
theorem Th23:
  for R being non empty RelStr, N being net of R
  holds rng the mapping of (inf_net N) =
  the set of all "/\"({N.i where i is Element of N: i >= j},R) where
  j is Element of N
proof
  let R be non empty RelStr, N be net of R;
  consider f being Function of N,R such that
A1: inf_net N = N*'f and
A2: for i being Element of N holds f.i = "/\" ({N.k where k is Element
  of N: k >= i},R)
  by Def4;
A3: the RelStr of inf_net N = the RelStr of N by A1,Def3;
A4: the mapping of (inf_net N) = f by A1,Def3;
  then
A5: the carrier of inf_net N = dom f by FUNCT_2:def 1;
  thus rng the mapping of inf_net N c=
  the set of all "/\"({N.i where i is Element of N: i >= j},R) where
  j is Element of N
  proof
    let e be object;
    assume e in rng the mapping of inf_net N;
    then consider u being object such that
A6: u in dom f and
A7: e = f.u by A4,FUNCT_1:def 3;
    reconsider u as Element of N by A6;
    f.u = "/\"({N.k where k is Element of N: k >= u},R) by A2;
    hence thesis by A7;
  end;
  let e be object;
  assume e in the set of all
"/\"({N.i where i is Element of N: i >= j},R) where
  j is Element of N;
  then consider j being Element of N such that
A8: e = "/\"({N.i where i is Element of N: i >= j},R);
  e = (the mapping of inf_net N).j by A2,A4,A8;
  hence thesis by A3,A4,A5,FUNCT_1:def 3;
end;
