let V be RealLinearSpace; :: thesis: for f, g, h being VECTOR of (V *') holds

( h = f + g iff for x being VECTOR of V holds h . x = (f . x) + (g . x) )

let f, g, h be VECTOR of (V *'); :: thesis: ( h = f + g iff for x being VECTOR of V holds h . x = (f . x) + (g . x) )

consider Y being VectSp of F_Real such that

AS1: ( Y = RLSp2RVSp V & V *' = RVSp2RLSp (Y *') ) by def2;

reconsider f1 = f, g1 = g, h1 = h as linear-Functional of Y by AS1, HAHNBAN1:def 10;

( h = f + g iff for x being VECTOR of V holds h . x = (f . x) + (g . x) )

let f, g, h be VECTOR of (V *'); :: thesis: ( h = f + g iff for x being VECTOR of V holds h . x = (f . x) + (g . x) )

consider Y being VectSp of F_Real such that

AS1: ( Y = RLSp2RVSp V & V *' = RVSp2RLSp (Y *') ) by def2;

reconsider f1 = f, g1 = g, h1 = h as linear-Functional of Y by AS1, HAHNBAN1:def 10;

A2: now :: thesis: ( h = f + g implies for x being Element of V holds h . x = (f . x) + (g . x) )

assume A3:
h = f + g
; :: thesis: for x being Element of V holds h . x = (f . x) + (g . x)

let x be Element of V; :: thesis: h . x = (f . x) + (g . x)

reconsider x1 = x as Element of Y by AS1;

h1 = f1 + g1 by A3, AS1, HAHNBAN1:def 10;

then h1 . x1 = (f1 . x1) + (g1 . x1) by HAHNBAN1:def 3;

hence h . x = (f . x) + (g . x) ; :: thesis: verum

end;let x be Element of V; :: thesis: h . x = (f . x) + (g . x)

reconsider x1 = x as Element of Y by AS1;

h1 = f1 + g1 by A3, AS1, HAHNBAN1:def 10;

then h1 . x1 = (f1 . x1) + (g1 . x1) by HAHNBAN1:def 3;

hence h . x = (f . x) + (g . x) ; :: thesis: verum

now :: thesis: ( ( for x being Element of V holds h . x = (f . x) + (g . x) ) implies h = f + g )

hence
( h = f + g iff for x being VECTOR of V holds h . x = (f . x) + (g . x) )
by A2; :: thesis: verumassume
for x being Element of V holds h . x = (f . x) + (g . x)
; :: thesis: h = f + g

then for x being Element of Y holds h1 . x = (f1 . x) + (g1 . x) by AS1;

then h1 = f1 + g1 by HAHNBAN1:def 3;

hence h = f + g by AS1, HAHNBAN1:def 10; :: thesis: verum

end;then for x being Element of Y holds h1 . x = (f1 . x) + (g1 . x) by AS1;

then h1 = f1 + g1 by HAHNBAN1:def 3;

hence h = f + g by AS1, HAHNBAN1:def 10; :: thesis: verum