
theorem
  for X being non empty compact TopSpace
  for f, g, h being Function of the carrier of X,COMPLEX
  for F, G, H being Point of C_Normed_Algebra_of_ContinuousFunctions X st
    f = F & g = G & h = H holds
      ( H = F - G iff for x being Element of X holds h.x = (f.x)-(g.x))
proof
  let X be non empty compact TopSpace;
  let f, g, h be Function of the carrier of X,COMPLEX;
  let F, G, H be Point of C_Normed_Algebra_of_ContinuousFunctions X;
  assume
A1: f = F & g = G & h = H;
A2:now
    assume H = F - G;
    then
      H + G = F - (G - G) by RLVECT_1:29;
    then
      H + G = F - (0.C_Normed_Algebra_of_ContinuousFunctions X)
                                                          by RLVECT_1:15;
    then
A3:   H + G = F by RLVECT_1:13;
    now
      let x be Element of X;
      f.x = (h.x) + (g.x) by A1,A3,Th22;
      hence (f.x) - (g.x) = h.x;
    end;
    hence for x being Element of X holds h . x = (f . x) - (g . x);
  end;
  now
    assume
A4:   for x being Element of X holds h.x = (f.x) - (g.x);
    now
      let x be Element of X;
      h . x = (f . x) - (g . x) by A4;
      hence (h . x) + (g . x) = f . x;
    end;
    then F = H + G by A1,Th22;
    then F - G = H + (G - G) by RLVECT_1:def 3;
    then F - G = H + (0.C_Normed_Algebra_of_ContinuousFunctions X)
                                                   by RLVECT_1:15;
    hence F - G = H by RLVECT_1:4;
  end;
  hence H = F - G iff for x being Element of X holds h.x = (f.x)-(g.x)
                                                     by A2;
end;
