
theorem Th24:
  for R being up-complete /\-complete LATTICE, N being net of R holds
  sup inf_net N = lim_inf N
proof
  let R be up-complete /\-complete LATTICE, N be net of R;
  defpred P[set] means not contradiction;
  deffunc F(Element of N) = "/\"({N.l where l is Element of N: l >= $1},R);
  sup inf_net N = Sup the mapping of (inf_net N) by WAYBEL_2:def 1
    .= sup rng the mapping of (inf_net N) by YELLOW_2:def 5
    .= lim_inf N by Th23;
  hence thesis;
end;
