reserve T for TopSpace;

theorem Th5:
  for A being Subset of T st A c= Cl Int A holds A \/ Int Cl A c=
  Cl Int(A \/ Int Cl A)
proof
  let A be Subset of T;
  assume
A1: A c= Cl Int A;
A2: Int Cl A c= Cl Int Cl A by PRE_TOPC:18;
A3: Int A c= Int Cl A by PRE_TOPC:18,TOPS_1:19;
  then Cl(Int A) c= Cl(Int Cl A) by PRE_TOPC:19;
  then A c= Cl(Int Cl A) by A1;
  then A \/ Int Cl A c= (Cl Int Cl A) \/ (Cl Int Cl A) by A2,XBOOLE_1:13;
  then
A4: A \/ Int Cl A c= Cl((Int A) \/ (Int Cl A)) by A3,XBOOLE_1:12;
  (Int A) \/ (Int Int Cl A) c= Int (A \/ Int Cl A) by TOPS_1:20;
  then Cl((Int A) \/ (Int Cl A)) c= Cl(Int (A \/ Int Cl A)) by PRE_TOPC:19;
  hence thesis by A4;
end;
