
theorem Th4:
  for I being set holds [#]OrderedFIN I is directed
  proof
    let I be set;
A1: the carrier of OrderedFIN I= Fin I by YELLOW_1:1;
    then
A2: [#]OrderedFIN I= Fin I by STRUCT_0:def 3;
    now
      let a,b be Element of OrderedFIN I;
      assume that
      a in [#]OrderedFIN I and
      b in [#]OrderedFIN I;
      reconsider z=a\/b as Element of OrderedFIN I by A1,FINSUB_1:def 1;
      take z;
      thus z in [#]OrderedFIN I &
      a <= z & b <= z by A1,A2,YELLOW_1:3,XBOOLE_1:7;
    end;
    hence thesis by WAYBEL_0:def 1;
  end;
