
theorem Th38:
  for T being complete LATTICE for N being net of T, M being subnet of N
  for e being Embedding of M, N
  st for i being Element of N, j being Element of M st e.j <= i
  ex j9 being Element of M st j9 >= j & N.i >= M.j9
  holds lim_inf N = lim_inf M
proof
  let T be complete LATTICE;
  let N be net of T, M be subnet of N;
  let e be Embedding of M, N such that
A1: for i being Element of N, j being Element of M st e.j <= i
  ex j9 being Element of M st j9 >= j & N.i >= M.j9;
  deffunc infy(net of T) = the set of all "/\" ({$1.i where i is Element of $1:
  i >= j}, T) where j is Element of $1;
  "\/"(infy(N), T) is_>=_than infy(M)
  proof
    let t be Element of T;
    assume t in infy(M);
    then consider j being Element of M such that
A2: t = "/\"({M.i where i is Element of M: i >= j}, T);
    reconsider j as Element of M;
    set j9 = e.j;
    set X = {N.i where i is Element of N: i >= j9};
    set Y = {M.i where i is Element of M: i >= j};
    t is_<=_than X
    proof
      let x be Element of T;
      assume x in X;
      then consider i being Element of N such that
A3:   x = N.i and
A4:   i >= j9;
      reconsider i as Element of N;
      consider k being Element of M such that
A5:   k >= j and
A6:   N.i >= M.k by A1,A4;
      M.k in Y by A5;
      then M.k >= t by A2,YELLOW_2:22;
      hence thesis by A3,A6,YELLOW_0:def 2;
    end;
    then
A7: t <= "/\"(X, T) by YELLOW_0:33;
    "/\"(X, T) in infy(N);
    then "/\"(X, T) <= "\/"(infy(N), T) by YELLOW_2:22;
    hence t <= "\/"(infy(N), T) by A7,YELLOW_0:def 2;
  end;
  then "\/"(infy(N), T) >= "\/"(infy(M), T) by YELLOW_0:32;
  then lim_inf N >= "\/"(infy(M), T) by WAYBEL11:def 6;
  then
A8: lim_inf N >= lim_inf M by WAYBEL11:def 6;
  lim_inf M >= lim_inf N by Th37;
  hence thesis by A8,YELLOW_0:def 3;
end;
