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 Th27:
  for a being Point of X holds (Stone-retraction(X,X0))" {(
  Stone-retraction(X,X0)).a} = MaxADSet(a)
proof
  let a be Point of X;
  reconsider A = the carrier of X0 as Subset of X by TSEP_1:1;
  set r = Stone-retraction(X,X0);
  r.a in A;
  then reconsider b = r.a as Point of X;
A1: r.a in MaxADSet(a) by Def11;
  r" {r.a} = MaxADSet(b) by Th25;
  hence thesis by A1,TEX_4:21;
end;
