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

theorem Th10:
  for A,B being Subset of T holds Int(Cl(A \/ (Int(Cl B) \/ B)))
  \/ (A \/ (Int(Cl B) \/ B)) = Int Cl(A \/ B) \/ (A \/ B)
proof
  let A,B be Subset of T;
  Cl B c= Cl (A \/ B) by PRE_TOPC:19,XBOOLE_1:7;
  then
A1: Int Cl B c= Int Cl(A \/ B) by TOPS_1:19;
  A \/ B c= A \/ (Int(Cl B) \/ B) by XBOOLE_1:7,9;
  then Cl(A \/ B) c= Cl(A \/ (Int(Cl B) \/ B)) by PRE_TOPC:19;
  then
A2: Int Cl(A \/ B) c= Int(Cl(A \/ (Int(Cl B) \/ B))) by TOPS_1:19;
  Int(Cl(A \/ (Int(Cl B) \/ B))) c= Int(Cl(A \/ (Int(Cl B) \/ B))) \/ Int(
  Cl B) by XBOOLE_1:7;
  then
A3: Int Cl(A \/ B) c= Int(Cl(A \/ (Int(Cl B) \/ B))) \/ Int(Cl B) by A2;
  Int(Cl(A \/ (Int(Cl B) \/ B))) = Int(Cl((A \/ Int(Cl B)) \/ B)) by XBOOLE_1:4
    .= Int(Cl(A \/ Int(Cl B)) \/ Cl B) by PRE_TOPC:20
    .= Int(Cl A \/ Cl Int Cl B \/ Cl B) by PRE_TOPC:20
    .= Int(Cl A \/ (Cl Int Cl B \/ Cl B)) by XBOOLE_1:4
    .= Int(Cl A \/ Cl B) by Th3,XBOOLE_1:12
    .= Int Cl(A \/ B) by PRE_TOPC:20;
  then Int(Cl(A \/ (Int(Cl B) \/ B))) \/ Int(Cl B) c= Int Cl(A \/ B) by A1,
XBOOLE_1:8;
  then
A4: Int(Cl(A \/ (Int(Cl B) \/ B))) \/ Int(Cl B) = Int Cl(A \/ B) by A3;
  A \/ (Int(Cl B) \/ B) = Int(Cl B) \/ (A \/ B) by XBOOLE_1:4;
  hence thesis by A4,XBOOLE_1:4;
end;
