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

theorem Th19:
  for L being non empty 1-sorted, N being non empty NetStr over L
  holds rng the mapping of N =
  the set of all  N.i where i is Element of N
proof
  let L be non empty 1-sorted, N be non empty NetStr over L;
  set X = the set of all  N.i where i is Element of N;
A1: the carrier of N = dom the mapping of N by FUNCT_2:def 1;
  thus rng the mapping of N c=
  the set of all  N.i where i is Element of N
  proof
    let e be object;
    assume e in rng the mapping of N;
    then consider u being object such that
A2: u in dom the mapping of N and
A3: e = (the mapping of N).u by FUNCT_1:def 3;
    reconsider u as Element of N by A2;
    e = N.u by A3;
    hence thesis;
  end;
  let e be object;
  assume e in X;
  then ex i being Element of N st ( e = N.i);
  hence thesis by A1,FUNCT_1:def 3;
end;
