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, B being Subset of X holds (Stone-retraction(X,X0)).:(A \/ B) =
  (Stone-retraction(X,X0)).:(A) \/ (Stone-retraction(X,X0)).:(B)
proof
  reconsider M = the carrier of X0 as Subset of X by TSEP_1:1;
  set r = Stone-retraction(X,X0);
  let A, B be Subset of X;
  r.: (A \/ B) = M /\ (MaxADSet(A \/ B)) by Def12
    .= M /\ (MaxADSet(A) \/ MaxADSet(B)) by TEX_4:36
    .= (M /\ MaxADSet(A)) \/ (M /\ MaxADSet(B)) by XBOOLE_1:23
    .= (r.: A) \/ (M /\ MaxADSet(B)) by Def12
    .= (r.: A) \/ (r.: B) by Def12;
  hence thesis;
end;
