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 Subset of X holds (Stone-retraction(X,X0)).: A = (
  Stone-retraction(X,X0)).: MaxADSet(A)
proof
  let A be Subset of X;
  set r = Stone-retraction(X,X0);
A1: rng r = r.: the carrier of X by RELSET_1:22;
  r.: r" (r.: A) = r.: MaxADSet(A) by Th29;
  hence thesis by A1,FUNCT_1:77,RELAT_1:123;
end;
