reserve x for set;

theorem Th11:
  for L being complete LATTICE, N being net of L holds lim_inf N =
  "\/"((the set of all inf (N|i) where i is Element of N), L)
proof
  let L be complete LATTICE, N be net of L;
  set X =the set of all
"/\"({N.i where i is Element of N:i >= j},L) where j is Element of N;
  set Y =the set of all inf (N|i) where i is Element of N;
  for x being object st x in X holds x in Y
  proof
    let x be object;
    assume x in X;
    then consider j being Element of N such that
A1: x = "/\"({N.i where i is Element of N : i >= j},L);
    reconsider x as Element of L by A1;
    set S = {N.i where i is Element of N : i >= j};
    reconsider i=j as Element of N;
    for b being object st b in rng the mapping of (N|i) holds b in S
    proof
      let b being object;
      assume b in rng the mapping of (N|i);
      then b in the set of all (N|i).k where k is Element of N|i by
WAYBEL11:19;
      then consider k being Element of N|i such that
A2:   b = (N|i).k;
      the carrier of N|i c= the carrier of N by WAYBEL_9:13;
      then reconsider l = k as Element of N;
      k in the carrier of N|i;
      then k in { y where y is Element of N : i <= y} by WAYBEL_9:12;
      then
A3:   ex y being Element of N st k = y & i <= y;
      reconsider k as Element of N|i;
      N.l = (N|i).k by WAYBEL_9:16;
      hence thesis by A2,A3;
    end;
    then
A4: rng the mapping of (N|i) c= S;
A5: ex_inf_of S,L by YELLOW_0:17;
    then
A6: S is_>=_than x by A1,YELLOW_0:def 10;
A7: rng the mapping of (N|i) is_>=_than x
    by A6,A4;
    for b being object st b in S holds b in rng the mapping of (N|i)
    proof
      let b being object;
      assume b in S;
      then consider k being Element of N such that
A8:   b = N.k and
A9:   k >= j;
      reconsider l = k as Element of N;
      l in { y where y is Element of N : i <= y} by A9;
      then reconsider k as Element of N|i by WAYBEL_9:12;
      reconsider k as Element of N|i;
      N.l = (N|i).k by WAYBEL_9:16;
      then
      b in the set of all (N|i).m where m is Element of N|i by A8;
      hence thesis by WAYBEL11:19;
    end;
    then S c= rng the mapping of (N|i);
    then S = rng the mapping of (N|i) by A4;
    then for a being Element of L st rng the mapping of (N|i) is_>=_than a
    holds a <= x by A1,A5,YELLOW_0:def 10;
    then consider i being Element of N such that
A10: ex_inf_of rng the mapping of (N|i),L & rng the mapping of (N|i)
is_>=_than x & for a being Element of L st rng the mapping of (N|i) is_>=_than
    a holds a <= x by A7,YELLOW_0:17;
A11: inf (N|i) = Inf the mapping of (N|i) by WAYBEL_9:def 2
      .= "/\"(rng the mapping of (N|i),L) by YELLOW_2:def 6;
    x = "/\"(rng the mapping of (N|i),L) by A10,YELLOW_0:def 10;
    hence thesis by A11;
  end;
  then
A12: lim_inf N = "\/"(X,L) & X c= Y by WAYBEL11:def 6;
  for x being object st x in Y holds x in X
  proof
    let x being object;
    assume x in Y;
    then consider i being Element of N such that
A13: x = inf (N|i);
    set S = {N.j where j is Element of N : j >= i};
    for b being object st b in rng the mapping of (N|i) holds b in S
    proof
      let b being object;
      assume b in rng the mapping of (N|i);
      then b in the set of all (N|i).k where k is Element of N|i by
WAYBEL11:19;
      then consider k being Element of N|i such that
A14:  b = (N|i).k;
      the carrier of N|i c= the carrier of N by WAYBEL_9:13;
      then reconsider l = k as Element of N;
      k in the carrier of N|i;
      then k in { y where y is Element of N : i <= y} by WAYBEL_9:12;
      then
A15:  ex y being Element of N st k = y & i <= y;
      reconsider k as Element of N|i;
      N.l = (N|i).k by WAYBEL_9:16;
      hence thesis by A14,A15;
    end;
    then
A16: rng the mapping of (N|i) c= S;
    reconsider x as Element of L by A13;
A17: x = Inf the mapping of (N|i) by A13,WAYBEL_9:def 2
      .= "/\"(rng the mapping of (N|i),L) by YELLOW_2:def 6;
    for b being object st b in S holds b in rng the mapping of (N|i)
    proof
      let b being object;
      assume b in S;
      then consider k being Element of N such that
A18:  b = N.k and
A19:  k >= i;
      reconsider l = k as Element of N;
      l in { y where y is Element of N : i <= y} by A19;
      then reconsider k as Element of N|i by WAYBEL_9:12;
      reconsider k as Element of N|i;
      N.l = (N|i).k by WAYBEL_9:16;
      then b in the set of all (N|i).m where m is Element of N|i by A18;
      hence thesis by WAYBEL11:19;
    end;
    then S c= rng the mapping of (N|i);
    then rng the mapping of (N|i) = S by A16;
    hence thesis by A17;
  end;
  then Y c= X;
  hence thesis by A12,XBOOLE_0:def 10;
end;
