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 a being Point of X holds Im(Stone-retraction(X,X0),a) = (
  Stone-retraction(X,X0)).: MaxADSet(a)
proof
  let a be Point of X;
  set r = Stone-retraction(X,X0);
A1: dom r = [#]X by FUNCT_2:def 1;
A2: r.: r" {r.a} c= {r.a} by FUNCT_1:75;
A3: r"{r.a} = MaxADSet(a) by Th27;
  then r.: r" {r.a} <> {} by A1,RELAT_1:119;
  then r.: r" {r.a} = {r.a} by A2,ZFMISC_1:33;
  hence thesis by A1,A3,FUNCT_1:59;
end;
