reserve T for 1-sorted;
reserve T for TopSpace;

theorem Th12:
  for A,B being Subset of T holds Cl(Int(A /\ (Cl(Int B) /\ B)))
  /\ (A /\ (Cl(Int B) /\ B)) = Cl Int(A /\ B) /\ (A /\ B)
proof
  let A,B be Subset of T;
  Int (A /\ B) c= Int B by TOPS_1:19,XBOOLE_1:17;
  then
A1: Cl Int(A /\ B) c= Cl(Int B) by PRE_TOPC:19;
  Cl(Int(A /\ (Cl(Int B) /\ B))) = Cl(Int((A /\ Cl(Int B)) /\ B)) by
XBOOLE_1:16
    .= Cl(Int(A /\ Cl(Int B)) /\ Int B) by TOPS_1:17
    .= Cl(Int A /\ Int Cl Int B /\ Int B) by TOPS_1:17
    .= Cl(Int A /\ (Int Cl Int B /\ Int B)) by XBOOLE_1:16
    .= Cl(Int A /\ Int B) by Th4,XBOOLE_1:28
    .= Cl Int(A /\ B) by TOPS_1:17;
  then
A2: Cl Int(A /\ B) c= Cl(Int(A /\ (Cl(Int B) /\ B))) /\ Cl(Int B) by A1,
XBOOLE_1:19;
  A /\ (Cl(Int B) /\ B) c= A /\ B by XBOOLE_1:17,26;
  then Int(A /\ (Cl(Int B) /\ B)) c= Int(A /\ B) by TOPS_1:19;
  then
A3: Cl(Int(A /\ (Cl(Int B) /\ B))) c= Cl Int(A /\ B) by PRE_TOPC:19;
  Cl(Int(A /\ (Cl(Int B) /\ B))) /\ Cl(Int B) c= Cl(Int(A /\ (Cl(Int B) /\
  B))) by XBOOLE_1:17;
  then Cl(Int(A /\ (Cl(Int B) /\ B))) /\ Cl(Int B) c= Cl Int(A /\ B) by A3;
  then
A4: Cl(Int(A /\ (Cl(Int B) /\ B))) /\ Cl(Int B) = Cl Int(A /\ B) by A2;
  A /\ (Cl(Int B) /\ B) = Cl(Int B) /\ (A /\ B) by XBOOLE_1:16;
  hence thesis by A4,XBOOLE_1:16;
end;
