id	sid	tid	token	lemma	pos
ap-1410	1	1	acta	acta	PROPN
ap-1410	1	2	polytechnica	polytechnica	PROPN
ap-1410	1	3	vol	vol	NOUN
ap-1410	1	4	.	.	PUNCT
ap-1410	2	1	51	51	NUM
ap-1410	2	2	no	no	INTJ
ap-1410	2	3	.	.	PUNCT
ap-1410	3	1	4/2011	4/2011	NUM
ap-1410	3	2	extensions	extension	NOUN
ap-1410	3	3	of	of	ADP
ap-1410	3	4	effect	effect	NOUN
ap-1410	3	5	algebra	algebra	NOUN
ap-1410	3	6	operations	operation	NOUN
ap-1410	3	7	z.	z.	PROPN
ap-1410	3	8	riečanová	riečanová	PROPN
ap-1410	3	9	,	,	PUNCT
ap-1410	3	10	m.	m.	PROPN
ap-1410	3	11	zajac	zajac	PROPN
ap-1410	3	12	abstract	abstract	PROPN
ap-1410	3	13	we	we	PRON
ap-1410	3	14	study	study	VERB
ap-1410	3	15	the	the	DET
ap-1410	3	16	set	set	NOUN
ap-1410	3	17	of	of	ADP
ap-1410	3	18	all	all	DET
ap-1410	3	19	positive	positive	ADJ
ap-1410	3	20	linear	linear	NOUN
ap-1410	3	21	operators	operator	NOUN
ap-1410	3	22	densely	densely	ADV
ap-1410	3	23	defined	define	VERB
ap-1410	3	24	in	in	ADP
ap-1410	3	25	an	an	DET
ap-1410	3	26	infinite	infinite	ADJ
ap-1410	3	27	-	-	PUNCT
ap-1410	3	28	dimensional	dimensional	ADJ
ap-1410	3	29	complex	complex	ADJ
ap-1410	3	30	hilbert	hilbert	NOUN
ap-1410	3	31	space	space	NOUN
ap-1410	3	32	.	.	PUNCT
ap-1410	4	1	we	we	PRON
ap-1410	4	2	equip	equip	VERB
ap-1410	4	3	this	this	DET
ap-1410	4	4	set	set	NOUN
ap-1410	4	5	with	with	ADP
ap-1410	4	6	various	various	ADJ
ap-1410	4	7	effect	effect	NOUN
ap-1410	4	8	algebraic	algebraic	ADJ
ap-1410	4	9	operations	operation	NOUN
ap-1410	4	10	making	make	VERB
ap-1410	4	11	it	it	PRON
ap-1410	4	12	a	a	DET
ap-1410	4	13	generalized	generalized	ADJ
ap-1410	4	14	effect	effect	NOUN
ap-1410	4	15	algebra	algebra	NOUN
ap-1410	4	16	.	.	PUNCT
ap-1410	5	1	further	far	ADV
ap-1410	5	2	,	,	PUNCT
ap-1410	5	3	sub	sub	ADJ
ap-1410	5	4	-	-	ADJ
ap-1410	5	5	generalized	generalized	ADJ
ap-1410	5	6	effect	effect	NOUN
ap-1410	5	7	algebras	algebra	NOUN
ap-1410	5	8	and	and	CCONJ
ap-1410	5	9	interval	interval	NOUN
ap-1410	5	10	effect	effect	NOUN
ap-1410	5	11	algebras	algebra	VERB
ap-1410	5	12	with	with	ADP
ap-1410	5	13	respect	respect	NOUN
ap-1410	5	14	of	of	ADP
ap-1410	5	15	these	these	DET
ap-1410	5	16	operations	operation	NOUN
ap-1410	5	17	are	be	AUX
ap-1410	5	18	investigated	investigate	VERB
ap-1410	5	19	.	.	PUNCT
ap-1410	6	1	keywords	keyword	NOUN
ap-1410	6	2	:	:	PUNCT
ap-1410	6	3	generalized	generalized	ADJ
ap-1410	6	4	effect	effect	NOUN
ap-1410	6	5	algebra	algebra	NOUN
ap-1410	6	6	,	,	PUNCT
ap-1410	6	7	effect	effect	NOUN
ap-1410	6	8	algebra	algebra	NOUN
ap-1410	6	9	,	,	PUNCT
ap-1410	6	10	hilbert	hilbert	NOUN
ap-1410	6	11	space	space	NOUN
ap-1410	6	12	,	,	PUNCT
ap-1410	6	13	densely	densely	ADV
ap-1410	6	14	defined	define	VERB
ap-1410	6	15	linear	linear	PROPN
ap-1410	6	16	operators	operator	NOUN
ap-1410	6	17	,	,	PUNCT
ap-1410	6	18	extension	extension	NOUN
ap-1410	6	19	of	of	ADP
ap-1410	6	20	operations	operation	NOUN
ap-1410	6	21	.	.	PUNCT
ap-1410	7	1	1	1	NUM
ap-1410	7	2	introduction	introduction	NOUN
ap-1410	7	3	and	and	CCONJ
ap-1410	7	4	some	some	DET
ap-1410	7	5	basic	basic	ADJ
ap-1410	7	6	definitions	definition	NOUN
ap-1410	7	7	and	and	CCONJ
ap-1410	7	8	facts	fact	VERB
ap-1410	7	9	the	the	DET
ap-1410	7	10	aim	aim	NOUN
ap-1410	7	11	of	of	ADP
ap-1410	7	12	this	this	DET
ap-1410	7	13	paper	paper	NOUN
ap-1410	7	14	is	be	AUX
ap-1410	7	15	to	to	PART
ap-1410	7	16	show	show	VERB
ap-1410	7	17	that	that	SCONJ
ap-1410	7	18	(	(	PUNCT
ap-1410	7	19	generalized	generalized	ADJ
ap-1410	7	20	)	)	PUNCT
ap-1410	7	21	effect	effect	NOUN
ap-1410	7	22	algebras	algebra	NOUN
ap-1410	7	23	may	may	AUX
ap-1410	7	24	be	be	AUX
ap-1410	7	25	suitable	suitable	ADJ
ap-1410	7	26	and	and	CCONJ
ap-1410	7	27	natural	natural	ADJ
ap-1410	7	28	algebraic	algebraic	ADJ
ap-1410	7	29	structures	structure	NOUN
ap-1410	7	30	for	for	ADP
ap-1410	7	31	sets	set	NOUN
ap-1410	7	32	of	of	ADP
ap-1410	7	33	linear	linear	PROPN
ap-1410	7	34	operators	operator	NOUN
ap-1410	7	35	(	(	PUNCT
ap-1410	7	36	including	include	VERB
ap-1410	7	37	unbounded	unbounded	ADJ
ap-1410	7	38	ones	one	NOUN
ap-1410	7	39	)	)	PUNCT
ap-1410	7	40	densely	densely	ADV
ap-1410	7	41	defined	define	VERB
ap-1410	7	42	on	on	ADP
ap-1410	7	43	an	an	DET
ap-1410	7	44	infinitedimensional	infinitedimensional	ADJ
ap-1410	7	45	complex	complex	ADJ
ap-1410	7	46	hilbert	hilbert	NOUN
ap-1410	7	47	space	space	NOUN
ap-1410	7	48	h.	h.	PROPN
ap-1410	7	49	in	in	ADP
ap-1410	7	50	all	all	DET
ap-1410	7	51	cases	case	NOUN
ap-1410	7	52	,	,	PUNCT
ap-1410	7	53	if	if	SCONJ
ap-1410	7	54	the	the	DET
ap-1410	7	55	effect	effect	NOUN
ap-1410	7	56	algebraic	algebraic	ADJ
ap-1410	7	57	sum	sum	NOUN
ap-1410	7	58	of	of	ADP
ap-1410	7	59	operators	operator	NOUN
ap-1410	7	60	a	a	PRON
ap-1410	7	61	,	,	PUNCT
ap-1410	7	62	b	b	PROPN
ap-1410	7	63	is	be	AUX
ap-1410	7	64	defined	define	VERB
ap-1410	7	65	then	then	ADV
ap-1410	7	66	it	it	PRON
ap-1410	7	67	coincides	coincide	VERB
ap-1410	7	68	with	with	ADP
ap-1410	7	69	the	the	DET
ap-1410	7	70	usual	usual	ADJ
ap-1410	7	71	sum	sum	NOUN
ap-1410	7	72	of	of	ADP
ap-1410	7	73	operators	operator	NOUN
ap-1410	7	74	in	in	ADP
ap-1410	7	75	h.	h.	PROPN
ap-1410	7	76	effect	effect	PROPN
ap-1410	7	77	algebras	algebra	NOUN
ap-1410	7	78	were	be	AUX
ap-1410	7	79	introduced	introduce	VERB
ap-1410	7	80	by	by	ADP
ap-1410	7	81	d.	d.	PROPN
ap-1410	7	82	foulis	foulis	PROPN
ap-1410	7	83	and	and	CCONJ
ap-1410	7	84	m.	m.	PROPN
ap-1410	7	85	k.	k.	PROPN
ap-1410	7	86	bennet	bennet	PROPN
ap-1410	7	87	in	in	ADP
ap-1410	7	88	1994	1994	NUM
ap-1410	8	1	[	[	X
ap-1410	8	2	2	2	NUM
ap-1410	8	3	]	]	PUNCT
ap-1410	8	4	.	.	PUNCT
ap-1410	9	1	the	the	DET
ap-1410	9	2	prototype	prototype	NOUN
ap-1410	9	3	for	for	ADP
ap-1410	9	4	the	the	DET
ap-1410	9	5	abstract	abstract	ADJ
ap-1410	9	6	definition	definition	NOUN
ap-1410	9	7	of	of	ADP
ap-1410	9	8	an	an	DET
ap-1410	9	9	effect	effect	NOUN
ap-1410	9	10	algebra	algebra	NOUN
ap-1410	9	11	was	be	AUX
ap-1410	9	12	the	the	DET
ap-1410	9	13	set	set	NOUN
ap-1410	9	14	e(h	e(h	PROPN
ap-1410	9	15	)	)	PUNCT
ap-1410	9	16	(	(	PUNCT
ap-1410	9	17	hilbert	hilbert	NOUN
ap-1410	9	18	space	space	NOUN
ap-1410	9	19	effects	effect	NOUN
ap-1410	9	20	)	)	PUNCT
ap-1410	9	21	of	of	ADP
ap-1410	9	22	all	all	DET
ap-1410	9	23	selfadjoint	selfadjoint	NOUN
ap-1410	9	24	operators	operator	NOUN
ap-1410	9	25	between	between	ADP
ap-1410	9	26	null	null	ADJ
ap-1410	9	27	and	and	CCONJ
ap-1410	9	28	identity	identity	NOUN
ap-1410	9	29	operators	operator	NOUN
ap-1410	9	30	in	in	ADP
ap-1410	9	31	a	a	DET
ap-1410	9	32	complex	complex	ADJ
ap-1410	9	33	hilbert	hilbert	NOUN
ap-1410	9	34	space	space	NOUN
ap-1410	9	35	h.	h.	PROPN
ap-1410	9	36	if	if	SCONJ
ap-1410	9	37	a	a	DET
ap-1410	9	38	quantum	quantum	ADJ
ap-1410	9	39	mechanical	mechanical	ADJ
ap-1410	9	40	system	system	NOUN
ap-1410	9	41	is	be	AUX
ap-1410	9	42	represented	represent	VERB
ap-1410	9	43	in	in	ADP
ap-1410	9	44	the	the	DET
ap-1410	9	45	usual	usual	ADJ
ap-1410	9	46	way	way	NOUN
ap-1410	9	47	by	by	ADP
ap-1410	9	48	a	a	DET
ap-1410	9	49	complex	complex	ADJ
ap-1410	9	50	hilbert	hilbert	NOUN
ap-1410	9	51	space	space	NOUN
ap-1410	9	52	h	h	NOUN
ap-1410	9	53	,	,	PUNCT
ap-1410	9	54	then	then	ADV
ap-1410	9	55	self	self	NOUN
ap-1410	9	56	-	-	PUNCT
ap-1410	9	57	adjoint	adjoint	NOUN
ap-1410	9	58	operators	operator	NOUN
ap-1410	9	59	from	from	ADP
ap-1410	9	60	e(h	e(h	PROPN
ap-1410	9	61	)	)	PUNCT
ap-1410	9	62	represent	represent	VERB
ap-1410	9	63	yes	yes	NOUN
ap-1410	9	64	-	-	PUNCT
ap-1410	9	65	no	no	PRON
ap-1410	9	66	measurements	measurement	NOUN
ap-1410	9	67	that	that	PRON
ap-1410	9	68	may	may	AUX
ap-1410	9	69	be	be	AUX
ap-1410	9	70	unsharp	unsharp	ADJ
ap-1410	9	71	.	.	PUNCT
ap-1410	10	1	the	the	DET
ap-1410	10	2	subset	subset	NOUN
ap-1410	10	3	p(h	p(h	NOUN
ap-1410	10	4	)	)	PUNCT
ap-1410	10	5	of	of	ADP
ap-1410	10	6	e(h	e(h	PROPN
ap-1410	10	7	)	)	PUNCT
ap-1410	10	8	consisting	consist	VERB
ap-1410	10	9	of	of	ADP
ap-1410	10	10	orthogonal	orthogonal	ADJ
ap-1410	10	11	projections	projection	NOUN
ap-1410	10	12	represents	represent	VERB
ap-1410	10	13	yes	yes	NOUN
ap-1410	10	14	-	-	PUNCT
ap-1410	10	15	no	no	DET
ap-1410	10	16	measurements	measurement	NOUN
ap-1410	10	17	that	that	PRON
ap-1410	10	18	are	be	AUX
ap-1410	10	19	sharp	sharp	ADJ
ap-1410	10	20	.	.	PUNCT
ap-1410	11	1	the	the	DET
ap-1410	11	2	abstract	abstract	ADJ
ap-1410	11	3	definition	definition	NOUN
ap-1410	11	4	of	of	ADP
ap-1410	11	5	an	an	DET
ap-1410	11	6	effect	effect	NOUN
ap-1410	11	7	algebra	algebra	NOUN
ap-1410	11	8	follows	follow	VERB
ap-1410	11	9	the	the	DET
ap-1410	11	10	properties	property	NOUN
ap-1410	11	11	of	of	ADP
ap-1410	11	12	the	the	DET
ap-1410	11	13	usual	usual	ADJ
ap-1410	11	14	sum	sum	NOUN
ap-1410	11	15	of	of	ADP
ap-1410	11	16	operators	operator	NOUN
ap-1410	11	17	in	in	ADP
ap-1410	11	18	the	the	DET
ap-1410	11	19	interval	interval	NOUN
ap-1410	11	20	[	[	X
ap-1410	11	21	0	0	NUM
ap-1410	11	22	,	,	PUNCT
ap-1410	11	23	i	i	PRON
ap-1410	11	24	]	]	X
ap-1410	11	25	(	(	PUNCT
ap-1410	11	26	i.e.	i.e.	X
ap-1410	11	27	between	between	ADP
ap-1410	11	28	null	null	ADJ
ap-1410	11	29	and	and	CCONJ
ap-1410	11	30	identity	identity	NOUN
ap-1410	11	31	operators	operator	NOUN
ap-1410	11	32	in	in	ADP
ap-1410	11	33	h	h	NOUN
ap-1410	11	34	)	)	PUNCT
ap-1410	11	35	and	and	CCONJ
ap-1410	11	36	it	it	PRON
ap-1410	11	37	is	be	AUX
ap-1410	11	38	the	the	DET
ap-1410	11	39	following	following	NOUN
ap-1410	11	40	.	.	PUNCT
ap-1410	12	1	definition	definition	NOUN
ap-1410	12	2	1	1	NUM
ap-1410	12	3	(	(	PUNCT
ap-1410	12	4	foulis	foulis	PROPN
ap-1410	12	5	,	,	PUNCT
ap-1410	12	6	bennet	bennet	PROPN
ap-1410	13	1	[	[	X
ap-1410	13	2	2	2	NUM
ap-1410	13	3	]	]	PUNCT
ap-1410	13	4	)	)	PUNCT
ap-1410	13	5	a	a	DET
ap-1410	13	6	partial	partial	ADJ
ap-1410	13	7	algebra	algebra	NOUN
ap-1410	13	8	(	(	PUNCT
ap-1410	13	9	e;⊕	e;⊕	ADJ
ap-1410	13	10	,	,	PUNCT
ap-1410	13	11	0	0	NUM
ap-1410	13	12	,	,	PUNCT
ap-1410	13	13	1	1	NUM
ap-1410	13	14	)	)	PUNCT
ap-1410	13	15	is	be	AUX
ap-1410	13	16	called	call	VERB
ap-1410	13	17	an	an	DET
ap-1410	13	18	effect	effect	NOUN
ap-1410	13	19	algebra	algebra	NOUN
ap-1410	13	20	if	if	SCONJ
ap-1410	13	21	0,1	0,1	NUM
ap-1410	13	22	are	be	AUX
ap-1410	13	23	two	two	NUM
ap-1410	13	24	distinguished	distinguished	ADJ
ap-1410	13	25	elements	element	NOUN
ap-1410	13	26	and	and	CCONJ
ap-1410	13	27	⊕	⊕	PROPN
ap-1410	13	28	is	be	AUX
ap-1410	13	29	a	a	DET
ap-1410	13	30	partially	partially	ADV
ap-1410	13	31	defined	define	VERB
ap-1410	13	32	binary	binary	ADJ
ap-1410	13	33	operation	operation	NOUN
ap-1410	13	34	on	on	ADP
ap-1410	13	35	e	e	PROPN
ap-1410	13	36	which	which	PRON
ap-1410	13	37	satisfy	satisfy	VERB
ap-1410	13	38	the	the	DET
ap-1410	13	39	following	follow	VERB
ap-1410	13	40	conditions	condition	NOUN
ap-1410	13	41	for	for	ADP
ap-1410	13	42	any	any	DET
ap-1410	13	43	x	x	NOUN
ap-1410	13	44	,	,	PUNCT
ap-1410	13	45	y	y	PROPN
ap-1410	13	46	,	,	PUNCT
ap-1410	13	47	z	z	NOUN
ap-1410	13	48	∈	∈	PROPN
ap-1410	14	1	e	e	NOUN
ap-1410	14	2	:	:	PUNCT
ap-1410	14	3	(	(	PUNCT
ap-1410	14	4	e1	e1	NOUN
ap-1410	14	5	)	)	PUNCT
ap-1410	14	6	x	x	PUNCT
ap-1410	14	7	⊕	⊕	NOUN
ap-1410	14	8	y	y	NOUN
ap-1410	14	9	=	=	SYM
ap-1410	14	10	y	y	PROPN
ap-1410	14	11	⊕	⊕	PROPN
ap-1410	14	12	x	x	PUNCT
ap-1410	15	1	if	if	SCONJ
ap-1410	15	2	x	x	PROPN
ap-1410	15	3	⊕	⊕	NOUN
ap-1410	15	4	y	y	PROPN
ap-1410	15	5	is	be	AUX
ap-1410	15	6	defined	define	VERB
ap-1410	15	7	,	,	PUNCT
ap-1410	15	8	(	(	PUNCT
ap-1410	15	9	e2	e2	PROPN
ap-1410	15	10	)	)	PUNCT
ap-1410	15	11	(	(	PUNCT
ap-1410	15	12	x	x	PROPN
ap-1410	15	13	⊕	⊕	PROPN
ap-1410	15	14	y	y	PROPN
ap-1410	15	15	)	)	PUNCT
ap-1410	15	16	⊕	⊕	PROPN
ap-1410	15	17	z	z	PUNCT
ap-1410	16	1	=	=	PUNCT
ap-1410	16	2	x	x	SYM
ap-1410	16	3	⊕	⊕	PROPN
ap-1410	16	4	(	(	PUNCT
ap-1410	16	5	y	y	PROPN
ap-1410	16	6	⊕	⊕	PROPN
ap-1410	16	7	z	z	PROPN
ap-1410	16	8	)	)	PUNCT
ap-1410	16	9	if	if	SCONJ
ap-1410	16	10	one	one	NUM
ap-1410	16	11	side	side	NOUN
ap-1410	16	12	is	be	AUX
ap-1410	16	13	defined	define	VERB
ap-1410	16	14	,	,	PUNCT
ap-1410	16	15	(	(	PUNCT
ap-1410	16	16	e3	e3	NOUN
ap-1410	16	17	)	)	PUNCT
ap-1410	16	18	for	for	ADP
ap-1410	16	19	every	every	DET
ap-1410	16	20	x	x	SYM
ap-1410	16	21	∈	∈	PROPN
ap-1410	16	22	e	e	NOUN
ap-1410	16	23	there	there	PRON
ap-1410	16	24	exists	exist	VERB
ap-1410	16	25	a	a	DET
ap-1410	16	26	unique	unique	ADJ
ap-1410	16	27	y	y	PROPN
ap-1410	16	28	∈	∈	PROPN
ap-1410	16	29	e	e	NOUN
ap-1410	16	30	such	such	ADJ
ap-1410	16	31	that	that	SCONJ
ap-1410	16	32	x	x	PROPN
ap-1410	16	33	⊕	⊕	NOUN
ap-1410	16	34	y	y	NOUN
ap-1410	16	35	=	=	SYM
ap-1410	16	36	1	1	NUM
ap-1410	16	37	(	(	PUNCT
ap-1410	16	38	we	we	PRON
ap-1410	16	39	put	put	VERB
ap-1410	16	40	x′	x′	PROPN
ap-1410	16	41	=	=	SYM
ap-1410	16	42	y	y	PROPN
ap-1410	16	43	)	)	PUNCT
ap-1410	16	44	,	,	PUNCT
ap-1410	16	45	(	(	PUNCT
ap-1410	16	46	e4	e4	PROPN
ap-1410	16	47	)	)	PUNCT
ap-1410	16	48	if	if	SCONJ
ap-1410	16	49	1	1	NUM
ap-1410	16	50	⊕	⊕	PROPN
ap-1410	16	51	x	x	PUNCT
ap-1410	16	52	is	be	AUX
ap-1410	16	53	defined	define	VERB
ap-1410	16	54	then	then	ADV
ap-1410	16	55	x	x	X
ap-1410	16	56	=	=	SYM
ap-1410	16	57	0	0	X
ap-1410	16	58	.	.	PUNCT
ap-1410	17	1	immediately	immediately	ADV
ap-1410	17	2	in	in	ADP
ap-1410	17	3	1994	1994	NUM
ap-1410	17	4	the	the	DET
ap-1410	17	5	study	study	NOUN
ap-1410	17	6	of	of	ADP
ap-1410	17	7	generalizations	generalization	NOUN
ap-1410	17	8	of	of	ADP
ap-1410	17	9	effect	effect	NOUN
ap-1410	17	10	algebras	algebra	NOUN
ap-1410	17	11	(	(	PUNCT
ap-1410	17	12	without	without	ADP
ap-1410	17	13	the	the	DET
ap-1410	17	14	top	top	ADJ
ap-1410	17	15	element	element	NOUN
ap-1410	17	16	1	1	NUM
ap-1410	17	17	)	)	PUNCT
ap-1410	17	18	was	be	AUX
ap-1410	17	19	started	start	VERB
ap-1410	17	20	by	by	ADP
ap-1410	17	21	several	several	ADJ
ap-1410	17	22	authors	author	NOUN
ap-1410	17	23	(	(	PUNCT
ap-1410	17	24	foulis	foulis	PROPN
ap-1410	17	25	and	and	CCONJ
ap-1410	17	26	bennet	bennet	NOUN
ap-1410	18	1	[	[	X
ap-1410	18	2	2	2	NUM
ap-1410	18	3	]	]	PUNCT
ap-1410	18	4	,	,	PUNCT
ap-1410	18	5	kalmbach	kalmbach	PROPN
ap-1410	18	6	and	and	CCONJ
ap-1410	18	7	riečanová	riečanová	PROPN
ap-1410	19	1	[	[	X
ap-1410	19	2	4	4	NUM
ap-1410	19	3	]	]	PUNCT
ap-1410	19	4	,	,	PUNCT
ap-1410	19	5	hedĺıková	hedĺıková	PROPN
ap-1410	19	6	and	and	CCONJ
ap-1410	19	7	pulmannová	pulmannová	ADJ
ap-1410	20	1	[	[	X
ap-1410	20	2	3	3	NUM
ap-1410	20	3	]	]	PUNCT
ap-1410	20	4	,	,	PUNCT
ap-1410	20	5	kôpka	kôpka	NOUN
ap-1410	20	6	and	and	CCONJ
ap-1410	20	7	chovanec	chovanec	NOUN
ap-1410	21	1	[	[	X
ap-1410	21	2	5	5	NUM
ap-1410	21	3	]	]	PUNCT
ap-1410	21	4	)	)	PUNCT
ap-1410	21	5	.	.	PUNCT
ap-1410	22	1	it	it	PRON
ap-1410	22	2	was	be	AUX
ap-1410	22	3	found	find	VERB
ap-1410	22	4	that	that	SCONJ
ap-1410	22	5	all	all	DET
ap-1410	22	6	these	these	DET
ap-1410	22	7	generalizations	generalization	NOUN
ap-1410	22	8	coincide	coincide	NOUN
ap-1410	22	9	,	,	PUNCT
ap-1410	22	10	and	and	CCONJ
ap-1410	22	11	their	their	PRON
ap-1410	22	12	common	common	ADJ
ap-1410	22	13	definition	definition	NOUN
ap-1410	22	14	is	be	AUX
ap-1410	22	15	the	the	DET
ap-1410	22	16	following	following	NOUN
ap-1410	22	17	:	:	PUNCT
ap-1410	22	18	definition	definition	NOUN
ap-1410	22	19	2	2	NUM
ap-1410	22	20	a	a	DET
ap-1410	22	21	generalized	generalized	ADJ
ap-1410	22	22	effect	effect	NOUN
ap-1410	22	23	algebra	algebra	NOUN
ap-1410	22	24	(	(	PUNCT
ap-1410	22	25	e;⊕	e;⊕	ADJ
ap-1410	22	26	,	,	PUNCT
ap-1410	22	27	0	0	NUM
ap-1410	22	28	)	)	PUNCT
ap-1410	22	29	is	be	AUX
ap-1410	22	30	a	a	DET
ap-1410	22	31	set	set	NOUN
ap-1410	22	32	e	e	NOUN
ap-1410	22	33	with	with	ADP
ap-1410	22	34	element	element	NOUN
ap-1410	22	35	0	0	NUM
ap-1410	22	36	∈	∈	PROPN
ap-1410	22	37	e	e	NOUN
ap-1410	22	38	and	and	CCONJ
ap-1410	22	39	partial	partial	ADJ
ap-1410	22	40	binary	binary	ADJ
ap-1410	22	41	operation	operation	NOUN
ap-1410	22	42	⊕	⊕	PROPN
ap-1410	22	43	satisfying	satisfy	VERB
ap-1410	22	44	for	for	ADP
ap-1410	22	45	any	any	DET
ap-1410	22	46	x	x	NOUN
ap-1410	22	47	,	,	PUNCT
ap-1410	22	48	y	y	PROPN
ap-1410	22	49	,	,	PUNCT
ap-1410	22	50	z	z	NOUN
ap-1410	22	51	∈	∈	PROPN
ap-1410	22	52	e	e	NOUN
ap-1410	22	53	the	the	DET
ap-1410	22	54	conditions	condition	NOUN
ap-1410	22	55	(	(	PUNCT
ap-1410	22	56	ge1	ge1	NOUN
ap-1410	22	57	)	)	PUNCT
ap-1410	22	58	x	x	PUNCT
ap-1410	22	59	⊕	⊕	NOUN
ap-1410	22	60	y	y	NOUN
ap-1410	22	61	=	=	SYM
ap-1410	22	62	y	y	PROPN
ap-1410	22	63	⊕	⊕	PROPN
ap-1410	22	64	x	x	PUNCT
ap-1410	23	1	if	if	SCONJ
ap-1410	23	2	one	one	NUM
ap-1410	23	3	side	side	NOUN
ap-1410	23	4	is	be	AUX
ap-1410	23	5	defined	define	VERB
ap-1410	23	6	,	,	PUNCT
ap-1410	23	7	(	(	PUNCT
ap-1410	23	8	ge2	ge2	PROPN
ap-1410	23	9	)	)	PUNCT
ap-1410	23	10	(	(	PUNCT
ap-1410	23	11	x⊕	x⊕	PROPN
ap-1410	23	12	y)⊕	y)⊕	NOUN
ap-1410	23	13	z	z	NOUN
ap-1410	23	14	=	=	SYM
ap-1410	23	15	x⊕	x⊕	PROPN
ap-1410	23	16	(	(	PUNCT
ap-1410	23	17	y⊕	y⊕	PROPN
ap-1410	23	18	z	z	PROPN
ap-1410	23	19	)	)	PUNCT
ap-1410	23	20	if	if	SCONJ
ap-1410	23	21	one	one	NUM
ap-1410	23	22	side	side	NOUN
ap-1410	23	23	is	be	AUX
ap-1410	23	24	defined	define	VERB
ap-1410	23	25	,	,	PUNCT
ap-1410	23	26	(	(	PUNCT
ap-1410	23	27	ge3	ge3	NOUN
ap-1410	23	28	)	)	PUNCT
ap-1410	24	1	if	if	SCONJ
ap-1410	24	2	x	x	PROPN
ap-1410	24	3	⊕	⊕	NOUN
ap-1410	24	4	y	y	NOUN
ap-1410	24	5	=	=	SYM
ap-1410	24	6	x	x	SYM
ap-1410	24	7	⊕	⊕	PROPN
ap-1410	24	8	z	z	NOUN
ap-1410	25	1	then	then	ADV
ap-1410	25	2	y	y	PROPN
ap-1410	25	3	=	=	SYM
ap-1410	25	4	z	z	PROPN
ap-1410	25	5	,	,	PUNCT
ap-1410	25	6	(	(	PUNCT
ap-1410	25	7	ge4	ge4	PROPN
ap-1410	25	8	)	)	PUNCT
ap-1410	25	9	if	if	SCONJ
ap-1410	25	10	x	x	PROPN
ap-1410	25	11	⊕	⊕	NOUN
ap-1410	25	12	y	y	NOUN
ap-1410	26	1	=	=	PUNCT
ap-1410	26	2	0	0	PUNCT
ap-1410	26	3	then	then	ADV
ap-1410	26	4	x	x	X
ap-1410	26	5	=	=	SYM
ap-1410	26	6	y	y	PROPN
ap-1410	26	7	=	=	SYM
ap-1410	26	8	0	0	PROPN
ap-1410	26	9	,	,	PUNCT
ap-1410	26	10	(	(	PUNCT
ap-1410	26	11	ge5	ge5	PROPN
ap-1410	26	12	)	)	PUNCT
ap-1410	26	13	x	x	PUNCT
ap-1410	26	14	⊕	⊕	NOUN
ap-1410	26	15	0	0	PUNCT
ap-1410	27	1	=	=	PUNCT
ap-1410	27	2	x	x	PUNCT
ap-1410	27	3	for	for	ADP
ap-1410	27	4	all	all	DET
ap-1410	27	5	x	x	SYM
ap-1410	27	6	∈	∈	PROPN
ap-1410	27	7	e.	e.	PROPN
ap-1410	27	8	in	in	ADP
ap-1410	27	9	every	every	DET
ap-1410	27	10	(	(	PUNCT
ap-1410	27	11	generalized	generalized	ADJ
ap-1410	27	12	)	)	PUNCT
ap-1410	27	13	effect	effect	NOUN
ap-1410	27	14	algebra	algebra	NOUN
ap-1410	27	15	e	e	PRON
ap-1410	27	16	a	a	DET
ap-1410	27	17	partial	partial	ADJ
ap-1410	27	18	order	order	NOUN
ap-1410	27	19	≤	≤	NOUN
ap-1410	27	20	and	and	CCONJ
ap-1410	27	21	a	a	DET
ap-1410	27	22	binary	binary	ADJ
ap-1410	27	23	operation	operation	NOUN
ap-1410	27	24	�	�	PROPN
ap-1410	27	25	can	can	AUX
ap-1410	27	26	be	be	AUX
ap-1410	27	27	introduced	introduce	VERB
ap-1410	27	28	as	as	SCONJ
ap-1410	27	29	follows	follow	VERB
ap-1410	27	30	:	:	PUNCT
ap-1410	27	31	for	for	SCONJ
ap-1410	27	32	any	any	PRON
ap-1410	27	33	a	a	NOUN
ap-1410	27	34	,	,	PUNCT
ap-1410	27	35	b	b	X
ap-1410	27	36	∈	∈	PROPN
ap-1410	27	37	e	e	NOUN
ap-1410	27	38	,	,	PUNCT
ap-1410	27	39	a	a	DET
ap-1410	27	40	≤	≤	PROPN
ap-1410	27	41	b	b	NOUN
ap-1410	27	42	and	and	CCONJ
ap-1410	27	43	b	b	PROPN
ap-1410	27	44	�	�	PROPN
ap-1410	27	45	a	a	PRON
ap-1410	27	46	=	=	X
ap-1410	27	47	c	c	PROPN
ap-1410	27	48	iff	iff	PROPN
ap-1410	27	49	a	a	DET
ap-1410	27	50	⊕	⊕	PROPN
ap-1410	27	51	c	c	PROPN
ap-1410	27	52	is	be	AUX
ap-1410	27	53	defined	define	VERB
ap-1410	27	54	and	and	CCONJ
ap-1410	27	55	a	a	DET
ap-1410	27	56	⊕	⊕	PROPN
ap-1410	27	57	c	c	X
ap-1410	27	58	=	=	PUNCT
ap-1410	27	59	b.	b.	PROPN
ap-1410	28	1	if	if	SCONJ
ap-1410	28	2	the	the	DET
ap-1410	28	3	elements	element	NOUN
ap-1410	28	4	of	of	ADP
ap-1410	28	5	a	a	DET
ap-1410	28	6	(	(	PUNCT
ap-1410	28	7	generalized	generalized	ADJ
ap-1410	28	8	)	)	PUNCT
ap-1410	28	9	effect	effect	NOUN
ap-1410	28	10	algebra	algebra	NOUN
ap-1410	28	11	e	e	NOUN
ap-1410	28	12	are	be	AUX
ap-1410	28	13	positive	positive	ADJ
ap-1410	28	14	linear	linear	ADJ
ap-1410	28	15	operators	operator	NOUN
ap-1410	28	16	in	in	ADP
ap-1410	28	17	a	a	DET
ap-1410	28	18	given	give	VERB
ap-1410	28	19	infinitedimensional	infinitedimensional	ADJ
ap-1410	28	20	complex	complex	ADJ
ap-1410	28	21	hilbert	hilbert	NOUN
ap-1410	28	22	space	space	NOUN
ap-1410	28	23	,	,	PUNCT
ap-1410	28	24	then	then	ADV
ap-1410	28	25	e	e	PROPN
ap-1410	28	26	is	be	AUX
ap-1410	28	27	called	call	VERB
ap-1410	28	28	an	an	DET
ap-1410	28	29	operator	operator	NOUN
ap-1410	28	30	(	(	PUNCT
ap-1410	28	31	generalized	generalized	ADJ
ap-1410	28	32	)	)	PUNCT
ap-1410	28	33	effect	effect	NOUN
ap-1410	28	34	algebra	algebra	NOUN
ap-1410	28	35	.	.	PUNCT
ap-1410	29	1	throughout	throughout	ADP
ap-1410	29	2	the	the	DET
ap-1410	29	3	paper	paper	NOUN
ap-1410	29	4	we	we	PRON
ap-1410	29	5	assume	assume	VERB
ap-1410	29	6	that	that	SCONJ
ap-1410	29	7	h	h	NOUN
ap-1410	29	8	is	be	AUX
ap-1410	29	9	an	an	DET
ap-1410	29	10	infinite	infinite	ADJ
ap-1410	29	11	-	-	PUNCT
ap-1410	29	12	dimensional	dimensional	ADJ
ap-1410	29	13	complex	complex	ADJ
ap-1410	29	14	hilbert	hilbert	NOUN
ap-1410	29	15	space	space	NOUN
ap-1410	29	16	,	,	PUNCT
ap-1410	29	17	i.e.	i.e.	X
ap-1410	29	18	,	,	PUNCT
ap-1410	29	19	a	a	DET
ap-1410	29	20	linear	linear	ADJ
ap-1410	29	21	space	space	NOUN
ap-1410	29	22	with	with	ADP
ap-1410	29	23	inner	inner	ADJ
ap-1410	29	24	product	product	NOUN
ap-1410	29	25	(	(	PUNCT
ap-1410	29	26	·	·	PUNCT
ap-1410	29	27	,	,	PUNCT
ap-1410	29	28	·	·	PUNCT
ap-1410	29	29	)	)	PUNCT
ap-1410	29	30	which	which	PRON
ap-1410	29	31	is	be	AUX
ap-1410	29	32	complete	complete	ADJ
ap-1410	29	33	in	in	ADP
ap-1410	29	34	the	the	DET
ap-1410	29	35	induced	induce	VERB
ap-1410	29	36	metric	metric	NOUN
ap-1410	29	37	.	.	PUNCT
ap-1410	30	1	recall	recall	VERB
ap-1410	30	2	that	that	SCONJ
ap-1410	30	3	here	here	ADV
ap-1410	30	4	for	for	ADP
ap-1410	30	5	any	any	DET
ap-1410	30	6	x	x	NOUN
ap-1410	30	7	,	,	PUNCT
ap-1410	30	8	y	y	PROPN
ap-1410	30	9	∈	∈	PROPN
ap-1410	30	10	h	h	NOUN
ap-1410	30	11	we	we	PRON
ap-1410	30	12	have	have	VERB
ap-1410	30	13	(	(	PUNCT
ap-1410	31	1	x	x	NOUN
ap-1410	31	2	,	,	PUNCT
ap-1410	31	3	y	y	NOUN
ap-1410	31	4	)	)	PUNCT
ap-1410	31	5	∈	∈	PROPN
ap-1410	31	6	c	c	NOUN
ap-1410	31	7	(	(	PUNCT
ap-1410	31	8	the	the	DET
ap-1410	31	9	set	set	NOUN
ap-1410	31	10	of	of	ADP
ap-1410	31	11	all	all	DET
ap-1410	31	12	complex	complex	ADJ
ap-1410	31	13	numbers	number	NOUN
ap-1410	31	14	)	)	PUNCT
ap-1410	31	15	such	such	ADJ
ap-1410	31	16	that	that	SCONJ
ap-1410	31	17	(	(	PUNCT
ap-1410	31	18	x	x	NOUN
ap-1410	31	19	,	,	PUNCT
ap-1410	31	20	αy+βz	αy+βz	NUM
ap-1410	31	21	)	)	PUNCT
ap-1410	31	22	=	=	SYM
ap-1410	31	23	α(x	α(x	PROPN
ap-1410	31	24	,	,	PUNCT
ap-1410	31	25	y)+β(x	y)+β(x	NOUN
ap-1410	31	26	,	,	PUNCT
ap-1410	31	27	z	z	NOUN
ap-1410	31	28	)	)	PUNCT
ap-1410	31	29	for	for	ADP
ap-1410	31	30	all	all	DET
ap-1410	31	31	α	α	NOUN
ap-1410	31	32	,	,	PUNCT
ap-1410	31	33	β	β	X
ap-1410	31	34	∈	∈	PROPN
ap-1410	31	35	c	c	NOUN
ap-1410	31	36	and	and	CCONJ
ap-1410	31	37	x	x	PROPN
ap-1410	31	38	,	,	PUNCT
ap-1410	31	39	y	y	PROPN
ap-1410	31	40	,	,	PUNCT
ap-1410	31	41	z	z	PROPN
ap-1410	31	42	∈	∈	PROPN
ap-1410	31	43	h.	h.	NOUN
ap-1410	32	1	moreover	moreover	ADV
ap-1410	32	2	(	(	PUNCT
ap-1410	32	3	x	x	NOUN
ap-1410	32	4	,	,	PUNCT
ap-1410	32	5	y	y	NOUN
ap-1410	32	6	)	)	PUNCT
ap-1410	32	7	=	=	SYM
ap-1410	32	8	(	(	PUNCT
ap-1410	32	9	y	y	PROPN
ap-1410	32	10	,	,	PUNCT
ap-1410	32	11	x	x	NOUN
ap-1410	32	12	)	)	PUNCT
ap-1410	32	13	and	and	CCONJ
ap-1410	32	14	(	(	PUNCT
ap-1410	32	15	x	x	X
ap-1410	32	16	,	,	PUNCT
ap-1410	32	17	x	x	X
ap-1410	32	18	)	)	PUNCT
ap-1410	32	19	≥	≥	NOUN
ap-1410	32	20	0	0	NUM
ap-1410	32	21	with	with	ADP
ap-1410	32	22	(	(	PUNCT
ap-1410	32	23	x	x	NOUN
ap-1410	32	24	,	,	PUNCT
ap-1410	32	25	x	x	X
ap-1410	32	26	)	)	PUNCT
ap-1410	32	27	=	=	SYM
ap-1410	32	28	0	0	NUM
ap-1410	32	29	iff	iff	NOUN
ap-1410	32	30	x	x	PROPN
ap-1410	32	31	=	=	NOUN
ap-1410	32	32	0	0	NUM
ap-1410	32	33	.	.	PUNCT
ap-1410	33	1	the	the	DET
ap-1410	33	2	term	term	NOUN
ap-1410	33	3	dimension	dimension	NOUN
ap-1410	33	4	of	of	ADP
ap-1410	33	5	h	h	NOUN
ap-1410	33	6	in	in	ADP
ap-1410	33	7	the	the	DET
ap-1410	33	8	following	following	NOUN
ap-1410	33	9	always	always	ADV
ap-1410	33	10	means	mean	VERB
ap-1410	33	11	the	the	DET
ap-1410	33	12	hilbertian	hilbertian	ADJ
ap-1410	33	13	dimension	dimension	NOUN
ap-1410	33	14	,	,	PUNCT
ap-1410	33	15	i.e.	i.e.	X
ap-1410	33	16	the	the	DET
ap-1410	33	17	cardinality	cardinality	NOUN
ap-1410	33	18	of	of	ADP
ap-1410	33	19	any	any	DET
ap-1410	33	20	orthonormal	orthonormal	ADJ
ap-1410	33	21	basis	basis	NOUN
ap-1410	33	22	of	of	ADP
ap-1410	33	23	h	h	PROPN
ap-1410	33	24	(	(	PUNCT
ap-1410	33	25	see	see	VERB
ap-1410	33	26	[	[	X
ap-1410	33	27	1	1	NUM
ap-1410	33	28	,	,	PUNCT
ap-1410	33	29	p.	p.	NOUN
ap-1410	33	30	44	44	NUM
ap-1410	33	31	]	]	PUNCT
ap-1410	33	32	)	)	PUNCT
ap-1410	33	33	.	.	PUNCT
ap-1410	34	1	for	for	ADP
ap-1410	34	2	notions	notion	NOUN
ap-1410	34	3	and	and	CCONJ
ap-1410	34	4	results	result	NOUN
ap-1410	34	5	on	on	ADP
ap-1410	34	6	hilbert	hilbert	NOUN
ap-1410	34	7	space	space	NOUN
ap-1410	34	8	operators	operator	NOUN
ap-1410	34	9	we	we	PRON
ap-1410	34	10	refer	refer	VERB
ap-1410	34	11	the	the	DET
ap-1410	34	12	reader	reader	NOUN
ap-1410	34	13	to	to	ADP
ap-1410	34	14	[	[	X
ap-1410	34	15	1	1	NUM
ap-1410	34	16	]	]	PUNCT
ap-1410	34	17	.	.	PUNCT
ap-1410	35	1	we	we	PRON
ap-1410	35	2	will	will	AUX
ap-1410	35	3	assume	assume	VERB
ap-1410	35	4	that	that	SCONJ
ap-1410	35	5	the	the	DET
ap-1410	35	6	domains	domain	NOUN
ap-1410	35	7	d(a	d(a	PROPN
ap-1410	35	8	)	)	PUNCT
ap-1410	35	9	of	of	ADP
ap-1410	35	10	all	all	PRON
ap-1410	35	11	considered	consider	VERB
ap-1410	35	12	linear	linear	PROPN
ap-1410	35	13	operators	operator	NOUN
ap-1410	35	14	a	a	PRON
ap-1410	35	15	are	be	AUX
ap-1410	35	16	dense	dense	ADJ
ap-1410	35	17	linear	linear	ADJ
ap-1410	35	18	subspaces	subspace	NOUN
ap-1410	35	19	of	of	ADP
ap-1410	35	20	h	h	NOUN
ap-1410	35	21	(	(	PUNCT
ap-1410	35	22	in	in	ADP
ap-1410	35	23	the	the	DET
ap-1410	35	24	metric	metric	ADJ
ap-1410	35	25	topology	topology	NOUN
ap-1410	35	26	induced	induce	VERB
ap-1410	35	27	by	by	ADP
ap-1410	35	28	inner	inner	ADJ
ap-1410	35	29	product	product	NOUN
ap-1410	35	30	)	)	PUNCT
ap-1410	35	31	.	.	PUNCT
ap-1410	36	1	we	we	PRON
ap-1410	36	2	say	say	VERB
ap-1410	36	3	that	that	SCONJ
ap-1410	36	4	operators	operator	NOUN
ap-1410	36	5	a	a	PRON
ap-1410	36	6	are	be	AUX
ap-1410	36	7	densely	densely	ADV
ap-1410	36	8	defined	define	VERB
ap-1410	36	9	on	on	ADP
ap-1410	36	10	h.	h.	PROPN
ap-1410	36	11	the	the	DET
ap-1410	36	12	set	set	NOUN
ap-1410	36	13	of	of	ADP
ap-1410	36	14	all	all	DET
ap-1410	36	15	densely	densely	ADV
ap-1410	36	16	defined	define	VERB
ap-1410	36	17	linear	linear	NOUN
ap-1410	36	18	operators	operator	NOUN
ap-1410	36	19	on	on	ADP
ap-1410	36	20	h	h	NOUN
ap-1410	36	21	will	will	AUX
ap-1410	36	22	be	be	AUX
ap-1410	36	23	denoted	denote	VERB
ap-1410	36	24	by	by	ADP
ap-1410	36	25	l(h	l(h	PROPN
ap-1410	36	26	)	)	PUNCT
ap-1410	36	27	.	.	PUNCT
ap-1410	37	1	73	73	NUM
ap-1410	37	2	acta	acta	PROPN
ap-1410	37	3	polytechnica	polytechnica	PROPN
ap-1410	37	4	vol	vol	NOUN
ap-1410	37	5	.	.	PUNCT
ap-1410	38	1	51	51	NUM
ap-1410	38	2	no	no	NOUN
ap-1410	38	3	.	.	PUNCT
ap-1410	39	1	4/2011	4/2011	NUM
ap-1410	39	2	recall	recall	NOUN
ap-1410	39	3	that	that	SCONJ
ap-1410	39	4	a	a	DET
ap-1410	39	5	:	:	PUNCT
ap-1410	39	6	d(a	d(a	PROPN
ap-1410	39	7	)	)	PUNCT
ap-1410	39	8	→	→	SYM
ap-1410	39	9	h	h	NOUN
ap-1410	39	10	is	be	AUX
ap-1410	39	11	a	a	DET
ap-1410	39	12	bounded	bounded	ADJ
ap-1410	39	13	operator	operator	NOUN
ap-1410	39	14	if	if	SCONJ
ap-1410	39	15	there	there	PRON
ap-1410	39	16	exists	exist	VERB
ap-1410	39	17	a	a	DET
ap-1410	39	18	real	real	ADJ
ap-1410	39	19	constant	constant	ADJ
ap-1410	39	20	c	c	NOUN
ap-1410	39	21	>	>	X
ap-1410	39	22	0	0	NUM
ap-1410	39	23	such	such	ADJ
ap-1410	39	24	that	that	SCONJ
ap-1410	39	25	‖ax‖	‖ax‖	ADJ
ap-1410	39	26	≤	≤	NOUN
ap-1410	39	27	c‖x‖	c‖x‖	NOUN
ap-1410	39	28	for	for	ADP
ap-1410	39	29	all	all	DET
ap-1410	39	30	x	x	SYM
ap-1410	39	31	∈	∈	PROPN
ap-1410	39	32	d(a	d(a	PROPN
ap-1410	39	33	)	)	PUNCT
ap-1410	39	34	.	.	PUNCT
ap-1410	40	1	if	if	SCONJ
ap-1410	40	2	a	a	PRON
ap-1410	40	3	is	be	AUX
ap-1410	40	4	not	not	PART
ap-1410	40	5	bounded	bound	VERB
ap-1410	40	6	then	then	ADV
ap-1410	40	7	it	it	PRON
ap-1410	40	8	is	be	AUX
ap-1410	40	9	called	call	VERB
ap-1410	40	10	unbounded	unbounded	ADJ
ap-1410	40	11	.	.	PUNCT
ap-1410	41	1	let	let	VERB
ap-1410	41	2	t	t	PROPN
ap-1410	41	3	∈	∈	PROPN
ap-1410	41	4	l(h	l(h	PROPN
ap-1410	41	5	)	)	PUNCT
ap-1410	41	6	.	.	PUNCT
ap-1410	42	1	since	since	SCONJ
ap-1410	42	2	d(t	d(t	PROPN
ap-1410	42	3	)	)	PUNCT
ap-1410	42	4	is	be	AUX
ap-1410	42	5	dense	dense	ADJ
ap-1410	42	6	in	in	ADP
ap-1410	42	7	h	h	NOUN
ap-1410	42	8	,	,	PUNCT
ap-1410	42	9	for	for	ADP
ap-1410	42	10	any	any	DET
ap-1410	42	11	y	y	PROPN
ap-1410	42	12	∈	∈	PROPN
ap-1410	42	13	h	h	NOUN
ap-1410	42	14	there	there	PRON
ap-1410	42	15	is	be	VERB
ap-1410	42	16	at	at	ADP
ap-1410	42	17	most	most	ADV
ap-1410	42	18	one	one	NUM
ap-1410	42	19	y∗	y∗	NOUN
ap-1410	42	20	∈	∈	NOUN
ap-1410	42	21	h	h	NOUN
ap-1410	42	22	satisfying	satisfy	VERB
ap-1410	42	23	(	(	PUNCT
ap-1410	42	24	y	y	NOUN
ap-1410	42	25	,	,	PUNCT
ap-1410	42	26	tx	tx	PROPN
ap-1410	42	27	)	)	PUNCT
ap-1410	42	28	=	=	SYM
ap-1410	43	1	(	(	PUNCT
ap-1410	43	2	y∗	y∗	ADV
ap-1410	43	3	,	,	PUNCT
ap-1410	43	4	x	x	NOUN
ap-1410	43	5	)	)	PUNCT
ap-1410	43	6	for	for	ADP
ap-1410	43	7	all	all	DET
ap-1410	43	8	x	x	SYM
ap-1410	43	9	∈	∈	PROPN
ap-1410	43	10	d(t	d(t	PROPN
ap-1410	43	11	)	)	PUNCT
ap-1410	43	12	.	.	PUNCT
ap-1410	44	1	this	this	PRON
ap-1410	44	2	allows	allow	VERB
ap-1410	44	3	us	we	PRON
ap-1410	44	4	to	to	PART
ap-1410	44	5	define	define	VERB
ap-1410	44	6	the	the	DET
ap-1410	44	7	adjoint	adjoint	PROPN
ap-1410	44	8	t	t	PROPN
ap-1410	44	9	∗	∗	NOUN
ap-1410	44	10	of	of	ADP
ap-1410	44	11	t	t	NOUN
ap-1410	44	12	by	by	ADP
ap-1410	44	13	putting	put	VERB
ap-1410	44	14	d(t	d(t	PROPN
ap-1410	44	15	∗	∗	NOUN
ap-1410	44	16	)	)	PUNCT
ap-1410	45	1	=	=	PRON
ap-1410	45	2	{	{	PUNCT
ap-1410	45	3	y	y	PROPN
ap-1410	45	4	∈	∈	PROPN
ap-1410	45	5	h	h	NOUN
ap-1410	46	1	|	|	ADV
ap-1410	46	2	there	there	PRON
ap-1410	46	3	exists	exist	VERB
ap-1410	46	4	y∗	y∗	PROPN
ap-1410	46	5	∈	∈	PROPN
ap-1410	46	6	h	h	NOUN
ap-1410	46	7	such	such	ADJ
ap-1410	46	8	that	that	SCONJ
ap-1410	46	9	(	(	PUNCT
ap-1410	46	10	y	y	NOUN
ap-1410	46	11	,	,	PUNCT
ap-1410	46	12	tx	tx	PROPN
ap-1410	46	13	)	)	PUNCT
ap-1410	46	14	=	=	SYM
ap-1410	46	15	(	(	PUNCT
ap-1410	46	16	y∗	y∗	ADV
ap-1410	46	17	,	,	PUNCT
ap-1410	46	18	x	x	NOUN
ap-1410	46	19	)	)	PUNCT
ap-1410	46	20	for	for	ADP
ap-1410	46	21	all	all	DET
ap-1410	46	22	x	x	SYM
ap-1410	46	23	∈	∈	PROPN
ap-1410	46	24	d(t	d(t	PROPN
ap-1410	46	25	)	)	PUNCT
ap-1410	46	26	}	}	PUNCT
ap-1410	46	27	and	and	CCONJ
ap-1410	46	28	t	t	X
ap-1410	46	29	∗y	∗y	PROPN
ap-1410	46	30	=	=	SYM
ap-1410	46	31	y∗.	y∗.	NUM
ap-1410	46	32	operator	operator	NOUN
ap-1410	46	33	t	t	PROPN
ap-1410	46	34	is	be	AUX
ap-1410	46	35	said	say	VERB
ap-1410	46	36	to	to	PART
ap-1410	46	37	be	be	AUX
ap-1410	46	38	self	self	NOUN
ap-1410	46	39	-	-	PUNCT
ap-1410	46	40	adjoint	adjoint	NOUN
ap-1410	46	41	if	if	SCONJ
ap-1410	46	42	t	t	PROPN
ap-1410	46	43	=	=	SYM
ap-1410	46	44	t	t	PROPN
ap-1410	46	45	∗.	∗.	PROPN
ap-1410	46	46	an	an	DET
ap-1410	46	47	operator	operator	NOUN
ap-1410	46	48	t	t	PROPN
ap-1410	46	49	∈	∈	PROPN
ap-1410	46	50	l(h	l(h	PROPN
ap-1410	46	51	)	)	PUNCT
ap-1410	46	52	is	be	AUX
ap-1410	46	53	called	call	VERB
ap-1410	46	54	symmetric	symmetric	ADJ
ap-1410	46	55	,	,	PUNCT
ap-1410	46	56	if	if	SCONJ
ap-1410	46	57	t	t	PROPN
ap-1410	46	58	(	(	PUNCT
ap-1410	46	59	x	x	PROPN
ap-1410	46	60	,	,	PUNCT
ap-1410	46	61	y	y	NOUN
ap-1410	46	62	)	)	PUNCT
ap-1410	46	63	=	=	SYM
ap-1410	47	1	(	(	PUNCT
ap-1410	47	2	x	x	PROPN
ap-1410	47	3	,	,	PUNCT
ap-1410	47	4	t	t	PROPN
ap-1410	47	5	y	y	PROPN
ap-1410	47	6	)	)	PUNCT
ap-1410	47	7	for	for	ADP
ap-1410	47	8	all	all	DET
ap-1410	47	9	x	x	NOUN
ap-1410	47	10	,	,	PUNCT
ap-1410	47	11	y	y	PROPN
ap-1410	47	12	∈	∈	PROPN
ap-1410	47	13	d(a	d(a	PROPN
ap-1410	47	14	)	)	PUNCT
ap-1410	47	15	.	.	PUNCT
ap-1410	48	1	it	it	PRON
ap-1410	48	2	is	be	AUX
ap-1410	48	3	well	well	ADV
ap-1410	48	4	-	-	PUNCT
ap-1410	48	5	known	know	VERB
ap-1410	48	6	that	that	SCONJ
ap-1410	48	7	this	this	PRON
ap-1410	48	8	is	be	AUX
ap-1410	48	9	equivalent	equivalent	ADJ
ap-1410	48	10	with	with	ADP
ap-1410	48	11	(	(	PUNCT
ap-1410	48	12	tx	tx	INTJ
ap-1410	48	13	,	,	PUNCT
ap-1410	48	14	x	x	NOUN
ap-1410	48	15	)	)	PUNCT
ap-1410	48	16	∈	∈	NOUN
ap-1410	48	17	r	r	NOUN
ap-1410	48	18	for	for	ADP
ap-1410	48	19	all	all	DET
ap-1410	48	20	x	x	SYM
ap-1410	48	21	∈	∈	PROPN
ap-1410	48	22	d(t	d(t	PROPN
ap-1410	48	23	)	)	PUNCT
ap-1410	48	24	.	.	PUNCT
ap-1410	49	1	clearly	clearly	ADV
ap-1410	49	2	every	every	DET
ap-1410	49	3	self	self	NOUN
ap-1410	49	4	-	-	PUNCT
ap-1410	49	5	adjoint	adjoint	NOUN
ap-1410	49	6	operator	operator	NOUN
ap-1410	49	7	is	be	AUX
ap-1410	49	8	symmetric	symmetric	ADJ
ap-1410	49	9	but	but	CCONJ
ap-1410	49	10	the	the	DET
ap-1410	49	11	converse	converse	NOUN
ap-1410	49	12	need	need	AUX
ap-1410	49	13	not	not	PART
ap-1410	49	14	hold	hold	VERB
ap-1410	49	15	for	for	ADP
ap-1410	49	16	unbounded	unbounded	ADJ
ap-1410	49	17	operators	operator	NOUN
ap-1410	49	18	(	(	PUNCT
ap-1410	49	19	see	see	VERB
ap-1410	49	20	[	[	X
ap-1410	49	21	1	1	NUM
ap-1410	49	22	]	]	PUNCT
ap-1410	49	23	,	,	PUNCT
ap-1410	49	24	p.	p.	NOUN
ap-1410	49	25	98	98	NUM
ap-1410	49	26	)	)	PUNCT
ap-1410	49	27	.	.	PUNCT
ap-1410	50	1	since	since	SCONJ
ap-1410	50	2	every	every	DET
ap-1410	50	3	(	(	PUNCT
ap-1410	50	4	generalized	generalized	ADJ
ap-1410	50	5	)	)	PUNCT
ap-1410	50	6	effect	effect	NOUN
ap-1410	50	7	algebra	algebra	NOUN
ap-1410	50	8	includes	include	VERB
ap-1410	50	9	the	the	DET
ap-1410	50	10	zero	zero	NUM
ap-1410	50	11	element	element	NOUN
ap-1410	50	12	0	0	PUNCT
ap-1410	50	13	as	as	ADP
ap-1410	50	14	the	the	DET
ap-1410	50	15	least	least	ADJ
ap-1410	50	16	element	element	NOUN
ap-1410	50	17	of	of	ADP
ap-1410	50	18	e	e	NOUN
ap-1410	50	19	,	,	PUNCT
ap-1410	50	20	we	we	PRON
ap-1410	50	21	will	will	AUX
ap-1410	50	22	assume	assume	VERB
ap-1410	50	23	that	that	SCONJ
ap-1410	50	24	all	all	PRON
ap-1410	50	25	considered	consider	VERB
ap-1410	50	26	operators	operator	NOUN
ap-1410	50	27	are	be	AUX
ap-1410	50	28	positive	positive	ADJ
ap-1410	50	29	(	(	PUNCT
ap-1410	50	30	written	write	VERB
ap-1410	50	31	a	a	DET
ap-1410	50	32	≥	≥	NOUN
ap-1410	50	33	0	0	NUM
ap-1410	50	34	)	)	PUNCT
ap-1410	50	35	.	.	PUNCT
ap-1410	51	1	this	this	PRON
ap-1410	51	2	means	mean	VERB
ap-1410	51	3	that	that	SCONJ
ap-1410	51	4	(	(	PUNCT
ap-1410	51	5	ax	ax	NOUN
ap-1410	51	6	,	,	PUNCT
ap-1410	51	7	x	x	NOUN
ap-1410	51	8	)	)	PUNCT
ap-1410	51	9	≥	≥	X
ap-1410	51	10	0	0	NUM
ap-1410	51	11	for	for	ADP
ap-1410	51	12	all	all	DET
ap-1410	51	13	x	x	SYM
ap-1410	51	14	∈	∈	PROPN
ap-1410	51	15	d(a	d(a	PROPN
ap-1410	51	16	)	)	PUNCT
ap-1410	51	17	and	and	CCONJ
ap-1410	51	18	hence	hence	ADV
ap-1410	51	19	a	a	PRON
ap-1410	51	20	is	be	AUX
ap-1410	51	21	also	also	ADV
ap-1410	51	22	symmetric	symmetric	ADJ
ap-1410	51	23	,	,	PUNCT
ap-1410	51	24	i.e.	i.e.	X
ap-1410	51	25	(	(	PUNCT
ap-1410	51	26	ax	ax	NOUN
ap-1410	51	27	,	,	PUNCT
ap-1410	51	28	y	y	NOUN
ap-1410	51	29	)	)	PUNCT
ap-1410	51	30	=	=	SYM
ap-1410	51	31	(	(	PUNCT
ap-1410	51	32	x	x	NOUN
ap-1410	51	33	,	,	PUNCT
ap-1410	51	34	ay	ay	NOUN
ap-1410	51	35	)	)	PUNCT
ap-1410	51	36	for	for	ADP
ap-1410	51	37	all	all	DET
ap-1410	51	38	x	x	NOUN
ap-1410	51	39	,	,	PUNCT
ap-1410	51	40	y	y	PROPN
ap-1410	51	41	∈	∈	PROPN
ap-1410	51	42	d(a	d(a	PROPN
ap-1410	51	43	)	)	PUNCT
ap-1410	51	44	(	(	PUNCT
ap-1410	51	45	see	see	VERB
ap-1410	51	46	[	[	X
ap-1410	51	47	1	1	NUM
ap-1410	51	48	,	,	PUNCT
ap-1410	51	49	pp	pp	ADJ
ap-1410	51	50	.	.	PUNCT
ap-1410	52	1	68	68	NUM
ap-1410	52	2	and	and	CCONJ
ap-1410	52	3	94	94	NUM
ap-1410	52	4	]	]	PUNCT
ap-1410	52	5	)	)	PUNCT
ap-1410	52	6	.	.	PUNCT
ap-1410	53	1	for	for	ADP
ap-1410	53	2	two	two	NUM
ap-1410	53	3	operators	operator	NOUN
ap-1410	53	4	a	a	DET
ap-1410	53	5	:	:	PUNCT
ap-1410	53	6	d(a	d(a	PROPN
ap-1410	53	7	)	)	PUNCT
ap-1410	53	8	→	→	SYM
ap-1410	53	9	h	h	NOUN
ap-1410	53	10	and	and	CCONJ
ap-1410	53	11	b	b	NOUN
ap-1410	53	12	:	:	PUNCT
ap-1410	53	13	d(b	d(b	X
ap-1410	53	14	)	)	PUNCT
ap-1410	53	15	→	→	SYM
ap-1410	54	1	h	h	NOUN
ap-1410	54	2	we	we	PRON
ap-1410	54	3	write	write	VERB
ap-1410	54	4	a	a	DET
ap-1410	54	5	⊂	⊂	PROPN
ap-1410	54	6	b	b	PROPN
ap-1410	54	7	iff	iff	PROPN
ap-1410	54	8	d(a	d(a	PROPN
ap-1410	54	9	)	)	PUNCT
ap-1410	54	10	⊂	⊂	PROPN
ap-1410	54	11	d(b	d(b	X
ap-1410	54	12	)	)	PUNCT
ap-1410	54	13	and	and	CCONJ
ap-1410	54	14	ax	ax	NOUN
ap-1410	54	15	=	=	PUNCT
ap-1410	54	16	bx	bx	PROPN
ap-1410	54	17	for	for	ADP
ap-1410	54	18	all	all	DET
ap-1410	54	19	x	x	PROPN
ap-1410	54	20	∈	∈	PROPN
ap-1410	54	21	d(a	d(a	PROPN
ap-1410	54	22	)	)	PUNCT
ap-1410	54	23	.	.	PUNCT
ap-1410	55	1	then	then	ADV
ap-1410	55	2	b	b	PROPN
ap-1410	55	3	is	be	AUX
ap-1410	55	4	called	call	VERB
ap-1410	55	5	an	an	DET
ap-1410	55	6	extension	extension	NOUN
ap-1410	55	7	of	of	ADP
ap-1410	55	8	a.	a.	NOUN
ap-1410	55	9	we	we	PRON
ap-1410	55	10	show	show	VERB
ap-1410	55	11	some	some	DET
ap-1410	55	12	examples	example	NOUN
ap-1410	55	13	of	of	ADP
ap-1410	55	14	partial	partial	ADJ
ap-1410	55	15	binary	binary	ADJ
ap-1410	55	16	operations	operation	NOUN
ap-1410	55	17	(	(	PUNCT
ap-1410	55	18	sums	sum	NOUN
ap-1410	55	19	)	)	PUNCT
ap-1410	55	20	on	on	ADP
ap-1410	55	21	the	the	DET
ap-1410	55	22	set	set	NOUN
ap-1410	55	23	v(h	v(h	NOUN
ap-1410	55	24	)	)	PUNCT
ap-1410	55	25	of	of	ADP
ap-1410	55	26	all	all	DET
ap-1410	55	27	positive	positive	ADJ
ap-1410	55	28	linear	linear	PROPN
ap-1410	55	29	operators	operator	NOUN
ap-1410	55	30	densely	densely	ADV
ap-1410	55	31	defined	define	VERB
ap-1410	55	32	on	on	ADP
ap-1410	55	33	infinite	infinite	ADJ
ap-1410	55	34	-	-	PUNCT
ap-1410	55	35	dimensional	dimensional	ADJ
ap-1410	55	36	hilbert	hilbert	NOUN
ap-1410	55	37	space	space	NOUN
ap-1410	55	38	h.	h.	PROPN
ap-1410	55	39	our	our	PRON
ap-1410	55	40	main	main	ADJ
ap-1410	55	41	goal	goal	NOUN
ap-1410	55	42	is	be	AUX
ap-1410	55	43	to	to	PART
ap-1410	55	44	study	study	VERB
ap-1410	55	45	the	the	DET
ap-1410	55	46	properties	property	NOUN
ap-1410	55	47	of	of	ADP
ap-1410	55	48	sub	sub	ADJ
ap-1410	55	49	-	-	ADJ
ap-1410	55	50	generalized	generalized	ADJ
ap-1410	55	51	effect	effect	NOUN
ap-1410	55	52	algebras	algebra	NOUN
ap-1410	55	53	and	and	CCONJ
ap-1410	55	54	effect	effect	NOUN
ap-1410	55	55	algebras	algebra	NOUN
ap-1410	55	56	being	be	AUX
ap-1410	55	57	intervals	interval	NOUN
ap-1410	55	58	in	in	ADP
ap-1410	55	59	v(h	v(h	NOUN
ap-1410	55	60	)	)	PUNCT
ap-1410	55	61	if	if	SCONJ
ap-1410	55	62	v(h	v(h	NOUN
ap-1410	55	63	)	)	PUNCT
ap-1410	55	64	is	be	AUX
ap-1410	55	65	equipped	equip	VERB
ap-1410	55	66	with	with	ADP
ap-1410	55	67	two	two	NUM
ap-1410	55	68	different	different	ADJ
ap-1410	55	69	partial	partial	ADJ
ap-1410	55	70	sums	sum	NOUN
ap-1410	55	71	such	such	ADJ
ap-1410	55	72	that	that	SCONJ
ap-1410	55	73	one	one	NUM
ap-1410	55	74	of	of	ADP
ap-1410	55	75	them	they	PRON
ap-1410	55	76	is	be	AUX
ap-1410	55	77	an	an	DET
ap-1410	55	78	extension	extension	NOUN
ap-1410	55	79	of	of	ADP
ap-1410	55	80	the	the	DET
ap-1410	55	81	other	other	ADJ
ap-1410	55	82	.	.	PUNCT
ap-1410	56	1	2	2	NUM
ap-1410	56	2	some	some	DET
ap-1410	56	3	properties	property	NOUN
ap-1410	56	4	of	of	ADP
ap-1410	56	5	unbounded	unbounded	ADJ
ap-1410	56	6	operators	operator	NOUN
ap-1410	56	7	in	in	ADP
ap-1410	56	8	complex	complex	ADJ
ap-1410	56	9	hilbert	hilbert	NOUN
ap-1410	56	10	spaces	space	NOUN
ap-1410	56	11	before	before	ADP
ap-1410	56	12	defining	define	VERB
ap-1410	56	13	(	(	PUNCT
ap-1410	56	14	generalized	generalized	ADJ
ap-1410	56	15	)	)	PUNCT
ap-1410	56	16	effect	effect	NOUN
ap-1410	56	17	algebras	algebra	NOUN
ap-1410	56	18	consisting	consist	VERB
ap-1410	56	19	of	of	ADP
ap-1410	56	20	operators	operator	NOUN
ap-1410	56	21	a	a	DET
ap-1410	56	22	∈	∈	NOUN
ap-1410	56	23	l(h	l(h	PROPN
ap-1410	56	24	)	)	PUNCT
ap-1410	56	25	,	,	PUNCT
ap-1410	56	26	we	we	PRON
ap-1410	56	27	review	review	VERB
ap-1410	56	28	some	some	DET
ap-1410	56	29	necessary	necessary	ADJ
ap-1410	56	30	results	result	NOUN
ap-1410	56	31	on	on	ADP
ap-1410	56	32	hilbert	hilbert	NOUN
ap-1410	56	33	space	space	NOUN
ap-1410	56	34	operators	operator	NOUN
ap-1410	56	35	[	[	X
ap-1410	56	36	1	1	NUM
ap-1410	56	37	,	,	PUNCT
ap-1410	56	38	chapter	chapter	NOUN
ap-1410	56	39	4	4	NUM
ap-1410	56	40	]	]	PUNCT
ap-1410	56	41	.	.	PUNCT
ap-1410	57	1	theorem	theorem	ADJ
ap-1410	57	2	3	3	NUM
ap-1410	57	3	let	let	VERB
ap-1410	57	4	d1	d1	PROPN
ap-1410	57	5	⊂	⊂	PROPN
ap-1410	57	6	d2	d2	PROPN
ap-1410	57	7	⊂	⊂	PROPN
ap-1410	57	8	h	h	PROPN
ap-1410	57	9	be	be	AUX
ap-1410	57	10	linear	linear	ADJ
ap-1410	57	11	subspaces	subspace	NOUN
ap-1410	57	12	,	,	PUNCT
ap-1410	58	1	d1	d1	PROPN
ap-1410	58	2	=	=	SYM
ap-1410	58	3	d2	d2	PROPN
ap-1410	58	4	=	=	SYM
ap-1410	58	5	h.	h.	PROPN
ap-1410	58	6	let	let	VERB
ap-1410	58	7	a	a	DET
ap-1410	58	8	∈	∈	NOUN
ap-1410	58	9	l(h	l(h	PROPN
ap-1410	58	10	)	)	PUNCT
ap-1410	58	11	,	,	PUNCT
ap-1410	58	12	d(a	d(a	PROPN
ap-1410	58	13	)	)	PUNCT
ap-1410	58	14	=	=	SYM
ap-1410	58	15	d2	d2	PROPN
ap-1410	58	16	and	and	CCONJ
ap-1410	58	17	its	its	PRON
ap-1410	58	18	restriction	restriction	NOUN
ap-1410	58	19	a1	a1	NOUN
ap-1410	58	20	=	=	PUNCT
ap-1410	58	21	a|d1	a|d1	NOUN
ap-1410	58	22	=	=	SYM
ap-1410	58	23	0	0	NUM
ap-1410	58	24	and	and	CCONJ
ap-1410	58	25	(	(	PUNCT
ap-1410	58	26	ax	ax	NOUN
ap-1410	58	27	,	,	PUNCT
ap-1410	58	28	x	x	NOUN
ap-1410	58	29	)	)	PUNCT
ap-1410	58	30	∈	∈	NOUN
ap-1410	58	31	r	r	NOUN
ap-1410	58	32	for	for	ADP
ap-1410	58	33	all	all	DET
ap-1410	58	34	x	x	SYM
ap-1410	58	35	∈	∈	PROPN
ap-1410	58	36	d2	d2	PROPN
ap-1410	58	37	.	.	PUNCT
ap-1410	59	1	then	then	ADV
ap-1410	59	2	a	a	DET
ap-1410	59	3	=	=	NOUN
ap-1410	59	4	0	0	NUM
ap-1410	59	5	.	.	PUNCT
ap-1410	60	1	proof	proof	NOUN
ap-1410	60	2	.	.	PUNCT
ap-1410	61	1	if	if	SCONJ
ap-1410	61	2	a	a	DET
ap-1410	61	3	�	�	PROPN
ap-1410	61	4	=	=	SYM
ap-1410	61	5	0	0	NUM
ap-1410	61	6	,	,	PUNCT
ap-1410	61	7	then	then	ADV
ap-1410	61	8	∃e	∃e	NUM
ap-1410	61	9	∈	∈	PROPN
ap-1410	61	10	d2	d2	NOUN
ap-1410	61	11	\	\	PROPN
ap-1410	61	12	d1	d1	PROPN
ap-1410	61	13	for	for	ADP
ap-1410	61	14	which	which	PRON
ap-1410	61	15	ae	ae	PROPN
ap-1410	61	16	�	�	PROPN
ap-1410	61	17	=	=	PROPN
ap-1410	61	18	0	0	PROPN
ap-1410	61	19	.	.	PUNCT
ap-1410	62	1	for	for	ADP
ap-1410	62	2	all	all	DET
ap-1410	62	3	d	d	PROPN
ap-1410	62	4	∈	∈	NOUN
ap-1410	62	5	d1	d1	NOUN
ap-1410	62	6	and	and	CCONJ
ap-1410	62	7	all	all	DET
ap-1410	62	8	λ	λ	X
ap-1410	62	9	∈	∈	NOUN
ap-1410	62	10	c	c	NOUN
ap-1410	62	11	ad	ad	NOUN
ap-1410	62	12	=	=	SYM
ap-1410	62	13	0	0	NUM
ap-1410	62	14	,	,	PUNCT
ap-1410	62	15	d	d	NOUN
ap-1410	62	16	+	+	CCONJ
ap-1410	62	17	λe	λe	ADP
ap-1410	62	18	∈	∈	PROPN
ap-1410	62	19	d2	d2	PROPN
ap-1410	62	20	,	,	PUNCT
ap-1410	62	21	(	(	PUNCT
ap-1410	62	22	1	1	X
ap-1410	62	23	)	)	PUNCT
ap-1410	62	24	(	(	PUNCT
ap-1410	62	25	a(d	a(d	PROPN
ap-1410	62	26	+	+	PROPN
ap-1410	62	27	λe	λe	PROPN
ap-1410	62	28	)	)	PUNCT
ap-1410	62	29	,	,	PUNCT
ap-1410	62	30	d	d	PROPN
ap-1410	62	31	+	+	CCONJ
ap-1410	62	32	λe	λe	ADP
ap-1410	62	33	)	)	PUNCT
ap-1410	62	34	=	=	SYM
ap-1410	63	1	λ(ae	λ(ae	NOUN
ap-1410	63	2	,	,	PUNCT
ap-1410	63	3	d	d	NOUN
ap-1410	63	4	)	)	PUNCT
ap-1410	63	5	+	+	NUM
ap-1410	63	6	|λ|2(ae	|λ|2(ae	NOUN
ap-1410	63	7	,	,	PUNCT
ap-1410	63	8	e	e	X
ap-1410	63	9	)	)	PUNCT
ap-1410	63	10	∈	∈	PROPN
ap-1410	63	11	r	r	NOUN
ap-1410	63	12	.	.	PUNCT
ap-1410	64	1	since	since	SCONJ
ap-1410	64	2	d⊥	d⊥	NOUN
ap-1410	64	3	1	1	NUM
ap-1410	64	4	=	=	SYM
ap-1410	64	5	{	{	PUNCT
ap-1410	64	6	0	0	NUM
ap-1410	64	7	}	}	PUNCT
ap-1410	64	8	and	and	CCONJ
ap-1410	64	9	ae	ae	PROPN
ap-1410	64	10	�	�	PROPN
ap-1410	64	11	=	=	PROPN
ap-1410	64	12	0	0	PROPN
ap-1410	64	13	,	,	PUNCT
ap-1410	64	14	we	we	PRON
ap-1410	64	15	can	can	AUX
ap-1410	64	16	choose	choose	VERB
ap-1410	64	17	d1	d1	PROPN
ap-1410	64	18	∈	∈	PROPN
ap-1410	64	19	d1	d1	PROPN
ap-1410	64	20	for	for	ADP
ap-1410	64	21	which	which	PRON
ap-1410	64	22	(	(	PUNCT
ap-1410	64	23	ae	ae	PROPN
ap-1410	64	24	,	,	PUNCT
ap-1410	64	25	d1	d1	PROPN
ap-1410	64	26	)	)	PUNCT
ap-1410	64	27	�	�	PROPN
ap-1410	64	28	=	=	SYM
ap-1410	64	29	0	0	NUM
ap-1410	64	30	.	.	PUNCT
ap-1410	65	1	then	then	ADV
ap-1410	65	2	,	,	PUNCT
ap-1410	65	3	by	by	ADP
ap-1410	65	4	(	(	PUNCT
ap-1410	65	5	1	1	NUM
ap-1410	65	6	)	)	PUNCT
ap-1410	65	7	,	,	PUNCT
ap-1410	65	8	∀λ	∀λ	X
ap-1410	65	9	∈	∈	PROPN
ap-1410	65	10	c	c	PROPN
ap-1410	65	11	,	,	PUNCT
ap-1410	65	12	λ(ae	λ(ae	PROPN
ap-1410	65	13	,	,	PUNCT
ap-1410	65	14	d1	d1	NOUN
ap-1410	65	15	)	)	PUNCT
ap-1410	65	16	+	+	NUM
ap-1410	65	17	|λ|2(ae	|λ|2(ae	NOUN
ap-1410	65	18	,	,	PUNCT
ap-1410	65	19	e	e	X
ap-1410	65	20	)	)	PUNCT
ap-1410	65	21	∈	∈	PROPN
ap-1410	65	22	r	r	NOUN
ap-1410	65	23	.	.	PUNCT
ap-1410	66	1	since	since	SCONJ
ap-1410	66	2	the	the	DET
ap-1410	66	3	second	second	ADJ
ap-1410	66	4	summand	summand	NOUN
ap-1410	66	5	is	be	AUX
ap-1410	66	6	real	real	ADJ
ap-1410	66	7	,	,	PUNCT
ap-1410	66	8	the	the	DET
ap-1410	66	9	first	first	ADJ
ap-1410	66	10	one	one	NUM
ap-1410	66	11	must	must	AUX
ap-1410	66	12	be	be	AUX
ap-1410	66	13	real	real	ADJ
ap-1410	66	14	for	for	ADP
ap-1410	66	15	all	all	DET
ap-1410	66	16	λ	λ	PROPN
ap-1410	66	17	∈	∈	PROPN
ap-1410	66	18	c.	c.	NOUN
ap-1410	66	19	however	however	ADV
ap-1410	66	20	,	,	PUNCT
ap-1410	66	21	this	this	PRON
ap-1410	66	22	is	be	AUX
ap-1410	66	23	possible	possible	ADJ
ap-1410	66	24	only	only	ADV
ap-1410	66	25	if	if	SCONJ
ap-1410	66	26	(	(	PUNCT
ap-1410	66	27	ae	ae	PROPN
ap-1410	66	28	,	,	PUNCT
ap-1410	66	29	d1	d1	PROPN
ap-1410	66	30	)	)	PUNCT
ap-1410	66	31	=	=	SYM
ap-1410	66	32	0	0	NUM
ap-1410	66	33	,	,	PUNCT
ap-1410	66	34	a	a	DET
ap-1410	66	35	contradiction	contradiction	NOUN
ap-1410	66	36	.	.	PUNCT
ap-1410	67	1	corollary	corollary	ADJ
ap-1410	67	2	4	4	NUM
ap-1410	67	3	if	if	SCONJ
ap-1410	67	4	a	a	DET
ap-1410	67	5	∈	∈	NOUN
ap-1410	67	6	l(h	l(h	PROPN
ap-1410	67	7	)	)	PUNCT
ap-1410	67	8	,	,	PUNCT
ap-1410	68	1	d	d	PROPN
ap-1410	68	2	=	=	SYM
ap-1410	68	3	d(a	d(a	PROPN
ap-1410	68	4	)	)	PUNCT
ap-1410	68	5	�	�	PROPN
ap-1410	68	6	=	=	PROPN
ap-1410	68	7	h.	h.	PROPN
ap-1410	68	8	is	be	AUX
ap-1410	68	9	a	a	DET
ap-1410	68	10	symmetric	symmetric	ADJ
ap-1410	68	11	bounded	bound	VERB
ap-1410	68	12	operator	operator	NOUN
ap-1410	68	13	and	and	CCONJ
ap-1410	68	14	b	b	NOUN
ap-1410	68	15	∈	∈	PROPN
ap-1410	68	16	l(h	l(h	PROPN
ap-1410	68	17	)	)	PUNCT
ap-1410	68	18	is	be	AUX
ap-1410	68	19	its	its	PRON
ap-1410	68	20	symmetric	symmetric	ADJ
ap-1410	68	21	extension	extension	NOUN
ap-1410	68	22	,	,	PUNCT
ap-1410	68	23	then	then	ADV
ap-1410	68	24	b	b	PROPN
ap-1410	68	25	is	be	AUX
ap-1410	68	26	also	also	ADV
ap-1410	68	27	bounded	bound	VERB
ap-1410	68	28	.	.	PUNCT
ap-1410	69	1	proof	proof	NOUN
ap-1410	69	2	.	.	PUNCT
ap-1410	70	1	let	let	VERB
ap-1410	70	2	b	b	X
ap-1410	70	3	be	be	AUX
ap-1410	70	4	a	a	DET
ap-1410	70	5	proper	proper	ADJ
ap-1410	70	6	symmetric	symmetric	ADJ
ap-1410	70	7	extension	extension	NOUN
ap-1410	70	8	of	of	ADP
ap-1410	70	9	a	a	PRON
ap-1410	70	10	and	and	CCONJ
ap-1410	70	11	let	let	VERB
ap-1410	70	12	ã	ã	PROPN
ap-1410	70	13	be	be	AUX
ap-1410	70	14	the	the	DET
ap-1410	70	15	unique	unique	ADJ
ap-1410	70	16	bounded	bounded	ADJ
ap-1410	70	17	extension	extension	NOUN
ap-1410	70	18	of	of	ADP
ap-1410	70	19	a.	a.	NOUN
ap-1410	70	20	then	then	ADV
ap-1410	70	21	(	(	PUNCT
ap-1410	70	22	b	b	X
ap-1410	70	23	−	−	PROPN
ap-1410	70	24	ã)|d	ã)|d	PROPN
ap-1410	70	25	=	=	PUNCT
ap-1410	70	26	0	0	PROPN
ap-1410	71	1	and	and	CCONJ
ap-1410	71	2	,	,	PUNCT
ap-1410	71	3	by	by	ADP
ap-1410	71	4	theorem	theorem	NOUN
ap-1410	71	5	3	3	NUM
ap-1410	71	6	,	,	PUNCT
ap-1410	71	7	(	(	PUNCT
ap-1410	71	8	b−ã)|d(b	b−ã)|d(b	PROPN
ap-1410	71	9	)	)	PUNCT
ap-1410	71	10	=	=	SYM
ap-1410	71	11	0	0	X
ap-1410	71	12	.	.	PUNCT
ap-1410	72	1	it	it	PRON
ap-1410	72	2	follows	follow	VERB
ap-1410	72	3	that	that	PRON
ap-1410	72	4	b	b	PROPN
ap-1410	72	5	=	=	SYM
ap-1410	72	6	b−ã|d(b)+	b−ã|d(b)+	X
ap-1410	72	7	ã|d(b	ã|d(b	X
ap-1410	72	8	)	)	PUNCT
ap-1410	72	9	=	=	PUNCT
ap-1410	72	10	ã|d(b	ã|d(b	X
ap-1410	72	11	)	)	PUNCT
ap-1410	72	12	is	be	AUX
ap-1410	72	13	a	a	DET
ap-1410	72	14	bounded	bounded	ADJ
ap-1410	72	15	linear	linear	ADJ
ap-1410	72	16	operator	operator	NOUN
ap-1410	72	17	.	.	PUNCT
ap-1410	73	1	theorem	theorem	NOUN
ap-1410	73	2	5	5	NUM
ap-1410	73	3	let	let	VERB
ap-1410	73	4	a	a	DET
ap-1410	73	5	∈	∈	NOUN
ap-1410	73	6	l(h	l(h	PROPN
ap-1410	73	7	)	)	PUNCT
ap-1410	73	8	and	and	CCONJ
ap-1410	73	9	there	there	PRON
ap-1410	73	10	exists	exist	VERB
ap-1410	73	11	k	k	PROPN
ap-1410	73	12	>	>	X
ap-1410	73	13	0	0	NUM
ap-1410	74	1	such	such	ADJ
ap-1410	74	2	that	that	SCONJ
ap-1410	74	3	∀x	∀x	VERB
ap-1410	74	4	∈	∈	PROPN
ap-1410	74	5	d(a	d(a	PROPN
ap-1410	74	6	)	)	PUNCT
ap-1410	74	7	|(ax	|(ax	NUM
ap-1410	74	8	,	,	PUNCT
ap-1410	74	9	x)|	x)|	PROPN
ap-1410	74	10	≤	≤	PUNCT
ap-1410	74	11	k‖x‖2	k‖x‖2	X
ap-1410	74	12	then	then	ADV
ap-1410	74	13	a	a	PRON
ap-1410	74	14	is	be	AUX
ap-1410	74	15	bounded	bound	VERB
ap-1410	74	16	.	.	PUNCT
ap-1410	75	1	proof	proof	NOUN
ap-1410	75	2	.	.	PUNCT
ap-1410	76	1	let	let	VERB
ap-1410	76	2	there	there	PRON
ap-1410	76	3	exist	exist	VERB
ap-1410	76	4	k	k	PROPN
ap-1410	76	5	>	>	X
ap-1410	76	6	0	0	NUM
ap-1410	76	7	such	such	ADJ
ap-1410	76	8	that	that	SCONJ
ap-1410	76	9	∀x	∀x	VERB
ap-1410	76	10	∈	∈	PROPN
ap-1410	76	11	d(a	d(a	PROPN
ap-1410	76	12	)	)	PUNCT
ap-1410	76	13	|(ax	|(ax	NUM
ap-1410	76	14	,	,	PUNCT
ap-1410	76	15	x)|	x)|	PROPN
ap-1410	76	16	≤	≤	PUNCT
ap-1410	76	17	k‖x‖2	k‖x‖2	PROPN
ap-1410	76	18	using	use	VERB
ap-1410	76	19	the	the	DET
ap-1410	76	20	polarization	polarization	NOUN
ap-1410	76	21	identity	identity	NOUN
ap-1410	76	22	:	:	PUNCT
ap-1410	76	23	(	(	PUNCT
ap-1410	76	24	ax	ax	NOUN
ap-1410	76	25	,	,	PUNCT
ap-1410	76	26	y	y	PROPN
ap-1410	76	27	)	)	PUNCT
ap-1410	76	28	=	=	SYM
ap-1410	76	29	1	1	NUM
ap-1410	76	30	4	4	NUM
ap-1410	76	31	{	{	PUNCT
ap-1410	76	32	(	(	PUNCT
ap-1410	76	33	a(x	a(x	PROPN
ap-1410	76	34	+	+	PROPN
ap-1410	76	35	y	y	NOUN
ap-1410	76	36	)	)	PUNCT
ap-1410	76	37	,	,	PUNCT
ap-1410	76	38	x	x	X
ap-1410	77	1	+	+	NUM
ap-1410	77	2	y	y	NOUN
ap-1410	77	3	)	)	PUNCT
ap-1410	77	4	−	−	PROPN
ap-1410	77	5	(	(	PUNCT
ap-1410	77	6	a(x	a(x	PROPN
ap-1410	77	7	−	−	PROPN
ap-1410	77	8	y	y	PROPN
ap-1410	77	9	)	)	PUNCT
ap-1410	77	10	,	,	PUNCT
ap-1410	77	11	x	x	PUNCT
ap-1410	77	12	−	−	PROPN
ap-1410	77	13	y	y	PROPN
ap-1410	77	14	)	)	PUNCT
ap-1410	78	1	+	+	NUM
ap-1410	78	2	i[(a(ix	i[(a(ix	NOUN
ap-1410	78	3	+	+	CCONJ
ap-1410	78	4	y	y	NOUN
ap-1410	78	5	)	)	PUNCT
ap-1410	78	6	,	,	PUNCT
ap-1410	78	7	ix	ix	PROPN
ap-1410	79	1	+	+	CCONJ
ap-1410	79	2	y	y	NOUN
ap-1410	79	3	)	)	PUNCT
ap-1410	79	4	−	−	PROPN
ap-1410	79	5	(	(	PUNCT
ap-1410	79	6	a(ix	a(ix	NOUN
ap-1410	79	7	−	−	PROPN
ap-1410	79	8	y	y	PROPN
ap-1410	79	9	)	)	PUNCT
ap-1410	79	10	,	,	PUNCT
ap-1410	79	11	ix	ix	ADP
ap-1410	79	12	−	−	PROPN
ap-1410	79	13	y	y	PROPN
ap-1410	79	14	)	)	PUNCT
ap-1410	79	15	]	]	PUNCT
ap-1410	80	1	}	}	PUNCT
ap-1410	80	2	,	,	PUNCT
ap-1410	80	3	we	we	PRON
ap-1410	80	4	obtain	obtain	VERB
ap-1410	80	5	for	for	ADP
ap-1410	80	6	any	any	DET
ap-1410	80	7	x	x	NOUN
ap-1410	80	8	,	,	PUNCT
ap-1410	80	9	y	y	PROPN
ap-1410	80	10	∈	∈	PROPN
ap-1410	80	11	d(a	d(a	PROPN
ap-1410	80	12	)	)	PUNCT
ap-1410	80	13	,	,	PUNCT
ap-1410	80	14	‖x‖	‖x‖	PROPN
ap-1410	80	15	=	=	SYM
ap-1410	81	1	‖y‖	‖y‖	PROPN
ap-1410	81	2	=	=	SYM
ap-1410	81	3	1	1	NUM
ap-1410	81	4	,	,	PUNCT
ap-1410	81	5	that	that	DET
ap-1410	81	6	|(ax	|(ax	NOUN
ap-1410	81	7	,	,	PUNCT
ap-1410	81	8	y)|	y)|	PROPN
ap-1410	81	9	≤	≤	NUM
ap-1410	81	10	1	1	NUM
ap-1410	81	11	4	4	NUM
ap-1410	81	12	{	{	PUNCT
ap-1410	81	13	|(a(x	|(a(x	PRON
ap-1410	81	14	+	+	NUM
ap-1410	81	15	y	y	NOUN
ap-1410	81	16	)	)	PUNCT
ap-1410	81	17	,	,	PUNCT
ap-1410	81	18	x	x	PUNCT
ap-1410	82	1	+	+	CCONJ
ap-1410	82	2	y)|	y)|	PROPN
ap-1410	82	3	+	+	CCONJ
ap-1410	82	4	|(a(x	|(a(x	ADP
ap-1410	82	5	−	−	PROPN
ap-1410	82	6	y	y	PROPN
ap-1410	82	7	)	)	PUNCT
ap-1410	82	8	,	,	PUNCT
ap-1410	82	9	x	x	X
ap-1410	82	10	−	−	NOUN
ap-1410	82	11	y)|	y)|	NOUN
ap-1410	82	12	+	+	CCONJ
ap-1410	82	13	|(a(ix	|(a(ix	PROPN
ap-1410	82	14	+	+	NUM
ap-1410	82	15	y	y	NOUN
ap-1410	82	16	)	)	PUNCT
ap-1410	82	17	,	,	PUNCT
ap-1410	82	18	ix	ix	PROPN
ap-1410	83	1	+	+	CCONJ
ap-1410	83	2	y)|	y)|	PROPN
ap-1410	83	3	+	+	CCONJ
ap-1410	83	4	|(a(ix	|(a(ix	PROPN
ap-1410	83	5	−	−	PROPN
ap-1410	83	6	y	y	PROPN
ap-1410	83	7	)	)	PUNCT
ap-1410	83	8	,	,	PUNCT
ap-1410	83	9	ix	ix	ADP
ap-1410	83	10	−	−	PROPN
ap-1410	83	11	y)|	y)|	PROPN
ap-1410	83	12	}	}	PUNCT
ap-1410	83	13	≤	≤	NOUN
ap-1410	83	14	(	(	PUNCT
ap-1410	83	15	2	2	X
ap-1410	83	16	)	)	PUNCT
ap-1410	83	17	k	k	NOUN
ap-1410	83	18	4	4	NUM
ap-1410	83	19	{	{	PUNCT
ap-1410	83	20	‖x	‖x	NOUN
ap-1410	83	21	+	+	NUM
ap-1410	83	22	y‖2	y‖2	X
ap-1410	83	23	+	+	CCONJ
ap-1410	83	24	‖x	‖x	NOUN
ap-1410	83	25	−	−	PROPN
ap-1410	84	1	y‖2	y‖2	X
ap-1410	84	2	+	+	CCONJ
ap-1410	84	3	‖ix	‖ix	NUM
ap-1410	84	4	+	+	NUM
ap-1410	84	5	y‖2	y‖2	X
ap-1410	84	6	+	+	CCONJ
ap-1410	84	7	‖ix	‖ix	NUM
ap-1410	84	8	−	−	NUM
ap-1410	84	9	y‖2	y‖2	NOUN
ap-1410	84	10	}	}	PUNCT
ap-1410	84	11	=	=	PUNCT
ap-1410	85	1	k	k	NOUN
ap-1410	85	2	4	4	NUM
ap-1410	85	3	{	{	PUNCT
ap-1410	85	4	(	(	PUNCT
ap-1410	85	5	x	x	SYM
ap-1410	85	6	+	+	NUM
ap-1410	85	7	y	y	NOUN
ap-1410	85	8	,	,	PUNCT
ap-1410	85	9	x	x	PUNCT
ap-1410	86	1	+	+	NUM
ap-1410	86	2	y	y	NOUN
ap-1410	86	3	)	)	PUNCT
ap-1410	87	1	+	+	CCONJ
ap-1410	87	2	(	(	PUNCT
ap-1410	87	3	x	x	X
ap-1410	87	4	−	−	PROPN
ap-1410	87	5	y	y	PROPN
ap-1410	87	6	,	,	PUNCT
ap-1410	87	7	x	x	PUNCT
ap-1410	87	8	−	−	PROPN
ap-1410	87	9	y	y	PROPN
ap-1410	87	10	)	)	PUNCT
ap-1410	88	1	+	+	CCONJ
ap-1410	88	2	(	(	PUNCT
ap-1410	88	3	ix	ix	PROPN
ap-1410	88	4	+	+	CCONJ
ap-1410	88	5	y	y	NOUN
ap-1410	88	6	,	,	PUNCT
ap-1410	88	7	ix	ix	X
ap-1410	88	8	+	+	CCONJ
ap-1410	88	9	y	y	NOUN
ap-1410	88	10	)	)	PUNCT
ap-1410	89	1	+	+	CCONJ
ap-1410	89	2	(	(	PUNCT
ap-1410	89	3	ix	ix	ADP
ap-1410	89	4	−	−	PROPN
ap-1410	89	5	y	y	PROPN
ap-1410	89	6	,	,	PUNCT
ap-1410	89	7	ix	ix	ADP
ap-1410	89	8	−	−	PROPN
ap-1410	89	9	y	y	PROPN
ap-1410	89	10	)	)	PUNCT
ap-1410	89	11	}	}	PUNCT
ap-1410	89	12	=	=	PUNCT
ap-1410	90	1	k	k	NOUN
ap-1410	90	2	4	4	NUM
ap-1410	90	3	{	{	PUNCT
ap-1410	90	4	2‖x‖2	2‖x‖2	PROPN
ap-1410	91	1	+	+	CCONJ
ap-1410	91	2	2‖y‖2	2‖y‖2	NOUN
ap-1410	91	3	+	+	CCONJ
ap-1410	92	1	2‖ix‖2	2‖ix‖2	NUM
ap-1410	92	2	+	+	NUM
ap-1410	92	3	2‖y‖2	2‖y‖2	NOUN
ap-1410	92	4	}	}	PUNCT
ap-1410	92	5	=	=	SYM
ap-1410	92	6	k(‖x‖2	k(‖x‖2	PROPN
ap-1410	92	7	+	+	CCONJ
ap-1410	92	8	‖y‖2	‖y‖2	X
ap-1410	92	9	)	)	PUNCT
ap-1410	93	1	=	=	SYM
ap-1410	93	2	2k	2k	NUM
ap-1410	93	3	.	.	PUNCT
ap-1410	94	1	this	this	PRON
ap-1410	94	2	means	mean	VERB
ap-1410	94	3	that	that	SCONJ
ap-1410	94	4	the	the	DET
ap-1410	94	5	quadratic	quadratic	ADJ
ap-1410	94	6	form	form	NOUN
ap-1410	94	7	(	(	PUNCT
ap-1410	94	8	ax	ax	NOUN
ap-1410	94	9	,	,	PUNCT
ap-1410	94	10	y	y	PROPN
ap-1410	94	11	)	)	PUNCT
ap-1410	94	12	is	be	AUX
ap-1410	94	13	bounded	bound	VERB
ap-1410	94	14	,	,	PUNCT
ap-1410	94	15	i.e.	i.e.	X
ap-1410	94	16	∀x	∀x	X
ap-1410	94	17	,	,	PUNCT
ap-1410	94	18	y	y	PROPN
ap-1410	94	19	∈	∈	PROPN
ap-1410	94	20	d(a	d(a	PROPN
ap-1410	94	21	)	)	PUNCT
ap-1410	94	22	|(ax	|(ax	NUM
ap-1410	94	23	,	,	PUNCT
ap-1410	94	24	y)|	y)|	NOUN
ap-1410	94	25	≤	≤	VERB
ap-1410	94	26	2k‖x‖	2k‖x‖	NUM
ap-1410	94	27	‖y‖	‖y‖	PROPN
ap-1410	94	28	.	.	PUNCT
ap-1410	95	1	(	(	PUNCT
ap-1410	95	2	3	3	X
ap-1410	95	3	)	)	PUNCT
ap-1410	95	4	it	it	PRON
ap-1410	95	5	follows	follow	VERB
ap-1410	95	6	that	that	SCONJ
ap-1410	95	7	for	for	ADP
ap-1410	95	8	any	any	DET
ap-1410	95	9	fixed	fix	VERB
ap-1410	95	10	x	x	SYM
ap-1410	95	11	∈	∈	PROPN
ap-1410	95	12	d(a	d(a	PROPN
ap-1410	95	13	)	)	PUNCT
ap-1410	95	14	the	the	DET
ap-1410	95	15	linear	linear	ADJ
ap-1410	95	16	functional	functional	ADJ
ap-1410	95	17	ϕ(y	ϕ(y	PROPN
ap-1410	95	18	)	)	PUNCT
ap-1410	96	1	=	=	PRON
ap-1410	96	2	(	(	PUNCT
ap-1410	96	3	ax	ax	NOUN
ap-1410	96	4	,	,	PUNCT
ap-1410	96	5	y	y	PROPN
ap-1410	96	6	)	)	PUNCT
ap-1410	96	7	is	be	AUX
ap-1410	96	8	bounded	bound	VERB
ap-1410	96	9	and	and	CCONJ
ap-1410	96	10	defined	define	VERB
ap-1410	96	11	on	on	ADP
ap-1410	96	12	d(a	d(a	PROPN
ap-1410	96	13	)	)	PUNCT
ap-1410	96	14	therefore	therefore	ADV
ap-1410	96	15	ϕ̃	ϕ̃	PROPN
ap-1410	96	16	:	:	PUNCT
ap-1410	97	1	d(a	d(a	PROPN
ap-1410	97	2	)	)	PUNCT
ap-1410	98	1	=	=	SYM
ap-1410	98	2	h	h	NOUN
ap-1410	98	3	→	→	SYM
ap-1410	98	4	c	c	X
ap-1410	98	5	,	,	PUNCT
ap-1410	98	6	ϕ̃(y	ϕ̃(y	PROPN
ap-1410	98	7	)	)	PUNCT
ap-1410	98	8	=	=	PUNCT
ap-1410	99	1	(	(	PUNCT
ap-1410	99	2	ax	ax	NOUN
ap-1410	99	3	,	,	PUNCT
ap-1410	99	4	y	y	PROPN
ap-1410	99	5	)	)	PUNCT
ap-1410	99	6	for	for	ADP
ap-1410	99	7	all	all	DET
ap-1410	99	8	y	y	PROPN
ap-1410	99	9	∈	∈	PROPN
ap-1410	99	10	h	h	NOUN
ap-1410	99	11	is	be	AUX
ap-1410	99	12	its	its	PRON
ap-1410	99	13	unique	unique	ADJ
ap-1410	99	14	bounded	bound	VERB
ap-1410	99	15	linear	linear	ADJ
ap-1410	99	16	extension	extension	NOUN
ap-1410	99	17	and	and	CCONJ
ap-1410	99	18	‖ϕ̃‖	‖ϕ̃‖	NOUN
ap-1410	99	19	=	=	SYM
ap-1410	99	20	‖ϕ‖	‖ϕ‖	PROPN
ap-1410	99	21	≤	≤	NOUN
ap-1410	99	22	2k‖x‖.	2k‖x‖.	PROPN
ap-1410	99	23	now	now	ADV
ap-1410	99	24	putting	put	VERB
ap-1410	99	25	y	y	NOUN
ap-1410	99	26	=	=	PUNCT
ap-1410	100	1	ax	ax	NOUN
ap-1410	100	2	we	we	PRON
ap-1410	100	3	obtain	obtain	VERB
ap-1410	100	4	,	,	PUNCT
ap-1410	100	5	by	by	ADP
ap-1410	100	6	(	(	PUNCT
ap-1410	100	7	3	3	NUM
ap-1410	100	8	)	)	PUNCT
ap-1410	100	9	,	,	PUNCT
ap-1410	100	10	‖ax‖2	‖ax‖2	PROPN
ap-1410	100	11	=	=	PUNCT
ap-1410	100	12	(	(	PUNCT
ap-1410	100	13	ax	ax	NOUN
ap-1410	100	14	,	,	PUNCT
ap-1410	100	15	ax	ax	NOUN
ap-1410	100	16	)	)	PUNCT
ap-1410	100	17	=	=	SYM
ap-1410	100	18	ϕ̃(ax	ϕ̃(ax	NOUN
ap-1410	100	19	)	)	PUNCT
ap-1410	100	20	≤	≤	NOUN
ap-1410	101	1	2k‖x‖‖ax‖	2k‖x‖‖ax‖	NUM
ap-1410	101	2	=	=	NOUN
ap-1410	101	3	⇒	⇒	NOUN
ap-1410	101	4	‖ax‖	‖ax‖	ADJ
ap-1410	101	5	≤	≤	NOUN
ap-1410	101	6	2k‖x‖	2k‖x‖	NUM
ap-1410	101	7	.	.	PUNCT
ap-1410	102	1	74	74	NUM
ap-1410	102	2	acta	acta	PROPN
ap-1410	102	3	polytechnica	polytechnica	PROPN
ap-1410	102	4	vol	vol	NOUN
ap-1410	102	5	.	.	PUNCT
ap-1410	103	1	51	51	NUM
ap-1410	103	2	no	no	INTJ
ap-1410	103	3	.	.	PUNCT
ap-1410	104	1	4/2011	4/2011	NUM
ap-1410	104	2	corollary	corollary	NOUN
ap-1410	104	3	6	6	NUM
ap-1410	104	4	let	let	VERB
ap-1410	104	5	a	a	DET
ap-1410	104	6	,	,	PUNCT
ap-1410	104	7	b	b	PROPN
ap-1410	104	8	be	be	AUX
ap-1410	104	9	nonnegative	nonnegative	ADJ
ap-1410	104	10	densely	densely	ADV
ap-1410	104	11	defined	define	VERB
ap-1410	104	12	linear	linear	PROPN
ap-1410	104	13	operators	operator	NOUN
ap-1410	104	14	having	have	VERB
ap-1410	104	15	the	the	DET
ap-1410	104	16	same	same	ADJ
ap-1410	104	17	domain	domain	NOUN
ap-1410	104	18	d.	d.	NOUN
ap-1410	104	19	if	if	SCONJ
ap-1410	104	20	a	a	DET
ap-1410	104	21	+	+	NOUN
ap-1410	104	22	b	b	NOUN
ap-1410	104	23	is	be	AUX
ap-1410	104	24	bounded	bound	VERB
ap-1410	104	25	then	then	ADV
ap-1410	104	26	both	both	CCONJ
ap-1410	104	27	a	a	PRON
ap-1410	104	28	,	,	PUNCT
ap-1410	104	29	b	b	PROPN
ap-1410	104	30	are	be	AUX
ap-1410	104	31	bounded	bound	VERB
ap-1410	104	32	.	.	PUNCT
ap-1410	105	1	proof	proof	NOUN
ap-1410	105	2	.	.	PUNCT
ap-1410	106	1	it	it	PRON
ap-1410	106	2	suffices	suffice	VERB
ap-1410	106	3	to	to	PART
ap-1410	106	4	observe	observe	VERB
ap-1410	106	5	that	that	SCONJ
ap-1410	106	6	0	0	NUM
ap-1410	106	7	≤	≤	NOUN
ap-1410	106	8	(	(	PUNCT
ap-1410	106	9	ax	ax	NOUN
ap-1410	106	10	,	,	PUNCT
ap-1410	106	11	x	x	NOUN
ap-1410	106	12	)	)	PUNCT
ap-1410	106	13	≤	≤	NOUN
ap-1410	106	14	(	(	PUNCT
ap-1410	106	15	ax	ax	NOUN
ap-1410	106	16	,	,	PUNCT
ap-1410	106	17	x	x	NOUN
ap-1410	106	18	)	)	PUNCT
ap-1410	107	1	+	+	CCONJ
ap-1410	107	2	(	(	PUNCT
ap-1410	107	3	bx	bx	X
ap-1410	107	4	,	,	PUNCT
ap-1410	107	5	x	x	NOUN
ap-1410	107	6	)	)	PUNCT
ap-1410	107	7	=	=	SYM
ap-1410	107	8	(	(	PUNCT
ap-1410	107	9	(	(	PUNCT
ap-1410	107	10	a	a	DET
ap-1410	107	11	+	+	X
ap-1410	107	12	b)x	b)x	ADJ
ap-1410	107	13	,	,	PUNCT
ap-1410	107	14	x	x	NOUN
ap-1410	107	15	)	)	PUNCT
ap-1410	107	16	≤	≤	NOUN
ap-1410	107	17	‖a	‖a	NOUN
ap-1410	107	18	+	+	CCONJ
ap-1410	107	19	b||‖x‖2	b||‖x‖2	ADJ
ap-1410	107	20	and	and	CCONJ
ap-1410	107	21	then	then	ADV
ap-1410	107	22	,	,	PUNCT
ap-1410	107	23	by	by	ADP
ap-1410	107	24	theorem	theorem	NOUN
ap-1410	107	25	5	5	NUM
ap-1410	107	26	,	,	PUNCT
ap-1410	107	27	a	a	PRON
ap-1410	107	28	is	be	AUX
ap-1410	107	29	bounded	bound	VERB
ap-1410	107	30	.	.	PUNCT
ap-1410	108	1	by	by	ADP
ap-1410	108	2	the	the	DET
ap-1410	108	3	same	same	ADJ
ap-1410	108	4	reasoning	reasoning	NOUN
ap-1410	108	5	we	we	PRON
ap-1410	108	6	obtain	obtain	VERB
ap-1410	108	7	that	that	PRON
ap-1410	108	8	b	b	NOUN
ap-1410	108	9	is	be	AUX
ap-1410	108	10	bounded	bound	VERB
ap-1410	108	11	.	.	PUNCT
ap-1410	108	12	3	3	NUM
ap-1410	108	13	extensions	extension	NOUN
ap-1410	108	14	of	of	ADP
ap-1410	108	15	effect	effect	NOUN
ap-1410	108	16	algebra	algebra	NOUN
ap-1410	108	17	operations	operation	NOUN
ap-1410	108	18	it	it	PRON
ap-1410	108	19	is	be	AUX
ap-1410	108	20	well	well	ADV
ap-1410	108	21	known	know	VERB
ap-1410	108	22	that	that	SCONJ
ap-1410	108	23	if	if	SCONJ
ap-1410	108	24	a	a	DET
ap-1410	108	25	set	set	NOUN
ap-1410	108	26	e	e	NOUN
ap-1410	108	27	includes	include	VERB
ap-1410	108	28	two	two	NUM
ap-1410	108	29	distinguished	distinguished	ADJ
ap-1410	108	30	elements	element	NOUN
ap-1410	108	31	0	0	NUM
ap-1410	108	32	,	,	PUNCT
ap-1410	108	33	1	1	NUM
ap-1410	108	34	then	then	ADV
ap-1410	108	35	there	there	PRON
ap-1410	108	36	may	may	AUX
ap-1410	108	37	exist	exist	VERB
ap-1410	108	38	more	more	ADJ
ap-1410	108	39	than	than	ADP
ap-1410	108	40	one	one	NUM
ap-1410	108	41	partial	partial	ADJ
ap-1410	108	42	binary	binary	NOUN
ap-1410	108	43	operations	operation	NOUN
ap-1410	108	44	⊕	⊕	PROPN
ap-1410	108	45	on	on	ADP
ap-1410	108	46	e	e	NOUN
ap-1410	108	47	making	make	VERB
ap-1410	108	48	e	e	NOUN
ap-1410	108	49	an	an	DET
ap-1410	108	50	effect	effect	NOUN
ap-1410	108	51	algebra	algebra	NOUN
ap-1410	108	52	.	.	PUNCT
ap-1410	109	1	example	example	NOUN
ap-1410	109	2	7	7	NUM
ap-1410	109	3	let	let	VERB
ap-1410	109	4	e	e	NOUN
ap-1410	109	5	=	=	PRON
ap-1410	109	6	{	{	PUNCT
ap-1410	109	7	0	0	NUM
ap-1410	109	8	,	,	PUNCT
ap-1410	109	9	a	a	DET
ap-1410	109	10	,	,	PUNCT
ap-1410	109	11	b	b	NOUN
ap-1410	109	12	,	,	PUNCT
ap-1410	109	13	1	1	NUM
ap-1410	109	14	}	}	PUNCT
ap-1410	109	15	and	and	CCONJ
ap-1410	109	16	let	let	VERB
ap-1410	109	17	us	we	PRON
ap-1410	109	18	define	define	VERB
ap-1410	109	19	operations	operation	NOUN
ap-1410	109	20	⊕1	⊕1	ADJ
ap-1410	109	21	,	,	PUNCT
ap-1410	109	22	⊕2	⊕2	NOUN
ap-1410	109	23	on	on	ADP
ap-1410	109	24	e	e	PROPN
ap-1410	109	25	as	as	SCONJ
ap-1410	109	26	follows	follow	VERB
ap-1410	109	27	:	:	PUNCT
ap-1410	110	1	⊕1	⊕1	NUM
ap-1410	110	2	:	:	PUNCT
ap-1410	110	3	a	a	DET
ap-1410	110	4	⊕1	⊕1	PROPN
ap-1410	110	5	b	b	NOUN
ap-1410	110	6	=	=	SYM
ap-1410	110	7	b	b	PROPN
ap-1410	110	8	⊕1	⊕1	PROPN
ap-1410	110	9	a	a	DET
ap-1410	110	10	=	=	SYM
ap-1410	110	11	1	1	NUM
ap-1410	110	12	and	and	CCONJ
ap-1410	110	13	0	0	NUM
ap-1410	110	14	⊕1	⊕1	NUM
ap-1410	110	15	x	x	X
ap-1410	111	1	=	=	PUNCT
ap-1410	111	2	x	x	PUNCT
ap-1410	111	3	⊕1	⊕1	NOUN
ap-1410	111	4	0	0	NUM
ap-1410	112	1	=	=	PUNCT
ap-1410	113	1	x	x	PROPN
ap-1410	114	1	for	for	ADP
ap-1410	114	2	all	all	DET
ap-1410	114	3	x	x	SYM
ap-1410	114	4	∈	∈	PROPN
ap-1410	114	5	e	e	NOUN
ap-1410	114	6	,	,	PUNCT
ap-1410	114	7	⊕2	⊕2	PROPN
ap-1410	114	8	:	:	PUNCT
ap-1410	114	9	a	a	DET
ap-1410	114	10	⊕2	⊕2	PROPN
ap-1410	114	11	a	a	DET
ap-1410	114	12	=	=	SYM
ap-1410	114	13	b	b	PROPN
ap-1410	114	14	⊕2	⊕2	PROPN
ap-1410	114	15	b	b	PROPN
ap-1410	114	16	=	=	SYM
ap-1410	114	17	1	1	NUM
ap-1410	114	18	and	and	CCONJ
ap-1410	114	19	0	0	NUM
ap-1410	114	20	⊕2	⊕2	NOUN
ap-1410	114	21	x	x	PUNCT
ap-1410	114	22	=	=	PUNCT
ap-1410	114	23	x	x	SYM
ap-1410	114	24	⊕2	⊕2	PROPN
ap-1410	114	25	0	0	PUNCT
ap-1410	114	26	=	=	SYM
ap-1410	114	27	x	x	PROPN
ap-1410	114	28	for	for	ADP
ap-1410	114	29	all	all	DET
ap-1410	114	30	x	x	SYM
ap-1410	114	31	∈	∈	PROPN
ap-1410	114	32	e.	e.	PROPN
ap-1410	114	33	then	then	ADV
ap-1410	114	34	(	(	PUNCT
ap-1410	114	35	e;⊕1	e;⊕1	PROPN
ap-1410	114	36	,	,	PUNCT
ap-1410	114	37	0	0	NUM
ap-1410	114	38	,	,	PUNCT
ap-1410	114	39	1	1	NUM
ap-1410	114	40	)	)	PUNCT
ap-1410	114	41	,	,	PUNCT
ap-1410	114	42	(	(	PUNCT
ap-1410	114	43	e;⊕2	e;⊕2	PROPN
ap-1410	114	44	,	,	PUNCT
ap-1410	114	45	0	0	NUM
ap-1410	114	46	,	,	PUNCT
ap-1410	114	47	1	1	NUM
ap-1410	114	48	)	)	PUNCT
ap-1410	114	49	are	be	AUX
ap-1410	114	50	effect	effect	NOUN
ap-1410	114	51	algebras	algebra	NOUN
ap-1410	114	52	with	with	ADP
ap-1410	114	53	the	the	DET
ap-1410	114	54	same	same	ADJ
ap-1410	114	55	set	set	NOUN
ap-1410	114	56	of	of	ADP
ap-1410	114	57	elements	element	NOUN
ap-1410	114	58	.	.	PUNCT
ap-1410	115	1	on	on	ADP
ap-1410	115	2	the	the	DET
ap-1410	115	3	other	other	ADJ
ap-1410	115	4	hand	hand	NOUN
ap-1410	115	5	(	(	PUNCT
ap-1410	115	6	e;⊕1	e;⊕1	PROPN
ap-1410	115	7	,	,	PUNCT
ap-1410	115	8	0	0	NUM
ap-1410	115	9	,	,	PUNCT
ap-1410	115	10	1	1	NUM
ap-1410	115	11	)	)	PUNCT
ap-1410	115	12	is	be	AUX
ap-1410	115	13	a	a	DET
ap-1410	115	14	boolean	boolean	ADJ
ap-1410	115	15	algebra	algebra	NOUN
ap-1410	115	16	,	,	PUNCT
ap-1410	115	17	while	while	SCONJ
ap-1410	115	18	(	(	PUNCT
ap-1410	115	19	e;⊕2	e;⊕2	PROPN
ap-1410	115	20	,	,	PUNCT
ap-1410	115	21	0	0	NUM
ap-1410	115	22	,	,	PUNCT
ap-1410	115	23	1	1	NUM
ap-1410	115	24	)	)	PUNCT
ap-1410	115	25	is	be	AUX
ap-1410	115	26	a	a	DET
ap-1410	115	27	horizontal	horizontal	ADJ
ap-1410	115	28	sum	sum	NOUN
ap-1410	115	29	of	of	ADP
ap-1410	115	30	two	two	NUM
ap-1410	115	31	chains	chain	NOUN
ap-1410	115	32	{	{	PUNCT
ap-1410	115	33	0	0	NUM
ap-1410	115	34	,	,	PUNCT
ap-1410	115	35	a	a	PRON
ap-1410	115	36	,	,	PUNCT
ap-1410	115	37	a	a	DET
ap-1410	115	38	⊕2	⊕2	PROPN
ap-1410	115	39	a	a	DET
ap-1410	115	40	=	=	NOUN
ap-1410	115	41	1	1	NUM
ap-1410	115	42	}	}	PUNCT
ap-1410	115	43	and	and	CCONJ
ap-1410	115	44	{	{	PUNCT
ap-1410	115	45	0	0	NUM
ap-1410	115	46	,	,	PUNCT
ap-1410	115	47	b	b	NOUN
ap-1410	115	48	,	,	PUNCT
ap-1410	115	49	b⊕2	b⊕2	PROPN
ap-1410	115	50	b	b	PROPN
ap-1410	115	51	=	=	NOUN
ap-1410	115	52	1	1	NUM
ap-1410	115	53	}	}	PUNCT
ap-1410	115	54	,	,	PUNCT
ap-1410	115	55	hence	hence	ADV
ap-1410	115	56	in	in	ADP
ap-1410	115	57	this	this	DET
ap-1410	115	58	case	case	NOUN
ap-1410	115	59	elements	element	VERB
ap-1410	115	60	a	a	DET
ap-1410	115	61	,	,	PUNCT
ap-1410	115	62	b	b	NOUN
ap-1410	115	63	are	be	AUX
ap-1410	115	64	noncompatible	noncompatible	ADJ
ap-1410	115	65	.	.	PUNCT
ap-1410	116	1	we	we	PRON
ap-1410	116	2	obtain	obtain	VERB
ap-1410	116	3	special	special	ADJ
ap-1410	116	4	cases	case	NOUN
ap-1410	116	5	of	of	ADP
ap-1410	116	6	two	two	NUM
ap-1410	116	7	operations	operation	NOUN
ap-1410	116	8	⊕1	⊕1	PROPN
ap-1410	116	9	�	�	NOUN
ap-1410	116	10	=	=	SYM
ap-1410	116	11	⊕2	⊕2	PROPN
ap-1410	116	12	on	on	ADP
ap-1410	116	13	the	the	DET
ap-1410	116	14	same	same	ADJ
ap-1410	116	15	underlying	underlying	ADJ
ap-1410	116	16	set	set	NOUN
ap-1410	116	17	e	e	NOUN
ap-1410	116	18	,	,	PUNCT
ap-1410	116	19	if	if	SCONJ
ap-1410	116	20	⊕2	⊕2	PROPN
ap-1410	116	21	extends	extend	VERB
ap-1410	116	22	⊕1	⊕1	NUM
ap-1410	116	23	:	:	PUNCT
ap-1410	116	24	definition	definition	NOUN
ap-1410	116	25	8	8	NUM
ap-1410	116	26	let	let	VERB
ap-1410	116	27	(	(	PUNCT
ap-1410	116	28	e;⊕1	e;⊕1	PROPN
ap-1410	116	29	,	,	PUNCT
ap-1410	116	30	0	0	NUM
ap-1410	116	31	)	)	PUNCT
ap-1410	116	32	and	and	CCONJ
ap-1410	116	33	(	(	PUNCT
ap-1410	116	34	e;⊕2	e;⊕2	PROPN
ap-1410	116	35	,	,	PUNCT
ap-1410	116	36	0	0	NUM
ap-1410	116	37	)	)	PUNCT
ap-1410	116	38	be	be	AUX
ap-1410	116	39	generalized	generalized	ADJ
ap-1410	116	40	effect	effect	NOUN
ap-1410	116	41	algebras	algebra	NOUN
ap-1410	116	42	.	.	PUNCT
ap-1410	117	1	we	we	PRON
ap-1410	117	2	say	say	VERB
ap-1410	117	3	that	that	SCONJ
ap-1410	117	4	the	the	DET
ap-1410	117	5	operation	operation	NOUN
ap-1410	117	6	⊕2	⊕2	PROPN
ap-1410	117	7	extends	extend	VERB
ap-1410	117	8	⊕1	⊕1	PROPN
ap-1410	117	9	(	(	PUNCT
ap-1410	117	10	written	write	VERB
ap-1410	117	11	⊕1	⊕1	PROPN
ap-1410	117	12	⊂	⊂	PROPN
ap-1410	117	13	⊕2	⊕2	PROPN
ap-1410	117	14	)	)	PUNCT
ap-1410	117	15	if	if	SCONJ
ap-1410	117	16	for	for	ADP
ap-1410	117	17	any	any	DET
ap-1410	117	18	a	a	NOUN
ap-1410	117	19	,	,	PUNCT
ap-1410	117	20	b	b	X
ap-1410	117	21	∈	∈	PROPN
ap-1410	117	22	e	e	X
ap-1410	117	23	the	the	DET
ap-1410	117	24	existence	existence	NOUN
ap-1410	117	25	of	of	ADP
ap-1410	117	26	a	a	DET
ap-1410	117	27	⊕1	⊕1	PROPN
ap-1410	117	28	b	b	NOUN
ap-1410	117	29	implies	imply	VERB
ap-1410	117	30	that	that	SCONJ
ap-1410	117	31	a	a	DET
ap-1410	117	32	⊕2	⊕2	PROPN
ap-1410	117	33	b	b	PROPN
ap-1410	117	34	exists	exist	VERB
ap-1410	117	35	and	and	CCONJ
ap-1410	117	36	a	a	DET
ap-1410	117	37	⊕2	⊕2	PROPN
ap-1410	117	38	b	b	PROPN
ap-1410	117	39	=	=	PROPN
ap-1410	117	40	a	a	DET
ap-1410	117	41	⊕1	⊕1	PROPN
ap-1410	117	42	b.	b.	PROPN
ap-1410	117	43	lemma	lemma	PROPN
ap-1410	117	44	9	9	NUM
ap-1410	117	45	let	let	VERB
ap-1410	117	46	e1	e1	NOUN
ap-1410	117	47	=	=	SYM
ap-1410	117	48	(	(	PUNCT
ap-1410	117	49	e;⊕1	e;⊕1	PROPN
ap-1410	117	50	,	,	PUNCT
ap-1410	117	51	0	0	NUM
ap-1410	117	52	)	)	PUNCT
ap-1410	117	53	,	,	PUNCT
ap-1410	117	54	e2	e2	PROPN
ap-1410	117	55	=	=	PUNCT
ap-1410	117	56	(	(	PUNCT
ap-1410	117	57	e;⊕2	e;⊕2	PROPN
ap-1410	117	58	,	,	PUNCT
ap-1410	117	59	0	0	NUM
ap-1410	117	60	)	)	PUNCT
ap-1410	117	61	be	be	AUX
ap-1410	117	62	generalized	generalized	ADJ
ap-1410	117	63	effect	effect	NOUN
ap-1410	117	64	algebras	algebra	NOUN
ap-1410	117	65	and	and	CCONJ
ap-1410	117	66	⊕1	⊕1	PROPN
ap-1410	117	67	⊂	⊂	PROPN
ap-1410	117	68	⊕2	⊕2	PROPN
ap-1410	117	69	.	.	PUNCT
ap-1410	118	1	then	then	ADV
ap-1410	118	2	(	(	PUNCT
ap-1410	118	3	i	i	NOUN
ap-1410	118	4	)	)	PUNCT
ap-1410	118	5	if	if	SCONJ
ap-1410	118	6	g	g	PROPN
ap-1410	118	7	⊆	⊆	NUM
ap-1410	118	8	e	e	NOUN
ap-1410	118	9	is	be	AUX
ap-1410	118	10	a	a	DET
ap-1410	118	11	sub	sub	ADJ
ap-1410	118	12	-	-	ADJ
ap-1410	118	13	generalized	generalized	ADJ
ap-1410	118	14	effect	effect	NOUN
ap-1410	118	15	algebra	algebra	NOUN
ap-1410	118	16	of	of	ADP
ap-1410	118	17	e2	e2	PROPN
ap-1410	118	18	,	,	PUNCT
ap-1410	118	19	then	then	ADV
ap-1410	118	20	g	g	PROPN
ap-1410	118	21	is	be	AUX
ap-1410	118	22	also	also	ADV
ap-1410	118	23	a	a	DET
ap-1410	118	24	sub	sub	ADJ
ap-1410	118	25	-	-	ADJ
ap-1410	118	26	generalized	generalized	ADJ
ap-1410	118	27	effect	effect	NOUN
ap-1410	118	28	algebra	algebra	NOUN
ap-1410	118	29	of	of	ADP
ap-1410	118	30	e1	e1	PROPN
ap-1410	118	31	.	.	PUNCT
ap-1410	119	1	(	(	PUNCT
ap-1410	119	2	ii	ii	NOUN
ap-1410	119	3	)	)	PUNCT
ap-1410	119	4	if	if	SCONJ
ap-1410	119	5	≤1	≤1	PROPN
ap-1410	119	6	and	and	CCONJ
ap-1410	119	7	≤2	≤2	NOUN
ap-1410	119	8	are	be	AUX
ap-1410	119	9	the	the	DET
ap-1410	119	10	partial	partial	ADJ
ap-1410	119	11	orders	order	NOUN
ap-1410	119	12	on	on	ADP
ap-1410	119	13	e	e	NOUN
ap-1410	119	14	derived	derive	VERB
ap-1410	119	15	from	from	ADP
ap-1410	119	16	⊕1	⊕1	PROPN
ap-1410	119	17	and	and	CCONJ
ap-1410	119	18	⊕2	⊕2	PROPN
ap-1410	119	19	,	,	PUNCT
ap-1410	119	20	respectively	respectively	ADV
ap-1410	119	21	,	,	PUNCT
ap-1410	119	22	then	then	ADV
ap-1410	119	23	,	,	PUNCT
ap-1410	119	24	for	for	ADP
ap-1410	119	25	a	a	DET
ap-1410	119	26	,	,	PUNCT
ap-1410	119	27	b	b	PROPN
ap-1410	119	28	∈	∈	PROPN
ap-1410	119	29	e	e	NOUN
ap-1410	119	30	,	,	PUNCT
ap-1410	119	31	if	if	SCONJ
ap-1410	119	32	a	a	DET
ap-1410	119	33	≤1	≤1	PROPN
ap-1410	119	34	b	b	PROPN
ap-1410	119	35	then	then	ADV
ap-1410	119	36	a	a	DET
ap-1410	119	37	≤2	≤2	NOUN
ap-1410	119	38	b.	b.	PROPN
ap-1410	119	39	(	(	PUNCT
ap-1410	119	40	iii	iii	NOUN
ap-1410	119	41	)	)	PUNCT
ap-1410	119	42	for	for	ADP
ap-1410	119	43	intervals	interval	NOUN
ap-1410	119	44	in	in	ADP
ap-1410	119	45	e1	e1	NOUN
ap-1410	119	46	,	,	PUNCT
ap-1410	119	47	e2	e2	VERB
ap-1410	119	48	the	the	DET
ap-1410	119	49	following	follow	VERB
ap-1410	119	50	inclusion	inclusion	NOUN
ap-1410	119	51	holds	hold	VERB
ap-1410	119	52	:	:	PUNCT
ap-1410	120	1	[	[	X
ap-1410	120	2	0	0	NUM
ap-1410	120	3	,	,	PUNCT
ap-1410	120	4	q]e1	q]e1	VERB
ap-1410	120	5	⊆	⊆	NUM
ap-1410	120	6	[	[	X
ap-1410	120	7	0	0	NUM
ap-1410	120	8	,	,	PUNCT
ap-1410	120	9	q]e2	q]e2	VERB
ap-1410	120	10	for	for	ADP
ap-1410	120	11	any	any	DET
ap-1410	120	12	nonzero	nonzero	NOUN
ap-1410	120	13	q	q	X
ap-1410	120	14	∈	∈	PROPN
ap-1410	120	15	e.	e.	PROPN
ap-1410	120	16	proof	proof	PROPN
ap-1410	120	17	.	.	PUNCT
ap-1410	121	1	the	the	DET
ap-1410	121	2	proof	proof	NOUN
ap-1410	121	3	obviously	obviously	ADV
ap-1410	121	4	follows	follow	VERB
ap-1410	121	5	from	from	ADP
ap-1410	121	6	the	the	DET
ap-1410	121	7	fact	fact	NOUN
ap-1410	121	8	that	that	SCONJ
ap-1410	121	9	,	,	PUNCT
ap-1410	121	10	for	for	ADP
ap-1410	121	11	any	any	DET
ap-1410	121	12	a	a	NOUN
ap-1410	121	13	,	,	PUNCT
ap-1410	121	14	b	b	X
ap-1410	121	15	∈	∈	PROPN
ap-1410	121	16	e	e	NOUN
ap-1410	121	17	,	,	PUNCT
ap-1410	121	18	the	the	DET
ap-1410	121	19	existence	existence	NOUN
ap-1410	121	20	of	of	ADP
ap-1410	121	21	a	a	DET
ap-1410	121	22	⊕1	⊕1	PROPN
ap-1410	121	23	b	b	PROPN
ap-1410	121	24	implies	imply	VERB
ap-1410	121	25	a	a	DET
ap-1410	121	26	⊕2	⊕2	PROPN
ap-1410	121	27	b	b	PROPN
ap-1410	121	28	=	=	PROPN
ap-1410	121	29	a	a	DET
ap-1410	121	30	⊕1	⊕1	PROPN
ap-1410	121	31	b.	b.	NOUN
ap-1410	121	32	let	let	VERB
ap-1410	121	33	us	we	PRON
ap-1410	121	34	prove	prove	VERB
ap-1410	121	35	,	,	PUNCT
ap-1410	121	36	e.g.	e.g.	ADV
ap-1410	121	37	,	,	PUNCT
ap-1410	121	38	(	(	PUNCT
ap-1410	121	39	i	i	NOUN
ap-1410	121	40	):	):	PUNCT
ap-1410	121	41	let	let	VERB
ap-1410	121	42	a	a	DET
ap-1410	121	43	,	,	PUNCT
ap-1410	121	44	b	b	NOUN
ap-1410	121	45	,	,	PUNCT
ap-1410	121	46	c	c	PROPN
ap-1410	121	47	∈	∈	PROPN
ap-1410	121	48	e	e	X
ap-1410	121	49	with	with	ADP
ap-1410	121	50	a	a	DET
ap-1410	121	51	⊕1	⊕1	PROPN
ap-1410	121	52	b	b	NOUN
ap-1410	121	53	=	=	SYM
ap-1410	121	54	c	c	PROPN
ap-1410	121	55	and	and	CCONJ
ap-1410	121	56	assume	assume	VERB
ap-1410	121	57	that	that	SCONJ
ap-1410	121	58	at	at	ADV
ap-1410	121	59	least	least	ADV
ap-1410	121	60	two	two	NUM
ap-1410	121	61	out	out	ADP
ap-1410	121	62	of	of	ADP
ap-1410	121	63	elements	element	NOUN
ap-1410	121	64	a	a	DET
ap-1410	121	65	,	,	PUNCT
ap-1410	121	66	b	b	NOUN
ap-1410	121	67	,	,	PUNCT
ap-1410	121	68	c	c	PROPN
ap-1410	121	69	are	be	AUX
ap-1410	121	70	in	in	ADP
ap-1410	121	71	g.	g.	NOUN
ap-1410	121	72	since	since	SCONJ
ap-1410	121	73	⊕1	⊕1	PROPN
ap-1410	121	74	⊂	⊂	PROPN
ap-1410	121	75	⊕2	⊕2	PROPN
ap-1410	121	76	implies	imply	VERB
ap-1410	121	77	a	a	DET
ap-1410	121	78	⊕1	⊕1	PROPN
ap-1410	121	79	b	b	NOUN
ap-1410	121	80	=	=	SYM
ap-1410	121	81	c	c	NOUN
ap-1410	121	82	=	=	PUNCT
ap-1410	121	83	a	a	DET
ap-1410	121	84	⊕2	⊕2	PROPN
ap-1410	121	85	b	b	PROPN
ap-1410	121	86	=	=	SYM
ap-1410	121	87	c	c	PROPN
ap-1410	121	88	and	and	CCONJ
ap-1410	121	89	g	g	PROPN
ap-1410	121	90	is	be	AUX
ap-1410	121	91	a	a	DET
ap-1410	121	92	sub	sub	ADJ
ap-1410	121	93	-	-	ADJ
ap-1410	121	94	effect	effect	ADJ
ap-1410	121	95	algebra	algebra	NOUN
ap-1410	121	96	of	of	ADP
ap-1410	121	97	e2	e2	PROPN
ap-1410	121	98	we	we	PRON
ap-1410	121	99	obtain	obtain	VERB
ap-1410	121	100	a	a	DET
ap-1410	121	101	,	,	PUNCT
ap-1410	121	102	b	b	NOUN
ap-1410	121	103	,	,	PUNCT
ap-1410	121	104	c	c	PROPN
ap-1410	121	105	∈	∈	PROPN
ap-1410	121	106	g.	g.	NOUN
ap-1410	121	107	hence	hence	ADV
ap-1410	121	108	g	g	PROPN
ap-1410	121	109	is	be	AUX
ap-1410	121	110	a	a	DET
ap-1410	121	111	sub	sub	ADJ
ap-1410	121	112	-	-	ADJ
ap-1410	121	113	generalized	generalized	ADJ
ap-1410	121	114	effect	effect	NOUN
ap-1410	121	115	algebra	algebra	NOUN
ap-1410	121	116	of	of	ADP
ap-1410	121	117	e1	e1	PROPN
ap-1410	121	118	.	.	PUNCT
ap-1410	122	1	the	the	DET
ap-1410	122	2	following	follow	VERB
ap-1410	122	3	example	example	NOUN
ap-1410	122	4	shows	show	VERB
ap-1410	122	5	that	that	SCONJ
ap-1410	122	6	the	the	DET
ap-1410	122	7	converses	converse	NOUN
ap-1410	122	8	of	of	ADP
ap-1410	122	9	assertions	assertion	NOUN
ap-1410	122	10	(	(	PUNCT
ap-1410	122	11	i)–(iii	i)–(iii	NOUN
ap-1410	122	12	)	)	PUNCT
ap-1410	122	13	do	do	AUX
ap-1410	122	14	not	not	PART
ap-1410	122	15	hold	hold	VERB
ap-1410	122	16	.	.	PUNCT
ap-1410	123	1	example	example	NOUN
ap-1410	123	2	10	10	NUM
ap-1410	123	3	let	let	VERB
ap-1410	123	4	e	e	NOUN
ap-1410	123	5	=	=	PRON
ap-1410	123	6	{	{	PUNCT
ap-1410	123	7	0	0	NUM
ap-1410	123	8	,	,	PUNCT
ap-1410	123	9	1	1	NUM
ap-1410	123	10	,	,	PUNCT
ap-1410	123	11	2	2	NUM
ap-1410	123	12	,	,	PUNCT
ap-1410	123	13	.	.	PUNCT
ap-1410	123	14	.	.	PUNCT
ap-1410	124	1	.	.	PUNCT
ap-1410	124	2	}	}	PUNCT
ap-1410	125	1	and	and	CCONJ
ap-1410	125	2	g	g	NOUN
ap-1410	125	3	=	=	SYM
ap-1410	125	4	{	{	PUNCT
ap-1410	125	5	0	0	NUM
ap-1410	125	6	,	,	PUNCT
ap-1410	125	7	1	1	NUM
ap-1410	125	8	,	,	PUNCT
ap-1410	125	9	2	2	NUM
ap-1410	125	10	,	,	PUNCT
ap-1410	125	11	4	4	NUM
ap-1410	125	12	,	,	PUNCT
ap-1410	125	13	6	6	NUM
ap-1410	125	14	,	,	PUNCT
ap-1410	125	15	.	.	PUNCT
ap-1410	125	16	.	.	PUNCT
ap-1410	125	17	.	.	PUNCT
ap-1410	125	18	}	}	PUNCT
ap-1410	125	19	.	.	PUNCT
ap-1410	126	1	define	define	VERB
ap-1410	126	2	the	the	DET
ap-1410	126	3	partial	partial	ADJ
ap-1410	126	4	binary	binary	ADJ
ap-1410	126	5	operations	operation	NOUN
ap-1410	126	6	⊕1	⊕1	PROPN
ap-1410	126	7	and	and	CCONJ
ap-1410	126	8	⊕2	⊕2	PROPN
ap-1410	126	9	for	for	ADP
ap-1410	126	10	a	a	DET
ap-1410	126	11	,	,	PUNCT
ap-1410	126	12	b	b	X
ap-1410	126	13	∈	∈	PROPN
ap-1410	126	14	e	e	NOUN
ap-1410	126	15	a	a	DET
ap-1410	126	16	⊕1	⊕1	PROPN
ap-1410	126	17	b	b	NOUN
ap-1410	126	18	=	=	PUNCT
ap-1410	126	19	{	{	PUNCT
ap-1410	126	20	a	a	DET
ap-1410	126	21	+	+	X
ap-1410	126	22	b	b	NOUN
ap-1410	126	23	,	,	PUNCT
ap-1410	126	24	if	if	SCONJ
ap-1410	126	25	a	a	PRON
ap-1410	126	26	=	=	NOUN
ap-1410	126	27	0	0	NUM
ap-1410	126	28	or	or	CCONJ
ap-1410	126	29	both	both	CCONJ
ap-1410	126	30	a	a	PRON
ap-1410	126	31	,	,	PUNCT
ap-1410	126	32	b	b	NOUN
ap-1410	126	33	are	be	AUX
ap-1410	126	34	even	even	ADV
ap-1410	126	35	,	,	PUNCT
ap-1410	126	36	not	not	PART
ap-1410	126	37	defined	define	VERB
ap-1410	126	38	,	,	PUNCT
ap-1410	126	39	otherwise	otherwise	ADV
ap-1410	126	40	,	,	PUNCT
ap-1410	126	41	(	(	PUNCT
ap-1410	126	42	4	4	X
ap-1410	126	43	)	)	PUNCT
ap-1410	127	1	a	a	DET
ap-1410	127	2	⊕2	⊕2	PROPN
ap-1410	127	3	b	b	PROPN
ap-1410	127	4	=	=	PUNCT
ap-1410	127	5	a	a	PROPN
ap-1410	127	6	+	+	X
ap-1410	127	7	b	b	NOUN
ap-1410	127	8	for	for	ADP
ap-1410	127	9	all	all	DET
ap-1410	127	10	a	a	PRON
ap-1410	127	11	,	,	PUNCT
ap-1410	127	12	b	b	X
ap-1410	127	13	∈	∈	PROPN
ap-1410	127	14	e	e	X
ap-1410	127	15	.	.	PUNCT
ap-1410	128	1	(	(	PUNCT
ap-1410	128	2	5	5	NUM
ap-1410	128	3	)	)	PUNCT
ap-1410	128	4	and	and	CCONJ
ap-1410	128	5	let	let	VERB
ap-1410	128	6	≤1	≤1	PROPN
ap-1410	128	7	and	and	CCONJ
ap-1410	128	8	≤2	≤2	NOUN
ap-1410	128	9	be	be	AUX
ap-1410	128	10	the	the	DET
ap-1410	128	11	corresponding	corresponding	ADJ
ap-1410	128	12	derived	derive	VERB
ap-1410	128	13	partial	partial	ADJ
ap-1410	128	14	orders	order	NOUN
ap-1410	128	15	.	.	PUNCT
ap-1410	129	1	then	then	ADV
ap-1410	129	2	obviously	obviously	ADV
ap-1410	129	3	⊕1	⊕1	PROPN
ap-1410	129	4	⊂	⊂	ADJ
ap-1410	129	5	⊕2	⊕2	PROPN
ap-1410	129	6	and	and	CCONJ
ap-1410	129	7	(	(	PUNCT
ap-1410	129	8	i	i	NOUN
ap-1410	129	9	)	)	PUNCT
ap-1410	129	10	e1	e1	PROPN
ap-1410	129	11	=	=	SYM
ap-1410	129	12	(	(	PUNCT
ap-1410	129	13	e,⊕1	e,⊕1	PROPN
ap-1410	129	14	,	,	PUNCT
ap-1410	129	15	0	0	NUM
ap-1410	129	16	)	)	PUNCT
ap-1410	129	17	and	and	CCONJ
ap-1410	129	18	e2	e2	PROPN
ap-1410	129	19	=	=	SYM
ap-1410	129	20	(	(	PUNCT
ap-1410	129	21	e,⊕2	e,⊕2	PROPN
ap-1410	129	22	,	,	PUNCT
ap-1410	129	23	0	0	NUM
ap-1410	129	24	)	)	PUNCT
ap-1410	129	25	are	be	AUX
ap-1410	129	26	generalized	generalized	ADJ
ap-1410	129	27	effect	effect	NOUN
ap-1410	129	28	algebras	algebra	NOUN
ap-1410	129	29	.	.	PUNCT
ap-1410	130	1	(	(	PUNCT
ap-1410	130	2	ii	ii	NOUN
ap-1410	130	3	)	)	PUNCT
ap-1410	130	4	g	g	NOUN
ap-1410	130	5	is	be	AUX
ap-1410	130	6	a	a	DET
ap-1410	130	7	sub	sub	ADJ
ap-1410	130	8	-	-	ADJ
ap-1410	130	9	generalized	generalized	ADJ
ap-1410	130	10	effect	effect	NOUN
ap-1410	130	11	algebra	algebra	NOUN
ap-1410	130	12	of	of	ADP
ap-1410	130	13	e1	e1	PROPN
ap-1410	130	14	.	.	PUNCT
ap-1410	131	1	(	(	PUNCT
ap-1410	131	2	iii	iii	X
ap-1410	131	3	)	)	PUNCT
ap-1410	131	4	g	g	NOUN
ap-1410	131	5	is	be	AUX
ap-1410	131	6	not	not	PART
ap-1410	131	7	a	a	DET
ap-1410	131	8	sub	sub	ADJ
ap-1410	131	9	-	-	ADJ
ap-1410	131	10	generalized	generalized	ADJ
ap-1410	131	11	effect	effect	NOUN
ap-1410	131	12	algebra	algebra	NOUN
ap-1410	131	13	of	of	ADP
ap-1410	131	14	e2	e2	PROPN
ap-1410	131	15	.	.	PUNCT
ap-1410	132	1	(	(	PUNCT
ap-1410	132	2	iv	iv	X
ap-1410	132	3	)	)	PUNCT
ap-1410	132	4	there	there	PRON
ap-1410	132	5	exist	exist	VERB
ap-1410	132	6	a	a	DET
ap-1410	132	7	,	,	PUNCT
ap-1410	132	8	b	b	X
ap-1410	132	9	∈	∈	PROPN
ap-1410	132	10	e	e	NOUN
ap-1410	132	11	for	for	ADP
ap-1410	132	12	which	which	PRON
ap-1410	132	13	a	a	DET
ap-1410	132	14	≤2	≤2	NOUN
ap-1410	132	15	b	b	NOUN
ap-1410	132	16	but	but	CCONJ
ap-1410	132	17	a	a	DET
ap-1410	132	18	�	�	PROPN
ap-1410	132	19	≤1	≤1	PROPN
ap-1410	132	20	b.	b.	PROPN
ap-1410	132	21	(	(	PUNCT
ap-1410	132	22	v	v	NOUN
ap-1410	132	23	)	)	PUNCT
ap-1410	132	24	there	there	PRON
ap-1410	132	25	exist	exist	VERB
ap-1410	132	26	q	q	PROPN
ap-1410	132	27	∈	∈	PROPN
ap-1410	132	28	e	e	NOUN
ap-1410	132	29	for	for	ADP
ap-1410	132	30	which	which	PRON
ap-1410	132	31	[	[	X
ap-1410	132	32	0	0	NUM
ap-1410	132	33	,	,	PUNCT
ap-1410	132	34	q]e2	q]e2	VERB
ap-1410	132	35	�	�	NOUN
ap-1410	132	36	⊆	⊆	NUM
ap-1410	132	37	[	[	X
ap-1410	132	38	0	0	NUM
ap-1410	132	39	,	,	PUNCT
ap-1410	132	40	q]e1	q]e1	NOUN
ap-1410	132	41	.	.	PUNCT
ap-1410	133	1	let	let	VERB
ap-1410	133	2	us	we	PRON
ap-1410	133	3	prove	prove	VERB
ap-1410	133	4	conditions	condition	NOUN
ap-1410	133	5	(	(	PUNCT
ap-1410	133	6	i)–(v	i)–(v	NOUN
ap-1410	133	7	)	)	PUNCT
ap-1410	133	8	.	.	PUNCT
ap-1410	134	1	(	(	PUNCT
ap-1410	134	2	i	i	NOUN
ap-1410	134	3	)	)	PUNCT
ap-1410	134	4	let	let	VERB
ap-1410	134	5	us	we	PRON
ap-1410	134	6	show	show	VERB
ap-1410	134	7	that	that	SCONJ
ap-1410	134	8	⊕1	⊕1	PROPN
ap-1410	134	9	is	be	AUX
ap-1410	134	10	associative	associative	ADJ
ap-1410	134	11	.	.	PUNCT
ap-1410	135	1	to	to	PART
ap-1410	135	2	show	show	VERB
ap-1410	135	3	this	this	PRON
ap-1410	135	4	,	,	PUNCT
ap-1410	135	5	suppose	suppose	VERB
ap-1410	135	6	that	that	SCONJ
ap-1410	135	7	for	for	ADP
ap-1410	135	8	a	a	DET
ap-1410	135	9	,	,	PUNCT
ap-1410	135	10	b	b	NOUN
ap-1410	135	11	,	,	PUNCT
ap-1410	135	12	c	c	PROPN
ap-1410	135	13	∈	∈	PROPN
ap-1410	135	14	e	e	X
ap-1410	135	15	the	the	DET
ap-1410	135	16	sum	sum	NOUN
ap-1410	135	17	(	(	PUNCT
ap-1410	135	18	a⊕1	a⊕1	INTJ
ap-1410	135	19	b)⊕1	b)⊕1	PROPN
ap-1410	135	20	c	c	PROPN
ap-1410	135	21	exists	exist	VERB
ap-1410	135	22	.	.	PUNCT
ap-1410	136	1	first	first	ADV
ap-1410	136	2	,	,	PUNCT
ap-1410	136	3	if	if	SCONJ
ap-1410	136	4	c	c	NOUN
ap-1410	136	5	=	=	SYM
ap-1410	136	6	0	0	PUNCT
ap-1410	137	1	then	then	ADV
ap-1410	137	2	(	(	PUNCT
ap-1410	137	3	a	a	DET
ap-1410	137	4	⊕1	⊕1	PROPN
ap-1410	137	5	b	b	NOUN
ap-1410	137	6	)	)	PUNCT
ap-1410	137	7	⊕1	⊕1	PROPN
ap-1410	137	8	c	c	NOUN
ap-1410	137	9	=	=	PUNCT
ap-1410	137	10	a	a	DET
ap-1410	137	11	⊕1	⊕1	PROPN
ap-1410	137	12	b	b	NOUN
ap-1410	137	13	=	=	PUNCT
ap-1410	137	14	a	a	DET
ap-1410	137	15	⊕1	⊕1	PROPN
ap-1410	137	16	(	(	PUNCT
ap-1410	137	17	b	b	NOUN
ap-1410	137	18	⊕1	⊕1	PROPN
ap-1410	137	19	c	c	NOUN
ap-1410	137	20	)	)	PUNCT
ap-1410	137	21	.	.	PUNCT
ap-1410	138	1	next	next	ADV
ap-1410	138	2	,	,	PUNCT
ap-1410	138	3	if	if	SCONJ
ap-1410	138	4	c	c	PROPN
ap-1410	138	5	�	�	PROPN
ap-1410	138	6	=	=	NOUN
ap-1410	138	7	0	0	NUM
ap-1410	139	1	then	then	ADV
ap-1410	139	2	either	either	CCONJ
ap-1410	139	3	a	a	DET
ap-1410	139	4	=	=	SYM
ap-1410	139	5	b	b	NOUN
ap-1410	139	6	=	=	SYM
ap-1410	139	7	0	0	NUM
ap-1410	139	8	or	or	CCONJ
ap-1410	139	9	both	both	DET
ap-1410	139	10	c	c	PROPN
ap-1410	139	11	and	and	CCONJ
ap-1410	139	12	a	a	DET
ap-1410	139	13	+	+	NOUN
ap-1410	139	14	b	b	NOUN
ap-1410	139	15	are	be	AUX
ap-1410	139	16	even	even	ADV
ap-1410	139	17	.	.	PUNCT
ap-1410	140	1	if	if	SCONJ
ap-1410	140	2	a	a	DET
ap-1410	140	3	=	=	SYM
ap-1410	140	4	b	b	NOUN
ap-1410	140	5	=	=	SYM
ap-1410	140	6	0	0	PROPN
ap-1410	141	1	then	then	ADV
ap-1410	141	2	(	(	PUNCT
ap-1410	141	3	a⊕1	a⊕1	PROPN
ap-1410	141	4	b	b	X
ap-1410	141	5	)	)	PUNCT
ap-1410	141	6	⊕1	⊕1	PROPN
ap-1410	141	7	c	c	NOUN
ap-1410	141	8	=	=	SYM
ap-1410	141	9	a⊕1	a⊕1	PROPN
ap-1410	141	10	(	(	PUNCT
ap-1410	141	11	b⊕1	b⊕1	NOUN
ap-1410	141	12	c	c	NOUN
ap-1410	141	13	)	)	PUNCT
ap-1410	141	14	is	be	AUX
ap-1410	141	15	obvious	obvious	ADJ
ap-1410	141	16	.	.	PUNCT
ap-1410	142	1	a	a	DET
ap-1410	142	2	+	+	NUM
ap-1410	142	3	b	b	NOUN
ap-1410	142	4	is	be	AUX
ap-1410	142	5	even	even	ADV
ap-1410	142	6	if	if	SCONJ
ap-1410	142	7	both	both	DET
ap-1410	142	8	a	a	PRON
ap-1410	142	9	,	,	PUNCT
ap-1410	142	10	b	b	NOUN
ap-1410	142	11	are	be	AUX
ap-1410	142	12	odd	odd	ADJ
ap-1410	142	13	,	,	PUNCT
ap-1410	142	14	but	but	CCONJ
ap-1410	142	15	this	this	PRON
ap-1410	142	16	is	be	AUX
ap-1410	142	17	impossible	impossible	ADJ
ap-1410	142	18	because	because	SCONJ
ap-1410	142	19	a⊕1	a⊕1	PROPN
ap-1410	142	20	b	b	PROPN
ap-1410	142	21	exists	exist	VERB
ap-1410	142	22	.	.	PUNCT
ap-1410	143	1	so	so	ADV
ap-1410	143	2	,	,	PUNCT
ap-1410	143	3	both	both	CCONJ
ap-1410	143	4	a	a	DET
ap-1410	143	5	,	,	PUNCT
ap-1410	143	6	b	b	NOUN
ap-1410	143	7	are	be	AUX
ap-1410	143	8	even	even	ADV
ap-1410	143	9	and	and	CCONJ
ap-1410	143	10	,	,	PUNCT
ap-1410	143	11	then	then	ADV
ap-1410	143	12	,	,	PUNCT
ap-1410	143	13	again	again	ADV
ap-1410	143	14	,	,	PUNCT
ap-1410	143	15	(	(	PUNCT
ap-1410	143	16	a	a	DET
ap-1410	143	17	⊕1	⊕1	PROPN
ap-1410	143	18	b	b	NOUN
ap-1410	143	19	)	)	PUNCT
ap-1410	143	20	⊕1	⊕1	PROPN
ap-1410	143	21	c	c	NOUN
ap-1410	143	22	=	=	PUNCT
ap-1410	143	23	a	a	DET
ap-1410	143	24	⊕1	⊕1	PROPN
ap-1410	143	25	(	(	PUNCT
ap-1410	143	26	b	b	NOUN
ap-1410	143	27	⊕1	⊕1	PROPN
ap-1410	143	28	c	c	NOUN
ap-1410	143	29	)	)	PUNCT
ap-1410	143	30	is	be	AUX
ap-1410	143	31	obvious	obvious	ADJ
ap-1410	143	32	.	.	PUNCT
ap-1410	144	1	the	the	DET
ap-1410	144	2	rest	rest	NOUN
ap-1410	144	3	of	of	ADP
ap-1410	144	4	the	the	DET
ap-1410	144	5	proof	proof	NOUN
ap-1410	144	6	of	of	ADP
ap-1410	144	7	(	(	PUNCT
ap-1410	144	8	i	i	NOUN
ap-1410	144	9	)	)	PUNCT
ap-1410	144	10	is	be	AUX
ap-1410	144	11	obvious	obvious	ADJ
ap-1410	144	12	.	.	PUNCT
ap-1410	145	1	(	(	PUNCT
ap-1410	145	2	ii	ii	NOUN
ap-1410	145	3	)	)	PUNCT
ap-1410	145	4	suppose	suppose	VERB
ap-1410	145	5	that	that	SCONJ
ap-1410	145	6	a	a	DET
ap-1410	145	7	,	,	PUNCT
ap-1410	145	8	b	b	NOUN
ap-1410	145	9	,	,	PUNCT
ap-1410	145	10	c	c	PROPN
ap-1410	145	11	∈	∈	PROPN
ap-1410	145	12	e	e	NOUN
ap-1410	145	13	satisfy	satisfy	VERB
ap-1410	145	14	a	a	DET
ap-1410	145	15	⊕1	⊕1	PROPN
ap-1410	145	16	b	b	NOUN
ap-1410	145	17	=	=	SYM
ap-1410	145	18	c.	c.	NOUN
ap-1410	145	19	if	if	SCONJ
ap-1410	145	20	a	a	DET
ap-1410	145	21	,	,	PUNCT
ap-1410	145	22	b	b	X
ap-1410	145	23	∈	∈	PROPN
ap-1410	145	24	g	g	NOUN
ap-1410	145	25	then	then	ADV
ap-1410	145	26	clearly	clearly	ADV
ap-1410	145	27	c	c	VERB
ap-1410	145	28	∈	∈	PROPN
ap-1410	145	29	g	g	PROPN
ap-1410	145	30	as	as	ADV
ap-1410	145	31	well	well	ADV
ap-1410	145	32	.	.	PUNCT
ap-1410	146	1	if	if	SCONJ
ap-1410	146	2	a	a	PRON
ap-1410	146	3	,	,	PUNCT
ap-1410	146	4	c	c	PROPN
ap-1410	146	5	∈	∈	PROPN
ap-1410	146	6	g	g	PROPN
ap-1410	146	7	and	and	CCONJ
ap-1410	146	8	a	a	DET
ap-1410	146	9	=	=	NOUN
ap-1410	146	10	0	0	NUM
ap-1410	146	11	,	,	PUNCT
ap-1410	146	12	then	then	ADV
ap-1410	146	13	b	b	X
ap-1410	146	14	=	=	SYM
ap-1410	146	15	c	c	NOUN
ap-1410	146	16	,	,	PUNCT
ap-1410	146	17	hence	hence	ADV
ap-1410	146	18	b	b	PROPN
ap-1410	146	19	∈	∈	PROPN
ap-1410	146	20	g.	g.	PROPN
ap-1410	146	21	a	a	NOUN
ap-1410	146	22	,	,	PUNCT
ap-1410	146	23	c	c	PROPN
ap-1410	146	24	∈	∈	PROPN
ap-1410	146	25	g	g	PROPN
ap-1410	146	26	are	be	AUX
ap-1410	146	27	nonzero	nonzero	NOUN
ap-1410	146	28	then	then	ADV
ap-1410	146	29	both	both	PRON
ap-1410	146	30	are	be	AUX
ap-1410	146	31	even	even	ADV
ap-1410	146	32	and	and	CCONJ
ap-1410	146	33	then	then	ADV
ap-1410	146	34	b	b	X
ap-1410	146	35	is	be	AUX
ap-1410	146	36	even	even	ADV
ap-1410	146	37	,	,	PUNCT
ap-1410	146	38	as	as	ADV
ap-1410	146	39	well	well	ADV
ap-1410	146	40	.	.	PUNCT
ap-1410	147	1	so	so	ADV
ap-1410	147	2	,	,	PUNCT
ap-1410	147	3	if	if	SCONJ
ap-1410	147	4	two	two	NUM
ap-1410	147	5	elements	element	NOUN
ap-1410	147	6	out	out	ADP
ap-1410	147	7	of	of	ADP
ap-1410	147	8	a	a	DET
ap-1410	147	9	,	,	PUNCT
ap-1410	147	10	b	b	NOUN
ap-1410	147	11	,	,	PUNCT
ap-1410	147	12	c	c	PROPN
ap-1410	147	13	are	be	AUX
ap-1410	147	14	in	in	ADP
ap-1410	147	15	g	g	NOUN
ap-1410	147	16	,	,	PUNCT
ap-1410	147	17	then	then	ADV
ap-1410	147	18	all	all	DET
ap-1410	147	19	three	three	NUM
ap-1410	147	20	are	be	AUX
ap-1410	147	21	in	in	ADP
ap-1410	147	22	g.	g.	PROPN
ap-1410	147	23	this	this	PRON
ap-1410	147	24	proves	prove	VERB
ap-1410	147	25	(	(	PUNCT
ap-1410	147	26	ii	ii	NOUN
ap-1410	147	27	)	)	PUNCT
ap-1410	147	28	.	.	PUNCT
ap-1410	148	1	(	(	PUNCT
ap-1410	148	2	iii	iii	X
ap-1410	148	3	)	)	PUNCT
ap-1410	148	4	put	put	VERB
ap-1410	148	5	a	a	DET
ap-1410	148	6	=	=	NOUN
ap-1410	148	7	1	1	NUM
ap-1410	148	8	,	,	PUNCT
ap-1410	148	9	b	b	NOUN
ap-1410	148	10	=	=	SYM
ap-1410	148	11	2	2	X
ap-1410	148	12	.	.	PUNCT
ap-1410	148	13	now	now	ADV
ap-1410	148	14	a	a	DET
ap-1410	148	15	,	,	PUNCT
ap-1410	148	16	b	b	PROPN
ap-1410	148	17	∈	∈	PROPN
ap-1410	148	18	g	g	NOUN
ap-1410	148	19	and	and	CCONJ
ap-1410	148	20	a⊕2	a⊕2	NOUN
ap-1410	148	21	b	b	X
ap-1410	148	22	=	=	SYM
ap-1410	148	23	3	3	NUM
ap-1410	148	24	/∈	/∈	NOUN
ap-1410	148	25	g	g	NOUN
ap-1410	148	26	shows	show	VERB
ap-1410	148	27	that	that	SCONJ
ap-1410	148	28	g	g	PROPN
ap-1410	148	29	is	be	AUX
ap-1410	148	30	not	not	PART
ap-1410	148	31	a	a	DET
ap-1410	148	32	sub	sub	ADJ
ap-1410	148	33	-	-	ADJ
ap-1410	148	34	generalized	generalized	ADJ
ap-1410	148	35	effect	effect	NOUN
ap-1410	148	36	algebra	algebra	NOUN
ap-1410	148	37	of	of	ADP
ap-1410	148	38	e2	e2	PROPN
ap-1410	148	39	.	.	PUNCT
ap-1410	149	1	(	(	PUNCT
ap-1410	149	2	v	v	NOUN
ap-1410	149	3	)	)	PUNCT
ap-1410	149	4	clearly	clearly	ADV
ap-1410	149	5	,	,	PUNCT
ap-1410	149	6	[	[	X
ap-1410	149	7	0	0	NUM
ap-1410	149	8	,	,	PUNCT
ap-1410	149	9	q]e1	q]e1	NOUN
ap-1410	149	10	=	=	PUNCT
ap-1410	149	11	{	{	PUNCT
ap-1410	149	12	0	0	NUM
ap-1410	149	13	,	,	PUNCT
ap-1410	149	14	q	q	NOUN
ap-1410	149	15	}	}	PUNCT
ap-1410	149	16	for	for	ADP
ap-1410	149	17	any	any	DET
ap-1410	149	18	odd	odd	ADJ
ap-1410	149	19	q.	q.	NOUN
ap-1410	150	1	so	so	ADV
ap-1410	150	2	,	,	PUNCT
ap-1410	150	3	e.g.	e.g.	ADV
ap-1410	150	4	[	[	X
ap-1410	150	5	0	0	NUM
ap-1410	150	6	,	,	PUNCT
ap-1410	150	7	3]e2	3]e2	NUM
ap-1410	150	8	=	=	SYM
ap-1410	150	9	{	{	PUNCT
ap-1410	150	10	0	0	NUM
ap-1410	150	11	,	,	PUNCT
ap-1410	150	12	1	1	NUM
ap-1410	150	13	,	,	PUNCT
ap-1410	150	14	2	2	NUM
ap-1410	150	15	,	,	PUNCT
ap-1410	150	16	3	3	NUM
ap-1410	150	17	}	}	PUNCT
ap-1410	150	18	�	�	NOUN
ap-1410	150	19	⊆	⊆	NUM
ap-1410	150	20	[	[	X
ap-1410	150	21	0	0	NUM
ap-1410	150	22	,	,	PUNCT
ap-1410	150	23	3]e1	3]e1	NUM
ap-1410	150	24	.	.	PUNCT
ap-1410	151	1	(	(	PUNCT
ap-1410	151	2	iv	iv	X
ap-1410	151	3	)	)	PUNCT
ap-1410	151	4	obviously	obviously	ADV
ap-1410	151	5	(	(	PUNCT
ap-1410	151	6	v	v	NOUN
ap-1410	151	7	)	)	PUNCT
ap-1410	151	8	implies	imply	VERB
ap-1410	151	9	(	(	PUNCT
ap-1410	151	10	iv	iv	X
ap-1410	151	11	)	)	PUNCT
ap-1410	151	12	.	.	PUNCT
ap-1410	152	1	4	4	NUM
ap-1410	152	2	extensions	extension	NOUN
ap-1410	152	3	of	of	ADP
ap-1410	152	4	operator	operator	NOUN
ap-1410	152	5	effect	effect	NOUN
ap-1410	152	6	algebra	algebra	NOUN
ap-1410	152	7	operations	operation	NOUN
ap-1410	152	8	we	we	PRON
ap-1410	152	9	introduce	introduce	VERB
ap-1410	152	10	examples	example	NOUN
ap-1410	152	11	of	of	ADP
ap-1410	152	12	operator	operator	NOUN
ap-1410	152	13	generalized	generalize	VERB
ap-1410	152	14	effect	effect	NOUN
ap-1410	152	15	algebra	algebra	NOUN
ap-1410	152	16	with	with	ADP
ap-1410	152	17	the	the	DET
ap-1410	152	18	same	same	ADJ
ap-1410	152	19	set	set	NOUN
ap-1410	152	20	of	of	ADP
ap-1410	152	21	elements	element	NOUN
ap-1410	152	22	and	and	CCONJ
ap-1410	152	23	different	different	ADJ
ap-1410	152	24	operations	operation	NOUN
ap-1410	152	25	.	.	PUNCT
ap-1410	153	1	moreover	moreover	ADV
ap-1410	153	2	,	,	PUNCT
ap-1410	153	3	we	we	PRON
ap-1410	153	4	consider	consider	VERB
ap-1410	153	5	intervals	interval	NOUN
ap-1410	153	6	in	in	ADP
ap-1410	153	7	these	these	DET
ap-1410	153	8	algebras	algebra	NOUN
ap-1410	153	9	.	.	PUNCT
ap-1410	154	1	75	75	NUM
ap-1410	154	2	acta	acta	PROPN
ap-1410	154	3	polytechnica	polytechnica	PROPN
ap-1410	154	4	vol	vol	NOUN
ap-1410	154	5	.	.	PUNCT
ap-1410	155	1	51	51	NUM
ap-1410	155	2	no	no	INTJ
ap-1410	155	3	.	.	PUNCT
ap-1410	156	1	4/2011	4/2011	NUM
ap-1410	156	2	a	a	DET
ap-1410	156	3	generalized	generalized	ADJ
ap-1410	156	4	effect	effect	NOUN
ap-1410	156	5	algebra	algebra	NOUN
ap-1410	156	6	whose	whose	DET
ap-1410	156	7	elements	element	NOUN
ap-1410	156	8	are	be	AUX
ap-1410	156	9	positive	positive	ADJ
ap-1410	156	10	linear	linear	ADJ
ap-1410	156	11	operators	operator	NOUN
ap-1410	156	12	on	on	ADP
ap-1410	156	13	a	a	DET
ap-1410	156	14	complex	complex	ADJ
ap-1410	156	15	hilbert	hilbert	NOUN
ap-1410	156	16	space	space	NOUN
ap-1410	156	17	h	h	NOUN
ap-1410	156	18	is	be	AUX
ap-1410	156	19	called	call	VERB
ap-1410	156	20	an	an	DET
ap-1410	156	21	operator	operator	NOUN
ap-1410	156	22	generalized	generalize	VERB
ap-1410	156	23	effect	effect	NOUN
ap-1410	156	24	algebra	algebra	NOUN
ap-1410	156	25	.	.	PUNCT
ap-1410	157	1	definition	definition	NOUN
ap-1410	157	2	11	11	NUM
ap-1410	157	3	assume	assume	VERB
ap-1410	157	4	that	that	SCONJ
ap-1410	157	5	h	h	NOUN
ap-1410	157	6	is	be	AUX
ap-1410	157	7	an	an	DET
ap-1410	157	8	infinitedimensional	infinitedimensional	ADJ
ap-1410	157	9	hilbert	hilbert	NOUN
ap-1410	157	10	space	space	NOUN
ap-1410	157	11	and	and	CCONJ
ap-1410	157	12	that	that	SCONJ
ap-1410	157	13	a	a	DET
ap-1410	157	14	∈	∈	PROPN
ap-1410	157	15	l(h	l(h	PROPN
ap-1410	157	16	)	)	PUNCT
ap-1410	157	17	is	be	AUX
ap-1410	157	18	positive	positive	ADJ
ap-1410	157	19	.	.	PUNCT
ap-1410	158	1	let	let	VERB
ap-1410	158	2	d	d	PART
ap-1410	158	3	denote	denote	VERB
ap-1410	158	4	the	the	DET
ap-1410	158	5	set	set	NOUN
ap-1410	158	6	of	of	ADP
ap-1410	158	7	all	all	DET
ap-1410	158	8	dense	dense	ADJ
ap-1410	158	9	linear	linear	ADJ
ap-1410	158	10	subspaces	subspace	NOUN
ap-1410	158	11	of	of	ADP
ap-1410	158	12	h	h	NOUN
ap-1410	158	13	and	and	CCONJ
ap-1410	158	14	(	(	PUNCT
ap-1410	158	15	i	i	NOUN
ap-1410	158	16	)	)	PUNCT
ap-1410	158	17	v(h	v(h	NOUN
ap-1410	158	18	)	)	PUNCT
ap-1410	159	1	=	=	PRON
ap-1410	159	2	{	{	PUNCT
ap-1410	159	3	a	a	DET
ap-1410	159	4	∈	∈	NOUN
ap-1410	159	5	l(h	l(h	PROPN
ap-1410	159	6	)	)	PUNCT
ap-1410	159	7	|	|	ADV
ap-1410	159	8	a	a	DET
ap-1410	159	9	≥	≥	NOUN
ap-1410	159	10	0	0	NUM
ap-1410	159	11	,	,	PUNCT
ap-1410	159	12	d(a	d(a	PROPN
ap-1410	159	13	)	)	PUNCT
ap-1410	160	1	=	=	SYM
ap-1410	160	2	h	h	NOUN
ap-1410	160	3	if	if	SCONJ
ap-1410	160	4	a	a	PRON
ap-1410	160	5	is	be	AUX
ap-1410	160	6	bounded	bound	VERB
ap-1410	160	7	and	and	CCONJ
ap-1410	160	8	d(a	d(a	PROPN
ap-1410	160	9	)	)	PUNCT
ap-1410	160	10	∈	∈	PROPN
ap-1410	161	1	d	d	NOUN
ap-1410	161	2	if	if	SCONJ
ap-1410	161	3	a	a	PRON
ap-1410	161	4	is	be	AUX
ap-1410	161	5	unbounded	unbounded	ADJ
ap-1410	161	6	}	}	PUNCT
ap-1410	161	7	.	.	PUNCT
ap-1410	162	1	(	(	PUNCT
ap-1410	162	2	ii	ii	NOUN
ap-1410	162	3	)	)	PUNCT
ap-1410	162	4	gd(h	gd(h	NOUN
ap-1410	162	5	)	)	PUNCT
ap-1410	163	1	=	=	PRON
ap-1410	163	2	{	{	PUNCT
ap-1410	163	3	a	a	DET
ap-1410	163	4	∈	∈	PROPN
ap-1410	163	5	v(h	v(h	NOUN
ap-1410	163	6	)	)	PUNCT
ap-1410	163	7	|	|	ADV
ap-1410	163	8	a	a	PRON
ap-1410	163	9	is	be	AUX
ap-1410	163	10	bounded	bound	VERB
ap-1410	163	11	or	or	CCONJ
ap-1410	163	12	d(a	d(a	PROPN
ap-1410	163	13	)	)	PUNCT
ap-1410	164	1	=	=	PUNCT
ap-1410	164	2	d	d	NOUN
ap-1410	164	3	if	if	SCONJ
ap-1410	164	4	a	a	PRON
ap-1410	164	5	is	be	AUX
ap-1410	164	6	unbounded	unbounded	ADJ
ap-1410	164	7	}	}	PUNCT
ap-1410	164	8	,	,	PUNCT
ap-1410	164	9	d	d	PROPN
ap-1410	164	10	∈	∈	PROPN
ap-1410	164	11	d.	d.	PROPN
ap-1410	164	12	(	(	PUNCT
ap-1410	164	13	iii	iii	NOUN
ap-1410	164	14	)	)	PUNCT
ap-1410	164	15	let	let	VERB
ap-1410	164	16	⊕	⊕	NOUN
ap-1410	164	17	be	be	AUX
ap-1410	164	18	a	a	DET
ap-1410	164	19	partial	partial	ADJ
ap-1410	164	20	binary	binary	ADJ
ap-1410	164	21	operation	operation	NOUN
ap-1410	164	22	on	on	ADP
ap-1410	164	23	v(h	v(h	NOUN
ap-1410	164	24	)	)	PUNCT
ap-1410	164	25	defined	define	VERB
ap-1410	164	26	by	by	ADP
ap-1410	164	27	:	:	PUNCT
ap-1410	164	28	for	for	ADP
ap-1410	164	29	a	a	DET
ap-1410	164	30	,	,	PUNCT
ap-1410	164	31	b	b	PROPN
ap-1410	164	32	∈	∈	PROPN
ap-1410	164	33	v(h	v(h	NOUN
ap-1410	164	34	)	)	PUNCT
ap-1410	164	35	,	,	PUNCT
ap-1410	164	36	a⊕b	a⊕b	NOUN
ap-1410	164	37	is	be	AUX
ap-1410	164	38	defined	define	VERB
ap-1410	164	39	and	and	CCONJ
ap-1410	164	40	a⊕b	a⊕b	NOUN
ap-1410	164	41	=	=	PUNCT
ap-1410	165	1	a	a	DET
ap-1410	165	2	+	+	NUM
ap-1410	165	3	b	b	NOUN
ap-1410	165	4	(	(	PUNCT
ap-1410	165	5	the	the	DET
ap-1410	165	6	usual	usual	ADJ
ap-1410	165	7	sum	sum	NOUN
ap-1410	165	8	)	)	PUNCT
ap-1410	165	9	iff	iff	NOUN
ap-1410	165	10	at	at	ADV
ap-1410	165	11	least	least	ADV
ap-1410	165	12	one	one	NUM
ap-1410	165	13	out	out	ADP
ap-1410	165	14	of	of	ADP
ap-1410	165	15	operators	operator	NOUN
ap-1410	165	16	a	a	PRON
ap-1410	165	17	,	,	PUNCT
ap-1410	165	18	b	b	PROPN
ap-1410	165	19	is	be	AUX
ap-1410	165	20	bounded	bound	VERB
ap-1410	165	21	.	.	PUNCT
ap-1410	166	1	the	the	DET
ap-1410	166	2	triple	triple	ADJ
ap-1410	166	3	(	(	PUNCT
ap-1410	166	4	v(h);⊕	v(h);⊕	NOUN
ap-1410	166	5	,	,	PUNCT
ap-1410	166	6	0	0	NUM
ap-1410	166	7	)	)	PUNCT
ap-1410	166	8	will	will	AUX
ap-1410	166	9	be	be	AUX
ap-1410	166	10	denoted	denote	VERB
ap-1410	166	11	by	by	ADP
ap-1410	166	12	v(h	v(h	NOUN
ap-1410	166	13	)	)	PUNCT
ap-1410	166	14	for	for	ADP
ap-1410	166	15	short	short	ADJ
ap-1410	166	16	.	.	PUNCT
ap-1410	167	1	(	(	PUNCT
ap-1410	167	2	iv	iv	X
ap-1410	167	3	)	)	PUNCT
ap-1410	167	4	let	let	VERB
ap-1410	167	5	⊕d	⊕d	NOUN
ap-1410	167	6	be	be	AUX
ap-1410	167	7	a	a	DET
ap-1410	167	8	partial	partial	ADJ
ap-1410	167	9	binary	binary	ADJ
ap-1410	167	10	operation	operation	NOUN
ap-1410	167	11	on	on	ADP
ap-1410	167	12	v(h	v(h	NOUN
ap-1410	167	13	)	)	PUNCT
ap-1410	167	14	defined	define	VERB
ap-1410	167	15	by	by	ADP
ap-1410	167	16	:	:	PUNCT
ap-1410	167	17	for	for	ADP
ap-1410	167	18	a	a	DET
ap-1410	167	19	,	,	PUNCT
ap-1410	167	20	b	b	PROPN
ap-1410	167	21	∈	∈	PROPN
ap-1410	167	22	v(h	v(h	NOUN
ap-1410	167	23	)	)	PUNCT
ap-1410	168	1	,	,	PUNCT
ap-1410	168	2	a	a	DET
ap-1410	168	3	⊕d	⊕d	NOUN
ap-1410	168	4	b	b	NOUN
ap-1410	168	5	is	be	AUX
ap-1410	168	6	defined	define	VERB
ap-1410	168	7	and	and	CCONJ
ap-1410	168	8	a	a	DET
ap-1410	168	9	⊕d	⊕d	NOUN
ap-1410	168	10	b	b	X
ap-1410	168	11	=	=	SYM
ap-1410	168	12	a+b	a+b	X
ap-1410	168	13	(	(	PUNCT
ap-1410	168	14	the	the	DET
ap-1410	168	15	usual	usual	ADJ
ap-1410	168	16	sum	sum	NOUN
ap-1410	168	17	)	)	PUNCT
ap-1410	168	18	iff	iff	NOUN
ap-1410	168	19	either	either	CCONJ
ap-1410	168	20	at	at	ADV
ap-1410	168	21	least	least	ADV
ap-1410	168	22	one	one	NUM
ap-1410	168	23	out	out	ADP
ap-1410	168	24	of	of	ADP
ap-1410	168	25	a	a	PRON
ap-1410	168	26	,	,	PUNCT
ap-1410	168	27	b	b	PROPN
ap-1410	168	28	is	be	AUX
ap-1410	168	29	bounded	bound	VERB
ap-1410	168	30	,	,	PUNCT
ap-1410	168	31	or	or	CCONJ
ap-1410	168	32	d(a	d(a	PROPN
ap-1410	168	33	)	)	PUNCT
ap-1410	168	34	=	=	SYM
ap-1410	169	1	d(b	d(b	X
ap-1410	169	2	)	)	PUNCT
ap-1410	169	3	∈	∈	PROPN
ap-1410	170	1	d	d	NOUN
ap-1410	170	2	if	if	SCONJ
ap-1410	170	3	both	both	PRON
ap-1410	170	4	are	be	AUX
ap-1410	170	5	unbounded	unbounde	VERB
ap-1410	170	6	.	.	PUNCT
ap-1410	171	1	the	the	DET
ap-1410	171	2	triple	triple	ADJ
ap-1410	171	3	(	(	PUNCT
ap-1410	171	4	v(h);⊕d	v(h);⊕d	ADJ
ap-1410	171	5	,	,	PUNCT
ap-1410	171	6	0	0	NUM
ap-1410	171	7	)	)	PUNCT
ap-1410	171	8	will	will	AUX
ap-1410	171	9	be	be	AUX
ap-1410	171	10	denoted	denote	VERB
ap-1410	171	11	by	by	ADP
ap-1410	171	12	vd(h	vd(h	NOUN
ap-1410	171	13	)	)	PUNCT
ap-1410	171	14	for	for	ADP
ap-1410	171	15	short	short	ADJ
ap-1410	171	16	.	.	PUNCT
ap-1410	172	1	from	from	ADP
ap-1410	172	2	the	the	DET
ap-1410	172	3	above	above	ADJ
ap-1410	172	4	definition	definition	NOUN
ap-1410	172	5	it	it	PRON
ap-1410	172	6	is	be	AUX
ap-1410	172	7	clear	clear	ADJ
ap-1410	172	8	that	that	SCONJ
ap-1410	172	9	v(h	v(h	NOUN
ap-1410	172	10	)	)	PUNCT
ap-1410	173	1	=	=	VERB
ap-1410	173	2	⋃	⋃	NOUN
ap-1410	173	3	{	{	PUNCT
ap-1410	173	4	gd(h	gd(h	NOUN
ap-1410	173	5	)	)	PUNCT
ap-1410	174	1	|	|	ADV
ap-1410	174	2	d	d	X
ap-1410	174	3	∈	∈	PROPN
ap-1410	174	4	d	d	NOUN
ap-1410	174	5	}	}	PUNCT
ap-1410	174	6	and	and	CCONJ
ap-1410	174	7	for	for	ADP
ap-1410	174	8	d1	d1	PROPN
ap-1410	174	9	�	�	NOUN
ap-1410	174	10	=	=	SYM
ap-1410	174	11	d2	d2	PROPN
ap-1410	174	12	it	it	PRON
ap-1410	174	13	holds	hold	VERB
ap-1410	174	14	gd1	gd1	PROPN
ap-1410	174	15	(	(	PUNCT
ap-1410	174	16	h	h	NOUN
ap-1410	174	17	)	)	PUNCT
ap-1410	174	18	∩	∩	ADJ
ap-1410	174	19	gd1(h	gd1(h	NOUN
ap-1410	174	20	)	)	PUNCT
ap-1410	174	21	=	=	SYM
ap-1410	174	22	b+(h	b+(h	PROPN
ap-1410	174	23	)	)	PUNCT
ap-1410	174	24	=	=	PRON
ap-1410	174	25	{	{	PUNCT
ap-1410	174	26	a	a	DET
ap-1410	174	27	∈	∈	PROPN
ap-1410	174	28	v(h	v(h	NOUN
ap-1410	174	29	)	)	PUNCT
ap-1410	174	30	|	|	ADV
ap-1410	174	31	a	a	PRON
ap-1410	174	32	is	be	AUX
ap-1410	174	33	bounded	bound	VERB
ap-1410	174	34	with	with	ADP
ap-1410	174	35	d(a	d(a	PROPN
ap-1410	174	36	)	)	PUNCT
ap-1410	174	37	=	=	SYM
ap-1410	174	38	h	h	NOUN
ap-1410	174	39	}	}	PUNCT
ap-1410	174	40	.	.	PUNCT
ap-1410	175	1	moreover	moreover	ADV
ap-1410	175	2	,	,	PUNCT
ap-1410	175	3	for	for	ADP
ap-1410	175	4	any	any	DET
ap-1410	175	5	a	a	PRON
ap-1410	175	6	,	,	PUNCT
ap-1410	175	7	b	b	PROPN
ap-1410	175	8	∈	∈	PROPN
ap-1410	175	9	v(h	v(h	NOUN
ap-1410	175	10	)	)	PUNCT
ap-1410	175	11	if	if	SCONJ
ap-1410	175	12	a⊕b	a⊕b	NOUN
ap-1410	175	13	is	be	AUX
ap-1410	175	14	defined	define	VERB
ap-1410	175	15	then	then	ADV
ap-1410	175	16	a⊕db	a⊕db	PROPN
ap-1410	175	17	is	be	AUX
ap-1410	175	18	defined	define	VERB
ap-1410	175	19	and	and	CCONJ
ap-1410	175	20	a⊕db	a⊕db	NOUN
ap-1410	175	21	=	=	SYM
ap-1410	175	22	a⊕b	a⊕b	PROPN
ap-1410	175	23	=	=	SYM
ap-1410	175	24	a+b	a+b	NUM
ap-1410	175	25	.	.	PUNCT
ap-1410	176	1	hence	hence	ADV
ap-1410	176	2	⊕	⊕	PROPN
ap-1410	176	3	⊂	⊂	PROPN
ap-1410	176	4	⊕d	⊕d	PROPN
ap-1410	176	5	.	.	PUNCT
ap-1410	177	1	recently	recently	ADV
ap-1410	177	2	in	in	ADP
ap-1410	177	3	[	[	X
ap-1410	177	4	6	6	NUM
ap-1410	177	5	,	,	PUNCT
ap-1410	177	6	8	8	NUM
ap-1410	177	7	]	]	PUNCT
ap-1410	177	8	the	the	DET
ap-1410	177	9	following	follow	VERB
ap-1410	177	10	theorems	theorem	NOUN
ap-1410	177	11	were	be	AUX
ap-1410	177	12	proved	prove	VERB
ap-1410	177	13	.	.	PUNCT
ap-1410	178	1	theorem	theorem	VERB
ap-1410	178	2	12	12	NUM
ap-1410	179	1	[	[	X
ap-1410	179	2	6	6	NUM
ap-1410	179	3	]	]	PUNCT
ap-1410	179	4	for	for	ADP
ap-1410	179	5	v(h	v(h	NOUN
ap-1410	179	6	)	)	PUNCT
ap-1410	179	7	from	from	ADP
ap-1410	179	8	definition	definition	NOUN
ap-1410	179	9	11	11	NUM
ap-1410	179	10	it	it	PRON
ap-1410	179	11	holds	hold	VERB
ap-1410	179	12	:	:	PUNCT
ap-1410	179	13	(	(	PUNCT
ap-1410	179	14	i	i	NOUN
ap-1410	179	15	)	)	PUNCT
ap-1410	179	16	(	(	PUNCT
ap-1410	179	17	v(h);⊕	v(h);⊕	VERB
ap-1410	179	18	,	,	PUNCT
ap-1410	179	19	0	0	NUM
ap-1410	179	20	)	)	PUNCT
ap-1410	179	21	is	be	AUX
ap-1410	179	22	a	a	DET
ap-1410	179	23	generalized	generalized	ADJ
ap-1410	179	24	effect	effect	NOUN
ap-1410	179	25	algebra	algebra	NOUN
ap-1410	179	26	.	.	PUNCT
ap-1410	180	1	(	(	PUNCT
ap-1410	180	2	ii	ii	NOUN
ap-1410	180	3	)	)	PUNCT
ap-1410	180	4	if	if	SCONJ
ap-1410	180	5	sp(h	sp(h	NOUN
ap-1410	180	6	)	)	PUNCT
ap-1410	180	7	=	=	SYM
ap-1410	180	8	{	{	PUNCT
ap-1410	180	9	a	a	DET
ap-1410	180	10	∈	∈	PROPN
ap-1410	180	11	v(h	v(h	NOUN
ap-1410	180	12	)	)	PUNCT
ap-1410	180	13	|	|	ADV
ap-1410	180	14	a	a	DET
ap-1410	180	15	=	=	NOUN
ap-1410	180	16	a∗	a∗	NOUN
ap-1410	180	17	}	}	PUNCT
ap-1410	180	18	is	be	AUX
ap-1410	180	19	equipped	equip	VERB
ap-1410	180	20	with	with	ADP
ap-1410	180	21	⊕|sp(h	⊕|sp(h	NOUN
ap-1410	180	22	)	)	PUNCT
ap-1410	180	23	,	,	PUNCT
ap-1410	180	24	i.e.	i.e.	X
ap-1410	180	25	,	,	PUNCT
ap-1410	180	26	for	for	ADP
ap-1410	180	27	a	a	DET
ap-1410	180	28	,	,	PUNCT
ap-1410	180	29	b	b	PROPN
ap-1410	180	30	∈	∈	PROPN
ap-1410	180	31	sp(h	sp(h	X
ap-1410	180	32	)	)	PUNCT
ap-1410	180	33	there	there	PRON
ap-1410	180	34	exists	exist	VERB
ap-1410	180	35	a	a	DET
ap-1410	180	36	⊕|sp(h)b	⊕|sp(h)b	NOUN
ap-1410	180	37	=	=	PUNCT
ap-1410	180	38	a	a	DET
ap-1410	180	39	⊕	⊕	PROPN
ap-1410	180	40	b	b	PROPN
ap-1410	180	41	iff	iff	PROPN
ap-1410	180	42	there	there	PRON
ap-1410	180	43	exists	exist	VERB
ap-1410	180	44	a	a	DET
ap-1410	180	45	⊕	⊕	PROPN
ap-1410	180	46	b	b	PROPN
ap-1410	180	47	in	in	ADP
ap-1410	180	48	v(h	v(h	NOUN
ap-1410	180	49	)	)	PUNCT
ap-1410	180	50	,	,	PUNCT
ap-1410	180	51	then	then	ADV
ap-1410	180	52	(	(	PUNCT
ap-1410	180	53	sp(h	sp(h	X
ap-1410	180	54	)	)	PUNCT
ap-1410	180	55	;	;	PUNCT
ap-1410	181	1	⊕|sp(h	⊕|sp(h	PRON
ap-1410	181	2	)	)	PUNCT
ap-1410	181	3	,	,	PUNCT
ap-1410	181	4	0	0	X
ap-1410	181	5	)	)	PUNCT
ap-1410	181	6	is	be	AUX
ap-1410	181	7	a	a	DET
ap-1410	181	8	subgeneralized	subgeneralize	VERB
ap-1410	181	9	effect	effect	NOUN
ap-1410	181	10	algebra	algebra	NOUN
ap-1410	181	11	of	of	ADP
ap-1410	181	12	(	(	PUNCT
ap-1410	181	13	v(h);⊕	v(h);⊕	NOUN
ap-1410	181	14	,	,	PUNCT
ap-1410	181	15	0	0	NUM
ap-1410	181	16	)	)	PUNCT
ap-1410	181	17	.	.	PUNCT
ap-1410	182	1	theorem	theorem	VERB
ap-1410	182	2	13	13	NUM
ap-1410	183	1	[	[	NOUN
ap-1410	183	2	8	8	NUM
ap-1410	183	3	]	]	PUNCT
ap-1410	183	4	using	use	VERB
ap-1410	183	5	the	the	DET
ap-1410	183	6	notation	notation	NOUN
ap-1410	183	7	from	from	ADP
ap-1410	183	8	definition	definition	NOUN
ap-1410	183	9	11	11	NUM
ap-1410	183	10	we	we	PRON
ap-1410	183	11	obtain	obtain	VERB
ap-1410	183	12	:	:	PUNCT
ap-1410	183	13	(	(	PUNCT
ap-1410	183	14	i	i	NOUN
ap-1410	183	15	)	)	PUNCT
ap-1410	183	16	(	(	PUNCT
ap-1410	183	17	v(h);⊕d	v(h);⊕d	ADJ
ap-1410	183	18	,	,	PUNCT
ap-1410	183	19	0	0	NUM
ap-1410	183	20	)	)	PUNCT
ap-1410	183	21	is	be	AUX
ap-1410	183	22	a	a	DET
ap-1410	183	23	generalized	generalized	ADJ
ap-1410	183	24	effect	effect	NOUN
ap-1410	183	25	algebra	algebra	NOUN
ap-1410	183	26	.	.	PUNCT
ap-1410	184	1	(	(	PUNCT
ap-1410	184	2	ii	ii	NOUN
ap-1410	184	3	)	)	PUNCT
ap-1410	184	4	let	let	VERB
ap-1410	184	5	d	d	X
ap-1410	184	6	∈	∈	PROPN
ap-1410	184	7	d	d	AUX
ap-1410	184	8	be	be	AUX
ap-1410	184	9	a	a	DET
ap-1410	184	10	fixed	fix	VERB
ap-1410	184	11	dense	dense	ADJ
ap-1410	184	12	subspace	subspace	NOUN
ap-1410	184	13	of	of	ADP
ap-1410	184	14	h.	h.	PROPN
ap-1410	184	15	let	let	VERB
ap-1410	184	16	gd	gd	NOUN
ap-1410	184	17	,	,	PUNCT
ap-1410	184	18	d(h	d(h	PROPN
ap-1410	184	19	)	)	PUNCT
ap-1410	185	1	=	=	PRON
ap-1410	185	2	{	{	PUNCT
ap-1410	185	3	a	a	DET
ap-1410	185	4	∈	∈	PROPN
ap-1410	185	5	v(h	v(h	NOUN
ap-1410	185	6	)	)	PUNCT
ap-1410	185	7	|	|	ADV
ap-1410	185	8	either	either	CCONJ
ap-1410	185	9	a	a	PRON
ap-1410	185	10	is	be	AUX
ap-1410	185	11	bounded	bound	VERB
ap-1410	185	12	with	with	ADP
ap-1410	185	13	d(a	d(a	PROPN
ap-1410	185	14	)	)	PUNCT
ap-1410	185	15	=	=	SYM
ap-1410	185	16	h	h	NOUN
ap-1410	185	17	or	or	CCONJ
ap-1410	185	18	a	a	PRON
ap-1410	185	19	is	be	AUX
ap-1410	185	20	unbounded	unbounded	ADJ
ap-1410	185	21	with	with	ADP
ap-1410	185	22	d(a	d(a	PROPN
ap-1410	185	23	)	)	PUNCT
ap-1410	185	24	=	=	PUNCT
ap-1410	186	1	d	d	X
ap-1410	186	2	}	}	PUNCT
ap-1410	186	3	.	.	PUNCT
ap-1410	187	1	let	let	VERB
ap-1410	187	2	for	for	ADP
ap-1410	187	3	a	a	DET
ap-1410	187	4	,	,	PUNCT
ap-1410	187	5	b	b	PROPN
ap-1410	187	6	∈	∈	PROPN
ap-1410	187	7	gd	gd	NOUN
ap-1410	187	8	,	,	PUNCT
ap-1410	187	9	d(h	d(h	PROPN
ap-1410	187	10	)	)	PUNCT
ap-1410	187	11	the	the	DET
ap-1410	187	12	sum	sum	NOUN
ap-1410	187	13	a	a	DET
ap-1410	187	14	⊕d|gd	⊕d|gd	NOUN
ap-1410	187	15	,	,	PUNCT
ap-1410	187	16	d(h)b	d(h)b	PROPN
ap-1410	187	17	=	=	PUNCT
ap-1410	187	18	a	a	DET
ap-1410	187	19	⊕d	⊕d	NOUN
ap-1410	187	20	b	b	NOUN
ap-1410	188	1	if	if	SCONJ
ap-1410	188	2	there	there	PRON
ap-1410	188	3	exists	exist	VERB
ap-1410	188	4	a	a	DET
ap-1410	188	5	⊕d	⊕d	NOUN
ap-1410	188	6	b	b	PROPN
ap-1410	188	7	∈	∈	PROPN
ap-1410	188	8	gd	gd	PROPN
ap-1410	188	9	,	,	PUNCT
ap-1410	188	10	d(h	d(h	PROPN
ap-1410	188	11	)	)	PUNCT
ap-1410	188	12	,	,	PUNCT
ap-1410	188	13	otherwise	otherwise	ADV
ap-1410	188	14	a	a	DET
ap-1410	188	15	⊕d|gd	⊕d|gd	NOUN
ap-1410	188	16	,	,	PUNCT
ap-1410	188	17	d(h)b	d(h)b	PROPN
ap-1410	188	18	is	be	AUX
ap-1410	188	19	not	not	PART
ap-1410	188	20	defined	define	VERB
ap-1410	188	21	.	.	PUNCT
ap-1410	189	1	then	then	ADV
ap-1410	189	2	(	(	PUNCT
ap-1410	189	3	gd	gd	NOUN
ap-1410	189	4	,	,	PUNCT
ap-1410	189	5	d(h	d(h	PROPN
ap-1410	189	6	)	)	PUNCT
ap-1410	189	7	;	;	PUNCT
ap-1410	189	8	a	a	DET
ap-1410	189	9	⊕d|gd	⊕d|gd	NOUN
ap-1410	189	10	,	,	PUNCT
ap-1410	189	11	d(h	d(h	PROPN
ap-1410	189	12	)	)	PUNCT
ap-1410	189	13	,	,	PUNCT
ap-1410	189	14	0	0	X
ap-1410	189	15	)	)	PUNCT
ap-1410	189	16	is	be	AUX
ap-1410	189	17	a	a	DET
ap-1410	189	18	sub	sub	ADJ
ap-1410	189	19	-	-	ADJ
ap-1410	189	20	generalized	generalized	ADJ
ap-1410	189	21	effect	effect	NOUN
ap-1410	189	22	algebra	algebra	NOUN
ap-1410	189	23	of	of	ADP
ap-1410	189	24	(	(	PUNCT
ap-1410	189	25	v(h),⊕d	v(h),⊕d	NOUN
ap-1410	189	26	,	,	PUNCT
ap-1410	189	27	0	0	NUM
ap-1410	189	28	)	)	PUNCT
ap-1410	189	29	.	.	PUNCT
ap-1410	190	1	now	now	ADV
ap-1410	190	2	,	,	PUNCT
ap-1410	190	3	because	because	SCONJ
ap-1410	190	4	⊕	⊕	PROPN
ap-1410	190	5	⊂	⊂	PROPN
ap-1410	190	6	⊕d	⊕d	PROPN
ap-1410	190	7	on	on	ADP
ap-1410	190	8	v(h	v(h	NOUN
ap-1410	190	9	)	)	PUNCT
ap-1410	190	10	and	and	CCONJ
ap-1410	190	11	,	,	PUNCT
ap-1410	190	12	for	for	ADP
ap-1410	190	13	every	every	DET
ap-1410	190	14	fixed	fix	VERB
ap-1410	190	15	d	d	PROPN
ap-1410	190	16	∈	∈	PROPN
ap-1410	190	17	d	d	PROPN
ap-1410	190	18	,	,	PUNCT
ap-1410	190	19	gd	gd	NOUN
ap-1410	190	20	,	,	PUNCT
ap-1410	190	21	d(h	d(h	PROPN
ap-1410	190	22	)	)	PUNCT
ap-1410	190	23	is	be	AUX
ap-1410	190	24	a	a	DET
ap-1410	190	25	sub	sub	ADJ
ap-1410	190	26	-	-	ADJ
ap-1410	190	27	generalized	generalized	ADJ
ap-1410	190	28	effect	effect	NOUN
ap-1410	190	29	algebra	algebra	NOUN
ap-1410	190	30	of	of	ADP
ap-1410	190	31	(	(	PUNCT
ap-1410	190	32	v(h);⊕d	v(h);⊕d	ADJ
ap-1410	190	33	,	,	PUNCT
ap-1410	190	34	0	0	NUM
ap-1410	190	35	)	)	PUNCT
ap-1410	190	36	we	we	PRON
ap-1410	190	37	obtain	obtain	VERB
ap-1410	190	38	that	that	SCONJ
ap-1410	190	39	gd	gd	NOUN
ap-1410	190	40	,	,	PUNCT
ap-1410	190	41	d(h	d(h	PROPN
ap-1410	190	42	)	)	PUNCT
ap-1410	190	43	is	be	AUX
ap-1410	190	44	also	also	ADV
ap-1410	190	45	a	a	DET
ap-1410	190	46	subgeneralized	subgeneralize	VERB
ap-1410	190	47	effect	effect	NOUN
ap-1410	190	48	algebra	algebra	NOUN
ap-1410	190	49	of	of	ADP
ap-1410	190	50	(	(	PUNCT
ap-1410	190	51	v(h);⊕	v(h);⊕	NOUN
ap-1410	190	52	,	,	PUNCT
ap-1410	190	53	0	0	NUM
ap-1410	190	54	)	)	PUNCT
ap-1410	190	55	.	.	PUNCT
ap-1410	191	1	clearly	clearly	ADV
ap-1410	191	2	,	,	PUNCT
ap-1410	191	3	the	the	DET
ap-1410	191	4	intersection	intersection	NOUN
ap-1410	191	5	of	of	ADP
ap-1410	191	6	two	two	NUM
ap-1410	191	7	sub	sub	ADJ
ap-1410	191	8	-	-	ADJ
ap-1410	191	9	generalized	generalized	ADJ
ap-1410	191	10	effect	effect	NOUN
ap-1410	191	11	algebras	algebra	NOUN
ap-1410	191	12	of	of	ADP
ap-1410	191	13	(	(	PUNCT
ap-1410	191	14	v(h);⊕	v(h);⊕	NOUN
ap-1410	191	15	,	,	PUNCT
ap-1410	191	16	0	0	NUM
ap-1410	191	17	)	)	PUNCT
ap-1410	191	18	is	be	AUX
ap-1410	191	19	again	again	ADV
ap-1410	191	20	its	its	PRON
ap-1410	191	21	sub	sub	ADJ
ap-1410	191	22	-	-	ADJ
ap-1410	191	23	generalized	generalized	ADJ
ap-1410	191	24	effect	effect	NOUN
ap-1410	191	25	algebra	algebra	NOUN
ap-1410	191	26	.	.	PUNCT
ap-1410	192	1	thus	thus	ADV
ap-1410	192	2	we	we	PRON
ap-1410	192	3	obtain	obtain	VERB
ap-1410	192	4	the	the	DET
ap-1410	192	5	following	follow	VERB
ap-1410	192	6	corollary	corollary	NOUN
ap-1410	192	7	of	of	ADP
ap-1410	192	8	theorems	theorem	NOUN
ap-1410	192	9	12	12	NUM
ap-1410	192	10	and	and	CCONJ
ap-1410	192	11	13	13	NUM
ap-1410	192	12	.	.	PUNCT
ap-1410	192	13	theorem	theorem	NOUN
ap-1410	192	14	14	14	NUM
ap-1410	192	15	let	let	VERB
ap-1410	192	16	for	for	ADP
ap-1410	192	17	every	every	DET
ap-1410	192	18	fixed	fix	VERB
ap-1410	192	19	d	d	PROPN
ap-1410	192	20	∈	∈	PROPN
ap-1410	192	21	d	d	PROPN
ap-1410	192	22	,	,	PUNCT
ap-1410	192	23	sp	sp	NOUN
ap-1410	192	24	,	,	PUNCT
ap-1410	192	25	d(h	d(h	PROPN
ap-1410	192	26	)	)	PUNCT
ap-1410	193	1	=	=	PRON
ap-1410	193	2	{	{	PUNCT
ap-1410	193	3	sp(h	sp(h	NOUN
ap-1410	193	4	)	)	PUNCT
ap-1410	193	5	∩	∩	ADJ
ap-1410	193	6	gd	gd	NOUN
ap-1410	193	7	,	,	PUNCT
ap-1410	193	8	d(h	d(h	PROPN
ap-1410	193	9	)	)	PUNCT
ap-1410	193	10	=	=	PRON
ap-1410	193	11	{	{	PUNCT
ap-1410	193	12	a	a	DET
ap-1410	193	13	∈	∈	PROPN
ap-1410	193	14	v(h	v(h	NOUN
ap-1410	193	15	)	)	PUNCT
ap-1410	193	16	|	|	ADV
ap-1410	193	17	a	a	DET
ap-1410	193	18	=	=	X
ap-1410	193	19	a∗	a∗	NOUN
ap-1410	193	20	and	and	CCONJ
ap-1410	193	21	either	either	PRON
ap-1410	193	22	d(a	d(a	PROPN
ap-1410	193	23	)	)	PUNCT
ap-1410	193	24	=	=	SYM
ap-1410	194	1	h	h	NOUN
ap-1410	194	2	,	,	PUNCT
ap-1410	194	3	if	if	SCONJ
ap-1410	194	4	a	a	PRON
ap-1410	194	5	is	be	AUX
ap-1410	194	6	bounded	bound	VERB
ap-1410	194	7	,	,	PUNCT
ap-1410	194	8	or	or	CCONJ
ap-1410	194	9	d(a	d(a	PROPN
ap-1410	194	10	)	)	PUNCT
ap-1410	195	1	=	=	PUNCT
ap-1410	195	2	d	d	NOUN
ap-1410	195	3	if	if	SCONJ
ap-1410	195	4	a	a	PRON
ap-1410	195	5	is	be	AUX
ap-1410	195	6	unbounded	unbounded	ADJ
ap-1410	195	7	}	}	PUNCT
ap-1410	195	8	.	.	PUNCT
ap-1410	196	1	let	let	VERB
ap-1410	196	2	⊕|sp	⊕|sp	ADJ
ap-1410	196	3	,	,	PUNCT
ap-1410	196	4	d(h	d(h	PROPN
ap-1410	196	5	)	)	PUNCT
ap-1410	196	6	on	on	ADP
ap-1410	196	7	sp	sp	NOUN
ap-1410	196	8	,	,	PUNCT
ap-1410	196	9	d(h	d(h	PROPN
ap-1410	196	10	)	)	PUNCT
ap-1410	196	11	be	be	AUX
ap-1410	196	12	defined	define	VERB
ap-1410	196	13	as	as	SCONJ
ap-1410	196	14	follows	follow	VERB
ap-1410	196	15	:	:	PUNCT
ap-1410	196	16	for	for	ADP
ap-1410	196	17	a	a	DET
ap-1410	196	18	,	,	PUNCT
ap-1410	196	19	b	b	PROPN
ap-1410	196	20	∈	∈	PROPN
ap-1410	196	21	sp	sp	NOUN
ap-1410	196	22	,	,	PUNCT
ap-1410	196	23	d(h	d(h	PROPN
ap-1410	196	24	)	)	PUNCT
ap-1410	196	25	,	,	PUNCT
ap-1410	196	26	a	a	DET
ap-1410	196	27	⊕|sp	⊕|sp	ADJ
ap-1410	196	28	,	,	PUNCT
ap-1410	196	29	d(h)b	d(h)b	PROPN
ap-1410	196	30	=	=	PUNCT
ap-1410	196	31	a	a	DET
ap-1410	196	32	⊕	⊕	PROPN
ap-1410	196	33	b	b	PROPN
ap-1410	196	34	iff	iff	PROPN
ap-1410	196	35	a	a	PROPN
ap-1410	196	36	⊕	⊕	PROPN
ap-1410	196	37	b	b	PROPN
ap-1410	196	38	is	be	AUX
ap-1410	196	39	defined	define	VERB
ap-1410	196	40	in	in	ADP
ap-1410	196	41	v(h	v(h	NOUN
ap-1410	196	42	)	)	PUNCT
ap-1410	196	43	.	.	PUNCT
ap-1410	197	1	then	then	ADV
ap-1410	197	2	(	(	PUNCT
ap-1410	197	3	sp	sp	NOUN
ap-1410	197	4	,	,	PUNCT
ap-1410	197	5	d(h	d(h	PROPN
ap-1410	197	6	)	)	PUNCT
ap-1410	197	7	;	;	PUNCT
ap-1410	197	8	⊕|sp	⊕|sp	NUM
ap-1410	197	9	,	,	PUNCT
ap-1410	197	10	d(h	d(h	PROPN
ap-1410	197	11	)	)	PUNCT
ap-1410	197	12	,	,	PUNCT
ap-1410	197	13	0	0	X
ap-1410	197	14	)	)	PUNCT
ap-1410	197	15	is	be	AUX
ap-1410	197	16	a	a	DET
ap-1410	197	17	sub	sub	ADJ
ap-1410	197	18	-	-	ADJ
ap-1410	197	19	generalized	generalized	ADJ
ap-1410	197	20	effect	effect	NOUN
ap-1410	197	21	algebra	algebra	NOUN
ap-1410	197	22	of	of	ADP
ap-1410	197	23	(	(	PUNCT
ap-1410	197	24	sp(h);⊕	sp(h);⊕	PROPN
ap-1410	197	25	,	,	PUNCT
ap-1410	197	26	0	0	NUM
ap-1410	197	27	)	)	PUNCT
ap-1410	197	28	.	.	PUNCT
ap-1410	198	1	the	the	DET
ap-1410	198	2	above	above	ADJ
ap-1410	198	3	observations	observation	NOUN
ap-1410	198	4	show	show	VERB
ap-1410	198	5	that	that	SCONJ
ap-1410	198	6	generalized	generalized	ADJ
ap-1410	198	7	effect	effect	NOUN
ap-1410	198	8	algebra	algebra	NOUN
ap-1410	198	9	vd(h	vd(h	PUNCT
ap-1410	198	10	)	)	PUNCT
ap-1410	198	11	is	be	AUX
ap-1410	198	12	a	a	DET
ap-1410	198	13	pasting	pasting	NOUN
ap-1410	198	14	of	of	ADP
ap-1410	198	15	its	its	PRON
ap-1410	198	16	sub	sub	ADJ
ap-1410	198	17	-	-	ADJ
ap-1410	198	18	generalized	generalized	ADJ
ap-1410	198	19	effect	effect	NOUN
ap-1410	198	20	algebras	algebras	X
ap-1410	198	21	gd	gd	PROPN
ap-1410	198	22	,	,	PUNCT
ap-1410	198	23	d(h	d(h	PROPN
ap-1410	198	24	)	)	PUNCT
ap-1410	198	25	through	through	ADP
ap-1410	198	26	the	the	DET
ap-1410	198	27	sub	sub	ADJ
ap-1410	198	28	-	-	ADJ
ap-1410	198	29	generalized	generalized	ADJ
ap-1410	198	30	effect	effect	NOUN
ap-1410	198	31	algebra	algebra	NOUN
ap-1410	198	32	b+(h	b+(h	PROPN
ap-1410	198	33	)	)	PUNCT
ap-1410	198	34	=	=	PRON
ap-1410	198	35	{	{	PUNCT
ap-1410	198	36	a	a	DET
ap-1410	198	37	∈	∈	PROPN
ap-1410	198	38	v(h	v(h	NOUN
ap-1410	198	39	)	)	PUNCT
ap-1410	198	40	|	|	ADV
ap-1410	198	41	a	a	PRON
ap-1410	198	42	is	be	AUX
ap-1410	198	43	bounded	bound	VERB
ap-1410	198	44	}	}	PUNCT
ap-1410	198	45	.	.	PUNCT
ap-1410	199	1	consequently	consequently	ADV
ap-1410	199	2	,	,	PUNCT
ap-1410	199	3	the	the	DET
ap-1410	199	4	generalized	generalized	ADJ
ap-1410	199	5	effect	effect	NOUN
ap-1410	199	6	algebra	algebra	NOUN
ap-1410	199	7	sp(h	sp(h	NOUN
ap-1410	199	8	)	)	PUNCT
ap-1410	199	9	is	be	AUX
ap-1410	199	10	a	a	DET
ap-1410	199	11	pasting	pasting	NOUN
ap-1410	199	12	of	of	ADP
ap-1410	199	13	its	its	PRON
ap-1410	199	14	sub	sub	ADJ
ap-1410	199	15	-	-	ADJ
ap-1410	199	16	generalized	generalized	ADJ
ap-1410	199	17	effect	effect	NOUN
ap-1410	199	18	algebras	algebras	PROPN
ap-1410	199	19	sp	sp	PROPN
ap-1410	199	20	,	,	PUNCT
ap-1410	199	21	d(h	d(h	PROPN
ap-1410	199	22	)	)	PUNCT
ap-1410	199	23	through	through	ADP
ap-1410	199	24	the	the	DET
ap-1410	199	25	sub	sub	ADJ
ap-1410	199	26	-	-	ADJ
ap-1410	199	27	generalized	generalized	ADJ
ap-1410	199	28	effect	effect	NOUN
ap-1410	199	29	algebra	algebra	PROPN
ap-1410	199	30	b+(h	b+(h	PROPN
ap-1410	199	31	)	)	PUNCT
ap-1410	199	32	.	.	PUNCT
ap-1410	200	1	more	more	ADV
ap-1410	200	2	precisely	precisely	ADV
ap-1410	200	3	:	:	PUNCT
ap-1410	200	4	theorem	theorem	ADJ
ap-1410	200	5	15	15	NUM
ap-1410	200	6	(	(	PUNCT
ap-1410	200	7	pasting	pasting	NOUN
ap-1410	200	8	theorem	theorem	NOUN
ap-1410	200	9	)	)	PUNCT
ap-1410	200	10	(	(	PUNCT
ap-1410	200	11	i	i	NOUN
ap-1410	200	12	)	)	PUNCT
ap-1410	200	13	for	for	ADP
ap-1410	200	14	generalized	generalized	ADJ
ap-1410	200	15	effect	effect	NOUN
ap-1410	200	16	algebra	algebra	NOUN
ap-1410	200	17	vd(h	vd(h	PUNCT
ap-1410	200	18	)	)	PUNCT
ap-1410	201	1	=	=	SYM
ap-1410	201	2	{	{	PUNCT
ap-1410	201	3	v(h);⊕d	v(h);⊕d	ADJ
ap-1410	201	4	,	,	PUNCT
ap-1410	201	5	0	0	NUM
ap-1410	201	6	)	)	PUNCT
ap-1410	201	7	and	and	CCONJ
ap-1410	201	8	its	its	PRON
ap-1410	201	9	sub	sub	ADJ
ap-1410	201	10	-	-	ADJ
ap-1410	201	11	generalized	generalized	ADJ
ap-1410	201	12	effect	effect	NOUN
ap-1410	201	13	algebras	algebras	X
ap-1410	201	14	gd	gd	PROPN
ap-1410	201	15	,	,	PUNCT
ap-1410	201	16	d(h	d(h	PROPN
ap-1410	201	17	)	)	PUNCT
ap-1410	201	18	,	,	PUNCT
ap-1410	201	19	d	d	PROPN
ap-1410	201	20	∈	∈	PROPN
ap-1410	201	21	d	d	NOUN
ap-1410	201	22	,	,	PUNCT
ap-1410	201	23	and	and	CCONJ
ap-1410	201	24	b+(h	b+(h	NUM
ap-1410	201	25	)	)	PUNCT
ap-1410	201	26	(	(	PUNCT
ap-1410	201	27	1	1	X
ap-1410	201	28	)	)	PUNCT
ap-1410	201	29	gd	gd	NOUN
ap-1410	201	30	,	,	PUNCT
ap-1410	201	31	d1(h	d1(h	NOUN
ap-1410	201	32	)	)	PUNCT
ap-1410	201	33	∩	∩	ADJ
ap-1410	201	34	gd	gd	NOUN
ap-1410	201	35	,	,	PUNCT
ap-1410	201	36	d2(h	d2(h	PROPN
ap-1410	201	37	)	)	PUNCT
ap-1410	201	38	=	=	SYM
ap-1410	201	39	b+(h	b+(h	PROPN
ap-1410	201	40	)	)	PUNCT
ap-1410	201	41	for	for	ADP
ap-1410	201	42	every	every	DET
ap-1410	201	43	d1	d1	NOUN
ap-1410	201	44	,	,	PUNCT
ap-1410	201	45	d2	d2	PROPN
ap-1410	201	46	∈	∈	PROPN
ap-1410	201	47	d	d	NOUN
ap-1410	201	48	,	,	PUNCT
ap-1410	201	49	d1	d1	PROPN
ap-1410	201	50	�	�	PROPN
ap-1410	201	51	=	=	SYM
ap-1410	201	52	d2	d2	PROPN
ap-1410	201	53	.	.	PUNCT
ap-1410	202	1	(	(	PUNCT
ap-1410	202	2	2	2	NUM
ap-1410	202	3	)	)	PUNCT
ap-1410	202	4	vd(h	vd(h	PUNCT
ap-1410	202	5	)	)	PUNCT
ap-1410	203	1	=	=	SYM
ap-1410	203	2	⋃	⋃	NOUN
ap-1410	203	3	{	{	PUNCT
ap-1410	203	4	gd	gd	NOUN
ap-1410	203	5	,	,	PUNCT
ap-1410	203	6	d(h	d(h	PROPN
ap-1410	203	7	)	)	PUNCT
ap-1410	204	1	|	|	ADV
ap-1410	204	2	d	d	X
ap-1410	204	3	∈	∈	PROPN
ap-1410	204	4	d	d	NOUN
ap-1410	204	5	}	}	PUNCT
ap-1410	204	6	.	.	PUNCT
ap-1410	205	1	(	(	PUNCT
ap-1410	205	2	ii	ii	NOUN
ap-1410	205	3	)	)	PUNCT
ap-1410	205	4	for	for	ADP
ap-1410	205	5	generalized	generalized	ADJ
ap-1410	205	6	effect	effect	NOUN
ap-1410	205	7	algebra	algebra	NOUN
ap-1410	205	8	sp(h	sp(h	NOUN
ap-1410	205	9	)	)	PUNCT
ap-1410	205	10	and	and	CCONJ
ap-1410	205	11	its	its	PRON
ap-1410	205	12	subgeneralized	subgeneralize	VERB
ap-1410	205	13	effect	effect	NOUN
ap-1410	205	14	algebras	algebras	PROPN
ap-1410	205	15	sp	sp	PROPN
ap-1410	205	16	,	,	PUNCT
ap-1410	205	17	d(h	d(h	PROPN
ap-1410	205	18	)	)	PUNCT
ap-1410	205	19	,	,	PUNCT
ap-1410	206	1	d	d	PROPN
ap-1410	206	2	∈	∈	PROPN
ap-1410	206	3	d	d	X
ap-1410	206	4	(	(	PUNCT
ap-1410	206	5	1	1	NUM
ap-1410	206	6	)	)	PUNCT
ap-1410	206	7	spd1(h	spd1(h	NOUN
ap-1410	206	8	)	)	PUNCT
ap-1410	206	9	∩	∩	ADJ
ap-1410	206	10	spd2	spd2	PROPN
ap-1410	206	11	(	(	PUNCT
ap-1410	206	12	h	h	NOUN
ap-1410	206	13	)	)	PUNCT
ap-1410	206	14	=	=	SYM
ap-1410	206	15	b+(h	b+(h	PROPN
ap-1410	206	16	)	)	PUNCT
ap-1410	206	17	for	for	ADP
ap-1410	206	18	every	every	DET
ap-1410	206	19	d1	d1	NOUN
ap-1410	206	20	,	,	PUNCT
ap-1410	206	21	d2	d2	PROPN
ap-1410	206	22	∈	∈	PROPN
ap-1410	206	23	d	d	NOUN
ap-1410	206	24	,	,	PUNCT
ap-1410	206	25	d1	d1	PROPN
ap-1410	206	26	�	�	PROPN
ap-1410	206	27	=	=	SYM
ap-1410	206	28	d2	d2	PROPN
ap-1410	206	29	.	.	PUNCT
ap-1410	207	1	(	(	PUNCT
ap-1410	207	2	2	2	NUM
ap-1410	207	3	)	)	PUNCT
ap-1410	207	4	sp(h	sp(h	NOUN
ap-1410	207	5	)	)	PUNCT
ap-1410	208	1	=	=	SYM
ap-1410	208	2	⋃	⋃	NOUN
ap-1410	208	3	{	{	PUNCT
ap-1410	208	4	sp	sp	NOUN
ap-1410	208	5	,	,	PUNCT
ap-1410	208	6	d(h	d(h	PROPN
ap-1410	208	7	)	)	PUNCT
ap-1410	209	1	|	|	ADV
ap-1410	209	2	d	d	X
ap-1410	209	3	∈	∈	PROPN
ap-1410	209	4	d	d	NOUN
ap-1410	209	5	}	}	PUNCT
ap-1410	209	6	.	.	PUNCT
ap-1410	210	1	remark	remark	NOUN
ap-1410	210	2	16	16	NUM
ap-1410	211	1	it	it	PRON
ap-1410	211	2	is	be	AUX
ap-1410	211	3	worth	worth	ADJ
ap-1410	211	4	noting	note	VERB
ap-1410	211	5	that	that	SCONJ
ap-1410	211	6	the	the	DET
ap-1410	211	7	operations	operation	NOUN
ap-1410	211	8	⊕	⊕	PROPN
ap-1410	211	9	,	,	PUNCT
ap-1410	211	10	⊕d	⊕d	PROPN
ap-1410	211	11	are	be	AUX
ap-1410	211	12	the	the	DET
ap-1410	211	13	usual	usual	ADJ
ap-1410	211	14	sum	sum	NOUN
ap-1410	211	15	of	of	ADP
ap-1410	211	16	operators	operator	NOUN
ap-1410	211	17	in	in	ADP
ap-1410	211	18	h	h	NOUN
ap-1410	211	19	,	,	PUNCT
ap-1410	211	20	but	but	CCONJ
ap-1410	211	21	only	only	ADV
ap-1410	211	22	partially	partially	ADV
ap-1410	211	23	applied	apply	VERB
ap-1410	211	24	on	on	ADP
ap-1410	211	25	pairs	pair	NOUN
ap-1410	211	26	a	a	DET
ap-1410	211	27	,	,	PUNCT
ap-1410	211	28	b	b	PROPN
ap-1410	211	29	∈	∈	PROPN
ap-1410	211	30	v(h	v(h	NOUN
ap-1410	211	31	)	)	PUNCT
ap-1410	211	32	.	.	PUNCT
ap-1410	212	1	therefore	therefore	ADV
ap-1410	212	2	all	all	DET
ap-1410	212	3	effect	effect	NOUN
ap-1410	212	4	algebra	algebra	NOUN
ap-1410	212	5	operations	operation	NOUN
ap-1410	212	6	⊕	⊕	PROPN
ap-1410	212	7	,	,	PUNCT
ap-1410	212	8	⊕d	⊕d	NOUN
ap-1410	212	9	,	,	PUNCT
ap-1410	212	10	⊕|sp(h	⊕|sp(h	NOUN
ap-1410	212	11	)	)	PUNCT
ap-1410	212	12	,	,	PUNCT
ap-1410	212	13	⊕d|gd	⊕d|gd	NOUN
ap-1410	212	14	,	,	PUNCT
ap-1410	212	15	d(h	d(h	PROPN
ap-1410	212	16	)	)	PUNCT
ap-1410	212	17	,	,	PUNCT
ap-1410	212	18	⊕|sp	⊕|sp	NOUN
ap-1410	212	19	,	,	PUNCT
ap-1410	212	20	d(h	d(h	PROPN
ap-1410	212	21	)	)	PUNCT
ap-1410	212	22	coincide	coincide	NOUN
ap-1410	212	23	with	with	ADP
ap-1410	212	24	the	the	DET
ap-1410	212	25	usual	usual	ADJ
ap-1410	212	26	sum	sum	NOUN
ap-1410	212	27	of	of	ADP
ap-1410	212	28	operators	operator	NOUN
ap-1410	212	29	a+b	a+b	NUM
ap-1410	212	30	if	if	SCONJ
ap-1410	212	31	the	the	DET
ap-1410	212	32	corresponding	corresponding	ADJ
ap-1410	212	33	effect	effect	NOUN
ap-1410	212	34	algebra	algebra	NOUN
ap-1410	212	35	sum	sum	NOUN
ap-1410	212	36	of	of	ADP
ap-1410	212	37	a	a	DET
ap-1410	212	38	and	and	CCONJ
ap-1410	212	39	b	b	NOUN
ap-1410	212	40	exists	exist	NOUN
ap-1410	212	41	.	.	PUNCT
ap-1410	213	1	5	5	NUM
ap-1410	213	2	intervals	interval	NOUN
ap-1410	213	3	in	in	ADP
ap-1410	213	4	generalized	generalized	ADJ
ap-1410	213	5	effect	effect	NOUN
ap-1410	213	6	algebras	algebra	NOUN
ap-1410	213	7	of	of	ADP
ap-1410	213	8	self	self	NOUN
ap-1410	213	9	-	-	PUNCT
ap-1410	213	10	adjoint	adjoint	NOUN
ap-1410	213	11	operators	operator	NOUN
ap-1410	213	12	assume	assume	VERB
ap-1410	213	13	that	that	SCONJ
ap-1410	213	14	(	(	PUNCT
ap-1410	213	15	e;⊕	e;⊕	ADJ
ap-1410	213	16	,	,	PUNCT
ap-1410	213	17	0	0	NUM
ap-1410	213	18	)	)	PUNCT
ap-1410	213	19	is	be	AUX
ap-1410	213	20	a	a	DET
ap-1410	213	21	generalized	generalized	ADJ
ap-1410	213	22	effect	effect	NOUN
ap-1410	213	23	algebra	algebra	NOUN
ap-1410	213	24	.	.	PUNCT
ap-1410	214	1	for	for	ADP
ap-1410	214	2	any	any	DET
ap-1410	214	3	q	q	NOUN
ap-1410	214	4	∈	∈	PROPN
ap-1410	214	5	e	e	NOUN
ap-1410	214	6	,	,	PUNCT
ap-1410	214	7	q	q	PROPN
ap-1410	214	8	�	�	PROPN
ap-1410	214	9	=	=	SYM
ap-1410	214	10	0	0	NUM
ap-1410	214	11	,	,	PUNCT
ap-1410	214	12	let	let	VERB
ap-1410	214	13	[	[	X
ap-1410	214	14	0	0	NUM
ap-1410	214	15	,	,	PUNCT
ap-1410	214	16	q]e	q]e	ADJ
ap-1410	214	17	=	=	PUNCT
ap-1410	214	18	{	{	PUNCT
ap-1410	214	19	a	a	DET
ap-1410	214	20	∈	∈	PROPN
ap-1410	214	21	e	e	NOUN
ap-1410	214	22	|	|	ADV
ap-1410	214	23	there	there	PRON
ap-1410	214	24	exists	exist	VERB
ap-1410	214	25	b	b	PROPN
ap-1410	214	26	∈	∈	PROPN
ap-1410	214	27	e	e	NOUN
ap-1410	214	28	with	with	ADP
ap-1410	214	29	a⊕	a⊕	PROPN
ap-1410	214	30	b	b	PROPN
ap-1410	214	31	=	=	PUNCT
ap-1410	214	32	q	q	AUX
ap-1410	214	33	}	}	PUNCT
ap-1410	214	34	be	be	AUX
ap-1410	214	35	an	an	DET
ap-1410	214	36	interval	interval	NOUN
ap-1410	214	37	in	in	ADP
ap-1410	214	38	(	(	PUNCT
ap-1410	214	39	e;⊕	e;⊕	ADJ
ap-1410	214	40	,	,	PUNCT
ap-1410	214	41	0	0	NUM
ap-1410	214	42	]	]	PUNCT
ap-1410	214	43	.	.	PUNCT
ap-1410	215	1	we	we	PRON
ap-1410	215	2	will	will	AUX
ap-1410	215	3	denote	denote	VERB
ap-1410	215	4	by	by	ADP
ap-1410	215	5	⊕|[0,q]e	⊕|[0,q]e	X
ap-1410	215	6	the	the	DET
ap-1410	215	7	partial	partial	ADJ
ap-1410	215	8	binary	binary	ADJ
ap-1410	215	9	operation	operation	NOUN
ap-1410	215	10	on	on	ADP
ap-1410	215	11	[	[	X
ap-1410	215	12	0	0	NUM
ap-1410	215	13	,	,	PUNCT
ap-1410	215	14	q]e	q]e	ADP
ap-1410	215	15	defined	define	VERB
ap-1410	215	16	as	as	SCONJ
ap-1410	215	17	follows	follow	VERB
ap-1410	215	18	:	:	PUNCT
ap-1410	215	19	76	76	NUM
ap-1410	215	20	acta	acta	PROPN
ap-1410	215	21	polytechnica	polytechnica	PROPN
ap-1410	215	22	vol	vol	NOUN
ap-1410	215	23	.	.	PUNCT
ap-1410	216	1	51	51	NUM
ap-1410	216	2	no	no	INTJ
ap-1410	216	3	.	.	PUNCT
ap-1410	217	1	4/2011	4/2011	NUM
ap-1410	217	2	for	for	ADP
ap-1410	217	3	a	a	DET
ap-1410	217	4	,	,	PUNCT
ap-1410	217	5	b	b	PROPN
ap-1410	217	6	∈	∈	PROPN
ap-1410	218	1	[	[	X
ap-1410	218	2	0	0	NUM
ap-1410	218	3	,	,	PUNCT
ap-1410	218	4	q]e	q]e	ADP
ap-1410	218	5	the	the	DET
ap-1410	218	6	sum	sum	NOUN
ap-1410	218	7	a	a	DET
ap-1410	218	8	⊕|[0,q]e	⊕|[0,q]e	NOUN
ap-1410	218	9	b	b	NOUN
ap-1410	218	10	=	=	SYM
ap-1410	218	11	a⊕	a⊕	PROPN
ap-1410	218	12	b	b	PROPN
ap-1410	218	13	iff	iff	PROPN
ap-1410	218	14	a⊕	a⊕	PROPN
ap-1410	218	15	b	b	PROPN
ap-1410	218	16	is	be	AUX
ap-1410	218	17	defined	define	VERB
ap-1410	218	18	in	in	ADP
ap-1410	218	19	e	e	NOUN
ap-1410	218	20	and	and	CCONJ
ap-1410	218	21	a	a	DET
ap-1410	218	22	⊕	⊕	PROPN
ap-1410	218	23	b	b	PROPN
ap-1410	218	24	∈	∈	PROPN
ap-1410	218	25	[	[	X
ap-1410	218	26	0	0	NUM
ap-1410	218	27	,	,	PUNCT
ap-1410	218	28	q]e	q]e	ADV
ap-1410	218	29	.	.	PUNCT
ap-1410	219	1	it	it	PRON
ap-1410	219	2	is	be	AUX
ap-1410	219	3	known	know	VERB
ap-1410	219	4	that	that	SCONJ
ap-1410	219	5	[	[	X
ap-1410	219	6	0	0	NUM
ap-1410	219	7	,	,	PUNCT
ap-1410	219	8	q]e	q]e	ADP
ap-1410	219	9	equipped	equip	VERB
ap-1410	219	10	with	with	ADP
ap-1410	219	11	⊕|[0,q]e	⊕|[0,q]e	NOUN
ap-1410	219	12	is	be	AUX
ap-1410	219	13	an	an	DET
ap-1410	219	14	effect	effect	NOUN
ap-1410	219	15	algebra	algebra	NOUN
ap-1410	219	16	(	(	PUNCT
ap-1410	219	17	see	see	VERB
ap-1410	219	18	,	,	PUNCT
ap-1410	219	19	e.g.	e.g.	ADV
ap-1410	220	1	[	[	X
ap-1410	220	2	7	7	NUM
ap-1410	220	3	]	]	NUM
ap-1410	220	4	)	)	PUNCT
ap-1410	220	5	.	.	PUNCT
ap-1410	221	1	in	in	ADP
ap-1410	221	2	this	this	DET
ap-1410	221	3	way	way	NOUN
ap-1410	221	4	,	,	PUNCT
ap-1410	221	5	for	for	ADP
ap-1410	221	6	all	all	DET
ap-1410	221	7	nonzero	nonzero	ADJ
ap-1410	221	8	q	q	X
ap-1410	221	9	∈	∈	PROPN
ap-1410	221	10	v(h	v(h	NOUN
ap-1410	221	11	)	)	PUNCT
ap-1410	221	12	,	,	PUNCT
ap-1410	221	13	we	we	PRON
ap-1410	221	14	obtain	obtain	VERB
ap-1410	221	15	the	the	DET
ap-1410	221	16	operator	operator	NOUN
ap-1410	221	17	effect	effect	NOUN
ap-1410	221	18	algebras	algebra	NOUN
ap-1410	221	19	(	(	PUNCT
ap-1410	221	20	[	[	X
ap-1410	221	21	0	0	NUM
ap-1410	221	22	,	,	PUNCT
ap-1410	221	23	q]v(h	q]v(h	NOUN
ap-1410	221	24	)	)	PUNCT
ap-1410	221	25	;	;	PUNCT
ap-1410	221	26	⊕|[0,q]v(h	⊕|[0,q]v(h	NOUN
ap-1410	221	27	)	)	PUNCT
ap-1410	221	28	,	,	PUNCT
ap-1410	221	29	0	0	NUM
ap-1410	221	30	,	,	PUNCT
ap-1410	221	31	q	q	PUNCT
ap-1410	221	32	)	)	PUNCT
ap-1410	221	33	and	and	CCONJ
ap-1410	221	34	(	(	PUNCT
ap-1410	221	35	[	[	X
ap-1410	221	36	0	0	NUM
ap-1410	221	37	,	,	PUNCT
ap-1410	221	38	q]vd(h	q]vd(h	PROPN
ap-1410	221	39	)	)	PUNCT
ap-1410	221	40	;	;	PUNCT
ap-1410	221	41	⊕d|[0,q]vd(h	⊕d|[0,q]vd(h	NUM
ap-1410	221	42	)	)	PUNCT
ap-1410	221	43	,	,	PUNCT
ap-1410	221	44	0	0	NUM
ap-1410	221	45	,	,	PUNCT
ap-1410	221	46	q	q	NOUN
ap-1410	221	47	)	)	PUNCT
ap-1410	221	48	(	(	PUNCT
ap-1410	221	49	see	see	VERB
ap-1410	221	50	[	[	X
ap-1410	221	51	8	8	NUM
ap-1410	221	52	]	]	NUM
ap-1410	221	53	)	)	PUNCT
ap-1410	221	54	.	.	PUNCT
ap-1410	222	1	by	by	ADP
ap-1410	222	2	definition	definition	NOUN
ap-1410	222	3	of	of	ADP
ap-1410	222	4	⊕	⊕	PROPN
ap-1410	222	5	,	,	PUNCT
ap-1410	222	6	⊕d	⊕d	NOUN
ap-1410	222	7	and	and	CCONJ
ap-1410	222	8	the	the	DET
ap-1410	222	9	results	result	NOUN
ap-1410	222	10	of	of	ADP
ap-1410	222	11	section	section	NOUN
ap-1410	222	12	2	2	NUM
ap-1410	222	13	,	,	PUNCT
ap-1410	222	14	it	it	PRON
ap-1410	222	15	is	be	AUX
ap-1410	222	16	clear	clear	ADJ
ap-1410	222	17	that	that	SCONJ
ap-1410	222	18	for	for	ADP
ap-1410	222	19	q	q	PROPN
ap-1410	222	20	∈	∈	PROPN
ap-1410	222	21	v(h	v(h	NOUN
ap-1410	222	22	)	)	PUNCT
ap-1410	222	23	with	with	ADP
ap-1410	222	24	d(q	d(q	NOUN
ap-1410	222	25	)	)	PUNCT
ap-1410	222	26	=	=	PUNCT
ap-1410	222	27	d	d	PUNCT
ap-1410	222	28	∈	∈	PROPN
ap-1410	223	1	d	d	X
ap-1410	223	2	we	we	PRON
ap-1410	223	3	have	have	VERB
ap-1410	223	4	[	[	X
ap-1410	223	5	0	0	NUM
ap-1410	223	6	,	,	PUNCT
ap-1410	223	7	q]v(h	q]v(h	ADJ
ap-1410	223	8	)	)	PUNCT
ap-1410	223	9	⊂	⊂	PROPN
ap-1410	223	10	gd(h	gd(h	PROPN
ap-1410	223	11	)	)	PUNCT
ap-1410	224	1	and	and	CCONJ
ap-1410	225	1	[	[	X
ap-1410	225	2	0	0	NUM
ap-1410	225	3	,	,	PUNCT
ap-1410	225	4	q]vd(h	q]vd(h	PRON
ap-1410	225	5	)	)	PUNCT
ap-1410	225	6	⊂	⊂	PROPN
ap-1410	225	7	gd	gd	NOUN
ap-1410	225	8	,	,	PUNCT
ap-1410	225	9	d(h	d(h	PROPN
ap-1410	225	10	)	)	PUNCT
ap-1410	225	11	.	.	PUNCT
ap-1410	226	1	the	the	DET
ap-1410	226	2	same	same	ADJ
ap-1410	226	3	is	be	AUX
ap-1410	226	4	true	true	ADJ
ap-1410	226	5	for	for	ADP
ap-1410	226	6	any	any	DET
ap-1410	226	7	q	q	NOUN
ap-1410	226	8	∈	∈	PROPN
ap-1410	226	9	sp(h	sp(h	NOUN
ap-1410	226	10	)	)	PUNCT
ap-1410	226	11	,	,	PUNCT
ap-1410	226	12	q	q	PROPN
ap-1410	226	13	�	�	PROPN
ap-1410	226	14	=	=	SYM
ap-1410	226	15	0	0	NUM
ap-1410	226	16	,	,	PUNCT
ap-1410	226	17	d(q	d(q	PROPN
ap-1410	226	18	)	)	PUNCT
ap-1410	226	19	=	=	SYM
ap-1410	226	20	d.	d.	PROPN
ap-1410	227	1	namely	namely	ADV
ap-1410	227	2	,	,	PUNCT
ap-1410	227	3	[	[	X
ap-1410	227	4	0	0	NUM
ap-1410	227	5	,	,	PUNCT
ap-1410	227	6	q]sp(h	q]sp(h	ADV
ap-1410	227	7	)	)	PUNCT
ap-1410	227	8	=	=	PUNCT
ap-1410	228	1	[	[	X
ap-1410	228	2	0	0	NUM
ap-1410	228	3	,	,	PUNCT
ap-1410	228	4	q]sp	q]sp	PROPN
ap-1410	228	5	,	,	PUNCT
ap-1410	228	6	d(h	d(h	PROPN
ap-1410	228	7	)	)	PUNCT
ap-1410	228	8	and	and	CCONJ
ap-1410	228	9	(	(	PUNCT
ap-1410	228	10	[	[	X
ap-1410	228	11	0	0	NUM
ap-1410	228	12	,	,	PUNCT
ap-1410	228	13	q]sp	q]sp	PROPN
ap-1410	228	14	,	,	PUNCT
ap-1410	228	15	d(h	d(h	PROPN
ap-1410	228	16	)	)	PUNCT
ap-1410	228	17	;	;	PUNCT
ap-1410	228	18	⊕|[0,q]sp	⊕|[0,q]sp	NUM
ap-1410	228	19	,	,	PUNCT
ap-1410	228	20	d(h	d(h	PROPN
ap-1410	228	21	)	)	PUNCT
ap-1410	228	22	,	,	PUNCT
ap-1410	228	23	0	0	NUM
ap-1410	228	24	,	,	PUNCT
ap-1410	228	25	q	q	NOUN
ap-1410	228	26	)	)	PUNCT
ap-1410	228	27	,	,	PUNCT
ap-1410	229	1	d	d	PROPN
ap-1410	229	2	∈	∈	PROPN
ap-1410	229	3	d	d	X
ap-1410	229	4	,	,	PUNCT
ap-1410	229	5	are	be	AUX
ap-1410	229	6	effect	effect	NOUN
ap-1410	229	7	algebras	algebra	NOUN
ap-1410	229	8	of	of	ADP
ap-1410	229	9	positive	positive	ADJ
ap-1410	229	10	self	self	NOUN
ap-1410	229	11	-	-	PUNCT
ap-1410	229	12	adjoint	adjoint	NOUN
ap-1410	229	13	operators	operator	NOUN
ap-1410	229	14	a	a	DET
ap-1410	229	15	≤d	≤d	NOUN
ap-1410	229	16	q	q	NOUN
ap-1410	229	17	,	,	PUNCT
ap-1410	229	18	where	where	SCONJ
ap-1410	229	19	≤d	≤d	NOUN
ap-1410	229	20	is	be	AUX
ap-1410	229	21	the	the	DET
ap-1410	229	22	partial	partial	ADJ
ap-1410	229	23	order	order	NOUN
ap-1410	229	24	on	on	ADP
ap-1410	229	25	sp(h	sp(h	NOUN
ap-1410	229	26	)	)	PUNCT
ap-1410	229	27	derived	derive	VERB
ap-1410	229	28	from	from	ADP
ap-1410	229	29	operation	operation	NOUN
ap-1410	229	30	⊕|sp	⊕|sp	PROPN
ap-1410	229	31	,	,	PUNCT
ap-1410	229	32	d(h	d(h	PROPN
ap-1410	229	33	)	)	PUNCT
ap-1410	229	34	.	.	PUNCT
ap-1410	230	1	since	since	SCONJ
ap-1410	230	2	d(q	d(q	NUM
ap-1410	230	3	)	)	PUNCT
ap-1410	230	4	=	=	PUNCT
ap-1410	231	1	d	d	NOUN
ap-1410	231	2	is	be	AUX
ap-1410	231	3	dense	dense	ADJ
ap-1410	231	4	in	in	ADP
ap-1410	231	5	h	h	NOUN
ap-1410	231	6	,	,	PUNCT
ap-1410	231	7	the	the	DET
ap-1410	231	8	next	next	ADJ
ap-1410	231	9	theorem	theorem	NOUN
ap-1410	231	10	18	18	NUM
ap-1410	231	11	about	about	ADP
ap-1410	231	12	states	state	NOUN
ap-1410	231	13	on	on	ADP
ap-1410	231	14	intervals	interval	NOUN
ap-1410	231	15	is	be	AUX
ap-1410	231	16	sp(h	sp(h	NOUN
ap-1410	231	17	)	)	PUNCT
ap-1410	231	18	(	(	PUNCT
ap-1410	231	19	hence	hence	ADV
ap-1410	231	20	in	in	ADP
ap-1410	231	21	sp	sp	NOUN
ap-1410	231	22	,	,	PUNCT
ap-1410	231	23	d(h	d(h	PROPN
ap-1410	231	24	)	)	PUNCT
ap-1410	231	25	)	)	PUNCT
ap-1410	231	26	can	can	AUX
ap-1410	231	27	be	be	AUX
ap-1410	231	28	proved	prove	VERB
ap-1410	231	29	by	by	ADP
ap-1410	231	30	the	the	DET
ap-1410	231	31	same	same	ADJ
ap-1410	231	32	argument	argument	NOUN
ap-1410	231	33	as	as	SCONJ
ap-1410	231	34	theorem	theorem	VERB
ap-1410	231	35	7	7	NUM
ap-1410	231	36	in	in	ADP
ap-1410	231	37	[	[	X
ap-1410	231	38	8	8	NUM
ap-1410	231	39	]	]	PUNCT
ap-1410	231	40	for	for	ADP
ap-1410	231	41	states	state	NOUN
ap-1410	231	42	on	on	ADP
ap-1410	231	43	intervals	interval	NOUN
ap-1410	231	44	in	in	ADP
ap-1410	231	45	gd	gd	NOUN
ap-1410	231	46	,	,	PUNCT
ap-1410	231	47	d(h	d(h	PROPN
ap-1410	231	48	)	)	PUNCT
ap-1410	231	49	.	.	PUNCT
ap-1410	232	1	definition	definition	NOUN
ap-1410	232	2	17	17	NUM
ap-1410	232	3	let	let	VERB
ap-1410	232	4	e	e	PRON
ap-1410	232	5	be	be	AUX
ap-1410	232	6	an	an	DET
ap-1410	232	7	effect	effect	NOUN
ap-1410	232	8	algebra	algebra	NOUN
ap-1410	232	9	.	.	PUNCT
ap-1410	233	1	(	(	PUNCT
ap-1410	233	2	i	i	NOUN
ap-1410	233	3	)	)	PUNCT
ap-1410	233	4	a	a	DET
ap-1410	233	5	map	map	NOUN
ap-1410	233	6	ω	ω	NOUN
ap-1410	233	7	:	:	PUNCT
ap-1410	233	8	e	e	X
ap-1410	233	9	→	→	PUNCT
ap-1410	233	10	[	[	X
ap-1410	233	11	0	0	NUM
ap-1410	233	12	,	,	PUNCT
ap-1410	233	13	1	1	NUM
ap-1410	233	14	]	]	PUNCT
ap-1410	233	15	is	be	AUX
ap-1410	233	16	called	call	VERB
ap-1410	233	17	a	a	DET
ap-1410	233	18	state	state	NOUN
ap-1410	233	19	on	on	ADP
ap-1410	233	20	e	e	NOUN
ap-1410	233	21	if	if	SCONJ
ap-1410	233	22	1	1	NUM
ap-1410	233	23	.	.	PUNCT
ap-1410	234	1	ω(0	ω(0	NUM
ap-1410	234	2	)	)	PUNCT
ap-1410	234	3	=	=	SYM
ap-1410	234	4	0	0	NUM
ap-1410	234	5	,	,	PUNCT
ap-1410	234	6	ω(1	ω(1	NOUN
ap-1410	234	7	)	)	PUNCT
ap-1410	234	8	=	=	SYM
ap-1410	234	9	1	1	NUM
ap-1410	234	10	,	,	PUNCT
ap-1410	234	11	2	2	NUM
ap-1410	234	12	.	.	NOUN
ap-1410	234	13	ω(a⊕	ω(a⊕	PROPN
ap-1410	234	14	b	b	X
ap-1410	234	15	)	)	PUNCT
ap-1410	234	16	=	=	SYM
ap-1410	234	17	ω(a	ω(a	PROPN
ap-1410	234	18	)	)	PUNCT
ap-1410	234	19	+	+	NUM
ap-1410	234	20	ω(b	ω(b	NOUN
ap-1410	234	21	)	)	PUNCT
ap-1410	234	22	for	for	ADP
ap-1410	234	23	all	all	DET
ap-1410	234	24	a	a	DET
ap-1410	234	25	,	,	PUNCT
ap-1410	234	26	b	b	X
ap-1410	234	27	∈	∈	PROPN
ap-1410	234	28	e	e	NOUN
ap-1410	234	29	with	with	ADP
ap-1410	234	30	a	a	DET
ap-1410	234	31	⊕	⊕	PROPN
ap-1410	234	32	b	b	PROPN
ap-1410	234	33	defined	define	VERB
ap-1410	234	34	in	in	ADP
ap-1410	234	35	e.	e.	PROPN
ap-1410	234	36	3	3	NUM
ap-1410	234	37	.	.	PUNCT
ap-1410	235	1	a	a	DET
ap-1410	235	2	state	state	NOUN
ap-1410	235	3	ω	ω	PROPN
ap-1410	235	4	is	be	AUX
ap-1410	235	5	faithful	faithful	ADJ
ap-1410	235	6	if	if	SCONJ
ap-1410	235	7	ω(a	ω(a	NUM
ap-1410	235	8	)	)	PUNCT
ap-1410	235	9	=	=	SYM
ap-1410	235	10	0	0	NUM
ap-1410	235	11	implies	imply	VERB
ap-1410	235	12	a	a	DET
ap-1410	235	13	=	=	NOUN
ap-1410	235	14	0	0	NUM
ap-1410	235	15	.	.	NOUN
ap-1410	235	16	4	4	NUM
ap-1410	235	17	.	.	X
ap-1410	236	1	a	a	DET
ap-1410	236	2	set	set	NOUN
ap-1410	236	3	m	m	NOUN
ap-1410	236	4	of	of	ADP
ap-1410	236	5	states	state	NOUN
ap-1410	236	6	is	be	AUX
ap-1410	236	7	called	call	VERB
ap-1410	236	8	an	an	DET
ap-1410	236	9	ordering	ordering	NOUN
ap-1410	236	10	set	set	NOUN
ap-1410	236	11	of	of	ADP
ap-1410	236	12	states	state	NOUN
ap-1410	236	13	on	on	ADP
ap-1410	236	14	e	e	PROPN
ap-1410	236	15	if	if	SCONJ
ap-1410	236	16	a	a	DET
ap-1410	236	17	≤	≤	NUM
ap-1410	236	18	b	b	NUM
ap-1410	236	19	iff	iff	PROPN
ap-1410	236	20	ω(a	ω(a	PROPN
ap-1410	236	21	)	)	PUNCT
ap-1410	236	22	≤	≤	NOUN
ap-1410	236	23	ω(b	ω(b	NOUN
ap-1410	236	24	)	)	PUNCT
ap-1410	236	25	for	for	ADP
ap-1410	236	26	all	all	DET
ap-1410	236	27	ω	ω	NUM
ap-1410	236	28	∈	∈	PROPN
ap-1410	236	29	m	m	PROPN
ap-1410	236	30	,	,	PUNCT
ap-1410	236	31	a	a	PRON
ap-1410	236	32	,	,	PUNCT
ap-1410	236	33	b	b	PROPN
ap-1410	236	34	∈	∈	PROPN
ap-1410	236	35	e.	e.	PROPN
ap-1410	236	36	theorem	theorem	VERB
ap-1410	236	37	18	18	NUM
ap-1410	236	38	let	let	VERB
ap-1410	236	39	d	d	X
ap-1410	236	40	∈	∈	PROPN
ap-1410	236	41	d	d	NOUN
ap-1410	236	42	and	and	CCONJ
ap-1410	236	43	q	q	PROPN
ap-1410	236	44	∈	∈	PROPN
ap-1410	236	45	sp	sp	NOUN
ap-1410	236	46	,	,	PUNCT
ap-1410	236	47	d(h	d(h	PROPN
ap-1410	236	48	)	)	PUNCT
ap-1410	236	49	,	,	PUNCT
ap-1410	236	50	q	q	PROPN
ap-1410	236	51	�	�	PROPN
ap-1410	236	52	=	=	NOUN
ap-1410	236	53	0	0	PROPN
ap-1410	236	54	.	.	PUNCT
ap-1410	237	1	then	then	ADV
ap-1410	237	2	(	(	PUNCT
ap-1410	237	3	i	i	NOUN
ap-1410	237	4	)	)	PUNCT
ap-1410	237	5	there	there	PRON
ap-1410	237	6	exists	exist	VERB
ap-1410	237	7	x̃	x̃	PROPN
ap-1410	237	8	∈	∈	PROPN
ap-1410	237	9	d(q	d(q	PROPN
ap-1410	237	10	)	)	PUNCT
ap-1410	237	11	such	such	ADJ
ap-1410	237	12	that	that	SCONJ
ap-1410	237	13	cx̃	cx̃	ADP
ap-1410	237	14	=	=	SYM
ap-1410	237	15	(	(	PUNCT
ap-1410	237	16	qx̃	qx̃	PROPN
ap-1410	237	17	,	,	PUNCT
ap-1410	237	18	x̃	x̃	PROPN
ap-1410	237	19	)	)	PUNCT
ap-1410	237	20	>	>	X
ap-1410	238	1	0	0	X
ap-1410	238	2	.	.	PUNCT
ap-1410	238	3	(	(	PUNCT
ap-1410	238	4	ii	ii	NOUN
ap-1410	238	5	)	)	PUNCT
ap-1410	238	6	the	the	DET
ap-1410	238	7	mapping	mapping	NOUN
ap-1410	238	8	ωx̃	ωx̃	PUNCT
ap-1410	238	9	:	:	PUNCT
ap-1410	238	10	[	[	X
ap-1410	238	11	0	0	NUM
ap-1410	238	12	,	,	PUNCT
ap-1410	238	13	q]sp	q]sp	PROPN
ap-1410	238	14	,	,	PUNCT
ap-1410	238	15	d(h	d(h	PROPN
ap-1410	238	16	)	)	PUNCT
ap-1410	238	17	→	→	PUNCT
ap-1410	239	1	[	[	X
ap-1410	239	2	0	0	NUM
ap-1410	239	3	,	,	PUNCT
ap-1410	239	4	1	1	NUM
ap-1410	239	5	]	]	PUNCT
ap-1410	239	6	⊂	⊂	X
ap-1410	239	7	r	r	NOUN
ap-1410	239	8	given	give	VERB
ap-1410	239	9	by	by	ADP
ap-1410	239	10	ωx̃(a	ωx̃(a	NOUN
ap-1410	239	11	)	)	PUNCT
ap-1410	239	12	=	=	SYM
ap-1410	239	13	1	1	NUM
ap-1410	239	14	cx̃	cx̃	X
ap-1410	239	15	(	(	PUNCT
ap-1410	239	16	ax̃	ax̃	PROPN
ap-1410	239	17	,	,	PUNCT
ap-1410	239	18	x̃	x̃	PROPN
ap-1410	239	19	)	)	PUNCT
ap-1410	239	20	for	for	ADP
ap-1410	239	21	every	every	DET
ap-1410	239	22	a	a	DET
ap-1410	239	23	∈	∈	NOUN
ap-1410	239	24	[	[	X
ap-1410	239	25	0	0	NUM
ap-1410	239	26	,	,	PUNCT
ap-1410	239	27	q]sp	q]sp	PROPN
ap-1410	239	28	,	,	PUNCT
ap-1410	239	29	d(h	d(h	PROPN
ap-1410	239	30	)	)	PUNCT
ap-1410	239	31	is	be	AUX
ap-1410	239	32	a	a	DET
ap-1410	239	33	state	state	NOUN
ap-1410	239	34	.	.	PUNCT
ap-1410	240	1	(	(	PUNCT
ap-1410	240	2	iii	iii	X
ap-1410	240	3	)	)	PUNCT
ap-1410	240	4	if	if	SCONJ
ap-1410	240	5	d0	d0	NOUN
ap-1410	240	6	=	=	SYM
ap-1410	240	7	{	{	PUNCT
ap-1410	240	8	x	x	PROPN
ap-1410	240	9	∈	∈	PROPN
ap-1410	240	10	d(q	d(q	PROPN
ap-1410	240	11	)	)	PUNCT
ap-1410	240	12	|	|	ADV
ap-1410	240	13	cx	cx	NOUN
ap-1410	241	1	=	=	SYM
ap-1410	241	2	(	(	PUNCT
ap-1410	241	3	qx	qx	INTJ
ap-1410	241	4	,	,	PUNCT
ap-1410	241	5	x	x	NOUN
ap-1410	241	6	)	)	PUNCT
ap-1410	241	7	>	>	X
ap-1410	241	8	0	0	NUM
ap-1410	241	9	}	}	PUNCT
ap-1410	241	10	then	then	ADV
ap-1410	241	11	m	m	VERB
ap-1410	241	12	=	=	PUNCT
ap-1410	241	13	{	{	PUNCT
ap-1410	241	14	ωx	ωx	VERB
ap-1410	241	15	|	|	ADV
ap-1410	241	16	x	x	SYM
ap-1410	241	17	∈	∈	NOUN
ap-1410	241	18	d0	d0	NOUN
ap-1410	241	19	}	}	PUNCT
ap-1410	241	20	is	be	AUX
ap-1410	241	21	an	an	DET
ap-1410	241	22	ordering	ordering	NOUN
ap-1410	241	23	set	set	NOUN
ap-1410	241	24	of	of	ADP
ap-1410	241	25	states	state	NOUN
ap-1410	241	26	on	on	ADP
ap-1410	241	27	[	[	X
ap-1410	241	28	0	0	NUM
ap-1410	241	29	,	,	PUNCT
ap-1410	241	30	q]sp	q]sp	PROPN
ap-1410	241	31	,	,	PUNCT
ap-1410	241	32	d(h	d(h	PROPN
ap-1410	241	33	)	)	PUNCT
ap-1410	241	34	.	.	PUNCT
ap-1410	242	1	(	(	PUNCT
ap-1410	242	2	iv	iv	X
ap-1410	242	3	)	)	PUNCT
ap-1410	242	4	if	if	SCONJ
ap-1410	242	5	h	h	NOUN
ap-1410	242	6	is	be	AUX
ap-1410	242	7	separable	separable	ADJ
ap-1410	242	8	,	,	PUNCT
ap-1410	242	9	then	then	ADV
ap-1410	242	10	there	there	PRON
ap-1410	242	11	exists	exist	VERB
ap-1410	242	12	a	a	DET
ap-1410	242	13	faithful	faithful	ADJ
ap-1410	242	14	state	state	NOUN
ap-1410	242	15	ω	ω	NOUN
ap-1410	242	16	:	:	PUNCT
ap-1410	243	1	[	[	X
ap-1410	243	2	0	0	NUM
ap-1410	243	3	,	,	PUNCT
ap-1410	243	4	q]sp	q]sp	PROPN
ap-1410	243	5	,	,	PUNCT
ap-1410	243	6	d(h	d(h	PROPN
ap-1410	243	7	)	)	PUNCT
ap-1410	243	8	→	→	PUNCT
ap-1410	244	1	[	[	X
ap-1410	244	2	0	0	NUM
ap-1410	244	3	,	,	PUNCT
ap-1410	244	4	1	1	NUM
ap-1410	244	5	]	]	PUNCT
ap-1410	244	6	.	.	PUNCT
ap-1410	245	1	acknowledgement	acknowledgement	NOUN
ap-1410	245	2	supported	support	VERB
ap-1410	245	3	by	by	ADP
ap-1410	245	4	grants	grant	NOUN
ap-1410	245	5	vega	vega	PROPN
ap-1410	245	6	1/0297/11	1/0297/11	PROPN
ap-1410	245	7	and	and	CCONJ
ap-1410	245	8	vega	vega	PROPN
ap-1410	245	9	1/0021/10	1/0021/10	NUM
ap-1410	245	10	of	of	ADP
ap-1410	245	11	the	the	DET
ap-1410	245	12	ministry	ministry	PROPN
ap-1410	245	13	of	of	ADP
ap-1410	245	14	education	education	NOUN
ap-1410	245	15	of	of	ADP
ap-1410	245	16	the	the	DET
ap-1410	245	17	slovak	slovak	ADJ
ap-1410	245	18	republic	republic	NOUN
ap-1410	245	19	.	.	PUNCT
ap-1410	246	1	references	reference	NOUN
ap-1410	246	2	[	[	X
ap-1410	246	3	1	1	NUM
ap-1410	246	4	]	]	X
ap-1410	246	5	blank	blank	PROPN
ap-1410	246	6	,	,	PUNCT
ap-1410	246	7	j.	j.	PROPN
ap-1410	246	8	,	,	PUNCT
ap-1410	246	9	exner	exner	PROPN
ap-1410	246	10	,	,	PUNCT
ap-1410	246	11	p.	p.	PROPN
ap-1410	246	12	,	,	PUNCT
ap-1410	246	13	havĺıček	havĺıček	PROPN
ap-1410	246	14	,	,	PUNCT
ap-1410	246	15	m.	m.	NOUN
ap-1410	246	16	:	:	PUNCT
ap-1410	246	17	hilbert	hilbert	NOUN
ap-1410	246	18	space	space	NOUN
ap-1410	246	19	operators	operator	NOUN
ap-1410	246	20	in	in	ADP
ap-1410	246	21	quantum	quantum	ADJ
ap-1410	246	22	physics	physics	NOUN
ap-1410	246	23	(	(	PUNCT
ap-1410	246	24	second	second	ADJ
ap-1410	246	25	edition	edition	NOUN
ap-1410	246	26	)	)	PUNCT
ap-1410	246	27	.	.	PUNCT
ap-1410	247	1	springer	springer	NOUN
ap-1410	247	2	,	,	PUNCT
ap-1410	247	3	2008	2008	NUM
ap-1410	247	4	.	.	PUNCT
ap-1410	248	1	[	[	X
ap-1410	248	2	2	2	NUM
ap-1410	248	3	]	]	PUNCT
ap-1410	248	4	foulis	foulis	PROPN
ap-1410	248	5	,	,	PUNCT
ap-1410	248	6	d.	d.	PROPN
ap-1410	248	7	j.	j.	PROPN
ap-1410	248	8	,	,	PUNCT
ap-1410	248	9	bennet	bennet	PROPN
ap-1410	248	10	,	,	PUNCT
ap-1410	248	11	m.	m.	PROPN
ap-1410	248	12	k.	k.	PROPN
ap-1410	248	13	:	:	PUNCT
ap-1410	248	14	effect	effect	NOUN
ap-1410	248	15	algebras	algebra	NOUN
ap-1410	248	16	and	and	CCONJ
ap-1410	248	17	unsharp	unsharp	ADJ
ap-1410	248	18	quantum	quantum	ADJ
ap-1410	248	19	logics	logic	NOUN
ap-1410	248	20	,	,	PUNCT
ap-1410	248	21	found	find	VERB
ap-1410	248	22	.	.	PUNCT
ap-1410	249	1	phys	phy	NOUN
ap-1410	249	2	.	.	PUNCT
ap-1410	250	1	24	24	NUM
ap-1410	250	2	(	(	PUNCT
ap-1410	250	3	1994	1994	NUM
ap-1410	250	4	)	)	PUNCT
ap-1410	250	5	,	,	PUNCT
ap-1410	250	6	1	1	NUM
ap-1410	250	7	331–1	331–1	NUM
ap-1410	250	8	352	352	NUM
ap-1410	250	9	.	.	PUNCT
ap-1410	251	1	[	[	X
ap-1410	251	2	3	3	NUM
ap-1410	251	3	]	]	X
ap-1410	251	4	hedĺıková	hedĺıková	PROPN
ap-1410	251	5	,	,	PUNCT
ap-1410	251	6	j.	j.	PROPN
ap-1410	251	7	,	,	PUNCT
ap-1410	251	8	pulmannová	pulmannová	ADV
ap-1410	251	9	,	,	PUNCT
ap-1410	251	10	s.	s.	PROPN
ap-1410	251	11	:	:	PUNCT
ap-1410	251	12	generalized	generalized	ADJ
ap-1410	251	13	difference	difference	NOUN
ap-1410	251	14	posets	poset	NOUN
ap-1410	251	15	and	and	CCONJ
ap-1410	251	16	orthoalgebras	orthoalgebra	NOUN
ap-1410	251	17	,	,	PUNCT
ap-1410	251	18	acta	acta	PROPN
ap-1410	251	19	math	math	PROPN
ap-1410	251	20	.	.	PUNCT
ap-1410	252	1	univ	univ	PROPN
ap-1410	252	2	.	.	PROPN
ap-1410	252	3	comenianae	comenianae	PROPN
ap-1410	252	4	45	45	NUM
ap-1410	252	5	(	(	PUNCT
ap-1410	252	6	1996	1996	NUM
ap-1410	252	7	)	)	PUNCT
ap-1410	252	8	,	,	PUNCT
ap-1410	252	9	247–279	247–279	NUM
ap-1410	252	10	.	.	PUNCT
ap-1410	253	1	[	[	X
ap-1410	253	2	4	4	X
ap-1410	253	3	]	]	X
ap-1410	253	4	kalmbach	kalmbach	PROPN
ap-1410	253	5	,	,	PUNCT
ap-1410	253	6	g.	g.	PROPN
ap-1410	253	7	,	,	PUNCT
ap-1410	253	8	riečanová	riečanová	PROPN
ap-1410	253	9	,	,	PUNCT
ap-1410	253	10	z.	z.	PROPN
ap-1410	253	11	:	:	PUNCT
ap-1410	253	12	an	an	DET
ap-1410	253	13	axiomatization	axiomatization	NOUN
ap-1410	253	14	for	for	ADP
ap-1410	253	15	abelian	abelian	PROPN
ap-1410	253	16	relative	relative	PROPN
ap-1410	253	17	inverses	inverses	PROPN
ap-1410	253	18	,	,	PUNCT
ap-1410	253	19	demonstratio	demonstratio	PROPN
ap-1410	253	20	math	math	PROPN
ap-1410	253	21	.	.	PUNCT
ap-1410	254	1	27	27	NUM
ap-1410	254	2	(	(	PUNCT
ap-1410	254	3	1996	1996	NUM
ap-1410	254	4	)	)	PUNCT
ap-1410	254	5	,	,	PUNCT
ap-1410	254	6	769–780	769–780	NUM
ap-1410	254	7	.	.	PUNCT
ap-1410	255	1	[	[	X
ap-1410	255	2	5	5	NUM
ap-1410	255	3	]	]	X
ap-1410	255	4	kôpka	kôpka	NOUN
ap-1410	255	5	,	,	PUNCT
ap-1410	255	6	f.	f.	PROPN
ap-1410	255	7	,	,	PUNCT
ap-1410	255	8	chovanec	chovanec	PROPN
ap-1410	255	9	,	,	PUNCT
ap-1410	255	10	f.	f.	PROPN
ap-1410	255	11	:	:	PUNCT
ap-1410	255	12	d	d	X
ap-1410	255	13	-	-	PUNCT
ap-1410	255	14	posets	poset	NOUN
ap-1410	255	15	,	,	PUNCT
ap-1410	255	16	math	math	NOUN
ap-1410	255	17	.	.	PUNCT
ap-1410	256	1	slovaca	slovaca	NOUN
ap-1410	256	2	44	44	NUM
ap-1410	256	3	(	(	PUNCT
ap-1410	256	4	1994	1994	NUM
ap-1410	256	5	)	)	PUNCT
ap-1410	256	6	,	,	PUNCT
ap-1410	256	7	21–34	21–34	NUM
ap-1410	256	8	.	.	PUNCT
ap-1410	257	1	[	[	X
ap-1410	257	2	6	6	NUM
ap-1410	257	3	]	]	SYM
ap-1410	257	4	riečanová	riečanová	PROPN
ap-1410	257	5	,	,	PUNCT
ap-1410	257	6	z.	z.	PROPN
ap-1410	257	7	:	:	PUNCT
ap-1410	257	8	effect	effect	NOUN
ap-1410	257	9	algebras	algebra	NOUN
ap-1410	257	10	of	of	ADP
ap-1410	257	11	positive	positive	ADJ
ap-1410	257	12	selfadjoint	selfadjoint	NOUN
ap-1410	257	13	operators	operator	NOUN
ap-1410	257	14	densely	densely	ADV
ap-1410	257	15	defined	define	VERB
ap-1410	257	16	on	on	ADP
ap-1410	257	17	hilbert	hilbert	NOUN
ap-1410	257	18	spaces	space	NOUN
ap-1410	257	19	,	,	PUNCT
ap-1410	257	20	preprint	preprint	NOUN
ap-1410	257	21	.	.	PUNCT
ap-1410	258	1	[	[	X
ap-1410	258	2	7	7	NUM
ap-1410	258	3	]	]	SYM
ap-1410	258	4	riečanová	riečanová	PROPN
ap-1410	258	5	,	,	PUNCT
ap-1410	258	6	z.	z.	PROPN
ap-1410	258	7	:	:	PUNCT
ap-1410	259	1	subalgebras	subalgebras	PROPN
ap-1410	259	2	,	,	PUNCT
ap-1410	259	3	intervals	interval	NOUN
ap-1410	259	4	and	and	CCONJ
ap-1410	259	5	central	central	ADJ
ap-1410	259	6	elements	element	NOUN
ap-1410	259	7	of	of	ADP
ap-1410	259	8	generalized	generalized	ADJ
ap-1410	259	9	effect	effect	NOUN
ap-1410	259	10	algebras	algebra	NOUN
ap-1410	259	11	.	.	PUNCT
ap-1410	260	1	international	international	ADJ
ap-1410	260	2	journal	journal	PROPN
ap-1410	260	3	of	of	ADP
ap-1410	260	4	theoretic	theoretic	ADJ
ap-1410	260	5	physics	physics	NOUN
ap-1410	260	6	38	38	NUM
ap-1410	260	7	(	(	PUNCT
ap-1410	260	8	1999	1999	NUM
ap-1410	260	9	)	)	PUNCT
ap-1410	260	10	,	,	PUNCT
ap-1410	260	11	3	3	NUM
ap-1410	260	12	209–3	209–3	NUM
ap-1410	260	13	220	220	NUM
ap-1410	260	14	.	.	PUNCT
ap-1410	261	1	[	[	X
ap-1410	261	2	8	8	NUM
ap-1410	261	3	]	]	SYM
ap-1410	261	4	riečanová	riečanová	PROPN
ap-1410	261	5	,	,	PUNCT
ap-1410	261	6	z.	z.	PROPN
ap-1410	261	7	,	,	PUNCT
ap-1410	261	8	zajac	zajac	PROPN
ap-1410	261	9	,	,	PUNCT
ap-1410	261	10	m.	m.	NOUN
ap-1410	261	11	,	,	PUNCT
ap-1410	261	12	pulmannová	pulmannová	ADJ
ap-1410	261	13	,	,	PUNCT
ap-1410	261	14	s.	s.	PROPN
ap-1410	261	15	:	:	PUNCT
ap-1410	261	16	effect	effect	NOUN
ap-1410	261	17	algebras	algebra	NOUN
ap-1410	261	18	of	of	ADP
ap-1410	261	19	positive	positive	ADJ
ap-1410	261	20	linear	linear	PROPN
ap-1410	261	21	operators	operator	NOUN
ap-1410	261	22	densely	densely	ADV
ap-1410	261	23	defined	define	VERB
ap-1410	261	24	on	on	ADP
ap-1410	261	25	hilbert	hilbert	PROPN
ap-1410	261	26	spaces	space	NOUN
ap-1410	261	27	,	,	PUNCT
ap-1410	261	28	reports	report	NOUN
ap-1410	261	29	of	of	ADP
ap-1410	261	30	mathematical	mathematical	ADJ
ap-1410	261	31	physics	physics	NOUN
ap-1410	261	32	,	,	PUNCT
ap-1410	261	33	to	to	PART
ap-1410	261	34	appear	appear	VERB
ap-1410	261	35	.	.	PUNCT
ap-1410	262	1	zdenka	zdenka	PROPN
ap-1410	262	2	riečanová	riečanová	PROPN
ap-1410	262	3	e	e	PROPN
ap-1410	262	4	-	-	NOUN
ap-1410	262	5	mail	mail	NOUN
ap-1410	262	6	:	:	PUNCT
ap-1410	262	7	zdenka.riecanova@stuba.sk	zdenka.riecanova@stuba.sk	PROPN
ap-1410	262	8	m.	m.	NOUN
ap-1410	262	9	zajac	zajac	PROPN
ap-1410	262	10	e	e	PROPN
ap-1410	262	11	-	-	NOUN
ap-1410	262	12	mail	mail	NOUN
ap-1410	262	13	:	:	PUNCT
ap-1410	262	14	zajacm@stuba.sk	zajacm@stuba.sk	PROPN
ap-1410	262	15	department	department	PROPN
ap-1410	262	16	of	of	ADP
ap-1410	262	17	mathematics	mathematics	PROPN
ap-1410	262	18	faculty	faculty	NOUN
ap-1410	262	19	of	of	ADP
ap-1410	262	20	electrical	electrical	ADJ
ap-1410	262	21	engineering	engineering	NOUN
ap-1410	262	22	and	and	CCONJ
ap-1410	262	23	information	information	NOUN
ap-1410	262	24	technology	technology	NOUN
ap-1410	262	25	stu	stu	PROPN
ap-1410	262	26	ilkovičova	ilkovičova	VERB
ap-1410	262	27	3	3	NUM
ap-1410	262	28	,	,	PUNCT
ap-1410	262	29	sk-81219	sk-81219	ADJ
ap-1410	262	30	bratislava	bratislava	PROPN
ap-1410	262	31	77	77	NUM