
theorem
  for X be compact non empty TopSpace
  for F,G,H be Point of R_Normed_Algebra_of_ContinuousFunctions(X)
  for f,g,h be RealMap of X holds
  f=F & g=G & h=H implies
  (H = F-G iff (for x be Element of X holds h.x = f.x - g.x))
proof
  let X be compact non empty TopSpace;
  let F,G,H be Point of R_Normed_Algebra_of_ContinuousFunctions(X);
  let f,g,h be RealMap of 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.R_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,Th15;
      hence f.x-g.x=h.x;
    end;
    hence for x be Element of X holds h.x = f.x - g.x;
  end;
  now assume
A4: for x be 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,Th15; then
    F-G=H+(G-G) by RLVECT_1:def 3; then
    F-G=H+0.R_Normed_Algebra_of_ContinuousFunctions(X) by RLVECT_1:15;
    hence F-G=H by RLVECT_1:4;
  end;
  hence thesis by A2;
end;
