
theorem SecondRough:
  for T being with_equivalence naturally_generated non empty TopRelStr,
      A being Subset of T holds
    A is 2nd_class iff A is rough
  proof
    let T be with_equivalence naturally_generated non empty TopRelStr,
        A be Subset of T;
    thus A is 2nd_class implies A is rough;
    assume
C1: A is rough;
    LAp A <> UAp A
    proof
      assume LAp A = UAp A; then
      LAp A = A by ROUGHS_1:13,12;
      hence thesis by C1;
    end; then
    LAp A c< UAp A by ROUGHS_1:14;
    hence thesis by SecondClass;
  end;
