
theorem Th43:
  for L being with_infima Poset for X being non empty upper Subset of L holds
  X is filtered iff for Y being finite Subset of X st Y <> {} holds "/\"
  (Y,L) in X
proof
  let L be with_infima Poset, X be non empty upper Subset of L;
  thus X is filtered implies
  for Y being finite Subset of X st Y <> {} holds "/\"(Y,L) in X
  proof
    assume
A1: X is filtered;
    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,Th2;
    Y c= the carrier of L by XBOOLE_1:1;
    then ex_inf_of Y,L by A2,YELLOW_0:55;
    then "/\"(Y,L) >= z by A4,YELLOW_0:31;
    hence thesis by A3,Def20;
  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_inf_of Y,L by A6,YELLOW_0:55;
      then "/\"(Y,L) is_<=_than Y by YELLOW_0:31;
      hence ex x being Element of L st x in X & x is_<=_than Y by A5,A6;
    end;
  end;
  hence thesis by Th2;
end;
