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 st A is closed or B is closed 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;
  assume
A1: A is closed or B is closed;
  r.: (A /\ B) = M /\ (MaxADSet(A /\ B)) by Def12
    .= (M /\ M) /\ (MaxADSet(A) /\ MaxADSet(B)) by A1,TEX_4:64
    .= M /\ (M /\ (MaxADSet(A) /\ MaxADSet(B))) by XBOOLE_1:16
    .= ((M /\ MaxADSet(A)) /\ MaxADSet(B)) /\ M by XBOOLE_1:16
    .= (M /\ MaxADSet(A)) /\ (M /\ MaxADSet(B)) by XBOOLE_1:16
    .= (r.: A) /\ (M /\ MaxADSet(B)) by Def12
    .= (r.: A) /\ (r.: B) by Def12;
  hence thesis;
end;
