id	sid	tid	token	lemma	pos
ejpam-5390	1	1	european	european	PROPN
ejpam-5390	1	2	journal	journal	PROPN
ejpam-5390	1	3	of	of	ADP
ejpam-5390	1	4	pure	pure	ADJ
ejpam-5390	1	5	and	and	CCONJ
ejpam-5390	1	6	applied	apply	VERB
ejpam-5390	1	7	mathematics	mathematic	NOUN
ejpam-5390	1	8	vol	vol	NOUN
ejpam-5390	1	9	.	.	PROPN
ejpam-5390	2	1	17	17	NUM
ejpam-5390	2	2	,	,	PUNCT
ejpam-5390	2	3	no	no	INTJ
ejpam-5390	2	4	.	.	NOUN
ejpam-5390	2	5	4	4	NUM
ejpam-5390	2	6	,	,	PUNCT
ejpam-5390	2	7	2024	2024	NUM
ejpam-5390	2	8	,	,	PUNCT
ejpam-5390	2	9	3399	3399	NUM
ejpam-5390	2	10	-	-	SYM
ejpam-5390	2	11	3414	3414	NUM
ejpam-5390	2	12	issn	issn	PROPN
ejpam-5390	2	13	1307	1307	NUM
ejpam-5390	2	14	-	-	SYM
ejpam-5390	2	15	5543	5543	NUM
ejpam-5390	2	16	–	–	PUNCT
ejpam-5390	2	17	ejpam.com	ejpam.com	X
ejpam-5390	2	18	published	publish	VERB
ejpam-5390	2	19	by	by	ADP
ejpam-5390	2	20	new	new	PROPN
ejpam-5390	2	21	york	york	PROPN
ejpam-5390	2	22	business	business	PROPN
ejpam-5390	2	23	global	global	PROPN
ejpam-5390	2	24	nonlinear	nonlinear	PROPN
ejpam-5390	2	25	mixed	mixed	PROPN
ejpam-5390	2	26	λ	λ	PROPN
ejpam-5390	2	27	-	-	PROPN
ejpam-5390	2	28	jordan	jordan	PROPN
ejpam-5390	2	29	triple	triple	ADJ
ejpam-5390	2	30	derivation	derivation	NOUN
ejpam-5390	2	31	on	on	ADP
ejpam-5390	2	32	∗-algebras	∗-algebra	NOUN
ejpam-5390	2	33	amal	amal	PROPN
ejpam-5390	2	34	s.	s.	PROPN
ejpam-5390	2	35	alali1	alali1	PROPN
ejpam-5390	2	36	,	,	PUNCT
ejpam-5390	3	1	junaid	junaid	VERB
ejpam-5390	3	2	nisar2,∗	nisar2,∗	PROPN
ejpam-5390	3	3	,	,	PUNCT
ejpam-5390	3	4	nadeem	nadeem	PROPN
ejpam-5390	3	5	ur	ur	INTJ
ejpam-5390	3	6	rehman3	rehman3	PROPN
ejpam-5390	3	7	,	,	PUNCT
ejpam-5390	3	8	hafedh	hafedh	NOUN
ejpam-5390	3	9	m.	m.	NOUN
ejpam-5390	3	10	alnoghashi4	alnoghashi4	NOUN
ejpam-5390	3	11	1	1	NUM
ejpam-5390	3	12	department	department	NOUN
ejpam-5390	3	13	of	of	ADP
ejpam-5390	3	14	mathematical	mathematical	ADJ
ejpam-5390	3	15	sciences	sciences	PROPN
ejpam-5390	3	16	,	,	PUNCT
ejpam-5390	3	17	college	college	NOUN
ejpam-5390	3	18	of	of	ADP
ejpam-5390	3	19	science	science	NOUN
ejpam-5390	3	20	,	,	PUNCT
ejpam-5390	3	21	princess	princess	PROPN
ejpam-5390	3	22	nourah	nourah	PROPN
ejpam-5390	3	23	bint	bint	PROPN
ejpam-5390	3	24	abdulrahman	abdulrahman	PROPN
ejpam-5390	3	25	university	university	PROPN
ejpam-5390	3	26	,	,	PUNCT
ejpam-5390	3	27	p.	p.	PROPN
ejpam-5390	3	28	o.	o.	PROPN
ejpam-5390	3	29	box	box	PROPN
ejpam-5390	3	30	84428	84428	NUM
ejpam-5390	3	31	,	,	PUNCT
ejpam-5390	3	32	riyadh	riyadh	PROPN
ejpam-5390	3	33	11671	11671	NUM
ejpam-5390	3	34	,	,	PUNCT
ejpam-5390	3	35	saudi	saudi	PROPN
ejpam-5390	3	36	arabia	arabia	PROPN
ejpam-5390	3	37	2	2	NUM
ejpam-5390	3	38	department	department	NOUN
ejpam-5390	3	39	of	of	ADP
ejpam-5390	3	40	applied	apply	VERB
ejpam-5390	3	41	sciences	science	NOUN
ejpam-5390	3	42	,	,	PUNCT
ejpam-5390	3	43	symbiosis	symbiosis	NOUN
ejpam-5390	3	44	institute	institute	PROPN
ejpam-5390	3	45	of	of	ADP
ejpam-5390	3	46	technology	technology	PROPN
ejpam-5390	3	47	,	,	PUNCT
ejpam-5390	3	48	symbiosis	symbiosis	NOUN
ejpam-5390	3	49	international	international	ADJ
ejpam-5390	3	50	(	(	PUNCT
ejpam-5390	3	51	deemed	deem	VERB
ejpam-5390	3	52	)	)	PUNCT
ejpam-5390	3	53	university	university	NOUN
ejpam-5390	3	54	,	,	PUNCT
ejpam-5390	3	55	lavale	lavale	NOUN
ejpam-5390	3	56	,	,	PUNCT
ejpam-5390	3	57	pune	pune	NOUN
ejpam-5390	3	58	,	,	PUNCT
ejpam-5390	3	59	india	india	PROPN
ejpam-5390	3	60	3	3	NUM
ejpam-5390	3	61	department	department	NOUN
ejpam-5390	3	62	of	of	ADP
ejpam-5390	3	63	mathematics	mathematics	PROPN
ejpam-5390	3	64	,	,	PUNCT
ejpam-5390	3	65	aligarh	aligarh	PROPN
ejpam-5390	3	66	muslim	muslim	PROPN
ejpam-5390	3	67	university	university	PROPN
ejpam-5390	3	68	,	,	PUNCT
ejpam-5390	3	69	aligarh-202002	aligarh-202002	NOUN
ejpam-5390	3	70	india	india	PROPN
ejpam-5390	3	71	4	4	NUM
ejpam-5390	3	72	department	department	NOUN
ejpam-5390	3	73	of	of	ADP
ejpam-5390	3	74	computer	computer	NOUN
ejpam-5390	3	75	science	science	NOUN
ejpam-5390	3	76	,	,	PUNCT
ejpam-5390	3	77	college	college	NOUN
ejpam-5390	3	78	of	of	ADP
ejpam-5390	3	79	engineering	engineering	NOUN
ejpam-5390	3	80	and	and	CCONJ
ejpam-5390	3	81	information	information	NOUN
ejpam-5390	3	82	technology	technology	NOUN
ejpam-5390	3	83	,	,	PUNCT
ejpam-5390	3	84	amran	amran	ADJ
ejpam-5390	3	85	university	university	NOUN
ejpam-5390	3	86	,	,	PUNCT
ejpam-5390	3	87	amran	amran	PROPN
ejpam-5390	3	88	,	,	PUNCT
ejpam-5390	3	89	yemen	yemen	PROPN
ejpam-5390	3	90	.	.	PUNCT
ejpam-5390	4	1	abstract	abstract	ADJ
ejpam-5390	4	2	.	.	PUNCT
ejpam-5390	5	1	let	let	VERB
ejpam-5390	5	2	a	a	DET
ejpam-5390	5	3	be	be	AUX
ejpam-5390	5	4	a	a	DET
ejpam-5390	5	5	∗-algebra	∗-algebra	NOUN
ejpam-5390	5	6	with	with	ADP
ejpam-5390	5	7	unit	unit	NOUN
ejpam-5390	5	8	i	i	PRON
ejpam-5390	5	9	and	and	CCONJ
ejpam-5390	5	10	p1	p1	PROPN
ejpam-5390	5	11	and	and	CCONJ
ejpam-5390	5	12	p2	p2	PROPN
ejpam-5390	5	13	=	=	SYM
ejpam-5390	6	1	i	i	PRON
ejpam-5390	6	2	−	−	PROPN
ejpam-5390	6	3	p1	p1	PROPN
ejpam-5390	6	4	includes	include	VERB
ejpam-5390	6	5	a	a	DET
ejpam-5390	6	6	non	non	ADJ
ejpam-5390	6	7	-	-	ADJ
ejpam-5390	6	8	trivial	trivial	ADJ
ejpam-5390	6	9	projections	projection	NOUN
ejpam-5390	6	10	,	,	PUNCT
ejpam-5390	6	11	and	and	CCONJ
ejpam-5390	6	12	let	let	VERB
ejpam-5390	6	13	λ	λ	X
ejpam-5390	6	14	∈	∈	NOUN
ejpam-5390	6	15	c	c	X
ejpam-5390	6	16	\	\	X
ejpam-5390	6	17	{	{	PUNCT
ejpam-5390	6	18	0,−1	0,−1	NOUN
ejpam-5390	6	19	}	}	PUNCT
ejpam-5390	6	20	.	.	PUNCT
ejpam-5390	7	1	in	in	ADP
ejpam-5390	7	2	this	this	DET
ejpam-5390	7	3	paper	paper	NOUN
ejpam-5390	7	4	,	,	PUNCT
ejpam-5390	7	5	we	we	PRON
ejpam-5390	7	6	aim	aim	VERB
ejpam-5390	7	7	to	to	PART
ejpam-5390	7	8	study	study	VERB
ejpam-5390	7	9	the	the	DET
ejpam-5390	7	10	characterization	characterization	NOUN
ejpam-5390	7	11	of	of	ADP
ejpam-5390	7	12	nonlinear	nonlinear	ADJ
ejpam-5390	7	13	mixed	mixed	ADJ
ejpam-5390	7	14	λ	λ	PROPN
ejpam-5390	7	15	-	-	PROPN
ejpam-5390	7	16	jordan	jordan	PROPN
ejpam-5390	7	17	triple	triple	ADJ
ejpam-5390	7	18	derivation	derivation	NOUN
ejpam-5390	7	19	on	on	ADP
ejpam-5390	7	20	∗-algebras	∗-algebra	NOUN
ejpam-5390	7	21	.	.	PUNCT
ejpam-5390	8	1	as	as	ADP
ejpam-5390	8	2	an	an	DET
ejpam-5390	8	3	application	application	NOUN
ejpam-5390	8	4	,	,	PUNCT
ejpam-5390	8	5	we	we	PRON
ejpam-5390	8	6	can	can	AUX
ejpam-5390	8	7	also	also	ADV
ejpam-5390	8	8	apply	apply	VERB
ejpam-5390	8	9	our	our	PRON
ejpam-5390	8	10	results	result	NOUN
ejpam-5390	8	11	on	on	ADP
ejpam-5390	8	12	prime	prime	ADJ
ejpam-5390	8	13	∗-algebras	∗-algebra	NOUN
ejpam-5390	8	14	,	,	PUNCT
ejpam-5390	8	15	factor	factor	NOUN
ejpam-5390	8	16	von	von	PROPN
ejpam-5390	8	17	-	-	PUNCT
ejpam-5390	8	18	neumann	neumann	PROPN
ejpam-5390	8	19	algebras	algebras	PROPN
ejpam-5390	8	20	and	and	CCONJ
ejpam-5390	8	21	standard	standard	ADJ
ejpam-5390	8	22	operator	operator	NOUN
ejpam-5390	8	23	algebras	algebra	NOUN
ejpam-5390	8	24	.	.	PUNCT
ejpam-5390	9	1	2020	2020	NUM
ejpam-5390	9	2	mathematics	mathematics	PROPN
ejpam-5390	9	3	subject	subject	NOUN
ejpam-5390	9	4	classifications	classification	NOUN
ejpam-5390	9	5	:	:	PUNCT
ejpam-5390	9	6	16w10	16w10	NUM
ejpam-5390	9	7	,	,	PUNCT
ejpam-5390	9	8	47b47	47b47	NOUN
ejpam-5390	9	9	,	,	PUNCT
ejpam-5390	9	10	46k15	46k15	NUM
ejpam-5390	9	11	.	.	PUNCT
ejpam-5390	10	1	key	key	ADJ
ejpam-5390	10	2	words	word	NOUN
ejpam-5390	10	3	and	and	CCONJ
ejpam-5390	10	4	phrases	phrase	NOUN
ejpam-5390	10	5	:	:	PUNCT
ejpam-5390	10	6	λ	λ	ADJ
ejpam-5390	10	7	-	-	PUNCT
ejpam-5390	10	8	mixed	mixed	ADJ
ejpam-5390	10	9	jordan	jordan	PROPN
ejpam-5390	10	10	triple	triple	ADJ
ejpam-5390	10	11	derivation	derivation	NOUN
ejpam-5390	10	12	,	,	PUNCT
ejpam-5390	10	13	∗-derivation	∗-derivation	NOUN
ejpam-5390	10	14	,	,	PUNCT
ejpam-5390	10	15	∗algebra	∗algebra	NOUN
ejpam-5390	10	16	.	.	NOUN
ejpam-5390	11	1	1	1	X
ejpam-5390	11	2	.	.	X
ejpam-5390	11	3	introduction	introduction	NOUN
ejpam-5390	11	4	consider	consider	VERB
ejpam-5390	11	5	an	an	DET
ejpam-5390	11	6	∗-algebra	∗-algebra	NOUN
ejpam-5390	11	7	a	a	DET
ejpam-5390	11	8	defined	define	VERB
ejpam-5390	11	9	over	over	ADP
ejpam-5390	11	10	the	the	DET
ejpam-5390	11	11	complex	complex	ADJ
ejpam-5390	11	12	field	field	NOUN
ejpam-5390	11	13	c.	c.	NOUN
ejpam-5390	11	14	introducing	introduce	VERB
ejpam-5390	11	15	the	the	DET
ejpam-5390	11	16	λ	λ	PROPN
ejpam-5390	11	17	-	-	PROPN
ejpam-5390	11	18	jordan	jordan	PROPN
ejpam-5390	11	19	product	product	PROPN
ejpam-5390	11	20	u	u	PROPN
ejpam-5390	11	21	♢	♢	PROPN
ejpam-5390	11	22	λv	λv	X
ejpam-5390	11	23	=	=	PUNCT
ejpam-5390	11	24	uv	uv	PROPN
ejpam-5390	12	1	+	+	CCONJ
ejpam-5390	12	2	λv	λv	ADP
ejpam-5390	12	3	u	u	NOUN
ejpam-5390	12	4	and	and	CCONJ
ejpam-5390	12	5	the	the	DET
ejpam-5390	12	6	skew	skew	ADJ
ejpam-5390	12	7	lie	lie	NOUN
ejpam-5390	12	8	product	product	NOUN
ejpam-5390	13	1	[	[	X
ejpam-5390	13	2	u	u	NOUN
ejpam-5390	13	3	,	,	PUNCT
ejpam-5390	13	4	v	v	NOUN
ejpam-5390	13	5	]	]	PUNCT
ejpam-5390	13	6	∗	∗	NOUN
ejpam-5390	13	7	=	=	PUNCT
ejpam-5390	14	1	uv	uv	NOUN
ejpam-5390	14	2	−	−	PROPN
ejpam-5390	14	3	v	v	NOUN
ejpam-5390	14	4	u∗	u∗	NOUN
ejpam-5390	14	5	for	for	ADP
ejpam-5390	14	6	nonzero	nonzero	PROPN
ejpam-5390	14	7	scalar	scalar	ADJ
ejpam-5390	14	8	λ	λ	PROPN
ejpam-5390	14	9	,	,	PUNCT
ejpam-5390	14	10	these	these	DET
ejpam-5390	14	11	algebraic	algebraic	ADJ
ejpam-5390	14	12	structures	structure	NOUN
ejpam-5390	14	13	have	have	AUX
ejpam-5390	14	14	gained	gain	VERB
ejpam-5390	14	15	significant	significant	ADJ
ejpam-5390	14	16	attention	attention	NOUN
ejpam-5390	14	17	in	in	ADP
ejpam-5390	14	18	various	various	ADJ
ejpam-5390	14	19	research	research	NOUN
ejpam-5390	14	20	domains	domain	NOUN
ejpam-5390	14	21	,	,	PUNCT
ejpam-5390	14	22	as	as	SCONJ
ejpam-5390	14	23	evidenced	evidence	VERB
ejpam-5390	14	24	by	by	ADP
ejpam-5390	14	25	studies	study	NOUN
ejpam-5390	14	26	such	such	ADJ
ejpam-5390	14	27	as	as	ADP
ejpam-5390	14	28	[	[	X
ejpam-5390	14	29	1–5	1–5	NUM
ejpam-5390	14	30	,	,	PUNCT
ejpam-5390	14	31	7	7	NUM
ejpam-5390	14	32	,	,	PUNCT
ejpam-5390	14	33	8	8	NUM
ejpam-5390	14	34	,	,	PUNCT
ejpam-5390	14	35	12	12	NUM
ejpam-5390	14	36	]	]	PUNCT
ejpam-5390	14	37	.	.	PUNCT
ejpam-5390	15	1	in	in	ADP
ejpam-5390	15	2	the	the	DET
ejpam-5390	15	3	context	context	NOUN
ejpam-5390	15	4	of	of	ADP
ejpam-5390	15	5	additive	additive	ADJ
ejpam-5390	15	6	mappings	mapping	NOUN
ejpam-5390	15	7	,	,	PUNCT
ejpam-5390	15	8	an	an	DET
ejpam-5390	15	9	additive	additive	ADJ
ejpam-5390	15	10	derivation	derivation	NOUN
ejpam-5390	15	11	is	be	AUX
ejpam-5390	15	12	characterized	characterize	VERB
ejpam-5390	15	13	by	by	ADP
ejpam-5390	15	14	π(uv	π(uv	NOUN
ejpam-5390	15	15	)	)	PUNCT
ejpam-5390	16	1	=	=	PUNCT
ejpam-5390	17	1	π(u)v	π(u)v	X
ejpam-5390	17	2	+	+	PUNCT
ejpam-5390	17	3	uπ(v	uπ(v	X
ejpam-5390	17	4	)	)	PUNCT
ejpam-5390	17	5	for	for	ADP
ejpam-5390	17	6	all	all	DET
ejpam-5390	17	7	u	u	NOUN
ejpam-5390	17	8	,	,	PUNCT
ejpam-5390	17	9	v	v	NOUN
ejpam-5390	17	10	∈	∈	NOUN
ejpam-5390	17	11	a.	a.	NOUN
ejpam-5390	17	12	if	if	SCONJ
ejpam-5390	17	13	the	the	DET
ejpam-5390	17	14	additional	additional	ADJ
ejpam-5390	17	15	condition	condition	NOUN
ejpam-5390	17	16	π(u∗	π(u∗	PRON
ejpam-5390	17	17	)	)	PUNCT
ejpam-5390	17	18	=	=	SYM
ejpam-5390	18	1	π(u)∗	π(u)∗	NOUN
ejpam-5390	18	2	holds	hold	VERB
ejpam-5390	18	3	for	for	ADP
ejpam-5390	18	4	all	all	PRON
ejpam-5390	18	5	u	u	NOUN
ejpam-5390	18	6	∈	∈	PROPN
ejpam-5390	18	7	a	a	PRON
ejpam-5390	18	8	,	,	PUNCT
ejpam-5390	18	9	then	then	ADV
ejpam-5390	18	10	π	π	PROPN
ejpam-5390	18	11	is	be	AUX
ejpam-5390	18	12	termed	term	VERB
ejpam-5390	18	13	an	an	DET
ejpam-5390	18	14	additive	additive	ADJ
ejpam-5390	18	15	∗-derivation	∗-derivation	NOUN
ejpam-5390	18	16	.	.	PUNCT
ejpam-5390	19	1	now	now	ADV
ejpam-5390	19	2	,	,	PUNCT
ejpam-5390	19	3	let	let	VERB
ejpam-5390	19	4	π	π	PRON
ejpam-5390	19	5	:	:	PUNCT
ejpam-5390	19	6	a	a	PRON
ejpam-5390	19	7	→	→	X
ejpam-5390	19	8	a	a	DET
ejpam-5390	19	9	be	be	AUX
ejpam-5390	19	10	a	a	DET
ejpam-5390	19	11	map	map	NOUN
ejpam-5390	19	12	without	without	ADP
ejpam-5390	19	13	assuming	assume	VERB
ejpam-5390	19	14	additivity	additivity	NOUN
ejpam-5390	19	15	.	.	PUNCT
ejpam-5390	20	1	the	the	DET
ejpam-5390	20	2	concept	concept	NOUN
ejpam-5390	20	3	of	of	ADP
ejpam-5390	20	4	a	a	DET
ejpam-5390	20	5	nonlinear	nonlinear	ADJ
ejpam-5390	20	6	skew	skew	ADJ
ejpam-5390	20	7	lie	lie	NOUN
ejpam-5390	20	8	derivation	derivation	NOUN
ejpam-5390	20	9	is	be	AUX
ejpam-5390	20	10	introduced	introduce	VERB
ejpam-5390	20	11	,	,	PUNCT
ejpam-5390	20	12	defined	define	VERB
ejpam-5390	20	13	by	by	ADP
ejpam-5390	20	14	the	the	DET
ejpam-5390	20	15	relation	relation	NOUN
ejpam-5390	20	16	π([u	π([u	PROPN
ejpam-5390	20	17	,	,	PUNCT
ejpam-5390	20	18	v	v	NOUN
ejpam-5390	20	19	]	]	PUNCT
ejpam-5390	20	20	∗	∗	NOUN
ejpam-5390	20	21	)	)	PUNCT
ejpam-5390	21	1	=	=	PUNCT
ejpam-5390	22	1	[	[	X
ejpam-5390	22	2	π(u	π(u	NOUN
ejpam-5390	22	3	)	)	PUNCT
ejpam-5390	22	4	,	,	PUNCT
ejpam-5390	22	5	v	v	X
ejpam-5390	22	6	]	]	PUNCT
ejpam-5390	22	7	∗	∗	NOUN
ejpam-5390	22	8	+	+	X
ejpam-5390	23	1	[	[	X
ejpam-5390	23	2	u	u	NOUN
ejpam-5390	23	3	,	,	PUNCT
ejpam-5390	23	4	π(v	π(v	NOUN
ejpam-5390	23	5	)	)	PUNCT
ejpam-5390	23	6	]	]	PUNCT
ejpam-5390	23	7	∗	∗	NOUN
ejpam-5390	23	8	for	for	ADP
ejpam-5390	23	9	all	all	DET
ejpam-5390	23	10	u	u	NOUN
ejpam-5390	23	11	,	,	PUNCT
ejpam-5390	23	12	v	v	PROPN
ejpam-5390	23	13	∈	∈	PROPN
ejpam-5390	23	14	a.	a.	NOUN
ejpam-5390	23	15	notably	notably	ADV
ejpam-5390	23	16	,	,	PUNCT
ejpam-5390	23	17	kong	kong	PROPN
ejpam-5390	23	18	and	and	CCONJ
ejpam-5390	23	19	zhang	zhang	PROPN
ejpam-5390	24	1	[	[	X
ejpam-5390	24	2	3	3	NUM
ejpam-5390	24	3	]	]	PUNCT
ejpam-5390	24	4	established	establish	VERB
ejpam-5390	24	5	the	the	DET
ejpam-5390	24	6	result	result	NOUN
ejpam-5390	24	7	that	that	SCONJ
ejpam-5390	24	8	every	every	DET
ejpam-5390	24	9	nonlinear	nonlinear	ADJ
ejpam-5390	24	10	skew	skew	ADJ
ejpam-5390	24	11	lie	lie	NOUN
ejpam-5390	24	12	derivation	derivation	NOUN
ejpam-5390	24	13	is	be	AUX
ejpam-5390	24	14	,	,	PUNCT
ejpam-5390	24	15	in	in	ADP
ejpam-5390	24	16	fact	fact	NOUN
ejpam-5390	24	17	,	,	PUNCT
ejpam-5390	24	18	an	an	DET
ejpam-5390	24	19	additive	additive	ADJ
ejpam-5390	24	20	∗-derivation	∗-derivation	NOUN
ejpam-5390	24	21	.	.	PUNCT
ejpam-5390	25	1	∗corresponding	∗corresponde	VERB
ejpam-5390	25	2	author	author	NOUN
ejpam-5390	25	3	.	.	PUNCT
ejpam-5390	26	1	doi	doi	NOUN
ejpam-5390	26	2	:	:	PUNCT
ejpam-5390	26	3	https://doi.org/10.29020/nybg.ejpam.v17i4.5390	https://doi.org/10.29020/nybg.ejpam.v17i4.5390	PRON
ejpam-5390	26	4	email	email	NOUN
ejpam-5390	26	5	addresses	address	NOUN
ejpam-5390	26	6	:	:	PUNCT
ejpam-5390	26	7	asalali@pnu.edu.sa	asalali@pnu.edu.sa	PROPN
ejpam-5390	26	8	(	(	PUNCT
ejpam-5390	26	9	a.	a.	NOUN
ejpam-5390	26	10	alali	alali	PROPN
ejpam-5390	26	11	)	)	PUNCT
ejpam-5390	26	12	,	,	PUNCT
ejpam-5390	26	13	junaidnisar73@gmail.com	junaidnisar73@gmail.com	X
ejpam-5390	26	14	(	(	PUNCT
ejpam-5390	26	15	j.	j.	PROPN
ejpam-5390	26	16	nisar	nisar	PROPN
ejpam-5390	26	17	)	)	PUNCT
ejpam-5390	26	18	,	,	PUNCT
ejpam-5390	26	19	nu.rehman.mm@amu.ac.in	nu.rehman.mm@amu.ac.in	PROPN
ejpam-5390	26	20	(	(	PUNCT
ejpam-5390	26	21	n.	n.	PROPN
ejpam-5390	26	22	rehman	rehman	PROPN
ejpam-5390	26	23	)	)	PUNCT
ejpam-5390	26	24	,	,	PUNCT
ejpam-5390	26	25	halnoghashi@gmail.com	halnoghashi@gmail.com	X
ejpam-5390	27	1	(	(	PUNCT
ejpam-5390	27	2	h.	h.	PROPN
ejpam-5390	27	3	alnoghashi	alnoghashi	PROPN
ejpam-5390	27	4	)	)	PUNCT
ejpam-5390	27	5	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5390	28	1	3399	3399	NUM
ejpam-5390	28	2	copyright	copyright	NOUN
ejpam-5390	28	3	:	:	PUNCT
ejpam-5390	28	4	©	©	PROPN
ejpam-5390	28	5	2024	2024	NUM
ejpam-5390	28	6	the	the	DET
ejpam-5390	28	7	author(s	author(s	NOUN
ejpam-5390	28	8	)	)	PUNCT
ejpam-5390	28	9	.	.	PUNCT
ejpam-5390	29	1	(	(	PUNCT
ejpam-5390	29	2	cc	cc	NOUN
ejpam-5390	29	3	by	by	ADP
ejpam-5390	29	4	-	-	PUNCT
ejpam-5390	29	5	nc	nc	PROPN
ejpam-5390	29	6	4.0	4.0	NUM
ejpam-5390	29	7	)	)	PUNCT
ejpam-5390	29	8	.	.	PUNCT
ejpam-5390	30	1	nisar	nisar	PROPN
ejpam-5390	30	2	et	et	PROPN
ejpam-5390	30	3	al	al	PROPN
ejpam-5390	30	4	.	.	PUNCT
ejpam-5390	30	5	/	/	SYM
ejpam-5390	30	6	eur	eur	PROPN
ejpam-5390	30	7	.	.	PUNCT
ejpam-5390	31	1	j.	j.	PROPN
ejpam-5390	31	2	pure	pure	PROPN
ejpam-5390	31	3	appl	appl	PROPN
ejpam-5390	31	4	.	.	PROPN
ejpam-5390	31	5	math	math	PROPN
ejpam-5390	31	6	,	,	PUNCT
ejpam-5390	31	7	17	17	NUM
ejpam-5390	31	8	(	(	PUNCT
ejpam-5390	31	9	4	4	NUM
ejpam-5390	31	10	)	)	PUNCT
ejpam-5390	31	11	(	(	PUNCT
ejpam-5390	31	12	2024	2024	NUM
ejpam-5390	31	13	)	)	PUNCT
ejpam-5390	31	14	,	,	PUNCT
ejpam-5390	31	15	3399	3399	NUM
ejpam-5390	31	16	-	-	SYM
ejpam-5390	31	17	3414	3414	NUM
ejpam-5390	31	18	3400	3400	NUM
ejpam-5390	31	19	similarly	similarly	ADV
ejpam-5390	31	20	,	,	PUNCT
ejpam-5390	31	21	a	a	DET
ejpam-5390	31	22	map	map	NOUN
ejpam-5390	32	1	π	π	X
ejpam-5390	32	2	:	:	PUNCT
ejpam-5390	32	3	a	a	DET
ejpam-5390	32	4	→	→	SYM
ejpam-5390	32	5	a	a	PRON
ejpam-5390	32	6	is	be	AUX
ejpam-5390	32	7	termed	term	VERB
ejpam-5390	32	8	a	a	DET
ejpam-5390	32	9	nonlinear	nonlinear	ADJ
ejpam-5390	32	10	skew	skew	NOUN
ejpam-5390	32	11	lie	lie	NOUN
ejpam-5390	32	12	triple	triple	ADJ
ejpam-5390	32	13	derivation	derivation	NOUN
ejpam-5390	32	14	if	if	SCONJ
ejpam-5390	32	15	it	it	PRON
ejpam-5390	32	16	satisfies	satisfy	VERB
ejpam-5390	32	17	the	the	DET
ejpam-5390	32	18	equation	equation	NOUN
ejpam-5390	32	19	π([[u	π([[u	NOUN
ejpam-5390	32	20	,	,	PUNCT
ejpam-5390	32	21	v	v	X
ejpam-5390	32	22	]	]	PUNCT
ejpam-5390	32	23	∗,w	∗,w	ADJ
ejpam-5390	32	24	]	]	PUNCT
ejpam-5390	32	25	∗	∗	NOUN
ejpam-5390	32	26	)	)	PUNCT
ejpam-5390	32	27	=	=	PUNCT
ejpam-5390	33	1	[	[	X
ejpam-5390	33	2	[	[	X
ejpam-5390	33	3	π(u	π(u	NOUN
ejpam-5390	33	4	)	)	PUNCT
ejpam-5390	33	5	,	,	PUNCT
ejpam-5390	33	6	v	v	X
ejpam-5390	33	7	]	]	PUNCT
ejpam-5390	33	8	∗,w	∗,w	VERB
ejpam-5390	33	9	]	]	PUNCT
ejpam-5390	33	10	∗	∗	NOUN
ejpam-5390	33	11	+	+	X
ejpam-5390	34	1	[	[	X
ejpam-5390	34	2	[	[	X
ejpam-5390	34	3	u	u	NOUN
ejpam-5390	34	4	,	,	PUNCT
ejpam-5390	34	5	π(v	π(v	NOUN
ejpam-5390	34	6	)	)	PUNCT
ejpam-5390	34	7	]	]	PUNCT
ejpam-5390	34	8	∗,w	∗,w	VERB
ejpam-5390	34	9	]	]	PUNCT
ejpam-5390	34	10	∗	∗	NOUN
ejpam-5390	34	11	+	+	X
ejpam-5390	35	1	[	[	X
ejpam-5390	35	2	[	[	X
ejpam-5390	35	3	u	u	NOUN
ejpam-5390	35	4	,	,	PUNCT
ejpam-5390	35	5	v	v	NOUN
ejpam-5390	35	6	]	]	PUNCT
ejpam-5390	35	7	∗,π(w	∗,π(w	ADJ
ejpam-5390	35	8	)	)	PUNCT
ejpam-5390	35	9	]	]	PUNCT
ejpam-5390	35	10	∗	∗	NOUN
ejpam-5390	35	11	for	for	ADP
ejpam-5390	35	12	all	all	DET
ejpam-5390	35	13	u	u	NOUN
ejpam-5390	35	14	,	,	PUNCT
ejpam-5390	35	15	v	v	NOUN
ejpam-5390	35	16	,	,	PUNCT
ejpam-5390	35	17	w	w	PROPN
ejpam-5390	35	18	∈	∈	PROPN
ejpam-5390	35	19	a.	a.	NOUN
ejpam-5390	35	20	several	several	ADJ
ejpam-5390	35	21	recent	recent	ADJ
ejpam-5390	35	22	studies	study	NOUN
ejpam-5390	35	23	have	have	AUX
ejpam-5390	35	24	delved	delve	VERB
ejpam-5390	35	25	into	into	ADP
ejpam-5390	35	26	the	the	DET
ejpam-5390	35	27	exploration	exploration	NOUN
ejpam-5390	35	28	of	of	ADP
ejpam-5390	35	29	derivations	derivation	NOUN
ejpam-5390	35	30	and	and	CCONJ
ejpam-5390	35	31	isomorphisms	isomorphism	NOUN
ejpam-5390	35	32	associated	associate	VERB
ejpam-5390	35	33	with	with	ADP
ejpam-5390	35	34	innovative	innovative	ADJ
ejpam-5390	35	35	products	product	NOUN
ejpam-5390	35	36	resulting	result	VERB
ejpam-5390	35	37	from	from	ADP
ejpam-5390	35	38	the	the	DET
ejpam-5390	35	39	combination	combination	NOUN
ejpam-5390	35	40	of	of	ADP
ejpam-5390	35	41	lie	lie	NOUN
ejpam-5390	35	42	and	and	CCONJ
ejpam-5390	35	43	skew	skew	ADJ
ejpam-5390	35	44	lie	lie	NOUN
ejpam-5390	35	45	products	product	NOUN
ejpam-5390	35	46	,	,	PUNCT
ejpam-5390	35	47	as	as	SCONJ
ejpam-5390	35	48	evidenced	evidence	VERB
ejpam-5390	35	49	by	by	ADP
ejpam-5390	35	50	works	work	NOUN
ejpam-5390	35	51	such	such	ADJ
ejpam-5390	35	52	as	as	ADP
ejpam-5390	35	53	[	[	X
ejpam-5390	35	54	6	6	NUM
ejpam-5390	35	55	,	,	PUNCT
ejpam-5390	35	56	9	9	NUM
ejpam-5390	35	57	,	,	PUNCT
ejpam-5390	35	58	11	11	NUM
ejpam-5390	35	59	]	]	PUNCT
ejpam-5390	35	60	.	.	PUNCT
ejpam-5390	36	1	notably	notably	ADV
ejpam-5390	36	2	,	,	PUNCT
ejpam-5390	36	3	zhou	zhou	PROPN
ejpam-5390	36	4	et	et	PROPN
ejpam-5390	36	5	al	al	PROPN
ejpam-5390	36	6	.	.	PUNCT
ejpam-5390	37	1	[	[	X
ejpam-5390	37	2	13	13	NUM
ejpam-5390	37	3	]	]	PUNCT
ejpam-5390	37	4	established	establish	VERB
ejpam-5390	37	5	the	the	DET
ejpam-5390	37	6	result	result	NOUN
ejpam-5390	37	7	that	that	SCONJ
ejpam-5390	37	8	every	every	DET
ejpam-5390	37	9	nonlinear	nonlinear	ADJ
ejpam-5390	37	10	mixed	mixed	ADJ
ejpam-5390	37	11	lie	lie	NOUN
ejpam-5390	37	12	triple	triple	ADJ
ejpam-5390	37	13	derivation	derivation	NOUN
ejpam-5390	37	14	on	on	ADP
ejpam-5390	37	15	a	a	DET
ejpam-5390	37	16	prime	prime	ADJ
ejpam-5390	37	17	∗-algebra	∗-algebra	NOUN
ejpam-5390	37	18	is	be	AUX
ejpam-5390	37	19	,	,	PUNCT
ejpam-5390	37	20	in	in	ADP
ejpam-5390	37	21	fact	fact	NOUN
ejpam-5390	37	22	,	,	PUNCT
ejpam-5390	37	23	an	an	DET
ejpam-5390	37	24	additive	additive	ADJ
ejpam-5390	37	25	∗-derivation	∗-derivation	NOUN
ejpam-5390	37	26	.	.	PUNCT
ejpam-5390	38	1	additionally	additionally	ADV
ejpam-5390	38	2	,	,	PUNCT
ejpam-5390	38	3	pang	pang	NOUN
ejpam-5390	38	4	et	et	PROPN
ejpam-5390	38	5	al	al	PROPN
ejpam-5390	38	6	.	.	PUNCT
ejpam-5390	39	1	[	[	X
ejpam-5390	39	2	10	10	NUM
ejpam-5390	39	3	]	]	PUNCT
ejpam-5390	39	4	demonstrated	demonstrate	VERB
ejpam-5390	39	5	that	that	SCONJ
ejpam-5390	39	6	every	every	DET
ejpam-5390	39	7	second	second	ADJ
ejpam-5390	39	8	nonlinear	nonlinear	ADJ
ejpam-5390	39	9	mixed	mixed	ADJ
ejpam-5390	39	10	jordan	jordan	PROPN
ejpam-5390	39	11	triple	triple	ADJ
ejpam-5390	39	12	derivable	derivable	ADJ
ejpam-5390	39	13	mapping	mapping	NOUN
ejpam-5390	39	14	on	on	ADP
ejpam-5390	39	15	factor	factor	NOUN
ejpam-5390	39	16	von	von	PROPN
ejpam-5390	39	17	neumann	neumann	PROPN
ejpam-5390	39	18	algebras	algebras	PROPN
ejpam-5390	39	19	also	also	ADV
ejpam-5390	39	20	an	an	DET
ejpam-5390	39	21	additive	additive	ADJ
ejpam-5390	39	22	∗-derivation	∗-derivation	NOUN
ejpam-5390	39	23	.	.	PUNCT
ejpam-5390	40	1	motivated	motivate	VERB
ejpam-5390	40	2	by	by	ADP
ejpam-5390	40	3	these	these	DET
ejpam-5390	40	4	previous	previous	ADJ
ejpam-5390	40	5	works	work	NOUN
ejpam-5390	40	6	,	,	PUNCT
ejpam-5390	40	7	our	our	PRON
ejpam-5390	40	8	paper	paper	NOUN
ejpam-5390	40	9	introduces	introduce	VERB
ejpam-5390	40	10	the	the	DET
ejpam-5390	40	11	λ	λ	PROPN
ejpam-5390	40	12	-	-	PROPN
ejpam-5390	40	13	jordan	jordan	PROPN
ejpam-5390	40	14	product	product	NOUN
ejpam-5390	40	15	defined	define	VERB
ejpam-5390	40	16	as	as	ADP
ejpam-5390	40	17	u	u	NOUN
ejpam-5390	40	18	♢	♢	PROPN
ejpam-5390	40	19	λv	λv	X
ejpam-5390	41	1	=	=	PUNCT
ejpam-5390	41	2	uv	uv	PROPN
ejpam-5390	42	1	+	+	PROPN
ejpam-5390	42	2	λv	λv	NOUN
ejpam-5390	42	3	u	u	NOUN
ejpam-5390	42	4	.	.	PUNCT
ejpam-5390	43	1	we	we	PRON
ejpam-5390	43	2	specifically	specifically	ADV
ejpam-5390	43	3	focus	focus	VERB
ejpam-5390	43	4	on	on	ADP
ejpam-5390	43	5	the	the	DET
ejpam-5390	43	6	derivation	derivation	NOUN
ejpam-5390	43	7	corresponding	correspond	VERB
ejpam-5390	43	8	to	to	ADP
ejpam-5390	43	9	the	the	DET
ejpam-5390	43	10	novel	novel	ADJ
ejpam-5390	43	11	product	product	NOUN
ejpam-5390	43	12	obtained	obtain	VERB
ejpam-5390	43	13	by	by	ADP
ejpam-5390	43	14	combining	combine	VERB
ejpam-5390	43	15	the	the	DET
ejpam-5390	43	16	skew	skew	ADJ
ejpam-5390	43	17	lie	lie	NOUN
ejpam-5390	43	18	product	product	NOUN
ejpam-5390	43	19	and	and	CCONJ
ejpam-5390	43	20	the	the	DET
ejpam-5390	43	21	λ	λ	PROPN
ejpam-5390	43	22	-	-	PROPN
ejpam-5390	43	23	jordan	jordan	PROPN
ejpam-5390	43	24	product	product	NOUN
ejpam-5390	43	25	.	.	PUNCT
ejpam-5390	44	1	in	in	ADP
ejpam-5390	44	2	this	this	DET
ejpam-5390	44	3	context	context	NOUN
ejpam-5390	44	4	,	,	PUNCT
ejpam-5390	44	5	we	we	PRON
ejpam-5390	44	6	define	define	VERB
ejpam-5390	44	7	a	a	DET
ejpam-5390	44	8	map	map	NOUN
ejpam-5390	44	9	π	π	NOUN
ejpam-5390	44	10	:	:	PUNCT
ejpam-5390	44	11	a	a	DET
ejpam-5390	44	12	→	→	X
ejpam-5390	44	13	a	a	PRON
ejpam-5390	44	14	as	as	ADP
ejpam-5390	44	15	a	a	DET
ejpam-5390	44	16	mixed	mixed	ADJ
ejpam-5390	44	17	λ	λ	PROPN
ejpam-5390	44	18	-	-	PROPN
ejpam-5390	44	19	jordan	jordan	PROPN
ejpam-5390	44	20	triple	triple	ADJ
ejpam-5390	44	21	derivation	derivation	NOUN
ejpam-5390	44	22	if	if	SCONJ
ejpam-5390	44	23	it	it	PRON
ejpam-5390	44	24	satisfies	satisfy	VERB
ejpam-5390	44	25	the	the	DET
ejpam-5390	44	26	equation	equation	NOUN
ejpam-5390	44	27	π([u	π([u	PROPN
ejpam-5390	44	28	,	,	PUNCT
ejpam-5390	44	29	v	v	NOUN
ejpam-5390	44	30	]	]	PUNCT
ejpam-5390	44	31	∗	∗	PROPN
ejpam-5390	44	32	♢	♢	PROPN
ejpam-5390	44	33	λw	λw	NOUN
ejpam-5390	44	34	)	)	PUNCT
ejpam-5390	44	35	=	=	PUNCT
ejpam-5390	45	1	[	[	X
ejpam-5390	45	2	π(u	π(u	NOUN
ejpam-5390	45	3	)	)	PUNCT
ejpam-5390	45	4	,	,	PUNCT
ejpam-5390	45	5	v	v	X
ejpam-5390	45	6	]	]	PUNCT
ejpam-5390	45	7	∗	∗	NOUN
ejpam-5390	45	8	⋄λ	⋄λ	NOUN
ejpam-5390	45	9	w	w	NOUN
ejpam-5390	45	10	+	+	CCONJ
ejpam-5390	46	1	[	[	X
ejpam-5390	46	2	u	u	NOUN
ejpam-5390	46	3	,	,	PUNCT
ejpam-5390	46	4	π(v	π(v	NOUN
ejpam-5390	46	5	)	)	PUNCT
ejpam-5390	46	6	]	]	PUNCT
ejpam-5390	46	7	∗	∗	PROPN
ejpam-5390	46	8	♢	♢	PROPN
ejpam-5390	46	9	λw	λw	X
ejpam-5390	46	10	+	+	PROPN
ejpam-5390	46	11	[	[	X
ejpam-5390	46	12	u	u	NOUN
ejpam-5390	46	13	,	,	PUNCT
ejpam-5390	46	14	v	v	NOUN
ejpam-5390	46	15	]	]	PUNCT
ejpam-5390	46	16	∗	∗	X
ejpam-5390	46	17	♢	♢	PROPN
ejpam-5390	46	18	λπ(w	λπ(w	NUM
ejpam-5390	46	19	)	)	PUNCT
ejpam-5390	46	20	for	for	ADP
ejpam-5390	46	21	all	all	DET
ejpam-5390	46	22	u	u	NOUN
ejpam-5390	46	23	,	,	PUNCT
ejpam-5390	46	24	v	v	NOUN
ejpam-5390	46	25	,	,	PUNCT
ejpam-5390	46	26	w	w	PROPN
ejpam-5390	46	27	∈	∈	PROPN
ejpam-5390	46	28	a.	a.	NOUN
ejpam-5390	46	29	our	our	PRON
ejpam-5390	46	30	main	main	ADJ
ejpam-5390	46	31	result	result	NOUN
ejpam-5390	46	32	establishes	establish	VERB
ejpam-5390	46	33	that	that	SCONJ
ejpam-5390	46	34	π	π	PROPN
ejpam-5390	46	35	is	be	AUX
ejpam-5390	46	36	a	a	DET
ejpam-5390	46	37	nonlinear	nonlinear	ADJ
ejpam-5390	46	38	mixed	mix	VERB
ejpam-5390	46	39	λ	λ	PROPN
ejpam-5390	46	40	-	-	PROPN
ejpam-5390	46	41	jordan	jordan	PROPN
ejpam-5390	46	42	triple	triple	ADJ
ejpam-5390	46	43	derivation	derivation	NOUN
ejpam-5390	46	44	on	on	ADP
ejpam-5390	46	45	∗-algebras	∗-algebra	NOUN
ejpam-5390	46	46	if	if	SCONJ
ejpam-5390	46	47	and	and	CCONJ
ejpam-5390	46	48	only	only	ADV
ejpam-5390	46	49	if	if	SCONJ
ejpam-5390	46	50	π	π	PROPN
ejpam-5390	46	51	is	be	AUX
ejpam-5390	46	52	an	an	DET
ejpam-5390	46	53	additive	additive	ADJ
ejpam-5390	46	54	∗-derivation	∗-derivation	NOUN
ejpam-5390	46	55	.	.	PUNCT
ejpam-5390	47	1	2	2	NUM
ejpam-5390	47	2	.	.	X
ejpam-5390	47	3	main	main	ADJ
ejpam-5390	47	4	result	result	NOUN
ejpam-5390	47	5	theorem	theorem	VERB
ejpam-5390	47	6	2.1	2.1	NUM
ejpam-5390	47	7	.	.	PUNCT
ejpam-5390	48	1	let	let	VERB
ejpam-5390	48	2	a	a	PRON
ejpam-5390	48	3	be	be	AUX
ejpam-5390	48	4	a	a	DET
ejpam-5390	48	5	unital	unital	ADJ
ejpam-5390	48	6	∗-algebra	∗-algebra	NOUN
ejpam-5390	48	7	with	with	ADP
ejpam-5390	48	8	unity	unity	NOUN
ejpam-5390	48	9	i	i	NOUN
ejpam-5390	48	10	containing	contain	VERB
ejpam-5390	48	11	a	a	DET
ejpam-5390	48	12	non	non	ADJ
ejpam-5390	48	13	-	-	ADJ
ejpam-5390	48	14	trivial	trivial	ADJ
ejpam-5390	48	15	projection	projection	NOUN
ejpam-5390	48	16	p	p	NOUN
ejpam-5390	48	17	satisfies	satisfy	VERB
ejpam-5390	48	18	xap	xap	X
ejpam-5390	49	1	=	=	SYM
ejpam-5390	49	2	0	0	PUNCT
ejpam-5390	50	1	=	=	NOUN
ejpam-5390	50	2	⇒	⇒	NOUN
ejpam-5390	50	3	x	x	PUNCT
ejpam-5390	50	4	=	=	SYM
ejpam-5390	50	5	0	0	NUM
ejpam-5390	50	6	(	(	PUNCT
ejpam-5390	50	7	▲	▲	PUNCT
ejpam-5390	50	8	)	)	PUNCT
ejpam-5390	50	9	and	and	CCONJ
ejpam-5390	50	10	xa(i	xa(i	NUM
ejpam-5390	51	1	−	−	PROPN
ejpam-5390	51	2	p	p	NOUN
ejpam-5390	51	3	)	)	PUNCT
ejpam-5390	51	4	=	=	SYM
ejpam-5390	51	5	0	0	PUNCT
ejpam-5390	52	1	=	=	NOUN
ejpam-5390	52	2	⇒	⇒	NOUN
ejpam-5390	52	3	x	x	PUNCT
ejpam-5390	53	1	=	=	SYM
ejpam-5390	53	2	0	0	PROPN
ejpam-5390	53	3	.	.	PUNCT
ejpam-5390	54	1	(	(	PUNCT
ejpam-5390	54	2	▼	▼	NOUN
ejpam-5390	54	3	)	)	PUNCT
ejpam-5390	54	4	define	define	VERB
ejpam-5390	54	5	a	a	DET
ejpam-5390	54	6	map	map	NOUN
ejpam-5390	55	1	π	π	NOUN
ejpam-5390	55	2	:	:	PUNCT
ejpam-5390	55	3	a	a	PRON
ejpam-5390	55	4	→	→	X
ejpam-5390	55	5	a	a	DET
ejpam-5390	55	6	such	such	ADJ
ejpam-5390	55	7	that	that	DET
ejpam-5390	55	8	π([u	π([u	PROPN
ejpam-5390	55	9	,	,	PUNCT
ejpam-5390	55	10	v	v	NOUN
ejpam-5390	55	11	]	]	PUNCT
ejpam-5390	55	12	∗	∗	PROPN
ejpam-5390	55	13	♢	♢	PROPN
ejpam-5390	55	14	λw	λw	NOUN
ejpam-5390	55	15	)	)	PUNCT
ejpam-5390	55	16	=	=	PUNCT
ejpam-5390	56	1	[	[	X
ejpam-5390	56	2	π(u	π(u	NOUN
ejpam-5390	56	3	)	)	PUNCT
ejpam-5390	56	4	,	,	PUNCT
ejpam-5390	56	5	v	v	X
ejpam-5390	56	6	]	]	PUNCT
ejpam-5390	56	7	∗	∗	PROPN
ejpam-5390	56	8	♢	♢	PROPN
ejpam-5390	56	9	λw	λw	NOUN
ejpam-5390	56	10	)	)	PUNCT
ejpam-5390	56	11	+	+	PUNCT
ejpam-5390	57	1	[	[	X
ejpam-5390	57	2	u	u	NOUN
ejpam-5390	57	3	,	,	PUNCT
ejpam-5390	57	4	π(v	π(v	NOUN
ejpam-5390	57	5	)	)	PUNCT
ejpam-5390	57	6	]	]	PUNCT
ejpam-5390	57	7	∗	∗	PROPN
ejpam-5390	57	8	♢	♢	PROPN
ejpam-5390	57	9	λw	λw	X
ejpam-5390	57	10	+	+	PROPN
ejpam-5390	57	11	[	[	X
ejpam-5390	57	12	u	u	NOUN
ejpam-5390	57	13	,	,	PUNCT
ejpam-5390	57	14	v	v	NOUN
ejpam-5390	57	15	]	]	PUNCT
ejpam-5390	57	16	∗	∗	X
ejpam-5390	57	17	♢	♢	PROPN
ejpam-5390	57	18	λπ(w	λπ(w	NUM
ejpam-5390	57	19	)	)	PUNCT
ejpam-5390	57	20	.	.	PUNCT
ejpam-5390	58	1	then	then	ADV
ejpam-5390	58	2	π	π	PROPN
ejpam-5390	58	3	is	be	AUX
ejpam-5390	58	4	an	an	DET
ejpam-5390	58	5	additive	additive	ADJ
ejpam-5390	58	6	∗-derivation	∗-derivation	NOUN
ejpam-5390	58	7	.	.	PUNCT
ejpam-5390	59	1	consider	consider	VERB
ejpam-5390	59	2	a	a	DET
ejpam-5390	59	3	non	non	ADJ
ejpam-5390	59	4	-	-	ADJ
ejpam-5390	59	5	trivial	trivial	ADJ
ejpam-5390	59	6	projection	projection	NOUN
ejpam-5390	59	7	p	p	NOUN
ejpam-5390	59	8	=	=	PROPN
ejpam-5390	59	9	p1	p1	NOUN
ejpam-5390	59	10	in	in	ADP
ejpam-5390	59	11	the	the	DET
ejpam-5390	59	12	algebra	algebra	NOUN
ejpam-5390	59	13	a	a	PRON
ejpam-5390	59	14	,	,	PUNCT
ejpam-5390	59	15	and	and	CCONJ
ejpam-5390	59	16	let	let	VERB
ejpam-5390	59	17	p2	p2	PROPN
ejpam-5390	59	18	=	=	PUNCT
ejpam-5390	60	1	i	i	PRON
ejpam-5390	60	2	−p1	−p1	PROPN
ejpam-5390	60	3	,	,	PUNCT
ejpam-5390	60	4	where	where	SCONJ
ejpam-5390	60	5	i	i	PRON
ejpam-5390	60	6	is	be	AUX
ejpam-5390	60	7	the	the	DET
ejpam-5390	60	8	unity	unity	NOUN
ejpam-5390	60	9	element	element	NOUN
ejpam-5390	60	10	of	of	ADP
ejpam-5390	60	11	the	the	DET
ejpam-5390	60	12	algebra	algebra	NOUN
ejpam-5390	60	13	.	.	PUNCT
ejpam-5390	61	1	utilizing	utilize	VERB
ejpam-5390	61	2	the	the	DET
ejpam-5390	61	3	peirce	peirce	NOUN
ejpam-5390	61	4	decomposition	decomposition	NOUN
ejpam-5390	61	5	of	of	ADP
ejpam-5390	61	6	a	a	PRON
ejpam-5390	61	7	,	,	PUNCT
ejpam-5390	61	8	we	we	PRON
ejpam-5390	61	9	express	express	VERB
ejpam-5390	61	10	a	a	PRON
ejpam-5390	61	11	as	as	ADP
ejpam-5390	61	12	the	the	DET
ejpam-5390	61	13	direct	direct	ADJ
ejpam-5390	61	14	sum	sum	NOUN
ejpam-5390	61	15	a	a	DET
ejpam-5390	61	16	=	=	SYM
ejpam-5390	61	17	p1ap1⊕p1ap2⊕p2ap1⊕p2ap2	p1ap1⊕p1ap2⊕p2ap1⊕p2ap2	NOUN
ejpam-5390	61	18	.	.	PUNCT
ejpam-5390	62	1	denoting	denote	VERB
ejpam-5390	62	2	the	the	DET
ejpam-5390	62	3	corresponding	corresponding	ADJ
ejpam-5390	62	4	subspaces	subspace	NOUN
ejpam-5390	62	5	as	as	ADP
ejpam-5390	62	6	a11	a11	PROPN
ejpam-5390	62	7	=	=	SYM
ejpam-5390	62	8	p1ap1	p1ap1	PROPN
ejpam-5390	62	9	,	,	PUNCT
ejpam-5390	62	10	a12	a12	NOUN
ejpam-5390	62	11	=	=	SYM
ejpam-5390	62	12	p1ap2	p1ap2	ADJ
ejpam-5390	62	13	,	,	PUNCT
ejpam-5390	62	14	a21	a21	NOUN
ejpam-5390	62	15	=	=	SYM
ejpam-5390	62	16	p2ap1	p2ap1	ADJ
ejpam-5390	62	17	,	,	PUNCT
ejpam-5390	62	18	and	and	CCONJ
ejpam-5390	62	19	a22	a22	PROPN
ejpam-5390	62	20	=	=	SYM
ejpam-5390	62	21	p2ap2	p2ap2	PROPN
ejpam-5390	62	22	,	,	PUNCT
ejpam-5390	62	23	we	we	PRON
ejpam-5390	62	24	can	can	AUX
ejpam-5390	62	25	represent	represent	VERB
ejpam-5390	62	26	any	any	DET
ejpam-5390	62	27	element	element	NOUN
ejpam-5390	62	28	u	u	NOUN
ejpam-5390	62	29	∈	∈	PROPN
ejpam-5390	62	30	a	a	PRON
ejpam-5390	62	31	as	as	ADP
ejpam-5390	62	32	the	the	DET
ejpam-5390	62	33	sum	sum	NOUN
ejpam-5390	62	34	u	u	NOUN
ejpam-5390	62	35	=	=	X
ejpam-5390	62	36	u11	u11	PROPN
ejpam-5390	62	37	+	+	NUM
ejpam-5390	62	38	u12	u12	PROPN
ejpam-5390	62	39	+	+	CCONJ
ejpam-5390	62	40	u21	u21	PROPN
ejpam-5390	62	41	+	+	CCONJ
ejpam-5390	62	42	u22	u22	NOUN
ejpam-5390	62	43	,	,	PUNCT
ejpam-5390	62	44	where	where	SCONJ
ejpam-5390	62	45	uij	uij	PRON
ejpam-5390	62	46	∈	∈	PROPN
ejpam-5390	62	47	aij	aij	PROPN
ejpam-5390	62	48	and	and	CCONJ
ejpam-5390	62	49	u∗	u∗	ADJ
ejpam-5390	62	50	ij	ij	NOUN
ejpam-5390	62	51	∈	∈	PROPN
ejpam-5390	62	52	aji	aji	NOUN
ejpam-5390	62	53	for	for	ADP
ejpam-5390	62	54	i	i	PROPN
ejpam-5390	62	55	,	,	PUNCT
ejpam-5390	62	56	j	j	PROPN
ejpam-5390	63	1	=	=	SYM
ejpam-5390	64	1	1	1	NUM
ejpam-5390	64	2	,	,	PUNCT
ejpam-5390	64	3	2	2	NUM
ejpam-5390	64	4	.	.	PUNCT
ejpam-5390	64	5	before	before	ADP
ejpam-5390	64	6	proving	prove	VERB
ejpam-5390	64	7	theorem	theorem	VERB
ejpam-5390	64	8	2.1	2.1	NUM
ejpam-5390	64	9	,	,	PUNCT
ejpam-5390	64	10	we	we	PRON
ejpam-5390	64	11	need	need	VERB
ejpam-5390	64	12	several	several	ADJ
ejpam-5390	64	13	lemmas	lemma	NOUN
ejpam-5390	64	14	and	and	CCONJ
ejpam-5390	64	15	remarks	remark	NOUN
ejpam-5390	64	16	.	.	PUNCT
ejpam-5390	64	17	.	.	PUNCT
ejpam-5390	65	1	nisar	nisar	PROPN
ejpam-5390	65	2	et	et	PROPN
ejpam-5390	65	3	al	al	PROPN
ejpam-5390	65	4	.	.	PUNCT
ejpam-5390	65	5	/	/	SYM
ejpam-5390	65	6	eur	eur	PROPN
ejpam-5390	65	7	.	.	PUNCT
ejpam-5390	66	1	j.	j.	PROPN
ejpam-5390	66	2	pure	pure	PROPN
ejpam-5390	66	3	appl	appl	PROPN
ejpam-5390	66	4	.	.	PROPN
ejpam-5390	66	5	math	math	PROPN
ejpam-5390	66	6	,	,	PUNCT
ejpam-5390	66	7	17	17	NUM
ejpam-5390	66	8	(	(	PUNCT
ejpam-5390	66	9	4	4	NUM
ejpam-5390	66	10	)	)	PUNCT
ejpam-5390	66	11	(	(	PUNCT
ejpam-5390	66	12	2024	2024	NUM
ejpam-5390	66	13	)	)	PUNCT
ejpam-5390	66	14	,	,	PUNCT
ejpam-5390	66	15	3399	3399	NUM
ejpam-5390	66	16	-	-	SYM
ejpam-5390	66	17	3414	3414	NUM
ejpam-5390	66	18	3401	3401	NUM
ejpam-5390	66	19	lemma	lemma	PROPN
ejpam-5390	66	20	2.1	2.1	NUM
ejpam-5390	66	21	.	.	PUNCT
ejpam-5390	67	1	π(0	π(0	NOUN
ejpam-5390	67	2	)	)	PUNCT
ejpam-5390	67	3	=	=	SYM
ejpam-5390	68	1	0	0	X
ejpam-5390	68	2	.	.	PUNCT
ejpam-5390	68	3	proof	proof	NOUN
ejpam-5390	68	4	.	.	PUNCT
ejpam-5390	69	1	it	it	PRON
ejpam-5390	69	2	is	be	AUX
ejpam-5390	69	3	obvious	obvious	ADJ
ejpam-5390	69	4	that	that	SCONJ
ejpam-5390	69	5	π(0	π(0	NOUN
ejpam-5390	69	6	)	)	PUNCT
ejpam-5390	69	7	=	=	SYM
ejpam-5390	69	8	π([0	π([0	NOUN
ejpam-5390	69	9	,	,	PUNCT
ejpam-5390	69	10	0]∗	0]∗	NOUN
ejpam-5390	69	11	♢	♢	NOUN
ejpam-5390	69	12	λ0	λ0	NOUN
ejpam-5390	69	13	)	)	PUNCT
ejpam-5390	69	14	=	=	SYM
ejpam-5390	70	1	[	[	X
ejpam-5390	70	2	π(0	π(0	NOUN
ejpam-5390	70	3	)	)	PUNCT
ejpam-5390	70	4	,	,	PUNCT
ejpam-5390	70	5	0]∗	0]∗	NOUN
ejpam-5390	70	6	♢	♢	NOUN
ejpam-5390	70	7	λ0	λ0	NOUN
ejpam-5390	70	8	+	+	CCONJ
ejpam-5390	70	9	[	[	X
ejpam-5390	70	10	0,π(0)]∗	0,π(0)]∗	NOUN
ejpam-5390	70	11	♢	♢	NOUN
ejpam-5390	70	12	λ0	λ0	NOUN
ejpam-5390	70	13	+	+	CCONJ
ejpam-5390	71	1	[	[	X
ejpam-5390	71	2	0	0	NUM
ejpam-5390	71	3	,	,	PUNCT
ejpam-5390	71	4	0]∗	0]∗	PROPN
ejpam-5390	71	5	♢	♢	PROPN
ejpam-5390	71	6	λπ(0	λπ(0	PROPN
ejpam-5390	71	7	)	)	PUNCT
ejpam-5390	71	8	=	=	SYM
ejpam-5390	71	9	0	0	X
ejpam-5390	71	10	.	.	PUNCT
ejpam-5390	72	1	lemma	lemma	PROPN
ejpam-5390	72	2	2.2	2.2	NUM
ejpam-5390	72	3	.	.	PUNCT
ejpam-5390	73	1	let	let	VERB
ejpam-5390	73	2	u12	u12	PROPN
ejpam-5390	73	3	∈	∈	PROPN
ejpam-5390	73	4	a12	a12	NOUN
ejpam-5390	73	5	and	and	CCONJ
ejpam-5390	73	6	u21	u21	PROPN
ejpam-5390	73	7	∈	∈	PROPN
ejpam-5390	73	8	a21	a21	NOUN
ejpam-5390	73	9	.	.	PUNCT
ejpam-5390	74	1	then	then	ADV
ejpam-5390	74	2	π(u12	π(u12	VERB
ejpam-5390	74	3	+	+	CCONJ
ejpam-5390	74	4	u21	u21	NOUN
ejpam-5390	74	5	)	)	PUNCT
ejpam-5390	74	6	=	=	SYM
ejpam-5390	74	7	π(u12	π(u12	NOUN
ejpam-5390	74	8	)	)	PUNCT
ejpam-5390	74	9	+	+	CCONJ
ejpam-5390	74	10	π(u21	π(u21	NOUN
ejpam-5390	74	11	)	)	PUNCT
ejpam-5390	74	12	.	.	PUNCT
ejpam-5390	75	1	proof	proof	NOUN
ejpam-5390	75	2	.	.	PUNCT
ejpam-5390	76	1	let	let	VERB
ejpam-5390	76	2	t	t	NOUN
ejpam-5390	76	3	=	=	SYM
ejpam-5390	76	4	π(u12	π(u12	PROPN
ejpam-5390	76	5	+	+	CCONJ
ejpam-5390	76	6	u21	u21	NOUN
ejpam-5390	76	7	)	)	PUNCT
ejpam-5390	76	8	−	−	NOUN
ejpam-5390	76	9	π(u12	π(u12	NOUN
ejpam-5390	76	10	)	)	PUNCT
ejpam-5390	76	11	−	−	PROPN
ejpam-5390	76	12	π(u21	π(u21	PROPN
ejpam-5390	76	13	)	)	PUNCT
ejpam-5390	76	14	.	.	PUNCT
ejpam-5390	77	1	since	since	SCONJ
ejpam-5390	77	2	[	[	X
ejpam-5390	77	3	u12	u12	NOUN
ejpam-5390	77	4	,	,	PUNCT
ejpam-5390	77	5	p1]∗	p1]∗	PROPN
ejpam-5390	77	6	♢	♢	NOUN
ejpam-5390	77	7	λp2	λp2	NOUN
ejpam-5390	77	8	=	=	SYM
ejpam-5390	77	9	0	0	NUM
ejpam-5390	77	10	and	and	CCONJ
ejpam-5390	77	11	by	by	ADP
ejpam-5390	77	12	using	use	VERB
ejpam-5390	77	13	lemma	lemma	PROPN
ejpam-5390	77	14	2.1	2.1	NUM
ejpam-5390	77	15	,	,	PUNCT
ejpam-5390	77	16	we	we	PRON
ejpam-5390	77	17	have	have	VERB
ejpam-5390	77	18	π([u12	π([u12	NUM
ejpam-5390	77	19	+	+	CCONJ
ejpam-5390	77	20	v21	v21	NOUN
ejpam-5390	77	21	,	,	PUNCT
ejpam-5390	77	22	p1]∗	p1]∗	PROPN
ejpam-5390	77	23	♢	♢	NOUN
ejpam-5390	77	24	λp2	λp2	NOUN
ejpam-5390	77	25	)	)	PUNCT
ejpam-5390	77	26	=	=	SYM
ejpam-5390	77	27	π([u12	π([u12	PROPN
ejpam-5390	77	28	,	,	PUNCT
ejpam-5390	77	29	p1]∗	p1]∗	PROPN
ejpam-5390	77	30	♢	♢	NOUN
ejpam-5390	77	31	λp2	λp2	NOUN
ejpam-5390	77	32	)	)	PUNCT
ejpam-5390	78	1	+	+	CCONJ
ejpam-5390	78	2	π([v21	π([v21	NUM
ejpam-5390	78	3	,	,	PUNCT
ejpam-5390	78	4	p1]∗	p1]∗	PROPN
ejpam-5390	78	5	♢	♢	NOUN
ejpam-5390	78	6	λp2	λp2	NOUN
ejpam-5390	78	7	)	)	PUNCT
ejpam-5390	78	8	=	=	PUNCT
ejpam-5390	79	1	[	[	X
ejpam-5390	79	2	π(u12	π(u12	NOUN
ejpam-5390	79	3	)	)	PUNCT
ejpam-5390	79	4	,	,	PUNCT
ejpam-5390	79	5	p1]∗	p1]∗	PROPN
ejpam-5390	79	6	♢	♢	NOUN
ejpam-5390	79	7	λp2	λp2	NOUN
ejpam-5390	79	8	+	+	CCONJ
ejpam-5390	80	1	[	[	X
ejpam-5390	80	2	u12,π(p1)]∗	u12,π(p1)]∗	X
ejpam-5390	80	3	♢	♢	NOUN
ejpam-5390	80	4	λp2	λp2	NOUN
ejpam-5390	80	5	+	+	CCONJ
ejpam-5390	80	6	[	[	X
ejpam-5390	80	7	u12	u12	NOUN
ejpam-5390	80	8	,	,	PUNCT
ejpam-5390	80	9	p1]∗	p1]∗	PROPN
ejpam-5390	80	10	♢	♢	NOUN
ejpam-5390	80	11	λπ(p2	λπ(p2	NOUN
ejpam-5390	80	12	)	)	PUNCT
ejpam-5390	80	13	+	+	NOUN
ejpam-5390	80	14	[	[	X
ejpam-5390	80	15	π(v21	π(v21	NUM
ejpam-5390	80	16	)	)	PUNCT
ejpam-5390	80	17	,	,	PUNCT
ejpam-5390	80	18	p1]∗	p1]∗	PROPN
ejpam-5390	80	19	♢	♢	NOUN
ejpam-5390	80	20	λp2	λp2	NOUN
ejpam-5390	80	21	+	+	NOUN
ejpam-5390	80	22	[	[	X
ejpam-5390	80	23	v21,π(p1)]∗	v21,π(p1)]∗	NOUN
ejpam-5390	80	24	♢	♢	NOUN
ejpam-5390	80	25	λp2	λp2	NOUN
ejpam-5390	80	26	+	+	NUM
ejpam-5390	80	27	[	[	X
ejpam-5390	80	28	v21	v21	NOUN
ejpam-5390	80	29	,	,	PUNCT
ejpam-5390	80	30	p1]∗	p1]∗	PROPN
ejpam-5390	80	31	♢	♢	NOUN
ejpam-5390	80	32	λπ(p2	λπ(p2	NOUN
ejpam-5390	80	33	)	)	PUNCT
ejpam-5390	80	34	.	.	PUNCT
ejpam-5390	81	1	on	on	ADP
ejpam-5390	81	2	the	the	DET
ejpam-5390	81	3	other	other	ADJ
ejpam-5390	81	4	hand	hand	NOUN
ejpam-5390	81	5	,	,	PUNCT
ejpam-5390	81	6	we	we	PRON
ejpam-5390	81	7	find	find	VERB
ejpam-5390	81	8	π([u12	π([u12	NUM
ejpam-5390	81	9	+	+	CCONJ
ejpam-5390	81	10	v21	v21	NOUN
ejpam-5390	81	11	,	,	PUNCT
ejpam-5390	81	12	p1]∗	p1]∗	PROPN
ejpam-5390	81	13	♢	♢	NOUN
ejpam-5390	81	14	λp2	λp2	NOUN
ejpam-5390	81	15	=	=	SYM
ejpam-5390	82	1	[	[	X
ejpam-5390	82	2	π(u12	π(u12	X
ejpam-5390	82	3	+	+	CCONJ
ejpam-5390	82	4	v21	v21	NOUN
ejpam-5390	82	5	)	)	PUNCT
ejpam-5390	82	6	,	,	PUNCT
ejpam-5390	82	7	p1]∗	p1]∗	PROPN
ejpam-5390	82	8	♢	♢	NOUN
ejpam-5390	82	9	λp2	λp2	NOUN
ejpam-5390	82	10	+	+	NUM
ejpam-5390	83	1	[	[	X
ejpam-5390	83	2	u12	u12	X
ejpam-5390	83	3	+	+	CCONJ
ejpam-5390	83	4	v21,π(p1)]∗	v21,π(p1)]∗	NOUN
ejpam-5390	83	5	♢	♢	NOUN
ejpam-5390	83	6	λp2	λp2	NOUN
ejpam-5390	83	7	+	+	PROPN
ejpam-5390	83	8	[	[	X
ejpam-5390	83	9	u12	u12	X
ejpam-5390	83	10	+	+	CCONJ
ejpam-5390	83	11	v21	v21	NOUN
ejpam-5390	83	12	,	,	PUNCT
ejpam-5390	83	13	p1]∗	p1]∗	PROPN
ejpam-5390	83	14	♢	♢	NOUN
ejpam-5390	83	15	λπ(p2	λπ(p2	NOUN
ejpam-5390	83	16	)	)	PUNCT
ejpam-5390	83	17	.	.	PUNCT
ejpam-5390	84	1	from	from	ADP
ejpam-5390	84	2	the	the	DET
ejpam-5390	84	3	above	above	ADJ
ejpam-5390	84	4	two	two	NUM
ejpam-5390	84	5	equations	equation	NOUN
ejpam-5390	84	6	,	,	PUNCT
ejpam-5390	84	7	we	we	PRON
ejpam-5390	84	8	get	get	VERB
ejpam-5390	84	9	[	[	X
ejpam-5390	84	10	t	t	NOUN
ejpam-5390	84	11	,	,	PUNCT
ejpam-5390	84	12	p1]∗	p1]∗	PROPN
ejpam-5390	84	13	♢	♢	NOUN
ejpam-5390	84	14	λp2	λp2	NOUN
ejpam-5390	84	15	=	=	NOUN
ejpam-5390	84	16	0	0	X
ejpam-5390	84	17	.	.	PUNCT
ejpam-5390	85	1	that	that	PRON
ejpam-5390	85	2	means−p1	means−p1	PROPN
ejpam-5390	85	3	t	t	PROPN
ejpam-5390	85	4	∗p2+λp2tp1	∗p2+λp2tp1	PROPN
ejpam-5390	85	5	=	=	SYM
ejpam-5390	85	6	0	0	X
ejpam-5390	85	7	.	.	X
ejpam-5390	85	8	multiplying	multiply	VERB
ejpam-5390	85	9	by	by	ADP
ejpam-5390	85	10	p2	p2	PROPN
ejpam-5390	85	11	from	from	ADP
ejpam-5390	85	12	the	the	DET
ejpam-5390	85	13	left	left	NOUN
ejpam-5390	85	14	and	and	CCONJ
ejpam-5390	85	15	since	since	SCONJ
ejpam-5390	85	16	λ	λ	PROPN
ejpam-5390	85	17	̸=	̸=	PROPN
ejpam-5390	85	18	0	0	NUM
ejpam-5390	85	19	,	,	PUNCT
ejpam-5390	85	20	we	we	PRON
ejpam-5390	85	21	get	get	VERB
ejpam-5390	85	22	p2tp1	p2tp1	ADJ
ejpam-5390	85	23	=	=	SYM
ejpam-5390	85	24	0	0	NUM
ejpam-5390	85	25	.	.	PUNCT
ejpam-5390	86	1	similarly	similarly	ADV
ejpam-5390	86	2	,	,	PUNCT
ejpam-5390	86	3	one	one	PRON
ejpam-5390	86	4	can	can	AUX
ejpam-5390	86	5	show	show	VERB
ejpam-5390	86	6	that	that	DET
ejpam-5390	86	7	p1tp2	p1tp2	NOUN
ejpam-5390	86	8	=	=	NOUN
ejpam-5390	86	9	0	0	X
ejpam-5390	86	10	.	.	PUNCT
ejpam-5390	87	1	now	now	ADV
ejpam-5390	87	2	,	,	PUNCT
ejpam-5390	87	3	for	for	ADP
ejpam-5390	87	4	every	every	DET
ejpam-5390	87	5	x21	x21	PROPN
ejpam-5390	87	6	∈	∈	PROPN
ejpam-5390	87	7	a21	a21	NOUN
ejpam-5390	87	8	,	,	PUNCT
ejpam-5390	87	9	it	it	PRON
ejpam-5390	87	10	follows	follow	VERB
ejpam-5390	87	11	from	from	ADP
ejpam-5390	87	12	[	[	X
ejpam-5390	87	13	x21	x21	PROPN
ejpam-5390	87	14	,	,	PUNCT
ejpam-5390	87	15	u12]∗	u12]∗	PROPN
ejpam-5390	87	16	♢	♢	NOUN
ejpam-5390	87	17	λp1	λp1	NOUN
ejpam-5390	87	18	=	=	SYM
ejpam-5390	87	19	0	0	PUNCT
ejpam-5390	87	20	and	and	CCONJ
ejpam-5390	87	21	using	use	VERB
ejpam-5390	87	22	lemma	lemma	PROPN
ejpam-5390	87	23	2.1	2.1	NUM
ejpam-5390	87	24	that	that	DET
ejpam-5390	87	25	π([x21	π([x21	NUM
ejpam-5390	87	26	,	,	PUNCT
ejpam-5390	87	27	u12	u12	PROPN
ejpam-5390	87	28	+	+	CCONJ
ejpam-5390	87	29	v21]∗	v21]∗	VERB
ejpam-5390	87	30	♢	♢	PROPN
ejpam-5390	87	31	λp1	λp1	NOUN
ejpam-5390	87	32	)	)	PUNCT
ejpam-5390	87	33	=	=	SYM
ejpam-5390	87	34	π([x21	π([x21	PROPN
ejpam-5390	87	35	,	,	PUNCT
ejpam-5390	87	36	u12]∗	u12]∗	PROPN
ejpam-5390	87	37	♢	♢	PROPN
ejpam-5390	87	38	λp1	λp1	NOUN
ejpam-5390	87	39	)	)	PUNCT
ejpam-5390	88	1	+	+	NUM
ejpam-5390	89	1	π([x21	π([x21	PROPN
ejpam-5390	89	2	,	,	PUNCT
ejpam-5390	89	3	v21]∗	v21]∗	VERB
ejpam-5390	89	4	♢	♢	PROPN
ejpam-5390	89	5	λp1	λp1	NOUN
ejpam-5390	89	6	)	)	PUNCT
ejpam-5390	89	7	=	=	PUNCT
ejpam-5390	90	1	[	[	X
ejpam-5390	90	2	π(x21	π(x21	NOUN
ejpam-5390	90	3	)	)	PUNCT
ejpam-5390	90	4	,	,	PUNCT
ejpam-5390	90	5	u12]∗	u12]∗	PROPN
ejpam-5390	90	6	♢	♢	PROPN
ejpam-5390	90	7	λp1	λp1	PROPN
ejpam-5390	90	8	+	+	X
ejpam-5390	91	1	[	[	X
ejpam-5390	91	2	x21,π(u12)]∗	x21,π(u12)]∗	X
ejpam-5390	91	3	♢	♢	X
ejpam-5390	91	4	λp1	λp1	NOUN
ejpam-5390	92	1	+	+	PROPN
ejpam-5390	92	2	[	[	X
ejpam-5390	92	3	x21	x21	NUM
ejpam-5390	92	4	,	,	PUNCT
ejpam-5390	92	5	u12]∗	u12]∗	PROPN
ejpam-5390	92	6	♢	♢	PROPN
ejpam-5390	92	7	λπ(p1	λπ(p1	PROPN
ejpam-5390	92	8	)	)	PUNCT
ejpam-5390	93	1	+	+	CCONJ
ejpam-5390	93	2	[	[	X
ejpam-5390	93	3	π(x21	π(x21	NOUN
ejpam-5390	93	4	)	)	PUNCT
ejpam-5390	93	5	,	,	PUNCT
ejpam-5390	93	6	v21]∗	v21]∗	VERB
ejpam-5390	93	7	♢	♢	NOUN
ejpam-5390	93	8	λp1	λp1	NOUN
ejpam-5390	94	1	+	+	PROPN
ejpam-5390	95	1	[	[	X
ejpam-5390	95	2	x21,π(v21)]∗	x21,π(v21)]∗	X
ejpam-5390	95	3	♢	♢	PROPN
ejpam-5390	95	4	λp1	λp1	NOUN
ejpam-5390	95	5	+	+	X
ejpam-5390	96	1	[	[	X
ejpam-5390	96	2	x21	x21	NUM
ejpam-5390	96	3	,	,	PUNCT
ejpam-5390	96	4	v21]∗	v21]∗	VERB
ejpam-5390	96	5	♢	♢	PROPN
ejpam-5390	96	6	λπ(p1	λπ(p1	PROPN
ejpam-5390	96	7	)	)	PUNCT
ejpam-5390	96	8	.	.	PUNCT
ejpam-5390	97	1	on	on	ADP
ejpam-5390	97	2	the	the	DET
ejpam-5390	97	3	other	other	ADJ
ejpam-5390	97	4	hand	hand	NOUN
ejpam-5390	97	5	,	,	PUNCT
ejpam-5390	97	6	we	we	PRON
ejpam-5390	97	7	have	have	VERB
ejpam-5390	97	8	π([x21	π([x21	NUM
ejpam-5390	97	9	,	,	PUNCT
ejpam-5390	97	10	u12	u12	PROPN
ejpam-5390	97	11	+	+	CCONJ
ejpam-5390	97	12	v21]∗	v21]∗	VERB
ejpam-5390	97	13	♢	♢	PROPN
ejpam-5390	97	14	λp1	λp1	NOUN
ejpam-5390	97	15	)	)	PUNCT
ejpam-5390	97	16	=	=	PUNCT
ejpam-5390	98	1	[	[	X
ejpam-5390	98	2	π(x21	π(x21	NOUN
ejpam-5390	98	3	)	)	PUNCT
ejpam-5390	98	4	,	,	PUNCT
ejpam-5390	98	5	u12	u12	PROPN
ejpam-5390	98	6	+	+	CCONJ
ejpam-5390	98	7	v21]∗	v21]∗	VERB
ejpam-5390	98	8	♢	♢	NOUN
ejpam-5390	98	9	λp1	λp1	NOUN
ejpam-5390	98	10	+	+	X
ejpam-5390	99	1	[	[	X
ejpam-5390	99	2	x21,π(u12	x21,π(u12	X
ejpam-5390	99	3	+	+	CCONJ
ejpam-5390	99	4	v21)]∗	v21)]∗	PROPN
ejpam-5390	99	5	♢	♢	NOUN
ejpam-5390	99	6	λp1	λp1	NOUN
ejpam-5390	99	7	+	+	PROPN
ejpam-5390	100	1	[	[	X
ejpam-5390	100	2	x21	x21	NUM
ejpam-5390	100	3	,	,	PUNCT
ejpam-5390	100	4	u12	u12	PROPN
ejpam-5390	100	5	+	+	CCONJ
ejpam-5390	100	6	v21]∗	v21]∗	VERB
ejpam-5390	100	7	♢	♢	PROPN
ejpam-5390	100	8	λπ(p1	λπ(p1	PROPN
ejpam-5390	100	9	)	)	PUNCT
ejpam-5390	100	10	.	.	PUNCT
ejpam-5390	101	1	from	from	ADP
ejpam-5390	101	2	the	the	DET
ejpam-5390	101	3	above	above	ADJ
ejpam-5390	101	4	expressions	expression	NOUN
ejpam-5390	101	5	,	,	PUNCT
ejpam-5390	101	6	we	we	PRON
ejpam-5390	101	7	find	find	VERB
ejpam-5390	101	8	that	that	SCONJ
ejpam-5390	102	1	[	[	X
ejpam-5390	102	2	x21	x21	PROPN
ejpam-5390	102	3	,	,	PUNCT
ejpam-5390	102	4	t	t	PROPN
ejpam-5390	102	5	]	]	PUNCT
ejpam-5390	102	6	∗	∗	X
ejpam-5390	102	7	♢	♢	NOUN
ejpam-5390	102	8	λp1	λp1	NOUN
ejpam-5390	102	9	=	=	SYM
ejpam-5390	102	10	0	0	NUM
ejpam-5390	102	11	.	.	PUNCT
ejpam-5390	102	12	that	that	PRON
ejpam-5390	102	13	means	mean	VERB
ejpam-5390	102	14	x21tp1	x21tp1	PROPN
ejpam-5390	102	15	−	−	PROPN
ejpam-5390	102	16	λp1tx	λp1tx	NUM
ejpam-5390	102	17	∗	∗	NOUN
ejpam-5390	102	18	21	21	NUM
ejpam-5390	102	19	=	=	SYM
ejpam-5390	102	20	0	0	X
ejpam-5390	102	21	.	.	PUNCT
ejpam-5390	103	1	multiplying	multiply	VERB
ejpam-5390	103	2	both	both	DET
ejpam-5390	103	3	sides	side	NOUN
ejpam-5390	103	4	by	by	ADP
ejpam-5390	103	5	p1	p1	PROPN
ejpam-5390	103	6	from	from	ADP
ejpam-5390	103	7	right	right	ADJ
ejpam-5390	103	8	,	,	PUNCT
ejpam-5390	103	9	we	we	PRON
ejpam-5390	103	10	get	get	VERB
ejpam-5390	103	11	x21tp1	x21tp1	PROPN
ejpam-5390	103	12	=	=	SYM
ejpam-5390	103	13	0	0	X
ejpam-5390	103	14	.	.	PUNCT
ejpam-5390	104	1	by	by	ADP
ejpam-5390	104	2	using	use	VERB
ejpam-5390	104	3	(	(	PUNCT
ejpam-5390	104	4	▲	▲	PUNCT
ejpam-5390	104	5	)	)	PUNCT
ejpam-5390	104	6	and	and	CCONJ
ejpam-5390	104	7	(	(	PUNCT
ejpam-5390	104	8	▼	▼	NOUN
ejpam-5390	104	9	)	)	PUNCT
ejpam-5390	104	10	,	,	PUNCT
ejpam-5390	104	11	we	we	PRON
ejpam-5390	104	12	have	have	VERB
ejpam-5390	104	13	p1tp1	p1tp1	NOUN
ejpam-5390	104	14	=	=	SYM
ejpam-5390	104	15	0	0	X
ejpam-5390	104	16	.	.	PUNCT
ejpam-5390	105	1	similarly	similarly	ADV
ejpam-5390	105	2	,	,	PUNCT
ejpam-5390	105	3	we	we	PRON
ejpam-5390	105	4	can	can	AUX
ejpam-5390	105	5	show	show	VERB
ejpam-5390	105	6	that	that	SCONJ
ejpam-5390	105	7	p2tp2	p2tp2	NOUN
ejpam-5390	105	8	=	=	NOUN
ejpam-5390	105	9	0	0	X
ejpam-5390	105	10	.	.	PUNCT
ejpam-5390	106	1	hence	hence	ADV
ejpam-5390	106	2	,	,	PUNCT
ejpam-5390	106	3	t	t	PROPN
ejpam-5390	106	4	=	=	SYM
ejpam-5390	106	5	0	0	PUNCT
ejpam-5390	106	6	i.e.	i.e.	X
ejpam-5390	106	7	,	,	PUNCT
ejpam-5390	106	8	π(u12	π(u12	X
ejpam-5390	106	9	+	+	CCONJ
ejpam-5390	106	10	u21	u21	NOUN
ejpam-5390	106	11	)	)	PUNCT
ejpam-5390	106	12	=	=	SYM
ejpam-5390	106	13	π(u12	π(u12	NOUN
ejpam-5390	106	14	)	)	PUNCT
ejpam-5390	106	15	+	+	CCONJ
ejpam-5390	106	16	π(u21	π(u21	NOUN
ejpam-5390	106	17	)	)	PUNCT
ejpam-5390	106	18	.	.	PUNCT
ejpam-5390	107	1	lemma	lemma	PROPN
ejpam-5390	107	2	2.3	2.3	NUM
ejpam-5390	107	3	.	.	PUNCT
ejpam-5390	108	1	for	for	ADP
ejpam-5390	108	2	any	any	DET
ejpam-5390	108	3	uij	uij	PROPN
ejpam-5390	108	4	∈	∈	PROPN
ejpam-5390	108	5	aij	aij	PROPN
ejpam-5390	108	6	,	,	PUNCT
ejpam-5390	108	7	1	1	NUM
ejpam-5390	108	8	≤	≤	NUM
ejpam-5390	108	9	i	i	PRON
ejpam-5390	108	10	,	,	PUNCT
ejpam-5390	108	11	j	j	PROPN
ejpam-5390	108	12	≤	≤	PROPN
ejpam-5390	108	13	2	2	NUM
ejpam-5390	108	14	,	,	PUNCT
ejpam-5390	108	15	we	we	PRON
ejpam-5390	108	16	have	have	VERB
ejpam-5390	108	17	π	π	PROPN
ejpam-5390	108	18	(	(	PUNCT
ejpam-5390	108	19	2∑	2∑	NUM
ejpam-5390	108	20	i	i	NOUN
ejpam-5390	108	21	,	,	PUNCT
ejpam-5390	108	22	j=1	j=1	PROPN
ejpam-5390	108	23	uij	uij	PRON
ejpam-5390	108	24	)	)	PUNCT
ejpam-5390	109	1	=	=	PUNCT
ejpam-5390	110	1	2∑	2∑	NUM
ejpam-5390	110	2	i	i	PRON
ejpam-5390	110	3	,	,	PUNCT
ejpam-5390	110	4	j=1	j=1	ADJ
ejpam-5390	110	5	π(uij	π(uij	PROPN
ejpam-5390	110	6	)	)	PUNCT
ejpam-5390	110	7	.	.	PUNCT
ejpam-5390	110	8	.	.	PUNCT
ejpam-5390	111	1	nisar	nisar	PROPN
ejpam-5390	111	2	et	et	PROPN
ejpam-5390	111	3	al	al	PROPN
ejpam-5390	111	4	.	.	PUNCT
ejpam-5390	111	5	/	/	SYM
ejpam-5390	111	6	eur	eur	PROPN
ejpam-5390	111	7	.	.	PUNCT
ejpam-5390	112	1	j.	j.	PROPN
ejpam-5390	112	2	pure	pure	PROPN
ejpam-5390	112	3	appl	appl	PROPN
ejpam-5390	112	4	.	.	PROPN
ejpam-5390	112	5	math	math	PROPN
ejpam-5390	112	6	,	,	PUNCT
ejpam-5390	112	7	17	17	NUM
ejpam-5390	112	8	(	(	PUNCT
ejpam-5390	112	9	4	4	NUM
ejpam-5390	112	10	)	)	PUNCT
ejpam-5390	112	11	(	(	PUNCT
ejpam-5390	112	12	2024	2024	NUM
ejpam-5390	112	13	)	)	PUNCT
ejpam-5390	112	14	,	,	PUNCT
ejpam-5390	112	15	3399	3399	NUM
ejpam-5390	112	16	-	-	SYM
ejpam-5390	112	17	3414	3414	NUM
ejpam-5390	112	18	3402	3402	NUM
ejpam-5390	112	19	proof	proof	NOUN
ejpam-5390	112	20	.	.	PUNCT
ejpam-5390	113	1	let	let	VERB
ejpam-5390	113	2	t	t	NOUN
ejpam-5390	113	3	=	=	SYM
ejpam-5390	113	4	π(u11	π(u11	ADJ
ejpam-5390	113	5	+	+	NUM
ejpam-5390	113	6	u12	u12	NOUN
ejpam-5390	113	7	+	+	CCONJ
ejpam-5390	113	8	u21	u21	NOUN
ejpam-5390	113	9	+	+	CCONJ
ejpam-5390	113	10	u22)−π(u11)−π(u12)−π(u21)−π(u22	u22)−π(u11)−π(u12)−π(u21)−π(u22	NOUN
ejpam-5390	113	11	)	)	PUNCT
ejpam-5390	113	12	.	.	PUNCT
ejpam-5390	114	1	for	for	ADP
ejpam-5390	114	2	every	every	DET
ejpam-5390	114	3	x12	x12	NUM
ejpam-5390	114	4	∈	∈	PROPN
ejpam-5390	114	5	a12	a12	NOUN
ejpam-5390	114	6	,	,	PUNCT
ejpam-5390	114	7	also	also	ADV
ejpam-5390	114	8	[	[	X
ejpam-5390	114	9	p1	p1	NOUN
ejpam-5390	114	10	,	,	PUNCT
ejpam-5390	114	11	u11]∗	u11]∗	PROPN
ejpam-5390	114	12	♢	♢	PROPN
ejpam-5390	115	1	λx12	λx12	PROPN
ejpam-5390	115	2	=	=	PUNCT
ejpam-5390	115	3	[	[	X
ejpam-5390	115	4	p1	p1	NOUN
ejpam-5390	115	5	,	,	PUNCT
ejpam-5390	115	6	u22]∗	u22]∗	NOUN
ejpam-5390	115	7	♢	♢	NOUN
ejpam-5390	115	8	λx12	λx12	PROPN
ejpam-5390	115	9	=	=	SYM
ejpam-5390	115	10	0	0	PUNCT
ejpam-5390	115	11	and	and	CCONJ
ejpam-5390	115	12	using	use	VERB
ejpam-5390	115	13	lemmas	lemmas	PROPN
ejpam-5390	115	14	2.1	2.1	NUM
ejpam-5390	115	15	and	and	CCONJ
ejpam-5390	115	16	2.2	2.2	NUM
ejpam-5390	115	17	,	,	PUNCT
ejpam-5390	115	18	we	we	PRON
ejpam-5390	115	19	get	get	VERB
ejpam-5390	115	20	π([p1	π([p1	NOUN
ejpam-5390	115	21	,	,	PUNCT
ejpam-5390	115	22	u11	u11	PROPN
ejpam-5390	115	23	+	+	CCONJ
ejpam-5390	115	24	u12	u12	PROPN
ejpam-5390	115	25	+	+	CCONJ
ejpam-5390	115	26	u21	u21	NOUN
ejpam-5390	115	27	+	+	CCONJ
ejpam-5390	115	28	u22]∗	u22]∗	NOUN
ejpam-5390	115	29	♢	♢	NOUN
ejpam-5390	115	30	λx12	λx12	PROPN
ejpam-5390	115	31	)	)	PUNCT
ejpam-5390	115	32	=	=	SYM
ejpam-5390	115	33	π([p1	π([p1	NOUN
ejpam-5390	115	34	,	,	PUNCT
ejpam-5390	115	35	u11]∗	u11]∗	PROPN
ejpam-5390	115	36	♢	♢	PROPN
ejpam-5390	115	37	λx12	λx12	PROPN
ejpam-5390	115	38	)	)	PUNCT
ejpam-5390	116	1	+	+	NUM
ejpam-5390	116	2	π([p1	π([p1	PROPN
ejpam-5390	116	3	,	,	PUNCT
ejpam-5390	116	4	u12]∗	u12]∗	PROPN
ejpam-5390	116	5	♢	♢	PROPN
ejpam-5390	116	6	λx12	λx12	PROPN
ejpam-5390	116	7	)	)	PUNCT
ejpam-5390	116	8	+	+	NOUN
ejpam-5390	116	9	π([p1	π([p1	NOUN
ejpam-5390	116	10	,	,	PUNCT
ejpam-5390	116	11	u21]∗	u21]∗	PROPN
ejpam-5390	116	12	♢	♢	PROPN
ejpam-5390	116	13	λx12	λx12	PROPN
ejpam-5390	116	14	)	)	PUNCT
ejpam-5390	116	15	+	+	NUM
ejpam-5390	116	16	π([p1	π([p1	NOUN
ejpam-5390	116	17	,	,	PUNCT
ejpam-5390	116	18	u22]∗	u22]∗	PROPN
ejpam-5390	116	19	♢	♢	NOUN
ejpam-5390	116	20	λx12	λx12	PROPN
ejpam-5390	116	21	)	)	PUNCT
ejpam-5390	116	22	=	=	PUNCT
ejpam-5390	117	1	[	[	X
ejpam-5390	117	2	π(p1	π(p1	NOUN
ejpam-5390	117	3	)	)	PUNCT
ejpam-5390	117	4	,	,	PUNCT
ejpam-5390	117	5	u11]∗	u11]∗	PROPN
ejpam-5390	117	6	♢	♢	PROPN
ejpam-5390	117	7	λx12	λx12	PROPN
ejpam-5390	117	8	+	+	PROPN
ejpam-5390	117	9	[	[	X
ejpam-5390	117	10	p1,π(u11)]∗	p1,π(u11)]∗	X
ejpam-5390	117	11	♢	♢	PROPN
ejpam-5390	117	12	λx12	λx12	PROPN
ejpam-5390	117	13	+	+	PROPN
ejpam-5390	117	14	[	[	X
ejpam-5390	117	15	p1	p1	NOUN
ejpam-5390	117	16	,	,	PUNCT
ejpam-5390	117	17	u11]∗	u11]∗	PROPN
ejpam-5390	117	18	♢	♢	PROPN
ejpam-5390	117	19	λπ(x12	λπ(x12	PROPN
ejpam-5390	117	20	)	)	PUNCT
ejpam-5390	117	21	+	+	CCONJ
ejpam-5390	118	1	[	[	X
ejpam-5390	118	2	π(p1	π(p1	NOUN
ejpam-5390	118	3	)	)	PUNCT
ejpam-5390	118	4	,	,	PUNCT
ejpam-5390	118	5	u12]∗	u12]∗	PROPN
ejpam-5390	118	6	♢	♢	PROPN
ejpam-5390	118	7	λx12	λx12	PROPN
ejpam-5390	118	8	+	+	PROPN
ejpam-5390	118	9	[	[	X
ejpam-5390	118	10	p1,π(u12)]∗	p1,π(u12)]∗	NOUN
ejpam-5390	118	11	♢	♢	NOUN
ejpam-5390	118	12	λx12	λx12	PROPN
ejpam-5390	118	13	+	+	PROPN
ejpam-5390	119	1	[	[	X
ejpam-5390	119	2	p1	p1	NOUN
ejpam-5390	119	3	,	,	PUNCT
ejpam-5390	119	4	u12]∗	u12]∗	PROPN
ejpam-5390	119	5	♢	♢	PROPN
ejpam-5390	119	6	λπ(x12	λπ(x12	PROPN
ejpam-5390	119	7	)	)	PUNCT
ejpam-5390	119	8	+	+	PROPN
ejpam-5390	119	9	[	[	X
ejpam-5390	119	10	π(p1	π(p1	NOUN
ejpam-5390	119	11	)	)	PUNCT
ejpam-5390	119	12	,	,	PUNCT
ejpam-5390	119	13	u21]∗	u21]∗	NOUN
ejpam-5390	119	14	♢	♢	PROPN
ejpam-5390	119	15	λx12	λx12	PROPN
ejpam-5390	119	16	+	+	PROPN
ejpam-5390	120	1	[	[	X
ejpam-5390	120	2	p1,π(u21)]∗	p1,π(u21)]∗	X
ejpam-5390	120	3	♢	♢	PROPN
ejpam-5390	120	4	λx12	λx12	PROPN
ejpam-5390	120	5	+	+	PROPN
ejpam-5390	120	6	[	[	X
ejpam-5390	120	7	p1	p1	NOUN
ejpam-5390	120	8	,	,	PUNCT
ejpam-5390	120	9	u21]∗	u21]∗	PROPN
ejpam-5390	120	10	♢	♢	PROPN
ejpam-5390	120	11	λπ(x12	λπ(x12	PUNCT
ejpam-5390	120	12	)	)	PUNCT
ejpam-5390	120	13	+	+	CCONJ
ejpam-5390	120	14	[	[	X
ejpam-5390	120	15	π(p1	π(p1	NOUN
ejpam-5390	120	16	)	)	PUNCT
ejpam-5390	120	17	,	,	PUNCT
ejpam-5390	120	18	u22]∗	u22]∗	PROPN
ejpam-5390	120	19	♢	♢	NOUN
ejpam-5390	120	20	λx12	λx12	PROPN
ejpam-5390	120	21	+	+	PROPN
ejpam-5390	120	22	[	[	X
ejpam-5390	120	23	p1,π(u22)]∗	p1,π(u22)]∗	NOUN
ejpam-5390	120	24	♢	♢	NOUN
ejpam-5390	120	25	λx12	λx12	PROPN
ejpam-5390	120	26	+	+	PROPN
ejpam-5390	120	27	[	[	X
ejpam-5390	120	28	p1	p1	NOUN
ejpam-5390	120	29	,	,	PUNCT
ejpam-5390	120	30	u22]∗	u22]∗	PROPN
ejpam-5390	120	31	♢	♢	PROPN
ejpam-5390	120	32	λπ(x12	λπ(x12	NOUN
ejpam-5390	120	33	)	)	PUNCT
ejpam-5390	120	34	.	.	PUNCT
ejpam-5390	121	1	on	on	ADP
ejpam-5390	121	2	the	the	DET
ejpam-5390	121	3	other	other	ADJ
ejpam-5390	121	4	hand	hand	NOUN
ejpam-5390	121	5	,	,	PUNCT
ejpam-5390	121	6	we	we	PRON
ejpam-5390	121	7	have	have	VERB
ejpam-5390	121	8	π([p1	π([p1	PROPN
ejpam-5390	121	9	,	,	PUNCT
ejpam-5390	121	10	u11	u11	PROPN
ejpam-5390	121	11	+	+	CCONJ
ejpam-5390	121	12	u12	u12	PROPN
ejpam-5390	121	13	+	+	CCONJ
ejpam-5390	121	14	u21	u21	NOUN
ejpam-5390	121	15	+	+	CCONJ
ejpam-5390	121	16	u22]∗	u22]∗	NOUN
ejpam-5390	121	17	♢	♢	NOUN
ejpam-5390	121	18	λx12	λx12	PROPN
ejpam-5390	121	19	)	)	PUNCT
ejpam-5390	121	20	=	=	PUNCT
ejpam-5390	122	1	[	[	X
ejpam-5390	122	2	π(p1	π(p1	NOUN
ejpam-5390	122	3	)	)	PUNCT
ejpam-5390	122	4	,	,	PUNCT
ejpam-5390	122	5	u11	u11	PROPN
ejpam-5390	122	6	+	+	NUM
ejpam-5390	122	7	u12	u12	PROPN
ejpam-5390	122	8	+	+	CCONJ
ejpam-5390	122	9	u21	u21	NOUN
ejpam-5390	122	10	+	+	CCONJ
ejpam-5390	122	11	u22]∗	u22]∗	NOUN
ejpam-5390	122	12	♢	♢	NOUN
ejpam-5390	122	13	λx12	λx12	PROPN
ejpam-5390	122	14	+	+	PROPN
ejpam-5390	122	15	[	[	X
ejpam-5390	122	16	p1,π(u11	p1,π(u11	ADJ
ejpam-5390	122	17	+	+	CCONJ
ejpam-5390	122	18	u12	u12	PROPN
ejpam-5390	122	19	+	+	CCONJ
ejpam-5390	122	20	u21	u21	PROPN
ejpam-5390	122	21	+	+	SYM
ejpam-5390	122	22	u22)]∗	u22)]∗	PROPN
ejpam-5390	122	23	♢	♢	PROPN
ejpam-5390	122	24	λx12	λx12	PROPN
ejpam-5390	122	25	+	+	PROPN
ejpam-5390	122	26	[	[	X
ejpam-5390	122	27	p1	p1	NOUN
ejpam-5390	122	28	,	,	PUNCT
ejpam-5390	122	29	u11	u11	PROPN
ejpam-5390	122	30	+	+	CCONJ
ejpam-5390	122	31	u12	u12	PROPN
ejpam-5390	122	32	+	+	CCONJ
ejpam-5390	122	33	u21	u21	NOUN
ejpam-5390	122	34	+	+	CCONJ
ejpam-5390	122	35	u22]∗	u22]∗	PROPN
ejpam-5390	122	36	♢	♢	PROPN
ejpam-5390	122	37	λπ(x12	λπ(x12	NOUN
ejpam-5390	122	38	)	)	PUNCT
ejpam-5390	122	39	.	.	PUNCT
ejpam-5390	123	1	by	by	ADP
ejpam-5390	123	2	comparing	compare	VERB
ejpam-5390	123	3	the	the	DET
ejpam-5390	123	4	above	above	ADJ
ejpam-5390	123	5	two	two	NUM
ejpam-5390	123	6	equations	equation	NOUN
ejpam-5390	123	7	,	,	PUNCT
ejpam-5390	123	8	we	we	PRON
ejpam-5390	123	9	get	get	VERB
ejpam-5390	124	1	[	[	X
ejpam-5390	124	2	p1,m	p1,m	NOUN
ejpam-5390	124	3	]	]	PUNCT
ejpam-5390	124	4	∗	∗	X
ejpam-5390	124	5	♢	♢	NOUN
ejpam-5390	124	6	λx12	λx12	PROPN
ejpam-5390	124	7	=	=	SYM
ejpam-5390	124	8	0	0	NUM
ejpam-5390	124	9	from	from	ADP
ejpam-5390	124	10	which	which	PRON
ejpam-5390	124	11	we	we	PRON
ejpam-5390	124	12	obtain	obtain	VERB
ejpam-5390	124	13	p1tx12	p1tx12	PROPN
ejpam-5390	124	14	−	−	PROPN
ejpam-5390	124	15	tx12	tx12	PROPN
ejpam-5390	124	16	−	−	PROPN
ejpam-5390	124	17	λx12tp1	λx12tp1	ADJ
ejpam-5390	124	18	=	=	NOUN
ejpam-5390	124	19	0	0	X
ejpam-5390	124	20	.	.	PUNCT
ejpam-5390	124	21	multiplying	multiply	VERB
ejpam-5390	124	22	p2	p2	PROPN
ejpam-5390	124	23	from	from	ADP
ejpam-5390	124	24	left	left	ADJ
ejpam-5390	124	25	and	and	CCONJ
ejpam-5390	124	26	right	right	ADJ
ejpam-5390	124	27	,	,	PUNCT
ejpam-5390	124	28	we	we	PRON
ejpam-5390	124	29	get	get	VERB
ejpam-5390	124	30	p2tx12	p2tx12	NOUN
ejpam-5390	124	31	=	=	SYM
ejpam-5390	124	32	0	0	NUM
ejpam-5390	124	33	.	.	PUNCT
ejpam-5390	125	1	by	by	ADP
ejpam-5390	125	2	using	use	VERB
ejpam-5390	125	3	(	(	PUNCT
ejpam-5390	125	4	▲	▲	PUNCT
ejpam-5390	125	5	)	)	PUNCT
ejpam-5390	125	6	and	and	CCONJ
ejpam-5390	125	7	(	(	PUNCT
ejpam-5390	125	8	▼	▼	NOUN
ejpam-5390	125	9	)	)	PUNCT
ejpam-5390	125	10	,	,	PUNCT
ejpam-5390	125	11	we	we	PRON
ejpam-5390	125	12	have	have	VERB
ejpam-5390	125	13	p2tp1	p2tp1	NOUN
ejpam-5390	125	14	=	=	SYM
ejpam-5390	125	15	0	0	NUM
ejpam-5390	125	16	.	.	PUNCT
ejpam-5390	126	1	similarly	similarly	ADV
ejpam-5390	126	2	,	,	PUNCT
ejpam-5390	126	3	we	we	PRON
ejpam-5390	126	4	can	can	AUX
ejpam-5390	126	5	show	show	VERB
ejpam-5390	126	6	that	that	DET
ejpam-5390	126	7	p1tp2	p1tp2	NOUN
ejpam-5390	126	8	=	=	NOUN
ejpam-5390	126	9	0	0	X
ejpam-5390	126	10	.	.	PUNCT
ejpam-5390	126	11	again	again	ADV
ejpam-5390	126	12	forx12	forx12	PROPN
ejpam-5390	126	13	∈	∈	PROPN
ejpam-5390	126	14	a12	a12	PROPN
ejpam-5390	126	15	,	,	PUNCT
ejpam-5390	126	16	it	it	PRON
ejpam-5390	126	17	follows	follow	VERB
ejpam-5390	126	18	from	from	ADP
ejpam-5390	126	19	[	[	X
ejpam-5390	126	20	x12	x12	NUM
ejpam-5390	126	21	,	,	PUNCT
ejpam-5390	126	22	u11]∗	u11]∗	NOUN
ejpam-5390	126	23	♢	♢	PROPN
ejpam-5390	126	24	λp2	λp2	NOUN
ejpam-5390	126	25	=	=	SYM
ejpam-5390	127	1	[	[	X
ejpam-5390	127	2	x12	x12	NUM
ejpam-5390	127	3	,	,	PUNCT
ejpam-5390	127	4	u12]∗	u12]∗	NOUN
ejpam-5390	127	5	♢	♢	PROPN
ejpam-5390	127	6	λp2	λp2	NOUN
ejpam-5390	127	7	=	=	PUNCT
ejpam-5390	128	1	[	[	X
ejpam-5390	128	2	x12	x12	NUM
ejpam-5390	128	3	,	,	PUNCT
ejpam-5390	128	4	u12]∗	u12]∗	NOUN
ejpam-5390	128	5	♢	♢	NOUN
ejpam-5390	128	6	λp2	λp2	NOUN
ejpam-5390	128	7	=	=	SYM
ejpam-5390	128	8	0	0	NUM
ejpam-5390	128	9	that	that	PRON
ejpam-5390	128	10	π([x12	π([x12	NUM
ejpam-5390	128	11	,	,	PUNCT
ejpam-5390	128	12	u11	u11	PROPN
ejpam-5390	128	13	+	+	CCONJ
ejpam-5390	128	14	u12	u12	PROPN
ejpam-5390	128	15	+	+	CCONJ
ejpam-5390	128	16	u21	u21	NOUN
ejpam-5390	128	17	+	+	CCONJ
ejpam-5390	128	18	u22]∗	u22]∗	NOUN
ejpam-5390	128	19	♢	♢	NOUN
ejpam-5390	128	20	λp2	λp2	NOUN
ejpam-5390	128	21	)	)	PUNCT
ejpam-5390	128	22	=	=	SYM
ejpam-5390	128	23	π([x12	π([x12	NUM
ejpam-5390	128	24	,	,	PUNCT
ejpam-5390	128	25	u11]∗	u11]∗	PROPN
ejpam-5390	128	26	♢	♢	PROPN
ejpam-5390	128	27	λp2	λp2	NOUN
ejpam-5390	128	28	)	)	PUNCT
ejpam-5390	129	1	+	+	NUM
ejpam-5390	129	2	π([x12	π([x12	NUM
ejpam-5390	129	3	,	,	PUNCT
ejpam-5390	129	4	u12]∗	u12]∗	NOUN
ejpam-5390	129	5	♢	♢	PROPN
ejpam-5390	129	6	λp2	λp2	NOUN
ejpam-5390	129	7	)	)	PUNCT
ejpam-5390	129	8	+	+	NOUN
ejpam-5390	129	9	π([x12	π([x12	NUM
ejpam-5390	129	10	,	,	PUNCT
ejpam-5390	129	11	u21]∗	u21]∗	NOUN
ejpam-5390	129	12	♢	♢	NOUN
ejpam-5390	129	13	λp2	λp2	NOUN
ejpam-5390	129	14	)	)	PUNCT
ejpam-5390	130	1	+	+	NUM
ejpam-5390	130	2	π([x12	π([x12	NUM
ejpam-5390	130	3	,	,	PUNCT
ejpam-5390	130	4	u22]∗	u22]∗	NOUN
ejpam-5390	130	5	♢	♢	NOUN
ejpam-5390	130	6	λp2	λp2	NOUN
ejpam-5390	130	7	)	)	PUNCT
ejpam-5390	130	8	=	=	PUNCT
ejpam-5390	131	1	[	[	X
ejpam-5390	131	2	π(x12	π(x12	NUM
ejpam-5390	131	3	)	)	PUNCT
ejpam-5390	131	4	,	,	PUNCT
ejpam-5390	131	5	u11]∗	u11]∗	PROPN
ejpam-5390	131	6	♢	♢	VERB
ejpam-5390	131	7	λp2	λp2	NOUN
ejpam-5390	131	8	+	+	SYM
ejpam-5390	132	1	[	[	X
ejpam-5390	132	2	x12,π(u11)]∗	x12,π(u11)]∗	X
ejpam-5390	132	3	♢	♢	PROPN
ejpam-5390	132	4	λp2	λp2	NOUN
ejpam-5390	132	5	+	+	PROPN
ejpam-5390	132	6	[	[	X
ejpam-5390	132	7	x12	x12	NUM
ejpam-5390	132	8	,	,	PUNCT
ejpam-5390	132	9	u11]∗	u11]∗	NOUN
ejpam-5390	132	10	♢	♢	PROPN
ejpam-5390	132	11	λπ(p2	λπ(p2	X
ejpam-5390	132	12	)	)	PUNCT
ejpam-5390	133	1	+	+	CCONJ
ejpam-5390	134	1	[	[	X
ejpam-5390	134	2	π(x12	π(x12	NUM
ejpam-5390	134	3	)	)	PUNCT
ejpam-5390	134	4	,	,	PUNCT
ejpam-5390	134	5	u12]∗	u12]∗	PROPN
ejpam-5390	134	6	♢	♢	PROPN
ejpam-5390	134	7	λp2	λp2	NOUN
ejpam-5390	134	8	+	+	PROPN
ejpam-5390	134	9	[	[	X
ejpam-5390	134	10	x12,π(u12)]∗	x12,π(u12)]∗	X
ejpam-5390	134	11	♢	♢	PROPN
ejpam-5390	134	12	λp2	λp2	NOUN
ejpam-5390	134	13	+	+	SYM
ejpam-5390	135	1	[	[	X
ejpam-5390	135	2	x12	x12	NUM
ejpam-5390	135	3	,	,	PUNCT
ejpam-5390	135	4	u12]∗	u12]∗	PROPN
ejpam-5390	135	5	♢	♢	NOUN
ejpam-5390	135	6	λπ(p2	λπ(p2	X
ejpam-5390	135	7	)	)	PUNCT
ejpam-5390	136	1	+	+	NOUN
ejpam-5390	136	2	[	[	X
ejpam-5390	136	3	π(x12	π(x12	NUM
ejpam-5390	136	4	)	)	PUNCT
ejpam-5390	136	5	,	,	PUNCT
ejpam-5390	136	6	u21]∗	u21]∗	PROPN
ejpam-5390	136	7	♢	♢	PROPN
ejpam-5390	137	1	λp2	λp2	NOUN
ejpam-5390	137	2	+	+	X
ejpam-5390	138	1	[	[	X
ejpam-5390	138	2	x12,π(u21)]∗	x12,π(u21)]∗	X
ejpam-5390	138	3	♢	♢	PROPN
ejpam-5390	138	4	λp2	λp2	NOUN
ejpam-5390	139	1	+	+	PROPN
ejpam-5390	139	2	[	[	X
ejpam-5390	139	3	x12	x12	NUM
ejpam-5390	139	4	,	,	PUNCT
ejpam-5390	139	5	u21]∗	u21]∗	NOUN
ejpam-5390	139	6	♢	♢	NOUN
ejpam-5390	139	7	λπ(p2	λπ(p2	X
ejpam-5390	139	8	)	)	PUNCT
ejpam-5390	139	9	+	+	CCONJ
ejpam-5390	140	1	[	[	X
ejpam-5390	140	2	π(x12	π(x12	NUM
ejpam-5390	140	3	)	)	PUNCT
ejpam-5390	140	4	,	,	PUNCT
ejpam-5390	140	5	u22]∗	u22]∗	NOUN
ejpam-5390	140	6	♢	♢	NOUN
ejpam-5390	140	7	λp2	λp2	NOUN
ejpam-5390	140	8	+	+	PROPN
ejpam-5390	140	9	[	[	X
ejpam-5390	140	10	x12,π(u22)]∗	x12,π(u22)]∗	X
ejpam-5390	140	11	♢	♢	PROPN
ejpam-5390	140	12	λp2	λp2	NOUN
ejpam-5390	140	13	+	+	NUM
ejpam-5390	141	1	[	[	X
ejpam-5390	141	2	x12	x12	NUM
ejpam-5390	141	3	,	,	PUNCT
ejpam-5390	141	4	u22]∗	u22]∗	NOUN
ejpam-5390	141	5	♢	♢	NOUN
ejpam-5390	141	6	λπ(p2	λπ(p2	NOUN
ejpam-5390	141	7	)	)	PUNCT
ejpam-5390	141	8	.	.	PUNCT
ejpam-5390	142	1	on	on	ADP
ejpam-5390	142	2	the	the	DET
ejpam-5390	142	3	other	other	ADJ
ejpam-5390	142	4	hand	hand	NOUN
ejpam-5390	142	5	,	,	PUNCT
ejpam-5390	142	6	we	we	PRON
ejpam-5390	142	7	get	get	VERB
ejpam-5390	142	8	π([x12	π([x12	NUM
ejpam-5390	142	9	,	,	PUNCT
ejpam-5390	142	10	u11	u11	PROPN
ejpam-5390	142	11	+	+	CCONJ
ejpam-5390	142	12	u12	u12	PROPN
ejpam-5390	142	13	+	+	CCONJ
ejpam-5390	142	14	u21	u21	NOUN
ejpam-5390	142	15	+	+	CCONJ
ejpam-5390	142	16	u22]∗	u22]∗	NOUN
ejpam-5390	142	17	♢	♢	NOUN
ejpam-5390	142	18	λp2	λp2	NOUN
ejpam-5390	142	19	)	)	PUNCT
ejpam-5390	142	20	=	=	PUNCT
ejpam-5390	143	1	[	[	X
ejpam-5390	143	2	π(x12	π(x12	NUM
ejpam-5390	143	3	)	)	PUNCT
ejpam-5390	143	4	,	,	PUNCT
ejpam-5390	143	5	u11	u11	PROPN
ejpam-5390	143	6	+	+	NUM
ejpam-5390	143	7	u12	u12	PROPN
ejpam-5390	143	8	+	+	CCONJ
ejpam-5390	143	9	u21	u21	NOUN
ejpam-5390	143	10	+	+	CCONJ
ejpam-5390	143	11	u22]∗	u22]∗	NOUN
ejpam-5390	143	12	♢	♢	NOUN
ejpam-5390	143	13	λp2	λp2	NOUN
ejpam-5390	143	14	+	+	PROPN
ejpam-5390	143	15	[	[	X
ejpam-5390	143	16	x12,π(u11	x12,π(u11	ADJ
ejpam-5390	143	17	+	+	NUM
ejpam-5390	143	18	u12	u12	PROPN
ejpam-5390	143	19	+	+	CCONJ
ejpam-5390	143	20	u21	u21	PROPN
ejpam-5390	143	21	+	+	SYM
ejpam-5390	143	22	u22)]∗	u22)]∗	PROPN
ejpam-5390	143	23	♢	♢	PROPN
ejpam-5390	143	24	λp2	λp2	NOUN
ejpam-5390	143	25	+	+	PROPN
ejpam-5390	143	26	[	[	X
ejpam-5390	143	27	x12	x12	NUM
ejpam-5390	143	28	,	,	PUNCT
ejpam-5390	143	29	u11	u11	PROPN
ejpam-5390	143	30	+	+	CCONJ
ejpam-5390	143	31	u12	u12	PROPN
ejpam-5390	143	32	+	+	CCONJ
ejpam-5390	143	33	u21	u21	NOUN
ejpam-5390	143	34	+	+	CCONJ
ejpam-5390	143	35	u22]∗	u22]∗	NOUN
ejpam-5390	143	36	♢	♢	NOUN
ejpam-5390	143	37	λπ(p2	λπ(p2	NOUN
ejpam-5390	143	38	)	)	PUNCT
ejpam-5390	143	39	.	.	PUNCT
ejpam-5390	144	1	by	by	ADP
ejpam-5390	144	2	the	the	DET
ejpam-5390	144	3	above	above	ADJ
ejpam-5390	144	4	two	two	NUM
ejpam-5390	144	5	equations	equation	NOUN
ejpam-5390	144	6	,	,	PUNCT
ejpam-5390	144	7	we	we	PRON
ejpam-5390	144	8	get	get	VERB
ejpam-5390	144	9	[	[	X
ejpam-5390	144	10	x12	x12	NUM
ejpam-5390	144	11	,	,	PUNCT
ejpam-5390	144	12	t	t	X
ejpam-5390	144	13	]	]	PUNCT
ejpam-5390	144	14	∗	∗	X
ejpam-5390	144	15	♢	♢	NOUN
ejpam-5390	144	16	λp2	λp2	NOUN
ejpam-5390	144	17	=	=	NOUN
ejpam-5390	144	18	0	0	X
ejpam-5390	144	19	.	.	PUNCT
ejpam-5390	145	1	that	that	PRON
ejpam-5390	145	2	means	mean	VERB
ejpam-5390	145	3	that	that	SCONJ
ejpam-5390	145	4	x12tp2	x12tp2	PROPN
ejpam-5390	146	1	−	−	PROPN
ejpam-5390	146	2	λp2tx	λp2tx	NUM
ejpam-5390	146	3	∗	∗	NOUN
ejpam-5390	146	4	12	12	NUM
ejpam-5390	146	5	=	=	SYM
ejpam-5390	146	6	0	0	NUM
ejpam-5390	146	7	.	.	PUNCT
ejpam-5390	147	1	when	when	SCONJ
ejpam-5390	147	2	we	we	PRON
ejpam-5390	147	3	multiply	multiply	VERB
ejpam-5390	147	4	both	both	DET
ejpam-5390	147	5	sides	side	NOUN
ejpam-5390	147	6	by	by	ADP
ejpam-5390	147	7	p1	p1	PROPN
ejpam-5390	147	8	on	on	ADP
ejpam-5390	147	9	the	the	DET
ejpam-5390	147	10	left	left	NOUN
ejpam-5390	147	11	,	,	PUNCT
ejpam-5390	147	12	the	the	DET
ejpam-5390	147	13	result	result	NOUN
ejpam-5390	147	14	is	be	AUX
ejpam-5390	147	15	x12tp2	x12tp2	PROPN
ejpam-5390	147	16	=	=	PROPN
ejpam-5390	147	17	0	0	PROPN
ejpam-5390	147	18	.	.	PUNCT
ejpam-5390	147	19	.	.	PUNCT
ejpam-5390	148	1	nisar	nisar	PROPN
ejpam-5390	148	2	et	et	PROPN
ejpam-5390	148	3	al	al	PROPN
ejpam-5390	148	4	.	.	PUNCT
ejpam-5390	148	5	/	/	SYM
ejpam-5390	148	6	eur	eur	PROPN
ejpam-5390	148	7	.	.	PUNCT
ejpam-5390	149	1	j.	j.	PROPN
ejpam-5390	149	2	pure	pure	PROPN
ejpam-5390	149	3	appl	appl	PROPN
ejpam-5390	149	4	.	.	PROPN
ejpam-5390	149	5	math	math	PROPN
ejpam-5390	149	6	,	,	PUNCT
ejpam-5390	149	7	17	17	NUM
ejpam-5390	149	8	(	(	PUNCT
ejpam-5390	149	9	4	4	NUM
ejpam-5390	149	10	)	)	PUNCT
ejpam-5390	149	11	(	(	PUNCT
ejpam-5390	149	12	2024	2024	NUM
ejpam-5390	149	13	)	)	PUNCT
ejpam-5390	149	14	,	,	PUNCT
ejpam-5390	149	15	3399	3399	NUM
ejpam-5390	149	16	-	-	SYM
ejpam-5390	149	17	3414	3414	NUM
ejpam-5390	149	18	3403	3403	NUM
ejpam-5390	149	19	by	by	ADP
ejpam-5390	149	20	using	use	VERB
ejpam-5390	149	21	(	(	PUNCT
ejpam-5390	149	22	▲	▲	PUNCT
ejpam-5390	149	23	)	)	PUNCT
ejpam-5390	149	24	and	and	CCONJ
ejpam-5390	149	25	(	(	PUNCT
ejpam-5390	149	26	▼	▼	NOUN
ejpam-5390	149	27	)	)	PUNCT
ejpam-5390	150	1	,	,	PUNCT
ejpam-5390	150	2	we	we	PRON
ejpam-5390	150	3	have	have	VERB
ejpam-5390	150	4	p2tp2	p2tp2	NOUN
ejpam-5390	150	5	=	=	SYM
ejpam-5390	150	6	0	0	X
ejpam-5390	150	7	.	.	PUNCT
ejpam-5390	151	1	similarly	similarly	ADV
ejpam-5390	151	2	,	,	PUNCT
ejpam-5390	151	3	p1tp1	p1tp1	NOUN
ejpam-5390	151	4	=	=	SYM
ejpam-5390	151	5	0	0	NUM
ejpam-5390	151	6	.	.	PUNCT
ejpam-5390	152	1	hence	hence	ADV
ejpam-5390	152	2	,	,	PUNCT
ejpam-5390	152	3	t	t	PROPN
ejpam-5390	152	4	=	=	SYM
ejpam-5390	152	5	0	0	PUNCT
ejpam-5390	152	6	i.e.	i.e.	X
ejpam-5390	152	7	,	,	PUNCT
ejpam-5390	152	8	π(u11	π(u11	ADJ
ejpam-5390	152	9	+	+	NUM
ejpam-5390	152	10	u12	u12	NOUN
ejpam-5390	152	11	+	+	CCONJ
ejpam-5390	152	12	u21	u21	PROPN
ejpam-5390	152	13	+	+	CCONJ
ejpam-5390	152	14	u22	u22	NOUN
ejpam-5390	152	15	)	)	PUNCT
ejpam-5390	153	1	=	=	PRON
ejpam-5390	153	2	π(u11	π(u11	ADJ
ejpam-5390	153	3	)	)	PUNCT
ejpam-5390	153	4	+	+	NUM
ejpam-5390	153	5	π(u12	π(u12	NOUN
ejpam-5390	153	6	)	)	PUNCT
ejpam-5390	153	7	+	+	NUM
ejpam-5390	153	8	π(u21	π(u21	X
ejpam-5390	153	9	)	)	PUNCT
ejpam-5390	153	10	+	+	CCONJ
ejpam-5390	153	11	π(u22	π(u22	NOUN
ejpam-5390	153	12	)	)	PUNCT
ejpam-5390	153	13	.	.	PUNCT
ejpam-5390	154	1	lemma	lemma	PROPN
ejpam-5390	154	2	2.4	2.4	NUM
ejpam-5390	154	3	.	.	PUNCT
ejpam-5390	155	1	for	for	ADP
ejpam-5390	155	2	any	any	DET
ejpam-5390	155	3	uij	uij	NOUN
ejpam-5390	155	4	,	,	PUNCT
ejpam-5390	155	5	vij	vij	PROPN
ejpam-5390	155	6	∈	∈	PROPN
ejpam-5390	155	7	aij	aij	PROPN
ejpam-5390	155	8	with	with	ADP
ejpam-5390	155	9	(	(	PUNCT
ejpam-5390	155	10	1	1	NUM
ejpam-5390	155	11	≤	≤	NUM
ejpam-5390	155	12	i	i	PRON
ejpam-5390	155	13	̸=	̸=	PROPN
ejpam-5390	155	14	j	j	PROPN
ejpam-5390	155	15	≤	≤	ADV
ejpam-5390	155	16	2	2	NUM
ejpam-5390	155	17	)	)	PUNCT
ejpam-5390	155	18	,	,	PUNCT
ejpam-5390	155	19	we	we	PRON
ejpam-5390	155	20	have	have	VERB
ejpam-5390	155	21	π(uij	π(uij	PROPN
ejpam-5390	155	22	+	+	CCONJ
ejpam-5390	155	23	vij	vij	NOUN
ejpam-5390	155	24	)	)	PUNCT
ejpam-5390	155	25	=	=	SYM
ejpam-5390	156	1	π(uij	π(uij	PROPN
ejpam-5390	156	2	)	)	PUNCT
ejpam-5390	157	1	+	+	CCONJ
ejpam-5390	157	2	π(vij	π(vij	ADV
ejpam-5390	157	3	)	)	PUNCT
ejpam-5390	157	4	.	.	PUNCT
ejpam-5390	158	1	proof	proof	NOUN
ejpam-5390	158	2	.	.	PUNCT
ejpam-5390	159	1	initially	initially	ADV
ejpam-5390	159	2	,	,	PUNCT
ejpam-5390	159	3	we	we	PRON
ejpam-5390	159	4	establish	establish	VERB
ejpam-5390	159	5	the	the	DET
ejpam-5390	159	6	result	result	NOUN
ejpam-5390	159	7	for	for	ADP
ejpam-5390	159	8	i	i	PROPN
ejpam-5390	159	9	=	=	SYM
ejpam-5390	159	10	1	1	NUM
ejpam-5390	159	11	and	and	CCONJ
ejpam-5390	159	12	j	j	NOUN
ejpam-5390	159	13	=	=	NOUN
ejpam-5390	159	14	2	2	X
ejpam-5390	159	15	.	.	PUNCT
ejpam-5390	159	16	let	let	VERB
ejpam-5390	159	17	t	t	NOUN
ejpam-5390	159	18	=	=	SYM
ejpam-5390	159	19	π(u12	π(u12	NOUN
ejpam-5390	159	20	+	+	CCONJ
ejpam-5390	159	21	v12	v12	ADJ
ejpam-5390	159	22	)	)	PUNCT
ejpam-5390	159	23	−	−	PROPN
ejpam-5390	159	24	π(u12)−π(v12	π(u12)−π(v12	VERB
ejpam-5390	159	25	)	)	PUNCT
ejpam-5390	159	26	.	.	PUNCT
ejpam-5390	160	1	since	since	SCONJ
ejpam-5390	160	2	[	[	X
ejpam-5390	160	3	x12	x12	NUM
ejpam-5390	160	4	,	,	PUNCT
ejpam-5390	160	5	u12]∗	u12]∗	NOUN
ejpam-5390	160	6	♢	♢	PROPN
ejpam-5390	160	7	λp2	λp2	NOUN
ejpam-5390	160	8	=	=	SYM
ejpam-5390	160	9	0	0	NUM
ejpam-5390	160	10	,	,	PUNCT
ejpam-5390	160	11	and	and	CCONJ
ejpam-5390	160	12	using	use	VERB
ejpam-5390	160	13	lemma	lemma	PROPN
ejpam-5390	160	14	2.1	2.1	NUM
ejpam-5390	160	15	,	,	PUNCT
ejpam-5390	160	16	we	we	PRON
ejpam-5390	160	17	get	get	VERB
ejpam-5390	160	18	π([x12	π([x12	NUM
ejpam-5390	160	19	,	,	PUNCT
ejpam-5390	160	20	u12	u12	PROPN
ejpam-5390	160	21	+	+	CCONJ
ejpam-5390	160	22	v12]∗	v12]∗	NOUN
ejpam-5390	160	23	♢	♢	NOUN
ejpam-5390	160	24	λp2	λp2	NOUN
ejpam-5390	160	25	)	)	PUNCT
ejpam-5390	160	26	=	=	SYM
ejpam-5390	160	27	π([x12	π([x12	NUM
ejpam-5390	160	28	,	,	PUNCT
ejpam-5390	160	29	u12]∗	u12]∗	NOUN
ejpam-5390	160	30	♢	♢	NOUN
ejpam-5390	160	31	λp2	λp2	NOUN
ejpam-5390	160	32	)	)	PUNCT
ejpam-5390	161	1	+	+	NUM
ejpam-5390	161	2	π([x12	π([x12	NUM
ejpam-5390	161	3	,	,	PUNCT
ejpam-5390	161	4	v12]∗	v12]∗	NOUN
ejpam-5390	161	5	♢	♢	PROPN
ejpam-5390	161	6	λp2	λp2	NOUN
ejpam-5390	161	7	)	)	PUNCT
ejpam-5390	161	8	=	=	PUNCT
ejpam-5390	162	1	[	[	X
ejpam-5390	162	2	π(x12	π(x12	NUM
ejpam-5390	162	3	)	)	PUNCT
ejpam-5390	162	4	,	,	PUNCT
ejpam-5390	162	5	u12]∗	u12]∗	PROPN
ejpam-5390	162	6	♢	♢	PROPN
ejpam-5390	162	7	λp2	λp2	NOUN
ejpam-5390	162	8	+	+	PROPN
ejpam-5390	163	1	[	[	X
ejpam-5390	163	2	x12,π(u12)]∗	x12,π(u12)]∗	X
ejpam-5390	163	3	♢	♢	PROPN
ejpam-5390	163	4	λp2	λp2	X
ejpam-5390	163	5	+	+	PROPN
ejpam-5390	163	6	[	[	X
ejpam-5390	163	7	x12	x12	NUM
ejpam-5390	163	8	,	,	PUNCT
ejpam-5390	163	9	u12]∗	u12]∗	NOUN
ejpam-5390	163	10	♢	♢	NOUN
ejpam-5390	163	11	λπ(p2	λπ(p2	X
ejpam-5390	163	12	)	)	PUNCT
ejpam-5390	163	13	+	+	CCONJ
ejpam-5390	164	1	[	[	X
ejpam-5390	164	2	π(x12	π(x12	NUM
ejpam-5390	164	3	)	)	PUNCT
ejpam-5390	164	4	,	,	PUNCT
ejpam-5390	164	5	v12]∗	v12]∗	NOUN
ejpam-5390	164	6	♢	♢	PROPN
ejpam-5390	164	7	λp2	λp2	NOUN
ejpam-5390	164	8	+	+	PROPN
ejpam-5390	164	9	[	[	X
ejpam-5390	164	10	x12,π(v12)]∗	x12,π(v12)]∗	X
ejpam-5390	164	11	♢	♢	PROPN
ejpam-5390	164	12	λp2	λp2	NOUN
ejpam-5390	164	13	+	+	SYM
ejpam-5390	165	1	[	[	X
ejpam-5390	165	2	x12	x12	NUM
ejpam-5390	165	3	,	,	PUNCT
ejpam-5390	165	4	v12]∗	v12]∗	NOUN
ejpam-5390	165	5	♢	♢	PROPN
ejpam-5390	165	6	λπ(p2	λπ(p2	NOUN
ejpam-5390	165	7	)	)	PUNCT
ejpam-5390	165	8	.	.	PUNCT
ejpam-5390	166	1	on	on	ADP
ejpam-5390	166	2	the	the	DET
ejpam-5390	166	3	other	other	ADJ
ejpam-5390	166	4	hand	hand	NOUN
ejpam-5390	166	5	,	,	PUNCT
ejpam-5390	166	6	we	we	PRON
ejpam-5390	166	7	have	have	VERB
ejpam-5390	166	8	π([x12	π([x12	NUM
ejpam-5390	166	9	,	,	PUNCT
ejpam-5390	166	10	u12	u12	PROPN
ejpam-5390	166	11	+	+	CCONJ
ejpam-5390	166	12	v12]∗	v12]∗	NOUN
ejpam-5390	166	13	♢	♢	NOUN
ejpam-5390	166	14	λp2	λp2	NOUN
ejpam-5390	166	15	)	)	PUNCT
ejpam-5390	166	16	=	=	PUNCT
ejpam-5390	167	1	[	[	X
ejpam-5390	167	2	π(x12	π(x12	NUM
ejpam-5390	167	3	)	)	PUNCT
ejpam-5390	167	4	,	,	PUNCT
ejpam-5390	167	5	u12	u12	PROPN
ejpam-5390	167	6	+	+	CCONJ
ejpam-5390	167	7	v12]∗	v12]∗	PROPN
ejpam-5390	167	8	♢	♢	NOUN
ejpam-5390	167	9	λp2	λp2	NOUN
ejpam-5390	167	10	+	+	PROPN
ejpam-5390	168	1	[	[	X
ejpam-5390	168	2	x12,π(u12	x12,π(u12	X
ejpam-5390	168	3	+	+	CCONJ
ejpam-5390	168	4	v12)]∗	v12)]∗	PROPN
ejpam-5390	168	5	♢	♢	NOUN
ejpam-5390	168	6	λp2	λp2	NOUN
ejpam-5390	169	1	+	+	PROPN
ejpam-5390	169	2	[	[	X
ejpam-5390	169	3	x12	x12	NUM
ejpam-5390	169	4	,	,	PUNCT
ejpam-5390	169	5	u12	u12	PROPN
ejpam-5390	169	6	+	+	CCONJ
ejpam-5390	169	7	v12]∗	v12]∗	PROPN
ejpam-5390	169	8	♢	♢	PROPN
ejpam-5390	169	9	λπ(p2	λπ(p2	NOUN
ejpam-5390	169	10	)	)	PUNCT
ejpam-5390	169	11	.	.	PUNCT
ejpam-5390	170	1	by	by	ADP
ejpam-5390	170	2	comparing	compare	VERB
ejpam-5390	170	3	the	the	DET
ejpam-5390	170	4	last	last	ADJ
ejpam-5390	170	5	two	two	NUM
ejpam-5390	170	6	expressions	expression	NOUN
ejpam-5390	170	7	,	,	PUNCT
ejpam-5390	170	8	we	we	PRON
ejpam-5390	170	9	get	get	VERB
ejpam-5390	170	10	[	[	X
ejpam-5390	170	11	x12	x12	NUM
ejpam-5390	170	12	,	,	PUNCT
ejpam-5390	170	13	t	t	X
ejpam-5390	170	14	]	]	PUNCT
ejpam-5390	170	15	∗	∗	X
ejpam-5390	170	16	♢	♢	NOUN
ejpam-5390	170	17	λp2	λp2	NOUN
ejpam-5390	170	18	=	=	NOUN
ejpam-5390	170	19	0	0	X
ejpam-5390	170	20	.	.	PUNCT
ejpam-5390	171	1	that	that	PRON
ejpam-5390	171	2	means	mean	VERB
ejpam-5390	171	3	x12tp2−	x12tp2−	PROPN
ejpam-5390	171	4	p2tx	p2tx	PROPN
ejpam-5390	171	5	∗	∗	NOUN
ejpam-5390	171	6	12	12	NUM
ejpam-5390	171	7	=	=	SYM
ejpam-5390	171	8	0	0	NUM
ejpam-5390	171	9	.	.	PUNCT
ejpam-5390	172	1	by	by	ADP
ejpam-5390	172	2	left	left	ADJ
ejpam-5390	172	3	-	-	PUNCT
ejpam-5390	172	4	multiplying	multiply	VERB
ejpam-5390	172	5	both	both	DET
ejpam-5390	172	6	sides	side	NOUN
ejpam-5390	172	7	of	of	ADP
ejpam-5390	172	8	the	the	DET
ejpam-5390	172	9	preceding	precede	VERB
ejpam-5390	172	10	equation	equation	NOUN
ejpam-5390	172	11	by	by	ADP
ejpam-5390	172	12	p1	p1	NOUN
ejpam-5390	172	13	and	and	CCONJ
ejpam-5390	172	14	utilizing	utilize	VERB
ejpam-5390	172	15	(	(	PUNCT
ejpam-5390	172	16	▲	▲	PUNCT
ejpam-5390	172	17	)	)	PUNCT
ejpam-5390	172	18	and	and	CCONJ
ejpam-5390	172	19	(	(	PUNCT
ejpam-5390	172	20	▼	▼	NOUN
ejpam-5390	172	21	)	)	PUNCT
ejpam-5390	172	22	,	,	PUNCT
ejpam-5390	172	23	we	we	PRON
ejpam-5390	172	24	obtain	obtain	VERB
ejpam-5390	172	25	p2tp2	p2tp2	NOUN
ejpam-5390	172	26	=	=	NOUN
ejpam-5390	172	27	0	0	X
ejpam-5390	172	28	.	.	PUNCT
ejpam-5390	173	1	similarly	similarly	ADV
ejpam-5390	173	2	,	,	PUNCT
ejpam-5390	173	3	we	we	PRON
ejpam-5390	173	4	can	can	AUX
ejpam-5390	173	5	show	show	VERB
ejpam-5390	173	6	that	that	DET
ejpam-5390	173	7	p1tp1	p1tp1	NOUN
ejpam-5390	173	8	=	=	SYM
ejpam-5390	173	9	0	0	X
ejpam-5390	173	10	.	.	PUNCT
ejpam-5390	174	1	now	now	ADV
ejpam-5390	174	2	,	,	PUNCT
ejpam-5390	174	3	again	again	ADV
ejpam-5390	174	4	for	for	ADP
ejpam-5390	174	5	any	any	DET
ejpam-5390	174	6	x12	x12	NUM
ejpam-5390	174	7	∈	∈	PROPN
ejpam-5390	174	8	a12	a12	NOUN
ejpam-5390	174	9	.	.	PUNCT
ejpam-5390	175	1	since	since	SCONJ
ejpam-5390	175	2	[	[	X
ejpam-5390	175	3	p1	p1	NOUN
ejpam-5390	175	4	,	,	PUNCT
ejpam-5390	175	5	u12]∗	u12]∗	PROPN
ejpam-5390	175	6	♢	♢	PROPN
ejpam-5390	175	7	λx12	λx12	PROPN
ejpam-5390	175	8	=	=	SYM
ejpam-5390	175	9	0	0	PUNCT
ejpam-5390	175	10	and	and	CCONJ
ejpam-5390	175	11	using	use	VERB
ejpam-5390	175	12	lemma	lemma	PROPN
ejpam-5390	175	13	2.1	2.1	NUM
ejpam-5390	175	14	,	,	PUNCT
ejpam-5390	175	15	we	we	PRON
ejpam-5390	175	16	have	have	VERB
ejpam-5390	175	17	π([p1	π([p1	PROPN
ejpam-5390	175	18	,	,	PUNCT
ejpam-5390	175	19	u12	u12	PROPN
ejpam-5390	175	20	+	+	CCONJ
ejpam-5390	175	21	v12]∗	v12]∗	PROPN
ejpam-5390	175	22	♢	♢	PROPN
ejpam-5390	175	23	λx12	λx12	PROPN
ejpam-5390	175	24	)	)	PUNCT
ejpam-5390	175	25	=	=	SYM
ejpam-5390	175	26	π([p1	π([p1	PROPN
ejpam-5390	175	27	,	,	PUNCT
ejpam-5390	175	28	u12]∗	u12]∗	PROPN
ejpam-5390	175	29	♢	♢	PROPN
ejpam-5390	175	30	λx12	λx12	PROPN
ejpam-5390	175	31	)	)	PUNCT
ejpam-5390	176	1	+	+	NUM
ejpam-5390	176	2	π([p1	π([p1	NOUN
ejpam-5390	176	3	,	,	PUNCT
ejpam-5390	176	4	v12]∗	v12]∗	NOUN
ejpam-5390	176	5	♢	♢	PROPN
ejpam-5390	176	6	λx12	λx12	PROPN
ejpam-5390	176	7	)	)	PUNCT
ejpam-5390	176	8	=	=	PUNCT
ejpam-5390	177	1	[	[	X
ejpam-5390	177	2	π(p1	π(p1	NOUN
ejpam-5390	177	3	)	)	PUNCT
ejpam-5390	177	4	,	,	PUNCT
ejpam-5390	177	5	u12]∗	u12]∗	PROPN
ejpam-5390	177	6	♢	♢	PROPN
ejpam-5390	177	7	λx12	λx12	PROPN
ejpam-5390	177	8	+	+	PROPN
ejpam-5390	178	1	[	[	X
ejpam-5390	178	2	p1,π(u12)]∗	p1,π(u12)]∗	X
ejpam-5390	178	3	♢	♢	NOUN
ejpam-5390	178	4	λx12	λx12	PROPN
ejpam-5390	178	5	+	+	PROPN
ejpam-5390	178	6	[	[	X
ejpam-5390	178	7	p1	p1	NOUN
ejpam-5390	178	8	,	,	PUNCT
ejpam-5390	178	9	u12]∗	u12]∗	PROPN
ejpam-5390	178	10	♢	♢	PROPN
ejpam-5390	178	11	λπ(x12	λπ(x12	PROPN
ejpam-5390	178	12	)	)	PUNCT
ejpam-5390	178	13	+	+	CCONJ
ejpam-5390	178	14	[	[	X
ejpam-5390	178	15	π(p1	π(p1	NOUN
ejpam-5390	178	16	)	)	PUNCT
ejpam-5390	178	17	,	,	PUNCT
ejpam-5390	178	18	v12]∗	v12]∗	VERB
ejpam-5390	178	19	♢	♢	PROPN
ejpam-5390	178	20	λx12	λx12	PROPN
ejpam-5390	178	21	+	+	PROPN
ejpam-5390	178	22	[	[	X
ejpam-5390	178	23	p1,π(v12)]∗	p1,π(v12)]∗	X
ejpam-5390	178	24	♢	♢	PROPN
ejpam-5390	178	25	λx12	λx12	PROPN
ejpam-5390	178	26	+	+	PROPN
ejpam-5390	179	1	[	[	X
ejpam-5390	179	2	p1	p1	NOUN
ejpam-5390	179	3	,	,	PUNCT
ejpam-5390	179	4	u12]∗	u12]∗	PROPN
ejpam-5390	179	5	♢	♢	PROPN
ejpam-5390	179	6	λπ(x12	λπ(x12	PROPN
ejpam-5390	179	7	)	)	PUNCT
ejpam-5390	179	8	.	.	PUNCT
ejpam-5390	180	1	on	on	ADP
ejpam-5390	180	2	the	the	DET
ejpam-5390	180	3	other	other	ADJ
ejpam-5390	180	4	hand	hand	NOUN
ejpam-5390	180	5	,	,	PUNCT
ejpam-5390	180	6	we	we	PRON
ejpam-5390	180	7	find	find	VERB
ejpam-5390	180	8	π([p1	π([p1	PROPN
ejpam-5390	180	9	,	,	PUNCT
ejpam-5390	180	10	u12	u12	PROPN
ejpam-5390	180	11	+	+	CCONJ
ejpam-5390	180	12	v12]∗	v12]∗	PROPN
ejpam-5390	180	13	♢	♢	PROPN
ejpam-5390	180	14	λx12	λx12	PROPN
ejpam-5390	180	15	)	)	PUNCT
ejpam-5390	180	16	=	=	PUNCT
ejpam-5390	181	1	[	[	X
ejpam-5390	181	2	π(p1	π(p1	NOUN
ejpam-5390	181	3	)	)	PUNCT
ejpam-5390	181	4	,	,	PUNCT
ejpam-5390	181	5	u12	u12	PROPN
ejpam-5390	181	6	+	+	CCONJ
ejpam-5390	181	7	v12]∗	v12]∗	PROPN
ejpam-5390	181	8	♢	♢	PROPN
ejpam-5390	182	1	λx12	λx12	PROPN
ejpam-5390	182	2	+	+	X
ejpam-5390	183	1	[	[	X
ejpam-5390	183	2	p1,π(u12	p1,π(u12	PROPN
ejpam-5390	183	3	+	+	CCONJ
ejpam-5390	183	4	v12)]∗	v12)]∗	PROPN
ejpam-5390	183	5	♢	♢	PROPN
ejpam-5390	183	6	λx12	λx12	PROPN
ejpam-5390	183	7	+	+	PROPN
ejpam-5390	183	8	[	[	X
ejpam-5390	183	9	p1	p1	NOUN
ejpam-5390	183	10	,	,	PUNCT
ejpam-5390	183	11	u12	u12	PROPN
ejpam-5390	183	12	+	+	CCONJ
ejpam-5390	183	13	v12]∗	v12]∗	PROPN
ejpam-5390	183	14	♢	♢	PROPN
ejpam-5390	183	15	λπ(x12	λπ(x12	NOUN
ejpam-5390	183	16	)	)	PUNCT
ejpam-5390	183	17	.	.	PUNCT
ejpam-5390	184	1	from	from	ADP
ejpam-5390	184	2	the	the	DET
ejpam-5390	184	3	last	last	ADJ
ejpam-5390	184	4	two	two	NUM
ejpam-5390	184	5	expressions	expression	NOUN
ejpam-5390	184	6	,	,	PUNCT
ejpam-5390	184	7	we	we	PRON
ejpam-5390	184	8	find	find	VERB
ejpam-5390	184	9	[	[	X
ejpam-5390	184	10	p1	p1	NOUN
ejpam-5390	184	11	,	,	PUNCT
ejpam-5390	184	12	t	t	X
ejpam-5390	184	13	]	]	PUNCT
ejpam-5390	184	14	∗	∗	NOUN
ejpam-5390	184	15	♢	♢	NOUN
ejpam-5390	184	16	λx12	λx12	PROPN
ejpam-5390	184	17	=	=	SYM
ejpam-5390	184	18	0	0	PROPN
ejpam-5390	184	19	.	.	PUNCT
ejpam-5390	185	1	that	that	PRON
ejpam-5390	185	2	means	mean	VERB
ejpam-5390	185	3	p1tx12−tx12−	p1tx12−tx12−	X
ejpam-5390	185	4	λx12tp1	λx12tp1	ADJ
ejpam-5390	185	5	=	=	SYM
ejpam-5390	185	6	0	0	X
ejpam-5390	185	7	.	.	PUNCT
ejpam-5390	186	1	multiplying	multiply	VERB
ejpam-5390	186	2	both	both	DET
ejpam-5390	186	3	sides	side	NOUN
ejpam-5390	186	4	by	by	ADP
ejpam-5390	186	5	p1	p1	PROPN
ejpam-5390	186	6	from	from	ADP
ejpam-5390	186	7	right	right	ADV
ejpam-5390	186	8	and	and	CCONJ
ejpam-5390	186	9	since	since	SCONJ
ejpam-5390	186	10	λ	λ	PROPN
ejpam-5390	186	11	̸=	̸=	PROPN
ejpam-5390	186	12	0	0	NUM
ejpam-5390	186	13	,	,	PUNCT
ejpam-5390	186	14	we	we	PRON
ejpam-5390	186	15	havex12tp1	havex12tp1	AUX
ejpam-5390	186	16	=	=	SYM
ejpam-5390	186	17	0	0	NUM
ejpam-5390	186	18	.	.	PUNCT
ejpam-5390	187	1	thus	thus	ADV
ejpam-5390	187	2	,	,	PUNCT
ejpam-5390	187	3	p2tp1	p2tp1	ADJ
ejpam-5390	187	4	=	=	SYM
ejpam-5390	187	5	0	0	NUM
ejpam-5390	187	6	follows	follow	VERB
ejpam-5390	187	7	from	from	ADP
ejpam-5390	187	8	(	(	PUNCT
ejpam-5390	187	9	▲	▲	PUNCT
ejpam-5390	187	10	)	)	PUNCT
ejpam-5390	187	11	and	and	CCONJ
ejpam-5390	187	12	(	(	PUNCT
ejpam-5390	187	13	▼	▼	NOUN
ejpam-5390	187	14	)	)	PUNCT
ejpam-5390	187	15	.	.	PUNCT
ejpam-5390	188	1	similarly	similarly	ADV
ejpam-5390	188	2	,	,	PUNCT
ejpam-5390	188	3	we	we	PRON
ejpam-5390	188	4	can	can	AUX
ejpam-5390	188	5	show	show	VERB
ejpam-5390	188	6	that	that	DET
ejpam-5390	188	7	p1tp2	p1tp2	NOUN
ejpam-5390	188	8	=	=	NOUN
ejpam-5390	188	9	0	0	X
ejpam-5390	188	10	.	.	PUNCT
ejpam-5390	189	1	hence	hence	ADV
ejpam-5390	189	2	,	,	PUNCT
ejpam-5390	189	3	t	t	PROPN
ejpam-5390	189	4	=	=	SYM
ejpam-5390	189	5	0	0	PUNCT
ejpam-5390	189	6	i.e.	i.e.	X
ejpam-5390	189	7	,	,	PUNCT
ejpam-5390	189	8	π(u12	π(u12	X
ejpam-5390	189	9	+	+	CCONJ
ejpam-5390	189	10	v12	v12	ADJ
ejpam-5390	189	11	)	)	PUNCT
ejpam-5390	189	12	=	=	SYM
ejpam-5390	189	13	π(u12	π(u12	NOUN
ejpam-5390	189	14	)	)	PUNCT
ejpam-5390	189	15	+	+	CCONJ
ejpam-5390	189	16	π(v12	π(v12	VERB
ejpam-5390	189	17	)	)	PUNCT
ejpam-5390	189	18	.	.	PUNCT
ejpam-5390	190	1	by	by	ADP
ejpam-5390	190	2	using	use	VERB
ejpam-5390	190	3	the	the	DET
ejpam-5390	190	4	same	same	ADJ
ejpam-5390	190	5	technique	technique	NOUN
ejpam-5390	190	6	as	as	ADP
ejpam-5390	190	7	above	above	ADV
ejpam-5390	190	8	,	,	PUNCT
ejpam-5390	190	9	one	one	PRON
ejpam-5390	190	10	can	can	AUX
ejpam-5390	190	11	show	show	VERB
ejpam-5390	190	12	that	that	SCONJ
ejpam-5390	190	13	π(u21	π(u21	NOUN
ejpam-5390	190	14	+	+	CCONJ
ejpam-5390	190	15	v21	v21	NUM
ejpam-5390	190	16	)	)	PUNCT
ejpam-5390	190	17	=	=	PUNCT
ejpam-5390	190	18	π(u21	π(u21	X
ejpam-5390	190	19	)	)	PUNCT
ejpam-5390	190	20	+	+	NUM
ejpam-5390	190	21	π(v21	π(v21	NUM
ejpam-5390	190	22	)	)	PUNCT
ejpam-5390	190	23	.	.	PUNCT
ejpam-5390	190	24	.	.	PUNCT
ejpam-5390	191	1	nisar	nisar	PROPN
ejpam-5390	191	2	et	et	PROPN
ejpam-5390	191	3	al	al	PROPN
ejpam-5390	191	4	.	.	PUNCT
ejpam-5390	191	5	/	/	SYM
ejpam-5390	191	6	eur	eur	PROPN
ejpam-5390	191	7	.	.	PUNCT
ejpam-5390	192	1	j.	j.	PROPN
ejpam-5390	192	2	pure	pure	PROPN
ejpam-5390	192	3	appl	appl	PROPN
ejpam-5390	192	4	.	.	PROPN
ejpam-5390	192	5	math	math	PROPN
ejpam-5390	192	6	,	,	PUNCT
ejpam-5390	192	7	17	17	NUM
ejpam-5390	192	8	(	(	PUNCT
ejpam-5390	192	9	4	4	NUM
ejpam-5390	192	10	)	)	PUNCT
ejpam-5390	192	11	(	(	PUNCT
ejpam-5390	192	12	2024	2024	NUM
ejpam-5390	192	13	)	)	PUNCT
ejpam-5390	192	14	,	,	PUNCT
ejpam-5390	192	15	3399	3399	NUM
ejpam-5390	192	16	-	-	SYM
ejpam-5390	192	17	3414	3414	NUM
ejpam-5390	192	18	3404	3404	NUM
ejpam-5390	192	19	lemma	lemma	PROPN
ejpam-5390	192	20	2.5	2.5	NUM
ejpam-5390	192	21	.	.	PUNCT
ejpam-5390	193	1	for	for	ADP
ejpam-5390	193	2	any	any	DET
ejpam-5390	193	3	u11	u11	ADJ
ejpam-5390	193	4	,	,	PUNCT
ejpam-5390	193	5	v11	v11	NOUN
ejpam-5390	193	6	∈	∈	PROPN
ejpam-5390	193	7	a11	a11	PROPN
ejpam-5390	193	8	and	and	CCONJ
ejpam-5390	193	9	u22	u22	PROPN
ejpam-5390	193	10	,	,	PUNCT
ejpam-5390	193	11	v22	v22	PROPN
ejpam-5390	193	12	∈	∈	PROPN
ejpam-5390	193	13	a22	a22	PROPN
ejpam-5390	193	14	,	,	PUNCT
ejpam-5390	193	15	we	we	PRON
ejpam-5390	193	16	have	have	VERB
ejpam-5390	193	17	(	(	PUNCT
ejpam-5390	193	18	i	i	NOUN
ejpam-5390	193	19	)	)	PUNCT
ejpam-5390	193	20	π(u11	π(u11	VERB
ejpam-5390	194	1	+	+	CCONJ
ejpam-5390	194	2	v11	v11	NOUN
ejpam-5390	194	3	)	)	PUNCT
ejpam-5390	194	4	=	=	VERB
ejpam-5390	194	5	π(u11	π(u11	PRON
ejpam-5390	194	6	)	)	PUNCT
ejpam-5390	195	1	+	+	CCONJ
ejpam-5390	195	2	π(v11	π(v11	ADJ
ejpam-5390	195	3	)	)	PUNCT
ejpam-5390	195	4	.	.	PUNCT
ejpam-5390	196	1	(	(	PUNCT
ejpam-5390	196	2	ii	ii	NOUN
ejpam-5390	196	3	)	)	PUNCT
ejpam-5390	196	4	π(u22	π(u22	NOUN
ejpam-5390	196	5	+	+	CCONJ
ejpam-5390	196	6	v22	v22	NOUN
ejpam-5390	196	7	)	)	PUNCT
ejpam-5390	196	8	=	=	SYM
ejpam-5390	196	9	π(u22	π(u22	NOUN
ejpam-5390	196	10	)	)	PUNCT
ejpam-5390	196	11	+	+	CCONJ
ejpam-5390	196	12	π(v22	π(v22	NOUN
ejpam-5390	196	13	)	)	PUNCT
ejpam-5390	196	14	.	.	PUNCT
ejpam-5390	197	1	proof	proof	NOUN
ejpam-5390	197	2	.	.	PUNCT
ejpam-5390	198	1	let	let	VERB
ejpam-5390	198	2	t	t	NOUN
ejpam-5390	198	3	=	=	SYM
ejpam-5390	198	4	π(u11	π(u11	ADJ
ejpam-5390	198	5	+	+	CCONJ
ejpam-5390	198	6	v11)−π(u11)−π(v11	v11)−π(u11)−π(v11	ADJ
ejpam-5390	198	7	)	)	PUNCT
ejpam-5390	198	8	.	.	PUNCT
ejpam-5390	199	1	on	on	ADP
ejpam-5390	199	2	the	the	DET
ejpam-5390	199	3	one	one	NUM
ejpam-5390	199	4	hand	hand	NOUN
ejpam-5390	199	5	,	,	PUNCT
ejpam-5390	199	6	we	we	PRON
ejpam-5390	199	7	have	have	VERB
ejpam-5390	199	8	π([p2	π([p2	NOUN
ejpam-5390	199	9	,	,	PUNCT
ejpam-5390	199	10	u11	u11	ADJ
ejpam-5390	199	11	+	+	CCONJ
ejpam-5390	199	12	v11]∗	v11]∗	PROPN
ejpam-5390	199	13	♢	♢	PROPN
ejpam-5390	199	14	λp1	λp1	NOUN
ejpam-5390	199	15	)	)	PUNCT
ejpam-5390	199	16	=	=	PUNCT
ejpam-5390	200	1	[	[	X
ejpam-5390	200	2	π(p2	π(p2	ADJ
ejpam-5390	200	3	)	)	PUNCT
ejpam-5390	200	4	,	,	PUNCT
ejpam-5390	200	5	u11	u11	PROPN
ejpam-5390	200	6	+	+	CCONJ
ejpam-5390	200	7	v11]∗	v11]∗	ADJ
ejpam-5390	200	8	♢	♢	NOUN
ejpam-5390	200	9	λp1	λp1	NOUN
ejpam-5390	200	10	+	+	X
ejpam-5390	201	1	[	[	X
ejpam-5390	201	2	p2,π(u11	p2,π(u11	X
ejpam-5390	201	3	+	+	CCONJ
ejpam-5390	201	4	v11)]∗	v11)]∗	PROPN
ejpam-5390	201	5	♢	♢	NOUN
ejpam-5390	201	6	λp1	λp1	NOUN
ejpam-5390	201	7	+	+	PROPN
ejpam-5390	201	8	[	[	X
ejpam-5390	201	9	p2	p2	NOUN
ejpam-5390	201	10	,	,	PUNCT
ejpam-5390	201	11	u11	u11	ADJ
ejpam-5390	201	12	+	+	CCONJ
ejpam-5390	201	13	v11]∗	v11]∗	PROPN
ejpam-5390	201	14	♢	♢	PROPN
ejpam-5390	201	15	λπ(p1	λπ(p1	PROPN
ejpam-5390	201	16	)	)	PUNCT
ejpam-5390	201	17	.	.	PUNCT
ejpam-5390	202	1	on	on	ADP
ejpam-5390	202	2	the	the	DET
ejpam-5390	202	3	other	other	ADJ
ejpam-5390	202	4	hand	hand	NOUN
ejpam-5390	202	5	,	,	PUNCT
ejpam-5390	202	6	it	it	PRON
ejpam-5390	202	7	follows	follow	VERB
ejpam-5390	202	8	from	from	ADP
ejpam-5390	202	9	[	[	X
ejpam-5390	202	10	p2	p2	NOUN
ejpam-5390	202	11	,	,	PUNCT
ejpam-5390	202	12	u11]∗	u11]∗	NOUN
ejpam-5390	202	13	♢	♢	NOUN
ejpam-5390	202	14	λp1	λp1	NOUN
ejpam-5390	202	15	=	=	SYM
ejpam-5390	202	16	0	0	NUM
ejpam-5390	202	17	that	that	SCONJ
ejpam-5390	202	18	π([p2	π([p2	NOUN
ejpam-5390	202	19	,	,	PUNCT
ejpam-5390	202	20	u11	u11	ADJ
ejpam-5390	202	21	+	+	CCONJ
ejpam-5390	202	22	v11]∗	v11]∗	PROPN
ejpam-5390	202	23	♢	♢	PROPN
ejpam-5390	202	24	λp1	λp1	NOUN
ejpam-5390	202	25	)	)	PUNCT
ejpam-5390	202	26	=	=	SYM
ejpam-5390	202	27	π([p2	π([p2	NOUN
ejpam-5390	202	28	,	,	PUNCT
ejpam-5390	202	29	u11]∗	u11]∗	PROPN
ejpam-5390	202	30	♢	♢	PROPN
ejpam-5390	202	31	λp1	λp1	NOUN
ejpam-5390	202	32	)	)	PUNCT
ejpam-5390	203	1	+	+	CCONJ
ejpam-5390	203	2	π([p2	π([p2	X
ejpam-5390	203	3	,	,	PUNCT
ejpam-5390	203	4	v11]∗	v11]∗	PROPN
ejpam-5390	203	5	♢	♢	NOUN
ejpam-5390	203	6	λp1	λp1	NOUN
ejpam-5390	203	7	)	)	PUNCT
ejpam-5390	203	8	=	=	PUNCT
ejpam-5390	204	1	[	[	X
ejpam-5390	204	2	π(p2	π(p2	ADJ
ejpam-5390	204	3	)	)	PUNCT
ejpam-5390	204	4	,	,	PUNCT
ejpam-5390	204	5	u11]∗	u11]∗	PROPN
ejpam-5390	204	6	♢	♢	PROPN
ejpam-5390	204	7	λp1	λp1	NOUN
ejpam-5390	204	8	+	+	X
ejpam-5390	205	1	[	[	X
ejpam-5390	205	2	p2,π(u11)]∗	p2,π(u11)]∗	X
ejpam-5390	205	3	♢	♢	NOUN
ejpam-5390	205	4	λp1	λp1	NOUN
ejpam-5390	205	5	+	+	CCONJ
ejpam-5390	206	1	[	[	X
ejpam-5390	206	2	p2	p2	NOUN
ejpam-5390	206	3	,	,	PUNCT
ejpam-5390	206	4	u11]∗	u11]∗	PROPN
ejpam-5390	206	5	♢	♢	PROPN
ejpam-5390	206	6	λπ(p1	λπ(p1	PROPN
ejpam-5390	206	7	)	)	PUNCT
ejpam-5390	207	1	+	+	PROPN
ejpam-5390	207	2	[	[	X
ejpam-5390	207	3	π(p2	π(p2	ADJ
ejpam-5390	207	4	)	)	PUNCT
ejpam-5390	207	5	,	,	PUNCT
ejpam-5390	207	6	v11]∗	v11]∗	PROPN
ejpam-5390	207	7	♢	♢	NOUN
ejpam-5390	207	8	λp1	λp1	NOUN
ejpam-5390	207	9	+	+	X
ejpam-5390	208	1	[	[	X
ejpam-5390	208	2	p2,π(v11)]∗	p2,π(v11)]∗	X
ejpam-5390	208	3	♢	♢	NOUN
ejpam-5390	208	4	λp1	λp1	NOUN
ejpam-5390	208	5	+	+	CCONJ
ejpam-5390	209	1	[	[	X
ejpam-5390	209	2	p2	p2	NOUN
ejpam-5390	209	3	,	,	PUNCT
ejpam-5390	209	4	v11]∗	v11]∗	PROPN
ejpam-5390	209	5	♢	♢	PROPN
ejpam-5390	209	6	λπ(p1	λπ(p1	PROPN
ejpam-5390	209	7	)	)	PUNCT
ejpam-5390	209	8	.	.	PUNCT
ejpam-5390	210	1	by	by	ADP
ejpam-5390	210	2	comparing	compare	VERB
ejpam-5390	210	3	the	the	DET
ejpam-5390	210	4	last	last	ADJ
ejpam-5390	210	5	two	two	NUM
ejpam-5390	210	6	equations	equation	NOUN
ejpam-5390	210	7	,	,	PUNCT
ejpam-5390	210	8	we	we	PRON
ejpam-5390	210	9	find	find	VERB
ejpam-5390	210	10	[	[	X
ejpam-5390	210	11	p2	p2	NOUN
ejpam-5390	210	12	,	,	PUNCT
ejpam-5390	210	13	t	t	X
ejpam-5390	210	14	]	]	PUNCT
ejpam-5390	210	15	∗	∗	X
ejpam-5390	210	16	♢	♢	NOUN
ejpam-5390	210	17	λp1	λp1	NOUN
ejpam-5390	210	18	=	=	SYM
ejpam-5390	210	19	0	0	X
ejpam-5390	210	20	.	.	PUNCT
ejpam-5390	211	1	this	this	PRON
ejpam-5390	211	2	gives	give	VERB
ejpam-5390	211	3	p2tp1	p2tp1	PROPN
ejpam-5390	211	4	−	−	NOUN
ejpam-5390	211	5	λp1tp2	λp1tp2	NOUN
ejpam-5390	211	6	=	=	PUNCT
ejpam-5390	211	7	0	0	NUM
ejpam-5390	211	8	and	and	CCONJ
ejpam-5390	211	9	hence	hence	ADV
ejpam-5390	211	10	,	,	PUNCT
ejpam-5390	211	11	p2tp1	p2tp1	ADJ
ejpam-5390	211	12	=	=	SYM
ejpam-5390	211	13	p1tp2	p1tp2	NOUN
ejpam-5390	211	14	=	=	NOUN
ejpam-5390	211	15	0	0	X
ejpam-5390	211	16	.	.	PUNCT
ejpam-5390	212	1	again	again	ADV
ejpam-5390	212	2	for	for	ADP
ejpam-5390	212	3	any	any	DET
ejpam-5390	212	4	x12	x12	NUM
ejpam-5390	212	5	∈	∈	PROPN
ejpam-5390	212	6	a12	a12	NOUN
ejpam-5390	212	7	and	and	CCONJ
ejpam-5390	212	8	since	since	SCONJ
ejpam-5390	212	9	[	[	X
ejpam-5390	212	10	x12	x12	NUM
ejpam-5390	212	11	,	,	PUNCT
ejpam-5390	212	12	u11]∗	u11]∗	NOUN
ejpam-5390	212	13	♢	♢	PROPN
ejpam-5390	212	14	λp2	λp2	NOUN
ejpam-5390	212	15	=	=	SYM
ejpam-5390	212	16	0	0	NUM
ejpam-5390	212	17	,	,	PUNCT
ejpam-5390	212	18	we	we	PRON
ejpam-5390	212	19	find	find	VERB
ejpam-5390	212	20	π([x12	π([x12	NUM
ejpam-5390	212	21	,	,	PUNCT
ejpam-5390	212	22	u11	u11	ADJ
ejpam-5390	212	23	+	+	CCONJ
ejpam-5390	212	24	v11]∗	v11]∗	ADJ
ejpam-5390	212	25	♢	♢	NOUN
ejpam-5390	212	26	λp2	λp2	NOUN
ejpam-5390	212	27	)	)	PUNCT
ejpam-5390	212	28	=	=	SYM
ejpam-5390	212	29	π([x12	π([x12	NUM
ejpam-5390	212	30	,	,	PUNCT
ejpam-5390	212	31	u11]∗	u11]∗	PROPN
ejpam-5390	212	32	♢	♢	PROPN
ejpam-5390	212	33	λp2	λp2	NOUN
ejpam-5390	212	34	)	)	PUNCT
ejpam-5390	213	1	+	+	NUM
ejpam-5390	213	2	π([x12	π([x12	NUM
ejpam-5390	213	3	,	,	PUNCT
ejpam-5390	213	4	v11]∗	v11]∗	ADJ
ejpam-5390	213	5	♢	♢	PROPN
ejpam-5390	213	6	λp2	λp2	NOUN
ejpam-5390	213	7	)	)	PUNCT
ejpam-5390	213	8	=	=	PUNCT
ejpam-5390	214	1	[	[	X
ejpam-5390	214	2	π(x12	π(x12	NUM
ejpam-5390	214	3	)	)	PUNCT
ejpam-5390	214	4	,	,	PUNCT
ejpam-5390	214	5	u11]∗	u11]∗	PROPN
ejpam-5390	214	6	♢	♢	VERB
ejpam-5390	214	7	λp2	λp2	NOUN
ejpam-5390	214	8	+	+	SYM
ejpam-5390	215	1	[	[	X
ejpam-5390	215	2	x12,π(u11)]∗	x12,π(u11)]∗	X
ejpam-5390	215	3	♢	♢	PROPN
ejpam-5390	215	4	λp2	λp2	NOUN
ejpam-5390	215	5	+	+	PROPN
ejpam-5390	215	6	[	[	X
ejpam-5390	215	7	x12	x12	NUM
ejpam-5390	215	8	,	,	PUNCT
ejpam-5390	215	9	u11]∗	u11]∗	NOUN
ejpam-5390	215	10	♢	♢	PROPN
ejpam-5390	215	11	λπ(p2	λπ(p2	X
ejpam-5390	215	12	)	)	PUNCT
ejpam-5390	216	1	+	+	CCONJ
ejpam-5390	217	1	[	[	X
ejpam-5390	217	2	π(x12	π(x12	NUM
ejpam-5390	217	3	)	)	PUNCT
ejpam-5390	217	4	,	,	PUNCT
ejpam-5390	217	5	v11]∗	v11]∗	PROPN
ejpam-5390	217	6	♢	♢	NOUN
ejpam-5390	217	7	λp2	λp2	NOUN
ejpam-5390	217	8	+	+	PROPN
ejpam-5390	217	9	[	[	X
ejpam-5390	217	10	x12,π(v11)]∗	x12,π(v11)]∗	X
ejpam-5390	217	11	♢	♢	VERB
ejpam-5390	217	12	λp2	λp2	NOUN
ejpam-5390	217	13	+	+	CCONJ
ejpam-5390	218	1	[	[	X
ejpam-5390	218	2	x12	x12	NUM
ejpam-5390	218	3	,	,	PUNCT
ejpam-5390	218	4	v11]∗	v11]∗	ADJ
ejpam-5390	218	5	♢	♢	NOUN
ejpam-5390	218	6	λπ(p2	λπ(p2	NOUN
ejpam-5390	218	7	)	)	PUNCT
ejpam-5390	218	8	.	.	PUNCT
ejpam-5390	219	1	from	from	ADP
ejpam-5390	219	2	the	the	DET
ejpam-5390	219	3	other	other	ADJ
ejpam-5390	219	4	side	side	NOUN
ejpam-5390	219	5	,	,	PUNCT
ejpam-5390	219	6	we	we	PRON
ejpam-5390	219	7	get	get	VERB
ejpam-5390	219	8	π([x12	π([x12	NUM
ejpam-5390	219	9	,	,	PUNCT
ejpam-5390	219	10	u11	u11	ADJ
ejpam-5390	219	11	+	+	CCONJ
ejpam-5390	219	12	v11]∗	v11]∗	ADJ
ejpam-5390	219	13	♢	♢	NOUN
ejpam-5390	219	14	λp2	λp2	NOUN
ejpam-5390	219	15	)	)	PUNCT
ejpam-5390	219	16	=	=	PUNCT
ejpam-5390	220	1	[	[	X
ejpam-5390	220	2	π(x12	π(x12	NUM
ejpam-5390	220	3	)	)	PUNCT
ejpam-5390	220	4	,	,	PUNCT
ejpam-5390	220	5	u11	u11	PROPN
ejpam-5390	220	6	+	+	CCONJ
ejpam-5390	220	7	v11]∗	v11]∗	ADJ
ejpam-5390	220	8	♢	♢	NOUN
ejpam-5390	220	9	λp2	λp2	NOUN
ejpam-5390	220	10	+	+	CCONJ
ejpam-5390	221	1	[	[	X
ejpam-5390	221	2	x12,π(u11	x12,π(u11	X
ejpam-5390	221	3	+	+	NUM
ejpam-5390	221	4	v11)]∗	v11)]∗	NOUN
ejpam-5390	221	5	♢	♢	NOUN
ejpam-5390	221	6	λp2	λp2	NOUN
ejpam-5390	222	1	+	+	PROPN
ejpam-5390	222	2	[	[	X
ejpam-5390	222	3	x12	x12	NUM
ejpam-5390	222	4	,	,	PUNCT
ejpam-5390	222	5	u11	u11	ADJ
ejpam-5390	222	6	+	+	CCONJ
ejpam-5390	222	7	v11]∗	v11]∗	ADJ
ejpam-5390	222	8	♢	♢	NOUN
ejpam-5390	222	9	λπ(p2	λπ(p2	NOUN
ejpam-5390	222	10	)	)	PUNCT
ejpam-5390	222	11	.	.	PUNCT
ejpam-5390	223	1	from	from	ADP
ejpam-5390	223	2	the	the	DET
ejpam-5390	223	3	last	last	ADJ
ejpam-5390	223	4	two	two	NUM
ejpam-5390	223	5	equations	equation	NOUN
ejpam-5390	223	6	,	,	PUNCT
ejpam-5390	223	7	we	we	PRON
ejpam-5390	223	8	get	get	VERB
ejpam-5390	223	9	[	[	X
ejpam-5390	223	10	x12	x12	NUM
ejpam-5390	223	11	,	,	PUNCT
ejpam-5390	223	12	t	t	X
ejpam-5390	223	13	]	]	PUNCT
ejpam-5390	223	14	∗	∗	X
ejpam-5390	223	15	♢	♢	NOUN
ejpam-5390	223	16	λp2	λp2	NOUN
ejpam-5390	223	17	=	=	NOUN
ejpam-5390	223	18	0	0	X
ejpam-5390	223	19	.	.	PUNCT
ejpam-5390	224	1	that	that	PRON
ejpam-5390	224	2	means	mean	VERB
ejpam-5390	224	3	x12tp2−λp2tx	x12tp2−λp2tx	PROPN
ejpam-5390	224	4	∗	∗	NOUN
ejpam-5390	224	5	12	12	NUM
ejpam-5390	224	6	=	=	SYM
ejpam-5390	224	7	0	0	NUM
ejpam-5390	224	8	.	.	PUNCT
ejpam-5390	225	1	thus	thus	ADV
ejpam-5390	225	2	,	,	PUNCT
ejpam-5390	225	3	x12tp2	x12tp2	PROPN
ejpam-5390	225	4	=	=	NOUN
ejpam-5390	226	1	0	0	X
ejpam-5390	226	2	.	.	PUNCT
ejpam-5390	226	3	by	by	ADP
ejpam-5390	226	4	using	use	VERB
ejpam-5390	226	5	(	(	PUNCT
ejpam-5390	226	6	▲	▲	PUNCT
ejpam-5390	226	7	)	)	PUNCT
ejpam-5390	226	8	and	and	CCONJ
ejpam-5390	226	9	(	(	PUNCT
ejpam-5390	226	10	▼	▼	NOUN
ejpam-5390	226	11	)	)	PUNCT
ejpam-5390	226	12	,	,	PUNCT
ejpam-5390	226	13	we	we	PRON
ejpam-5390	226	14	get	get	VERB
ejpam-5390	226	15	p2tp2	p2tp2	NOUN
ejpam-5390	226	16	=	=	NOUN
ejpam-5390	226	17	0	0	X
ejpam-5390	226	18	.	.	PUNCT
ejpam-5390	227	1	similarly	similarly	ADV
ejpam-5390	227	2	,	,	PUNCT
ejpam-5390	227	3	we	we	PRON
ejpam-5390	227	4	can	can	AUX
ejpam-5390	227	5	show	show	VERB
ejpam-5390	227	6	that	that	DET
ejpam-5390	227	7	p1tp1	p1tp1	NOUN
ejpam-5390	227	8	=	=	SYM
ejpam-5390	227	9	0	0	NUM
ejpam-5390	227	10	.	.	PUNCT
ejpam-5390	228	1	hence	hence	ADV
ejpam-5390	228	2	,	,	PUNCT
ejpam-5390	228	3	t	t	PROPN
ejpam-5390	228	4	=	=	SYM
ejpam-5390	228	5	0	0	PUNCT
ejpam-5390	229	1	i	i	PRON
ejpam-5390	229	2	,	,	PUNCT
ejpam-5390	229	3	e.	e.	PROPN
ejpam-5390	229	4	π(u11	π(u11	VERB
ejpam-5390	229	5	+	+	NUM
ejpam-5390	230	1	v11	v11	NOUN
ejpam-5390	230	2	)	)	PUNCT
ejpam-5390	230	3	=	=	VERB
ejpam-5390	230	4	π(u11	π(u11	PRON
ejpam-5390	230	5	)	)	PUNCT
ejpam-5390	231	1	+	+	CCONJ
ejpam-5390	231	2	π(v11	π(v11	ADJ
ejpam-5390	231	3	)	)	PUNCT
ejpam-5390	231	4	.	.	PUNCT
ejpam-5390	232	1	(	(	PUNCT
ejpam-5390	232	2	ii	ii	NOUN
ejpam-5390	232	3	)	)	PUNCT
ejpam-5390	232	4	.	.	PUNCT
ejpam-5390	233	1	by	by	ADP
ejpam-5390	233	2	using	use	VERB
ejpam-5390	233	3	the	the	DET
ejpam-5390	233	4	same	same	ADJ
ejpam-5390	233	5	argument	argument	NOUN
ejpam-5390	233	6	as	as	ADP
ejpam-5390	233	7	in	in	ADP
ejpam-5390	233	8	(	(	PUNCT
ejpam-5390	233	9	i	i	NOUN
ejpam-5390	233	10	)	)	PUNCT
ejpam-5390	233	11	,	,	PUNCT
ejpam-5390	233	12	one	one	PRON
ejpam-5390	233	13	can	can	AUX
ejpam-5390	233	14	show	show	VERB
ejpam-5390	233	15	that	that	SCONJ
ejpam-5390	233	16	π(u22	π(u22	NOUN
ejpam-5390	233	17	+	+	CCONJ
ejpam-5390	233	18	v22	v22	NOUN
ejpam-5390	233	19	)	)	PUNCT
ejpam-5390	233	20	=	=	SYM
ejpam-5390	233	21	π(u22	π(u22	NOUN
ejpam-5390	233	22	)	)	PUNCT
ejpam-5390	233	23	+	+	CCONJ
ejpam-5390	233	24	π(v22	π(v22	NOUN
ejpam-5390	233	25	)	)	PUNCT
ejpam-5390	233	26	.	.	PUNCT
ejpam-5390	234	1	lemma	lemma	PROPN
ejpam-5390	234	2	2.6	2.6	NUM
ejpam-5390	234	3	.	.	PUNCT
ejpam-5390	235	1	π	π	PROPN
ejpam-5390	235	2	is	be	AUX
ejpam-5390	235	3	an	an	DET
ejpam-5390	235	4	additive	additive	ADJ
ejpam-5390	235	5	map	map	NOUN
ejpam-5390	235	6	.	.	PUNCT
ejpam-5390	235	7	.	.	PUNCT
ejpam-5390	236	1	nisar	nisar	PROPN
ejpam-5390	236	2	et	et	PROPN
ejpam-5390	236	3	al	al	PROPN
ejpam-5390	236	4	.	.	PUNCT
ejpam-5390	236	5	/	/	SYM
ejpam-5390	236	6	eur	eur	PROPN
ejpam-5390	236	7	.	.	PUNCT
ejpam-5390	237	1	j.	j.	PROPN
ejpam-5390	237	2	pure	pure	PROPN
ejpam-5390	237	3	appl	appl	PROPN
ejpam-5390	237	4	.	.	PROPN
ejpam-5390	237	5	math	math	PROPN
ejpam-5390	237	6	,	,	PUNCT
ejpam-5390	237	7	17	17	NUM
ejpam-5390	237	8	(	(	PUNCT
ejpam-5390	237	9	4	4	NUM
ejpam-5390	237	10	)	)	PUNCT
ejpam-5390	237	11	(	(	PUNCT
ejpam-5390	237	12	2024	2024	NUM
ejpam-5390	237	13	)	)	PUNCT
ejpam-5390	237	14	,	,	PUNCT
ejpam-5390	237	15	3399	3399	NUM
ejpam-5390	237	16	-	-	SYM
ejpam-5390	237	17	3414	3414	NUM
ejpam-5390	237	18	3405	3405	NUM
ejpam-5390	237	19	proof	proof	NOUN
ejpam-5390	237	20	.	.	PUNCT
ejpam-5390	238	1	for	for	ADP
ejpam-5390	238	2	any	any	DET
ejpam-5390	238	3	u	u	NOUN
ejpam-5390	238	4	,	,	PUNCT
ejpam-5390	238	5	v	v	ADP
ejpam-5390	238	6	∈	∈	PROPN
ejpam-5390	238	7	a	a	PRON
ejpam-5390	238	8	,	,	PUNCT
ejpam-5390	238	9	we	we	PRON
ejpam-5390	238	10	write	write	VERB
ejpam-5390	238	11	u	u	NOUN
ejpam-5390	238	12	=	=	SYM
ejpam-5390	238	13	u11	u11	PROPN
ejpam-5390	238	14	+	+	NUM
ejpam-5390	238	15	u12	u12	PROPN
ejpam-5390	238	16	+	+	CCONJ
ejpam-5390	238	17	u21	u21	PROPN
ejpam-5390	238	18	+	+	CCONJ
ejpam-5390	238	19	u22	u22	PROPN
ejpam-5390	238	20	and	and	CCONJ
ejpam-5390	238	21	v	v	NOUN
ejpam-5390	238	22	=	=	SYM
ejpam-5390	238	23	v11	v11	NOUN
ejpam-5390	238	24	+	+	CCONJ
ejpam-5390	238	25	v12	v12	ADJ
ejpam-5390	238	26	+	+	CCONJ
ejpam-5390	238	27	v21	v21	NOUN
ejpam-5390	238	28	+	+	CCONJ
ejpam-5390	238	29	v22	v22	NOUN
ejpam-5390	238	30	.	.	PUNCT
ejpam-5390	239	1	by	by	ADP
ejpam-5390	239	2	using	use	VERB
ejpam-5390	239	3	lemmas	lemmas	PROPN
ejpam-5390	239	4	2.3	2.3	NUM
ejpam-5390	239	5	2.5	2.5	NUM
ejpam-5390	239	6	,	,	PUNCT
ejpam-5390	239	7	we	we	PRON
ejpam-5390	239	8	get	get	VERB
ejpam-5390	239	9	π(u	π(u	PROPN
ejpam-5390	239	10	+	+	CCONJ
ejpam-5390	239	11	v	v	NOUN
ejpam-5390	239	12	)	)	PUNCT
ejpam-5390	239	13	=	=	VERB
ejpam-5390	240	1	π(u11	π(u11	ADJ
ejpam-5390	240	2	+	+	NUM
ejpam-5390	240	3	u12	u12	NOUN
ejpam-5390	240	4	+	+	CCONJ
ejpam-5390	240	5	u21	u21	PROPN
ejpam-5390	240	6	+	+	CCONJ
ejpam-5390	240	7	u22	u22	PROPN
ejpam-5390	240	8	+	+	CCONJ
ejpam-5390	240	9	v11	v11	NOUN
ejpam-5390	240	10	+	+	CCONJ
ejpam-5390	240	11	v12	v12	ADJ
ejpam-5390	240	12	+	+	CCONJ
ejpam-5390	240	13	v21	v21	NOUN
ejpam-5390	240	14	+	+	CCONJ
ejpam-5390	240	15	v22	v22	NOUN
ejpam-5390	240	16	)	)	PUNCT
ejpam-5390	241	1	=	=	VERB
ejpam-5390	241	2	π(u11	π(u11	ADJ
ejpam-5390	241	3	+	+	CCONJ
ejpam-5390	241	4	v11	v11	NOUN
ejpam-5390	241	5	)	)	PUNCT
ejpam-5390	242	1	+	+	NUM
ejpam-5390	242	2	π(u12	π(u12	NOUN
ejpam-5390	242	3	+	+	CCONJ
ejpam-5390	242	4	v12	v12	VERB
ejpam-5390	242	5	)	)	PUNCT
ejpam-5390	242	6	+	+	CCONJ
ejpam-5390	242	7	π(u21	π(u21	NOUN
ejpam-5390	242	8	+	+	CCONJ
ejpam-5390	242	9	v21	v21	NOUN
ejpam-5390	242	10	)	)	PUNCT
ejpam-5390	242	11	+	+	CCONJ
ejpam-5390	242	12	π(u22	π(u22	NOUN
ejpam-5390	242	13	+	+	CCONJ
ejpam-5390	242	14	v22	v22	NOUN
ejpam-5390	242	15	)	)	PUNCT
ejpam-5390	243	1	=	=	SYM
ejpam-5390	243	2	π(u11	π(u11	PRON
ejpam-5390	243	3	)	)	PUNCT
ejpam-5390	244	1	+	+	CCONJ
ejpam-5390	244	2	π(v11	π(v11	ADJ
ejpam-5390	244	3	)	)	PUNCT
ejpam-5390	244	4	+	+	NUM
ejpam-5390	244	5	π(u12	π(u12	NOUN
ejpam-5390	244	6	)	)	PUNCT
ejpam-5390	245	1	+	+	CCONJ
ejpam-5390	245	2	π(v12	π(v12	VERB
ejpam-5390	245	3	)	)	PUNCT
ejpam-5390	246	1	+	+	CCONJ
ejpam-5390	246	2	π(u21	π(u21	X
ejpam-5390	246	3	)	)	PUNCT
ejpam-5390	246	4	+	+	NUM
ejpam-5390	246	5	π(v21	π(v21	NUM
ejpam-5390	246	6	)	)	PUNCT
ejpam-5390	246	7	+	+	CCONJ
ejpam-5390	246	8	π(u22	π(u22	NOUN
ejpam-5390	246	9	)	)	PUNCT
ejpam-5390	246	10	+	+	CCONJ
ejpam-5390	246	11	π(v22	π(v22	NOUN
ejpam-5390	246	12	)	)	PUNCT
ejpam-5390	246	13	=	=	VERB
ejpam-5390	246	14	π(u11	π(u11	ADJ
ejpam-5390	246	15	+	+	NUM
ejpam-5390	246	16	u12	u12	NOUN
ejpam-5390	246	17	+	+	CCONJ
ejpam-5390	246	18	u21	u21	PROPN
ejpam-5390	246	19	+	+	CCONJ
ejpam-5390	246	20	u22	u22	NOUN
ejpam-5390	246	21	)	)	PUNCT
ejpam-5390	247	1	+	+	CCONJ
ejpam-5390	247	2	π(v11	π(v11	ADJ
ejpam-5390	247	3	+	+	CCONJ
ejpam-5390	247	4	v12	v12	ADJ
ejpam-5390	247	5	+	+	CCONJ
ejpam-5390	247	6	v21	v21	NOUN
ejpam-5390	247	7	+	+	CCONJ
ejpam-5390	247	8	v22	v22	NOUN
ejpam-5390	247	9	)	)	PUNCT
ejpam-5390	247	10	=	=	SYM
ejpam-5390	247	11	π(u	π(u	PROPN
ejpam-5390	247	12	)	)	PUNCT
ejpam-5390	248	1	+	+	CCONJ
ejpam-5390	248	2	π(v	π(v	NOUN
ejpam-5390	248	3	)	)	PUNCT
ejpam-5390	248	4	.	.	PUNCT
ejpam-5390	249	1	lemma	lemma	PROPN
ejpam-5390	249	2	2.7	2.7	NUM
ejpam-5390	249	3	.	.	PUNCT
ejpam-5390	250	1	(	(	PUNCT
ejpam-5390	250	2	i	i	NOUN
ejpam-5390	250	3	)	)	PUNCT
ejpam-5390	250	4	π(p1	π(p1	NOUN
ejpam-5390	250	5	)	)	PUNCT
ejpam-5390	250	6	∗	∗	NOUN
ejpam-5390	250	7	=	=	SYM
ejpam-5390	250	8	π(p1	π(p1	NOUN
ejpam-5390	250	9	)	)	PUNCT
ejpam-5390	250	10	,	,	PUNCT
ejpam-5390	250	11	π(p2	π(p2	ADJ
ejpam-5390	250	12	)	)	PUNCT
ejpam-5390	250	13	∗	∗	NOUN
ejpam-5390	250	14	=	=	SYM
ejpam-5390	250	15	π(p2	π(p2	PROPN
ejpam-5390	250	16	)	)	PUNCT
ejpam-5390	250	17	.	.	PUNCT
ejpam-5390	251	1	(	(	PUNCT
ejpam-5390	251	2	ii	ii	NOUN
ejpam-5390	251	3	)	)	PUNCT
ejpam-5390	251	4	p1π(p1)p2	p1π(p1)p2	NOUN
ejpam-5390	252	1	=	=	PROPN
ejpam-5390	252	2	−p1π(p2)p2	−p1π(p2)p2	PROPN
ejpam-5390	252	3	.	.	PROPN
ejpam-5390	252	4	(	(	PUNCT
ejpam-5390	252	5	iii	iii	NOUN
ejpam-5390	252	6	)	)	PUNCT
ejpam-5390	252	7	p2π(p2)p1	p2π(p2)p1	NOUN
ejpam-5390	252	8	=	=	PUNCT
ejpam-5390	252	9	−p2π(p1)p1	−p2π(p1)p1	ADJ
ejpam-5390	252	10	.	.	PUNCT
ejpam-5390	253	1	proof	proof	NOUN
ejpam-5390	253	2	.	.	PUNCT
ejpam-5390	254	1	(	(	PUNCT
ejpam-5390	254	2	i	i	NOUN
ejpam-5390	254	3	)	)	PUNCT
ejpam-5390	254	4	in	in	ADP
ejpam-5390	254	5	the	the	DET
ejpam-5390	254	6	view	view	NOUN
ejpam-5390	254	7	of	of	ADP
ejpam-5390	254	8	[	[	X
ejpam-5390	254	9	p1	p1	NOUN
ejpam-5390	254	10	,	,	PUNCT
ejpam-5390	254	11	p1]∗	p1]∗	PROPN
ejpam-5390	254	12	♢	♢	PROPN
ejpam-5390	254	13	λi	λi	NOUN
ejpam-5390	254	14	=	=	NOUN
ejpam-5390	254	15	0	0	NUM
ejpam-5390	254	16	and	and	CCONJ
ejpam-5390	254	17	using	use	VERB
ejpam-5390	254	18	lemma	lemma	PROPN
ejpam-5390	254	19	2.1	2.1	NUM
ejpam-5390	254	20	,	,	PUNCT
ejpam-5390	254	21	we	we	PRON
ejpam-5390	254	22	have	have	VERB
ejpam-5390	254	23	0	0	NUM
ejpam-5390	254	24	=	=	SYM
ejpam-5390	254	25	π([p1	π([p1	NOUN
ejpam-5390	254	26	,	,	PUNCT
ejpam-5390	254	27	p1]∗	p1]∗	PROPN
ejpam-5390	254	28	♢	♢	PROPN
ejpam-5390	254	29	λi	λi	NUM
ejpam-5390	254	30	)	)	PUNCT
ejpam-5390	254	31	=	=	PUNCT
ejpam-5390	255	1	[	[	X
ejpam-5390	255	2	π(p1	π(p1	NOUN
ejpam-5390	255	3	)	)	PUNCT
ejpam-5390	255	4	,	,	PUNCT
ejpam-5390	255	5	p1]∗	p1]∗	PROPN
ejpam-5390	255	6	♢	♢	PROPN
ejpam-5390	255	7	λi	λi	X
ejpam-5390	255	8	+	+	X
ejpam-5390	256	1	[	[	X
ejpam-5390	256	2	p1,π(p1)]∗	p1,π(p1)]∗	X
ejpam-5390	256	3	♢	♢	X
ejpam-5390	256	4	λi	λi	NOUN
ejpam-5390	256	5	+	+	PROPN
ejpam-5390	257	1	[	[	X
ejpam-5390	257	2	p1	p1	NOUN
ejpam-5390	257	3	,	,	PUNCT
ejpam-5390	257	4	p1]∗	p1]∗	PROPN
ejpam-5390	257	5	♢	♢	NOUN
ejpam-5390	257	6	λπ(i	λπ(i	NOUN
ejpam-5390	257	7	)	)	PUNCT
ejpam-5390	257	8	=	=	PUNCT
ejpam-5390	257	9	(	(	PUNCT
ejpam-5390	257	10	1	1	NUM
ejpam-5390	257	11	+	+	SYM
ejpam-5390	257	12	λ)(p1π(p1	λ)(p1π(p1	NOUN
ejpam-5390	257	13	)	)	PUNCT
ejpam-5390	257	14	∗	∗	NOUN
ejpam-5390	257	15	−	−	PROPN
ejpam-5390	257	16	p1π(p1	p1π(p1	PROPN
ejpam-5390	257	17	)	)	PUNCT
ejpam-5390	257	18	)	)	PUNCT
ejpam-5390	257	19	.	.	PUNCT
ejpam-5390	258	1	since	since	SCONJ
ejpam-5390	258	2	λ	λ	PROPN
ejpam-5390	258	3	̸=	̸=	PROPN
ejpam-5390	258	4	−1	−1	NOUN
ejpam-5390	258	5	,	,	PUNCT
ejpam-5390	258	6	we	we	PRON
ejpam-5390	258	7	have	have	VERB
ejpam-5390	258	8	p1π(p1	p1π(p1	NOUN
ejpam-5390	258	9	)	)	PUNCT
ejpam-5390	258	10	∗	∗	NOUN
ejpam-5390	258	11	=	=	SYM
ejpam-5390	258	12	p1π(p1	p1π(p1	NOUN
ejpam-5390	258	13	)	)	PUNCT
ejpam-5390	258	14	.	.	PUNCT
ejpam-5390	259	1	that	that	PRON
ejpam-5390	259	2	means	mean	VERB
ejpam-5390	259	3	p1π(p1	p1π(p1	PROPN
ejpam-5390	259	4	)	)	PUNCT
ejpam-5390	259	5	∗p1	∗p1	PROPN
ejpam-5390	260	1	=	=	PUNCT
ejpam-5390	260	2	p1π(p1)p1	p1π(p1)p1	ADJ
ejpam-5390	260	3	.	.	PUNCT
ejpam-5390	261	1	(	(	PUNCT
ejpam-5390	261	2	2.1	2.1	NUM
ejpam-5390	261	3	)	)	PUNCT
ejpam-5390	261	4	p1π(p1	p1π(p1	NOUN
ejpam-5390	261	5	)	)	PUNCT
ejpam-5390	261	6	∗p2	∗p2	PROPN
ejpam-5390	261	7	=	=	SYM
ejpam-5390	261	8	p1π(p1)p2	p1π(p1)p2	NOUN
ejpam-5390	261	9	(	(	PUNCT
ejpam-5390	261	10	2.2	2.2	NUM
ejpam-5390	261	11	)	)	PUNCT
ejpam-5390	261	12	p2π(p1	p2π(p1	PROPN
ejpam-5390	261	13	)	)	PUNCT
ejpam-5390	261	14	∗p1	∗p1	PROPN
ejpam-5390	261	15	=	=	SYM
ejpam-5390	261	16	p2π(p1)p1	p2π(p1)p1	NOUN
ejpam-5390	261	17	(	(	PUNCT
ejpam-5390	261	18	2.3	2.3	NUM
ejpam-5390	261	19	)	)	PUNCT
ejpam-5390	261	20	also	also	ADV
ejpam-5390	261	21	,	,	PUNCT
ejpam-5390	261	22	[	[	X
ejpam-5390	261	23	p1	p1	NOUN
ejpam-5390	261	24	,	,	PUNCT
ejpam-5390	261	25	p2]∗	p2]∗	NOUN
ejpam-5390	261	26	♢	♢	NOUN
ejpam-5390	261	27	λi	λi	NOUN
ejpam-5390	261	28	=	=	NOUN
ejpam-5390	261	29	0	0	NUM
ejpam-5390	261	30	and	and	CCONJ
ejpam-5390	261	31	π(0	π(0	PROPN
ejpam-5390	261	32	)	)	PUNCT
ejpam-5390	261	33	=	=	SYM
ejpam-5390	261	34	0	0	NUM
ejpam-5390	261	35	,	,	PUNCT
ejpam-5390	261	36	we	we	PRON
ejpam-5390	261	37	have	have	VERB
ejpam-5390	261	38	0	0	NUM
ejpam-5390	261	39	=	=	SYM
ejpam-5390	261	40	π([p1	π([p1	NOUN
ejpam-5390	261	41	,	,	PUNCT
ejpam-5390	261	42	p2]∗	p2]∗	NOUN
ejpam-5390	261	43	♢	♢	NOUN
ejpam-5390	261	44	λi	λi	NOUN
ejpam-5390	261	45	=	=	NOUN
ejpam-5390	261	46	0	0	NUM
ejpam-5390	261	47	)	)	PUNCT
ejpam-5390	261	48	=	=	PUNCT
ejpam-5390	262	1	[	[	X
ejpam-5390	262	2	π(p1	π(p1	NOUN
ejpam-5390	262	3	)	)	PUNCT
ejpam-5390	262	4	,	,	PUNCT
ejpam-5390	262	5	p2]∗	p2]∗	PROPN
ejpam-5390	262	6	♢	♢	NOUN
ejpam-5390	262	7	λi	λi	NOUN
ejpam-5390	262	8	+	+	X
ejpam-5390	263	1	[	[	X
ejpam-5390	263	2	p1,π(p2)]∗	p1,π(p2)]∗	PROPN
ejpam-5390	263	3	♢	♢	X
ejpam-5390	263	4	λi	λi	NOUN
ejpam-5390	263	5	+	+	PROPN
ejpam-5390	264	1	[	[	X
ejpam-5390	264	2	p1	p1	NOUN
ejpam-5390	264	3	,	,	PUNCT
ejpam-5390	264	4	p2]∗	p2]∗	NOUN
ejpam-5390	264	5	♢	♢	NOUN
ejpam-5390	264	6	λπ(i	λπ(i	NOUN
ejpam-5390	264	7	)	)	PUNCT
ejpam-5390	264	8	=	=	SYM
ejpam-5390	264	9	(	(	PUNCT
ejpam-5390	264	10	1	1	NUM
ejpam-5390	264	11	+	+	NUM
ejpam-5390	264	12	λ)(p2π(p1)p2	λ)(p2π(p1)p2	NOUN
ejpam-5390	264	13	−	−	PROPN
ejpam-5390	264	14	p2π(p1	p2π(p1	PROPN
ejpam-5390	264	15	)	)	PUNCT
ejpam-5390	264	16	∗p2	∗p2	NOUN
ejpam-5390	264	17	)	)	PUNCT
ejpam-5390	264	18	.	.	PUNCT
ejpam-5390	265	1	that	that	PRON
ejpam-5390	265	2	is	is	ADV
ejpam-5390	265	3	,	,	PUNCT
ejpam-5390	265	4	p2π(p1	p2π(p1	PROPN
ejpam-5390	265	5	)	)	PUNCT
ejpam-5390	265	6	∗p2	∗p2	NOUN
ejpam-5390	265	7	=	=	PUNCT
ejpam-5390	265	8	p2π(p1)p2	p2π(p1)p2	NOUN
ejpam-5390	265	9	.	.	PUNCT
ejpam-5390	266	1	(	(	PUNCT
ejpam-5390	266	2	2.4	2.4	NUM
ejpam-5390	266	3	)	)	PUNCT
ejpam-5390	266	4	from	from	ADP
ejpam-5390	266	5	equations	equation	NOUN
ejpam-5390	266	6	(	(	PUNCT
ejpam-5390	266	7	2.1)-(2.4	2.1)-(2.4	NUM
ejpam-5390	266	8	)	)	PUNCT
ejpam-5390	266	9	,	,	PUNCT
ejpam-5390	266	10	we	we	PRON
ejpam-5390	266	11	conclude	conclude	VERB
ejpam-5390	266	12	π(p1	π(p1	NOUN
ejpam-5390	266	13	)	)	PUNCT
ejpam-5390	266	14	∗	∗	NOUN
ejpam-5390	266	15	=	=	SYM
ejpam-5390	266	16	π(p1	π(p1	NOUN
ejpam-5390	266	17	)	)	PUNCT
ejpam-5390	266	18	.	.	PUNCT
ejpam-5390	267	1	similarly	similarly	ADV
ejpam-5390	267	2	,	,	PUNCT
ejpam-5390	267	3	by	by	ADP
ejpam-5390	267	4	using	use	VERB
ejpam-5390	267	5	the	the	DET
ejpam-5390	267	6	same	same	ADJ
ejpam-5390	267	7	technique	technique	NOUN
ejpam-5390	267	8	as	as	ADP
ejpam-5390	267	9	above	above	ADV
ejpam-5390	267	10	,	,	PUNCT
ejpam-5390	267	11	we	we	PRON
ejpam-5390	267	12	get	get	VERB
ejpam-5390	267	13	π(p2	π(p2	ADJ
ejpam-5390	267	14	)	)	PUNCT
ejpam-5390	267	15	∗	∗	NOUN
ejpam-5390	267	16	=	=	SYM
ejpam-5390	267	17	π(p2	π(p2	PROPN
ejpam-5390	267	18	)	)	PUNCT
ejpam-5390	267	19	.	.	PUNCT
ejpam-5390	268	1	(	(	PUNCT
ejpam-5390	268	2	ii	ii	NOUN
ejpam-5390	268	3	)	)	PUNCT
ejpam-5390	268	4	again	again	ADV
ejpam-5390	269	1	[	[	X
ejpam-5390	269	2	p1	p1	NOUN
ejpam-5390	269	3	,	,	PUNCT
ejpam-5390	269	4	p2]∗	p2]∗	NOUN
ejpam-5390	269	5	♢	♢	NOUN
ejpam-5390	269	6	λp2	λp2	NOUN
ejpam-5390	269	7	=	=	SYM
ejpam-5390	269	8	0	0	NUM
ejpam-5390	269	9	and	and	CCONJ
ejpam-5390	269	10	π(0	π(0	PROPN
ejpam-5390	269	11	)	)	PUNCT
ejpam-5390	269	12	=	=	SYM
ejpam-5390	269	13	0	0	NUM
ejpam-5390	269	14	,	,	PUNCT
ejpam-5390	269	15	we	we	PRON
ejpam-5390	269	16	find	find	VERB
ejpam-5390	269	17	0	0	NUM
ejpam-5390	269	18	=	=	SYM
ejpam-5390	269	19	π([p1	π([p1	NOUN
ejpam-5390	269	20	,	,	PUNCT
ejpam-5390	269	21	p2]∗	p2]∗	NOUN
ejpam-5390	269	22	♢	♢	NOUN
ejpam-5390	269	23	λp2	λp2	NOUN
ejpam-5390	269	24	)	)	PUNCT
ejpam-5390	269	25	.	.	PUNCT
ejpam-5390	270	1	nisar	nisar	PROPN
ejpam-5390	270	2	et	et	PROPN
ejpam-5390	270	3	al	al	PROPN
ejpam-5390	270	4	.	.	PUNCT
ejpam-5390	270	5	/	/	SYM
ejpam-5390	270	6	eur	eur	PROPN
ejpam-5390	270	7	.	.	PUNCT
ejpam-5390	271	1	j.	j.	PROPN
ejpam-5390	271	2	pure	pure	PROPN
ejpam-5390	271	3	appl	appl	PROPN
ejpam-5390	271	4	.	.	PROPN
ejpam-5390	271	5	math	math	PROPN
ejpam-5390	271	6	,	,	PUNCT
ejpam-5390	271	7	17	17	NUM
ejpam-5390	271	8	(	(	PUNCT
ejpam-5390	271	9	4	4	NUM
ejpam-5390	271	10	)	)	PUNCT
ejpam-5390	271	11	(	(	PUNCT
ejpam-5390	271	12	2024	2024	NUM
ejpam-5390	271	13	)	)	PUNCT
ejpam-5390	271	14	,	,	PUNCT
ejpam-5390	271	15	3399	3399	NUM
ejpam-5390	271	16	-	-	SYM
ejpam-5390	271	17	3414	3414	NUM
ejpam-5390	271	18	3406	3406	NUM
ejpam-5390	271	19	=	=	SYM
ejpam-5390	271	20	[	[	X
ejpam-5390	271	21	π(p1	π(p1	NOUN
ejpam-5390	271	22	)	)	PUNCT
ejpam-5390	271	23	,	,	PUNCT
ejpam-5390	271	24	p2]∗	p2]∗	NOUN
ejpam-5390	271	25	♢	♢	NOUN
ejpam-5390	271	26	λp2	λp2	NOUN
ejpam-5390	271	27	+	+	NOUN
ejpam-5390	272	1	[	[	X
ejpam-5390	272	2	p1,π(p2)]∗	p1,π(p2)]∗	X
ejpam-5390	272	3	♢	♢	X
ejpam-5390	272	4	λp2	λp2	NOUN
ejpam-5390	272	5	+	+	SYM
ejpam-5390	272	6	[	[	X
ejpam-5390	272	7	p1	p1	NOUN
ejpam-5390	272	8	,	,	PUNCT
ejpam-5390	272	9	p2]∗	p2]∗	NOUN
ejpam-5390	272	10	♢	♢	NOUN
ejpam-5390	272	11	λπ(p2	λπ(p2	NOUN
ejpam-5390	272	12	)	)	PUNCT
ejpam-5390	272	13	=	=	PUNCT
ejpam-5390	272	14	π(p1)p2	π(p1)p2	X
ejpam-5390	272	15	−	−	PROPN
ejpam-5390	272	16	p2π(p1	p2π(p1	PROPN
ejpam-5390	272	17	)	)	PUNCT
ejpam-5390	272	18	∗p2	∗p2	PROPN
ejpam-5390	272	19	+	+	NUM
ejpam-5390	272	20	λp2π(p1)p2	λp2π(p1)p2	NOUN
ejpam-5390	272	21	−	−	PROPN
ejpam-5390	272	22	λp2π(p1	λp2π(p1	PROPN
ejpam-5390	272	23	)	)	PUNCT
ejpam-5390	272	24	∗	∗	NOUN
ejpam-5390	272	25	+	+	NUM
ejpam-5390	272	26	p1π(p2)p2	p1π(p2)p2	NOUN
ejpam-5390	272	27	−	−	NOUN
ejpam-5390	272	28	λp2π(p2)p1	λp2π(p2)p1	PROPN
ejpam-5390	272	29	.	.	PUNCT
ejpam-5390	273	1	multiplying	multiply	VERB
ejpam-5390	273	2	above	above	ADP
ejpam-5390	273	3	equation	equation	NOUN
ejpam-5390	273	4	by	by	ADP
ejpam-5390	273	5	p1	p1	PROPN
ejpam-5390	273	6	from	from	ADP
ejpam-5390	273	7	left	left	ADJ
ejpam-5390	273	8	,	,	PUNCT
ejpam-5390	273	9	we	we	PRON
ejpam-5390	273	10	get	get	VERB
ejpam-5390	273	11	p1π(p1)p2	p1π(p1)p2	NOUN
ejpam-5390	274	1	=	=	NOUN
ejpam-5390	274	2	−p1π(p2)p2	−p1π(p2)p2	PROPN
ejpam-5390	274	3	.	.	PROPN
ejpam-5390	274	4	(	(	PUNCT
ejpam-5390	274	5	iii	iii	X
ejpam-5390	274	6	)	)	PUNCT
ejpam-5390	274	7	by	by	ADP
ejpam-5390	274	8	using	use	VERB
ejpam-5390	274	9	the	the	DET
ejpam-5390	274	10	same	same	ADJ
ejpam-5390	274	11	technique	technique	NOUN
ejpam-5390	274	12	as	as	ADP
ejpam-5390	274	13	in	in	ADP
ejpam-5390	274	14	(	(	PUNCT
ejpam-5390	274	15	ii	ii	NOUN
ejpam-5390	274	16	)	)	PUNCT
ejpam-5390	274	17	,	,	PUNCT
ejpam-5390	274	18	we	we	PRON
ejpam-5390	274	19	can	can	AUX
ejpam-5390	274	20	show	show	VERB
ejpam-5390	274	21	that	that	SCONJ
ejpam-5390	274	22	p2π(p2)p1	p2π(p2)p1	NOUN
ejpam-5390	274	23	=	=	PUNCT
ejpam-5390	274	24	−p2π(p1)p1	−p2π(p1)p1	PROPN
ejpam-5390	274	25	.	.	PUNCT
ejpam-5390	275	1	lemma	lemma	PROPN
ejpam-5390	275	2	2.8	2.8	NUM
ejpam-5390	275	3	.	.	PUNCT
ejpam-5390	276	1	for	for	ADP
ejpam-5390	276	2	every	every	DET
ejpam-5390	276	3	uij	uij	PROPN
ejpam-5390	276	4	∈	∈	NOUN
ejpam-5390	276	5	aij(1	aij(1	NOUN
ejpam-5390	276	6	≤	≤	PUNCT
ejpam-5390	277	1	i	i	PRON
ejpam-5390	277	2	̸=	̸=	PROPN
ejpam-5390	277	3	j	j	PROPN
ejpam-5390	277	4	≤	≤	ADV
ejpam-5390	277	5	2	2	NUM
ejpam-5390	277	6	)	)	PUNCT
ejpam-5390	277	7	,	,	PUNCT
ejpam-5390	277	8	we	we	PRON
ejpam-5390	277	9	have	have	VERB
ejpam-5390	277	10	pjπ(λuij)pi	pjπ(λuij)pi	NOUN
ejpam-5390	277	11	=	=	SYM
ejpam-5390	277	12	0	0	X
ejpam-5390	277	13	.	.	PUNCT
ejpam-5390	278	1	proof	proof	NOUN
ejpam-5390	278	2	.	.	PUNCT
ejpam-5390	279	1	to	to	PART
ejpam-5390	279	2	begin	begin	VERB
ejpam-5390	279	3	,	,	PUNCT
ejpam-5390	279	4	we	we	PRON
ejpam-5390	279	5	establish	establish	VERB
ejpam-5390	279	6	the	the	DET
ejpam-5390	279	7	result	result	NOUN
ejpam-5390	279	8	for	for	ADP
ejpam-5390	279	9	i	i	PROPN
ejpam-5390	279	10	=	=	SYM
ejpam-5390	279	11	1	1	NUM
ejpam-5390	279	12	and	and	CCONJ
ejpam-5390	279	13	j	j	NOUN
ejpam-5390	279	14	=	=	NOUN
ejpam-5390	279	15	2	2	X
ejpam-5390	279	16	.	.	X
ejpam-5390	280	1	for	for	ADP
ejpam-5390	280	2	any	any	DET
ejpam-5390	280	3	u12	u12	PROPN
ejpam-5390	280	4	∈	∈	PROPN
ejpam-5390	280	5	a12	a12	NOUN
ejpam-5390	280	6	,	,	PUNCT
ejpam-5390	280	7	we	we	PRON
ejpam-5390	280	8	get	get	VERB
ejpam-5390	280	9	π(λu12	π(λu12	PRON
ejpam-5390	280	10	)	)	PUNCT
ejpam-5390	280	11	=	=	SYM
ejpam-5390	281	1	π([p1	π([p1	NOUN
ejpam-5390	281	2	,	,	PUNCT
ejpam-5390	281	3	λu12]∗	λu12]∗	NOUN
ejpam-5390	281	4	♢	♢	NOUN
ejpam-5390	281	5	λp2	λp2	NOUN
ejpam-5390	281	6	)	)	PUNCT
ejpam-5390	281	7	=	=	PUNCT
ejpam-5390	282	1	[	[	X
ejpam-5390	282	2	π(p1	π(p1	NOUN
ejpam-5390	282	3	)	)	PUNCT
ejpam-5390	282	4	,	,	PUNCT
ejpam-5390	282	5	λu12]∗	λu12]∗	NOUN
ejpam-5390	282	6	♢	♢	NOUN
ejpam-5390	282	7	λp2	λp2	NOUN
ejpam-5390	282	8	+	+	SYM
ejpam-5390	283	1	[	[	X
ejpam-5390	283	2	p1,π(λu12)]∗	p1,π(λu12)]∗	X
ejpam-5390	283	3	♢	♢	X
ejpam-5390	283	4	λp2	λp2	NOUN
ejpam-5390	283	5	+	+	SYM
ejpam-5390	283	6	[	[	X
ejpam-5390	283	7	p1	p1	NOUN
ejpam-5390	283	8	,	,	PUNCT
ejpam-5390	283	9	λu12]∗	λu12]∗	NOUN
ejpam-5390	283	10	♢	♢	NOUN
ejpam-5390	283	11	λπ(p2	λπ(p2	NOUN
ejpam-5390	283	12	)	)	PUNCT
ejpam-5390	283	13	=	=	PUNCT
ejpam-5390	284	1	λπ(p1)u12	λπ(p1)u12	PROPN
ejpam-5390	284	2	−	−	NOUN
ejpam-5390	284	3	λu12π(p1)p2	λu12π(p1)p2	X
ejpam-5390	285	1	+	+	CCONJ
ejpam-5390	285	2	λ2p2π(p1)u12	λ2p2π(p1)u12	PUNCT
ejpam-5390	285	3	+	+	NUM
ejpam-5390	285	4	p1π(λu12)p2	p1π(λu12)p2	NOUN
ejpam-5390	285	5	−λp2π(λu12)p1	−λp2π(λu12)p1	NOUN
ejpam-5390	285	6	+	+	X
ejpam-5390	285	7	λu12π(p2	λu12π(p2	PROPN
ejpam-5390	285	8	)	)	PUNCT
ejpam-5390	285	9	+	+	NUM
ejpam-5390	285	10	λ2π(p2)u12	λ2π(p2)u12	NOUN
ejpam-5390	285	11	.	.	PUNCT
ejpam-5390	286	1	by	by	ADP
ejpam-5390	286	2	left	left	ADV
ejpam-5390	286	3	-	-	PUNCT
ejpam-5390	286	4	multiplying	multiplying	NOUN
ejpam-5390	286	5	the	the	DET
ejpam-5390	286	6	above	above	ADJ
ejpam-5390	286	7	equation	equation	NOUN
ejpam-5390	286	8	with	with	ADP
ejpam-5390	286	9	p2	p2	PROPN
ejpam-5390	286	10	and	and	CCONJ
ejpam-5390	286	11	right	right	ADV
ejpam-5390	286	12	-	-	PUNCT
ejpam-5390	286	13	multiplying	multiplying	NOUN
ejpam-5390	286	14	with	with	ADP
ejpam-5390	286	15	p1	p1	PROPN
ejpam-5390	286	16	,	,	PUNCT
ejpam-5390	286	17	we	we	PRON
ejpam-5390	286	18	obtain	obtain	VERB
ejpam-5390	286	19	(	(	PUNCT
ejpam-5390	286	20	1	1	NUM
ejpam-5390	286	21	+	+	CCONJ
ejpam-5390	286	22	λ)p2π(λu12)p1	λ)p2π(λu12)p1	PROPN
ejpam-5390	286	23	=	=	SYM
ejpam-5390	286	24	0	0	NUM
ejpam-5390	286	25	.	.	PUNCT
ejpam-5390	287	1	(	(	PUNCT
ejpam-5390	287	2	2.5	2.5	NUM
ejpam-5390	287	3	)	)	PUNCT
ejpam-5390	287	4	since	since	SCONJ
ejpam-5390	287	5	λ	λ	X
ejpam-5390	287	6	̸=	̸=	PROPN
ejpam-5390	287	7	−1	−1	NOUN
ejpam-5390	287	8	,	,	PUNCT
ejpam-5390	287	9	we	we	PRON
ejpam-5390	287	10	find	find	VERB
ejpam-5390	287	11	p2π(λu12)p1	p2π(λu12)p1	NOUN
ejpam-5390	287	12	=	=	SYM
ejpam-5390	287	13	0	0	X
ejpam-5390	287	14	.	.	PUNCT
ejpam-5390	288	1	similarly	similarly	ADV
ejpam-5390	288	2	,	,	PUNCT
ejpam-5390	288	3	by	by	ADP
ejpam-5390	288	4	using	use	VERB
ejpam-5390	288	5	the	the	DET
ejpam-5390	288	6	same	same	ADJ
ejpam-5390	288	7	technique	technique	NOUN
ejpam-5390	288	8	for	for	ADP
ejpam-5390	288	9	i	i	PROPN
ejpam-5390	288	10	=	=	SYM
ejpam-5390	288	11	2	2	NUM
ejpam-5390	288	12	,	,	PUNCT
ejpam-5390	288	13	j	j	PROPN
ejpam-5390	288	14	=	=	SYM
ejpam-5390	288	15	1	1	NUM
ejpam-5390	288	16	,	,	PUNCT
ejpam-5390	288	17	one	one	PRON
ejpam-5390	288	18	can	can	AUX
ejpam-5390	288	19	show	show	VERB
ejpam-5390	288	20	that	that	SCONJ
ejpam-5390	288	21	p1π(λu21)p2	p1π(λu21)p2	NOUN
ejpam-5390	288	22	=	=	SYM
ejpam-5390	288	23	0	0	X
ejpam-5390	288	24	.	.	PUNCT
ejpam-5390	289	1	lemma	lemma	PROPN
ejpam-5390	289	2	2.9	2.9	NUM
ejpam-5390	289	3	.	.	PUNCT
ejpam-5390	290	1	(	(	PUNCT
ejpam-5390	290	2	i	i	NOUN
ejpam-5390	290	3	)	)	PUNCT
ejpam-5390	290	4	p1π(p2)p1	p1π(p2)p1	NOUN
ejpam-5390	290	5	=	=	PUNCT
ejpam-5390	290	6	p2π(p1)p2	p2π(p1)p2	X
ejpam-5390	291	1	=	=	PUNCT
ejpam-5390	291	2	0	0	X
ejpam-5390	291	3	.	.	PUNCT
ejpam-5390	291	4	(	(	PUNCT
ejpam-5390	291	5	ii	ii	NOUN
ejpam-5390	291	6	)	)	PUNCT
ejpam-5390	291	7	p1π(p1)p1	p1π(p1)p1	ADJ
ejpam-5390	291	8	=	=	PUNCT
ejpam-5390	291	9	p2π(p2)p2	p2π(p2)p2	NOUN
ejpam-5390	291	10	=	=	SYM
ejpam-5390	291	11	0	0	X
ejpam-5390	291	12	.	.	PUNCT
ejpam-5390	291	13	proof	proof	NOUN
ejpam-5390	291	14	.	.	PUNCT
ejpam-5390	292	1	(	(	PUNCT
ejpam-5390	292	2	i	i	NOUN
ejpam-5390	292	3	)	)	PUNCT
ejpam-5390	292	4	for	for	ADP
ejpam-5390	292	5	every	every	DET
ejpam-5390	292	6	x12	x12	NUM
ejpam-5390	292	7	∈	∈	PROPN
ejpam-5390	292	8	a12	a12	NOUN
ejpam-5390	292	9	,	,	PUNCT
ejpam-5390	292	10	it	it	PRON
ejpam-5390	292	11	follows	follow	VERB
ejpam-5390	292	12	from	from	ADP
ejpam-5390	292	13	[	[	X
ejpam-5390	292	14	x12	x12	NUM
ejpam-5390	292	15	,	,	PUNCT
ejpam-5390	292	16	p1]∗	p1]∗	PROPN
ejpam-5390	292	17	♢	♢	NOUN
ejpam-5390	292	18	λp1	λp1	NOUN
ejpam-5390	292	19	=	=	SYM
ejpam-5390	292	20	0	0	PUNCT
ejpam-5390	292	21	that	that	SCONJ
ejpam-5390	292	22	0	0	X
ejpam-5390	292	23	=	=	SYM
ejpam-5390	292	24	π([x12	π([x12	NUM
ejpam-5390	292	25	,	,	PUNCT
ejpam-5390	292	26	p1]∗	p1]∗	PROPN
ejpam-5390	292	27	♢	♢	NOUN
ejpam-5390	292	28	λp1	λp1	NOUN
ejpam-5390	292	29	)	)	PUNCT
ejpam-5390	292	30	=	=	PUNCT
ejpam-5390	293	1	[	[	X
ejpam-5390	293	2	π(x12	π(x12	NUM
ejpam-5390	293	3	)	)	PUNCT
ejpam-5390	293	4	,	,	PUNCT
ejpam-5390	293	5	p1]∗	p1]∗	PROPN
ejpam-5390	293	6	♢	♢	NOUN
ejpam-5390	293	7	λp1	λp1	NOUN
ejpam-5390	293	8	+	+	X
ejpam-5390	294	1	[	[	X
ejpam-5390	294	2	x12,π(p1)]∗	x12,π(p1)]∗	NUM
ejpam-5390	294	3	♢	♢	NOUN
ejpam-5390	294	4	λp1	λp1	NOUN
ejpam-5390	294	5	+	+	X
ejpam-5390	295	1	[	[	X
ejpam-5390	295	2	x12	x12	NUM
ejpam-5390	295	3	,	,	PUNCT
ejpam-5390	295	4	p1]∗	p1]∗	PROPN
ejpam-5390	295	5	♢	♢	PROPN
ejpam-5390	295	6	λπ(p1	λπ(p1	PROPN
ejpam-5390	295	7	)	)	PUNCT
ejpam-5390	295	8	=	=	SYM
ejpam-5390	295	9	π(x12)p1	π(x12)p1	PROPN
ejpam-5390	295	10	−	−	NOUN
ejpam-5390	295	11	p1π(x12	p1π(x12	SYM
ejpam-5390	295	12	)	)	PUNCT
ejpam-5390	295	13	∗p1	∗p1	PROPN
ejpam-5390	295	14	+	+	CCONJ
ejpam-5390	295	15	λp1π(x12)p1	λp1π(x12)p1	PROPN
ejpam-5390	295	16	−	−	PROPN
ejpam-5390	295	17	λp1π(x12	λp1π(x12	PROPN
ejpam-5390	295	18	)	)	PUNCT
ejpam-5390	295	19	∗	∗	NOUN
ejpam-5390	296	1	+	+	PROPN
ejpam-5390	296	2	x12π(p1)p1	x12π(p1)p1	VERB
ejpam-5390	296	3	−π(p1)x	−π(p1)x	NOUN
ejpam-5390	296	4	∗	∗	VERB
ejpam-5390	296	5	12	12	NUM
ejpam-5390	297	1	+	+	CCONJ
ejpam-5390	297	2	λx12π(p1)−	λx12π(p1)−	PUNCT
ejpam-5390	297	3	λp1π(p1)x	λp1π(p1)x	ADJ
ejpam-5390	297	4	∗	∗	NOUN
ejpam-5390	297	5	12	12	NUM
ejpam-5390	297	6	.	.	PUNCT
ejpam-5390	298	1	by	by	ADP
ejpam-5390	298	2	left	left	NOUN
ejpam-5390	298	3	-	-	PUNCT
ejpam-5390	298	4	multiplying	multiply	VERB
ejpam-5390	298	5	the	the	DET
ejpam-5390	298	6	preceding	precede	VERB
ejpam-5390	298	7	equation	equation	NOUN
ejpam-5390	298	8	by	by	ADP
ejpam-5390	298	9	p1	p1	NOUN
ejpam-5390	298	10	and	and	CCONJ
ejpam-5390	298	11	right	right	NOUN
ejpam-5390	298	12	-	-	PUNCT
ejpam-5390	298	13	multiplying	multiplying	NOUN
ejpam-5390	298	14	by	by	ADP
ejpam-5390	298	15	p2	p2	NOUN
ejpam-5390	298	16	,	,	PUNCT
ejpam-5390	298	17	and	and	CCONJ
ejpam-5390	298	18	considering	consider	VERB
ejpam-5390	298	19	the	the	DET
ejpam-5390	298	20	fact	fact	NOUN
ejpam-5390	298	21	that	that	SCONJ
ejpam-5390	298	22	λ	λ	PROPN
ejpam-5390	298	23	̸=	̸=	PROPN
ejpam-5390	298	24	0	0	NUM
ejpam-5390	298	25	,	,	PUNCT
ejpam-5390	298	26	we	we	PRON
ejpam-5390	298	27	obtain	obtain	VERB
ejpam-5390	298	28	x12π(p1)p2	x12π(p1)p2	PUNCT
ejpam-5390	298	29	=	=	PUNCT
ejpam-5390	298	30	p1π(x12	p1π(x12	X
ejpam-5390	298	31	)	)	PUNCT
ejpam-5390	298	32	∗p2	∗p2	NOUN
ejpam-5390	298	33	.	.	PUNCT
ejpam-5390	299	1	(	(	PUNCT
ejpam-5390	299	2	2.6	2.6	NUM
ejpam-5390	299	3	)	)	PUNCT
ejpam-5390	299	4	that	that	PRON
ejpam-5390	299	5	means	mean	VERB
ejpam-5390	299	6	,	,	PUNCT
ejpam-5390	299	7	p2π(p1)x	p2π(p1)x	NOUN
ejpam-5390	299	8	∗	∗	NOUN
ejpam-5390	299	9	12	12	NUM
ejpam-5390	299	10	=	=	SYM
ejpam-5390	299	11	p2π(x12)p1	p2π(x12)p1	NOUN
ejpam-5390	299	12	.	.	PUNCT
ejpam-5390	299	13	.	.	PUNCT
ejpam-5390	300	1	nisar	nisar	PROPN
ejpam-5390	300	2	et	et	PROPN
ejpam-5390	300	3	al	al	PROPN
ejpam-5390	300	4	.	.	PUNCT
ejpam-5390	300	5	/	/	SYM
ejpam-5390	300	6	eur	eur	PROPN
ejpam-5390	300	7	.	.	PUNCT
ejpam-5390	301	1	j.	j.	PROPN
ejpam-5390	301	2	pure	pure	PROPN
ejpam-5390	301	3	appl	appl	PROPN
ejpam-5390	301	4	.	.	PROPN
ejpam-5390	301	5	math	math	PROPN
ejpam-5390	301	6	,	,	PUNCT
ejpam-5390	301	7	17	17	NUM
ejpam-5390	301	8	(	(	PUNCT
ejpam-5390	301	9	4	4	NUM
ejpam-5390	301	10	)	)	PUNCT
ejpam-5390	301	11	(	(	PUNCT
ejpam-5390	301	12	2024	2024	NUM
ejpam-5390	301	13	)	)	PUNCT
ejpam-5390	301	14	,	,	PUNCT
ejpam-5390	301	15	3399	3399	NUM
ejpam-5390	301	16	-	-	SYM
ejpam-5390	301	17	3414	3414	NUM
ejpam-5390	301	18	3407	3407	NUM
ejpam-5390	301	19	also	also	ADV
ejpam-5390	301	20	,	,	PUNCT
ejpam-5390	301	21	[	[	X
ejpam-5390	301	22	p1	p1	NOUN
ejpam-5390	301	23	,	,	PUNCT
ejpam-5390	301	24	x12]∗	x12]∗	PROPN
ejpam-5390	301	25	♢	♢	NOUN
ejpam-5390	301	26	λx12	λx12	PROPN
ejpam-5390	301	27	=	=	SYM
ejpam-5390	301	28	0	0	PUNCT
ejpam-5390	301	29	and	and	CCONJ
ejpam-5390	301	30	using	use	VERB
ejpam-5390	301	31	lemma	lemma	PROPN
ejpam-5390	301	32	2.1	2.1	NUM
ejpam-5390	301	33	,	,	PUNCT
ejpam-5390	301	34	we	we	PRON
ejpam-5390	301	35	get	get	VERB
ejpam-5390	301	36	0	0	NUM
ejpam-5390	301	37	=	=	SYM
ejpam-5390	301	38	π([p1	π([p1	NOUN
ejpam-5390	301	39	,	,	PUNCT
ejpam-5390	301	40	x12]∗	x12]∗	PROPN
ejpam-5390	301	41	♢	♢	PROPN
ejpam-5390	301	42	λx12	λx12	PROPN
ejpam-5390	301	43	)	)	PUNCT
ejpam-5390	301	44	=	=	PUNCT
ejpam-5390	302	1	[	[	X
ejpam-5390	302	2	π(p1	π(p1	NOUN
ejpam-5390	302	3	)	)	PUNCT
ejpam-5390	302	4	,	,	PUNCT
ejpam-5390	302	5	x12]∗	x12]∗	PROPN
ejpam-5390	302	6	♢	♢	PROPN
ejpam-5390	302	7	λx12	λx12	PROPN
ejpam-5390	302	8	+	+	PROPN
ejpam-5390	303	1	[	[	X
ejpam-5390	303	2	p1,π(x12)]∗	p1,π(x12)]∗	X
ejpam-5390	303	3	♢	♢	PROPN
ejpam-5390	303	4	λx12	λx12	PROPN
ejpam-5390	303	5	+	+	PROPN
ejpam-5390	303	6	[	[	X
ejpam-5390	303	7	p1	p1	NOUN
ejpam-5390	303	8	,	,	PUNCT
ejpam-5390	303	9	x12]∗	x12]∗	PROPN
ejpam-5390	303	10	♢	♢	NOUN
ejpam-5390	303	11	λπ(x12	λπ(x12	PROPN
ejpam-5390	303	12	)	)	PUNCT
ejpam-5390	303	13	=	=	SYM
ejpam-5390	303	14	−x12π(p1	−x12π(p1	PROPN
ejpam-5390	303	15	)	)	PUNCT
ejpam-5390	303	16	∗x12	∗x12	PUNCT
ejpam-5390	303	17	+	+	PUNCT
ejpam-5390	303	18	λx12π(p1)x12	λx12π(p1)x12	X
ejpam-5390	303	19	+	+	CCONJ
ejpam-5390	303	20	p1π(x12)x12	p1π(x12)x12	NUM
ejpam-5390	303	21	−π(x12)x12	−π(x12)x12	NOUN
ejpam-5390	303	22	−	−	PROPN
ejpam-5390	303	23	λx12π(x12)p1	λx12π(x12)p1	PROPN
ejpam-5390	303	24	+	+	PROPN
ejpam-5390	303	25	x12π(x12	x12π(x12	X
ejpam-5390	303	26	)	)	PUNCT
ejpam-5390	303	27	+	+	SYM
ejpam-5390	303	28	λx12π(x12	λx12π(x12	NOUN
ejpam-5390	303	29	)	)	PUNCT
ejpam-5390	303	30	.	.	PUNCT
ejpam-5390	304	1	multiplying	multiply	VERB
ejpam-5390	304	2	the	the	DET
ejpam-5390	304	3	above	above	ADJ
ejpam-5390	304	4	equation	equation	NOUN
ejpam-5390	304	5	by	by	ADP
ejpam-5390	304	6	p1	p1	PROPN
ejpam-5390	304	7	from	from	ADP
ejpam-5390	304	8	left	left	ADJ
ejpam-5390	304	9	and	and	CCONJ
ejpam-5390	304	10	right	right	ADJ
ejpam-5390	304	11	,	,	PUNCT
ejpam-5390	304	12	we	we	PRON
ejpam-5390	304	13	find	find	VERB
ejpam-5390	304	14	x12π(x12)p1	x12π(x12)p1	PROPN
ejpam-5390	304	15	=	=	SYM
ejpam-5390	304	16	0	0	X
ejpam-5390	304	17	.	.	PUNCT
ejpam-5390	305	1	by	by	ADP
ejpam-5390	305	2	using	use	VERB
ejpam-5390	305	3	(	(	PUNCT
ejpam-5390	305	4	▲	▲	PUNCT
ejpam-5390	305	5	)	)	PUNCT
ejpam-5390	305	6	and	and	CCONJ
ejpam-5390	305	7	(	(	PUNCT
ejpam-5390	305	8	▼	▼	NOUN
ejpam-5390	305	9	)	)	PUNCT
ejpam-5390	305	10	,	,	PUNCT
ejpam-5390	305	11	we	we	PRON
ejpam-5390	305	12	get	get	VERB
ejpam-5390	305	13	p2π(x12)p1	p2π(x12)p1	ADV
ejpam-5390	305	14	=	=	NOUN
ejpam-5390	305	15	0	0	NUM
ejpam-5390	305	16	.	.	PUNCT
ejpam-5390	306	1	that	that	PRON
ejpam-5390	306	2	means	mean	VERB
ejpam-5390	306	3	p1π(x12	p1π(x12	NOUN
ejpam-5390	306	4	)	)	PUNCT
ejpam-5390	306	5	∗p2	∗p2	NOUN
ejpam-5390	306	6	=	=	SYM
ejpam-5390	306	7	0	0	NUM
ejpam-5390	306	8	.	.	PUNCT
ejpam-5390	306	9	from	from	ADP
ejpam-5390	306	10	equation	equation	NOUN
ejpam-5390	306	11	(	(	PUNCT
ejpam-5390	306	12	2.6	2.6	NUM
ejpam-5390	306	13	)	)	PUNCT
ejpam-5390	306	14	,	,	PUNCT
ejpam-5390	306	15	(	(	PUNCT
ejpam-5390	306	16	▲	▲	PUNCT
ejpam-5390	306	17	)	)	PUNCT
ejpam-5390	306	18	and	and	CCONJ
ejpam-5390	306	19	(	(	PUNCT
ejpam-5390	306	20	▼	▼	NOUN
ejpam-5390	306	21	)	)	PUNCT
ejpam-5390	306	22	,	,	PUNCT
ejpam-5390	306	23	we	we	PRON
ejpam-5390	306	24	have	have	VERB
ejpam-5390	306	25	p2π(p1)p2	p2π(p1)p2	NOUN
ejpam-5390	307	1	=	=	PUNCT
ejpam-5390	307	2	0	0	X
ejpam-5390	307	3	.	.	PUNCT
ejpam-5390	307	4	similarly	similarly	ADV
ejpam-5390	307	5	,	,	PUNCT
ejpam-5390	307	6	we	we	PRON
ejpam-5390	307	7	can	can	AUX
ejpam-5390	307	8	show	show	VERB
ejpam-5390	307	9	that	that	SCONJ
ejpam-5390	307	10	p1π(p2)p1	p1π(p2)p1	NOUN
ejpam-5390	307	11	=	=	NOUN
ejpam-5390	307	12	0	0	X
ejpam-5390	307	13	.	.	PUNCT
ejpam-5390	307	14	(	(	PUNCT
ejpam-5390	307	15	ii	ii	NOUN
ejpam-5390	307	16	)	)	PUNCT
ejpam-5390	307	17	for	for	ADP
ejpam-5390	307	18	any	any	DET
ejpam-5390	307	19	x21	x21	PROPN
ejpam-5390	307	20	∈	∈	PROPN
ejpam-5390	307	21	a21	a21	NOUN
ejpam-5390	307	22	and	and	CCONJ
ejpam-5390	307	23	using	use	VERB
ejpam-5390	307	24	lemma	lemma	PROPN
ejpam-5390	307	25	2.6	2.6	NUM
ejpam-5390	307	26	,	,	PUNCT
ejpam-5390	307	27	we	we	PRON
ejpam-5390	307	28	have	have	VERB
ejpam-5390	307	29	π([x21	π([x21	NUM
ejpam-5390	307	30	,	,	PUNCT
ejpam-5390	307	31	p1]∗	p1]∗	PROPN
ejpam-5390	307	32	♢	♢	NOUN
ejpam-5390	307	33	λp1	λp1	NOUN
ejpam-5390	307	34	)	)	PUNCT
ejpam-5390	308	1	=	=	NOUN
ejpam-5390	308	2	π(x21)−π(λx∗	π(x21)−π(λx∗	NOUN
ejpam-5390	308	3	21	21	NUM
ejpam-5390	308	4	)	)	PUNCT
ejpam-5390	308	5	.	.	PUNCT
ejpam-5390	309	1	on	on	ADP
ejpam-5390	309	2	the	the	DET
ejpam-5390	309	3	other	other	ADJ
ejpam-5390	309	4	hand	hand	NOUN
ejpam-5390	309	5	,	,	PUNCT
ejpam-5390	309	6	we	we	PRON
ejpam-5390	309	7	have	have	VERB
ejpam-5390	309	8	π([x21	π([x21	NUM
ejpam-5390	309	9	,	,	PUNCT
ejpam-5390	309	10	p1]∗	p1]∗	PROPN
ejpam-5390	309	11	♢	♢	NOUN
ejpam-5390	309	12	λp1	λp1	NOUN
ejpam-5390	309	13	)	)	PUNCT
ejpam-5390	309	14	=	=	PUNCT
ejpam-5390	310	1	[	[	X
ejpam-5390	310	2	π(x21	π(x21	NOUN
ejpam-5390	310	3	)	)	PUNCT
ejpam-5390	310	4	,	,	PUNCT
ejpam-5390	310	5	p1]∗	p1]∗	PROPN
ejpam-5390	310	6	♢	♢	NOUN
ejpam-5390	310	7	λp1	λp1	NOUN
ejpam-5390	310	8	+	+	X
ejpam-5390	311	1	[	[	X
ejpam-5390	311	2	x21,π(p1)]∗	x21,π(p1)]∗	NUM
ejpam-5390	311	3	♢	♢	NOUN
ejpam-5390	311	4	λp1	λp1	NOUN
ejpam-5390	311	5	+	+	X
ejpam-5390	312	1	[	[	X
ejpam-5390	312	2	x21	x21	NUM
ejpam-5390	312	3	,	,	PUNCT
ejpam-5390	312	4	p1]∗	p1]∗	PROPN
ejpam-5390	312	5	♢	♢	PROPN
ejpam-5390	312	6	λπ(p1	λπ(p1	PROPN
ejpam-5390	312	7	)	)	PUNCT
ejpam-5390	312	8	=	=	PUNCT
ejpam-5390	312	9	π(x21)p1	π(x21)p1	ADJ
ejpam-5390	312	10	−	−	PROPN
ejpam-5390	312	11	p1π(x21	p1π(x21	NUM
ejpam-5390	312	12	)	)	PUNCT
ejpam-5390	312	13	∗p1	∗p1	PROPN
ejpam-5390	313	1	+	+	CCONJ
ejpam-5390	313	2	λp1π(x21)p1	λp1π(x21)p1	VERB
ejpam-5390	313	3	−	−	PROPN
ejpam-5390	313	4	λp1π(x21	λp1π(x21	NOUN
ejpam-5390	313	5	)	)	PUNCT
ejpam-5390	313	6	∗	∗	NOUN
ejpam-5390	314	1	+	+	NOUN
ejpam-5390	314	2	x21π(p1)p1	x21π(p1)p1	NOUN
ejpam-5390	314	3	−	−	NOUN
ejpam-5390	314	4	λp1π(p1)x	λp1π(p1)x	ADJ
ejpam-5390	314	5	∗	∗	NOUN
ejpam-5390	314	6	21	21	NUM
ejpam-5390	315	1	+	+	NOUN
ejpam-5390	315	2	x21π(p1	x21π(p1	PROPN
ejpam-5390	315	3	)	)	PUNCT
ejpam-5390	316	1	+	+	PUNCT
ejpam-5390	317	1	λπ(p1)x21	λπ(p1)x21	PROPN
ejpam-5390	317	2	.	.	PUNCT
ejpam-5390	317	3	by	by	ADP
ejpam-5390	317	4	comparing	compare	VERB
ejpam-5390	317	5	the	the	DET
ejpam-5390	317	6	aforementioned	aforementioned	ADJ
ejpam-5390	317	7	two	two	NUM
ejpam-5390	317	8	equations	equation	NOUN
ejpam-5390	317	9	and	and	CCONJ
ejpam-5390	317	10	subsequently	subsequently	ADV
ejpam-5390	317	11	left	left	ADJ
ejpam-5390	317	12	-	-	PUNCT
ejpam-5390	317	13	multiplying	multiplying	NOUN
ejpam-5390	317	14	by	by	ADP
ejpam-5390	317	15	p2	p2	PROPN
ejpam-5390	317	16	and	and	CCONJ
ejpam-5390	317	17	right	right	ADV
ejpam-5390	317	18	-	-	PUNCT
ejpam-5390	317	19	multiplying	multiplying	NOUN
ejpam-5390	317	20	by	by	ADP
ejpam-5390	317	21	p1	p1	PROPN
ejpam-5390	317	22	,	,	PUNCT
ejpam-5390	317	23	we	we	PRON
ejpam-5390	317	24	obtain	obtain	VERB
ejpam-5390	317	25	2x21π(p1)p1	2x21π(p1)p1	PROPN
ejpam-5390	318	1	+	+	CCONJ
ejpam-5390	318	2	λp2π(p1)x21	λp2π(p1)x21	PROPN
ejpam-5390	318	3	+	+	CCONJ
ejpam-5390	318	4	p2π(λx	p2π(λx	NOUN
ejpam-5390	318	5	∗	∗	NOUN
ejpam-5390	318	6	21)p1	21)p1	NUM
ejpam-5390	319	1	=	=	SYM
ejpam-5390	319	2	0	0	PUNCT
ejpam-5390	319	3	now	now	ADV
ejpam-5390	319	4	,	,	PUNCT
ejpam-5390	319	5	by	by	ADP
ejpam-5390	319	6	using	use	VERB
ejpam-5390	319	7	lemma	lemma	PROPN
ejpam-5390	319	8	2.8	2.8	NUM
ejpam-5390	319	9	and	and	CCONJ
ejpam-5390	319	10	lemma	lemma	PROPN
ejpam-5390	319	11	2.9(i	2.9(i	NUM
ejpam-5390	319	12	)	)	PUNCT
ejpam-5390	319	13	,	,	PUNCT
ejpam-5390	319	14	we	we	PRON
ejpam-5390	319	15	getx21π(p1)p1	getx21π(p1)p1	VERB
ejpam-5390	319	16	=	=	PUNCT
ejpam-5390	320	1	0	0	X
ejpam-5390	320	2	.	.	PUNCT
ejpam-5390	321	1	hence	hence	ADV
ejpam-5390	321	2	,	,	PUNCT
ejpam-5390	321	3	p1π(p1)p1	p1π(p1)p1	ADJ
ejpam-5390	321	4	=	=	NOUN
ejpam-5390	321	5	0	0	X
ejpam-5390	321	6	.	.	PUNCT
ejpam-5390	322	1	similarly	similarly	ADV
ejpam-5390	322	2	,	,	PUNCT
ejpam-5390	322	3	we	we	PRON
ejpam-5390	322	4	can	can	AUX
ejpam-5390	322	5	show	show	VERB
ejpam-5390	322	6	that	that	SCONJ
ejpam-5390	322	7	p2π(p2)p2	p2π(p2)p2	NOUN
ejpam-5390	322	8	=	=	NOUN
ejpam-5390	322	9	0	0	X
ejpam-5390	322	10	.	.	PUNCT
ejpam-5390	323	1	now	now	ADV
ejpam-5390	323	2	,	,	PUNCT
ejpam-5390	323	3	let	let	VERB
ejpam-5390	323	4	m	m	PRON
ejpam-5390	323	5	=	=	VERB
ejpam-5390	323	6	p1π(p1)p2	p1π(p1)p2	NOUN
ejpam-5390	323	7	−	−	NOUN
ejpam-5390	323	8	p2π(p1)p1	p2π(p1)p1	NOUN
ejpam-5390	323	9	.	.	PUNCT
ejpam-5390	324	1	then	then	ADV
ejpam-5390	324	2	m	m	VERB
ejpam-5390	324	3	=	=	PUNCT
ejpam-5390	324	4	−m∗.	−m∗.	NOUN
ejpam-5390	324	5	we	we	PRON
ejpam-5390	324	6	define	define	VERB
ejpam-5390	324	7	a	a	DET
ejpam-5390	324	8	mapping	mapping	NOUN
ejpam-5390	324	9	∆	∆	PROPN
ejpam-5390	324	10	:	:	PUNCT
ejpam-5390	324	11	a	a	DET
ejpam-5390	324	12	→	→	SYM
ejpam-5390	324	13	a	a	DET
ejpam-5390	324	14	as	as	ADP
ejpam-5390	324	15	∆(u	∆(u	NOUN
ejpam-5390	324	16	)	)	PUNCT
ejpam-5390	324	17	=	=	SYM
ejpam-5390	324	18	π(u)−(um−mu	π(u)−(um−mu	NOUN
ejpam-5390	324	19	)	)	PUNCT
ejpam-5390	324	20	for	for	ADP
ejpam-5390	324	21	all	all	DET
ejpam-5390	324	22	u	u	PROPN
ejpam-5390	324	23	∈	∈	NOUN
ejpam-5390	324	24	a.	a.	NOUN
ejpam-5390	324	25	it	it	PRON
ejpam-5390	324	26	can	can	AUX
ejpam-5390	324	27	be	be	AUX
ejpam-5390	324	28	easily	easily	ADV
ejpam-5390	324	29	verified	verify	VERB
ejpam-5390	324	30	that	that	SCONJ
ejpam-5390	324	31	for	for	ADP
ejpam-5390	324	32	all	all	DET
ejpam-5390	324	33	u	u	NOUN
ejpam-5390	324	34	,	,	PUNCT
ejpam-5390	324	35	v	v	NOUN
ejpam-5390	324	36	,	,	PUNCT
ejpam-5390	324	37	w	w	PROPN
ejpam-5390	324	38	∈	∈	PROPN
ejpam-5390	324	39	a	a	PRON
ejpam-5390	324	40	,	,	PUNCT
ejpam-5390	324	41	∆([u	∆([u	VERB
ejpam-5390	324	42	,	,	PUNCT
ejpam-5390	324	43	v	v	X
ejpam-5390	324	44	]	]	PUNCT
ejpam-5390	324	45	∗	∗	PROPN
ejpam-5390	324	46	♢	♢	PROPN
ejpam-5390	324	47	λw	λw	NOUN
ejpam-5390	324	48	)	)	PUNCT
ejpam-5390	324	49	=	=	PUNCT
ejpam-5390	325	1	[	[	X
ejpam-5390	325	2	∆(u	∆(u	NOUN
ejpam-5390	325	3	)	)	PUNCT
ejpam-5390	325	4	,	,	PUNCT
ejpam-5390	325	5	v	v	X
ejpam-5390	325	6	]	]	PUNCT
ejpam-5390	325	7	∗	∗	PROPN
ejpam-5390	325	8	♢	♢	PROPN
ejpam-5390	325	9	λw	λw	NOUN
ejpam-5390	325	10	)	)	PUNCT
ejpam-5390	325	11	+	+	CCONJ
ejpam-5390	326	1	[	[	X
ejpam-5390	326	2	u,∆(v	u,∆(v	X
ejpam-5390	326	3	)	)	PUNCT
ejpam-5390	326	4	]	]	PUNCT
ejpam-5390	326	5	∗	∗	PROPN
ejpam-5390	326	6	♢	♢	PROPN
ejpam-5390	326	7	λw	λw	ADP
ejpam-5390	326	8	+	+	PROPN
ejpam-5390	326	9	[	[	X
ejpam-5390	326	10	u	u	NOUN
ejpam-5390	326	11	,	,	PUNCT
ejpam-5390	326	12	v	v	NOUN
ejpam-5390	326	13	]	]	PUNCT
ejpam-5390	326	14	∗	∗	PROPN
ejpam-5390	326	15	♢	♢	PROPN
ejpam-5390	326	16	λ∆(w	λ∆(w	X
ejpam-5390	326	17	)	)	PUNCT
ejpam-5390	326	18	.	.	PUNCT
ejpam-5390	327	1	remark	remark	PROPN
ejpam-5390	327	2	2.1	2.1	NUM
ejpam-5390	327	3	.	.	PUNCT
ejpam-5390	328	1	the	the	DET
ejpam-5390	328	2	mapping	mapping	NOUN
ejpam-5390	328	3	∆	∆	PROPN
ejpam-5390	328	4	possesses	possess	VERB
ejpam-5390	328	5	the	the	DET
ejpam-5390	328	6	following	follow	VERB
ejpam-5390	328	7	properties	property	NOUN
ejpam-5390	328	8	:	:	PUNCT
ejpam-5390	328	9	(	(	PUNCT
ejpam-5390	328	10	i	i	NOUN
ejpam-5390	328	11	)	)	PUNCT
ejpam-5390	328	12	∆	∆	PROPN
ejpam-5390	328	13	is	be	AUX
ejpam-5390	328	14	additive	additive	ADJ
ejpam-5390	328	15	.	.	PUNCT
ejpam-5390	329	1	(	(	PUNCT
ejpam-5390	329	2	ii	ii	NOUN
ejpam-5390	329	3	)	)	PUNCT
ejpam-5390	329	4	∆(p1	∆(p1	NUM
ejpam-5390	329	5	)	)	PUNCT
ejpam-5390	329	6	=	=	SYM
ejpam-5390	329	7	∆(p2	∆(p2	X
ejpam-5390	329	8	)	)	PUNCT
ejpam-5390	329	9	=	=	SYM
ejpam-5390	329	10	0	0	X
ejpam-5390	329	11	.	.	PUNCT
ejpam-5390	330	1	(	(	PUNCT
ejpam-5390	330	2	iii	iii	X
ejpam-5390	330	3	)	)	PUNCT
ejpam-5390	330	4	∆(i	∆(i	NOUN
ejpam-5390	330	5	)	)	PUNCT
ejpam-5390	331	1	=	=	SYM
ejpam-5390	331	2	0	0	NUM
ejpam-5390	331	3	(	(	PUNCT
ejpam-5390	331	4	iv	iv	X
ejpam-5390	331	5	)	)	PUNCT
ejpam-5390	331	6	for	for	ADP
ejpam-5390	331	7	every	every	DET
ejpam-5390	331	8	uij	uij	PROPN
ejpam-5390	331	9	∈	∈	NOUN
ejpam-5390	331	10	aij(1	aij(1	NOUN
ejpam-5390	331	11	≤	≤	PUNCT
ejpam-5390	332	1	i	i	PRON
ejpam-5390	332	2	̸=	̸=	PROPN
ejpam-5390	332	3	j	j	PROPN
ejpam-5390	332	4	≤	≤	ADV
ejpam-5390	332	5	2	2	NUM
ejpam-5390	332	6	)	)	PUNCT
ejpam-5390	332	7	,	,	PUNCT
ejpam-5390	332	8	we	we	PRON
ejpam-5390	332	9	have	have	VERB
ejpam-5390	332	10	pj∆(λuij)pi	pj∆(λuij)pi	PROPN
ejpam-5390	332	11	=	=	SYM
ejpam-5390	332	12	0	0	PUNCT
ejpam-5390	332	13	.	.	PUNCT
ejpam-5390	333	1	nisar	nisar	PROPN
ejpam-5390	333	2	et	et	PROPN
ejpam-5390	333	3	al	al	PROPN
ejpam-5390	333	4	.	.	PUNCT
ejpam-5390	333	5	/	/	SYM
ejpam-5390	333	6	eur	eur	PROPN
ejpam-5390	333	7	.	.	PUNCT
ejpam-5390	334	1	j.	j.	PROPN
ejpam-5390	334	2	pure	pure	PROPN
ejpam-5390	334	3	appl	appl	PROPN
ejpam-5390	334	4	.	.	PROPN
ejpam-5390	334	5	math	math	PROPN
ejpam-5390	334	6	,	,	PUNCT
ejpam-5390	334	7	17	17	NUM
ejpam-5390	334	8	(	(	PUNCT
ejpam-5390	334	9	4	4	NUM
ejpam-5390	334	10	)	)	PUNCT
ejpam-5390	334	11	(	(	PUNCT
ejpam-5390	334	12	2024	2024	NUM
ejpam-5390	334	13	)	)	PUNCT
ejpam-5390	334	14	,	,	PUNCT
ejpam-5390	334	15	3399	3399	NUM
ejpam-5390	334	16	-	-	SYM
ejpam-5390	334	17	3414	3414	NUM
ejpam-5390	334	18	3408	3408	NUM
ejpam-5390	334	19	(	(	PUNCT
ejpam-5390	334	20	v	v	NOUN
ejpam-5390	334	21	)	)	PUNCT
ejpam-5390	334	22	∆	∆	PROPN
ejpam-5390	334	23	is	be	AUX
ejpam-5390	334	24	a	a	DET
ejpam-5390	334	25	∗-derivation	∗-derivation	NOUN
ejpam-5390	334	26	if	if	SCONJ
ejpam-5390	335	1	and	and	CCONJ
ejpam-5390	335	2	only	only	ADV
ejpam-5390	335	3	if	if	SCONJ
ejpam-5390	335	4	π	π	PROPN
ejpam-5390	335	5	is	be	AUX
ejpam-5390	335	6	an	an	DET
ejpam-5390	335	7	∗	∗	NOUN
ejpam-5390	335	8	-derivation	-derivation	NOUN
ejpam-5390	335	9	.	.	PUNCT
ejpam-5390	336	1	proof	proof	NOUN
ejpam-5390	336	2	.	.	PUNCT
ejpam-5390	337	1	(	(	PUNCT
ejpam-5390	337	2	i	i	NOUN
ejpam-5390	337	3	)	)	PUNCT
ejpam-5390	337	4	since	since	SCONJ
ejpam-5390	337	5	[	[	X
ejpam-5390	337	6	u	u	NOUN
ejpam-5390	337	7	,	,	PUNCT
ejpam-5390	337	8	m	m	VERB
ejpam-5390	337	9	]	]	PUNCT
ejpam-5390	337	10	is	be	AUX
ejpam-5390	337	11	additive	additive	ADJ
ejpam-5390	337	12	and	and	CCONJ
ejpam-5390	337	13	also	also	ADV
ejpam-5390	337	14	using	use	VERB
ejpam-5390	337	15	lemma	lemma	PROPN
ejpam-5390	337	16	2.6	2.6	NUM
ejpam-5390	337	17	,	,	PUNCT
ejpam-5390	337	18	it	it	PRON
ejpam-5390	337	19	is	be	AUX
ejpam-5390	337	20	clear	clear	ADJ
ejpam-5390	337	21	that	that	SCONJ
ejpam-5390	337	22	∆	∆	PROPN
ejpam-5390	337	23	is	be	AUX
ejpam-5390	337	24	additve	additve	ADJ
ejpam-5390	337	25	.	.	PUNCT
ejpam-5390	338	1	(	(	PUNCT
ejpam-5390	338	2	ii	ii	NOUN
ejpam-5390	338	3	)	)	PUNCT
ejpam-5390	338	4	by	by	ADP
ejpam-5390	338	5	using	use	VERB
ejpam-5390	338	6	lemma	lemma	PROPN
ejpam-5390	338	7	2.9	2.9	NUM
ejpam-5390	338	8	,	,	PUNCT
ejpam-5390	338	9	we	we	PRON
ejpam-5390	338	10	have	have	VERB
ejpam-5390	338	11	∆(p1	∆(p1	VERB
ejpam-5390	338	12	)	)	PUNCT
ejpam-5390	339	1	=	=	SYM
ejpam-5390	340	1	π(p1)−	π(p1)−	PROPN
ejpam-5390	340	2	p1π(p1)p2	p1π(p1)p2	NOUN
ejpam-5390	340	3	−	−	NOUN
ejpam-5390	340	4	p2π(p1)p1	p2π(p1)p1	NOUN
ejpam-5390	340	5	=	=	SYM
ejpam-5390	340	6	p1π(p1)p2	p1π(p1)p2	NOUN
ejpam-5390	340	7	+	+	NOUN
ejpam-5390	340	8	p2π(p1)p1	p2π(p1)p1	NOUN
ejpam-5390	340	9	−	−	PROPN
ejpam-5390	340	10	p1π(p1)p2	p1π(p1)p2	NOUN
ejpam-5390	340	11	−	−	NOUN
ejpam-5390	340	12	p2π(p1)p1	p2π(p1)p1	NOUN
ejpam-5390	340	13	=	=	NOUN
ejpam-5390	340	14	0	0	X
ejpam-5390	340	15	.	.	PUNCT
ejpam-5390	341	1	similarly	similarly	ADV
ejpam-5390	341	2	,	,	PUNCT
ejpam-5390	341	3	we	we	PRON
ejpam-5390	341	4	can	can	AUX
ejpam-5390	341	5	show	show	VERB
ejpam-5390	341	6	that	that	SCONJ
ejpam-5390	341	7	∆(p2	∆(p2	VERB
ejpam-5390	341	8	)	)	PUNCT
ejpam-5390	341	9	=	=	SYM
ejpam-5390	341	10	0	0	X
ejpam-5390	341	11	.	.	PUNCT
ejpam-5390	342	1	(	(	PUNCT
ejpam-5390	342	2	iii	iii	NOUN
ejpam-5390	342	3	)	)	PUNCT
ejpam-5390	342	4	by	by	ADP
ejpam-5390	342	5	using	use	VERB
ejpam-5390	342	6	additivity	additivity	NOUN
ejpam-5390	342	7	of	of	ADP
ejpam-5390	342	8	∆	∆	PROPN
ejpam-5390	342	9	,	,	PUNCT
ejpam-5390	342	10	we	we	PRON
ejpam-5390	342	11	have	have	VERB
ejpam-5390	342	12	∆(i	∆(i	X
ejpam-5390	342	13	)	)	PUNCT
ejpam-5390	342	14	=	=	SYM
ejpam-5390	342	15	∆(p1	∆(p1	NOUN
ejpam-5390	342	16	+	+	CCONJ
ejpam-5390	342	17	p2	p2	NOUN
ejpam-5390	342	18	)	)	PUNCT
ejpam-5390	342	19	=	=	SYM
ejpam-5390	342	20	∆(p1	∆(p1	VERB
ejpam-5390	342	21	)	)	PUNCT
ejpam-5390	343	1	+	+	NUM
ejpam-5390	343	2	∆(p2	∆(p2	NOUN
ejpam-5390	343	3	)	)	PUNCT
ejpam-5390	343	4	=	=	SYM
ejpam-5390	343	5	0	0	X
ejpam-5390	343	6	.	.	PUNCT
ejpam-5390	343	7	(	(	PUNCT
ejpam-5390	343	8	iv	iv	X
ejpam-5390	343	9	)	)	PUNCT
ejpam-5390	343	10	for	for	ADP
ejpam-5390	343	11	i	i	PROPN
ejpam-5390	343	12	=	=	SYM
ejpam-5390	343	13	1	1	NUM
ejpam-5390	343	14	and	and	CCONJ
ejpam-5390	343	15	j	j	NOUN
ejpam-5390	343	16	=	=	SYM
ejpam-5390	343	17	2	2	NUM
ejpam-5390	343	18	,	,	PUNCT
ejpam-5390	343	19	it	it	PRON
ejpam-5390	343	20	can	can	AUX
ejpam-5390	343	21	be	be	AUX
ejpam-5390	343	22	inferred	infer	VERB
ejpam-5390	343	23	from	from	ADP
ejpam-5390	343	24	lemma	lemma	PROPN
ejpam-5390	343	25	2.8	2.8	NUM
ejpam-5390	343	26	that	that	PRON
ejpam-5390	343	27	p2∆(λu12)p1	p2∆(λu12)p1	VERB
ejpam-5390	343	28	=	=	PRON
ejpam-5390	343	29	p2(π(λu12)−	p2(π(λu12)−	NOUN
ejpam-5390	343	30	λu12	λu12	PROPN
ejpam-5390	343	31	m	m	VERB
ejpam-5390	343	32	+	+	NOUN
ejpam-5390	343	33	mλu12)p1	mλu12)p1	NOUN
ejpam-5390	343	34	=	=	NOUN
ejpam-5390	343	35	0	0	X
ejpam-5390	343	36	.	.	PUNCT
ejpam-5390	344	1	in	in	ADP
ejpam-5390	344	2	the	the	DET
ejpam-5390	344	3	same	same	ADJ
ejpam-5390	344	4	way	way	NOUN
ejpam-5390	344	5	,	,	PUNCT
ejpam-5390	344	6	one	one	PRON
ejpam-5390	344	7	can	can	AUX
ejpam-5390	344	8	show	show	VERB
ejpam-5390	344	9	for	for	ADP
ejpam-5390	344	10	i	i	PRON
ejpam-5390	344	11	=	=	SYM
ejpam-5390	344	12	2	2	NUM
ejpam-5390	344	13	,	,	PUNCT
ejpam-5390	344	14	j	j	PROPN
ejpam-5390	344	15	=	=	SYM
ejpam-5390	344	16	1	1	NUM
ejpam-5390	344	17	,	,	PUNCT
ejpam-5390	344	18	i.e.	i.e.	X
ejpam-5390	344	19	,	,	PUNCT
ejpam-5390	344	20	p1∆(λu21)p2	p1∆(λu21)p2	X
ejpam-5390	344	21	=	=	NOUN
ejpam-5390	344	22	0	0	X
ejpam-5390	344	23	.	.	PUNCT
ejpam-5390	345	1	(	(	PUNCT
ejpam-5390	345	2	v	v	NOUN
ejpam-5390	345	3	)	)	PUNCT
ejpam-5390	345	4	since	since	SCONJ
ejpam-5390	345	5	[	[	X
ejpam-5390	345	6	u	u	NOUN
ejpam-5390	345	7	,	,	PUNCT
ejpam-5390	345	8	m	m	VERB
ejpam-5390	345	9	]	]	PUNCT
ejpam-5390	346	1	=	=	PUNCT
ejpam-5390	346	2	um	um	INTJ
ejpam-5390	346	3	−	−	PUNCT
ejpam-5390	346	4	mu	mu	PROPN
ejpam-5390	346	5	is	be	AUX
ejpam-5390	346	6	an	an	DET
ejpam-5390	346	7	additive	additive	ADJ
ejpam-5390	346	8	∗-derivation	∗-derivation	NOUN
ejpam-5390	346	9	.	.	PUNCT
ejpam-5390	347	1	therefore	therefore	ADV
ejpam-5390	347	2	,	,	PUNCT
ejpam-5390	347	3	∆	∆	PROPN
ejpam-5390	347	4	qualifies	qualify	VERB
ejpam-5390	347	5	as	as	ADP
ejpam-5390	347	6	a	a	DET
ejpam-5390	347	7	∗-derivation	∗-derivation	NOUN
ejpam-5390	347	8	if	if	SCONJ
ejpam-5390	347	9	and	and	CCONJ
ejpam-5390	347	10	only	only	ADV
ejpam-5390	347	11	if	if	SCONJ
ejpam-5390	347	12	π	π	PROPN
ejpam-5390	347	13	is	be	AUX
ejpam-5390	347	14	a	a	DET
ejpam-5390	347	15	∗-derivation	∗-derivation	NOUN
ejpam-5390	347	16	.	.	PUNCT
ejpam-5390	348	1	lemma	lemma	PROPN
ejpam-5390	348	2	2.10	2.10	NUM
ejpam-5390	348	3	.	.	PUNCT
ejpam-5390	349	1	∆(uij	∆(uij	PROPN
ejpam-5390	349	2	)	)	PUNCT
ejpam-5390	350	1	⊆	⊆	NUM
ejpam-5390	350	2	uij	uij	NOUN
ejpam-5390	350	3	,	,	PUNCT
ejpam-5390	350	4	i	i	PRON
ejpam-5390	350	5	,	,	PUNCT
ejpam-5390	350	6	j	j	PROPN
ejpam-5390	350	7	=	=	SYM
ejpam-5390	350	8	1	1	NUM
ejpam-5390	350	9	,	,	PUNCT
ejpam-5390	350	10	2	2	NUM
ejpam-5390	350	11	.	.	PUNCT
ejpam-5390	350	12	proof	proof	NOUN
ejpam-5390	350	13	.	.	PUNCT
ejpam-5390	351	1	first	first	ADV
ejpam-5390	351	2	,	,	PUNCT
ejpam-5390	351	3	we	we	PRON
ejpam-5390	351	4	prove	prove	VERB
ejpam-5390	351	5	for	for	ADP
ejpam-5390	351	6	i	i	PRON
ejpam-5390	351	7	=	=	SYM
ejpam-5390	351	8	1	1	NUM
ejpam-5390	351	9	,	,	PUNCT
ejpam-5390	351	10	j	j	NOUN
ejpam-5390	351	11	=	=	SYM
ejpam-5390	351	12	1	1	X
ejpam-5390	351	13	.	.	X
ejpam-5390	352	1	for	for	ADP
ejpam-5390	352	2	every	every	DET
ejpam-5390	352	3	u11	u11	PROPN
ejpam-5390	352	4	∈	∈	PROPN
ejpam-5390	352	5	a11	a11	PROPN
ejpam-5390	352	6	,	,	PUNCT
ejpam-5390	352	7	it	it	PRON
ejpam-5390	352	8	follows	follow	VERB
ejpam-5390	352	9	from	from	ADP
ejpam-5390	352	10	remark	remark	NOUN
ejpam-5390	352	11	2.1	2.1	NUM
ejpam-5390	352	12	that	that	PRON
ejpam-5390	352	13	0	0	X
ejpam-5390	352	14	=	=	SYM
ejpam-5390	352	15	∆([p1	∆([p1	PROPN
ejpam-5390	352	16	,	,	PUNCT
ejpam-5390	352	17	u11]∗	u11]∗	PROPN
ejpam-5390	352	18	♢	♢	PROPN
ejpam-5390	352	19	λp1	λp1	PROPN
ejpam-5390	352	20	)	)	PUNCT
ejpam-5390	352	21	=	=	PUNCT
ejpam-5390	353	1	[	[	X
ejpam-5390	353	2	p1,∆(u11)]∗	p1,∆(u11)]∗	X
ejpam-5390	353	3	♢	♢	NOUN
ejpam-5390	353	4	λp1	λp1	NOUN
ejpam-5390	353	5	=	=	SYM
ejpam-5390	353	6	p1∆(u11)p1	p1∆(u11)p1	PROPN
ejpam-5390	353	7	−∆(u11)p1	−∆(u11)p1	PROPN
ejpam-5390	353	8	+	+	CCONJ
ejpam-5390	353	9	λp1∆(u11)−	λp1∆(u11)−	PROPN
ejpam-5390	353	10	λp1∆(u11)p1	λp1∆(u11)p1	NOUN
ejpam-5390	353	11	.	.	PUNCT
ejpam-5390	354	1	left	leave	VERB
ejpam-5390	354	2	multiplying	multiply	VERB
ejpam-5390	354	3	the	the	DET
ejpam-5390	354	4	above	above	ADJ
ejpam-5390	354	5	equation	equation	NOUN
ejpam-5390	354	6	by	by	ADP
ejpam-5390	354	7	p2	p2	PROPN
ejpam-5390	354	8	,	,	PUNCT
ejpam-5390	354	9	we	we	PRON
ejpam-5390	354	10	find	find	VERB
ejpam-5390	354	11	p2∆(u11)p1	p2∆(u11)p1	NOUN
ejpam-5390	354	12	=	=	NOUN
ejpam-5390	354	13	0	0	X
ejpam-5390	354	14	.	.	PUNCT
ejpam-5390	355	1	(	(	PUNCT
ejpam-5390	355	2	2.7	2.7	NUM
ejpam-5390	355	3	)	)	PUNCT
ejpam-5390	355	4	similarly	similarly	ADV
ejpam-5390	355	5	,	,	PUNCT
ejpam-5390	355	6	again	again	ADV
ejpam-5390	355	7	by	by	ADP
ejpam-5390	355	8	using	use	VERB
ejpam-5390	355	9	remark	remark	NOUN
ejpam-5390	355	10	2.1	2.1	NUM
ejpam-5390	355	11	,	,	PUNCT
ejpam-5390	355	12	we	we	PRON
ejpam-5390	355	13	have	have	VERB
ejpam-5390	355	14	0	0	NUM
ejpam-5390	355	15	=	=	SYM
ejpam-5390	355	16	∆([p2	∆([p2	PROPN
ejpam-5390	355	17	,	,	PUNCT
ejpam-5390	355	18	u11]∗	u11]∗	PROPN
ejpam-5390	355	19	♢	♢	PROPN
ejpam-5390	355	20	λp1	λp1	PROPN
ejpam-5390	355	21	)	)	PUNCT
ejpam-5390	355	22	=	=	PUNCT
ejpam-5390	356	1	[	[	X
ejpam-5390	356	2	p2,∆(u11)]∗	p2,∆(u11)]∗	X
ejpam-5390	356	3	♢	♢	PROPN
ejpam-5390	356	4	λp1	λp1	NOUN
ejpam-5390	356	5	=	=	SYM
ejpam-5390	356	6	p2∆(u11)p1	p2∆(u11)p1	NOUN
ejpam-5390	356	7	−	−	PROPN
ejpam-5390	356	8	λp1∆(u11)p2	λp1∆(u11)p2	PROPN
ejpam-5390	356	9	.	.	PUNCT
ejpam-5390	357	1	multiplying	multiply	VERB
ejpam-5390	357	2	p2	p2	PROPN
ejpam-5390	357	3	on	on	ADP
ejpam-5390	357	4	the	the	DET
ejpam-5390	357	5	right	right	NOUN
ejpam-5390	357	6	and	and	CCONJ
ejpam-5390	357	7	since	since	SCONJ
ejpam-5390	357	8	λ	λ	PROPN
ejpam-5390	357	9	̸=	̸=	PROPN
ejpam-5390	357	10	0	0	NUM
ejpam-5390	357	11	,	,	PUNCT
ejpam-5390	357	12	we	we	PRON
ejpam-5390	357	13	get	get	VERB
ejpam-5390	357	14	p1∆(u11)p2	p1∆(u11)p2	NOUN
ejpam-5390	357	15	=	=	SYM
ejpam-5390	357	16	0	0	NUM
ejpam-5390	357	17	(	(	PUNCT
ejpam-5390	357	18	2.8	2.8	NUM
ejpam-5390	357	19	)	)	PUNCT
ejpam-5390	357	20	.	.	PUNCT
ejpam-5390	358	1	nisar	nisar	PROPN
ejpam-5390	358	2	et	et	PROPN
ejpam-5390	358	3	al	al	PROPN
ejpam-5390	358	4	.	.	PUNCT
ejpam-5390	358	5	/	/	SYM
ejpam-5390	358	6	eur	eur	PROPN
ejpam-5390	358	7	.	.	PUNCT
ejpam-5390	359	1	j.	j.	PROPN
ejpam-5390	359	2	pure	pure	PROPN
ejpam-5390	359	3	appl	appl	PROPN
ejpam-5390	359	4	.	.	PROPN
ejpam-5390	359	5	math	math	PROPN
ejpam-5390	359	6	,	,	PUNCT
ejpam-5390	359	7	17	17	NUM
ejpam-5390	359	8	(	(	PUNCT
ejpam-5390	359	9	4	4	NUM
ejpam-5390	359	10	)	)	PUNCT
ejpam-5390	359	11	(	(	PUNCT
ejpam-5390	359	12	2024	2024	NUM
ejpam-5390	359	13	)	)	PUNCT
ejpam-5390	359	14	,	,	PUNCT
ejpam-5390	359	15	3399	3399	NUM
ejpam-5390	359	16	-	-	SYM
ejpam-5390	359	17	3414	3414	NUM
ejpam-5390	359	18	3409	3409	NUM
ejpam-5390	359	19	now	now	ADV
ejpam-5390	359	20	,	,	PUNCT
ejpam-5390	359	21	for	for	ADP
ejpam-5390	359	22	every	every	DET
ejpam-5390	359	23	x12	x12	NUM
ejpam-5390	359	24	∈	∈	PROPN
ejpam-5390	359	25	a12	a12	NOUN
ejpam-5390	359	26	and	and	CCONJ
ejpam-5390	359	27	∆(p2	∆(p2	NUM
ejpam-5390	359	28	)	)	PUNCT
ejpam-5390	359	29	=	=	SYM
ejpam-5390	359	30	0	0	NUM
ejpam-5390	360	1	,	,	PUNCT
ejpam-5390	360	2	it	it	PRON
ejpam-5390	360	3	follows	follow	VERB
ejpam-5390	360	4	that	that	SCONJ
ejpam-5390	360	5	0	0	NUM
ejpam-5390	360	6	=	=	SYM
ejpam-5390	360	7	∆([x12	∆([x12	PROPN
ejpam-5390	360	8	,	,	PUNCT
ejpam-5390	360	9	u11]∗	u11]∗	NOUN
ejpam-5390	360	10	♢	♢	NOUN
ejpam-5390	360	11	λp2	λp2	NOUN
ejpam-5390	360	12	)	)	PUNCT
ejpam-5390	360	13	=	=	PUNCT
ejpam-5390	361	1	[	[	X
ejpam-5390	361	2	∆(x12	∆(x12	ADV
ejpam-5390	361	3	)	)	PUNCT
ejpam-5390	361	4	,	,	PUNCT
ejpam-5390	361	5	u11]∗	u11]∗	PROPN
ejpam-5390	361	6	♢	♢	VERB
ejpam-5390	361	7	λp2	λp2	NOUN
ejpam-5390	361	8	+	+	CCONJ
ejpam-5390	362	1	[	[	X
ejpam-5390	362	2	x12,∆(u11)]∗	x12,∆(u11)]∗	X
ejpam-5390	362	3	♢	♢	PROPN
ejpam-5390	362	4	λp2	λp2	NOUN
ejpam-5390	362	5	=	=	SYM
ejpam-5390	362	6	−u11∆(x12	−u11∆(x12	NUM
ejpam-5390	362	7	)	)	PUNCT
ejpam-5390	362	8	∗p2	∗p2	PROPN
ejpam-5390	362	9	+	+	PROPN
ejpam-5390	362	10	λp2∆(x12)u11	λp2∆(x12)u11	PROPN
ejpam-5390	362	11	+	+	PROPN
ejpam-5390	362	12	x12∆(u11)p2	x12∆(u11)p2	PROPN
ejpam-5390	362	13	−λp2∆(u11)x	−λp2∆(u11)x	PROPN
ejpam-5390	362	14	∗	∗	NOUN
ejpam-5390	362	15	12	12	NUM
ejpam-5390	362	16	.	.	PUNCT
ejpam-5390	363	1	multiplying	multiply	VERB
ejpam-5390	363	2	p1	p1	PROPN
ejpam-5390	363	3	from	from	ADP
ejpam-5390	363	4	left	left	ADJ
ejpam-5390	363	5	and	and	CCONJ
ejpam-5390	363	6	p2	p2	NOUN
ejpam-5390	363	7	from	from	ADP
ejpam-5390	363	8	right	right	ADV
ejpam-5390	363	9	and	and	CCONJ
ejpam-5390	363	10	using	use	VERB
ejpam-5390	363	11	lemma	lemma	PROPN
ejpam-5390	363	12	2.8	2.8	NUM
ejpam-5390	363	13	,	,	PUNCT
ejpam-5390	363	14	we	we	PRON
ejpam-5390	363	15	get	get	VERB
ejpam-5390	363	16	x12∆(u11)p2	x12∆(u11)p2	PROPN
ejpam-5390	363	17	=	=	SYM
ejpam-5390	363	18	u11∆(x12	u11∆(x12	NOUN
ejpam-5390	363	19	)	)	PUNCT
ejpam-5390	363	20	∗p2	∗p2	NOUN
ejpam-5390	363	21	=	=	SYM
ejpam-5390	363	22	u11(p2∆(x12)p1	u11(p2∆(x12)p1	PROPN
ejpam-5390	363	23	)	)	PUNCT
ejpam-5390	363	24	∗	∗	NOUN
ejpam-5390	363	25	=	=	SYM
ejpam-5390	363	26	0	0	NUM
ejpam-5390	363	27	.	.	PUNCT
ejpam-5390	364	1	thus	thus	ADV
ejpam-5390	364	2	,	,	PUNCT
ejpam-5390	364	3	x12∆(u11)p2	x12∆(u11)p2	PROPN
ejpam-5390	364	4	=	=	SYM
ejpam-5390	364	5	0	0	X
ejpam-5390	364	6	.	.	PUNCT
ejpam-5390	364	7	by	by	ADP
ejpam-5390	364	8	using	use	VERB
ejpam-5390	364	9	(	(	PUNCT
ejpam-5390	364	10	▲	▲	PUNCT
ejpam-5390	364	11	)	)	PUNCT
ejpam-5390	364	12	and	and	CCONJ
ejpam-5390	364	13	(	(	PUNCT
ejpam-5390	364	14	▼	▼	NOUN
ejpam-5390	364	15	)	)	PUNCT
ejpam-5390	364	16	,	,	PUNCT
ejpam-5390	364	17	we	we	PRON
ejpam-5390	364	18	have	have	VERB
ejpam-5390	364	19	p2∆(u11)p2	p2∆(u11)p2	NOUN
ejpam-5390	364	20	=	=	SYM
ejpam-5390	364	21	0	0	NUM
ejpam-5390	364	22	.	.	PUNCT
ejpam-5390	365	1	(	(	PUNCT
ejpam-5390	365	2	2.9	2.9	NUM
ejpam-5390	365	3	)	)	PUNCT
ejpam-5390	365	4	from	from	ADP
ejpam-5390	365	5	equations	equation	NOUN
ejpam-5390	365	6	(	(	PUNCT
ejpam-5390	365	7	2.7)-(2.9	2.7)-(2.9	NUM
ejpam-5390	365	8	)	)	PUNCT
ejpam-5390	365	9	,	,	PUNCT
ejpam-5390	365	10	we	we	PRON
ejpam-5390	365	11	have	have	AUX
ejpam-5390	365	12	∆(u11	∆(u11	VERB
ejpam-5390	365	13	)	)	PUNCT
ejpam-5390	366	1	⊆	⊆	NUM
ejpam-5390	366	2	u11	u11	PROPN
ejpam-5390	366	3	.	.	PUNCT
ejpam-5390	367	1	similarly	similarly	ADV
ejpam-5390	367	2	,	,	PUNCT
ejpam-5390	367	3	we	we	PRON
ejpam-5390	367	4	can	can	AUX
ejpam-5390	367	5	show	show	VERB
ejpam-5390	367	6	that	that	SCONJ
ejpam-5390	367	7	∆(u22	∆(u22	VERB
ejpam-5390	367	8	)	)	PUNCT
ejpam-5390	367	9	⊆	⊆	NUM
ejpam-5390	367	10	u22	u22	NOUN
ejpam-5390	367	11	.	.	PUNCT
ejpam-5390	368	1	next	next	ADV
ejpam-5390	368	2	,	,	PUNCT
ejpam-5390	368	3	we	we	PRON
ejpam-5390	368	4	establish	establish	VERB
ejpam-5390	368	5	the	the	DET
ejpam-5390	368	6	result	result	NOUN
ejpam-5390	368	7	for	for	ADP
ejpam-5390	368	8	i	i	PROPN
ejpam-5390	368	9	=	=	SYM
ejpam-5390	368	10	1	1	NUM
ejpam-5390	368	11	,	,	PUNCT
ejpam-5390	368	12	j	j	PROPN
ejpam-5390	368	13	=	=	SYM
ejpam-5390	368	14	2	2	X
ejpam-5390	368	15	.	.	PUNCT
ejpam-5390	368	16	additionally	additionally	ADV
ejpam-5390	368	17	,	,	PUNCT
ejpam-5390	368	18	for	for	ADP
ejpam-5390	368	19	any	any	DET
ejpam-5390	368	20	u12	u12	PROPN
ejpam-5390	368	21	∈	∈	PROPN
ejpam-5390	368	22	a12	a12	NOUN
ejpam-5390	368	23	and	and	CCONJ
ejpam-5390	368	24	∆(p2	∆(p2	NUM
ejpam-5390	368	25	)	)	PUNCT
ejpam-5390	369	1	=	=	SYM
ejpam-5390	369	2	0	0	NUM
ejpam-5390	369	3	,	,	PUNCT
ejpam-5390	369	4	we	we	PRON
ejpam-5390	369	5	have	have	AUX
ejpam-5390	369	6	∆(u12	∆(u12	NOUN
ejpam-5390	369	7	)	)	PUNCT
ejpam-5390	370	1	=	=	SYM
ejpam-5390	370	2	∆([p1	∆([p1	PROPN
ejpam-5390	370	3	,	,	PUNCT
ejpam-5390	370	4	u12]∗	u12]∗	NOUN
ejpam-5390	370	5	♢	♢	NOUN
ejpam-5390	370	6	λp2	λp2	PROPN
ejpam-5390	370	7	)	)	PUNCT
ejpam-5390	370	8	=	=	PUNCT
ejpam-5390	371	1	[	[	X
ejpam-5390	371	2	p1,∆(u12)]∗	p1,∆(u12)]∗	X
ejpam-5390	371	3	♢	♢	PROPN
ejpam-5390	371	4	λp2	λp2	NOUN
ejpam-5390	371	5	=	=	SYM
ejpam-5390	371	6	p1∆(u12)p2	p1∆(u12)p2	NOUN
ejpam-5390	371	7	−	−	NOUN
ejpam-5390	371	8	λp2∆(u12)p1	λp2∆(u12)p1	NOUN
ejpam-5390	371	9	.	.	PUNCT
ejpam-5390	372	1	by	by	ADP
ejpam-5390	372	2	left	left	ADJ
ejpam-5390	372	3	and	and	CCONJ
ejpam-5390	372	4	right	right	ADJ
ejpam-5390	372	5	multiplying	multiply	VERB
ejpam-5390	372	6	the	the	DET
ejpam-5390	372	7	above	above	ADJ
ejpam-5390	372	8	equation	equation	NOUN
ejpam-5390	372	9	by	by	ADP
ejpam-5390	372	10	p1	p1	PROPN
ejpam-5390	372	11	,	,	PUNCT
ejpam-5390	372	12	we	we	PRON
ejpam-5390	372	13	obtain	obtain	VERB
ejpam-5390	372	14	p1∆(u12)p1	p1∆(u12)p1	NOUN
ejpam-5390	372	15	=	=	SYM
ejpam-5390	372	16	0	0	NUM
ejpam-5390	372	17	.	.	PUNCT
ejpam-5390	373	1	(	(	PUNCT
ejpam-5390	373	2	2.10	2.10	NUM
ejpam-5390	373	3	)	)	PUNCT
ejpam-5390	373	4	additionally	additionally	ADV
ejpam-5390	373	5	,	,	PUNCT
ejpam-5390	373	6	by	by	ADP
ejpam-5390	373	7	left	left	ADJ
ejpam-5390	373	8	and	and	CCONJ
ejpam-5390	373	9	right	right	ADV
ejpam-5390	373	10	-	-	PUNCT
ejpam-5390	373	11	multiplying	multiplying	NOUN
ejpam-5390	373	12	by	by	ADP
ejpam-5390	373	13	p2	p2	NOUN
ejpam-5390	373	14	,	,	PUNCT
ejpam-5390	373	15	we	we	PRON
ejpam-5390	373	16	find	find	VERB
ejpam-5390	373	17	p2∆(u12)p2	p2∆(u12)p2	NOUN
ejpam-5390	373	18	=	=	SYM
ejpam-5390	373	19	0	0	X
ejpam-5390	373	20	.	.	PUNCT
ejpam-5390	374	1	(	(	PUNCT
ejpam-5390	374	2	2.11	2.11	NUM
ejpam-5390	374	3	)	)	PUNCT
ejpam-5390	374	4	similarly	similarly	ADV
ejpam-5390	374	5	,	,	PUNCT
ejpam-5390	374	6	multiplying	multiply	VERB
ejpam-5390	374	7	p2	p2	NOUN
ejpam-5390	374	8	from	from	ADP
ejpam-5390	374	9	left	left	ADJ
ejpam-5390	374	10	and	and	CCONJ
ejpam-5390	374	11	p1	p1	NOUN
ejpam-5390	374	12	from	from	ADP
ejpam-5390	374	13	right	right	ADV
ejpam-5390	374	14	and	and	CCONJ
ejpam-5390	374	15	since	since	SCONJ
ejpam-5390	374	16	λ	λ	PROPN
ejpam-5390	374	17	̸=	̸=	PROPN
ejpam-5390	374	18	−1	−1	NOUN
ejpam-5390	374	19	,	,	PUNCT
ejpam-5390	374	20	we	we	PRON
ejpam-5390	374	21	get	get	VERB
ejpam-5390	374	22	p2∆(u12)p1	p2∆(u12)p1	NOUN
ejpam-5390	374	23	=	=	SYM
ejpam-5390	374	24	0	0	NUM
ejpam-5390	375	1	(	(	PUNCT
ejpam-5390	375	2	2.12	2.12	NUM
ejpam-5390	375	3	)	)	PUNCT
ejpam-5390	375	4	from	from	ADP
ejpam-5390	375	5	equations	equation	NOUN
ejpam-5390	375	6	(	(	PUNCT
ejpam-5390	375	7	2.10)(2.12	2.10)(2.12	NUM
ejpam-5390	375	8	)	)	PUNCT
ejpam-5390	375	9	,	,	PUNCT
ejpam-5390	375	10	we	we	PRON
ejpam-5390	375	11	get	get	VERB
ejpam-5390	375	12	∆(u12	∆(u12	NOUN
ejpam-5390	375	13	)	)	PUNCT
ejpam-5390	375	14	⊆	⊆	NUM
ejpam-5390	375	15	u12	u12	NOUN
ejpam-5390	375	16	.	.	PUNCT
ejpam-5390	376	1	similarly	similarly	ADV
ejpam-5390	376	2	,	,	PUNCT
ejpam-5390	376	3	by	by	ADP
ejpam-5390	376	4	using	use	VERB
ejpam-5390	376	5	the	the	DET
ejpam-5390	376	6	same	same	ADJ
ejpam-5390	376	7	technique	technique	NOUN
ejpam-5390	376	8	as	as	ADP
ejpam-5390	376	9	above	above	ADV
ejpam-5390	376	10	,	,	PUNCT
ejpam-5390	376	11	one	one	PRON
ejpam-5390	376	12	can	can	AUX
ejpam-5390	376	13	prove	prove	VERB
ejpam-5390	376	14	that	that	SCONJ
ejpam-5390	376	15	∆(u21	∆(u21	NOUN
ejpam-5390	376	16	)	)	PUNCT
ejpam-5390	376	17	⊆	⊆	NUM
ejpam-5390	376	18	u21	u21	NOUN
ejpam-5390	376	19	.	.	PUNCT
ejpam-5390	377	1	lemma	lemma	PROPN
ejpam-5390	377	2	2.11	2.11	NUM
ejpam-5390	377	3	.	.	PUNCT
ejpam-5390	378	1	for	for	ADP
ejpam-5390	378	2	any	any	DET
ejpam-5390	378	3	ui	ui	PROPN
ejpam-5390	378	4	,	,	PUNCT
ejpam-5390	378	5	j	j	PROPN
ejpam-5390	378	6	,	,	PUNCT
ejpam-5390	378	7	vi	vi	PROPN
ejpam-5390	378	8	,	,	PUNCT
ejpam-5390	378	9	j	j	PROPN
ejpam-5390	378	10	∈	∈	PROPN
ejpam-5390	378	11	aij	aij	PROPN
ejpam-5390	378	12	,	,	PUNCT
ejpam-5390	378	13	1	1	NUM
ejpam-5390	378	14	≤	≤	NUM
ejpam-5390	378	15	i	i	PRON
ejpam-5390	378	16	,	,	PUNCT
ejpam-5390	378	17	j	j	PROPN
ejpam-5390	378	18	≤	≤	PROPN
ejpam-5390	378	19	2	2	NUM
ejpam-5390	378	20	,	,	PUNCT
ejpam-5390	378	21	we	we	PRON
ejpam-5390	378	22	have	have	VERB
ejpam-5390	378	23	(	(	PUNCT
ejpam-5390	378	24	i	i	NOUN
ejpam-5390	378	25	)	)	PUNCT
ejpam-5390	378	26	∆(u11v12	∆(u11v12	PROPN
ejpam-5390	378	27	)	)	PUNCT
ejpam-5390	378	28	=	=	PUNCT
ejpam-5390	379	1	∆(u11)v12	∆(u11)v12	VERB
ejpam-5390	379	2	+	+	CCONJ
ejpam-5390	379	3	u11∆(v12	u11∆(v12	ADJ
ejpam-5390	379	4	)	)	PUNCT
ejpam-5390	379	5	and	and	CCONJ
ejpam-5390	379	6	∆(u22v21	∆(u22v21	NOUN
ejpam-5390	379	7	)	)	PUNCT
ejpam-5390	380	1	=	=	SYM
ejpam-5390	380	2	∆(u22)v21	∆(u22)v21	PROPN
ejpam-5390	380	3	+	+	NUM
ejpam-5390	380	4	u22∆(v21	u22∆(v21	NOUN
ejpam-5390	380	5	)	)	PUNCT
ejpam-5390	380	6	.	.	PUNCT
ejpam-5390	381	1	(	(	PUNCT
ejpam-5390	381	2	ii	ii	NOUN
ejpam-5390	381	3	)	)	PUNCT
ejpam-5390	381	4	∆(u12v21	∆(u12v21	PROPN
ejpam-5390	381	5	)	)	PUNCT
ejpam-5390	381	6	=	=	SYM
ejpam-5390	382	1	∆(u12)v21	∆(u12)v21	NOUN
ejpam-5390	382	2	+	+	CCONJ
ejpam-5390	382	3	u12∆(v21	u12∆(v21	ADJ
ejpam-5390	382	4	)	)	PUNCT
ejpam-5390	382	5	and	and	CCONJ
ejpam-5390	382	6	∆(u21v12	∆(u21v12	PROPN
ejpam-5390	382	7	)	)	PUNCT
ejpam-5390	382	8	=	=	SYM
ejpam-5390	383	1	∆(u21)v12	∆(u21)v12	VERB
ejpam-5390	383	2	+	+	CCONJ
ejpam-5390	383	3	u21∆(v12	u21∆(v12	ADJ
ejpam-5390	383	4	)	)	PUNCT
ejpam-5390	383	5	.	.	PUNCT
ejpam-5390	384	1	(	(	PUNCT
ejpam-5390	384	2	iii	iii	X
ejpam-5390	384	3	)	)	PUNCT
ejpam-5390	384	4	∆(u11v11	∆(u11v11	NOUN
ejpam-5390	384	5	)	)	PUNCT
ejpam-5390	385	1	=	=	PUNCT
ejpam-5390	386	1	∆(u11)v11	∆(u11)v11	ADJ
ejpam-5390	386	2	+	+	CCONJ
ejpam-5390	386	3	u11∆(v11	u11∆(v11	NOUN
ejpam-5390	386	4	)	)	PUNCT
ejpam-5390	386	5	and	and	CCONJ
ejpam-5390	386	6	∆(u22v22	∆(u22v22	NOUN
ejpam-5390	386	7	)	)	PUNCT
ejpam-5390	387	1	=	=	PUNCT
ejpam-5390	388	1	∆(u22)v22	∆(u22)v22	NOUN
ejpam-5390	388	2	+	+	NUM
ejpam-5390	388	3	u22∆(v22	u22∆(v22	NOUN
ejpam-5390	388	4	)	)	PUNCT
ejpam-5390	388	5	.	.	PUNCT
ejpam-5390	388	6	.	.	PUNCT
ejpam-5390	389	1	nisar	nisar	PROPN
ejpam-5390	389	2	et	et	PROPN
ejpam-5390	389	3	al	al	PROPN
ejpam-5390	389	4	.	.	PUNCT
ejpam-5390	389	5	/	/	SYM
ejpam-5390	389	6	eur	eur	PROPN
ejpam-5390	389	7	.	.	PUNCT
ejpam-5390	390	1	j.	j.	PROPN
ejpam-5390	390	2	pure	pure	PROPN
ejpam-5390	390	3	appl	appl	PROPN
ejpam-5390	390	4	.	.	PROPN
ejpam-5390	390	5	math	math	PROPN
ejpam-5390	390	6	,	,	PUNCT
ejpam-5390	390	7	17	17	NUM
ejpam-5390	390	8	(	(	PUNCT
ejpam-5390	390	9	4	4	NUM
ejpam-5390	390	10	)	)	PUNCT
ejpam-5390	390	11	(	(	PUNCT
ejpam-5390	390	12	2024	2024	NUM
ejpam-5390	390	13	)	)	PUNCT
ejpam-5390	390	14	,	,	PUNCT
ejpam-5390	390	15	3399	3399	NUM
ejpam-5390	390	16	-	-	SYM
ejpam-5390	390	17	3414	3414	NUM
ejpam-5390	390	18	3410	3410	NUM
ejpam-5390	390	19	(	(	PUNCT
ejpam-5390	390	20	iv	iv	X
ejpam-5390	390	21	)	)	PUNCT
ejpam-5390	390	22	∆(u12v22	∆(u12v22	PROPN
ejpam-5390	390	23	)	)	PUNCT
ejpam-5390	390	24	=	=	SYM
ejpam-5390	391	1	∆(u12)v22	∆(u12)v22	PROPN
ejpam-5390	391	2	+	+	CCONJ
ejpam-5390	391	3	u12∆(v22	u12∆(v22	NOUN
ejpam-5390	391	4	)	)	PUNCT
ejpam-5390	391	5	and	and	CCONJ
ejpam-5390	391	6	∆(u21v11	∆(u21v11	NOUN
ejpam-5390	391	7	)	)	PUNCT
ejpam-5390	391	8	=	=	PUNCT
ejpam-5390	392	1	∆(u21)v11	∆(u21)v11	PROPN
ejpam-5390	392	2	+	+	CCONJ
ejpam-5390	392	3	u21∆(v11	u21∆(v11	ADJ
ejpam-5390	392	4	)	)	PUNCT
ejpam-5390	392	5	.	.	PUNCT
ejpam-5390	393	1	proof	proof	NOUN
ejpam-5390	393	2	.	.	PUNCT
ejpam-5390	394	1	(	(	PUNCT
ejpam-5390	394	2	i	i	NOUN
ejpam-5390	394	3	)	)	PUNCT
ejpam-5390	394	4	using	use	VERB
ejpam-5390	394	5	lemma	lemma	PROPN
ejpam-5390	394	6	2.10	2.10	NUM
ejpam-5390	394	7	and	and	CCONJ
ejpam-5390	394	8	∆(p2	∆(p2	NUM
ejpam-5390	394	9	)	)	PUNCT
ejpam-5390	394	10	=	=	SYM
ejpam-5390	394	11	0	0	NUM
ejpam-5390	394	12	,	,	PUNCT
ejpam-5390	394	13	we	we	PRON
ejpam-5390	394	14	get	get	VERB
ejpam-5390	394	15	∆([u11	∆([u11	PROPN
ejpam-5390	394	16	,	,	PUNCT
ejpam-5390	394	17	v12]∗	v12]∗	NOUN
ejpam-5390	394	18	♢	♢	PROPN
ejpam-5390	394	19	λp2	λp2	NOUN
ejpam-5390	394	20	)	)	PUNCT
ejpam-5390	394	21	=	=	PUNCT
ejpam-5390	395	1	[	[	X
ejpam-5390	395	2	∆(u11	∆(u11	X
ejpam-5390	395	3	)	)	PUNCT
ejpam-5390	395	4	,	,	PUNCT
ejpam-5390	395	5	v12]∗	v12]∗	NOUN
ejpam-5390	395	6	♢	♢	PROPN
ejpam-5390	395	7	λp2	λp2	NOUN
ejpam-5390	395	8	+	+	CCONJ
ejpam-5390	396	1	[	[	X
ejpam-5390	396	2	u11,∆(v12)]∗	u11,∆(v12)]∗	X
ejpam-5390	396	3	♢	♢	NOUN
ejpam-5390	396	4	λp2	λp2	NOUN
ejpam-5390	396	5	=	=	PUNCT
ejpam-5390	396	6	∆(u11)v12	∆(u11)v12	VERB
ejpam-5390	396	7	+	+	CCONJ
ejpam-5390	396	8	u11∆(v12	u11∆(v12	ADJ
ejpam-5390	396	9	)	)	PUNCT
ejpam-5390	396	10	.	.	PUNCT
ejpam-5390	397	1	on	on	ADP
ejpam-5390	397	2	the	the	DET
ejpam-5390	397	3	other	other	ADJ
ejpam-5390	397	4	side	side	NOUN
ejpam-5390	397	5	,	,	PUNCT
ejpam-5390	397	6	we	we	PRON
ejpam-5390	397	7	get	get	AUX
ejpam-5390	397	8	∆([u11	∆([u11	PROPN
ejpam-5390	397	9	,	,	PUNCT
ejpam-5390	397	10	v12]∗	v12]∗	NOUN
ejpam-5390	397	11	♢	♢	PROPN
ejpam-5390	397	12	λp2	λp2	NOUN
ejpam-5390	397	13	)	)	PUNCT
ejpam-5390	397	14	=	=	PUNCT
ejpam-5390	397	15	∆(u11v12	∆(u11v12	PROPN
ejpam-5390	397	16	)	)	PUNCT
ejpam-5390	397	17	.	.	PUNCT
ejpam-5390	398	1	by	by	ADP
ejpam-5390	398	2	comparing	compare	VERB
ejpam-5390	398	3	the	the	DET
ejpam-5390	398	4	above	above	ADJ
ejpam-5390	398	5	two	two	NUM
ejpam-5390	398	6	equations	equation	NOUN
ejpam-5390	398	7	,	,	PUNCT
ejpam-5390	398	8	we	we	PRON
ejpam-5390	398	9	get	get	AUX
ejpam-5390	398	10	∆(u11v12	∆(u11v12	VERB
ejpam-5390	398	11	)	)	PUNCT
ejpam-5390	399	1	=	=	PUNCT
ejpam-5390	399	2	∆(u11)v12	∆(u11)v12	VERB
ejpam-5390	399	3	+	+	CCONJ
ejpam-5390	399	4	u11∆(v12	u11∆(v12	ADJ
ejpam-5390	399	5	)	)	PUNCT
ejpam-5390	399	6	.	.	PUNCT
ejpam-5390	400	1	similarly	similarly	ADV
ejpam-5390	400	2	,	,	PUNCT
ejpam-5390	400	3	we	we	PRON
ejpam-5390	400	4	can	can	AUX
ejpam-5390	400	5	show	show	VERB
ejpam-5390	400	6	that	that	SCONJ
ejpam-5390	400	7	∆(u22v21	∆(u22v21	NOUN
ejpam-5390	400	8	)	)	PUNCT
ejpam-5390	401	1	=	=	SYM
ejpam-5390	401	2	∆(u22)v21	∆(u22)v21	PROPN
ejpam-5390	401	3	+	+	NUM
ejpam-5390	401	4	u22∆(v21	u22∆(v21	NOUN
ejpam-5390	401	5	)	)	PUNCT
ejpam-5390	401	6	.	.	PUNCT
ejpam-5390	402	1	(	(	PUNCT
ejpam-5390	402	2	ii	ii	NOUN
ejpam-5390	402	3	)	)	PUNCT
ejpam-5390	402	4	for	for	ADP
ejpam-5390	402	5	any	any	DET
ejpam-5390	402	6	x12	x12	NUM
ejpam-5390	402	7	∈	∈	PROPN
ejpam-5390	402	8	a12	a12	NOUN
ejpam-5390	402	9	and	and	CCONJ
ejpam-5390	402	10	by	by	ADP
ejpam-5390	402	11	using	use	VERB
ejpam-5390	402	12	lemma	lemma	PROPN
ejpam-5390	402	13	2.11	2.11	NUM
ejpam-5390	402	14	(	(	PUNCT
ejpam-5390	402	15	1	1	NUM
ejpam-5390	402	16	)	)	PUNCT
ejpam-5390	402	17	,	,	PUNCT
ejpam-5390	402	18	we	we	PRON
ejpam-5390	402	19	have	have	VERB
ejpam-5390	402	20	∆([u12	∆([u12	NOUN
ejpam-5390	402	21	,	,	PUNCT
ejpam-5390	402	22	v21]∗	v21]∗	VERB
ejpam-5390	402	23	♢	♢	PROPN
ejpam-5390	402	24	λx12	λx12	PROPN
ejpam-5390	402	25	)	)	PUNCT
ejpam-5390	402	26	=	=	SYM
ejpam-5390	402	27	∆(u12v21x21	∆(u12v21x21	NOUN
ejpam-5390	402	28	)	)	PUNCT
ejpam-5390	402	29	=	=	SYM
ejpam-5390	402	30	∆(u12v21)x12	∆(u12v21)x12	X
ejpam-5390	402	31	+	+	PUNCT
ejpam-5390	402	32	u12v21∆(x12	u12v21∆(x12	X
ejpam-5390	402	33	)	)	PUNCT
ejpam-5390	402	34	.	.	PUNCT
ejpam-5390	403	1	on	on	ADP
ejpam-5390	403	2	the	the	DET
ejpam-5390	403	3	other	other	ADJ
ejpam-5390	403	4	hand	hand	NOUN
ejpam-5390	403	5	,	,	PUNCT
ejpam-5390	403	6	we	we	PRON
ejpam-5390	403	7	have	have	VERB
ejpam-5390	403	8	∆([u12	∆([u12	NOUN
ejpam-5390	403	9	,	,	PUNCT
ejpam-5390	403	10	v21]∗	v21]∗	VERB
ejpam-5390	403	11	♢	♢	PROPN
ejpam-5390	403	12	λx12	λx12	PROPN
ejpam-5390	403	13	)	)	PUNCT
ejpam-5390	403	14	=	=	PUNCT
ejpam-5390	404	1	[	[	X
ejpam-5390	404	2	∆(u12	∆(u12	NOUN
ejpam-5390	404	3	)	)	PUNCT
ejpam-5390	404	4	,	,	PUNCT
ejpam-5390	404	5	v21]∗	v21]∗	VERB
ejpam-5390	404	6	♢	♢	PROPN
ejpam-5390	404	7	λx12	λx12	PROPN
ejpam-5390	404	8	+	+	X
ejpam-5390	405	1	[	[	X
ejpam-5390	405	2	u12,∆(v21)]∗	u12,∆(v21)]∗	X
ejpam-5390	405	3	♢	♢	VERB
ejpam-5390	405	4	λx12	λx12	PROPN
ejpam-5390	405	5	+	+	PROPN
ejpam-5390	405	6	[	[	X
ejpam-5390	405	7	u12	u12	NOUN
ejpam-5390	405	8	,	,	PUNCT
ejpam-5390	405	9	v21]∗	v21]∗	VERB
ejpam-5390	405	10	♢	♢	PROPN
ejpam-5390	405	11	λ∆(x12	λ∆(x12	NOUN
ejpam-5390	405	12	)	)	PUNCT
ejpam-5390	405	13	=	=	PUNCT
ejpam-5390	405	14	∆(u12)v21x12	∆(u12)v21x12	X
ejpam-5390	405	15	+	+	NUM
ejpam-5390	405	16	u12∆(v21)x12	u12∆(v21)x12	NOUN
ejpam-5390	405	17	+	+	CCONJ
ejpam-5390	405	18	u12v21∆(x12	u12v21∆(x12	X
ejpam-5390	405	19	)	)	PUNCT
ejpam-5390	405	20	.	.	PUNCT
ejpam-5390	406	1	from	from	ADP
ejpam-5390	406	2	the	the	DET
ejpam-5390	406	3	above	above	ADJ
ejpam-5390	406	4	two	two	NUM
ejpam-5390	406	5	expressions	expression	NOUN
ejpam-5390	406	6	,	,	PUNCT
ejpam-5390	406	7	we	we	PRON
ejpam-5390	406	8	get	get	VERB
ejpam-5390	406	9	(	(	PUNCT
ejpam-5390	406	10	∆(u12v21)−∆(u12)v21	∆(u12v21)−∆(u12)v21	PROPN
ejpam-5390	406	11	−	−	PROPN
ejpam-5390	406	12	u12∆(v21))x12	u12∆(v21))x12	NOUN
ejpam-5390	406	13	=	=	SYM
ejpam-5390	407	1	0	0	X
ejpam-5390	407	2	.	.	PUNCT
ejpam-5390	408	1	thus	thus	ADV
ejpam-5390	408	2	,	,	PUNCT
ejpam-5390	408	3	by	by	ADP
ejpam-5390	408	4	(	(	PUNCT
ejpam-5390	408	5	▲	▲	PUNCT
ejpam-5390	408	6	)	)	PUNCT
ejpam-5390	408	7	and	and	CCONJ
ejpam-5390	408	8	(	(	PUNCT
ejpam-5390	408	9	▼	▼	NOUN
ejpam-5390	408	10	)	)	PUNCT
ejpam-5390	408	11	,	,	PUNCT
ejpam-5390	408	12	∆(u12v21	∆(u12v21	NOUN
ejpam-5390	408	13	)	)	PUNCT
ejpam-5390	408	14	=	=	SYM
ejpam-5390	408	15	∆(u12)v21	∆(u12)v21	PROPN
ejpam-5390	408	16	+	+	CCONJ
ejpam-5390	408	17	u12∆(v21	u12∆(v21	ADJ
ejpam-5390	408	18	)	)	PUNCT
ejpam-5390	408	19	.	.	PUNCT
ejpam-5390	409	1	similarly	similarly	ADV
ejpam-5390	409	2	,	,	PUNCT
ejpam-5390	409	3	we	we	PRON
ejpam-5390	409	4	can	can	AUX
ejpam-5390	409	5	show	show	VERB
ejpam-5390	409	6	that	that	SCONJ
ejpam-5390	409	7	∆(u21v12	∆(u21v12	PROPN
ejpam-5390	409	8	)	)	PUNCT
ejpam-5390	409	9	=	=	SYM
ejpam-5390	410	1	∆(u21)v12	∆(u21)v12	VERB
ejpam-5390	410	2	+	+	CCONJ
ejpam-5390	410	3	u21∆(v12	u21∆(v12	ADJ
ejpam-5390	410	4	)	)	PUNCT
ejpam-5390	410	5	.	.	PUNCT
ejpam-5390	411	1	(	(	PUNCT
ejpam-5390	411	2	iii	iii	X
ejpam-5390	411	3	)	)	PUNCT
ejpam-5390	411	4	for	for	ADP
ejpam-5390	411	5	any	any	DET
ejpam-5390	411	6	x12	x12	NUM
ejpam-5390	411	7	∈	∈	PROPN
ejpam-5390	411	8	a12	a12	NOUN
ejpam-5390	411	9	,	,	PUNCT
ejpam-5390	411	10	it	it	PRON
ejpam-5390	411	11	follows	follow	VERB
ejpam-5390	411	12	from	from	ADP
ejpam-5390	411	13	lemma	lemma	PROPN
ejpam-5390	411	14	2.11(i	2.11(i	PROPN
ejpam-5390	411	15	)	)	PUNCT
ejpam-5390	411	16	that	that	SCONJ
ejpam-5390	411	17	∆(u11v11x12	∆(u11v11x12	VERB
ejpam-5390	411	18	)	)	PUNCT
ejpam-5390	411	19	=	=	SYM
ejpam-5390	411	20	∆(u11v11)x12	∆(u11v11)x12	X
ejpam-5390	411	21	+	+	CCONJ
ejpam-5390	411	22	u11v11∆(x12	u11v11∆(x12	NOUN
ejpam-5390	411	23	)	)	PUNCT
ejpam-5390	411	24	.	.	PUNCT
ejpam-5390	412	1	again	again	ADV
ejpam-5390	412	2	using	use	VERB
ejpam-5390	412	3	lemma	lemma	PROPN
ejpam-5390	412	4	2.11	2.11	NUM
ejpam-5390	412	5	from	from	ADP
ejpam-5390	412	6	the	the	DET
ejpam-5390	412	7	other	other	ADJ
ejpam-5390	412	8	side	side	NOUN
ejpam-5390	412	9	,	,	PUNCT
ejpam-5390	412	10	we	we	PRON
ejpam-5390	412	11	have	have	VERB
ejpam-5390	412	12	∆(u11v11x12	∆(u11v11x12	NUM
ejpam-5390	412	13	)	)	PUNCT
ejpam-5390	412	14	=	=	SYM
ejpam-5390	412	15	∆(u11)v11x12	∆(u11)v11x12	X
ejpam-5390	412	16	+	+	CCONJ
ejpam-5390	412	17	u11∆(v11x12	u11∆(v11x12	NOUN
ejpam-5390	412	18	)	)	PUNCT
ejpam-5390	412	19	=	=	PUNCT
ejpam-5390	412	20	∆(u11)v11x12	∆(u11)v11x12	X
ejpam-5390	412	21	+	+	X
ejpam-5390	412	22	u11∆(v11)x12	u11∆(v11)x12	ADJ
ejpam-5390	412	23	+	+	CCONJ
ejpam-5390	412	24	u11v11∆(x12	u11v11∆(x12	NOUN
ejpam-5390	412	25	)	)	PUNCT
ejpam-5390	412	26	.	.	PUNCT
ejpam-5390	413	1	by	by	ADP
ejpam-5390	413	2	comparing	compare	VERB
ejpam-5390	413	3	the	the	DET
ejpam-5390	413	4	aforementioned	aforementioned	ADJ
ejpam-5390	413	5	two	two	NUM
ejpam-5390	413	6	equations	equation	NOUN
ejpam-5390	413	7	,	,	PUNCT
ejpam-5390	413	8	we	we	PRON
ejpam-5390	413	9	obtain	obtain	VERB
ejpam-5390	413	10	(	(	PUNCT
ejpam-5390	413	11	∆(u11v11	∆(u11v11	NOUN
ejpam-5390	413	12	)	)	PUNCT
ejpam-5390	413	13	−	−	PROPN
ejpam-5390	414	1	∆(u11)v11	∆(u11)v11	ADJ
ejpam-5390	414	2	−	−	NOUN
ejpam-5390	414	3	u11∆(v11))x12	u11∆(v11))x12	PUNCT
ejpam-5390	414	4	=	=	SYM
ejpam-5390	414	5	0	0	X
ejpam-5390	414	6	.	.	PUNCT
ejpam-5390	415	1	therefore	therefore	ADV
ejpam-5390	415	2	,	,	PUNCT
ejpam-5390	415	3	utilizing	utilize	VERB
ejpam-5390	415	4	(	(	PUNCT
ejpam-5390	415	5	▲	▲	PUNCT
ejpam-5390	415	6	)	)	PUNCT
ejpam-5390	415	7	and	and	CCONJ
ejpam-5390	415	8	(	(	PUNCT
ejpam-5390	415	9	▼	▼	NOUN
ejpam-5390	415	10	)	)	PUNCT
ejpam-5390	415	11	,	,	PUNCT
ejpam-5390	415	12	we	we	PRON
ejpam-5390	415	13	conclude	conclude	VERB
ejpam-5390	415	14	that	that	PRON
ejpam-5390	415	15	∆(u11v11	∆(u11v11	NOUN
ejpam-5390	415	16	)	)	PUNCT
ejpam-5390	415	17	=	=	PUNCT
ejpam-5390	416	1	∆(u11)v11	∆(u11)v11	ADJ
ejpam-5390	416	2	+	+	CCONJ
ejpam-5390	416	3	u11∆(v11	u11∆(v11	NOUN
ejpam-5390	416	4	)	)	PUNCT
ejpam-5390	416	5	.	.	PUNCT
ejpam-5390	417	1	similarly	similarly	ADV
ejpam-5390	417	2	,	,	PUNCT
ejpam-5390	417	3	one	one	PRON
ejpam-5390	417	4	can	can	AUX
ejpam-5390	417	5	show	show	VERB
ejpam-5390	417	6	that	that	SCONJ
ejpam-5390	417	7	∆(u22v22	∆(u22v22	NOUN
ejpam-5390	417	8	)	)	PUNCT
ejpam-5390	417	9	=	=	PUNCT
ejpam-5390	418	1	∆(u22)v22	∆(u22)v22	NOUN
ejpam-5390	418	2	+	+	NUM
ejpam-5390	418	3	u22∆(v22	u22∆(v22	NOUN
ejpam-5390	418	4	)	)	PUNCT
ejpam-5390	418	5	.	.	PUNCT
ejpam-5390	419	1	(	(	PUNCT
ejpam-5390	419	2	iv	iv	X
ejpam-5390	419	3	)	)	PUNCT
ejpam-5390	419	4	for	for	ADP
ejpam-5390	419	5	any	any	DET
ejpam-5390	419	6	x21	x21	PROPN
ejpam-5390	419	7	∈	∈	PROPN
ejpam-5390	419	8	a21	a21	NOUN
ejpam-5390	419	9	.	.	PUNCT
ejpam-5390	420	1	it	it	PRON
ejpam-5390	420	2	follows	follow	VERB
ejpam-5390	420	3	from	from	ADP
ejpam-5390	420	4	lemma	lemma	PROPN
ejpam-5390	420	5	2.11(2	2.11(2	NUM
ejpam-5390	420	6	)	)	PUNCT
ejpam-5390	420	7	that	that	DET
ejpam-5390	420	8	∆(u12v22x21	∆(u12v22x21	X
ejpam-5390	420	9	)	)	PUNCT
ejpam-5390	420	10	=	=	SYM
ejpam-5390	420	11	∆(u12v22)x21	∆(u12v22)x21	NOUN
ejpam-5390	420	12	+	+	CCONJ
ejpam-5390	420	13	u12v22∆(x21	u12v22∆(x21	NUM
ejpam-5390	420	14	)	)	PUNCT
ejpam-5390	420	15	.	.	PUNCT
ejpam-5390	420	16	.	.	PUNCT
ejpam-5390	421	1	nisar	nisar	PROPN
ejpam-5390	421	2	et	et	PROPN
ejpam-5390	421	3	al	al	PROPN
ejpam-5390	421	4	.	.	PUNCT
ejpam-5390	421	5	/	/	SYM
ejpam-5390	421	6	eur	eur	PROPN
ejpam-5390	421	7	.	.	PUNCT
ejpam-5390	422	1	j.	j.	PROPN
ejpam-5390	422	2	pure	pure	PROPN
ejpam-5390	422	3	appl	appl	PROPN
ejpam-5390	422	4	.	.	PROPN
ejpam-5390	422	5	math	math	PROPN
ejpam-5390	422	6	,	,	PUNCT
ejpam-5390	422	7	17	17	NUM
ejpam-5390	422	8	(	(	PUNCT
ejpam-5390	422	9	4	4	NUM
ejpam-5390	422	10	)	)	PUNCT
ejpam-5390	422	11	(	(	PUNCT
ejpam-5390	422	12	2024	2024	NUM
ejpam-5390	422	13	)	)	PUNCT
ejpam-5390	422	14	,	,	PUNCT
ejpam-5390	422	15	3399	3399	NUM
ejpam-5390	422	16	-	-	SYM
ejpam-5390	422	17	3414	3414	NUM
ejpam-5390	422	18	3411	3411	NUM
ejpam-5390	422	19	again	again	ADV
ejpam-5390	422	20	on	on	ADP
ejpam-5390	422	21	the	the	DET
ejpam-5390	422	22	other	other	ADJ
ejpam-5390	422	23	side	side	NOUN
ejpam-5390	422	24	,	,	PUNCT
ejpam-5390	422	25	it	it	PRON
ejpam-5390	422	26	follows	follow	VERB
ejpam-5390	422	27	from	from	ADP
ejpam-5390	422	28	lemma	lemma	PROPN
ejpam-5390	422	29	2.11(i	2.11(i	PROPN
ejpam-5390	422	30	)	)	PUNCT
ejpam-5390	422	31	and	and	CCONJ
ejpam-5390	422	32	lemma	lemma	PROPN
ejpam-5390	422	33	2.11(ii	2.11(ii	NUM
ejpam-5390	422	34	)	)	PUNCT
ejpam-5390	422	35	that	that	PRON
ejpam-5390	422	36	∆(u12v22x21	∆(u12v22x21	X
ejpam-5390	422	37	)	)	PUNCT
ejpam-5390	422	38	=	=	SYM
ejpam-5390	422	39	∆(u12)v22x21	∆(u12)v22x21	PROPN
ejpam-5390	422	40	+	+	CCONJ
ejpam-5390	422	41	u12∆(v22x21	u12∆(v22x21	PROPN
ejpam-5390	422	42	)	)	PUNCT
ejpam-5390	422	43	=	=	SYM
ejpam-5390	422	44	∆(u12)v22x21	∆(u12)v22x21	PROPN
ejpam-5390	422	45	+	+	NUM
ejpam-5390	422	46	u12∆(v22)x21	u12∆(v22)x21	NOUN
ejpam-5390	422	47	+	+	CCONJ
ejpam-5390	422	48	u12v22∆(x21	u12v22∆(x21	NUM
ejpam-5390	422	49	)	)	PUNCT
ejpam-5390	422	50	.	.	PUNCT
ejpam-5390	423	1	from	from	ADP
ejpam-5390	423	2	the	the	DET
ejpam-5390	423	3	above	above	ADJ
ejpam-5390	423	4	two	two	NUM
ejpam-5390	423	5	equations	equation	NOUN
ejpam-5390	423	6	and	and	CCONJ
ejpam-5390	423	7	using	use	VERB
ejpam-5390	423	8	(	(	PUNCT
ejpam-5390	423	9	▲	▲	PUNCT
ejpam-5390	423	10	)	)	PUNCT
ejpam-5390	423	11	and	and	CCONJ
ejpam-5390	423	12	(	(	PUNCT
ejpam-5390	423	13	▼	▼	NOUN
ejpam-5390	423	14	)	)	PUNCT
ejpam-5390	423	15	,	,	PUNCT
ejpam-5390	423	16	we	we	PRON
ejpam-5390	423	17	find	find	VERB
ejpam-5390	423	18	∆(u12v22	∆(u12v22	PROPN
ejpam-5390	423	19	)	)	PUNCT
ejpam-5390	423	20	=	=	SYM
ejpam-5390	424	1	∆(u12)v22	∆(u12)v22	PROPN
ejpam-5390	424	2	+	+	CCONJ
ejpam-5390	424	3	u12∆(v22	u12∆(v22	NOUN
ejpam-5390	424	4	)	)	PUNCT
ejpam-5390	424	5	.	.	PUNCT
ejpam-5390	425	1	similarly	similarly	ADV
ejpam-5390	425	2	,	,	PUNCT
ejpam-5390	425	3	one	one	PRON
ejpam-5390	425	4	can	can	AUX
ejpam-5390	425	5	show	show	VERB
ejpam-5390	425	6	that	that	SCONJ
ejpam-5390	425	7	∆(u21v11	∆(u21v11	NOUN
ejpam-5390	425	8	)	)	PUNCT
ejpam-5390	425	9	=	=	PUNCT
ejpam-5390	426	1	∆(u21)v11	∆(u21)v11	PROPN
ejpam-5390	426	2	+	+	CCONJ
ejpam-5390	426	3	u21∆(v11	u21∆(v11	ADJ
ejpam-5390	426	4	)	)	PUNCT
ejpam-5390	426	5	.	.	PUNCT
ejpam-5390	427	1	lemma	lemma	PROPN
ejpam-5390	427	2	2.12	2.12	NUM
ejpam-5390	427	3	.	.	PUNCT
ejpam-5390	428	1	∆(u∗	∆(u∗	NOUN
ejpam-5390	428	2	)	)	PUNCT
ejpam-5390	429	1	=	=	SYM
ejpam-5390	429	2	∆(u)∗	∆(u)∗	NOUN
ejpam-5390	429	3	for	for	ADP
ejpam-5390	429	4	all	all	DET
ejpam-5390	429	5	u	u	NOUN
ejpam-5390	429	6	∈	∈	NOUN
ejpam-5390	429	7	a.	a.	NOUN
ejpam-5390	429	8	proof	proof	NOUN
ejpam-5390	429	9	.	.	PUNCT
ejpam-5390	430	1	for	for	ADP
ejpam-5390	430	2	any	any	DET
ejpam-5390	430	3	x12	x12	NUM
ejpam-5390	430	4	∈	∈	PROPN
ejpam-5390	430	5	a12	a12	NOUN
ejpam-5390	430	6	,	,	PUNCT
ejpam-5390	430	7	it	it	PRON
ejpam-5390	430	8	follows	follow	VERB
ejpam-5390	430	9	from	from	ADP
ejpam-5390	430	10	remark	remark	NOUN
ejpam-5390	430	11	2.1	2.1	NUM
ejpam-5390	430	12	and	and	CCONJ
ejpam-5390	430	13	lemma	lemma	PROPN
ejpam-5390	430	14	2.11(i	2.11(i	PROPN
ejpam-5390	430	15	)	)	PUNCT
ejpam-5390	430	16	that	that	PRON
ejpam-5390	430	17	∆([u11	∆([u11	PROPN
ejpam-5390	430	18	,	,	PUNCT
ejpam-5390	430	19	p1]∗	p1]∗	PROPN
ejpam-5390	430	20	♢	♢	PROPN
ejpam-5390	430	21	λx12	λx12	PROPN
ejpam-5390	430	22	)	)	PUNCT
ejpam-5390	431	1	=	=	SYM
ejpam-5390	431	2	∆(u11x12)−∆(u∗	∆(u11x12)−∆(u∗	NOUN
ejpam-5390	431	3	11x12	11x12	NUM
ejpam-5390	431	4	)	)	PUNCT
ejpam-5390	431	5	=	=	SYM
ejpam-5390	432	1	∆(u11)x12	∆(u11)x12	PROPN
ejpam-5390	432	2	+	+	NUM
ejpam-5390	432	3	u11∆(x12)−∆(u∗	u11∆(x12)−∆(u∗	ADJ
ejpam-5390	432	4	11)x12	11)x12	NUM
ejpam-5390	432	5	−	−	NOUN
ejpam-5390	432	6	u∗	u∗	ADJ
ejpam-5390	432	7	11∆(x12	11∆(x12	NUM
ejpam-5390	432	8	)	)	PUNCT
ejpam-5390	432	9	.	.	PUNCT
ejpam-5390	433	1	alternatively	alternatively	ADV
ejpam-5390	433	2	,	,	PUNCT
ejpam-5390	433	3	it	it	PRON
ejpam-5390	433	4	can	can	AUX
ejpam-5390	433	5	be	be	AUX
ejpam-5390	433	6	deduced	deduce	VERB
ejpam-5390	433	7	from	from	ADP
ejpam-5390	433	8	∆(p1	∆(p1	VERB
ejpam-5390	433	9	)	)	PUNCT
ejpam-5390	434	1	=	=	SYM
ejpam-5390	434	2	0	0	NUM
ejpam-5390	434	3	that	that	PRON
ejpam-5390	434	4	∆([u11	∆([u11	PROPN
ejpam-5390	434	5	,	,	PUNCT
ejpam-5390	434	6	p1]∗	p1]∗	PROPN
ejpam-5390	434	7	♢	♢	PROPN
ejpam-5390	434	8	λx12	λx12	PROPN
ejpam-5390	434	9	)	)	PUNCT
ejpam-5390	434	10	=	=	PUNCT
ejpam-5390	435	1	[	[	X
ejpam-5390	435	2	∆(u11	∆(u11	X
ejpam-5390	435	3	)	)	PUNCT
ejpam-5390	435	4	,	,	PUNCT
ejpam-5390	435	5	p1]∗	p1]∗	PROPN
ejpam-5390	435	6	♢	♢	NOUN
ejpam-5390	435	7	λx12	λx12	PROPN
ejpam-5390	435	8	+	+	PROPN
ejpam-5390	436	1	[	[	X
ejpam-5390	436	2	u11	u11	ADJ
ejpam-5390	436	3	,	,	PUNCT
ejpam-5390	436	4	p1]∗	p1]∗	PROPN
ejpam-5390	436	5	♢	♢	NOUN
ejpam-5390	436	6	λ∆(x12	λ∆(x12	NOUN
ejpam-5390	436	7	)	)	PUNCT
ejpam-5390	436	8	=	=	PUNCT
ejpam-5390	436	9	∆(u11)x12	∆(u11)x12	PROPN
ejpam-5390	436	10	−∆(u11	−∆(u11	NUM
ejpam-5390	436	11	)	)	PUNCT
ejpam-5390	436	12	∗x12	∗x12	PUNCT
ejpam-5390	436	13	+	+	CCONJ
ejpam-5390	436	14	u11∆(x12)−	u11∆(x12)−	PROPN
ejpam-5390	436	15	u∗	u∗	ADJ
ejpam-5390	436	16	11∆(x12	11∆(x12	NUM
ejpam-5390	436	17	)	)	PUNCT
ejpam-5390	436	18	.	.	PUNCT
ejpam-5390	437	1	from	from	ADP
ejpam-5390	437	2	the	the	DET
ejpam-5390	437	3	above	above	ADJ
ejpam-5390	437	4	two	two	NUM
ejpam-5390	437	5	equations	equation	NOUN
ejpam-5390	437	6	,	,	PUNCT
ejpam-5390	437	7	we	we	PRON
ejpam-5390	437	8	have	have	VERB
ejpam-5390	437	9	(	(	PUNCT
ejpam-5390	437	10	∆(u11	∆(u11	PROPN
ejpam-5390	437	11	)	)	PUNCT
ejpam-5390	437	12	∗	∗	NOUN
ejpam-5390	438	1	−∆(u∗	−∆(u∗	PROPN
ejpam-5390	438	2	11))x12	11))x12	NUM
ejpam-5390	438	3	=	=	SYM
ejpam-5390	438	4	0	0	PROPN
ejpam-5390	438	5	.	.	PUNCT
ejpam-5390	439	1	now	now	ADV
ejpam-5390	439	2	,	,	PUNCT
ejpam-5390	439	3	by	by	ADP
ejpam-5390	439	4	using	use	VERB
ejpam-5390	439	5	(	(	PUNCT
ejpam-5390	439	6	▲	▲	PUNCT
ejpam-5390	439	7	)	)	PUNCT
ejpam-5390	439	8	and	and	CCONJ
ejpam-5390	439	9	(	(	PUNCT
ejpam-5390	439	10	▼	▼	NOUN
ejpam-5390	439	11	)	)	PUNCT
ejpam-5390	439	12	,	,	PUNCT
ejpam-5390	439	13	we	we	PRON
ejpam-5390	439	14	get	get	VERB
ejpam-5390	439	15	∆(u11	∆(u11	VERB
ejpam-5390	439	16	)	)	PUNCT
ejpam-5390	439	17	∗	∗	NOUN
ejpam-5390	439	18	=	=	PUNCT
ejpam-5390	440	1	∆(u∗	∆(u∗	ADP
ejpam-5390	440	2	11	11	NUM
ejpam-5390	440	3	)	)	PUNCT
ejpam-5390	440	4	.	.	PUNCT
ejpam-5390	441	1	(	(	PUNCT
ejpam-5390	441	2	2.13	2.13	NUM
ejpam-5390	441	3	)	)	PUNCT
ejpam-5390	441	4	similarly	similarly	ADV
ejpam-5390	441	5	,	,	PUNCT
ejpam-5390	441	6	by	by	ADP
ejpam-5390	441	7	using	use	VERB
ejpam-5390	441	8	the	the	DET
ejpam-5390	441	9	same	same	ADJ
ejpam-5390	441	10	technique	technique	NOUN
ejpam-5390	441	11	,	,	PUNCT
ejpam-5390	441	12	one	one	PRON
ejpam-5390	441	13	can	can	AUX
ejpam-5390	441	14	show	show	VERB
ejpam-5390	441	15	that	that	SCONJ
ejpam-5390	441	16	∆(u22	∆(u22	NOUN
ejpam-5390	441	17	)	)	PUNCT
ejpam-5390	441	18	∗	∗	NOUN
ejpam-5390	441	19	=	=	PUNCT
ejpam-5390	442	1	∆(u∗	∆(u∗	NOUN
ejpam-5390	442	2	22	22	NUM
ejpam-5390	442	3	)	)	PUNCT
ejpam-5390	442	4	.	.	PUNCT
ejpam-5390	443	1	(	(	PUNCT
ejpam-5390	443	2	2.14	2.14	NUM
ejpam-5390	443	3	)	)	PUNCT
ejpam-5390	443	4	again	again	ADV
ejpam-5390	443	5	,	,	PUNCT
ejpam-5390	443	6	it	it	PRON
ejpam-5390	443	7	follows	follow	VERB
ejpam-5390	443	8	from	from	ADP
ejpam-5390	443	9	lemma	lemma	PROPN
ejpam-5390	443	10	2.10	2.10	NUM
ejpam-5390	443	11	and	and	CCONJ
ejpam-5390	443	12	∆(p2	∆(p2	NUM
ejpam-5390	443	13	)	)	PUNCT
ejpam-5390	443	14	=	=	SYM
ejpam-5390	443	15	0	0	NUM
ejpam-5390	443	16	that	that	SCONJ
ejpam-5390	443	17	∆([u12	∆([u12	NOUN
ejpam-5390	443	18	,	,	PUNCT
ejpam-5390	443	19	p2]∗	p2]∗	NOUN
ejpam-5390	443	20	♢	♢	NOUN
ejpam-5390	443	21	λx12	λx12	PROPN
ejpam-5390	443	22	)	)	PUNCT
ejpam-5390	443	23	=	=	PUNCT
ejpam-5390	444	1	[	[	X
ejpam-5390	444	2	∆(u12	∆(u12	NOUN
ejpam-5390	444	3	)	)	PUNCT
ejpam-5390	444	4	,	,	PUNCT
ejpam-5390	444	5	p2]∗	p2]∗	VERB
ejpam-5390	444	6	♢	♢	NOUN
ejpam-5390	444	7	λx12	λx12	PROPN
ejpam-5390	444	8	+	+	PROPN
ejpam-5390	445	1	[	[	X
ejpam-5390	445	2	u12	u12	NOUN
ejpam-5390	445	3	,	,	PUNCT
ejpam-5390	445	4	p2]∗	p2]∗	NOUN
ejpam-5390	445	5	♢	♢	NOUN
ejpam-5390	445	6	λ∆(x12	λ∆(x12	NOUN
ejpam-5390	445	7	)	)	PUNCT
ejpam-5390	445	8	=	=	SYM
ejpam-5390	445	9	−∆(u12	−∆(u12	X
ejpam-5390	445	10	)	)	PUNCT
ejpam-5390	445	11	∗x12	∗x12	PUNCT
ejpam-5390	445	12	−	−	PUNCT
ejpam-5390	445	13	λx12∆(u12	λx12∆(u12	NUM
ejpam-5390	445	14	)	)	PUNCT
ejpam-5390	445	15	∗	∗	NOUN
ejpam-5390	445	16	−	−	PROPN
ejpam-5390	446	1	u∗	u∗	ADJ
ejpam-5390	446	2	12∆(x12)−	12∆(x12)−	NUM
ejpam-5390	446	3	λ∆(x12)u	λ∆(x12)u	PROPN
ejpam-5390	446	4	∗	∗	PROPN
ejpam-5390	446	5	12	12	NUM
ejpam-5390	446	6	.	.	PUNCT
ejpam-5390	447	1	on	on	ADP
ejpam-5390	447	2	the	the	DET
ejpam-5390	447	3	other	other	ADJ
ejpam-5390	447	4	hand	hand	NOUN
ejpam-5390	447	5	,	,	PUNCT
ejpam-5390	447	6	using	use	VERB
ejpam-5390	447	7	lemma	lemma	PROPN
ejpam-5390	447	8	2.11	2.11	NUM
ejpam-5390	447	9	,	,	PUNCT
ejpam-5390	447	10	we	we	PRON
ejpam-5390	447	11	have	have	VERB
ejpam-5390	447	12	∆([u12	∆([u12	NOUN
ejpam-5390	447	13	,	,	PUNCT
ejpam-5390	447	14	p2]∗	p2]∗	NOUN
ejpam-5390	447	15	♢	♢	NOUN
ejpam-5390	447	16	λx12	λx12	PROPN
ejpam-5390	447	17	)	)	PUNCT
ejpam-5390	447	18	=	=	PUNCT
ejpam-5390	448	1	∆(−u∗	∆(−u∗	NOUN
ejpam-5390	448	2	12x12	12x12	NUM
ejpam-5390	448	3	−	−	NOUN
ejpam-5390	448	4	λx12u	λx12u	PUNCT
ejpam-5390	448	5	∗	∗	PROPN
ejpam-5390	448	6	12	12	NUM
ejpam-5390	448	7	)	)	PUNCT
ejpam-5390	449	1	=	=	SYM
ejpam-5390	449	2	−∆(u∗	−∆(u∗	ADV
ejpam-5390	449	3	12x12)−∆(x12λu	12x12)−∆(x12λu	NUM
ejpam-5390	449	4	∗	∗	NOUN
ejpam-5390	449	5	12	12	NUM
ejpam-5390	449	6	)	)	PUNCT
ejpam-5390	449	7	=	=	PUNCT
ejpam-5390	450	1	−∆(u∗	−∆(u∗	ADV
ejpam-5390	450	2	12)x12	12)x12	NUM
ejpam-5390	450	3	−	−	NOUN
ejpam-5390	450	4	u∗	u∗	ADJ
ejpam-5390	450	5	12∆(x12)−	12∆(x12)−	NUM
ejpam-5390	450	6	λ∆(x12)u	λ∆(x12)u	NOUN
ejpam-5390	450	7	∗	∗	VERB
ejpam-5390	450	8	12	12	NUM
ejpam-5390	450	9	−x12∆(λu∗	−x12∆(λu∗	SYM
ejpam-5390	450	10	12	12	NUM
ejpam-5390	450	11	)	)	PUNCT
ejpam-5390	450	12	.	.	PUNCT
ejpam-5390	451	1	by	by	ADP
ejpam-5390	451	2	comparing	compare	VERB
ejpam-5390	451	3	the	the	DET
ejpam-5390	451	4	above	above	ADJ
ejpam-5390	451	5	two	two	NUM
ejpam-5390	451	6	equations	equation	NOUN
ejpam-5390	451	7	,	,	PUNCT
ejpam-5390	451	8	we	we	PRON
ejpam-5390	451	9	get	get	VERB
ejpam-5390	451	10	(	(	PUNCT
ejpam-5390	451	11	∆(u12	∆(u12	NOUN
ejpam-5390	451	12	)	)	PUNCT
ejpam-5390	451	13	∗	∗	NOUN
ejpam-5390	451	14	−∆(u∗	−∆(u∗	ADV
ejpam-5390	451	15	12))x12	12))x12	NUM
ejpam-5390	451	16	+	+	NUM
ejpam-5390	451	17	λx12∆(u12	λx12∆(u12	NUM
ejpam-5390	451	18	)	)	PUNCT
ejpam-5390	451	19	∗	∗	NOUN
ejpam-5390	451	20	−x12∆(λu∗	−x12∆(λu∗	PUNCT
ejpam-5390	451	21	12	12	NUM
ejpam-5390	451	22	)	)	PUNCT
ejpam-5390	451	23	=	=	SYM
ejpam-5390	452	1	0	0	X
ejpam-5390	452	2	.	.	PUNCT
ejpam-5390	452	3	by	by	ADP
ejpam-5390	452	4	left	left	ADJ
ejpam-5390	452	5	-	-	PUNCT
ejpam-5390	452	6	multiplying	multiply	VERB
ejpam-5390	452	7	both	both	DET
ejpam-5390	452	8	sides	side	NOUN
ejpam-5390	452	9	by	by	ADP
ejpam-5390	452	10	p2	p2	PROPN
ejpam-5390	452	11	and	and	CCONJ
ejpam-5390	452	12	utilizing	utilize	VERB
ejpam-5390	452	13	(	(	PUNCT
ejpam-5390	452	14	▲	▲	PUNCT
ejpam-5390	452	15	)	)	PUNCT
ejpam-5390	452	16	and	and	CCONJ
ejpam-5390	452	17	(	(	PUNCT
ejpam-5390	452	18	▼	▼	NOUN
ejpam-5390	452	19	)	)	PUNCT
ejpam-5390	452	20	,	,	PUNCT
ejpam-5390	452	21	we	we	PRON
ejpam-5390	452	22	obtain	obtain	VERB
ejpam-5390	452	23	∆(u12	∆(u12	NOUN
ejpam-5390	452	24	)	)	PUNCT
ejpam-5390	452	25	∗	∗	NOUN
ejpam-5390	452	26	=	=	PUNCT
ejpam-5390	453	1	∆(u∗	∆(u∗	NOUN
ejpam-5390	453	2	12	12	NUM
ejpam-5390	453	3	)	)	PUNCT
ejpam-5390	453	4	.	.	PUNCT
ejpam-5390	454	1	(	(	PUNCT
ejpam-5390	454	2	2.15	2.15	NUM
ejpam-5390	454	3	)	)	PUNCT
ejpam-5390	454	4	.	.	PUNCT
ejpam-5390	455	1	nisar	nisar	PROPN
ejpam-5390	455	2	et	et	PROPN
ejpam-5390	455	3	al	al	PROPN
ejpam-5390	455	4	.	.	PUNCT
ejpam-5390	455	5	/	/	SYM
ejpam-5390	455	6	eur	eur	PROPN
ejpam-5390	455	7	.	.	PUNCT
ejpam-5390	456	1	j.	j.	PROPN
ejpam-5390	456	2	pure	pure	PROPN
ejpam-5390	456	3	appl	appl	PROPN
ejpam-5390	456	4	.	.	PROPN
ejpam-5390	456	5	math	math	PROPN
ejpam-5390	456	6	,	,	PUNCT
ejpam-5390	456	7	17	17	NUM
ejpam-5390	456	8	(	(	PUNCT
ejpam-5390	456	9	4	4	NUM
ejpam-5390	456	10	)	)	PUNCT
ejpam-5390	456	11	(	(	PUNCT
ejpam-5390	456	12	2024	2024	NUM
ejpam-5390	456	13	)	)	PUNCT
ejpam-5390	456	14	,	,	PUNCT
ejpam-5390	456	15	3399	3399	NUM
ejpam-5390	456	16	-	-	SYM
ejpam-5390	456	17	3414	3414	NUM
ejpam-5390	456	18	3412	3412	NUM
ejpam-5390	456	19	similarly	similarly	ADV
ejpam-5390	456	20	,	,	PUNCT
ejpam-5390	456	21	we	we	PRON
ejpam-5390	456	22	can	can	AUX
ejpam-5390	456	23	show	show	VERB
ejpam-5390	456	24	that	that	SCONJ
ejpam-5390	456	25	∆(u21	∆(u21	PROPN
ejpam-5390	456	26	)	)	PUNCT
ejpam-5390	456	27	∗	∗	NOUN
ejpam-5390	456	28	=	=	PUNCT
ejpam-5390	457	1	∆(u∗	∆(u∗	NOUN
ejpam-5390	457	2	21	21	NUM
ejpam-5390	457	3	)	)	PUNCT
ejpam-5390	457	4	.	.	PUNCT
ejpam-5390	458	1	(	(	PUNCT
ejpam-5390	458	2	2.16	2.16	NUM
ejpam-5390	458	3	)	)	PUNCT
ejpam-5390	458	4	from	from	ADP
ejpam-5390	458	5	equations	equation	NOUN
ejpam-5390	458	6	(	(	PUNCT
ejpam-5390	458	7	2.13)-(2.16	2.13)-(2.16	NUM
ejpam-5390	458	8	)	)	PUNCT
ejpam-5390	458	9	and	and	CCONJ
ejpam-5390	458	10	using	use	VERB
ejpam-5390	458	11	additivity	additivity	NOUN
ejpam-5390	458	12	of	of	ADP
ejpam-5390	458	13	∆	∆	PROPN
ejpam-5390	458	14	,	,	PUNCT
ejpam-5390	458	15	we	we	PRON
ejpam-5390	458	16	get	get	VERB
ejpam-5390	458	17	∆(u∗	∆(u∗	NOUN
ejpam-5390	458	18	)	)	PUNCT
ejpam-5390	458	19	=	=	PUNCT
ejpam-5390	459	1	∆(u)∗.	∆(u)∗.	PROPN
ejpam-5390	459	2	proof	proof	NOUN
ejpam-5390	459	3	of	of	ADP
ejpam-5390	459	4	theorem	theorem	ADJ
ejpam-5390	459	5	2.1	2.1	NUM
ejpam-5390	459	6	for	for	ADP
ejpam-5390	459	7	every	every	DET
ejpam-5390	459	8	u	u	NOUN
ejpam-5390	459	9	,	,	PUNCT
ejpam-5390	459	10	v	v	ADP
ejpam-5390	459	11	∈	∈	PROPN
ejpam-5390	459	12	a	a	PRON
ejpam-5390	459	13	,	,	PUNCT
ejpam-5390	459	14	we	we	PRON
ejpam-5390	459	15	can	can	AUX
ejpam-5390	459	16	write	write	VERB
ejpam-5390	459	17	u	u	NOUN
ejpam-5390	459	18	=	=	PROPN
ejpam-5390	459	19	u11+u12+u21+u22	u11+u12+u21+u22	PROPN
ejpam-5390	459	20	and	and	CCONJ
ejpam-5390	459	21	v	v	NOUN
ejpam-5390	459	22	=	=	SYM
ejpam-5390	459	23	v11	v11	NOUN
ejpam-5390	459	24	+	+	CCONJ
ejpam-5390	459	25	v12	v12	ADJ
ejpam-5390	459	26	+	+	CCONJ
ejpam-5390	459	27	v21	v21	NOUN
ejpam-5390	459	28	+	+	CCONJ
ejpam-5390	459	29	v22	v22	NOUN
ejpam-5390	459	30	.	.	PUNCT
ejpam-5390	460	1	since	since	ADV
ejpam-5390	460	2	,	,	PUNCT
ejpam-5390	460	3	∆	∆	PROPN
ejpam-5390	460	4	is	be	AUX
ejpam-5390	460	5	additive	additive	ADJ
ejpam-5390	460	6	and	and	CCONJ
ejpam-5390	460	7	using	use	VERB
ejpam-5390	460	8	lemma	lemma	PROPN
ejpam-5390	460	9	2.11	2.11	NUM
ejpam-5390	460	10	,	,	PUNCT
ejpam-5390	460	11	we	we	PRON
ejpam-5390	460	12	get	get	VERB
ejpam-5390	460	13	∆(uv	∆(uv	NOUN
ejpam-5390	460	14	)	)	PUNCT
ejpam-5390	461	1	=	=	PUNCT
ejpam-5390	462	1	∆(u11v11	∆(u11v11	PROPN
ejpam-5390	462	2	+	+	CCONJ
ejpam-5390	462	3	u11v12	u11v12	ADJ
ejpam-5390	463	1	+	+	CCONJ
ejpam-5390	463	2	u12v21	u12v21	ADJ
ejpam-5390	463	3	+	+	CCONJ
ejpam-5390	463	4	u12v22	u12v22	NOUN
ejpam-5390	463	5	+	+	CCONJ
ejpam-5390	463	6	u21v11	u21v11	ADJ
ejpam-5390	463	7	+	+	CCONJ
ejpam-5390	463	8	u21v12	u21v12	ADJ
ejpam-5390	463	9	+	+	CCONJ
ejpam-5390	463	10	u22v21	u22v21	ADJ
ejpam-5390	463	11	+	+	CCONJ
ejpam-5390	463	12	u22v22	u22v22	NOUN
ejpam-5390	463	13	)	)	PUNCT
ejpam-5390	463	14	=	=	SYM
ejpam-5390	463	15	∆(u11v11	∆(u11v11	PROPN
ejpam-5390	463	16	)	)	PUNCT
ejpam-5390	464	1	+	+	CCONJ
ejpam-5390	464	2	∆(u11v12	∆(u11v12	ADJ
ejpam-5390	464	3	)	)	PUNCT
ejpam-5390	465	1	+	+	CCONJ
ejpam-5390	465	2	∆(u12v21	∆(u12v21	NOUN
ejpam-5390	465	3	)	)	PUNCT
ejpam-5390	466	1	+	+	CCONJ
ejpam-5390	466	2	∆(u12v22	∆(u12v22	PROPN
ejpam-5390	466	3	)	)	PUNCT
ejpam-5390	467	1	+	+	NOUN
ejpam-5390	467	2	∆(u21v11	∆(u21v11	NOUN
ejpam-5390	467	3	)	)	PUNCT
ejpam-5390	467	4	+	+	CCONJ
ejpam-5390	467	5	∆(u21v12	∆(u21v12	PROPN
ejpam-5390	467	6	)	)	PUNCT
ejpam-5390	467	7	+	+	CCONJ
ejpam-5390	467	8	∆(u22v21	∆(u22v21	NOUN
ejpam-5390	467	9	)	)	PUNCT
ejpam-5390	468	1	+	+	CCONJ
ejpam-5390	468	2	∆(u22v22	∆(u22v22	NOUN
ejpam-5390	468	3	)	)	PUNCT
ejpam-5390	468	4	=	=	PRON
ejpam-5390	469	1	∆(u11	∆(u11	PROPN
ejpam-5390	469	2	+	+	CCONJ
ejpam-5390	469	3	u12	u12	PROPN
ejpam-5390	469	4	+	+	CCONJ
ejpam-5390	469	5	u21	u21	NOUN
ejpam-5390	469	6	+	+	CCONJ
ejpam-5390	469	7	u22)(v11	u22)(v11	ADJ
ejpam-5390	469	8	+	+	CCONJ
ejpam-5390	469	9	v12	v12	ADJ
ejpam-5390	469	10	+	+	CCONJ
ejpam-5390	469	11	v21	v21	NOUN
ejpam-5390	469	12	+	+	CCONJ
ejpam-5390	469	13	v22	v22	NOUN
ejpam-5390	469	14	)	)	PUNCT
ejpam-5390	470	1	+	+	PROPN
ejpam-5390	470	2	(	(	PUNCT
ejpam-5390	470	3	u11	u11	PROPN
ejpam-5390	470	4	+	+	NUM
ejpam-5390	470	5	u12	u12	PROPN
ejpam-5390	470	6	+	+	CCONJ
ejpam-5390	470	7	u21	u21	PROPN
ejpam-5390	470	8	+	+	CCONJ
ejpam-5390	470	9	u22	u22	PROPN
ejpam-5390	470	10	)	)	PUNCT
ejpam-5390	470	11	∆(v11	∆(v11	NOUN
ejpam-5390	470	12	+	+	CCONJ
ejpam-5390	470	13	v12	v12	ADJ
ejpam-5390	470	14	+	+	CCONJ
ejpam-5390	470	15	v21	v21	NOUN
ejpam-5390	470	16	+	+	CCONJ
ejpam-5390	470	17	v22	v22	NOUN
ejpam-5390	470	18	)	)	PUNCT
ejpam-5390	471	1	=	=	PUNCT
ejpam-5390	471	2	∆(u)v	∆(u)v	X
ejpam-5390	472	1	+	+	CCONJ
ejpam-5390	472	2	u∆(v	u∆(v	X
ejpam-5390	472	3	)	)	PUNCT
ejpam-5390	472	4	.	.	PUNCT
ejpam-5390	473	1	so	so	ADV
ejpam-5390	473	2	,	,	PUNCT
ejpam-5390	473	3	∆	∆	PROPN
ejpam-5390	473	4	is	be	AUX
ejpam-5390	473	5	a	a	DET
ejpam-5390	473	6	derivation	derivation	NOUN
ejpam-5390	473	7	.	.	PUNCT
ejpam-5390	474	1	by	by	ADP
ejpam-5390	474	2	using	use	VERB
ejpam-5390	474	3	lemma	lemma	PROPN
ejpam-5390	474	4	2.12	2.12	NUM
ejpam-5390	474	5	,	,	PUNCT
ejpam-5390	474	6	∆	∆	PROPN
ejpam-5390	474	7	is	be	AUX
ejpam-5390	474	8	an	an	DET
ejpam-5390	474	9	additive	additive	ADJ
ejpam-5390	474	10	∗-derivation	∗-derivation	NOUN
ejpam-5390	474	11	.	.	PUNCT
ejpam-5390	475	1	hence	hence	ADV
ejpam-5390	475	2	,	,	PUNCT
ejpam-5390	475	3	by	by	ADP
ejpam-5390	475	4	remark	remark	NOUN
ejpam-5390	475	5	2.1	2.1	NUM
ejpam-5390	475	6	,	,	PUNCT
ejpam-5390	475	7	π	π	PROPN
ejpam-5390	475	8	is	be	AUX
ejpam-5390	475	9	an	an	DET
ejpam-5390	475	10	additive	additive	ADJ
ejpam-5390	475	11	∗-derivation	∗-derivation	NOUN
ejpam-5390	475	12	.	.	PUNCT
ejpam-5390	476	1	this	this	PRON
ejpam-5390	476	2	completes	complete	VERB
ejpam-5390	476	3	the	the	DET
ejpam-5390	476	4	proof	proof	NOUN
ejpam-5390	476	5	of	of	ADP
ejpam-5390	476	6	theorem	theorem	ADJ
ejpam-5390	476	7	2.1	2.1	NUM
ejpam-5390	476	8	.	.	PUNCT
ejpam-5390	477	1	the	the	DET
ejpam-5390	477	2	corollaries	corollary	NOUN
ejpam-5390	477	3	following	follow	VERB
ejpam-5390	477	4	directly	directly	ADV
ejpam-5390	477	5	from	from	ADP
ejpam-5390	477	6	theorem	theorem	ADJ
ejpam-5390	477	7	2.1	2.1	NUM
ejpam-5390	477	8	are	be	AUX
ejpam-5390	477	9	as	as	SCONJ
ejpam-5390	477	10	follows	follow	VERB
ejpam-5390	477	11	:	:	PUNCT
ejpam-5390	477	12	corollary	corollary	ADJ
ejpam-5390	477	13	2.1	2.1	NUM
ejpam-5390	477	14	.	.	PUNCT
ejpam-5390	478	1	let	let	VERB
ejpam-5390	478	2	a	a	PRON
ejpam-5390	478	3	be	be	AUX
ejpam-5390	478	4	a	a	DET
ejpam-5390	478	5	standard	standard	ADJ
ejpam-5390	478	6	operator	operator	NOUN
ejpam-5390	478	7	algebra	algebra	NOUN
ejpam-5390	478	8	on	on	ADP
ejpam-5390	478	9	an	an	DET
ejpam-5390	478	10	infinite	infinite	ADJ
ejpam-5390	478	11	dimensional	dimensional	ADJ
ejpam-5390	478	12	complex	complex	ADJ
ejpam-5390	478	13	hilbert	hilbert	NOUN
ejpam-5390	478	14	space	space	NOUN
ejpam-5390	478	15	h	h	NOUN
ejpam-5390	478	16	containing	contain	VERB
ejpam-5390	478	17	identity	identity	NOUN
ejpam-5390	478	18	operator	operator	NOUN
ejpam-5390	478	19	i.	i.	NOUN
ejpam-5390	478	20	suppose	suppose	VERB
ejpam-5390	478	21	that	that	SCONJ
ejpam-5390	478	22	a	a	PRON
ejpam-5390	478	23	is	be	AUX
ejpam-5390	478	24	closed	close	VERB
ejpam-5390	478	25	under	under	ADP
ejpam-5390	478	26	adjoint	adjoint	NOUN
ejpam-5390	478	27	operation	operation	NOUN
ejpam-5390	478	28	.	.	PUNCT
ejpam-5390	479	1	define	define	VERB
ejpam-5390	479	2	π	π	NOUN
ejpam-5390	479	3	:	:	PUNCT
ejpam-5390	479	4	a	a	DET
ejpam-5390	479	5	→	→	X
ejpam-5390	479	6	a	a	DET
ejpam-5390	479	7	such	such	ADJ
ejpam-5390	479	8	that	that	DET
ejpam-5390	479	9	π([u	π([u	PROPN
ejpam-5390	479	10	,	,	PUNCT
ejpam-5390	479	11	v	v	NOUN
ejpam-5390	479	12	]	]	PUNCT
ejpam-5390	479	13	∗	∗	PROPN
ejpam-5390	479	14	♢	♢	PROPN
ejpam-5390	479	15	λw	λw	NOUN
ejpam-5390	479	16	)	)	PUNCT
ejpam-5390	479	17	=	=	PUNCT
ejpam-5390	480	1	[	[	X
ejpam-5390	480	2	π(u	π(u	NOUN
ejpam-5390	480	3	)	)	PUNCT
ejpam-5390	480	4	,	,	PUNCT
ejpam-5390	480	5	v	v	X
ejpam-5390	480	6	]	]	PUNCT
ejpam-5390	480	7	∗	∗	PROPN
ejpam-5390	480	8	♢	♢	PROPN
ejpam-5390	480	9	λw	λw	NOUN
ejpam-5390	480	10	)	)	PUNCT
ejpam-5390	480	11	+	+	PUNCT
ejpam-5390	481	1	[	[	X
ejpam-5390	481	2	u	u	NOUN
ejpam-5390	481	3	,	,	PUNCT
ejpam-5390	481	4	π(v	π(v	NOUN
ejpam-5390	481	5	)	)	PUNCT
ejpam-5390	481	6	]	]	PUNCT
ejpam-5390	481	7	∗	∗	PROPN
ejpam-5390	481	8	♢	♢	PROPN
ejpam-5390	481	9	λw	λw	X
ejpam-5390	481	10	+	+	PROPN
ejpam-5390	481	11	[	[	X
ejpam-5390	481	12	u	u	NOUN
ejpam-5390	481	13	,	,	PUNCT
ejpam-5390	481	14	v	v	NOUN
ejpam-5390	481	15	]	]	PUNCT
ejpam-5390	481	16	∗	∗	X
ejpam-5390	481	17	♢	♢	PROPN
ejpam-5390	481	18	λπ(w	λπ(w	NUM
ejpam-5390	481	19	)	)	PUNCT
ejpam-5390	481	20	for	for	ADP
ejpam-5390	481	21	all	all	DET
ejpam-5390	481	22	u	u	NOUN
ejpam-5390	481	23	,	,	PUNCT
ejpam-5390	481	24	v	v	NOUN
ejpam-5390	481	25	,	,	PUNCT
ejpam-5390	481	26	w	w	PROPN
ejpam-5390	481	27	∈	∈	PROPN
ejpam-5390	481	28	a	a	PRON
ejpam-5390	481	29	.	.	PUNCT
ejpam-5390	482	1	then	then	ADV
ejpam-5390	482	2	π	π	PROPN
ejpam-5390	482	3	is	be	AUX
ejpam-5390	482	4	an	an	DET
ejpam-5390	482	5	additive	additive	ADJ
ejpam-5390	482	6	∗-derivation	∗-derivation	NOUN
ejpam-5390	482	7	.	.	PUNCT
ejpam-5390	483	1	corollary	corollary	ADJ
ejpam-5390	483	2	2.2	2.2	NUM
ejpam-5390	483	3	.	.	PUNCT
ejpam-5390	484	1	let	let	VERB
ejpam-5390	484	2	a	a	DET
ejpam-5390	484	3	ba	ba	PROPN
ejpam-5390	484	4	a	a	DET
ejpam-5390	484	5	factor	factor	NOUN
ejpam-5390	484	6	von	von	PROPN
ejpam-5390	484	7	neumann	neumann	PROPN
ejpam-5390	484	8	algebra	algebra	PROPN
ejpam-5390	484	9	with	with	ADP
ejpam-5390	484	10	dimm	dimm	NOUN
ejpam-5390	484	11	≥	≥	NOUN
ejpam-5390	484	12	2	2	NUM
ejpam-5390	484	13	.	.	PUNCT
ejpam-5390	485	1	define	define	VERB
ejpam-5390	485	2	π	π	NOUN
ejpam-5390	485	3	:	:	PUNCT
ejpam-5390	485	4	m	m	VERB
ejpam-5390	485	5	→	→	NOUN
ejpam-5390	485	6	m	m	VERB
ejpam-5390	485	7	such	such	ADJ
ejpam-5390	485	8	that	that	PRON
ejpam-5390	485	9	π([u	π([u	PROPN
ejpam-5390	485	10	,	,	PUNCT
ejpam-5390	485	11	v	v	NOUN
ejpam-5390	485	12	]	]	PUNCT
ejpam-5390	485	13	∗	∗	PROPN
ejpam-5390	485	14	♢	♢	PROPN
ejpam-5390	485	15	λw	λw	NOUN
ejpam-5390	485	16	)	)	PUNCT
ejpam-5390	485	17	=	=	PUNCT
ejpam-5390	486	1	[	[	X
ejpam-5390	486	2	π(u	π(u	NOUN
ejpam-5390	486	3	)	)	PUNCT
ejpam-5390	486	4	,	,	PUNCT
ejpam-5390	486	5	v	v	X
ejpam-5390	486	6	]	]	PUNCT
ejpam-5390	486	7	∗	∗	PROPN
ejpam-5390	486	8	♢	♢	PROPN
ejpam-5390	486	9	λw	λw	NOUN
ejpam-5390	486	10	)	)	PUNCT
ejpam-5390	486	11	+	+	PUNCT
ejpam-5390	487	1	[	[	X
ejpam-5390	487	2	u	u	NOUN
ejpam-5390	487	3	,	,	PUNCT
ejpam-5390	487	4	π(v	π(v	NOUN
ejpam-5390	487	5	)	)	PUNCT
ejpam-5390	487	6	]	]	PUNCT
ejpam-5390	487	7	∗	∗	PROPN
ejpam-5390	487	8	♢	♢	PROPN
ejpam-5390	487	9	λw	λw	X
ejpam-5390	487	10	+	+	PROPN
ejpam-5390	487	11	[	[	X
ejpam-5390	487	12	u	u	NOUN
ejpam-5390	487	13	,	,	PUNCT
ejpam-5390	487	14	v	v	NOUN
ejpam-5390	487	15	]	]	PUNCT
ejpam-5390	487	16	∗	∗	X
ejpam-5390	487	17	♢	♢	PROPN
ejpam-5390	487	18	λπ(w	λπ(w	NUM
ejpam-5390	487	19	)	)	PUNCT
ejpam-5390	487	20	for	for	ADP
ejpam-5390	487	21	all	all	DET
ejpam-5390	487	22	u	u	NOUN
ejpam-5390	487	23	,	,	PUNCT
ejpam-5390	487	24	v	v	NOUN
ejpam-5390	487	25	,	,	PUNCT
ejpam-5390	487	26	w	w	PROPN
ejpam-5390	487	27	∈	∈	PROPN
ejpam-5390	487	28	a	a	PRON
ejpam-5390	487	29	.	.	PUNCT
ejpam-5390	488	1	then	then	ADV
ejpam-5390	488	2	π	π	PROPN
ejpam-5390	488	3	is	be	AUX
ejpam-5390	488	4	an	an	DET
ejpam-5390	488	5	additive	additive	ADJ
ejpam-5390	488	6	∗-derivation	∗-derivation	NOUN
ejpam-5390	488	7	.	.	PUNCT
ejpam-5390	489	1	corollary	corollary	ADJ
ejpam-5390	489	2	2.3	2.3	NUM
ejpam-5390	489	3	.	.	PUNCT
ejpam-5390	490	1	let	let	VERB
ejpam-5390	490	2	a	a	PRON
ejpam-5390	490	3	be	be	AUX
ejpam-5390	490	4	a	a	DET
ejpam-5390	490	5	prime	prime	ADJ
ejpam-5390	490	6	∗-algebra	∗-algebra	NOUN
ejpam-5390	490	7	with	with	ADP
ejpam-5390	490	8	unit	unit	NOUN
ejpam-5390	490	9	i	i	PRON
ejpam-5390	490	10	containing	contain	VERB
ejpam-5390	490	11	non	non	ADJ
ejpam-5390	490	12	-	-	ADJ
ejpam-5390	490	13	trivial	trivial	ADJ
ejpam-5390	490	14	projection	projection	NOUN
ejpam-5390	490	15	p	p	NOUN
ejpam-5390	490	16	.	.	PUNCT
ejpam-5390	491	1	a	a	DET
ejpam-5390	491	2	map	map	NOUN
ejpam-5390	491	3	π	π	X
ejpam-5390	491	4	:	:	PUNCT
ejpam-5390	491	5	a	a	DET
ejpam-5390	491	6	→	→	X
ejpam-5390	491	7	a	a	DET
ejpam-5390	491	8	satisfies	satisfie	NOUN
ejpam-5390	491	9	π([u	π([u	PROPN
ejpam-5390	491	10	,	,	PUNCT
ejpam-5390	491	11	v	v	NOUN
ejpam-5390	491	12	]	]	PUNCT
ejpam-5390	491	13	∗	∗	PROPN
ejpam-5390	491	14	♢	♢	PROPN
ejpam-5390	491	15	λw	λw	NOUN
ejpam-5390	491	16	)	)	PUNCT
ejpam-5390	491	17	=	=	PUNCT
ejpam-5390	492	1	[	[	X
ejpam-5390	492	2	π(u	π(u	NOUN
ejpam-5390	492	3	)	)	PUNCT
ejpam-5390	492	4	,	,	PUNCT
ejpam-5390	492	5	v	v	X
ejpam-5390	492	6	]	]	PUNCT
ejpam-5390	492	7	∗	∗	PROPN
ejpam-5390	492	8	♢	♢	PROPN
ejpam-5390	492	9	λw	λw	NOUN
ejpam-5390	492	10	)	)	PUNCT
ejpam-5390	492	11	+	+	PUNCT
ejpam-5390	493	1	[	[	X
ejpam-5390	493	2	u	u	NOUN
ejpam-5390	493	3	,	,	PUNCT
ejpam-5390	493	4	π(v	π(v	NOUN
ejpam-5390	493	5	)	)	PUNCT
ejpam-5390	493	6	]	]	PUNCT
ejpam-5390	493	7	∗	∗	PROPN
ejpam-5390	493	8	♢	♢	PROPN
ejpam-5390	493	9	λw	λw	X
ejpam-5390	493	10	+	+	PROPN
ejpam-5390	493	11	[	[	X
ejpam-5390	493	12	u	u	NOUN
ejpam-5390	493	13	,	,	PUNCT
ejpam-5390	493	14	v	v	NOUN
ejpam-5390	493	15	]	]	PUNCT
ejpam-5390	493	16	∗	∗	X
ejpam-5390	493	17	♢	♢	PROPN
ejpam-5390	493	18	λπ(w	λπ(w	NUM
ejpam-5390	493	19	)	)	PUNCT
ejpam-5390	493	20	for	for	ADP
ejpam-5390	493	21	all	all	DET
ejpam-5390	493	22	u	u	NOUN
ejpam-5390	493	23	,	,	PUNCT
ejpam-5390	493	24	v	v	NOUN
ejpam-5390	493	25	,	,	PUNCT
ejpam-5390	493	26	w	w	PROPN
ejpam-5390	493	27	∈	∈	PROPN
ejpam-5390	493	28	a	a	PRON
ejpam-5390	493	29	.	.	PUNCT
ejpam-5390	494	1	then	then	ADV
ejpam-5390	494	2	π	π	PROPN
ejpam-5390	494	3	is	be	AUX
ejpam-5390	494	4	an	an	DET
ejpam-5390	494	5	additive	additive	ADJ
ejpam-5390	494	6	∗-derivation	∗-derivation	NOUN
ejpam-5390	494	7	.	.	PUNCT
ejpam-5390	495	1	references	reference	NOUN
ejpam-5390	495	2	3413	3413	NUM
ejpam-5390	495	3	acknowledgements	acknowledgement	NOUN
ejpam-5390	495	4	the	the	DET
ejpam-5390	495	5	authors	author	NOUN
ejpam-5390	495	6	extended	extend	VERB
ejpam-5390	495	7	their	their	PRON
ejpam-5390	495	8	appreciation	appreciation	NOUN
ejpam-5390	495	9	to	to	PART
ejpam-5390	495	10	princess	princess	VERB
ejpam-5390	495	11	nourah	nourah	PROPN
ejpam-5390	495	12	bint	bint	PROPN
ejpam-5390	495	13	abdulrahman	abdulrahman	PROPN
ejpam-5390	495	14	university	university	PROPN
ejpam-5390	495	15	for	for	ADP
ejpam-5390	495	16	funding	fund	VERB
ejpam-5390	495	17	this	this	DET
ejpam-5390	495	18	research	research	NOUN
ejpam-5390	495	19	under	under	ADP
ejpam-5390	495	20	researchers	researcher	NOUN
ejpam-5390	495	21	supporting	support	VERB
ejpam-5390	495	22	project	project	NOUN
ejpam-5390	495	23	number	number	NOUN
ejpam-5390	495	24	(	(	PUNCT
ejpam-5390	495	25	pnursp2024r231	pnursp2024r231	PROPN
ejpam-5390	495	26	)	)	PUNCT
ejpam-5390	495	27	,	,	PUNCT
ejpam-5390	495	28	princess	princess	PROPN
ejpam-5390	495	29	nourah	nourah	PROPN
ejpam-5390	495	30	bint	bint	PROPN
ejpam-5390	495	31	abdulrahman	abdulrahman	PROPN
ejpam-5390	495	32	university	university	PROPN
ejpam-5390	495	33	,	,	PUNCT
ejpam-5390	495	34	riyadh	riyadh	PROPN
ejpam-5390	495	35	saudi	saudi	PROPN
ejpam-5390	495	36	arabia	arabia	PROPN
ejpam-5390	495	37	.	.	PUNCT
ejpam-5390	496	1	conflicts	conflict	NOUN
ejpam-5390	496	2	of	of	ADP
ejpam-5390	496	3	interest	interest	NOUN
ejpam-5390	496	4	:	:	PUNCT
ejpam-5390	496	5	the	the	DET
ejpam-5390	496	6	authors	author	NOUN
ejpam-5390	496	7	declare	declare	VERB
ejpam-5390	496	8	no	no	DET
ejpam-5390	496	9	conflict	conflict	NOUN
ejpam-5390	496	10	of	of	ADP
ejpam-5390	496	11	interest	interest	NOUN
ejpam-5390	496	12	.	.	PUNCT
ejpam-5390	497	1	references	reference	NOUN
ejpam-5390	497	2	[	[	X
ejpam-5390	497	3	1	1	NUM
ejpam-5390	497	4	]	]	PUNCT
ejpam-5390	497	5	m.	m.	NOUN
ejpam-5390	497	6	ashraf	ashraf	PROPN
ejpam-5390	497	7	,	,	PUNCT
ejpam-5390	497	8	md	md	PROPN
ejpam-5390	497	9	.	.	PROPN
ejpam-5390	497	10	shamim	shamim	PROPN
ejpam-5390	497	11	akhter	akhter	PROPN
ejpam-5390	497	12	,	,	PUNCT
ejpam-5390	497	13	and	and	CCONJ
ejpam-5390	497	14	m.	m.	NOUN
ejpam-5390	497	15	ansari	ansari	PROPN
ejpam-5390	497	16	.	.	PUNCT
ejpam-5390	498	1	nonlinear	nonlinear	ADJ
ejpam-5390	498	2	bi	bi	ADJ
ejpam-5390	498	3	-	-	ADJ
ejpam-5390	498	4	skew	skew	ADJ
ejpam-5390	498	5	jordan	jordan	PROPN
ejpam-5390	498	6	-	-	PUNCT
ejpam-5390	498	7	type	type	NOUN
ejpam-5390	498	8	derivations	derivation	NOUN
ejpam-5390	498	9	on	on	ADP
ejpam-5390	498	10	factor	factor	NOUN
ejpam-5390	498	11	von	von	PROPN
ejpam-5390	498	12	neumann	neumann	PROPN
ejpam-5390	498	13	algebras	algebras	PROPN
ejpam-5390	498	14	.	.	PUNCT
ejpam-5390	499	1	filomat	filomat	PROPN
ejpam-5390	499	2	,	,	PUNCT
ejpam-5390	499	3	37(17):5591–5599	37(17):5591–5599	NUM
ejpam-5390	499	4	,	,	PUNCT
ejpam-5390	499	5	2023	2023	NUM
ejpam-5390	499	6	.	.	PUNCT
ejpam-5390	500	1	[	[	X
ejpam-5390	500	2	2	2	X
ejpam-5390	500	3	]	]	X
ejpam-5390	500	4	d.	d.	PROPN
ejpam-5390	500	5	huo	huo	PROPN
ejpam-5390	500	6	,	,	PUNCT
ejpam-5390	500	7	b.	b.	PROPN
ejpam-5390	500	8	zheng	zheng	PROPN
ejpam-5390	500	9	,	,	PUNCT
ejpam-5390	500	10	j.	j.	PROPN
ejpam-5390	500	11	xu	xu	PROPN
ejpam-5390	500	12	,	,	PUNCT
ejpam-5390	500	13	and	and	CCONJ
ejpam-5390	500	14	h.	h.	PROPN
ejpam-5390	500	15	liu	liu	PROPN
ejpam-5390	500	16	.	.	PUNCT
ejpam-5390	501	1	nonlinear	nonlinear	ADJ
ejpam-5390	501	2	mappings	mapping	NOUN
ejpam-5390	501	3	preserving	preserve	VERB
ejpam-5390	501	4	jordan	jordan	PROPN
ejpam-5390	501	5	multiple	multiple	ADJ
ejpam-5390	501	6	–	–	PUNCT
ejpam-5390	501	7	product	product	NOUN
ejpam-5390	501	8	on	on	ADP
ejpam-5390	501	9	factor	factor	NOUN
ejpam-5390	501	10	von	von	PROPN
ejpam-5390	501	11	neumann	neumann	PROPN
ejpam-5390	501	12	algebras	algebras	PROPN
ejpam-5390	501	13	.	.	PUNCT
ejpam-5390	502	1	linear	linear	PROPN
ejpam-5390	502	2	and	and	CCONJ
ejpam-5390	502	3	multilinear	multilinear	PROPN
ejpam-5390	502	4	algebra	algebra	PROPN
ejpam-5390	502	5	,	,	PUNCT
ejpam-5390	502	6	63(5):1026	63(5):1026	NUM
ejpam-5390	502	7	–	–	PUNCT
ejpam-5390	502	8	1036	1036	NUM
ejpam-5390	502	9	,	,	PUNCT
ejpam-5390	502	10	2015	2015	NUM
ejpam-5390	502	11	.	.	PUNCT
ejpam-5390	503	1	[	[	X
ejpam-5390	503	2	3	3	X
ejpam-5390	503	3	]	]	X
ejpam-5390	503	4	l.	l.	PROPN
ejpam-5390	503	5	kong	kong	PROPN
ejpam-5390	503	6	and	and	CCONJ
ejpam-5390	503	7	j.	j.	PROPN
ejpam-5390	503	8	zhang	zhang	PROPN
ejpam-5390	503	9	.	.	PUNCT
ejpam-5390	504	1	nonlinear	nonlinear	PROPN
ejpam-5390	504	2	skew	skew	ADJ
ejpam-5390	504	3	lie	lie	NOUN
ejpam-5390	504	4	derivations	derivation	NOUN
ejpam-5390	504	5	on	on	ADP
ejpam-5390	504	6	prime	prime	ADJ
ejpam-5390	504	7	-	-	PUNCT
ejpam-5390	504	8	rings	ring	NOUN
ejpam-5390	504	9	.	.	PUNCT
ejpam-5390	505	1	indian	indian	PROPN
ejpam-5390	505	2	journal	journal	PROPN
ejpam-5390	505	3	of	of	ADP
ejpam-5390	505	4	pure	pure	ADJ
ejpam-5390	505	5	and	and	CCONJ
ejpam-5390	505	6	applied	applied	ADJ
ejpam-5390	505	7	mathematics	mathematic	NOUN
ejpam-5390	505	8	,	,	PUNCT
ejpam-5390	505	9	54(2):475–484	54(2):475–484	PROPN
ejpam-5390	505	10	,	,	PUNCT
ejpam-5390	505	11	2023	2023	NUM
ejpam-5390	505	12	.	.	PUNCT
ejpam-5390	506	1	[	[	X
ejpam-5390	506	2	4	4	NUM
ejpam-5390	506	3	]	]	X
ejpam-5390	506	4	c.	c.	PROPN
ejpam-5390	506	5	li	li	PROPN
ejpam-5390	506	6	,	,	PUNCT
ejpam-5390	506	7	q.	q.	PROPN
ejpam-5390	506	8	chen	chen	PROPN
ejpam-5390	506	9	,	,	PUNCT
ejpam-5390	506	10	and	and	CCONJ
ejpam-5390	506	11	t.	t.	PROPN
ejpam-5390	506	12	wang	wang	PROPN
ejpam-5390	506	13	.	.	PUNCT
ejpam-5390	507	1	nonlinear	nonlinear	ADJ
ejpam-5390	507	2	maps	map	NOUN
ejpam-5390	507	3	preserving	preserve	VERB
ejpam-5390	507	4	the	the	DET
ejpam-5390	507	5	jordan	jordan	PROPN
ejpam-5390	507	6	triple*product	triple*product	PROPN
ejpam-5390	507	7	on	on	ADP
ejpam-5390	507	8	factor	factor	NOUN
ejpam-5390	507	9	von	von	PROPN
ejpam-5390	507	10	neumann	neumann	PROPN
ejpam-5390	507	11	algebras	algebras	PROPN
ejpam-5390	507	12	.	.	PUNCT
ejpam-5390	508	1	chinese	chinese	ADJ
ejpam-5390	508	2	annals	annal	NOUN
ejpam-5390	508	3	of	of	ADP
ejpam-5390	508	4	mathematics	mathematic	NOUN
ejpam-5390	508	5	,	,	PUNCT
ejpam-5390	508	6	series	series	NOUN
ejpam-5390	508	7	b	b	PROPN
ejpam-5390	508	8	,	,	PUNCT
ejpam-5390	508	9	39(4):633–642	39(4):633–642	PROPN
ejpam-5390	508	10	,	,	PUNCT
ejpam-5390	508	11	2018	2018	NUM
ejpam-5390	508	12	.	.	PUNCT
ejpam-5390	509	1	[	[	X
ejpam-5390	509	2	5	5	X
ejpam-5390	509	3	]	]	PUNCT
ejpam-5390	509	4	c.	c.	PROPN
ejpam-5390	509	5	li	li	PROPN
ejpam-5390	509	6	and	and	CCONJ
ejpam-5390	509	7	f.	f.	PROPN
ejpam-5390	509	8	lu	lu	PROPN
ejpam-5390	509	9	.	.	PUNCT
ejpam-5390	510	1	nonlinear	nonlinear	ADJ
ejpam-5390	510	2	maps	map	NOUN
ejpam-5390	510	3	preserving	preserve	VERB
ejpam-5390	510	4	the	the	DET
ejpam-5390	510	5	jordan	jordan	PROPN
ejpam-5390	510	6	triple	triple	ADJ
ejpam-5390	510	7	1-*-product	1-*-product	NUM
ejpam-5390	510	8	on	on	ADP
ejpam-5390	510	9	von	von	PROPN
ejpam-5390	510	10	neumann	neumann	PROPN
ejpam-5390	510	11	algebras	algebras	PROPN
ejpam-5390	510	12	.	.	PUNCT
ejpam-5390	511	1	complex	complex	ADJ
ejpam-5390	511	2	analysis	analysis	NOUN
ejpam-5390	511	3	and	and	CCONJ
ejpam-5390	511	4	operator	operator	NOUN
ejpam-5390	511	5	theory	theory	NOUN
ejpam-5390	511	6	,	,	PUNCT
ejpam-5390	511	7	11:109–117	11:109–117	PROPN
ejpam-5390	511	8	,	,	PUNCT
ejpam-5390	511	9	2017	2017	NUM
ejpam-5390	511	10	.	.	PUNCT
ejpam-5390	512	1	[	[	X
ejpam-5390	512	2	6	6	NUM
ejpam-5390	512	3	]	]	PUNCT
ejpam-5390	512	4	c.	c.	PROPN
ejpam-5390	512	5	li	li	PROPN
ejpam-5390	512	6	and	and	CCONJ
ejpam-5390	512	7	d.	d.	PROPN
ejpam-5390	512	8	zhang	zhang	PROPN
ejpam-5390	512	9	.	.	PUNCT
ejpam-5390	513	1	nonlinear	nonlinear	PROPN
ejpam-5390	513	2	mixed	mixed	PROPN
ejpam-5390	513	3	jordan	jordan	PROPN
ejpam-5390	513	4	triple	triple	ADJ
ejpam-5390	513	5	-	-	PUNCT
ejpam-5390	513	6	derivations	derivation	NOUN
ejpam-5390	513	7	on	on	ADP
ejpam-5390	513	8	-	-	PUNCT
ejpam-5390	513	9	algebras	algebras	X
ejpam-5390	513	10	.	.	PUNCT
ejpam-5390	514	1	siberian	siberian	PROPN
ejpam-5390	514	2	mathematical	mathematical	ADJ
ejpam-5390	514	3	journal	journal	NOUN
ejpam-5390	514	4	,	,	PUNCT
ejpam-5390	514	5	63(4):735–742	63(4):735–742	NUM
ejpam-5390	514	6	,	,	PUNCT
ejpam-5390	514	7	2022	2022	NUM
ejpam-5390	514	8	.	.	PUNCT
ejpam-5390	515	1	[	[	X
ejpam-5390	515	2	7	7	X
ejpam-5390	515	3	]	]	X
ejpam-5390	515	4	c.	c.	PROPN
ejpam-5390	515	5	li	li	PROPN
ejpam-5390	515	6	,	,	PUNCT
ejpam-5390	515	7	f.	f.	PROPN
ejpam-5390	515	8	zhao	zhao	PROPN
ejpam-5390	515	9	,	,	PUNCT
ejpam-5390	515	10	and	and	CCONJ
ejpam-5390	515	11	q.	q.	PROPN
ejpam-5390	515	12	chen	chen	PROPN
ejpam-5390	515	13	.	.	PUNCT
ejpam-5390	516	1	nonlinear	nonlinear	PROPN
ejpam-5390	516	2	skew	skew	NOUN
ejpam-5390	516	3	lie	lie	VERB
ejpam-5390	516	4	triple	triple	ADJ
ejpam-5390	516	5	derivations	derivation	NOUN
ejpam-5390	516	6	between	between	ADP
ejpam-5390	516	7	factors	factor	NOUN
ejpam-5390	516	8	.	.	PUNCT
ejpam-5390	517	1	acta	acta	PROPN
ejpam-5390	517	2	mathematica	mathematica	PROPN
ejpam-5390	517	3	sinica	sinica	PROPN
ejpam-5390	517	4	,	,	PUNCT
ejpam-5390	517	5	english	english	ADJ
ejpam-5390	517	6	series	series	NOUN
ejpam-5390	517	7	,	,	PUNCT
ejpam-5390	517	8	32(7):821–830	32(7):821–830	PROPN
ejpam-5390	517	9	,	,	PUNCT
ejpam-5390	517	10	2016	2016	NUM
ejpam-5390	517	11	.	.	PUNCT
ejpam-5390	518	1	[	[	X
ejpam-5390	518	2	8	8	NUM
ejpam-5390	518	3	]	]	X
ejpam-5390	518	4	c.	c.	PROPN
ejpam-5390	518	5	li	li	PROPN
ejpam-5390	518	6	,	,	PUNCT
ejpam-5390	518	7	y.	y.	PROPN
ejpam-5390	518	8	zhao	zhao	PROPN
ejpam-5390	518	9	,	,	PUNCT
ejpam-5390	518	10	and	and	CCONJ
ejpam-5390	518	11	f.	f.	PROPN
ejpam-5390	518	12	zhao	zhao	PROPN
ejpam-5390	518	13	.	.	PUNCT
ejpam-5390	519	1	nonlinear*-jordan	nonlinear*-jordan	ADJ
ejpam-5390	519	2	-	-	PUNCT
ejpam-5390	519	3	type	type	NOUN
ejpam-5390	519	4	derivations	derivation	NOUN
ejpam-5390	519	5	on*-algebras	on*-algebras	ADJ
ejpam-5390	519	6	.	.	PUNCT
ejpam-5390	520	1	rocky	rocky	ADJ
ejpam-5390	520	2	mountain	mountain	PROPN
ejpam-5390	520	3	journal	journal	NOUN
ejpam-5390	520	4	of	of	ADP
ejpam-5390	520	5	mathematics	mathematic	NOUN
ejpam-5390	520	6	,	,	PUNCT
ejpam-5390	520	7	51(2):601–612	51(2):601–612	PROPN
ejpam-5390	520	8	,	,	PUNCT
ejpam-5390	520	9	2021	2021	NUM
ejpam-5390	520	10	.	.	PUNCT
ejpam-5390	521	1	[	[	X
ejpam-5390	521	2	9	9	NUM
ejpam-5390	521	3	]	]	X
ejpam-5390	521	4	y.	y.	PROPN
ejpam-5390	521	5	liang	liang	PROPN
ejpam-5390	521	6	and	and	CCONJ
ejpam-5390	521	7	j.	j.	PROPN
ejpam-5390	521	8	zhang	zhang	PROPN
ejpam-5390	521	9	.	.	PUNCT
ejpam-5390	522	1	nonlinear	nonlinear	PROPN
ejpam-5390	522	2	mixed	mix	VERB
ejpam-5390	522	3	lie	lie	NOUN
ejpam-5390	522	4	triple	triple	ADJ
ejpam-5390	522	5	derivations	derivation	NOUN
ejpam-5390	522	6	on	on	ADP
ejpam-5390	522	7	factor	factor	NOUN
ejpam-5390	522	8	von	von	PROPN
ejpam-5390	522	9	neumann	neumann	PROPN
ejpam-5390	522	10	algebras	algebras	PROPN
ejpam-5390	522	11	.	.	PUNCT
ejpam-5390	523	1	acta	acta	PROPN
ejpam-5390	523	2	math	math	PROPN
ejpam-5390	523	3	sci	sci	PROPN
ejpam-5390	523	4	chinese	chinese	PROPN
ejpam-5390	523	5	series	series	PROPN
ejpam-5390	523	6	,	,	PUNCT
ejpam-5390	523	7	62:1–13	62:1–13	NUM
ejpam-5390	523	8	,	,	PUNCT
ejpam-5390	523	9	2019	2019	NUM
ejpam-5390	523	10	.	.	PUNCT
ejpam-5390	524	1	[	[	X
ejpam-5390	524	2	10	10	NUM
ejpam-5390	524	3	]	]	X
ejpam-5390	524	4	y.	y.	PROPN
ejpam-5390	524	5	pang	pang	PROPN
ejpam-5390	524	6	,	,	PUNCT
ejpam-5390	524	7	d.	d.	PROPN
ejpam-5390	524	8	zhang	zhang	PROPN
ejpam-5390	524	9	,	,	PUNCT
ejpam-5390	524	10	and	and	CCONJ
ejpam-5390	524	11	d.	d.	PROPN
ejpam-5390	524	12	ma	ma	PROPN
ejpam-5390	524	13	.	.	PUNCT
ejpam-5390	525	1	the	the	DET
ejpam-5390	525	2	second	second	ADJ
ejpam-5390	525	3	nonlinear	nonlinear	ADJ
ejpam-5390	525	4	mixed	mixed	ADJ
ejpam-5390	525	5	jordan	jordan	PROPN
ejpam-5390	525	6	triple	triple	ADJ
ejpam-5390	525	7	derivable	derivable	ADJ
ejpam-5390	525	8	mapping	mapping	NOUN
ejpam-5390	525	9	on	on	ADP
ejpam-5390	525	10	factor	factor	NOUN
ejpam-5390	525	11	von	von	PROPN
ejpam-5390	525	12	neumann	neumann	PROPN
ejpam-5390	525	13	algebras	algebras	PROPN
ejpam-5390	525	14	.	.	PUNCT
ejpam-5390	526	1	bulletin	bulletin	NOUN
ejpam-5390	526	2	of	of	ADP
ejpam-5390	526	3	the	the	DET
ejpam-5390	526	4	iranian	iranian	PROPN
ejpam-5390	526	5	mathematical	mathematical	PROPN
ejpam-5390	526	6	society	society	NOUN
ejpam-5390	526	7	,	,	PUNCT
ejpam-5390	526	8	48(3):951–962	48(3):951–962	PROPN
ejpam-5390	526	9	,	,	PUNCT
ejpam-5390	526	10	2022	2022	NUM
ejpam-5390	526	11	.	.	PUNCT
ejpam-5390	527	1	[	[	X
ejpam-5390	527	2	11	11	NUM
ejpam-5390	527	3	]	]	X
ejpam-5390	527	4	n.	n.	PROPN
ejpam-5390	527	5	rehman	rehman	PROPN
ejpam-5390	527	6	,	,	PUNCT
ejpam-5390	527	7	j.	j.	PROPN
ejpam-5390	527	8	nisar	nisar	PROPN
ejpam-5390	527	9	,	,	PUNCT
ejpam-5390	527	10	and	and	CCONJ
ejpam-5390	527	11	m.	m.	PROPN
ejpam-5390	527	12	nazim	nazim	PROPN
ejpam-5390	527	13	.	.	PUNCT
ejpam-5390	528	1	a	a	DET
ejpam-5390	528	2	note	note	NOUN
ejpam-5390	528	3	on	on	ADP
ejpam-5390	528	4	nonlinear	nonlinear	ADJ
ejpam-5390	528	5	mixed	mix	VERB
ejpam-5390	528	6	jordan	jordan	PROPN
ejpam-5390	528	7	triple	triple	ADJ
ejpam-5390	528	8	derivation	derivation	NOUN
ejpam-5390	528	9	on*-algebras	on*-algebra	NOUN
ejpam-5390	528	10	.	.	PUNCT
ejpam-5390	529	1	communications	communication	NOUN
ejpam-5390	529	2	in	in	ADP
ejpam-5390	529	3	algebra	algebra	NOUN
ejpam-5390	529	4	,	,	PUNCT
ejpam-5390	529	5	51(4):1334–1343	51(4):1334–1343	PROPN
ejpam-5390	529	6	,	,	PUNCT
ejpam-5390	529	7	2023	2023	NUM
ejpam-5390	529	8	.	.	PUNCT
ejpam-5390	530	1	references	reference	NOUN
ejpam-5390	530	2	3414	3414	NUM
ejpam-5390	531	1	[	[	X
ejpam-5390	531	2	12	12	NUM
ejpam-5390	531	3	]	]	PUNCT
ejpam-5390	531	4	f.	f.	PROPN
ejpam-5390	531	5	zhang	zhang	PROPN
ejpam-5390	531	6	.	.	PUNCT
ejpam-5390	532	1	nonlinear	nonlinear	PROPN
ejpam-5390	532	2	η	η	PROPN
ejpam-5390	532	3	-	-	PROPN
ejpam-5390	532	4	jordan	jordan	PROPN
ejpam-5390	532	5	triple	triple	ADJ
ejpam-5390	532	6	∗-derivation	∗-derivation	NOUN
ejpam-5390	532	7	on	on	ADP
ejpam-5390	532	8	prime	prime	ADJ
ejpam-5390	532	9	∗-algebras	∗-algebra	NOUN
ejpam-5390	532	10	.	.	PUNCT
ejpam-5390	533	1	rocky	rocky	ADJ
ejpam-5390	533	2	mountain	mountain	PROPN
ejpam-5390	533	3	j.	j.	PROPN
ejpam-5390	533	4	math	math	PROPN
ejpam-5390	533	5	,	,	PUNCT
ejpam-5390	533	6	52:323–333	52:323–333	PROPN
ejpam-5390	533	7	,	,	PUNCT
ejpam-5390	533	8	2022	2022	NUM
ejpam-5390	533	9	.	.	PUNCT
ejpam-5390	534	1	[	[	X
ejpam-5390	534	2	13	13	NUM
ejpam-5390	534	3	]	]	PUNCT
ejpam-5390	534	4	z.	z.	PROPN
ejpam-5390	534	5	zhou	zhou	PROPN
ejpam-5390	534	6	,	,	PUNCT
ejpam-5390	534	7	y.	y.	PROPN
ejpam-5390	534	8	zhujun	zhujun	PROPN
ejpam-5390	534	9	,	,	PUNCT
ejpam-5390	534	10	and	and	CCONJ
ejpam-5390	534	11	z	z	PROPN
ejpam-5390	534	12	jianhua	jianhua	PROPN
ejpam-5390	534	13	.	.	PUNCT
ejpam-5390	535	1	nonlinear	nonlinear	ADJ
ejpam-5390	535	2	mixed	mixed	ADJ
ejpam-5390	535	3	lie	lie	NOUN
ejpam-5390	535	4	triple	triple	ADJ
ejpam-5390	535	5	derivations	derivation	NOUN
ejpam-5390	535	6	on	on	ADP
ejpam-5390	535	7	prime	prime	ADJ
ejpam-5390	535	8	*	*	PUNCT
ejpam-5390	535	9	-algebras	-algebra	NOUN
ejpam-5390	535	10	.	.	PUNCT
ejpam-5390	536	1	communications	communication	NOUN
ejpam-5390	536	2	in	in	ADP
ejpam-5390	536	3	algebra	algebra	NOUN
ejpam-5390	536	4	,	,	PUNCT
ejpam-5390	536	5	47(11):4791–4796	47(11):4791–4796	NUM
ejpam-5390	536	6	,	,	PUNCT
ejpam-5390	536	7	2019	2019	NUM
ejpam-5390	536	8	.	.	PUNCT