
theorem Th9:
  for R being /\-complete Semilattice, N being monotone reflexive net of R
  holds lim_inf N = sup N
proof
  let R be /\-complete Semilattice, N be monotone reflexive net of R;
  deffunc F(Element of N) = "/\"({N.i where i is Element of N: i >= $1},R);
  deffunc G(Element of N) = N.$1;
  defpred P[set] means not contradiction;
  set X = {F(j) where j is Element of N: P[j]};
A1: for j being Element of N holds G(j) = F(j)
  proof
    let j be Element of N;
    defpred P[Element of N] means $1 >= j;
    set Y = {G(i) where i is Element of N: P[i]};
    j <= j;
    then
A2: N.j in Y;
    Y is Subset of R from DOMAIN_1:sch 8;
    then
A3: ex_inf_of Y,R by A2,WAYBEL_0:76;
A4: 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 WAYBEL11:def 9;
    end;
    for b being Element of R st b is_<=_than Y holds N.j >= b
    proof
      let b be Element of R;
      assume
A5:   b is_<=_than Y;
      reconsider j9 = j as Element of N;
      j9 <= j9;
      then N.j9 in Y;
      hence thesis by A5;
    end;
    hence thesis by A3,A4,YELLOW_0:def 10;
  end;
  rng the mapping of N = {G(j) where j is Element of N: P[j]} by WAYBEL11:19
    .= 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;
