reserve T for TopSpace;

theorem Th6:
  for A being Subset of T st Int Cl A c= A holds Int Cl(A /\ Cl Int
  A) c= A /\ Cl Int A
proof
  let A be Subset of T;
  assume
A1: Int Cl A c= A;
A2: Int Cl Int A c= Cl Int A by TOPS_1:16;
A3: Cl Int A c= Cl A by PRE_TOPC:19,TOPS_1:16;
  then Int(Cl Int A) c= Int(Cl A) by TOPS_1:19;
  then Int Cl Int A c= A by A1;
  then (Int Cl Int A) /\ (Int Cl Int A) c= A /\ Cl Int A by A2,XBOOLE_1:27;
  then
A4: Int((Cl A) /\ (Cl Int A)) c= A /\ Cl Int A by A3,XBOOLE_1:28;
  Cl (A /\ Cl Int A) c= (Cl A) /\ (Cl Cl Int A) by PRE_TOPC:21;
  then Int(Cl (A /\ Cl Int A)) c= Int((Cl A) /\ (Cl Int A)) by TOPS_1:19;
  hence thesis by A4;
end;
