
theorem SecondClass:
  for T being with_equivalence naturally_generated non empty TopRelStr,
      A being Subset of T holds
    A is 2nd_class iff LAp A c< UAp A
  proof
    let T be with_equivalence naturally_generated non empty TopRelStr,
        A be Subset of T;
a2: LAp UAp A = UAp UAp A by ROUGHS_1:36 .= UAp A;
    UAp LAp A = LAp LAp A by ROUGHS_1:34 .= LAp A;
    hence thesis by a2;
  end;
