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 Th29:
  for A being Subset of X holds (Stone-retraction(X,X0))"((
  Stone-retraction(X,X0)).: A) = MaxADSet(A)
proof
  let A be Subset of X;
  reconsider C = the carrier of X0 as non empty Subset of X by TSEP_1:1;
  set r = Stone-retraction(X,X0);
  C c= the carrier of X;
  then reconsider B = r.: A as Subset of X by XBOOLE_1:1;
  C /\ MaxADSet(A) = B by Def12;
  then
A1: B c= MaxADSet(A) by XBOOLE_1:17;
A2: r" (r.: A) = MaxADSet(B) by Th26;
  then A c= MaxADSet(B) by FUNCT_2:42;
  hence thesis by A2,A1,TEX_4:35;
end;
