id	sid	tid	token	lemma	pos
ap-1408	1	1	acta	acta	PROPN
ap-1408	1	2	polytechnica	polytechnica	PROPN
ap-1408	1	3	vol	vol	NOUN
ap-1408	1	4	.	.	PUNCT
ap-1408	2	1	51	51	NUM
ap-1408	2	2	no	no	INTJ
ap-1408	2	3	.	.	PUNCT
ap-1408	3	1	4/2011	4/2011	NUM
ap-1408	3	2	more	more	ADJ
ap-1408	3	3	on	on	ADP
ap-1408	3	4	pt	pt	PRON
ap-1408	3	5	-symmetry	-symmetry	NOUN
ap-1408	3	6	in	in	ADP
ap-1408	3	7	(	(	PUNCT
ap-1408	3	8	generalized	generalized	ADJ
ap-1408	3	9	)	)	PUNCT
ap-1408	3	10	effect	effect	NOUN
ap-1408	3	11	algebras	algebra	NOUN
ap-1408	3	12	and	and	CCONJ
ap-1408	3	13	partial	partial	ADJ
ap-1408	3	14	groups	group	NOUN
ap-1408	3	15	j.	j.	PROPN
ap-1408	3	16	paseka	paseka	PROPN
ap-1408	3	17	,	,	PUNCT
ap-1408	3	18	j.	j.	PROPN
ap-1408	3	19	janda	janda	PROPN
ap-1408	3	20	abstract	abstract	PROPN
ap-1408	3	21	we	we	PRON
ap-1408	3	22	continue	continue	VERB
ap-1408	3	23	in	in	ADP
ap-1408	3	24	the	the	DET
ap-1408	3	25	direction	direction	NOUN
ap-1408	3	26	of	of	ADP
ap-1408	3	27	our	our	PRON
ap-1408	3	28	paper	paper	NOUN
ap-1408	3	29	on	on	ADP
ap-1408	3	30	pt	pt	PROPN
ap-1408	3	31	-symmetry	-symmetry	PROPN
ap-1408	3	32	in	in	ADP
ap-1408	3	33	(	(	PUNCT
ap-1408	3	34	generalized	generalized	ADJ
ap-1408	3	35	)	)	PUNCT
ap-1408	3	36	effect	effect	NOUN
ap-1408	3	37	algebras	algebra	NOUN
ap-1408	3	38	and	and	CCONJ
ap-1408	3	39	partial	partial	ADJ
ap-1408	3	40	groups	group	NOUN
ap-1408	3	41	.	.	PUNCT
ap-1408	4	1	namely	namely	ADV
ap-1408	4	2	we	we	PRON
ap-1408	4	3	extend	extend	VERB
ap-1408	4	4	our	our	PRON
ap-1408	4	5	considerations	consideration	NOUN
ap-1408	4	6	to	to	ADP
ap-1408	4	7	the	the	DET
ap-1408	4	8	setting	setting	NOUN
ap-1408	4	9	of	of	ADP
ap-1408	4	10	weakly	weakly	ADJ
ap-1408	4	11	ordered	order	VERB
ap-1408	4	12	partial	partial	ADJ
ap-1408	4	13	groups	group	NOUN
ap-1408	4	14	.	.	PUNCT
ap-1408	5	1	in	in	ADP
ap-1408	5	2	this	this	DET
ap-1408	5	3	setting	setting	NOUN
ap-1408	5	4	,	,	PUNCT
ap-1408	5	5	any	any	DET
ap-1408	5	6	operator	operator	NOUN
ap-1408	5	7	weakly	weakly	ADV
ap-1408	5	8	ordered	order	VERB
ap-1408	5	9	partial	partial	ADJ
ap-1408	5	10	group	group	NOUN
ap-1408	5	11	is	be	AUX
ap-1408	5	12	a	a	DET
ap-1408	5	13	pasting	pasting	NOUN
ap-1408	5	14	of	of	ADP
ap-1408	5	15	its	its	PRON
ap-1408	5	16	partially	partially	ADV
ap-1408	5	17	ordered	order	VERB
ap-1408	5	18	commutative	commutative	ADJ
ap-1408	5	19	subgroups	subgroup	NOUN
ap-1408	5	20	of	of	ADP
ap-1408	5	21	linear	linear	PROPN
ap-1408	5	22	operators	operator	NOUN
ap-1408	5	23	with	with	ADP
ap-1408	5	24	a	a	DET
ap-1408	5	25	fixed	fix	VERB
ap-1408	5	26	dense	dense	ADJ
ap-1408	5	27	domain	domain	NOUN
ap-1408	5	28	over	over	ADP
ap-1408	5	29	bounded	bounded	ADJ
ap-1408	5	30	operators	operator	NOUN
ap-1408	5	31	.	.	PUNCT
ap-1408	6	1	moreover	moreover	ADV
ap-1408	6	2	,	,	PUNCT
ap-1408	6	3	applications	application	NOUN
ap-1408	6	4	of	of	ADP
ap-1408	6	5	our	our	PRON
ap-1408	6	6	approach	approach	NOUN
ap-1408	6	7	for	for	ADP
ap-1408	6	8	generalized	generalized	ADJ
ap-1408	6	9	effect	effect	NOUN
ap-1408	6	10	algebras	algebra	NOUN
ap-1408	6	11	are	be	AUX
ap-1408	6	12	mentioned	mention	VERB
ap-1408	6	13	.	.	PUNCT
ap-1408	7	1	keywords	keyword	NOUN
ap-1408	7	2	:	:	PUNCT
ap-1408	7	3	(	(	PUNCT
ap-1408	7	4	generalized	generalized	ADJ
ap-1408	7	5	)	)	PUNCT
ap-1408	7	6	effect	effect	NOUN
ap-1408	7	7	algebra	algebra	NOUN
ap-1408	7	8	,	,	PUNCT
ap-1408	7	9	partially	partially	ADV
ap-1408	7	10	ordered	order	VERB
ap-1408	7	11	commutative	commutative	ADJ
ap-1408	7	12	group	group	NOUN
ap-1408	7	13	,	,	PUNCT
ap-1408	7	14	weakly	weakly	ADV
ap-1408	7	15	ordered	order	VERB
ap-1408	7	16	partial	partial	ADJ
ap-1408	7	17	group	group	NOUN
ap-1408	7	18	,	,	PUNCT
ap-1408	7	19	hilbert	hilbert	NOUN
ap-1408	7	20	space	space	NOUN
ap-1408	7	21	,	,	PUNCT
ap-1408	7	22	(	(	PUNCT
ap-1408	7	23	unbounded	unbounded	ADJ
ap-1408	7	24	)	)	PUNCT
ap-1408	7	25	linear	linear	PROPN
ap-1408	7	26	operators	operator	NOUN
ap-1408	7	27	,	,	PUNCT
ap-1408	7	28	pt	pt	PROPN
ap-1408	7	29	-symmetry	-symmetry	NOUN
ap-1408	7	30	,	,	PUNCT
ap-1408	7	31	pseudo	pseudo	NOUN
ap-1408	7	32	-	-	ADJ
ap-1408	7	33	hermitian	hermitian	ADJ
ap-1408	7	34	quantum	quantum	NOUN
ap-1408	7	35	mechanics	mechanic	NOUN
ap-1408	7	36	.	.	PUNCT
ap-1408	8	1	1	1	NUM
ap-1408	8	2	introduction	introduction	NOUN
ap-1408	8	3	it	it	PRON
ap-1408	8	4	is	be	AUX
ap-1408	8	5	a	a	DET
ap-1408	8	6	well	well	ADV
ap-1408	8	7	known	know	VERB
ap-1408	8	8	fact	fact	NOUN
ap-1408	8	9	that	that	SCONJ
ap-1408	8	10	unbounded	unbounded	ADJ
ap-1408	8	11	linear	linear	PROPN
ap-1408	8	12	operators	operator	NOUN
ap-1408	8	13	play	play	VERB
ap-1408	8	14	the	the	DET
ap-1408	8	15	role	role	NOUN
ap-1408	8	16	of	of	ADP
ap-1408	8	17	the	the	DET
ap-1408	8	18	observable	observable	ADJ
ap-1408	8	19	in	in	ADP
ap-1408	8	20	the	the	DET
ap-1408	8	21	mathematical	mathematical	ADJ
ap-1408	8	22	formulation	formulation	NOUN
ap-1408	8	23	of	of	ADP
ap-1408	8	24	quantum	quantum	ADJ
ap-1408	8	25	mechanics	mechanic	NOUN
ap-1408	8	26	.	.	PUNCT
ap-1408	9	1	examples	example	NOUN
ap-1408	9	2	of	of	ADP
ap-1408	9	3	such	such	ADJ
ap-1408	9	4	observables	observable	NOUN
ap-1408	9	5	corresponding	correspond	VERB
ap-1408	9	6	to	to	ADP
ap-1408	9	7	the	the	DET
ap-1408	9	8	momentum	momentum	NOUN
ap-1408	9	9	and	and	CCONJ
ap-1408	9	10	position	position	NOUN
ap-1408	9	11	observables	observable	NOUN
ap-1408	9	12	,	,	PUNCT
ap-1408	9	13	respectively	respectively	ADV
ap-1408	9	14	,	,	PUNCT
ap-1408	9	15	are	be	AUX
ap-1408	9	16	the	the	DET
ap-1408	9	17	following	follow	VERB
ap-1408	9	18	self	self	NOUN
ap-1408	9	19	-	-	PUNCT
ap-1408	9	20	adjoint	adjoint	NOUN
ap-1408	9	21	unbounded	unbounded	ADJ
ap-1408	9	22	linear	linear	PROPN
ap-1408	9	23	operators	operator	NOUN
ap-1408	9	24	on	on	ADP
ap-1408	9	25	the	the	DET
ap-1408	9	26	hilbert	hilbert	PROPN
ap-1408	9	27	space	space	NOUN
ap-1408	9	28	l2(r	l2(r	PROPN
ap-1408	9	29	):	):	PUNCT
ap-1408	9	30	(	(	PUNCT
ap-1408	9	31	i	i	NOUN
ap-1408	9	32	)	)	PUNCT
ap-1408	9	33	the	the	DET
ap-1408	9	34	differential	differential	ADJ
ap-1408	9	35	operator	operator	NOUN
ap-1408	9	36	a	a	PRON
ap-1408	9	37	defined	define	VERB
ap-1408	9	38	by	by	ADP
ap-1408	9	39	(	(	PUNCT
ap-1408	9	40	af)(x	af)(x	PROPN
ap-1408	9	41	)	)	PUNCT
ap-1408	10	1	=	=	PUNCT
ap-1408	11	1	i	i	PRON
ap-1408	11	2	d	d	X
ap-1408	11	3	dx	dx	PROPN
ap-1408	11	4	f(x	f(x	PROPN
ap-1408	11	5	)	)	PUNCT
ap-1408	11	6	where	where	SCONJ
ap-1408	11	7	i	i	PRON
ap-1408	11	8	is	be	AUX
ap-1408	11	9	the	the	DET
ap-1408	11	10	imaginary	imaginary	ADJ
ap-1408	11	11	unit	unit	NOUN
ap-1408	11	12	and	and	CCONJ
ap-1408	11	13	f	f	PROPN
ap-1408	11	14	is	be	AUX
ap-1408	11	15	a	a	DET
ap-1408	11	16	differentiable	differentiable	ADJ
ap-1408	11	17	function	function	NOUN
ap-1408	11	18	with	with	ADP
ap-1408	11	19	compact	compact	ADJ
ap-1408	11	20	support	support	NOUN
ap-1408	11	21	.	.	PUNCT
ap-1408	12	1	then	then	ADV
ap-1408	12	2	d(a	d(a	PROPN
ap-1408	12	3	)	)	PUNCT
ap-1408	12	4	�	�	PROPN
ap-1408	12	5	=	=	SYM
ap-1408	12	6	l2(r	l2(r	PROPN
ap-1408	12	7	)	)	PUNCT
ap-1408	12	8	,	,	PUNCT
ap-1408	12	9	since	since	SCONJ
ap-1408	12	10	otherwise	otherwise	ADV
ap-1408	12	11	the	the	DET
ap-1408	12	12	derivative	derivative	ADJ
ap-1408	12	13	need	need	NOUN
ap-1408	12	14	not	not	PART
ap-1408	12	15	exist	exist	VERB
ap-1408	12	16	.	.	PUNCT
ap-1408	13	1	(	(	PUNCT
ap-1408	13	2	ii	ii	NOUN
ap-1408	13	3	)	)	PUNCT
ap-1408	13	4	(	(	PUNCT
ap-1408	13	5	bf)(x	bf)(x	PROPN
ap-1408	13	6	)	)	PUNCT
ap-1408	13	7	=	=	PUNCT
ap-1408	14	1	xf(x	xf(x	NUM
ap-1408	14	2	)	)	PUNCT
ap-1408	14	3	,	,	PUNCT
ap-1408	14	4	multiplication	multiplication	NOUN
ap-1408	14	5	by	by	ADP
ap-1408	14	6	x	x	PUNCT
ap-1408	14	7	and	and	CCONJ
ap-1408	14	8	again	again	ADV
ap-1408	14	9	d(b	d(b	PROPN
ap-1408	14	10	)	)	PUNCT
ap-1408	14	11	�	�	PROPN
ap-1408	14	12	=	=	SYM
ap-1408	14	13	l2(r	l2(r	PROPN
ap-1408	14	14	)	)	PUNCT
ap-1408	14	15	,	,	PUNCT
ap-1408	14	16	since	since	SCONJ
ap-1408	14	17	otherwise	otherwise	ADV
ap-1408	14	18	xf(x	xf(x	PRON
ap-1408	14	19	)	)	PUNCT
ap-1408	14	20	need	need	AUX
ap-1408	14	21	not	not	PART
ap-1408	14	22	be	be	AUX
ap-1408	14	23	square	square	ADJ
ap-1408	14	24	integrable	integrable	ADJ
ap-1408	14	25	.	.	PUNCT
ap-1408	15	1	note	note	VERB
ap-1408	15	2	that	that	SCONJ
ap-1408	15	3	in	in	ADP
ap-1408	15	4	both	both	DET
ap-1408	15	5	cases	case	NOUN
ap-1408	15	6	the	the	DET
ap-1408	15	7	possible	possible	ADJ
ap-1408	15	8	domains	domain	NOUN
ap-1408	15	9	are	be	AUX
ap-1408	15	10	dense	dense	ADJ
ap-1408	15	11	sub	sub	NOUN
ap-1408	15	12	-	-	NOUN
ap-1408	15	13	spaces	space	NOUN
ap-1408	15	14	of	of	ADP
ap-1408	15	15	l2(r	l2(r	NOUN
ap-1408	15	16	)	)	PUNCT
ap-1408	15	17	,	,	PUNCT
ap-1408	15	18	i.e.	i.e.	X
ap-1408	15	19	,	,	PUNCT
ap-1408	15	20	d(a	d(a	PROPN
ap-1408	15	21	)	)	PUNCT
ap-1408	15	22	=	=	SYM
ap-1408	15	23	d(b	d(b	X
ap-1408	15	24	)	)	PUNCT
ap-1408	16	1	=	=	SYM
ap-1408	16	2	l2(r	l2(r	NOUN
ap-1408	16	3	)	)	PUNCT
ap-1408	16	4	.	.	PUNCT
ap-1408	17	1	the	the	DET
ap-1408	17	2	same	same	ADJ
ap-1408	17	3	is	be	AUX
ap-1408	17	4	true	true	ADJ
ap-1408	17	5	in	in	ADP
ap-1408	17	6	general	general	ADJ
ap-1408	17	7	,	,	PUNCT
ap-1408	17	8	since	since	SCONJ
ap-1408	17	9	for	for	ADP
ap-1408	17	10	any	any	DET
ap-1408	17	11	unbounded	unbounded	ADJ
ap-1408	17	12	linear	linear	NOUN
ap-1408	17	13	operator	operator	NOUN
ap-1408	17	14	there	there	PRON
ap-1408	17	15	is	be	VERB
ap-1408	17	16	no	no	DET
ap-1408	17	17	standard	standard	ADJ
ap-1408	17	18	way	way	NOUN
ap-1408	17	19	to	to	PART
ap-1408	17	20	extend	extend	VERB
ap-1408	17	21	it	it	PRON
ap-1408	17	22	to	to	ADP
ap-1408	17	23	the	the	DET
ap-1408	17	24	whole	whole	ADJ
ap-1408	17	25	space	space	NOUN
ap-1408	17	26	h.	h.	NOUN
ap-1408	17	27	by	by	ADP
ap-1408	17	28	the	the	DET
ap-1408	17	29	hellingertoeplitz	hellingertoeplitz	PROPN
ap-1408	17	30	theorem	theorem	PROPN
ap-1408	17	31	,	,	PUNCT
ap-1408	17	32	every	every	DET
ap-1408	17	33	symmetric	symmetric	ADJ
ap-1408	17	34	operator	operator	NOUN
ap-1408	17	35	a	a	PRON
ap-1408	17	36	with	with	ADP
ap-1408	17	37	d(a	d(a	PROPN
ap-1408	17	38	)	)	PUNCT
ap-1408	18	1	=	=	SYM
ap-1408	18	2	h	h	NOUN
ap-1408	18	3	is	be	AUX
ap-1408	18	4	bounded	bound	VERB
ap-1408	18	5	.	.	PUNCT
ap-1408	19	1	an	an	DET
ap-1408	19	2	important	important	ADJ
ap-1408	19	3	attempt	attempt	NOUN
ap-1408	19	4	at	at	ADP
ap-1408	19	5	an	an	DET
ap-1408	19	6	alternative	alternative	ADJ
ap-1408	19	7	formulation	formulation	NOUN
ap-1408	19	8	of	of	ADP
ap-1408	19	9	quantum	quantum	ADJ
ap-1408	19	10	mechanics	mechanic	NOUN
ap-1408	19	11	started	start	VERB
ap-1408	19	12	in	in	ADP
ap-1408	19	13	the	the	DET
ap-1408	19	14	seminal	seminal	ADJ
ap-1408	19	15	paper	paper	NOUN
ap-1408	19	16	[	[	X
ap-1408	19	17	1	1	X
ap-1408	19	18	]	]	PUNCT
ap-1408	19	19	by	by	ADP
ap-1408	19	20	bender	bender	NOUN
ap-1408	19	21	and	and	CCONJ
ap-1408	19	22	boettcher	boettcher	NOUN
ap-1408	19	23	in	in	ADP
ap-1408	19	24	1998	1998	NUM
ap-1408	19	25	.	.	PUNCT
ap-1408	20	1	bender	bender	NOUN
ap-1408	20	2	and	and	CCONJ
ap-1408	20	3	others	other	NOUN
ap-1408	20	4	adopted	adopt	VERB
ap-1408	20	5	all	all	DET
ap-1408	20	6	the	the	DET
ap-1408	20	7	axioms	axiom	NOUN
ap-1408	20	8	of	of	ADP
ap-1408	20	9	quantum	quantum	ADJ
ap-1408	20	10	mechanics	mechanic	NOUN
ap-1408	20	11	except	except	SCONJ
ap-1408	20	12	the	the	DET
ap-1408	20	13	axiom	axiom	NOUN
ap-1408	20	14	that	that	PRON
ap-1408	20	15	restricted	restrict	VERB
ap-1408	20	16	the	the	DET
ap-1408	20	17	hamiltonian	hamiltonian	NOUN
ap-1408	20	18	to	to	PART
ap-1408	20	19	be	be	AUX
ap-1408	20	20	hermitian	hermitian	ADJ
ap-1408	20	21	.	.	PUNCT
ap-1408	21	1	they	they	PRON
ap-1408	21	2	replaced	replace	VERB
ap-1408	21	3	this	this	DET
ap-1408	21	4	condition	condition	NOUN
ap-1408	21	5	with	with	ADP
ap-1408	21	6	the	the	DET
ap-1408	21	7	requirement	requirement	NOUN
ap-1408	21	8	that	that	SCONJ
ap-1408	21	9	the	the	DET
ap-1408	21	10	hamiltonian	hamiltonian	NOUN
ap-1408	21	11	must	must	AUX
ap-1408	21	12	have	have	VERB
ap-1408	21	13	an	an	DET
ap-1408	21	14	exact	exact	ADJ
ap-1408	21	15	pt	pt	X
ap-1408	21	16	-symmetry	-symmetry	NOUN
ap-1408	21	17	.	.	PUNCT
ap-1408	22	1	later	later	ADV
ap-1408	22	2	,	,	PUNCT
ap-1408	22	3	a.	a.	NOUN
ap-1408	22	4	mostafazadeh	mostafazadeh	NOUN
ap-1408	23	1	[	[	X
ap-1408	23	2	6	6	NUM
ap-1408	23	3	]	]	PUNCT
ap-1408	23	4	showed	show	VERB
ap-1408	23	5	that	that	SCONJ
ap-1408	23	6	pt	pt	NOUN
ap-1408	23	7	-symmetric	-symmetric	ADJ
ap-1408	23	8	quantum	quantum	NOUN
ap-1408	23	9	mechanics	mechanic	NOUN
ap-1408	23	10	is	be	AUX
ap-1408	23	11	an	an	DET
ap-1408	23	12	example	example	NOUN
ap-1408	23	13	of	of	ADP
ap-1408	23	14	a	a	DET
ap-1408	23	15	more	more	ADV
ap-1408	23	16	general	general	ADJ
ap-1408	23	17	class	class	NOUN
ap-1408	23	18	of	of	ADP
ap-1408	23	19	theories	theory	NOUN
ap-1408	23	20	,	,	PUNCT
ap-1408	23	21	called	call	VERB
ap-1408	23	22	pseudo	pseudo	NOUN
ap-1408	23	23	-	-	ADJ
ap-1408	23	24	hermitian	hermitian	ADJ
ap-1408	23	25	quantum	quantum	NOUN
ap-1408	23	26	mechanics	mechanic	NOUN
ap-1408	23	27	.	.	PUNCT
ap-1408	24	1	in	in	ADP
ap-1408	24	2	[	[	X
ap-1408	24	3	4	4	NUM
ap-1408	24	4	]	]	X
ap-1408	24	5	foulis	foulis	PROPN
ap-1408	24	6	and	and	CCONJ
ap-1408	24	7	bennett	bennett	PROPN
ap-1408	24	8	introduced	introduce	VERB
ap-1408	24	9	the	the	DET
ap-1408	24	10	notion	notion	NOUN
ap-1408	24	11	of	of	ADP
ap-1408	24	12	effect	effect	NOUN
ap-1408	24	13	algebras	algebra	VERB
ap-1408	24	14	that	that	SCONJ
ap-1408	24	15	generalized	generalize	VERB
ap-1408	24	16	the	the	DET
ap-1408	24	17	algebraic	algebraic	ADJ
ap-1408	24	18	structure	structure	NOUN
ap-1408	24	19	of	of	ADP
ap-1408	24	20	the	the	DET
ap-1408	24	21	set	set	NOUN
ap-1408	24	22	e(h	e(h	PROPN
ap-1408	24	23	)	)	PUNCT
ap-1408	24	24	of	of	ADP
ap-1408	24	25	hilbert	hilbert	NOUN
ap-1408	24	26	space	space	NOUN
ap-1408	24	27	effects	effect	NOUN
ap-1408	24	28	.	.	PUNCT
ap-1408	25	1	in	in	ADP
ap-1408	25	2	such	such	DET
ap-1408	25	3	a	a	DET
ap-1408	25	4	case	case	NOUN
ap-1408	25	5	the	the	DET
ap-1408	25	6	set	set	NOUN
ap-1408	25	7	e(h	e(h	PROPN
ap-1408	25	8	)	)	PUNCT
ap-1408	25	9	of	of	ADP
ap-1408	25	10	effects	effect	NOUN
ap-1408	25	11	is	be	AUX
ap-1408	25	12	the	the	DET
ap-1408	25	13	set	set	NOUN
ap-1408	25	14	of	of	ADP
ap-1408	25	15	all	all	DET
ap-1408	25	16	self	self	NOUN
ap-1408	25	17	-	-	PUNCT
ap-1408	25	18	adjoint	adjoint	NOUN
ap-1408	25	19	operators	operator	NOUN
ap-1408	26	1	a	a	PRON
ap-1408	26	2	on	on	ADP
ap-1408	26	3	a	a	DET
ap-1408	26	4	hilbert	hilbert	NOUN
ap-1408	26	5	space	space	NOUN
ap-1408	26	6	h	h	NOUN
ap-1408	26	7	between	between	ADP
ap-1408	26	8	the	the	DET
ap-1408	26	9	null	null	ADJ
ap-1408	26	10	operator	operator	NOUN
ap-1408	26	11	0	0	NUM
ap-1408	26	12	and	and	CCONJ
ap-1408	26	13	the	the	DET
ap-1408	26	14	identity	identity	NOUN
ap-1408	26	15	operator	operator	NOUN
ap-1408	26	16	1	1	NUM
ap-1408	26	17	and	and	CCONJ
ap-1408	26	18	endowed	endow	VERB
ap-1408	26	19	with	with	ADP
ap-1408	26	20	the	the	DET
ap-1408	26	21	partial	partial	ADJ
ap-1408	26	22	operation	operation	NOUN
ap-1408	26	23	+	+	CCONJ
ap-1408	26	24	defined	define	VERB
ap-1408	26	25	iff	iff	PROPN
ap-1408	26	26	a	a	DET
ap-1408	26	27	+	+	NOUN
ap-1408	26	28	b	b	NOUN
ap-1408	26	29	is	be	AUX
ap-1408	26	30	in	in	ADP
ap-1408	26	31	e(h	e(h	PROPN
ap-1408	26	32	)	)	PUNCT
ap-1408	26	33	,	,	PUNCT
ap-1408	26	34	where	where	SCONJ
ap-1408	26	35	+	+	PRON
ap-1408	26	36	is	be	AUX
ap-1408	26	37	the	the	DET
ap-1408	26	38	usual	usual	ADJ
ap-1408	26	39	operator	operator	NOUN
ap-1408	26	40	sum	sum	NOUN
ap-1408	26	41	.	.	PUNCT
ap-1408	27	1	recently	recently	ADV
ap-1408	27	2	,	,	PUNCT
ap-1408	27	3	m.	m.	NOUN
ap-1408	27	4	polakovič	polakovič	NOUN
ap-1408	27	5	and	and	CCONJ
ap-1408	27	6	z.	z.	PROPN
ap-1408	27	7	riečanová	riečanová	PROPN
ap-1408	28	1	[	[	X
ap-1408	28	2	8	8	NUM
ap-1408	28	3	]	]	PUNCT
ap-1408	28	4	established	establish	VERB
ap-1408	28	5	new	new	ADJ
ap-1408	28	6	examples	example	NOUN
ap-1408	28	7	of	of	ADP
ap-1408	28	8	generalized	generalized	ADJ
ap-1408	28	9	effect	effect	NOUN
ap-1408	28	10	algebras	algebra	NOUN
ap-1408	28	11	of	of	ADP
ap-1408	28	12	positive	positive	ADJ
ap-1408	28	13	operators	operator	NOUN
ap-1408	28	14	on	on	ADP
ap-1408	28	15	a	a	DET
ap-1408	28	16	hilbert	hilbert	NOUN
ap-1408	28	17	space	space	NOUN
ap-1408	28	18	.	.	PUNCT
ap-1408	29	1	in	in	ADP
ap-1408	29	2	[	[	X
ap-1408	29	3	7	7	X
ap-1408	29	4	]	]	PUNCT
ap-1408	29	5	we	we	PRON
ap-1408	29	6	showed	show	VERB
ap-1408	29	7	how	how	SCONJ
ap-1408	29	8	the	the	DET
ap-1408	29	9	standard	standard	ADJ
ap-1408	29	10	effect	effect	NOUN
ap-1408	29	11	algebra	algebra	VERB
ap-1408	29	12	e(h	e(h	PROPN
ap-1408	29	13	)	)	PUNCT
ap-1408	29	14	and	and	CCONJ
ap-1408	29	15	the	the	DET
ap-1408	29	16	latter	latter	ADJ
ap-1408	29	17	generalized	generalized	ADJ
ap-1408	29	18	effect	effect	NOUN
ap-1408	29	19	algebras	algebra	NOUN
ap-1408	29	20	of	of	ADP
ap-1408	29	21	positive	positive	ADJ
ap-1408	29	22	operators	operator	NOUN
ap-1408	29	23	are	be	AUX
ap-1408	29	24	related	relate	VERB
ap-1408	29	25	to	to	ADP
ap-1408	29	26	the	the	DET
ap-1408	29	27	type	type	NOUN
ap-1408	29	28	of	of	ADP
ap-1408	29	29	attempt	attempt	NOUN
ap-1408	29	30	mentioned	mention	VERB
ap-1408	29	31	above	above	ADV
ap-1408	29	32	.	.	PUNCT
ap-1408	30	1	as	as	ADP
ap-1408	30	2	a	a	DET
ap-1408	30	3	by	by	NOUN
ap-1408	30	4	-	-	PUNCT
ap-1408	30	5	product	product	NOUN
ap-1408	30	6	,	,	PUNCT
ap-1408	30	7	we	we	PRON
ap-1408	30	8	placed	place	VERB
ap-1408	30	9	some	some	PRON
ap-1408	30	10	of	of	ADP
ap-1408	30	11	the	the	DET
ap-1408	30	12	results	result	NOUN
ap-1408	30	13	from	from	ADP
ap-1408	30	14	[	[	X
ap-1408	30	15	8	8	NUM
ap-1408	30	16	]	]	PUNCT
ap-1408	30	17	under	under	ADP
ap-1408	30	18	the	the	DET
ap-1408	30	19	common	common	ADJ
ap-1408	30	20	roof	roof	NOUN
ap-1408	30	21	of	of	ADP
ap-1408	30	22	partially	partially	ADV
ap-1408	30	23	ordered	order	VERB
ap-1408	30	24	commutative	commutative	ADJ
ap-1408	30	25	groups	group	NOUN
ap-1408	30	26	.	.	PUNCT
ap-1408	31	1	the	the	DET
ap-1408	31	2	aim	aim	NOUN
ap-1408	31	3	of	of	ADP
ap-1408	31	4	the	the	DET
ap-1408	31	5	present	present	ADJ
ap-1408	31	6	note	note	NOUN
ap-1408	31	7	is	be	AUX
ap-1408	31	8	to	to	PART
ap-1408	31	9	continue	continue	VERB
ap-1408	31	10	in	in	ADP
ap-1408	31	11	this	this	DET
ap-1408	31	12	direction	direction	NOUN
ap-1408	31	13	.	.	PUNCT
ap-1408	32	1	the	the	DET
ap-1408	32	2	paper	paper	NOUN
ap-1408	32	3	is	be	AUX
ap-1408	32	4	organized	organize	VERB
ap-1408	32	5	in	in	ADP
ap-1408	32	6	the	the	DET
ap-1408	32	7	following	following	ADJ
ap-1408	32	8	way	way	NOUN
ap-1408	32	9	.	.	PUNCT
ap-1408	33	1	in	in	ADP
ap-1408	33	2	section	section	NOUN
ap-1408	33	3	1	1	NUM
ap-1408	33	4	we	we	PRON
ap-1408	33	5	recall	recall	VERB
ap-1408	33	6	the	the	DET
ap-1408	33	7	basic	basic	ADJ
ap-1408	33	8	notions	notion	NOUN
ap-1408	33	9	concerning	concern	VERB
ap-1408	33	10	the	the	DET
ap-1408	33	11	theory	theory	NOUN
ap-1408	33	12	of	of	ADP
ap-1408	33	13	(	(	PUNCT
ap-1408	33	14	generalized	generalized	ADJ
ap-1408	33	15	)	)	PUNCT
ap-1408	33	16	effect	effect	NOUN
ap-1408	33	17	algebras	algebra	NOUN
ap-1408	33	18	and	and	CCONJ
ap-1408	33	19	partially	partially	ADV
ap-1408	33	20	ordered	order	VERB
ap-1408	33	21	commutative	commutative	ADJ
ap-1408	33	22	groups	group	NOUN
ap-1408	33	23	.	.	PUNCT
ap-1408	34	1	in	in	ADP
ap-1408	34	2	section	section	NOUN
ap-1408	34	3	2	2	NUM
ap-1408	34	4	we	we	PRON
ap-1408	34	5	show	show	VERB
ap-1408	34	6	that	that	SCONJ
ap-1408	34	7	the	the	DET
ap-1408	34	8	linear	linear	PROPN
ap-1408	34	9	operators	operator	NOUN
ap-1408	34	10	on	on	ADP
ap-1408	34	11	h	h	NOUN
ap-1408	34	12	and	and	CCONJ
ap-1408	34	13	symmetric	symmetric	ADJ
ap-1408	34	14	linear	linear	PROPN
ap-1408	34	15	operators	operator	NOUN
ap-1408	34	16	on	on	ADP
ap-1408	34	17	h	h	NOUN
ap-1408	34	18	are	be	AUX
ap-1408	34	19	equipped	equip	VERB
ap-1408	34	20	with	with	ADP
ap-1408	34	21	the	the	DET
ap-1408	34	22	structure	structure	NOUN
ap-1408	34	23	of	of	ADP
ap-1408	34	24	a	a	DET
ap-1408	34	25	weakly	weakly	ADJ
ap-1408	34	26	ordered	order	VERB
ap-1408	34	27	commutative	commutative	ADJ
ap-1408	34	28	partial	partial	ADJ
ap-1408	34	29	group	group	NOUN
ap-1408	34	30	.	.	PUNCT
ap-1408	35	1	in	in	ADP
ap-1408	35	2	section	section	NOUN
ap-1408	35	3	3	3	NUM
ap-1408	35	4	we	we	PRON
ap-1408	35	5	manifest	manifest	VERB
ap-1408	35	6	the	the	DET
ap-1408	35	7	fact	fact	NOUN
ap-1408	35	8	that	that	SCONJ
ap-1408	35	9	each	each	PRON
ap-1408	35	10	of	of	ADP
ap-1408	35	11	these	these	DET
ap-1408	35	12	operator	operator	NOUN
ap-1408	35	13	structures	structure	NOUN
ap-1408	35	14	is	be	AUX
ap-1408	35	15	a	a	DET
ap-1408	35	16	pasting	pasting	NOUN
ap-1408	35	17	of	of	ADP
ap-1408	35	18	partially	partially	ADV
ap-1408	35	19	ordered	order	VERB
ap-1408	35	20	commutative	commutative	ADJ
ap-1408	35	21	groups	group	NOUN
ap-1408	35	22	of	of	ADP
ap-1408	35	23	the	the	DET
ap-1408	35	24	respective	respective	ADJ
ap-1408	35	25	operators	operator	NOUN
ap-1408	35	26	with	with	ADP
ap-1408	35	27	a	a	DET
ap-1408	35	28	fixed	fix	VERB
ap-1408	35	29	dense	dense	ADJ
ap-1408	35	30	domain	domain	NOUN
ap-1408	35	31	.	.	PUNCT
ap-1408	36	1	in	in	ADP
ap-1408	36	2	the	the	DET
ap-1408	36	3	last	last	ADJ
ap-1408	36	4	section	section	NOUN
ap-1408	36	5	we	we	PRON
ap-1408	36	6	show	show	VERB
ap-1408	36	7	that	that	SCONJ
ap-1408	36	8	our	our	PRON
ap-1408	36	9	results	result	NOUN
ap-1408	36	10	concerning	concern	VERB
ap-1408	36	11	the	the	DET
ap-1408	36	12	application	application	NOUN
ap-1408	36	13	of	of	ADP
ap-1408	36	14	renormalization	renormalization	NOUN
ap-1408	36	15	due	due	ADP
ap-1408	36	16	to	to	ADP
ap-1408	36	17	the	the	DET
ap-1408	36	18	pt	pt	PROPN
ap-1408	36	19	-symmetry	-symmetry	NOUN
ap-1408	36	20	of	of	ADP
ap-1408	36	21	an	an	DET
ap-1408	36	22	operator	operator	NOUN
ap-1408	36	23	from	from	ADP
ap-1408	36	24	[	[	X
ap-1408	36	25	7	7	NUM
ap-1408	36	26	]	]	PUNCT
ap-1408	36	27	remain	remain	VERB
ap-1408	36	28	true	true	ADJ
ap-1408	36	29	for	for	ADP
ap-1408	36	30	weakly	weakly	ADJ
ap-1408	36	31	ordered	order	VERB
ap-1408	36	32	commutative	commutative	ADJ
ap-1408	36	33	partial	partial	ADJ
ap-1408	36	34	groups	group	NOUN
ap-1408	36	35	.	.	PUNCT
ap-1408	37	1	65	65	NUM
ap-1408	37	2	acta	acta	PROPN
ap-1408	37	3	polytechnica	polytechnica	PROPN
ap-1408	37	4	vol	vol	NOUN
ap-1408	37	5	.	.	PUNCT
ap-1408	38	1	51	51	NUM
ap-1408	38	2	no	no	INTJ
ap-1408	38	3	.	.	PUNCT
ap-1408	39	1	4/2011	4/2011	NUM
ap-1408	39	2	2	2	NUM
ap-1408	39	3	basic	basic	ADJ
ap-1408	39	4	definitions	definition	NOUN
ap-1408	39	5	and	and	CCONJ
ap-1408	39	6	some	some	DET
ap-1408	39	7	known	know	VERB
ap-1408	39	8	facts	fact	NOUN
ap-1408	39	9	the	the	DET
ap-1408	39	10	basic	basic	ADJ
ap-1408	39	11	reference	reference	NOUN
ap-1408	39	12	for	for	ADP
ap-1408	39	13	the	the	DET
ap-1408	39	14	present	present	ADJ
ap-1408	39	15	text	text	NOUN
ap-1408	39	16	is	be	AUX
ap-1408	39	17	the	the	DET
ap-1408	39	18	classic	classic	ADJ
ap-1408	39	19	book	book	NOUN
ap-1408	39	20	by	by	ADP
ap-1408	39	21	a.	a.	NOUN
ap-1408	39	22	dvurečenskij	dvurečenskij	PROPN
ap-1408	39	23	and	and	CCONJ
ap-1408	39	24	s.	s.	PROPN
ap-1408	39	25	pulmannová	pulmannová	PROPN
ap-1408	40	1	[	[	X
ap-1408	40	2	3	3	NUM
ap-1408	40	3	]	]	PUNCT
ap-1408	40	4	,	,	PUNCT
ap-1408	40	5	where	where	SCONJ
ap-1408	40	6	the	the	DET
ap-1408	40	7	interested	interested	ADJ
ap-1408	40	8	reader	reader	NOUN
ap-1408	40	9	can	can	AUX
ap-1408	40	10	find	find	VERB
ap-1408	40	11	unexplained	unexplained	ADJ
ap-1408	40	12	terms	term	NOUN
ap-1408	40	13	and	and	CCONJ
ap-1408	40	14	notation	notation	NOUN
ap-1408	40	15	concerning	concern	VERB
ap-1408	40	16	the	the	DET
ap-1408	40	17	subject	subject	NOUN
ap-1408	40	18	.	.	PUNCT
ap-1408	41	1	we	we	PRON
ap-1408	41	2	now	now	ADV
ap-1408	41	3	review	review	VERB
ap-1408	41	4	some	some	DET
ap-1408	41	5	terminology	terminology	NOUN
ap-1408	41	6	concerning	concern	VERB
ap-1408	41	7	(	(	PUNCT
ap-1408	41	8	generalized	generalized	ADJ
ap-1408	41	9	)	)	PUNCT
ap-1408	41	10	effect	effect	NOUN
ap-1408	41	11	algebras	algebra	NOUN
ap-1408	41	12	and	and	CCONJ
ap-1408	41	13	weakly	weakly	ADJ
ap-1408	41	14	ordered	order	VERB
ap-1408	41	15	partial	partial	ADJ
ap-1408	41	16	commutative	commutative	ADJ
ap-1408	41	17	groups	group	NOUN
ap-1408	41	18	.	.	PUNCT
ap-1408	42	1	definition	definition	NOUN
ap-1408	42	2	1	1	NUM
ap-1408	42	3	(	(	PUNCT
ap-1408	42	4	[	[	X
ap-1408	42	5	4	4	NUM
ap-1408	42	6	]	]	PUNCT
ap-1408	42	7	)	)	PUNCT
ap-1408	42	8	a	a	DET
ap-1408	42	9	partial	partial	ADJ
ap-1408	42	10	algebra	algebra	NOUN
ap-1408	42	11	(	(	PUNCT
ap-1408	42	12	e	e	NOUN
ap-1408	42	13	;	;	PUNCT
ap-1408	42	14	+	+	ADJ
ap-1408	42	15	,	,	PUNCT
ap-1408	42	16	0	0	NUM
ap-1408	42	17	,	,	PUNCT
ap-1408	42	18	1	1	NUM
ap-1408	42	19	)	)	PUNCT
ap-1408	42	20	is	be	AUX
ap-1408	42	21	called	call	VERB
ap-1408	42	22	an	an	DET
ap-1408	42	23	effect	effect	NOUN
ap-1408	42	24	algebra	algebra	NOUN
ap-1408	42	25	if	if	SCONJ
ap-1408	42	26	0	0	NUM
ap-1408	42	27	,	,	PUNCT
ap-1408	42	28	1	1	NUM
ap-1408	42	29	are	be	AUX
ap-1408	42	30	two	two	NUM
ap-1408	42	31	distinct	distinct	ADJ
ap-1408	42	32	elements	element	NOUN
ap-1408	42	33	and	and	CCONJ
ap-1408	42	34	+	+	CCONJ
ap-1408	42	35	is	be	AUX
ap-1408	42	36	a	a	DET
ap-1408	42	37	partially	partially	ADV
ap-1408	42	38	defined	define	VERB
ap-1408	42	39	binary	binary	ADJ
ap-1408	42	40	operation	operation	NOUN
ap-1408	42	41	on	on	ADP
ap-1408	42	42	e	e	PROPN
ap-1408	42	43	which	which	PRON
ap-1408	42	44	satisfy	satisfy	VERB
ap-1408	42	45	the	the	DET
ap-1408	42	46	following	follow	VERB
ap-1408	42	47	conditions	condition	NOUN
ap-1408	42	48	for	for	ADP
ap-1408	42	49	any	any	DET
ap-1408	42	50	x	x	NOUN
ap-1408	42	51	,	,	PUNCT
ap-1408	42	52	y	y	PROPN
ap-1408	42	53	,	,	PUNCT
ap-1408	42	54	z	z	NOUN
ap-1408	42	55	∈	∈	PROPN
ap-1408	43	1	e	e	NOUN
ap-1408	43	2	:	:	PUNCT
ap-1408	43	3	(	(	PUNCT
ap-1408	43	4	ei	ei	NOUN
ap-1408	43	5	)	)	PUNCT
ap-1408	43	6	x	x	PUNCT
ap-1408	44	1	+	+	PUNCT
ap-1408	44	2	y	y	PROPN
ap-1408	44	3	=	=	SYM
ap-1408	44	4	y	y	PROPN
ap-1408	45	1	+	+	NOUN
ap-1408	45	2	x	x	SYM
ap-1408	45	3	if	if	SCONJ
ap-1408	45	4	x	x	PRON
ap-1408	45	5	+	+	CCONJ
ap-1408	45	6	y	y	NOUN
ap-1408	45	7	is	be	AUX
ap-1408	45	8	defined	define	VERB
ap-1408	45	9	,	,	PUNCT
ap-1408	45	10	(	(	PUNCT
ap-1408	45	11	eii	eii	PROPN
ap-1408	45	12	)	)	PUNCT
ap-1408	45	13	(	(	PUNCT
ap-1408	45	14	x	x	X
ap-1408	46	1	+	+	NUM
ap-1408	46	2	y	y	NOUN
ap-1408	46	3	)	)	PUNCT
ap-1408	47	1	+	+	PUNCT
ap-1408	47	2	z	z	NOUN
ap-1408	47	3	=	=	PUNCT
ap-1408	48	1	x	x	X
ap-1408	49	1	+	+	PUNCT
ap-1408	49	2	(	(	PUNCT
ap-1408	49	3	y	y	PROPN
ap-1408	49	4	+	+	PROPN
ap-1408	49	5	z	z	X
ap-1408	49	6	)	)	PUNCT
ap-1408	49	7	if	if	SCONJ
ap-1408	49	8	one	one	NUM
ap-1408	49	9	side	side	NOUN
ap-1408	49	10	is	be	AUX
ap-1408	49	11	defined	define	VERB
ap-1408	49	12	,	,	PUNCT
ap-1408	49	13	(	(	PUNCT
ap-1408	49	14	eiii	eiii	PROPN
ap-1408	49	15	)	)	PUNCT
ap-1408	49	16	for	for	ADP
ap-1408	49	17	every	every	DET
ap-1408	49	18	x	x	SYM
ap-1408	49	19	∈	∈	PROPN
ap-1408	49	20	e	e	NOUN
ap-1408	49	21	there	there	PRON
ap-1408	49	22	exists	exist	VERB
ap-1408	49	23	a	a	DET
ap-1408	49	24	unique	unique	ADJ
ap-1408	49	25	y	y	PROPN
ap-1408	49	26	∈	∈	PROPN
ap-1408	49	27	e	e	NOUN
ap-1408	49	28	such	such	ADJ
ap-1408	49	29	that	that	SCONJ
ap-1408	49	30	x	x	X
ap-1408	50	1	+	+	NUM
ap-1408	50	2	y	y	NOUN
ap-1408	50	3	=	=	SYM
ap-1408	50	4	1	1	NUM
ap-1408	50	5	(	(	PUNCT
ap-1408	50	6	we	we	PRON
ap-1408	50	7	put	put	VERB
ap-1408	50	8	x′	x′	PROPN
ap-1408	50	9	=	=	SYM
ap-1408	50	10	y	y	PROPN
ap-1408	50	11	)	)	PUNCT
ap-1408	50	12	,	,	PUNCT
ap-1408	50	13	(	(	PUNCT
ap-1408	50	14	eiv	eiv	PROPN
ap-1408	50	15	)	)	PUNCT
ap-1408	50	16	if	if	SCONJ
ap-1408	50	17	1	1	NUM
ap-1408	50	18	+	+	CCONJ
ap-1408	50	19	x	x	AUX
ap-1408	50	20	is	be	AUX
ap-1408	50	21	defined	define	VERB
ap-1408	50	22	then	then	ADV
ap-1408	50	23	x	x	X
ap-1408	50	24	=	=	SYM
ap-1408	50	25	0	0	PROPN
ap-1408	50	26	.	.	PUNCT
ap-1408	51	1	definition	definition	NOUN
ap-1408	51	2	2	2	NUM
ap-1408	51	3	(	(	PUNCT
ap-1408	51	4	[	[	X
ap-1408	51	5	5	5	NUM
ap-1408	51	6	]	]	PUNCT
ap-1408	51	7	)	)	PUNCT
ap-1408	51	8	a	a	DET
ap-1408	51	9	partial	partial	ADJ
ap-1408	51	10	algebra	algebra	NOUN
ap-1408	51	11	(	(	PUNCT
ap-1408	51	12	e	e	NOUN
ap-1408	51	13	;	;	PUNCT
ap-1408	51	14	+	+	ADJ
ap-1408	51	15	,	,	PUNCT
ap-1408	51	16	0	0	NUM
ap-1408	51	17	)	)	PUNCT
ap-1408	51	18	is	be	AUX
ap-1408	51	19	called	call	VERB
ap-1408	51	20	a	a	DET
ap-1408	51	21	generalized	generalized	ADJ
ap-1408	51	22	effect	effect	NOUN
ap-1408	51	23	algebra	algebra	NOUN
ap-1408	51	24	if	if	SCONJ
ap-1408	51	25	0	0	NUM
ap-1408	51	26	∈	∈	NOUN
ap-1408	51	27	e	e	NOUN
ap-1408	51	28	is	be	AUX
ap-1408	51	29	a	a	DET
ap-1408	51	30	distinguished	distinguished	ADJ
ap-1408	51	31	element	element	NOUN
ap-1408	51	32	and	and	CCONJ
ap-1408	51	33	+	+	CCONJ
ap-1408	51	34	is	be	AUX
ap-1408	51	35	a	a	DET
ap-1408	51	36	partially	partially	ADV
ap-1408	51	37	defined	define	VERB
ap-1408	51	38	binary	binary	ADJ
ap-1408	51	39	operation	operation	NOUN
ap-1408	51	40	on	on	ADP
ap-1408	51	41	e	e	PROPN
ap-1408	51	42	which	which	PRON
ap-1408	51	43	satisfies	satisfy	VERB
ap-1408	51	44	the	the	DET
ap-1408	51	45	following	follow	VERB
ap-1408	51	46	conditions	condition	NOUN
ap-1408	51	47	for	for	ADP
ap-1408	51	48	any	any	DET
ap-1408	51	49	x	x	NOUN
ap-1408	51	50	,	,	PUNCT
ap-1408	51	51	y	y	PROPN
ap-1408	51	52	,	,	PUNCT
ap-1408	51	53	z	z	NOUN
ap-1408	51	54	∈	∈	PROPN
ap-1408	52	1	e	e	NOUN
ap-1408	52	2	:	:	PUNCT
ap-1408	52	3	(	(	PUNCT
ap-1408	52	4	gei	gei	NOUN
ap-1408	52	5	)	)	PUNCT
ap-1408	52	6	x	x	PUNCT
ap-1408	53	1	+	+	PUNCT
ap-1408	53	2	y	y	PROPN
ap-1408	53	3	=	=	SYM
ap-1408	53	4	y	y	PROPN
ap-1408	54	1	+	+	CCONJ
ap-1408	54	2	x	x	X
ap-1408	54	3	,	,	PUNCT
ap-1408	54	4	if	if	SCONJ
ap-1408	54	5	one	one	NUM
ap-1408	54	6	side	side	NOUN
ap-1408	54	7	is	be	AUX
ap-1408	54	8	defined	define	VERB
ap-1408	54	9	,	,	PUNCT
ap-1408	54	10	(	(	PUNCT
ap-1408	54	11	geii	geii	NOUN
ap-1408	54	12	)	)	PUNCT
ap-1408	54	13	(	(	PUNCT
ap-1408	54	14	x+y)+z	x+y)+z	PROPN
ap-1408	54	15	=	=	SYM
ap-1408	54	16	x+(y+z	x+(y+z	PROPN
ap-1408	54	17	)	)	PUNCT
ap-1408	54	18	,	,	PUNCT
ap-1408	54	19	if	if	SCONJ
ap-1408	54	20	one	one	NUM
ap-1408	54	21	side	side	NOUN
ap-1408	54	22	is	be	AUX
ap-1408	54	23	defined	define	VERB
ap-1408	54	24	,	,	PUNCT
ap-1408	54	25	(	(	PUNCT
ap-1408	54	26	geiii	geiii	PROPN
ap-1408	54	27	)	)	PUNCT
ap-1408	54	28	x	x	PUNCT
ap-1408	55	1	+	+	PUNCT
ap-1408	55	2	0	0	NUM
ap-1408	55	3	=	=	SYM
ap-1408	55	4	x	x	NOUN
ap-1408	55	5	,	,	PUNCT
ap-1408	55	6	(	(	PUNCT
ap-1408	55	7	geiv	geiv	NOUN
ap-1408	55	8	)	)	PUNCT
ap-1408	55	9	x	x	PUNCT
ap-1408	56	1	+	+	PUNCT
ap-1408	56	2	y	y	NOUN
ap-1408	56	3	=	=	PUNCT
ap-1408	57	1	x	x	PUNCT
ap-1408	58	1	+	+	CCONJ
ap-1408	58	2	z	z	NOUN
ap-1408	58	3	implies	imply	VERB
ap-1408	58	4	y	y	PROPN
ap-1408	58	5	=	=	SYM
ap-1408	58	6	z	z	PROPN
ap-1408	58	7	(	(	PUNCT
ap-1408	58	8	cancellation	cancellation	NOUN
ap-1408	58	9	law	law	NOUN
ap-1408	58	10	)	)	PUNCT
ap-1408	58	11	,	,	PUNCT
ap-1408	58	12	(	(	PUNCT
ap-1408	58	13	gev	gev	NOUN
ap-1408	58	14	)	)	PUNCT
ap-1408	58	15	x	x	PUNCT
ap-1408	59	1	+	+	CCONJ
ap-1408	59	2	y	y	PROPN
ap-1408	59	3	=	=	SYM
ap-1408	59	4	0	0	NUM
ap-1408	59	5	implies	imply	VERB
ap-1408	59	6	x	x	PUNCT
ap-1408	59	7	=	=	SYM
ap-1408	59	8	y	y	PROPN
ap-1408	59	9	=	=	SYM
ap-1408	59	10	0	0	PROPN
ap-1408	59	11	.	.	PUNCT
ap-1408	60	1	in	in	ADP
ap-1408	60	2	every	every	DET
ap-1408	60	3	generalized	generalized	ADJ
ap-1408	60	4	effect	effect	NOUN
ap-1408	60	5	algebra	algebra	NOUN
ap-1408	60	6	e	e	NOUN
ap-1408	60	7	the	the	DET
ap-1408	60	8	partial	partial	ADJ
ap-1408	60	9	binary	binary	PROPN
ap-1408	60	10	operation	operation	NOUN
ap-1408	60	11	�	�	PROPN
ap-1408	60	12	and	and	CCONJ
ap-1408	60	13	relation	relation	NOUN
ap-1408	60	14	≤	≤	NOUN
ap-1408	60	15	can	can	AUX
ap-1408	60	16	be	be	AUX
ap-1408	60	17	defined	define	VERB
ap-1408	60	18	by	by	ADP
ap-1408	60	19	(	(	PUNCT
ap-1408	60	20	ed	ed	NOUN
ap-1408	60	21	)	)	PUNCT
ap-1408	60	22	x	x	SYM
ap-1408	60	23	≤	≤	ADJ
ap-1408	60	24	y	y	PROPN
ap-1408	60	25	and	and	CCONJ
ap-1408	60	26	y	y	PROPN
ap-1408	60	27	�	�	PROPN
ap-1408	60	28	x	x	PUNCT
ap-1408	61	1	=	=	PUNCT
ap-1408	61	2	z	z	SYM
ap-1408	61	3	iff	iff	NOUN
ap-1408	61	4	x	x	PROPN
ap-1408	62	1	+	+	CCONJ
ap-1408	62	2	z	z	NOUN
ap-1408	62	3	is	be	AUX
ap-1408	62	4	defined	define	VERB
ap-1408	62	5	and	and	CCONJ
ap-1408	62	6	x	x	PUNCT
ap-1408	62	7	+	+	CCONJ
ap-1408	62	8	z	z	NOUN
ap-1408	62	9	=	=	SYM
ap-1408	62	10	y.	y.	NOUN
ap-1408	62	11	then	then	ADV
ap-1408	62	12	≤	≤	NUM
ap-1408	62	13	is	be	AUX
ap-1408	62	14	a	a	DET
ap-1408	62	15	partial	partial	ADJ
ap-1408	62	16	order	order	NOUN
ap-1408	62	17	on	on	ADP
ap-1408	62	18	e	e	NOUN
ap-1408	62	19	under	under	ADP
ap-1408	62	20	which	which	PRON
ap-1408	62	21	0	0	NUM
ap-1408	62	22	is	be	AUX
ap-1408	62	23	the	the	DET
ap-1408	62	24	least	least	ADJ
ap-1408	62	25	element	element	NOUN
ap-1408	62	26	of	of	ADP
ap-1408	62	27	e.	e.	PROPN
ap-1408	62	28	note	note	PROPN
ap-1408	62	29	that	that	SCONJ
ap-1408	62	30	every	every	DET
ap-1408	62	31	effect	effect	NOUN
ap-1408	62	32	algebra	algebra	NOUN
ap-1408	62	33	satisfies	satisfy	VERB
ap-1408	62	34	the	the	DET
ap-1408	62	35	axioms	axiom	NOUN
ap-1408	62	36	of	of	ADP
ap-1408	62	37	a	a	DET
ap-1408	62	38	generalized	generalized	ADJ
ap-1408	62	39	effect	effect	NOUN
ap-1408	62	40	algebra	algebra	NOUN
ap-1408	62	41	,	,	PUNCT
ap-1408	62	42	and	and	CCONJ
ap-1408	62	43	if	if	SCONJ
ap-1408	62	44	a	a	DET
ap-1408	62	45	generalized	generalized	ADJ
ap-1408	62	46	effect	effect	NOUN
ap-1408	62	47	algebra	algebra	NOUN
ap-1408	62	48	has	have	VERB
ap-1408	62	49	the	the	DET
ap-1408	62	50	greatest	great	ADJ
ap-1408	62	51	element	element	NOUN
ap-1408	62	52	then	then	ADV
ap-1408	62	53	it	it	PRON
ap-1408	62	54	is	be	AUX
ap-1408	62	55	an	an	DET
ap-1408	62	56	effect	effect	NOUN
ap-1408	62	57	algebra	algebra	NOUN
ap-1408	62	58	.	.	PUNCT
ap-1408	63	1	definition	definition	NOUN
ap-1408	63	2	3	3	NUM
ap-1408	63	3	a	a	DET
ap-1408	63	4	partial	partial	ADJ
ap-1408	63	5	algebra	algebra	NOUN
ap-1408	63	6	(	(	PUNCT
ap-1408	63	7	g	g	NOUN
ap-1408	63	8	;	;	PUNCT
ap-1408	63	9	+	+	ADJ
ap-1408	63	10	,	,	PUNCT
ap-1408	63	11	0	0	NUM
ap-1408	63	12	)	)	PUNCT
ap-1408	63	13	is	be	AUX
ap-1408	63	14	called	call	VERB
ap-1408	63	15	a	a	DET
ap-1408	63	16	commutative	commutative	ADJ
ap-1408	63	17	partial	partial	ADJ
ap-1408	63	18	group	group	NOUN
ap-1408	63	19	if	if	SCONJ
ap-1408	63	20	0	0	NUM
ap-1408	63	21	∈	∈	NOUN
ap-1408	63	22	e	e	NOUN
ap-1408	63	23	is	be	AUX
ap-1408	63	24	a	a	DET
ap-1408	63	25	distinguished	distinguished	ADJ
ap-1408	63	26	element	element	NOUN
ap-1408	63	27	and	and	CCONJ
ap-1408	63	28	+	+	CCONJ
ap-1408	63	29	is	be	AUX
ap-1408	63	30	a	a	DET
ap-1408	63	31	partially	partially	ADV
ap-1408	63	32	defined	define	VERB
ap-1408	63	33	binary	binary	ADJ
ap-1408	63	34	operation	operation	NOUN
ap-1408	63	35	on	on	ADP
ap-1408	63	36	e	e	PROPN
ap-1408	63	37	which	which	PRON
ap-1408	63	38	satisfy	satisfy	VERB
ap-1408	63	39	the	the	DET
ap-1408	63	40	following	follow	VERB
ap-1408	63	41	conditions	condition	NOUN
ap-1408	63	42	for	for	ADP
ap-1408	63	43	any	any	DET
ap-1408	63	44	x	x	NOUN
ap-1408	63	45	,	,	PUNCT
ap-1408	63	46	y	y	PROPN
ap-1408	63	47	,	,	PUNCT
ap-1408	63	48	z	z	NOUN
ap-1408	63	49	∈	∈	PROPN
ap-1408	64	1	e	e	NOUN
ap-1408	64	2	:	:	PUNCT
ap-1408	64	3	(	(	PUNCT
ap-1408	64	4	gi	gi	INTJ
ap-1408	64	5	)	)	PUNCT
ap-1408	64	6	x	x	PUNCT
ap-1408	65	1	+	+	PUNCT
ap-1408	65	2	y	y	PROPN
ap-1408	65	3	=	=	SYM
ap-1408	65	4	y	y	PROPN
ap-1408	66	1	+	+	NOUN
ap-1408	66	2	x	x	SYM
ap-1408	66	3	if	if	SCONJ
ap-1408	66	4	x	x	PRON
ap-1408	66	5	+	+	CCONJ
ap-1408	66	6	y	y	NOUN
ap-1408	66	7	is	be	AUX
ap-1408	66	8	defined	define	VERB
ap-1408	66	9	,	,	PUNCT
ap-1408	66	10	(	(	PUNCT
ap-1408	66	11	gii	gii	NOUN
ap-1408	66	12	)	)	PUNCT
ap-1408	66	13	(	(	PUNCT
ap-1408	66	14	x	x	X
ap-1408	66	15	+	+	NUM
ap-1408	66	16	y	y	NOUN
ap-1408	66	17	)	)	PUNCT
ap-1408	67	1	+	+	PUNCT
ap-1408	67	2	z	z	NOUN
ap-1408	67	3	=	=	PUNCT
ap-1408	68	1	x	x	X
ap-1408	69	1	+	+	PUNCT
ap-1408	69	2	(	(	PUNCT
ap-1408	69	3	y	y	PROPN
ap-1408	69	4	+	+	PROPN
ap-1408	69	5	z	z	X
ap-1408	69	6	)	)	PUNCT
ap-1408	69	7	if	if	SCONJ
ap-1408	69	8	both	both	DET
ap-1408	69	9	sides	side	NOUN
ap-1408	69	10	are	be	AUX
ap-1408	69	11	defined	define	VERB
ap-1408	69	12	,	,	PUNCT
ap-1408	69	13	(	(	PUNCT
ap-1408	69	14	giii	giii	NOUN
ap-1408	69	15	)	)	PUNCT
ap-1408	69	16	x	x	PUNCT
ap-1408	70	1	+	+	CCONJ
ap-1408	70	2	0	0	NUM
ap-1408	70	3	is	be	AUX
ap-1408	70	4	defined	define	VERB
ap-1408	70	5	and	and	CCONJ
ap-1408	70	6	x	x	SYM
ap-1408	71	1	+	+	NOUN
ap-1408	71	2	0	0	NUM
ap-1408	71	3	=	=	SYM
ap-1408	71	4	x	x	NOUN
ap-1408	71	5	,	,	PUNCT
ap-1408	71	6	(	(	PUNCT
ap-1408	71	7	giv	giv	NOUN
ap-1408	71	8	)	)	PUNCT
ap-1408	71	9	for	for	ADP
ap-1408	71	10	every	every	DET
ap-1408	71	11	x	x	SYM
ap-1408	71	12	∈	∈	PROPN
ap-1408	71	13	e	e	NOUN
ap-1408	71	14	there	there	PRON
ap-1408	71	15	exists	exist	VERB
ap-1408	71	16	a	a	DET
ap-1408	71	17	unique	unique	ADJ
ap-1408	71	18	y	y	PROPN
ap-1408	71	19	∈	∈	PROPN
ap-1408	71	20	e	e	NOUN
ap-1408	71	21	such	such	ADJ
ap-1408	71	22	that	that	SCONJ
ap-1408	71	23	x	x	X
ap-1408	72	1	+	+	NUM
ap-1408	72	2	y	y	PROPN
ap-1408	72	3	=	=	SYM
ap-1408	72	4	0	0	PUNCT
ap-1408	73	1	(	(	PUNCT
ap-1408	73	2	we	we	PRON
ap-1408	73	3	put	put	VERB
ap-1408	73	4	−x	−x	NOUN
ap-1408	73	5	=	=	SYM
ap-1408	73	6	y	y	NOUN
ap-1408	73	7	)	)	PUNCT
ap-1408	73	8	,	,	PUNCT
ap-1408	73	9	(	(	PUNCT
ap-1408	73	10	gv	gv	ADP
ap-1408	73	11	)	)	PUNCT
ap-1408	73	12	x	x	PUNCT
ap-1408	74	1	+	+	PUNCT
ap-1408	74	2	y	y	NOUN
ap-1408	74	3	=	=	PUNCT
ap-1408	75	1	x	x	PUNCT
ap-1408	76	1	+	+	CCONJ
ap-1408	76	2	z	z	NOUN
ap-1408	76	3	implies	imply	VERB
ap-1408	76	4	y	y	PROPN
ap-1408	76	5	=	=	PUNCT
ap-1408	76	6	z.	z.	PROPN
ap-1408	76	7	we	we	PRON
ap-1408	76	8	will	will	AUX
ap-1408	76	9	put	put	VERB
ap-1408	76	10	⊥g	⊥g	NOUN
ap-1408	76	11	=	=	PUNCT
ap-1408	76	12	{	{	PUNCT
ap-1408	76	13	(	(	PUNCT
ap-1408	76	14	x	x	NOUN
ap-1408	76	15	,	,	PUNCT
ap-1408	76	16	y	y	NOUN
ap-1408	76	17	)	)	PUNCT
ap-1408	76	18	∈	∈	PROPN
ap-1408	77	1	g	g	ADP
ap-1408	77	2	×	×	NOUN
ap-1408	77	3	g	g	NOUN
ap-1408	77	4	|	|	ADV
ap-1408	77	5	x	x	PUNCT
ap-1408	78	1	+	+	CCONJ
ap-1408	78	2	y	y	NOUN
ap-1408	78	3	is	be	AUX
ap-1408	78	4	defined	define	VERB
ap-1408	78	5	}	}	PUNCT
ap-1408	78	6	.	.	PUNCT
ap-1408	79	1	a	a	DET
ap-1408	79	2	commutative	commutative	ADJ
ap-1408	79	3	partial	partial	ADJ
ap-1408	79	4	group	group	NOUN
ap-1408	79	5	(	(	PUNCT
ap-1408	79	6	g	g	NOUN
ap-1408	79	7	;	;	PUNCT
ap-1408	79	8	+	+	ADJ
ap-1408	79	9	,	,	PUNCT
ap-1408	79	10	0	0	NUM
ap-1408	79	11	)	)	PUNCT
ap-1408	79	12	is	be	AUX
ap-1408	79	13	called	call	VERB
ap-1408	79	14	weakly	weakly	ADV
ap-1408	79	15	ordered	order	VERB
ap-1408	79	16	(	(	PUNCT
ap-1408	79	17	shortly	shortly	ADV
ap-1408	79	18	a	a	DET
ap-1408	79	19	wop	wop	NOUN
ap-1408	79	20	-	-	PUNCT
ap-1408	79	21	group	group	NOUN
ap-1408	79	22	)	)	PUNCT
ap-1408	79	23	with	with	ADP
ap-1408	79	24	respect	respect	NOUN
ap-1408	79	25	to	to	ADP
ap-1408	79	26	a	a	DET
ap-1408	79	27	reflexive	reflexive	ADJ
ap-1408	79	28	and	and	CCONJ
ap-1408	79	29	antisymmetric	antisymmetric	ADJ
ap-1408	79	30	relation	relation	NOUN
ap-1408	79	31	≤	≤	NOUN
ap-1408	79	32	on	on	ADP
ap-1408	79	33	g	g	PROPN
ap-1408	79	34	if	if	SCONJ
ap-1408	79	35	≤	≤	PROPN
ap-1408	79	36	is	be	AUX
ap-1408	79	37	compatible	compatible	ADJ
ap-1408	79	38	w.r.t	w.r.t	NOUN
ap-1408	79	39	.	.	PUNCT
ap-1408	80	1	partial	partial	ADJ
ap-1408	80	2	addition	addition	NOUN
ap-1408	80	3	,	,	PUNCT
ap-1408	80	4	i.e.	i.e.	X
ap-1408	80	5	,	,	PUNCT
ap-1408	80	6	for	for	ADP
ap-1408	80	7	all	all	DET
ap-1408	80	8	x	x	NOUN
ap-1408	80	9	,	,	PUNCT
ap-1408	80	10	y	y	PROPN
ap-1408	80	11	,	,	PUNCT
ap-1408	80	12	z	z	PROPN
ap-1408	80	13	∈	∈	PROPN
ap-1408	80	14	g	g	PROPN
ap-1408	80	15	,	,	PUNCT
ap-1408	80	16	x	x	PUNCT
ap-1408	80	17	≤	≤	ADJ
ap-1408	80	18	y	y	NOUN
ap-1408	80	19	and	and	CCONJ
ap-1408	81	1	both	both	DET
ap-1408	81	2	x	x	X
ap-1408	81	3	+	+	CCONJ
ap-1408	81	4	z	z	NOUN
ap-1408	81	5	and	and	CCONJ
ap-1408	81	6	y	y	PROPN
ap-1408	82	1	+	+	CCONJ
ap-1408	82	2	z	z	NOUN
ap-1408	82	3	are	be	AUX
ap-1408	82	4	defined	define	VERB
ap-1408	82	5	implies	imply	VERB
ap-1408	82	6	x+z	x+z	NUM
ap-1408	82	7	≤	≤	NUM
ap-1408	83	1	y+z	y+z	PROPN
ap-1408	83	2	.	.	PUNCT
ap-1408	84	1	we	we	PRON
ap-1408	84	2	will	will	AUX
ap-1408	84	3	denote	denote	VERB
ap-1408	84	4	by	by	ADP
ap-1408	84	5	pos(g	pos(g	PROPN
ap-1408	84	6	)	)	PUNCT
ap-1408	84	7	the	the	DET
ap-1408	84	8	set	set	NOUN
ap-1408	84	9	{	{	PUNCT
ap-1408	84	10	x	x	SYM
ap-1408	84	11	∈	∈	PROPN
ap-1408	84	12	g	g	NOUN
ap-1408	84	13	|	|	ADV
ap-1408	84	14	x	x	NOUN
ap-1408	84	15	≥	≥	NOUN
ap-1408	84	16	0	0	NUM
ap-1408	84	17	}	}	PUNCT
ap-1408	84	18	.	.	PUNCT
ap-1408	85	1	recall	recall	VERB
ap-1408	85	2	that	that	SCONJ
ap-1408	85	3	wop	wop	NOUN
ap-1408	85	4	-	-	PUNCT
ap-1408	85	5	groups	group	NOUN
ap-1408	85	6	equipped	equip	VERB
ap-1408	85	7	with	with	ADP
ap-1408	85	8	a	a	DET
ap-1408	85	9	total	total	ADJ
ap-1408	85	10	operation	operation	NOUN
ap-1408	85	11	+	+	CCONJ
ap-1408	85	12	such	such	ADJ
ap-1408	85	13	that	that	SCONJ
ap-1408	85	14	≤	≤	NOUN
ap-1408	85	15	is	be	AUX
ap-1408	85	16	an	an	DET
ap-1408	85	17	order	order	NOUN
ap-1408	86	1	are	be	AUX
ap-1408	86	2	exactly	exactly	ADV
ap-1408	86	3	partially	partially	ADV
ap-1408	86	4	ordered	order	VERB
ap-1408	86	5	commutative	commutative	ADJ
ap-1408	86	6	groups	group	NOUN
ap-1408	86	7	.	.	PUNCT
ap-1408	87	1	throughout	throughout	ADP
ap-1408	87	2	the	the	DET
ap-1408	87	3	paper	paper	NOUN
ap-1408	87	4	we	we	PRON
ap-1408	87	5	assume	assume	VERB
ap-1408	87	6	that	that	SCONJ
ap-1408	87	7	h	h	NOUN
ap-1408	87	8	is	be	AUX
ap-1408	87	9	an	an	DET
ap-1408	87	10	infinite	infinite	ADJ
ap-1408	87	11	-	-	PUNCT
ap-1408	87	12	dimensional	dimensional	ADJ
ap-1408	87	13	complex	complex	ADJ
ap-1408	87	14	hilbert	hilbert	NOUN
ap-1408	87	15	space	space	NOUN
ap-1408	87	16	,	,	PUNCT
ap-1408	87	17	i.e.	i.e.	X
ap-1408	87	18	,	,	PUNCT
ap-1408	87	19	a	a	DET
ap-1408	87	20	linear	linear	ADJ
ap-1408	87	21	space	space	NOUN
ap-1408	87	22	with	with	ADP
ap-1408	87	23	inner	inner	ADJ
ap-1408	87	24	product	product	NOUN
ap-1408	87	25	〈	〈	PROPN
ap-1408	87	26	·	·	PUNCT
ap-1408	87	27	,	,	PUNCT
ap-1408	87	28	·	·	PUNCT
ap-1408	87	29	〉	〉	NOUN
ap-1408	87	30	which	which	PRON
ap-1408	87	31	is	be	AUX
ap-1408	87	32	complete	complete	ADJ
ap-1408	87	33	in	in	ADP
ap-1408	87	34	the	the	DET
ap-1408	87	35	induced	induce	VERB
ap-1408	87	36	metric	metric	NOUN
ap-1408	87	37	.	.	PUNCT
ap-1408	88	1	recall	recall	VERB
ap-1408	88	2	that	that	SCONJ
ap-1408	88	3	here	here	ADV
ap-1408	88	4	for	for	ADP
ap-1408	88	5	any	any	DET
ap-1408	88	6	x	x	NOUN
ap-1408	88	7	,	,	PUNCT
ap-1408	88	8	y	y	PROPN
ap-1408	88	9	∈	∈	PROPN
ap-1408	88	10	h	h	NOUN
ap-1408	88	11	we	we	PRON
ap-1408	88	12	have	have	VERB
ap-1408	88	13	〈	〈	PROPN
ap-1408	88	14	x	x	X
ap-1408	88	15	,	,	PUNCT
ap-1408	89	1	y	y	PROPN
ap-1408	89	2	〉	〉	NUM
ap-1408	89	3	∈	∈	PROPN
ap-1408	89	4	c	c	NOUN
ap-1408	89	5	(	(	PUNCT
ap-1408	89	6	the	the	DET
ap-1408	89	7	set	set	NOUN
ap-1408	89	8	of	of	ADP
ap-1408	89	9	complex	complex	ADJ
ap-1408	89	10	numbers	number	NOUN
ap-1408	89	11	)	)	PUNCT
ap-1408	89	12	such	such	ADJ
ap-1408	89	13	that	that	SCONJ
ap-1408	89	14	〈	〈	PROPN
ap-1408	89	15	x	x	X
ap-1408	89	16	,	,	PUNCT
ap-1408	89	17	αy+βz	αy+βz	PROPN
ap-1408	89	18	〉	〉	NOUN
ap-1408	89	19	=	=	SYM
ap-1408	89	20	α〈x	α〈x	NUM
ap-1408	89	21	,	,	PUNCT
ap-1408	89	22	y〉+β〈x	y〉+β〈x	PROPN
ap-1408	89	23	,	,	PUNCT
ap-1408	89	24	z	z	PROPN
ap-1408	89	25	〉	〉	NOUN
ap-1408	89	26	for	for	ADP
ap-1408	89	27	all	all	DET
ap-1408	89	28	α	α	NOUN
ap-1408	89	29	,	,	PUNCT
ap-1408	89	30	β	β	X
ap-1408	89	31	∈	∈	PROPN
ap-1408	89	32	c	c	NOUN
ap-1408	89	33	and	and	CCONJ
ap-1408	89	34	x	x	PROPN
ap-1408	89	35	,	,	PUNCT
ap-1408	89	36	y	y	PROPN
ap-1408	89	37	,	,	PUNCT
ap-1408	89	38	z	z	PROPN
ap-1408	89	39	∈	∈	PROPN
ap-1408	89	40	h.	h.	PROPN
ap-1408	90	1	moreover	moreover	ADV
ap-1408	90	2	,	,	PUNCT
ap-1408	90	3	〈	〈	PROPN
ap-1408	90	4	x	x	X
ap-1408	90	5	,	,	PUNCT
ap-1408	90	6	y	y	PROPN
ap-1408	90	7	〉	〉	NUM
ap-1408	90	8	=	=	SYM
ap-1408	90	9	〈	〈	PROPN
ap-1408	90	10	y	y	PROPN
ap-1408	90	11	,	,	PUNCT
ap-1408	90	12	x	x	NOUN
ap-1408	90	13	〉	〉	NUM
ap-1408	90	14	and	and	CCONJ
ap-1408	90	15	finally	finally	ADV
ap-1408	90	16	〈	〈	PROPN
ap-1408	90	17	x	x	X
ap-1408	90	18	,	,	PUNCT
ap-1408	90	19	x	x	PROPN
ap-1408	90	20	〉	〉	NUM
ap-1408	90	21	≥	≥	NOUN
ap-1408	90	22	0	0	NUM
ap-1408	90	23	at	at	ADP
ap-1408	90	24	which	which	PRON
ap-1408	90	25	〈	〈	PROPN
ap-1408	90	26	x	x	X
ap-1408	90	27	,	,	PUNCT
ap-1408	90	28	x	x	NOUN
ap-1408	90	29	〉	〉	NOUN
ap-1408	90	30	=	=	SYM
ap-1408	90	31	0	0	PUNCT
ap-1408	90	32	iff	iff	NOUN
ap-1408	90	33	x	x	PROPN
ap-1408	90	34	=	=	NOUN
ap-1408	90	35	0	0	NUM
ap-1408	90	36	.	.	PUNCT
ap-1408	91	1	the	the	DET
ap-1408	91	2	term	term	NOUN
ap-1408	91	3	dimension	dimension	NOUN
ap-1408	91	4	of	of	ADP
ap-1408	91	5	h	h	NOUN
ap-1408	91	6	in	in	ADP
ap-1408	91	7	the	the	DET
ap-1408	91	8	following	following	NOUN
ap-1408	91	9	always	always	ADV
ap-1408	91	10	means	mean	VERB
ap-1408	91	11	the	the	DET
ap-1408	91	12	hilbertian	hilbertian	ADJ
ap-1408	91	13	dimension	dimension	NOUN
ap-1408	91	14	defined	define	VERB
ap-1408	91	15	as	as	ADP
ap-1408	91	16	the	the	DET
ap-1408	91	17	cardinality	cardinality	NOUN
ap-1408	91	18	of	of	ADP
ap-1408	91	19	any	any	DET
ap-1408	91	20	orthonormal	orthonormal	ADJ
ap-1408	91	21	basis	basis	NOUN
ap-1408	91	22	of	of	ADP
ap-1408	91	23	h	h	PROPN
ap-1408	91	24	(	(	PUNCT
ap-1408	91	25	see	see	VERB
ap-1408	91	26	[	[	X
ap-1408	91	27	2	2	NUM
ap-1408	91	28	]	]	NUM
ap-1408	91	29	)	)	PUNCT
ap-1408	91	30	.	.	PUNCT
ap-1408	92	1	moreover	moreover	ADV
ap-1408	92	2	,	,	PUNCT
ap-1408	92	3	we	we	PRON
ap-1408	92	4	will	will	AUX
ap-1408	92	5	assume	assume	VERB
ap-1408	92	6	that	that	SCONJ
ap-1408	92	7	all	all	PRON
ap-1408	92	8	considered	consider	VERB
ap-1408	92	9	linear	linear	PROPN
ap-1408	92	10	operators	operator	NOUN
ap-1408	92	11	a	a	DET
ap-1408	92	12	(	(	PUNCT
ap-1408	92	13	i.e.	i.e.	X
ap-1408	92	14	,	,	PUNCT
ap-1408	92	15	linear	linear	PROPN
ap-1408	92	16	maps	map	VERB
ap-1408	92	17	a	a	DET
ap-1408	92	18	:	:	PUNCT
ap-1408	92	19	d(a	d(a	PROPN
ap-1408	92	20	)	)	PUNCT
ap-1408	92	21	→	→	SYM
ap-1408	92	22	h	h	X
ap-1408	92	23	)	)	PUNCT
ap-1408	92	24	have	have	VERB
ap-1408	92	25	a	a	DET
ap-1408	92	26	domain	domain	NOUN
ap-1408	92	27	d(a	d(a	PROPN
ap-1408	92	28	)	)	PUNCT
ap-1408	92	29	a	a	DET
ap-1408	92	30	linear	linear	ADJ
ap-1408	92	31	subspace	subspace	NOUN
ap-1408	92	32	dense	dense	ADJ
ap-1408	92	33	in	in	ADP
ap-1408	92	34	h	h	NOUN
ap-1408	92	35	with	with	ADP
ap-1408	92	36	respect	respect	NOUN
ap-1408	92	37	to	to	ADP
ap-1408	92	38	the	the	DET
ap-1408	92	39	metric	metric	ADJ
ap-1408	92	40	topology	topology	NOUN
ap-1408	92	41	induced	induce	VERB
ap-1408	92	42	by	by	ADP
ap-1408	92	43	the	the	DET
ap-1408	92	44	inner	inner	ADJ
ap-1408	92	45	product	product	NOUN
ap-1408	92	46	,	,	PUNCT
ap-1408	92	47	so	so	ADV
ap-1408	92	48	d(a	d(a	PROPN
ap-1408	92	49	)	)	PUNCT
ap-1408	93	1	=	=	SYM
ap-1408	93	2	h	h	NOUN
ap-1408	93	3	(	(	PUNCT
ap-1408	93	4	we	we	PRON
ap-1408	93	5	say	say	VERB
ap-1408	93	6	that	that	SCONJ
ap-1408	93	7	a	a	PRON
ap-1408	93	8	is	be	AUX
ap-1408	93	9	densely	densely	ADV
ap-1408	93	10	defined	define	VERB
ap-1408	93	11	)	)	PUNCT
ap-1408	93	12	.	.	PUNCT
ap-1408	94	1	we	we	PRON
ap-1408	94	2	denote	denote	VERB
ap-1408	94	3	by	by	ADP
ap-1408	94	4	d	d	PROPN
ap-1408	94	5	the	the	DET
ap-1408	94	6	set	set	NOUN
ap-1408	94	7	of	of	ADP
ap-1408	94	8	all	all	DET
ap-1408	94	9	dense	dense	ADJ
ap-1408	94	10	linear	linear	ADJ
ap-1408	94	11	subspaces	subspace	NOUN
ap-1408	94	12	of	of	ADP
ap-1408	94	13	h.	h.	NOUN
ap-1408	94	14	moreover	moreover	ADV
ap-1408	94	15	,	,	PUNCT
ap-1408	94	16	by	by	ADP
ap-1408	94	17	positive	positive	ADJ
ap-1408	94	18	linear	linear	PROPN
ap-1408	94	19	operators	operator	NOUN
ap-1408	94	20	a	a	PRON
ap-1408	94	21	,	,	PUNCT
ap-1408	94	22	(	(	PUNCT
ap-1408	94	23	denoted	denote	VERB
ap-1408	94	24	by	by	ADP
ap-1408	94	25	a	a	DET
ap-1408	94	26	≥	≥	NOUN
ap-1408	94	27	0	0	NUM
ap-1408	94	28	)	)	PUNCT
ap-1408	94	29	it	it	PRON
ap-1408	94	30	means	mean	VERB
ap-1408	94	31	that	that	SCONJ
ap-1408	94	32	〈	〈	PROPN
ap-1408	94	33	ax	ax	NOUN
ap-1408	94	34	,	,	PUNCT
ap-1408	94	35	x	x	PROPN
ap-1408	94	36	〉	〉	NUM
ap-1408	94	37	≥	≥	NOUN
ap-1408	94	38	0	0	NUM
ap-1408	94	39	for	for	ADP
ap-1408	94	40	all	all	DET
ap-1408	94	41	x	x	SYM
ap-1408	94	42	∈	∈	PROPN
ap-1408	94	43	d(a	d(a	PROPN
ap-1408	94	44	)	)	PUNCT
ap-1408	94	45	,	,	PUNCT
ap-1408	94	46	therefore	therefore	ADV
ap-1408	94	47	operators	operator	VERB
ap-1408	94	48	a	a	PRON
ap-1408	94	49	are	be	AUX
ap-1408	94	50	also	also	ADV
ap-1408	94	51	symmetric	symmetric	ADJ
ap-1408	94	52	,	,	PUNCT
ap-1408	94	53	i.e.	i.e.	X
ap-1408	94	54	,	,	PUNCT
ap-1408	94	55	〈	〈	PROPN
ap-1408	94	56	y	y	PROPN
ap-1408	94	57	,	,	PUNCT
ap-1408	95	1	ax	ax	NOUN
ap-1408	95	2	〉	〉	NOUN
ap-1408	95	3	=	=	SYM
ap-1408	95	4	〈	〈	PROPN
ap-1408	95	5	ay	ay	NOUN
ap-1408	95	6	,	,	PUNCT
ap-1408	95	7	x	x	NOUN
ap-1408	95	8	〉	〉	NOUN
ap-1408	95	9	for	for	ADP
ap-1408	95	10	all	all	DET
ap-1408	95	11	x	x	NOUN
ap-1408	95	12	,	,	PUNCT
ap-1408	95	13	y	y	PROPN
ap-1408	95	14	∈	∈	PROPN
ap-1408	95	15	d(a	d(a	PROPN
ap-1408	95	16	)	)	PUNCT
ap-1408	95	17	(	(	PUNCT
ap-1408	95	18	for	for	ADP
ap-1408	95	19	more	more	ADJ
ap-1408	95	20	details	detail	NOUN
ap-1408	95	21	see	see	VERB
ap-1408	95	22	[	[	X
ap-1408	95	23	2	2	NUM
ap-1408	95	24	]	]	PUNCT
ap-1408	95	25	)	)	PUNCT
ap-1408	95	26	.	.	PUNCT
ap-1408	96	1	to	to	ADP
ap-1408	96	2	every	every	DET
ap-1408	96	3	linear	linear	ADJ
ap-1408	96	4	operator	operator	NOUN
ap-1408	96	5	a	a	DET
ap-1408	96	6	:	:	PUNCT
ap-1408	96	7	d(a	d(a	PROPN
ap-1408	96	8	)	)	PUNCT
ap-1408	96	9	→	→	SYM
ap-1408	96	10	h	h	NOUN
ap-1408	96	11	with	with	ADP
ap-1408	96	12	d(a	d(a	PROPN
ap-1408	96	13	)	)	PUNCT
ap-1408	97	1	=	=	NOUN
ap-1408	98	1	h	h	NOUN
ap-1408	98	2	there	there	PRON
ap-1408	98	3	exists	exist	VERB
ap-1408	98	4	the	the	DET
ap-1408	98	5	adjoint	adjoint	NOUN
ap-1408	98	6	operator	operator	NOUN
ap-1408	98	7	a∗	a∗	NOUN
ap-1408	98	8	of	of	ADP
ap-1408	98	9	a	a	DET
ap-1408	98	10	such	such	ADJ
ap-1408	98	11	that	that	DET
ap-1408	98	12	d(a∗	d(a∗	NOUN
ap-1408	98	13	)	)	PUNCT
ap-1408	98	14	=	=	SYM
ap-1408	99	1	{	{	PUNCT
ap-1408	99	2	y	y	PROPN
ap-1408	99	3	∈	∈	PROPN
ap-1408	99	4	h	h	NOUN
ap-1408	99	5	|	|	ADV
ap-1408	99	6	there	there	PRON
ap-1408	99	7	exists	exist	VERB
ap-1408	99	8	y∗	y∗	PROPN
ap-1408	99	9	∈	∈	PROPN
ap-1408	99	10	h	h	NOUN
ap-1408	99	11	such	such	ADJ
ap-1408	99	12	that	that	PRON
ap-1408	99	13	(	(	PUNCT
ap-1408	99	14	y∗	y∗	ADV
ap-1408	99	15	,	,	PUNCT
ap-1408	99	16	x	x	NOUN
ap-1408	99	17	)	)	PUNCT
ap-1408	99	18	=	=	SYM
ap-1408	100	1	(	(	PUNCT
ap-1408	100	2	y	y	NOUN
ap-1408	100	3	,	,	PUNCT
ap-1408	100	4	ax	ax	NOUN
ap-1408	100	5	)	)	PUNCT
ap-1408	100	6	for	for	ADP
ap-1408	100	7	every	every	DET
ap-1408	100	8	x	x	PROPN
ap-1408	100	9	∈	∈	PROPN
ap-1408	100	10	d(a	d(a	PROPN
ap-1408	100	11	)	)	PUNCT
ap-1408	100	12	}	}	PUNCT
ap-1408	100	13	and	and	CCONJ
ap-1408	100	14	a∗y	a∗y	NUM
ap-1408	100	15	=	=	PUNCT
ap-1408	101	1	y∗	y∗	ADV
ap-1408	101	2	for	for	ADP
ap-1408	101	3	every	every	DET
ap-1408	101	4	y	y	PROPN
ap-1408	101	5	∈	∈	PROPN
ap-1408	101	6	d(a∗	d(a∗	NUM
ap-1408	101	7	)	)	PUNCT
ap-1408	101	8	.	.	PUNCT
ap-1408	102	1	if	if	SCONJ
ap-1408	102	2	a∗	a∗	PROPN
ap-1408	102	3	=	=	PUNCT
ap-1408	102	4	a	a	PRON
ap-1408	102	5	then	then	ADV
ap-1408	102	6	a	a	PRON
ap-1408	102	7	is	be	AUX
ap-1408	102	8	called	call	VERB
ap-1408	102	9	self	self	NOUN
ap-1408	102	10	-	-	PUNCT
ap-1408	102	11	adjoint	adjoint	NOUN
ap-1408	102	12	.	.	PUNCT
ap-1408	103	1	recall	recall	VERB
ap-1408	103	2	that	that	PRON
ap-1408	103	3	a	a	DET
ap-1408	103	4	:	:	PUNCT
ap-1408	103	5	d(a	d(a	PROPN
ap-1408	103	6	)	)	PUNCT
ap-1408	103	7	→	→	SYM
ap-1408	103	8	h	h	NOUN
ap-1408	103	9	is	be	AUX
ap-1408	103	10	called	call	VERB
ap-1408	103	11	a	a	DET
ap-1408	103	12	bounded	bounded	ADJ
ap-1408	103	13	operator	operator	NOUN
ap-1408	103	14	if	if	SCONJ
ap-1408	103	15	there	there	PRON
ap-1408	103	16	exists	exist	VERB
ap-1408	103	17	a	a	DET
ap-1408	103	18	real	real	ADJ
ap-1408	103	19	constant	constant	ADJ
ap-1408	103	20	c	c	PROPN
ap-1408	103	21	≥	≥	NOUN
ap-1408	103	22	0	0	NUM
ap-1408	103	23	such	such	ADJ
ap-1408	103	24	that	that	SCONJ
ap-1408	103	25	‖ax‖	‖ax‖	ADJ
ap-1408	103	26	≤	≤	NOUN
ap-1408	103	27	c‖x‖	c‖x‖	NOUN
ap-1408	103	28	for	for	ADP
ap-1408	103	29	all	all	DET
ap-1408	103	30	x	x	SYM
ap-1408	103	31	∈	∈	PROPN
ap-1408	103	32	d(a	d(a	PROPN
ap-1408	103	33	)	)	PUNCT
ap-1408	103	34	and	and	CCONJ
ap-1408	103	35	hence	hence	ADV
ap-1408	103	36	a	a	PRON
ap-1408	103	37	is	be	AUX
ap-1408	103	38	an	an	DET
ap-1408	103	39	unbounded	unbounded	ADJ
ap-1408	103	40	operator	operator	NOUN
ap-1408	103	41	if	if	SCONJ
ap-1408	103	42	to	to	ADP
ap-1408	103	43	every	every	DET
ap-1408	103	44	c	c	NOUN
ap-1408	103	45	∈	∈	PROPN
ap-1408	103	46	r	r	NOUN
ap-1408	103	47	,	,	PUNCT
ap-1408	103	48	c	c	X
ap-1408	103	49	≥	≥	NOUN
ap-1408	103	50	0	0	NUM
ap-1408	103	51	there	there	PRON
ap-1408	103	52	exists	exist	VERB
ap-1408	103	53	xc	xc	PROPN
ap-1408	103	54	∈	∈	PROPN
ap-1408	103	55	d(a	d(a	PROPN
ap-1408	103	56	)	)	PUNCT
ap-1408	103	57	with	with	ADP
ap-1408	103	58	‖axc‖	‖axc‖	ADV
ap-1408	103	59	>	>	X
ap-1408	103	60	c‖xc‖.	c‖xc‖.	NOUN
ap-1408	103	61	the	the	DET
ap-1408	103	62	set	set	NOUN
ap-1408	103	63	of	of	ADP
ap-1408	103	64	all	all	DET
ap-1408	103	65	bounded	bounded	ADJ
ap-1408	103	66	operators	operator	NOUN
ap-1408	103	67	on	on	ADP
ap-1408	103	68	h	h	NOUN
ap-1408	103	69	is	be	AUX
ap-1408	103	70	denoted	denote	VERB
ap-1408	103	71	by	by	ADP
ap-1408	103	72	b(h	b(h	NOUN
ap-1408	103	73	)	)	PUNCT
ap-1408	103	74	.	.	PUNCT
ap-1408	104	1	for	for	ADP
ap-1408	104	2	every	every	DET
ap-1408	104	3	bounded	bounded	ADJ
ap-1408	104	4	operator	operator	NOUN
ap-1408	104	5	a	a	DET
ap-1408	104	6	:	:	PUNCT
ap-1408	104	7	d(a	d(a	PROPN
ap-1408	104	8	)	)	PUNCT
ap-1408	104	9	→	→	SYM
ap-1408	104	10	h	h	NOUN
ap-1408	104	11	densely	densely	ADV
ap-1408	104	12	defined	define	VERB
ap-1408	104	13	on	on	ADP
ap-1408	104	14	d(a	d(a	PROPN
ap-1408	104	15	)	)	PUNCT
ap-1408	104	16	=	=	PUNCT
ap-1408	105	1	d	d	PROPN
ap-1408	105	2	⊂	⊂	PROPN
ap-1408	105	3	h	h	PROPN
ap-1408	105	4	exists	exist	VERB
ap-1408	105	5	a	a	DET
ap-1408	105	6	unique	unique	ADJ
ap-1408	105	7	extension	extension	NOUN
ap-1408	105	8	b	b	NOUN
ap-1408	105	9	such	such	ADJ
ap-1408	105	10	as	as	ADP
ap-1408	105	11	d(b	d(b	X
ap-1408	105	12	)	)	PUNCT
ap-1408	106	1	=	=	SYM
ap-1408	106	2	h	h	NOUN
ap-1408	106	3	and	and	CCONJ
ap-1408	106	4	ax	ax	NOUN
ap-1408	106	5	=	=	PROPN
ap-1408	106	6	bx	bx	PROPN
ap-1408	106	7	for	for	ADP
ap-1408	106	8	every	every	DET
ap-1408	106	9	x	x	PROPN
ap-1408	106	10	∈	∈	PROPN
ap-1408	106	11	d(a	d(a	PROPN
ap-1408	106	12	)	)	PUNCT
ap-1408	106	13	.	.	PUNCT
ap-1408	107	1	we	we	PRON
ap-1408	107	2	will	will	AUX
ap-1408	107	3	denote	denote	VERB
ap-1408	107	4	this	this	DET
ap-1408	107	5	extension	extension	NOUN
ap-1408	107	6	b	b	NOUN
ap-1408	107	7	=	=	SYM
ap-1408	107	8	ab	ab	PROPN
ap-1408	107	9	(	(	PUNCT
ap-1408	107	10	for	for	SCONJ
ap-1408	107	11	more	more	ADJ
ap-1408	107	12	details	detail	NOUN
ap-1408	107	13	see	see	VERB
ap-1408	107	14	[	[	X
ap-1408	107	15	2	2	NUM
ap-1408	107	16	]	]	NUM
ap-1408	107	17	)	)	PUNCT
ap-1408	107	18	.	.	PUNCT
ap-1408	108	1	bounded	bound	VERB
ap-1408	108	2	and	and	CCONJ
ap-1408	108	3	symmetric	symmetric	ADJ
ap-1408	108	4	operators	operator	NOUN
ap-1408	108	5	are	be	AUX
ap-1408	108	6	called	call	VERB
ap-1408	108	7	hermitian	hermitian	ADJ
ap-1408	108	8	operators	operator	NOUN
ap-1408	108	9	.	.	PUNCT
ap-1408	109	1	we	we	PRON
ap-1408	109	2	also	also	ADV
ap-1408	109	3	write	write	VERB
ap-1408	109	4	,	,	PUNCT
ap-1408	109	5	for	for	ADP
ap-1408	109	6	linear	linear	PROPN
ap-1408	109	7	operators	operator	NOUN
ap-1408	109	8	a	a	DET
ap-1408	109	9	:	:	PUNCT
ap-1408	109	10	d(a	d(a	PROPN
ap-1408	109	11	)	)	PUNCT
ap-1408	109	12	→	→	SYM
ap-1408	109	13	h	h	NOUN
ap-1408	109	14	and	and	CCONJ
ap-1408	109	15	b	b	NOUN
ap-1408	109	16	:	:	PUNCT
ap-1408	109	17	d(b	d(b	X
ap-1408	109	18	)	)	PUNCT
ap-1408	109	19	→	→	SYM
ap-1408	109	20	h	h	NOUN
ap-1408	109	21	,	,	PUNCT
ap-1408	109	22	a	a	DET
ap-1408	109	23	⊂	⊂	PROPN
ap-1408	109	24	b	b	PROPN
ap-1408	109	25	iff	iff	PROPN
ap-1408	109	26	d(a	d(a	PROPN
ap-1408	109	27	)	)	PUNCT
ap-1408	109	28	⊆	⊆	NUM
ap-1408	109	29	d(b	d(b	NOUN
ap-1408	109	30	)	)	PUNCT
ap-1408	109	31	and	and	CCONJ
ap-1408	109	32	ax	ax	NOUN
ap-1408	109	33	=	=	NOUN
ap-1408	109	34	bx	bx	PROPN
ap-1408	109	35	for	for	ADP
ap-1408	109	36	every	every	DET
ap-1408	109	37	x	x	PROPN
ap-1408	109	38	∈	∈	PROPN
ap-1408	109	39	d(a	d(a	PROPN
ap-1408	109	40	)	)	PUNCT
ap-1408	109	41	.	.	PUNCT
ap-1408	110	1	66	66	NUM
ap-1408	110	2	acta	acta	PROPN
ap-1408	110	3	polytechnica	polytechnica	PROPN
ap-1408	110	4	vol	vol	NOUN
ap-1408	110	5	.	.	PUNCT
ap-1408	111	1	51	51	NUM
ap-1408	111	2	no	no	INTJ
ap-1408	111	3	.	.	PUNCT
ap-1408	112	1	4/2011	4/2011	NUM
ap-1408	112	2	3	3	NUM
ap-1408	112	3	operator	operator	NOUN
ap-1408	112	4	wop	wop	NOUN
ap-1408	112	5	-	-	PUNCT
ap-1408	112	6	groups	group	NOUN
ap-1408	112	7	as	as	ADP
ap-1408	112	8	a	a	DET
ap-1408	112	9	pasting	pasting	NOUN
ap-1408	112	10	of	of	ADP
ap-1408	112	11	operator	operator	NOUN
ap-1408	112	12	sub	sub	NOUN
ap-1408	112	13	-	-	NOUN
ap-1408	112	14	groups	group	NOUN
ap-1408	112	15	equipped	equip	VERB
ap-1408	112	16	with	with	ADP
ap-1408	112	17	the	the	DET
ap-1408	112	18	usual	usual	ADJ
ap-1408	112	19	sum	sum	NOUN
ap-1408	112	20	of	of	ADP
ap-1408	112	21	operators	operator	NOUN
ap-1408	112	22	definition	definition	NOUN
ap-1408	112	23	4	4	NUM
ap-1408	112	24	let	let	VERB
ap-1408	112	25	h	h	NOUN
ap-1408	112	26	be	be	AUX
ap-1408	112	27	an	an	DET
ap-1408	112	28	infinite	infinite	ADJ
ap-1408	112	29	-	-	PUNCT
ap-1408	112	30	dimensional	dimensional	ADJ
ap-1408	112	31	complex	complex	ADJ
ap-1408	112	32	hilbert	hilbert	NOUN
ap-1408	112	33	space	space	NOUN
ap-1408	112	34	.	.	PUNCT
ap-1408	113	1	let	let	VERB
ap-1408	113	2	us	we	PRON
ap-1408	113	3	define	define	VERB
ap-1408	113	4	the	the	DET
ap-1408	113	5	following	follow	VERB
ap-1408	113	6	set	set	NOUN
ap-1408	113	7	of	of	ADP
ap-1408	113	8	linear	linear	PROPN
ap-1408	113	9	operators	operator	NOUN
ap-1408	113	10	densely	densely	ADV
ap-1408	113	11	defined	define	VERB
ap-1408	113	12	in	in	ADP
ap-1408	113	13	h	h	NOUN
ap-1408	113	14	:	:	PUNCT
ap-1408	113	15	gr(h	gr(h	X
ap-1408	113	16	)	)	PUNCT
ap-1408	114	1	=	=	PRON
ap-1408	114	2	{	{	PUNCT
ap-1408	114	3	a	a	X
ap-1408	114	4	:	:	PUNCT
ap-1408	114	5	d(a	d(a	PROPN
ap-1408	114	6	)	)	PUNCT
ap-1408	114	7	→	→	SYM
ap-1408	114	8	h	h	NOUN
ap-1408	114	9	|	|	ADV
ap-1408	114	10	d(a	d(a	ADJ
ap-1408	114	11	)	)	PUNCT
ap-1408	114	12	=	=	SYM
ap-1408	114	13	h	h	NOUN
ap-1408	114	14	and	and	CCONJ
ap-1408	114	15	d(a	d(a	PROPN
ap-1408	114	16	)	)	PUNCT
ap-1408	115	1	=	=	SYM
ap-1408	115	2	h	h	NOUN
ap-1408	115	3	if	if	SCONJ
ap-1408	115	4	a	a	PRON
ap-1408	115	5	is	be	AUX
ap-1408	115	6	bounded	bound	VERB
ap-1408	115	7	}	}	PUNCT
ap-1408	115	8	.	.	PUNCT
ap-1408	116	1	theorem	theorem	NOUN
ap-1408	116	2	1	1	NUM
ap-1408	116	3	let	let	VERB
ap-1408	116	4	h	h	NOUN
ap-1408	116	5	be	be	AUX
ap-1408	116	6	an	an	DET
ap-1408	116	7	infinite	infinite	ADJ
ap-1408	116	8	-	-	PUNCT
ap-1408	116	9	dimensional	dimensional	ADJ
ap-1408	116	10	complex	complex	ADJ
ap-1408	116	11	hilbert	hilbert	NOUN
ap-1408	116	12	space	space	NOUN
ap-1408	116	13	.	.	PUNCT
ap-1408	117	1	let	let	VERB
ap-1408	117	2	⊕d	⊕d	NOUN
ap-1408	117	3	be	be	AUX
ap-1408	117	4	a	a	DET
ap-1408	117	5	partial	partial	ADJ
ap-1408	117	6	operation	operation	NOUN
ap-1408	117	7	on	on	ADP
ap-1408	117	8	gr(h	gr(h	NOUN
ap-1408	117	9	)	)	PUNCT
ap-1408	117	10	defined	define	VERB
ap-1408	117	11	for	for	ADP
ap-1408	117	12	a	a	DET
ap-1408	117	13	,	,	PUNCT
ap-1408	117	14	b	b	PROPN
ap-1408	117	15	∈	∈	PROPN
ap-1408	117	16	gr(h	gr(h	NOUN
ap-1408	117	17	)	)	PUNCT
ap-1408	117	18	by	by	ADP
ap-1408	117	19	a	a	DET
ap-1408	117	20	⊕d	⊕d	NOUN
ap-1408	117	21	b	b	PROPN
ap-1408	117	22	=	=	SYM
ap-1408	117	23	⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩	⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩	PROPN
ap-1408	117	24	a+b	a+b	X
ap-1408	117	25	(	(	PUNCT
ap-1408	117	26	the	the	DET
ap-1408	117	27	usual	usual	ADJ
ap-1408	117	28	sum	sum	NOUN
ap-1408	117	29	)	)	PUNCT
ap-1408	117	30	if	if	SCONJ
ap-1408	117	31	a+b	a+b	NUM
ap-1408	117	32	is	be	AUX
ap-1408	117	33	unbounded	unbounded	ADJ
ap-1408	117	34	and	and	CCONJ
ap-1408	117	35	(	(	PUNCT
ap-1408	117	36	d(a	d(a	PROPN
ap-1408	117	37	)	)	PUNCT
ap-1408	117	38	=	=	SYM
ap-1408	118	1	d(b	d(b	X
ap-1408	118	2	)	)	PUNCT
ap-1408	118	3	or	or	CCONJ
ap-1408	118	4	one	one	NUM
ap-1408	118	5	out	out	ADP
ap-1408	118	6	of	of	ADP
ap-1408	118	7	a	a	PRON
ap-1408	118	8	,	,	PUNCT
ap-1408	118	9	b	b	PROPN
ap-1408	118	10	is	be	AUX
ap-1408	118	11	bounded	bound	VERB
ap-1408	118	12	)	)	PUNCT
ap-1408	118	13	,	,	PUNCT
ap-1408	118	14	(	(	PUNCT
ap-1408	118	15	a+b)b	a+b)b	VERB
ap-1408	118	16	if	if	SCONJ
ap-1408	118	17	a+b	a+b	NUM
ap-1408	118	18	is	be	AUX
ap-1408	118	19	bounded	bound	VERB
ap-1408	118	20	and	and	CCONJ
ap-1408	118	21	d(a	d(a	PROPN
ap-1408	118	22	)	)	PUNCT
ap-1408	118	23	=	=	PUNCT
ap-1408	118	24	d(b	d(b	PROPN
ap-1408	118	25	)	)	PUNCT
ap-1408	118	26	,	,	PUNCT
ap-1408	118	27	undefined	undefined	ADJ
ap-1408	118	28	otherwise	otherwise	ADV
ap-1408	118	29	and	and	CCONJ
ap-1408	118	30	≤	≤	NUM
ap-1408	118	31	be	be	VERB
ap-1408	118	32	a	a	DET
ap-1408	118	33	relation	relation	NOUN
ap-1408	118	34	on	on	ADP
ap-1408	118	35	gr(h	gr(h	NOUN
ap-1408	118	36	)	)	PUNCT
ap-1408	118	37	defined	define	VERB
ap-1408	118	38	for	for	ADP
ap-1408	118	39	a	a	DET
ap-1408	118	40	,	,	PUNCT
ap-1408	118	41	b	b	PROPN
ap-1408	118	42	∈	∈	PROPN
ap-1408	118	43	gr(h	gr(h	NOUN
ap-1408	118	44	)	)	PUNCT
ap-1408	118	45	by	by	ADP
ap-1408	118	46	a	a	DET
ap-1408	118	47	≤	≤	PROPN
ap-1408	118	48	b	b	SYM
ap-1408	118	49	iff	iff	PROPN
ap-1408	118	50	there	there	PRON
ap-1408	118	51	is	be	VERB
ap-1408	118	52	a	a	DET
ap-1408	118	53	positive	positive	ADJ
ap-1408	118	54	linear	linear	NOUN
ap-1408	118	55	operator	operator	NOUN
ap-1408	118	56	c	c	PROPN
ap-1408	118	57	∈	∈	PROPN
ap-1408	118	58	gr(h	gr(h	NOUN
ap-1408	118	59	)	)	PUNCT
ap-1408	118	60	such	such	ADJ
ap-1408	118	61	that	that	PRON
ap-1408	118	62	b	b	X
ap-1408	118	63	=	=	PUNCT
ap-1408	118	64	a	a	DET
ap-1408	118	65	⊕d	⊕d	NOUN
ap-1408	118	66	c.	c.	PROPN
ap-1408	118	67	then	then	ADV
ap-1408	118	68	gr(h	gr(h	PUNCT
ap-1408	118	69	)	)	PUNCT
ap-1408	119	1	=	=	SYM
ap-1408	119	2	(	(	PUNCT
ap-1408	119	3	gr(h);⊕d	gr(h);⊕d	PROPN
ap-1408	119	4	,	,	PUNCT
ap-1408	119	5	0	0	NUM
ap-1408	119	6	)	)	PUNCT
ap-1408	119	7	is	be	AUX
ap-1408	119	8	a	a	DET
ap-1408	119	9	wop	wop	NOUN
ap-1408	119	10	-	-	PUNCT
ap-1408	119	11	group	group	NOUN
ap-1408	119	12	with	with	ADP
ap-1408	119	13	respect	respect	NOUN
ap-1408	119	14	to	to	ADP
ap-1408	119	15	≤.	≤.	NOUN
ap-1408	119	16	proof	proof	NOUN
ap-1408	119	17	.	.	PUNCT
ap-1408	120	1	let	let	VERB
ap-1408	120	2	a	a	DET
ap-1408	120	3	,	,	PUNCT
ap-1408	120	4	b	b	NOUN
ap-1408	120	5	,	,	PUNCT
ap-1408	120	6	c	c	PROPN
ap-1408	120	7	∈	∈	PROPN
ap-1408	120	8	gr(h	gr(h	NOUN
ap-1408	120	9	)	)	PUNCT
ap-1408	120	10	.	.	PUNCT
ap-1408	121	1	then	then	ADV
ap-1408	121	2	(	(	PUNCT
ap-1408	121	3	gi	gi	INTJ
ap-1408	121	4	)	)	PUNCT
ap-1408	121	5	is	be	AUX
ap-1408	121	6	valid	valid	ADJ
ap-1408	121	7	since	since	SCONJ
ap-1408	121	8	a	a	DET
ap-1408	121	9	⊕d	⊕d	NOUN
ap-1408	121	10	b	b	NOUN
ap-1408	121	11	is	be	AUX
ap-1408	121	12	defined	define	VERB
ap-1408	121	13	iff	iff	PROPN
ap-1408	121	14	b	b	PROPN
ap-1408	121	15	⊕d	⊕d	PROPN
ap-1408	121	16	a	a	PRON
ap-1408	121	17	is	be	AUX
ap-1408	121	18	defined	define	VERB
ap-1408	121	19	and	and	CCONJ
ap-1408	121	20	because	because	SCONJ
ap-1408	121	21	the	the	DET
ap-1408	121	22	usual	usual	ADJ
ap-1408	121	23	sum	sum	NOUN
ap-1408	121	24	is	be	AUX
ap-1408	121	25	commutative	commutative	ADJ
ap-1408	121	26	we	we	PRON
ap-1408	121	27	get	get	VERB
ap-1408	121	28	that	that	PRON
ap-1408	121	29	a	a	DET
ap-1408	121	30	⊕d	⊕d	NOUN
ap-1408	121	31	b	b	NOUN
ap-1408	121	32	=	=	SYM
ap-1408	121	33	b	b	PROPN
ap-1408	121	34	⊕d	⊕d	NOUN
ap-1408	121	35	a.	a.	NOUN
ap-1408	121	36	moreover	moreover	ADV
ap-1408	121	37	,	,	PUNCT
ap-1408	121	38	(	(	PUNCT
ap-1408	121	39	giii	giii	NOUN
ap-1408	121	40	)	)	PUNCT
ap-1408	121	41	is	be	AUX
ap-1408	121	42	valid	valid	ADJ
ap-1408	121	43	since	since	SCONJ
ap-1408	121	44	a	a	DET
ap-1408	121	45	⊕d	⊕d	NOUN
ap-1408	121	46	0	0	PUNCT
ap-1408	121	47	is	be	AUX
ap-1408	121	48	always	always	ADV
ap-1408	121	49	defined	define	VERB
ap-1408	121	50	and	and	CCONJ
ap-1408	121	51	a	a	DET
ap-1408	121	52	⊕d	⊕d	NOUN
ap-1408	121	53	0	0	PUNCT
ap-1408	122	1	=	=	SYM
ap-1408	122	2	a.	a.	NOUN
ap-1408	122	3	clearly	clearly	ADV
ap-1408	122	4	,	,	PUNCT
ap-1408	122	5	(	(	PUNCT
ap-1408	122	6	giv	giv	NOUN
ap-1408	122	7	)	)	PUNCT
ap-1408	122	8	follows	follow	VERB
ap-1408	122	9	from	from	ADP
ap-1408	122	10	the	the	DET
ap-1408	122	11	fact	fact	NOUN
ap-1408	122	12	that	that	SCONJ
ap-1408	122	13	for	for	ADP
ap-1408	122	14	every	every	DET
ap-1408	122	15	a	a	DET
ap-1408	122	16	∈	∈	PROPN
ap-1408	122	17	gr(h	gr(h	NOUN
ap-1408	122	18	)	)	PUNCT
ap-1408	122	19	there	there	PRON
ap-1408	122	20	exists	exist	VERB
ap-1408	122	21	a	a	DET
ap-1408	122	22	unique	unique	ADJ
ap-1408	122	23	b	b	NOUN
ap-1408	122	24	∈	∈	PROPN
ap-1408	122	25	gr(h	gr(h	NOUN
ap-1408	122	26	)	)	PUNCT
ap-1408	122	27	such	such	ADJ
ap-1408	122	28	that	that	SCONJ
ap-1408	122	29	a	a	DET
ap-1408	122	30	⊕d	⊕d	NOUN
ap-1408	122	31	b	b	NOUN
ap-1408	122	32	=	=	SYM
ap-1408	122	33	0	0	PROPN
ap-1408	123	1	(	(	PUNCT
ap-1408	123	2	namely	namely	ADV
ap-1408	123	3	we	we	PRON
ap-1408	123	4	put	put	VERB
ap-1408	123	5	−a	−a	NOUN
ap-1408	123	6	=	=	SYM
ap-1408	123	7	b	b	NOUN
ap-1408	123	8	and	and	CCONJ
ap-1408	123	9	evidently	evidently	ADV
ap-1408	123	10	a	a	DET
ap-1408	123	11	+	+	NOUN
ap-1408	123	12	b	b	NOUN
ap-1408	123	13	=	=	SYM
ap-1408	123	14	0	0	NUM
ap-1408	123	15	/	/	SYM
ap-1408	123	16	d(a	d(a	PROPN
ap-1408	123	17	)	)	PUNCT
ap-1408	123	18	yields	yield	VERB
ap-1408	123	19	0b	0b	PROPN
ap-1408	123	20	/	/	SYM
ap-1408	123	21	d(a	d(a	PROPN
ap-1408	123	22	)	)	PUNCT
ap-1408	123	23	=	=	PUNCT
ap-1408	123	24	0	0	NUM
ap-1408	123	25	)	)	PUNCT
ap-1408	123	26	.	.	PUNCT
ap-1408	124	1	it	it	PRON
ap-1408	124	2	remains	remain	VERB
ap-1408	124	3	to	to	PART
ap-1408	124	4	check	check	VERB
ap-1408	124	5	(	(	PUNCT
ap-1408	124	6	gii	gii	NOUN
ap-1408	124	7	)	)	PUNCT
ap-1408	124	8	and	and	CCONJ
ap-1408	124	9	(	(	PUNCT
ap-1408	124	10	gv	gv	NOUN
ap-1408	124	11	)	)	PUNCT
ap-1408	124	12	.	.	PUNCT
ap-1408	125	1	this	this	PRON
ap-1408	125	2	will	will	AUX
ap-1408	125	3	be	be	AUX
ap-1408	125	4	proved	prove	VERB
ap-1408	125	5	by	by	ADP
ap-1408	125	6	cases	case	NOUN
ap-1408	125	7	.	.	PUNCT
ap-1408	126	1	assume	assume	VERB
ap-1408	126	2	that	that	SCONJ
ap-1408	126	3	(	(	PUNCT
ap-1408	126	4	a⊕d	a⊕d	NOUN
ap-1408	126	5	b)⊕d	b)⊕d	ADP
ap-1408	126	6	c	c	NOUN
ap-1408	126	7	is	be	AUX
ap-1408	126	8	defined	define	VERB
ap-1408	126	9	and	and	CCONJ
ap-1408	126	10	a	a	DET
ap-1408	126	11	⊕d	⊕d	NOUN
ap-1408	126	12	(	(	PUNCT
ap-1408	126	13	b	b	NOUN
ap-1408	126	14	⊕d	⊕d	NOUN
ap-1408	126	15	c	c	NOUN
ap-1408	126	16	)	)	PUNCT
ap-1408	126	17	is	be	AUX
ap-1408	126	18	defined	define	VERB
ap-1408	126	19	.	.	PUNCT
ap-1408	127	1	first	first	ADV
ap-1408	127	2	,	,	PUNCT
ap-1408	127	3	let	let	VERB
ap-1408	127	4	(	(	PUNCT
ap-1408	127	5	a⊕db)⊕dc	a⊕db)⊕dc	NOUN
ap-1408	127	6	be	be	AUX
ap-1408	127	7	of	of	ADP
ap-1408	127	8	the	the	DET
ap-1408	127	9	form	form	NOUN
ap-1408	127	10	(	(	PUNCT
ap-1408	127	11	a+b)+c	a+b)+c	NOUN
ap-1408	127	12	,	,	PUNCT
ap-1408	127	13	hence	hence	ADV
ap-1408	127	14	(	(	PUNCT
ap-1408	127	15	a	a	DET
ap-1408	127	16	+	+	NOUN
ap-1408	127	17	b	b	NOUN
ap-1408	127	18	)	)	PUNCT
ap-1408	128	1	+	+	CCONJ
ap-1408	128	2	c	c	NOUN
ap-1408	128	3	is	be	AUX
ap-1408	128	4	unbounded	unbounded	ADJ
ap-1408	128	5	.	.	PUNCT
ap-1408	129	1	then	then	ADV
ap-1408	129	2	d(a	d(a	PROPN
ap-1408	129	3	+	+	PROPN
ap-1408	129	4	b	b	X
ap-1408	129	5	)	)	PUNCT
ap-1408	129	6	=	=	SYM
ap-1408	129	7	d(a)∩d(b	d(a)∩d(b	X
ap-1408	129	8	)	)	PUNCT
ap-1408	129	9	∈	∈	PROPN
ap-1408	129	10	{	{	PUNCT
ap-1408	129	11	d(a	d(a	PROPN
ap-1408	129	12	)	)	PUNCT
ap-1408	129	13	,	,	PUNCT
ap-1408	129	14	d(b	d(b	PROPN
ap-1408	129	15	)	)	PUNCT
ap-1408	129	16	}	}	PUNCT
ap-1408	129	17	and	and	CCONJ
ap-1408	129	18	d((a+b)+c	d((a+b)+c	NOUN
ap-1408	129	19	)	)	PUNCT
ap-1408	129	20	=	=	SYM
ap-1408	129	21	d(a	d(a	PROPN
ap-1408	129	22	+	+	CCONJ
ap-1408	129	23	b	b	NOUN
ap-1408	129	24	)	)	PUNCT
ap-1408	129	25	∩	∩	ADJ
ap-1408	129	26	d(c	d(c	PROPN
ap-1408	129	27	)	)	PUNCT
ap-1408	129	28	∈	∈	PROPN
ap-1408	129	29	{	{	PUNCT
ap-1408	129	30	d(a	d(a	PROPN
ap-1408	129	31	)	)	PUNCT
ap-1408	129	32	,	,	PUNCT
ap-1408	129	33	d(b	d(b	PROPN
ap-1408	129	34	)	)	PUNCT
ap-1408	129	35	,	,	PUNCT
ap-1408	129	36	d(c	d(c	PROPN
ap-1408	129	37	)	)	PUNCT
ap-1408	129	38	}	}	PUNCT
ap-1408	129	39	.	.	PUNCT
ap-1408	130	1	assume	assume	VERB
ap-1408	130	2	for	for	ADP
ap-1408	130	3	the	the	DET
ap-1408	130	4	moment	moment	NOUN
ap-1408	130	5	that	that	DET
ap-1408	130	6	d((a	d((a	PROPN
ap-1408	131	1	+	+	CCONJ
ap-1408	132	1	b	b	X
ap-1408	132	2	)	)	PUNCT
ap-1408	132	3	+	+	NOUN
ap-1408	133	1	c	c	X
ap-1408	133	2	)	)	PUNCT
ap-1408	133	3	=	=	SYM
ap-1408	133	4	d(a	d(a	PROPN
ap-1408	133	5	)	)	PUNCT
ap-1408	133	6	�	�	PROPN
ap-1408	133	7	=	=	SYM
ap-1408	133	8	h	h	PROPN
ap-1408	133	9	(	(	PUNCT
ap-1408	133	10	the	the	DET
ap-1408	133	11	other	other	ADJ
ap-1408	133	12	cases	case	NOUN
ap-1408	133	13	follow	follow	VERB
ap-1408	133	14	by	by	ADP
ap-1408	133	15	a	a	DET
ap-1408	133	16	symmetric	symmetric	ADJ
ap-1408	133	17	argument	argument	NOUN
ap-1408	133	18	)	)	PUNCT
ap-1408	133	19	.	.	PUNCT
ap-1408	134	1	we	we	PRON
ap-1408	134	2	have	have	VERB
ap-1408	134	3	the	the	DET
ap-1408	134	4	following	follow	VERB
ap-1408	134	5	possibilities	possibility	NOUN
ap-1408	134	6	:	:	PUNCT
ap-1408	134	7	(	(	PUNCT
ap-1408	134	8	α1	α1	PROPN
ap-1408	134	9	):	):	PUNCT
ap-1408	134	10	d(a	d(a	PROPN
ap-1408	134	11	)	)	PUNCT
ap-1408	134	12	=	=	PUNCT
ap-1408	134	13	d(b	d(b	PROPN
ap-1408	134	14	)	)	PUNCT
ap-1408	134	15	,	,	PUNCT
ap-1408	134	16	hence	hence	ADV
ap-1408	134	17	a	a	DET
ap-1408	134	18	+	+	NOUN
ap-1408	134	19	b	b	NOUN
ap-1408	134	20	is	be	AUX
ap-1408	134	21	unbounded	unbounded	ADJ
ap-1408	134	22	and	and	CCONJ
ap-1408	134	23	d((a	d((a	PROPN
ap-1408	135	1	+	+	CCONJ
ap-1408	136	1	b	b	X
ap-1408	136	2	)	)	PUNCT
ap-1408	136	3	+	+	NOUN
ap-1408	137	1	c	c	X
ap-1408	137	2	)	)	PUNCT
ap-1408	137	3	=	=	SYM
ap-1408	138	1	d(a	d(a	PROPN
ap-1408	138	2	+	+	CCONJ
ap-1408	138	3	b	b	X
ap-1408	138	4	)	)	PUNCT
ap-1408	138	5	=	=	SYM
ap-1408	138	6	d(a	d(a	PROPN
ap-1408	138	7	)	)	PUNCT
ap-1408	138	8	=	=	PUNCT
ap-1408	138	9	d(b	d(b	PROPN
ap-1408	138	10	)	)	PUNCT
ap-1408	138	11	.	.	PUNCT
ap-1408	139	1	then	then	ADV
ap-1408	139	2	either	either	CCONJ
ap-1408	139	3	c	c	PROPN
ap-1408	139	4	is	be	AUX
ap-1408	139	5	unbounded	unbounded	ADJ
ap-1408	139	6	and	and	CCONJ
ap-1408	139	7	d(a	d(a	PROPN
ap-1408	139	8	+	+	PROPN
ap-1408	139	9	b	b	X
ap-1408	139	10	)	)	PUNCT
ap-1408	139	11	=	=	SYM
ap-1408	139	12	d(c	d(c	PROPN
ap-1408	139	13	)	)	PUNCT
ap-1408	139	14	or	or	CCONJ
ap-1408	139	15	c	c	PROPN
ap-1408	139	16	is	be	AUX
ap-1408	139	17	bounded	bound	VERB
ap-1408	139	18	and	and	CCONJ
ap-1408	139	19	d(a	d(a	PROPN
ap-1408	140	1	+	+	PROPN
ap-1408	140	2	b	b	X
ap-1408	140	3	)	)	PUNCT
ap-1408	140	4	⊂	⊂	PROPN
ap-1408	140	5	d(c	d(c	PROPN
ap-1408	140	6	)	)	PUNCT
ap-1408	140	7	.	.	PUNCT
ap-1408	141	1	in	in	ADP
ap-1408	141	2	both	both	DET
ap-1408	141	3	cases	case	NOUN
ap-1408	141	4	we	we	PRON
ap-1408	141	5	have	have	VERB
ap-1408	141	6	that	that	PRON
ap-1408	141	7	d(b	d(b	PROPN
ap-1408	141	8	+	+	CCONJ
ap-1408	141	9	c	c	X
ap-1408	141	10	)	)	PUNCT
ap-1408	141	11	=	=	SYM
ap-1408	141	12	d(b	d(b	X
ap-1408	141	13	)	)	PUNCT
ap-1408	142	1	=	=	SYM
ap-1408	143	1	d(a	d(a	PROPN
ap-1408	143	2	)	)	PUNCT
ap-1408	143	3	.	.	PUNCT
ap-1408	144	1	but	but	CCONJ
ap-1408	144	2	this	this	DET
ap-1408	144	3	yields	yield	NOUN
ap-1408	144	4	that	that	PRON
ap-1408	144	5	(	(	PUNCT
ap-1408	144	6	a	a	DET
ap-1408	144	7	+	+	NOUN
ap-1408	144	8	b	b	NOUN
ap-1408	144	9	)	)	PUNCT
ap-1408	144	10	+	+	NOUN
ap-1408	144	11	c	c	X
ap-1408	144	12	=	=	PUNCT
ap-1408	144	13	a	a	PRON
ap-1408	144	14	+	+	X
ap-1408	144	15	(	(	PUNCT
ap-1408	144	16	b	b	X
ap-1408	144	17	+	+	CCONJ
ap-1408	144	18	c	c	X
ap-1408	144	19	)	)	PUNCT
ap-1408	144	20	=	=	NOUN
ap-1408	145	1	a	a	DET
ap-1408	145	2	⊕d	⊕d	NOUN
ap-1408	145	3	(	(	PUNCT
ap-1408	145	4	b	b	NOUN
ap-1408	145	5	⊕d	⊕d	NOUN
ap-1408	145	6	c	c	NOUN
ap-1408	145	7	)	)	PUNCT
ap-1408	145	8	.	.	PUNCT
ap-1408	146	1	(	(	PUNCT
ap-1408	146	2	β1	β1	NOUN
ap-1408	146	3	):	):	PUNCT
ap-1408	146	4	d(a	d(a	PROPN
ap-1408	146	5	)	)	PUNCT
ap-1408	146	6	�	�	PROPN
ap-1408	146	7	=	=	SYM
ap-1408	146	8	d(b	d(b	PROPN
ap-1408	146	9	)	)	PUNCT
ap-1408	146	10	,	,	PUNCT
ap-1408	146	11	hence	hence	ADV
ap-1408	146	12	a	a	DET
ap-1408	146	13	+	+	NOUN
ap-1408	146	14	b	b	NOUN
ap-1408	146	15	is	be	AUX
ap-1408	146	16	unbounded	unbounde	VERB
ap-1408	146	17	,	,	PUNCT
ap-1408	146	18	d(a	d(a	PROPN
ap-1408	146	19	+	+	PROPN
ap-1408	146	20	b	b	X
ap-1408	146	21	)	)	PUNCT
ap-1408	146	22	=	=	SYM
ap-1408	147	1	d(a	d(a	PROPN
ap-1408	147	2	)	)	PUNCT
ap-1408	147	3	and	and	CCONJ
ap-1408	147	4	b	b	PROPN
ap-1408	147	5	is	be	AUX
ap-1408	147	6	bounded	bound	VERB
ap-1408	147	7	.	.	PUNCT
ap-1408	148	1	as	as	ADP
ap-1408	148	2	in	in	ADP
ap-1408	148	3	(	(	PUNCT
ap-1408	148	4	α1	α1	PROPN
ap-1408	148	5	)	)	PUNCT
ap-1408	148	6	,	,	PUNCT
ap-1408	148	7	either	either	CCONJ
ap-1408	148	8	c	c	NOUN
ap-1408	148	9	is	be	AUX
ap-1408	148	10	unbounded	unbounded	ADJ
ap-1408	148	11	and	and	CCONJ
ap-1408	148	12	d(a+b	d(a+b	NUM
ap-1408	148	13	)	)	PUNCT
ap-1408	148	14	=	=	SYM
ap-1408	148	15	d(c	d(c	PROPN
ap-1408	148	16	)	)	PUNCT
ap-1408	148	17	or	or	CCONJ
ap-1408	148	18	c	c	PROPN
ap-1408	148	19	is	be	AUX
ap-1408	148	20	bounded	bound	VERB
ap-1408	148	21	and	and	CCONJ
ap-1408	148	22	d(a+b	d(a+b	NOUN
ap-1408	148	23	)	)	PUNCT
ap-1408	149	1	⊂	⊂	PROPN
ap-1408	149	2	d(c	d(c	PROPN
ap-1408	149	3	)	)	PUNCT
ap-1408	149	4	,	,	PUNCT
ap-1408	149	5	d(b	d(b	X
ap-1408	149	6	)	)	PUNCT
ap-1408	149	7	=	=	SYM
ap-1408	149	8	d(c	d(c	PROPN
ap-1408	149	9	)	)	PUNCT
ap-1408	150	1	=	=	SYM
ap-1408	151	1	h.	h.	PROPN
ap-1408	151	2	in	in	ADP
ap-1408	151	3	both	both	DET
ap-1408	151	4	cases	case	NOUN
ap-1408	151	5	we	we	PRON
ap-1408	151	6	have	have	VERB
ap-1408	151	7	that	that	PRON
ap-1408	151	8	d(b	d(b	PROPN
ap-1408	151	9	+	+	CCONJ
ap-1408	151	10	c	c	X
ap-1408	151	11	)	)	PUNCT
ap-1408	151	12	=	=	SYM
ap-1408	151	13	d(c	d(c	PROPN
ap-1408	151	14	)	)	PUNCT
ap-1408	151	15	⊇	⊇	PROPN
ap-1408	151	16	d(a	d(a	PROPN
ap-1408	151	17	)	)	PUNCT
ap-1408	151	18	.	.	PUNCT
ap-1408	152	1	hence	hence	ADV
ap-1408	152	2	again	again	ADV
ap-1408	152	3	(	(	PUNCT
ap-1408	152	4	a	a	DET
ap-1408	152	5	+	+	NOUN
ap-1408	152	6	b	b	NOUN
ap-1408	152	7	)	)	PUNCT
ap-1408	152	8	+	+	NOUN
ap-1408	153	1	c	c	X
ap-1408	153	2	=	=	PUNCT
ap-1408	153	3	a	a	DET
ap-1408	153	4	+	+	X
ap-1408	153	5	(	(	PUNCT
ap-1408	153	6	b	b	X
ap-1408	153	7	+	+	CCONJ
ap-1408	153	8	c	c	X
ap-1408	153	9	)	)	PUNCT
ap-1408	153	10	=	=	NOUN
ap-1408	154	1	a	a	DET
ap-1408	154	2	⊕d	⊕d	NOUN
ap-1408	154	3	(	(	PUNCT
ap-1408	154	4	b	b	NOUN
ap-1408	154	5	⊕d	⊕d	NOUN
ap-1408	154	6	c	c	NOUN
ap-1408	154	7	)	)	PUNCT
ap-1408	154	8	.	.	PUNCT
ap-1408	155	1	similarly	similarly	ADV
ap-1408	155	2	,	,	PUNCT
ap-1408	155	3	let	let	VERB
ap-1408	155	4	(	(	PUNCT
ap-1408	155	5	a	a	DET
ap-1408	155	6	⊕d	⊕d	NOUN
ap-1408	155	7	b	b	X
ap-1408	155	8	)	)	PUNCT
ap-1408	155	9	⊕d	⊕d	NOUN
ap-1408	155	10	c	c	NOUN
ap-1408	155	11	be	be	AUX
ap-1408	155	12	of	of	ADP
ap-1408	155	13	the	the	DET
ap-1408	155	14	form	form	NOUN
ap-1408	155	15	(	(	PUNCT
ap-1408	155	16	(	(	PUNCT
ap-1408	155	17	a	a	DET
ap-1408	155	18	+	+	NOUN
ap-1408	155	19	b	b	NOUN
ap-1408	155	20	)	)	PUNCT
ap-1408	156	1	+	+	CCONJ
ap-1408	156	2	c)b	c)b	NOUN
ap-1408	156	3	,	,	PUNCT
ap-1408	156	4	hence	hence	ADV
ap-1408	156	5	(	(	PUNCT
ap-1408	156	6	a	a	DET
ap-1408	156	7	+	+	NOUN
ap-1408	156	8	b	b	NOUN
ap-1408	156	9	)	)	PUNCT
ap-1408	156	10	+	+	CCONJ
ap-1408	156	11	c	c	NOUN
ap-1408	156	12	is	be	AUX
ap-1408	156	13	bounded	bound	VERB
ap-1408	156	14	.	.	PUNCT
ap-1408	157	1	because	because	SCONJ
ap-1408	157	2	(	(	PUNCT
ap-1408	157	3	a+b	a+b	NUM
ap-1408	157	4	)	)	PUNCT
ap-1408	157	5	is	be	AUX
ap-1408	157	6	unbounded	unbounde	VERB
ap-1408	157	7	,	,	PUNCT
ap-1408	157	8	c	c	PROPN
ap-1408	157	9	is	be	AUX
ap-1408	157	10	also	also	ADV
ap-1408	157	11	unbounded	unbounded	ADJ
ap-1408	157	12	.	.	PUNCT
ap-1408	158	1	then	then	ADV
ap-1408	158	2	d(a	d(a	PROPN
ap-1408	158	3	+	+	PROPN
ap-1408	158	4	b	b	X
ap-1408	158	5	)	)	PUNCT
ap-1408	158	6	=	=	SYM
ap-1408	158	7	d(a	d(a	ADJ
ap-1408	158	8	)	)	PUNCT
ap-1408	158	9	∩	∩	NOUN
ap-1408	158	10	d(b	d(b	X
ap-1408	158	11	)	)	PUNCT
ap-1408	158	12	∈	∈	PROPN
ap-1408	158	13	{	{	PUNCT
ap-1408	158	14	d(a	d(a	PROPN
ap-1408	158	15	)	)	PUNCT
ap-1408	158	16	,	,	PUNCT
ap-1408	158	17	d(b	d(b	PROPN
ap-1408	158	18	)	)	PUNCT
ap-1408	158	19	}	}	PUNCT
ap-1408	158	20	and	and	CCONJ
ap-1408	158	21	d(a	d(a	PROPN
ap-1408	158	22	+	+	PROPN
ap-1408	158	23	b	b	X
ap-1408	158	24	)	)	PUNCT
ap-1408	158	25	=	=	SYM
ap-1408	158	26	d(c	d(c	PROPN
ap-1408	158	27	)	)	PUNCT
ap-1408	158	28	.	.	PUNCT
ap-1408	159	1	assume	assume	VERB
ap-1408	159	2	that	that	SCONJ
ap-1408	159	3	d(a	d(a	PROPN
ap-1408	159	4	)	)	PUNCT
ap-1408	159	5	�	�	PROPN
ap-1408	159	6	=	=	SYM
ap-1408	159	7	h	h	PROPN
ap-1408	159	8	(	(	PUNCT
ap-1408	159	9	the	the	DET
ap-1408	159	10	other	other	ADJ
ap-1408	159	11	case	case	NOUN
ap-1408	159	12	where	where	SCONJ
ap-1408	159	13	d(b	d(b	NOUN
ap-1408	159	14	)	)	PUNCT
ap-1408	159	15	�	�	NOUN
ap-1408	159	16	=	=	SYM
ap-1408	159	17	h	h	NOUN
ap-1408	159	18	is	be	AUX
ap-1408	159	19	symmetric	symmetric	ADJ
ap-1408	159	20	)	)	PUNCT
ap-1408	159	21	.	.	PUNCT
ap-1408	160	1	we	we	PRON
ap-1408	160	2	will	will	AUX
ap-1408	160	3	distinguish	distinguish	VERB
ap-1408	160	4	the	the	DET
ap-1408	160	5	following	follow	VERB
ap-1408	160	6	cases	case	NOUN
ap-1408	160	7	:	:	PUNCT
ap-1408	160	8	(	(	PUNCT
ap-1408	160	9	α2	α2	ADV
ap-1408	160	10	):	):	PUNCT
ap-1408	160	11	d(a	d(a	PROPN
ap-1408	160	12	)	)	PUNCT
ap-1408	160	13	=	=	PUNCT
ap-1408	160	14	d(b	d(b	PROPN
ap-1408	160	15	)	)	PUNCT
ap-1408	160	16	,	,	PUNCT
ap-1408	160	17	then	then	ADV
ap-1408	160	18	d(a	d(a	PROPN
ap-1408	160	19	+	+	PROPN
ap-1408	160	20	b	b	X
ap-1408	160	21	)	)	PUNCT
ap-1408	160	22	=	=	SYM
ap-1408	160	23	d(a	d(a	PROPN
ap-1408	160	24	)	)	PUNCT
ap-1408	160	25	=	=	SYM
ap-1408	160	26	d(b	d(b	X
ap-1408	160	27	)	)	PUNCT
ap-1408	160	28	and	and	CCONJ
ap-1408	160	29	because	because	SCONJ
ap-1408	160	30	d(c	d(c	PROPN
ap-1408	160	31	)	)	PUNCT
ap-1408	161	1	=	=	PUNCT
ap-1408	162	1	d(a	d(a	PROPN
ap-1408	162	2	+	+	CCONJ
ap-1408	162	3	b	b	X
ap-1408	162	4	)	)	PUNCT
ap-1408	162	5	we	we	PRON
ap-1408	162	6	have	have	VERB
ap-1408	162	7	d(a	d(a	PROPN
ap-1408	162	8	)	)	PUNCT
ap-1408	162	9	=	=	SYM
ap-1408	162	10	d(b	d(b	X
ap-1408	162	11	)	)	PUNCT
ap-1408	162	12	=	=	SYM
ap-1408	162	13	d(c	d(c	PROPN
ap-1408	162	14	)	)	PUNCT
ap-1408	162	15	.	.	PUNCT
ap-1408	163	1	so	so	ADV
ap-1408	163	2	(	(	PUNCT
ap-1408	163	3	(	(	PUNCT
ap-1408	163	4	a	a	DET
ap-1408	163	5	+	+	NOUN
ap-1408	163	6	b	b	NOUN
ap-1408	163	7	)	)	PUNCT
ap-1408	163	8	+	+	NUM
ap-1408	163	9	c)b	c)b	NOUN
ap-1408	163	10	=	=	SYM
ap-1408	163	11	(	(	PUNCT
ap-1408	163	12	a	a	DET
ap-1408	163	13	+	+	X
ap-1408	163	14	(	(	PUNCT
ap-1408	163	15	b	b	NOUN
ap-1408	163	16	+	+	NUM
ap-1408	163	17	c))b	c))b	NOUN
ap-1408	163	18	=	=	PUNCT
ap-1408	163	19	a	a	DET
ap-1408	163	20	⊕d	⊕d	NOUN
ap-1408	163	21	(	(	PUNCT
ap-1408	163	22	b	b	NOUN
ap-1408	163	23	⊕d	⊕d	NOUN
ap-1408	163	24	c	c	NOUN
ap-1408	163	25	)	)	PUNCT
ap-1408	163	26	.	.	PUNCT
ap-1408	164	1	(	(	PUNCT
ap-1408	164	2	β2	β2	NOUN
ap-1408	164	3	):	):	PUNCT
ap-1408	164	4	d(a	d(a	PROPN
ap-1408	164	5	)	)	PUNCT
ap-1408	164	6	�	�	PROPN
ap-1408	164	7	=	=	SYM
ap-1408	164	8	d(b	d(b	PROPN
ap-1408	164	9	)	)	PUNCT
ap-1408	164	10	,	,	PUNCT
ap-1408	164	11	then	then	ADV
ap-1408	164	12	b	b	PROPN
ap-1408	164	13	is	be	AUX
ap-1408	164	14	bounded	bound	VERB
ap-1408	164	15	and	and	CCONJ
ap-1408	164	16	d(b	d(b	NOUN
ap-1408	164	17	)	)	PUNCT
ap-1408	165	1	=	=	SYM
ap-1408	165	2	h	h	NOUN
ap-1408	165	3	,	,	PUNCT
ap-1408	165	4	hence	hence	ADV
ap-1408	165	5	d(a	d(a	PROPN
ap-1408	165	6	)	)	PUNCT
ap-1408	165	7	=	=	SYM
ap-1408	166	1	d(a	d(a	PROPN
ap-1408	166	2	+	+	CCONJ
ap-1408	166	3	b	b	X
ap-1408	166	4	)	)	PUNCT
ap-1408	166	5	=	=	SYM
ap-1408	166	6	d(c	d(c	PROPN
ap-1408	166	7	)	)	PUNCT
ap-1408	166	8	.	.	PUNCT
ap-1408	167	1	then	then	ADV
ap-1408	167	2	d(b	d(b	PROPN
ap-1408	167	3	+	+	CCONJ
ap-1408	167	4	c	c	X
ap-1408	167	5	)	)	PUNCT
ap-1408	167	6	=	=	SYM
ap-1408	167	7	d(c	d(c	PROPN
ap-1408	167	8	)	)	PUNCT
ap-1408	168	1	=	=	SYM
ap-1408	168	2	d(a	d(a	PROPN
ap-1408	168	3	)	)	PUNCT
ap-1408	168	4	,	,	PUNCT
ap-1408	168	5	which	which	PRON
ap-1408	168	6	yields	yield	VERB
ap-1408	168	7	that	that	PRON
ap-1408	168	8	(	(	PUNCT
ap-1408	168	9	(	(	PUNCT
ap-1408	168	10	a+b)+c)b	a+b)+c)b	VERB
ap-1408	168	11	=	=	SYM
ap-1408	168	12	(	(	PUNCT
ap-1408	168	13	a+(b	a+(b	PUNCT
ap-1408	168	14	+	+	NOUN
ap-1408	168	15	c))b	c))b	NOUN
ap-1408	168	16	=	=	SYM
ap-1408	168	17	a⊕d	a⊕d	NOUN
ap-1408	168	18	(	(	PUNCT
ap-1408	168	19	b⊕d	b⊕d	PROPN
ap-1408	168	20	c	c	NOUN
ap-1408	168	21	)	)	PUNCT
ap-1408	168	22	.	.	PUNCT
ap-1408	169	1	let	let	VERB
ap-1408	169	2	(	(	PUNCT
ap-1408	169	3	a⊕db)⊕dc	a⊕db)⊕dc	NOUN
ap-1408	169	4	be	be	AUX
ap-1408	169	5	of	of	ADP
ap-1408	169	6	the	the	DET
ap-1408	169	7	form	form	NOUN
ap-1408	169	8	(	(	PUNCT
ap-1408	169	9	(	(	PUNCT
ap-1408	169	10	a+b)b+c)b	a+b)b+c)b	ADV
ap-1408	169	11	,	,	PUNCT
ap-1408	169	12	hence	hence	ADV
ap-1408	169	13	(	(	PUNCT
ap-1408	169	14	a	a	DET
ap-1408	169	15	+	+	X
ap-1408	169	16	b)b	b)b	NOUN
ap-1408	170	1	+	+	CCONJ
ap-1408	170	2	c	c	X
ap-1408	170	3	is	be	AUX
ap-1408	170	4	bounded	bound	VERB
ap-1408	170	5	a	a	DET
ap-1408	170	6	+	+	NUM
ap-1408	170	7	b	b	NOUN
ap-1408	170	8	is	be	AUX
ap-1408	170	9	bounded	bound	VERB
ap-1408	170	10	and	and	CCONJ
ap-1408	170	11	then	then	ADV
ap-1408	170	12	c	c	PROPN
ap-1408	170	13	is	be	AUX
ap-1408	170	14	also	also	ADV
ap-1408	170	15	bounded	bound	VERB
ap-1408	170	16	.	.	PUNCT
ap-1408	171	1	we	we	PRON
ap-1408	171	2	will	will	AUX
ap-1408	171	3	verify	verify	VERB
ap-1408	171	4	each	each	PRON
ap-1408	171	5	of	of	ADP
ap-1408	171	6	the	the	DET
ap-1408	171	7	following	following	ADJ
ap-1408	171	8	cases	case	NOUN
ap-1408	171	9	:	:	PUNCT
ap-1408	171	10	(	(	PUNCT
ap-1408	171	11	α3	α3	NOUN
ap-1408	171	12	):	):	PUNCT
ap-1408	171	13	d(a	d(a	PROPN
ap-1408	171	14	)	)	PUNCT
ap-1408	171	15	=	=	SYM
ap-1408	171	16	d(b	d(b	PROPN
ap-1408	171	17	)	)	PUNCT
ap-1408	171	18	�	�	PROPN
ap-1408	171	19	=	=	SYM
ap-1408	171	20	h	h	PROPN
ap-1408	171	21	,	,	PUNCT
ap-1408	171	22	then	then	ADV
ap-1408	171	23	d(b	d(b	PROPN
ap-1408	171	24	+	+	CCONJ
ap-1408	171	25	c	c	X
ap-1408	171	26	)	)	PUNCT
ap-1408	171	27	=	=	SYM
ap-1408	171	28	d(b	d(b	X
ap-1408	171	29	)	)	PUNCT
ap-1408	172	1	=	=	SYM
ap-1408	172	2	d(a	d(a	PROPN
ap-1408	172	3	)	)	PUNCT
ap-1408	172	4	.	.	PUNCT
ap-1408	173	1	and	and	CCONJ
ap-1408	173	2	then	then	ADV
ap-1408	173	3	(	(	PUNCT
ap-1408	173	4	(	(	PUNCT
ap-1408	173	5	a	a	DET
ap-1408	173	6	+	+	X
ap-1408	173	7	b)b	b)b	NOUN
ap-1408	173	8	+	+	CCONJ
ap-1408	173	9	c)b	c)b	X
ap-1408	174	1	=	=	SYM
ap-1408	174	2	(	(	PUNCT
ap-1408	174	3	a	a	PRON
ap-1408	174	4	+	+	X
ap-1408	174	5	(	(	PUNCT
ap-1408	174	6	b	b	NOUN
ap-1408	174	7	+	+	NUM
ap-1408	174	8	c))b	c))b	NOUN
ap-1408	174	9	=	=	PUNCT
ap-1408	174	10	a	a	DET
ap-1408	174	11	⊕d	⊕d	NOUN
ap-1408	174	12	(	(	PUNCT
ap-1408	174	13	b	b	NOUN
ap-1408	174	14	⊕d	⊕d	NOUN
ap-1408	174	15	c	c	NOUN
ap-1408	174	16	)	)	PUNCT
ap-1408	174	17	.	.	PUNCT
ap-1408	175	1	(	(	PUNCT
ap-1408	175	2	β3	β3	VERB
ap-1408	175	3	):	):	PUNCT
ap-1408	175	4	d(a	d(a	PROPN
ap-1408	175	5	)	)	PUNCT
ap-1408	175	6	=	=	SYM
ap-1408	175	7	d(b	d(b	X
ap-1408	175	8	)	)	PUNCT
ap-1408	176	1	=	=	SYM
ap-1408	176	2	h	h	NOUN
ap-1408	176	3	,	,	PUNCT
ap-1408	176	4	therefore	therefore	ADV
ap-1408	176	5	d(b	d(b	PROPN
ap-1408	176	6	+	+	CCONJ
ap-1408	176	7	c	c	X
ap-1408	176	8	)	)	PUNCT
ap-1408	176	9	=	=	SYM
ap-1408	177	1	h	h	NOUN
ap-1408	177	2	=	=	SYM
ap-1408	177	3	d(a	d(a	PROPN
ap-1408	177	4	)	)	PUNCT
ap-1408	177	5	and	and	CCONJ
ap-1408	177	6	(	(	PUNCT
ap-1408	177	7	(	(	PUNCT
ap-1408	177	8	a+b)b+c)b	a+b)b+c)b	ADP
ap-1408	177	9	=	=	SYM
ap-1408	177	10	(	(	PUNCT
ap-1408	177	11	a+(b+c)b)b	a+(b+c)b)b	NOUN
ap-1408	177	12	=	=	PRON
ap-1408	177	13	a	a	DET
ap-1408	177	14	⊕d	⊕d	NOUN
ap-1408	177	15	(	(	PUNCT
ap-1408	177	16	b	b	NOUN
ap-1408	177	17	⊕d	⊕d	NOUN
ap-1408	177	18	c	c	NOUN
ap-1408	177	19	)	)	PUNCT
ap-1408	177	20	.	.	PUNCT
ap-1408	178	1	and	and	CCONJ
ap-1408	178	2	in	in	ADP
ap-1408	178	3	the	the	DET
ap-1408	178	4	last	last	ADJ
ap-1408	178	5	case	case	NOUN
ap-1408	178	6	,	,	PUNCT
ap-1408	178	7	let	let	VERB
ap-1408	178	8	(	(	PUNCT
ap-1408	178	9	a⊕d	a⊕d	NOUN
ap-1408	178	10	b)⊕d	b)⊕d	ADP
ap-1408	178	11	c	c	NOUN
ap-1408	178	12	be	be	AUX
ap-1408	178	13	of	of	ADP
ap-1408	178	14	the	the	DET
ap-1408	178	15	form	form	NOUN
ap-1408	178	16	(	(	PUNCT
ap-1408	178	17	(	(	PUNCT
ap-1408	178	18	a	a	DET
ap-1408	178	19	+	+	X
ap-1408	178	20	b)b	b)b	NOUN
ap-1408	179	1	+	+	CCONJ
ap-1408	179	2	c	c	X
ap-1408	179	3	)	)	PUNCT
ap-1408	179	4	.	.	PUNCT
ap-1408	180	1	that	that	PRON
ap-1408	180	2	is	be	AUX
ap-1408	180	3	,	,	PUNCT
ap-1408	180	4	(	(	PUNCT
ap-1408	180	5	a	a	DET
ap-1408	180	6	+	+	NOUN
ap-1408	180	7	b	b	NOUN
ap-1408	180	8	)	)	PUNCT
ap-1408	180	9	is	be	AUX
ap-1408	180	10	bounded	bound	VERB
ap-1408	180	11	and	and	CCONJ
ap-1408	180	12	c	c	NOUN
ap-1408	180	13	is	be	AUX
ap-1408	180	14	unbounded	unbounded	ADJ
ap-1408	180	15	.	.	PUNCT
ap-1408	181	1	then	then	ADV
ap-1408	181	2	we	we	PRON
ap-1408	181	3	prove	prove	VERB
ap-1408	181	4	:	:	PUNCT
ap-1408	181	5	(	(	PUNCT
ap-1408	181	6	α4	α4	NOUN
ap-1408	181	7	):	):	PUNCT
ap-1408	181	8	d(a	d(a	PROPN
ap-1408	181	9	)	)	PUNCT
ap-1408	181	10	=	=	SYM
ap-1408	181	11	d(b	d(b	X
ap-1408	181	12	)	)	PUNCT
ap-1408	181	13	=	=	SYM
ap-1408	182	1	h	h	NOUN
ap-1408	182	2	i.e.	i.e.	X
ap-1408	182	3	a	a	PRON
ap-1408	182	4	and	and	CCONJ
ap-1408	182	5	b	b	NOUN
ap-1408	182	6	are	be	AUX
ap-1408	182	7	bounded	bound	VERB
ap-1408	182	8	.	.	PUNCT
ap-1408	183	1	then	then	ADV
ap-1408	183	2	d((a	d((a	PROPN
ap-1408	183	3	+	+	CCONJ
ap-1408	183	4	b)b	b)b	NOUN
ap-1408	183	5	+	+	CCONJ
ap-1408	183	6	c	c	X
ap-1408	183	7	)	)	PUNCT
ap-1408	183	8	=	=	SYM
ap-1408	183	9	d(c	d(c	PROPN
ap-1408	183	10	)	)	PUNCT
ap-1408	183	11	and	and	CCONJ
ap-1408	183	12	d(c	d(c	PROPN
ap-1408	183	13	)	)	PUNCT
ap-1408	184	1	=	=	PUNCT
ap-1408	185	1	d(b	d(b	PROPN
ap-1408	185	2	+	+	CCONJ
ap-1408	185	3	c	c	X
ap-1408	185	4	)	)	PUNCT
ap-1408	185	5	=	=	SYM
ap-1408	186	1	d(a	d(a	PROPN
ap-1408	186	2	+	+	X
ap-1408	186	3	(	(	PUNCT
ap-1408	186	4	b	b	X
ap-1408	186	5	+	+	CCONJ
ap-1408	186	6	c	c	NOUN
ap-1408	186	7	)	)	PUNCT
ap-1408	186	8	)	)	PUNCT
ap-1408	186	9	.	.	PUNCT
ap-1408	187	1	hence	hence	ADV
ap-1408	187	2	(	(	PUNCT
ap-1408	187	3	(	(	PUNCT
ap-1408	187	4	a	a	DET
ap-1408	187	5	+	+	X
ap-1408	187	6	b)b	b)b	NOUN
ap-1408	187	7	+	+	CCONJ
ap-1408	187	8	c	c	X
ap-1408	187	9	)	)	PUNCT
ap-1408	187	10	=	=	NOUN
ap-1408	187	11	(	(	PUNCT
ap-1408	187	12	a	a	PRON
ap-1408	187	13	+	+	X
ap-1408	187	14	(	(	PUNCT
ap-1408	187	15	b	b	NOUN
ap-1408	187	16	+	+	CCONJ
ap-1408	187	17	c	c	NOUN
ap-1408	187	18	)	)	PUNCT
ap-1408	187	19	)	)	PUNCT
ap-1408	188	1	=	=	PRON
ap-1408	188	2	a⊕d	a⊕d	NOUN
ap-1408	188	3	(	(	PUNCT
ap-1408	188	4	b	b	NOUN
ap-1408	188	5	⊕d	⊕d	NOUN
ap-1408	188	6	c	c	NOUN
ap-1408	188	7	)	)	PUNCT
ap-1408	188	8	.	.	PUNCT
ap-1408	189	1	(	(	PUNCT
ap-1408	189	2	β4	β4	PROPN
ap-1408	189	3	):	):	PUNCT
ap-1408	189	4	d(a	d(a	PROPN
ap-1408	189	5	)	)	PUNCT
ap-1408	189	6	=	=	SYM
ap-1408	189	7	d(b	d(b	PROPN
ap-1408	189	8	)	)	PUNCT
ap-1408	189	9	�	�	PROPN
ap-1408	189	10	=	=	SYM
ap-1408	189	11	h	h	PROPN
ap-1408	189	12	,	,	PUNCT
ap-1408	189	13	i.e.	i.e.	X
ap-1408	189	14	a	a	PRON
ap-1408	189	15	and	and	CCONJ
ap-1408	189	16	b	b	NOUN
ap-1408	189	17	are	be	AUX
ap-1408	189	18	unbounded	unbounded	ADJ
ap-1408	189	19	.	.	PUNCT
ap-1408	190	1	then	then	ADV
ap-1408	190	2	if	if	SCONJ
ap-1408	190	3	d(b	d(b	NOUN
ap-1408	190	4	)	)	PUNCT
ap-1408	190	5	�	�	PROPN
ap-1408	190	6	=	=	SYM
ap-1408	190	7	d(c	d(c	PROPN
ap-1408	190	8	)	)	PUNCT
ap-1408	190	9	,	,	PUNCT
ap-1408	190	10	(	(	PUNCT
ap-1408	190	11	(	(	PUNCT
ap-1408	190	12	a⊕d	a⊕d	NOUN
ap-1408	190	13	b)⊕d	b)⊕d	NOUN
ap-1408	190	14	c	c	NOUN
ap-1408	190	15	)	)	PUNCT
ap-1408	190	16	=	=	SYM
ap-1408	191	1	(	(	PUNCT
ap-1408	191	2	(	(	PUNCT
ap-1408	191	3	a	a	DET
ap-1408	191	4	+	+	X
ap-1408	191	5	b)b	b)b	NOUN
ap-1408	191	6	+	+	CCONJ
ap-1408	191	7	c	c	X
ap-1408	191	8	)	)	PUNCT
ap-1408	191	9	is	be	AUX
ap-1408	191	10	defined	define	VERB
ap-1408	191	11	and	and	CCONJ
ap-1408	191	12	d((a	d((a	NOUN
ap-1408	191	13	+	+	CCONJ
ap-1408	191	14	b)b	b)b	NOUN
ap-1408	191	15	+	+	CCONJ
ap-1408	191	16	c	c	X
ap-1408	191	17	)	)	PUNCT
ap-1408	191	18	=	=	SYM
ap-1408	191	19	d(c	d(c	PROPN
ap-1408	191	20	)	)	PUNCT
ap-1408	191	21	,	,	PUNCT
ap-1408	191	22	but	but	CCONJ
ap-1408	191	23	(	(	PUNCT
ap-1408	191	24	b⊕dc	b⊕dc	PROPN
ap-1408	191	25	)	)	PUNCT
ap-1408	191	26	is	be	AUX
ap-1408	191	27	not	not	PART
ap-1408	191	28	defined	define	VERB
ap-1408	191	29	,	,	PUNCT
ap-1408	191	30	so	so	CCONJ
ap-1408	191	31	(	(	PUNCT
ap-1408	191	32	a⊕d(b⊕dc	a⊕d(b⊕dc	NOUN
ap-1408	191	33	)	)	PUNCT
ap-1408	191	34	)	)	PUNCT
ap-1408	191	35	is	be	AUX
ap-1408	191	36	not	not	PART
ap-1408	191	37	defined	define	VERB
ap-1408	191	38	.	.	PUNCT
ap-1408	192	1	in	in	ADP
ap-1408	192	2	the	the	DET
ap-1408	192	3	case	case	NOUN
ap-1408	192	4	that	that	SCONJ
ap-1408	192	5	d(b	d(b	PRON
ap-1408	192	6	)	)	PUNCT
ap-1408	192	7	=	=	SYM
ap-1408	192	8	d(c	d(c	PROPN
ap-1408	192	9	)	)	PUNCT
ap-1408	192	10	we	we	PRON
ap-1408	192	11	have	have	VERB
ap-1408	192	12	d(b	d(b	X
ap-1408	192	13	)	)	PUNCT
ap-1408	193	1	=	=	SYM
ap-1408	193	2	d(c	d(c	PROPN
ap-1408	193	3	)	)	PUNCT
ap-1408	194	1	=	=	SYM
ap-1408	194	2	d(a	d(a	PROPN
ap-1408	194	3	)	)	PUNCT
ap-1408	194	4	,	,	PUNCT
ap-1408	194	5	so	so	CCONJ
ap-1408	194	6	(	(	PUNCT
ap-1408	194	7	(	(	PUNCT
ap-1408	194	8	a	a	DET
ap-1408	194	9	+	+	X
ap-1408	194	10	b)b	b)b	NOUN
ap-1408	194	11	+	+	CCONJ
ap-1408	194	12	c	c	X
ap-1408	194	13	)	)	PUNCT
ap-1408	194	14	=	=	NOUN
ap-1408	194	15	(	(	PUNCT
ap-1408	194	16	a	a	DET
ap-1408	194	17	+	+	X
ap-1408	194	18	(	(	PUNCT
ap-1408	194	19	b	b	NOUN
ap-1408	194	20	+	+	CCONJ
ap-1408	194	21	c	c	NOUN
ap-1408	194	22	)	)	PUNCT
ap-1408	194	23	)	)	PUNCT
ap-1408	195	1	=	=	PUNCT
ap-1408	195	2	a	a	DET
ap-1408	195	3	⊕d	⊕d	NOUN
ap-1408	195	4	(	(	PUNCT
ap-1408	195	5	b	b	NOUN
ap-1408	195	6	⊕d	⊕d	NOUN
ap-1408	195	7	c	c	NOUN
ap-1408	195	8	)	)	PUNCT
ap-1408	195	9	.	.	PUNCT
ap-1408	196	1	67	67	NUM
ap-1408	196	2	acta	acta	PROPN
ap-1408	196	3	polytechnica	polytechnica	PROPN
ap-1408	196	4	vol	vol	NOUN
ap-1408	196	5	.	.	PUNCT
ap-1408	197	1	51	51	NUM
ap-1408	197	2	no	no	INTJ
ap-1408	197	3	.	.	PUNCT
ap-1408	198	1	4/2011	4/2011	NUM
ap-1408	198	2	table	table	NOUN
ap-1408	198	3	1	1	NUM
ap-1408	198	4	:	:	PUNCT
ap-1408	198	5	d(c1	d(c1	NOUN
ap-1408	198	6	)	)	PUNCT
ap-1408	198	7	d(c2	d(c2	NOUN
ap-1408	198	8	)	)	PUNCT
ap-1408	198	9	c1	c1	PROPN
ap-1408	198	10	⊕d	⊕d	PROPN
ap-1408	198	11	c2	c2	PROPN
ap-1408	198	12	d(c1	d(c1	PROPN
ap-1408	198	13	⊕d	⊕d	PROPN
ap-1408	198	14	c2	c2	PROPN
ap-1408	198	15	)	)	PUNCT
ap-1408	199	1	=	=	SYM
ap-1408	199	2	h	h	NOUN
ap-1408	200	1	=	=	NOUN
ap-1408	200	2	h	h	PROPN
ap-1408	200	3	(	(	PUNCT
ap-1408	200	4	c1	c1	PROPN
ap-1408	200	5	+	+	CCONJ
ap-1408	200	6	c2	c2	PROPN
ap-1408	200	7	)	)	PUNCT
ap-1408	200	8	b	b	NOUN
ap-1408	200	9	=	=	SYM
ap-1408	200	10	h	h	NOUN
ap-1408	200	11	=	=	PUNCT
ap-1408	200	12	d(b	d(b	X
ap-1408	200	13	)	)	PUNCT
ap-1408	201	1	=	=	SYM
ap-1408	201	2	h	h	PROPN
ap-1408	201	3	c1	c1	PROPN
ap-1408	201	4	+	+	CCONJ
ap-1408	201	5	c2	c2	PROPN
ap-1408	201	6	=	=	PUNCT
ap-1408	201	7	d(c1	d(c1	PROPN
ap-1408	201	8	)	)	PUNCT
ap-1408	201	9	=	=	PUNCT
ap-1408	201	10	d(b	d(b	X
ap-1408	201	11	)	)	PUNCT
ap-1408	202	1	=	=	SYM
ap-1408	202	2	h	h	NOUN
ap-1408	203	1	=	=	SYM
ap-1408	203	2	d(a	d(a	PROPN
ap-1408	203	3	)	)	PUNCT
ap-1408	203	4	c1	c1	NOUN
ap-1408	203	5	+	+	CCONJ
ap-1408	203	6	c2	c2	PROPN
ap-1408	203	7	=	=	SYM
ap-1408	203	8	d(c2	d(c2	NOUN
ap-1408	203	9	)	)	PUNCT
ap-1408	203	10	=	=	SYM
ap-1408	203	11	d(a	d(a	PROPN
ap-1408	203	12	)	)	PUNCT
ap-1408	203	13	=	=	SYM
ap-1408	203	14	d(b	d(b	X
ap-1408	203	15	)	)	PUNCT
ap-1408	204	1	=	=	SYM
ap-1408	204	2	d(a	d(a	ADJ
ap-1408	204	3	)	)	PUNCT
ap-1408	204	4	=	=	SYM
ap-1408	204	5	d(c1	d(c1	PROPN
ap-1408	204	6	)	)	PUNCT
ap-1408	204	7	c1	c1	NOUN
ap-1408	204	8	+	+	CCONJ
ap-1408	204	9	c2	c2	PROPN
ap-1408	204	10	if	if	SCONJ
ap-1408	204	11	c1	c1	PROPN
ap-1408	204	12	+	+	CCONJ
ap-1408	204	13	c2	c2	PROPN
ap-1408	204	14	is	be	AUX
ap-1408	204	15	unbounded	unbounded	ADJ
ap-1408	204	16	�	�	NOUN
ap-1408	204	17	=	=	SYM
ap-1408	204	18	h	h	PROPN
ap-1408	204	19	(	(	PUNCT
ap-1408	204	20	c1	c1	NOUN
ap-1408	204	21	+	+	CCONJ
ap-1408	204	22	c2)b	c2)b	NOUN
ap-1408	204	23	if	if	SCONJ
ap-1408	204	24	c1	c1	PROPN
ap-1408	204	25	+	+	CCONJ
ap-1408	204	26	c2	c2	PROPN
ap-1408	204	27	is	be	AUX
ap-1408	204	28	bounded	bound	VERB
ap-1408	204	29	=	=	PUNCT
ap-1408	204	30	d(a	d(a	PROPN
ap-1408	204	31	)	)	PUNCT
ap-1408	204	32	=	=	SYM
ap-1408	204	33	d(b	d(b	NOUN
ap-1408	204	34	)	)	PUNCT
ap-1408	204	35	now	now	ADV
ap-1408	204	36	,	,	PUNCT
ap-1408	204	37	assume	assume	VERB
ap-1408	204	38	that	that	SCONJ
ap-1408	204	39	a	a	DET
ap-1408	204	40	⊕d	⊕d	NOUN
ap-1408	204	41	b	b	X
ap-1408	204	42	=	=	PUNCT
ap-1408	204	43	a	a	DET
ap-1408	204	44	⊕d	⊕d	NOUN
ap-1408	204	45	c.	c.	PROPN
ap-1408	204	46	first	first	ADV
ap-1408	204	47	,	,	PUNCT
ap-1408	204	48	assume	assume	VERB
ap-1408	204	49	that	that	SCONJ
ap-1408	204	50	a	a	PRON
ap-1408	204	51	is	be	AUX
ap-1408	204	52	bounded	bound	VERB
ap-1408	204	53	.	.	PUNCT
ap-1408	205	1	then	then	ADV
ap-1408	205	2	d(b	d(b	NUM
ap-1408	205	3	)	)	PUNCT
ap-1408	205	4	=	=	SYM
ap-1408	205	5	d(c	d(c	PROPN
ap-1408	205	6	)	)	PUNCT
ap-1408	206	1	⊆	⊆	NUM
ap-1408	206	2	d(a	d(a	PROPN
ap-1408	206	3	)	)	PUNCT
ap-1408	206	4	.	.	PUNCT
ap-1408	207	1	hence	hence	ADV
ap-1408	207	2	a	a	DET
ap-1408	207	3	+	+	NOUN
ap-1408	207	4	b	b	NOUN
ap-1408	207	5	=	=	SYM
ap-1408	207	6	a	a	DET
ap-1408	207	7	+	+	CCONJ
ap-1408	207	8	c.	c.	NOUN
ap-1408	207	9	this	this	DET
ap-1408	207	10	yields	yield	NOUN
ap-1408	207	11	by	by	ADP
ap-1408	207	12	(	(	PUNCT
ap-1408	207	13	gii	gii	NOUN
ap-1408	207	14	)	)	PUNCT
ap-1408	207	15	that	that	PRON
ap-1408	207	16	c	c	AUX
ap-1408	207	17	=	=	PUNCT
ap-1408	207	18	−a	−a	NOUN
ap-1408	207	19	+	+	CCONJ
ap-1408	207	20	(	(	PUNCT
ap-1408	207	21	a	a	DET
ap-1408	207	22	+	+	NOUN
ap-1408	207	23	c	c	NOUN
ap-1408	207	24	)	)	PUNCT
ap-1408	207	25	=	=	SYM
ap-1408	207	26	−a	−a	NOUN
ap-1408	207	27	+	+	CCONJ
ap-1408	207	28	(	(	PUNCT
ap-1408	207	29	a	a	DET
ap-1408	207	30	+	+	NOUN
ap-1408	207	31	b	b	NOUN
ap-1408	207	32	)	)	PUNCT
ap-1408	207	33	=	=	SYM
ap-1408	207	34	b.	b.	PROPN
ap-1408	208	1	now	now	ADV
ap-1408	208	2	,	,	PUNCT
ap-1408	208	3	assume	assume	VERB
ap-1408	208	4	that	that	SCONJ
ap-1408	208	5	a	a	PRON
ap-1408	208	6	is	be	AUX
ap-1408	208	7	unbounded	unbounded	ADJ
ap-1408	208	8	.	.	PUNCT
ap-1408	209	1	if	if	SCONJ
ap-1408	209	2	b	b	NOUN
ap-1408	209	3	is	be	AUX
ap-1408	209	4	unbounded	unbounded	ADJ
ap-1408	209	5	then	then	ADV
ap-1408	209	6	d(b	d(b	NOUN
ap-1408	209	7	)	)	PUNCT
ap-1408	210	1	=	=	SYM
ap-1408	210	2	d(a	d(a	PROPN
ap-1408	210	3	)	)	PUNCT
ap-1408	210	4	and	and	CCONJ
ap-1408	210	5	we	we	PRON
ap-1408	210	6	will	will	AUX
ap-1408	210	7	distinguish	distinguish	VERB
ap-1408	210	8	the	the	DET
ap-1408	210	9	following	follow	VERB
ap-1408	210	10	cases	case	NOUN
ap-1408	210	11	:	:	PUNCT
ap-1408	210	12	(	(	PUNCT
ap-1408	210	13	γ1	γ1	NOUN
ap-1408	210	14	):	):	PUNCT
ap-1408	210	15	a	a	DET
ap-1408	210	16	+	+	NUM
ap-1408	210	17	b	b	NOUN
ap-1408	210	18	is	be	AUX
ap-1408	210	19	bounded	bound	VERB
ap-1408	210	20	and	and	CCONJ
ap-1408	210	21	hence	hence	ADV
ap-1408	210	22	also	also	ADV
ap-1408	210	23	a	a	PRON
ap-1408	210	24	+	+	NOUN
ap-1408	210	25	c	c	NOUN
ap-1408	210	26	is	be	AUX
ap-1408	210	27	bounded	bound	VERB
ap-1408	210	28	.	.	PUNCT
ap-1408	211	1	it	it	PRON
ap-1408	211	2	follows	follow	VERB
ap-1408	211	3	that	that	SCONJ
ap-1408	211	4	c	c	PROPN
ap-1408	211	5	is	be	AUX
ap-1408	211	6	unbounded	unbounded	ADJ
ap-1408	211	7	and	and	CCONJ
ap-1408	211	8	hence	hence	ADV
ap-1408	211	9	d(c	d(c	PROPN
ap-1408	211	10	)	)	PUNCT
ap-1408	212	1	=	=	SYM
ap-1408	212	2	d(a	d(a	PROPN
ap-1408	212	3	)	)	PUNCT
ap-1408	212	4	.	.	PUNCT
ap-1408	213	1	therefore	therefore	ADV
ap-1408	213	2	also	also	ADV
ap-1408	213	3	a	a	DET
ap-1408	213	4	+	+	X
ap-1408	213	5	b	b	NOUN
ap-1408	213	6	=	=	SYM
ap-1408	213	7	a	a	DET
ap-1408	213	8	+	+	X
ap-1408	213	9	c.	c.	NOUN
ap-1408	213	10	(	(	PUNCT
ap-1408	213	11	δ1	δ1	NOUN
ap-1408	213	12	):	):	PUNCT
ap-1408	213	13	a	a	DET
ap-1408	213	14	+	+	NUM
ap-1408	213	15	b	b	NOUN
ap-1408	213	16	is	be	AUX
ap-1408	213	17	unbounded	unbounded	ADJ
ap-1408	213	18	and	and	CCONJ
ap-1408	213	19	hence	hence	ADV
ap-1408	213	20	also	also	ADV
ap-1408	213	21	a	a	PRON
ap-1408	213	22	+	+	NOUN
ap-1408	213	23	c	c	NOUN
ap-1408	213	24	is	be	AUX
ap-1408	213	25	unbounded	unbounded	ADJ
ap-1408	213	26	.	.	PUNCT
ap-1408	214	1	we	we	PRON
ap-1408	214	2	get	get	VERB
ap-1408	214	3	that	that	DET
ap-1408	214	4	d(c	d(c	PROPN
ap-1408	214	5	)	)	PUNCT
ap-1408	215	1	=	=	SYM
ap-1408	215	2	d(a	d(a	PROPN
ap-1408	215	3	)	)	PUNCT
ap-1408	215	4	,	,	PUNCT
ap-1408	215	5	i.e.	i.e.	X
ap-1408	215	6	a	a	DET
ap-1408	215	7	+	+	X
ap-1408	215	8	b	b	NOUN
ap-1408	215	9	=	=	SYM
ap-1408	215	10	a	a	PRON
ap-1408	215	11	+	+	X
ap-1408	215	12	c.	c.	NOUN
ap-1408	215	13	in	in	ADP
ap-1408	215	14	both	both	DET
ap-1408	215	15	cases	case	NOUN
ap-1408	215	16	we	we	PRON
ap-1408	215	17	get	get	VERB
ap-1408	215	18	as	as	ADV
ap-1408	215	19	above	above	ADV
ap-1408	215	20	from	from	ADP
ap-1408	215	21	(	(	PUNCT
ap-1408	215	22	gii	gii	NOUN
ap-1408	215	23	)	)	PUNCT
ap-1408	215	24	that	that	PRON
ap-1408	215	25	c	c	AUX
ap-1408	215	26	=	=	PUNCT
ap-1408	215	27	b.	b.	PROPN
ap-1408	216	1	if	if	SCONJ
ap-1408	216	2	b	b	PROPN
ap-1408	216	3	is	be	AUX
ap-1408	216	4	bounded	bound	VERB
ap-1408	216	5	we	we	PRON
ap-1408	216	6	have	have	VERB
ap-1408	216	7	that	that	DET
ap-1408	216	8	a+b	a+b	NUM
ap-1408	217	1	is	be	AUX
ap-1408	217	2	unbounded	unbounded	ADJ
ap-1408	217	3	.	.	PUNCT
ap-1408	218	1	this	this	PRON
ap-1408	218	2	implies	imply	VERB
ap-1408	218	3	that	that	SCONJ
ap-1408	218	4	a+c	a+c	PROPN
ap-1408	218	5	is	be	AUX
ap-1408	218	6	unbounded	unbounded	ADJ
ap-1408	218	7	as	as	ADV
ap-1408	218	8	well	well	ADV
ap-1408	218	9	.	.	PUNCT
ap-1408	219	1	therefore	therefore	ADV
ap-1408	219	2	d(a	d(a	PROPN
ap-1408	219	3	+	+	PROPN
ap-1408	219	4	b	b	X
ap-1408	219	5	)	)	PUNCT
ap-1408	219	6	=	=	SYM
ap-1408	219	7	d(a	d(a	PROPN
ap-1408	219	8	)	)	PUNCT
ap-1408	219	9	⊆	⊆	NUM
ap-1408	219	10	d(b	d(b	NOUN
ap-1408	219	11	)	)	PUNCT
ap-1408	219	12	and	and	CCONJ
ap-1408	219	13	d(a	d(a	PROPN
ap-1408	219	14	)	)	PUNCT
ap-1408	219	15	⊆	⊆	NUM
ap-1408	219	16	d(c	d(c	PROPN
ap-1408	219	17	)	)	PUNCT
ap-1408	219	18	.	.	PUNCT
ap-1408	220	1	we	we	PRON
ap-1408	220	2	then	then	ADV
ap-1408	220	3	have	have	VERB
ap-1408	220	4	a	a	DET
ap-1408	220	5	+	+	NOUN
ap-1408	220	6	b	b	NOUN
ap-1408	220	7	=	=	SYM
ap-1408	220	8	a	a	PRON
ap-1408	220	9	+	+	NOUN
ap-1408	220	10	c.	c.	NOUN
ap-1408	220	11	it	it	PRON
ap-1408	220	12	follows	follow	VERB
ap-1408	220	13	again	again	ADV
ap-1408	220	14	by	by	ADP
ap-1408	220	15	(	(	PUNCT
ap-1408	220	16	gii	gii	NOUN
ap-1408	220	17	)	)	PUNCT
ap-1408	220	18	that	that	SCONJ
ap-1408	220	19	c	c	X
ap-1408	220	20	/	/	SYM
ap-1408	220	21	d(a	d(a	NOUN
ap-1408	220	22	)	)	PUNCT
ap-1408	220	23	is	be	AUX
ap-1408	220	24	bounded	bound	VERB
ap-1408	220	25	and	and	CCONJ
ap-1408	220	26	b	b	X
ap-1408	220	27	/	/	SYM
ap-1408	220	28	d(a	d(a	PROPN
ap-1408	220	29	)	)	PUNCT
ap-1408	220	30	=	=	SYM
ap-1408	221	1	c	c	X
ap-1408	221	2	/	/	SYM
ap-1408	221	3	d(a	d(a	PROPN
ap-1408	221	4	)	)	PUNCT
ap-1408	221	5	.	.	PUNCT
ap-1408	222	1	therefore	therefore	ADV
ap-1408	222	2	b	b	X
ap-1408	222	3	=	=	PROPN
ap-1408	222	4	c.	c.	PROPN
ap-1408	222	5	let	let	VERB
ap-1408	222	6	us	we	PRON
ap-1408	222	7	check	check	VERB
ap-1408	222	8	that	that	SCONJ
ap-1408	222	9	≤	≤	NOUN
ap-1408	222	10	is	be	AUX
ap-1408	222	11	reflexive	reflexive	ADJ
ap-1408	222	12	and	and	CCONJ
ap-1408	222	13	antisymmetric	antisymmetric	ADJ
ap-1408	222	14	.	.	PUNCT
ap-1408	223	1	let	let	VERB
ap-1408	223	2	a	a	DET
ap-1408	223	3	,	,	PUNCT
ap-1408	223	4	b	b	PROPN
ap-1408	223	5	∈	∈	PROPN
ap-1408	223	6	gr(h	gr(h	NOUN
ap-1408	223	7	)	)	PUNCT
ap-1408	223	8	.	.	PUNCT
ap-1408	224	1	evidently	evidently	ADV
ap-1408	224	2	,	,	PUNCT
ap-1408	224	3	a	a	DET
ap-1408	224	4	≤	≤	NOUN
ap-1408	224	5	a	a	PRON
ap-1408	224	6	since	since	SCONJ
ap-1408	224	7	a	a	DET
ap-1408	224	8	=	=	NOUN
ap-1408	224	9	a	a	DET
ap-1408	224	10	⊕d	⊕d	NOUN
ap-1408	224	11	0	0	PUNCT
ap-1408	225	1	and	and	CCONJ
ap-1408	225	2	0	0	NUM
ap-1408	225	3	is	be	AUX
ap-1408	225	4	a	a	DET
ap-1408	225	5	positive	positive	ADJ
ap-1408	225	6	bounded	bounded	ADJ
ap-1408	225	7	linear	linear	ADJ
ap-1408	225	8	operator	operator	NOUN
ap-1408	225	9	on	on	ADP
ap-1408	225	10	h.	h.	PROPN
ap-1408	225	11	now	now	ADV
ap-1408	225	12	,	,	PUNCT
ap-1408	225	13	assume	assume	VERB
ap-1408	225	14	that	that	SCONJ
ap-1408	225	15	b	b	X
ap-1408	225	16	=	=	PUNCT
ap-1408	225	17	a	a	DET
ap-1408	225	18	⊕d	⊕d	NOUN
ap-1408	225	19	c1	c1	NOUN
ap-1408	225	20	and	and	CCONJ
ap-1408	225	21	a	a	DET
ap-1408	225	22	=	=	SYM
ap-1408	225	23	b	b	PROPN
ap-1408	225	24	⊕d	⊕d	NOUN
ap-1408	225	25	c2	c2	PROPN
ap-1408	225	26	for	for	ADP
ap-1408	225	27	some	some	DET
ap-1408	225	28	positive	positive	ADJ
ap-1408	225	29	linear	linear	PROPN
ap-1408	225	30	operators	operator	NOUN
ap-1408	225	31	c1	c1	PROPN
ap-1408	225	32	,	,	PUNCT
ap-1408	225	33	c2	c2	PROPN
ap-1408	225	34	on	on	ADP
ap-1408	225	35	h.	h.	PROPN
ap-1408	225	36	by	by	ADP
ap-1408	225	37	cases	case	NOUN
ap-1408	225	38	we	we	PRON
ap-1408	225	39	have	have	VERB
ap-1408	225	40	that	that	PRON
ap-1408	225	41	(	(	PUNCT
ap-1408	225	42	a+c1	a+c1	PROPN
ap-1408	225	43	=	=	SYM
ap-1408	225	44	b	b	PROPN
ap-1408	225	45	with	with	ADP
ap-1408	225	46	b	b	NOUN
ap-1408	225	47	unbounded	unbounded	ADJ
ap-1408	225	48	or	or	CCONJ
ap-1408	225	49	(	(	PUNCT
ap-1408	225	50	a	a	DET
ap-1408	225	51	+	+	X
ap-1408	225	52	c1)b	c1)b	NOUN
ap-1408	225	53	=	=	SYM
ap-1408	225	54	b	b	PROPN
ap-1408	225	55	with	with	ADP
ap-1408	225	56	b	b	NOUN
ap-1408	225	57	bounded	bound	VERB
ap-1408	225	58	)	)	PUNCT
ap-1408	225	59	and	and	CCONJ
ap-1408	225	60	(	(	PUNCT
ap-1408	225	61	b+c2	b+c2	PROPN
ap-1408	225	62	=	=	PUNCT
ap-1408	225	63	a	a	NOUN
ap-1408	225	64	with	with	ADP
ap-1408	225	65	a	a	DET
ap-1408	225	66	unbounded	unbounded	ADJ
ap-1408	225	67	or	or	CCONJ
ap-1408	225	68	(	(	PUNCT
ap-1408	225	69	b+c2)b	b+c2)b	NOUN
ap-1408	225	70	=	=	NOUN
ap-1408	225	71	a	a	NOUN
ap-1408	225	72	with	with	ADP
ap-1408	225	73	a	a	DET
ap-1408	225	74	bounded	bound	VERB
ap-1408	225	75	)	)	PUNCT
ap-1408	225	76	.	.	PUNCT
ap-1408	226	1	assume	assume	VERB
ap-1408	226	2	first	first	ADV
ap-1408	226	3	that	that	SCONJ
ap-1408	226	4	a+c1	a+c1	PROPN
ap-1408	226	5	=	=	SYM
ap-1408	226	6	b	b	PROPN
ap-1408	226	7	with	with	ADP
ap-1408	226	8	b	b	NOUN
ap-1408	226	9	unbounded	unbounded	ADJ
ap-1408	226	10	and	and	CCONJ
ap-1408	226	11	b	b	NOUN
ap-1408	226	12	+	+	CCONJ
ap-1408	226	13	c2	c2	PROPN
ap-1408	226	14	=	=	PUNCT
ap-1408	226	15	a	a	PRON
ap-1408	226	16	with	with	ADP
ap-1408	226	17	a	a	DET
ap-1408	226	18	unbounded	unbounded	ADJ
ap-1408	226	19	.	.	PUNCT
ap-1408	227	1	we	we	PRON
ap-1408	227	2	have	have	VERB
ap-1408	227	3	the	the	DET
ap-1408	227	4	following	follow	VERB
ap-1408	227	5	possibilities	possibility	NOUN
ap-1408	227	6	for	for	ADP
ap-1408	227	7	c1	c1	PROPN
ap-1408	227	8	and	and	CCONJ
ap-1408	227	9	c2	c2	PROPN
ap-1408	227	10	(	(	PUNCT
ap-1408	227	11	see	see	VERB
ap-1408	227	12	tab	tab	NOUN
ap-1408	227	13	.	.	PUNCT
ap-1408	227	14	1	1	NUM
ap-1408	227	15	)	)	PUNCT
ap-1408	227	16	.	.	PUNCT
ap-1408	228	1	now	now	ADV
ap-1408	228	2	assume	assume	VERB
ap-1408	228	3	that	that	SCONJ
ap-1408	228	4	a+c1	a+c1	PROPN
ap-1408	228	5	=	=	SYM
ap-1408	228	6	b	b	PROPN
ap-1408	228	7	with	with	ADP
ap-1408	228	8	b	b	NOUN
ap-1408	228	9	unbounded	unbounded	ADJ
ap-1408	228	10	and	and	CCONJ
ap-1408	228	11	(	(	PUNCT
ap-1408	228	12	b	b	NOUN
ap-1408	228	13	+	+	CCONJ
ap-1408	228	14	c2)b	c2)b	NOUN
ap-1408	228	15	=	=	PUNCT
ap-1408	228	16	a	a	NOUN
ap-1408	228	17	with	with	ADP
ap-1408	228	18	a	a	DET
ap-1408	228	19	bounded	bound	VERB
ap-1408	228	20	.	.	PUNCT
ap-1408	229	1	then	then	ADV
ap-1408	229	2	c1	c1	PROPN
ap-1408	229	3	has	have	VERB
ap-1408	229	4	to	to	PART
ap-1408	229	5	be	be	AUX
ap-1408	229	6	unbounded	unbounde	VERB
ap-1408	229	7	with	with	ADP
ap-1408	229	8	d(c1	d(c1	NOUN
ap-1408	229	9	)	)	PUNCT
ap-1408	229	10	=	=	PUNCT
ap-1408	229	11	d(b	d(b	X
ap-1408	229	12	)	)	PUNCT
ap-1408	229	13	and	and	CCONJ
ap-1408	229	14	c2	c2	PROPN
ap-1408	229	15	can	can	AUX
ap-1408	229	16	only	only	ADV
ap-1408	229	17	also	also	ADV
ap-1408	229	18	be	be	AUX
ap-1408	229	19	unbounded	unbounded	ADJ
ap-1408	229	20	with	with	ADP
ap-1408	229	21	d(c2	d(c2	NOUN
ap-1408	229	22	)	)	PUNCT
ap-1408	229	23	=	=	PUNCT
ap-1408	229	24	d(b	d(b	PROPN
ap-1408	229	25	)	)	PUNCT
ap-1408	229	26	.	.	PUNCT
ap-1408	230	1	when	when	SCONJ
ap-1408	230	2	c1+c2	c1+c2	PROPN
ap-1408	230	3	is	be	AUX
ap-1408	230	4	bounded	bound	VERB
ap-1408	230	5	then	then	ADV
ap-1408	230	6	c1⊕d	c1⊕d	ADJ
ap-1408	230	7	c2	c2	PROPN
ap-1408	230	8	=	=	PUNCT
ap-1408	230	9	(	(	PUNCT
ap-1408	230	10	c1+c2)b	c1+c2)b	NOUN
ap-1408	230	11	and	and	CCONJ
ap-1408	230	12	d(c1	d(c1	NOUN
ap-1408	230	13	⊕d	⊕d	PROPN
ap-1408	230	14	c2	c2	PROPN
ap-1408	230	15	)	)	PUNCT
ap-1408	231	1	=	=	SYM
ap-1408	231	2	h.	h.	PROPN
ap-1408	231	3	for	for	ADP
ap-1408	231	4	c1	c1	PROPN
ap-1408	231	5	+	+	CCONJ
ap-1408	231	6	c2	c2	PROPN
ap-1408	231	7	unbounded	unbounde	VERB
ap-1408	231	8	we	we	PRON
ap-1408	231	9	have	have	VERB
ap-1408	231	10	d(c1	d(c1	NOUN
ap-1408	231	11	⊕d	⊕d	PROPN
ap-1408	231	12	c2	c2	PROPN
ap-1408	231	13	)	)	PUNCT
ap-1408	231	14	=	=	PUNCT
ap-1408	232	1	d(b	d(b	PROPN
ap-1408	232	2	)	)	PUNCT
ap-1408	232	3	.	.	PUNCT
ap-1408	233	1	the	the	DET
ap-1408	233	2	situation	situation	NOUN
ap-1408	233	3	for	for	ADP
ap-1408	233	4	(	(	PUNCT
ap-1408	233	5	a+c1	a+c1	PROPN
ap-1408	233	6	)	)	PUNCT
ap-1408	233	7	b	b	NOUN
ap-1408	233	8	=	=	SYM
ap-1408	233	9	b	b	PROPN
ap-1408	233	10	with	with	ADP
ap-1408	233	11	b	b	NOUN
ap-1408	233	12	bounded	bound	VERB
ap-1408	233	13	and	and	CCONJ
ap-1408	233	14	(	(	PUNCT
ap-1408	233	15	b	b	PROPN
ap-1408	233	16	+	+	CCONJ
ap-1408	233	17	c2	c2	PROPN
ap-1408	233	18	)	)	PUNCT
ap-1408	234	1	=	=	NOUN
ap-1408	235	1	a	a	PRON
ap-1408	235	2	with	with	ADP
ap-1408	235	3	a	a	DET
ap-1408	235	4	unbounded	unbounded	ADJ
ap-1408	235	5	is	be	AUX
ap-1408	235	6	symmetric	symmetric	ADJ
ap-1408	235	7	to	to	ADP
ap-1408	235	8	the	the	DET
ap-1408	235	9	previous	previous	ADJ
ap-1408	235	10	case	case	NOUN
ap-1408	235	11	with	with	ADP
ap-1408	235	12	d(c1	d(c1	NOUN
ap-1408	235	13	⊕d	⊕d	PROPN
ap-1408	235	14	c2	c2	PROPN
ap-1408	235	15	)	)	PUNCT
ap-1408	236	1	=	=	PUNCT
ap-1408	236	2	h	h	NOUN
ap-1408	236	3	when	when	SCONJ
ap-1408	236	4	c1	c1	PROPN
ap-1408	236	5	+	+	CCONJ
ap-1408	236	6	c2	c2	PROPN
ap-1408	236	7	is	be	AUX
ap-1408	236	8	bounded	bound	VERB
ap-1408	236	9	and	and	CCONJ
ap-1408	236	10	d(c1	d(c1	NOUN
ap-1408	236	11	⊕d	⊕d	PROPN
ap-1408	236	12	c2	c2	PROPN
ap-1408	236	13	)	)	PUNCT
ap-1408	237	1	=	=	SYM
ap-1408	237	2	d(a	d(a	PROPN
ap-1408	237	3	)	)	PUNCT
ap-1408	237	4	when	when	SCONJ
ap-1408	237	5	c1	c1	PROPN
ap-1408	237	6	+	+	CCONJ
ap-1408	237	7	c2	c2	PROPN
ap-1408	237	8	is	be	AUX
ap-1408	237	9	unbounded	unbounded	ADJ
ap-1408	237	10	.	.	PUNCT
ap-1408	238	1	the	the	DET
ap-1408	238	2	last	last	ADJ
ap-1408	238	3	case	case	NOUN
ap-1408	238	4	is	be	AUX
ap-1408	238	5	(	(	PUNCT
ap-1408	238	6	a	a	DET
ap-1408	238	7	+	+	X
ap-1408	238	8	c1)b	c1)b	NOUN
ap-1408	238	9	=	=	SYM
ap-1408	238	10	b	b	PROPN
ap-1408	238	11	with	with	ADP
ap-1408	238	12	b	b	NOUN
ap-1408	238	13	bounded	bound	VERB
ap-1408	238	14	and	and	CCONJ
ap-1408	238	15	(	(	PUNCT
ap-1408	238	16	b	b	NOUN
ap-1408	238	17	+	+	CCONJ
ap-1408	238	18	c2)b	c2)b	NOUN
ap-1408	238	19	=	=	PUNCT
ap-1408	238	20	a	a	NOUN
ap-1408	238	21	with	with	ADP
ap-1408	238	22	a	a	DET
ap-1408	238	23	bounded	bound	VERB
ap-1408	238	24	too	too	ADV
ap-1408	238	25	.	.	PUNCT
ap-1408	239	1	hence	hence	ADV
ap-1408	239	2	c1	c1	PROPN
ap-1408	239	3	and	and	CCONJ
ap-1408	239	4	c2	c2	PROPN
ap-1408	239	5	are	be	AUX
ap-1408	239	6	bounded	bound	VERB
ap-1408	239	7	as	as	ADV
ap-1408	239	8	well	well	ADV
ap-1408	239	9	and	and	CCONJ
ap-1408	239	10	(	(	PUNCT
ap-1408	239	11	c1	c1	PROPN
ap-1408	239	12	⊕d	⊕d	PROPN
ap-1408	239	13	c2	c2	PROPN
ap-1408	239	14	)	)	PUNCT
ap-1408	240	1	=	=	PRON
ap-1408	240	2	(	(	PUNCT
ap-1408	240	3	c1	c1	NOUN
ap-1408	240	4	+	+	CCONJ
ap-1408	240	5	c2)b	c2)b	NOUN
ap-1408	240	6	with	with	ADP
ap-1408	240	7	d(c1	d(c1	NOUN
ap-1408	240	8	⊕d	⊕d	PROPN
ap-1408	240	9	c2	c2	PROPN
ap-1408	240	10	)	)	PUNCT
ap-1408	241	1	=	=	PUNCT
ap-1408	241	2	h.	h.	NOUN
ap-1408	242	1	this	this	PRON
ap-1408	242	2	yields	yield	VERB
ap-1408	242	3	that	that	SCONJ
ap-1408	242	4	a	a	DET
ap-1408	242	5	⊕d	⊕d	NOUN
ap-1408	242	6	(	(	PUNCT
ap-1408	242	7	c1	c1	PROPN
ap-1408	242	8	⊕d	⊕d	PROPN
ap-1408	242	9	c2	c2	PROPN
ap-1408	242	10	)	)	PUNCT
ap-1408	242	11	is	be	AUX
ap-1408	242	12	defined	define	VERB
ap-1408	242	13	and	and	CCONJ
ap-1408	242	14	a⊕d	a⊕d	NOUN
ap-1408	242	15	(	(	PUNCT
ap-1408	242	16	c1⊕dc2	c1⊕dc2	NOUN
ap-1408	242	17	)	)	PUNCT
ap-1408	242	18	=	=	SYM
ap-1408	242	19	(	(	PUNCT
ap-1408	242	20	a⊕dc1)⊕dc2	a⊕dc1)⊕dc2	NOUN
ap-1408	242	21	=	=	SYM
ap-1408	242	22	b⊕dc2	b⊕dc2	NOUN
ap-1408	242	23	=	=	PUNCT
ap-1408	242	24	a.	a.	NOUN
ap-1408	242	25	hence	hence	ADV
ap-1408	242	26	by	by	ADP
ap-1408	242	27	(	(	PUNCT
ap-1408	242	28	giii	giii	NOUN
ap-1408	242	29	)	)	PUNCT
ap-1408	242	30	and	and	CCONJ
ap-1408	242	31	(	(	PUNCT
ap-1408	242	32	giv	giv	PROPN
ap-1408	242	33	)	)	PUNCT
ap-1408	242	34	we	we	PRON
ap-1408	242	35	obtain	obtain	VERB
ap-1408	242	36	that	that	DET
ap-1408	242	37	c1⊕dc2	c1⊕dc2	NOUN
ap-1408	242	38	=	=	SYM
ap-1408	242	39	0	0	NUM
ap-1408	242	40	,	,	PUNCT
ap-1408	242	41	hence	hence	ADV
ap-1408	242	42	c1	c1	PROPN
ap-1408	242	43	+	+	CCONJ
ap-1408	242	44	c2	c2	PROPN
ap-1408	242	45	=	=	SYM
ap-1408	242	46	0	0	NUM
ap-1408	242	47	/	/	SYM
ap-1408	242	48	d(c1	d(c1	NOUN
ap-1408	242	49	)	)	PUNCT
ap-1408	242	50	.	.	PUNCT
ap-1408	243	1	by	by	ADP
ap-1408	243	2	[	[	X
ap-1408	243	3	7	7	NUM
ap-1408	243	4	,	,	PUNCT
ap-1408	243	5	theorem	theorem	VERB
ap-1408	243	6	2	2	NUM
ap-1408	243	7	]	]	PUNCT
ap-1408	243	8	we	we	PRON
ap-1408	243	9	have	have	VERB
ap-1408	243	10	that	that	DET
ap-1408	243	11	c1	c1	PROPN
ap-1408	243	12	=	=	PROPN
ap-1408	243	13	c2	c2	PROPN
ap-1408	243	14	=	=	PUNCT
ap-1408	244	1	0	0	PROPN
ap-1408	244	2	.	.	PUNCT
ap-1408	245	1	so	so	ADV
ap-1408	245	2	,	,	PUNCT
ap-1408	245	3	it	it	PRON
ap-1408	245	4	remains	remain	VERB
ap-1408	245	5	to	to	PART
ap-1408	245	6	check	check	VERB
ap-1408	245	7	that	that	SCONJ
ap-1408	245	8	≤	≤	NOUN
ap-1408	245	9	is	be	AUX
ap-1408	245	10	compatible	compatible	ADJ
ap-1408	245	11	with	with	ADP
ap-1408	245	12	addition	addition	NOUN
ap-1408	245	13	,	,	PUNCT
ap-1408	245	14	i.e.	i.e.	X
ap-1408	245	15	,	,	PUNCT
ap-1408	245	16	for	for	SCONJ
ap-1408	245	17	all	all	DET
ap-1408	245	18	a	a	DET
ap-1408	245	19	,	,	PUNCT
ap-1408	245	20	b	b	NOUN
ap-1408	245	21	,	,	PUNCT
ap-1408	245	22	c	c	PROPN
ap-1408	245	23	∈	∈	PROPN
ap-1408	245	24	gr(h	gr(h	NOUN
ap-1408	245	25	)	)	PUNCT
ap-1408	245	26	such	such	ADJ
ap-1408	245	27	that	that	SCONJ
ap-1408	245	28	a	a	DET
ap-1408	245	29	≤	≤	PROPN
ap-1408	245	30	b	b	NOUN
ap-1408	245	31	,	,	PUNCT
ap-1408	245	32	c	c	PROPN
ap-1408	245	33	⊕d	⊕d	PROPN
ap-1408	245	34	a	a	PRON
ap-1408	245	35	and	and	CCONJ
ap-1408	245	36	c	c	PROPN
ap-1408	245	37	⊕d	⊕d	PROPN
ap-1408	245	38	b	b	PROPN
ap-1408	245	39	are	be	AUX
ap-1408	245	40	defined	define	VERB
ap-1408	245	41	we	we	PRON
ap-1408	245	42	have	have	VERB
ap-1408	245	43	that	that	PRON
ap-1408	245	44	c	c	PROPN
ap-1408	245	45	⊕d	⊕d	VERB
ap-1408	245	46	a	a	DET
ap-1408	245	47	≤	≤	PROPN
ap-1408	245	48	c	c	PROPN
ap-1408	245	49	⊕d	⊕d	PROPN
ap-1408	245	50	b.	b.	PROPN
ap-1408	245	51	again	again	ADV
ap-1408	245	52	by	by	ADP
ap-1408	245	53	cases	case	NOUN
ap-1408	245	54	we	we	PRON
ap-1408	245	55	have	have	VERB
ap-1408	245	56	that	that	PRON
ap-1408	245	57	(	(	PUNCT
ap-1408	245	58	c	c	NOUN
ap-1408	245	59	⊕d	⊕d	VERB
ap-1408	245	60	a	a	DET
ap-1408	245	61	=	=	SYM
ap-1408	245	62	c	c	PROPN
ap-1408	246	1	+	+	NOUN
ap-1408	246	2	a	a	PRON
ap-1408	246	3	with	with	ADP
ap-1408	246	4	c	c	NOUN
ap-1408	246	5	+	+	CCONJ
ap-1408	246	6	a	a	DET
ap-1408	246	7	unbounded	unbounded	ADJ
ap-1408	246	8	or	or	CCONJ
ap-1408	246	9	c	c	NOUN
ap-1408	246	10	⊕d	⊕d	NOUN
ap-1408	246	11	a	a	DET
ap-1408	246	12	=	=	X
ap-1408	246	13	(	(	PUNCT
ap-1408	246	14	c	c	NOUN
ap-1408	246	15	+	+	CCONJ
ap-1408	246	16	a)b	a)b	X
ap-1408	246	17	with	with	ADP
ap-1408	246	18	c	c	PROPN
ap-1408	247	1	+	+	CCONJ
ap-1408	247	2	a	a	DET
ap-1408	247	3	bounded	bound	VERB
ap-1408	247	4	)	)	PUNCT
ap-1408	247	5	and	and	CCONJ
ap-1408	247	6	(	(	PUNCT
ap-1408	247	7	c	c	NOUN
ap-1408	247	8	⊕d	⊕d	PROPN
ap-1408	248	1	b	b	PROPN
ap-1408	248	2	=	=	SYM
ap-1408	248	3	c	c	PROPN
ap-1408	249	1	+	+	NOUN
ap-1408	249	2	b	b	NOUN
ap-1408	249	3	with	with	ADP
ap-1408	249	4	c	c	NOUN
ap-1408	249	5	+	+	CCONJ
ap-1408	249	6	b	b	NOUN
ap-1408	249	7	unbounded	unbounded	ADJ
ap-1408	249	8	or	or	CCONJ
ap-1408	249	9	c	c	NOUN
ap-1408	250	1	⊕d	⊕d	NOUN
ap-1408	250	2	b	b	PROPN
ap-1408	250	3	=	=	PUNCT
ap-1408	250	4	(	(	PUNCT
ap-1408	250	5	c	c	NOUN
ap-1408	250	6	+	+	X
ap-1408	250	7	b)b	b)b	X
ap-1408	250	8	with	with	ADP
ap-1408	250	9	c	c	PROPN
ap-1408	250	10	+	+	PROPN
ap-1408	250	11	b	b	X
ap-1408	250	12	bounded	bound	VERB
ap-1408	250	13	)	)	PUNCT
ap-1408	250	14	.	.	PUNCT
ap-1408	251	1	since	since	SCONJ
ap-1408	251	2	a	a	DET
ap-1408	251	3	≤	≤	PROPN
ap-1408	251	4	b	b	NOUN
ap-1408	251	5	there	there	PRON
ap-1408	251	6	is	be	VERB
ap-1408	251	7	e	e	NOUN
ap-1408	251	8	∈	∈	PROPN
ap-1408	251	9	gr(h	gr(h	NOUN
ap-1408	251	10	)	)	PUNCT
ap-1408	251	11	,	,	PUNCT
ap-1408	252	1	e	e	X
ap-1408	252	2	positive	positive	ADJ
ap-1408	252	3	such	such	ADJ
ap-1408	252	4	that	that	SCONJ
ap-1408	252	5	a	a	DET
ap-1408	252	6	⊕d	⊕d	NOUN
ap-1408	252	7	e	e	X
ap-1408	252	8	=	=	PROPN
ap-1408	252	9	b.	b.	PROPN
ap-1408	253	1	but	but	CCONJ
ap-1408	253	2	c	c	PROPN
ap-1408	253	3	⊕d	⊕d	PROPN
ap-1408	253	4	b	b	X
ap-1408	253	5	=	=	SYM
ap-1408	253	6	c	c	PROPN
ap-1408	253	7	⊕d	⊕d	NOUN
ap-1408	253	8	(	(	PUNCT
ap-1408	253	9	a	a	DET
ap-1408	253	10	⊕d	⊕d	NOUN
ap-1408	253	11	e	e	NOUN
ap-1408	253	12	)	)	PUNCT
ap-1408	253	13	.	.	PUNCT
ap-1408	254	1	for	for	ADP
ap-1408	254	2	the	the	DET
ap-1408	254	3	case	case	NOUN
ap-1408	254	4	when	when	SCONJ
ap-1408	254	5	e	e	PROPN
ap-1408	254	6	is	be	AUX
ap-1408	254	7	bounded	bound	VERB
ap-1408	254	8	,	,	PUNCT
ap-1408	254	9	clearly	clearly	ADV
ap-1408	254	10	,	,	PUNCT
ap-1408	254	11	(	(	PUNCT
ap-1408	254	12	c	c	NOUN
ap-1408	254	13	⊕d	⊕d	PROPN
ap-1408	254	14	a	a	X
ap-1408	254	15	)	)	PUNCT
ap-1408	254	16	⊕d	⊕d	NOUN
ap-1408	254	17	e	e	NOUN
ap-1408	254	18	is	be	AUX
ap-1408	254	19	always	always	ADV
ap-1408	254	20	defined	define	VERB
ap-1408	254	21	.	.	PUNCT
ap-1408	255	1	now	now	ADV
ap-1408	255	2	assume	assume	VERB
ap-1408	255	3	that	that	SCONJ
ap-1408	255	4	e	e	PRON
ap-1408	255	5	is	be	AUX
ap-1408	255	6	unbounded	unbounded	ADJ
ap-1408	255	7	.	.	PUNCT
ap-1408	256	1	in	in	ADP
ap-1408	256	2	the	the	DET
ap-1408	256	3	case	case	NOUN
ap-1408	256	4	when	when	SCONJ
ap-1408	256	5	a	a	PRON
ap-1408	256	6	is	be	AUX
ap-1408	256	7	unbounded	unbounded	ADJ
ap-1408	256	8	and	and	CCONJ
ap-1408	256	9	b	b	NOUN
ap-1408	256	10	is	be	AUX
ap-1408	256	11	bounded	bound	VERB
ap-1408	256	12	we	we	PRON
ap-1408	256	13	get	get	VERB
ap-1408	256	14	that	that	DET
ap-1408	256	15	d(e	d(e	NOUN
ap-1408	256	16	)	)	PUNCT
ap-1408	257	1	=	=	SYM
ap-1408	257	2	d(a	d(a	PROPN
ap-1408	257	3	)	)	PUNCT
ap-1408	257	4	and	and	CCONJ
ap-1408	257	5	d((e	d((e	NOUN
ap-1408	257	6	+	+	CCONJ
ap-1408	257	7	a)b	a)b	X
ap-1408	257	8	)	)	PUNCT
ap-1408	258	1	=	=	VERB
ap-1408	259	1	h.	h.	NOUN
ap-1408	259	2	we	we	PRON
ap-1408	259	3	have	have	VERB
ap-1408	259	4	the	the	DET
ap-1408	259	5	following	follow	VERB
ap-1408	259	6	possibilities	possibility	NOUN
ap-1408	259	7	:	:	PUNCT
ap-1408	259	8	(	(	PUNCT
ap-1408	259	9	a	a	X
ap-1408	259	10	):	):	PUNCT
ap-1408	259	11	c	c	PROPN
ap-1408	259	12	is	be	AUX
ap-1408	259	13	bounded	bound	VERB
ap-1408	259	14	.	.	PUNCT
ap-1408	260	1	then	then	ADV
ap-1408	260	2	d(c	d(c	PROPN
ap-1408	260	3	+	+	NOUN
ap-1408	260	4	a	a	X
ap-1408	260	5	)	)	PUNCT
ap-1408	260	6	=	=	SYM
ap-1408	260	7	d(a	d(a	PROPN
ap-1408	260	8	)	)	PUNCT
ap-1408	260	9	=	=	SYM
ap-1408	260	10	d(e	d(e	PROPN
ap-1408	260	11	)	)	PUNCT
ap-1408	260	12	.	.	PUNCT
ap-1408	261	1	(	(	PUNCT
ap-1408	261	2	b	b	X
ap-1408	261	3	):	):	PUNCT
ap-1408	261	4	c	c	PROPN
ap-1408	261	5	is	be	AUX
ap-1408	261	6	unbounded	unbounded	ADJ
ap-1408	261	7	.	.	PUNCT
ap-1408	262	1	then	then	ADV
ap-1408	262	2	d(c	d(c	PROPN
ap-1408	262	3	)	)	PUNCT
ap-1408	263	1	=	=	SYM
ap-1408	263	2	d(a	d(a	PROPN
ap-1408	263	3	)	)	PUNCT
ap-1408	263	4	=	=	SYM
ap-1408	263	5	d(e	d(e	PROPN
ap-1408	263	6	)	)	PUNCT
ap-1408	263	7	.	.	PUNCT
ap-1408	264	1	in	in	ADP
ap-1408	264	2	the	the	DET
ap-1408	264	3	case	case	NOUN
ap-1408	264	4	when	when	SCONJ
ap-1408	264	5	both	both	DET
ap-1408	264	6	a	a	PRON
ap-1408	264	7	,	,	PUNCT
ap-1408	264	8	b	b	NOUN
ap-1408	264	9	are	be	AUX
ap-1408	264	10	unbounded	unbounded	ADJ
ap-1408	264	11	we	we	PRON
ap-1408	264	12	obtain	obtain	VERB
ap-1408	265	1	that	that	SCONJ
ap-1408	265	2	d(e	d(e	PROPN
ap-1408	265	3	+	+	CCONJ
ap-1408	265	4	a	a	X
ap-1408	265	5	)	)	PUNCT
ap-1408	265	6	=	=	SYM
ap-1408	265	7	d(b	d(b	X
ap-1408	265	8	)	)	PUNCT
ap-1408	265	9	=	=	SYM
ap-1408	265	10	d(a	d(a	PROPN
ap-1408	265	11	)	)	PUNCT
ap-1408	265	12	=	=	SYM
ap-1408	265	13	d(e	d(e	PROPN
ap-1408	265	14	)	)	PUNCT
ap-1408	265	15	.	.	PUNCT
ap-1408	266	1	we	we	PRON
ap-1408	266	2	distinguish	distinguish	VERB
ap-1408	266	3	:	:	PUNCT
ap-1408	266	4	(	(	PUNCT
ap-1408	266	5	c	c	X
ap-1408	266	6	):	):	PUNCT
ap-1408	266	7	c	c	PROPN
ap-1408	266	8	is	be	AUX
ap-1408	266	9	bounded	bound	VERB
ap-1408	266	10	.	.	PUNCT
ap-1408	267	1	then	then	ADV
ap-1408	267	2	d(c	d(c	PROPN
ap-1408	267	3	+	+	CCONJ
ap-1408	267	4	a	a	X
ap-1408	267	5	)	)	PUNCT
ap-1408	267	6	=	=	SYM
ap-1408	267	7	d(a	d(a	PROPN
ap-1408	267	8	)	)	PUNCT
ap-1408	267	9	=	=	SYM
ap-1408	267	10	d(e	d(e	PROPN
ap-1408	267	11	)	)	PUNCT
ap-1408	267	12	.	.	PUNCT
ap-1408	268	1	(	(	PUNCT
ap-1408	268	2	d	d	X
ap-1408	268	3	):	):	PUNCT
ap-1408	268	4	c	c	NOUN
ap-1408	268	5	is	be	AUX
ap-1408	268	6	unbounded	unbounded	ADJ
ap-1408	268	7	.	.	PUNCT
ap-1408	269	1	then	then	ADV
ap-1408	269	2	d(c	d(c	PROPN
ap-1408	269	3	)	)	PUNCT
ap-1408	270	1	=	=	SYM
ap-1408	270	2	d(a	d(a	PROPN
ap-1408	270	3	)	)	PUNCT
ap-1408	270	4	=	=	SYM
ap-1408	270	5	d(e	d(e	PROPN
ap-1408	270	6	)	)	PUNCT
ap-1408	270	7	.	.	PUNCT
ap-1408	271	1	in	in	ADP
ap-1408	271	2	the	the	DET
ap-1408	271	3	last	last	ADJ
ap-1408	271	4	case	case	NOUN
ap-1408	271	5	assume	assume	VERB
ap-1408	271	6	that	that	SCONJ
ap-1408	271	7	a	a	PRON
ap-1408	271	8	is	be	AUX
ap-1408	271	9	bounded	bound	VERB
ap-1408	271	10	and	and	CCONJ
ap-1408	271	11	b	b	NOUN
ap-1408	271	12	is	be	AUX
ap-1408	271	13	unbounded	unbounded	ADJ
ap-1408	271	14	,	,	PUNCT
ap-1408	271	15	hence	hence	ADV
ap-1408	271	16	d(e	d(e	PROPN
ap-1408	271	17	+	+	CCONJ
ap-1408	271	18	a	a	X
ap-1408	271	19	)	)	PUNCT
ap-1408	271	20	=	=	SYM
ap-1408	271	21	d(b	d(b	X
ap-1408	271	22	)	)	PUNCT
ap-1408	271	23	=	=	SYM
ap-1408	272	1	d(e	d(e	NOUN
ap-1408	272	2	)	)	PUNCT
ap-1408	272	3	.	.	PUNCT
ap-1408	273	1	(	(	PUNCT
ap-1408	273	2	e	e	X
ap-1408	273	3	):	):	PUNCT
ap-1408	273	4	c	c	PROPN
ap-1408	273	5	is	be	AUX
ap-1408	273	6	bounded	bound	VERB
ap-1408	273	7	.	.	PUNCT
ap-1408	274	1	then	then	ADV
ap-1408	274	2	d(c	d(c	PROPN
ap-1408	274	3	+	+	CCONJ
ap-1408	274	4	a	a	X
ap-1408	274	5	)	)	PUNCT
ap-1408	275	1	=	=	SYM
ap-1408	275	2	h.	h.	NOUN
ap-1408	275	3	(	(	PUNCT
ap-1408	275	4	f	f	X
ap-1408	275	5	):	):	PUNCT
ap-1408	275	6	c	c	PROPN
ap-1408	275	7	is	be	AUX
ap-1408	275	8	unbounded	unbounded	ADJ
ap-1408	275	9	.	.	PUNCT
ap-1408	276	1	then	then	ADV
ap-1408	276	2	d(c	d(c	PROPN
ap-1408	276	3	+	+	CCONJ
ap-1408	276	4	a	a	X
ap-1408	276	5	)	)	PUNCT
ap-1408	276	6	=	=	SYM
ap-1408	276	7	d(c	d(c	PROPN
ap-1408	276	8	)	)	PUNCT
ap-1408	276	9	=	=	PUNCT
ap-1408	276	10	d(b	d(b	X
ap-1408	276	11	)	)	PUNCT
ap-1408	276	12	=	=	SYM
ap-1408	276	13	d(e	d(e	NOUN
ap-1408	276	14	)	)	PUNCT
ap-1408	276	15	.	.	PUNCT
ap-1408	277	1	hence	hence	ADV
ap-1408	277	2	in	in	ADP
ap-1408	277	3	all	all	DET
ap-1408	277	4	cases	case	NOUN
ap-1408	277	5	(	(	PUNCT
ap-1408	277	6	c	c	X
ap-1408	277	7	⊕d	⊕d	PROPN
ap-1408	277	8	a	a	X
ap-1408	277	9	)	)	PUNCT
ap-1408	277	10	⊕d	⊕d	NOUN
ap-1408	277	11	e	e	NOUN
ap-1408	277	12	is	be	AUX
ap-1408	277	13	defined	define	VERB
ap-1408	277	14	.	.	PUNCT
ap-1408	278	1	recall	recall	VERB
ap-1408	278	2	that	that	SCONJ
ap-1408	278	3	we	we	PRON
ap-1408	278	4	have	have	VERB
ap-1408	278	5	the	the	DET
ap-1408	278	6	following	following	ADJ
ap-1408	278	7	result	result	NOUN
ap-1408	278	8	of	of	ADP
ap-1408	278	9	[	[	X
ap-1408	278	10	9	9	NUM
ap-1408	278	11	]	]	PUNCT
ap-1408	278	12	.	.	PUNCT
ap-1408	279	1	68	68	NUM
ap-1408	279	2	acta	acta	PROPN
ap-1408	279	3	polytechnica	polytechnica	PROPN
ap-1408	279	4	vol	vol	NOUN
ap-1408	279	5	.	.	PUNCT
ap-1408	280	1	51	51	NUM
ap-1408	280	2	no	no	NOUN
ap-1408	280	3	.	.	PUNCT
ap-1408	281	1	4/2011	4/2011	PROPN
ap-1408	281	2	theorem	theorem	NOUN
ap-1408	281	3	2	2	NUM
ap-1408	281	4	[	[	SYM
ap-1408	281	5	9	9	NUM
ap-1408	281	6	,	,	PUNCT
ap-1408	281	7	theorem	theorem	VERB
ap-1408	281	8	1	1	NUM
ap-1408	281	9	]	]	PUNCT
ap-1408	281	10	let	let	VERB
ap-1408	281	11	h	h	PRON
ap-1408	281	12	be	be	AUX
ap-1408	281	13	an	an	DET
ap-1408	281	14	infinitedimensional	infinitedimensional	ADJ
ap-1408	281	15	complex	complex	ADJ
ap-1408	281	16	hilbert	hilbert	NOUN
ap-1408	281	17	space	space	NOUN
ap-1408	281	18	.	.	PUNCT
ap-1408	282	1	let	let	VERB
ap-1408	282	2	us	we	PRON
ap-1408	282	3	define	define	VERB
ap-1408	282	4	the	the	DET
ap-1408	282	5	following	follow	VERB
ap-1408	282	6	set	set	NOUN
ap-1408	282	7	of	of	ADP
ap-1408	282	8	positive	positive	ADJ
ap-1408	282	9	linear	linear	PROPN
ap-1408	282	10	operators	operator	NOUN
ap-1408	282	11	densely	densely	ADV
ap-1408	282	12	defined	define	VERB
ap-1408	282	13	in	in	ADP
ap-1408	282	14	h	h	NOUN
ap-1408	282	15	:	:	PUNCT
ap-1408	282	16	v(h	v(h	NOUN
ap-1408	282	17	)	)	PUNCT
ap-1408	283	1	=	=	PRON
ap-1408	283	2	{	{	PUNCT
ap-1408	283	3	a	a	X
ap-1408	283	4	:	:	PUNCT
ap-1408	283	5	d(a	d(a	PROPN
ap-1408	283	6	)	)	PUNCT
ap-1408	283	7	→	→	SYM
ap-1408	283	8	h	h	NOUN
ap-1408	283	9	|	|	ADV
ap-1408	283	10	a	a	DET
ap-1408	283	11	≥	≥	NOUN
ap-1408	283	12	0	0	NUM
ap-1408	283	13	,	,	PUNCT
ap-1408	283	14	d(a	d(a	PROPN
ap-1408	283	15	)	)	PUNCT
ap-1408	283	16	=	=	SYM
ap-1408	283	17	h	h	NOUN
ap-1408	283	18	and	and	CCONJ
ap-1408	283	19	d(a	d(a	PROPN
ap-1408	283	20	)	)	PUNCT
ap-1408	284	1	=	=	SYM
ap-1408	284	2	h	h	NOUN
ap-1408	284	3	if	if	SCONJ
ap-1408	284	4	a	a	PRON
ap-1408	284	5	is	be	AUX
ap-1408	284	6	bounded	bound	VERB
ap-1408	284	7	}	}	PUNCT
ap-1408	284	8	.	.	PUNCT
ap-1408	285	1	let	let	VERB
ap-1408	285	2	⊕d	⊕d	NOUN
ap-1408	285	3	be	be	AUX
ap-1408	285	4	defined	define	VERB
ap-1408	285	5	for	for	ADP
ap-1408	285	6	a	a	DET
ap-1408	285	7	,	,	PUNCT
ap-1408	285	8	b	b	PROPN
ap-1408	285	9	∈	∈	PROPN
ap-1408	285	10	v(h	v(h	NOUN
ap-1408	285	11	)	)	PUNCT
ap-1408	285	12	by	by	ADP
ap-1408	285	13	a	a	DET
ap-1408	285	14	⊕d	⊕d	NOUN
ap-1408	285	15	b	b	X
ap-1408	285	16	=	=	PUNCT
ap-1408	285	17	a	a	DET
ap-1408	285	18	+	+	X
ap-1408	285	19	b	b	NOUN
ap-1408	285	20	(	(	PUNCT
ap-1408	285	21	the	the	DET
ap-1408	285	22	usual	usual	ADJ
ap-1408	285	23	sum	sum	NOUN
ap-1408	285	24	)	)	PUNCT
ap-1408	285	25	iff	iff	PROPN
ap-1408	285	26	1	1	NUM
ap-1408	285	27	.	.	PUNCT
ap-1408	285	28	either	either	CCONJ
ap-1408	285	29	at	at	ADV
ap-1408	285	30	least	least	ADV
ap-1408	285	31	one	one	NUM
ap-1408	285	32	out	out	ADP
ap-1408	285	33	of	of	ADP
ap-1408	285	34	a	a	PRON
ap-1408	285	35	,	,	PUNCT
ap-1408	285	36	b	b	PROPN
ap-1408	285	37	is	be	AUX
ap-1408	285	38	bounded	bound	VERB
ap-1408	285	39	2	2	NUM
ap-1408	285	40	.	.	PUNCT
ap-1408	286	1	or	or	CCONJ
ap-1408	286	2	both	both	CCONJ
ap-1408	286	3	a	a	DET
ap-1408	286	4	,	,	PUNCT
ap-1408	286	5	b	b	NOUN
ap-1408	286	6	are	be	AUX
ap-1408	286	7	unbounded	unbounded	ADJ
ap-1408	286	8	and	and	CCONJ
ap-1408	286	9	d(a	d(a	ADJ
ap-1408	286	10	)	)	PUNCT
ap-1408	286	11	=	=	PUNCT
ap-1408	286	12	d(b	d(b	PROPN
ap-1408	286	13	)	)	PUNCT
ap-1408	286	14	.	.	PUNCT
ap-1408	287	1	then	then	ADV
ap-1408	287	2	vd(h	vd(h	PUNCT
ap-1408	287	3	)	)	PUNCT
ap-1408	287	4	=	=	SYM
ap-1408	287	5	(	(	PUNCT
ap-1408	287	6	v(h);⊕d	v(h);⊕d	ADJ
ap-1408	287	7	,	,	PUNCT
ap-1408	287	8	0	0	NUM
ap-1408	287	9	)	)	PUNCT
ap-1408	287	10	is	be	AUX
ap-1408	287	11	a	a	DET
ap-1408	287	12	generalized	generalized	ADJ
ap-1408	287	13	effect	effect	NOUN
ap-1408	287	14	algebra	algebra	NOUN
ap-1408	287	15	such	such	ADJ
ap-1408	287	16	that	that	SCONJ
ap-1408	287	17	⊕d	⊕d	NOUN
ap-1408	287	18	extends	extend	VERB
ap-1408	287	19	the	the	DET
ap-1408	287	20	operation	operation	NOUN
ap-1408	287	21	⊕.	⊕.	ADV
ap-1408	287	22	then	then	ADV
ap-1408	287	23	pos(gr(h	pos(gr(h	PROPN
ap-1408	287	24	)	)	PUNCT
ap-1408	287	25	)	)	PUNCT
ap-1408	288	1	=	=	X
ap-1408	289	1	v(h	v(h	NOUN
ap-1408	289	2	)	)	PUNCT
ap-1408	289	3	,	,	PUNCT
ap-1408	289	4	hence	hence	ADV
ap-1408	289	5	the	the	DET
ap-1408	289	6	generalized	generalized	ADJ
ap-1408	289	7	effect	effect	NOUN
ap-1408	289	8	algebra	algebra	NOUN
ap-1408	289	9	v(h	v(h	NOUN
ap-1408	289	10	)	)	PUNCT
ap-1408	289	11	is	be	AUX
ap-1408	289	12	the	the	DET
ap-1408	289	13	positive	positive	ADJ
ap-1408	289	14	cone	cone	NOUN
ap-1408	289	15	of	of	ADP
ap-1408	289	16	gr(h	gr(h	NOUN
ap-1408	289	17	)	)	PUNCT
ap-1408	289	18	and	and	CCONJ
ap-1408	289	19	,	,	PUNCT
ap-1408	289	20	for	for	ADP
ap-1408	289	21	all	all	DET
ap-1408	289	22	a	a	DET
ap-1408	289	23	,	,	PUNCT
ap-1408	289	24	b	b	NOUN
ap-1408	289	25	,	,	PUNCT
ap-1408	289	26	c	c	PROPN
ap-1408	289	27	∈	∈	PROPN
ap-1408	289	28	v(h	v(h	NOUN
ap-1408	289	29	)	)	PUNCT
ap-1408	289	30	,	,	PUNCT
ap-1408	289	31	(	(	PUNCT
ap-1408	289	32	a	a	DET
ap-1408	289	33	⊕d	⊕d	NOUN
ap-1408	289	34	b	b	X
ap-1408	289	35	)	)	PUNCT
ap-1408	289	36	⊕d	⊕d	PROPN
ap-1408	289	37	c	c	PROPN
ap-1408	289	38	exists	exist	VERB
ap-1408	289	39	iff	iff	PROPN
ap-1408	289	40	a	a	DET
ap-1408	289	41	⊕d	⊕d	NOUN
ap-1408	289	42	(	(	PUNCT
ap-1408	289	43	b	b	NOUN
ap-1408	289	44	⊕d	⊕d	NOUN
ap-1408	289	45	c	c	NOUN
ap-1408	289	46	)	)	PUNCT
ap-1408	289	47	exists	exist	VERB
ap-1408	289	48	.	.	PUNCT
ap-1408	290	1	definition	definition	NOUN
ap-1408	290	2	5	5	NUM
ap-1408	290	3	let	let	VERB
ap-1408	290	4	(	(	PUNCT
ap-1408	290	5	g	g	NOUN
ap-1408	290	6	,	,	PUNCT
ap-1408	290	7	+	+	ADJ
ap-1408	290	8	,	,	PUNCT
ap-1408	290	9	0	0	NUM
ap-1408	290	10	)	)	PUNCT
ap-1408	290	11	be	be	AUX
ap-1408	290	12	a	a	DET
ap-1408	290	13	commutative	commutative	ADJ
ap-1408	290	14	partial	partial	ADJ
ap-1408	290	15	group	group	NOUN
ap-1408	290	16	and	and	CCONJ
ap-1408	290	17	let	let	VERB
ap-1408	290	18	s	s	PRON
ap-1408	290	19	be	be	AUX
ap-1408	290	20	a	a	DET
ap-1408	290	21	subset	subset	NOUN
ap-1408	290	22	of	of	ADP
ap-1408	290	23	g	g	NOUN
ap-1408	290	24	such	such	ADJ
ap-1408	290	25	as	as	ADP
ap-1408	290	26	:	:	PUNCT
ap-1408	290	27	(	(	PUNCT
ap-1408	290	28	si	si	NOUN
ap-1408	290	29	)	)	PUNCT
ap-1408	290	30	0	0	PUNCT
ap-1408	291	1	∈	∈	PROPN
ap-1408	291	2	s	s	NOUN
ap-1408	291	3	,	,	PUNCT
ap-1408	291	4	(	(	PUNCT
ap-1408	291	5	sii	sii	ADJ
ap-1408	291	6	)	)	PUNCT
ap-1408	291	7	−x	−x	NOUN
ap-1408	291	8	∈	∈	PROPN
ap-1408	291	9	s	s	NOUN
ap-1408	291	10	for	for	ADP
ap-1408	291	11	all	all	DET
ap-1408	291	12	x	x	SYM
ap-1408	291	13	∈	∈	PROPN
ap-1408	291	14	s	s	NOUN
ap-1408	291	15	,	,	PUNCT
ap-1408	291	16	(	(	PUNCT
ap-1408	291	17	siii	siii	NOUN
ap-1408	291	18	)	)	PUNCT
ap-1408	291	19	for	for	ADP
ap-1408	291	20	every	every	DET
ap-1408	291	21	x	x	PROPN
ap-1408	291	22	,	,	PUNCT
ap-1408	291	23	y	y	PROPN
ap-1408	291	24	∈	∈	PROPN
ap-1408	291	25	s	s	VERB
ap-1408	291	26	such	such	ADJ
ap-1408	291	27	x	x	PUNCT
ap-1408	292	1	+	+	NUM
ap-1408	292	2	y	y	NOUN
ap-1408	292	3	is	be	AUX
ap-1408	292	4	defined	define	VERB
ap-1408	292	5	also	also	ADV
ap-1408	292	6	x	x	X
ap-1408	293	1	+	+	CCONJ
ap-1408	293	2	y	y	PROPN
ap-1408	293	3	∈	∈	PROPN
ap-1408	293	4	s.	s.	PROPN
ap-1408	293	5	then	then	ADV
ap-1408	293	6	we	we	PRON
ap-1408	293	7	call	call	VERB
ap-1408	293	8	s	s	VERB
ap-1408	293	9	a	a	DET
ap-1408	293	10	commutative	commutative	ADJ
ap-1408	293	11	partial	partial	ADJ
ap-1408	293	12	subgroup	subgroup	NOUN
ap-1408	293	13	of	of	ADP
ap-1408	293	14	g.	g.	PROPN
ap-1408	293	15	let	let	VERB
ap-1408	293	16	g	g	NOUN
ap-1408	293	17	be	be	AUX
ap-1408	293	18	a	a	DET
ap-1408	293	19	wop	wop	NOUN
ap-1408	293	20	-	-	PUNCT
ap-1408	293	21	group	group	NOUN
ap-1408	293	22	with	with	ADP
ap-1408	293	23	respect	respect	NOUN
ap-1408	293	24	to	to	ADP
ap-1408	293	25	a	a	DET
ap-1408	293	26	partial	partial	ADJ
ap-1408	293	27	order	order	NOUN
ap-1408	293	28	≤g	≤g	NOUN
ap-1408	293	29	and	and	CCONJ
ap-1408	293	30	let	let	VERB
ap-1408	293	31	≤s	≤s	PROPN
ap-1408	293	32	be	be	AUX
ap-1408	293	33	a	a	DET
ap-1408	293	34	partial	partial	ADJ
ap-1408	293	35	order	order	NOUN
ap-1408	293	36	on	on	ADP
ap-1408	293	37	a	a	DET
ap-1408	293	38	commutative	commutative	ADJ
ap-1408	293	39	partial	partial	ADJ
ap-1408	293	40	subgroup	subgroup	NOUN
ap-1408	293	41	s	s	PART
ap-1408	293	42	⊆	⊆	NUM
ap-1408	293	43	g.	g.	NOUN
ap-1408	293	44	if	if	SCONJ
ap-1408	293	45	for	for	ADP
ap-1408	293	46	all	all	DET
ap-1408	293	47	x	x	NOUN
ap-1408	293	48	,	,	PUNCT
ap-1408	293	49	y	y	PROPN
ap-1408	293	50	∈	∈	PROPN
ap-1408	293	51	s	s	AUX
ap-1408	293	52	holds	hold	VERB
ap-1408	293	53	:	:	PUNCT
ap-1408	294	1	x	x	X
ap-1408	294	2	≤s	≤s	NOUN
ap-1408	294	3	y	y	PROPN
ap-1408	294	4	if	if	SCONJ
ap-1408	294	5	and	and	CCONJ
ap-1408	294	6	only	only	ADV
ap-1408	294	7	if	if	SCONJ
ap-1408	294	8	x	x	SYM
ap-1408	294	9	≤g	≤g	NOUN
ap-1408	294	10	y	y	PROPN
ap-1408	294	11	,	,	PUNCT
ap-1408	294	12	we	we	PRON
ap-1408	294	13	call	call	VERB
ap-1408	294	14	s	s	VERB
ap-1408	294	15	a	a	DET
ap-1408	294	16	wop	wop	NOUN
ap-1408	294	17	-	-	PUNCT
ap-1408	294	18	subgroup	subgroup	NOUN
ap-1408	294	19	of	of	ADP
ap-1408	294	20	g.	g.	PROPN
ap-1408	294	21	for	for	ADP
ap-1408	294	22	a	a	DET
ap-1408	294	23	commutative	commutative	ADJ
ap-1408	294	24	partial	partial	ADJ
ap-1408	294	25	group	group	NOUN
ap-1408	294	26	g	g	NOUN
ap-1408	294	27	=	=	PUNCT
ap-1408	294	28	(	(	PUNCT
ap-1408	294	29	g	g	PROPN
ap-1408	294	30	,	,	PUNCT
ap-1408	294	31	+	+	ADJ
ap-1408	294	32	,	,	PUNCT
ap-1408	294	33	0	0	NUM
ap-1408	294	34	)	)	PUNCT
ap-1408	294	35	and	and	CCONJ
ap-1408	294	36	a	a	DET
ap-1408	294	37	commutative	commutative	ADJ
ap-1408	294	38	partial	partial	ADJ
ap-1408	294	39	subgroup	subgroup	NOUN
ap-1408	294	40	s	s	PART
ap-1408	294	41	,	,	PUNCT
ap-1408	294	42	we	we	PRON
ap-1408	294	43	denote	denote	VERB
ap-1408	294	44	+	+	PROPN
ap-1408	294	45	s	s	NOUN
ap-1408	294	46	=	=	X
ap-1408	294	47	+	+	NOUN
ap-1408	294	48	/s2	/s2	NOUN
ap-1408	294	49	.	.	PUNCT
ap-1408	295	1	we	we	PRON
ap-1408	295	2	will	will	AUX
ap-1408	295	3	omit	omit	VERB
ap-1408	295	4	an	an	DET
ap-1408	295	5	index	index	NOUN
ap-1408	295	6	and	and	CCONJ
ap-1408	295	7	we	we	PRON
ap-1408	295	8	will	will	AUX
ap-1408	295	9	write	write	VERB
ap-1408	295	10	s	s	NOUN
ap-1408	295	11	=	=	PUNCT
ap-1408	295	12	(	(	PUNCT
ap-1408	295	13	s	s	X
ap-1408	295	14	,	,	PUNCT
ap-1408	295	15	+	+	ADJ
ap-1408	295	16	,	,	PUNCT
ap-1408	295	17	0	0	NUM
ap-1408	295	18	)	)	PUNCT
ap-1408	295	19	instead	instead	ADV
ap-1408	295	20	of	of	ADP
ap-1408	295	21	(	(	PUNCT
ap-1408	295	22	s	s	X
ap-1408	295	23	,	,	PUNCT
ap-1408	295	24	+	+	NOUN
ap-1408	295	25	s	s	NOUN
ap-1408	295	26	,	,	PUNCT
ap-1408	295	27	0	0	NUM
ap-1408	295	28	)	)	PUNCT
ap-1408	295	29	where	where	SCONJ
ap-1408	295	30	no	no	DET
ap-1408	295	31	confusion	confusion	NOUN
ap-1408	295	32	can	can	AUX
ap-1408	295	33	result	result	VERB
ap-1408	295	34	.	.	PUNCT
ap-1408	296	1	lemma	lemma	PROPN
ap-1408	296	2	1	1	NUM
ap-1408	296	3	let	let	VERB
ap-1408	296	4	g	g	NOUN
ap-1408	296	5	=	=	PUNCT
ap-1408	296	6	(	(	PUNCT
ap-1408	296	7	g	g	PROPN
ap-1408	296	8	,	,	PUNCT
ap-1408	296	9	+	+	ADJ
ap-1408	296	10	,	,	PUNCT
ap-1408	296	11	0	0	NUM
ap-1408	296	12	)	)	PUNCT
ap-1408	296	13	be	be	AUX
ap-1408	296	14	a	a	DET
ap-1408	296	15	commutative	commutative	ADJ
ap-1408	296	16	partial	partial	ADJ
ap-1408	296	17	group	group	NOUN
ap-1408	296	18	and	and	CCONJ
ap-1408	296	19	let	let	VERB
ap-1408	296	20	s	s	PRON
ap-1408	296	21	be	be	AUX
ap-1408	296	22	a	a	DET
ap-1408	296	23	commutative	commutative	ADJ
ap-1408	296	24	partial	partial	ADJ
ap-1408	296	25	subgroup	subgroup	NOUN
ap-1408	296	26	of	of	ADP
ap-1408	296	27	g.	g.	PROPN
ap-1408	296	28	then	then	ADV
ap-1408	296	29	(	(	PUNCT
ap-1408	296	30	s	s	X
ap-1408	296	31	,	,	PUNCT
ap-1408	296	32	+	+	ADJ
ap-1408	296	33	,	,	PUNCT
ap-1408	296	34	0	0	NUM
ap-1408	296	35	)	)	PUNCT
ap-1408	296	36	is	be	AUX
ap-1408	296	37	a	a	DET
ap-1408	296	38	commutative	commutative	ADJ
ap-1408	296	39	partial	partial	ADJ
ap-1408	296	40	group	group	NOUN
ap-1408	296	41	.	.	PUNCT
ap-1408	297	1	let	let	VERB
ap-1408	297	2	g	g	PRON
ap-1408	297	3	be	be	AUX
ap-1408	297	4	a	a	DET
ap-1408	297	5	wop	wop	NOUN
ap-1408	297	6	-	-	PUNCT
ap-1408	297	7	group	group	NOUN
ap-1408	297	8	and	and	CCONJ
ap-1408	297	9	let	let	VERB
ap-1408	297	10	s	s	PRON
ap-1408	297	11	be	be	AUX
ap-1408	297	12	a	a	DET
ap-1408	297	13	wop	wop	NOUN
ap-1408	297	14	-	-	PUNCT
ap-1408	297	15	subgroup	subgroup	NOUN
ap-1408	297	16	of	of	ADP
ap-1408	297	17	g.	g.	PROPN
ap-1408	298	1	then	then	ADV
ap-1408	298	2	s	s	VERB
ap-1408	298	3	is	be	AUX
ap-1408	298	4	a	a	DET
ap-1408	298	5	wop	wop	NOUN
ap-1408	298	6	-	-	PUNCT
ap-1408	298	7	group	group	NOUN
ap-1408	298	8	.	.	PUNCT
ap-1408	299	1	proof	proof	NOUN
ap-1408	299	2	.	.	PUNCT
ap-1408	300	1	conditions	condition	NOUN
ap-1408	300	2	(	(	PUNCT
ap-1408	300	3	gi	gi	NOUN
ap-1408	300	4	)	)	PUNCT
ap-1408	300	5	,	,	PUNCT
ap-1408	300	6	(	(	PUNCT
ap-1408	300	7	gii	gii	NOUN
ap-1408	300	8	)	)	PUNCT
ap-1408	300	9	and	and	CCONJ
ap-1408	300	10	(	(	PUNCT
ap-1408	300	11	gv	gv	PART
ap-1408	300	12	)	)	PUNCT
ap-1408	300	13	follow	follow	VERB
ap-1408	300	14	immediately	immediately	ADV
ap-1408	300	15	from	from	ADP
ap-1408	300	16	(	(	PUNCT
ap-1408	300	17	siii	siii	NOUN
ap-1408	300	18	)	)	PUNCT
ap-1408	300	19	.	.	PUNCT
ap-1408	301	1	condition	condition	NOUN
ap-1408	301	2	(	(	PUNCT
ap-1408	301	3	giii	giii	NOUN
ap-1408	301	4	)	)	PUNCT
ap-1408	301	5	follows	follow	VERB
ap-1408	301	6	from	from	ADP
ap-1408	301	7	(	(	PUNCT
ap-1408	301	8	si	si	NOUN
ap-1408	301	9	)	)	PUNCT
ap-1408	301	10	and	and	CCONJ
ap-1408	301	11	(	(	PUNCT
ap-1408	301	12	siii	siii	NOUN
ap-1408	301	13	)	)	PUNCT
ap-1408	301	14	and	and	CCONJ
ap-1408	301	15	condition	condition	NOUN
ap-1408	301	16	(	(	PUNCT
ap-1408	301	17	giv	giv	NOUN
ap-1408	301	18	)	)	PUNCT
ap-1408	301	19	follows	follow	VERB
ap-1408	301	20	from	from	ADP
ap-1408	301	21	(	(	PUNCT
ap-1408	301	22	sii	sii	PROPN
ap-1408	301	23	)	)	PUNCT
ap-1408	301	24	.	.	PUNCT
ap-1408	302	1	assume	assume	VERB
ap-1408	302	2	now	now	ADV
ap-1408	302	3	that	that	SCONJ
ap-1408	302	4	g	g	PROPN
ap-1408	302	5	is	be	AUX
ap-1408	302	6	a	a	DET
ap-1408	302	7	wop	wop	NOUN
ap-1408	302	8	-	-	PUNCT
ap-1408	302	9	group	group	NOUN
ap-1408	302	10	such	such	ADJ
ap-1408	303	1	that	that	PRON
ap-1408	303	2	s	s	PART
ap-1408	303	3	is	be	AUX
ap-1408	303	4	a	a	DET
ap-1408	303	5	wop	wop	NOUN
ap-1408	303	6	-	-	PUNCT
ap-1408	303	7	subgroup	subgroup	NOUN
ap-1408	303	8	of	of	ADP
ap-1408	303	9	g.	g.	PROPN
ap-1408	303	10	if	if	SCONJ
ap-1408	303	11	x	x	PROPN
ap-1408	303	12	,	,	PUNCT
ap-1408	303	13	y	y	PROPN
ap-1408	303	14	,	,	PUNCT
ap-1408	303	15	z	z	PROPN
ap-1408	303	16	∈	∈	PROPN
ap-1408	303	17	s	s	PART
ap-1408	303	18	,	,	PUNCT
ap-1408	303	19	x	x	SYM
ap-1408	303	20	≤s	≤s	PROPN
ap-1408	303	21	y	y	PROPN
ap-1408	303	22	and	and	CCONJ
ap-1408	303	23	x	x	PROPN
ap-1408	304	1	+	+	CCONJ
ap-1408	304	2	z	z	PROPN
ap-1408	304	3	,	,	PUNCT
ap-1408	304	4	y	y	PROPN
ap-1408	304	5	+	+	CCONJ
ap-1408	304	6	z	z	NOUN
ap-1408	304	7	are	be	AUX
ap-1408	304	8	defined	define	VERB
ap-1408	304	9	,	,	PUNCT
ap-1408	304	10	then	then	ADV
ap-1408	304	11	x	x	X
ap-1408	305	1	+	+	CCONJ
ap-1408	305	2	z	z	PROPN
ap-1408	305	3	,	,	PUNCT
ap-1408	305	4	y	y	PROPN
ap-1408	305	5	+	+	NUM
ap-1408	305	6	z	z	PROPN
ap-1408	305	7	∈	∈	PROPN
ap-1408	305	8	s	s	PART
ap-1408	305	9	and	and	CCONJ
ap-1408	305	10	x	x	X
ap-1408	306	1	+	+	CCONJ
ap-1408	306	2	z	z	AUX
ap-1408	306	3	≤g	≤g	PROPN
ap-1408	306	4	y	y	PROPN
ap-1408	307	1	+	+	CCONJ
ap-1408	307	2	z	z	NOUN
ap-1408	307	3	hence	hence	ADV
ap-1408	307	4	x	x	PUNCT
ap-1408	308	1	+	+	CCONJ
ap-1408	308	2	z	z	VERB
ap-1408	308	3	≤s	≤s	PROPN
ap-1408	308	4	y	y	PROPN
ap-1408	309	1	+	+	PROPN
ap-1408	309	2	z.	z.	PROPN
ap-1408	309	3	lemma	lemma	PROPN
ap-1408	309	4	2	2	NUM
ap-1408	309	5	let	let	VERB
ap-1408	309	6	g	g	NOUN
ap-1408	309	7	=	=	PUNCT
ap-1408	309	8	(	(	PUNCT
ap-1408	309	9	g	g	PROPN
ap-1408	309	10	,	,	PUNCT
ap-1408	309	11	+	+	ADJ
ap-1408	309	12	,	,	PUNCT
ap-1408	309	13	0	0	NUM
ap-1408	309	14	)	)	PUNCT
ap-1408	309	15	be	be	AUX
ap-1408	309	16	a	a	DET
ap-1408	309	17	commutative	commutative	ADJ
ap-1408	309	18	partial	partial	ADJ
ap-1408	309	19	group	group	NOUN
ap-1408	309	20	and	and	CCONJ
ap-1408	309	21	s1	s1	NOUN
ap-1408	309	22	,	,	PUNCT
ap-1408	309	23	s2	s2	NOUN
ap-1408	309	24	commutative	commutative	ADJ
ap-1408	309	25	partial	partial	ADJ
ap-1408	309	26	subgroups	subgroup	NOUN
ap-1408	309	27	of	of	ADP
ap-1408	309	28	g.	g.	PROPN
ap-1408	310	1	then	then	ADV
ap-1408	310	2	s	s	VERB
ap-1408	310	3	=	=	NOUN
ap-1408	310	4	s1∩s2	s1∩s2	NOUN
ap-1408	310	5	is	be	AUX
ap-1408	310	6	also	also	ADV
ap-1408	310	7	a	a	DET
ap-1408	310	8	commutative	commutative	ADJ
ap-1408	310	9	partial	partial	ADJ
ap-1408	310	10	subgroup	subgroup	NOUN
ap-1408	310	11	of	of	ADP
ap-1408	310	12	g.	g.	PROPN
ap-1408	310	13	proof	proof	PROPN
ap-1408	310	14	.	.	PUNCT
ap-1408	311	1	condition	condition	NOUN
ap-1408	311	2	(	(	PUNCT
ap-1408	311	3	si	si	X
ap-1408	311	4	)	)	PUNCT
ap-1408	311	5	is	be	AUX
ap-1408	311	6	clear	clear	ADJ
ap-1408	311	7	.	.	PUNCT
ap-1408	312	1	(	(	PUNCT
ap-1408	312	2	sii	sii	ADV
ap-1408	312	3	):	):	PUNCT
ap-1408	312	4	if	if	SCONJ
ap-1408	312	5	x	x	SYM
ap-1408	312	6	∈	∈	PROPN
ap-1408	312	7	s	s	VERB
ap-1408	312	8	then	then	ADV
ap-1408	312	9	x	x	SYM
ap-1408	312	10	∈	∈	PROPN
ap-1408	312	11	s1	s1	NOUN
ap-1408	312	12	and	and	CCONJ
ap-1408	312	13	x	x	PUNCT
ap-1408	312	14	∈	∈	PROPN
ap-1408	312	15	s2	s2	NOUN
ap-1408	312	16	hence	hence	ADV
ap-1408	312	17	−x	−x	NOUN
ap-1408	312	18	∈	∈	PROPN
ap-1408	312	19	s1	s1	NOUN
ap-1408	312	20	and	and	CCONJ
ap-1408	312	21	−x	−x	PROPN
ap-1408	312	22	∈	∈	PROPN
ap-1408	312	23	s2	s2	PROPN
ap-1408	312	24	.	.	PUNCT
ap-1408	313	1	therefore	therefore	ADV
ap-1408	313	2	−x	−x	PROPN
ap-1408	313	3	∈	∈	PROPN
ap-1408	313	4	s.	s.	PROPN
ap-1408	313	5	(	(	PUNCT
ap-1408	313	6	siii	siii	PROPN
ap-1408	313	7	):	):	PUNCT
ap-1408	313	8	assume	assume	VERB
ap-1408	313	9	that	that	SCONJ
ap-1408	313	10	x	x	X
ap-1408	313	11	,	,	PUNCT
ap-1408	313	12	y	y	PROPN
ap-1408	313	13	∈	∈	PROPN
ap-1408	313	14	s	s	VERB
ap-1408	313	15	such	such	ADJ
ap-1408	313	16	that	that	SCONJ
ap-1408	313	17	x+y	x+y	PROPN
ap-1408	313	18	is	be	AUX
ap-1408	313	19	defined	define	VERB
ap-1408	313	20	.	.	PUNCT
ap-1408	314	1	then	then	ADV
ap-1408	314	2	x	x	X
ap-1408	314	3	,	,	PUNCT
ap-1408	314	4	y	y	PROPN
ap-1408	314	5	∈	∈	PROPN
ap-1408	314	6	s1	s1	PROPN
ap-1408	314	7	and	and	CCONJ
ap-1408	314	8	x	x	NOUN
ap-1408	314	9	,	,	PUNCT
ap-1408	314	10	y	y	PROPN
ap-1408	314	11	∈	∈	PROPN
ap-1408	314	12	s2	s2	PROPN
ap-1408	314	13	.	.	PUNCT
ap-1408	315	1	hence	hence	ADV
ap-1408	315	2	x+y	x+y	NUM
ap-1408	315	3	∈	∈	PROPN
ap-1408	315	4	s1	s1	NOUN
ap-1408	315	5	and	and	CCONJ
ap-1408	315	6	x	x	PUNCT
ap-1408	315	7	+	+	CCONJ
ap-1408	315	8	y	y	PROPN
ap-1408	315	9	∈	∈	PROPN
ap-1408	315	10	s2	s2	PROPN
ap-1408	315	11	.	.	PUNCT
ap-1408	316	1	this	this	DET
ap-1408	316	2	yields	yield	VERB
ap-1408	316	3	x	x	PUNCT
ap-1408	317	1	+	+	CCONJ
ap-1408	317	2	y	y	PROPN
ap-1408	317	3	∈	∈	PROPN
ap-1408	317	4	s.	s.	PROPN
ap-1408	317	5	definition	definition	NOUN
ap-1408	317	6	6	6	NUM
ap-1408	317	7	let	let	VERB
ap-1408	317	8	g1	g1	PROPN
ap-1408	317	9	=	=	SYM
ap-1408	317	10	(	(	PUNCT
ap-1408	317	11	g1	g1	PROPN
ap-1408	317	12	,	,	PUNCT
ap-1408	317	13	+1	+1	PROPN
ap-1408	317	14	,	,	PUNCT
ap-1408	317	15	01	01	NUM
ap-1408	317	16	)	)	PUNCT
ap-1408	317	17	and	and	CCONJ
ap-1408	317	18	g2	g2	PROPN
ap-1408	317	19	=	=	SYM
ap-1408	317	20	(	(	PUNCT
ap-1408	317	21	g2	g2	PROPN
ap-1408	317	22	,	,	PUNCT
ap-1408	317	23	+2	+2	PROPN
ap-1408	317	24	,	,	PUNCT
ap-1408	317	25	02	02	NUM
ap-1408	317	26	)	)	PUNCT
ap-1408	317	27	be	be	AUX
ap-1408	317	28	commutative	commutative	ADJ
ap-1408	317	29	partial	partial	ADJ
ap-1408	317	30	groups	group	NOUN
ap-1408	317	31	.	.	PUNCT
ap-1408	318	1	a	a	DET
ap-1408	318	2	morphism	morphism	NOUN
ap-1408	318	3	is	be	AUX
ap-1408	318	4	a	a	DET
ap-1408	318	5	map	map	NOUN
ap-1408	318	6	ϕ	ϕ	NOUN
ap-1408	318	7	:	:	PUNCT
ap-1408	318	8	g1	g1	PROPN
ap-1408	318	9	→	→	PUNCT
ap-1408	318	10	g2	g2	PROPN
ap-1408	318	11	such	such	ADJ
ap-1408	318	12	that	that	SCONJ
ap-1408	318	13	,	,	PUNCT
ap-1408	318	14	for	for	ADP
ap-1408	318	15	any	any	DET
ap-1408	318	16	x	x	NOUN
ap-1408	318	17	,	,	PUNCT
ap-1408	318	18	y	y	PROPN
ap-1408	318	19	∈	∈	PROPN
ap-1408	318	20	g1	g1	NOUN
ap-1408	318	21	,	,	PUNCT
ap-1408	318	22	whenever	whenever	SCONJ
ap-1408	318	23	x	x	PRON
ap-1408	318	24	+1	+1	ADJ
ap-1408	318	25	y	y	PROPN
ap-1408	318	26	exists	exist	VERB
ap-1408	318	27	then	then	ADV
ap-1408	318	28	ϕ(x	ϕ(x	X
ap-1408	318	29	)	)	PUNCT
ap-1408	318	30	+2	+2	ADP
ap-1408	318	31	ϕ(y	ϕ(y	PROPN
ap-1408	318	32	)	)	PUNCT
ap-1408	318	33	exists	exist	VERB
ap-1408	318	34	,	,	PUNCT
ap-1408	318	35	in	in	ADP
ap-1408	318	36	which	which	DET
ap-1408	318	37	case	case	NOUN
ap-1408	318	38	ϕ(x	ϕ(x	VERB
ap-1408	318	39	+1	+1	PROPN
ap-1408	318	40	y	y	NOUN
ap-1408	318	41	)	)	PUNCT
ap-1408	318	42	=	=	PUNCT
ap-1408	318	43	ϕ(x	ϕ(x	PROPN
ap-1408	318	44	)	)	PUNCT
ap-1408	318	45	+2	+2	PRON
ap-1408	318	46	ϕ(y	ϕ(y	NUM
ap-1408	318	47	)	)	PUNCT
ap-1408	318	48	.	.	PUNCT
ap-1408	319	1	if	if	SCONJ
ap-1408	319	2	ϕ	ϕ	NOUN
ap-1408	319	3	is	be	AUX
ap-1408	319	4	a	a	DET
ap-1408	319	5	bijection	bijection	NOUN
ap-1408	319	6	such	such	ADJ
ap-1408	319	7	that	that	DET
ap-1408	319	8	ϕ	ϕ	NOUN
ap-1408	319	9	and	and	CCONJ
ap-1408	319	10	ϕ−1	ϕ−1	PROPN
ap-1408	319	11	are	be	AUX
ap-1408	319	12	morphisms	morphism	NOUN
ap-1408	319	13	we	we	PRON
ap-1408	319	14	say	say	VERB
ap-1408	319	15	that	that	SCONJ
ap-1408	319	16	ϕ	ϕ	PROPN
ap-1408	319	17	is	be	AUX
ap-1408	319	18	an	an	DET
ap-1408	319	19	isomorphism	isomorphism	NOUN
ap-1408	319	20	of	of	ADP
ap-1408	319	21	commutative	commutative	ADJ
ap-1408	319	22	partial	partial	ADJ
ap-1408	319	23	groups	group	NOUN
ap-1408	319	24	and	and	CCONJ
ap-1408	319	25	g1	g1	PROPN
ap-1408	319	26	and	and	CCONJ
ap-1408	319	27	g2	g2	PROPN
ap-1408	319	28	are	be	AUX
ap-1408	319	29	isomorphic	isomorphic	ADJ
ap-1408	319	30	.	.	PUNCT
ap-1408	320	1	moreover	moreover	ADV
ap-1408	320	2	,	,	PUNCT
ap-1408	320	3	let	let	VERB
ap-1408	320	4	≤1	≤1	PROPN
ap-1408	320	5	on	on	ADP
ap-1408	320	6	g1	g1	NOUN
ap-1408	320	7	and	and	CCONJ
ap-1408	320	8	≤2	≤2	NOUN
ap-1408	320	9	on	on	ADP
ap-1408	320	10	g2	g2	PROPN
ap-1408	320	11	be	be	AUX
ap-1408	320	12	partial	partial	ADJ
ap-1408	320	13	orders	order	NOUN
ap-1408	320	14	such	such	ADJ
ap-1408	320	15	that	that	SCONJ
ap-1408	320	16	g1	g1	NOUN
ap-1408	320	17	and	and	CCONJ
ap-1408	320	18	g2	g2	PROPN
ap-1408	320	19	are	be	AUX
ap-1408	320	20	wop	wop	NOUN
ap-1408	320	21	-	-	PUNCT
ap-1408	320	22	groups	group	NOUN
ap-1408	320	23	.	.	PUNCT
ap-1408	321	1	let	let	VERB
ap-1408	321	2	ϕ	ϕ	NOUN
ap-1408	321	3	:	:	PUNCT
ap-1408	321	4	g1	g1	PROPN
ap-1408	321	5	→	→	PUNCT
ap-1408	321	6	g2	g2	PROPN
ap-1408	321	7	be	be	AUX
ap-1408	321	8	a	a	DET
ap-1408	321	9	morphism	morphism	NOUN
ap-1408	321	10	between	between	ADP
ap-1408	321	11	commutative	commutative	ADJ
ap-1408	321	12	partial	partial	ADJ
ap-1408	321	13	groups	group	NOUN
ap-1408	321	14	.	.	PUNCT
ap-1408	322	1	if	if	SCONJ
ap-1408	322	2	for	for	ADP
ap-1408	322	3	every	every	DET
ap-1408	322	4	x	x	NOUN
ap-1408	322	5	,	,	PUNCT
ap-1408	322	6	y	y	PROPN
ap-1408	322	7	∈	∈	PROPN
ap-1408	322	8	g1	g1	PROPN
ap-1408	322	9	:	:	PUNCT
ap-1408	322	10	x	x	X
ap-1408	322	11	≤1	≤1	PRON
ap-1408	322	12	y	y	PROPN
ap-1408	322	13	implies	imply	VERB
ap-1408	322	14	ϕ(x	ϕ(x	X
ap-1408	322	15	)	)	PUNCT
ap-1408	322	16	≤2	≤2	VERB
ap-1408	322	17	ϕ(y	ϕ(y	PROPN
ap-1408	322	18	)	)	PUNCT
ap-1408	322	19	,	,	PUNCT
ap-1408	322	20	then	then	ADV
ap-1408	322	21	ϕ	ϕ	PROPN
ap-1408	322	22	is	be	AUX
ap-1408	322	23	a	a	DET
ap-1408	322	24	morphism	morphism	NOUN
ap-1408	322	25	between	between	ADP
ap-1408	322	26	wopgroups	wopgroup	NOUN
ap-1408	322	27	.	.	PUNCT
ap-1408	323	1	if	if	SCONJ
ap-1408	323	2	ϕ	ϕ	NOUN
ap-1408	323	3	is	be	AUX
ap-1408	323	4	a	a	DET
ap-1408	323	5	bijection	bijection	NOUN
ap-1408	323	6	,	,	PUNCT
ap-1408	323	7	ϕ	ϕ	NOUN
ap-1408	323	8	and	and	CCONJ
ap-1408	323	9	ϕ−1	ϕ−1	PROPN
ap-1408	323	10	are	be	AUX
ap-1408	323	11	morphisms	morphism	NOUN
ap-1408	323	12	we	we	PRON
ap-1408	323	13	say	say	VERB
ap-1408	323	14	that	that	SCONJ
ap-1408	323	15	ϕ	ϕ	PROPN
ap-1408	323	16	is	be	AUX
ap-1408	323	17	an	an	DET
ap-1408	323	18	isomorphism	isomorphism	NOUN
ap-1408	323	19	of	of	ADP
ap-1408	323	20	wop	wop	NOUN
ap-1408	323	21	-	-	PUNCT
ap-1408	323	22	groups	group	NOUN
ap-1408	323	23	and	and	CCONJ
ap-1408	323	24	g1	g1	PROPN
ap-1408	323	25	and	and	CCONJ
ap-1408	323	26	g2	g2	PROPN
ap-1408	323	27	are	be	AUX
ap-1408	323	28	isomorphic	isomorphic	ADJ
ap-1408	323	29	as	as	ADP
ap-1408	323	30	wop	wop	NOUN
ap-1408	323	31	-	-	PUNCT
ap-1408	323	32	groups	group	NOUN
ap-1408	323	33	.	.	PUNCT
ap-1408	324	1	definition	definition	NOUN
ap-1408	324	2	7	7	NUM
ap-1408	324	3	let	let	VERB
ap-1408	324	4	h	h	NOUN
ap-1408	324	5	be	be	AUX
ap-1408	324	6	an	an	DET
ap-1408	324	7	infinite	infinite	ADJ
ap-1408	324	8	-	-	PUNCT
ap-1408	324	9	dimensional	dimensional	ADJ
ap-1408	324	10	complex	complex	ADJ
ap-1408	324	11	hilbert	hilbert	NOUN
ap-1408	324	12	space	space	NOUN
ap-1408	324	13	.	.	PUNCT
ap-1408	325	1	let	let	VERB
ap-1408	325	2	us	we	PRON
ap-1408	325	3	define	define	VERB
ap-1408	325	4	the	the	DET
ap-1408	325	5	following	follow	VERB
ap-1408	325	6	sets	set	NOUN
ap-1408	325	7	of	of	ADP
ap-1408	325	8	linear	linear	PROPN
ap-1408	325	9	operators	operator	NOUN
ap-1408	325	10	densely	densely	ADV
ap-1408	325	11	defined	define	VERB
ap-1408	325	12	in	in	ADP
ap-1408	325	13	h	h	NOUN
ap-1408	325	14	:	:	PUNCT
ap-1408	325	15	sgr(h	sgr(h	X
ap-1408	325	16	)	)	PUNCT
ap-1408	326	1	=	=	PRON
ap-1408	326	2	{	{	PUNCT
ap-1408	326	3	a	a	DET
ap-1408	326	4	∈	∈	PROPN
ap-1408	326	5	gr(h	gr(h	NOUN
ap-1408	326	6	)	)	PUNCT
ap-1408	327	1	|	|	ADV
ap-1408	327	2	a	a	DET
ap-1408	327	3	⊂	⊂	PROPN
ap-1408	327	4	a∗	a∗	PROPN
ap-1408	327	5	}	}	PUNCT
ap-1408	327	6	hgr(h	hgr(h	PROPN
ap-1408	327	7	)	)	PUNCT
ap-1408	327	8	=	=	PRON
ap-1408	327	9	{	{	PUNCT
ap-1408	327	10	a	a	DET
ap-1408	327	11	∈	∈	PROPN
ap-1408	327	12	gr(h	gr(h	NOUN
ap-1408	327	13	)	)	PUNCT
ap-1408	327	14	|	|	ADV
ap-1408	327	15	a	a	DET
ap-1408	327	16	⊂	⊂	PROPN
ap-1408	327	17	a∗	a∗	PROPN
ap-1408	327	18	,	,	PUNCT
ap-1408	327	19	d(a	d(a	PROPN
ap-1408	327	20	)	)	PUNCT
ap-1408	327	21	=	=	SYM
ap-1408	328	1	h	h	NOUN
ap-1408	328	2	}	}	PUNCT
ap-1408	328	3	.	.	PUNCT
ap-1408	329	1	i.e.	i.e.	X
ap-1408	329	2	sgr(h	sgr(h	NOUN
ap-1408	329	3	)	)	PUNCT
ap-1408	329	4	is	be	AUX
ap-1408	329	5	the	the	DET
ap-1408	329	6	set	set	NOUN
ap-1408	329	7	of	of	ADP
ap-1408	329	8	all	all	DET
ap-1408	329	9	symmetric	symmetric	ADJ
ap-1408	329	10	operators	operator	NOUN
ap-1408	329	11	and	and	CCONJ
ap-1408	329	12	hgr(h	hgr(h	PROPN
ap-1408	329	13	)	)	PUNCT
ap-1408	329	14	is	be	AUX
ap-1408	329	15	the	the	DET
ap-1408	329	16	set	set	NOUN
ap-1408	329	17	of	of	ADP
ap-1408	329	18	all	all	DET
ap-1408	329	19	hermitian	hermitian	ADJ
ap-1408	329	20	operators	operator	NOUN
ap-1408	329	21	.	.	PUNCT
ap-1408	330	1	from	from	ADP
ap-1408	330	2	the	the	DET
ap-1408	330	3	definition	definition	NOUN
ap-1408	330	4	we	we	PRON
ap-1408	330	5	can	can	AUX
ap-1408	330	6	see	see	VERB
ap-1408	330	7	that	that	DET
ap-1408	330	8	hgr(h	hgr(h	PROPN
ap-1408	330	9	)	)	PUNCT
ap-1408	330	10	⊆	⊆	NUM
ap-1408	330	11	sgr(h	sgr(h	NOUN
ap-1408	330	12	)	)	PUNCT
ap-1408	330	13	.	.	PUNCT
ap-1408	331	1	it	it	PRON
ap-1408	331	2	is	be	AUX
ap-1408	331	3	a	a	DET
ap-1408	331	4	well	well	ADV
ap-1408	331	5	known	know	VERB
ap-1408	331	6	fact	fact	NOUN
ap-1408	331	7	that	that	SCONJ
ap-1408	331	8	every	every	DET
ap-1408	331	9	positive	positive	ADJ
ap-1408	331	10	operator	operator	NOUN
ap-1408	331	11	is	be	AUX
ap-1408	331	12	symmetric	symmetric	ADJ
ap-1408	331	13	and	and	CCONJ
ap-1408	331	14	every	every	DET
ap-1408	331	15	positive	positive	ADJ
ap-1408	331	16	bounded	bounded	ADJ
ap-1408	331	17	operator	operator	NOUN
ap-1408	331	18	is	be	AUX
ap-1408	331	19	both	both	DET
ap-1408	331	20	self	self	NOUN
ap-1408	331	21	-	-	PUNCT
ap-1408	331	22	adjoint	adjoint	NOUN
ap-1408	331	23	and	and	CCONJ
ap-1408	331	24	hermitian	hermitian	PROPN
ap-1408	331	25	(	(	PUNCT
ap-1408	331	26	see	see	VERB
ap-1408	331	27	[	[	X
ap-1408	331	28	2	2	NUM
ap-1408	331	29	]	]	NUM
ap-1408	331	30	)	)	PUNCT
ap-1408	331	31	.	.	PUNCT
ap-1408	332	1	theorem	theorem	NOUN
ap-1408	332	2	3	3	NUM
ap-1408	332	3	let	let	VERB
ap-1408	332	4	h	h	NOUN
ap-1408	332	5	be	be	AUX
ap-1408	332	6	an	an	DET
ap-1408	332	7	infinite	infinite	ADJ
ap-1408	332	8	-	-	PUNCT
ap-1408	332	9	dimensional	dimensional	ADJ
ap-1408	332	10	complex	complex	ADJ
ap-1408	332	11	hilbert	hilbert	NOUN
ap-1408	332	12	space	space	NOUN
ap-1408	332	13	.	.	PUNCT
ap-1408	333	1	let	let	VERB
ap-1408	333	2	≤s	≤s	PROPN
ap-1408	333	3	be	be	AUX
ap-1408	333	4	a	a	DET
ap-1408	333	5	relation	relation	NOUN
ap-1408	333	6	on	on	ADP
ap-1408	333	7	sgr(h	sgr(h	PROPN
ap-1408	333	8	)	)	PUNCT
ap-1408	333	9	defined	define	VERB
ap-1408	333	10	for	for	ADP
ap-1408	333	11	a	a	DET
ap-1408	333	12	,	,	PUNCT
ap-1408	333	13	b	b	PROPN
ap-1408	333	14	∈	∈	PROPN
ap-1408	333	15	sgr(h	sgr(h	PROPN
ap-1408	333	16	)	)	PUNCT
ap-1408	333	17	by	by	ADP
ap-1408	333	18	a	a	DET
ap-1408	333	19	≤s	≤s	PROPN
ap-1408	333	20	b	b	PROPN
ap-1408	334	1	if	if	SCONJ
ap-1408	334	2	and	and	CCONJ
ap-1408	334	3	only	only	ADV
ap-1408	334	4	if	if	SCONJ
ap-1408	334	5	there	there	PRON
ap-1408	334	6	exists	exist	VERB
ap-1408	334	7	a	a	DET
ap-1408	334	8	positive	positive	ADJ
ap-1408	334	9	operator	operator	NOUN
ap-1408	334	10	c	c	PROPN
ap-1408	334	11	∈	∈	PROPN
ap-1408	334	12	sgr(h	sgr(h	PROPN
ap-1408	334	13	)	)	PUNCT
ap-1408	334	14	such	such	ADJ
ap-1408	334	15	as	as	ADP
ap-1408	334	16	a	a	DET
ap-1408	334	17	⊕d	⊕d	NOUN
ap-1408	334	18	c	c	PROPN
ap-1408	334	19	=	=	PROPN
ap-1408	334	20	b.	b.	PROPN
ap-1408	334	21	then	then	ADV
ap-1408	334	22	(	(	PUNCT
ap-1408	334	23	sgr(h);⊕d	sgr(h);⊕d	ADJ
ap-1408	334	24	,	,	PUNCT
ap-1408	334	25	0	0	NUM
ap-1408	334	26	)	)	PUNCT
ap-1408	334	27	equipped	equip	VERB
ap-1408	334	28	with	with	ADP
ap-1408	334	29	≤s	≤s	PROPN
ap-1408	334	30	forms	form	VERB
ap-1408	334	31	a	a	DET
ap-1408	334	32	wop	wop	NOUN
ap-1408	334	33	-	-	PUNCT
ap-1408	334	34	subgroup	subgroup	NOUN
ap-1408	334	35	of	of	ADP
ap-1408	334	36	gr(h	gr(h	PROPN
ap-1408	334	37	)	)	PUNCT
ap-1408	334	38	.	.	PUNCT
ap-1408	335	1	proof	proof	NOUN
ap-1408	335	2	.	.	PUNCT
ap-1408	336	1	conditions	condition	NOUN
ap-1408	336	2	(	(	PUNCT
ap-1408	336	3	si	si	NOUN
ap-1408	336	4	)	)	PUNCT
ap-1408	336	5	and	and	CCONJ
ap-1408	336	6	(	(	PUNCT
ap-1408	336	7	sii	sii	PROPN
ap-1408	336	8	)	)	PUNCT
ap-1408	336	9	are	be	AUX
ap-1408	336	10	clearly	clearly	ADV
ap-1408	336	11	satisfied	satisfied	ADJ
ap-1408	336	12	.	.	PUNCT
ap-1408	337	1	we	we	PRON
ap-1408	337	2	have	have	VERB
ap-1408	337	3	to	to	PART
ap-1408	337	4	verify	verify	VERB
ap-1408	337	5	that	that	DET
ap-1408	337	6	sgr(h	sgr(h	NOUN
ap-1408	337	7	)	)	PUNCT
ap-1408	337	8	is	be	AUX
ap-1408	337	9	closed	close	VERB
ap-1408	337	10	under	under	ADP
ap-1408	337	11	addition	addition	NOUN
ap-1408	337	12	.	.	PUNCT
ap-1408	338	1	let	let	VERB
ap-1408	338	2	a	a	PRON
ap-1408	338	3	,	,	PUNCT
ap-1408	338	4	b	b	PROPN
ap-1408	338	5	∈	∈	PROPN
ap-1408	338	6	sgr(h	sgr(h	PROPN
ap-1408	338	7	)	)	PUNCT
ap-1408	338	8	be	be	AUX
ap-1408	338	9	bounded	bound	VERB
ap-1408	338	10	,	,	PUNCT
ap-1408	338	11	then	then	ADV
ap-1408	338	12	they	they	PRON
ap-1408	338	13	are	be	AUX
ap-1408	338	14	hermitian	hermitian	ADJ
ap-1408	338	15	and	and	CCONJ
ap-1408	338	16	it	it	PRON
ap-1408	338	17	is	be	AUX
ap-1408	338	18	well	well	ADV
ap-1408	338	19	known	know	VERB
ap-1408	338	20	that	that	SCONJ
ap-1408	338	21	the	the	DET
ap-1408	338	22	sum	sum	NOUN
ap-1408	338	23	of	of	ADP
ap-1408	338	24	two	two	NUM
ap-1408	338	25	hermitian	hermitian	ADJ
ap-1408	338	26	operators	operator	NOUN
ap-1408	338	27	is	be	AUX
ap-1408	338	28	also	also	ADV
ap-1408	338	29	hermitian	hermitian	PROPN
ap-1408	338	30	.	.	PUNCT
ap-1408	339	1	recall	recall	VERB
ap-1408	339	2	that	that	SCONJ
ap-1408	339	3	a	a	DET
ap-1408	339	4	⊂	⊂	PROPN
ap-1408	339	5	a∗	a∗	PROPN
ap-1408	339	6	iff	iff	PROPN
ap-1408	339	7	for	for	ADP
ap-1408	339	8	all	all	DET
ap-1408	339	9	x	x	NOUN
ap-1408	339	10	,	,	PUNCT
ap-1408	339	11	y	y	PROPN
ap-1408	339	12	∈	∈	PROPN
ap-1408	339	13	d(a	d(a	PROPN
ap-1408	339	14	)	)	PUNCT
ap-1408	339	15	:	:	PUNCT
ap-1408	340	1	〈	〈	PROPN
ap-1408	340	2	x	x	X
ap-1408	340	3	,	,	PUNCT
ap-1408	340	4	ay	ay	NOUN
ap-1408	340	5	〉	〉	NOUN
ap-1408	340	6	=	=	SYM
ap-1408	340	7	〈	〈	NOUN
ap-1408	340	8	ax	ax	NOUN
ap-1408	340	9	,	,	PUNCT
ap-1408	340	10	y	y	PROPN
ap-1408	340	11	〉	〉	PROPN
ap-1408	340	12	.	.	PUNCT
ap-1408	341	1	if	if	SCONJ
ap-1408	341	2	a	a	PRON
ap-1408	341	3	is	be	AUX
ap-1408	341	4	bounded	bound	VERB
ap-1408	341	5	and	and	CCONJ
ap-1408	341	6	b	b	NOUN
ap-1408	341	7	is	be	AUX
ap-1408	341	8	unbounded	unbounded	ADJ
ap-1408	341	9	then	then	ADV
ap-1408	341	10	d(a	d(a	PROPN
ap-1408	341	11	+	+	PROPN
ap-1408	341	12	b	b	X
ap-1408	341	13	)	)	PUNCT
ap-1408	341	14	=	=	SYM
ap-1408	341	15	d(b	d(b	X
ap-1408	341	16	)	)	PUNCT
ap-1408	341	17	and	and	CCONJ
ap-1408	341	18	,	,	PUNCT
ap-1408	341	19	for	for	ADP
ap-1408	341	20	all	all	DET
ap-1408	341	21	x	x	NOUN
ap-1408	341	22	,	,	PUNCT
ap-1408	341	23	y	y	PROPN
ap-1408	341	24	∈	∈	PROPN
ap-1408	341	25	d(b	d(b	PROPN
ap-1408	341	26	)	)	PUNCT
ap-1408	341	27	,	,	PUNCT
ap-1408	341	28	it	it	PRON
ap-1408	341	29	holds	hold	VERB
ap-1408	341	30	〈	〈	PROPN
ap-1408	341	31	x	x	X
ap-1408	341	32	,	,	PUNCT
ap-1408	341	33	(	(	PUNCT
ap-1408	341	34	a	a	DET
ap-1408	341	35	+	+	NUM
ap-1408	341	36	b)y	b)y	NOUN
ap-1408	341	37	〉	〉	NOUN
ap-1408	341	38	=	=	SYM
ap-1408	341	39	〈	〈	PROPN
ap-1408	341	40	x	x	X
ap-1408	341	41	,	,	PUNCT
ap-1408	341	42	ay	ay	NOUN
ap-1408	341	43	〉	〉	NOUN
ap-1408	341	44	+	+	CCONJ
ap-1408	341	45	〈	〈	PROPN
ap-1408	341	46	x	x	X
ap-1408	341	47	,	,	PUNCT
ap-1408	341	48	by	by	ADP
ap-1408	341	49	〉	〉	NOUN
ap-1408	341	50	=	=	SYM
ap-1408	341	51	69	69	NUM
ap-1408	341	52	acta	acta	PROPN
ap-1408	341	53	polytechnica	polytechnica	PROPN
ap-1408	341	54	vol	vol	NOUN
ap-1408	341	55	.	.	PUNCT
ap-1408	342	1	51	51	NUM
ap-1408	342	2	no	no	NOUN
ap-1408	342	3	.	.	PUNCT
ap-1408	343	1	4/2011	4/2011	NUM
ap-1408	343	2	〈	〈	NOUN
ap-1408	343	3	ax	ax	NOUN
ap-1408	343	4	,	,	PUNCT
ap-1408	343	5	y	y	PROPN
ap-1408	343	6	〉	〉	PROPN
ap-1408	343	7	+	+	CCONJ
ap-1408	343	8	〈	〈	PROPN
ap-1408	343	9	bx	bx	X
ap-1408	343	10	,	,	PUNCT
ap-1408	343	11	y	y	PROPN
ap-1408	343	12	〉	〉	NUM
ap-1408	343	13	=	=	SYM
ap-1408	343	14	〈	〈	PROPN
ap-1408	343	15	(	(	PUNCT
ap-1408	343	16	a	a	DET
ap-1408	343	17	+	+	X
ap-1408	343	18	b)x	b)x	X
ap-1408	343	19	,	,	PUNCT
ap-1408	343	20	y	y	PROPN
ap-1408	343	21	〉	〉	PROPN
ap-1408	343	22	hence	hence	ADV
ap-1408	343	23	(	(	PUNCT
ap-1408	343	24	a	a	DET
ap-1408	343	25	+	+	NUM
ap-1408	343	26	b	b	NOUN
ap-1408	343	27	)	)	PUNCT
ap-1408	343	28	⊂	⊂	PROPN
ap-1408	343	29	(	(	PUNCT
ap-1408	343	30	a	a	PRON
ap-1408	343	31	+	+	X
ap-1408	343	32	b)∗.	b)∗.	PROPN
ap-1408	343	33	a	a	DET
ap-1408	343	34	similar	similar	ADJ
ap-1408	343	35	argument	argument	NOUN
ap-1408	343	36	holds	hold	VERB
ap-1408	343	37	for	for	ADP
ap-1408	343	38	a	a	DET
ap-1408	343	39	unbounded	unbounded	ADJ
ap-1408	343	40	,	,	PUNCT
ap-1408	343	41	b	b	NOUN
ap-1408	343	42	unbounded	unbounded	ADJ
ap-1408	343	43	and	and	CCONJ
ap-1408	343	44	a	a	DET
ap-1408	343	45	+	+	NOUN
ap-1408	343	46	b	b	NOUN
ap-1408	343	47	unbounded	unbounded	ADJ
ap-1408	343	48	,	,	PUNCT
ap-1408	343	49	where	where	SCONJ
ap-1408	343	50	d(a	d(a	PROPN
ap-1408	343	51	+	+	PROPN
ap-1408	343	52	b	b	NOUN
ap-1408	343	53	)	)	PUNCT
ap-1408	343	54	=	=	SYM
ap-1408	343	55	d(a	d(a	PROPN
ap-1408	343	56	)	)	PUNCT
ap-1408	343	57	=	=	PUNCT
ap-1408	343	58	d(b	d(b	PROPN
ap-1408	343	59	)	)	PUNCT
ap-1408	343	60	.	.	PUNCT
ap-1408	344	1	if	if	SCONJ
ap-1408	344	2	a	a	PRON
ap-1408	344	3	and	and	CCONJ
ap-1408	344	4	b	b	NOUN
ap-1408	344	5	are	be	AUX
ap-1408	344	6	unbounded	unbounded	ADJ
ap-1408	344	7	and	and	CCONJ
ap-1408	344	8	a+b	a+b	NUM
ap-1408	344	9	is	be	AUX
ap-1408	344	10	bounded	bound	VERB
ap-1408	344	11	,	,	PUNCT
ap-1408	344	12	then	then	ADV
ap-1408	344	13	,	,	PUNCT
ap-1408	344	14	for	for	ADP
ap-1408	344	15	all	all	DET
ap-1408	344	16	x	x	NOUN
ap-1408	344	17	,	,	PUNCT
ap-1408	344	18	y	y	PROPN
ap-1408	344	19	∈	∈	PROPN
ap-1408	344	20	d(a	d(a	PROPN
ap-1408	344	21	)	)	PUNCT
ap-1408	344	22	=	=	PUNCT
ap-1408	345	1	d(b	d(b	PROPN
ap-1408	345	2	)	)	PUNCT
ap-1408	345	3	,	,	PUNCT
ap-1408	345	4	we	we	PRON
ap-1408	345	5	have	have	VERB
ap-1408	345	6	〈	〈	PROPN
ap-1408	345	7	x	x	X
ap-1408	345	8	,	,	PUNCT
ap-1408	345	9	(	(	PUNCT
ap-1408	345	10	a	a	DET
ap-1408	345	11	+	+	NUM
ap-1408	345	12	b)y	b)y	NOUN
ap-1408	345	13	〉	〉	NOUN
ap-1408	345	14	=	=	SYM
ap-1408	345	15	〈	〈	PROPN
ap-1408	345	16	x	x	X
ap-1408	345	17	,	,	PUNCT
ap-1408	345	18	ay	ay	NOUN
ap-1408	345	19	〉	〉	NOUN
ap-1408	345	20	+	+	CCONJ
ap-1408	345	21	〈	〈	PROPN
ap-1408	345	22	x	x	X
ap-1408	345	23	,	,	PUNCT
ap-1408	345	24	by	by	ADP
ap-1408	345	25	〉	〉	NOUN
ap-1408	345	26	=	=	SYM
ap-1408	345	27	〈	〈	NOUN
ap-1408	345	28	ax	ax	NOUN
ap-1408	345	29	,	,	PUNCT
ap-1408	345	30	y	y	PROPN
ap-1408	345	31	〉	〉	PROPN
ap-1408	345	32	+	+	CCONJ
ap-1408	345	33	〈	〈	PROPN
ap-1408	345	34	bx	bx	X
ap-1408	345	35	,	,	PUNCT
ap-1408	345	36	y	y	PROPN
ap-1408	345	37	〉	〉	NUM
ap-1408	345	38	=	=	SYM
ap-1408	345	39	〈	〈	PROPN
ap-1408	345	40	(	(	PUNCT
ap-1408	345	41	a	a	DET
ap-1408	345	42	+	+	X
ap-1408	345	43	b)x	b)x	X
ap-1408	345	44	,	,	PUNCT
ap-1408	345	45	y	y	PROPN
ap-1408	345	46	〉	〉	PROPN
ap-1408	345	47	.	.	PUNCT
ap-1408	346	1	therefore	therefore	ADV
ap-1408	346	2	(	(	PUNCT
ap-1408	346	3	a	a	DET
ap-1408	346	4	+	+	NUM
ap-1408	346	5	b	b	NOUN
ap-1408	346	6	)	)	PUNCT
ap-1408	346	7	⊂	⊂	PROPN
ap-1408	346	8	(	(	PUNCT
ap-1408	346	9	a	a	PRON
ap-1408	346	10	+	+	X
ap-1408	346	11	b)∗.	b)∗.	NOUN
ap-1408	346	12	from	from	ADP
ap-1408	346	13	(	(	PUNCT
ap-1408	346	14	a	a	DET
ap-1408	346	15	+	+	NUM
ap-1408	346	16	b	b	NOUN
ap-1408	346	17	)	)	PUNCT
ap-1408	346	18	⊂	⊂	PROPN
ap-1408	346	19	(	(	PUNCT
ap-1408	346	20	a	a	DET
ap-1408	346	21	+	+	X
ap-1408	346	22	b)b	b)b	NOUN
ap-1408	346	23	we	we	PRON
ap-1408	346	24	get	get	VERB
ap-1408	346	25	(	(	PUNCT
ap-1408	346	26	(	(	PUNCT
ap-1408	346	27	a	a	DET
ap-1408	346	28	+	+	X
ap-1408	346	29	b)b)∗	b)b)∗	NOUN
ap-1408	346	30	⊂	⊂	PROPN
ap-1408	346	31	(	(	PUNCT
ap-1408	346	32	a	a	X
ap-1408	346	33	+	+	X
ap-1408	346	34	b)∗	b)∗	PROPN
ap-1408	346	35	and	and	CCONJ
ap-1408	346	36	then	then	ADV
ap-1408	346	37	h	h	PROPN
ap-1408	346	38	=	=	SYM
ap-1408	346	39	d((a	d((a	PROPN
ap-1408	346	40	+	+	NUM
ap-1408	346	41	b)b∗	b)b∗	NUM
ap-1408	346	42	)	)	PUNCT
ap-1408	346	43	=	=	SYM
ap-1408	346	44	d((a	d((a	PROPN
ap-1408	346	45	+	+	NUM
ap-1408	346	46	b)∗	b)∗	PROPN
ap-1408	346	47	)	)	PUNCT
ap-1408	346	48	.	.	PUNCT
ap-1408	347	1	hence	hence	ADV
ap-1408	347	2	(	(	PUNCT
ap-1408	347	3	a	a	DET
ap-1408	347	4	+	+	X
ap-1408	347	5	b)∗	b)∗	NOUN
ap-1408	347	6	=	=	SYM
ap-1408	347	7	(	(	PUNCT
ap-1408	347	8	(	(	PUNCT
ap-1408	347	9	a	a	DET
ap-1408	347	10	+	+	X
ap-1408	347	11	b)b)∗.	b)b)∗.	NOUN
ap-1408	347	12	because	because	SCONJ
ap-1408	347	13	(	(	PUNCT
ap-1408	347	14	a	a	DET
ap-1408	347	15	+	+	NOUN
ap-1408	347	16	b)∗	b)∗	PROPN
ap-1408	347	17	is	be	AUX
ap-1408	347	18	symmetric	symmetric	ADJ
ap-1408	347	19	one	one	NUM
ap-1408	347	20	obtains	obtain	VERB
ap-1408	347	21	that	that	PRON
ap-1408	347	22	(	(	PUNCT
ap-1408	347	23	a	a	DET
ap-1408	347	24	+	+	X
ap-1408	347	25	b)∗	b)∗	NOUN
ap-1408	347	26	=	=	SYM
ap-1408	347	27	(	(	PUNCT
ap-1408	347	28	(	(	PUNCT
ap-1408	347	29	a	a	DET
ap-1408	347	30	+	+	X
ap-1408	347	31	b)b)∗	b)b)∗	NOUN
ap-1408	347	32	=	=	SYM
ap-1408	347	33	(	(	PUNCT
ap-1408	347	34	a	a	DET
ap-1408	347	35	+	+	NOUN
ap-1408	347	36	b)b	b)b	NOUN
ap-1408	347	37	.	.	PUNCT
ap-1408	348	1	hence	hence	ADV
ap-1408	348	2	(	(	PUNCT
ap-1408	348	3	a	a	DET
ap-1408	348	4	+	+	X
ap-1408	348	5	b)b	b)b	NOUN
ap-1408	348	6	=	=	SYM
ap-1408	348	7	(	(	PUNCT
ap-1408	348	8	a	a	DET
ap-1408	348	9	⊕d	⊕d	NOUN
ap-1408	348	10	b	b	X
ap-1408	348	11	)	)	PUNCT
ap-1408	348	12	∈	∈	PROPN
ap-1408	348	13	sgr(h	sgr(h	PROPN
ap-1408	348	14	)	)	PUNCT
ap-1408	348	15	.	.	PUNCT
ap-1408	349	1	now	now	ADV
ap-1408	349	2	let	let	VERB
ap-1408	349	3	a	a	DET
ap-1408	349	4	⊕d	⊕d	NOUN
ap-1408	349	5	c	c	NOUN
ap-1408	349	6	=	=	SYM
ap-1408	349	7	b	b	PROPN
ap-1408	349	8	where	where	SCONJ
ap-1408	349	9	a	a	PRON
ap-1408	349	10	,	,	PUNCT
ap-1408	349	11	b	b	PROPN
ap-1408	349	12	∈	∈	PROPN
ap-1408	349	13	sgr(h	sgr(h	PROPN
ap-1408	349	14	)	)	PUNCT
ap-1408	349	15	and	and	CCONJ
ap-1408	349	16	c	c	NOUN
ap-1408	349	17	∈	∈	PROPN
ap-1408	349	18	gr(h	gr(h	NOUN
ap-1408	349	19	)	)	PUNCT
ap-1408	349	20	,	,	PUNCT
ap-1408	349	21	c	c	NOUN
ap-1408	349	22	positive	positive	ADJ
ap-1408	349	23	.	.	PUNCT
ap-1408	350	1	since	since	SCONJ
ap-1408	350	2	c	c	PROPN
ap-1408	350	3	is	be	AUX
ap-1408	350	4	a	a	DET
ap-1408	350	5	positive	positive	ADJ
ap-1408	350	6	operator	operator	NOUN
ap-1408	350	7	we	we	PRON
ap-1408	350	8	get	get	VERB
ap-1408	350	9	that	that	PRON
ap-1408	350	10	c	c	PROPN
ap-1408	350	11	∈	∈	PROPN
ap-1408	350	12	sgr(h	sgr(h	PROPN
ap-1408	350	13	)	)	PUNCT
ap-1408	350	14	.	.	PUNCT
ap-1408	351	1	therefore	therefore	ADV
ap-1408	351	2	≤s=≤/sgr(h)2	≤s=≤/sgr(h)2	NOUN
ap-1408	351	3	.	.	PUNCT
ap-1408	352	1	4	4	NUM
ap-1408	352	2	operator	operator	NOUN
ap-1408	352	3	weakly	weakly	ADV
ap-1408	352	4	ordered	order	VERB
ap-1408	352	5	partial	partial	ADJ
ap-1408	352	6	groups	group	NOUN
ap-1408	352	7	as	as	ADP
ap-1408	352	8	a	a	DET
ap-1408	352	9	pasting	pasting	NOUN
ap-1408	352	10	of	of	ADP
ap-1408	352	11	operator	operator	NOUN
ap-1408	352	12	sub	sub	NOUN
ap-1408	352	13	-	-	NOUN
ap-1408	352	14	groups	group	NOUN
ap-1408	352	15	equipped	equip	VERB
ap-1408	352	16	with	with	ADP
ap-1408	352	17	the	the	DET
ap-1408	352	18	usual	usual	ADJ
ap-1408	352	19	sum	sum	NOUN
ap-1408	352	20	of	of	ADP
ap-1408	352	21	operators	operator	NOUN
ap-1408	352	22	let	let	VERB
ap-1408	352	23	us	we	PRON
ap-1408	352	24	recall	recall	VERB
ap-1408	352	25	the	the	DET
ap-1408	352	26	following	following	NOUN
ap-1408	352	27	theorem	theorem	NOUN
ap-1408	352	28	from	from	ADP
ap-1408	352	29	[	[	X
ap-1408	352	30	7	7	X
ap-1408	352	31	]	]	PUNCT
ap-1408	352	32	that	that	PRON
ap-1408	352	33	was	be	AUX
ap-1408	352	34	our	our	PRON
ap-1408	352	35	basic	basic	ADJ
ap-1408	352	36	motivation	motivation	NOUN
ap-1408	352	37	for	for	ADP
ap-1408	352	38	investigating	investigate	VERB
ap-1408	352	39	the	the	DET
ap-1408	352	40	set	set	NOUN
ap-1408	352	41	of	of	ADP
ap-1408	352	42	linear	linear	PROPN
ap-1408	352	43	operators	operator	NOUN
ap-1408	352	44	on	on	ADP
ap-1408	352	45	a	a	DET
ap-1408	352	46	hilbert	hilbert	NOUN
ap-1408	352	47	space	space	NOUN
ap-1408	352	48	.	.	PUNCT
ap-1408	353	1	theorem	theorem	NOUN
ap-1408	353	2	4	4	NUM
ap-1408	353	3	let	let	VERB
ap-1408	353	4	h	h	NOUN
ap-1408	353	5	be	be	AUX
ap-1408	353	6	an	an	DET
ap-1408	353	7	infinite	infinite	ADJ
ap-1408	353	8	-	-	PUNCT
ap-1408	353	9	dimensional	dimensional	ADJ
ap-1408	353	10	complex	complex	ADJ
ap-1408	353	11	hilbert	hilbert	NOUN
ap-1408	353	12	space	space	NOUN
ap-1408	353	13	and	and	CCONJ
ap-1408	353	14	let	let	VERB
ap-1408	353	15	d	d	PROPN
ap-1408	353	16	∈	∈	PROPN
ap-1408	353	17	d.	d.	PROPN
ap-1408	353	18	let	let	VERB
ap-1408	353	19	lind(h	lind(h	NOUN
ap-1408	353	20	)	)	PUNCT
ap-1408	354	1	=	=	PRON
ap-1408	354	2	{	{	PUNCT
ap-1408	354	3	a	a	DET
ap-1408	354	4	:	:	PUNCT
ap-1408	354	5	d	d	X
ap-1408	354	6	→	→	SYM
ap-1408	354	7	h	h	NOUN
ap-1408	354	8	|	|	ADV
ap-1408	354	9	a	a	PRON
ap-1408	354	10	is	be	AUX
ap-1408	354	11	a	a	DET
ap-1408	354	12	linear	linear	ADJ
ap-1408	354	13	operator	operator	NOUN
ap-1408	354	14	defined	define	VERB
ap-1408	354	15	on	on	ADP
ap-1408	354	16	d	d	NOUN
ap-1408	354	17	}	}	PUNCT
ap-1408	354	18	.	.	PUNCT
ap-1408	355	1	then	then	ADV
ap-1408	355	2	(	(	PUNCT
ap-1408	355	3	lind(h	lind(h	NOUN
ap-1408	355	4	)	)	PUNCT
ap-1408	355	5	;	;	PUNCT
ap-1408	356	1	+	+	X
ap-1408	356	2	,	,	PUNCT
ap-1408	356	3	≤	≤	NOUN
ap-1408	356	4	,	,	PUNCT
ap-1408	356	5	0	0	NUM
ap-1408	356	6	)	)	PUNCT
ap-1408	356	7	is	be	AUX
ap-1408	356	8	a	a	DET
ap-1408	356	9	partially	partially	ADV
ap-1408	356	10	ordered	order	VERB
ap-1408	356	11	commutative	commutative	ADJ
ap-1408	356	12	group	group	NOUN
ap-1408	356	13	where	where	SCONJ
ap-1408	356	14	0	0	NUM
ap-1408	356	15	is	be	AUX
ap-1408	356	16	the	the	DET
ap-1408	356	17	null	null	ADJ
ap-1408	356	18	operator	operator	NOUN
ap-1408	356	19	,	,	PUNCT
ap-1408	356	20	+	+	CCONJ
ap-1408	356	21	is	be	AUX
ap-1408	356	22	the	the	DET
ap-1408	356	23	usual	usual	ADJ
ap-1408	356	24	sum	sum	NOUN
ap-1408	356	25	of	of	ADP
ap-1408	356	26	operators	operator	NOUN
ap-1408	356	27	defined	define	VERB
ap-1408	356	28	on	on	ADP
ap-1408	356	29	d	d	PROPN
ap-1408	356	30	and	and	CCONJ
ap-1408	356	31	≤	≤	NUM
ap-1408	356	32	is	be	AUX
ap-1408	356	33	defined	define	VERB
ap-1408	356	34	for	for	ADP
ap-1408	356	35	all	all	DET
ap-1408	356	36	a	a	PRON
ap-1408	356	37	,	,	PUNCT
ap-1408	356	38	b	b	PROPN
ap-1408	356	39	∈	∈	ADP
ap-1408	356	40	lind(h	lind(h	NOUN
ap-1408	356	41	)	)	PUNCT
ap-1408	356	42	by	by	ADP
ap-1408	356	43	a	a	DET
ap-1408	356	44	≤	≤	PROPN
ap-1408	356	45	b	b	PROPN
ap-1408	357	1	iff	iff	PROPN
ap-1408	357	2	b	b	PROPN
ap-1408	357	3	−	−	PROPN
ap-1408	357	4	a	a	PRON
ap-1408	357	5	is	be	AUX
ap-1408	357	6	positive	positive	ADJ
ap-1408	357	7	.	.	PUNCT
ap-1408	358	1	definition	definition	NOUN
ap-1408	358	2	8	8	NUM
ap-1408	358	3	let	let	VERB
ap-1408	358	4	h	h	NOUN
ap-1408	358	5	be	be	AUX
ap-1408	358	6	an	an	DET
ap-1408	358	7	infinite	infinite	ADJ
ap-1408	358	8	-	-	PUNCT
ap-1408	358	9	dimensional	dimensional	ADJ
ap-1408	358	10	complex	complex	ADJ
ap-1408	358	11	hilbert	hilbert	NOUN
ap-1408	358	12	space	space	NOUN
ap-1408	358	13	and	and	CCONJ
ap-1408	358	14	let	let	VERB
ap-1408	358	15	d	d	PROPN
ap-1408	358	16	∈	∈	PROPN
ap-1408	358	17	d.	d.	PROPN
ap-1408	358	18	let	let	VERB
ap-1408	358	19	grd(h	grd(h	NOUN
ap-1408	358	20	)	)	PUNCT
ap-1408	359	1	=	=	PRON
ap-1408	359	2	{	{	PUNCT
ap-1408	359	3	a	a	DET
ap-1408	359	4	∈	∈	PROPN
ap-1408	359	5	gr(h	gr(h	NOUN
ap-1408	359	6	)	)	PUNCT
ap-1408	359	7	|	|	ADV
ap-1408	359	8	d(a	d(a	PROPN
ap-1408	359	9	)	)	PUNCT
ap-1408	360	1	=	=	SYM
ap-1408	360	2	d	d	PROPN
ap-1408	360	3	or	or	CCONJ
ap-1408	360	4	a	a	PRON
ap-1408	360	5	is	be	AUX
ap-1408	360	6	bounded	bound	VERB
ap-1408	360	7	}	}	PUNCT
ap-1408	360	8	.	.	PUNCT
ap-1408	361	1	sgrd(h	sgrd(h	NOUN
ap-1408	361	2	)	)	PUNCT
ap-1408	362	1	=	=	PRON
ap-1408	362	2	{	{	PUNCT
ap-1408	362	3	a	a	DET
ap-1408	362	4	∈	∈	PROPN
ap-1408	362	5	sgr(h	sgr(h	NOUN
ap-1408	362	6	)	)	PUNCT
ap-1408	362	7	|	|	ADV
ap-1408	362	8	d(a	d(a	PROPN
ap-1408	362	9	)	)	PUNCT
ap-1408	363	1	=	=	SYM
ap-1408	363	2	d	d	PROPN
ap-1408	363	3	or	or	CCONJ
ap-1408	363	4	a	a	PRON
ap-1408	363	5	is	be	AUX
ap-1408	363	6	bounded	bound	VERB
ap-1408	363	7	}	}	PUNCT
ap-1408	363	8	.	.	PUNCT
ap-1408	364	1	now	now	ADV
ap-1408	364	2	,	,	PUNCT
ap-1408	364	3	we	we	PRON
ap-1408	364	4	are	be	AUX
ap-1408	364	5	going	go	VERB
ap-1408	364	6	to	to	PART
ap-1408	364	7	show	show	VERB
ap-1408	364	8	that	that	SCONJ
ap-1408	364	9	the	the	DET
ap-1408	364	10	set	set	NOUN
ap-1408	364	11	grd(h	grd(h	NOUN
ap-1408	364	12	)	)	PUNCT
ap-1408	364	13	equipped	equip	VERB
ap-1408	364	14	with	with	ADP
ap-1408	364	15	the	the	DET
ap-1408	364	16	prescription	prescription	NOUN
ap-1408	364	17	⊕d	⊕d	NOUN
ap-1408	364	18	=	=	PUNCT
ap-1408	364	19	⊕d/(grd(h))2	⊕d/(grd(h))2	NOUN
ap-1408	364	20	and	and	CCONJ
ap-1408	364	21	the	the	DET
ap-1408	364	22	relation	relation	NOUN
ap-1408	364	23	≤d=≤/(grd(h))2	≤d=≤/(grd(h))2	NOUN
ap-1408	364	24	is	be	AUX
ap-1408	364	25	a	a	DET
ap-1408	364	26	partially	partially	ADV
ap-1408	364	27	ordered	order	VERB
ap-1408	364	28	commutative	commutative	ADJ
ap-1408	364	29	group	group	NOUN
ap-1408	364	30	isomorphic	isomorphic	ADJ
ap-1408	364	31	to	to	ADP
ap-1408	364	32	lind(h	lind(h	PROPN
ap-1408	364	33	)	)	PUNCT
ap-1408	364	34	.	.	PUNCT
ap-1408	365	1	theorem	theorem	NOUN
ap-1408	365	2	5	5	NUM
ap-1408	365	3	let	let	VERB
ap-1408	365	4	h	h	NOUN
ap-1408	365	5	be	be	AUX
ap-1408	365	6	an	an	DET
ap-1408	365	7	infinite	infinite	ADJ
ap-1408	365	8	-	-	PUNCT
ap-1408	365	9	dimensional	dimensional	ADJ
ap-1408	365	10	complex	complex	ADJ
ap-1408	365	11	hilbert	hilbert	NOUN
ap-1408	365	12	space	space	NOUN
ap-1408	365	13	and	and	CCONJ
ap-1408	365	14	let	let	VERB
ap-1408	365	15	d	d	PROPN
ap-1408	365	16	∈	∈	PROPN
ap-1408	365	17	d.	d.	PROPN
ap-1408	365	18	then	then	ADV
ap-1408	365	19	grd(h	grd(h	PROPN
ap-1408	365	20	)	)	PUNCT
ap-1408	366	1	=	=	SYM
ap-1408	366	2	(	(	PUNCT
ap-1408	366	3	grd(h);⊕d	grd(h);⊕d	NOUN
ap-1408	366	4	,	,	PUNCT
ap-1408	366	5	0	0	NUM
ap-1408	366	6	)	)	PUNCT
ap-1408	366	7	with	with	ADP
ap-1408	366	8	respect	respect	NOUN
ap-1408	366	9	to	to	ADP
ap-1408	366	10	≤d	≤d	NOUN
ap-1408	366	11	is	be	AUX
ap-1408	366	12	a	a	DET
ap-1408	366	13	wopsubgroup	wopsubgroup	NOUN
ap-1408	366	14	of	of	ADP
ap-1408	366	15	gr(h	gr(h	NOUN
ap-1408	366	16	)	)	PUNCT
ap-1408	366	17	such	such	ADJ
ap-1408	366	18	that	that	SCONJ
ap-1408	366	19	the	the	DET
ap-1408	366	20	induced	induced	ADJ
ap-1408	366	21	operation	operation	NOUN
ap-1408	366	22	⊕d	⊕d	NOUN
ap-1408	366	23	is	be	AUX
ap-1408	366	24	total	total	ADJ
ap-1408	366	25	.	.	PUNCT
ap-1408	367	1	moreover	moreover	ADV
ap-1408	367	2	,	,	PUNCT
ap-1408	367	3	grd(h	grd(h	NOUN
ap-1408	367	4	)	)	PUNCT
ap-1408	367	5	is	be	AUX
ap-1408	367	6	isomorphic	isomorphic	ADJ
ap-1408	367	7	to	to	ADP
ap-1408	367	8	lind(h	lind(h	PROPN
ap-1408	367	9	)	)	PUNCT
ap-1408	367	10	and	and	CCONJ
ap-1408	367	11	hence	hence	ADV
ap-1408	367	12	a	a	DET
ap-1408	367	13	partially	partially	ADV
ap-1408	367	14	ordered	order	VERB
ap-1408	367	15	commutative	commutative	ADJ
ap-1408	367	16	group	group	NOUN
ap-1408	367	17	.	.	PUNCT
ap-1408	368	1	proof	proof	NOUN
ap-1408	368	2	.	.	PUNCT
ap-1408	369	1	conditions	condition	NOUN
ap-1408	369	2	(	(	PUNCT
ap-1408	369	3	si	si	NOUN
ap-1408	369	4	)	)	PUNCT
ap-1408	369	5	and	and	CCONJ
ap-1408	369	6	(	(	PUNCT
ap-1408	369	7	sii	sii	PROPN
ap-1408	369	8	)	)	PUNCT
ap-1408	369	9	are	be	AUX
ap-1408	369	10	clearly	clearly	ADV
ap-1408	369	11	satisfied	satisfied	ADJ
ap-1408	369	12	.	.	PUNCT
ap-1408	370	1	let	let	VERB
ap-1408	370	2	us	we	PRON
ap-1408	370	3	check	check	VERB
ap-1408	370	4	condition	condition	NOUN
ap-1408	370	5	(	(	PUNCT
ap-1408	370	6	siii	siii	NOUN
ap-1408	370	7	)	)	PUNCT
ap-1408	370	8	.	.	PUNCT
ap-1408	371	1	for	for	ADP
ap-1408	371	2	a	a	DET
ap-1408	371	3	,	,	PUNCT
ap-1408	371	4	b	b	PROPN
ap-1408	371	5	∈	∈	PROPN
ap-1408	371	6	grd(h	grd(h	PROPN
ap-1408	371	7	)	)	PUNCT
ap-1408	371	8	,	,	PUNCT
ap-1408	371	9	first	first	ADV
ap-1408	371	10	assume	assume	VERB
ap-1408	371	11	that	that	SCONJ
ap-1408	371	12	d(a	d(a	PROPN
ap-1408	371	13	)	)	PUNCT
ap-1408	371	14	=	=	SYM
ap-1408	371	15	d(b	d(b	X
ap-1408	371	16	)	)	PUNCT
ap-1408	371	17	∈	∈	PROPN
ap-1408	371	18	{	{	PUNCT
ap-1408	371	19	d	d	NOUN
ap-1408	371	20	,	,	PUNCT
ap-1408	371	21	h	h	NOUN
ap-1408	371	22	}	}	PUNCT
ap-1408	371	23	.	.	PUNCT
ap-1408	372	1	then	then	ADV
ap-1408	372	2	d(a	d(a	PROPN
ap-1408	372	3	)	)	PUNCT
ap-1408	372	4	=	=	SYM
ap-1408	372	5	d(b	d(b	X
ap-1408	372	6	)	)	PUNCT
ap-1408	373	1	=	=	PUNCT
ap-1408	374	1	d(a	d(a	PROPN
ap-1408	374	2	+	+	CCONJ
ap-1408	374	3	b	b	X
ap-1408	374	4	)	)	PUNCT
ap-1408	374	5	∈	∈	PROPN
ap-1408	374	6	{	{	PUNCT
ap-1408	374	7	d	d	NOUN
ap-1408	374	8	,	,	PUNCT
ap-1408	374	9	h	h	NOUN
ap-1408	374	10	}	}	PUNCT
ap-1408	374	11	,	,	PUNCT
ap-1408	374	12	hence	hence	ADV
ap-1408	374	13	a	a	DET
ap-1408	374	14	⊕d	⊕d	NOUN
ap-1408	374	15	b	b	NOUN
ap-1408	374	16	exists	exist	VERB
ap-1408	374	17	in	in	ADP
ap-1408	374	18	grd(h	grd(h	NOUN
ap-1408	374	19	)	)	PUNCT
ap-1408	374	20	.	.	PUNCT
ap-1408	375	1	on	on	ADP
ap-1408	375	2	the	the	DET
ap-1408	375	3	other	other	ADJ
ap-1408	375	4	hand	hand	NOUN
ap-1408	375	5	,	,	PUNCT
ap-1408	375	6	let	let	VERB
ap-1408	375	7	d(a	d(a	PROPN
ap-1408	375	8	)	)	PUNCT
ap-1408	375	9	�	�	PROPN
ap-1408	375	10	=	=	SYM
ap-1408	375	11	d(b	d(b	PROPN
ap-1408	375	12	)	)	PUNCT
ap-1408	375	13	.	.	PUNCT
ap-1408	376	1	then	then	ADV
ap-1408	376	2	either	either	CCONJ
ap-1408	376	3	d(a	d(a	PROPN
ap-1408	376	4	)	)	PUNCT
ap-1408	376	5	⊂	⊂	X
ap-1408	376	6	d(b	d(b	X
ap-1408	376	7	)	)	PUNCT
ap-1408	377	1	=	=	SYM
ap-1408	377	2	h	h	NOUN
ap-1408	377	3	,	,	PUNCT
ap-1408	377	4	in	in	ADP
ap-1408	377	5	which	which	DET
ap-1408	377	6	case	case	NOUN
ap-1408	377	7	d(a	d(a	ADJ
ap-1408	377	8	)	)	PUNCT
ap-1408	377	9	=	=	SYM
ap-1408	378	1	d(a	d(a	PROPN
ap-1408	378	2	+	+	CCONJ
ap-1408	378	3	b	b	X
ap-1408	378	4	)	)	PUNCT
ap-1408	378	5	=	=	SYM
ap-1408	378	6	d	d	PROPN
ap-1408	378	7	or	or	CCONJ
ap-1408	378	8	d(b	d(b	NOUN
ap-1408	378	9	)	)	PUNCT
ap-1408	379	1	⊂	⊂	PROPN
ap-1408	380	1	d(a	d(a	PROPN
ap-1408	380	2	)	)	PUNCT
ap-1408	380	3	=	=	SYM
ap-1408	380	4	h	h	NOUN
ap-1408	380	5	with	with	ADP
ap-1408	380	6	d(b	d(b	NOUN
ap-1408	380	7	)	)	PUNCT
ap-1408	380	8	=	=	SYM
ap-1408	381	1	d(a	d(a	PROPN
ap-1408	381	2	+	+	CCONJ
ap-1408	381	3	b	b	X
ap-1408	381	4	)	)	PUNCT
ap-1408	381	5	=	=	SYM
ap-1408	382	1	d	d	NOUN
ap-1408	382	2	hence	hence	ADV
ap-1408	382	3	a	a	DET
ap-1408	382	4	⊕d	⊕d	NOUN
ap-1408	382	5	b	b	NOUN
ap-1408	382	6	also	also	ADV
ap-1408	382	7	exists	exist	VERB
ap-1408	382	8	in	in	ADP
ap-1408	382	9	grd(h	grd(h	NOUN
ap-1408	382	10	)	)	PUNCT
ap-1408	382	11	.	.	PUNCT
ap-1408	383	1	hence	hence	ADV
ap-1408	383	2	for	for	ADP
ap-1408	383	3	all	all	DET
ap-1408	383	4	a	a	DET
ap-1408	383	5	,	,	PUNCT
ap-1408	383	6	b	b	PROPN
ap-1408	383	7	∈	∈	PROPN
ap-1408	383	8	grd(h	grd(h	NOUN
ap-1408	383	9	)	)	PUNCT
ap-1408	384	1	we	we	PRON
ap-1408	384	2	have	have	VERB
ap-1408	384	3	that	that	PRON
ap-1408	384	4	a	a	DET
ap-1408	384	5	⊕d	⊕d	NOUN
ap-1408	384	6	b	b	PROPN
ap-1408	384	7	∈	∈	PROPN
ap-1408	384	8	grd(h	grd(h	PROPN
ap-1408	384	9	)	)	PUNCT
ap-1408	384	10	i.e.	i.e.	X
ap-1408	384	11	⊕d	⊕d	NOUN
ap-1408	384	12	is	be	AUX
ap-1408	384	13	a	a	DET
ap-1408	384	14	total	total	ADJ
ap-1408	384	15	operation	operation	NOUN
ap-1408	384	16	on	on	ADP
ap-1408	384	17	grd(h	grd(h	NOUN
ap-1408	384	18	)	)	PUNCT
ap-1408	384	19	.	.	PUNCT
ap-1408	385	1	now	now	ADV
ap-1408	385	2	let	let	VERB
ap-1408	385	3	a	a	DET
ap-1408	385	4	⊕d	⊕d	NOUN
ap-1408	385	5	b	b	NOUN
ap-1408	385	6	=	=	SYM
ap-1408	385	7	c	c	X
ap-1408	385	8	where	where	SCONJ
ap-1408	385	9	a	a	X
ap-1408	385	10	,	,	PUNCT
ap-1408	385	11	c	c	PROPN
ap-1408	385	12	∈	∈	PROPN
ap-1408	385	13	grd(h	grd(h	PROPN
ap-1408	385	14	)	)	PUNCT
ap-1408	385	15	and	and	CCONJ
ap-1408	385	16	b	b	X
ap-1408	385	17	∈	∈	PROPN
ap-1408	385	18	gr(h	gr(h	NOUN
ap-1408	385	19	)	)	PUNCT
ap-1408	385	20	,	,	PUNCT
ap-1408	386	1	b	b	X
ap-1408	386	2	positive	positive	ADJ
ap-1408	386	3	.	.	PUNCT
ap-1408	387	1	since	since	SCONJ
ap-1408	387	2	a⊕d	a⊕d	PROPN
ap-1408	387	3	b	b	PROPN
ap-1408	387	4	is	be	AUX
ap-1408	387	5	defined	define	VERB
ap-1408	387	6	and	and	CCONJ
ap-1408	387	7	b	b	NOUN
ap-1408	387	8	is	be	AUX
ap-1408	387	9	positive	positive	ADJ
ap-1408	387	10	we	we	PRON
ap-1408	387	11	have	have	VERB
ap-1408	387	12	that	that	DET
ap-1408	387	13	b	b	PROPN
ap-1408	387	14	∈	∈	PROPN
ap-1408	387	15	grd(h	grd(h	PROPN
ap-1408	387	16	)	)	PUNCT
ap-1408	387	17	∩	∩	NOUN
ap-1408	387	18	v(h	v(h	NOUN
ap-1408	387	19	)	)	PUNCT
ap-1408	387	20	.	.	PUNCT
ap-1408	388	1	we	we	PRON
ap-1408	388	2	can	can	AUX
ap-1408	388	3	define	define	VERB
ap-1408	388	4	a	a	DET
ap-1408	388	5	map	map	NOUN
ap-1408	388	6	ϕ	ϕ	NOUN
ap-1408	388	7	:	:	PUNCT
ap-1408	388	8	lind	lind	PROPN
ap-1408	388	9	→	→	SYM
ap-1408	388	10	grd(h	grd(h	PROPN
ap-1408	388	11	)	)	PUNCT
ap-1408	389	1	where	where	SCONJ
ap-1408	389	2	:	:	PUNCT
ap-1408	389	3	ϕ(a	ϕ(a	NOUN
ap-1408	389	4	)	)	PUNCT
ap-1408	389	5	=	=	PRON
ap-1408	389	6	{	{	PUNCT
ap-1408	389	7	a	a	PRON
ap-1408	389	8	if	if	SCONJ
ap-1408	389	9	a	a	PRON
ap-1408	389	10	is	be	AUX
ap-1408	389	11	unbounded	unbounded	ADJ
ap-1408	389	12	,	,	PUNCT
ap-1408	389	13	ab	ab	PROPN
ap-1408	389	14	if	if	SCONJ
ap-1408	389	15	a	a	PRON
ap-1408	389	16	is	be	AUX
ap-1408	389	17	bounded	bound	VERB
ap-1408	389	18	.	.	PUNCT
ap-1408	390	1	for	for	ADP
ap-1408	390	2	a	a	DET
ap-1408	390	3	∈	∈	PROPN
ap-1408	390	4	lind	lind	NOUN
ap-1408	390	5	unbounded	unbounded	ADJ
ap-1408	390	6	it	it	PRON
ap-1408	390	7	holds	hold	VERB
ap-1408	390	8	d(a	d(a	PROPN
ap-1408	390	9	)	)	PUNCT
ap-1408	390	10	=	=	SYM
ap-1408	390	11	d(ϕ(a	d(ϕ(a	NOUN
ap-1408	390	12	)	)	PUNCT
ap-1408	390	13	)	)	PUNCT
ap-1408	391	1	=	=	PUNCT
ap-1408	391	2	d	d	X
ap-1408	391	3	hence	hence	ADV
ap-1408	391	4	ϕ(a	ϕ(a	NOUN
ap-1408	391	5	)	)	PUNCT
ap-1408	391	6	∈	∈	PROPN
ap-1408	391	7	grd(h	grd(h	PROPN
ap-1408	391	8	)	)	PUNCT
ap-1408	391	9	.	.	PUNCT
ap-1408	392	1	for	for	ADP
ap-1408	392	2	a	a	DET
ap-1408	392	3	bounded	bounded	ADJ
ap-1408	392	4	we	we	PRON
ap-1408	392	5	have	have	VERB
ap-1408	392	6	d(ϕ(a	d(ϕ(a	NOUN
ap-1408	392	7	)	)	PUNCT
ap-1408	392	8	)	)	PUNCT
ap-1408	393	1	=	=	PUNCT
ap-1408	393	2	d(ab	d(ab	PROPN
ap-1408	393	3	)	)	PUNCT
ap-1408	393	4	=	=	SYM
ap-1408	394	1	h	h	NOUN
ap-1408	394	2	hence	hence	ADV
ap-1408	394	3	a	a	DET
ap-1408	394	4	∈	∈	PROPN
ap-1408	394	5	grd(h	grd(h	NOUN
ap-1408	394	6	)	)	PUNCT
ap-1408	394	7	.	.	PUNCT
ap-1408	395	1	we	we	PRON
ap-1408	395	2	can	can	AUX
ap-1408	395	3	define	define	VERB
ap-1408	395	4	ψ	ψ	X
ap-1408	395	5	:	:	PUNCT
ap-1408	395	6	grd(h	grd(h	NOUN
ap-1408	395	7	)	)	PUNCT
ap-1408	395	8	→	→	SYM
ap-1408	395	9	lind	lind	NOUN
ap-1408	395	10	as	as	ADP
ap-1408	395	11	ψ(b	ψ(b	NOUN
ap-1408	395	12	)	)	PUNCT
ap-1408	396	1	=	=	SYM
ap-1408	396	2	b	b	PROPN
ap-1408	396	3	for	for	ADP
ap-1408	396	4	b	b	NOUN
ap-1408	396	5	unbounded	unbounded	ADJ
ap-1408	396	6	and	and	CCONJ
ap-1408	396	7	ψ(b	ψ(b	NOUN
ap-1408	396	8	)	)	PUNCT
ap-1408	397	1	=	=	SYM
ap-1408	397	2	b	b	X
ap-1408	397	3	/	/	SYM
ap-1408	397	4	d	d	NOUN
ap-1408	397	5	for	for	ADP
ap-1408	397	6	b	b	PROPN
ap-1408	397	7	bounded	bound	VERB
ap-1408	397	8	.	.	PUNCT
ap-1408	398	1	then	then	ADV
ap-1408	398	2	clearly	clearly	ADV
ap-1408	398	3	ψ	ψ	ADP
ap-1408	398	4	◦	◦	NOUN
ap-1408	398	5	ϕ	ϕ	X
ap-1408	398	6	=	=	SYM
ap-1408	398	7	idlind	idlind	ADJ
ap-1408	398	8	and	and	CCONJ
ap-1408	398	9	ϕ	ϕ	NOUN
ap-1408	398	10	◦	◦	NOUN
ap-1408	398	11	ψ	ψ	X
ap-1408	398	12	=	=	SYM
ap-1408	398	13	idgrd(h	idgrd(h	PROPN
ap-1408	398	14	)	)	PUNCT
ap-1408	398	15	hence	hence	ADV
ap-1408	398	16	ϕ	ϕ	PROPN
ap-1408	398	17	is	be	AUX
ap-1408	398	18	a	a	DET
ap-1408	398	19	bijection	bijection	NOUN
ap-1408	398	20	.	.	PUNCT
ap-1408	399	1	it	it	PRON
ap-1408	399	2	is	be	AUX
ap-1408	399	3	evident	evident	ADJ
ap-1408	399	4	that	that	SCONJ
ap-1408	399	5	ϕ(0	ϕ(0	PRON
ap-1408	399	6	)	)	PUNCT
ap-1408	400	1	=	=	SYM
ap-1408	400	2	0	0	PUNCT
ap-1408	401	1	and	and	CCONJ
ap-1408	401	2	+	+	X
ap-1408	401	3	on	on	ADP
ap-1408	401	4	lind	lind	PROPN
ap-1408	401	5	is	be	AUX
ap-1408	401	6	total	total	ADJ
ap-1408	401	7	.	.	PUNCT
ap-1408	402	1	for	for	ADP
ap-1408	402	2	a	a	DET
ap-1408	402	3	,	,	PUNCT
ap-1408	402	4	b	b	PROPN
ap-1408	402	5	∈	∈	PROPN
ap-1408	402	6	lind	lind	NOUN
ap-1408	402	7	,	,	PUNCT
ap-1408	402	8	let	let	VERB
ap-1408	402	9	us	we	PRON
ap-1408	402	10	assume	assume	VERB
ap-1408	402	11	that	that	SCONJ
ap-1408	402	12	:	:	PUNCT
ap-1408	402	13	(	(	PUNCT
ap-1408	402	14	a	a	X
ap-1408	402	15	):	):	PUNCT
ap-1408	402	16	a	a	PRON
ap-1408	402	17	,	,	PUNCT
ap-1408	402	18	b	b	PROPN
ap-1408	402	19	be	be	AUX
ap-1408	402	20	bounded	bound	VERB
ap-1408	402	21	.	.	PUNCT
ap-1408	403	1	then	then	ADV
ap-1408	403	2	ϕ(a	ϕ(a	VERB
ap-1408	403	3	+	+	PUNCT
ap-1408	403	4	b	b	X
ap-1408	403	5	)	)	PUNCT
ap-1408	403	6	=	=	SYM
ap-1408	403	7	(	(	PUNCT
ap-1408	403	8	a	a	DET
ap-1408	403	9	+	+	X
ap-1408	403	10	b)b	b)b	NOUN
ap-1408	403	11	=	=	SYM
ap-1408	403	12	ab	ab	PROPN
ap-1408	404	1	+	+	CCONJ
ap-1408	404	2	bb	bb	NOUN
ap-1408	404	3	=	=	SYM
ap-1408	404	4	ϕ(a	ϕ(a	PROPN
ap-1408	404	5	)	)	PUNCT
ap-1408	404	6	⊕d	⊕d	NOUN
ap-1408	404	7	ϕ(b	ϕ(b	PROPN
ap-1408	404	8	)	)	PUNCT
ap-1408	404	9	.	.	PUNCT
ap-1408	405	1	(	(	PUNCT
ap-1408	405	2	b	b	X
ap-1408	405	3	):	):	PUNCT
ap-1408	405	4	a	a	PRON
ap-1408	405	5	be	be	AUX
ap-1408	405	6	bounded	bound	VERB
ap-1408	405	7	,	,	PUNCT
ap-1408	405	8	b	b	X
ap-1408	405	9	be	be	AUX
ap-1408	405	10	unbounded	unbounde	VERB
ap-1408	405	11	.	.	PUNCT
ap-1408	406	1	then	then	ADV
ap-1408	406	2	d(a	d(a	PROPN
ap-1408	406	3	+	+	PROPN
ap-1408	406	4	b	b	X
ap-1408	406	5	)	)	PUNCT
ap-1408	406	6	=	=	PUNCT
ap-1408	406	7	d(b	d(b	X
ap-1408	406	8	)	)	PUNCT
ap-1408	407	1	=	=	SYM
ap-1408	407	2	d	d	PROPN
ap-1408	407	3	and	and	CCONJ
ap-1408	407	4	ϕ(a	ϕ(a	PROPN
ap-1408	407	5	+	+	NUM
ap-1408	407	6	b	b	X
ap-1408	407	7	)	)	PUNCT
ap-1408	407	8	=	=	PUNCT
ap-1408	408	1	a	a	DET
ap-1408	408	2	+	+	NUM
ap-1408	408	3	b	b	NOUN
ap-1408	408	4	=	=	SYM
ap-1408	408	5	ab	ab	PROPN
ap-1408	408	6	+	+	PROPN
ap-1408	408	7	b	b	PROPN
ap-1408	408	8	=	=	SYM
ap-1408	408	9	ϕ(a	ϕ(a	PROPN
ap-1408	408	10	)	)	PUNCT
ap-1408	408	11	⊕d	⊕d	NOUN
ap-1408	408	12	ϕ(b	ϕ(b	PROPN
ap-1408	408	13	)	)	PUNCT
ap-1408	408	14	.	.	PUNCT
ap-1408	409	1	(	(	PUNCT
ap-1408	409	2	c	c	X
ap-1408	409	3	):	):	PUNCT
ap-1408	409	4	a	a	PRON
ap-1408	409	5	be	be	AUX
ap-1408	409	6	unbounded	unbounde	VERB
ap-1408	409	7	,	,	PUNCT
ap-1408	409	8	b	b	NOUN
ap-1408	409	9	be	be	AUX
ap-1408	409	10	unbounded	unbounde	VERB
ap-1408	409	11	,	,	PUNCT
ap-1408	409	12	a+b	a+b	NUM
ap-1408	409	13	be	be	AUX
ap-1408	409	14	unbounded	unbounde	VERB
ap-1408	409	15	.	.	PUNCT
ap-1408	410	1	then	then	ADV
ap-1408	410	2	ϕ(a+b	ϕ(a+b	NOUN
ap-1408	410	3	)	)	PUNCT
ap-1408	411	1	=	=	PRON
ap-1408	411	2	a+b	a+b	NUM
ap-1408	411	3	=	=	SYM
ap-1408	411	4	ϕ(a)⊕dϕ(b	ϕ(a)⊕dϕ(b	NOUN
ap-1408	411	5	)	)	PUNCT
ap-1408	411	6	(	(	PUNCT
ap-1408	411	7	d	d	X
ap-1408	411	8	):	):	PUNCT
ap-1408	411	9	a	a	PRON
ap-1408	411	10	be	be	AUX
ap-1408	411	11	unbounded	unbounde	VERB
ap-1408	411	12	,	,	PUNCT
ap-1408	411	13	b	b	X
ap-1408	411	14	be	be	AUX
ap-1408	411	15	unbounded	unbounde	VERB
ap-1408	411	16	,	,	PUNCT
ap-1408	411	17	a	a	PRON
ap-1408	411	18	+	+	X
ap-1408	411	19	b	b	AUX
ap-1408	411	20	be	be	AUX
ap-1408	411	21	bounded	bound	VERB
ap-1408	411	22	.	.	PUNCT
ap-1408	412	1	then	then	ADV
ap-1408	412	2	ϕ(a	ϕ(a	VERB
ap-1408	412	3	+	+	PUNCT
ap-1408	412	4	b	b	X
ap-1408	412	5	)	)	PUNCT
ap-1408	412	6	=	=	SYM
ap-1408	412	7	(	(	PUNCT
ap-1408	412	8	a	a	DET
ap-1408	412	9	+	+	X
ap-1408	412	10	b)b	b)b	NOUN
ap-1408	412	11	=	=	SYM
ap-1408	412	12	(	(	PUNCT
ap-1408	412	13	ϕ(a	ϕ(a	NOUN
ap-1408	412	14	)	)	PUNCT
ap-1408	413	1	+	+	CCONJ
ap-1408	413	2	ϕ(b))b	ϕ(b))b	NOUN
ap-1408	413	3	=	=	SYM
ap-1408	413	4	ϕ(a	ϕ(a	PROPN
ap-1408	413	5	)	)	PUNCT
ap-1408	413	6	⊕d	⊕d	NOUN
ap-1408	413	7	ϕ(b	ϕ(b	PROPN
ap-1408	413	8	)	)	PUNCT
ap-1408	413	9	.	.	PUNCT
ap-1408	414	1	now	now	ADV
ap-1408	414	2	,	,	PUNCT
ap-1408	414	3	we	we	PRON
ap-1408	414	4	should	should	AUX
ap-1408	414	5	verify	verify	VERB
ap-1408	414	6	order	order	NOUN
ap-1408	414	7	preservation	preservation	NOUN
ap-1408	414	8	,	,	PUNCT
ap-1408	414	9	but	but	CCONJ
ap-1408	414	10	it	it	PRON
ap-1408	414	11	is	be	AUX
ap-1408	414	12	clear	clear	ADJ
ap-1408	414	13	that	that	SCONJ
ap-1408	414	14	ϕ	ϕ	NOUN
ap-1408	414	15	and	and	CCONJ
ap-1408	414	16	ψ	ψ	VERB
ap-1408	414	17	preserve	preserve	VERB
ap-1408	414	18	order	order	NOUN
ap-1408	414	19	.	.	PUNCT
ap-1408	415	1	theorem	theorem	NOUN
ap-1408	415	2	6	6	NUM
ap-1408	415	3	let	let	VERB
ap-1408	415	4	h	h	NOUN
ap-1408	415	5	be	be	AUX
ap-1408	415	6	an	an	DET
ap-1408	415	7	infinite	infinite	ADJ
ap-1408	415	8	-	-	PUNCT
ap-1408	415	9	dimensional	dimensional	ADJ
ap-1408	415	10	complex	complex	ADJ
ap-1408	415	11	hilbert	hilbert	NOUN
ap-1408	415	12	space	space	NOUN
ap-1408	415	13	and	and	CCONJ
ap-1408	415	14	let	let	VERB
ap-1408	415	15	d	d	PROPN
ap-1408	415	16	∈	∈	PROPN
ap-1408	415	17	d.	d.	PROPN
ap-1408	415	18	then	then	ADV
ap-1408	415	19	sgrd(h	sgrd(h	PROPN
ap-1408	415	20	)	)	PUNCT
ap-1408	415	21	with	with	ADP
ap-1408	415	22	the	the	DET
ap-1408	415	23	induced	induced	ADJ
ap-1408	415	24	total	total	ADJ
ap-1408	415	25	operation	operation	NOUN
ap-1408	415	26	⊕d	⊕d	NOUN
ap-1408	415	27	and	and	CCONJ
ap-1408	415	28	the	the	DET
ap-1408	415	29	induced	induced	ADJ
ap-1408	415	30	partial	partial	ADJ
ap-1408	415	31	order	order	NOUN
ap-1408	415	32	≤sgrd(h	≤sgrd(h	PUNCT
ap-1408	415	33	)	)	PUNCT
ap-1408	415	34	is	be	AUX
ap-1408	415	35	a	a	DET
ap-1408	415	36	wop	wop	NOUN
ap-1408	415	37	-	-	PUNCT
ap-1408	415	38	subgroup	subgroup	NOUN
ap-1408	415	39	of	of	ADP
ap-1408	415	40	grd(h	grd(h	PROPN
ap-1408	415	41	)	)	PUNCT
ap-1408	415	42	and	and	CCONJ
ap-1408	415	43	hence	hence	ADV
ap-1408	415	44	a	a	DET
ap-1408	415	45	partially	partially	ADV
ap-1408	415	46	ordered	order	VERB
ap-1408	415	47	commutative	commutative	ADJ
ap-1408	415	48	subgroup	subgroup	NOUN
ap-1408	415	49	of	of	ADP
ap-1408	415	50	grd(h	grd(h	PROPN
ap-1408	415	51	)	)	PUNCT
ap-1408	415	52	.	.	PUNCT
ap-1408	416	1	proof	proof	NOUN
ap-1408	416	2	.	.	PUNCT
ap-1408	417	1	sgrd(h	sgrd(h	NOUN
ap-1408	417	2	)	)	PUNCT
ap-1408	417	3	is	be	AUX
ap-1408	417	4	a	a	DET
ap-1408	417	5	commutative	commutative	ADJ
ap-1408	417	6	subgroup	subgroup	NOUN
ap-1408	417	7	of	of	ADP
ap-1408	417	8	grd(h	grd(h	PROPN
ap-1408	417	9	)	)	PUNCT
ap-1408	417	10	because	because	SCONJ
ap-1408	417	11	of	of	ADP
ap-1408	417	12	lemma	lemma	PROPN
ap-1408	417	13	2	2	NUM
ap-1408	417	14	and	and	CCONJ
ap-1408	417	15	sgrd(h	sgrd(h	NOUN
ap-1408	417	16	)	)	PUNCT
ap-1408	418	1	=	=	SYM
ap-1408	418	2	70	70	NUM
ap-1408	418	3	acta	acta	PROPN
ap-1408	418	4	polytechnica	polytechnica	PROPN
ap-1408	418	5	vol	vol	NOUN
ap-1408	418	6	.	.	PUNCT
ap-1408	419	1	51	51	NUM
ap-1408	419	2	no	no	INTJ
ap-1408	419	3	.	.	PUNCT
ap-1408	420	1	4/2011	4/2011	NUM
ap-1408	420	2	grd(h)∩sgr(h	grd(h)∩sgr(h	NOUN
ap-1408	420	3	)	)	PUNCT
ap-1408	420	4	.	.	PUNCT
ap-1408	421	1	we	we	PRON
ap-1408	421	2	have	have	VERB
ap-1408	421	3	to	to	PART
ap-1408	421	4	check	check	VERB
ap-1408	421	5	order	order	NOUN
ap-1408	421	6	preservation	preservation	NOUN
ap-1408	421	7	.	.	PUNCT
ap-1408	422	1	for	for	ADP
ap-1408	422	2	any	any	DET
ap-1408	422	3	a	a	PRON
ap-1408	422	4	,	,	PUNCT
ap-1408	422	5	b	b	PROPN
ap-1408	422	6	∈	∈	PROPN
ap-1408	422	7	sgrd(h	sgrd(h	NOUN
ap-1408	422	8	)	)	PUNCT
ap-1408	423	1	such	such	ADJ
ap-1408	423	2	that	that	SCONJ
ap-1408	423	3	a	a	DET
ap-1408	423	4	≤grd(h	≤grd(h	PROPN
ap-1408	423	5	)	)	PUNCT
ap-1408	423	6	b	b	NOUN
ap-1408	423	7	we	we	PRON
ap-1408	423	8	have	have	VERB
ap-1408	423	9	that	that	SCONJ
ap-1408	423	10	there	there	PRON
ap-1408	423	11	exists	exist	VERB
ap-1408	423	12	positive	positive	ADJ
ap-1408	423	13	c	c	PROPN
ap-1408	423	14	∈	∈	PROPN
ap-1408	423	15	grd(h	grd(h	PROPN
ap-1408	423	16	)	)	PUNCT
ap-1408	424	1	such	such	ADJ
ap-1408	424	2	that	that	SCONJ
ap-1408	424	3	a	a	DET
ap-1408	424	4	+	+	NOUN
ap-1408	424	5	c	c	NOUN
ap-1408	424	6	=	=	SYM
ap-1408	424	7	b.	b.	PROPN
ap-1408	424	8	since	since	SCONJ
ap-1408	424	9	every	every	DET
ap-1408	424	10	positive	positive	ADJ
ap-1408	424	11	operator	operator	NOUN
ap-1408	424	12	is	be	AUX
ap-1408	424	13	symmetric	symmetric	ADJ
ap-1408	424	14	we	we	PRON
ap-1408	424	15	have	have	VERB
ap-1408	424	16	that	that	DET
ap-1408	424	17	c	c	PROPN
ap-1408	424	18	∈	∈	PROPN
ap-1408	424	19	sgrd(h	sgrd(h	NOUN
ap-1408	424	20	)	)	PUNCT
ap-1408	424	21	.	.	PUNCT
ap-1408	425	1	this	this	DET
ap-1408	425	2	yields	yield	VERB
ap-1408	425	3	that	that	PRON
ap-1408	425	4	≤sgrd(h)=	≤sgrd(h)=	NOUN
ap-1408	425	5	(	(	PUNCT
ap-1408	425	6	≤grd(h))/sgrd(h)2	≤grd(h))/sgrd(h)2	PROPN
ap-1408	425	7	.	.	PUNCT
ap-1408	426	1	theorem	theorem	VERB
ap-1408	426	2	7	7	NUM
ap-1408	426	3	(	(	PUNCT
ap-1408	426	4	the	the	DET
ap-1408	426	5	pasting	pasting	NOUN
ap-1408	426	6	theorem	theorem	NOUN
ap-1408	426	7	for	for	ADP
ap-1408	426	8	gr(h	gr(h	NOUN
ap-1408	426	9	)	)	PUNCT
ap-1408	426	10	)	)	PUNCT
ap-1408	427	1	let	let	VERB
ap-1408	427	2	h	h	NOUN
ap-1408	427	3	be	be	AUX
ap-1408	427	4	an	an	DET
ap-1408	427	5	infinite	infinite	ADJ
ap-1408	427	6	-	-	PUNCT
ap-1408	427	7	dimensional	dimensional	ADJ
ap-1408	427	8	complex	complex	ADJ
ap-1408	427	9	hilbert	hilbert	NOUN
ap-1408	427	10	space	space	NOUN
ap-1408	427	11	.	.	PUNCT
ap-1408	428	1	then	then	ADV
ap-1408	428	2	the	the	DET
ap-1408	428	3	wop	wop	NOUN
ap-1408	428	4	-	-	PUNCT
ap-1408	428	5	group	group	NOUN
ap-1408	428	6	gr(h	gr(h	NOUN
ap-1408	428	7	)	)	PUNCT
ap-1408	428	8	pastes	paste	NOUN
ap-1408	428	9	their	their	PRON
ap-1408	428	10	partially	partially	ADV
ap-1408	428	11	ordered	order	VERB
ap-1408	428	12	commutative	commutative	ADJ
ap-1408	428	13	subgroups	subgroup	NOUN
ap-1408	428	14	grd(h	grd(h	PROPN
ap-1408	428	15	)	)	PUNCT
ap-1408	428	16	,	,	PUNCT
ap-1408	429	1	d	d	PROPN
ap-1408	429	2	⊆	⊆	NUM
ap-1408	429	3	h	h	NOUN
ap-1408	429	4	a	a	DET
ap-1408	429	5	dense	dense	ADJ
ap-1408	429	6	linear	linear	NOUN
ap-1408	429	7	subspace	subspace	NOUN
ap-1408	429	8	of	of	ADP
ap-1408	429	9	h	h	NOUN
ap-1408	429	10	,	,	PUNCT
ap-1408	429	11	together	together	ADV
ap-1408	429	12	over	over	ADP
ap-1408	429	13	b(h	b(h	NOUN
ap-1408	429	14	)	)	PUNCT
ap-1408	429	15	,	,	PUNCT
ap-1408	429	16	i.e.	i.e.	X
ap-1408	429	17	grd1(h	grd1(h	NOUN
ap-1408	429	18	)	)	PUNCT
ap-1408	429	19	∩	∩	NOUN
ap-1408	429	20	grd2(h	grd2(h	NOUN
ap-1408	429	21	)	)	PUNCT
ap-1408	429	22	=	=	SYM
ap-1408	429	23	b(h	b(h	PROPN
ap-1408	429	24	)	)	PUNCT
ap-1408	429	25	for	for	ADP
ap-1408	429	26	every	every	DET
ap-1408	429	27	pair	pair	NOUN
ap-1408	429	28	d1	d1	NOUN
ap-1408	429	29	,	,	PUNCT
ap-1408	429	30	d2	d2	NOUN
ap-1408	429	31	of	of	ADP
ap-1408	429	32	dense	dense	ADJ
ap-1408	429	33	linear	linear	ADJ
ap-1408	429	34	subspaces	subspace	NOUN
ap-1408	429	35	of	of	ADP
ap-1408	429	36	h	h	NOUN
ap-1408	429	37	,	,	PUNCT
ap-1408	429	38	d1	d1	PROPN
ap-1408	429	39	�	�	PROPN
ap-1408	429	40	=	=	SYM
ap-1408	429	41	d2	d2	PROPN
ap-1408	429	42	,	,	PUNCT
ap-1408	429	43	and	and	CCONJ
ap-1408	429	44	gr(h	gr(h	NOUN
ap-1408	429	45	)	)	PUNCT
ap-1408	430	1	=	=	SYM
ap-1408	430	2	⋃	⋃	NOUN
ap-1408	430	3	{	{	PUNCT
ap-1408	430	4	grd(h	grd(h	NOUN
ap-1408	430	5	)	)	PUNCT
ap-1408	431	1	|	|	ADV
ap-1408	431	2	d	d	X
ap-1408	431	3	∈	∈	PROPN
ap-1408	431	4	d	d	NOUN
ap-1408	431	5	}	}	PUNCT
ap-1408	431	6	.	.	PUNCT
ap-1408	432	1	proof	proof	NOUN
ap-1408	432	2	.	.	PUNCT
ap-1408	433	1	straightforward	straightforward	ADJ
ap-1408	433	2	from	from	ADP
ap-1408	433	3	definition	definition	NOUN
ap-1408	433	4	,	,	PUNCT
ap-1408	433	5	for	for	ADP
ap-1408	433	6	d	d	PROPN
ap-1408	433	7	∈	∈	PROPN
ap-1408	433	8	d	d	X
ap-1408	433	9	,	,	PUNCT
ap-1408	433	10	every	every	PRON
ap-1408	433	11	bounded	bound	VERB
ap-1408	433	12	a	a	DET
ap-1408	433	13	∈	∈	PROPN
ap-1408	433	14	gr(h	gr(h	NOUN
ap-1408	433	15	)	)	PUNCT
ap-1408	433	16	lies	lie	VERB
ap-1408	433	17	in	in	ADP
ap-1408	433	18	grd(h	grd(h	NOUN
ap-1408	433	19	)	)	PUNCT
ap-1408	433	20	.	.	PUNCT
ap-1408	434	1	for	for	ADP
ap-1408	434	2	any	any	DET
ap-1408	434	3	unbounded	unbounded	ADJ
ap-1408	434	4	b	b	PROPN
ap-1408	434	5	∈	∈	PROPN
ap-1408	434	6	gr(h	gr(h	NOUN
ap-1408	434	7	)	)	PUNCT
ap-1408	434	8	,	,	PUNCT
ap-1408	434	9	b	b	X
ap-1408	434	10	∈	∈	PROPN
ap-1408	434	11	grd(h	grd(h	NOUN
ap-1408	434	12	)	)	PUNCT
ap-1408	434	13	if	if	SCONJ
ap-1408	434	14	and	and	CCONJ
ap-1408	434	15	only	only	ADV
ap-1408	434	16	if	if	SCONJ
ap-1408	434	17	d(b	d(b	X
ap-1408	434	18	)	)	PUNCT
ap-1408	434	19	=	=	PUNCT
ap-1408	435	1	d	d	NOUN
ap-1408	435	2	hence	hence	ADV
ap-1408	435	3	there	there	PRON
ap-1408	435	4	is	be	VERB
ap-1408	435	5	unique	unique	ADJ
ap-1408	435	6	grd(h	grd(h	NOUN
ap-1408	435	7	)	)	PUNCT
ap-1408	435	8	in	in	ADP
ap-1408	435	9	which	which	PRON
ap-1408	435	10	b	b	NOUN
ap-1408	435	11	lies	lie	VERB
ap-1408	435	12	.	.	PUNCT
ap-1408	436	1	hence	hence	ADV
ap-1408	436	2	grd1	grd1	PROPN
ap-1408	436	3	(	(	PUNCT
ap-1408	436	4	h	h	NOUN
ap-1408	436	5	)	)	PUNCT
ap-1408	436	6	∩	∩	ADJ
ap-1408	436	7	grd2	grd2	NOUN
ap-1408	436	8	(	(	PUNCT
ap-1408	436	9	h	h	NOUN
ap-1408	436	10	)	)	PUNCT
ap-1408	436	11	=	=	SYM
ap-1408	436	12	b(h	b(h	PROPN
ap-1408	436	13	)	)	PUNCT
ap-1408	436	14	for	for	ADP
ap-1408	436	15	all	all	DET
ap-1408	436	16	d1	d1	PROPN
ap-1408	436	17	�	�	NOUN
ap-1408	436	18	=	=	SYM
ap-1408	436	19	d2	d2	PROPN
ap-1408	436	20	,	,	PUNCT
ap-1408	436	21	d1	d1	PROPN
ap-1408	436	22	,	,	PUNCT
ap-1408	436	23	d2	d2	PROPN
ap-1408	436	24	∈	∈	PROPN
ap-1408	436	25	d.	d.	PROPN
ap-1408	436	26	and	and	CCONJ
ap-1408	436	27	because	because	SCONJ
ap-1408	436	28	grd(h	grd(h	NOUN
ap-1408	436	29	)	)	PUNCT
ap-1408	436	30	in	in	ADP
ap-1408	436	31	which	which	PRON
ap-1408	436	32	b	b	NOUN
ap-1408	436	33	lies	lie	NOUN
ap-1408	436	34	exists	exist	VERB
ap-1408	436	35	for	for	ADP
ap-1408	436	36	every	every	DET
ap-1408	436	37	b	b	PROPN
ap-1408	436	38	∈	∈	PROPN
ap-1408	436	39	gr(h	gr(h	NOUN
ap-1408	436	40	)	)	PUNCT
ap-1408	436	41	,	,	PUNCT
ap-1408	436	42	we	we	PRON
ap-1408	436	43	have	have	VERB
ap-1408	436	44	gr(h	gr(h	X
ap-1408	436	45	)	)	PUNCT
ap-1408	437	1	=	=	SYM
ap-1408	437	2	⋃	⋃	NOUN
ap-1408	437	3	{	{	PUNCT
ap-1408	437	4	grd(h	grd(h	NOUN
ap-1408	437	5	)	)	PUNCT
ap-1408	438	1	|	|	ADV
ap-1408	438	2	d	d	X
ap-1408	438	3	∈	∈	PROPN
ap-1408	438	4	d	d	NOUN
ap-1408	438	5	}	}	PUNCT
ap-1408	438	6	.	.	PUNCT
ap-1408	439	1	theorem	theorem	ADJ
ap-1408	439	2	8	8	NUM
ap-1408	439	3	(	(	PUNCT
ap-1408	439	4	the	the	DET
ap-1408	439	5	pasting	pasting	NOUN
ap-1408	439	6	theorem	theorem	NOUN
ap-1408	439	7	for	for	ADP
ap-1408	439	8	sgr(h	sgr(h	PROPN
ap-1408	439	9	)	)	PUNCT
ap-1408	439	10	)	)	PUNCT
ap-1408	440	1	let	let	VERB
ap-1408	440	2	h	h	NOUN
ap-1408	440	3	be	be	AUX
ap-1408	440	4	an	an	DET
ap-1408	440	5	infinite	infinite	ADJ
ap-1408	440	6	-	-	PUNCT
ap-1408	440	7	dimensional	dimensional	ADJ
ap-1408	440	8	complex	complex	ADJ
ap-1408	440	9	hilbert	hilbert	NOUN
ap-1408	440	10	space	space	NOUN
ap-1408	440	11	.	.	PUNCT
ap-1408	441	1	then	then	ADV
ap-1408	441	2	the	the	DET
ap-1408	441	3	wop	wop	NOUN
ap-1408	441	4	-	-	PUNCT
ap-1408	441	5	group	group	NOUN
ap-1408	441	6	sgr(h	sgr(h	NOUN
ap-1408	441	7	)	)	PUNCT
ap-1408	441	8	pastes	paste	NOUN
ap-1408	441	9	their	their	PRON
ap-1408	441	10	partially	partially	ADV
ap-1408	441	11	ordered	order	VERB
ap-1408	441	12	commutative	commutative	ADJ
ap-1408	441	13	subgroups	subgroup	NOUN
ap-1408	441	14	sgrd(h	sgrd(h	NOUN
ap-1408	441	15	)	)	PUNCT
ap-1408	441	16	,	,	PUNCT
ap-1408	442	1	d	d	PROPN
ap-1408	442	2	∈	∈	PROPN
ap-1408	442	3	d	d	NOUN
ap-1408	442	4	,	,	PUNCT
ap-1408	442	5	together	together	ADV
ap-1408	442	6	over	over	ADP
ap-1408	442	7	hgr(h	hgr(h	PROPN
ap-1408	442	8	)	)	PUNCT
ap-1408	442	9	,	,	PUNCT
ap-1408	442	10	i.e.	i.e.	X
ap-1408	442	11	,	,	PUNCT
ap-1408	442	12	for	for	ADP
ap-1408	442	13	every	every	DET
ap-1408	442	14	pair	pair	NOUN
ap-1408	442	15	d1	d1	NOUN
ap-1408	442	16	,	,	PUNCT
ap-1408	442	17	d2	d2	NOUN
ap-1408	442	18	of	of	ADP
ap-1408	442	19	dense	dense	ADJ
ap-1408	442	20	linear	linear	ADJ
ap-1408	442	21	subspaces	subspace	NOUN
ap-1408	442	22	of	of	ADP
ap-1408	442	23	h	h	NOUN
ap-1408	442	24	,	,	PUNCT
ap-1408	442	25	d1	d1	PROPN
ap-1408	442	26	�	�	PROPN
ap-1408	442	27	=	=	SYM
ap-1408	442	28	d2	d2	PROPN
ap-1408	442	29	,	,	PUNCT
ap-1408	442	30	sgrd1	sgrd1	NOUN
ap-1408	442	31	(	(	PUNCT
ap-1408	442	32	h	h	NOUN
ap-1408	442	33	)	)	PUNCT
ap-1408	442	34	∩	∩	NOUN
ap-1408	442	35	sgrd2(h	sgrd2(h	NOUN
ap-1408	442	36	)	)	PUNCT
ap-1408	442	37	=	=	SYM
ap-1408	442	38	hgr(h	hgr(h	PROPN
ap-1408	442	39	)	)	PUNCT
ap-1408	442	40	and	and	CCONJ
ap-1408	442	41	sgr(h	sgr(h	PROPN
ap-1408	442	42	)	)	PUNCT
ap-1408	442	43	=	=	SYM
ap-1408	442	44	⋃	⋃	NOUN
ap-1408	442	45	{	{	PUNCT
ap-1408	442	46	sgrd(h	sgrd(h	NOUN
ap-1408	442	47	)	)	PUNCT
ap-1408	443	1	|	|	ADV
ap-1408	443	2	d	d	X
ap-1408	443	3	∈	∈	PROPN
ap-1408	443	4	d	d	NOUN
ap-1408	443	5	}	}	PUNCT
ap-1408	443	6	.	.	PUNCT
ap-1408	444	1	proof	proof	NOUN
ap-1408	444	2	.	.	PUNCT
ap-1408	445	1	let	let	VERB
ap-1408	445	2	d	d	X
ap-1408	445	3	∈	∈	PROPN
ap-1408	445	4	d.	d.	PROPN
ap-1408	445	5	since	since	SCONJ
ap-1408	445	6	sgrd(h	sgrd(h	PROPN
ap-1408	445	7	)	)	PUNCT
ap-1408	445	8	=	=	PUNCT
ap-1408	445	9	grd(h	grd(h	NOUN
ap-1408	445	10	)	)	PUNCT
ap-1408	445	11	∩	∩	X
ap-1408	445	12	sgr(h	sgr(h	PROPN
ap-1408	445	13	)	)	PUNCT
ap-1408	445	14	,	,	PUNCT
ap-1408	445	15	with	with	ADP
ap-1408	445	16	previous	previous	ADJ
ap-1408	445	17	theorem	theorem	ADJ
ap-1408	445	18	⋃	⋃	NOUN
ap-1408	445	19	d∈d	d∈d	NOUN
ap-1408	445	20	sgrd(h	sgrd(h	NOUN
ap-1408	445	21	)	)	PUNCT
ap-1408	446	1	=	=	SYM
ap-1408	446	2	⋃	⋃	NOUN
ap-1408	446	3	d∈d	d∈d	NOUN
ap-1408	446	4	(	(	PUNCT
ap-1408	446	5	grd(h	grd(h	NOUN
ap-1408	446	6	)	)	PUNCT
ap-1408	446	7	∩	∩	ADJ
ap-1408	446	8	sgr(h	sgr(h	NOUN
ap-1408	446	9	)	)	PUNCT
ap-1408	446	10	)	)	PUNCT
ap-1408	447	1	=	=	PRON
ap-1408	447	2	(	(	PUNCT
ap-1408	447	3	⋃	⋃	ADP
ap-1408	447	4	d∈d	d∈d	ADJ
ap-1408	447	5	grd(h	grd(h	NOUN
ap-1408	447	6	)	)	PUNCT
ap-1408	447	7	)	)	PUNCT
ap-1408	448	1	∩	∩	PROPN
ap-1408	448	2	sgr(h	sgr(h	PROPN
ap-1408	448	3	)	)	PUNCT
ap-1408	448	4	=	=	SYM
ap-1408	448	5	gr(h	gr(h	NOUN
ap-1408	448	6	)	)	PUNCT
ap-1408	448	7	∩sgr(h	∩sgr(h	PUNCT
ap-1408	448	8	)	)	PUNCT
ap-1408	448	9	=	=	SYM
ap-1408	448	10	sgr(h	sgr(h	PROPN
ap-1408	448	11	)	)	PUNCT
ap-1408	448	12	.	.	PUNCT
ap-1408	449	1	similarly	similarly	ADV
ap-1408	449	2	we	we	PRON
ap-1408	449	3	have	have	VERB
ap-1408	449	4	sgrd1(h)∩sgrd2(h	sgrd1(h)∩sgrd2(h	NOUN
ap-1408	449	5	)	)	PUNCT
ap-1408	450	1	=	=	PRON
ap-1408	450	2	(	(	PUNCT
ap-1408	450	3	sgr(h)∩grd1	sgr(h)∩grd1	X
ap-1408	450	4	(	(	PUNCT
ap-1408	450	5	h))∩	h))∩	X
ap-1408	450	6	(	(	PUNCT
ap-1408	450	7	sgr(h	sgr(h	NOUN
ap-1408	450	8	)	)	PUNCT
ap-1408	450	9	∩	∩	NOUN
ap-1408	450	10	grd2(h	grd2(h	NOUN
ap-1408	450	11	)	)	PUNCT
ap-1408	450	12	)	)	PUNCT
ap-1408	451	1	=	=	SYM
ap-1408	451	2	sgr(h	sgr(h	PROPN
ap-1408	451	3	)	)	PUNCT
ap-1408	451	4	∩	∩	NOUN
ap-1408	451	5	(	(	PUNCT
ap-1408	451	6	grd1	grd1	PROPN
ap-1408	451	7	(	(	PUNCT
ap-1408	451	8	h	h	NOUN
ap-1408	451	9	)	)	PUNCT
ap-1408	451	10	∩	∩	ADJ
ap-1408	451	11	grd2	grd2	NOUN
ap-1408	451	12	(	(	PUNCT
ap-1408	451	13	h	h	NOUN
ap-1408	451	14	)	)	PUNCT
ap-1408	451	15	)	)	PUNCT
ap-1408	452	1	=	=	SYM
ap-1408	452	2	sgr(h	sgr(h	PROPN
ap-1408	452	3	)	)	PUNCT
ap-1408	452	4	∩	∩	NOUN
ap-1408	452	5	b(h	b(h	PROPN
ap-1408	452	6	)	)	PUNCT
ap-1408	452	7	=	=	SYM
ap-1408	452	8	hgr(h	hgr(h	PROPN
ap-1408	452	9	)	)	PUNCT
ap-1408	452	10	for	for	ADP
ap-1408	452	11	all	all	DET
ap-1408	452	12	d1	d1	PROPN
ap-1408	452	13	�	�	NOUN
ap-1408	452	14	=	=	SYM
ap-1408	452	15	d2	d2	PROPN
ap-1408	452	16	,	,	PUNCT
ap-1408	452	17	d1	d1	PROPN
ap-1408	452	18	,	,	PUNCT
ap-1408	452	19	d2	d2	PROPN
ap-1408	452	20	∈	∈	PROPN
ap-1408	452	21	d.	d.	PROPN
ap-1408	452	22	5	5	NUM
ap-1408	452	23	pt	pt	NOUN
ap-1408	452	24	-symmetry	-symmetry	PROPN
ap-1408	452	25	and	and	CCONJ
ap-1408	452	26	related	related	ADJ
ap-1408	452	27	effect	effect	NOUN
ap-1408	452	28	algebras	algebra	NOUN
ap-1408	452	29	let	let	VERB
ap-1408	452	30	us	we	PRON
ap-1408	452	31	repeat	repeat	VERB
ap-1408	452	32	some	some	PRON
ap-1408	452	33	of	of	ADP
ap-1408	452	34	the	the	DET
ap-1408	452	35	notions	notion	NOUN
ap-1408	452	36	concerning	concern	VERB
ap-1408	452	37	the	the	DET
ap-1408	452	38	basics	basic	NOUN
ap-1408	452	39	of	of	ADP
ap-1408	452	40	pt	pt	PROPN
ap-1408	452	41	-symmetry	-symmetry	NOUN
ap-1408	452	42	from	from	ADP
ap-1408	452	43	[	[	X
ap-1408	452	44	7	7	NUM
ap-1408	452	45	]	]	PUNCT
ap-1408	452	46	.	.	PUNCT
ap-1408	453	1	let	let	VERB
ap-1408	453	2	h	h	PRON
ap-1408	453	3	be	be	AUX
ap-1408	453	4	a	a	DET
ap-1408	453	5	hilbert	hilbert	NOUN
ap-1408	453	6	space	space	NOUN
ap-1408	453	7	equipped	equip	VERB
ap-1408	453	8	with	with	ADP
ap-1408	453	9	an	an	DET
ap-1408	453	10	inner	inner	ADJ
ap-1408	453	11	product	product	NOUN
ap-1408	453	12	〈	〈	PROPN
ap-1408	453	13	ψ	ψ	NOUN
ap-1408	453	14	,	,	PUNCT
ap-1408	453	15	φ	φ	PROPN
ap-1408	453	16	〉	〉	NOUN
ap-1408	453	17	.	.	PUNCT
ap-1408	454	1	let	let	VERB
ap-1408	454	2	ω	ω	NOUN
ap-1408	454	3	:	:	PUNCT
ap-1408	454	4	h	h	NOUN
ap-1408	454	5	→	→	SYM
ap-1408	454	6	h	h	NOUN
ap-1408	454	7	be	be	AUX
ap-1408	454	8	an	an	DET
ap-1408	454	9	invertible	invertible	ADJ
ap-1408	454	10	linear	linear	NOUN
ap-1408	454	11	operator	operator	NOUN
ap-1408	454	12	.	.	PUNCT
ap-1408	455	1	then	then	ADV
ap-1408	455	2	we	we	PRON
ap-1408	455	3	obtain	obtain	VERB
ap-1408	455	4	a	a	DET
ap-1408	455	5	new	new	ADJ
ap-1408	455	6	inner	inner	ADJ
ap-1408	455	7	product	product	NOUN
ap-1408	455	8	〈	〈	PROPN
ap-1408	455	9	〈	〈	PROPN
ap-1408	455	10	−,−	−,−	PROPN
ap-1408	455	11	〉	〉	NOUN
ap-1408	455	12	〉	〉	NOUN
ap-1408	455	13	on	on	ADP
ap-1408	455	14	h	h	NOUN
ap-1408	455	15	which	which	PRON
ap-1408	455	16	will	will	AUX
ap-1408	455	17	have	have	VERB
ap-1408	455	18	the	the	DET
ap-1408	455	19	form	form	NOUN
ap-1408	455	20	:	:	PUNCT
ap-1408	456	1	〈	〈	PROPN
ap-1408	456	2	〈	〈	PROPN
ap-1408	456	3	ψ	ψ	X
ap-1408	456	4	,	,	PUNCT
ap-1408	456	5	ϕ	ϕ	PROPN
ap-1408	456	6	〉	〉	NOUN
ap-1408	456	7	〉	〉	NOUN
ap-1408	456	8	=	=	SYM
ap-1408	456	9	〈	〈	PROPN
ap-1408	456	10	ωψ	ωψ	ADV
ap-1408	456	11	,	,	PUNCT
ap-1408	456	12	ωϕ	ωϕ	PRON
ap-1408	456	13	〉	〉	NOUN
ap-1408	456	14	,	,	PUNCT
ap-1408	456	15	∀ψ	∀ψ	NOUN
ap-1408	456	16	,	,	PUNCT
ap-1408	456	17	ϕ	ϕ	PROPN
ap-1408	456	18	∈	∈	PROPN
ap-1408	456	19	h.	h.	PROPN
ap-1408	456	20	clearly	clearly	ADV
ap-1408	456	21	,	,	PUNCT
ap-1408	456	22	our	our	PRON
ap-1408	456	23	new	new	ADJ
ap-1408	456	24	inner	inner	ADJ
ap-1408	456	25	product	product	NOUN
ap-1408	456	26	space	space	NOUN
ap-1408	456	27	is	be	AUX
ap-1408	456	28	complete	complete	ADJ
ap-1408	456	29	with	with	ADP
ap-1408	456	30	respect	respect	NOUN
ap-1408	456	31	to	to	ADP
ap-1408	456	32	〈	〈	PROPN
ap-1408	456	33	〈	〈	PROPN
ap-1408	456	34	−,−	−,−	PROPN
ap-1408	456	35	〉	〉	PROPN
ap-1408	456	36	〉	〉	PROPN
ap-1408	456	37	.	.	PUNCT
ap-1408	457	1	let	let	VERB
ap-1408	457	2	us	we	PRON
ap-1408	457	3	denote	denote	VERB
ap-1408	457	4	hω	hω	ADP
ap-1408	457	5	the	the	DET
ap-1408	457	6	corresponding	corresponding	ADJ
ap-1408	457	7	hilbert	hilbert	NOUN
ap-1408	457	8	space	space	NOUN
ap-1408	457	9	.	.	PUNCT
ap-1408	458	1	hence	hence	ADV
ap-1408	458	2	ω	ω	NUM
ap-1408	458	3	:	:	PUNCT
ap-1408	458	4	hω	hω	ADP
ap-1408	458	5	→	→	SYM
ap-1408	458	6	h	h	NOUN
ap-1408	458	7	and	and	CCONJ
ap-1408	458	8	ω−1	ω−1	NOUN
ap-1408	458	9	:	:	PUNCT
ap-1408	458	10	h	h	NOUN
ap-1408	458	11	→	→	PUNCT
ap-1408	458	12	hω	hω	PART
ap-1408	458	13	provide	provide	VERB
ap-1408	458	14	a	a	DET
ap-1408	458	15	realization	realization	NOUN
ap-1408	458	16	of	of	ADP
ap-1408	458	17	the	the	DET
ap-1408	458	18	unitaryequivalence	unitaryequivalence	NOUN
ap-1408	458	19	of	of	ADP
ap-1408	458	20	the	the	DET
ap-1408	458	21	hilbert	hilbert	NOUN
ap-1408	458	22	spaces	space	NOUN
ap-1408	458	23	hω	hω	ADP
ap-1408	458	24	and	and	CCONJ
ap-1408	458	25	h.	h.	PROPN
ap-1408	458	26	let	let	VERB
ap-1408	458	27	us	we	PRON
ap-1408	458	28	define	define	VERB
ap-1408	458	29	a	a	DET
ap-1408	458	30	map	map	NOUN
ap-1408	458	31	(	(	PUNCT
ap-1408	458	32	·	·	PUNCT
ap-1408	458	33	)	)	PUNCT
ap-1408	459	1	d	d	X
ap-1408	459	2	ω	ω	NUM
ap-1408	459	3	:	:	PUNCT
ap-1408	459	4	grd(h	grd(h	NOUN
ap-1408	459	5	)	)	PUNCT
ap-1408	459	6	→	→	SYM
ap-1408	459	7	grω−1(d)(hω	grω−1(d)(hω	NOUN
ap-1408	459	8	)	)	PUNCT
ap-1408	459	9	by	by	ADP
ap-1408	459	10	aω	aω	NOUN
ap-1408	459	11	=	=	SYM
ap-1408	459	12	ω−1	ω−1	NOUN
ap-1408	459	13	◦	◦	NOUN
ap-1408	459	14	a	a	DET
ap-1408	459	15	◦	◦	NOUN
ap-1408	459	16	ω	ω	NOUN
ap-1408	459	17	for	for	ADP
ap-1408	459	18	a	a	DET
ap-1408	459	19	linear	linear	ADJ
ap-1408	459	20	map	map	NOUN
ap-1408	459	21	a	a	DET
ap-1408	459	22	∈	∈	PROPN
ap-1408	459	23	grd(h	grd(h	NOUN
ap-1408	459	24	)	)	PUNCT
ap-1408	459	25	,	,	PUNCT
ap-1408	459	26	d	d	PROPN
ap-1408	459	27	∈	∈	PROPN
ap-1408	459	28	d.	d.	NOUN
ap-1408	459	29	we	we	PRON
ap-1408	459	30	then	then	ADV
ap-1408	459	31	have	have	VERB
ap-1408	459	32	proposition	proposition	NOUN
ap-1408	459	33	1	1	NUM
ap-1408	460	1	[	[	X
ap-1408	460	2	7	7	NUM
ap-1408	460	3	,	,	PUNCT
ap-1408	460	4	proposition	proposition	NOUN
ap-1408	460	5	3	3	NUM
ap-1408	460	6	]	]	PUNCT
ap-1408	460	7	let	let	VERB
ap-1408	460	8	h	h	NOUN
ap-1408	460	9	be	be	AUX
ap-1408	460	10	an	an	DET
ap-1408	460	11	infinite	infinite	ADJ
ap-1408	460	12	-	-	PUNCT
ap-1408	460	13	dimensional	dimensional	ADJ
ap-1408	460	14	complex	complex	ADJ
ap-1408	460	15	hilbert	hilbert	NOUN
ap-1408	460	16	space	space	NOUN
ap-1408	460	17	.	.	PUNCT
ap-1408	461	1	assume	assume	VERB
ap-1408	461	2	moreover	moreover	ADV
ap-1408	461	3	that	that	SCONJ
ap-1408	461	4	ω	ω	NOUN
ap-1408	461	5	:	:	PUNCT
ap-1408	461	6	h	h	NOUN
ap-1408	461	7	→	→	SYM
ap-1408	461	8	h	h	NOUN
ap-1408	461	9	is	be	AUX
ap-1408	461	10	an	an	DET
ap-1408	461	11	invertible	invertible	ADJ
ap-1408	461	12	linear	linear	NOUN
ap-1408	461	13	operator	operator	NOUN
ap-1408	461	14	and	and	CCONJ
ap-1408	461	15	d	d	PROPN
ap-1408	461	16	∈	∈	PROPN
ap-1408	461	17	d.	d.	NOUN
ap-1408	461	18	then	then	ADV
ap-1408	461	19	1	1	X
ap-1408	461	20	.	.	PUNCT
ap-1408	461	21	(	(	PUNCT
ap-1408	461	22	a)d	a)d	PROPN
ap-1408	461	23	ω	ω	X
ap-1408	461	24	is	be	AUX
ap-1408	461	25	a	a	DET
ap-1408	461	26	positive	positive	ADJ
ap-1408	461	27	operator	operator	NOUN
ap-1408	461	28	on	on	ADP
ap-1408	462	1	hω	hω	ADP
ap-1408	462	2	iff	iff	PROPN
ap-1408	462	3	a	a	PRON
ap-1408	462	4	is	be	AUX
ap-1408	462	5	a	a	DET
ap-1408	462	6	positive	positive	ADJ
ap-1408	462	7	operator	operator	NOUN
ap-1408	462	8	on	on	ADP
ap-1408	462	9	h.	h.	PROPN
ap-1408	462	10	2	2	NUM
ap-1408	462	11	.	.	PUNCT
ap-1408	463	1	(	(	PUNCT
ap-1408	463	2	·	·	PUNCT
ap-1408	463	3	)	)	PUNCT
ap-1408	463	4	d	d	PROPN
ap-1408	463	5	ω	ω	PROPN
ap-1408	463	6	is	be	AUX
ap-1408	463	7	an	an	DET
ap-1408	463	8	isomorphism	isomorphism	NOUN
ap-1408	463	9	of	of	ADP
ap-1408	463	10	partially	partially	ADV
ap-1408	463	11	ordered	order	VERB
ap-1408	463	12	commutative	commutative	ADJ
ap-1408	463	13	groups	group	NOUN
ap-1408	463	14	.	.	PUNCT
ap-1408	464	1	3	3	X
ap-1408	464	2	.	.	X
ap-1408	464	3	(	(	PUNCT
ap-1408	464	4	a)d	a)d	PROPN
ap-1408	464	5	ω	ω	PROPN
ap-1408	464	6	is	be	AUX
ap-1408	464	7	a	a	DET
ap-1408	464	8	hermitian	hermitian	ADJ
ap-1408	464	9	operator	operator	NOUN
ap-1408	464	10	on	on	ADP
ap-1408	464	11	hω	hω	ADP
ap-1408	464	12	iff	iff	PROPN
ap-1408	464	13	a	a	PRON
ap-1408	464	14	is	be	AUX
ap-1408	464	15	a	a	DET
ap-1408	464	16	hermitian	hermitian	ADJ
ap-1408	464	17	operator	operator	NOUN
ap-1408	464	18	on	on	ADP
ap-1408	464	19	h.	h.	PROPN
ap-1408	464	20	4	4	NUM
ap-1408	464	21	.	.	PUNCT
ap-1408	465	1	(	(	PUNCT
ap-1408	465	2	i)d	i)d	X
ap-1408	465	3	ω	ω	X
ap-1408	465	4	=	=	SYM
ap-1408	465	5	i.	i.	NOUN
ap-1408	465	6	the	the	DET
ap-1408	465	7	preceding	precede	VERB
ap-1408	465	8	proposition	proposition	NOUN
ap-1408	465	9	immediately	immediately	ADV
ap-1408	465	10	yields	yield	VERB
ap-1408	465	11	that	that	PRON
ap-1408	465	12	theorem	theorem	VERB
ap-1408	465	13	9	9	NUM
ap-1408	465	14	let	let	VERB
ap-1408	465	15	h	h	NOUN
ap-1408	465	16	be	be	AUX
ap-1408	465	17	an	an	DET
ap-1408	465	18	infinite	infinite	ADJ
ap-1408	465	19	-	-	PUNCT
ap-1408	465	20	dimensional	dimensional	ADJ
ap-1408	465	21	complex	complex	ADJ
ap-1408	465	22	hilbert	hilbert	NOUN
ap-1408	465	23	space	space	NOUN
ap-1408	465	24	and	and	CCONJ
ap-1408	465	25	ω	ω	NUM
ap-1408	465	26	:	:	PUNCT
ap-1408	465	27	h	h	NOUN
ap-1408	465	28	→	→	SYM
ap-1408	465	29	h	h	NOUN
ap-1408	465	30	an	an	DET
ap-1408	465	31	invertible	invertible	ADJ
ap-1408	465	32	linear	linear	NOUN
ap-1408	465	33	operator	operator	NOUN
ap-1408	465	34	.	.	PUNCT
ap-1408	466	1	then	then	ADV
ap-1408	466	2	the	the	DET
ap-1408	466	3	map	map	NOUN
ap-1408	466	4	(	(	PUNCT
ap-1408	466	5	·	·	PUNCT
ap-1408	466	6	)	)	PUNCT
ap-1408	466	7	ω	ω	PROPN
ap-1408	466	8	:	:	PUNCT
ap-1408	466	9	gr(h	gr(h	X
ap-1408	466	10	)	)	PUNCT
ap-1408	466	11	→	→	SYM
ap-1408	466	12	gr(hω	gr(hω	PROPN
ap-1408	466	13	)	)	PUNCT
ap-1408	466	14	defined	define	VERB
ap-1408	466	15	by	by	ADP
ap-1408	466	16	(	(	PUNCT
ap-1408	466	17	a)ω	a)ω	X
ap-1408	466	18	:	:	PUNCT
ap-1408	466	19	=	=	SYM
ap-1408	466	20	(	(	PUNCT
ap-1408	466	21	a)d(a	a)d(a	PROPN
ap-1408	466	22	)	)	PUNCT
ap-1408	466	23	ω	ω	PROPN
ap-1408	466	24	for	for	ADP
ap-1408	466	25	all	all	DET
ap-1408	466	26	a	a	DET
ap-1408	466	27	∈	∈	PROPN
ap-1408	466	28	gr(h	gr(h	NOUN
ap-1408	466	29	)	)	PUNCT
ap-1408	466	30	is	be	AUX
ap-1408	466	31	an	an	DET
ap-1408	466	32	isomorphism	isomorphism	NOUN
ap-1408	466	33	of	of	ADP
ap-1408	466	34	wop	wop	NOUN
ap-1408	466	35	-	-	PUNCT
ap-1408	466	36	groups	group	NOUN
ap-1408	466	37	.	.	PUNCT
ap-1408	467	1	proof	proof	NOUN
ap-1408	467	2	.	.	PUNCT
ap-1408	468	1	it	it	PRON
ap-1408	468	2	follows	follow	VERB
ap-1408	468	3	from	from	ADP
ap-1408	468	4	proposition	proposition	NOUN
ap-1408	468	5	1	1	NUM
ap-1408	468	6	and	and	CCONJ
ap-1408	468	7	theorem	theorem	VERB
ap-1408	468	8	7	7	NUM
ap-1408	468	9	.	.	PUNCT
ap-1408	468	10	corollary	corollary	ADJ
ap-1408	468	11	1	1	NUM
ap-1408	468	12	let	let	VERB
ap-1408	468	13	h	h	NOUN
ap-1408	468	14	be	be	AUX
ap-1408	468	15	an	an	DET
ap-1408	468	16	infinite	infinite	ADJ
ap-1408	468	17	-	-	PUNCT
ap-1408	468	18	dimensional	dimensional	ADJ
ap-1408	468	19	complex	complex	ADJ
ap-1408	468	20	hilbert	hilbert	NOUN
ap-1408	468	21	space	space	NOUN
ap-1408	468	22	and	and	CCONJ
ap-1408	468	23	ω	ω	NUM
ap-1408	468	24	:	:	PUNCT
ap-1408	468	25	h	h	NOUN
ap-1408	468	26	→	→	SYM
ap-1408	468	27	h	h	NOUN
ap-1408	468	28	an	an	DET
ap-1408	468	29	invertible	invertible	ADJ
ap-1408	468	30	linear	linear	NOUN
ap-1408	468	31	operator	operator	NOUN
ap-1408	468	32	.	.	PUNCT
ap-1408	469	1	then	then	ADV
ap-1408	469	2	v(h	v(h	NOUN
ap-1408	469	3	)	)	PUNCT
ap-1408	469	4	and	and	CCONJ
ap-1408	469	5	v(hω	v(hω	NOUN
ap-1408	469	6	)	)	PUNCT
ap-1408	469	7	are	be	AUX
ap-1408	469	8	isomorphic	isomorphic	ADJ
ap-1408	469	9	generalized	generalized	ADJ
ap-1408	469	10	effect	effect	NOUN
ap-1408	469	11	algebras	algebra	NOUN
ap-1408	469	12	.	.	PUNCT
ap-1408	470	1	proof	proof	NOUN
ap-1408	470	2	.	.	PUNCT
ap-1408	471	1	it	it	PRON
ap-1408	471	2	follows	follow	VERB
ap-1408	471	3	immediately	immediately	ADV
ap-1408	471	4	from	from	ADP
ap-1408	471	5	the	the	DET
ap-1408	471	6	fact	fact	NOUN
ap-1408	471	7	that	that	SCONJ
ap-1408	471	8	(	(	PUNCT
ap-1408	471	9	·	·	PUNCT
ap-1408	471	10	)	)	PUNCT
ap-1408	471	11	ω	ω	PROPN
ap-1408	471	12	preserves	preserve	NOUN
ap-1408	471	13	and	and	CCONJ
ap-1408	471	14	reflects	reflect	VERB
ap-1408	471	15	positive	positive	ADJ
ap-1408	471	16	operators	operator	NOUN
ap-1408	471	17	and	and	CCONJ
ap-1408	471	18	from	from	ADP
ap-1408	471	19	theorem	theorem	ADJ
ap-1408	471	20	9	9	NUM
ap-1408	471	21	.	.	PUNCT
ap-1408	472	1	we	we	PRON
ap-1408	472	2	say	say	VERB
ap-1408	472	3	that	that	SCONJ
ap-1408	472	4	an	an	DET
ap-1408	472	5	operator	operator	NOUN
ap-1408	472	6	h	h	NOUN
ap-1408	472	7	:	:	PUNCT
ap-1408	473	1	d	d	X
ap-1408	473	2	→	→	SYM
ap-1408	473	3	h	h	NOUN
ap-1408	473	4	defined	define	VERB
ap-1408	473	5	on	on	ADP
ap-1408	473	6	a	a	DET
ap-1408	473	7	dense	dense	ADJ
ap-1408	473	8	linear	linear	NOUN
ap-1408	473	9	subspace	subspace	NOUN
ap-1408	473	10	d	d	PROPN
ap-1408	473	11	of	of	ADP
ap-1408	473	12	a	a	DET
ap-1408	473	13	hilbert	hilbert	NOUN
ap-1408	473	14	space	space	NOUN
ap-1408	473	15	h	h	NOUN
ap-1408	473	16	is	be	AUX
ap-1408	473	17	η+-pseudo	η+-pseudo	NOUN
ap-1408	473	18	-	-	ADJ
ap-1408	473	19	hermitian	hermitian	ADJ
ap-1408	473	20	and	and	CCONJ
ap-1408	473	21	η+	η+	NOUN
ap-1408	473	22	is	be	AUX
ap-1408	473	23	a	a	DET
ap-1408	473	24	metric	metric	ADJ
ap-1408	473	25	operator	operator	NOUN
ap-1408	473	26	if	if	SCONJ
ap-1408	473	27	η+	η+	NUM
ap-1408	473	28	:	:	PUNCT
ap-1408	473	29	h	h	NOUN
ap-1408	473	30	→	→	SYM
ap-1408	473	31	h	h	NOUN
ap-1408	473	32	is	be	AUX
ap-1408	473	33	a	a	DET
ap-1408	473	34	positive	positive	ADJ
ap-1408	473	35	,	,	PUNCT
ap-1408	473	36	hermitian	hermitian	ADJ
ap-1408	473	37	,	,	PUNCT
ap-1408	473	38	invertible	invertible	ADJ
ap-1408	473	39	,	,	PUNCT
ap-1408	473	40	linear	linear	ADJ
ap-1408	473	41	operator	operator	NOUN
ap-1408	473	42	such	such	ADJ
ap-1408	473	43	that	that	DET
ap-1408	473	44	h∗	h∗	PROPN
ap-1408	473	45	=	=	NOUN
ap-1408	473	46	η+hη−1	η+hη−1	PROPN
ap-1408	473	47	+	+	X
ap-1408	473	48	(	(	PUNCT
ap-1408	473	49	see	see	VERB
ap-1408	473	50	also	also	ADV
ap-1408	473	51	[	[	X
ap-1408	473	52	6	6	NUM
ap-1408	473	53	]	]	NUM
ap-1408	473	54	)	)	PUNCT
ap-1408	473	55	.	.	PUNCT
ap-1408	474	1	theorem	theorem	ADJ
ap-1408	474	2	10	10	NUM
ap-1408	474	3	let	let	VERB
ap-1408	474	4	h	h	NOUN
ap-1408	474	5	be	be	AUX
ap-1408	474	6	an	an	DET
ap-1408	474	7	infinite	infinite	ADJ
ap-1408	474	8	-	-	PUNCT
ap-1408	474	9	dimensional	dimensional	ADJ
ap-1408	474	10	complex	complex	ADJ
ap-1408	474	11	hilbert	hilbert	NOUN
ap-1408	474	12	space	space	NOUN
ap-1408	474	13	,	,	PUNCT
ap-1408	474	14	let	let	VERB
ap-1408	474	15	d	d	PRON
ap-1408	474	16	⊆	⊆	NUM
ap-1408	474	17	h	h	NOUN
ap-1408	474	18	be	be	AUX
ap-1408	474	19	a	a	DET
ap-1408	474	20	linear	linear	ADJ
ap-1408	474	21	subspace	subspace	NOUN
ap-1408	474	22	dense	dense	ADJ
ap-1408	474	23	in	in	ADP
ap-1408	474	24	h	h	NOUN
ap-1408	474	25	and	and	CCONJ
ap-1408	474	26	let	let	VERB
ap-1408	474	27	h	h	NOUN
ap-1408	474	28	:	:	PUNCT
ap-1408	474	29	d	d	X
ap-1408	474	30	→	→	SYM
ap-1408	474	31	h	h	NOUN
ap-1408	474	32	be	be	AUX
ap-1408	474	33	a	a	DET
ap-1408	474	34	η+pseudo	η+pseudo	ADJ
ap-1408	474	35	-	-	ADJ
ap-1408	474	36	hermitian	hermitian	ADJ
ap-1408	474	37	operator	operator	NOUN
ap-1408	474	38	for	for	ADP
ap-1408	474	39	some	some	DET
ap-1408	474	40	metric	metric	ADJ
ap-1408	474	41	operator	operator	NOUN
ap-1408	474	42	η+	η+	PUNCT
ap-1408	474	43	:	:	PUNCT
ap-1408	474	44	h	h	NOUN
ap-1408	474	45	→	→	PUNCT
ap-1408	474	46	h	h	NOUN
ap-1408	474	47	such	such	ADJ
ap-1408	474	48	that	that	SCONJ
ap-1408	474	49	η+	η+	ADJ
ap-1408	474	50	=	=	SYM
ap-1408	474	51	ρ2	ρ2	PROPN
ap-1408	474	52	+	+	NOUN
ap-1408	474	53	.	.	PUNCT
ap-1408	475	1	then	then	ADV
ap-1408	475	2	1	1	X
ap-1408	475	3	.	.	PUNCT
ap-1408	475	4	gr(hρ+	gr(hρ+	PROPN
ap-1408	475	5	)	)	PUNCT
ap-1408	475	6	and	and	CCONJ
ap-1408	475	7	gr(h	gr(h	NOUN
ap-1408	475	8	)	)	PUNCT
ap-1408	475	9	are	be	AUX
ap-1408	475	10	mutually	mutually	ADV
ap-1408	475	11	isomorphic	isomorphic	ADJ
ap-1408	475	12	wop	wop	NOUN
ap-1408	475	13	-	-	PUNCT
ap-1408	475	14	groups	group	NOUN
ap-1408	475	15	such	such	ADJ
ap-1408	475	16	that	that	SCONJ
ap-1408	475	17	h	h	NOUN
ap-1408	475	18	∈	∈	PROPN
ap-1408	475	19	gr(hρ+	gr(hρ+	PROPN
ap-1408	475	20	)	)	PUNCT
ap-1408	476	1	and	and	CCONJ
ap-1408	476	2	h	h	NOUN
ap-1408	476	3	is	be	AUX
ap-1408	476	4	a	a	DET
ap-1408	476	5	self	self	NOUN
ap-1408	476	6	-	-	PUNCT
ap-1408	476	7	adjoint	adjoint	NOUN
ap-1408	476	8	operator	operator	NOUN
ap-1408	476	9	with	with	ADP
ap-1408	476	10	respect	respect	NOUN
ap-1408	476	11	to	to	ADP
ap-1408	476	12	the	the	DET
ap-1408	476	13	positivedefinite	positivedefinite	ADJ
ap-1408	476	14	inner	inner	ADJ
ap-1408	476	15	product	product	NOUN
ap-1408	476	16	〈	〈	PROPN
ap-1408	476	17	〈	〈	PROPN
ap-1408	476	18	−,−	−,−	PROPN
ap-1408	476	19	〉	〉	NOUN
ap-1408	476	20	〉	〉	NOUN
ap-1408	476	21	=	=	SYM
ap-1408	476	22	〈	〈	PROPN
ap-1408	476	23	ρ+−	ρ+−	NUM
ap-1408	476	24	,	,	PUNCT
ap-1408	476	25	ρ+−	ρ+−	NUM
ap-1408	476	26	〉	〉	NOUN
ap-1408	476	27	on	on	ADP
ap-1408	476	28	hρ+	hρ+	NOUN
ap-1408	476	29	.	.	PUNCT
ap-1408	477	1	71	71	NUM
ap-1408	477	2	acta	acta	PROPN
ap-1408	477	3	polytechnica	polytechnica	PROPN
ap-1408	477	4	vol	vol	NOUN
ap-1408	477	5	.	.	PUNCT
ap-1408	478	1	51	51	NUM
ap-1408	478	2	no	no	INTJ
ap-1408	478	3	.	.	PUNCT
ap-1408	479	1	4/2011	4/2011	NUM
ap-1408	479	2	2	2	NUM
ap-1408	479	3	.	.	PUNCT
ap-1408	480	1	v(h	v(h	NOUN
ap-1408	480	2	)	)	PUNCT
ap-1408	480	3	and	and	CCONJ
ap-1408	480	4	v(hρ+	v(hρ+	NOUN
ap-1408	480	5	)	)	PUNCT
ap-1408	480	6	are	be	AUX
ap-1408	480	7	mutually	mutually	ADV
ap-1408	480	8	isomorphic	isomorphic	ADJ
ap-1408	480	9	generalized	generalized	ADJ
ap-1408	480	10	effect	effect	NOUN
ap-1408	480	11	algebras	algebra	NOUN
ap-1408	480	12	.	.	PUNCT
ap-1408	481	1	if	if	SCONJ
ap-1408	481	2	moreover	moreover	ADV
ap-1408	481	3	h	h	NOUN
ap-1408	481	4	is	be	AUX
ap-1408	481	5	a	a	DET
ap-1408	481	6	positive	positive	ADJ
ap-1408	481	7	operator	operator	NOUN
ap-1408	481	8	with	with	ADP
ap-1408	481	9	respect	respect	NOUN
ap-1408	481	10	to	to	ADP
ap-1408	481	11	〈	〈	PROPN
ap-1408	481	12	〈	〈	PROPN
ap-1408	481	13	−,−	−,−	PROPN
ap-1408	481	14	〉	〉	PROPN
ap-1408	481	15	〉	〉	PROPN
ap-1408	481	16	(	(	PUNCT
ap-1408	481	17	i.e.	i.e.	X
ap-1408	481	18	,	,	PUNCT
ap-1408	481	19	its	its	PRON
ap-1408	481	20	real	real	ADJ
ap-1408	481	21	spectrum	spectrum	NOUN
ap-1408	481	22	will	will	AUX
ap-1408	481	23	be	be	AUX
ap-1408	481	24	contained	contain	VERB
ap-1408	481	25	in	in	ADP
ap-1408	481	26	the	the	DET
ap-1408	481	27	interval	interval	NOUN
ap-1408	481	28	[	[	X
ap-1408	481	29	0,∞	0,∞	NOUN
ap-1408	481	30	)	)	PUNCT
ap-1408	481	31	)	)	PUNCT
ap-1408	482	1	then	then	ADV
ap-1408	482	2	h	h	PROPN
ap-1408	482	3	∈	∈	PROPN
ap-1408	482	4	v(hρ+	v(hρ+	NOUN
ap-1408	482	5	)	)	PUNCT
ap-1408	482	6	.	.	PUNCT
ap-1408	483	1	proof	proof	NOUN
ap-1408	483	2	.	.	PUNCT
ap-1408	484	1	it	it	PRON
ap-1408	484	2	follows	follow	VERB
ap-1408	484	3	from	from	ADP
ap-1408	484	4	the	the	DET
ap-1408	484	5	above	above	ADJ
ap-1408	484	6	considerations	consideration	NOUN
ap-1408	484	7	and	and	CCONJ
ap-1408	484	8	[	[	X
ap-1408	484	9	7	7	NUM
ap-1408	484	10	,	,	PUNCT
ap-1408	484	11	theorem	theorem	VERB
ap-1408	484	12	3	3	NUM
ap-1408	484	13	]	]	PUNCT
ap-1408	484	14	.	.	PUNCT
ap-1408	485	1	6	6	NUM
ap-1408	485	2	conclusion	conclusion	NOUN
ap-1408	485	3	in	in	ADP
ap-1408	485	4	this	this	DET
ap-1408	485	5	paper	paper	NOUN
ap-1408	485	6	we	we	PRON
ap-1408	485	7	have	have	AUX
ap-1408	485	8	shown	show	VERB
ap-1408	485	9	that	that	SCONJ
ap-1408	485	10	a	a	DET
ap-1408	485	11	η+-pseudohermitian	η+-pseudohermitian	ADJ
ap-1408	485	12	operator	operator	NOUN
ap-1408	485	13	for	for	ADP
ap-1408	485	14	some	some	DET
ap-1408	485	15	metric	metric	ADJ
ap-1408	485	16	operator	operator	NOUN
ap-1408	485	17	η+	η+	PUNCT
ap-1408	485	18	of	of	ADP
ap-1408	485	19	a	a	DET
ap-1408	485	20	quantum	quantum	ADJ
ap-1408	485	21	system	system	NOUN
ap-1408	485	22	described	describe	VERB
ap-1408	485	23	by	by	ADP
ap-1408	485	24	a	a	DET
ap-1408	485	25	hilbert	hilbert	NOUN
ap-1408	485	26	space	space	NOUN
ap-1408	485	27	h	h	PROPN
ap-1408	485	28	yields	yield	VERB
ap-1408	485	29	an	an	DET
ap-1408	485	30	isomorphism	isomorphism	NOUN
ap-1408	485	31	between	between	ADP
ap-1408	485	32	the	the	DET
ap-1408	485	33	weakly	weakly	ADJ
ap-1408	485	34	ordered	order	VERB
ap-1408	485	35	commutative	commutative	ADJ
ap-1408	485	36	partial	partial	ADJ
ap-1408	485	37	group	group	NOUN
ap-1408	485	38	of	of	ADP
ap-1408	485	39	linear	linear	PROPN
ap-1408	485	40	maps	map	NOUN
ap-1408	485	41	on	on	ADP
ap-1408	485	42	h	h	NOUN
ap-1408	485	43	and	and	CCONJ
ap-1408	485	44	the	the	DET
ap-1408	485	45	weakly	weakly	ADJ
ap-1408	485	46	ordered	order	VERB
ap-1408	485	47	commutative	commutative	ADJ
ap-1408	485	48	partial	partial	ADJ
ap-1408	485	49	group	group	NOUN
ap-1408	485	50	of	of	ADP
ap-1408	485	51	linear	linear	PROPN
ap-1408	485	52	maps	map	NOUN
ap-1408	485	53	on	on	ADP
ap-1408	485	54	hρ+	hρ+	NOUN
ap-1408	485	55	.	.	PUNCT
ap-1408	486	1	the	the	DET
ap-1408	486	2	same	same	ADJ
ap-1408	486	3	applies	apply	VERB
ap-1408	486	4	to	to	ADP
ap-1408	486	5	the	the	DET
ap-1408	486	6	generalized	generalized	ADJ
ap-1408	486	7	effect	effect	NOUN
ap-1408	486	8	algebras	algebra	NOUN
ap-1408	486	9	of	of	ADP
ap-1408	486	10	positive	positive	ADJ
ap-1408	486	11	operators	operator	NOUN
ap-1408	486	12	introduced	introduce	VERB
ap-1408	486	13	in	in	ADP
ap-1408	486	14	[	[	X
ap-1408	486	15	9	9	NUM
ap-1408	486	16	]	]	PUNCT
ap-1408	486	17	.	.	PUNCT
ap-1408	487	1	hence	hence	ADV
ap-1408	487	2	,	,	PUNCT
ap-1408	487	3	from	from	ADP
ap-1408	487	4	the	the	DET
ap-1408	487	5	standpoint	standpoint	NOUN
ap-1408	487	6	of	of	ADP
ap-1408	487	7	(	(	PUNCT
ap-1408	487	8	generalized	generalized	ADJ
ap-1408	487	9	)	)	PUNCT
ap-1408	487	10	effect	effect	NOUN
ap-1408	487	11	algebra	algebra	NOUN
ap-1408	487	12	theory	theory	NOUN
ap-1408	487	13	the	the	DET
ap-1408	487	14	two	two	NUM
ap-1408	487	15	representations	representation	NOUN
ap-1408	487	16	of	of	ADP
ap-1408	487	17	our	our	PRON
ap-1408	487	18	quantum	quantum	ADJ
ap-1408	487	19	system	system	NOUN
ap-1408	487	20	coincide	coincide	NOUN
ap-1408	487	21	.	.	PUNCT
ap-1408	488	1	acknowledgement	acknowledgement	NOUN
ap-1408	488	2	the	the	DET
ap-1408	488	3	work	work	NOUN
ap-1408	488	4	of	of	ADP
ap-1408	488	5	the	the	DET
ap-1408	488	6	author	author	NOUN
ap-1408	488	7	was	be	AUX
ap-1408	488	8	supported	support	VERB
ap-1408	488	9	by	by	ADP
ap-1408	488	10	the	the	DET
ap-1408	488	11	ministry	ministry	PROPN
ap-1408	488	12	of	of	ADP
ap-1408	488	13	education	education	NOUN
ap-1408	488	14	of	of	ADP
ap-1408	488	15	the	the	DET
ap-1408	488	16	czech	czech	PROPN
ap-1408	488	17	republic	republic	NOUN
ap-1408	488	18	under	under	ADP
ap-1408	488	19	project	project	NOUN
ap-1408	488	20	msm0021622409	msm0021622409	NOUN
ap-1408	488	21	and	and	CCONJ
ap-1408	488	22	by	by	ADP
ap-1408	488	23	grant	grant	NOUN
ap-1408	488	24	0964/2009	0964/2009	NUM
ap-1408	488	25	of	of	ADP
ap-1408	488	26	masaryk	masaryk	PROPN
ap-1408	488	27	university	university	PROPN
ap-1408	488	28	.	.	PUNCT
ap-1408	489	1	the	the	DET
ap-1408	489	2	second	second	ADJ
ap-1408	489	3	author	author	NOUN
ap-1408	489	4	was	be	AUX
ap-1408	489	5	supported	support	VERB
ap-1408	489	6	by	by	ADP
ap-1408	489	7	grant	grant	NOUN
ap-1408	489	8	0964/2009	0964/2009	NUM
ap-1408	489	9	of	of	ADP
ap-1408	489	10	masaryk	masaryk	PROPN
ap-1408	489	11	university	university	PROPN
ap-1408	489	12	.	.	PUNCT
ap-1408	490	1	references	reference	NOUN
ap-1408	490	2	[	[	X
ap-1408	490	3	1	1	NUM
ap-1408	490	4	]	]	X
ap-1408	490	5	bender	bender	NOUN
ap-1408	490	6	,	,	PUNCT
ap-1408	490	7	c.	c.	PROPN
ap-1408	490	8	m.	m.	PROPN
ap-1408	490	9	,	,	PUNCT
ap-1408	490	10	boettcher	boettcher	PROPN
ap-1408	490	11	,	,	PUNCT
ap-1408	490	12	s.	s.	PROPN
ap-1408	490	13	:	:	PUNCT
ap-1408	490	14	real	real	ADJ
ap-1408	490	15	spectra	spectra	NOUN
ap-1408	490	16	in	in	ADP
ap-1408	490	17	non	non	ADJ
ap-1408	490	18	-	-	ADJ
ap-1408	490	19	hermitian	hermitian	ADJ
ap-1408	490	20	hamiltonians	hamiltonian	NOUN
ap-1408	490	21	having	have	VERB
ap-1408	490	22	pt	pt	PROPN
ap-1408	490	23	symmetry	symmetry	NOUN
ap-1408	490	24	,	,	PUNCT
ap-1408	490	25	phys	phy	NOUN
ap-1408	490	26	.	.	PUNCT
ap-1408	491	1	rev	rev	PROPN
ap-1408	491	2	.	.	PROPN
ap-1408	491	3	lett	lett	PROPN
ap-1408	491	4	.	.	PROPN
ap-1408	492	1	80	80	NUM
ap-1408	492	2	(	(	PUNCT
ap-1408	492	3	1998	1998	NUM
ap-1408	492	4	)	)	PUNCT
ap-1408	492	5	,	,	PUNCT
ap-1408	492	6	5	5	NUM
ap-1408	492	7	243–5	243–5	NUM
ap-1408	492	8	246	246	NUM
ap-1408	492	9	.	.	PUNCT
ap-1408	493	1	[	[	X
ap-1408	493	2	2	2	NUM
ap-1408	493	3	]	]	SYM
ap-1408	493	4	blank	blank	NOUN
ap-1408	493	5	,	,	PUNCT
ap-1408	493	6	j.	j.	PROPN
ap-1408	493	7	,	,	PUNCT
ap-1408	493	8	exner	exner	PROPN
ap-1408	493	9	,	,	PUNCT
ap-1408	493	10	p.	p.	PROPN
ap-1408	493	11	,	,	PUNCT
ap-1408	493	12	havĺıček	havĺıček	PROPN
ap-1408	493	13	,	,	PUNCT
ap-1408	493	14	m.	m.	NOUN
ap-1408	493	15	:	:	PUNCT
ap-1408	493	16	hilbert	hilbert	NOUN
ap-1408	493	17	space	space	NOUN
ap-1408	493	18	operators	operator	NOUN
ap-1408	493	19	in	in	ADP
ap-1408	493	20	quantum	quantum	ADJ
ap-1408	493	21	physics	physic	NOUN
ap-1408	493	22	.	.	PUNCT
ap-1408	494	1	2nd	2nd	ADJ
ap-1408	494	2	edn	edn	PROPN
ap-1408	494	3	.	.	PUNCT
ap-1408	495	1	berlin	berlin	PROPN
ap-1408	495	2	:	:	PUNCT
ap-1408	495	3	springer	springer	NOUN
ap-1408	495	4	,	,	PUNCT
ap-1408	495	5	2008	2008	NUM
ap-1408	495	6	.	.	PUNCT
ap-1408	496	1	[	[	X
ap-1408	496	2	3	3	NUM
ap-1408	496	3	]	]	X
ap-1408	496	4	dvurečenskij	dvurečenskij	PROPN
ap-1408	496	5	,	,	PUNCT
ap-1408	496	6	a.	a.	NOUN
ap-1408	496	7	,	,	PUNCT
ap-1408	496	8	pulmannová	pulmannová	ADJ
ap-1408	496	9	,	,	PUNCT
ap-1408	496	10	s.	s.	PROPN
ap-1408	496	11	:	:	PUNCT
ap-1408	496	12	new	new	ADJ
ap-1408	496	13	trends	trend	NOUN
ap-1408	496	14	in	in	ADP
ap-1408	496	15	quantum	quantum	ADJ
ap-1408	496	16	structures	structure	NOUN
ap-1408	496	17	,	,	PUNCT
ap-1408	496	18	bratislava	bratislava	NOUN
ap-1408	496	19	:	:	PUNCT
ap-1408	496	20	kluwer	kluwer	PROPN
ap-1408	496	21	acad	acad	PROPN
ap-1408	496	22	.	.	PUNCT
ap-1408	497	1	publ	publ	PROPN
ap-1408	497	2	.	.	PUNCT
ap-1408	497	3	,	,	PUNCT
ap-1408	497	4	dordrecht	dordrecht	PROPN
ap-1408	497	5	/	/	SYM
ap-1408	497	6	ister	ist	ADJ
ap-1408	497	7	science	science	NOUN
ap-1408	497	8	,	,	PUNCT
ap-1408	497	9	2000	2000	NUM
ap-1408	497	10	.	.	PUNCT
ap-1408	498	1	[	[	X
ap-1408	498	2	4	4	NUM
ap-1408	498	3	]	]	X
ap-1408	498	4	foulis	foulis	PROPN
ap-1408	498	5	,	,	PUNCT
ap-1408	498	6	d.	d.	PROPN
ap-1408	498	7	j.	j.	PROPN
ap-1408	498	8	,	,	PUNCT
ap-1408	498	9	bennett	bennett	PROPN
ap-1408	498	10	,	,	PUNCT
ap-1408	498	11	m.	m.	PROPN
ap-1408	498	12	k.	k.	PROPN
ap-1408	498	13	:	:	PUNCT
ap-1408	498	14	effect	effect	NOUN
ap-1408	498	15	algebras	algebra	NOUN
ap-1408	498	16	and	and	CCONJ
ap-1408	498	17	unsharp	unsharp	ADJ
ap-1408	498	18	quantum	quantum	ADJ
ap-1408	498	19	logics	logic	NOUN
ap-1408	498	20	,	,	PUNCT
ap-1408	498	21	found	find	VERB
ap-1408	498	22	.	.	PUNCT
ap-1408	499	1	phys	phy	NOUN
ap-1408	499	2	.	.	PUNCT
ap-1408	500	1	24	24	NUM
ap-1408	500	2	(	(	PUNCT
ap-1408	500	3	1994	1994	NUM
ap-1408	500	4	)	)	PUNCT
ap-1408	500	5	,	,	PUNCT
ap-1408	500	6	1	1	NUM
ap-1408	500	7	331–1352	331–1352	NUM
ap-1408	500	8	.	.	PUNCT
ap-1408	501	1	[	[	X
ap-1408	501	2	5	5	NUM
ap-1408	501	3	]	]	PUNCT
ap-1408	501	4	hedĺıková	hedĺıková	PROPN
ap-1408	501	5	,	,	PUNCT
ap-1408	501	6	j.	j.	PROPN
ap-1408	501	7	,	,	PUNCT
ap-1408	501	8	pulmannová	pulmannová	ADV
ap-1408	501	9	,	,	PUNCT
ap-1408	501	10	s.	s.	PROPN
ap-1408	501	11	:	:	PUNCT
ap-1408	501	12	generalized	generalized	ADJ
ap-1408	501	13	difference	difference	NOUN
ap-1408	501	14	posets	poset	NOUN
ap-1408	501	15	and	and	CCONJ
ap-1408	501	16	orthoalgebras	orthoalgebra	NOUN
ap-1408	501	17	,	,	PUNCT
ap-1408	501	18	acta	acta	PROPN
ap-1408	501	19	math	math	PROPN
ap-1408	501	20	.	.	PUNCT
ap-1408	502	1	univ	univ	PROPN
ap-1408	502	2	.	.	PROPN
ap-1408	502	3	comenianae	comenianae	PROPN
ap-1408	502	4	45	45	NUM
ap-1408	502	5	(	(	PUNCT
ap-1408	502	6	1996	1996	NUM
ap-1408	502	7	)	)	PUNCT
ap-1408	502	8	,	,	PUNCT
ap-1408	502	9	247–279	247–279	NUM
ap-1408	502	10	.	.	PUNCT
ap-1408	503	1	[	[	X
ap-1408	503	2	6	6	NUM
ap-1408	503	3	]	]	X
ap-1408	503	4	mostafazadeh	mostafazadeh	NOUN
ap-1408	503	5	,	,	PUNCT
ap-1408	503	6	a.	a.	NOUN
ap-1408	503	7	:	:	PUNCT
ap-1408	503	8	pseudo	pseudo	NOUN
ap-1408	503	9	-	-	ADJ
ap-1408	503	10	hermitian	hermitian	ADJ
ap-1408	503	11	representation	representation	NOUN
ap-1408	503	12	of	of	ADP
ap-1408	503	13	quantum	quantum	ADJ
ap-1408	503	14	mechanics	mechanic	NOUN
ap-1408	503	15	,	,	PUNCT
ap-1408	503	16	int	int	NOUN
ap-1408	503	17	.	.	PUNCT
ap-1408	504	1	j.	j.	PROPN
ap-1408	504	2	geom	geom	PROPN
ap-1408	504	3	.	.	PUNCT
ap-1408	505	1	meth	meth	PROPN
ap-1408	505	2	.	.	PUNCT
ap-1408	506	1	mod	mod	PROPN
ap-1408	506	2	.	.	PUNCT
ap-1408	507	1	phys	phy	NOUN
ap-1408	507	2	7	7	NUM
ap-1408	507	3	(	(	PUNCT
ap-1408	507	4	2010	2010	NUM
ap-1408	507	5	)	)	PUNCT
ap-1408	507	6	,	,	PUNCT
ap-1408	507	7	1	1	NUM
ap-1408	507	8	191–1306	191–1306	NUM
ap-1408	507	9	.	.	PUNCT
ap-1408	508	1	[	[	X
ap-1408	508	2	7	7	NUM
ap-1408	508	3	]	]	PUNCT
ap-1408	508	4	paseka	paseka	NOUN
ap-1408	508	5	,	,	PUNCT
ap-1408	508	6	j.	j.	PROPN
ap-1408	508	7	:	:	PUNCT
ap-1408	508	8	pt	pt	PROPN
ap-1408	508	9	-	-	PUNCT
ap-1408	508	10	symmetry	symmetry	NOUN
ap-1408	508	11	in	in	ADP
ap-1408	508	12	(	(	PUNCT
ap-1408	508	13	generalized	generalized	ADJ
ap-1408	508	14	)	)	PUNCT
ap-1408	508	15	effect	effect	NOUN
ap-1408	508	16	algebras	algebra	NOUN
ap-1408	508	17	,	,	PUNCT
ap-1408	508	18	internat	internat	PROPN
ap-1408	508	19	.	.	PUNCT
ap-1408	509	1	j.	j.	PROPN
ap-1408	509	2	theoret	theoret	PROPN
ap-1408	509	3	.	.	PUNCT
ap-1408	510	1	phys	phy	NOUN
ap-1408	510	2	.	.	PUNCT
ap-1408	511	1	50	50	NUM
ap-1408	511	2	(	(	PUNCT
ap-1408	511	3	2011	2011	NUM
ap-1408	511	4	)	)	PUNCT
ap-1408	511	5	,	,	PUNCT
ap-1408	511	6	1	1	NUM
ap-1408	511	7	198–1205	198–1205	NUM
ap-1408	511	8	.	.	PUNCT
ap-1408	512	1	[	[	X
ap-1408	512	2	8	8	NUM
ap-1408	512	3	]	]	SYM
ap-1408	512	4	polakovič	polakovič	NOUN
ap-1408	512	5	,	,	PUNCT
ap-1408	512	6	m.	m.	NOUN
ap-1408	512	7	,	,	PUNCT
ap-1408	512	8	riečanová	riečanová	PROPN
ap-1408	512	9	,	,	PUNCT
ap-1408	512	10	z.	z.	PROPN
ap-1408	512	11	:	:	PUNCT
ap-1408	512	12	generalized	generalized	ADJ
ap-1408	512	13	effect	effect	NOUN
ap-1408	512	14	algebras	algebra	NOUN
ap-1408	512	15	of	of	ADP
ap-1408	512	16	positive	positive	ADJ
ap-1408	512	17	operators	operator	NOUN
ap-1408	512	18	densely	densely	ADV
ap-1408	512	19	defined	define	VERB
ap-1408	512	20	on	on	ADP
ap-1408	512	21	hilbert	hilbert	PROPN
ap-1408	512	22	spaces	space	NOUN
ap-1408	512	23	,	,	PUNCT
ap-1408	512	24	internat	internat	PROPN
ap-1408	512	25	.	.	PUNCT
ap-1408	513	1	j.	j.	PROPN
ap-1408	513	2	theoret	theoret	PROPN
ap-1408	513	3	.	.	PUNCT
ap-1408	514	1	phys	phy	NOUN
ap-1408	514	2	.	.	PUNCT
ap-1408	515	1	50	50	NUM
ap-1408	515	2	(	(	PUNCT
ap-1408	515	3	2011	2011	NUM
ap-1408	515	4	)	)	PUNCT
ap-1408	515	5	,	,	PUNCT
ap-1408	515	6	1	1	NUM
ap-1408	515	7	167–1	167–1	NUM
ap-1408	515	8	174	174	NUM
ap-1408	515	9	.	.	PUNCT
ap-1408	516	1	[	[	X
ap-1408	516	2	9	9	NUM
ap-1408	516	3	]	]	SYM
ap-1408	516	4	riečanová	riečanová	PROPN
ap-1408	516	5	,	,	PUNCT
ap-1408	516	6	z.	z.	PROPN
ap-1408	516	7	,	,	PUNCT
ap-1408	516	8	zajac	zajac	PROPN
ap-1408	516	9	,	,	PUNCT
ap-1408	516	10	m.	m.	NOUN
ap-1408	516	11	,	,	PUNCT
ap-1408	516	12	pulmannová	pulmannová	ADJ
ap-1408	516	13	,	,	PUNCT
ap-1408	516	14	s.	s.	PROPN
ap-1408	516	15	:	:	PUNCT
ap-1408	516	16	effect	effect	NOUN
ap-1408	516	17	algebras	algebra	NOUN
ap-1408	516	18	of	of	ADP
ap-1408	516	19	positive	positive	ADJ
ap-1408	516	20	operators	operator	NOUN
ap-1408	516	21	densely	densely	ADV
ap-1408	516	22	defined	define	VERB
ap-1408	516	23	on	on	ADP
ap-1408	516	24	hilbert	hilbert	PROPN
ap-1408	516	25	spaces	space	NOUN
ap-1408	516	26	,	,	PUNCT
ap-1408	516	27	reports	report	NOUN
ap-1408	516	28	on	on	ADP
ap-1408	516	29	mathematical	mathematical	ADJ
ap-1408	516	30	physics	physics	NOUN
ap-1408	516	31	,	,	PUNCT
ap-1408	516	32	(	(	PUNCT
ap-1408	516	33	2011	2011	NUM
ap-1408	516	34	)	)	PUNCT
ap-1408	516	35	,	,	PUNCT
ap-1408	516	36	accepted	accept	VERB
ap-1408	516	37	.	.	PUNCT
ap-1408	517	1	jan	jan	PROPN
ap-1408	517	2	paseka	paseka	SCONJ
ap-1408	517	3	e	e	NOUN
ap-1408	517	4	-	-	NOUN
ap-1408	517	5	mail	mail	NOUN
ap-1408	517	6	:	:	PUNCT
ap-1408	518	1	paseka@math.muni.cz	paseka@math.muni.cz	NOUN
ap-1408	518	2	jǐŕı	jǐŕı	PROPN
ap-1408	519	1	janda	janda	PROPN
ap-1408	519	2	e	e	PROPN
ap-1408	519	3	-	-	NOUN
ap-1408	519	4	mail	mail	NOUN
ap-1408	519	5	:	:	PUNCT
ap-1408	519	6	98599@mail.muni.cz	98599@mail.muni.cz	NUM
ap-1408	519	7	department	department	NOUN
ap-1408	519	8	of	of	ADP
ap-1408	519	9	mathematics	mathematics	PROPN
ap-1408	519	10	and	and	CCONJ
ap-1408	519	11	statistics	statistic	NOUN
ap-1408	519	12	faculty	faculty	NOUN
ap-1408	519	13	of	of	ADP
ap-1408	519	14	science	science	PROPN
ap-1408	519	15	masaryk	masaryk	PROPN
ap-1408	519	16	university	university	PROPN
ap-1408	519	17	kotlářská	kotlářská	PROPN
ap-1408	519	18	2	2	NUM
ap-1408	519	19	,	,	PUNCT
ap-1408	519	20	cz-611	cz-611	VERB
ap-1408	519	21	37	37	NUM
ap-1408	519	22	brno	brno	NOUN
ap-1408	519	23	72	72	NUM