reserve V for non empty RealLinearSpace;

theorem Th20b:
for X be RealLinearSpace, f,g,h be VECTOR of X*' holds
  h = f+g iff for x be VECTOR of X holds h.x = f.x + g.x
proof
  let X be RealLinearSpace, f,g,h be VECTOR of X*';
A1:X*' is Subspace of RealVectSpace(the carrier of X) by Th17,RSSPACE:11;
  then reconsider f1=f, g1=g, h1=h as VECTOR
         of RealVectSpace(the carrier of X) by RLSUB_1:10;
  hereby assume
A3: h = f+g;
    let x be Element of X;
    h1=f1+g1 by A1,A3,RLSUB_1:13;
    hence h.x=f.x+g.x by FUNCSDOM:1;
  end;
  assume for x be Element of X holds h.x=f.x+g.x;
  then h1=f1+g1 by FUNCSDOM:1;
  hence h =f+g by A1,RLSUB_1:13;
end;
