
theorem Th37:
  for T being complete LATTICE for N being net of T, M being subnet of N
  holds lim_inf N <= lim_inf M
proof
  let T be complete LATTICE;
  let N be net of T, M be subnet of N;
  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;
A1: "\/"(infy(M), T) is_>=_than infy(N)
  proof
    let t be Element of T;
    assume t in infy(N);
    then consider j being Element of N such that
A2: t = "/\"({N.i where i is Element of N: i >= j}, T);
    set e = the Embedding of M,N;
    reconsider j as Element of N;
    consider j9 being Element of M such that
A3: for p being Element of M st j9 <= p holds j <= e.p by Def3;
    set X = {N.i where i is Element of N: i >= j};
    set Y = {M.i where i is Element of M: i >= j9};
A4: ex_inf_of X, T by YELLOW_0:17;
A5: ex_inf_of Y, T by YELLOW_0:17;
    Y c= X
    proof
      let y be object;
      assume y in Y;
      then consider i being Element of M such that
A6:   y = M.i and
A7:   i >= j9;
      reconsider i as Element of M;
      e.i >= j by A3,A7;
      then N.(e.i) in X;
      hence thesis by A6,Th36;
    end;
    then
A8: t <= "/\"(Y, T) by A2,A4,A5,YELLOW_0:35;
    "/\"(Y, T) in infy(M);
    then "/\"(Y, T) <= "\/"(infy(M), T) by YELLOW_2:22;
    hence thesis by A8,YELLOW_0:def 2;
  end;
A9: lim_inf M = "\/"(infy(M), T) by WAYBEL11:def 6;
  lim_inf N = "\/"(infy(N), T) by WAYBEL11:def 6;
  hence thesis by A1,A9,YELLOW_0:32;
end;
