reserve T for TopSpace,
  A, B for Subset of T;

theorem Th41: :: Theorem 9
  A is 1st_class implies Int Cl A = Int Cl Int A & Cl Int A = Cl Int Cl A
proof
  Cl Int A c= Cl A by PRE_TOPC:19,TOPS_1:16;
  then
A1: Int Cl Int A c= Int Cl A by TOPS_1:19;
  Int A c= Int Cl A by PRE_TOPC:18,TOPS_1:19;
  then
A2: Cl Int A c= Cl Int Cl A by PRE_TOPC:19;
  assume A is 1st_class;
  then
A3: Int Cl A c= Cl Int A;
  then
A4: Cl Int Cl A c= Cl Cl Int A by PRE_TOPC:19;
  Int Int Cl A c= Int Cl Int A by A3,TOPS_1:19;
  hence thesis by A1,A4,A2,XBOOLE_0:def 10;
end;
