reserve R for non empty RelStr,
  N for net of R,
  i for Element of N;

theorem Th20:
  for R being non empty RelStr,
  S being non empty set, f be Function of S, the carrier of R
  st rng f is directed holds Net-Str(S,f) is directed
proof
  let R be non empty RelStr,
  S be non empty set, f be Function of S, the carrier of R such that
A1: rng f is directed;
  set N = Net-Str(S,f);
  let x,y be Element of N;
  f = the mapping of N by Def10;
  then
A2: rng f = the set of all  N.i where i is Element of N by Th19;
  then
A3: N.x in rng f;
  N.y in rng f by A2;
  then consider p being Element of R such that
A4: p in rng f and
A5: N.x <= p and
A6: N.y <= p by A1,A3;
  consider z being object such that
A7: z in dom f and
A8: p = f.z by A4,FUNCT_1:def 3;
  reconsider z as Element of N by A7,Def10;
  take z;
  p = N.z by A8,Def10;
  hence thesis by A5,A6,Def10;
end;
