reserve x,y,X for set;

theorem
  for S being non empty 1-sorted, N being net of S for F being finite
  non empty set st for A being Element of F holds A is Subset of S,N ex B being
  Subset of S,N st B c= meet F
proof
  let S be non empty 1-sorted, N be net of S;
  defpred P[object,object] means
  ex i being Element of N st $2 = i & $1 = rng the
  mapping of N|i;
  let F be finite non empty set such that
A1: for A being Element of F holds A is Subset of S,N;
A2: now
    let x be object;
    assume x in F;
    then reconsider A = x as Subset of S, N by A1;
    consider i being Element of N such that
A3: A = rng the mapping of N|i by Def2;
    reconsider y = i as object;
    take y;
    thus y in the carrier of N;
    thus P[x, y] by A3;
  end;
  consider f being Function such that
A4: dom f = F & rng f c= the carrier of N and
A5: for x being object st x in F holds P[x, f.x] from FUNCT_1:sch 6(A2);
  reconsider B = rng f as finite Subset of N by A4,FINSET_1:8;
  [#]N is directed by WAYBEL_0:def 6;
  then consider j being Element of N such that
  j in [#]N and
A6: j is_>=_than B by WAYBEL_0:1;
  reconsider C = rng the mapping of N|j as Subset of S, N by Def2;
  take C;
  let x be object;
A7: the carrier of N|j = {k where k is Element of N: j <= k} by WAYBEL_9:12;
  assume x in C;
  then consider y being object such that
A8: y in dom the mapping of N|j and
A9: x = (the mapping of N|j).y by FUNCT_1:def 3;
A10: y in the carrier of N|j by A8;
  reconsider y as Element of N|j by A8;
  consider k being Element of N such that
A11: y = k and
A12: j <= k by A10,A7;
  now
    let X;
    assume
A13: X in F;
    then consider i being Element of N such that
A14: f.X = i and
A15: X = rng the mapping of N|i by A5;
    i in B by A4,A13,A14,FUNCT_1:def 3;
    then i <= j by A6,LATTICE3:def 9;
    then i <= k by A12,ORDERS_2:3;
    then y in {l where l is Element of N: i <= l} by A11;
    then reconsider z = y as Element of N|i by WAYBEL_9:12;
    x = (N|j).y by A9
      .= N.k by A11,WAYBEL_9:16
      .= (N|i).z by A11,WAYBEL_9:16
      .= (the mapping of N|i).z;
    hence x in X by A15,FUNCT_2:4;
  end;
  hence thesis by SETFAM_1:def 1;
end;
