reserve T for TopSpace;

theorem Th4:
  for A, B being Subset of T st A is open holds Int(Cl(A \/ B)) = B
  implies A c= B
proof
  let A, B be Subset of T;
  assume
A1: A is open;
A2: A c= A \/ B by XBOOLE_1:7;
  A \/ B c= Cl(A \/ B) by PRE_TOPC:18;
  then A c= Cl(A \/ B) by A2;
  then
A3: Int A c= Int Cl(A \/ B) by TOPS_1:19;
  assume Int(Cl(A \/ B)) = B;
  hence thesis by A1,A3,TOPS_1:23;
end;
