
theorem Th26:
  for L being non empty reflexive antisymmetric RelStr, J being
set for f being Function of J,the carrier of L st ex_sup_of rng f,L & ex_sup_of
rng netmap (FinSups f,L),L & for x being Element of Fin J holds ex_sup_of f.:x,
  L holds Sup f = sup FinSups f
proof
  let L be non empty reflexive antisymmetric RelStr, J be set, f be Function
  of J,the carrier of L such that
A1: ex_sup_of rng f,L and
A2: ex_sup_of rng netmap (FinSups f,L),L and
A3: for x being Element of Fin J holds ex_sup_of f.:x,L;
  set h = netmap (FinSups f,L);
A4: "\/"(rng f,L) <= sup rng h by A1,A2,Th25,YELLOW_0:34;
  rng h is_<=_than "\/"(rng f,L)
  proof
    let a be Element of L;
    assume a in rng h;
    then consider x being object such that
A5: x in dom h and
A6: a = h.x by FUNCT_1:def 3;
A7: 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 A5;
    ex_sup_of f.:x,L by A3;
    then sup(f.:x) <= "\/"(rng f,L) by A1,RELAT_1:111,YELLOW_0:34;
    hence a <= "\/"(rng f,L) by A6,A7;
  end;
  then
A8: sup rng h <= "\/"(rng f,L) by A2,YELLOW_0:def 9;
  thus Sup f = "\/"(rng f,L) by YELLOW_2:def 5
    .= sup rng netmap (FinSups f,L) by A4,A8,ORDERS_2:2
    .= sup FinSups f by YELLOW_2:def 5;
end;
