reserve V for non empty RealLinearSpace;

theorem
for V be non empty RealLinearSpace, f,h be VECTOR of V*', a be Real
  holds h = a*f iff for x be VECTOR of V holds h.x = a * f.x
proof
  let V be non empty RealLinearSpace, f,h be VECTOR of V*', a be Real;
  reconsider a1=a as Element of F_Real by XREAL_0:def 1;
  consider Y be non empty 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 assume A3: h = a*f;
   hereby let x be Element of V;
    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;
   end;
  end;
  assume for x be Element of V holds h.x=a*f.x;
  then
  for x be 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;
end;
