 reserve
  S for non empty TopSpace,
  T for LinearTopSpace,
  X for non empty Subset of the carrier of S;
 reserve
    S,T for RealNormSpace,
    X for non empty Subset of the carrier of S;

theorem
  for S be non empty compact TopSpace,T be NormedLinearTopSpace
  for F,G,H be Point of R_NormSpace_of_ContinuousFunctions(S,T)
  for f,g,h be Function of S,T holds
  f=F & g=G & h=H implies
  (H = F-G iff (for x be Element of S holds h.x = f.x - g.x))
proof
  let S be non empty compact TopSpace,T be NormedLinearTopSpace;
  let F,G,H be Point of R_NormSpace_of_ContinuousFunctions(S,T);
  let f,g,h be Function of S,T;
  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
A3:    H+G=F-0.R_NormSpace_of_ContinuousFunctions(S,T) by RLVECT_1:15;
    now let x be Element of S;
      f.x -g.x = h.x + g.x -g.x by A1,A3,Th44;
      hence f.x-g.x=h.x by RLVECT_4:1;
    end;
    hence for x be Element of S holds h.x = f.x - g.x;
  end;
  now assume
A4: for x be Element of S holds h.x = f.x - g.x;
    now let x be Element of S;
      h.x = f.x - g.x by A4;
      hence h.x + g.x= f.x by RLVECT_4:1;
    end; then
    F=H+G by A1,Th44; then
    F-G=H+(G-G) by RLVECT_1:def 3; then
    F-G=H+0.R_NormSpace_of_ContinuousFunctions(S,T) by RLVECT_1:15;
    hence F-G=H;
  end;
  hence thesis by A2;
end;
