let V be RealLinearSpace; :: thesis: for f, h being VECTOR of (V *')
for a being Real holds
( h = a * f iff for x being VECTOR of V holds h . x = a * (f . x) )
let f, h be VECTOR of (V *'); :: thesis: for a being Real holds
( h = a * f iff for x being VECTOR of V holds h . x = a * (f . x) )
let a be Real; :: thesis: ( h = a * f iff for x being VECTOR of V holds h . x = a * (f . x) )
reconsider a1 = a as Element of F_Real by XREAL_0:def 1;
consider Y being VectSp of F_Real such that
AS1: ( Y = RLSp2RVSp V & V *' = RVSp2RLSp (Y *') ) by def2;
reconsider f1 = f, h1 = h as linear-Functional of Y by AS1, HAHNBAN1:def 10;

hereby :: thesis: ( ( for x being VECTOR of V holds h . x = a * (f . x) ) implies h = a * f )

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

hereby :: thesis: verum

let x be Element of V; :: thesis: h . x = a * (f . x)
reconsider x1 = x as Element of Y by AS1;
h1 = a1 * f1 by A3, AS1, HAHNBAN1:def 10;
then h1 . x1 = a1 * (f1 . x1) by HAHNBAN1:def 6;
hence h . x = a * (f . x) ; :: thesis: verum

end;

end;

assume for x being Element of V holds h . x = a * (f . x) ; :: thesis: h = a * f
then for x being Element of Y holds h1 . x = a1 * (f1 . x) by AS1;
then h1 = a1 * f1 by HAHNBAN1:def 6;
hence h = a * f by AS1, HAHNBAN1:def 10; :: thesis: verum