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 Th16:
  for F being DoubleIndexedSet of K, L holds Inf Sups F >= Sup Infs Frege F
proof
  let F be DoubleIndexedSet of K, L;
  set a = Sup Infs Frege F;
A1: for j being Element of J for f being Element of product doms F holds Sup
  (F.j) >= Inf((Frege F).f)
  proof
    let j be Element of J;
    let f be Element of product doms F;
A2: f in dom(Frege F) by Lm7;
    then reconsider k = f.j as Element of K.j by Lm6;
    (F.j).k = ((Frege F).f).j by A2,Lm5;
    then
A3: Inf((Frege F).f) <= (F.j).k by YELLOW_2:53;
    (F.j).k <= Sup(F.j) by YELLOW_2:53;
    hence thesis by A3,ORDERS_2:3;
  end;
  a is_<=_than rng Sups F
  proof
    let c;
    assume c in rng Sups F;
    then consider j being Element of J such that
A4: c = Sup(F.j) by Th14;
    for f being Function st f in dom(Frege F) holds //\((Frege F).f, L) <= c
    by A1,A4;
    hence a <= c by Th15;
  end;
  then a <= inf rng Sups F by YELLOW_0:33;
  hence thesis by YELLOW_2:def 6;
end;
