
theorem Th20:
  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, N being
prenet of L st n = id D & N = NetStr (#D,(the InternalRel of L)|_2 D,n#) holds
  N is eventually-directed
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, N be prenet of L such that
A1: n = id D and
A2: N = NetStr (#D,(the InternalRel of L)|_2 D,n#);
  for i being Element of N ex j being Element of N st for k being Element
  of N st j <= k holds N.i <= N.k
  proof
    let i be Element of N;
    take j = i;
    let k be Element of N such that
A3: j <= k;
    the InternalRel of N c= the InternalRel of L by A2,XBOOLE_1:17;
    then
A4: N is SubRelStr of L by A2,YELLOW_0:def 13;
    reconsider nj = n.j, nk = n.k as Element of L by A2,FUNCT_2:5;
    nj = j & nk = k by A1,A2;
    hence thesis by A2,A3,A4,YELLOW_0:59;
  end;
  hence thesis by WAYBEL_0:11;
end;
