let V be RealNormSpace; for V1 being Subset of V
for x, y being Point of V
for x1, y1 being Point of (NLin V1)
for a being Real st x = x1 & y = y1 holds
( ||.x.|| = ||.x1.|| & x + y = x1 + y1 & a * x = a * x1 )
let V1 be Subset of V; for x, y being Point of V
for x1, y1 being Point of (NLin V1)
for a being Real st x = x1 & y = y1 holds
( ||.x.|| = ||.x1.|| & x + y = x1 + y1 & a * x = a * x1 )
let x, y be Point of V; for x1, y1 being Point of (NLin V1)
for a being Real st x = x1 & y = y1 holds
( ||.x.|| = ||.x1.|| & x + y = x1 + y1 & a * x = a * x1 )
let x1, y1 be Point of (NLin V1); for a being Real st x = x1 & y = y1 holds
( ||.x.|| = ||.x1.|| & x + y = x1 + y1 & a * x = a * x1 )
let a be Real; ( x = x1 & y = y1 implies ( ||.x.|| = ||.x1.|| & x + y = x1 + y1 & a * x = a * x1 ) )
assume A1:
( x = x1 & y = y1 )
; ( ||.x.|| = ||.x1.|| & x + y = x1 + y1 & a * x = a * x1 )
set l = NLin V1;
A2:
the carrier of (NLin V1) c= the carrier of V
by RLSUB_1:def 2;
reconsider x2 = x1 as Point of (Lin V1) ;
reconsider y2 = y1 as Point of (Lin V1) ;
thus ||.x1.|| =
( the normF of V | the carrier of (NLin V1)) . x1
by A2, DefNorm
.=
||.x.||
by A1, FUNCT_1:49
; ( x + y = x1 + y1 & a * x = a * x1 )
thus x1 + y1 =
x2 + y2
.=
x + y
by A1, RLSUB_1:13
; a * x = a * x1
thus a * x1 =
a * x2
.=
a * x
by A1, RLSUB_1:14
; verum