
theorem Th24:
  for L being non empty RelStr, J being set for f being Function
  of J,the carrier of L st for x being set holds ex_sup_of f.:x,L holds rng
  netmap(FinSups f,L) c= finsups rng f
proof
  let L be non empty RelStr, J be set, f be Function of J,the carrier of L
  such that
A1: for x being set holds ex_sup_of f.:x,L;
  let q be object;
  set h = netmap(FinSups f,L);
  assume q in rng h;
  then consider x being object such that
A2: x in dom h and
A3: h.x = q by FUNCT_1:def 3;
A4: ex g being Function of Fin J, the carrier of L st for x being Element of
Fin J holds g.x = sup (f.:x) & FinSups f = NetStr (# Fin J, RelIncl Fin J, g #)
  by Def2;
  then reconsider x as Element of Fin J by A2;
A5: f.:x is finite Subset of rng f & ex_sup_of f.:x,L by A1,RELAT_1:111;
  h.x = sup (f.:x) by A4;
  hence thesis by A3,A5;
end;
