
theorem Th47:
  for L being complete antisymmetric non empty reflexive RelStr
  st for x being Element of L, N being prenet of L st N is eventually-directed
holds x "/\" sup N = sup ({x} "/\" rng netmap (N,L)) holds for x being Element
of L, J being set, f being Function of J,the carrier of L holds x "/\" Sup f =
  sup(x "/\" FinSups f)
proof
  let L be complete antisymmetric non empty reflexive RelStr such that
A1: for x being Element of L, N being prenet of L st N is
  eventually-directed holds x "/\" sup N = sup ({x} "/\" rng netmap (N,L));
  let x be Element of L, J be set, f be Function of J,the carrier of L;
  set F = FinSups f;
A2: for x being Element of Fin J holds ex_sup_of f.:x,L by YELLOW_0:17;
  ex_sup_of rng f,L & ex_sup_of rng netmap (FinSups f,L),L by YELLOW_0:17;
  hence x "/\" Sup f = x "/\" sup F by A2,Th26
    .= sup ({x} "/\" rng netmap (F,L)) by A1
    .= "\/" (rng the mapping of x "/\" F,L) by Th23
    .= sup (x "/\" F) by YELLOW_2:def 5;

end;
