
theorem Th18:
  for L being non empty RelStr for N being prenet of L st N is
  eventually-directed holds rng netmap (N,L) is directed
proof
  let L be non empty RelStr, N be prenet of L such that
A1: N is eventually-directed;
  set f = netmap (N,L);
  let x, y be Element of L such that
A2: x in rng f and
A3: y in rng f;
  consider a being object such that
A4: a in dom f and
A5: f.a = x by A2,FUNCT_1:def 3;
  consider b being object such that
A6: b in dom f and
A7: f.b = y by A3,FUNCT_1:def 3;
  reconsider a, b as Element of N by A4,A6;
  consider ja being Element of N such that
A8: for k being Element of N st ja <= k holds N.a <= N.k by A1,WAYBEL_0:11;
  consider jb being Element of N such that
A9: for k being Element of N st jb <= k holds N.b <= N.k by A1,WAYBEL_0:11;
  [#]N is directed by WAYBEL_0:def 6;
  then consider c being Element of N such that
  c in [#]N and
A10: ja <= c & jb <= c;
  take z = f.c;
  dom f = the carrier of N by FUNCT_2:def 1;
  hence z in rng f by FUNCT_1:def 3;
  N.c = f.c;
  hence thesis by A5,A7,A8,A9,A10;
end;
