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

theorem Th38:
  for L being with_suprema antisymmetric RelStr for F being
  Subset-Family of L st for X being Subset of L st X in F holds X is lower
  directed holds meet F is directed Subset of L
proof
  let L be with_suprema antisymmetric RelStr;
  let F be Subset-Family of L;
  assume
A1: for X being Subset of L st X in F holds X is lower directed;
  reconsider F9 = F as Subset-Family of L;
  reconsider M = meet F9 as Subset of L;
  per cases;
  suppose
A2: F = {};
    M is directed
    by A2,SETFAM_1:def 1;
    hence thesis;
  end;
  suppose
A3: F <> {};
    M is directed
    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 lower directed & x in X by A1,A4,A6,SETFAM_1:def 1;
        hence thesis by A7,WAYBEL_0:40;
      end;
      hence z in M by A3,SETFAM_1:def 1;
      thus thesis by YELLOW_0:22;
    end;
    hence thesis;
  end;
end;
