
theorem Th41:
  for T being complete LATTICE, N being net of T
  for i being Element of N holds lim_inf (N|i) = lim_inf N
proof
  let T be complete LATTICE, N be net of T;
  let i be Element of N;
  reconsider M = N|i as subnet of N;
  reconsider e = incl(M,N) as Embedding of M, N by Th40;
  M is full SubNetStr of N by WAYBEL_9:14;
  then
A1: M is full SubRelStr of N by YELLOW_6:def 7;
A2: incl(M,N) = id the carrier of M by WAYBEL_9:13,YELLOW_9:def 1;
  now
    let k be Element of N, j be Element of M;
    consider j9 being Element of N such that
A3: j9 = j and
A4: j9 >= i by WAYBEL_9:def 7;
    assume e.j <= k;
    then
A5: k >= j9 by A2,A3;
    then k >= i by A4,YELLOW_0:def 2;
    then reconsider k9 = k as Element of M by WAYBEL_9:def 7;
    take k9;
    thus k9 >= j by A1,A3,A5,YELLOW_0:60;
A6: M.k9 = N.(e.k9) by Th36;
    M.k9 <= M.k9;
    hence N.k >= M.k9 by A2,A6;
  end;
  hence thesis by Th38;
end;
