reserve X for non empty TopSpace;
reserve X for non empty TopSpace;
reserve X for non empty TopSpace,
  X0 for non empty maximal_Kolmogorov_subspace of X;

theorem
  for r being Function of X,X0 holds (for a being Point of X holds r.a
  in MaxADSet(a)) implies r is continuous Function of X,X0
proof
  let r be Function of X,X0;
  reconsider M = the carrier of X0 as Subset of X by TSEP_1:1;
  assume
A1: for a being Point of X holds r.a in MaxADSet(a);
  M is maximal_T_0 by Th11;
  then
A2: M is T_0;
A3: M c= the carrier of X;
  for a being Point of X holds M /\ MaxADSet(a) = {r.a}
  proof
    let a be Point of X;
    reconsider s = r.a as Point of X by A3,TARSKI:def 3;
A4: s in MaxADSet(a) by A1;
    M /\ MaxADSet(s) = {s} by A2;
    hence thesis by A4,TEX_4:21;
  end;
  hence thesis by Th18;
end;
