
theorem Th29:
  for S,T being /\-complete non empty Poset
  for X being non empty filtered Subset of S
  for f being monotone Function of S,T
  holds lim_inf (f*(X opp+id)) = inf (f.:X)
proof
  let S,T be /\-complete non empty Poset;
  let X be non empty filtered Subset of S;
  let f be monotone Function of S,T;
  set M = X opp+id, N = f*M;
  deffunc up(Element of N) = {N.i where i is Element of N: i >= $1};
  deffunc infy(Element of N) = "/\"(up($1), T);
A1: the RelStr of N = the RelStr of M by WAYBEL_9:def 8;
A2: the mapping of N = f*the mapping of M by WAYBEL_9:def 8;
A3: the carrier of M = X by YELLOW_9:7;
A4: the mapping of M = id X by WAYBEL19:27;
  defpred P[set] means not contradiction;
  deffunc G(set) = inf (f.:X);
A5: for j being Element of N st P[j] holds infy(j) = G(j)
  proof
    let j be Element of N;
    reconsider j as Element of N;
A6: [#]N is directed by WAYBEL_0:def 6;
    then consider i9 being Element of N such that
    i9 in [#]N and
A7: i9 >= j and i9 >= j;
A8: up(j) c= f.:X
    proof
      let a be object;
      assume a in up(j);
      then consider i being Element of N such that
A9:   a = N.i and i >= j;
      reconsider i as Element of N;
      reconsider i9 = i as Element of M by A1;
A10:  N.i = f.((id X).i) by A1,A2,A4,FUNCT_2:15
        .= f.i9 by A3;
      i9 in X by A3;
      hence thesis by A9,A10,FUNCT_2:35;
    end;
    then
A11: up(j) c= the carrier of T by XBOOLE_1:1;
    N.i9 in up(j) by A7;
    then
A12: ex_inf_of up(j), T by A11,WAYBEL_0:76;
A13: ex_inf_of f.:X, T by WAYBEL_0:76;
    then
A14: infy(j) >= inf (f.:X) by A8,A12,YELLOW_0:35;
    infy(j) is_<=_than f.:X
    proof
      let x be Element of T;
      assume x in f.:X;
      then consider y being object such that
A15:  y in the carrier of S and
A16:  y in X and
A17:  x = f.y by FUNCT_2:64;
      reconsider y as Element of N by A1,A16,YELLOW_9:7;
      consider i being Element of N such that
      i in [#]N and
A18:  i >= y and
A19:  i >= j by A6;
      i in X by A1,A3;
      then reconsider xi = i, xy = y as Element of S by A15;
      M is full SubRelStr of S opp by YELLOW_9:7;
      then N is full SubRelStr of S opp by A1,Th12;
      then xi~ >= xy~ by A18,YELLOW_0:59;
      then xi <= xy by LATTICE3:9;
      then
A20:  f.xi <= x by A17,WAYBEL_1:def 2;
      N.i = f.((id X).i) by A1,A2,A4,FUNCT_2:15
        .= f.xi by A1,A3;
      then f.xi in up(j) by A19;
      then f.xi >= infy(j) by A12,YELLOW_4:2;
      hence thesis by A20,ORDERS_2:3;
    end;
    then infy(j) <= inf (f.:X) by A13,YELLOW_0:31;
    hence thesis by A14,ORDERS_2:2;
  end;
A21: ex j being Element of N st P[j];
  {infy(j) where j is Element of N: P[j]} =
  {G(j) where j is Element of N: P[j]} from FRAENKEL:sch 6(A5)
    .= {inf (f.:X) where j is Element of N: P[j]}
    .= {inf (f.:X)} from LATTICE3:sch 1(A21);
  hence lim_inf N = sup {inf (f.:X)} by WAYBEL11:def 6
    .= inf (f.:X) by YELLOW_0:39;
end;
