id	sid	tid	token	lemma	pos
ap-1807	1	1	acta	acta	PROPN
ap-1807	1	2	polytechnica	polytechnica	PROPN
ap-1807	1	3	acta	acta	PROPN
ap-1807	1	4	polytechnica	polytechnica	PROPN
ap-1807	1	5	53(3):289–294	53(3):289–294	NUM
ap-1807	1	6	,	,	PUNCT
ap-1807	1	7	2013	2013	NUM
ap-1807	1	8	©	©	PROPN
ap-1807	1	9	czech	czech	PROPN
ap-1807	1	10	technical	technical	PROPN
ap-1807	1	11	university	university	PROPN
ap-1807	1	12	in	in	ADP
ap-1807	1	13	prague	prague	PROPN
ap-1807	1	14	,	,	PUNCT
ap-1807	1	15	2013	2013	NUM
ap-1807	1	16	available	available	ADJ
ap-1807	1	17	online	online	ADV
ap-1807	1	18	at	at	ADP
ap-1807	1	19	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-1807	1	20	weakly	weakly	ADV
ap-1807	1	21	ordered	order	VERB
ap-1807	1	22	a	a	DET
ap-1807	1	23	-	-	PUNCT
ap-1807	1	24	commutative	commutative	ADJ
ap-1807	1	25	partial	partial	ADJ
ap-1807	1	26	groups	group	NOUN
ap-1807	1	27	of	of	ADP
ap-1807	1	28	linear	linear	PROPN
ap-1807	1	29	operators	operator	NOUN
ap-1807	1	30	densely	densely	ADV
ap-1807	1	31	defined	define	VERB
ap-1807	1	32	on	on	ADP
ap-1807	1	33	hilbert	hilbert	NOUN
ap-1807	1	34	space	space	NOUN
ap-1807	1	35	jiří	jiří	NOUN
ap-1807	1	36	janda∗	janda∗	PROPN
ap-1807	1	37	department	department	PROPN
ap-1807	1	38	of	of	ADP
ap-1807	1	39	mathematics	mathematics	PROPN
ap-1807	1	40	and	and	CCONJ
ap-1807	1	41	statistics	statistic	NOUN
ap-1807	1	42	,	,	PUNCT
ap-1807	1	43	faculty	faculty	NOUN
ap-1807	1	44	of	of	ADP
ap-1807	1	45	science	science	NOUN
ap-1807	1	46	,	,	PUNCT
ap-1807	1	47	masaryk	masaryk	PROPN
ap-1807	1	48	university	university	NOUN
ap-1807	1	49	,	,	PUNCT
ap-1807	1	50	kotlářská	kotlářská	NOUN
ap-1807	1	51	2	2	NUM
ap-1807	1	52	,	,	PUNCT
ap-1807	1	53	cz-611	cz-611	VERB
ap-1807	1	54	37	37	NUM
ap-1807	1	55	brno	brno	NOUN
ap-1807	1	56	,	,	PUNCT
ap-1807	1	57	czech	czech	PROPN
ap-1807	1	58	republic	republic	NOUN
ap-1807	1	59	∗	∗	NOUN
ap-1807	1	60	corresponding	correspond	VERB
ap-1807	1	61	author	author	NOUN
ap-1807	1	62	:	:	PUNCT
ap-1807	1	63	98599@mail.muni.cz	98599@mail.muni.cz	NOUN
ap-1807	1	64	abstract	abstract	NOUN
ap-1807	1	65	.	.	PUNCT
ap-1807	2	1	the	the	DET
ap-1807	2	2	notion	notion	NOUN
ap-1807	2	3	of	of	ADP
ap-1807	2	4	a	a	DET
ap-1807	2	5	generalized	generalized	ADJ
ap-1807	2	6	effect	effect	NOUN
ap-1807	2	7	algebra	algebra	NOUN
ap-1807	2	8	is	be	AUX
ap-1807	2	9	presented	present	VERB
ap-1807	2	10	as	as	ADP
ap-1807	2	11	a	a	DET
ap-1807	2	12	generalization	generalization	NOUN
ap-1807	2	13	of	of	ADP
ap-1807	2	14	effect	effect	NOUN
ap-1807	2	15	algebra	algebra	NOUN
ap-1807	2	16	for	for	ADP
ap-1807	2	17	an	an	DET
ap-1807	2	18	algebraic	algebraic	ADJ
ap-1807	2	19	description	description	NOUN
ap-1807	2	20	of	of	ADP
ap-1807	2	21	the	the	DET
ap-1807	2	22	structure	structure	NOUN
ap-1807	2	23	of	of	ADP
ap-1807	2	24	the	the	DET
ap-1807	2	25	set	set	NOUN
ap-1807	2	26	of	of	ADP
ap-1807	2	27	all	all	DET
ap-1807	2	28	positive	positive	ADJ
ap-1807	2	29	linear	linear	NOUN
ap-1807	2	30	operators	operator	NOUN
ap-1807	2	31	densely	densely	ADV
ap-1807	2	32	defined	define	VERB
ap-1807	2	33	on	on	ADP
ap-1807	2	34	a	a	DET
ap-1807	2	35	hilbert	hilbert	NOUN
ap-1807	2	36	space	space	NOUN
ap-1807	2	37	with	with	ADP
ap-1807	2	38	the	the	DET
ap-1807	2	39	usual	usual	ADJ
ap-1807	2	40	sum	sum	NOUN
ap-1807	2	41	of	of	ADP
ap-1807	2	42	operators	operator	NOUN
ap-1807	2	43	.	.	PUNCT
ap-1807	3	1	the	the	DET
ap-1807	3	2	structure	structure	NOUN
ap-1807	3	3	of	of	ADP
ap-1807	3	4	the	the	DET
ap-1807	3	5	set	set	NOUN
ap-1807	3	6	of	of	ADP
ap-1807	3	7	not	not	PART
ap-1807	3	8	only	only	ADV
ap-1807	3	9	positive	positive	ADJ
ap-1807	3	10	linear	linear	NOUN
ap-1807	3	11	operators	operator	NOUN
ap-1807	3	12	can	can	AUX
ap-1807	3	13	be	be	AUX
ap-1807	3	14	described	describe	VERB
ap-1807	3	15	with	with	ADP
ap-1807	3	16	the	the	DET
ap-1807	3	17	notion	notion	NOUN
ap-1807	3	18	of	of	ADP
ap-1807	3	19	a	a	DET
ap-1807	3	20	weakly	weakly	ADJ
ap-1807	3	21	ordered	order	VERB
ap-1807	3	22	partial	partial	ADJ
ap-1807	3	23	commutative	commutative	ADJ
ap-1807	3	24	group	group	NOUN
ap-1807	3	25	(	(	PUNCT
ap-1807	3	26	wop	wop	NOUN
ap-1807	3	27	-	-	PUNCT
ap-1807	3	28	group	group	NOUN
ap-1807	3	29	)	)	PUNCT
ap-1807	3	30	.	.	PUNCT
ap-1807	4	1	due	due	ADP
ap-1807	4	2	to	to	ADP
ap-1807	4	3	the	the	DET
ap-1807	4	4	non	non	ADJ
ap-1807	4	5	-	-	ADJ
ap-1807	4	6	constructive	constructive	ADJ
ap-1807	4	7	algebraic	algebraic	ADJ
ap-1807	4	8	nature	nature	NOUN
ap-1807	4	9	of	of	ADP
ap-1807	4	10	the	the	DET
ap-1807	4	11	wop	wop	NOUN
ap-1807	4	12	-	-	PUNCT
ap-1807	4	13	group	group	NOUN
ap-1807	4	14	we	we	PRON
ap-1807	4	15	introduce	introduce	VERB
ap-1807	4	16	its	its	PRON
ap-1807	4	17	stronger	strong	ADJ
ap-1807	4	18	version	version	NOUN
ap-1807	4	19	called	call	VERB
ap-1807	4	20	a	a	DET
ap-1807	4	21	weakly	weakly	ADJ
ap-1807	4	22	ordered	order	VERB
ap-1807	4	23	partial	partial	ADJ
ap-1807	4	24	a	a	PRON
ap-1807	4	25	-	-	PUNCT
ap-1807	4	26	commutative	commutative	ADJ
ap-1807	4	27	group	group	NOUN
ap-1807	4	28	(	(	PUNCT
ap-1807	4	29	woa	woa	NOUN
ap-1807	4	30	-	-	PUNCT
ap-1807	4	31	group	group	NOUN
ap-1807	4	32	)	)	PUNCT
ap-1807	4	33	.	.	PUNCT
ap-1807	5	1	we	we	PRON
ap-1807	5	2	show	show	VERB
ap-1807	5	3	that	that	SCONJ
ap-1807	5	4	it	it	PRON
ap-1807	5	5	also	also	ADV
ap-1807	5	6	describes	describe	VERB
ap-1807	5	7	the	the	DET
ap-1807	5	8	structure	structure	NOUN
ap-1807	5	9	of	of	ADP
ap-1807	5	10	not	not	PART
ap-1807	5	11	only	only	ADV
ap-1807	5	12	positive	positive	ADJ
ap-1807	5	13	linear	linear	PROPN
ap-1807	5	14	operators	operator	NOUN
ap-1807	5	15	.	.	PUNCT
ap-1807	6	1	keywords	keyword	NOUN
ap-1807	6	2	:	:	PUNCT
ap-1807	6	3	(	(	PUNCT
ap-1807	6	4	generalized	generalized	ADJ
ap-1807	6	5	)	)	PUNCT
ap-1807	6	6	effect	effect	NOUN
ap-1807	6	7	algebra	algebra	NOUN
ap-1807	6	8	,	,	PUNCT
ap-1807	6	9	partial	partial	ADJ
ap-1807	6	10	group	group	NOUN
ap-1807	6	11	,	,	PUNCT
ap-1807	6	12	weakly	weakly	ADV
ap-1807	6	13	ordered	order	VERB
ap-1807	6	14	partial	partial	ADJ
ap-1807	6	15	group	group	NOUN
ap-1807	6	16	,	,	PUNCT
ap-1807	6	17	hilbert	hilbert	NOUN
ap-1807	6	18	space	space	NOUN
ap-1807	6	19	,	,	PUNCT
ap-1807	6	20	unbounded	unbounded	ADJ
ap-1807	6	21	linear	linear	NOUN
ap-1807	6	22	operator	operator	NOUN
ap-1807	6	23	,	,	PUNCT
ap-1807	6	24	self	self	NOUN
ap-1807	6	25	-	-	PUNCT
ap-1807	6	26	adjoint	adjoint	NOUN
ap-1807	6	27	linear	linear	NOUN
ap-1807	6	28	operator	operator	NOUN
ap-1807	6	29	.	.	PUNCT
ap-1807	7	1	ams	am	NOUN
ap-1807	7	2	mathematics	mathematics	PROPN
ap-1807	7	3	subject	subject	ADJ
ap-1807	7	4	classification	classification	NOUN
ap-1807	7	5	:	:	PUNCT
ap-1807	7	6	06f05	06f05	NUM
ap-1807	7	7	,	,	PUNCT
ap-1807	7	8	(	(	PUNCT
ap-1807	7	9	03g25	03g25	NOUN
ap-1807	7	10	,	,	PUNCT
ap-1807	7	11	81p10	81p10	NUM
ap-1807	7	12	,	,	PUNCT
ap-1807	7	13	08a55	08a55	NUM
ap-1807	7	14	)	)	PUNCT
ap-1807	7	15	.	.	PUNCT
ap-1807	8	1	1	1	X
ap-1807	8	2	.	.	X
ap-1807	8	3	introduction	introduction	NOUN
ap-1807	8	4	the	the	DET
ap-1807	8	5	notion	notion	NOUN
ap-1807	8	6	of	of	ADP
ap-1807	8	7	an	an	DET
ap-1807	8	8	effect	effect	NOUN
ap-1807	8	9	algebra	algebra	NOUN
ap-1807	8	10	was	be	AUX
ap-1807	8	11	presented	present	VERB
ap-1807	8	12	by	by	ADP
ap-1807	8	13	foulis	foulis	PROPN
ap-1807	8	14	and	and	CCONJ
ap-1807	8	15	bennett	bennett	PROPN
ap-1807	8	16	in	in	ADP
ap-1807	8	17	[	[	X
ap-1807	8	18	3	3	NUM
ap-1807	8	19	]	]	PUNCT
ap-1807	8	20	.	.	PUNCT
ap-1807	9	1	the	the	DET
ap-1807	9	2	definition	definition	NOUN
ap-1807	9	3	was	be	AUX
ap-1807	9	4	motivated	motivate	VERB
ap-1807	9	5	by	by	ADP
ap-1807	9	6	giving	give	VERB
ap-1807	9	7	an	an	DET
ap-1807	9	8	algebraic	algebraic	ADJ
ap-1807	9	9	description	description	NOUN
ap-1807	9	10	of	of	ADP
ap-1807	9	11	positive	positive	ADJ
ap-1807	9	12	self	self	NOUN
ap-1807	9	13	-	-	PUNCT
ap-1807	9	14	adjoint	adjoint	NOUN
ap-1807	9	15	linear	linear	PROPN
ap-1807	9	16	operators	operator	NOUN
ap-1807	9	17	between	between	ADP
ap-1807	9	18	the	the	DET
ap-1807	9	19	zero	zero	NUM
ap-1807	9	20	and	and	CCONJ
ap-1807	9	21	the	the	DET
ap-1807	9	22	identity	identity	NOUN
ap-1807	9	23	operator	operator	NOUN
ap-1807	9	24	in	in	ADP
ap-1807	9	25	a	a	DET
ap-1807	9	26	complex	complex	ADJ
ap-1807	9	27	hilbert	hilbert	NOUN
ap-1807	9	28	space	space	NOUN
ap-1807	9	29	h.	h.	PROPN
ap-1807	10	1	the	the	DET
ap-1807	10	2	notion	notion	NOUN
ap-1807	10	3	of	of	ADP
ap-1807	10	4	a	a	DET
ap-1807	10	5	generalized	generalized	ADJ
ap-1807	10	6	effect	effect	NOUN
ap-1807	10	7	algebra	algebra	NOUN
ap-1807	10	8	extends	extend	VERB
ap-1807	10	9	these	these	DET
ap-1807	10	10	ideas	idea	NOUN
ap-1807	10	11	on	on	ADP
ap-1807	10	12	unbounded	unbounded	ADJ
ap-1807	10	13	sets	set	NOUN
ap-1807	10	14	of	of	ADP
ap-1807	10	15	positive	positive	ADJ
ap-1807	10	16	linear	linear	PROPN
ap-1807	10	17	operators	operator	NOUN
ap-1807	10	18	.	.	PUNCT
ap-1807	11	1	to	to	PART
ap-1807	11	2	answer	answer	VERB
ap-1807	11	3	the	the	DET
ap-1807	11	4	natural	natural	ADJ
ap-1807	11	5	question	question	NOUN
ap-1807	11	6	concerning	concern	VERB
ap-1807	11	7	the	the	DET
ap-1807	11	8	structure	structure	NOUN
ap-1807	11	9	of	of	ADP
ap-1807	11	10	sets	set	NOUN
ap-1807	11	11	of	of	ADP
ap-1807	11	12	not	not	PART
ap-1807	11	13	only	only	ADV
ap-1807	11	14	positive	positive	ADJ
ap-1807	11	15	linear	linear	ADJ
ap-1807	11	16	operators	operator	NOUN
ap-1807	11	17	paseka	paseka	NOUN
ap-1807	11	18	started	start	VERB
ap-1807	11	19	to	to	PART
ap-1807	11	20	investigate	investigate	VERB
ap-1807	11	21	a	a	DET
ap-1807	11	22	partially	partially	ADV
ap-1807	11	23	ordered	order	VERB
ap-1807	11	24	commutative	commutative	ADJ
ap-1807	11	25	group	group	NOUN
ap-1807	11	26	of	of	ADP
ap-1807	11	27	operators	operator	NOUN
ap-1807	11	28	with	with	ADP
ap-1807	11	29	a	a	DET
ap-1807	11	30	fixed	fix	VERB
ap-1807	11	31	domain	domain	NOUN
ap-1807	11	32	[	[	X
ap-1807	11	33	5	5	NUM
ap-1807	11	34	]	]	PUNCT
ap-1807	11	35	.	.	PUNCT
ap-1807	12	1	in	in	ADP
ap-1807	12	2	[	[	X
ap-1807	12	3	6	6	NUM
ap-1807	12	4	]	]	PUNCT
ap-1807	12	5	paseka	paseka	NOUN
ap-1807	12	6	and	and	CCONJ
ap-1807	12	7	janda	janda	PROPN
ap-1807	12	8	introduced	introduce	VERB
ap-1807	12	9	the	the	DET
ap-1807	12	10	structure	structure	NOUN
ap-1807	12	11	of	of	ADP
ap-1807	12	12	a	a	DET
ap-1807	12	13	weakly	weakly	ADJ
ap-1807	12	14	ordered	order	VERB
ap-1807	12	15	partial	partial	ADJ
ap-1807	12	16	commutative	commutative	ADJ
ap-1807	12	17	group	group	NOUN
ap-1807	12	18	(	(	PUNCT
ap-1807	12	19	shortly	shortly	ADV
ap-1807	12	20	a	a	DET
ap-1807	12	21	wop	wop	NOUN
ap-1807	12	22	-	-	PUNCT
ap-1807	12	23	group	group	NOUN
ap-1807	12	24	)	)	PUNCT
ap-1807	12	25	.	.	PUNCT
ap-1807	13	1	they	they	PRON
ap-1807	13	2	also	also	ADV
ap-1807	13	3	showed	show	VERB
ap-1807	13	4	that	that	SCONJ
ap-1807	13	5	the	the	DET
ap-1807	13	6	set	set	NOUN
ap-1807	13	7	of	of	ADP
ap-1807	13	8	all	all	DET
ap-1807	13	9	linear	linear	ADJ
ap-1807	13	10	operators	operator	NOUN
ap-1807	13	11	on	on	ADP
ap-1807	13	12	complex	complex	ADJ
ap-1807	13	13	hilbert	hilbert	NOUN
ap-1807	13	14	space	space	NOUN
ap-1807	13	15	h	h	NOUN
ap-1807	13	16	with	with	ADP
ap-1807	13	17	the	the	DET
ap-1807	13	18	usual	usual	ADJ
ap-1807	13	19	sum	sum	NOUN
ap-1807	13	20	,	,	PUNCT
ap-1807	13	21	which	which	PRON
ap-1807	13	22	is	be	AUX
ap-1807	13	23	restricted	restrict	VERB
ap-1807	13	24	to	to	ADP
ap-1807	13	25	the	the	DET
ap-1807	13	26	same	same	ADJ
ap-1807	13	27	domain	domain	NOUN
ap-1807	13	28	for	for	ADP
ap-1807	13	29	unbounded	unbounded	ADJ
ap-1807	13	30	operators	operator	NOUN
ap-1807	13	31	(	(	PUNCT
ap-1807	13	32	partial	partial	ADJ
ap-1807	13	33	operation	operation	NOUN
ap-1807	13	34	⊕d	⊕d	NOUN
ap-1807	13	35	)	)	PUNCT
ap-1807	13	36	,	,	PUNCT
ap-1807	13	37	possesses	possess	VERB
ap-1807	13	38	this	this	DET
ap-1807	13	39	structure	structure	NOUN
ap-1807	13	40	.	.	PUNCT
ap-1807	14	1	in	in	ADP
ap-1807	14	2	[	[	X
ap-1807	14	3	4	4	X
ap-1807	14	4	]	]	PUNCT
ap-1807	14	5	we	we	PRON
ap-1807	14	6	considered	consider	VERB
ap-1807	14	7	the	the	DET
ap-1807	14	8	structure	structure	NOUN
ap-1807	14	9	on	on	ADP
ap-1807	14	10	the	the	DET
ap-1807	14	11	important	important	ADJ
ap-1807	14	12	subset	subset	NOUN
ap-1807	14	13	of	of	ADP
ap-1807	14	14	self	self	NOUN
ap-1807	14	15	-	-	PUNCT
ap-1807	14	16	adjoint	adjoint	NOUN
ap-1807	14	17	operators	operator	NOUN
ap-1807	14	18	,	,	PUNCT
ap-1807	14	19	showing	show	VERB
ap-1807	14	20	that	that	SCONJ
ap-1807	14	21	it	it	PRON
ap-1807	14	22	is	be	AUX
ap-1807	14	23	also	also	ADV
ap-1807	14	24	a	a	DET
ap-1807	14	25	wop	wop	NOUN
ap-1807	14	26	-	-	PUNCT
ap-1807	14	27	group	group	NOUN
ap-1807	14	28	.	.	PUNCT
ap-1807	15	1	wop	wop	NOUN
ap-1807	15	2	-	-	PUNCT
ap-1807	15	3	groups	group	NOUN
ap-1807	15	4	have	have	VERB
ap-1807	15	5	only	only	ADV
ap-1807	15	6	a	a	DET
ap-1807	15	7	non	non	ADJ
ap-1807	15	8	-	-	ADJ
ap-1807	15	9	constructive	constructive	ADJ
ap-1807	15	10	associativity	associativity	NOUN
ap-1807	15	11	(	(	PUNCT
ap-1807	15	12	the	the	DET
ap-1807	15	13	equation	equation	NOUN
ap-1807	15	14	holds	hold	VERB
ap-1807	15	15	if	if	SCONJ
ap-1807	15	16	and	and	CCONJ
ap-1807	15	17	only	only	ADV
ap-1807	15	18	if	if	SCONJ
ap-1807	15	19	both	both	DET
ap-1807	15	20	sides	side	NOUN
ap-1807	15	21	are	be	AUX
ap-1807	15	22	defined	define	VERB
ap-1807	15	23	)	)	PUNCT
ap-1807	15	24	.	.	PUNCT
ap-1807	16	1	it	it	PRON
ap-1807	16	2	has	have	AUX
ap-1807	16	3	been	be	AUX
ap-1807	16	4	shown	show	VERB
ap-1807	16	5	[	[	PUNCT
ap-1807	16	6	4	4	X
ap-1807	16	7	]	]	PUNCT
ap-1807	16	8	that	that	SCONJ
ap-1807	16	9	the	the	DET
ap-1807	16	10	set	set	NOUN
ap-1807	16	11	of	of	ADP
ap-1807	16	12	all	all	DET
ap-1807	16	13	linear	linear	PROPN
ap-1807	16	14	operators	operator	NOUN
ap-1807	16	15	has	have	VERB
ap-1807	16	16	generally	generally	ADV
ap-1807	16	17	stronger	strong	ADJ
ap-1807	16	18	algebraic	algebraic	ADJ
ap-1807	16	19	properties	property	NOUN
ap-1807	16	20	.	.	PUNCT
ap-1807	17	1	this	this	PRON
ap-1807	17	2	was	be	AUX
ap-1807	17	3	a	a	DET
ap-1807	17	4	motivation	motivation	NOUN
ap-1807	17	5	for	for	ADP
ap-1807	17	6	introducing	introduce	VERB
ap-1807	17	7	the	the	DET
ap-1807	17	8	notion	notion	NOUN
ap-1807	17	9	of	of	ADP
ap-1807	17	10	a	a	DET
ap-1807	17	11	weakly	weakly	ADJ
ap-1807	17	12	ordered	order	VERB
ap-1807	17	13	partial	partial	ADJ
ap-1807	17	14	a	a	PRON
ap-1807	17	15	-	-	PUNCT
ap-1807	17	16	commutative	commutative	ADJ
ap-1807	17	17	group	group	NOUN
ap-1807	17	18	(	(	PUNCT
ap-1807	17	19	woa	woa	NOUN
ap-1807	17	20	-	-	PUNCT
ap-1807	17	21	group	group	NOUN
ap-1807	17	22	)	)	PUNCT
ap-1807	17	23	where	where	SCONJ
ap-1807	17	24	the	the	DET
ap-1807	17	25	associative	associative	ADJ
ap-1807	17	26	law	law	NOUN
ap-1807	17	27	is	be	AUX
ap-1807	17	28	more	more	ADV
ap-1807	17	29	constructive	constructive	ADJ
ap-1807	17	30	.	.	PUNCT
ap-1807	18	1	also	also	ADV
ap-1807	18	2	a	a	DET
ap-1807	18	3	weak	weak	ADJ
ap-1807	18	4	order	order	NOUN
ap-1807	18	5	is	be	AUX
ap-1807	18	6	more	more	ADV
ap-1807	18	7	strongly	strongly	ADV
ap-1807	18	8	related	relate	VERB
ap-1807	18	9	to	to	ADP
ap-1807	18	10	the	the	DET
ap-1807	18	11	partial	partial	ADJ
ap-1807	18	12	operation	operation	NOUN
ap-1807	18	13	.	.	PUNCT
ap-1807	19	1	moreover	moreover	ADV
ap-1807	19	2	,	,	PUNCT
ap-1807	19	3	every	every	DET
ap-1807	19	4	positive	positive	ADJ
ap-1807	19	5	cone	cone	NOUN
ap-1807	19	6	of	of	ADP
ap-1807	19	7	a	a	DET
ap-1807	19	8	woa	woa	NOUN
ap-1807	19	9	-	-	PUNCT
ap-1807	19	10	group	group	NOUN
ap-1807	19	11	is	be	AUX
ap-1807	19	12	a	a	DET
ap-1807	19	13	generalized	generalized	ADJ
ap-1807	19	14	effect	effect	NOUN
ap-1807	19	15	algebra	algebra	NOUN
ap-1807	19	16	.	.	PUNCT
ap-1807	20	1	on	on	ADP
ap-1807	20	2	the	the	DET
ap-1807	20	3	other	other	ADJ
ap-1807	20	4	hand	hand	NOUN
ap-1807	20	5	,	,	PUNCT
ap-1807	20	6	we	we	PRON
ap-1807	20	7	present	present	VERB
ap-1807	20	8	a	a	DET
ap-1807	20	9	construction	construction	NOUN
ap-1807	20	10	showing	show	VERB
ap-1807	20	11	that	that	SCONJ
ap-1807	20	12	every	every	DET
ap-1807	20	13	generalized	generalized	ADJ
ap-1807	20	14	effect	effect	NOUN
ap-1807	20	15	algebra	algebra	NOUN
ap-1807	20	16	is	be	AUX
ap-1807	20	17	a	a	DET
ap-1807	20	18	positive	positive	ADJ
ap-1807	20	19	cone	cone	NOUN
ap-1807	20	20	of	of	ADP
ap-1807	20	21	some	some	DET
ap-1807	20	22	woa	woa	NOUN
ap-1807	20	23	-	-	PUNCT
ap-1807	20	24	group	group	NOUN
ap-1807	20	25	.	.	PUNCT
ap-1807	21	1	2	2	X
ap-1807	21	2	.	.	X
ap-1807	21	3	preliminaries	preliminary	NOUN
ap-1807	21	4	we	we	PRON
ap-1807	21	5	review	review	VERB
ap-1807	21	6	some	some	DET
ap-1807	21	7	basic	basic	ADJ
ap-1807	21	8	terminology	terminology	NOUN
ap-1807	21	9	,	,	PUNCT
ap-1807	21	10	definitions	definition	NOUN
ap-1807	21	11	and	and	CCONJ
ap-1807	21	12	statements	statement	NOUN
ap-1807	21	13	.	.	PUNCT
ap-1807	22	1	the	the	DET
ap-1807	22	2	basic	basic	ADJ
ap-1807	22	3	reference	reference	NOUN
ap-1807	22	4	for	for	ADP
ap-1807	22	5	this	this	DET
ap-1807	22	6	text	text	NOUN
ap-1807	22	7	is	be	AUX
ap-1807	22	8	the	the	DET
ap-1807	22	9	book	book	NOUN
ap-1807	22	10	by	by	ADP
ap-1807	22	11	dvurečenskij	dvurečenskij	NOUN
ap-1807	22	12	and	and	CCONJ
ap-1807	22	13	pulmannová	pulmannová	PROPN
ap-1807	23	1	[	[	X
ap-1807	23	2	2	2	NUM
ap-1807	23	3	]	]	PUNCT
ap-1807	23	4	.	.	PUNCT
ap-1807	24	1	definition	definition	NOUN
ap-1807	24	2	1	1	NUM
ap-1807	24	3	.	.	PUNCT
ap-1807	25	1	a	a	DET
ap-1807	25	2	partial	partial	ADJ
ap-1807	25	3	algebra	algebra	NOUN
ap-1807	25	4	(	(	PUNCT
ap-1807	25	5	e,+	e,+	ADJ
ap-1807	25	6	,	,	PUNCT
ap-1807	25	7	0	0	NUM
ap-1807	25	8	)	)	PUNCT
ap-1807	25	9	is	be	AUX
ap-1807	25	10	called	call	VERB
ap-1807	25	11	a	a	DET
ap-1807	25	12	generalized	generalized	ADJ
ap-1807	25	13	effect	effect	NOUN
ap-1807	25	14	algebra	algebra	NOUN
ap-1807	25	15	if	if	SCONJ
ap-1807	25	16	0	0	NUM
ap-1807	25	17	∈	∈	NOUN
ap-1807	25	18	e	e	NOUN
ap-1807	25	19	is	be	AUX
ap-1807	25	20	a	a	DET
ap-1807	25	21	distinguished	distinguished	ADJ
ap-1807	25	22	element	element	NOUN
ap-1807	25	23	and	and	CCONJ
ap-1807	25	24	+	+	CCONJ
ap-1807	25	25	is	be	AUX
ap-1807	25	26	a	a	DET
ap-1807	25	27	partially	partially	ADV
ap-1807	25	28	defined	define	VERB
ap-1807	25	29	binary	binary	ADJ
ap-1807	25	30	operation	operation	NOUN
ap-1807	25	31	on	on	ADP
ap-1807	25	32	e	e	PROPN
ap-1807	25	33	which	which	PRON
ap-1807	25	34	satisfies	satisfy	VERB
ap-1807	25	35	the	the	DET
ap-1807	25	36	following	follow	VERB
ap-1807	25	37	conditions	condition	NOUN
ap-1807	25	38	for	for	ADP
ap-1807	25	39	any	any	DET
ap-1807	25	40	x	x	NOUN
ap-1807	25	41	,	,	PUNCT
ap-1807	25	42	y	y	PROPN
ap-1807	25	43	,	,	PUNCT
ap-1807	25	44	z	z	NOUN
ap-1807	25	45	∈	∈	PROPN
ap-1807	26	1	e	e	NOUN
ap-1807	26	2	:	:	PUNCT
ap-1807	26	3	(	(	PUNCT
ap-1807	26	4	gei	gei	PROPN
ap-1807	26	5	)	)	PUNCT
ap-1807	26	6	x+	x+	NOUN
ap-1807	26	7	y	y	NOUN
ap-1807	26	8	=	=	SYM
ap-1807	26	9	y	y	PROPN
ap-1807	27	1	+	+	CCONJ
ap-1807	27	2	x	x	X
ap-1807	27	3	,	,	PUNCT
ap-1807	27	4	if	if	SCONJ
ap-1807	27	5	one	one	NUM
ap-1807	27	6	side	side	NOUN
ap-1807	27	7	is	be	AUX
ap-1807	27	8	defined	define	VERB
ap-1807	27	9	,	,	PUNCT
ap-1807	27	10	(	(	PUNCT
ap-1807	27	11	geii	geii	NOUN
ap-1807	27	12	)	)	PUNCT
ap-1807	28	1	(	(	PUNCT
ap-1807	28	2	x	x	X
ap-1807	28	3	+	+	NUM
ap-1807	28	4	y	y	NOUN
ap-1807	28	5	)	)	PUNCT
ap-1807	28	6	+	+	PUNCT
ap-1807	28	7	z	z	NOUN
ap-1807	28	8	=	=	PUNCT
ap-1807	29	1	x	x	X
ap-1807	30	1	+	+	PUNCT
ap-1807	30	2	(	(	PUNCT
ap-1807	30	3	y	y	PROPN
ap-1807	30	4	+	+	PROPN
ap-1807	30	5	z	z	NOUN
ap-1807	30	6	)	)	PUNCT
ap-1807	30	7	,	,	PUNCT
ap-1807	30	8	if	if	SCONJ
ap-1807	30	9	one	one	NUM
ap-1807	30	10	side	side	NOUN
ap-1807	30	11	is	be	AUX
ap-1807	30	12	defined	define	VERB
ap-1807	30	13	,	,	PUNCT
ap-1807	30	14	(	(	PUNCT
ap-1807	30	15	geiii	geiii	PROPN
ap-1807	30	16	)	)	PUNCT
ap-1807	30	17	x+	x+	SYM
ap-1807	30	18	0	0	NUM
ap-1807	30	19	=	=	SYM
ap-1807	30	20	x	x	NOUN
ap-1807	30	21	,	,	PUNCT
ap-1807	30	22	(	(	PUNCT
ap-1807	30	23	geiv	geiv	NOUN
ap-1807	30	24	)	)	PUNCT
ap-1807	30	25	x+	x+	PUNCT
ap-1807	30	26	y	y	NOUN
ap-1807	30	27	=	=	PUNCT
ap-1807	31	1	x+	x+	PROPN
ap-1807	31	2	z	z	NOUN
ap-1807	31	3	implies	imply	VERB
ap-1807	31	4	y	y	PROPN
ap-1807	31	5	=	=	SYM
ap-1807	31	6	z	z	PROPN
ap-1807	31	7	(	(	PUNCT
ap-1807	31	8	cancellation	cancellation	NOUN
ap-1807	31	9	law	law	NOUN
ap-1807	31	10	)	)	PUNCT
ap-1807	31	11	,	,	PUNCT
ap-1807	31	12	(	(	PUNCT
ap-1807	31	13	gev	gev	NOUN
ap-1807	31	14	)	)	PUNCT
ap-1807	31	15	x+	x+	NOUN
ap-1807	31	16	y	y	NOUN
ap-1807	31	17	=	=	SYM
ap-1807	31	18	0	0	NUM
ap-1807	31	19	implies	imply	VERB
ap-1807	31	20	x	x	PUNCT
ap-1807	31	21	=	=	SYM
ap-1807	31	22	y	y	PROPN
ap-1807	31	23	=	=	SYM
ap-1807	31	24	0	0	PROPN
ap-1807	31	25	.	.	PUNCT
ap-1807	32	1	in	in	ADP
ap-1807	32	2	every	every	DET
ap-1807	32	3	generalized	generalized	ADJ
ap-1807	32	4	effect	effect	NOUN
ap-1807	32	5	algebra	algebra	NOUN
ap-1807	32	6	e	e	NOUN
ap-1807	32	7	the	the	DET
ap-1807	32	8	partial	partial	ADJ
ap-1807	32	9	binary	binary	NOUN
ap-1807	32	10	operation	operation	NOUN
ap-1807	32	11	−	−	PROPN
ap-1807	32	12	and	and	CCONJ
ap-1807	32	13	relation	relation	NOUN
ap-1807	32	14	≤	≤	NOUN
ap-1807	32	15	can	can	AUX
ap-1807	32	16	be	be	AUX
ap-1807	32	17	defined	define	VERB
ap-1807	32	18	by	by	ADP
ap-1807	32	19	(	(	PUNCT
ap-1807	32	20	ed	ed	NOUN
ap-1807	32	21	)	)	PUNCT
ap-1807	32	22	x	x	SYM
ap-1807	32	23	≤	≤	ADJ
ap-1807	32	24	y	y	PROPN
ap-1807	32	25	and	and	CCONJ
ap-1807	32	26	y	y	PROPN
ap-1807	32	27	−	−	PROPN
ap-1807	33	1	x	x	SYM
ap-1807	34	1	=	=	PUNCT
ap-1807	34	2	z	z	SYM
ap-1807	34	3	iff	iff	PROPN
ap-1807	34	4	x+	x+	PROPN
ap-1807	34	5	z	z	PROPN
ap-1807	34	6	is	be	AUX
ap-1807	34	7	defined	define	VERB
ap-1807	34	8	and	and	CCONJ
ap-1807	34	9	x+	x+	ADJ
ap-1807	35	1	z	z	NOUN
ap-1807	35	2	=	=	PUNCT
ap-1807	35	3	y.	y.	NOUN
ap-1807	35	4	then	then	ADV
ap-1807	35	5	≤	≤	NUM
ap-1807	35	6	is	be	AUX
ap-1807	35	7	a	a	DET
ap-1807	35	8	partial	partial	ADJ
ap-1807	35	9	order	order	NOUN
ap-1807	35	10	on	on	ADP
ap-1807	35	11	e	e	NOUN
ap-1807	35	12	under	under	ADP
ap-1807	35	13	which	which	PRON
ap-1807	35	14	0	0	NUM
ap-1807	35	15	is	be	AUX
ap-1807	35	16	the	the	DET
ap-1807	35	17	least	least	ADJ
ap-1807	35	18	element	element	NOUN
ap-1807	35	19	of	of	ADP
ap-1807	35	20	e.	e.	PROPN
ap-1807	35	21	a	a	DET
ap-1807	35	22	generalized	generalized	ADJ
ap-1807	35	23	effect	effect	NOUN
ap-1807	35	24	algebra	algebra	NOUN
ap-1807	35	25	with	with	ADP
ap-1807	35	26	the	the	DET
ap-1807	35	27	top	top	ADJ
ap-1807	35	28	element	element	NOUN
ap-1807	35	29	1	1	NUM
ap-1807	35	30	∈	∈	NOUN
ap-1807	35	31	e	e	NOUN
ap-1807	35	32	is	be	AUX
ap-1807	35	33	called	call	VERB
ap-1807	35	34	an	an	DET
ap-1807	35	35	effect	effect	NOUN
ap-1807	35	36	algebra	algebra	NOUN
ap-1807	35	37	and	and	CCONJ
ap-1807	35	38	we	we	PRON
ap-1807	35	39	usually	usually	ADV
ap-1807	35	40	write	write	VERB
ap-1807	35	41	(	(	PUNCT
ap-1807	35	42	e,+	e,+	ADJ
ap-1807	35	43	,	,	PUNCT
ap-1807	35	44	0	0	NUM
ap-1807	35	45	,	,	PUNCT
ap-1807	35	46	1	1	NUM
ap-1807	35	47	)	)	PUNCT
ap-1807	35	48	.	.	PUNCT
ap-1807	36	1	a	a	DET
ap-1807	36	2	subset	subset	NOUN
ap-1807	36	3	s	s	NOUN
ap-1807	36	4	of	of	ADP
ap-1807	36	5	e	e	PROPN
ap-1807	36	6	is	be	AUX
ap-1807	36	7	called	call	VERB
ap-1807	36	8	a	a	DET
ap-1807	36	9	sub	sub	ADJ
ap-1807	36	10	-	-	ADJ
ap-1807	36	11	generalized	generalized	ADJ
ap-1807	36	12	effect	effect	NOUN
ap-1807	36	13	algebra	algebra	NOUN
ap-1807	36	14	(	(	PUNCT
ap-1807	36	15	sub	sub	ADJ
ap-1807	36	16	-	-	ADJ
ap-1807	36	17	effect	effect	ADJ
ap-1807	36	18	algebra	algebra	NOUN
ap-1807	36	19	)	)	PUNCT
ap-1807	36	20	of	of	ADP
ap-1807	36	21	e	e	PROPN
ap-1807	36	22	iff	iff	PROPN
ap-1807	36	23	(	(	PUNCT
ap-1807	36	24	i	i	NOUN
ap-1807	36	25	)	)	PUNCT
ap-1807	36	26	0	0	PUNCT
ap-1807	37	1	∈	∈	NOUN
ap-1807	37	2	s	s	X
ap-1807	37	3	(	(	PUNCT
ap-1807	37	4	1	1	NUM
ap-1807	37	5	∈	∈	PROPN
ap-1807	37	6	s	s	NOUN
ap-1807	37	7	)	)	PUNCT
ap-1807	37	8	,	,	PUNCT
ap-1807	37	9	(	(	PUNCT
ap-1807	37	10	ii	ii	NOUN
ap-1807	37	11	)	)	PUNCT
ap-1807	37	12	if	if	SCONJ
ap-1807	37	13	out	out	ADP
ap-1807	37	14	of	of	ADP
ap-1807	37	15	elements	element	NOUN
ap-1807	37	16	x	x	X
ap-1807	37	17	,	,	PUNCT
ap-1807	37	18	y	y	PROPN
ap-1807	37	19	,	,	PUNCT
ap-1807	37	20	z	z	NOUN
ap-1807	37	21	∈	∈	PROPN
ap-1807	37	22	e	e	NOUN
ap-1807	37	23	such	such	ADJ
ap-1807	37	24	that	that	SCONJ
ap-1807	37	25	x+	x+	ADJ
ap-1807	37	26	y	y	NOUN
ap-1807	37	27	=	=	SYM
ap-1807	37	28	z	z	NOUN
ap-1807	37	29	at	at	ADV
ap-1807	37	30	least	least	ADJ
ap-1807	37	31	two	two	NUM
ap-1807	37	32	are	be	AUX
ap-1807	37	33	in	in	ADP
ap-1807	37	34	s	s	PROPN
ap-1807	37	35	,	,	PUNCT
ap-1807	37	36	then	then	ADV
ap-1807	37	37	all	all	DET
ap-1807	37	38	x	x	NOUN
ap-1807	37	39	,	,	PUNCT
ap-1807	37	40	y	y	PROPN
ap-1807	37	41	,	,	PUNCT
ap-1807	37	42	z	z	PROPN
ap-1807	37	43	∈	∈	PROPN
ap-1807	37	44	s.	s.	PROPN
ap-1807	37	45	definition	definition	NOUN
ap-1807	37	46	2	2	NUM
ap-1807	37	47	.	.	PUNCT
ap-1807	38	1	[	[	X
ap-1807	38	2	6	6	NUM
ap-1807	38	3	]	]	PUNCT
ap-1807	38	4	a	a	DET
ap-1807	38	5	partial	partial	ADJ
ap-1807	38	6	algebra	algebra	NOUN
ap-1807	38	7	(	(	PUNCT
ap-1807	38	8	g,+	g,+	PROPN
ap-1807	38	9	,	,	PUNCT
ap-1807	38	10	0	0	NUM
ap-1807	38	11	)	)	PUNCT
ap-1807	38	12	is	be	AUX
ap-1807	38	13	called	call	VERB
ap-1807	38	14	a	a	DET
ap-1807	38	15	commutative	commutative	ADJ
ap-1807	38	16	partial	partial	ADJ
ap-1807	38	17	group	group	NOUN
ap-1807	38	18	if	if	SCONJ
ap-1807	38	19	0	0	NUM
ap-1807	38	20	∈	∈	NOUN
ap-1807	38	21	e	e	NOUN
ap-1807	38	22	is	be	AUX
ap-1807	38	23	a	a	DET
ap-1807	38	24	distinguished	distinguished	ADJ
ap-1807	38	25	289	289	NUM
ap-1807	38	26	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	ADJ
ap-1807	38	27	jiří	jiří	NOUN
ap-1807	38	28	janda	janda	PROPN
ap-1807	38	29	acta	acta	PROPN
ap-1807	38	30	polytechnica	polytechnica	PROPN
ap-1807	38	31	element	element	NOUN
ap-1807	38	32	and	and	CCONJ
ap-1807	38	33	+	+	CCONJ
ap-1807	38	34	is	be	AUX
ap-1807	38	35	a	a	DET
ap-1807	38	36	partially	partially	ADV
ap-1807	38	37	defined	define	VERB
ap-1807	38	38	binary	binary	ADJ
ap-1807	38	39	operation	operation	NOUN
ap-1807	38	40	on	on	ADP
ap-1807	38	41	e	e	PROPN
ap-1807	38	42	which	which	PRON
ap-1807	38	43	satisfies	satisfy	VERB
ap-1807	38	44	the	the	DET
ap-1807	38	45	following	follow	VERB
ap-1807	38	46	conditions	condition	NOUN
ap-1807	38	47	for	for	ADP
ap-1807	38	48	any	any	DET
ap-1807	38	49	x	x	NOUN
ap-1807	38	50	,	,	PUNCT
ap-1807	38	51	y	y	PROPN
ap-1807	38	52	,	,	PUNCT
ap-1807	38	53	z	z	NOUN
ap-1807	38	54	∈	∈	PROPN
ap-1807	39	1	e	e	NOUN
ap-1807	39	2	:	:	PUNCT
ap-1807	39	3	(	(	PUNCT
ap-1807	39	4	gpi	gpi	PROPN
ap-1807	39	5	)	)	PUNCT
ap-1807	39	6	x+	x+	PROPN
ap-1807	39	7	y	y	NOUN
ap-1807	39	8	=	=	SYM
ap-1807	39	9	y	y	PROPN
ap-1807	40	1	+	+	NOUN
ap-1807	40	2	x	x	SYM
ap-1807	40	3	if	if	SCONJ
ap-1807	40	4	x+	x+	ADJ
ap-1807	40	5	y	y	PROPN
ap-1807	40	6	is	be	AUX
ap-1807	40	7	defined	define	VERB
ap-1807	40	8	,	,	PUNCT
ap-1807	40	9	(	(	PUNCT
ap-1807	40	10	gpii	gpii	NOUN
ap-1807	40	11	)	)	PUNCT
ap-1807	40	12	(	(	PUNCT
ap-1807	40	13	x+	x+	X
ap-1807	40	14	y	y	X
ap-1807	40	15	)	)	PUNCT
ap-1807	40	16	+	+	NUM
ap-1807	40	17	z	z	NOUN
ap-1807	40	18	=	=	SYM
ap-1807	40	19	x+	x+	PUNCT
ap-1807	40	20	(	(	PUNCT
ap-1807	40	21	y	y	PROPN
ap-1807	40	22	+	+	PROPN
ap-1807	40	23	z	z	X
ap-1807	40	24	)	)	PUNCT
ap-1807	40	25	if	if	SCONJ
ap-1807	40	26	both	both	DET
ap-1807	40	27	sides	side	NOUN
ap-1807	40	28	are	be	AUX
ap-1807	40	29	defined	define	VERB
ap-1807	40	30	,	,	PUNCT
ap-1807	40	31	(	(	PUNCT
ap-1807	40	32	gpiii	gpiii	NOUN
ap-1807	40	33	)	)	PUNCT
ap-1807	40	34	x+	x+	SYM
ap-1807	40	35	0	0	NUM
ap-1807	40	36	is	be	AUX
ap-1807	40	37	defined	define	VERB
ap-1807	40	38	and	and	CCONJ
ap-1807	40	39	x+	x+	ADJ
ap-1807	40	40	0	0	NUM
ap-1807	41	1	=	=	SYM
ap-1807	41	2	x	x	NOUN
ap-1807	41	3	,	,	PUNCT
ap-1807	41	4	(	(	PUNCT
ap-1807	41	5	gpiv	gpiv	NOUN
ap-1807	41	6	)	)	PUNCT
ap-1807	41	7	for	for	ADP
ap-1807	41	8	every	every	DET
ap-1807	41	9	x	x	SYM
ap-1807	41	10	∈	∈	PROPN
ap-1807	41	11	e	e	NOUN
ap-1807	41	12	there	there	PRON
ap-1807	41	13	exists	exist	VERB
ap-1807	41	14	a	a	DET
ap-1807	41	15	unique	unique	ADJ
ap-1807	41	16	y	y	PROPN
ap-1807	41	17	∈	∈	PROPN
ap-1807	41	18	e	e	NOUN
ap-1807	41	19	such	such	ADJ
ap-1807	41	20	that	that	SCONJ
ap-1807	41	21	x+	x+	ADJ
ap-1807	41	22	y	y	PROPN
ap-1807	41	23	=	=	SYM
ap-1807	41	24	0	0	PUNCT
ap-1807	42	1	(	(	PUNCT
ap-1807	42	2	we	we	PRON
ap-1807	42	3	put	put	VERB
ap-1807	42	4	−x	−x	NOUN
ap-1807	42	5	=	=	SYM
ap-1807	42	6	y	y	NOUN
ap-1807	42	7	)	)	PUNCT
ap-1807	42	8	,	,	PUNCT
ap-1807	42	9	(	(	PUNCT
ap-1807	42	10	gpv	gpv	NOUN
ap-1807	42	11	)	)	PUNCT
ap-1807	42	12	x+	x+	PUNCT
ap-1807	43	1	y	y	NOUN
ap-1807	43	2	=	=	PUNCT
ap-1807	43	3	x+	x+	PROPN
ap-1807	43	4	z	z	NOUN
ap-1807	43	5	implies	imply	VERB
ap-1807	43	6	y	y	PROPN
ap-1807	43	7	=	=	SYM
ap-1807	43	8	z	z	PROPN
ap-1807	43	9	(	(	PUNCT
ap-1807	43	10	cancellation	cancellation	NOUN
ap-1807	43	11	law	law	NOUN
ap-1807	43	12	)	)	PUNCT
ap-1807	43	13	.	.	PUNCT
ap-1807	44	1	we	we	PRON
ap-1807	44	2	say	say	VERB
ap-1807	44	3	that	that	SCONJ
ap-1807	44	4	a	a	DET
ap-1807	44	5	commutative	commutative	ADJ
ap-1807	44	6	partial	partial	ADJ
ap-1807	44	7	group	group	NOUN
ap-1807	44	8	(	(	PUNCT
ap-1807	44	9	g,+	g,+	PROPN
ap-1807	44	10	,	,	PUNCT
ap-1807	44	11	0	0	NUM
ap-1807	44	12	)	)	PUNCT
ap-1807	44	13	is	be	AUX
ap-1807	44	14	weakly	weakly	ADV
ap-1807	44	15	ordered	order	VERB
ap-1807	44	16	(	(	PUNCT
ap-1807	44	17	shortly	shortly	ADV
ap-1807	44	18	a	a	DET
ap-1807	44	19	wop	wop	NOUN
ap-1807	44	20	-	-	PUNCT
ap-1807	44	21	group	group	NOUN
ap-1807	44	22	)	)	PUNCT
ap-1807	44	23	with	with	ADP
ap-1807	44	24	respect	respect	NOUN
ap-1807	44	25	to	to	ADP
ap-1807	44	26	a	a	DET
ap-1807	44	27	reflexive	reflexive	ADJ
ap-1807	44	28	and	and	CCONJ
ap-1807	44	29	antisymmetric	antisymmetric	ADJ
ap-1807	44	30	relation	relation	NOUN
ap-1807	44	31	≤	≤	NOUN
ap-1807	44	32	on	on	ADP
ap-1807	44	33	g	g	PROPN
ap-1807	44	34	if	if	SCONJ
ap-1807	44	35	≤	≤	PROPN
ap-1807	44	36	is	be	AUX
ap-1807	44	37	compatible	compatible	ADJ
ap-1807	44	38	w.r.t	w.r.t	NOUN
ap-1807	44	39	.	.	PUNCT
ap-1807	45	1	partial	partial	ADJ
ap-1807	45	2	addition	addition	NOUN
ap-1807	45	3	,	,	PUNCT
ap-1807	45	4	i.e.	i.e.	X
ap-1807	45	5	,	,	PUNCT
ap-1807	45	6	for	for	ADP
ap-1807	45	7	all	all	DET
ap-1807	45	8	x	x	NOUN
ap-1807	45	9	,	,	PUNCT
ap-1807	45	10	y	y	PROPN
ap-1807	45	11	,	,	PUNCT
ap-1807	45	12	z	z	PROPN
ap-1807	45	13	∈	∈	PROPN
ap-1807	45	14	g	g	PROPN
ap-1807	45	15	,	,	PUNCT
ap-1807	45	16	x	x	PUNCT
ap-1807	45	17	≤	≤	ADJ
ap-1807	45	18	y	y	NOUN
ap-1807	45	19	and	and	CCONJ
ap-1807	45	20	both	both	DET
ap-1807	45	21	x+z	x+z	NUM
ap-1807	45	22	and	and	CCONJ
ap-1807	45	23	y+z	y+z	PROPN
ap-1807	45	24	are	be	AUX
ap-1807	45	25	defined	define	VERB
ap-1807	45	26	implies	imply	VERB
ap-1807	45	27	x+	x+	PROPN
ap-1807	45	28	z	z	NOUN
ap-1807	45	29	≤	≤	NUM
ap-1807	45	30	y	y	PROPN
ap-1807	45	31	+	+	PROPN
ap-1807	45	32	z.	z.	PROPN
ap-1807	46	1	due	due	ADP
ap-1807	46	2	to	to	ADP
ap-1807	46	3	the	the	DET
ap-1807	46	4	non	non	ADJ
ap-1807	46	5	-	-	ADJ
ap-1807	46	6	constructive	constructive	ADJ
ap-1807	46	7	algebraic	algebraic	ADJ
ap-1807	46	8	nature	nature	NOUN
ap-1807	46	9	of	of	ADP
ap-1807	46	10	wop	wop	NOUN
ap-1807	46	11	-	-	PUNCT
ap-1807	46	12	groups	group	NOUN
ap-1807	46	13	,	,	PUNCT
ap-1807	46	14	we	we	PRON
ap-1807	46	15	will	will	AUX
ap-1807	46	16	introduce	introduce	VERB
ap-1807	46	17	a	a	DET
ap-1807	46	18	stronger	strong	ADJ
ap-1807	46	19	structure	structure	NOUN
ap-1807	46	20	with	with	ADP
ap-1807	46	21	the	the	DET
ap-1807	46	22	notion	notion	NOUN
ap-1807	46	23	of	of	ADP
ap-1807	46	24	a	a	DET
ap-1807	46	25	woa	woa	NOUN
ap-1807	46	26	-	-	PUNCT
ap-1807	46	27	group	group	NOUN
ap-1807	46	28	.	.	PUNCT
ap-1807	47	1	definition	definition	NOUN
ap-1807	47	2	3	3	NUM
ap-1807	47	3	.	.	PUNCT
ap-1807	48	1	a	a	DET
ap-1807	48	2	partial	partial	ADJ
ap-1807	48	3	algebra	algebra	NOUN
ap-1807	48	4	(	(	PUNCT
ap-1807	48	5	g,+	g,+	PROPN
ap-1807	48	6	,	,	PUNCT
ap-1807	48	7	0	0	NUM
ap-1807	48	8	)	)	PUNCT
ap-1807	48	9	is	be	AUX
ap-1807	48	10	called	call	VERB
ap-1807	48	11	an	an	DET
ap-1807	48	12	a	a	PRON
ap-1807	48	13	-	-	PUNCT
ap-1807	48	14	commutative	commutative	ADJ
ap-1807	48	15	partial	partial	ADJ
ap-1807	48	16	group	group	NOUN
ap-1807	48	17	if	if	SCONJ
ap-1807	48	18	0	0	NUM
ap-1807	48	19	∈	∈	PROPN
ap-1807	48	20	g	g	NOUN
ap-1807	48	21	is	be	AUX
ap-1807	48	22	a	a	DET
ap-1807	48	23	distinguished	distinguished	ADJ
ap-1807	48	24	element	element	NOUN
ap-1807	48	25	and	and	CCONJ
ap-1807	48	26	+	+	CCONJ
ap-1807	48	27	is	be	AUX
ap-1807	48	28	a	a	DET
ap-1807	48	29	partially	partially	ADV
ap-1807	48	30	defined	define	VERB
ap-1807	48	31	binary	binary	ADJ
ap-1807	48	32	operation	operation	NOUN
ap-1807	48	33	on	on	ADP
ap-1807	48	34	g	g	PROPN
ap-1807	48	35	which	which	PRON
ap-1807	48	36	satisfies	satisfy	VERB
ap-1807	48	37	the	the	DET
ap-1807	48	38	following	follow	VERB
ap-1807	48	39	conditions	condition	NOUN
ap-1807	48	40	for	for	ADP
ap-1807	48	41	any	any	DET
ap-1807	48	42	x	x	NOUN
ap-1807	48	43	,	,	PUNCT
ap-1807	48	44	y	y	PROPN
ap-1807	48	45	,	,	PUNCT
ap-1807	48	46	z	z	PROPN
ap-1807	48	47	∈	∈	PROPN
ap-1807	48	48	g	g	NOUN
ap-1807	48	49	:	:	PUNCT
ap-1807	48	50	(	(	PUNCT
ap-1807	48	51	gi	gi	INTJ
ap-1807	48	52	)	)	PUNCT
ap-1807	48	53	x+	x+	PUNCT
ap-1807	49	1	y	y	NOUN
ap-1807	49	2	=	=	SYM
ap-1807	49	3	y	y	PROPN
ap-1807	50	1	+	+	NOUN
ap-1807	50	2	x	x	SYM
ap-1807	50	3	if	if	SCONJ
ap-1807	50	4	x+	x+	ADJ
ap-1807	50	5	y	y	PROPN
ap-1807	50	6	is	be	AUX
ap-1807	50	7	defined	define	VERB
ap-1807	50	8	,	,	PUNCT
ap-1807	50	9	(	(	PUNCT
ap-1807	50	10	gii	gii	NOUN
ap-1807	50	11	)	)	PUNCT
ap-1807	50	12	x+	x+	X
ap-1807	50	13	0	0	NUM
ap-1807	50	14	is	be	AUX
ap-1807	50	15	defined	define	VERB
ap-1807	50	16	and	and	CCONJ
ap-1807	50	17	x+	x+	ADJ
ap-1807	50	18	0	0	NUM
ap-1807	51	1	=	=	SYM
ap-1807	51	2	x	x	NOUN
ap-1807	51	3	,	,	PUNCT
ap-1807	51	4	(	(	PUNCT
ap-1807	51	5	giii	giii	NOUN
ap-1807	51	6	)	)	PUNCT
ap-1807	51	7	for	for	ADP
ap-1807	51	8	every	every	DET
ap-1807	51	9	x	x	SYM
ap-1807	51	10	∈	∈	PROPN
ap-1807	51	11	e	e	NOUN
ap-1807	51	12	there	there	PRON
ap-1807	51	13	exists	exist	VERB
ap-1807	51	14	a	a	DET
ap-1807	51	15	unique	unique	ADJ
ap-1807	51	16	y	y	PROPN
ap-1807	51	17	∈	∈	PROPN
ap-1807	51	18	e	e	NOUN
ap-1807	51	19	such	such	ADJ
ap-1807	51	20	that	that	SCONJ
ap-1807	51	21	x+	x+	ADJ
ap-1807	51	22	y	y	PROPN
ap-1807	51	23	=	=	SYM
ap-1807	51	24	0	0	PUNCT
ap-1807	52	1	(	(	PUNCT
ap-1807	52	2	we	we	PRON
ap-1807	52	3	put	put	VERB
ap-1807	52	4	−x	−x	NOUN
ap-1807	52	5	=	=	SYM
ap-1807	52	6	y	y	NOUN
ap-1807	52	7	)	)	PUNCT
ap-1807	52	8	,	,	PUNCT
ap-1807	52	9	(	(	PUNCT
ap-1807	52	10	giv	giv	NOUN
ap-1807	52	11	)	)	PUNCT
ap-1807	52	12	if	if	SCONJ
ap-1807	52	13	(	(	PUNCT
ap-1807	52	14	x+	x+	X
ap-1807	52	15	y	y	NOUN
ap-1807	52	16	)	)	PUNCT
ap-1807	53	1	+	+	CCONJ
ap-1807	53	2	z	z	NOUN
ap-1807	53	3	and	and	CCONJ
ap-1807	53	4	(	(	PUNCT
ap-1807	53	5	y	y	PROPN
ap-1807	53	6	+	+	PROPN
ap-1807	53	7	z	z	X
ap-1807	53	8	)	)	PUNCT
ap-1807	53	9	are	be	AUX
ap-1807	53	10	defined	define	VERB
ap-1807	53	11	,	,	PUNCT
ap-1807	53	12	then	then	ADV
ap-1807	53	13	x+	x+	NUM
ap-1807	53	14	(	(	PUNCT
ap-1807	53	15	y+	y+	PROPN
ap-1807	53	16	z	z	NOUN
ap-1807	53	17	)	)	PUNCT
ap-1807	53	18	is	be	AUX
ap-1807	53	19	defined	define	VERB
ap-1807	53	20	and	and	CCONJ
ap-1807	53	21	(	(	PUNCT
ap-1807	53	22	x+y	x+y	NUM
ap-1807	53	23	)	)	PUNCT
ap-1807	54	1	+	+	CCONJ
ap-1807	54	2	z	z	X
ap-1807	54	3	=	=	SYM
ap-1807	54	4	x+	x+	X
ap-1807	54	5	(	(	PUNCT
ap-1807	54	6	y+	y+	PROPN
ap-1807	54	7	z	z	PROPN
ap-1807	54	8	)	)	PUNCT
ap-1807	54	9	.	.	PUNCT
ap-1807	55	1	an	an	DET
ap-1807	55	2	a	a	PRON
ap-1807	55	3	-	-	PUNCT
ap-1807	55	4	commutative	commutative	ADJ
ap-1807	55	5	partial	partial	ADJ
ap-1807	55	6	group	group	NOUN
ap-1807	55	7	(	(	PUNCT
ap-1807	55	8	g,+	g,+	PROPN
ap-1807	55	9	,	,	PUNCT
ap-1807	55	10	0	0	NUM
ap-1807	55	11	)	)	PUNCT
ap-1807	55	12	is	be	AUX
ap-1807	55	13	called	call	VERB
ap-1807	55	14	weakly	weakly	ADV
ap-1807	55	15	ordered	order	VERB
ap-1807	55	16	(	(	PUNCT
ap-1807	55	17	shortly	shortly	ADV
ap-1807	55	18	a	a	DET
ap-1807	55	19	woa	woa	NOUN
ap-1807	55	20	-	-	PUNCT
ap-1807	55	21	group	group	NOUN
ap-1807	55	22	)	)	PUNCT
ap-1807	55	23	with	with	ADP
ap-1807	55	24	respect	respect	NOUN
ap-1807	55	25	to	to	ADP
ap-1807	55	26	a	a	DET
ap-1807	55	27	reflexive	reflexive	ADJ
ap-1807	55	28	and	and	CCONJ
ap-1807	55	29	antisymmetric	antisymmetric	ADJ
ap-1807	55	30	relation	relation	NOUN
ap-1807	55	31	≤	≤	NOUN
ap-1807	55	32	on	on	ADP
ap-1807	55	33	g	g	PROPN
ap-1807	55	34	(	(	PUNCT
ap-1807	55	35	we	we	PRON
ap-1807	55	36	call	call	VERB
ap-1807	55	37	it	it	PRON
ap-1807	55	38	a	a	DET
ap-1807	55	39	weak	weak	ADJ
ap-1807	55	40	order	order	NOUN
ap-1807	55	41	)	)	PUNCT
ap-1807	55	42	if	if	SCONJ
ap-1807	55	43	(	(	PUNCT
ap-1807	55	44	ri	ri	NOUN
ap-1807	55	45	)	)	PUNCT
ap-1807	55	46	x	x	SYM
ap-1807	55	47	≤	≤	PROPN
ap-1807	55	48	y	y	PROPN
ap-1807	55	49	iff	iff	PROPN
ap-1807	55	50	there	there	PRON
ap-1807	55	51	exists	exist	VERB
ap-1807	55	52	0	0	NUM
ap-1807	55	53	≤	≤	NUM
ap-1807	56	1	z	z	NOUN
ap-1807	56	2	,	,	PUNCT
ap-1807	56	3	x	x	PUNCT
ap-1807	57	1	+	+	CCONJ
ap-1807	57	2	z	z	NOUN
ap-1807	57	3	is	be	AUX
ap-1807	57	4	defined	define	VERB
ap-1807	57	5	and	and	CCONJ
ap-1807	57	6	x+	x+	ADJ
ap-1807	57	7	z	z	NOUN
ap-1807	57	8	=	=	SYM
ap-1807	57	9	y	y	PROPN
ap-1807	57	10	,	,	PUNCT
ap-1807	57	11	(	(	PUNCT
ap-1807	57	12	rii	rii	PROPN
ap-1807	57	13	)	)	PUNCT
ap-1807	57	14	0	0	NUM
ap-1807	57	15	≤	≤	NUM
ap-1807	57	16	x	x	X
ap-1807	57	17	,	,	PUNCT
ap-1807	57	18	y	y	PROPN
ap-1807	57	19	and	and	CCONJ
ap-1807	57	20	x+	x+	PROPN
ap-1807	57	21	y	y	PROPN
ap-1807	57	22	defined	define	VERB
ap-1807	57	23	,	,	PUNCT
ap-1807	57	24	then	then	ADV
ap-1807	57	25	0	0	NUM
ap-1807	57	26	≤	≤	NUM
ap-1807	58	1	x+	x+	PROPN
ap-1807	58	2	y	y	PROPN
ap-1807	58	3	,	,	PUNCT
ap-1807	58	4	(	(	PUNCT
ap-1807	58	5	riii	riii	PROPN
ap-1807	58	6	)	)	PUNCT
ap-1807	58	7	0	0	NUM
ap-1807	58	8	≤	≤	NUM
ap-1807	58	9	x	x	X
ap-1807	58	10	,	,	PUNCT
ap-1807	58	11	0	0	NUM
ap-1807	58	12	≤	≤	NUM
ap-1807	59	1	z	z	NOUN
ap-1807	59	2	,	,	PUNCT
ap-1807	59	3	x	x	PUNCT
ap-1807	59	4	≤	≤	ADJ
ap-1807	59	5	y	y	PROPN
ap-1807	59	6	and	and	CCONJ
ap-1807	59	7	y+z	y+z	PROPN
ap-1807	59	8	defined	define	VERB
ap-1807	59	9	implies	imply	VERB
ap-1807	59	10	x+	x+	PROPN
ap-1807	59	11	z	z	NOUN
ap-1807	59	12	defined	define	VERB
ap-1807	59	13	.	.	PUNCT
ap-1807	60	1	note	note	VERB
ap-1807	60	2	that	that	SCONJ
ap-1807	60	3	in	in	ADP
ap-1807	60	4	the	the	DET
ap-1807	60	5	case	case	NOUN
ap-1807	60	6	of	of	ADP
ap-1807	60	7	generalized	generalized	ADJ
ap-1807	60	8	effect	effect	NOUN
ap-1807	60	9	algebras	algebra	VERB
ap-1807	60	10	the	the	DET
ap-1807	60	11	partial	partial	ADJ
ap-1807	60	12	order	order	NOUN
ap-1807	60	13	is	be	AUX
ap-1807	60	14	induced	induce	VERB
ap-1807	60	15	from	from	ADP
ap-1807	60	16	a	a	DET
ap-1807	60	17	partial	partial	ADJ
ap-1807	60	18	operation	operation	NOUN
ap-1807	60	19	.	.	PUNCT
ap-1807	61	1	on	on	ADP
ap-1807	61	2	the	the	DET
ap-1807	61	3	other	other	ADJ
ap-1807	61	4	hand	hand	NOUN
ap-1807	61	5	,	,	PUNCT
ap-1807	61	6	the	the	DET
ap-1807	61	7	weak	weak	ADJ
ap-1807	61	8	order	order	NOUN
ap-1807	61	9	for	for	ADP
ap-1807	61	10	woa	woa	NOUN
ap-1807	61	11	-	-	PUNCT
ap-1807	61	12	groups	group	NOUN
ap-1807	61	13	(	(	PUNCT
ap-1807	61	14	wop	wop	NOUN
ap-1807	61	15	-	-	PUNCT
ap-1807	61	16	groups	group	NOUN
ap-1807	61	17	)	)	PUNCT
ap-1807	61	18	can	can	AUX
ap-1807	61	19	be	be	AUX
ap-1807	61	20	chosen	choose	VERB
ap-1807	61	21	in	in	ADP
ap-1807	61	22	various	various	ADJ
ap-1807	61	23	ways	way	NOUN
ap-1807	61	24	.	.	PUNCT
ap-1807	62	1	we	we	PRON
ap-1807	62	2	will	will	AUX
ap-1807	62	3	show	show	VERB
ap-1807	62	4	that	that	SCONJ
ap-1807	62	5	the	the	DET
ap-1807	62	6	weak	weak	ADJ
ap-1807	62	7	order	order	NOUN
ap-1807	62	8	is	be	AUX
ap-1807	62	9	determined	determine	VERB
ap-1807	62	10	by	by	ADP
ap-1807	62	11	the	the	DET
ap-1807	62	12	partial	partial	ADJ
ap-1807	62	13	operation	operation	NOUN
ap-1807	62	14	and	and	CCONJ
ap-1807	62	15	the	the	DET
ap-1807	62	16	set	set	NOUN
ap-1807	62	17	of	of	ADP
ap-1807	62	18	positive	positive	ADJ
ap-1807	62	19	elements	element	NOUN
ap-1807	62	20	.	.	PUNCT
ap-1807	63	1	remark	remark	NOUN
ap-1807	63	2	1	1	NUM
ap-1807	63	3	.	.	PUNCT
ap-1807	64	1	using	use	VERB
ap-1807	64	2	the	the	DET
ap-1807	64	3	commutativity	commutativity	NOUN
ap-1807	64	4	(	(	PUNCT
ap-1807	64	5	gi	gi	NOUN
ap-1807	64	6	)	)	PUNCT
ap-1807	64	7	with	with	ADP
ap-1807	64	8	the	the	DET
ap-1807	64	9	axiom	axiom	NOUN
ap-1807	64	10	(	(	PUNCT
ap-1807	64	11	giv	giv	PROPN
ap-1807	64	12	)	)	PUNCT
ap-1807	64	13	,	,	PUNCT
ap-1807	64	14	we	we	PRON
ap-1807	64	15	can	can	AUX
ap-1807	64	16	obtain	obtain	VERB
ap-1807	64	17	similar	similar	ADJ
ap-1807	64	18	formulas	formula	NOUN
ap-1807	64	19	to	to	ADP
ap-1807	64	20	(	(	PUNCT
ap-1807	64	21	giv	giv	PROPN
ap-1807	64	22	)	)	PUNCT
ap-1807	64	23	for	for	ADP
ap-1807	64	24	any	any	DET
ap-1807	64	25	permutation	permutation	NOUN
ap-1807	64	26	of	of	ADP
ap-1807	64	27	variables	variable	NOUN
ap-1807	64	28	.	.	PUNCT
ap-1807	65	1	that	that	PRON
ap-1807	65	2	is	is	ADV
ap-1807	65	3	,	,	PUNCT
ap-1807	65	4	let	let	VERB
ap-1807	65	5	us	we	PRON
ap-1807	65	6	have	have	VERB
ap-1807	65	7	an	an	DET
ap-1807	65	8	a	a	PRON
ap-1807	65	9	-	-	PUNCT
ap-1807	65	10	commutative	commutative	ADJ
ap-1807	65	11	group	group	NOUN
ap-1807	65	12	(	(	PUNCT
ap-1807	65	13	g	g	NOUN
ap-1807	65	14	,	,	PUNCT
ap-1807	65	15	0,+	0,+	NUM
ap-1807	65	16	)	)	PUNCT
ap-1807	65	17	.	.	PUNCT
ap-1807	66	1	then	then	ADV
ap-1807	66	2	for	for	ADP
ap-1807	66	3	any	any	DET
ap-1807	66	4	x	x	NOUN
ap-1807	66	5	,	,	PUNCT
ap-1807	66	6	y	y	PROPN
ap-1807	66	7	,	,	PUNCT
ap-1807	66	8	z	z	PROPN
ap-1807	66	9	∈	∈	PROPN
ap-1807	66	10	g	g	PROPN
ap-1807	66	11	,	,	PUNCT
ap-1807	66	12	from	from	ADP
ap-1807	66	13	the	the	DET
ap-1807	66	14	existence	existence	NOUN
ap-1807	66	15	of	of	ADP
ap-1807	66	16	(	(	PUNCT
ap-1807	66	17	x	x	X
ap-1807	66	18	+	+	NUM
ap-1807	66	19	y	y	NOUN
ap-1807	66	20	)	)	PUNCT
ap-1807	67	1	+	+	CCONJ
ap-1807	67	2	z	z	NOUN
ap-1807	67	3	and	and	CCONJ
ap-1807	67	4	x	x	X
ap-1807	68	1	+	+	CCONJ
ap-1807	68	2	z	z	NOUN
ap-1807	68	3	we	we	PRON
ap-1807	68	4	have	have	VERB
ap-1807	68	5	(	(	PUNCT
ap-1807	68	6	x	x	X
ap-1807	68	7	+	+	NUM
ap-1807	68	8	y	y	NOUN
ap-1807	68	9	)	)	PUNCT
ap-1807	69	1	+	+	NUM
ap-1807	69	2	z	z	NOUN
ap-1807	69	3	=	=	SYM
ap-1807	69	4	(	(	PUNCT
ap-1807	69	5	x	x	X
ap-1807	69	6	+	+	NUM
ap-1807	69	7	z	z	NOUN
ap-1807	69	8	)	)	PUNCT
ap-1807	70	1	+	+	CCONJ
ap-1807	70	2	y.	y.	NOUN
ap-1807	70	3	or	or	CCONJ
ap-1807	70	4	similarly	similarly	ADV
ap-1807	70	5	,	,	PUNCT
ap-1807	70	6	the	the	DET
ap-1807	70	7	existence	existence	NOUN
ap-1807	70	8	of	of	ADP
ap-1807	70	9	x+	x+	X
ap-1807	70	10	(	(	PUNCT
ap-1807	70	11	y+	y+	PROPN
ap-1807	70	12	z	z	NOUN
ap-1807	70	13	)	)	PUNCT
ap-1807	70	14	and	and	CCONJ
ap-1807	70	15	x+	x+	NUM
ap-1807	70	16	y	y	PROPN
ap-1807	70	17	implies	imply	VERB
ap-1807	70	18	x+	x+	NUM
ap-1807	70	19	(	(	PUNCT
ap-1807	70	20	y	y	PROPN
ap-1807	70	21	+	+	PROPN
ap-1807	70	22	z	z	NOUN
ap-1807	70	23	)	)	PUNCT
ap-1807	70	24	=	=	SYM
ap-1807	71	1	(	(	PUNCT
ap-1807	71	2	x+	x+	X
ap-1807	71	3	y	y	NOUN
ap-1807	71	4	)	)	PUNCT
ap-1807	72	1	+	+	CCONJ
ap-1807	72	2	z	z	NOUN
ap-1807	72	3	and	and	CCONJ
ap-1807	72	4	so	so	ADV
ap-1807	72	5	on	on	ADV
ap-1807	72	6	.	.	PUNCT
ap-1807	73	1	in	in	ADP
ap-1807	73	2	the	the	DET
ap-1807	73	3	following	following	NOUN
ap-1807	73	4	,	,	PUNCT
ap-1807	73	5	we	we	PRON
ap-1807	73	6	will	will	AUX
ap-1807	73	7	be	be	AUX
ap-1807	73	8	using	use	VERB
ap-1807	73	9	(	(	PUNCT
ap-1807	73	10	giv	giv	NOUN
ap-1807	73	11	)	)	PUNCT
ap-1807	73	12	in	in	ADP
ap-1807	73	13	this	this	DET
ap-1807	73	14	more	more	ADV
ap-1807	73	15	general	general	ADJ
ap-1807	73	16	sense	sense	NOUN
ap-1807	73	17	and	and	CCONJ
ap-1807	73	18	we	we	PRON
ap-1807	73	19	omit	omit	VERB
ap-1807	73	20	mentioning	mention	VERB
ap-1807	73	21	the	the	DET
ap-1807	73	22	commutativity	commutativity	NOUN
ap-1807	73	23	.	.	PUNCT
ap-1807	74	1	definition	definition	NOUN
ap-1807	74	2	4	4	NUM
ap-1807	74	3	.	.	PUNCT
ap-1807	75	1	let	let	VERB
ap-1807	75	2	(	(	PUNCT
ap-1807	75	3	g,+	g,+	PROPN
ap-1807	75	4	,	,	PUNCT
ap-1807	75	5	0	0	NUM
ap-1807	75	6	)	)	PUNCT
ap-1807	75	7	be	be	AUX
ap-1807	75	8	an	an	DET
ap-1807	75	9	(	(	PUNCT
ap-1807	75	10	a-)commutative	a-)commutative	ADJ
ap-1807	75	11	partial	partial	ADJ
ap-1807	75	12	group	group	NOUN
ap-1807	75	13	and	and	CCONJ
ap-1807	75	14	let	let	VERB
ap-1807	75	15	s	s	PRON
ap-1807	75	16	be	be	AUX
ap-1807	75	17	a	a	DET
ap-1807	75	18	subset	subset	NOUN
ap-1807	75	19	of	of	ADP
ap-1807	75	20	g	g	NOUN
ap-1807	75	21	such	such	ADJ
ap-1807	75	22	as	as	ADP
ap-1807	75	23	(	(	PUNCT
ap-1807	75	24	si	si	NOUN
ap-1807	75	25	)	)	PUNCT
ap-1807	75	26	0	0	PUNCT
ap-1807	76	1	∈	∈	PROPN
ap-1807	76	2	s	s	NOUN
ap-1807	76	3	,	,	PUNCT
ap-1807	76	4	(	(	PUNCT
ap-1807	76	5	sii	sii	ADJ
ap-1807	76	6	)	)	PUNCT
ap-1807	76	7	−x	−x	NOUN
ap-1807	76	8	∈	∈	PROPN
ap-1807	76	9	s	s	NOUN
ap-1807	76	10	for	for	ADP
ap-1807	76	11	all	all	DET
ap-1807	76	12	x	x	SYM
ap-1807	76	13	∈	∈	PROPN
ap-1807	76	14	s	s	NOUN
ap-1807	76	15	,	,	PUNCT
ap-1807	76	16	(	(	PUNCT
ap-1807	76	17	siii	siii	NOUN
ap-1807	76	18	)	)	PUNCT
ap-1807	76	19	for	for	ADP
ap-1807	76	20	every	every	DET
ap-1807	76	21	x	x	PROPN
ap-1807	76	22	,	,	PUNCT
ap-1807	76	23	y	y	PROPN
ap-1807	76	24	∈	∈	PROPN
ap-1807	76	25	s	s	VERB
ap-1807	76	26	such	such	ADJ
ap-1807	76	27	that	that	SCONJ
ap-1807	76	28	x+	x+	PROPN
ap-1807	76	29	y	y	PROPN
ap-1807	76	30	is	be	AUX
ap-1807	76	31	defined	define	VERB
ap-1807	76	32	also	also	ADV
ap-1807	76	33	x+	x+	ADJ
ap-1807	76	34	y	y	PROPN
ap-1807	76	35	∈	∈	PROPN
ap-1807	76	36	s.	s.	PROPN
ap-1807	77	1	then	then	ADV
ap-1807	77	2	we	we	PRON
ap-1807	77	3	call	call	VERB
ap-1807	77	4	s	s	VERB
ap-1807	77	5	an	an	DET
ap-1807	77	6	(	(	PUNCT
ap-1807	77	7	a-)commutative	a-)commutative	ADJ
ap-1807	77	8	partial	partial	ADJ
ap-1807	77	9	subgroup	subgroup	NOUN
ap-1807	77	10	of	of	ADP
ap-1807	77	11	g.	g.	PROPN
ap-1807	77	12	let	let	VERB
ap-1807	77	13	g	g	NOUN
ap-1807	77	14	be	be	AUX
ap-1807	77	15	a	a	DET
ap-1807	77	16	wop	wop	NOUN
ap-1807	77	17	-	-	PUNCT
ap-1807	77	18	group	group	NOUN
ap-1807	77	19	(	(	PUNCT
ap-1807	77	20	woa	woa	NOUN
ap-1807	77	21	-	-	PUNCT
ap-1807	77	22	group	group	NOUN
ap-1807	77	23	)	)	PUNCT
ap-1807	77	24	with	with	ADP
ap-1807	77	25	respect	respect	NOUN
ap-1807	77	26	to	to	ADP
ap-1807	77	27	a	a	DET
ap-1807	77	28	relation	relation	NOUN
ap-1807	77	29	≤g	≤g	NOUN
ap-1807	77	30	and	and	CCONJ
ap-1807	77	31	let	let	VERB
ap-1807	77	32	≤s	≤s	PROPN
ap-1807	77	33	be	be	AUX
ap-1807	77	34	a	a	DET
ap-1807	77	35	relation	relation	NOUN
ap-1807	77	36	on	on	ADP
ap-1807	77	37	a	a	DET
ap-1807	77	38	(	(	PUNCT
ap-1807	77	39	a-)commutative	a-)commutative	ADJ
ap-1807	77	40	partial	partial	ADJ
ap-1807	77	41	subgroup	subgroup	NOUN
ap-1807	77	42	s	s	PART
ap-1807	77	43	⊆	⊆	NUM
ap-1807	77	44	g.	g.	NOUN
ap-1807	77	45	if	if	SCONJ
ap-1807	77	46	for	for	ADP
ap-1807	77	47	all	all	DET
ap-1807	77	48	x	x	NOUN
ap-1807	77	49	,	,	PUNCT
ap-1807	77	50	y	y	PROPN
ap-1807	77	51	∈	∈	PROPN
ap-1807	77	52	s	s	AUX
ap-1807	77	53	holds	hold	VERB
ap-1807	77	54	:	:	PUNCT
ap-1807	78	1	x	x	X
ap-1807	78	2	≤s	≤s	NOUN
ap-1807	78	3	y	y	PROPN
ap-1807	78	4	if	if	SCONJ
ap-1807	78	5	and	and	CCONJ
ap-1807	78	6	only	only	ADV
ap-1807	78	7	if	if	SCONJ
ap-1807	78	8	x	x	SYM
ap-1807	78	9	≤g	≤g	NOUN
ap-1807	78	10	y	y	PROPN
ap-1807	78	11	,	,	PUNCT
ap-1807	78	12	we	we	PRON
ap-1807	78	13	call	call	VERB
ap-1807	78	14	s	s	VERB
ap-1807	78	15	a	a	DET
ap-1807	78	16	wop	wop	NOUN
ap-1807	78	17	-	-	PUNCT
ap-1807	78	18	subgroup	subgroup	NOUN
ap-1807	78	19	(	(	PUNCT
ap-1807	78	20	woa	woa	NOUN
ap-1807	78	21	-	-	PUNCT
ap-1807	78	22	subgroup	subgroup	NOUN
ap-1807	78	23	)	)	PUNCT
ap-1807	78	24	of	of	ADP
ap-1807	78	25	g.	g.	PROPN
ap-1807	78	26	lemma	lemma	PROPN
ap-1807	79	1	1	1	X
ap-1807	79	2	.	.	PUNCT
ap-1807	80	1	let	let	VERB
ap-1807	80	2	(	(	PUNCT
ap-1807	80	3	g,+	g,+	PROPN
ap-1807	80	4	,	,	PUNCT
ap-1807	80	5	0	0	NUM
ap-1807	80	6	)	)	PUNCT
ap-1807	80	7	be	be	AUX
ap-1807	80	8	an	an	DET
ap-1807	80	9	a	a	PRON
ap-1807	80	10	-	-	PUNCT
ap-1807	80	11	commutative	commutative	ADJ
ap-1807	80	12	partial	partial	ADJ
ap-1807	80	13	group	group	NOUN
ap-1807	80	14	.	.	PUNCT
ap-1807	81	1	then	then	ADV
ap-1807	81	2	for	for	ADP
ap-1807	81	3	any	any	DET
ap-1807	81	4	a	a	DET
ap-1807	81	5	,	,	PUNCT
ap-1807	81	6	b	b	NOUN
ap-1807	81	7	,	,	PUNCT
ap-1807	81	8	c	c	NOUN
ap-1807	81	9	,	,	PUNCT
ap-1807	81	10	x	x	NOUN
ap-1807	81	11	,	,	PUNCT
ap-1807	81	12	y	y	PROPN
ap-1807	81	13	,	,	PUNCT
ap-1807	81	14	z	z	PROPN
ap-1807	81	15	∈	∈	PROPN
ap-1807	81	16	g	g	ADP
ap-1807	81	17	the	the	DET
ap-1807	81	18	following	follow	VERB
ap-1807	81	19	holds	hold	VERB
ap-1807	81	20	:	:	PUNCT
ap-1807	81	21	(	(	PUNCT
ap-1807	81	22	1	1	NUM
ap-1807	81	23	.	.	PUNCT
ap-1807	81	24	)	)	PUNCT
ap-1807	81	25	a+	a+	PUNCT
ap-1807	82	1	c	c	X
ap-1807	82	2	=	=	SYM
ap-1807	82	3	b	b	X
ap-1807	82	4	iff	iff	PROPN
ap-1807	82	5	c	c	PROPN
ap-1807	82	6	=	=	PRON
ap-1807	82	7	b+	b+	X
ap-1807	82	8	(	(	PUNCT
ap-1807	82	9	−a	−a	ADV
ap-1807	82	10	)	)	PUNCT
ap-1807	82	11	,	,	PUNCT
ap-1807	82	12	(	(	PUNCT
ap-1807	82	13	2	2	NUM
ap-1807	82	14	.	.	PUNCT
ap-1807	82	15	)	)	PUNCT
ap-1807	82	16	a+	a+	PUNCT
ap-1807	83	1	x	x	X
ap-1807	83	2	=	=	SYM
ap-1807	83	3	(	(	PUNCT
ap-1807	83	4	a+	a+	X
ap-1807	83	5	y	y	NOUN
ap-1807	83	6	)	)	PUNCT
ap-1807	84	1	+	+	CCONJ
ap-1807	84	2	z	z	NOUN
ap-1807	84	3	implies	imply	VERB
ap-1807	84	4	x	x	PUNCT
ap-1807	84	5	=	=	SYM
ap-1807	84	6	(	(	PUNCT
ap-1807	84	7	y	y	PROPN
ap-1807	84	8	+	+	PROPN
ap-1807	84	9	z	z	NOUN
ap-1807	84	10	)	)	PUNCT
ap-1807	84	11	,	,	PUNCT
ap-1807	84	12	(	(	PUNCT
ap-1807	84	13	3	3	X
ap-1807	84	14	.	.	PUNCT
ap-1807	84	15	)	)	PUNCT
ap-1807	85	1	whenever	whenever	SCONJ
ap-1807	85	2	a+	a+	PRON
ap-1807	85	3	b	b	NOUN
ap-1807	85	4	is	be	AUX
ap-1807	85	5	defined	define	VERB
ap-1807	85	6	,	,	PUNCT
ap-1807	85	7	then	then	ADV
ap-1807	85	8	(	(	PUNCT
ap-1807	85	9	−a	−a	ADV
ap-1807	85	10	)	)	PUNCT
ap-1807	86	1	+	+	CCONJ
ap-1807	86	2	(	(	PUNCT
ap-1807	86	3	−b	−b	ADJ
ap-1807	86	4	)	)	PUNCT
ap-1807	86	5	is	be	AUX
ap-1807	86	6	defined	define	VERB
ap-1807	86	7	and	and	CCONJ
ap-1807	86	8	(	(	PUNCT
ap-1807	86	9	−a	−a	ADV
ap-1807	86	10	)	)	PUNCT
ap-1807	87	1	+	+	CCONJ
ap-1807	87	2	(	(	PUNCT
ap-1807	87	3	−b	−b	ADJ
ap-1807	87	4	)	)	PUNCT
ap-1807	87	5	=	=	PUNCT
ap-1807	88	1	−(a+	−(a+	NUM
ap-1807	88	2	b	b	NOUN
ap-1807	88	3	)	)	PUNCT
ap-1807	88	4	.	.	PUNCT
ap-1807	89	1	proof	proof	NOUN
ap-1807	89	2	.	.	PUNCT
ap-1807	90	1	(	(	PUNCT
ap-1807	90	2	1	1	NUM
ap-1807	90	3	.	.	PUNCT
ap-1807	90	4	)	)	PUNCT
ap-1807	91	1	we	we	PRON
ap-1807	91	2	have	have	VERB
ap-1807	91	3	c	c	NOUN
ap-1807	91	4	=	=	SYM
ap-1807	91	5	c+	c+	X
ap-1807	91	6	(	(	PUNCT
ap-1807	91	7	a+	a+	X
ap-1807	91	8	(	(	PUNCT
ap-1807	91	9	−a	−a	NOUN
ap-1807	91	10	)	)	PUNCT
ap-1807	91	11	)	)	PUNCT
ap-1807	92	1	=	=	PUNCT
ap-1807	92	2	(	(	PUNCT
ap-1807	92	3	c+	c+	VERB
ap-1807	92	4	a	a	NOUN
ap-1807	92	5	)	)	PUNCT
ap-1807	92	6	+	+	CCONJ
ap-1807	92	7	(	(	PUNCT
ap-1807	92	8	−a	−a	ADJ
ap-1807	92	9	)	)	PUNCT
ap-1807	92	10	=	=	PUNCT
ap-1807	92	11	b+	b+	X
ap-1807	92	12	(	(	PUNCT
ap-1807	92	13	−a	−a	ADV
ap-1807	92	14	)	)	PUNCT
ap-1807	92	15	.	.	PUNCT
ap-1807	93	1	(	(	PUNCT
ap-1807	93	2	2	2	NUM
ap-1807	93	3	.	.	PUNCT
ap-1807	93	4	)	)	PUNCT
ap-1807	93	5	let	let	VERB
ap-1807	93	6	us	we	PRON
ap-1807	93	7	have	have	VERB
ap-1807	93	8	a	a	DET
ap-1807	93	9	+	+	NOUN
ap-1807	93	10	x	x	SYM
ap-1807	93	11	=	=	SYM
ap-1807	93	12	(	(	PUNCT
ap-1807	93	13	a	a	DET
ap-1807	93	14	+	+	NOUN
ap-1807	93	15	y	y	NOUN
ap-1807	93	16	)	)	PUNCT
ap-1807	94	1	+	+	CCONJ
ap-1807	95	1	z.	z.	PROPN
ap-1807	95	2	then	then	ADV
ap-1807	95	3	x	x	PROPN
ap-1807	95	4	=	=	SYM
ap-1807	95	5	(	(	PUNCT
ap-1807	95	6	(	(	PUNCT
ap-1807	95	7	a+	a+	X
ap-1807	95	8	y	y	NOUN
ap-1807	95	9	)	)	PUNCT
ap-1807	95	10	+	+	PUNCT
ap-1807	95	11	z	z	X
ap-1807	95	12	)	)	PUNCT
ap-1807	95	13	+	+	CCONJ
ap-1807	95	14	(	(	PUNCT
ap-1807	95	15	−a	−a	ADV
ap-1807	95	16	)	)	PUNCT
ap-1807	95	17	.	.	PUNCT
ap-1807	96	1	since	since	SCONJ
ap-1807	96	2	y	y	PROPN
ap-1807	96	3	+	+	CCONJ
ap-1807	96	4	a	a	PRON
ap-1807	96	5	and	and	CCONJ
ap-1807	96	6	y	y	PROPN
ap-1807	96	7	+	+	CCONJ
ap-1807	96	8	(	(	PUNCT
ap-1807	96	9	a+	a+	PUNCT
ap-1807	96	10	(	(	PUNCT
ap-1807	96	11	−a	−a	NOUN
ap-1807	96	12	)	)	PUNCT
ap-1807	96	13	)	)	PUNCT
ap-1807	96	14	are	be	AUX
ap-1807	96	15	defined	define	VERB
ap-1807	96	16	then	then	ADV
ap-1807	96	17	(	(	PUNCT
ap-1807	96	18	y	y	PROPN
ap-1807	96	19	+	+	NOUN
ap-1807	96	20	a	a	X
ap-1807	96	21	)	)	PUNCT
ap-1807	96	22	+	+	CCONJ
ap-1807	96	23	(	(	PUNCT
ap-1807	96	24	−a	−a	ADV
ap-1807	96	25	)	)	PUNCT
ap-1807	96	26	is	be	AUX
ap-1807	96	27	defined	define	VERB
ap-1807	96	28	.	.	PUNCT
ap-1807	97	1	we	we	PRON
ap-1807	97	2	have	have	VERB
ap-1807	97	3	x	x	NOUN
ap-1807	97	4	=	=	SYM
ap-1807	97	5	(	(	PUNCT
ap-1807	97	6	(	(	PUNCT
ap-1807	97	7	a+y)+z)+(−a	a+y)+z)+(−a	ADJ
ap-1807	97	8	)	)	PUNCT
ap-1807	97	9	=	=	SYM
ap-1807	97	10	(	(	PUNCT
ap-1807	97	11	(	(	PUNCT
ap-1807	97	12	(	(	PUNCT
ap-1807	97	13	a+y)+(−a))+z	a+y)+(−a))+z	NOUN
ap-1807	97	14	)	)	PUNCT
ap-1807	97	15	=	=	PUNCT
ap-1807	98	1	y+z	y+z	PROPN
ap-1807	98	2	.	.	PUNCT
ap-1807	99	1	(	(	PUNCT
ap-1807	99	2	3	3	NUM
ap-1807	99	3	.	.	PUNCT
ap-1807	99	4	)	)	PUNCT
ap-1807	99	5	let	let	VERB
ap-1807	99	6	us	we	PRON
ap-1807	99	7	have	have	VERB
ap-1807	99	8	a	a	DET
ap-1807	99	9	,	,	PUNCT
ap-1807	99	10	b	b	PROPN
ap-1807	99	11	∈	∈	PROPN
ap-1807	99	12	g	g	NOUN
ap-1807	99	13	such	such	DET
ap-1807	99	14	that	that	SCONJ
ap-1807	99	15	a	a	DET
ap-1807	99	16	+	+	NOUN
ap-1807	99	17	b	b	NOUN
ap-1807	99	18	∈	∈	ADJ
ap-1807	99	19	g.	g.	NOUN
ap-1807	99	20	we	we	PRON
ap-1807	99	21	have	have	VERB
ap-1807	99	22	a	a	DET
ap-1807	99	23	=	=	X
ap-1807	99	24	a+	a+	PUNCT
ap-1807	99	25	(	(	PUNCT
ap-1807	99	26	b+	b+	X
ap-1807	99	27	(	(	PUNCT
ap-1807	99	28	−b	−b	NOUN
ap-1807	99	29	)	)	PUNCT
ap-1807	99	30	)	)	PUNCT
ap-1807	100	1	=	=	PUNCT
ap-1807	100	2	(	(	PUNCT
ap-1807	100	3	a+	a+	NOUN
ap-1807	100	4	b	b	NOUN
ap-1807	100	5	)	)	PUNCT
ap-1807	100	6	+	+	CCONJ
ap-1807	100	7	(	(	PUNCT
ap-1807	100	8	−b	−b	ADJ
ap-1807	100	9	)	)	PUNCT
ap-1807	100	10	defined	define	VERB
ap-1807	100	11	.	.	PUNCT
ap-1807	101	1	then	then	ADV
ap-1807	101	2	by	by	ADP
ap-1807	101	3	(	(	PUNCT
ap-1807	101	4	1	1	NUM
ap-1807	101	5	.	.	NUM
ap-1807	101	6	)	)	PUNCT
ap-1807	102	1	(	(	PUNCT
ap-1807	102	2	−b	−b	NOUN
ap-1807	102	3	)	)	PUNCT
ap-1807	102	4	=	=	SYM
ap-1807	102	5	a+	a+	PUNCT
ap-1807	102	6	(	(	PUNCT
ap-1807	102	7	−(a+	−(a+	NOUN
ap-1807	102	8	b	b	NOUN
ap-1807	102	9	)	)	PUNCT
ap-1807	102	10	)	)	PUNCT
ap-1807	102	11	and	and	CCONJ
ap-1807	102	12	also	also	ADV
ap-1807	102	13	by	by	ADP
ap-1807	102	14	(	(	PUNCT
ap-1807	102	15	1	1	NUM
ap-1807	102	16	.	.	NUM
ap-1807	102	17	)	)	PUNCT
ap-1807	103	1	(	(	PUNCT
ap-1807	103	2	−a	−a	ADV
ap-1807	103	3	)	)	PUNCT
ap-1807	104	1	+	+	CCONJ
ap-1807	104	2	(	(	PUNCT
ap-1807	104	3	−b	−b	ADJ
ap-1807	104	4	)	)	PUNCT
ap-1807	104	5	=	=	PUNCT
ap-1807	104	6	(	(	PUNCT
ap-1807	104	7	−(a+	−(a+	PROPN
ap-1807	104	8	b	b	NOUN
ap-1807	104	9	)	)	PUNCT
ap-1807	104	10	)	)	PUNCT
ap-1807	104	11	.	.	PUNCT
ap-1807	105	1	lemma	lemma	PROPN
ap-1807	105	2	2	2	X
ap-1807	105	3	.	.	PUNCT
ap-1807	106	1	let	let	VERB
ap-1807	106	2	(	(	PUNCT
ap-1807	106	3	g,+	g,+	PROPN
ap-1807	106	4	,	,	PUNCT
ap-1807	106	5	0	0	NUM
ap-1807	106	6	)	)	PUNCT
ap-1807	106	7	w.r.t	w.r.t	NOUN
ap-1807	106	8	.	.	PUNCT
ap-1807	107	1	≤	≤	NUM
ap-1807	107	2	be	be	AUX
ap-1807	107	3	a	a	DET
ap-1807	107	4	woa	woa	NOUN
ap-1807	107	5	-	-	PUNCT
ap-1807	107	6	group	group	NOUN
ap-1807	107	7	.	.	PUNCT
ap-1807	108	1	then	then	ADV
ap-1807	108	2	for	for	ADP
ap-1807	108	3	any	any	DET
ap-1807	108	4	a	a	DET
ap-1807	108	5	,	,	PUNCT
ap-1807	108	6	b	b	NOUN
ap-1807	108	7	,	,	PUNCT
ap-1807	108	8	c	c	NOUN
ap-1807	108	9	,	,	PUNCT
ap-1807	108	10	x	x	NOUN
ap-1807	108	11	,	,	PUNCT
ap-1807	108	12	y	y	PROPN
ap-1807	108	13	,	,	PUNCT
ap-1807	108	14	z	z	PROPN
ap-1807	108	15	∈	∈	PROPN
ap-1807	108	16	g	g	ADP
ap-1807	108	17	the	the	DET
ap-1807	108	18	following	follow	VERB
ap-1807	108	19	holds	hold	VERB
ap-1807	108	20	:	:	PUNCT
ap-1807	108	21	(	(	PUNCT
ap-1807	108	22	1	1	NUM
ap-1807	108	23	.	.	PUNCT
ap-1807	108	24	)	)	PUNCT
ap-1807	109	1	a	a	DET
ap-1807	109	2	≤	≤	PROPN
ap-1807	109	3	b	b	X
ap-1807	109	4	iff	iff	PROPN
ap-1807	109	5	b+	b+	X
ap-1807	109	6	(	(	PUNCT
ap-1807	109	7	−a	−a	ADV
ap-1807	109	8	)	)	PUNCT
ap-1807	109	9	≥	≥	NOUN
ap-1807	109	10	0	0	NUM
ap-1807	109	11	,	,	PUNCT
ap-1807	109	12	(	(	PUNCT
ap-1807	109	13	2	2	NUM
ap-1807	109	14	.	.	PUNCT
ap-1807	109	15	)	)	PUNCT
ap-1807	110	1	a	a	DET
ap-1807	110	2	≤	≤	PROPN
ap-1807	110	3	b	b	X
ap-1807	110	4	iff	iff	NOUN
ap-1807	110	5	−b	−b	ADJ
ap-1807	110	6	≤	≤	PROPN
ap-1807	110	7	−a	−a	NOUN
ap-1807	110	8	.	.	PUNCT
ap-1807	111	1	proof	proof	NOUN
ap-1807	111	2	.	.	PUNCT
ap-1807	112	1	(	(	PUNCT
ap-1807	112	2	1	1	NUM
ap-1807	112	3	.	.	PUNCT
ap-1807	112	4	)	)	PUNCT
ap-1807	112	5	let	let	VERB
ap-1807	112	6	a	a	DET
ap-1807	112	7	≤	≤	PROPN
ap-1807	112	8	b.	b.	NOUN
ap-1807	113	1	then	then	ADV
ap-1807	113	2	there	there	PRON
ap-1807	113	3	exists	exist	VERB
ap-1807	113	4	c	c	PROPN
ap-1807	113	5	≥	≥	NUM
ap-1807	113	6	0	0	NUM
ap-1807	113	7	such	such	ADJ
ap-1807	113	8	that	that	SCONJ
ap-1807	113	9	a+	a+	PUNCT
ap-1807	113	10	c	c	NOUN
ap-1807	113	11	=	=	SYM
ap-1807	113	12	b	b	PROPN
ap-1807	113	13	and	and	CCONJ
ap-1807	113	14	from	from	ADP
ap-1807	113	15	lemma	lemma	PROPN
ap-1807	113	16	1	1	NUM
ap-1807	113	17	(	(	PUNCT
ap-1807	113	18	1	1	NUM
ap-1807	113	19	.	.	PUNCT
ap-1807	113	20	)	)	PUNCT
ap-1807	113	21	b+	b+	X
ap-1807	113	22	(	(	PUNCT
ap-1807	113	23	−a	−a	ADV
ap-1807	113	24	)	)	PUNCT
ap-1807	113	25	≥	≥	NOUN
ap-1807	113	26	0	0	NUM
ap-1807	113	27	.	.	PUNCT
ap-1807	114	1	on	on	ADP
ap-1807	114	2	the	the	DET
ap-1807	114	3	other	other	ADJ
ap-1807	114	4	hand	hand	NOUN
ap-1807	114	5	,	,	PUNCT
ap-1807	114	6	let	let	VERB
ap-1807	114	7	b	b	NOUN
ap-1807	114	8	+	+	CCONJ
ap-1807	114	9	(	(	PUNCT
ap-1807	114	10	−a	−a	ADJ
ap-1807	114	11	)	)	PUNCT
ap-1807	114	12	≥	≥	NOUN
ap-1807	114	13	0	0	NUM
ap-1807	114	14	.	.	PUNCT
ap-1807	115	1	then	then	ADV
ap-1807	115	2	b	b	X
ap-1807	115	3	=	=	PRON
ap-1807	115	4	b+	b+	X
ap-1807	115	5	(	(	PUNCT
ap-1807	115	6	a+	a+	X
ap-1807	115	7	(	(	PUNCT
ap-1807	115	8	−a	−a	NOUN
ap-1807	115	9	)	)	PUNCT
ap-1807	115	10	)	)	PUNCT
ap-1807	116	1	=	=	PRON
ap-1807	116	2	a+	a+	X
ap-1807	116	3	(	(	PUNCT
ap-1807	116	4	b+	b+	X
ap-1807	116	5	(	(	PUNCT
ap-1807	116	6	−a	−a	NOUN
ap-1807	116	7	)	)	PUNCT
ap-1807	116	8	)	)	PUNCT
ap-1807	116	9	hence	hence	ADV
ap-1807	116	10	by	by	ADP
ap-1807	116	11	(	(	PUNCT
ap-1807	116	12	ri	ri	NOUN
ap-1807	116	13	)	)	PUNCT
ap-1807	116	14	a	a	DET
ap-1807	116	15	≤	≤	PROPN
ap-1807	116	16	b.	b.	NOUN
ap-1807	116	17	(	(	PUNCT
ap-1807	116	18	2	2	NUM
ap-1807	116	19	.	.	PUNCT
ap-1807	116	20	)	)	PUNCT
ap-1807	116	21	let	let	VERB
ap-1807	116	22	a	a	DET
ap-1807	116	23	≤	≤	ADJ
ap-1807	116	24	b	b	NOUN
ap-1807	116	25	for	for	ADP
ap-1807	116	26	some	some	DET
ap-1807	116	27	a	a	PRON
ap-1807	116	28	,	,	PUNCT
ap-1807	116	29	b	b	PROPN
ap-1807	116	30	∈	∈	PROPN
ap-1807	116	31	g.	g.	NOUN
ap-1807	116	32	then	then	ADV
ap-1807	116	33	(	(	PUNCT
ap-1807	116	34	b+	b+	X
ap-1807	116	35	(	(	PUNCT
ap-1807	116	36	−a	−a	NOUN
ap-1807	116	37	)	)	PUNCT
ap-1807	116	38	)	)	PUNCT
ap-1807	117	1	≥	≥	NOUN
ap-1807	117	2	0	0	NUM
ap-1807	117	3	and	and	CCONJ
ap-1807	117	4	−a	−a	ADV
ap-1807	117	5	=	=	SYM
ap-1807	117	6	(	(	PUNCT
ap-1807	117	7	(	(	PUNCT
ap-1807	117	8	−b	−b	ADJ
ap-1807	117	9	)	)	PUNCT
ap-1807	118	1	+	+	SYM
ap-1807	119	1	b	b	X
ap-1807	119	2	)	)	PUNCT
ap-1807	119	3	+	+	CCONJ
ap-1807	119	4	(	(	PUNCT
ap-1807	119	5	−a	−a	ADJ
ap-1807	119	6	)	)	PUNCT
ap-1807	119	7	=	=	PUNCT
ap-1807	119	8	(	(	PUNCT
ap-1807	119	9	b+	b+	X
ap-1807	119	10	(	(	PUNCT
ap-1807	119	11	−a	−a	NOUN
ap-1807	119	12	)	)	PUNCT
ap-1807	119	13	)	)	PUNCT
ap-1807	120	1	+	+	CCONJ
ap-1807	120	2	(	(	PUNCT
ap-1807	120	3	−b	−b	ADJ
ap-1807	120	4	)	)	PUNCT
ap-1807	120	5	ie	ie	X
ap-1807	120	6	.	.	X
ap-1807	120	7	−b	−b	VERB
ap-1807	120	8	≤	≤	NUM
ap-1807	120	9	−a	−a	NOUN
ap-1807	120	10	.	.	PUNCT
ap-1807	121	1	lemma	lemma	PROPN
ap-1807	121	2	3	3	NUM
ap-1807	121	3	.	.	PUNCT
ap-1807	122	1	every	every	DET
ap-1807	122	2	a	a	DET
ap-1807	122	3	-	-	PUNCT
ap-1807	122	4	commutative	commutative	ADJ
ap-1807	122	5	partial	partial	ADJ
ap-1807	122	6	group	group	NOUN
ap-1807	122	7	(	(	PUNCT
ap-1807	122	8	g,+	g,+	PROPN
ap-1807	122	9	,	,	PUNCT
ap-1807	122	10	0	0	NUM
ap-1807	122	11	)	)	PUNCT
ap-1807	122	12	is	be	AUX
ap-1807	122	13	a	a	DET
ap-1807	122	14	partial	partial	ADJ
ap-1807	122	15	commutative	commutative	ADJ
ap-1807	122	16	group	group	NOUN
ap-1807	122	17	.	.	PUNCT
ap-1807	123	1	proof	proof	NOUN
ap-1807	123	2	.	.	PUNCT
ap-1807	124	1	the	the	DET
ap-1807	124	2	cancellation	cancellation	NOUN
ap-1807	124	3	law	law	NOUN
ap-1807	124	4	follows	follow	VERB
ap-1807	124	5	from	from	ADP
ap-1807	124	6	lemma	lemma	PROPN
ap-1807	124	7	1	1	NUM
ap-1807	124	8	(	(	PUNCT
ap-1807	124	9	2	2	NUM
ap-1807	124	10	.	.	PUNCT
ap-1807	124	11	)	)	PUNCT
ap-1807	125	1	choosing	choose	VERB
ap-1807	125	2	z	z	NOUN
ap-1807	125	3	=	=	SYM
ap-1807	125	4	0	0	NUM
ap-1807	125	5	.	.	NOUN
ap-1807	125	6	290	290	NUM
ap-1807	125	7	vol	vol	NOUN
ap-1807	125	8	.	.	PUNCT
ap-1807	126	1	53	53	NUM
ap-1807	126	2	no	no	NOUN
ap-1807	126	3	.	.	PUNCT
ap-1807	127	1	3/2013	3/2013	PROPN
ap-1807	127	2	weakly	weakly	ADV
ap-1807	127	3	ordered	order	VERB
ap-1807	127	4	a	a	DET
ap-1807	127	5	-	-	PUNCT
ap-1807	127	6	commutative	commutative	ADJ
ap-1807	127	7	partial	partial	ADJ
ap-1807	127	8	groups	group	NOUN
ap-1807	127	9	of	of	ADP
ap-1807	127	10	linear	linear	PROPN
ap-1807	127	11	operators	operator	NOUN
ap-1807	127	12	lemma	lemma	VERB
ap-1807	127	13	4	4	X
ap-1807	127	14	.	.	PUNCT
ap-1807	128	1	every	every	DET
ap-1807	128	2	woa	woa	NOUN
ap-1807	128	3	-	-	PUNCT
ap-1807	128	4	group	group	NOUN
ap-1807	128	5	(	(	PUNCT
ap-1807	128	6	g,+	g,+	PROPN
ap-1807	128	7	,	,	PUNCT
ap-1807	128	8	0	0	NUM
ap-1807	128	9	)	)	PUNCT
ap-1807	128	10	w.r.t	w.r.t	NOUN
ap-1807	128	11	.	.	PUNCT
ap-1807	129	1	≤	≤	PROPN
ap-1807	129	2	is	be	AUX
ap-1807	129	3	a	a	DET
ap-1807	129	4	wop	wop	NOUN
ap-1807	129	5	-	-	PUNCT
ap-1807	129	6	group	group	NOUN
ap-1807	129	7	w.r.t	w.r.t	NOUN
ap-1807	129	8	.	.	PUNCT
ap-1807	130	1	≤.	≤.	NOUN
ap-1807	130	2	proof	proof	NOUN
ap-1807	130	3	.	.	PUNCT
ap-1807	131	1	by	by	ADP
ap-1807	131	2	the	the	DET
ap-1807	131	3	previous	previous	ADJ
ap-1807	131	4	lemma	lemma	PROPN
ap-1807	131	5	,	,	PUNCT
ap-1807	131	6	we	we	PRON
ap-1807	131	7	have	have	VERB
ap-1807	131	8	(	(	PUNCT
ap-1807	131	9	g,+	g,+	PROPN
ap-1807	131	10	,	,	PUNCT
ap-1807	131	11	0	0	NUM
ap-1807	131	12	)	)	PUNCT
ap-1807	131	13	is	be	AUX
ap-1807	131	14	a	a	DET
ap-1807	131	15	partial	partial	ADJ
ap-1807	131	16	commutative	commutative	ADJ
ap-1807	131	17	group	group	NOUN
ap-1807	131	18	.	.	PUNCT
ap-1807	132	1	let	let	VERB
ap-1807	132	2	us	we	PRON
ap-1807	132	3	have	have	VERB
ap-1807	132	4	x	x	PROPN
ap-1807	132	5	,	,	PUNCT
ap-1807	132	6	y	y	PROPN
ap-1807	132	7	,	,	PUNCT
ap-1807	132	8	z	z	PROPN
ap-1807	132	9	∈	∈	PROPN
ap-1807	132	10	g	g	PROPN
ap-1807	132	11	,	,	PUNCT
ap-1807	132	12	x	x	PUNCT
ap-1807	132	13	≤	≤	NOUN
ap-1807	132	14	y	y	NOUN
ap-1807	132	15	,	,	PUNCT
ap-1807	132	16	x	x	PUNCT
ap-1807	133	1	+	+	CCONJ
ap-1807	133	2	z	z	X
ap-1807	133	3	,	,	PUNCT
ap-1807	133	4	y	y	PROPN
ap-1807	133	5	+	+	CCONJ
ap-1807	133	6	z	z	NOUN
ap-1807	133	7	defined	define	VERB
ap-1807	133	8	.	.	PUNCT
ap-1807	134	1	then	then	ADV
ap-1807	134	2	by	by	ADP
ap-1807	134	3	lemma	lemma	PROPN
ap-1807	134	4	2	2	NUM
ap-1807	134	5	(	(	PUNCT
ap-1807	134	6	1	1	NUM
ap-1807	134	7	.	.	PUNCT
ap-1807	134	8	)	)	PUNCT
ap-1807	134	9	0	0	NUM
ap-1807	135	1	≤	≤	NOUN
ap-1807	135	2	(	(	PUNCT
ap-1807	135	3	y	y	NOUN
ap-1807	135	4	+	+	CCONJ
ap-1807	135	5	(	(	PUNCT
ap-1807	135	6	−x	−x	NOUN
ap-1807	135	7	)	)	PUNCT
ap-1807	135	8	)	)	PUNCT
ap-1807	136	1	and	and	CCONJ
ap-1807	136	2	x	x	X
ap-1807	136	3	+	+	PUNCT
ap-1807	136	4	(	(	PUNCT
ap-1807	136	5	y	y	NOUN
ap-1807	136	6	+	+	CCONJ
ap-1807	136	7	(	(	PUNCT
ap-1807	136	8	−x	−x	NOUN
ap-1807	136	9	)	)	PUNCT
ap-1807	136	10	)	)	PUNCT
ap-1807	137	1	=	=	SYM
ap-1807	137	2	y	y	PROPN
ap-1807	137	3	hence	hence	ADV
ap-1807	137	4	y	y	PROPN
ap-1807	138	1	+	+	CCONJ
ap-1807	138	2	z	z	NOUN
ap-1807	138	3	=	=	SYM
ap-1807	138	4	(	(	PUNCT
ap-1807	138	5	x+	x+	X
ap-1807	138	6	(	(	PUNCT
ap-1807	138	7	y	y	NOUN
ap-1807	138	8	+	+	CCONJ
ap-1807	138	9	(	(	PUNCT
ap-1807	138	10	−x	−x	NOUN
ap-1807	138	11	)	)	PUNCT
ap-1807	138	12	)	)	PUNCT
ap-1807	139	1	+	+	CCONJ
ap-1807	139	2	z	z	X
ap-1807	139	3	=	=	SYM
ap-1807	139	4	(	(	PUNCT
ap-1807	139	5	x+	x+	ADJ
ap-1807	139	6	z	z	NOUN
ap-1807	139	7	)	)	PUNCT
ap-1807	140	1	+	+	CCONJ
ap-1807	140	2	(	(	PUNCT
ap-1807	140	3	y	y	NOUN
ap-1807	140	4	+	+	CCONJ
ap-1807	140	5	(	(	PUNCT
ap-1807	140	6	−x	−x	NOUN
ap-1807	140	7	)	)	PUNCT
ap-1807	140	8	)	)	PUNCT
ap-1807	141	1	and	and	CCONJ
ap-1807	141	2	according	accord	VERB
ap-1807	141	3	to	to	ADP
ap-1807	141	4	(	(	PUNCT
ap-1807	141	5	rii	rii	PROPN
ap-1807	141	6	)	)	PUNCT
ap-1807	141	7	x+	x+	PROPN
ap-1807	141	8	z	z	NOUN
ap-1807	141	9	≤	≤	NUM
ap-1807	142	1	y	y	PROPN
ap-1807	142	2	+	+	PROPN
ap-1807	142	3	z.	z.	PROPN
ap-1807	142	4	example	example	NOUN
ap-1807	142	5	1	1	X
ap-1807	142	6	.	.	PUNCT
ap-1807	143	1	let	let	VERB
ap-1807	143	2	g	g	NOUN
ap-1807	143	3	=	=	PUNCT
ap-1807	143	4	{	{	PUNCT
ap-1807	143	5	0	0	NUM
ap-1807	143	6	,	,	PUNCT
ap-1807	143	7	a	a	DET
ap-1807	143	8	,	,	PUNCT
ap-1807	143	9	b	b	NOUN
ap-1807	143	10	,	,	PUNCT
ap-1807	143	11	c,−a,−b,−c	c,−a,−b,−c	PROPN
ap-1807	143	12	}	}	PUNCT
ap-1807	143	13	be	be	AUX
ap-1807	143	14	a	a	DET
ap-1807	143	15	set	set	NOUN
ap-1807	143	16	with	with	ADP
ap-1807	143	17	partial	partial	ADJ
ap-1807	143	18	operation	operation	NOUN
ap-1807	143	19	+	+	CCONJ
ap-1807	143	20	defined	define	VERB
ap-1807	143	21	for	for	ADP
ap-1807	143	22	a+	a+	DET
ap-1807	143	23	b	b	PROPN
ap-1807	143	24	=	=	SYM
ap-1807	143	25	c	c	PROPN
ap-1807	143	26	and	and	CCONJ
ap-1807	143	27	0	0	NUM
ap-1807	144	1	+	+	CCONJ
ap-1807	144	2	x	x	SYM
ap-1807	144	3	=	=	SYM
ap-1807	144	4	x	x	NOUN
ap-1807	144	5	,	,	PUNCT
ap-1807	144	6	x	x	X
ap-1807	144	7	+	+	CCONJ
ap-1807	144	8	(	(	PUNCT
ap-1807	144	9	−x	−x	NOUN
ap-1807	144	10	)	)	PUNCT
ap-1807	144	11	=	=	SYM
ap-1807	144	12	0	0	NUM
ap-1807	144	13	for	for	ADP
ap-1807	144	14	all	all	DET
ap-1807	144	15	x	x	SYM
ap-1807	144	16	∈	∈	PROPN
ap-1807	144	17	g.	g.	NOUN
ap-1807	144	18	then	then	ADV
ap-1807	144	19	(	(	PUNCT
ap-1807	144	20	g,+	g,+	PROPN
ap-1807	144	21	,	,	PUNCT
ap-1807	144	22	0	0	NUM
ap-1807	144	23	)	)	PUNCT
ap-1807	144	24	is	be	AUX
ap-1807	144	25	a	a	DET
ap-1807	144	26	commutative	commutative	ADJ
ap-1807	144	27	partial	partial	ADJ
ap-1807	144	28	group	group	NOUN
ap-1807	144	29	,	,	PUNCT
ap-1807	144	30	but	but	CCONJ
ap-1807	144	31	it	it	PRON
ap-1807	144	32	is	be	AUX
ap-1807	144	33	not	not	PART
ap-1807	144	34	an	an	DET
ap-1807	144	35	a	a	PRON
ap-1807	144	36	-	-	PUNCT
ap-1807	144	37	commutative	commutative	ADJ
ap-1807	144	38	partial	partial	ADJ
ap-1807	144	39	group	group	NOUN
ap-1807	144	40	.	.	PUNCT
ap-1807	145	1	note	note	VERB
ap-1807	145	2	that	that	SCONJ
ap-1807	145	3	even	even	ADV
ap-1807	145	4	c+	c+	VERB
ap-1807	145	5	(	(	PUNCT
ap-1807	145	6	−b	−b	NOUN
ap-1807	145	7	)	)	PUNCT
ap-1807	145	8	=	=	PUNCT
ap-1807	145	9	(	(	PUNCT
ap-1807	145	10	a+	a+	NOUN
ap-1807	145	11	b	b	NOUN
ap-1807	145	12	)	)	PUNCT
ap-1807	145	13	+	+	CCONJ
ap-1807	145	14	(	(	PUNCT
ap-1807	145	15	−b	−b	ADJ
ap-1807	145	16	)	)	PUNCT
ap-1807	145	17	is	be	AUX
ap-1807	145	18	not	not	PART
ap-1807	145	19	defined	define	VERB
ap-1807	145	20	.	.	PUNCT
ap-1807	146	1	lemma	lemma	PROPN
ap-1807	146	2	5	5	X
ap-1807	146	3	.	.	PUNCT
ap-1807	147	1	let	let	VERB
ap-1807	147	2	(	(	PUNCT
ap-1807	147	3	g,+	g,+	PROPN
ap-1807	147	4	,	,	PUNCT
ap-1807	147	5	0	0	NUM
ap-1807	147	6	)	)	PUNCT
ap-1807	147	7	be	be	AUX
ap-1807	147	8	an	an	DET
ap-1807	147	9	a	a	PRON
ap-1807	147	10	-	-	PUNCT
ap-1807	147	11	commutative	commutative	ADJ
ap-1807	147	12	partial	partial	ADJ
ap-1807	147	13	group	group	NOUN
ap-1807	147	14	and	and	CCONJ
ap-1807	147	15	s	s	VERB
ap-1807	147	16	⊆	⊆	NUM
ap-1807	147	17	g	g	ADP
ap-1807	147	18	its	its	PRON
ap-1807	147	19	a	a	DET
ap-1807	147	20	-	-	PUNCT
ap-1807	147	21	commutative	commutative	ADJ
ap-1807	147	22	partial	partial	ADJ
ap-1807	147	23	subgroup	subgroup	NOUN
ap-1807	147	24	.	.	PUNCT
ap-1807	148	1	then	then	ADV
ap-1807	148	2	(	(	PUNCT
ap-1807	148	3	s,+/s	s,+/s	NOUN
ap-1807	148	4	,	,	PUNCT
ap-1807	148	5	0	0	NUM
ap-1807	148	6	)	)	PUNCT
ap-1807	148	7	is	be	AUX
ap-1807	148	8	an	an	DET
ap-1807	148	9	a	a	PRON
ap-1807	148	10	-	-	PUNCT
ap-1807	148	11	commutative	commutative	ADJ
ap-1807	148	12	partial	partial	ADJ
ap-1807	148	13	group	group	NOUN
ap-1807	148	14	.	.	PUNCT
ap-1807	149	1	proof	proof	NOUN
ap-1807	149	2	.	.	PUNCT
ap-1807	150	1	immediately	immediately	ADV
ap-1807	150	2	(	(	PUNCT
ap-1807	150	3	gi	gi	INTJ
ap-1807	150	4	)	)	PUNCT
ap-1807	150	5	and	and	CCONJ
ap-1807	150	6	(	(	PUNCT
ap-1807	150	7	giv	giv	NOUN
ap-1807	150	8	)	)	PUNCT
ap-1807	150	9	follows	follow	VERB
ap-1807	150	10	from	from	ADP
ap-1807	150	11	(	(	PUNCT
ap-1807	150	12	siii	siii	NOUN
ap-1807	150	13	)	)	PUNCT
ap-1807	150	14	,	,	PUNCT
ap-1807	150	15	(	(	PUNCT
ap-1807	150	16	gii	gii	NOUN
ap-1807	150	17	)	)	PUNCT
ap-1807	150	18	from	from	ADP
ap-1807	150	19	(	(	PUNCT
ap-1807	150	20	si	si	NOUN
ap-1807	150	21	)	)	PUNCT
ap-1807	150	22	,	,	PUNCT
ap-1807	150	23	(	(	PUNCT
ap-1807	150	24	giii	giii	NOUN
ap-1807	150	25	)	)	PUNCT
ap-1807	150	26	from	from	ADP
ap-1807	150	27	(	(	PUNCT
ap-1807	150	28	sii	sii	PROPN
ap-1807	150	29	)	)	PUNCT
ap-1807	150	30	.	.	PUNCT
ap-1807	151	1	lemma	lemma	PROPN
ap-1807	151	2	6	6	NUM
ap-1807	151	3	.	.	PUNCT
ap-1807	152	1	let	let	VERB
ap-1807	152	2	(	(	PUNCT
ap-1807	152	3	g,+	g,+	PROPN
ap-1807	152	4	,	,	PUNCT
ap-1807	152	5	0	0	NUM
ap-1807	152	6	)	)	PUNCT
ap-1807	152	7	be	be	AUX
ap-1807	152	8	a	a	DET
ap-1807	152	9	woa	woa	VERB
ap-1807	152	10	-	-	PUNCT
ap-1807	152	11	group	group	NOUN
ap-1807	152	12	w.r.t	w.r.t	NOUN
ap-1807	152	13	.	.	PUNCT
ap-1807	153	1	≤	≤	NUM
ap-1807	153	2	and	and	CCONJ
ap-1807	153	3	s	s	VERB
ap-1807	153	4	⊆	⊆	NUM
ap-1807	153	5	g	g	ADP
ap-1807	153	6	its	its	PRON
ap-1807	153	7	woa	woa	ADJ
ap-1807	153	8	-	-	PUNCT
ap-1807	153	9	subgroup	subgroup	NOUN
ap-1807	153	10	w.r.t	w.r.t	NOUN
ap-1807	153	11	.	.	PUNCT
ap-1807	154	1	≤s	≤s	PROPN
ap-1807	154	2	.	.	PUNCT
ap-1807	155	1	then	then	ADV
ap-1807	155	2	(	(	PUNCT
ap-1807	155	3	s,+/s	s,+/s	NOUN
ap-1807	155	4	,	,	PUNCT
ap-1807	155	5	0	0	X
ap-1807	155	6	)	)	PUNCT
ap-1807	155	7	w.r.t	w.r.t	NOUN
ap-1807	155	8	.	.	PUNCT
ap-1807	156	1	≤s	≤s	PROPN
ap-1807	156	2	is	be	AUX
ap-1807	156	3	a	a	DET
ap-1807	156	4	woa	woa	NOUN
ap-1807	156	5	-	-	PUNCT
ap-1807	156	6	group	group	NOUN
ap-1807	156	7	.	.	PUNCT
ap-1807	157	1	proof	proof	NOUN
ap-1807	157	2	.	.	PUNCT
ap-1807	158	1	by	by	ADP
ap-1807	158	2	the	the	DET
ap-1807	158	3	previous	previous	ADJ
ap-1807	158	4	lemma	lemma	PROPN
ap-1807	158	5	(	(	PUNCT
ap-1807	158	6	s,+/s	s,+/s	NOUN
ap-1807	158	7	,	,	PUNCT
ap-1807	158	8	0	0	NUM
ap-1807	158	9	)	)	PUNCT
ap-1807	158	10	is	be	AUX
ap-1807	158	11	an	an	DET
ap-1807	158	12	acommutative	acommutative	ADJ
ap-1807	158	13	partial	partial	ADJ
ap-1807	158	14	group	group	NOUN
ap-1807	158	15	.	.	PUNCT
ap-1807	159	1	for	for	ADP
ap-1807	159	2	(	(	PUNCT
ap-1807	159	3	ri	ri	NOUN
ap-1807	159	4	)	)	PUNCT
ap-1807	159	5	let	let	VERB
ap-1807	159	6	x	x	PRON
ap-1807	159	7	,	,	PUNCT
ap-1807	159	8	y	y	PROPN
ap-1807	159	9	∈	∈	PROPN
ap-1807	159	10	s	s	PROPN
ap-1807	159	11	,	,	PUNCT
ap-1807	159	12	x	x	SYM
ap-1807	159	13	≤s	≤s	PROPN
ap-1807	159	14	y.	y.	PROPN
ap-1807	159	15	then	then	ADV
ap-1807	159	16	by	by	ADP
ap-1807	159	17	lemma	lemma	PROPN
ap-1807	159	18	2	2	NUM
ap-1807	159	19	(	(	PUNCT
ap-1807	159	20	1	1	NUM
ap-1807	159	21	.	.	PUNCT
ap-1807	159	22	)	)	PUNCT
ap-1807	159	23	0	0	NUM
ap-1807	160	1	≤	≤	NOUN
ap-1807	160	2	y	y	PROPN
ap-1807	160	3	+	+	CCONJ
ap-1807	160	4	(	(	PUNCT
ap-1807	160	5	−x	−x	NOUN
ap-1807	160	6	)	)	PUNCT
ap-1807	160	7	is	be	AUX
ap-1807	160	8	defined	define	VERB
ap-1807	160	9	and	and	CCONJ
ap-1807	160	10	by	by	ADP
ap-1807	160	11	(	(	PUNCT
ap-1807	160	12	siii	siii	NOUN
ap-1807	160	13	)	)	PUNCT
ap-1807	160	14	y+(−x	y+(−x	PROPN
ap-1807	160	15	)	)	PUNCT
ap-1807	160	16	∈	∈	PROPN
ap-1807	160	17	s.	s.	PROPN
ap-1807	160	18	the	the	DET
ap-1807	160	19	other	other	ADJ
ap-1807	160	20	direction	direction	NOUN
ap-1807	160	21	is	be	AUX
ap-1807	160	22	straightforward	straightforward	ADJ
ap-1807	160	23	.	.	PUNCT
ap-1807	161	1	(	(	PUNCT
ap-1807	161	2	rii	rii	PROPN
ap-1807	161	3	)	)	PUNCT
ap-1807	161	4	is	be	AUX
ap-1807	161	5	clear	clear	ADJ
ap-1807	161	6	and	and	CCONJ
ap-1807	161	7	(	(	PUNCT
ap-1807	161	8	riii	riii	NOUN
ap-1807	161	9	)	)	PUNCT
ap-1807	161	10	follows	follow	VERB
ap-1807	161	11	from	from	ADP
ap-1807	161	12	(	(	PUNCT
ap-1807	161	13	siii	siii	NOUN
ap-1807	161	14	)	)	PUNCT
ap-1807	161	15	.	.	PUNCT
ap-1807	162	1	corollary	corollary	ADJ
ap-1807	162	2	1	1	NUM
ap-1807	162	3	.	.	PUNCT
ap-1807	163	1	let	let	VERB
ap-1807	163	2	(	(	PUNCT
ap-1807	163	3	g,+	g,+	PROPN
ap-1807	163	4	,	,	PUNCT
ap-1807	163	5	0	0	NUM
ap-1807	163	6	)	)	PUNCT
ap-1807	163	7	be	be	AUX
ap-1807	163	8	a	a	DET
ap-1807	163	9	wop	wop	NOUN
ap-1807	163	10	-	-	PUNCT
ap-1807	163	11	group	group	NOUN
ap-1807	163	12	w.r.t	w.r.t	NOUN
ap-1807	163	13	.	.	PUNCT
ap-1807	164	1	≤	≤	NUM
ap-1807	164	2	and	and	CCONJ
ap-1807	164	3	s	s	VERB
ap-1807	164	4	⊆	⊆	NUM
ap-1807	164	5	g	g	ADP
ap-1807	164	6	its	its	PRON
ap-1807	164	7	wop	wop	NOUN
ap-1807	164	8	-	-	PUNCT
ap-1807	164	9	subgroup	subgroup	NOUN
ap-1807	164	10	w.r.t	w.r.t	NOUN
ap-1807	164	11	.	.	PUNCT
ap-1807	165	1	≤s	≤s	PROPN
ap-1807	165	2	.	.	PUNCT
ap-1807	166	1	whenever	whenever	SCONJ
ap-1807	166	2	(	(	PUNCT
ap-1807	166	3	g,+	g,+	PROPN
ap-1807	166	4	,	,	PUNCT
ap-1807	166	5	0	0	NUM
ap-1807	166	6	)	)	PUNCT
ap-1807	166	7	w.r.t	w.r.t	NOUN
ap-1807	166	8	.	.	PUNCT
ap-1807	167	1	≤	≤	PROPN
ap-1807	167	2	is	be	AUX
ap-1807	167	3	also	also	ADV
ap-1807	167	4	a	a	DET
ap-1807	167	5	woa	woa	NOUN
ap-1807	167	6	-	-	PUNCT
ap-1807	167	7	group	group	NOUN
ap-1807	167	8	,	,	PUNCT
ap-1807	167	9	then	then	ADV
ap-1807	167	10	s	s	VERB
ap-1807	167	11	is	be	AUX
ap-1807	167	12	its	its	PRON
ap-1807	167	13	woa	woa	NOUN
ap-1807	167	14	-	-	PUNCT
ap-1807	167	15	subgroup	subgroup	NOUN
ap-1807	167	16	.	.	PUNCT
ap-1807	168	1	theorem	theorem	NOUN
ap-1807	168	2	1	1	NUM
ap-1807	168	3	.	.	PUNCT
ap-1807	169	1	let	let	VERB
ap-1807	169	2	(	(	PUNCT
ap-1807	169	3	g,+	g,+	PROPN
ap-1807	169	4	,	,	PUNCT
ap-1807	169	5	0	0	NUM
ap-1807	169	6	)	)	PUNCT
ap-1807	169	7	w.r.t	w.r.t	NOUN
ap-1807	169	8	.	.	PUNCT
ap-1807	170	1	≤	≤	NUM
ap-1807	170	2	be	be	AUX
ap-1807	170	3	a	a	DET
ap-1807	170	4	woa	woa	NOUN
ap-1807	170	5	-	-	PUNCT
ap-1807	170	6	group	group	NOUN
ap-1807	170	7	.	.	PUNCT
ap-1807	171	1	then	then	ADV
ap-1807	171	2	the	the	DET
ap-1807	171	3	set	set	NOUN
ap-1807	171	4	pos(g	pos(g	PROPN
ap-1807	171	5	)	)	PUNCT
ap-1807	171	6	=	=	PRON
ap-1807	172	1	{	{	PUNCT
ap-1807	172	2	x	x	PUNCT
ap-1807	172	3	∈	∈	PROPN
ap-1807	172	4	g	g	NOUN
ap-1807	172	5	|	|	ADV
ap-1807	172	6	0	0	NUM
ap-1807	172	7	≤	≤	NUM
ap-1807	172	8	x	x	X
ap-1807	172	9	}	}	PUNCT
ap-1807	172	10	with	with	ADP
ap-1807	172	11	the	the	DET
ap-1807	172	12	restriction	restriction	NOUN
ap-1807	172	13	of	of	ADP
ap-1807	172	14	the	the	DET
ap-1807	172	15	partial	partial	ADJ
ap-1807	172	16	operation	operation	NOUN
ap-1807	172	17	+	+	CCONJ
ap-1807	172	18	on	on	ADP
ap-1807	172	19	pos(g	pos(g	PROPN
ap-1807	172	20	)	)	PUNCT
ap-1807	172	21	,	,	PUNCT
ap-1807	172	22	i.e.	i.e.	X
ap-1807	172	23	,	,	PUNCT
ap-1807	172	24	(	(	PUNCT
ap-1807	172	25	pos(g),+/	pos(g),+/	PROPN
ap-1807	172	26	pos(g	pos(g	PROPN
ap-1807	172	27	)	)	PUNCT
ap-1807	172	28	,	,	PUNCT
ap-1807	172	29	0	0	X
ap-1807	172	30	)	)	PUNCT
ap-1807	172	31	forms	form	VERB
ap-1807	172	32	a	a	DET
ap-1807	172	33	generalized	generalized	ADJ
ap-1807	172	34	effect	effect	NOUN
ap-1807	172	35	algebra	algebra	NOUN
ap-1807	172	36	.	.	PUNCT
ap-1807	173	1	proof	proof	NOUN
ap-1807	173	2	.	.	PUNCT
ap-1807	174	1	(	(	PUNCT
ap-1807	174	2	gei	gei	PROPN
ap-1807	174	3	)	)	PUNCT
ap-1807	174	4	and	and	CCONJ
ap-1807	174	5	(	(	PUNCT
ap-1807	174	6	geiii	geiii	PROPN
ap-1807	174	7	)	)	PUNCT
ap-1807	174	8	hold	hold	NOUN
ap-1807	174	9	from	from	ADP
ap-1807	174	10	definition	definition	NOUN
ap-1807	174	11	and	and	CCONJ
ap-1807	174	12	the	the	DET
ap-1807	174	13	cancellation	cancellation	NOUN
ap-1807	174	14	law	law	NOUN
ap-1807	174	15	follows	follow	VERB
ap-1807	174	16	from	from	ADP
ap-1807	174	17	lemma	lemma	PROPN
ap-1807	174	18	4	4	NUM
ap-1807	174	19	.	.	PUNCT
ap-1807	174	20	for	for	ADP
ap-1807	174	21	the	the	DET
ap-1807	174	22	axiom	axiom	NOUN
ap-1807	174	23	(	(	PUNCT
ap-1807	174	24	gev	gev	NOUN
ap-1807	174	25	)	)	PUNCT
ap-1807	174	26	let	let	VERB
ap-1807	174	27	x	x	PRON
ap-1807	174	28	,	,	PUNCT
ap-1807	174	29	y	y	PROPN
ap-1807	174	30	∈	∈	PROPN
ap-1807	174	31	pos(g	pos(g	PROPN
ap-1807	174	32	)	)	PUNCT
ap-1807	174	33	such	such	ADJ
ap-1807	174	34	that	that	SCONJ
ap-1807	174	35	x+	x+	ADJ
ap-1807	174	36	y	y	PROPN
ap-1807	174	37	=	=	SYM
ap-1807	174	38	0	0	PROPN
ap-1807	174	39	.	.	PUNCT
ap-1807	175	1	then	then	ADV
ap-1807	175	2	x	x	X
ap-1807	175	3	=	=	PRON
ap-1807	175	4	(	(	PUNCT
ap-1807	175	5	−y	−y	NOUN
ap-1807	175	6	)	)	PUNCT
ap-1807	175	7	and	and	CCONJ
ap-1807	175	8	with	with	ADP
ap-1807	175	9	lemma	lemma	PROPN
ap-1807	175	10	2	2	NUM
ap-1807	175	11	(	(	PUNCT
ap-1807	175	12	2	2	NUM
ap-1807	175	13	.	.	PUNCT
ap-1807	175	14	)	)	PUNCT
ap-1807	176	1	we	we	PRON
ap-1807	176	2	have	have	AUX
ap-1807	176	3	−y	−y	VERB
ap-1807	176	4	≤	≤	NOUN
ap-1807	176	5	0	0	NUM
ap-1807	177	1	hence	hence	ADV
ap-1807	177	2	x	x	X
ap-1807	177	3	=	=	SYM
ap-1807	177	4	y	y	PROPN
ap-1807	177	5	=	=	SYM
ap-1807	177	6	0	0	X
ap-1807	177	7	.	.	PUNCT
ap-1807	178	1	we	we	PRON
ap-1807	178	2	will	will	AUX
ap-1807	178	3	verify	verify	VERB
ap-1807	178	4	(	(	PUNCT
ap-1807	178	5	geii	geii	NOUN
ap-1807	178	6	)	)	PUNCT
ap-1807	178	7	.	.	PUNCT
ap-1807	179	1	let	let	VERB
ap-1807	179	2	us	we	PRON
ap-1807	179	3	have	have	VERB
ap-1807	179	4	x	x	PROPN
ap-1807	179	5	,	,	PUNCT
ap-1807	179	6	y	y	PROPN
ap-1807	179	7	,	,	PUNCT
ap-1807	179	8	z	z	PROPN
ap-1807	179	9	∈	∈	PROPN
ap-1807	179	10	pos(g	pos(g	PROPN
ap-1807	179	11	)	)	PUNCT
ap-1807	179	12	such	such	ADJ
ap-1807	179	13	that	that	SCONJ
ap-1807	179	14	(	(	PUNCT
ap-1807	179	15	x+	x+	X
ap-1807	179	16	y	y	NOUN
ap-1807	179	17	)	)	PUNCT
ap-1807	180	1	+	+	CCONJ
ap-1807	180	2	z	z	NOUN
ap-1807	180	3	is	be	AUX
ap-1807	180	4	defined	define	VERB
ap-1807	180	5	.	.	PUNCT
ap-1807	181	1	therefore	therefore	ADV
ap-1807	181	2	y	y	PROPN
ap-1807	181	3	≤	≤	PROPN
ap-1807	181	4	x+	x+	PROPN
ap-1807	181	5	y	y	PROPN
ap-1807	181	6	,	,	PUNCT
ap-1807	181	7	(	(	PUNCT
ap-1807	181	8	x+	x+	X
ap-1807	181	9	y	y	NOUN
ap-1807	181	10	)	)	PUNCT
ap-1807	182	1	+	+	CCONJ
ap-1807	182	2	z	z	NOUN
ap-1807	182	3	exists	exist	VERB
ap-1807	182	4	and	and	CCONJ
ap-1807	182	5	(	(	PUNCT
ap-1807	182	6	riii	riii	PROPN
ap-1807	182	7	)	)	PUNCT
ap-1807	182	8	implies	imply	VERB
ap-1807	182	9	that	that	SCONJ
ap-1807	182	10	y	y	PROPN
ap-1807	182	11	+	+	NOUN
ap-1807	182	12	z	z	NOUN
ap-1807	182	13	exists	exist	VERB
ap-1807	182	14	.	.	PUNCT
ap-1807	183	1	using	use	VERB
ap-1807	183	2	(	(	PUNCT
ap-1807	183	3	giv	giv	X
ap-1807	183	4	)	)	PUNCT
ap-1807	183	5	we	we	PRON
ap-1807	183	6	have	have	VERB
ap-1807	183	7	(	(	PUNCT
ap-1807	183	8	x+	x+	X
ap-1807	183	9	y	y	NOUN
ap-1807	183	10	)	)	PUNCT
ap-1807	184	1	+	+	NUM
ap-1807	184	2	z	z	NOUN
ap-1807	184	3	=	=	SYM
ap-1807	184	4	x+	x+	PUNCT
ap-1807	184	5	(	(	PUNCT
ap-1807	184	6	y	y	PROPN
ap-1807	184	7	+	+	PROPN
ap-1807	184	8	z	z	NOUN
ap-1807	184	9	)	)	PUNCT
ap-1807	184	10	.	.	PUNCT
ap-1807	185	1	lemma	lemma	PROPN
ap-1807	185	2	7	7	X
ap-1807	185	3	.	.	PUNCT
ap-1807	186	1	let	let	VERB
ap-1807	186	2	(	(	PUNCT
ap-1807	186	3	g,+	g,+	PROPN
ap-1807	186	4	,	,	PUNCT
ap-1807	186	5	0	0	NUM
ap-1807	186	6	)	)	PUNCT
ap-1807	186	7	be	be	AUX
ap-1807	186	8	an	an	DET
ap-1807	186	9	a	a	PRON
ap-1807	186	10	-	-	PUNCT
ap-1807	186	11	commutative	commutative	ADJ
ap-1807	186	12	partial	partial	ADJ
ap-1807	186	13	group	group	NOUN
ap-1807	186	14	and	and	CCONJ
ap-1807	186	15	e	e	NOUN
ap-1807	186	16	⊆	⊆	NUM
ap-1807	186	17	g	g	ADP
ap-1807	186	18	a	a	DET
ap-1807	186	19	subset	subset	NOUN
ap-1807	186	20	closed	close	VERB
ap-1807	186	21	under	under	ADP
ap-1807	186	22	the	the	DET
ap-1807	186	23	+	+	ADJ
ap-1807	186	24	,	,	PUNCT
ap-1807	186	25	i.e.	i.e.	X
ap-1807	186	26	,	,	PUNCT
ap-1807	186	27	x	x	PRON
ap-1807	186	28	,	,	PUNCT
ap-1807	186	29	y	y	PROPN
ap-1807	186	30	∈	∈	PROPN
ap-1807	186	31	e	e	PROPN
ap-1807	186	32	,	,	PUNCT
ap-1807	186	33	x+y	x+y	PROPN
ap-1807	186	34	∈	∈	PROPN
ap-1807	186	35	g	g	PROPN
ap-1807	186	36	implies	imply	VERB
ap-1807	186	37	x+y	x+y	NUM
ap-1807	186	38	∈	∈	PROPN
ap-1807	186	39	e	e	NOUN
ap-1807	186	40	,	,	PUNCT
ap-1807	186	41	such	such	ADJ
ap-1807	186	42	that	that	SCONJ
ap-1807	186	43	0	0	NUM
ap-1807	186	44	∈	∈	PROPN
ap-1807	186	45	e	e	NOUN
ap-1807	186	46	and	and	CCONJ
ap-1807	186	47	(	(	PUNCT
ap-1807	186	48	e,+/e	e,+/e	X
ap-1807	186	49	,	,	PUNCT
ap-1807	186	50	0	0	X
ap-1807	186	51	)	)	PUNCT
ap-1807	186	52	forms	form	VERB
ap-1807	186	53	a	a	DET
ap-1807	186	54	generalized	generalized	ADJ
ap-1807	186	55	effect	effect	NOUN
ap-1807	186	56	algebra	algebra	NOUN
ap-1807	186	57	.	.	PUNCT
ap-1807	187	1	define	define	VERB
ap-1807	187	2	a	a	DET
ap-1807	187	3	relation	relation	NOUN
ap-1807	187	4	≤	≤	NUM
ap-1807	187	5	by	by	ADP
ap-1807	187	6	x	x	SYM
ap-1807	187	7	≤	≤	PROPN
ap-1807	187	8	y	y	PROPN
ap-1807	187	9	iff	iff	PROPN
ap-1807	187	10	(	(	PUNCT
ap-1807	187	11	−x	−x	NOUN
ap-1807	187	12	)	)	PUNCT
ap-1807	188	1	+	+	CCONJ
ap-1807	188	2	y	y	PROPN
ap-1807	188	3	is	be	AUX
ap-1807	188	4	defined	define	VERB
ap-1807	188	5	and	and	CCONJ
ap-1807	188	6	(	(	PUNCT
ap-1807	188	7	(	(	PUNCT
ap-1807	188	8	−x	−x	NOUN
ap-1807	188	9	)	)	PUNCT
ap-1807	189	1	+	+	NUM
ap-1807	189	2	y	y	X
ap-1807	189	3	)	)	PUNCT
ap-1807	189	4	∈	∈	PROPN
ap-1807	189	5	e.	e.	PROPN
ap-1807	189	6	then	then	ADV
ap-1807	189	7	(	(	PUNCT
ap-1807	189	8	g,+	g,+	PROPN
ap-1807	189	9	,	,	PUNCT
ap-1807	189	10	0	0	NUM
ap-1807	189	11	)	)	PUNCT
ap-1807	189	12	is	be	AUX
ap-1807	189	13	a	a	DET
ap-1807	189	14	woa	woa	VERB
ap-1807	189	15	-	-	PUNCT
ap-1807	189	16	group	group	NOUN
ap-1807	189	17	w.r.t	w.r.t	NOUN
ap-1807	189	18	.	.	PUNCT
ap-1807	190	1	≤	≤	NOUN
ap-1807	190	2	,	,	PUNCT
ap-1807	190	3	pos(g	pos(g	PROPN
ap-1807	190	4	)	)	PUNCT
ap-1807	190	5	=	=	SYM
ap-1807	190	6	e	e	NOUN
ap-1807	190	7	and	and	CCONJ
ap-1807	190	8	≤	≤	NUM
ap-1807	190	9	on	on	ADP
ap-1807	190	10	pos(g	pos(g	PROPN
ap-1807	190	11	)	)	PUNCT
ap-1807	190	12	coincides	coincide	VERB
ap-1807	190	13	with	with	ADP
ap-1807	190	14	induced	induced	ADJ
ap-1807	190	15	partial	partial	ADJ
ap-1807	190	16	order	order	NOUN
ap-1807	190	17	≤e	≤e	VERB
ap-1807	190	18	from	from	ADP
ap-1807	190	19	(	(	PUNCT
ap-1807	190	20	e,+/e	e,+/e	X
ap-1807	190	21	,	,	PUNCT
ap-1807	190	22	0	0	NUM
ap-1807	190	23	)	)	PUNCT
ap-1807	190	24	.	.	PUNCT
ap-1807	191	1	proof	proof	NOUN
ap-1807	191	2	.	.	PUNCT
ap-1807	192	1	reflexivity	reflexivity	NOUN
ap-1807	192	2	is	be	AUX
ap-1807	192	3	clear	clear	ADJ
ap-1807	192	4	since	since	SCONJ
ap-1807	192	5	0	0	NUM
ap-1807	192	6	∈	∈	PROPN
ap-1807	192	7	e.	e.	PROPN
ap-1807	192	8	let	let	VERB
ap-1807	192	9	x	x	SYM
ap-1807	192	10	≤	≤	VERB
ap-1807	192	11	y	y	PROPN
ap-1807	192	12	and	and	CCONJ
ap-1807	192	13	y	y	PROPN
ap-1807	192	14	≤	≤	PROPN
ap-1807	193	1	x	x	PUNCT
ap-1807	193	2	,	,	PUNCT
ap-1807	193	3	then	then	ADV
ap-1807	193	4	−x	−x	PROPN
ap-1807	193	5	+	+	CCONJ
ap-1807	193	6	y	y	PROPN
ap-1807	193	7	,	,	PUNCT
ap-1807	193	8	x	x	X
ap-1807	193	9	+	+	CCONJ
ap-1807	193	10	(	(	PUNCT
ap-1807	193	11	−y	−y	NOUN
ap-1807	193	12	)	)	PUNCT
ap-1807	193	13	∈	∈	PROPN
ap-1807	193	14	e	e	NOUN
ap-1807	193	15	and	and	CCONJ
ap-1807	193	16	with	with	ADP
ap-1807	193	17	(	(	PUNCT
ap-1807	193	18	giv	giv	NOUN
ap-1807	193	19	)	)	PUNCT
ap-1807	193	20	(	(	PUNCT
ap-1807	193	21	−x	−x	NOUN
ap-1807	193	22	+	+	CCONJ
ap-1807	193	23	x	x	X
ap-1807	193	24	)	)	PUNCT
ap-1807	193	25	+	+	CCONJ
ap-1807	193	26	(	(	PUNCT
ap-1807	193	27	−y	−y	VERB
ap-1807	193	28	+	+	CCONJ
ap-1807	193	29	y	y	NOUN
ap-1807	193	30	)	)	PUNCT
ap-1807	193	31	=	=	SYM
ap-1807	194	1	(	(	PUNCT
ap-1807	194	2	−x	−x	X
ap-1807	194	3	+	+	CCONJ
ap-1807	194	4	y	y	NOUN
ap-1807	194	5	)	)	PUNCT
ap-1807	195	1	+	+	CCONJ
ap-1807	195	2	(	(	PUNCT
ap-1807	195	3	x	x	SYM
ap-1807	195	4	+	+	CCONJ
ap-1807	195	5	(	(	PUNCT
ap-1807	195	6	−y	−y	NOUN
ap-1807	195	7	)	)	PUNCT
ap-1807	195	8	)	)	PUNCT
ap-1807	196	1	=	=	PUNCT
ap-1807	196	2	0	0	X
ap-1807	196	3	.	.	X
ap-1807	197	1	e	e	NOUN
ap-1807	197	2	is	be	AUX
ap-1807	197	3	a	a	DET
ap-1807	197	4	generalized	generalized	ADJ
ap-1807	197	5	effect	effect	NOUN
ap-1807	197	6	algebra	algebra	NOUN
ap-1807	197	7	hence	hence	ADV
ap-1807	197	8	by	by	ADP
ap-1807	197	9	(	(	PUNCT
ap-1807	197	10	gev	gev	NOUN
ap-1807	197	11	)	)	PUNCT
ap-1807	197	12	x	x	PUNCT
ap-1807	198	1	+	+	CCONJ
ap-1807	198	2	(	(	PUNCT
ap-1807	198	3	−y	−y	NOUN
ap-1807	198	4	)	)	PUNCT
ap-1807	198	5	=	=	SYM
ap-1807	198	6	−x	−x	NOUN
ap-1807	198	7	+	+	CCONJ
ap-1807	198	8	y	y	PROPN
ap-1807	198	9	=	=	PUNCT
ap-1807	198	10	0	0	PROPN
ap-1807	198	11	that	that	PRON
ap-1807	198	12	is	be	AUX
ap-1807	198	13	x	x	NOUN
ap-1807	198	14	=	=	SYM
ap-1807	198	15	y.	y.	NOUN
ap-1807	198	16	clearly	clearly	ADV
ap-1807	198	17	pos(g	pos(g	VERB
ap-1807	198	18	)	)	PUNCT
ap-1807	199	1	=	=	SYM
ap-1807	199	2	e.	e.	PROPN
ap-1807	199	3	then	then	ADV
ap-1807	199	4	(	(	PUNCT
ap-1807	199	5	ri	ri	NOUN
ap-1807	199	6	)	)	PUNCT
ap-1807	199	7	is	be	AUX
ap-1807	199	8	straightforward	straightforward	ADJ
ap-1807	199	9	using	use	VERB
ap-1807	199	10	the	the	DET
ap-1807	199	11	definition	definition	NOUN
ap-1807	199	12	of	of	ADP
ap-1807	199	13	≤	≤	NUM
ap-1807	199	14	and	and	CCONJ
ap-1807	199	15	lemma	lemma	PROPN
ap-1807	199	16	1	1	NUM
ap-1807	199	17	(	(	PUNCT
ap-1807	199	18	1	1	NUM
ap-1807	199	19	.	.	NUM
ap-1807	199	20	)	)	PUNCT
ap-1807	199	21	.	.	PUNCT
ap-1807	200	1	(	(	PUNCT
ap-1807	200	2	rii	rii	PROPN
ap-1807	200	3	)	)	PUNCT
ap-1807	200	4	holds	hold	VERB
ap-1807	200	5	because	because	SCONJ
ap-1807	200	6	we	we	PRON
ap-1807	200	7	want	want	VERB
ap-1807	200	8	e	e	NOUN
ap-1807	200	9	to	to	PART
ap-1807	200	10	be	be	AUX
ap-1807	200	11	closed	close	VERB
ap-1807	200	12	under	under	ADP
ap-1807	200	13	the	the	DET
ap-1807	200	14	+	+	NOUN
ap-1807	200	15	.	.	NOUN
ap-1807	201	1	for	for	ADP
ap-1807	201	2	(	(	PUNCT
ap-1807	201	3	riii	riii	PROPN
ap-1807	201	4	)	)	PUNCT
ap-1807	201	5	,	,	PUNCT
ap-1807	201	6	let	let	VERB
ap-1807	201	7	x	x	PRON
ap-1807	201	8	,	,	PUNCT
ap-1807	201	9	z	z	PROPN
ap-1807	201	10	∈	∈	PROPN
ap-1807	201	11	e	e	NOUN
ap-1807	201	12	,	,	PUNCT
ap-1807	201	13	x	x	PUNCT
ap-1807	201	14	≤	≤	ADJ
ap-1807	201	15	y	y	PROPN
ap-1807	201	16	and	and	CCONJ
ap-1807	201	17	y	y	PROPN
ap-1807	202	1	+	+	CCONJ
ap-1807	202	2	z	z	AUX
ap-1807	202	3	be	be	AUX
ap-1807	202	4	defined	define	VERB
ap-1807	202	5	.	.	PUNCT
ap-1807	203	1	then	then	ADV
ap-1807	203	2	y	y	PROPN
ap-1807	203	3	+	+	NOUN
ap-1807	203	4	z	z	NOUN
ap-1807	203	5	=	=	SYM
ap-1807	203	6	(	(	PUNCT
ap-1807	203	7	x+	x+	X
ap-1807	203	8	(	(	PUNCT
ap-1807	203	9	(	(	PUNCT
ap-1807	203	10	−x	−x	NOUN
ap-1807	203	11	)	)	PUNCT
ap-1807	203	12	+	+	NUM
ap-1807	203	13	y	y	NOUN
ap-1807	203	14	)	)	PUNCT
ap-1807	203	15	)	)	PUNCT
ap-1807	204	1	+	+	CCONJ
ap-1807	204	2	z	z	X
ap-1807	204	3	=	=	SYM
ap-1807	204	4	(	(	PUNCT
ap-1807	204	5	x+	x+	ADJ
ap-1807	204	6	z	z	NOUN
ap-1807	204	7	)	)	PUNCT
ap-1807	204	8	+	+	CCONJ
ap-1807	204	9	(	(	PUNCT
ap-1807	204	10	(	(	PUNCT
ap-1807	204	11	−x	−x	NOUN
ap-1807	204	12	)	)	PUNCT
ap-1807	204	13	+	+	NUM
ap-1807	204	14	y	y	NOUN
ap-1807	204	15	)	)	PUNCT
ap-1807	204	16	)	)	PUNCT
ap-1807	205	1	using	use	VERB
ap-1807	205	2	the	the	DET
ap-1807	205	3	associativity	associativity	NOUN
ap-1807	205	4	of	of	ADP
ap-1807	205	5	generalized	generalized	ADJ
ap-1807	205	6	effect	effect	NOUN
ap-1807	205	7	algebra	algebra	NOUN
ap-1807	205	8	e	e	NOUN
ap-1807	205	9	since	since	SCONJ
ap-1807	205	10	x	x	PROPN
ap-1807	205	11	,	,	PUNCT
ap-1807	205	12	z	z	PROPN
ap-1807	205	13	,	,	PUNCT
ap-1807	205	14	(	(	PUNCT
ap-1807	205	15	(	(	PUNCT
ap-1807	205	16	−x	−x	NOUN
ap-1807	205	17	)	)	PUNCT
ap-1807	205	18	+	+	NUM
ap-1807	205	19	y	y	X
ap-1807	205	20	)	)	PUNCT
ap-1807	205	21	∈	∈	PROPN
ap-1807	205	22	e.	e.	NOUN
ap-1807	205	23	we	we	PRON
ap-1807	205	24	show	show	VERB
ap-1807	205	25	the	the	DET
ap-1807	205	26	coincidence	coincidence	NOUN
ap-1807	205	27	.	.	PUNCT
ap-1807	206	1	let	let	VERB
ap-1807	206	2	us	we	PRON
ap-1807	206	3	have	have	VERB
ap-1807	206	4	−x+	−x+	PROPN
ap-1807	206	5	y	y	PROPN
ap-1807	206	6	,	,	PUNCT
ap-1807	206	7	x	x	PRON
ap-1807	206	8	,	,	PUNCT
ap-1807	206	9	y	y	PROPN
ap-1807	206	10	∈	∈	PROPN
ap-1807	206	11	e.	e.	PROPN
ap-1807	206	12	because	because	SCONJ
ap-1807	206	13	e	e	PROPN
ap-1807	206	14	is	be	AUX
ap-1807	206	15	closed	close	VERB
ap-1807	206	16	on	on	ADP
ap-1807	206	17	+	+	CCONJ
ap-1807	206	18	we	we	PRON
ap-1807	206	19	have	have	AUX
ap-1807	206	20	x+	x+	VERB
ap-1807	206	21	(	(	PUNCT
ap-1807	206	22	−x+	−x+	NOUN
ap-1807	206	23	y	y	NOUN
ap-1807	206	24	)	)	PUNCT
ap-1807	206	25	defined	define	VERB
ap-1807	206	26	and	and	CCONJ
ap-1807	206	27	x+	x+	NUM
ap-1807	206	28	(	(	PUNCT
ap-1807	206	29	−x+	−x+	NOUN
ap-1807	206	30	y	y	NOUN
ap-1807	206	31	)	)	PUNCT
ap-1807	207	1	=	=	SYM
ap-1807	207	2	y	y	PROPN
ap-1807	207	3	,	,	PUNCT
ap-1807	207	4	i.e.	i.e.	X
ap-1807	207	5	,	,	PUNCT
ap-1807	207	6	x	x	PUNCT
ap-1807	207	7	≤e	≤e	VERB
ap-1807	207	8	y.	y.	NOUN
ap-1807	207	9	on	on	ADP
ap-1807	207	10	the	the	DET
ap-1807	207	11	other	other	ADJ
ap-1807	207	12	hand	hand	NOUN
ap-1807	207	13	let	let	VERB
ap-1807	207	14	x	x	PRON
ap-1807	207	15	≤e	≤e	VERB
ap-1807	207	16	y	y	PROPN
ap-1807	207	17	,	,	PUNCT
ap-1807	207	18	x	x	PRON
ap-1807	207	19	,	,	PUNCT
ap-1807	207	20	y	y	PROPN
ap-1807	207	21	∈	∈	PROPN
ap-1807	207	22	e.	e.	PROPN
ap-1807	207	23	then	then	ADV
ap-1807	207	24	there	there	PRON
ap-1807	207	25	exists	exist	VERB
ap-1807	207	26	z	z	NOUN
ap-1807	207	27	∈	∈	PROPN
ap-1807	207	28	e	e	NOUN
ap-1807	207	29	such	such	ADJ
ap-1807	207	30	that	that	SCONJ
ap-1807	207	31	x	x	X
ap-1807	208	1	+	+	PUNCT
ap-1807	208	2	z	z	NOUN
ap-1807	208	3	=	=	SYM
ap-1807	208	4	y	y	PROPN
ap-1807	208	5	and	and	CCONJ
ap-1807	208	6	by	by	ADP
ap-1807	208	7	lemma	lemma	PROPN
ap-1807	208	8	1	1	NUM
ap-1807	208	9	(	(	PUNCT
ap-1807	208	10	1	1	NUM
ap-1807	208	11	.	.	PUNCT
ap-1807	208	12	)	)	PUNCT
ap-1807	209	1	z	z	NOUN
ap-1807	210	1	=	=	PUNCT
ap-1807	210	2	y	y	PROPN
ap-1807	210	3	+	+	CCONJ
ap-1807	210	4	(	(	PUNCT
ap-1807	210	5	−x	−x	NOUN
ap-1807	210	6	)	)	PUNCT
ap-1807	210	7	.	.	PUNCT
ap-1807	211	1	the	the	DET
ap-1807	211	2	previous	previous	ADJ
ap-1807	211	3	lemma	lemma	PROPN
ap-1807	211	4	formalizes	formalize	VERB
ap-1807	211	5	the	the	DET
ap-1807	211	6	idea	idea	NOUN
ap-1807	211	7	of	of	ADP
ap-1807	211	8	determining	determine	VERB
ap-1807	211	9	a	a	DET
ap-1807	211	10	weak	weak	ADJ
ap-1807	211	11	order	order	NOUN
ap-1807	211	12	by	by	ADP
ap-1807	211	13	the	the	DET
ap-1807	211	14	set	set	NOUN
ap-1807	211	15	of	of	ADP
ap-1807	211	16	positive	positive	ADJ
ap-1807	211	17	elements	element	NOUN
ap-1807	211	18	.	.	PUNCT
ap-1807	212	1	let	let	VERB
ap-1807	212	2	us	we	PRON
ap-1807	212	3	have	have	VERB
ap-1807	212	4	an	an	DET
ap-1807	212	5	a	a	PRON
ap-1807	212	6	-	-	PUNCT
ap-1807	212	7	commutative	commutative	ADJ
ap-1807	212	8	partial	partial	ADJ
ap-1807	212	9	group	group	NOUN
ap-1807	212	10	and	and	CCONJ
ap-1807	212	11	choose	choose	VERB
ap-1807	212	12	some	some	DET
ap-1807	212	13	elements	element	NOUN
ap-1807	212	14	to	to	PART
ap-1807	212	15	be	be	AUX
ap-1807	212	16	positive	positive	ADJ
ap-1807	212	17	.	.	PUNCT
ap-1807	213	1	consider	consider	VERB
ap-1807	213	2	the	the	DET
ap-1807	213	3	smallest	small	ADJ
ap-1807	213	4	set	set	NOUN
ap-1807	213	5	closed	close	VERB
ap-1807	213	6	under	under	ADP
ap-1807	213	7	partial	partial	ADJ
ap-1807	213	8	addition	addition	NOUN
ap-1807	213	9	containing	contain	VERB
ap-1807	213	10	zero	zero	NUM
ap-1807	213	11	and	and	CCONJ
ap-1807	213	12	the	the	DET
ap-1807	213	13	chosen	choose	VERB
ap-1807	213	14	elements	element	NOUN
ap-1807	213	15	.	.	PUNCT
ap-1807	214	1	if	if	SCONJ
ap-1807	214	2	the	the	DET
ap-1807	214	3	set	set	NOUN
ap-1807	214	4	has	have	VERB
ap-1807	214	5	the	the	DET
ap-1807	214	6	form	form	NOUN
ap-1807	214	7	of	of	ADP
ap-1807	214	8	a	a	DET
ap-1807	214	9	generalized	generalized	ADJ
ap-1807	214	10	effect	effect	NOUN
ap-1807	214	11	algebra	algebra	NOUN
ap-1807	214	12	,	,	PUNCT
ap-1807	214	13	then	then	ADV
ap-1807	214	14	there	there	PRON
ap-1807	214	15	exists	exist	VERB
ap-1807	214	16	such	such	DET
ap-1807	214	17	a	a	DET
ap-1807	214	18	weak	weak	ADJ
ap-1807	214	19	order	order	NOUN
ap-1807	214	20	that	that	SCONJ
ap-1807	214	21	our	our	PRON
ap-1807	214	22	set	set	NOUN
ap-1807	214	23	is	be	AUX
ap-1807	214	24	exactly	exactly	ADV
ap-1807	214	25	the	the	DET
ap-1807	214	26	set	set	NOUN
ap-1807	214	27	of	of	ADP
ap-1807	214	28	positive	positive	ADJ
ap-1807	214	29	elements	element	NOUN
ap-1807	214	30	.	.	PUNCT
ap-1807	215	1	on	on	ADP
ap-1807	215	2	the	the	DET
ap-1807	215	3	other	other	ADJ
ap-1807	215	4	hand	hand	NOUN
ap-1807	215	5	,	,	PUNCT
ap-1807	215	6	it	it	PRON
ap-1807	215	7	is	be	AUX
ap-1807	215	8	not	not	PART
ap-1807	215	9	hard	hard	ADJ
ap-1807	215	10	to	to	PART
ap-1807	215	11	show	show	VERB
ap-1807	215	12	that	that	SCONJ
ap-1807	215	13	if	if	SCONJ
ap-1807	215	14	the	the	DET
ap-1807	215	15	set	set	NOUN
ap-1807	215	16	is	be	AUX
ap-1807	215	17	not	not	PART
ap-1807	215	18	a	a	DET
ap-1807	215	19	generalized	generalized	ADJ
ap-1807	215	20	effect	effect	NOUN
ap-1807	215	21	algebra	algebra	NOUN
ap-1807	215	22	,	,	PUNCT
ap-1807	215	23	then	then	ADV
ap-1807	215	24	there	there	PRON
ap-1807	215	25	is	be	VERB
ap-1807	215	26	no	no	DET
ap-1807	215	27	such	such	ADJ
ap-1807	215	28	weak	weak	ADJ
ap-1807	215	29	order	order	NOUN
ap-1807	215	30	that	that	PRON
ap-1807	215	31	all	all	PRON
ap-1807	215	32	of	of	ADP
ap-1807	215	33	our	our	PRON
ap-1807	215	34	chosen	choose	VERB
ap-1807	215	35	elements	element	NOUN
ap-1807	215	36	are	be	AUX
ap-1807	215	37	positive	positive	ADJ
ap-1807	215	38	.	.	PUNCT
ap-1807	216	1	theorem	theorem	NOUN
ap-1807	216	2	2	2	NUM
ap-1807	216	3	.	.	PUNCT
ap-1807	217	1	let	let	VERB
ap-1807	217	2	(	(	PUNCT
ap-1807	217	3	e,+	e,+	ADJ
ap-1807	217	4	,	,	PUNCT
ap-1807	217	5	0	0	NUM
ap-1807	217	6	)	)	PUNCT
ap-1807	217	7	be	be	AUX
ap-1807	217	8	a	a	DET
ap-1807	217	9	generalized	generalized	ADJ
ap-1807	217	10	effect	effect	NOUN
ap-1807	217	11	algebra	algebra	NOUN
ap-1807	217	12	with	with	ADP
ap-1807	217	13	induced	induced	ADJ
ap-1807	217	14	order	order	NOUN
ap-1807	217	15	≤.	≤.	NOUN
ap-1807	217	16	then	then	ADV
ap-1807	217	17	there	there	PRON
ap-1807	217	18	exists	exist	VERB
ap-1807	217	19	a	a	DET
ap-1807	217	20	woa	woa	NOUN
ap-1807	217	21	-	-	PUNCT
ap-1807	217	22	group	group	NOUN
ap-1807	217	23	(	(	PUNCT
ap-1807	217	24	g,⊕	g,⊕	PROPN
ap-1807	217	25	,	,	PUNCT
ap-1807	217	26	0	0	NUM
ap-1807	217	27	)	)	PUNCT
ap-1807	217	28	w.r.t	w.r.t	NOUN
ap-1807	217	29	.	.	PUNCT
ap-1807	218	1	relation	relation	NOUN
ap-1807	218	2	≤g	≤g	PROPN
ap-1807	218	3	such	such	ADJ
ap-1807	218	4	that	that	SCONJ
ap-1807	218	5	(	(	PUNCT
ap-1807	218	6	pos(g	pos(g	PROPN
ap-1807	218	7	)	)	PUNCT
ap-1807	218	8	,	,	PUNCT
ap-1807	218	9	⊕/	⊕/	PRON
ap-1807	218	10	pos(g	pos(g	PROPN
ap-1807	218	11	)	)	PUNCT
ap-1807	218	12	,	,	PUNCT
ap-1807	218	13	0	0	X
ap-1807	218	14	)	)	PUNCT
ap-1807	218	15	=	=	NOUN
ap-1807	218	16	(	(	PUNCT
ap-1807	218	17	e,+	e,+	X
ap-1807	218	18	,	,	PUNCT
ap-1807	218	19	0	0	NUM
ap-1807	218	20	)	)	PUNCT
ap-1807	218	21	and	and	CCONJ
ap-1807	218	22	≤g	≤g	PROPN
ap-1807	218	23	/	/	SYM
ap-1807	218	24	pos(g)=≤.	pos(g)=≤.	NUM
ap-1807	218	25	proof	proof	NOUN
ap-1807	218	26	.	.	PUNCT
ap-1807	219	1	let	let	VERB
ap-1807	219	2	(	(	PUNCT
ap-1807	219	3	e,+	e,+	ADJ
ap-1807	219	4	,	,	PUNCT
ap-1807	219	5	0	0	NUM
ap-1807	219	6	)	)	PUNCT
ap-1807	219	7	be	be	AUX
ap-1807	219	8	a	a	DET
ap-1807	219	9	generalized	generalized	ADJ
ap-1807	219	10	effect	effect	NOUN
ap-1807	219	11	algebra	algebra	NOUN
ap-1807	219	12	with	with	ADP
ap-1807	219	13	induced	induced	ADJ
ap-1807	219	14	order	order	NOUN
ap-1807	219	15	≤.	≤.	NOUN
ap-1807	219	16	for	for	ADP
ap-1807	219	17	any	any	DET
ap-1807	219	18	a	a	PRON
ap-1807	219	19	,	,	PUNCT
ap-1807	219	20	b	b	X
ap-1807	219	21	∈	∈	PROPN
ap-1807	219	22	e	e	NOUN
ap-1807	219	23	,	,	PUNCT
ap-1807	219	24	a	a	DET
ap-1807	219	25	≤	≤	PROPN
ap-1807	219	26	b	b	NOUN
ap-1807	219	27	,	,	PUNCT
ap-1807	219	28	the	the	DET
ap-1807	219	29	symbol	symbol	NOUN
ap-1807	219	30	b−a	b−a	NOUN
ap-1807	219	31	denotes	denote	VERB
ap-1807	219	32	such	such	DET
ap-1807	219	33	an	an	DET
ap-1807	219	34	element	element	NOUN
ap-1807	219	35	that	that	SCONJ
ap-1807	219	36	a+(b−a	a+(b−a	PROPN
ap-1807	219	37	)	)	PUNCT
ap-1807	219	38	=	=	SYM
ap-1807	219	39	b.	b.	PROPN
ap-1807	219	40	let	let	VERB
ap-1807	219	41	e−	e−	NOUN
ap-1807	219	42	be	be	AUX
ap-1807	219	43	a	a	DET
ap-1807	219	44	set	set	NOUN
ap-1807	219	45	with	with	ADP
ap-1807	219	46	the	the	DET
ap-1807	219	47	same	same	ADJ
ap-1807	219	48	cardinality	cardinality	NOUN
ap-1807	219	49	disjoint	disjoint	NOUN
ap-1807	219	50	from	from	ADP
ap-1807	219	51	e.	e.	PROPN
ap-1807	219	52	consider	consider	VERB
ap-1807	219	53	a	a	DET
ap-1807	219	54	bijection	bijection	NOUN
ap-1807	219	55	ϕ	ϕ	NOUN
ap-1807	219	56	:	:	PUNCT
ap-1807	219	57	e	e	X
ap-1807	219	58	→	→	PUNCT
ap-1807	219	59	e−.	e−.	NOUN
ap-1807	219	60	we	we	PRON
ap-1807	219	61	set	set	VERB
ap-1807	219	62	a−	a−	PROPN
ap-1807	219	63	=	=	SYM
ap-1807	219	64	ϕ(a	ϕ(a	NOUN
ap-1807	219	65	)	)	PUNCT
ap-1807	219	66	for	for	ADP
ap-1807	219	67	a	a	DET
ap-1807	219	68	∈	∈	PROPN
ap-1807	219	69	e	e	NOUN
ap-1807	219	70	r	r	NOUN
ap-1807	219	71	{	{	PUNCT
ap-1807	219	72	0	0	NUM
ap-1807	219	73	}	}	PUNCT
ap-1807	219	74	and	and	CCONJ
ap-1807	219	75	0−	0−	NUM
ap-1807	219	76	=	=	SYM
ap-1807	219	77	0	0	X
ap-1807	219	78	.	.	PUNCT
ap-1807	220	1	let	let	VERB
ap-1807	220	2	g	g	NOUN
ap-1807	220	3	=	=	SYM
ap-1807	220	4	e	e	X
ap-1807	220	5	∪̇	∪̇	X
ap-1807	220	6	(	(	PUNCT
ap-1807	220	7	e−	e−	PROPN
ap-1807	220	8	r	r	NOUN
ap-1807	220	9	{	{	PUNCT
ap-1807	220	10	ϕ(0	ϕ(0	PROPN
ap-1807	220	11	)	)	PUNCT
ap-1807	220	12	}	}	PUNCT
ap-1807	220	13	)	)	PUNCT
ap-1807	221	1	be	be	AUX
ap-1807	221	2	a	a	DET
ap-1807	221	3	disjoint	disjoint	NOUN
ap-1807	221	4	union	union	NOUN
ap-1807	221	5	of	of	ADP
ap-1807	221	6	e	e	PROPN
ap-1807	221	7	and	and	CCONJ
ap-1807	221	8	e−	e−	PROPN
ap-1807	221	9	r	r	NOUN
ap-1807	221	10	{	{	PUNCT
ap-1807	221	11	ϕ(0	ϕ(0	PROPN
ap-1807	221	12	)	)	PUNCT
ap-1807	221	13	}	}	PUNCT
ap-1807	221	14	.	.	PUNCT
ap-1807	222	1	let	let	VERB
ap-1807	222	2	us	we	PRON
ap-1807	222	3	define	define	VERB
ap-1807	222	4	define	define	VERB
ap-1807	222	5	a	a	DET
ap-1807	222	6	partial	partial	ADJ
ap-1807	222	7	binary	binary	NOUN
ap-1807	222	8	operation	operation	NOUN
ap-1807	222	9	⊕	⊕	PROPN
ap-1807	222	10	on	on	ADP
ap-1807	222	11	g	g	PROPN
ap-1807	222	12	by	by	ADP
ap-1807	222	13	•	•	NUM
ap-1807	222	14	a⊕	a⊕	PROPN
ap-1807	222	15	b	b	PROPN
ap-1807	222	16	exists	exist	VERB
ap-1807	222	17	iff	iff	PROPN
ap-1807	222	18	a+	a+	SYM
ap-1807	222	19	b	b	PROPN
ap-1807	222	20	exists	exist	VERB
ap-1807	222	21	and	and	CCONJ
ap-1807	222	22	then	then	ADV
ap-1807	222	23	a⊕	a⊕	PROPN
ap-1807	222	24	b	b	PROPN
ap-1807	222	25	=	=	PRON
ap-1807	222	26	a+	a+	PUNCT
ap-1807	222	27	b	b	NOUN
ap-1807	222	28	for	for	ADP
ap-1807	222	29	all	all	DET
ap-1807	222	30	a	a	PRON
ap-1807	222	31	,	,	PUNCT
ap-1807	222	32	b	b	X
ap-1807	222	33	∈	∈	PROPN
ap-1807	222	34	e	e	NOUN
ap-1807	222	35	,	,	PUNCT
ap-1807	222	36	•	•	ADJ
ap-1807	222	37	a−	a−	PROPN
ap-1807	222	38	⊕	⊕	PROPN
ap-1807	222	39	b−	b−	PRON
ap-1807	222	40	exist	exist	VERB
ap-1807	222	41	iff	iff	PROPN
ap-1807	222	42	a+	a+	PUNCT
ap-1807	222	43	b	b	PROPN
ap-1807	222	44	exists	exist	VERB
ap-1807	222	45	and	and	CCONJ
ap-1807	222	46	then	then	ADV
ap-1807	222	47	a−	a−	PROPN
ap-1807	222	48	⊕	⊕	PROPN
ap-1807	222	49	b−	b−	PROPN
ap-1807	222	50	=	=	SYM
ap-1807	222	51	(	(	PUNCT
ap-1807	222	52	a+	a+	PUNCT
ap-1807	222	53	b)−	b)−	PROPN
ap-1807	222	54	for	for	ADP
ap-1807	222	55	all	all	DET
ap-1807	222	56	a	a	PRON
ap-1807	222	57	,	,	PUNCT
ap-1807	222	58	b	b	X
ap-1807	222	59	∈	∈	PROPN
ap-1807	222	60	e	e	NOUN
ap-1807	222	61	,	,	PUNCT
ap-1807	222	62	•	•	ADV
ap-1807	222	63	a⊕	a⊕	NOUN
ap-1807	222	64	b−	b−	NOUN
ap-1807	222	65	=	=	SYM
ap-1807	222	66	b−	b−	PROPN
ap-1807	222	67	⊕	⊕	PROPN
ap-1807	223	1	a	a	PRON
ap-1807	223	2	is	be	AUX
ap-1807	223	3	defined	define	VERB
ap-1807	223	4	iff	iff	PROPN
ap-1807	223	5	(	(	PUNCT
ap-1807	223	6	1	1	NUM
ap-1807	223	7	.	.	PUNCT
ap-1807	223	8	)	)	PUNCT
ap-1807	224	1	b	b	NOUN
ap-1807	224	2	≤	≤	ADV
ap-1807	224	3	a	a	DET
ap-1807	224	4	(	(	PUNCT
ap-1807	224	5	a−	a−	PROPN
ap-1807	224	6	b	b	PROPN
ap-1807	224	7	exists	exist	VERB
ap-1807	224	8	)	)	PUNCT
ap-1807	224	9	then	then	ADV
ap-1807	224	10	a⊕	a⊕	X
ap-1807	224	11	b−	b−	NOUN
ap-1807	224	12	=	=	SYM
ap-1807	224	13	a−	a−	PROPN
ap-1807	224	14	b	b	PROPN
ap-1807	224	15	or	or	CCONJ
ap-1807	224	16	(	(	PUNCT
ap-1807	224	17	2	2	NUM
ap-1807	224	18	.	.	PUNCT
ap-1807	224	19	)	)	PUNCT
ap-1807	225	1	a	a	DET
ap-1807	225	2	≤	≤	NUM
ap-1807	225	3	b	b	X
ap-1807	225	4	(	(	PUNCT
ap-1807	225	5	b−	b−	PROPN
ap-1807	225	6	a	a	PRON
ap-1807	225	7	exists	exist	NOUN
ap-1807	225	8	)	)	PUNCT
ap-1807	225	9	then	then	ADV
ap-1807	225	10	a⊕	a⊕	X
ap-1807	225	11	b−	b−	PROPN
ap-1807	225	12	=	=	SYM
ap-1807	225	13	(	(	PUNCT
ap-1807	225	14	b−	b−	PROPN
ap-1807	225	15	a)−	a)−	VERB
ap-1807	225	16	for	for	ADP
ap-1807	225	17	any	any	DET
ap-1807	225	18	nonzero	nonzero	NOUN
ap-1807	225	19	a	a	NOUN
ap-1807	225	20	,	,	PUNCT
ap-1807	225	21	b	b	PROPN
ap-1807	225	22	∈	∈	PROPN
ap-1807	225	23	e.	e.	PROPN
ap-1807	225	24	291	291	NUM
ap-1807	225	25	jiří	jiří	NOUN
ap-1807	225	26	janda	janda	PROPN
ap-1807	225	27	acta	acta	PROPN
ap-1807	225	28	polytechnica	polytechnica	PROPN
ap-1807	226	1	it	it	PRON
ap-1807	226	2	is	be	AUX
ap-1807	226	3	not	not	PART
ap-1807	226	4	hard	hard	ADJ
ap-1807	226	5	to	to	PART
ap-1807	226	6	show	show	VERB
ap-1807	226	7	that	that	SCONJ
ap-1807	226	8	the	the	DET
ap-1807	226	9	definition	definition	NOUN
ap-1807	226	10	is	be	AUX
ap-1807	226	11	correct	correct	ADJ
ap-1807	226	12	.	.	PUNCT
ap-1807	227	1	for	for	ADP
ap-1807	227	2	any	any	DET
ap-1807	227	3	a	a	PRON
ap-1807	227	4	,	,	PUNCT
ap-1807	227	5	b	b	X
ap-1807	227	6	∈	∈	PROPN
ap-1807	227	7	e	e	NOUN
ap-1807	227	8	it	it	PRON
ap-1807	227	9	holds	hold	VERB
ap-1807	227	10	that	that	SCONJ
ap-1807	227	11	(	(	PUNCT
ap-1807	227	12	α1)(a⊕	α1)(a⊕	ADV
ap-1807	227	13	b)−	b)−	PROPN
ap-1807	227	14	=	=	PUNCT
ap-1807	227	15	(	(	PUNCT
ap-1807	227	16	a+	a+	PUNCT
ap-1807	227	17	b)−	b)−	PROPN
ap-1807	227	18	=	=	SYM
ap-1807	227	19	(	(	PUNCT
ap-1807	227	20	a−	a−	PROPN
ap-1807	227	21	⊕	⊕	PROPN
ap-1807	227	22	b−	b−	NOUN
ap-1807	227	23	)	)	PUNCT
ap-1807	227	24	.	.	PUNCT
ap-1807	228	1	let	let	VERB
ap-1807	228	2	a⊕	a⊕	PROPN
ap-1807	228	3	b−	b−	PROPN
ap-1807	228	4	be	be	AUX
ap-1807	228	5	defined	define	VERB
ap-1807	228	6	and	and	CCONJ
ap-1807	228	7	(	(	PUNCT
ap-1807	228	8	β1)b	β1)b	PROPN
ap-1807	228	9	≤	≤	NUM
ap-1807	228	10	a	a	DET
ap-1807	228	11	(	(	PUNCT
ap-1807	228	12	a−	a−	PROPN
ap-1807	228	13	b	b	PROPN
ap-1807	228	14	exists	exist	VERB
ap-1807	228	15	)	)	PUNCT
ap-1807	228	16	,	,	PUNCT
ap-1807	228	17	then	then	ADV
ap-1807	228	18	(	(	PUNCT
ap-1807	228	19	a⊕	a⊕	X
ap-1807	228	20	b−)−	b−)−	NOUN
ap-1807	228	21	=	=	SYM
ap-1807	228	22	(	(	PUNCT
ap-1807	228	23	a−	a−	PROPN
ap-1807	228	24	b)−	b)−	PROPN
ap-1807	228	25	=	=	SYM
ap-1807	228	26	a−	a−	PROPN
ap-1807	228	27	⊕	⊕	PROPN
ap-1807	228	28	b.	b.	PROPN
ap-1807	229	1	we	we	PRON
ap-1807	229	2	show	show	VERB
ap-1807	229	3	that	that	SCONJ
ap-1807	229	4	(	(	PUNCT
ap-1807	229	5	g,⊕	g,⊕	NOUN
ap-1807	229	6	,	,	PUNCT
ap-1807	229	7	0	0	NUM
ap-1807	229	8	)	)	PUNCT
ap-1807	229	9	forms	form	VERB
ap-1807	229	10	an	an	DET
ap-1807	229	11	a	a	PRON
ap-1807	229	12	-	-	PUNCT
ap-1807	229	13	commutative	commutative	ADJ
ap-1807	229	14	partial	partial	ADJ
ap-1807	229	15	group	group	NOUN
ap-1807	229	16	.	.	PUNCT
ap-1807	230	1	commutativity	commutativity	NOUN
ap-1807	230	2	(	(	PUNCT
ap-1807	230	3	gi	gi	INTJ
ap-1807	230	4	)	)	PUNCT
ap-1807	230	5	is	be	AUX
ap-1807	230	6	clear	clear	ADJ
ap-1807	230	7	from	from	ADP
ap-1807	230	8	the	the	DET
ap-1807	230	9	definition	definition	NOUN
ap-1807	230	10	.	.	PUNCT
ap-1807	231	1	since	since	SCONJ
ap-1807	231	2	0	0	NUM
ap-1807	231	3	∈	∈	NOUN
ap-1807	231	4	e	e	NOUN
ap-1807	231	5	it	it	PRON
ap-1807	231	6	follows	follow	VERB
ap-1807	231	7	a⊕	a⊕	PROPN
ap-1807	231	8	0	0	PUNCT
ap-1807	232	1	=	=	SYM
ap-1807	232	2	(	(	PUNCT
ap-1807	232	3	a+	a+	X
ap-1807	232	4	0	0	NUM
ap-1807	232	5	)	)	PUNCT
ap-1807	232	6	=	=	SYM
ap-1807	232	7	a	a	PRON
ap-1807	232	8	and	and	CCONJ
ap-1807	232	9	(	(	PUNCT
ap-1807	232	10	a−	a−	PROPN
ap-1807	232	11	⊕	⊕	PROPN
ap-1807	232	12	0	0	NUM
ap-1807	232	13	)	)	PUNCT
ap-1807	232	14	=	=	SYM
ap-1807	232	15	(	(	PUNCT
ap-1807	232	16	a−	a−	PROPN
ap-1807	232	17	0)−	0)−	PUNCT
ap-1807	233	1	=	=	SYM
ap-1807	233	2	a−	a−	PROPN
ap-1807	233	3	for	for	ADP
ap-1807	233	4	all	all	DET
ap-1807	233	5	a	a	DET
ap-1807	233	6	∈	∈	ADJ
ap-1807	233	7	e	e	NOUN
ap-1807	233	8	,	,	PUNCT
ap-1807	233	9	that	that	ADV
ap-1807	233	10	is	is	ADV
ap-1807	233	11	(	(	PUNCT
ap-1807	233	12	gii	gii	NOUN
ap-1807	233	13	)	)	PUNCT
ap-1807	233	14	.	.	PUNCT
ap-1807	234	1	clearly	clearly	ADV
ap-1807	234	2	for	for	ADP
ap-1807	234	3	any	any	DET
ap-1807	234	4	a	a	DET
ap-1807	234	5	∈	∈	NOUN
ap-1807	234	6	e	e	NOUN
ap-1807	234	7	there	there	PRON
ap-1807	234	8	exists	exist	VERB
ap-1807	234	9	a−	a−	PROPN
ap-1807	234	10	∈	∈	PROPN
ap-1807	234	11	e−	e−	X
ap-1807	234	12	where	where	SCONJ
ap-1807	234	13	a⊕	a⊕	PROPN
ap-1807	234	14	a−	a−	PROPN
ap-1807	234	15	=	=	SYM
ap-1807	234	16	a−	a−	PROPN
ap-1807	234	17	a	a	NOUN
ap-1807	234	18	=	=	SYM
ap-1807	234	19	0	0	NUM
ap-1807	234	20	.	.	PUNCT
ap-1807	235	1	this	this	DET
ap-1807	235	2	defines	define	NOUN
ap-1807	235	3	also	also	ADV
ap-1807	235	4	inverse	inverse	VERB
ap-1807	235	5	elements	element	NOUN
ap-1807	235	6	for	for	ADP
ap-1807	235	7	any	any	DET
ap-1807	235	8	a−	a−	PROPN
ap-1807	235	9	∈	∈	PROPN
ap-1807	235	10	e−.	e−.	NOUN
ap-1807	235	11	let	let	VERB
ap-1807	235	12	us	we	PRON
ap-1807	235	13	verify	verify	VERB
ap-1807	235	14	the	the	DET
ap-1807	235	15	associativity	associativity	NOUN
ap-1807	235	16	case	case	NOUN
ap-1807	235	17	by	by	ADP
ap-1807	235	18	case	case	NOUN
ap-1807	235	19	.	.	PUNCT
ap-1807	236	1	we	we	PRON
ap-1807	236	2	assume	assume	VERB
ap-1807	236	3	that	that	SCONJ
ap-1807	236	4	x	x	NOUN
ap-1807	236	5	,	,	PUNCT
ap-1807	236	6	y	y	PROPN
ap-1807	236	7	,	,	PUNCT
ap-1807	236	8	z	z	PROPN
ap-1807	236	9	∈	∈	PROPN
ap-1807	236	10	e.	e.	PROPN
ap-1807	236	11	case	case	PROPN
ap-1807	236	12	i.	i.	PROPN
ap-1807	236	13	first	first	ADV
ap-1807	236	14	,	,	PUNCT
ap-1807	236	15	let	let	VERB
ap-1807	236	16	us	we	PRON
ap-1807	236	17	have	have	VERB
ap-1807	236	18	(	(	PUNCT
ap-1807	236	19	x⊕	x⊕	PROPN
ap-1807	236	20	y)⊕	y)⊕	PROPN
ap-1807	237	1	z	z	NOUN
ap-1807	237	2	defined	define	VERB
ap-1807	237	3	and	and	CCONJ
ap-1807	237	4	y⊕	y⊕	PROPN
ap-1807	237	5	z	z	PROPN
ap-1807	237	6	defined	define	VERB
ap-1807	237	7	.	.	PUNCT
ap-1807	238	1	then	then	ADV
ap-1807	238	2	(	(	PUNCT
ap-1807	238	3	x⊕y)⊕z	x⊕y)⊕z	X
ap-1807	238	4	=	=	SYM
ap-1807	238	5	(	(	PUNCT
ap-1807	238	6	x+y	x+y	NUM
ap-1807	238	7	)	)	PUNCT
ap-1807	239	1	+	+	NOUN
ap-1807	239	2	z	z	NOUN
ap-1807	239	3	=	=	SYM
ap-1807	239	4	x+	x+	X
ap-1807	239	5	(	(	PUNCT
ap-1807	239	6	y+z	y+z	PROPN
ap-1807	239	7	)	)	PUNCT
ap-1807	239	8	=	=	SYM
ap-1807	239	9	x⊕	x⊕	PROPN
ap-1807	239	10	(	(	PUNCT
ap-1807	239	11	y⊕	y⊕	PROPN
ap-1807	239	12	z	z	PROPN
ap-1807	239	13	)	)	PUNCT
ap-1807	239	14	where	where	SCONJ
ap-1807	239	15	the	the	DET
ap-1807	239	16	existence	existence	NOUN
ap-1807	239	17	and	and	CCONJ
ap-1807	239	18	the	the	DET
ap-1807	239	19	equation	equation	NOUN
ap-1807	239	20	follow	follow	VERB
ap-1807	239	21	from	from	ADP
ap-1807	239	22	the	the	DET
ap-1807	239	23	associativity	associativity	NOUN
ap-1807	239	24	of	of	ADP
ap-1807	239	25	the	the	DET
ap-1807	239	26	generalized	generalized	ADJ
ap-1807	239	27	effect	effect	NOUN
ap-1807	239	28	algebra	algebra	PROPN
ap-1807	239	29	e.	e.	PROPN
ap-1807	239	30	case	case	PROPN
ap-1807	239	31	ii	ii	PROPN
ap-1807	239	32	.	.	PUNCT
ap-1807	240	1	let	let	VERB
ap-1807	240	2	(	(	PUNCT
ap-1807	240	3	x⊕	x⊕	PROPN
ap-1807	240	4	y)⊕	y)⊕	NOUN
ap-1807	240	5	z−	z−	PROPN
ap-1807	240	6	and	and	CCONJ
ap-1807	240	7	y	y	PROPN
ap-1807	240	8	⊕	⊕	PROPN
ap-1807	240	9	z−	z−	PROPN
ap-1807	240	10	be	be	AUX
ap-1807	240	11	defined	define	VERB
ap-1807	240	12	.	.	PUNCT
ap-1807	241	1	and	and	CCONJ
ap-1807	241	2	let	let	VERB
ap-1807	241	3	(	(	PUNCT
ap-1807	241	4	α2	α2	ADJ
ap-1807	241	5	)	)	PUNCT
ap-1807	241	6	z	z	NOUN
ap-1807	242	1	≤	≤	NOUN
ap-1807	242	2	y	y	PROPN
ap-1807	242	3	(	(	PUNCT
ap-1807	242	4	i.e.	i.e.	X
ap-1807	242	5	,	,	PUNCT
ap-1807	242	6	y−	y−	NOUN
ap-1807	242	7	z	z	PROPN
ap-1807	242	8	is	be	AUX
ap-1807	242	9	defined	define	VERB
ap-1807	242	10	)	)	PUNCT
ap-1807	242	11	.	.	PUNCT
ap-1807	243	1	hence	hence	ADV
ap-1807	243	2	z	z	NOUN
ap-1807	243	3	≤	≤	NUM
ap-1807	243	4	(	(	PUNCT
ap-1807	243	5	x+	x+	X
ap-1807	243	6	y	y	NOUN
ap-1807	243	7	)	)	PUNCT
ap-1807	243	8	and	and	CCONJ
ap-1807	243	9	(	(	PUNCT
ap-1807	243	10	x	x	PROPN
ap-1807	243	11	⊕	⊕	PROPN
ap-1807	243	12	y	y	NOUN
ap-1807	243	13	)	)	PUNCT
ap-1807	243	14	⊕	⊕	PROPN
ap-1807	243	15	z−	z−	PROPN
ap-1807	243	16	=	=	PUNCT
ap-1807	244	1	(	(	PUNCT
ap-1807	244	2	x	x	X
ap-1807	244	3	+	+	NUM
ap-1807	244	4	y	y	NOUN
ap-1807	244	5	)	)	PUNCT
ap-1807	244	6	−	−	PROPN
ap-1807	244	7	z.	z.	PROPN
ap-1807	244	8	since	since	SCONJ
ap-1807	244	9	y	y	PROPN
ap-1807	244	10	−	−	PROPN
ap-1807	244	11	z	z	PROPN
ap-1807	244	12	exists	exist	VERB
ap-1807	244	13	,	,	PUNCT
ap-1807	244	14	we	we	PRON
ap-1807	244	15	have	have	VERB
ap-1807	244	16	y	y	NOUN
ap-1807	244	17	=	=	SYM
ap-1807	244	18	(	(	PUNCT
ap-1807	244	19	y	y	PROPN
ap-1807	244	20	−	−	PROPN
ap-1807	244	21	z	z	PROPN
ap-1807	244	22	)	)	PUNCT
ap-1807	245	1	+	+	CCONJ
ap-1807	245	2	z	z	NOUN
ap-1807	246	1	and	and	CCONJ
ap-1807	246	2	then	then	ADV
ap-1807	246	3	(	(	PUNCT
ap-1807	246	4	(	(	PUNCT
ap-1807	246	5	x	x	SYM
ap-1807	246	6	+	+	NUM
ap-1807	246	7	y	y	NOUN
ap-1807	246	8	)	)	PUNCT
ap-1807	246	9	−	−	PROPN
ap-1807	247	1	z	z	X
ap-1807	247	2	)	)	PUNCT
ap-1807	248	1	+	+	PUNCT
ap-1807	248	2	z	z	NOUN
ap-1807	248	3	=	=	SYM
ap-1807	248	4	x	x	PUNCT
ap-1807	249	1	+	+	NUM
ap-1807	249	2	y	y	NOUN
ap-1807	249	3	=	=	PUNCT
ap-1807	249	4	x	x	PROPN
ap-1807	250	1	+	+	PUNCT
ap-1807	250	2	(	(	PUNCT
ap-1807	250	3	(	(	PUNCT
ap-1807	250	4	y	y	PROPN
ap-1807	250	5	−	−	PROPN
ap-1807	250	6	z	z	NOUN
ap-1807	250	7	)	)	PUNCT
ap-1807	251	1	+	+	PUNCT
ap-1807	251	2	z	z	X
ap-1807	251	3	)	)	PUNCT
ap-1807	251	4	=	=	SYM
ap-1807	251	5	(	(	PUNCT
ap-1807	251	6	x	x	SYM
ap-1807	251	7	+	+	PUNCT
ap-1807	251	8	(	(	PUNCT
ap-1807	251	9	y	y	PROPN
ap-1807	251	10	−	−	PROPN
ap-1807	251	11	z	z	NOUN
ap-1807	251	12	)	)	PUNCT
ap-1807	251	13	)	)	PUNCT
ap-1807	252	1	+	+	CCONJ
ap-1807	253	1	z	z	X
ap-1807	253	2	,	,	PUNCT
ap-1807	253	3	where	where	SCONJ
ap-1807	253	4	we	we	PRON
ap-1807	253	5	used	use	VERB
ap-1807	253	6	the	the	DET
ap-1807	253	7	associativity	associativity	NOUN
ap-1807	253	8	of	of	ADP
ap-1807	253	9	the	the	DET
ap-1807	253	10	generalized	generalized	ADJ
ap-1807	253	11	effect	effect	NOUN
ap-1807	253	12	algebra	algebra	PROPN
ap-1807	253	13	e.	e.	PROPN
ap-1807	253	14	by	by	ADP
ap-1807	253	15	the	the	DET
ap-1807	253	16	cancellation	cancellation	NOUN
ap-1807	253	17	law	law	NOUN
ap-1807	253	18	we	we	PRON
ap-1807	253	19	have	have	VERB
ap-1807	253	20	(	(	PUNCT
ap-1807	253	21	x⊕y)⊕z−	x⊕y)⊕z−	X
ap-1807	253	22	=	=	PUNCT
ap-1807	253	23	(	(	PUNCT
ap-1807	253	24	x+y)−z	x+y)−z	ADJ
ap-1807	253	25	=	=	SYM
ap-1807	253	26	x+	x+	X
ap-1807	253	27	(	(	PUNCT
ap-1807	253	28	y−z	y−z	PROPN
ap-1807	253	29	)	)	PUNCT
ap-1807	254	1	=	=	SYM
ap-1807	254	2	x⊕	x⊕	PROPN
ap-1807	254	3	(	(	PUNCT
ap-1807	254	4	y⊕z−	y⊕z−	NUM
ap-1807	254	5	)	)	PUNCT
ap-1807	254	6	.	.	PUNCT
ap-1807	255	1	(	(	PUNCT
ap-1807	255	2	β2	β2	NOUN
ap-1807	255	3	)	)	PUNCT
ap-1807	255	4	y	y	PROPN
ap-1807	255	5	≤	≤	PROPN
ap-1807	255	6	z	z	NOUN
ap-1807	255	7	and	and	CCONJ
ap-1807	255	8	z	z	NOUN
ap-1807	255	9	≤	≤	NOUN
ap-1807	255	10	(	(	PUNCT
ap-1807	255	11	x	x	X
ap-1807	255	12	+	+	NUM
ap-1807	255	13	y	y	NOUN
ap-1807	255	14	)	)	PUNCT
ap-1807	255	15	(	(	PUNCT
ap-1807	255	16	that	that	PRON
ap-1807	255	17	is	be	AUX
ap-1807	255	18	z	z	NOUN
ap-1807	255	19	−	−	PROPN
ap-1807	255	20	y	y	PROPN
ap-1807	255	21	and	and	CCONJ
ap-1807	255	22	(	(	PUNCT
ap-1807	255	23	x	x	PROPN
ap-1807	255	24	+	+	NUM
ap-1807	255	25	y	y	NOUN
ap-1807	255	26	)	)	PUNCT
ap-1807	255	27	−	−	PROPN
ap-1807	255	28	z	z	NOUN
ap-1807	255	29	exist	exist	VERB
ap-1807	255	30	)	)	PUNCT
ap-1807	255	31	.	.	PUNCT
ap-1807	256	1	we	we	PRON
ap-1807	256	2	have	have	VERB
ap-1807	256	3	z	z	NOUN
ap-1807	256	4	=	=	SYM
ap-1807	256	5	(	(	PUNCT
ap-1807	256	6	z	z	NOUN
ap-1807	256	7	−	−	PROPN
ap-1807	256	8	y	y	PROPN
ap-1807	256	9	)	)	PUNCT
ap-1807	257	1	+	+	CCONJ
ap-1807	257	2	y	y	PROPN
ap-1807	257	3	and	and	CCONJ
ap-1807	257	4	also	also	ADV
ap-1807	257	5	(	(	PUNCT
ap-1807	257	6	(	(	PUNCT
ap-1807	257	7	x	x	SYM
ap-1807	257	8	+	+	NUM
ap-1807	257	9	y	y	NOUN
ap-1807	257	10	)	)	PUNCT
ap-1807	258	1	−	−	PROPN
ap-1807	259	1	z	z	X
ap-1807	259	2	)	)	PUNCT
ap-1807	260	1	+	+	PUNCT
ap-1807	260	2	z	z	NOUN
ap-1807	260	3	=	=	SYM
ap-1807	260	4	x	x	NOUN
ap-1807	261	1	+	+	CCONJ
ap-1807	261	2	y.	y.	NOUN
ap-1807	261	3	putting	put	VERB
ap-1807	261	4	together	together	ADV
ap-1807	261	5	(	(	PUNCT
ap-1807	261	6	(	(	PUNCT
ap-1807	261	7	z	z	NOUN
ap-1807	261	8	−	−	PROPN
ap-1807	261	9	y	y	PROPN
ap-1807	261	10	)	)	PUNCT
ap-1807	261	11	+	+	NOUN
ap-1807	261	12	y	y	X
ap-1807	261	13	)	)	PUNCT
ap-1807	262	1	+	+	CCONJ
ap-1807	262	2	(	(	PUNCT
ap-1807	262	3	(	(	PUNCT
ap-1807	262	4	x	x	SYM
ap-1807	262	5	+	+	NUM
ap-1807	262	6	y	y	NOUN
ap-1807	262	7	)	)	PUNCT
ap-1807	262	8	−	−	PROPN
ap-1807	263	1	z	z	X
ap-1807	263	2	)	)	PUNCT
ap-1807	263	3	=	=	PUNCT
ap-1807	264	1	x	x	PUNCT
ap-1807	265	1	+	+	NUM
ap-1807	265	2	y	y	PROPN
ap-1807	265	3	from	from	ADP
ap-1807	265	4	which	which	PRON
ap-1807	265	5	(	(	PUNCT
ap-1807	265	6	z−y)+((x+y)−z	z−y)+((x+y)−z	PROPN
ap-1807	265	7	)	)	PUNCT
ap-1807	266	1	=	=	PUNCT
ap-1807	267	1	x	x	X
ap-1807	267	2	hence	hence	ADV
ap-1807	267	3	(	(	PUNCT
ap-1807	267	4	x+y)−z	x+y)−z	PROPN
ap-1807	267	5	=	=	SYM
ap-1807	267	6	x−(z−y	x−(z−y	PROPN
ap-1807	267	7	)	)	PUNCT
ap-1807	267	8	.	.	PUNCT
ap-1807	268	1	so	so	ADV
ap-1807	268	2	we	we	PRON
ap-1807	268	3	have	have	VERB
ap-1807	268	4	(	(	PUNCT
ap-1807	268	5	x⊕	x⊕	PROPN
ap-1807	268	6	y)⊕	y)⊕	NOUN
ap-1807	268	7	z−	z−	NOUN
ap-1807	269	1	=	=	SYM
ap-1807	270	1	(	(	PUNCT
ap-1807	270	2	x+	x+	X
ap-1807	270	3	y)−	y)−	PROPN
ap-1807	270	4	z	z	NOUN
ap-1807	270	5	=	=	SYM
ap-1807	270	6	x−	x−	PROPN
ap-1807	270	7	(	(	PUNCT
ap-1807	270	8	z−	z−	PROPN
ap-1807	270	9	y	y	NOUN
ap-1807	270	10	)	)	PUNCT
ap-1807	270	11	=	=	SYM
ap-1807	270	12	x⊕	x⊕	PROPN
ap-1807	270	13	(	(	PUNCT
ap-1807	270	14	z⊕	z⊕	PROPN
ap-1807	270	15	y−)−	y−)−	NOUN
ap-1807	270	16	=	=	SYM
ap-1807	270	17	x⊕	x⊕	PROPN
ap-1807	270	18	(	(	PUNCT
ap-1807	270	19	z−⊕	z−⊕	NOUN
ap-1807	270	20	y	y	NOUN
ap-1807	270	21	)	)	PUNCT
ap-1807	270	22	.	.	PUNCT
ap-1807	271	1	the	the	DET
ap-1807	271	2	last	last	ADJ
ap-1807	271	3	equation	equation	NOUN
ap-1807	271	4	holds	hold	VERB
ap-1807	271	5	by	by	ADP
ap-1807	271	6	β1	β1	PROPN
ap-1807	271	7	.	.	PUNCT
ap-1807	272	1	(	(	PUNCT
ap-1807	272	2	γ2	γ2	ADJ
ap-1807	272	3	)	)	PUNCT
ap-1807	272	4	y	y	PROPN
ap-1807	272	5	≤	≤	PROPN
ap-1807	272	6	z	z	PROPN
ap-1807	272	7	and	and	CCONJ
ap-1807	272	8	(	(	PUNCT
ap-1807	272	9	x+y	x+y	NUM
ap-1807	272	10	)	)	PUNCT
ap-1807	272	11	≤	≤	NUM
ap-1807	273	1	z	z	NOUN
ap-1807	273	2	(	(	PUNCT
ap-1807	273	3	hence	hence	ADV
ap-1807	273	4	z−y	z−y	VERB
ap-1807	273	5	and	and	CCONJ
ap-1807	273	6	z−(x+y	z−(x+y	NOUN
ap-1807	273	7	)	)	PUNCT
ap-1807	273	8	exist	exist	VERB
ap-1807	273	9	)	)	PUNCT
ap-1807	273	10	.	.	PUNCT
ap-1807	274	1	from	from	ADP
ap-1807	274	2	(	(	PUNCT
ap-1807	274	3	z−y)+y	z−y)+y	NOUN
ap-1807	274	4	=	=	SYM
ap-1807	274	5	(	(	PUNCT
ap-1807	274	6	z−(x+y))+(x+y	z−(x+y))+(x+y	NUM
ap-1807	274	7	)	)	PUNCT
ap-1807	274	8	we	we	PRON
ap-1807	274	9	have	have	AUX
ap-1807	274	10	z−y	z−y	VERB
ap-1807	274	11	=	=	PUNCT
ap-1807	274	12	(	(	PUNCT
ap-1807	274	13	z−	z−	X
ap-1807	274	14	(	(	PUNCT
ap-1807	274	15	x+y))+x	x+y))+x	PROPN
ap-1807	274	16	hence	hence	ADV
ap-1807	274	17	(	(	PUNCT
ap-1807	274	18	z−y)−x	z−y)−x	PROPN
ap-1807	274	19	=	=	SYM
ap-1807	274	20	z−	z−	X
ap-1807	274	21	(	(	PUNCT
ap-1807	274	22	x+y	x+y	NUM
ap-1807	274	23	)	)	PUNCT
ap-1807	274	24	.	.	PUNCT
ap-1807	275	1	therefore	therefore	ADV
ap-1807	275	2	(	(	PUNCT
ap-1807	275	3	x⊕	x⊕	PROPN
ap-1807	275	4	y)⊕	y)⊕	NOUN
ap-1807	275	5	z−	z−	NOUN
ap-1807	275	6	=	=	PUNCT
ap-1807	275	7	(	(	PUNCT
ap-1807	275	8	z	z	NOUN
ap-1807	275	9	−	−	PROPN
ap-1807	275	10	(	(	PUNCT
ap-1807	275	11	x+	x+	ADJ
ap-1807	275	12	y))−	y))−	NOUN
ap-1807	275	13	=	=	SYM
ap-1807	275	14	(	(	PUNCT
ap-1807	275	15	(	(	PUNCT
ap-1807	275	16	z	z	NOUN
ap-1807	275	17	−	−	PROPN
ap-1807	275	18	y)−	y)−	PROPN
ap-1807	275	19	x)−	x)−	PROPN
ap-1807	275	20	=	=	SYM
ap-1807	275	21	(	(	PUNCT
ap-1807	275	22	x⊕	x⊕	PROPN
ap-1807	275	23	(	(	PUNCT
ap-1807	275	24	z	z	NOUN
ap-1807	275	25	−	−	PROPN
ap-1807	275	26	y)−	y)−	PROPN
ap-1807	275	27	)	)	PUNCT
ap-1807	275	28	=	=	SYM
ap-1807	275	29	x⊕	x⊕	PROPN
ap-1807	275	30	(	(	PUNCT
ap-1807	275	31	y	y	PROPN
ap-1807	275	32	⊕	⊕	PROPN
ap-1807	275	33	z−	z−	PROPN
ap-1807	275	34	)	)	PUNCT
ap-1807	275	35	.	.	PUNCT
ap-1807	276	1	case	case	NOUN
ap-1807	276	2	iii	iii	X
ap-1807	276	3	.	.	PUNCT
ap-1807	277	1	let	let	VERB
ap-1807	277	2	(	(	PUNCT
ap-1807	277	3	x−	x−	PROPN
ap-1807	277	4	⊕	⊕	PROPN
ap-1807	277	5	y)⊕	y)⊕	PROPN
ap-1807	277	6	z	z	PROPN
ap-1807	278	1	and	and	CCONJ
ap-1807	278	2	y	y	PROPN
ap-1807	278	3	⊕	⊕	PROPN
ap-1807	278	4	z	z	AUX
ap-1807	278	5	be	be	AUX
ap-1807	278	6	defined	define	VERB
ap-1807	278	7	.	.	PUNCT
ap-1807	279	1	let	let	VERB
ap-1807	279	2	(	(	PUNCT
ap-1807	279	3	α3	α3	ADJ
ap-1807	279	4	)	)	PUNCT
ap-1807	279	5	x	x	PUNCT
ap-1807	279	6	≤	≤	NUM
ap-1807	279	7	y	y	NOUN
ap-1807	279	8	(	(	PUNCT
ap-1807	279	9	i.e.	i.e.	X
ap-1807	279	10	,	,	PUNCT
ap-1807	279	11	y	y	PROPN
ap-1807	279	12	−	−	PROPN
ap-1807	279	13	x	x	VERB
ap-1807	279	14	is	be	AUX
ap-1807	279	15	defined	define	VERB
ap-1807	279	16	and	and	CCONJ
ap-1807	279	17	also	also	ADV
ap-1807	280	1	x	x	ADP
ap-1807	280	2	≤	≤	NUM
ap-1807	280	3	y	y	NOUN
ap-1807	280	4	≤	≤	NUM
ap-1807	280	5	y+z	y+z	PROPN
ap-1807	280	6	)	)	PUNCT
ap-1807	280	7	.	.	PUNCT
ap-1807	281	1	then	then	ADV
ap-1807	281	2	y	y	PROPN
ap-1807	281	3	=	=	PUNCT
ap-1807	281	4	(	(	PUNCT
ap-1807	281	5	y−x)+x	y−x)+x	ADV
ap-1807	281	6	and	and	CCONJ
ap-1807	281	7	y+z	y+z	NUM
ap-1807	281	8	=	=	PUNCT
ap-1807	282	1	(	(	PUNCT
ap-1807	282	2	(	(	PUNCT
ap-1807	282	3	y−x)+x)+z	y−x)+x)+z	NOUN
ap-1807	282	4	hence	hence	ADV
ap-1807	282	5	(	(	PUNCT
ap-1807	282	6	y+z)−x	y+z)−x	PROPN
ap-1807	282	7	=	=	SYM
ap-1807	282	8	(	(	PUNCT
ap-1807	282	9	y−x)+z	y−x)+z	NOUN
ap-1807	282	10	.	.	PUNCT
ap-1807	283	1	therefore	therefore	ADV
ap-1807	283	2	(	(	PUNCT
ap-1807	283	3	x−⊕y)⊕z	x−⊕y)⊕z	PROPN
ap-1807	283	4	=	=	SYM
ap-1807	283	5	(	(	PUNCT
ap-1807	283	6	y	y	PROPN
ap-1807	283	7	−	−	PROPN
ap-1807	283	8	x	x	X
ap-1807	283	9	)	)	PUNCT
ap-1807	283	10	+	+	NUM
ap-1807	283	11	z	z	NOUN
ap-1807	283	12	=	=	SYM
ap-1807	283	13	(	(	PUNCT
ap-1807	283	14	y	y	PROPN
ap-1807	283	15	+	+	PUNCT
ap-1807	283	16	z)−	z)−	PROPN
ap-1807	283	17	x	x	X
ap-1807	283	18	=	=	SYM
ap-1807	283	19	x−	x−	PROPN
ap-1807	283	20	⊕	⊕	PROPN
ap-1807	283	21	(	(	PUNCT
ap-1807	283	22	y	y	PROPN
ap-1807	283	23	⊕	⊕	PROPN
ap-1807	283	24	z	z	PROPN
ap-1807	283	25	)	)	PUNCT
ap-1807	283	26	.	.	PUNCT
ap-1807	284	1	(	(	PUNCT
ap-1807	284	2	β3	β3	ADJ
ap-1807	284	3	)	)	PUNCT
ap-1807	284	4	y	y	PROPN
ap-1807	284	5	≤	≤	NUM
ap-1807	284	6	x	x	PUNCT
ap-1807	284	7	and	and	CCONJ
ap-1807	284	8	(	(	PUNCT
ap-1807	284	9	x	x	X
ap-1807	284	10	−	−	PROPN
ap-1807	284	11	y	y	PROPN
ap-1807	284	12	)	)	PUNCT
ap-1807	284	13	≤	≤	NUM
ap-1807	284	14	z	z	NOUN
ap-1807	285	1	(	(	PUNCT
ap-1807	285	2	that	that	PRON
ap-1807	285	3	is	is	ADV
ap-1807	285	4	x	x	X
ap-1807	285	5	−	−	PROPN
ap-1807	285	6	y	y	PROPN
ap-1807	285	7	and	and	CCONJ
ap-1807	285	8	z−	z−	PROPN
ap-1807	285	9	(	(	PUNCT
ap-1807	285	10	x−y	x−y	PROPN
ap-1807	285	11	)	)	PUNCT
ap-1807	285	12	are	be	AUX
ap-1807	285	13	defined	define	VERB
ap-1807	285	14	)	)	PUNCT
ap-1807	285	15	.	.	PUNCT
ap-1807	286	1	since	since	SCONJ
ap-1807	286	2	z+y	z+y	PROPN
ap-1807	286	3	is	be	AUX
ap-1807	286	4	defined	define	VERB
ap-1807	286	5	,	,	PUNCT
ap-1807	286	6	we	we	PRON
ap-1807	286	7	have	have	VERB
ap-1807	286	8	from	from	ADP
ap-1807	286	9	z	z	NOUN
ap-1807	286	10	=	=	SYM
ap-1807	286	11	(	(	PUNCT
ap-1807	286	12	z	z	NOUN
ap-1807	286	13	−	−	PROPN
ap-1807	286	14	(	(	PUNCT
ap-1807	286	15	x−	x−	PROPN
ap-1807	286	16	y	y	PROPN
ap-1807	286	17	)	)	PUNCT
ap-1807	286	18	)	)	PUNCT
ap-1807	287	1	+	+	CCONJ
ap-1807	287	2	(	(	PUNCT
ap-1807	287	3	x−	x−	PROPN
ap-1807	287	4	y	y	PROPN
ap-1807	287	5	)	)	PUNCT
ap-1807	287	6	with	with	ADP
ap-1807	287	7	x	x	X
ap-1807	287	8	=	=	SYM
ap-1807	287	9	(	(	PUNCT
ap-1807	287	10	x−	x−	PROPN
ap-1807	287	11	y	y	PROPN
ap-1807	287	12	)	)	PUNCT
ap-1807	288	1	+	+	CCONJ
ap-1807	288	2	y	y	PRON
ap-1807	288	3	the	the	DET
ap-1807	288	4	equation	equation	NOUN
ap-1807	288	5	z+y	z+y	X
ap-1807	288	6	=	=	SYM
ap-1807	288	7	(	(	PUNCT
ap-1807	288	8	z−	z−	X
ap-1807	288	9	(	(	PUNCT
ap-1807	288	10	x−y))+x	x−y))+x	ADJ
ap-1807	288	11	and	and	CCONJ
ap-1807	288	12	(	(	PUNCT
ap-1807	288	13	z+y)−x	z+y)−x	NOUN
ap-1807	288	14	=	=	SYM
ap-1807	288	15	z	z	NOUN
ap-1807	289	1	−	−	PROPN
ap-1807	289	2	(	(	PUNCT
ap-1807	289	3	x	x	SYM
ap-1807	289	4	−	−	PROPN
ap-1807	289	5	y	y	PROPN
ap-1807	289	6	)	)	PUNCT
ap-1807	289	7	.	.	PUNCT
ap-1807	290	1	hence	hence	ADV
ap-1807	290	2	(	(	PUNCT
ap-1807	290	3	x−	x−	PROPN
ap-1807	290	4	⊕	⊕	PROPN
ap-1807	290	5	y	y	PROPN
ap-1807	290	6	)	)	PUNCT
ap-1807	290	7	⊕	⊕	PROPN
ap-1807	290	8	z	z	NOUN
ap-1807	291	1	=	=	PUNCT
ap-1807	291	2	(	(	PUNCT
ap-1807	291	3	x	x	X
ap-1807	291	4	−	−	PROPN
ap-1807	291	5	y)−	y)−	PROPN
ap-1807	291	6	⊕	⊕	PROPN
ap-1807	291	7	z	z	NOUN
ap-1807	291	8	=	=	PUNCT
ap-1807	292	1	z	z	X
ap-1807	292	2	−	−	PROPN
ap-1807	293	1	(	(	PUNCT
ap-1807	293	2	x−	x−	PROPN
ap-1807	293	3	y	y	PROPN
ap-1807	293	4	)	)	PUNCT
ap-1807	294	1	=	=	PRON
ap-1807	295	1	(	(	PUNCT
ap-1807	295	2	z	z	NOUN
ap-1807	295	3	+	+	NOUN
ap-1807	296	1	y)−	y)−	PROPN
ap-1807	296	2	x	x	SYM
ap-1807	296	3	=	=	SYM
ap-1807	296	4	x−	x−	PROPN
ap-1807	296	5	⊕	⊕	PROPN
ap-1807	296	6	(	(	PUNCT
ap-1807	296	7	y	y	PROPN
ap-1807	296	8	⊕	⊕	PROPN
ap-1807	296	9	z	z	PROPN
ap-1807	296	10	)	)	PUNCT
ap-1807	296	11	.	.	PUNCT
ap-1807	297	1	(	(	PUNCT
ap-1807	297	2	γ3	γ3	NOUN
ap-1807	297	3	)	)	PUNCT
ap-1807	297	4	y	y	PROPN
ap-1807	297	5	≤	≤	NUM
ap-1807	297	6	x	x	PUNCT
ap-1807	297	7	and	and	CCONJ
ap-1807	297	8	z	z	NOUN
ap-1807	297	9	≤	≤	NOUN
ap-1807	297	10	(	(	PUNCT
ap-1807	297	11	x	x	SYM
ap-1807	297	12	−	−	PROPN
ap-1807	297	13	y	y	PROPN
ap-1807	297	14	)	)	PUNCT
ap-1807	297	15	(	(	PUNCT
ap-1807	297	16	hence	hence	ADV
ap-1807	297	17	x	x	PUNCT
ap-1807	297	18	−	−	PROPN
ap-1807	297	19	y	y	PROPN
ap-1807	297	20	and	and	CCONJ
ap-1807	297	21	(	(	PUNCT
ap-1807	297	22	x	x	X
ap-1807	297	23	−	−	PROPN
ap-1807	297	24	y	y	PROPN
ap-1807	297	25	)	)	PUNCT
ap-1807	297	26	−	−	PROPN
ap-1807	298	1	z	z	NOUN
ap-1807	298	2	are	be	AUX
ap-1807	298	3	defined	define	VERB
ap-1807	298	4	)	)	PUNCT
ap-1807	298	5	.	.	PUNCT
ap-1807	299	1	we	we	PRON
ap-1807	299	2	have	have	VERB
ap-1807	299	3	(	(	PUNCT
ap-1807	299	4	x	x	SYM
ap-1807	299	5	−	−	PROPN
ap-1807	299	6	y	y	PROPN
ap-1807	299	7	)	)	PUNCT
ap-1807	299	8	−	−	PROPN
ap-1807	300	1	z	z	NOUN
ap-1807	300	2	=	=	SYM
ap-1807	300	3	(	(	PUNCT
ap-1807	300	4	x−	x−	PROPN
ap-1807	300	5	y)−	y)−	PROPN
ap-1807	300	6	z	z	VERB
ap-1807	300	7	from	from	ADP
ap-1807	300	8	which	which	PRON
ap-1807	300	9	x	x	SYM
ap-1807	300	10	=	=	SYM
ap-1807	300	11	(	(	PUNCT
ap-1807	300	12	(	(	PUNCT
ap-1807	300	13	x−	x−	PROPN
ap-1807	300	14	y)−	y)−	PROPN
ap-1807	300	15	z	z	PROPN
ap-1807	300	16	)	)	PUNCT
ap-1807	301	1	+	+	CCONJ
ap-1807	301	2	y	y	X
ap-1807	301	3	)	)	PUNCT
ap-1807	302	1	+	+	CCONJ
ap-1807	302	2	z	z	NOUN
ap-1807	302	3	hence	hence	ADV
ap-1807	302	4	x−	x−	PROPN
ap-1807	303	1	(	(	PUNCT
ap-1807	303	2	y	y	PROPN
ap-1807	303	3	+	+	PROPN
ap-1807	303	4	z	z	NOUN
ap-1807	303	5	)	)	PUNCT
ap-1807	303	6	=	=	SYM
ap-1807	303	7	(	(	PUNCT
ap-1807	303	8	x−	x−	PROPN
ap-1807	303	9	y)−	y)−	PROPN
ap-1807	303	10	z.	z.	PROPN
ap-1807	304	1	and	and	CCONJ
ap-1807	304	2	then	then	ADV
ap-1807	304	3	(	(	PUNCT
ap-1807	304	4	x−	x−	PROPN
ap-1807	304	5	⊕	⊕	PROPN
ap-1807	304	6	y)⊕	y)⊕	NOUN
ap-1807	304	7	z	z	NOUN
ap-1807	305	1	=	=	SYM
ap-1807	306	1	(	(	PUNCT
ap-1807	306	2	x	x	X
ap-1807	306	3	−	−	PROPN
ap-1807	306	4	y)−	y)−	PROPN
ap-1807	306	5	⊕	⊕	PROPN
ap-1807	306	6	z	z	NOUN
ap-1807	306	7	=	=	PUNCT
ap-1807	306	8	(	(	PUNCT
ap-1807	306	9	(	(	PUNCT
ap-1807	306	10	x	x	SYM
ap-1807	306	11	−	−	PROPN
ap-1807	306	12	y	y	PROPN
ap-1807	306	13	)	)	PUNCT
ap-1807	306	14	−	−	PROPN
ap-1807	307	1	z)−	z)−	NOUN
ap-1807	307	2	=	=	PUNCT
ap-1807	307	3	(	(	PUNCT
ap-1807	307	4	x	x	SYM
ap-1807	307	5	−	−	PROPN
ap-1807	307	6	(	(	PUNCT
ap-1807	307	7	y	y	PROPN
ap-1807	307	8	+	+	CCONJ
ap-1807	307	9	z))−	z))−	NUM
ap-1807	307	10	=	=	SYM
ap-1807	307	11	(	(	PUNCT
ap-1807	307	12	x	x	PROPN
ap-1807	307	13	⊕	⊕	PROPN
ap-1807	307	14	(	(	PUNCT
ap-1807	307	15	y	y	PROPN
ap-1807	307	16	⊕	⊕	PROPN
ap-1807	307	17	z)−)−	z)−)−	PROPN
ap-1807	308	1	=	=	SYM
ap-1807	308	2	x−	x−	PROPN
ap-1807	308	3	⊕	⊕	PROPN
ap-1807	308	4	(	(	PUNCT
ap-1807	308	5	y	y	PROPN
ap-1807	308	6	⊕	⊕	PROPN
ap-1807	308	7	z	z	PROPN
ap-1807	308	8	)	)	PUNCT
ap-1807	308	9	(	(	PUNCT
ap-1807	308	10	the	the	DET
ap-1807	308	11	last	last	ADJ
ap-1807	308	12	equation	equation	NOUN
ap-1807	308	13	follows	follow	VERB
ap-1807	308	14	from	from	ADP
ap-1807	308	15	β1	β1	PROPN
ap-1807	308	16	)	)	PUNCT
ap-1807	308	17	.	.	PUNCT
ap-1807	309	1	case	case	NOUN
ap-1807	309	2	iv	iv	X
ap-1807	309	3	.	.	PUNCT
ap-1807	310	1	let	let	VERB
ap-1807	310	2	(	(	PUNCT
ap-1807	310	3	x⊕	x⊕	X
ap-1807	310	4	y−)⊕	y−)⊕	PROPN
ap-1807	310	5	z	z	NOUN
ap-1807	310	6	and	and	CCONJ
ap-1807	310	7	y−	y−	PROPN
ap-1807	310	8	⊕	⊕	PROPN
ap-1807	310	9	z	z	PROPN
ap-1807	310	10	be	be	AUX
ap-1807	310	11	defined	define	VERB
ap-1807	310	12	.	.	PUNCT
ap-1807	311	1	let	let	VERB
ap-1807	311	2	(	(	PUNCT
ap-1807	311	3	α4	α4	VERB
ap-1807	311	4	)	)	PUNCT
ap-1807	311	5	y	y	PROPN
ap-1807	311	6	≤	≤	PROPN
ap-1807	311	7	x	x	PUNCT
ap-1807	311	8	and	and	CCONJ
ap-1807	311	9	y	y	PROPN
ap-1807	311	10	≤	≤	PROPN
ap-1807	311	11	z	z	NOUN
ap-1807	311	12	(	(	PUNCT
ap-1807	311	13	i.e.	i.e.	X
ap-1807	311	14	,	,	PUNCT
ap-1807	311	15	x	x	SYM
ap-1807	311	16	−	−	PROPN
ap-1807	311	17	y	y	PROPN
ap-1807	311	18	and	and	CCONJ
ap-1807	311	19	z	z	NOUN
ap-1807	311	20	−	−	PROPN
ap-1807	312	1	y	y	PROPN
ap-1807	312	2	are	be	AUX
ap-1807	312	3	defined	define	VERB
ap-1807	312	4	)	)	PUNCT
ap-1807	312	5	.	.	PUNCT
ap-1807	313	1	using	use	VERB
ap-1807	313	2	y	y	PROPN
ap-1807	313	3	=	=	SYM
ap-1807	313	4	x−(x−y	x−(x−y	PROPN
ap-1807	313	5	)	)	PUNCT
ap-1807	313	6	and	and	CCONJ
ap-1807	313	7	z	z	NOUN
ap-1807	313	8	=	=	SYM
ap-1807	313	9	(	(	PUNCT
ap-1807	313	10	z−y)+y	z−y)+y	NOUN
ap-1807	313	11	we	we	PRON
ap-1807	313	12	get	get	VERB
ap-1807	313	13	(	(	PUNCT
ap-1807	313	14	x−y)+z	x−y)+z	NOUN
ap-1807	313	15	=	=	SYM
ap-1807	313	16	(	(	PUNCT
ap-1807	313	17	x−y)+((z−y)+y	x−y)+((z−y)+y	NUM
ap-1807	313	18	)	)	PUNCT
ap-1807	314	1	=	=	PRON
ap-1807	314	2	(	(	PUNCT
ap-1807	314	3	z−y)+x	z−y)+x	NOUN
ap-1807	314	4	.	.	PUNCT
ap-1807	315	1	hence	hence	ADV
ap-1807	315	2	(	(	PUNCT
ap-1807	315	3	x⊕y−)⊕z	x⊕y−)⊕z	NOUN
ap-1807	315	4	=	=	SYM
ap-1807	315	5	(	(	PUNCT
ap-1807	315	6	x−y	x−y	PROPN
ap-1807	315	7	)	)	PUNCT
ap-1807	316	1	+	+	NOUN
ap-1807	316	2	z	z	NOUN
ap-1807	316	3	=	=	SYM
ap-1807	316	4	(	(	PUNCT
ap-1807	316	5	z−y	z−y	VERB
ap-1807	316	6	)	)	PUNCT
ap-1807	316	7	+	+	NOUN
ap-1807	316	8	x	x	SYM
ap-1807	316	9	=	=	SYM
ap-1807	316	10	x⊕	x⊕	PROPN
ap-1807	316	11	(	(	PUNCT
ap-1807	316	12	y−⊕z	y−⊕z	NUM
ap-1807	316	13	)	)	PUNCT
ap-1807	316	14	.	.	PUNCT
ap-1807	317	1	(	(	PUNCT
ap-1807	317	2	β4	β4	PROPN
ap-1807	317	3	)	)	PUNCT
ap-1807	317	4	y	y	PROPN
ap-1807	317	5	≤	≤	PROPN
ap-1807	317	6	x	x	PUNCT
ap-1807	317	7	and	and	CCONJ
ap-1807	317	8	z	z	NOUN
ap-1807	317	9	≤	≤	NOUN
ap-1807	318	1	y	y	PROPN
ap-1807	318	2	(	(	PUNCT
ap-1807	318	3	hence	hence	ADV
ap-1807	318	4	x	x	X
ap-1807	318	5	−	−	PROPN
ap-1807	318	6	y	y	PROPN
ap-1807	318	7	and	and	CCONJ
ap-1807	318	8	y	y	PROPN
ap-1807	318	9	−	−	PROPN
ap-1807	318	10	z	z	NOUN
ap-1807	318	11	are	be	AUX
ap-1807	318	12	defined	define	VERB
ap-1807	318	13	)	)	PUNCT
ap-1807	318	14	.	.	PUNCT
ap-1807	319	1	similarly	similarly	ADV
ap-1807	319	2	y	y	PROPN
ap-1807	319	3	=	=	PUNCT
ap-1807	319	4	(	(	PUNCT
ap-1807	319	5	y−	y−	PROPN
ap-1807	319	6	z	z	PROPN
ap-1807	319	7	)	)	PUNCT
ap-1807	320	1	+	+	CCONJ
ap-1807	320	2	z	z	NOUN
ap-1807	320	3	and	and	CCONJ
ap-1807	320	4	x	x	SYM
ap-1807	320	5	=	=	SYM
ap-1807	320	6	(	(	PUNCT
ap-1807	320	7	x−	x−	PROPN
ap-1807	320	8	y	y	PROPN
ap-1807	320	9	)	)	PUNCT
ap-1807	321	1	+	+	CCONJ
ap-1807	321	2	y	y	PROPN
ap-1807	322	1	and	and	CCONJ
ap-1807	322	2	then	then	ADV
ap-1807	322	3	x	x	X
ap-1807	322	4	=	=	PUNCT
ap-1807	322	5	(	(	PUNCT
ap-1807	322	6	x	x	SYM
ap-1807	322	7	−	−	PROPN
ap-1807	322	8	y	y	PROPN
ap-1807	322	9	)	)	PUNCT
ap-1807	323	1	+	+	CCONJ
ap-1807	323	2	(	(	PUNCT
ap-1807	323	3	(	(	PUNCT
ap-1807	323	4	y	y	PROPN
ap-1807	323	5	−	−	PROPN
ap-1807	323	6	z	z	NOUN
ap-1807	323	7	)	)	PUNCT
ap-1807	324	1	+	+	CCONJ
ap-1807	325	1	z	z	X
ap-1807	325	2	)	)	PUNCT
ap-1807	325	3	from	from	ADP
ap-1807	325	4	which	which	PRON
ap-1807	325	5	x	x	PUNCT
ap-1807	325	6	−	−	PROPN
ap-1807	325	7	(	(	PUNCT
ap-1807	325	8	y	y	PROPN
ap-1807	325	9	−	−	PROPN
ap-1807	325	10	z	z	NOUN
ap-1807	325	11	)	)	PUNCT
ap-1807	325	12	=	=	SYM
ap-1807	325	13	(	(	PUNCT
ap-1807	325	14	x	x	X
ap-1807	325	15	−	−	PROPN
ap-1807	325	16	y	y	PROPN
ap-1807	325	17	)	)	PUNCT
ap-1807	326	1	+	+	CCONJ
ap-1807	326	2	z.	z.	PROPN
ap-1807	326	3	hence	hence	ADV
ap-1807	326	4	(	(	PUNCT
ap-1807	326	5	x	x	PROPN
ap-1807	326	6	⊕	⊕	PROPN
ap-1807	326	7	y−	y−	PROPN
ap-1807	326	8	)	)	PUNCT
ap-1807	326	9	⊕	⊕	PROPN
ap-1807	326	10	z	z	NOUN
ap-1807	327	1	=	=	PUNCT
ap-1807	327	2	(	(	PUNCT
ap-1807	327	3	x−y)+z	x−y)+z	X
ap-1807	327	4	=	=	SYM
ap-1807	327	5	x−(y−z	x−(y−z	PROPN
ap-1807	327	6	)	)	PUNCT
ap-1807	327	7	=	=	PUNCT
ap-1807	327	8	x−(y⊕z−	x−(y⊕z−	PUNCT
ap-1807	327	9	)	)	PUNCT
ap-1807	328	1	=	=	PUNCT
ap-1807	328	2	x⊕(y⊕z−)−	x⊕(y⊕z−)−	PROPN
ap-1807	329	1	=	=	SYM
ap-1807	329	2	x⊕	x⊕	PROPN
ap-1807	329	3	(	(	PUNCT
ap-1807	329	4	y−	y−	PROPN
ap-1807	329	5	⊕	⊕	PROPN
ap-1807	329	6	z	z	PROPN
ap-1807	329	7	)	)	PUNCT
ap-1807	329	8	(	(	PUNCT
ap-1807	329	9	the	the	DET
ap-1807	329	10	last	last	ADJ
ap-1807	329	11	equation	equation	NOUN
ap-1807	329	12	follows	follow	VERB
ap-1807	329	13	from	from	ADP
ap-1807	329	14	β1	β1	PROPN
ap-1807	329	15	)	)	PUNCT
ap-1807	329	16	.	.	PUNCT
ap-1807	330	1	(	(	PUNCT
ap-1807	330	2	γ4	γ4	NOUN
ap-1807	330	3	)	)	PUNCT
ap-1807	330	4	x	x	SYM
ap-1807	331	1	≤	≤	NOUN
ap-1807	331	2	y	y	PROPN
ap-1807	331	3	and	and	CCONJ
ap-1807	331	4	y	y	PROPN
ap-1807	331	5	≤	≤	PROPN
ap-1807	331	6	z	z	NOUN
ap-1807	331	7	which	which	PRON
ap-1807	331	8	implies	imply	VERB
ap-1807	331	9	(	(	PUNCT
ap-1807	331	10	y	y	PROPN
ap-1807	331	11	−	−	PROPN
ap-1807	331	12	x	x	SYM
ap-1807	331	13	)	)	PUNCT
ap-1807	331	14	≤	≤	NUM
ap-1807	331	15	z	z	NOUN
ap-1807	331	16	(	(	PUNCT
ap-1807	331	17	hence	hence	ADV
ap-1807	331	18	y−	y−	PROPN
ap-1807	331	19	x	x	SYM
ap-1807	331	20	,	,	PUNCT
ap-1807	331	21	z	z	NOUN
ap-1807	331	22	−	−	PROPN
ap-1807	331	23	(	(	PUNCT
ap-1807	331	24	y−	y−	X
ap-1807	331	25	x	x	SYM
ap-1807	331	26	)	)	PUNCT
ap-1807	331	27	and	and	CCONJ
ap-1807	331	28	z	z	NOUN
ap-1807	331	29	−	−	PROPN
ap-1807	332	1	y	y	PROPN
ap-1807	332	2	are	be	AUX
ap-1807	332	3	defined	define	VERB
ap-1807	332	4	)	)	PUNCT
ap-1807	332	5	.	.	PUNCT
ap-1807	333	1	then	then	ADV
ap-1807	333	2	z	z	X
ap-1807	333	3	=	=	SYM
ap-1807	333	4	(	(	PUNCT
ap-1807	333	5	z	z	NOUN
ap-1807	333	6	−	−	PROPN
ap-1807	333	7	y	y	PROPN
ap-1807	333	8	)	)	PUNCT
ap-1807	334	1	+	+	NOUN
ap-1807	334	2	y	y	NOUN
ap-1807	334	3	=	=	SYM
ap-1807	334	4	(	(	PUNCT
ap-1807	334	5	z	z	NOUN
ap-1807	334	6	−	−	PROPN
ap-1807	334	7	y	y	PROPN
ap-1807	334	8	)	)	PUNCT
ap-1807	335	1	+	+	CCONJ
ap-1807	335	2	(	(	PUNCT
ap-1807	335	3	(	(	PUNCT
ap-1807	335	4	y	y	PROPN
ap-1807	335	5	−	−	PROPN
ap-1807	335	6	x	x	NOUN
ap-1807	335	7	)	)	PUNCT
ap-1807	336	1	+	+	CCONJ
ap-1807	337	1	x	x	X
ap-1807	337	2	)	)	PUNCT
ap-1807	337	3	from	from	ADP
ap-1807	337	4	which	which	PRON
ap-1807	337	5	z	z	NOUN
ap-1807	337	6	−	−	PROPN
ap-1807	338	1	(	(	PUNCT
ap-1807	338	2	y	y	PROPN
ap-1807	338	3	−	−	PROPN
ap-1807	338	4	x	x	X
ap-1807	338	5	)	)	PUNCT
ap-1807	338	6	=	=	SYM
ap-1807	339	1	(	(	PUNCT
ap-1807	339	2	z	z	NOUN
ap-1807	339	3	−	−	PROPN
ap-1807	339	4	y	y	PROPN
ap-1807	339	5	)	)	PUNCT
ap-1807	340	1	+	+	CCONJ
ap-1807	340	2	x	x	PUNCT
ap-1807	340	3	and	and	CCONJ
ap-1807	340	4	hence	hence	ADV
ap-1807	340	5	(	(	PUNCT
ap-1807	340	6	x⊕	x⊕	X
ap-1807	340	7	y−)⊕	y−)⊕	VERB
ap-1807	340	8	z	z	NOUN
ap-1807	340	9	=	=	PUNCT
ap-1807	340	10	(	(	PUNCT
ap-1807	340	11	y−x)−⊕z	y−x)−⊕z	NOUN
ap-1807	340	12	=	=	SYM
ap-1807	340	13	z−	z−	X
ap-1807	340	14	(	(	PUNCT
ap-1807	340	15	y−x	y−x	X
ap-1807	340	16	)	)	PUNCT
ap-1807	340	17	=	=	SYM
ap-1807	340	18	(	(	PUNCT
ap-1807	340	19	z−y	z−y	VERB
ap-1807	340	20	)	)	PUNCT
ap-1807	341	1	+	+	NOUN
ap-1807	341	2	x	x	SYM
ap-1807	341	3	=	=	SYM
ap-1807	341	4	x⊕	x⊕	PROPN
ap-1807	341	5	(	(	PUNCT
ap-1807	341	6	y−⊕z	y−⊕z	NUM
ap-1807	341	7	)	)	PUNCT
ap-1807	341	8	.	.	PUNCT
ap-1807	342	1	(	(	PUNCT
ap-1807	342	2	δ4	δ4	NOUN
ap-1807	342	3	)	)	PUNCT
ap-1807	342	4	x	x	SYM
ap-1807	342	5	≤	≤	NOUN
ap-1807	342	6	y	y	PROPN
ap-1807	342	7	,	,	PUNCT
ap-1807	342	8	(	(	PUNCT
ap-1807	342	9	y	y	PROPN
ap-1807	342	10	−	−	PROPN
ap-1807	342	11	x	x	SYM
ap-1807	342	12	)	)	PUNCT
ap-1807	342	13	≤	≤	NUM
ap-1807	342	14	z	z	NOUN
ap-1807	342	15	and	and	CCONJ
ap-1807	342	16	z	z	NOUN
ap-1807	342	17	≤	≤	NOUN
ap-1807	342	18	y	y	PROPN
ap-1807	342	19	(	(	PUNCT
ap-1807	342	20	i.e.	i.e.	X
ap-1807	342	21	,	,	PUNCT
ap-1807	342	22	y	y	PROPN
ap-1807	342	23	−	−	PROPN
ap-1807	342	24	x	x	SYM
ap-1807	342	25	,	,	PUNCT
ap-1807	342	26	z−(y−x	z−(y−x	X
ap-1807	342	27	)	)	PUNCT
ap-1807	342	28	and	and	CCONJ
ap-1807	342	29	y−z	y−z	PROPN
ap-1807	342	30	are	be	AUX
ap-1807	342	31	defined	define	VERB
ap-1807	342	32	)	)	PUNCT
ap-1807	342	33	.	.	PUNCT
ap-1807	343	1	then	then	ADV
ap-1807	343	2	y	y	PROPN
ap-1807	343	3	=	=	PUNCT
ap-1807	343	4	(	(	PUNCT
ap-1807	343	5	y−z)+z	y−z)+z	NOUN
ap-1807	343	6	=	=	SYM
ap-1807	343	7	(	(	PUNCT
ap-1807	343	8	y−	y−	PROPN
ap-1807	343	9	z	z	PROPN
ap-1807	343	10	)	)	PUNCT
ap-1807	344	1	+	+	CCONJ
ap-1807	344	2	(	(	PUNCT
ap-1807	344	3	(	(	PUNCT
ap-1807	344	4	z	z	NOUN
ap-1807	344	5	−	−	PROPN
ap-1807	344	6	(	(	PUNCT
ap-1807	344	7	y−	y−	X
ap-1807	344	8	x	x	NOUN
ap-1807	344	9	)	)	PUNCT
ap-1807	344	10	)	)	PUNCT
ap-1807	345	1	+	+	CCONJ
ap-1807	345	2	(	(	PUNCT
ap-1807	345	3	y−	y−	X
ap-1807	345	4	x	x	NOUN
ap-1807	345	5	)	)	PUNCT
ap-1807	345	6	)	)	PUNCT
ap-1807	345	7	=	=	SYM
ap-1807	346	1	(	(	PUNCT
ap-1807	346	2	y−	y−	X
ap-1807	346	3	x	x	SYM
ap-1807	346	4	)	)	PUNCT
ap-1807	347	1	+	+	CCONJ
ap-1807	347	2	x	x	SYM
ap-1807	347	3	hence	hence	ADV
ap-1807	347	4	(	(	PUNCT
ap-1807	347	5	y−	y−	PROPN
ap-1807	347	6	z	z	PROPN
ap-1807	347	7	)	)	PUNCT
ap-1807	348	1	+	+	CCONJ
ap-1807	348	2	(	(	PUNCT
ap-1807	348	3	z	z	NOUN
ap-1807	348	4	−	−	PROPN
ap-1807	348	5	(	(	PUNCT
ap-1807	348	6	y−	y−	X
ap-1807	348	7	x	x	NOUN
ap-1807	348	8	)	)	PUNCT
ap-1807	348	9	)	)	PUNCT
ap-1807	349	1	=	=	PUNCT
ap-1807	349	2	x	x	X
ap-1807	349	3	from	from	ADP
ap-1807	349	4	which	which	PRON
ap-1807	349	5	(	(	PUNCT
ap-1807	349	6	x⊕	x⊕	X
ap-1807	349	7	y−)⊕	y−)⊕	VERB
ap-1807	349	8	z	z	NOUN
ap-1807	349	9	=	=	PUNCT
ap-1807	349	10	(	(	PUNCT
ap-1807	349	11	y−x)−⊕z	y−x)−⊕z	NUM
ap-1807	349	12	=	=	SYM
ap-1807	349	13	z−(y−x	z−(y−x	NOUN
ap-1807	349	14	)	)	PUNCT
ap-1807	349	15	=	=	SYM
ap-1807	349	16	x−(y−z	x−(y−z	PROPN
ap-1807	349	17	)	)	PUNCT
ap-1807	349	18	=	=	SYM
ap-1807	349	19	x−(y⊕z−	x−(y⊕z−	PUNCT
ap-1807	349	20	)	)	PUNCT
ap-1807	350	1	=	=	SYM
ap-1807	350	2	x⊕	x⊕	PROPN
ap-1807	350	3	(	(	PUNCT
ap-1807	350	4	y−	y−	PROPN
ap-1807	350	5	⊕	⊕	PROPN
ap-1807	350	6	z	z	PROPN
ap-1807	350	7	)	)	PUNCT
ap-1807	350	8	(	(	PUNCT
ap-1807	350	9	using	use	VERB
ap-1807	350	10	β1	β1	PROPN
ap-1807	350	11	)	)	PUNCT
ap-1807	350	12	.	.	PUNCT
ap-1807	351	1	(	(	PUNCT
ap-1807	351	2	ε4	ε4	NOUN
ap-1807	351	3	)	)	PUNCT
ap-1807	351	4	x	x	PUNCT
ap-1807	351	5	≤	≤	NUM
ap-1807	351	6	y	y	NOUN
ap-1807	351	7	,	,	PUNCT
ap-1807	351	8	z	z	NOUN
ap-1807	351	9	≤	≤	NOUN
ap-1807	351	10	(	(	PUNCT
ap-1807	351	11	y	y	PROPN
ap-1807	351	12	−	−	PROPN
ap-1807	352	1	x	x	NOUN
ap-1807	352	2	)	)	PUNCT
ap-1807	353	1	and	and	CCONJ
ap-1807	353	2	z	z	NOUN
ap-1807	353	3	≤	≤	NOUN
ap-1807	353	4	y	y	PROPN
ap-1807	353	5	(	(	PUNCT
ap-1807	353	6	that	that	PRON
ap-1807	353	7	is	be	AUX
ap-1807	353	8	y	y	PROPN
ap-1807	353	9	−	−	PROPN
ap-1807	353	10	x	x	SYM
ap-1807	353	11	,	,	PUNCT
ap-1807	353	12	(	(	PUNCT
ap-1807	353	13	y−x)−z	y−x)−z	PROPN
ap-1807	353	14	and	and	CCONJ
ap-1807	353	15	y−z	y−z	PROPN
ap-1807	353	16	are	be	AUX
ap-1807	353	17	defined	define	VERB
ap-1807	353	18	)	)	PUNCT
ap-1807	353	19	.	.	PUNCT
ap-1807	354	1	then	then	ADV
ap-1807	354	2	y	y	PROPN
ap-1807	354	3	=	=	PUNCT
ap-1807	354	4	(	(	PUNCT
ap-1807	354	5	y−x)+x	y−x)+x	ADV
ap-1807	354	6	=	=	SYM
ap-1807	354	7	(	(	PUNCT
ap-1807	354	8	(	(	PUNCT
ap-1807	354	9	y−x)−z)+z)+x	y−x)−z)+z)+x	PROPN
ap-1807	354	10	=	=	SYM
ap-1807	354	11	(	(	PUNCT
ap-1807	354	12	y−z)+z	y−z)+z	NOUN
ap-1807	354	13	.	.	PUNCT
ap-1807	355	1	hence	hence	ADV
ap-1807	355	2	(	(	PUNCT
ap-1807	355	3	y−x)−z)+x	y−x)−z)+x	NOUN
ap-1807	355	4	=	=	SYM
ap-1807	355	5	(	(	PUNCT
ap-1807	355	6	y−z	y−z	PROPN
ap-1807	355	7	)	)	PUNCT
ap-1807	355	8	and	and	CCONJ
ap-1807	355	9	(	(	PUNCT
ap-1807	355	10	x⊕y−)⊕z	x⊕y−)⊕z	NOUN
ap-1807	355	11	=	=	SYM
ap-1807	355	12	(	(	PUNCT
ap-1807	355	13	y−x)−⊕z	y−x)−⊕z	NOUN
ap-1807	355	14	=	=	SYM
ap-1807	355	15	(	(	PUNCT
ap-1807	355	16	(	(	PUNCT
ap-1807	355	17	y−x)−z)−	y−x)−z)−	PROPN
ap-1807	355	18	=	=	SYM
ap-1807	355	19	(	(	PUNCT
ap-1807	355	20	(	(	PUNCT
ap-1807	355	21	y	y	PROPN
ap-1807	355	22	−	−	PROPN
ap-1807	355	23	z)−	z)−	PROPN
ap-1807	355	24	x)−	x)−	PROPN
ap-1807	355	25	=	=	PUNCT
ap-1807	355	26	(	(	PUNCT
ap-1807	355	27	x⊕	x⊕	PROPN
ap-1807	355	28	(	(	PUNCT
ap-1807	355	29	y	y	PROPN
ap-1807	355	30	−	−	PROPN
ap-1807	355	31	z)−	z)−	PROPN
ap-1807	355	32	)	)	PUNCT
ap-1807	355	33	=	=	SYM
ap-1807	355	34	x⊕	x⊕	PROPN
ap-1807	355	35	(	(	PUNCT
ap-1807	355	36	y−	y−	PROPN
ap-1807	355	37	⊕	⊕	PROPN
ap-1807	355	38	z	z	PROPN
ap-1807	355	39	)	)	PUNCT
ap-1807	355	40	.	.	PUNCT
ap-1807	356	1	until	until	ADP
ap-1807	356	2	now	now	ADV
ap-1807	356	3	,	,	PUNCT
ap-1807	356	4	the	the	DET
ap-1807	356	5	map	map	NOUN
ap-1807	356	6	−	−	NOUN
ap-1807	356	7	:	:	PUNCT
ap-1807	356	8	e	e	X
ap-1807	356	9	→	→	PUNCT
ap-1807	356	10	e−	e−	X
ap-1807	356	11	∪	∪	X
ap-1807	356	12	0	0	NUM
ap-1807	356	13	has	have	AUX
ap-1807	356	14	been	be	AUX
ap-1807	356	15	defined	define	VERB
ap-1807	356	16	only	only	ADV
ap-1807	356	17	for	for	ADP
ap-1807	356	18	elements	element	NOUN
ap-1807	356	19	of	of	ADP
ap-1807	356	20	e.	e.	PROPN
ap-1807	356	21	we	we	PRON
ap-1807	356	22	can	can	AUX
ap-1807	356	23	extend	extend	VERB
ap-1807	356	24	it	it	PRON
ap-1807	356	25	on	on	ADP
ap-1807	356	26	g	g	PROPN
ap-1807	356	27	by	by	ADP
ap-1807	356	28	(	(	PUNCT
ap-1807	356	29	a−)−	a−)−	PROPN
ap-1807	356	30	=	=	SYM
ap-1807	356	31	a	a	PROPN
ap-1807	356	32	,	,	PUNCT
ap-1807	356	33	which	which	PRON
ap-1807	356	34	gives	give	VERB
ap-1807	356	35	us	we	PRON
ap-1807	356	36	an	an	DET
ap-1807	356	37	involution	involution	NOUN
ap-1807	356	38	on	on	ADP
ap-1807	356	39	g.	g.	PROPN
ap-1807	356	40	then	then	ADV
ap-1807	356	41	for	for	ADP
ap-1807	356	42	any	any	DET
ap-1807	356	43	a	a	PRON
ap-1807	356	44	,	,	PUNCT
ap-1807	356	45	b	b	X
ap-1807	356	46	∈	∈	PROPN
ap-1807	356	47	e	e	X
ap-1807	356	48	whenever	whenever	SCONJ
ap-1807	356	49	b	b	X
ap-1807	356	50	−	−	PROPN
ap-1807	356	51	a	a	PRON
ap-1807	356	52	is	be	AUX
ap-1807	356	53	defined	define	VERB
ap-1807	356	54	it	it	PRON
ap-1807	356	55	holds	hold	VERB
ap-1807	356	56	(	(	PUNCT
ap-1807	356	57	a	a	DET
ap-1807	356	58	⊕	⊕	NOUN
ap-1807	356	59	b−)−	b−)−	NOUN
ap-1807	357	1	=	=	SYM
ap-1807	358	1	(	(	PUNCT
ap-1807	358	2	(	(	PUNCT
ap-1807	358	3	b	b	X
ap-1807	358	4	−	−	NOUN
ap-1807	358	5	a)−)−	a)−)−	NOUN
ap-1807	358	6	=	=	PUNCT
ap-1807	358	7	(	(	PUNCT
ap-1807	358	8	b	b	X
ap-1807	358	9	−	−	NOUN
ap-1807	358	10	a	a	NOUN
ap-1807	358	11	)	)	PUNCT
ap-1807	358	12	=	=	SYM
ap-1807	358	13	a−	a−	PROPN
ap-1807	358	14	⊕	⊕	PROPN
ap-1807	358	15	b.	b.	PROPN
ap-1807	359	1	together	together	ADV
ap-1807	359	2	with	with	ADP
ap-1807	359	3	(	(	PUNCT
ap-1807	359	4	α1	α1	PROPN
ap-1807	359	5	)	)	PUNCT
ap-1807	359	6	and	and	CCONJ
ap-1807	359	7	(	(	PUNCT
ap-1807	359	8	β1	β1	PROPN
ap-1807	359	9	)	)	PUNCT
ap-1807	359	10	we	we	PRON
ap-1807	359	11	have	have	VERB
ap-1807	359	12	(	(	PUNCT
ap-1807	359	13	γ1)(a⊕	γ1)(a⊕	ADV
ap-1807	359	14	b−)−	b−)−	NOUN
ap-1807	359	15	=	=	SYM
ap-1807	359	16	a−	a−	PROPN
ap-1807	359	17	⊕	⊕	PROPN
ap-1807	359	18	b	b	PROPN
ap-1807	359	19	for	for	ADP
ap-1807	359	20	all	all	DET
ap-1807	359	21	a	a	PRON
ap-1807	359	22	,	,	PUNCT
ap-1807	359	23	b	b	X
ap-1807	359	24	∈	∈	PROPN
ap-1807	359	25	e	e	NOUN
ap-1807	359	26	,	,	PUNCT
ap-1807	359	27	(	(	PUNCT
ap-1807	359	28	δ1)(a−⊕	δ1)(a−⊕	ADV
ap-1807	359	29	b−)−	b−)−	PROPN
ap-1807	359	30	=	=	SYM
ap-1807	359	31	(	(	PUNCT
ap-1807	359	32	(	(	PUNCT
ap-1807	359	33	a⊕	a⊕	PROPN
ap-1807	359	34	b)−)−	b)−)−	PROPN
ap-1807	360	1	=	=	SYM
ap-1807	360	2	a⊕	a⊕	PROPN
ap-1807	360	3	b	b	PROPN
ap-1807	360	4	for	for	ADP
ap-1807	360	5	all	all	DET
ap-1807	360	6	a	a	DET
ap-1807	360	7	,	,	PUNCT
ap-1807	360	8	b	b	X
ap-1807	360	9	∈	∈	PROPN
ap-1807	360	10	e	e	NOUN
ap-1807	360	11	case	case	NOUN
ap-1807	360	12	v.	v.	ADP
ap-1807	360	13	let	let	NOUN
ap-1807	360	14	(	(	PUNCT
ap-1807	360	15	x−⊕y−)⊕z	x−⊕y−)⊕z	PROPN
ap-1807	360	16	and	and	CCONJ
ap-1807	360	17	y−⊕z	y−⊕z	NOUN
ap-1807	360	18	be	be	AUX
ap-1807	360	19	defined	define	VERB
ap-1807	360	20	.	.	PUNCT
ap-1807	361	1	then	then	ADV
ap-1807	361	2	also	also	ADV
ap-1807	361	3	y	y	PROPN
ap-1807	361	4	⊕	⊕	PROPN
ap-1807	361	5	z−	z−	PROPN
ap-1807	361	6	is	be	AUX
ap-1807	361	7	defined	define	VERB
ap-1807	361	8	and	and	CCONJ
ap-1807	361	9	using	use	VERB
ap-1807	361	10	(	(	PUNCT
ap-1807	361	11	α1	α1	PROPN
ap-1807	361	12	)	)	PUNCT
ap-1807	361	13	,	,	PUNCT
ap-1807	361	14	(	(	PUNCT
ap-1807	361	15	γ1	γ1	PROPN
ap-1807	361	16	)	)	PUNCT
ap-1807	361	17	and	and	CCONJ
ap-1807	361	18	(	(	PUNCT
ap-1807	361	19	ii	ii	NOUN
ap-1807	361	20	)	)	PUNCT
ap-1807	361	21	we	we	PRON
ap-1807	361	22	get	get	VERB
ap-1807	361	23	(	(	PUNCT
ap-1807	361	24	x−⊕y−)⊕z	x−⊕y−)⊕z	X
ap-1807	361	25	=	=	SYM
ap-1807	361	26	(	(	PUNCT
ap-1807	361	27	(	(	PUNCT
ap-1807	361	28	x⊕y)⊕z−)−	x⊕y)⊕z−)−	PROPN
ap-1807	361	29	=	=	SYM
ap-1807	361	30	(	(	PUNCT
ap-1807	361	31	x⊕(y⊕z−))−	x⊕(y⊕z−))−	PROPN
ap-1807	361	32	=	=	SYM
ap-1807	361	33	(	(	PUNCT
ap-1807	361	34	x⊕	x⊕	PROPN
ap-1807	361	35	(	(	PUNCT
ap-1807	361	36	y−	y−	PROPN
ap-1807	361	37	⊕	⊕	PROPN
ap-1807	361	38	z	z	PROPN
ap-1807	361	39	)	)	PUNCT
ap-1807	361	40	)	)	PUNCT
ap-1807	361	41	.	.	PUNCT
ap-1807	362	1	case	case	NOUN
ap-1807	362	2	vi	vi	X
ap-1807	362	3	.	.	PUNCT
ap-1807	363	1	let	let	VERB
ap-1807	363	2	(	(	PUNCT
ap-1807	363	3	x⊕y−)⊕z−	x⊕y−)⊕z−	PROPN
ap-1807	363	4	and	and	CCONJ
ap-1807	363	5	y−⊕z−	y−⊕z−	VERB
ap-1807	363	6	be	be	AUX
ap-1807	363	7	defined	define	VERB
ap-1807	363	8	.	.	PUNCT
ap-1807	364	1	then	then	ADV
ap-1807	364	2	y	y	PROPN
ap-1807	364	3	⊕	⊕	PROPN
ap-1807	364	4	z	z	PROPN
ap-1807	364	5	is	be	AUX
ap-1807	364	6	defined	define	VERB
ap-1807	364	7	and	and	CCONJ
ap-1807	364	8	using	use	VERB
ap-1807	364	9	(	(	PUNCT
ap-1807	364	10	α1	α1	PROPN
ap-1807	364	11	)	)	PUNCT
ap-1807	364	12	,	,	PUNCT
ap-1807	364	13	(	(	PUNCT
ap-1807	364	14	γ1	γ1	PROPN
ap-1807	364	15	)	)	PUNCT
ap-1807	364	16	and	and	CCONJ
ap-1807	364	17	(	(	PUNCT
ap-1807	364	18	iii	iii	X
ap-1807	364	19	)	)	PUNCT
ap-1807	364	20	we	we	PRON
ap-1807	364	21	have	have	VERB
ap-1807	364	22	(	(	PUNCT
ap-1807	364	23	(	(	PUNCT
ap-1807	364	24	x⊕y−)⊕z−	x⊕y−)⊕z−	PROPN
ap-1807	364	25	)	)	PUNCT
ap-1807	365	1	=	=	SYM
ap-1807	365	2	(	(	PUNCT
ap-1807	365	3	(	(	PUNCT
ap-1807	365	4	x−⊕y)⊕z)−	x−⊕y)⊕z)−	PROPN
ap-1807	365	5	=	=	SYM
ap-1807	365	6	(	(	PUNCT
ap-1807	365	7	x−⊕	x−⊕	PROPN
ap-1807	365	8	(	(	PUNCT
ap-1807	365	9	y⊕z))−	y⊕z))−	NOUN
ap-1807	365	10	=	=	SYM
ap-1807	365	11	(	(	PUNCT
ap-1807	365	12	x⊕	x⊕	PROPN
ap-1807	365	13	(	(	PUNCT
ap-1807	365	14	y−	y−	PROPN
ap-1807	365	15	⊕	⊕	PROPN
ap-1807	365	16	z−	z−	PROPN
ap-1807	365	17	)	)	PUNCT
ap-1807	365	18	)	)	PUNCT
ap-1807	365	19	.	.	PUNCT
ap-1807	366	1	case	case	NOUN
ap-1807	366	2	vii	vii	PROPN
ap-1807	366	3	.	.	PUNCT
ap-1807	367	1	let	let	VERB
ap-1807	367	2	(	(	PUNCT
ap-1807	367	3	x−	x−	PROPN
ap-1807	367	4	⊕	⊕	PROPN
ap-1807	367	5	y)⊕	y)⊕	NOUN
ap-1807	367	6	z−	z−	PROPN
ap-1807	367	7	and	and	CCONJ
ap-1807	367	8	y	y	PROPN
ap-1807	367	9	⊕	⊕	PROPN
ap-1807	367	10	z−	z−	PROPN
ap-1807	367	11	be	be	AUX
ap-1807	367	12	defined	define	VERB
ap-1807	367	13	.	.	PUNCT
ap-1807	368	1	then	then	ADV
ap-1807	368	2	y−	y−	PROPN
ap-1807	368	3	⊕	⊕	PROPN
ap-1807	368	4	z	z	PROPN
ap-1807	368	5	is	be	AUX
ap-1807	368	6	defined	define	VERB
ap-1807	368	7	and	and	CCONJ
ap-1807	368	8	with	with	ADP
ap-1807	368	9	(	(	PUNCT
ap-1807	368	10	α1	α1	PROPN
ap-1807	368	11	)	)	PUNCT
ap-1807	368	12	,	,	PUNCT
ap-1807	368	13	(	(	PUNCT
ap-1807	368	14	γ1	γ1	PROPN
ap-1807	368	15	)	)	PUNCT
ap-1807	368	16	and	and	CCONJ
ap-1807	368	17	(	(	PUNCT
ap-1807	368	18	iv	iv	X
ap-1807	368	19	)	)	PUNCT
ap-1807	368	20	we	we	PRON
ap-1807	368	21	can	can	AUX
ap-1807	368	22	see	see	VERB
ap-1807	368	23	that	that	PRON
ap-1807	368	24	(	(	PUNCT
ap-1807	368	25	(	(	PUNCT
ap-1807	368	26	x−	x−	PROPN
ap-1807	368	27	⊕	⊕	PROPN
ap-1807	368	28	y)⊕	y)⊕	NOUN
ap-1807	368	29	z−	z−	PROPN
ap-1807	368	30	)	)	PUNCT
ap-1807	368	31	=	=	SYM
ap-1807	369	1	(	(	PUNCT
ap-1807	369	2	(	(	PUNCT
ap-1807	369	3	x⊕	x⊕	X
ap-1807	369	4	y−)⊕	y−)⊕	PROPN
ap-1807	369	5	z)−	z)−	PROPN
ap-1807	369	6	=	=	SYM
ap-1807	369	7	(	(	PUNCT
ap-1807	369	8	x⊕	x⊕	PROPN
ap-1807	369	9	(	(	PUNCT
ap-1807	369	10	y−	y−	PROPN
ap-1807	369	11	⊕	⊕	PROPN
ap-1807	369	12	z))−	z))−	NUM
ap-1807	369	13	=	=	SYM
ap-1807	369	14	(	(	PUNCT
ap-1807	369	15	x−	x−	PROPN
ap-1807	369	16	⊕	⊕	PROPN
ap-1807	369	17	(	(	PUNCT
ap-1807	369	18	y	y	PROPN
ap-1807	369	19	⊕	⊕	PROPN
ap-1807	369	20	z−	z−	PROPN
ap-1807	369	21	)	)	PUNCT
ap-1807	369	22	)	)	PUNCT
ap-1807	369	23	.	.	PUNCT
ap-1807	370	1	case	case	NOUN
ap-1807	370	2	viii	viii	VERB
ap-1807	370	3	.	.	PUNCT
ap-1807	371	1	let	let	VERB
ap-1807	371	2	(	(	PUNCT
ap-1807	371	3	x−⊕	x−⊕	NUM
ap-1807	371	4	y−)⊕	y−)⊕	PRON
ap-1807	371	5	z−	z−	PROPN
ap-1807	371	6	and	and	CCONJ
ap-1807	371	7	y−⊕	y−⊕	PROPN
ap-1807	371	8	z−	z−	PROPN
ap-1807	371	9	be	be	AUX
ap-1807	371	10	defined	define	VERB
ap-1807	371	11	.	.	PUNCT
ap-1807	372	1	then	then	ADV
ap-1807	372	2	y⊕z	y⊕z	NOUN
ap-1807	372	3	is	be	AUX
ap-1807	372	4	defined	define	VERB
ap-1807	372	5	and	and	CCONJ
ap-1807	372	6	(	(	PUNCT
ap-1807	372	7	x−⊕y−)⊕z−	x−⊕y−)⊕z−	X
ap-1807	372	8	=	=	SYM
ap-1807	372	9	(	(	PUNCT
ap-1807	372	10	x⊕y)−⊕	x⊕y)−⊕	PROPN
ap-1807	372	11	z−	z−	PROPN
ap-1807	372	12	=	=	PUNCT
ap-1807	372	13	(	(	PUNCT
ap-1807	372	14	(	(	PUNCT
ap-1807	372	15	x⊕y)⊕z)−	x⊕y)⊕z)−	X
ap-1807	372	16	=	=	SYM
ap-1807	372	17	(	(	PUNCT
ap-1807	372	18	x⊕(y⊕z))−	x⊕(y⊕z))−	NUM
ap-1807	372	19	=	=	SYM
ap-1807	372	20	x−⊕(y−⊕z−	x−⊕(y−⊕z−	PROPN
ap-1807	372	21	)	)	PUNCT
ap-1807	372	22	.	.	PUNCT
ap-1807	373	1	292	292	NUM
ap-1807	373	2	vol	vol	NOUN
ap-1807	373	3	.	.	PUNCT
ap-1807	374	1	53	53	NUM
ap-1807	374	2	no	no	NOUN
ap-1807	374	3	.	.	PUNCT
ap-1807	375	1	3/2013	3/2013	PROPN
ap-1807	375	2	weakly	weakly	ADV
ap-1807	375	3	ordered	order	VERB
ap-1807	375	4	a	a	DET
ap-1807	375	5	-	-	PUNCT
ap-1807	375	6	commutative	commutative	ADJ
ap-1807	375	7	partial	partial	ADJ
ap-1807	375	8	groups	group	NOUN
ap-1807	375	9	of	of	ADP
ap-1807	375	10	linear	linear	PROPN
ap-1807	375	11	operators	operator	NOUN
ap-1807	375	12	hence	hence	ADV
ap-1807	375	13	(	(	PUNCT
ap-1807	375	14	g,⊕	g,⊕	PROPN
ap-1807	375	15	,	,	PUNCT
ap-1807	375	16	0	0	NUM
ap-1807	375	17	)	)	PUNCT
ap-1807	375	18	forms	form	VERB
ap-1807	375	19	an	an	DET
ap-1807	375	20	a	a	PRON
ap-1807	375	21	-	-	PUNCT
ap-1807	375	22	commutative	commutative	ADJ
ap-1807	375	23	partial	partial	ADJ
ap-1807	375	24	group	group	NOUN
ap-1807	375	25	.	.	PUNCT
ap-1807	376	1	since	since	SCONJ
ap-1807	376	2	e	e	PROPN
ap-1807	376	3	is	be	AUX
ap-1807	376	4	a	a	DET
ap-1807	376	5	generalized	generalized	ADJ
ap-1807	376	6	effect	effect	NOUN
ap-1807	376	7	algebra	algebra	NOUN
ap-1807	376	8	,	,	PUNCT
ap-1807	376	9	by	by	ADP
ap-1807	376	10	lemma	lemma	PROPN
ap-1807	376	11	7	7	NUM
ap-1807	376	12	there	there	ADV
ap-1807	376	13	exists	exist	VERB
ap-1807	376	14	a	a	DET
ap-1807	376	15	relation	relation	NOUN
ap-1807	376	16	≤g	≤g	NOUN
ap-1807	376	17	such	such	ADJ
ap-1807	376	18	that	that	PRON
ap-1807	376	19	(	(	PUNCT
ap-1807	376	20	g,⊕	g,⊕	NOUN
ap-1807	376	21	,	,	PUNCT
ap-1807	376	22	0	0	NUM
ap-1807	376	23	)	)	PUNCT
ap-1807	376	24	w.r.t	w.r.t	NOUN
ap-1807	376	25	.	.	PUNCT
ap-1807	377	1	≤g	≤g	PROPN
ap-1807	377	2	forms	form	VERB
ap-1807	377	3	a	a	DET
ap-1807	377	4	woa	woa	NOUN
ap-1807	377	5	-	-	PUNCT
ap-1807	377	6	group	group	NOUN
ap-1807	377	7	.	.	PUNCT
ap-1807	377	8	example	example	NOUN
ap-1807	378	1	2	2	NUM
ap-1807	378	2	.	.	PUNCT
ap-1807	378	3	an	an	DET
ap-1807	378	4	interval	interval	NOUN
ap-1807	378	5	[	[	X
ap-1807	378	6	−1	−1	NOUN
ap-1807	378	7	,	,	PUNCT
ap-1807	378	8	1	1	NUM
ap-1807	378	9	]	]	PUNCT
ap-1807	378	10	with	with	ADP
ap-1807	378	11	⊕	⊕	PROPN
ap-1807	378	12	defined	define	VERB
ap-1807	378	13	for	for	ADP
ap-1807	378	14	0	0	NUM
ap-1807	378	15	≤	≤	NUM
ap-1807	378	16	x	x	X
ap-1807	378	17	,	,	PUNCT
ap-1807	378	18	y	y	PROPN
ap-1807	378	19	by	by	ADP
ap-1807	378	20	•	•	NUM
ap-1807	378	21	x⊕	x⊕	PROPN
ap-1807	378	22	y	y	PROPN
ap-1807	378	23	=	=	PUNCT
ap-1807	378	24	x+	x+	PROPN
ap-1807	378	25	y	y	PROPN
ap-1807	378	26	iff	iff	PROPN
ap-1807	378	27	x+	x+	PROPN
ap-1807	378	28	y	y	PROPN
ap-1807	378	29	≤	≤	PROPN
ap-1807	378	30	1	1	NUM
ap-1807	378	31	,	,	PUNCT
ap-1807	378	32	•	•	NUM
ap-1807	378	33	x⊕	x⊕	PROPN
ap-1807	378	34	(	(	PUNCT
ap-1807	378	35	−y	−y	PROPN
ap-1807	378	36	)	)	PUNCT
ap-1807	378	37	=	=	PUNCT
ap-1807	378	38	x−	x−	PROPN
ap-1807	378	39	y	y	PROPN
ap-1807	378	40	,	,	PUNCT
ap-1807	378	41	•	•	PRON
ap-1807	378	42	(	(	PUNCT
ap-1807	378	43	−x)⊕	−x)⊕	PROPN
ap-1807	378	44	(	(	PUNCT
ap-1807	378	45	−y	−y	NOUN
ap-1807	378	46	)	)	PUNCT
ap-1807	378	47	=	=	PUNCT
ap-1807	378	48	−(x⊕	−(x⊕	PROPN
ap-1807	378	49	y	y	PROPN
ap-1807	378	50	)	)	PUNCT
ap-1807	378	51	iff	iff	PROPN
ap-1807	378	52	(	(	PUNCT
ap-1807	378	53	x⊕	x⊕	PROPN
ap-1807	378	54	y	y	NOUN
ap-1807	378	55	)	)	PUNCT
ap-1807	378	56	exists	exist	VERB
ap-1807	378	57	and	and	CCONJ
ap-1807	378	58	relation	relation	NOUN
ap-1807	378	59	≤g	≤g	PROPN
ap-1807	378	60	defined	define	VERB
ap-1807	378	61	by	by	ADP
ap-1807	378	62	x	x	PROPN
ap-1807	378	63	≤g	≤g	PROPN
ap-1807	378	64	y	y	PROPN
ap-1807	378	65	iff	iff	PROPN
ap-1807	378	66	x	x	PROPN
ap-1807	378	67	≤	≤	PROPN
ap-1807	378	68	y	y	PROPN
ap-1807	378	69	and	and	CCONJ
ap-1807	379	1	y	y	PROPN
ap-1807	379	2	−	−	PROPN
ap-1807	379	3	x	x	SYM
ap-1807	379	4	≤	≤	ADV
ap-1807	379	5	1	1	NUM
ap-1807	379	6	for	for	ADP
ap-1807	379	7	all	all	DET
ap-1807	379	8	x	x	NOUN
ap-1807	379	9	,	,	PUNCT
ap-1807	379	10	y	y	PROPN
ap-1807	379	11	∈	∈	PROPN
ap-1807	380	1	[	[	X
ap-1807	380	2	−1	−1	NOUN
ap-1807	380	3	,	,	PUNCT
ap-1807	380	4	1	1	X
ap-1807	380	5	]	]	PUNCT
ap-1807	380	6	forms	form	VERB
ap-1807	380	7	a	a	DET
ap-1807	380	8	woa	woa	NOUN
ap-1807	380	9	-	-	PUNCT
ap-1807	380	10	group	group	NOUN
ap-1807	380	11	.	.	PUNCT
ap-1807	381	1	a	a	DET
ap-1807	381	2	positive	positive	ADJ
ap-1807	381	3	cone	cone	NOUN
ap-1807	381	4	(	(	PUNCT
ap-1807	381	5	[	[	X
ap-1807	381	6	0	0	NUM
ap-1807	381	7	,	,	PUNCT
ap-1807	381	8	1],⊕/[0,1	1],⊕/[0,1	NOUN
ap-1807	381	9	]	]	PUNCT
ap-1807	381	10	,	,	PUNCT
ap-1807	381	11	0	0	NUM
ap-1807	381	12	)	)	PUNCT
ap-1807	381	13	forms	form	VERB
ap-1807	381	14	a	a	DET
ap-1807	381	15	well	well	ADV
ap-1807	381	16	-	-	PUNCT
ap-1807	381	17	known	know	VERB
ap-1807	381	18	unit	unit	NOUN
ap-1807	381	19	interval	interval	NOUN
ap-1807	381	20	effect	effect	NOUN
ap-1807	381	21	algebra	algebra	NOUN
ap-1807	381	22	.	.	PUNCT
ap-1807	382	1	3	3	X
ap-1807	382	2	.	.	X
ap-1807	382	3	hilbert	hilbert	PROPN
ap-1807	382	4	spaces	space	NOUN
ap-1807	382	5	we	we	PRON
ap-1807	382	6	assume	assume	VERB
ap-1807	382	7	that	that	SCONJ
ap-1807	382	8	h	h	NOUN
ap-1807	382	9	is	be	AUX
ap-1807	382	10	an	an	DET
ap-1807	382	11	infinite	infinite	ADJ
ap-1807	382	12	-	-	PUNCT
ap-1807	382	13	dimensional	dimensional	ADJ
ap-1807	382	14	complex	complex	ADJ
ap-1807	382	15	hilbert	hilbert	NOUN
ap-1807	382	16	space	space	NOUN
ap-1807	382	17	,	,	PUNCT
ap-1807	382	18	i.e.	i.e.	X
ap-1807	382	19	,	,	PUNCT
ap-1807	382	20	a	a	DET
ap-1807	382	21	linear	linear	ADJ
ap-1807	382	22	space	space	NOUN
ap-1807	382	23	with	with	ADP
ap-1807	382	24	inner	inner	ADJ
ap-1807	382	25	product	product	NOUN
ap-1807	382	26	〈	〈	PROPN
ap-1807	382	27	·	·	PUNCT
ap-1807	382	28	,	,	PUNCT
ap-1807	382	29	·	·	PUNCT
ap-1807	382	30	〉	〉	NOUN
ap-1807	382	31	which	which	PRON
ap-1807	382	32	is	be	AUX
ap-1807	382	33	complete	complete	ADJ
ap-1807	382	34	in	in	ADP
ap-1807	382	35	the	the	DET
ap-1807	382	36	induced	induce	VERB
ap-1807	382	37	metric	metric	NOUN
ap-1807	382	38	.	.	PUNCT
ap-1807	383	1	the	the	DET
ap-1807	383	2	term	term	NOUN
ap-1807	383	3	dimension	dimension	NOUN
ap-1807	383	4	of	of	ADP
ap-1807	383	5	h	h	NOUN
ap-1807	383	6	in	in	ADP
ap-1807	383	7	the	the	DET
ap-1807	383	8	following	following	NOUN
ap-1807	383	9	always	always	ADV
ap-1807	383	10	means	mean	VERB
ap-1807	383	11	the	the	DET
ap-1807	383	12	hilbertian	hilbertian	ADJ
ap-1807	383	13	dimension	dimension	NOUN
ap-1807	383	14	defined	define	VERB
ap-1807	383	15	as	as	ADP
ap-1807	383	16	the	the	DET
ap-1807	383	17	cardinality	cardinality	NOUN
ap-1807	383	18	of	of	ADP
ap-1807	383	19	any	any	DET
ap-1807	383	20	orthonormal	orthonormal	ADJ
ap-1807	383	21	basis	basis	NOUN
ap-1807	383	22	of	of	ADP
ap-1807	383	23	h	h	PROPN
ap-1807	383	24	(	(	PUNCT
ap-1807	383	25	see	see	VERB
ap-1807	383	26	[	[	X
ap-1807	383	27	1	1	NUM
ap-1807	383	28	]	]	NUM
ap-1807	383	29	)	)	PUNCT
ap-1807	383	30	.	.	PUNCT
ap-1807	384	1	moreover	moreover	ADV
ap-1807	384	2	,	,	PUNCT
ap-1807	384	3	we	we	PRON
ap-1807	384	4	will	will	AUX
ap-1807	384	5	assume	assume	VERB
ap-1807	384	6	that	that	SCONJ
ap-1807	384	7	all	all	PRON
ap-1807	384	8	considered	consider	VERB
ap-1807	384	9	linear	linear	PROPN
ap-1807	384	10	operators	operator	NOUN
ap-1807	384	11	a	a	DET
ap-1807	384	12	(	(	PUNCT
ap-1807	384	13	i.e.	i.e.	X
ap-1807	384	14	,	,	PUNCT
ap-1807	384	15	linear	linear	PROPN
ap-1807	384	16	maps	map	VERB
ap-1807	384	17	a	a	DET
ap-1807	384	18	:	:	PUNCT
ap-1807	384	19	d(a)→	d(a)→	ADJ
ap-1807	384	20	h	h	X
ap-1807	384	21	)	)	PUNCT
ap-1807	384	22	have	have	VERB
ap-1807	384	23	a	a	DET
ap-1807	384	24	domain	domain	NOUN
ap-1807	384	25	d(a	d(a	PROPN
ap-1807	384	26	)	)	PUNCT
ap-1807	384	27	a	a	DET
ap-1807	384	28	linear	linear	ADJ
ap-1807	384	29	subspace	subspace	NOUN
ap-1807	384	30	dense	dense	ADJ
ap-1807	384	31	in	in	ADP
ap-1807	384	32	h	h	NOUN
ap-1807	384	33	with	with	ADP
ap-1807	384	34	respect	respect	NOUN
ap-1807	384	35	to	to	ADP
ap-1807	384	36	the	the	DET
ap-1807	384	37	metric	metric	ADJ
ap-1807	384	38	topology	topology	NOUN
ap-1807	384	39	induced	induce	VERB
ap-1807	384	40	by	by	ADP
ap-1807	384	41	the	the	DET
ap-1807	384	42	inner	inner	ADJ
ap-1807	384	43	product	product	NOUN
ap-1807	384	44	,	,	PUNCT
ap-1807	384	45	so	so	ADV
ap-1807	384	46	d(a	d(a	PROPN
ap-1807	384	47	)	)	PUNCT
ap-1807	385	1	=	=	SYM
ap-1807	385	2	h	h	NOUN
ap-1807	385	3	(	(	PUNCT
ap-1807	385	4	we	we	PRON
ap-1807	385	5	say	say	VERB
ap-1807	385	6	that	that	SCONJ
ap-1807	385	7	a	a	PRON
ap-1807	385	8	is	be	AUX
ap-1807	385	9	densely	densely	ADV
ap-1807	385	10	defined	define	VERB
ap-1807	385	11	)	)	PUNCT
ap-1807	385	12	.	.	PUNCT
ap-1807	386	1	we	we	PRON
ap-1807	386	2	denote	denote	VERB
ap-1807	386	3	by	by	ADP
ap-1807	386	4	d	d	PROPN
ap-1807	386	5	the	the	DET
ap-1807	386	6	set	set	NOUN
ap-1807	386	7	of	of	ADP
ap-1807	386	8	all	all	DET
ap-1807	386	9	dense	dense	ADJ
ap-1807	386	10	linear	linear	ADJ
ap-1807	386	11	subspaces	subspace	NOUN
ap-1807	386	12	of	of	ADP
ap-1807	386	13	h.	h.	NOUN
ap-1807	386	14	by	by	ADP
ap-1807	386	15	positive	positive	ADJ
ap-1807	386	16	linear	linear	PROPN
ap-1807	386	17	operators	operator	NOUN
ap-1807	386	18	a	a	PRON
ap-1807	386	19	,	,	PUNCT
ap-1807	386	20	(	(	PUNCT
ap-1807	386	21	denoted	denote	VERB
ap-1807	386	22	by	by	ADP
ap-1807	386	23	a	a	DET
ap-1807	386	24	≥	≥	NOUN
ap-1807	386	25	0	0	NUM
ap-1807	386	26	)	)	PUNCT
ap-1807	386	27	it	it	PRON
ap-1807	386	28	means	mean	VERB
ap-1807	386	29	that	that	SCONJ
ap-1807	386	30	〈	〈	PROPN
ap-1807	386	31	ax	ax	NOUN
ap-1807	386	32	,	,	PUNCT
ap-1807	386	33	x	x	PROPN
ap-1807	386	34	〉	〉	NUM
ap-1807	386	35	≥	≥	NOUN
ap-1807	386	36	0	0	NUM
ap-1807	386	37	for	for	ADP
ap-1807	386	38	all	all	DET
ap-1807	386	39	x	x	SYM
ap-1807	386	40	∈	∈	PROPN
ap-1807	386	41	d(a	d(a	PROPN
ap-1807	386	42	)	)	PUNCT
ap-1807	386	43	.	.	PUNCT
ap-1807	387	1	to	to	ADP
ap-1807	387	2	every	every	DET
ap-1807	387	3	linear	linear	ADJ
ap-1807	387	4	operator	operator	NOUN
ap-1807	387	5	a	a	DET
ap-1807	387	6	:	:	PUNCT
ap-1807	387	7	d(a	d(a	PROPN
ap-1807	387	8	)	)	PUNCT
ap-1807	387	9	→	→	SYM
ap-1807	387	10	h	h	NOUN
ap-1807	387	11	with	with	ADP
ap-1807	387	12	d(a	d(a	PROPN
ap-1807	387	13	)	)	PUNCT
ap-1807	388	1	=	=	NOUN
ap-1807	389	1	h	h	NOUN
ap-1807	389	2	there	there	PRON
ap-1807	389	3	exists	exist	VERB
ap-1807	389	4	the	the	DET
ap-1807	389	5	adjoint	adjoint	NOUN
ap-1807	389	6	operator	operator	NOUN
ap-1807	389	7	a∗	a∗	NOUN
ap-1807	389	8	of	of	ADP
ap-1807	389	9	a	a	DET
ap-1807	389	10	such	such	ADJ
ap-1807	389	11	that	that	DET
ap-1807	389	12	d(a∗	d(a∗	NOUN
ap-1807	389	13	)	)	PUNCT
ap-1807	389	14	=	=	SYM
ap-1807	389	15	{	{	PUNCT
ap-1807	390	1	y	y	PROPN
ap-1807	390	2	∈	∈	PROPN
ap-1807	390	3	h	h	NOUN
ap-1807	391	1	|	|	ADV
ap-1807	391	2	there	there	PRON
ap-1807	391	3	exists	exist	VERB
ap-1807	391	4	y∗	y∗	PROPN
ap-1807	391	5	∈	∈	PROPN
ap-1807	391	6	h	h	NOUN
ap-1807	391	7	such	such	ADJ
ap-1807	391	8	that	that	SCONJ
ap-1807	391	9	〈	〈	PROPN
ap-1807	391	10	y∗	y∗	PROPN
ap-1807	391	11	,	,	PUNCT
ap-1807	391	12	x	x	NOUN
ap-1807	391	13	〉	〉	NOUN
ap-1807	391	14	=	=	SYM
ap-1807	391	15	〈	〈	PROPN
ap-1807	391	16	y	y	PROPN
ap-1807	391	17	,	,	PUNCT
ap-1807	391	18	ax	ax	NOUN
ap-1807	391	19	〉	〉	NOUN
ap-1807	391	20	for	for	ADP
ap-1807	391	21	every	every	DET
ap-1807	391	22	x	x	PROPN
ap-1807	391	23	∈	∈	PROPN
ap-1807	391	24	d(a	d(a	PROPN
ap-1807	391	25	)	)	PUNCT
ap-1807	391	26	}	}	PUNCT
ap-1807	391	27	and	and	CCONJ
ap-1807	391	28	a∗y	a∗y	NUM
ap-1807	391	29	=	=	PUNCT
ap-1807	391	30	y∗	y∗	ADV
ap-1807	391	31	for	for	ADP
ap-1807	391	32	every	every	DET
ap-1807	391	33	y	y	PROPN
ap-1807	391	34	∈	∈	PROPN
ap-1807	391	35	d(a∗	d(a∗	NUM
ap-1807	391	36	)	)	PUNCT
ap-1807	391	37	.	.	PUNCT
ap-1807	392	1	when	when	SCONJ
ap-1807	392	2	a∗	a∗	PROPN
ap-1807	392	3	=	=	SYM
ap-1807	392	4	a	a	PROPN
ap-1807	392	5	,	,	PUNCT
ap-1807	392	6	a	a	PRON
ap-1807	392	7	is	be	AUX
ap-1807	392	8	called	call	VERB
ap-1807	392	9	self	self	NOUN
ap-1807	392	10	-	-	PUNCT
ap-1807	392	11	adjoint	adjoint	NOUN
ap-1807	392	12	(	(	PUNCT
ap-1807	392	13	for	for	SCONJ
ap-1807	392	14	more	more	ADJ
ap-1807	392	15	details	detail	NOUN
ap-1807	392	16	see	see	VERB
ap-1807	392	17	[	[	X
ap-1807	392	18	1	1	NUM
ap-1807	392	19	]	]	NUM
ap-1807	392	20	)	)	PUNCT
ap-1807	392	21	.	.	PUNCT
ap-1807	393	1	recall	recall	VERB
ap-1807	393	2	that	that	SCONJ
ap-1807	393	3	a	a	DET
ap-1807	393	4	linear	linear	ADJ
ap-1807	393	5	operator	operator	NOUN
ap-1807	393	6	a	a	DET
ap-1807	393	7	:	:	PUNCT
ap-1807	393	8	d(a	d(a	PROPN
ap-1807	393	9	)	)	PUNCT
ap-1807	393	10	→	→	SYM
ap-1807	393	11	h	h	NOUN
ap-1807	393	12	is	be	AUX
ap-1807	393	13	called	call	VERB
ap-1807	393	14	a	a	DET
ap-1807	393	15	bounded	bounded	ADJ
ap-1807	393	16	operator	operator	NOUN
ap-1807	393	17	if	if	SCONJ
ap-1807	393	18	there	there	PRON
ap-1807	393	19	exists	exist	VERB
ap-1807	393	20	a	a	DET
ap-1807	393	21	real	real	ADJ
ap-1807	393	22	constant	constant	ADJ
ap-1807	393	23	c	c	PROPN
ap-1807	393	24	≥	≥	NOUN
ap-1807	393	25	0	0	NUM
ap-1807	393	26	such	such	ADJ
ap-1807	393	27	that	that	SCONJ
ap-1807	393	28	‖ax‖	‖ax‖	ADJ
ap-1807	393	29	≤	≤	NOUN
ap-1807	393	30	c‖x‖	c‖x‖	NOUN
ap-1807	393	31	for	for	ADP
ap-1807	393	32	all	all	DET
ap-1807	393	33	x	x	SYM
ap-1807	393	34	∈	∈	PROPN
ap-1807	393	35	d(a	d(a	PROPN
ap-1807	393	36	)	)	PUNCT
ap-1807	393	37	and	and	CCONJ
ap-1807	393	38	hence	hence	ADV
ap-1807	393	39	a	a	PRON
ap-1807	393	40	is	be	AUX
ap-1807	393	41	an	an	DET
ap-1807	393	42	unbounded	unbounded	ADJ
ap-1807	393	43	operator	operator	NOUN
ap-1807	393	44	if	if	SCONJ
ap-1807	393	45	to	to	ADP
ap-1807	393	46	every	every	DET
ap-1807	393	47	c	c	NOUN
ap-1807	393	48	∈	∈	PROPN
ap-1807	393	49	r	r	NOUN
ap-1807	393	50	,	,	PUNCT
ap-1807	393	51	c	c	X
ap-1807	393	52	≥	≥	NOUN
ap-1807	393	53	0	0	NUM
ap-1807	393	54	there	there	PRON
ap-1807	393	55	exists	exist	VERB
ap-1807	393	56	xc	xc	PROPN
ap-1807	393	57	∈	∈	PROPN
ap-1807	393	58	d(a	d(a	PROPN
ap-1807	393	59	)	)	PUNCT
ap-1807	393	60	with	with	ADP
ap-1807	393	61	‖axc‖	‖axc‖	ADV
ap-1807	393	62	>	>	X
ap-1807	393	63	c‖xc‖.	c‖xc‖.	NOUN
ap-1807	393	64	by	by	ADP
ap-1807	393	65	symbol	symbol	NOUN
ap-1807	393	66	0	0	PUNCT
ap-1807	394	1	we	we	PRON
ap-1807	394	2	mean	mean	VERB
ap-1807	394	3	a	a	DET
ap-1807	394	4	null	null	ADJ
ap-1807	394	5	operator	operator	NOUN
ap-1807	394	6	and	and	CCONJ
ap-1807	394	7	it	it	PRON
ap-1807	394	8	is	be	AUX
ap-1807	394	9	a	a	DET
ap-1807	394	10	bounded	bounded	ADJ
ap-1807	394	11	operator	operator	NOUN
ap-1807	394	12	.	.	PUNCT
ap-1807	395	1	the	the	DET
ap-1807	395	2	set	set	NOUN
ap-1807	395	3	of	of	ADP
ap-1807	395	4	all	all	DET
ap-1807	395	5	bounded	bounded	ADJ
ap-1807	395	6	operators	operator	NOUN
ap-1807	395	7	on	on	ADP
ap-1807	395	8	h	h	NOUN
ap-1807	395	9	is	be	AUX
ap-1807	395	10	denoted	denote	VERB
ap-1807	395	11	by	by	ADP
ap-1807	395	12	b(h	b(h	NOUN
ap-1807	395	13	)	)	PUNCT
ap-1807	395	14	.	.	PUNCT
ap-1807	396	1	for	for	ADP
ap-1807	396	2	every	every	DET
ap-1807	396	3	bounded	bounded	ADJ
ap-1807	396	4	operator	operator	NOUN
ap-1807	396	5	a	a	PRON
ap-1807	396	6	:	:	PUNCT
ap-1807	396	7	d(a)→	d(a)→	PUNCT
ap-1807	396	8	h	h	PROPN
ap-1807	396	9	densely	densely	ADV
ap-1807	396	10	defined	define	VERB
ap-1807	396	11	on	on	ADP
ap-1807	396	12	d(a	d(a	PROPN
ap-1807	396	13	)	)	PUNCT
ap-1807	397	1	=	=	PUNCT
ap-1807	398	1	d	d	PROPN
ap-1807	398	2	⊂	⊂	PROPN
ap-1807	398	3	h	h	PROPN
ap-1807	398	4	exists	exist	VERB
ap-1807	398	5	a	a	DET
ap-1807	398	6	unique	unique	ADJ
ap-1807	398	7	extension	extension	NOUN
ap-1807	398	8	b	b	NOUN
ap-1807	398	9	such	such	ADJ
ap-1807	398	10	as	as	ADP
ap-1807	398	11	d(b	d(b	X
ap-1807	398	12	)	)	PUNCT
ap-1807	399	1	=	=	SYM
ap-1807	399	2	h	h	NOUN
ap-1807	399	3	and	and	CCONJ
ap-1807	399	4	ax	ax	NOUN
ap-1807	399	5	=	=	PROPN
ap-1807	399	6	bx	bx	PROPN
ap-1807	399	7	for	for	ADP
ap-1807	399	8	every	every	DET
ap-1807	399	9	x	x	PROPN
ap-1807	399	10	∈	∈	PROPN
ap-1807	399	11	d(a	d(a	PROPN
ap-1807	399	12	)	)	PUNCT
ap-1807	399	13	.	.	PUNCT
ap-1807	400	1	we	we	PRON
ap-1807	400	2	will	will	AUX
ap-1807	400	3	denote	denote	VERB
ap-1807	400	4	this	this	DET
ap-1807	400	5	extension	extension	NOUN
ap-1807	400	6	b	b	NOUN
ap-1807	400	7	=	=	SYM
ap-1807	400	8	ab	ab	PROPN
ap-1807	400	9	(	(	PUNCT
ap-1807	400	10	see	see	VERB
ap-1807	400	11	again	again	ADV
ap-1807	400	12	[	[	X
ap-1807	400	13	1	1	NUM
ap-1807	400	14	]	]	NUM
ap-1807	400	15	)	)	PUNCT
ap-1807	400	16	.	.	PUNCT
ap-1807	401	1	definition	definition	NOUN
ap-1807	401	2	5	5	NUM
ap-1807	401	3	.	.	PUNCT
ap-1807	402	1	for	for	ADP
ap-1807	402	2	an	an	DET
ap-1807	402	3	infinite	infinite	ADJ
ap-1807	402	4	-	-	PUNCT
ap-1807	402	5	dimensional	dimensional	ADJ
ap-1807	402	6	complex	complex	ADJ
ap-1807	402	7	hilbert	hilbert	NOUN
ap-1807	402	8	space	space	NOUN
ap-1807	402	9	h	h	NOUN
ap-1807	402	10	,	,	PUNCT
ap-1807	402	11	let	let	VERB
ap-1807	402	12	us	we	PRON
ap-1807	402	13	define	define	VERB
ap-1807	402	14	these	these	DET
ap-1807	402	15	sets	set	NOUN
ap-1807	402	16	of	of	ADP
ap-1807	402	17	operators	operator	NOUN
ap-1807	402	18	(	(	PUNCT
ap-1807	402	19	in	in	ADP
ap-1807	402	20	order	order	NOUN
ap-1807	402	21	to	to	ADP
ap-1807	402	22	[	[	X
ap-1807	402	23	4	4	NUM
ap-1807	402	24	,	,	PUNCT
ap-1807	402	25	6	6	NUM
ap-1807	402	26	]	]	SYM
ap-1807	402	27	):	):	PUNCT
ap-1807	402	28	•	•	NUM
ap-1807	402	29	gr(h	gr(h	NOUN
ap-1807	402	30	)	)	PUNCT
ap-1807	403	1	=	=	PRON
ap-1807	403	2	{	{	PUNCT
ap-1807	403	3	a	a	X
ap-1807	403	4	:	:	PUNCT
ap-1807	403	5	d(a)→	d(a)→	PUNCT
ap-1807	403	6	h	h	NOUN
ap-1807	403	7	|	|	ADV
ap-1807	403	8	d(a	d(a	PROPN
ap-1807	403	9	)	)	PUNCT
ap-1807	404	1	=	=	SYM
ap-1807	404	2	h	h	NOUN
ap-1807	404	3	and	and	CCONJ
ap-1807	404	4	d(a	d(a	PROPN
ap-1807	404	5	)	)	PUNCT
ap-1807	405	1	=	=	SYM
ap-1807	405	2	h	h	NOUN
ap-1807	405	3	if	if	SCONJ
ap-1807	405	4	a	a	PRON
ap-1807	405	5	is	be	AUX
ap-1807	405	6	bounded	bound	VERB
ap-1807	405	7	}	}	PUNCT
ap-1807	405	8	•	•	NUM
ap-1807	405	9	grd(h	grd(h	NOUN
ap-1807	405	10	)	)	PUNCT
ap-1807	406	1	=	=	PRON
ap-1807	406	2	{	{	PUNCT
ap-1807	406	3	a	a	DET
ap-1807	406	4	∈	∈	PROPN
ap-1807	406	5	gr(h	gr(h	NOUN
ap-1807	406	6	)	)	PUNCT
ap-1807	406	7	|	|	ADV
ap-1807	406	8	d(a	d(a	PROPN
ap-1807	406	9	)	)	PUNCT
ap-1807	407	1	=	=	SYM
ap-1807	407	2	d	d	PROPN
ap-1807	407	3	or	or	CCONJ
ap-1807	407	4	a	a	PRON
ap-1807	407	5	is	be	AUX
ap-1807	407	6	bounded	bound	VERB
ap-1807	407	7	}	}	PUNCT
ap-1807	407	8	•	•	NUM
ap-1807	407	9	sagr(h	sagr(h	NOUN
ap-1807	407	10	)	)	PUNCT
ap-1807	407	11	=	=	PRON
ap-1807	407	12	{	{	PUNCT
ap-1807	407	13	a	a	DET
ap-1807	407	14	∈	∈	PROPN
ap-1807	407	15	gr(h	gr(h	NOUN
ap-1807	407	16	)	)	PUNCT
ap-1807	407	17	|	|	ADV
ap-1807	407	18	a	a	DET
ap-1807	407	19	=	=	PUNCT
ap-1807	407	20	a∗	a∗	ADJ
ap-1807	407	21	}	}	PUNCT
ap-1807	407	22	•	•	NUM
ap-1807	407	23	sagrd(h	sagrd(h	NOUN
ap-1807	407	24	)	)	PUNCT
ap-1807	407	25	=	=	PRON
ap-1807	407	26	{	{	PUNCT
ap-1807	407	27	a	a	DET
ap-1807	407	28	∈	∈	PROPN
ap-1807	407	29	sagr(h	sagr(h	NOUN
ap-1807	407	30	)	)	PUNCT
ap-1807	407	31	|	|	ADV
ap-1807	407	32	d(a	d(a	PROPN
ap-1807	407	33	)	)	PUNCT
ap-1807	408	1	=	=	SYM
ap-1807	408	2	d	d	PROPN
ap-1807	408	3	or	or	CCONJ
ap-1807	408	4	a	a	PRON
ap-1807	408	5	is	be	AUX
ap-1807	408	6	bounded	bound	VERB
ap-1807	408	7	}	}	PUNCT
ap-1807	408	8	•	•	NOUN
ap-1807	408	9	v(h	v(h	NOUN
ap-1807	408	10	)	)	PUNCT
ap-1807	409	1	=	=	PRON
ap-1807	409	2	{	{	PUNCT
ap-1807	409	3	a	a	DET
ap-1807	409	4	∈	∈	PROPN
ap-1807	409	5	gr(h	gr(h	NOUN
ap-1807	409	6	)	)	PUNCT
ap-1807	409	7	|	|	ADV
ap-1807	409	8	a	a	DET
ap-1807	409	9	≥	≥	NOUN
ap-1807	409	10	0	0	NUM
ap-1807	409	11	}	}	PUNCT
ap-1807	409	12	.	.	PUNCT
ap-1807	410	1	we	we	PRON
ap-1807	410	2	also	also	ADV
ap-1807	410	3	define	define	VERB
ap-1807	410	4	an	an	DET
ap-1807	410	5	operation	operation	NOUN
ap-1807	410	6	⊕d	⊕d	NOUN
ap-1807	410	7	on	on	ADP
ap-1807	410	8	gr(h	gr(h	NOUN
ap-1807	410	9	)	)	PUNCT
ap-1807	410	10	by	by	ADP
ap-1807	410	11	a⊕db	a⊕db	NOUN
ap-1807	410	12	=	=	SYM
ap-1807	410	13			PROPN
ap-1807	410	14	a+b	a+b	NUM
ap-1807	410	15	if	if	SCONJ
ap-1807	410	16	a+b	a+b	NUM
ap-1807	410	17	is	be	AUX
ap-1807	410	18	unbounded	unbounded	ADJ
ap-1807	410	19	and	and	CCONJ
ap-1807	410	20	(	(	PUNCT
ap-1807	410	21	d(a	d(a	PROPN
ap-1807	410	22	)	)	PUNCT
ap-1807	410	23	=	=	SYM
ap-1807	410	24	d(b	d(b	X
ap-1807	410	25	)	)	PUNCT
ap-1807	410	26	or	or	CCONJ
ap-1807	410	27	one	one	NUM
ap-1807	410	28	out	out	ADP
ap-1807	410	29	of	of	ADP
ap-1807	410	30	a	a	PRON
ap-1807	410	31	,	,	PUNCT
ap-1807	410	32	b	b	PROPN
ap-1807	410	33	is	be	AUX
ap-1807	410	34	bounded	bound	VERB
ap-1807	410	35	)	)	PUNCT
ap-1807	410	36	,	,	PUNCT
ap-1807	410	37	(	(	PUNCT
ap-1807	410	38	a+b)b	a+b)b	VERB
ap-1807	410	39	if	if	SCONJ
ap-1807	410	40	a+b	a+b	NUM
ap-1807	410	41	is	be	AUX
ap-1807	410	42	bounded	bound	VERB
ap-1807	410	43	and	and	CCONJ
ap-1807	410	44	d(a	d(a	PROPN
ap-1807	410	45	)	)	PUNCT
ap-1807	410	46	=	=	PUNCT
ap-1807	410	47	d(b	d(b	PROPN
ap-1807	410	48	)	)	PUNCT
ap-1807	410	49	,	,	PUNCT
ap-1807	410	50	undefined	undefined	ADJ
ap-1807	410	51	otherwise	otherwise	ADV
ap-1807	410	52	,	,	PUNCT
ap-1807	410	53	and	and	CCONJ
ap-1807	410	54	operation	operation	NOUN
ap-1807	410	55	⊕u	⊕u	VERB
ap-1807	410	56	by	by	ADP
ap-1807	410	57	a⊕u	a⊕u	PROPN
ap-1807	410	58	b	b	NOUN
ap-1807	410	59	=	=	SYM
ap-1807	410	60	a⊕d	a⊕d	PROPN
ap-1807	410	61	b	b	PROPN
ap-1807	410	62	iff	iff	PROPN
ap-1807	410	63	at	at	ADV
ap-1807	410	64	least	least	ADJ
ap-1807	410	65	one	one	NUM
ap-1807	410	66	of	of	ADP
ap-1807	410	67	a	a	DET
ap-1807	410	68	,	,	PUNCT
ap-1807	410	69	b	b	PROPN
ap-1807	410	70	∈	∈	PROPN
ap-1807	410	71	gr(h	gr(h	NOUN
ap-1807	410	72	)	)	PUNCT
ap-1807	410	73	is	be	AUX
ap-1807	410	74	bounded	bound	VERB
ap-1807	410	75	or	or	CCONJ
ap-1807	410	76	a	a	DET
ap-1807	410	77	,	,	PUNCT
ap-1807	410	78	b	b	PROPN
ap-1807	410	79	∈	∈	PROPN
ap-1807	410	80	gr(h	gr(h	NOUN
ap-1807	410	81	)	)	PUNCT
ap-1807	410	82	are	be	AUX
ap-1807	410	83	both	both	ADV
ap-1807	410	84	unbounded	unbounded	ADJ
ap-1807	410	85	,	,	PUNCT
ap-1807	410	86	d(a	d(a	PROPN
ap-1807	410	87	)	)	PUNCT
ap-1807	410	88	=	=	SYM
ap-1807	410	89	d(b	d(b	X
ap-1807	410	90	)	)	PUNCT
ap-1807	410	91	and	and	CCONJ
ap-1807	410	92	there	there	PRON
ap-1807	410	93	exists	exist	VERB
ap-1807	410	94	a	a	DET
ap-1807	410	95	real	real	ADJ
ap-1807	410	96	number	number	NOUN
ap-1807	410	97	λa	λa	X
ap-1807	410	98	b	b	PROPN
ap-1807	410	99	6=	6=	PROPN
ap-1807	410	100	0	0	NUM
ap-1807	410	101	such	such	ADJ
ap-1807	410	102	that	that	DET
ap-1807	410	103	a−	a−	PROPN
ap-1807	410	104	λa	λa	X
ap-1807	410	105	bb	bb	PROPN
ap-1807	410	106	is	be	AUX
ap-1807	410	107	bounded[4	bounded[4	NOUN
ap-1807	410	108	]	]	X
ap-1807	410	109	.	.	PUNCT
ap-1807	411	1	for	for	ADP
ap-1807	411	2	an	an	DET
ap-1807	411	3	arbitrary	arbitrary	ADJ
ap-1807	411	4	subset	subset	NOUN
ap-1807	411	5	x	x	PUNCT
ap-1807	411	6	⊆	⊆	NUM
ap-1807	411	7	gr(h	gr(h	NOUN
ap-1807	411	8	)	)	PUNCT
ap-1807	411	9	let	let	VERB
ap-1807	411	10	us	we	PRON
ap-1807	411	11	define	define	VERB
ap-1807	411	12	a	a	DET
ap-1807	411	13	relation	relation	NOUN
ap-1807	411	14	≤x	≤x	PROPN
ap-1807	411	15	d	d	PROPN
ap-1807	411	16	(	(	PUNCT
ap-1807	411	17	resp	resp	NOUN
ap-1807	411	18	.	.	PUNCT
ap-1807	412	1	≤x	≤x	PROPN
ap-1807	412	2	u	u	NOUN
ap-1807	412	3	)	)	PUNCT
ap-1807	412	4	such	such	ADJ
ap-1807	412	5	that	that	PRON
ap-1807	412	6	for	for	ADP
ap-1807	412	7	any	any	DET
ap-1807	412	8	a	a	DET
ap-1807	412	9	,	,	PUNCT
ap-1807	412	10	b	b	PROPN
ap-1807	412	11	∈	∈	PROPN
ap-1807	412	12	x	x	NOUN
ap-1807	412	13	,	,	PUNCT
ap-1807	412	14	a	a	DET
ap-1807	412	15	≤x	≤x	PROPN
ap-1807	412	16	d	d	PROPN
ap-1807	412	17	b	b	PROPN
ap-1807	412	18	(	(	PUNCT
ap-1807	412	19	resp	resp	NOUN
ap-1807	412	20	.	.	PUNCT
ap-1807	413	1	a	a	DET
ap-1807	413	2	≤x	≤x	PROPN
ap-1807	413	3	u	u	PROPN
ap-1807	413	4	b	b	NOUN
ap-1807	413	5	)	)	PUNCT
ap-1807	413	6	iff	iff	PROPN
ap-1807	413	7	there	there	PRON
ap-1807	413	8	exists	exist	VERB
ap-1807	413	9	a	a	DET
ap-1807	413	10	positive	positive	ADJ
ap-1807	413	11	c	c	NOUN
ap-1807	413	12	∈	∈	PROPN
ap-1807	413	13	x	x	PUNCT
ap-1807	413	14	such	such	ADJ
ap-1807	413	15	that	that	SCONJ
ap-1807	413	16	a⊕d	a⊕d	NOUN
ap-1807	413	17	c	c	NOUN
ap-1807	413	18	=	=	SYM
ap-1807	413	19	b	b	PROPN
ap-1807	413	20	(	(	PUNCT
ap-1807	413	21	resp	resp	PROPN
ap-1807	413	22	.	.	PUNCT
ap-1807	413	23	a⊕u	a⊕u	PROPN
ap-1807	413	24	c	c	NOUN
ap-1807	413	25	=	=	SYM
ap-1807	413	26	b	b	PROPN
ap-1807	413	27	)	)	PUNCT
ap-1807	413	28	.	.	PUNCT
ap-1807	414	1	theorem	theorem	NOUN
ap-1807	414	2	3	3	X
ap-1807	414	3	.	.	PUNCT
ap-1807	415	1	let	let	VERB
ap-1807	415	2	h	h	PRON
ap-1807	415	3	be	be	AUX
ap-1807	415	4	an	an	DET
ap-1807	415	5	infinite	infinite	ADJ
ap-1807	415	6	-	-	PUNCT
ap-1807	415	7	dimensional	dimensional	ADJ
ap-1807	415	8	complex	complex	ADJ
ap-1807	415	9	hilbert	hilbert	NOUN
ap-1807	415	10	space	space	NOUN
ap-1807	415	11	.	.	PUNCT
ap-1807	416	1	then	then	ADV
ap-1807	416	2	(	(	PUNCT
ap-1807	416	3	gr(h),⊕d,0	gr(h),⊕d,0	NOUN
ap-1807	416	4	)	)	PUNCT
ap-1807	416	5	w.r.t	w.r.t	PROPN
ap-1807	416	6	.	.	PUNCT
ap-1807	417	1	relation	relation	NOUN
ap-1807	417	2	≤gr(h	≤gr(h	PROPN
ap-1807	417	3	)	)	PUNCT
ap-1807	418	1	d	d	NOUN
ap-1807	418	2	forms	form	VERB
ap-1807	418	3	a	a	DET
ap-1807	418	4	woa	woa	NOUN
ap-1807	418	5	-	-	PUNCT
ap-1807	418	6	group	group	NOUN
ap-1807	418	7	.	.	PUNCT
ap-1807	419	1	moreover	moreover	ADV
ap-1807	419	2	,	,	PUNCT
ap-1807	419	3	(	(	PUNCT
ap-1807	419	4	grd(h),⊕d	grd(h),⊕d	X
ap-1807	419	5	/	/	SYM
ap-1807	419	6	grd(h),0	grd(h),0	NOUN
ap-1807	419	7	)	)	PUNCT
ap-1807	419	8	w.r.t	w.r.t	NOUN
ap-1807	419	9	.	.	PUNCT
ap-1807	420	1	relation	relation	PROPN
ap-1807	420	2	≤grd(h	≤grd(h	PROPN
ap-1807	420	3	)	)	PUNCT
ap-1807	421	1	d	d	NOUN
ap-1807	421	2	forms	form	VERB
ap-1807	421	3	its	its	PRON
ap-1807	421	4	woa	woa	NOUN
ap-1807	421	5	-	-	PUNCT
ap-1807	421	6	subgroup	subgroup	NOUN
ap-1807	421	7	.	.	PUNCT
ap-1807	422	1	proof	proof	NOUN
ap-1807	422	2	.	.	PUNCT
ap-1807	423	1	it	it	PRON
ap-1807	423	2	has	have	AUX
ap-1807	423	3	been	be	AUX
ap-1807	423	4	shown	show	VERB
ap-1807	423	5	[	[	PUNCT
ap-1807	423	6	6	6	NUM
ap-1807	423	7	]	]	PUNCT
ap-1807	423	8	that	that	SCONJ
ap-1807	423	9	(	(	PUNCT
ap-1807	423	10	gr(h),⊕d,0	gr(h),⊕d,0	NOUN
ap-1807	423	11	)	)	PUNCT
ap-1807	423	12	w.r.t	w.r.t	PROPN
ap-1807	423	13	.	.	PUNCT
ap-1807	424	1	relation	relation	NOUN
ap-1807	424	2	≤gr(h	≤gr(h	PROPN
ap-1807	424	3	)	)	PUNCT
ap-1807	425	1	d	d	NOUN
ap-1807	425	2	forms	form	VERB
ap-1807	425	3	a	a	DET
ap-1807	425	4	wop	wop	NOUN
ap-1807	425	5	-	-	PUNCT
ap-1807	425	6	group	group	NOUN
ap-1807	425	7	.	.	PUNCT
ap-1807	426	1	moreover	moreover	ADV
ap-1807	426	2	,	,	PUNCT
ap-1807	426	3	in	in	ADP
ap-1807	426	4	[	[	PUNCT
ap-1807	426	5	4	4	NUM
ap-1807	426	6	,	,	PUNCT
ap-1807	426	7	lemma	lemma	PROPN
ap-1807	426	8	4	4	NUM
ap-1807	426	9	]	]	X
ap-1807	426	10	the	the	DET
ap-1807	426	11	axiom	axiom	NOUN
ap-1807	426	12	(	(	PUNCT
ap-1807	426	13	giv	giv	NOUN
ap-1807	426	14	)	)	PUNCT
ap-1807	426	15	is	be	AUX
ap-1807	426	16	proved	prove	VERB
ap-1807	426	17	.	.	PUNCT
ap-1807	427	1	axiom	axiom	NOUN
ap-1807	427	2	(	(	PUNCT
ap-1807	427	3	ri	ri	NOUN
ap-1807	427	4	)	)	PUNCT
ap-1807	427	5	holds	hold	VERB
ap-1807	427	6	by	by	ADP
ap-1807	427	7	definition	definition	NOUN
ap-1807	427	8	.	.	PUNCT
ap-1807	428	1	since	since	SCONJ
ap-1807	428	2	(	(	PUNCT
ap-1807	428	3	v(h),⊕d	v(h),⊕d	NOUN
ap-1807	428	4	/	/	SYM
ap-1807	428	5	v(h),0	v(h),0	NOUN
ap-1807	428	6	)	)	PUNCT
ap-1807	428	7	=(	=(	NOUN
ap-1807	428	8	pos(gr(h)),⊕d	pos(gr(h)),⊕d	PROPN
ap-1807	428	9	/	/	SYM
ap-1807	428	10	p	p	NOUN
ap-1807	428	11	os(gr(h)),0	os(gr(h)),0	NOUN
ap-1807	428	12	)	)	PUNCT
ap-1807	428	13	is	be	AUX
ap-1807	428	14	a	a	DET
ap-1807	428	15	generalized	generalized	ADJ
ap-1807	428	16	effect	effect	NOUN
ap-1807	428	17	algebra	algebra	NOUN
ap-1807	428	18	[	[	X
ap-1807	428	19	8	8	NUM
ap-1807	428	20	]	]	PUNCT
ap-1807	428	21	,	,	PUNCT
ap-1807	428	22	(	(	PUNCT
ap-1807	428	23	rii	rii	NOUN
ap-1807	428	24	)	)	PUNCT
ap-1807	428	25	and	and	CCONJ
ap-1807	428	26	(	(	PUNCT
ap-1807	428	27	riii	riii	PROPN
ap-1807	428	28	)	)	PUNCT
ap-1807	428	29	hold	hold	NOUN
ap-1807	428	30	.	.	PUNCT
ap-1807	429	1	according	accord	VERB
ap-1807	429	2	to	to	ADP
ap-1807	429	3	[	[	X
ap-1807	429	4	6	6	NUM
ap-1807	429	5	]	]	PUNCT
ap-1807	429	6	(	(	PUNCT
ap-1807	429	7	grd(h	grd(h	NOUN
ap-1807	429	8	)	)	PUNCT
ap-1807	429	9	,	,	PUNCT
ap-1807	429	10	⊕d	⊕d	NOUN
ap-1807	429	11	/	/	SYM
ap-1807	429	12	grd(h),0	grd(h),0	NOUN
ap-1807	429	13	)	)	PUNCT
ap-1807	429	14	w.r.t	w.r.t	NOUN
ap-1807	429	15	.	.	PUNCT
ap-1807	429	16	≤gr(h	≤gr(h	NOUN
ap-1807	429	17	)	)	PUNCT
ap-1807	430	1	d	d	NOUN
ap-1807	430	2	is	be	AUX
ap-1807	430	3	a	a	DET
ap-1807	430	4	wopsubgroup	wopsubgroup	NOUN
ap-1807	430	5	hence	hence	ADV
ap-1807	430	6	by	by	ADP
ap-1807	430	7	corollary	corollary	ADJ
ap-1807	430	8	1	1	NUM
ap-1807	430	9	it	it	PRON
ap-1807	430	10	is	be	AUX
ap-1807	430	11	also	also	ADV
ap-1807	430	12	a	a	DET
ap-1807	430	13	woasubgroup	woasubgroup	NOUN
ap-1807	430	14	.	.	PUNCT
ap-1807	431	1	note	note	VERB
ap-1807	431	2	that	that	SCONJ
ap-1807	431	3	since	since	SCONJ
ap-1807	431	4	the	the	DET
ap-1807	431	5	operation	operation	NOUN
ap-1807	431	6	⊕d	⊕d	NOUN
ap-1807	431	7	/	/	SYM
ap-1807	431	8	grd(h	grd(h	NOUN
ap-1807	431	9	)	)	PUNCT
ap-1807	431	10	is	be	AUX
ap-1807	431	11	total	total	ADJ
ap-1807	431	12	on	on	ADP
ap-1807	431	13	grd(h	grd(h	NOUN
ap-1807	431	14	)	)	PUNCT
ap-1807	431	15	,	,	PUNCT
ap-1807	431	16	it	it	PRON
ap-1807	431	17	is	be	AUX
ap-1807	431	18	also	also	ADV
ap-1807	431	19	a	a	DET
ap-1807	431	20	partially	partially	ADV
ap-1807	431	21	ordered	order	VERB
ap-1807	431	22	commutative	commutative	ADJ
ap-1807	431	23	group	group	NOUN
ap-1807	431	24	.	.	PUNCT
ap-1807	432	1	theorem	theorem	ADJ
ap-1807	432	2	4	4	NUM
ap-1807	432	3	.	.	PUNCT
ap-1807	433	1	let	let	VERB
ap-1807	433	2	h	h	PRON
ap-1807	433	3	be	be	AUX
ap-1807	433	4	an	an	DET
ap-1807	433	5	infinite	infinite	ADJ
ap-1807	433	6	-	-	PUNCT
ap-1807	433	7	dimensional	dimensional	ADJ
ap-1807	433	8	complex	complex	ADJ
ap-1807	433	9	hilbert	hilbert	NOUN
ap-1807	433	10	space	space	NOUN
ap-1807	433	11	.	.	PUNCT
ap-1807	434	1	then	then	ADV
ap-1807	434	2	(	(	PUNCT
ap-1807	434	3	gr(h),⊕u,0	gr(h),⊕u,0	PROPN
ap-1807	434	4	)	)	PUNCT
ap-1807	434	5	w.r.t	w.r.t	NOUN
ap-1807	434	6	.	.	PUNCT
ap-1807	435	1	relation	relation	NOUN
ap-1807	435	2	≤gr(h	≤gr(h	PROPN
ap-1807	435	3	)	)	PUNCT
ap-1807	436	1	u	u	NOUN
ap-1807	436	2	forms	form	VERB
ap-1807	436	3	a	a	DET
ap-1807	436	4	woa	woa	NOUN
ap-1807	436	5	-	-	PUNCT
ap-1807	436	6	group	group	NOUN
ap-1807	436	7	.	.	PUNCT
ap-1807	437	1	moreover	moreover	ADV
ap-1807	437	2	,	,	PUNCT
ap-1807	437	3	(	(	PUNCT
ap-1807	437	4	grd(h	grd(h	NOUN
ap-1807	437	5	)	)	PUNCT
ap-1807	437	6	,	,	PUNCT
ap-1807	437	7	⊕u	⊕u	ADP
ap-1807	437	8	/	/	SYM
ap-1807	437	9	grd(h),0	grd(h),0	NOUN
ap-1807	437	10	)	)	PUNCT
ap-1807	437	11	w.r.t	w.r.t	NOUN
ap-1807	437	12	.	.	PUNCT
ap-1807	438	1	relation	relation	PROPN
ap-1807	438	2	≤grd(h	≤grd(h	PROPN
ap-1807	438	3	)	)	PUNCT
ap-1807	438	4	u	u	NOUN
ap-1807	438	5	,	,	PUNCT
ap-1807	438	6	(	(	PUNCT
ap-1807	438	7	sagr(h),⊕u	sagr(h),⊕u	VERB
ap-1807	438	8	/	/	SYM
ap-1807	438	9	sagr(h),0	sagr(h),0	ADJ
ap-1807	438	10	)	)	PUNCT
ap-1807	438	11	w.r.t	w.r.t	NOUN
ap-1807	438	12	.	.	PUNCT
ap-1807	439	1	relation	relation	NOUN
ap-1807	439	2	≤sagr(h	≤sagr(h	PROPN
ap-1807	439	3	)	)	PUNCT
ap-1807	439	4	u	u	NOUN
ap-1807	439	5	and	and	CCONJ
ap-1807	439	6	(	(	PUNCT
ap-1807	439	7	sagrd(h),⊕u	sagrd(h),⊕u	NOUN
ap-1807	439	8	/	/	SYM
ap-1807	439	9	sagrd(h),0	sagrd(h),0	ADJ
ap-1807	439	10	)	)	PUNCT
ap-1807	439	11	w.r.t	w.r.t	NOUN
ap-1807	439	12	.	.	PUNCT
ap-1807	440	1	relation	relation	NOUN
ap-1807	440	2	≤sagrd(h	≤sagrd(h	NUM
ap-1807	440	3	)	)	PUNCT
ap-1807	440	4	u	u	PRON
ap-1807	440	5	form	form	VERB
ap-1807	440	6	its	its	PRON
ap-1807	440	7	woa	woa	NOUN
ap-1807	440	8	-	-	PUNCT
ap-1807	440	9	subgroups	subgroup	NOUN
ap-1807	440	10	.	.	PUNCT
ap-1807	441	1	proof	proof	NOUN
ap-1807	441	2	.	.	PUNCT
ap-1807	442	1	we	we	PRON
ap-1807	442	2	have	have	AUX
ap-1807	442	3	shown	show	VERB
ap-1807	442	4	[	[	PUNCT
ap-1807	442	5	6	6	NUM
ap-1807	442	6	]	]	PUNCT
ap-1807	442	7	that	that	SCONJ
ap-1807	442	8	(	(	PUNCT
ap-1807	442	9	gr(h),⊕u,0	gr(h),⊕u,0	PROPN
ap-1807	442	10	)	)	PUNCT
ap-1807	442	11	w.r.t	w.r.t	NOUN
ap-1807	442	12	.	.	PUNCT
ap-1807	443	1	relation	relation	NOUN
ap-1807	443	2	≤gr(h	≤gr(h	PROPN
ap-1807	443	3	)	)	PUNCT
ap-1807	443	4	u	u	NOUN
ap-1807	443	5	forms	form	VERB
ap-1807	443	6	a	a	DET
ap-1807	443	7	wop	wop	NOUN
ap-1807	443	8	-	-	PUNCT
ap-1807	443	9	group	group	NOUN
ap-1807	443	10	.	.	PUNCT
ap-1807	444	1	in	in	ADP
ap-1807	444	2	[	[	X
ap-1807	444	3	4	4	NUM
ap-1807	444	4	,	,	PUNCT
ap-1807	444	5	lemma	lemma	PROPN
ap-1807	444	6	6	6	NUM
ap-1807	444	7	]	]	PUNCT
ap-1807	444	8	.	.	PUNCT
ap-1807	445	1	is	be	AUX
ap-1807	445	2	proved	prove	VERB
ap-1807	445	3	the	the	DET
ap-1807	445	4	axiom	axiom	NOUN
ap-1807	445	5	(	(	PUNCT
ap-1807	445	6	giv	giv	PROPN
ap-1807	445	7	)	)	PUNCT
ap-1807	445	8	.	.	PUNCT
ap-1807	446	1	(	(	PUNCT
ap-1807	446	2	ri	ri	NOUN
ap-1807	446	3	)	)	PUNCT
ap-1807	446	4	holds	hold	VERB
ap-1807	446	5	by	by	ADP
ap-1807	446	6	definition	definition	NOUN
ap-1807	446	7	.	.	PUNCT
ap-1807	447	1	because	because	SCONJ
ap-1807	447	2	(	(	PUNCT
ap-1807	447	3	v(h)⊕u	v(h)⊕u	NOUN
ap-1807	447	4	/	/	SYM
ap-1807	447	5	v(h),0	v(h),0	NOUN
ap-1807	447	6	)	)	PUNCT
ap-1807	447	7	=(	=(	NOUN
ap-1807	447	8	pos(gr(h)),⊕u	pos(gr(h)),⊕u	NOUN
ap-1807	447	9	/	/	SYM
ap-1807	447	10	p	p	NOUN
ap-1807	447	11	os(gr(h)),0	os(gr(h)),0	NOUN
ap-1807	447	12	)	)	PUNCT
ap-1807	447	13	is	be	AUX
ap-1807	447	14	a	a	DET
ap-1807	447	15	generalized	generalized	ADJ
ap-1807	447	16	effect	effect	NOUN
ap-1807	447	17	algebra	algebra	NOUN
ap-1807	447	18	[	[	X
ap-1807	447	19	4	4	NUM
ap-1807	447	20	,	,	PUNCT
ap-1807	447	21	7	7	NUM
ap-1807	447	22	]	]	PUNCT
ap-1807	447	23	,	,	PUNCT
ap-1807	447	24	we	we	PRON
ap-1807	447	25	have	have	VERB
ap-1807	447	26	(	(	PUNCT
ap-1807	447	27	rii	rii	VERB
ap-1807	447	28	)	)	PUNCT
ap-1807	447	29	and	and	CCONJ
ap-1807	447	30	(	(	PUNCT
ap-1807	447	31	riii	riii	PROPN
ap-1807	447	32	)	)	PUNCT
ap-1807	447	33	.	.	PUNCT
ap-1807	448	1	according	accord	VERB
ap-1807	448	2	to	to	ADP
ap-1807	448	3	[	[	X
ap-1807	448	4	4	4	NUM
ap-1807	448	5	]	]	X
ap-1807	448	6	(	(	PUNCT
ap-1807	448	7	grd(h),⊕u	grd(h),⊕u	NOUN
ap-1807	448	8	/	/	SYM
ap-1807	448	9	grd(h),0	grd(h),0	NOUN
ap-1807	448	10	)	)	PUNCT
ap-1807	448	11	w.r.t	w.r.t	NOUN
ap-1807	448	12	.	.	PUNCT
ap-1807	449	1	relation	relation	PROPN
ap-1807	449	2	≤grd(h	≤grd(h	PROPN
ap-1807	449	3	)	)	PUNCT
ap-1807	449	4	u	u	NOUN
ap-1807	449	5	,	,	PUNCT
ap-1807	449	6	293	293	NUM
ap-1807	449	7	jiří	jiří	NOUN
ap-1807	449	8	janda	janda	PROPN
ap-1807	449	9	acta	acta	PROPN
ap-1807	449	10	polytechnica	polytechnica	PROPN
ap-1807	449	11	(	(	PUNCT
ap-1807	449	12	sagr(h),⊕u	sagr(h),⊕u	NOUN
ap-1807	449	13	/	/	SYM
ap-1807	449	14	sagr(h),0	sagr(h),0	ADJ
ap-1807	449	15	)	)	PUNCT
ap-1807	449	16	w.r.t	w.r.t	NOUN
ap-1807	449	17	.	.	PUNCT
ap-1807	450	1	relation	relation	NOUN
ap-1807	450	2	≤sagr(h	≤sagr(h	PROPN
ap-1807	450	3	)	)	PUNCT
ap-1807	450	4	u	u	NOUN
ap-1807	450	5	and	and	CCONJ
ap-1807	450	6	(	(	PUNCT
ap-1807	450	7	sagrd(h),⊕u	sagrd(h),⊕u	NOUN
ap-1807	450	8	/	/	SYM
ap-1807	450	9	sagrd(h),0	sagrd(h),0	ADJ
ap-1807	450	10	)	)	PUNCT
ap-1807	450	11	w.r.t	w.r.t	NOUN
ap-1807	450	12	.	.	PUNCT
ap-1807	451	1	≤sagrd(h	≤sagrd(h	PUNCT
ap-1807	451	2	)	)	PUNCT
ap-1807	451	3	u	u	PRON
ap-1807	451	4	are	be	AUX
ap-1807	451	5	wop	wop	NOUN
ap-1807	451	6	-	-	PUNCT
ap-1807	451	7	subgroups	subgroup	NOUN
ap-1807	451	8	hence	hence	ADV
ap-1807	451	9	by	by	ADP
ap-1807	451	10	using	use	VERB
ap-1807	451	11	corollary	corollary	ADJ
ap-1807	451	12	1	1	NUM
ap-1807	451	13	they	they	PRON
ap-1807	451	14	are	be	AUX
ap-1807	451	15	woa	woa	ADJ
ap-1807	451	16	-	-	PUNCT
ap-1807	451	17	subgroups	subgroup	NOUN
ap-1807	451	18	.	.	PUNCT
ap-1807	452	1	acknowledgements	acknowledgement	NOUN
ap-1807	452	2	the	the	DET
ap-1807	452	3	author	author	NOUN
ap-1807	452	4	gratefully	gratefully	ADV
ap-1807	452	5	acknowledges	acknowledge	VERB
ap-1807	452	6	financial	financial	ADJ
ap-1807	452	7	support	support	NOUN
ap-1807	452	8	from	from	ADP
ap-1807	452	9	masaryk	masaryk	PROPN
ap-1807	452	10	university	university	PROPN
ap-1807	452	11	under	under	ADP
ap-1807	452	12	grant	grant	NOUN
ap-1807	452	13	0964/2009	0964/2009	NUM
ap-1807	452	14	and	and	CCONJ
ap-1807	452	15	financial	financial	ADJ
ap-1807	452	16	support	support	NOUN
ap-1807	452	17	from	from	ADP
ap-1807	452	18	esf	esf	PROPN
ap-1807	452	19	project	project	PROPN
ap-1807	452	20	cz.1.07/2.3.00/20.0051	cz.1.07/2.3.00/20.0051	PROPN
ap-1807	452	21	algebraic	algebraic	ADJ
ap-1807	452	22	methods	method	NOUN
ap-1807	452	23	in	in	ADP
ap-1807	452	24	quantum	quantum	ADJ
ap-1807	452	25	logic	logic	NOUN
ap-1807	452	26	of	of	ADP
ap-1807	452	27	the	the	DET
ap-1807	452	28	masaryk	masaryk	PROPN
ap-1807	452	29	university	university	PROPN
ap-1807	452	30	.	.	PUNCT
ap-1807	453	1	references	reference	NOUN
ap-1807	453	2	[	[	X
ap-1807	453	3	1	1	NUM
ap-1807	453	4	]	]	PUNCT
ap-1807	453	5	blank	blank	PROPN
ap-1807	453	6	j.	j.	PROPN
ap-1807	453	7	,	,	PUNCT
ap-1807	453	8	exner	exner	PROPN
ap-1807	453	9	p.	p.	PROPN
ap-1807	453	10	,	,	PUNCT
ap-1807	453	11	havlíček	havlíček	NOUN
ap-1807	453	12	m.	m.	NOUN
ap-1807	453	13	,	,	PUNCT
ap-1807	453	14	hilbert	hilbert	NOUN
ap-1807	453	15	space	space	NOUN
ap-1807	453	16	operators	operator	NOUN
ap-1807	453	17	in	in	ADP
ap-1807	453	18	quantum	quantum	ADJ
ap-1807	453	19	physics	physics	NOUN
ap-1807	453	20	,	,	PUNCT
ap-1807	453	21	2nd	2nd	PROPN
ap-1807	453	22	edn	edn	PROPN
ap-1807	453	23	.	.	PUNCT
ap-1807	453	24	springer	springer	PROPN
ap-1807	453	25	,	,	PUNCT
ap-1807	453	26	berlin	berlin	PROPN
ap-1807	453	27	(	(	PUNCT
ap-1807	453	28	2008	2008	NUM
ap-1807	453	29	)	)	PUNCT
ap-1807	453	30	.	.	PUNCT
ap-1807	454	1	[	[	X
ap-1807	454	2	2	2	X
ap-1807	454	3	]	]	PUNCT
ap-1807	454	4	dvurečenskij	dvurečenskij	PROPN
ap-1807	454	5	a.	a.	PROPN
ap-1807	454	6	,	,	PUNCT
ap-1807	454	7	pulmannová	pulmannová	PROPN
ap-1807	454	8	s.	s.	PROPN
ap-1807	454	9	,	,	PUNCT
ap-1807	454	10	new	new	ADJ
ap-1807	454	11	trends	trend	NOUN
ap-1807	454	12	in	in	ADP
ap-1807	454	13	quantum	quantum	ADJ
ap-1807	454	14	structures	structure	NOUN
ap-1807	454	15	,	,	PUNCT
ap-1807	454	16	kluwer	kluwer	NOUN
ap-1807	454	17	acad	acad	PROPN
ap-1807	454	18	.	.	PUNCT
ap-1807	455	1	publ	publ	PROPN
ap-1807	455	2	.	.	PUNCT
ap-1807	455	3	,	,	PUNCT
ap-1807	455	4	dordrecht	dordrecht	PROPN
ap-1807	455	5	/	/	SYM
ap-1807	455	6	ister	ister	PROPN
ap-1807	455	7	science	science	NOUN
ap-1807	455	8	,	,	PUNCT
ap-1807	455	9	bratislava	bratislava	PROPN
ap-1807	455	10	,	,	PUNCT
ap-1807	455	11	2000	2000	NUM
ap-1807	455	12	.	.	PUNCT
ap-1807	456	1	[	[	X
ap-1807	456	2	3	3	X
ap-1807	456	3	]	]	X
ap-1807	456	4	foulis	foulis	PROPN
ap-1807	456	5	d.	d.	PROPN
ap-1807	456	6	j.	j.	PROPN
ap-1807	456	7	,	,	PUNCT
ap-1807	456	8	bennett	bennett	PROPN
ap-1807	456	9	m.	m.	PROPN
ap-1807	456	10	k.	k.	PROPN
ap-1807	456	11	,	,	PUNCT
ap-1807	456	12	effect	effect	NOUN
ap-1807	456	13	algebras	algebra	NOUN
ap-1807	456	14	and	and	CCONJ
ap-1807	456	15	unsharp	unsharp	ADJ
ap-1807	456	16	quantum	quantum	ADJ
ap-1807	456	17	logics	logic	NOUN
ap-1807	456	18	,	,	PUNCT
ap-1807	456	19	found	find	VERB
ap-1807	456	20	.	.	PUNCT
ap-1807	457	1	phys	phy	NOUN
ap-1807	457	2	.	.	PUNCT
ap-1807	458	1	24	24	NUM
ap-1807	458	2	(	(	PUNCT
ap-1807	458	3	1994	1994	NUM
ap-1807	458	4	)	)	PUNCT
ap-1807	458	5	,	,	PUNCT
ap-1807	458	6	1331–1352	1331–1352	NUM
ap-1807	458	7	.	.	PUNCT
ap-1807	459	1	[	[	X
ap-1807	459	2	4	4	NUM
ap-1807	459	3	]	]	PUNCT
ap-1807	459	4	janda	janda	PROPN
ap-1807	459	5	j.	j.	PROPN
ap-1807	459	6	,	,	PUNCT
ap-1807	459	7	weakly	weakly	ADV
ap-1807	459	8	ordered	order	VERB
ap-1807	459	9	partial	partial	ADJ
ap-1807	459	10	commutative	commutative	ADJ
ap-1807	459	11	group	group	NOUN
ap-1807	459	12	of	of	ADP
ap-1807	459	13	self	self	NOUN
ap-1807	459	14	-	-	PUNCT
ap-1807	459	15	adjoint	adjoint	NOUN
ap-1807	459	16	operators	operator	NOUN
ap-1807	459	17	densely	densely	ADV
ap-1807	459	18	defined	define	VERB
ap-1807	459	19	on	on	ADP
ap-1807	459	20	hilbert	hilbert	NOUN
ap-1807	459	21	space	space	NOUN
ap-1807	459	22	,	,	PUNCT
ap-1807	459	23	tatra	tatra	PROPN
ap-1807	459	24	mt	mt	PROPN
ap-1807	459	25	.	.	PROPN
ap-1807	459	26	math	math	PROPN
ap-1807	459	27	.	.	PUNCT
ap-1807	459	28	publ	publ	PROPN
ap-1807	459	29	.	.	PUNCT
ap-1807	459	30	,	,	PUNCT
ap-1807	459	31	50	50	NUM
ap-1807	459	32	(	(	PUNCT
ap-1807	459	33	2011	2011	NUM
ap-1807	459	34	)	)	PUNCT
ap-1807	459	35	,	,	PUNCT
ap-1807	459	36	1	1	NUM
ap-1807	459	37	-	-	SYM
ap-1807	459	38	16	16	NUM
ap-1807	459	39	.	.	PUNCT
ap-1807	460	1	[	[	X
ap-1807	460	2	5	5	NUM
ap-1807	460	3	]	]	PUNCT
ap-1807	460	4	paseka	paseka	ADP
ap-1807	460	5	j.	j.	PROPN
ap-1807	460	6	,	,	PUNCT
ap-1807	460	7	pt	pt	PROPN
ap-1807	460	8	-symmetry	-symmetry	NOUN
ap-1807	460	9	in	in	ADP
ap-1807	460	10	(	(	PUNCT
ap-1807	460	11	generalized	generalized	ADJ
ap-1807	460	12	)	)	PUNCT
ap-1807	460	13	effect	effect	NOUN
ap-1807	460	14	algebras	algebra	NOUN
ap-1807	460	15	,	,	PUNCT
ap-1807	460	16	internat	internat	PROPN
ap-1807	460	17	.	.	PUNCT
ap-1807	461	1	j.	j.	PROPN
ap-1807	461	2	theoret	theoret	PROPN
ap-1807	461	3	.	.	PUNCT
ap-1807	462	1	phys	phy	NOUN
ap-1807	462	2	.	.	PUNCT
ap-1807	462	3	,	,	PUNCT
ap-1807	462	4	50	50	NUM
ap-1807	462	5	(	(	PUNCT
ap-1807	462	6	2011	2011	NUM
ap-1807	462	7	)	)	PUNCT
ap-1807	462	8	,	,	PUNCT
ap-1807	462	9	1198–1205	1198–1205	NUM
ap-1807	462	10	.	.	PUNCT
ap-1807	463	1	[	[	X
ap-1807	463	2	6	6	NUM
ap-1807	463	3	]	]	PUNCT
ap-1807	463	4	paseka	paseka	ADP
ap-1807	463	5	j.	j.	PROPN
ap-1807	463	6	,	,	PUNCT
ap-1807	463	7	janda	janda	PROPN
ap-1807	463	8	j.	j.	PROPN
ap-1807	463	9	,	,	PUNCT
ap-1807	463	10	more	more	ADV
ap-1807	463	11	on	on	ADP
ap-1807	463	12	pt	pt	NOUN
ap-1807	463	13	-symmetry	-symmetry	NOUN
ap-1807	463	14	in	in	ADP
ap-1807	463	15	(	(	PUNCT
ap-1807	463	16	generalized	generalized	ADJ
ap-1807	463	17	)	)	PUNCT
ap-1807	463	18	effect	effect	NOUN
ap-1807	463	19	algebras	algebra	NOUN
ap-1807	463	20	and	and	CCONJ
ap-1807	463	21	partial	partial	ADJ
ap-1807	463	22	groups	group	NOUN
ap-1807	463	23	,	,	PUNCT
ap-1807	463	24	acta	acta	PROPN
ap-1807	463	25	polytechnica	polytechnica	PROPN
ap-1807	463	26	,	,	PUNCT
ap-1807	463	27	51	51	NUM
ap-1807	463	28	(	(	PUNCT
ap-1807	463	29	2011	2011	NUM
ap-1807	463	30	)	)	PUNCT
ap-1807	463	31	,	,	PUNCT
ap-1807	463	32	no	no	INTJ
ap-1807	463	33	.	.	NOUN
ap-1807	463	34	4	4	NUM
ap-1807	463	35	,	,	PUNCT
ap-1807	463	36	65–72	65–72	NUM
ap-1807	463	37	.	.	PUNCT
ap-1807	464	1	[	[	X
ap-1807	464	2	7	7	X
ap-1807	464	3	]	]	PUNCT
ap-1807	464	4	paseka	paseka	SCONJ
ap-1807	464	5	j.	j.	PROPN
ap-1807	464	6	,	,	PUNCT
ap-1807	464	7	riečanová	riečanová	PROPN
ap-1807	464	8	z.	z.	PROPN
ap-1807	464	9	,	,	PUNCT
ap-1807	464	10	considerable	considerable	ADJ
ap-1807	464	11	sets	set	NOUN
ap-1807	464	12	of	of	ADP
ap-1807	464	13	linear	linear	PROPN
ap-1807	464	14	operators	operator	NOUN
ap-1807	464	15	in	in	ADP
ap-1807	464	16	hilbert	hilbert	PROPN
ap-1807	464	17	spaces	space	NOUN
ap-1807	464	18	as	as	ADP
ap-1807	464	19	generalized	generalized	ADJ
ap-1807	464	20	effect	effect	NOUN
ap-1807	464	21	algebras	algebra	NOUN
ap-1807	464	22	.	.	PUNCT
ap-1807	464	23	found	find	VERB
ap-1807	464	24	.	.	PUNCT
ap-1807	465	1	phys	phy	NOUN
ap-1807	465	2	.	.	PUNCT
ap-1807	466	1	41	41	NUM
ap-1807	466	2	(	(	PUNCT
ap-1807	466	3	2011	2011	NUM
ap-1807	466	4	)	)	PUNCT
ap-1807	466	5	,	,	PUNCT
ap-1807	466	6	1634–1647	1634–1647	NUM
ap-1807	466	7	.	.	PUNCT
ap-1807	467	1	[	[	X
ap-1807	467	2	8	8	NUM
ap-1807	467	3	]	]	X
ap-1807	467	4	riečanová	riečanová	PROPN
ap-1807	467	5	z.	z.	PROPN
ap-1807	467	6	,	,	PUNCT
ap-1807	467	7	zajac	zajac	PROPN
ap-1807	467	8	m.	m.	PROPN
ap-1807	467	9	and	and	CCONJ
ap-1807	467	10	pulmannová	pulmannová	PROPN
ap-1807	467	11	s.	s.	PROPN
ap-1807	467	12	,	,	PUNCT
ap-1807	467	13	effect	effect	NOUN
ap-1807	467	14	algebras	algebra	NOUN
ap-1807	467	15	of	of	ADP
ap-1807	467	16	positive	positive	ADJ
ap-1807	467	17	linear	linear	PROPN
ap-1807	467	18	operators	operator	NOUN
ap-1807	467	19	densely	densely	ADV
ap-1807	467	20	defined	define	VERB
ap-1807	467	21	on	on	ADP
ap-1807	467	22	hilbert	hilbert	PROPN
ap-1807	467	23	spaces	space	NOUN
ap-1807	467	24	,	,	PUNCT
ap-1807	467	25	reports	report	NOUN
ap-1807	467	26	on	on	ADP
ap-1807	467	27	mathematical	mathematical	ADJ
ap-1807	467	28	physics	physics	NOUN
ap-1807	467	29	68	68	NUM
ap-1807	467	30	,	,	PUNCT
ap-1807	467	31	(	(	PUNCT
ap-1807	467	32	2011	2011	NUM
ap-1807	467	33	)	)	PUNCT
ap-1807	467	34	,	,	PUNCT
ap-1807	467	35	261–270	261–270	NUM
ap-1807	467	36	.	.	PUNCT
ap-1807	468	1	294	294	NUM
ap-1807	468	2	acta	acta	PROPN
ap-1807	468	3	polytechnica	polytechnica	PROPN
ap-1807	468	4	53(3):289–294	53(3):289–294	NOUN
ap-1807	468	5	,	,	PUNCT
ap-1807	468	6	2013	2013	NUM
ap-1807	468	7	1	1	NUM
ap-1807	468	8	introduction	introduction	NOUN
ap-1807	468	9	2	2	NUM
ap-1807	468	10	preliminaries	preliminary	NOUN
ap-1807	468	11	3	3	NUM
ap-1807	468	12	hilbert	hilbert	NOUN
ap-1807	468	13	spaces	space	VERB
ap-1807	468	14	acknowledgements	acknowledgement	NOUN
ap-1807	468	15	references	reference	NOUN