
theorem Th28:
  for R being /\-complete non empty Poset
  for F being non empty filtered Subset of R holds lim_inf (F opp+id) = inf F
proof
  let R be /\-complete non empty Poset;
  let F be non empty filtered Subset of R;
  set N = F opp+id;
  defpred P[set] means not contradiction;
  deffunc F(Element of N) = inf F;
  deffunc G(Element of N) = "/\"({N.i where i is Element of N: i >= $1}, R);
A1: for v being Element of N st P[v] holds F(v) = G(v)
  proof
    let v be Element of N;
    set X = {N.i where i is Element of N: i >= v};
A2: the carrier of N = F by YELLOW_9:7;
    then v in F;
    then reconsider j = v as Element of R;
    reconsider vv = v as Element of N;
A3: X c= F
    proof
      let x be object;
      assume x in X;
      then consider i being Element of N such that
A4:   x = N.i and i >= v;
      reconsider i as Element of N;
      x = i by A4,YELLOW_9:7;
      hence thesis by A2;
    end;
    vv <= vv;
    then N.v in X;
    then reconsider X as non empty Subset of R by A3,XBOOLE_1:1;
A5: ex_inf_of F, R by WAYBEL_0:76;
A6: ex_inf_of X, R by WAYBEL_0:76;
    then
A7: inf X >= inf F by A3,A5,YELLOW_0:35;
    F is_>=_than inf X
    proof
      let a be Element of R;
      assume a in F;
      then consider b being Element of R such that
A8:   b in F and
A9:   a >= b and
A10:  j >= b by A2,WAYBEL_0:def 2;
      reconsider k = b as Element of N by A8,YELLOW_9:7;
A11:  N is full SubRelStr of R opp by YELLOW_9:7;
A12:  j~ <= b~ by A10,LATTICE3:9;
A13:  N.k = b by YELLOW_9:7;
      k >= vv by A11,A12,YELLOW_0:60;
      then b in X by A13;
      then
A14:  {b} c= X by ZFMISC_1:31;
A15:  ex_inf_of {b}, R by YELLOW_0:38;
      inf {b} = b by YELLOW_0:39;
      then b >= inf X by A6,A14,A15,YELLOW_0:35;
      hence thesis by A9,YELLOW_0:def 2;
    end;
    then inf F >= "/\"(X, R) by A5,YELLOW_0:31;
    hence thesis by A7,ORDERS_2:2;
  end;
A16: {F(j) where j is Element of N: P[j]}
  = {G(k) where k is Element of N: P[k]} from FRAENKEL:sch 6(A1);
A17: ex j being Element of N st P[j];
  {inf F where j is Element of N: P[j]} = {inf F} from LATTICE3:sch 1(
  A17);
  hence lim_inf N = "\/"({inf F}, R) by A16,WAYBEL11:def 6
    .= inf F by YELLOW_0:39;
end;
