
theorem Th25:
  for L being non empty reflexive antisymmetric RelStr, J being
  set for f being Function of J,the carrier of L holds rng f c= rng netmap (
  FinSups f,L)
proof
  let L be non empty reflexive antisymmetric RelStr, J be set, f be Function
  of J,the carrier of L;
  per cases;
  suppose
A1: J is non empty;
    let a be object;
    assume a in rng f;
    then consider b being object such that
A2: b in dom f and
A3: a = f.b by FUNCT_1:def 3;
    reconsider b as Element of J by A2;
    f.b in rng f by A2,FUNCT_1:def 3;
    then reconsider fb = f.b as Element of L;
A4: Im(f,b) = {fb} by A2,FUNCT_1:59;
    {b} c= J by A1,ZFMISC_1:31;
    then reconsider x = {b} as Element of Fin J by FINSUB_1:def 5;
    consider g being Function of Fin J, the carrier of L such that
A5: for x being Element of Fin J holds g.x = sup (f.:x) & FinSups f =
    NetStr (# Fin J, RelIncl Fin J, g #) by Def2;
    dom g = Fin J by FUNCT_2:def 1;
    then
A6: x in dom g;
    g.{b} = sup (f.:x) by A5
      .= a by A3,A4,YELLOW_0:39;
    hence thesis by A5,A6,FUNCT_1:def 3;
  end;
  suppose
A7: J is empty;
    rng f = {} by A7;
    hence thesis;
  end;
end;
