
theorem Th26:
  for X being non empty set,
      f, g, h being Function of X,COMPLEX,
      F, G, H being Point of C_Normed_Algebra_of_BoundedFunctions(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 set,
      f, g, h be Function of X,COMPLEX,
      F, G, H be Point of C_Normed_Algebra_of_BoundedFunctions(X);
  assume
A1: f = F & g = G & h = H;
A2:now
    assume
A3: 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 A3;
     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_BoundedFunctions X)
                                                         by RLVECT_1:15;
    hence F - G = H by RLVECT_1:4;
   end;
   now
    assume H = F - G;
    then H + G = F - (G - G) by RLVECT_1:29;
    then H + G = F - (0.C_Normed_Algebra_of_BoundedFunctions X)
                                                by RLVECT_1:15; then
A4: H + G = F by RLVECT_1:13;
    now
     let x be Element of X;
     f.x = (h.x) + (g.x) by A1,A4,Th22;
     hence (f.x) - (g.x) = h.x;
    end;
    hence for x being Element of X holds h.x = (f.x) - (g.x);
   end;
  hence thesis by A2;
end;
