejpam-4548	640	2	journal	journal	PROPN
ejpam-4548	640	3	of	of	ADP
ejpam-4548	640	4	pure	pure	ADJ
ejpam-4548	640	5	and	and	CCONJ
ejpam-4548	640	6	applied	applied	ADJ
ejpam-4548	640	7	mathematics	mathematic	NOUN
ejpam-4548	640	8	,	,	PUNCT
ejpam-4548	640	9	10(2):255–271	10(2):255–271	PROPN
ejpam-4548	640	10	,	,	PUNCT
ejpam-4548	640	11	2017	2017	NUM
ejpam-4548	640	12	.	.	PUNCT
ejpam-4548	641	1	references	reference	NOUN
ejpam-4548	641	2	2115	2115	NUM
ejpam-4548	642	1	[	[	X
ejpam-4548	642	2	8	8	NUM
ejpam-4548	642	3	]	]	X
ejpam-4548	642	4	j	j	PROPN
ejpam-4548	642	5	a	a	DET
ejpam-4548	642	6	goguen	goguen	NOUN
ejpam-4548	642	7	.	.	PUNCT
ejpam-4548	643	1	l	l	ADJ
ejpam-4548	643	2	-	-	ADJ
ejpam-4548	643	3	fuzzy	fuzzy	ADJ
ejpam-4548	643	4	sets	set	NOUN
ejpam-4548	643	5	.	.	PUNCT
ejpam-4548	644	1	j.	j.	PROPN
ejpam-4548	644	2	math	math	PROPN
ejpam-4548	644	3	.	.	PUNCT
ejpam-4548	645	1	anal	anal	PROPN
ejpam-4548	645	2	.	.	PUNCT
ejpam-4548	645	3	appl	appl	PROPN
ejpam-4548	645	4	.	.	PROPN
ejpam-4548	645	5	,	,	PUNCT
ejpam-4548	645	6	18(1):145–17	18(1):145–17	NUM
ejpam-4548	645	7	,	,	PUNCT
ejpam-4548	645	8	1967	1967	NUM
ejpam-4548	645	9	.	.	PUNCT
ejpam-4548	646	1	[	[	X
ejpam-4548	646	2	9	9	NUM
ejpam-4548	646	3	]	]	PUNCT
ejpam-4548	646	4	t	t	PROPN
ejpam-4548	646	5	head	head	NOUN
ejpam-4548	646	6	.	.	PUNCT
ejpam-4548	647	1	a	a	DET
ejpam-4548	647	2	metatheorem	metatheorem	NOUN
ejpam-4548	647	3	for	for	ADP
ejpam-4548	647	4	deriving	derive	VERB
ejpam-4548	647	5	fuzzy	fuzzy	ADJ
ejpam-4548	647	6	theorems	theorem	NOUN
ejpam-4548	647	7	from	from	ADP
ejpam-4548	647	8	crisp	crisp	ADJ
ejpam-4548	647	9	versions	version	NOUN
ejpam-4548	647	10	.	.	PUNCT
ejpam-4548	648	1	fuzzy	fuzzy	ADJ
ejpam-4548	648	2	sets	set	NOUN
ejpam-4548	648	3	and	and	CCONJ
ejpam-4548	648	4	system	system	NOUN
ejpam-4548	648	5	,	,	PUNCT
ejpam-4548	648	6	73(3):349–358	73(3):349–358	PROPN
ejpam-4548	648	7	,	,	PUNCT
ejpam-4548	648	8	1995	1995	NUM
ejpam-4548	648	9	.	.	PUNCT
ejpam-4548	649	1	[	[	X
ejpam-4548	649	2	10	10	NUM
ejpam-4548	649	3	]	]	X
ejpam-4548	649	4	i	i	PRON
ejpam-4548	649	5	jahan	jahan	PROPN
ejpam-4548	649	6	,	,	PUNCT
ejpam-4548	649	7	b	b	PROPN
ejpam-4548	649	8	davvaz	davvaz	NOUN
ejpam-4548	649	9	,	,	PUNCT
ejpam-4548	649	10	and	and	CCONJ
ejpam-4548	649	11	n	n	ADV
ejpam-4548	649	12	ajmal	ajmal	ADJ
ejpam-4548	649	13	.	.	PUNCT
ejpam-4548	650	1	nilpotent	nilpotent	ADJ
ejpam-4548	650	2	l	l	NOUN
ejpam-4548	650	3	-	-	NOUN
ejpam-4548	650	4	subgroups	subgroup	NOUN
ejpam-4548	650	5	satisfy	satisfy	VERB
ejpam-4548	650	6	the	the	DET
ejpam-4548	650	7	normalizer	normalizer	PROPN
ejpam-4548	650	8	condition	condition	NOUN
ejpam-4548	650	9	.	.	PUNCT
ejpam-4548	651	1	journal	journal	NOUN
ejpam-4548	651	2	of	of	ADP
ejpam-4548	651	3	intelligent	intelligent	ADJ
ejpam-4548	651	4	and	and	CCONJ
ejpam-4548	651	5	fuzzy	fuzzy	ADJ
ejpam-4548	651	6	systems	system	NOUN
ejpam-4548	651	7	,	,	PUNCT
ejpam-4548	651	8	33(3):1841–1854	33(3):1841–1854	NUM
ejpam-4548	651	9	,	,	PUNCT
ejpam-4548	651	10	2017	2017	NUM
ejpam-4548	651	11	.	.	PUNCT
ejpam-4548	652	1	[	[	X
ejpam-4548	652	2	11	11	NUM
ejpam-4548	652	3	]	]	X
ejpam-4548	652	4	j	j	PROPN
ejpam-4548	652	5	c	c	PROPN
ejpam-4548	652	6	lennox	lennox	PROPN
ejpam-4548	652	7	and	and	CCONJ
ejpam-4548	652	8	s	s	PROPN
ejpam-4548	652	9	stonehewer	stonehewer	NOUN
ejpam-4548	652	10	.	.	PUNCT
ejpam-4548	653	1	subnormal	subnormal	ADJ
ejpam-4548	653	2	subgroups	subgroup	NOUN
ejpam-4548	653	3	of	of	ADP
ejpam-4548	653	4	groups	group	NOUN
ejpam-4548	653	5	.	.	PUNCT
ejpam-4548	654	1	oxford	oxford	PROPN
ejpam-4548	654	2	university	university	PROPN
ejpam-4548	654	3	press	press	NOUN
ejpam-4548	654	4	,	,	PUNCT
ejpam-4548	654	5	new	new	PROPN
ejpam-4548	654	6	york	york	PROPN
ejpam-4548	654	7	,	,	PUNCT
ejpam-4548	654	8	1987	1987	NUM
ejpam-4548	654	9	.	.	PUNCT
ejpam-4548	655	1	[	[	X
ejpam-4548	655	2	12	12	NUM
ejpam-4548	655	3	]	]	PUNCT
ejpam-4548	655	4	a	a	DET
ejpam-4548	655	5	razzaque	razzaque	NOUN
ejpam-4548	655	6	and	and	CCONJ
ejpam-4548	655	7	a	a	DET
ejpam-4548	655	8	razaq	razaq	NOUN
ejpam-4548	655	9	.	.	PUNCT
ejpam-4548	656	1	on	on	ADP
ejpam-4548	656	2	q	q	ADJ
ejpam-4548	656	3	-	-	PUNCT
ejpam-4548	656	4	rung	rung	ADJ
ejpam-4548	656	5	orthopair	orthopair	ADJ
ejpam-4548	656	6	fuzzy	fuzzy	ADJ
ejpam-4548	656	7	subgroups	subgroup	NOUN
ejpam-4548	656	8	.	.	PUNCT
ejpam-4548	657	1	journal	journal	NOUN
ejpam-4548	657	2	of	of	ADP
ejpam-4548	657	3	function	function	NOUN
ejpam-4548	657	4	spaces	space	NOUN
ejpam-4548	657	5	,	,	PUNCT
ejpam-4548	657	6	2022(1):1–9	2022(1):1–9	NOUN
ejpam-4548	657	7	,	,	PUNCT
ejpam-4548	657	8	2022	2022	NUM
ejpam-4548	657	9	.	.	PUNCT
ejpam-4548	658	1	[	[	X
ejpam-4548	658	2	13	13	NUM
ejpam-4548	658	3	]	]	X
ejpam-4548	658	4	d	d	X
ejpam-4548	658	5	j	j	PROPN
ejpam-4548	658	6	s	s	PROPN
ejpam-4548	658	7	robinson	robinson	PROPN
ejpam-4548	658	8	.	.	PUNCT
ejpam-4548	659	1	a	a	DET
ejpam-4548	659	2	course	course	NOUN
ejpam-4548	659	3	in	in	ADP
ejpam-4548	659	4	the	the	DET
ejpam-4548	659	5	theory	theory	NOUN
ejpam-4548	659	6	of	of	ADP
ejpam-4548	659	7	groups	group	NOUN
ejpam-4548	659	8	.	.	PUNCT
ejpam-4548	660	1	springer	springer	NOUN
ejpam-4548	660	2	-	-	PUNCT
ejpam-4548	660	3	verlag	verlag	PROPN
ejpam-4548	660	4	,	,	PUNCT
ejpam-4548	660	5	new	new	PROPN
ejpam-4548	660	6	york	york	PROPN
ejpam-4548	660	7	,	,	PUNCT
ejpam-4548	660	8	1980	1980	NUM
ejpam-4548	660	9	.	.	PUNCT
ejpam-4548	661	1	[	[	X
ejpam-4548	661	2	14	14	NUM
ejpam-4548	661	3	]	]	PUNCT
ejpam-4548	661	4	a	a	DET
ejpam-4548	661	5	rosenfeld	rosenfeld	PROPN
ejpam-4548	661	6	.	.	PUNCT
ejpam-4548	662	1	fuzzy	fuzzy	ADJ
ejpam-4548	662	2	groups	group	NOUN
ejpam-4548	662	3	.	.	PUNCT
ejpam-4548	663	1	j.	j.	PROPN
ejpam-4548	663	2	math	math	PROPN
ejpam-4548	663	3	.	.	PUNCT
ejpam-4548	664	1	anal	anal	PROPN
ejpam-4548	664	2	.	.	PUNCT
ejpam-4548	665	1	appl	appl	PROPN
ejpam-4548	665	2	.	.	PROPN
ejpam-4548	666	1	,	,	PUNCT
ejpam-4548	666	2	35(3):512–517	35(3):512–517	PROPN
ejpam-4548	666	3	,	,	PUNCT
ejpam-4548	666	4	1971	1971	NUM
ejpam-4548	666	5	.	.	PUNCT
ejpam-4548	667	1	[	[	X
ejpam-4548	667	2	15	15	NUM
ejpam-4548	667	3	]	]	X
ejpam-4548	667	4	y	y	PROPN
ejpam-4548	667	5	yu	yu	PROPN
ejpam-4548	667	6	,	,	PUNCT
ejpam-4548	667	7	s	s	PROPN
ejpam-4548	667	8	c	c	PROPN
ejpam-4548	667	9	cheng	cheng	PROPN
ejpam-4548	667	10	,	,	PUNCT
ejpam-4548	667	11	and	and	CCONJ
ejpam-4548	667	12	j	j	PROPN
ejpam-4548	667	13	n	n	PRON
ejpam-4548	667	14	mordeson	mordeson	NOUN
ejpam-4548	667	15	.	.	PUNCT
ejpam-4548	668	1	elements	element	NOUN
ejpam-4548	668	2	of	of	ADP
ejpam-4548	668	3	algebra(lecture	algebra(lecture	PROPN
ejpam-4548	668	4	notes	note	NOUN
ejpam-4548	668	5	in	in	ADP
ejpam-4548	668	6	fuzzy	fuzzy	ADJ
ejpam-4548	668	7	mathematics	mathematic	NOUN
ejpam-4548	668	8	and	and	CCONJ
ejpam-4548	668	9	computer	computer	NOUN
ejpam-4548	668	10	science	science	NOUN
ejpam-4548	668	11	)	)	PUNCT
ejpam-4548	668	12	.	.	PUNCT
ejpam-4548	669	1	center	center	NOUN
ejpam-4548	669	2	for	for	ADP
ejpam-4548	669	3	research	research	NOUN
ejpam-4548	669	4	in	in	ADP
ejpam-4548	669	5	fuzzy	fuzzy	ADJ
ejpam-4548	669	6	mathematics	mathematic	NOUN
ejpam-4548	669	7	and	and	CCONJ
ejpam-4548	669	8	computer	computer	NOUN
ejpam-4548	669	9	science	science	NOUN
ejpam-4548	669	10	,	,	PUNCT
ejpam-4548	669	11	creighton	creighton	PROPN
ejpam-4548	669	12	university	university	PROPN
ejpam-4548	669	13	,	,	PUNCT
ejpam-4548	669	14	usa	usa	PROPN
ejpam-4548	669	15	,	,	PUNCT
ejpam-4548	669	16	1994	1994	NUM
ejpam-4548	669	17	.	.	PUNCT