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

theorem Th42: :: Theorem 9
  (Int Cl A = Int Cl Int A or Cl Int A = Cl Int Cl A) implies A is 1st_class
proof
  assume
A1: Int Cl A = Int Cl Int A or Cl Int A = Cl Int Cl A;
  per cases by A1;
  suppose
A2: Int Cl A = Int Cl Int A;
    Int Cl Int A c= Cl Int A by TOPS_1:16;
    hence thesis by A2;
  end;
  suppose
A3: Cl Int A = Cl Int Cl A;
    Int Cl A c= Cl Int Cl A by PRE_TOPC:18;
    hence thesis by A3;
  end;
end;
