
theorem Th40:
for L being with_suprema antisymmetric RelStr, X being lower Subset of L holds
  X is directed iff for x,y being Element of L st x in X & y in X holds x"\/"
  y in X
proof
  let L be with_suprema antisymmetric RelStr, X be lower Subset of L;
  thus X is directed implies
  for x,y being Element of L st x in X & y in X holds x"\/"y in X
  proof
    assume
A1: for x,y being Element of L st x in X & y in X
    ex z being Element of L st z in X & x <= z & y <= z;
    let x,y be Element of L;
    assume that
A2: x in X and
A3: y in X;
    consider z being Element of L such that
A4: z in X and
A5: x <= z and
A6: y <= z by A1,A2,A3;
    x"\/"y <= z by A5,A6,YELLOW_0:22;
    hence thesis by A4,Def19;
  end;
  assume
A7: for x,y being Element of L st x in X & y in X holds x"\/"y in X;
  let x,y be Element of L;
  assume that
A8: x in X and
A9: y in X;
A10: x <= x"\/"y by YELLOW_0:22;
  y <= x"\/"y by YELLOW_0:22;
  hence thesis by A7,A8,A9,A10;
end;
