reserve R for non empty RelStr,
  N for net of R,
  i for Element of N;

theorem Th22:
  for R being complete LATTICE, N being monotone reflexive net of R
  holds lim_inf N = sup N
proof
  let R be complete LATTICE, N be monotone reflexive net of R;
  deffunc F(Element of N) = "/\"({N.i where i is Element of N: i >= $1},R);
  set X = {F(j) where j is Element of N: P[j]};
  deffunc G(Element of N) = N.$1;
A1: for j being Element of N holds G(j) = F(j)
  proof
    let j be Element of N;
    set Y = {N.i where i is Element of N: i >= j};
A2: N.j is_<=_than Y
    proof
      let y be Element of R;
      assume y in Y;
      then ex i being Element of N st y = N.i & j <= i;
      hence N.j <= y by Def9;
    end;
    for b being Element of R st b is_<=_than Y holds N.j >= b
    proof
      let b be Element of R;
      assume
A3:   b is_<=_than Y;
      reconsider j9 = j as Element of N;
      j9 <= j9;
      then N.j9 in Y;
      hence thesis by A3;
    end;
    hence thesis by A2,YELLOW_0:33;
  end;
  rng the mapping of N = { G(j) where j is Element of N: P[j]} by Th19
    .= X from FRAENKEL:sch 5(A1);
  hence lim_inf N = Sup the mapping of N by YELLOW_2:def 5
    .= sup N by WAYBEL_2:def 1;
end;
