reserve x, X, Y for set;
reserve L for complete LATTICE,
  a for Element of L;

theorem
  for L being with_infima antisymmetric RelStr for F being Subset-Family
of L st for X being Subset of L st X in F holds X is upper filtered holds meet
  F is filtered Subset of L
proof
  let L be with_infima antisymmetric RelStr;
  let F be Subset-Family of L;
  assume
A1: for X being Subset of L st X in F holds X is upper filtered;
  reconsider F9 = F as Subset-Family of L;
  reconsider M = meet F9 as Subset of L;
  per cases;
  suppose
A2: F = {};
    M is filtered
    by A2,SETFAM_1:def 1;
    hence thesis;
  end;
  suppose
A3: F <> {};
    M is filtered
    proof
      let x, y be Element of L such that
A4:   x in M and
A5:   y in M;
      take z = x"/\"y;
      for Y being set st Y in F holds z in Y
      proof
        let Y be set;
        assume
A6:     Y in F;
        then reconsider X = Y as Subset of L;
A7:     y in X by A5,A6,SETFAM_1:def 1;
        X is upper filtered & x in X by A1,A4,A6,SETFAM_1:def 1;
        hence thesis by A7,WAYBEL_0:41;
      end;
      hence z in M by A3,SETFAM_1:def 1;
      thus thesis by YELLOW_0:23;
    end;
    hence thesis;
  end;
end;
