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

theorem Th40: :: Theorem 8
  A is 3rd_class iff A` is 3rd_class
proof
  (Int Cl A)` = Cl (Cl A)` by TDLAT_3:2
    .= Cl Int A` by TDLAT_3:3;
  then
A1: Int Cl A = (Cl Int A`)`;
  (Cl Int A)` = Int (Int A)` by TDLAT_3:3
    .= Int Cl A` by TDLAT_3:2;
  then
A2: Cl Int A = (Int Cl A`)`;
A3: A` is 3rd_class implies A is 3rd_class
  proof
    assume A` is 3rd_class;
    then
A4: Cl Int A`, Int Cl A` are_c=-incomparable;
    then not Int Cl A` c= Cl Int A` by XBOOLE_0:def 9;
    then
A5: not Int Cl A c= Cl Int A by A2,A1,SUBSET_1:12;
    not Cl Int A` c= Int Cl A` by A4,XBOOLE_0:def 9;
    then not Cl Int A c= Int Cl A by A2,A1,SUBSET_1:12;
    then Cl Int A, Int Cl A are_c=-incomparable by A5,XBOOLE_0:def 9;
    hence thesis;
  end;
  A is 3rd_class implies A` is 3rd_class
  proof
    assume A is 3rd_class;
    then
A6: Cl Int A, Int Cl A are_c=-incomparable;
    then not Int Cl A c= Cl Int A by XBOOLE_0:def 9;
    then
A7: not Int Cl A` c= Cl Int A` by A2,A1,SUBSET_1:12;
    not Cl Int A c= Int Cl A by A6,XBOOLE_0:def 9;
    then not Cl Int A` c= Int Cl A` by A2,A1,SUBSET_1:12;
    then Cl Int A`, Int Cl A` are_c=-incomparable by A7,XBOOLE_0:def 9;
    hence thesis;
  end;
  hence thesis by A3;
end;
