
theorem Th30:
  for L being non empty transitive reflexive RelStr, X be Subset of L holds
  X is directed iff downarrow X is directed
proof
  let L be non empty transitive reflexive RelStr, X be Subset of L;
  thus X is directed implies downarrow X is directed
  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 downarrow X and
A3: y in downarrow X;
    consider x9 being Element of L such that
A4: x <= x9 and
A5: x9 in X by A2,Def15;
    consider y9 being Element of L such that
A6: y <= y9 and
A7: y9 in X by A3,Def15;
    consider z being Element of L such that
A8: z in X and
A9: x9 <= z and
A10: y9 <= z by A1,A5,A7;
    take z;
    z <= z;
    hence z in downarrow X by A8,Def15;
    thus thesis by A4,A6,A9,A10,ORDERS_2:3;
  end;
  set Y = downarrow X;
  assume
A11: for x,y being Element of L st x in Y & y in Y
  ex z being Element of L st z in Y & x <= z & y <= z;
  let x,y be Element of L;
  assume that
A12: x in X and
A13: y in X;
A14: x <= x;
A15: y <= y;
A16: x in Y by A12,A14,Def15;
  y in Y by A13,A15,Def15;
  then consider z being Element of L such that
A17: z in Y and
A18: x <= z and
A19: y <= z by A11,A16;
  consider z9 being Element of L such that
A20: z <= z9 and
A21: z9 in X by A17,Def15;
  take z9;
  thus z9 in X by A21;
  thus thesis by A18,A19,A20,ORDERS_2:3;
end;
