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

theorem Th39: :: Theorem 8
  A is 2nd_class iff A` is 2nd_class
proof
A1: for A being Subset of T st A` is 2nd_class holds A is 2nd_class
  proof
    let A be Subset of T;
    assume A` is 2nd_class;
    then
A2: Cl Int A` c< Int Cl A`;
    then Cl Int A` c= Int Cl A` by XBOOLE_0:def 8;
    then Cl Int A` c= Int (Int A)` by TDLAT_3:2;
    then Cl Int A` c= (Cl Int A)` by TDLAT_3:3;
    then Cl (Cl A)` c= (Cl Int A)` by TDLAT_3:3;
    then (Int Cl A)` c= (Cl Int A)` by TDLAT_3:2;
    then
A3: Cl Int A c= Int Cl A by SUBSET_1:12;
    Cl (Cl A)` <> Int Cl A` by A2,TDLAT_3:3;
    then Cl (Cl A)` <> Int (Int A)` by TDLAT_3:2;
    then (Cl Int A)` <> Cl (Cl A)` by TDLAT_3:3;
    then Cl Int A <> Int Cl A by TDLAT_3:2;
    then Cl Int A c< Int Cl A by A3,XBOOLE_0:def 8;
    hence thesis;
  end;
  A is 2nd_class implies A` is 2nd_class
  proof
    assume A is 2nd_class;
    then A`` is 2nd_class;
    hence thesis by A1;
  end;
  hence thesis by A1;
end;
