
theorem :: Theorem 1.3.2. (d)
  for T being TopSpace, A, B being Subset of T holds Fr (A /\ B) c= (Cl
  A /\ Fr B) \/ (Fr A /\ Cl B)
proof
  let T be TopSpace, A, B be Subset of T;
A1: Cl A /\ Cl B /\ ((Cl A`) \/ Cl B`) = (Cl A /\ Cl B /\ (Cl A`)) \/ (Cl A
  /\ Cl B /\ Cl B`) by XBOOLE_1:23
    .= (Cl A /\ Cl A` /\ Cl B) \/ (Cl A /\ Cl B /\ Cl B`) by XBOOLE_1:16
    .= (Fr A /\ Cl B) \/ (Cl A /\ Cl B /\ Cl B`) by TOPS_1:def 2
    .= (Fr A /\ Cl B) \/ (Cl A /\ (Cl B /\ Cl B`)) by XBOOLE_1:16
    .= (Fr A /\ Cl B) \/ (Cl A /\ Fr B) by TOPS_1:def 2;
  Fr (A /\ B) = Cl (A /\ B) /\ Cl (A /\ B)` by TOPS_1:def 2
    .= Cl (A /\ B) /\ Cl (A` \/ B`) by XBOOLE_1:54
    .= Cl (A /\ B) /\ ((Cl A`) \/ Cl B`) by PRE_TOPC:20;
  hence thesis by A1,PRE_TOPC:21,XBOOLE_1:26;
end;
