
theorem Th19:
  for L being non empty reflexive RelStr, D being non empty
  directed Subset of L for n being Function of D, the carrier of L holds NetStr
  (#D,(the InternalRel of L)|_2 D,n#) is prenet of L
proof
  let L be non empty reflexive RelStr, D be non empty directed Subset of L, n
  be Function of D, the carrier of L;
  set N = NetStr (#D,(the InternalRel of L)|_2 D,n#);
  N is directed
  proof
    let x, y be Element of N;
    assume that
    x in [#]N and
    y in [#]N;
    reconsider x1 = x, y1 = y as Element of D;
    consider z1 being Element of L such that
A1: z1 in D and
A2: x1 <= z1 & y1 <= z1 by WAYBEL_0:def 1;
    reconsider z = z1 as Element of N by A1;
    take z;
    thus z in [#]N;
    the InternalRel of N c= the InternalRel of L by XBOOLE_1:17;
    then reconsider N as SubRelStr of L by YELLOW_0:def 13;
    N is full;
    hence thesis by A2,YELLOW_0:60;
  end;
  hence thesis;
end;
