
theorem Th42:
  for L being with_suprema Poset for X being non empty lower Subset of L holds
  X is directed iff for Y being finite Subset of X st Y <> {} holds "\/"
  (Y,L) in X
proof
  let L be with_suprema Poset;
  let X be non empty lower Subset of L;
  thus X is directed implies
  for Y being finite Subset of X st Y <> {} holds "\/"(Y,L) in X
  proof
    assume
A1: X is directed;
    let Y be finite Subset of X such that
A2: Y <> {};
    consider z being Element of L such that
A3: z in X and
A4: Y is_<=_than z by A1,Th1;
    Y c= the carrier of L by XBOOLE_1:1;
    then ex_sup_of Y,L by A2,YELLOW_0:54;
    then "\/"(Y,L) <= z by A4,YELLOW_0:30;
    hence thesis by A3,Def19;
  end;
  assume
A5: for Y being finite Subset of X st Y <> {} holds "\/"(Y,L) in X;
  set x = the Element of X;
  reconsider x as Element of L;
  now
    let Y be finite Subset of X;
    per cases;
    suppose Y = {};
      then x is_>=_than Y;
      hence ex x being Element of L st x in X & x is_>=_than Y;
    end;
    suppose
A6:   Y <> {};
      Y c= the carrier of L by XBOOLE_1:1;
      then ex_sup_of Y,L by A6,YELLOW_0:54;
      then "\/"(Y,L) is_>=_than Y by YELLOW_0:30;
      hence ex x being Element of L st x in X & x is_>=_than Y by A5,A6;
    end;
  end;
  hence thesis by Th1;
end;
