
theorem FirstIsExact:
  for T being with_equivalence naturally_generated non empty TopRelStr,
      A being Subset of T holds
    A is 1st_class iff A is exact
  proof
    let T be with_equivalence naturally_generated non empty TopRelStr,
        A be Subset of T;
    thus A is 1st_class implies A is exact
    proof
      assume A is 1st_class; then
      LAp A = UAp A by ROUGHS_1:14,FirstApprox; then
      LAp A = A by ROUGHS_1:13,12;
      hence thesis;
    end;
    assume A is exact;
    hence thesis by ROUGHS_1:16;
  end;
