reserve n,k for Nat,
  X for non empty set,
  S for SigmaField of X;

theorem Th2:
  for F be SetSequence of X, n be Nat holds
  (superior_setsequence F).n = union rng (F^\n) &
  (inferior_setsequence F).n = meet rng (F^\n)
proof
  let F be SetSequence of X, n be Nat;
  {F.k where k is Nat: n <= k} = rng (F^\n) by SETLIM_1:6;
  hence (superior_setsequence F).n = union rng (F ^\ n) by SETLIM_1:def 3;
  (inferior_setsequence F).n = meet({F.k where k: n <= k}) by SETLIM_1:def 2;
  hence thesis by SETLIM_1:6;
end;
