reserve x, y, i for object,
  L for up-complete Semilattice;
reserve L for complete LATTICE,
  a, b, c for Element of L,
  J for non empty set,
  K for non-empty ManySortedSet of J;

theorem Th15:
  for F being Function-yielding Function holds (for f being
  Function st f in dom Frege F holds //\((Frege F).f, L) <= a) implies Sup /\\(
  Frege F, L) <= a
proof
  let F be Function-yielding Function;
  assume
A1: for f being Function st f in dom(Frege F) holds //\((Frege F).f, L) <= a;
  rng /\\(Frege F, L) is_<=_than a
  proof
    let c;
    assume c in rng /\\(Frege F, L);
    then consider f being object such that
A2: f in dom(Frege F) and
A3: c = //\((Frege F).f, L) by Th13;
    reconsider f as Function by A2;
    f in dom(Frege F) by A2;
    hence c <= a by A1,A3;
  end;
  then sup rng /\\(Frege F, L) <= a by YELLOW_0:32;
  hence thesis by YELLOW_2:def 5;
end;
