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

theorem Th38: :: Theorem 8
  A is 1st_class iff A` is 1st_class
proof
A1: A` is 1st_class implies A is 1st_class
  proof
    assume A` is 1st_class;
    then Int Cl A` c= Cl Int A`;
    then Int (Int A)` c= Cl Int A` by TDLAT_3:2;
    then (Cl Int A)` c= Cl Int A` by TDLAT_3:3;
    then (Cl Int A)` c= Cl (Cl A)` by TDLAT_3:3;
    then (Cl Int A)` c= (Int Cl A)` by TDLAT_3:2;
    then Int Cl A c= Cl Int A by SUBSET_1:12;
    hence thesis;
  end;
  A is 1st_class implies A` is 1st_class
  proof
    assume A is 1st_class;
    then Int Cl A c= Cl Int A;
    then (Cl Int A)` c= (Int Cl A)` by SUBSET_1:12;
    then Int (Int A)` c= (Int Cl A)` by TDLAT_3:3;
    then Int (Int A)` c= Cl (Cl A)` by TDLAT_3:2;
    then Int (Int A)` c= Cl Int A` by TDLAT_3:3;
    then Int Cl A` c= Cl Int A` by TDLAT_3:2;
    hence thesis;
  end;
  hence thesis by A1;
end;
