reserve V for non empty RealLinearSpace;

theorem
  for V be non empty RealLinearSpace, f,g,h be VECTOR of V*' holds
    h = f+g iff for x be VECTOR of V holds h.x = f.x + g.x
proof
  let V be non empty RealLinearSpace;
  let f,g,h be VECTOR of V*';
  consider Y be non empty 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 assume A3: h = f+g;
   let x be Element of V;
   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;
  end;
  now
   assume for x be Element of V holds h.x=f.x+g.x;
   then
   for x be 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;
  end;
  hence thesis by A2;
end;
