reserve n,m,k for Nat,
  x,X for set,
  A for Subset of X,
  A1,A2 for SetSequence of X;

theorem Th2:
  (superior_setsequence(A1)).n = Union (A1 ^\n)
proof
  reconsider Y = {A1.k: n <= k} as Subset-Family of X by SETLIM_1:28;
  (superior_setsequence(A1)).n = union Y by SETLIM_1:def 3
    .= union rng (A1 ^\ n) by SETLIM_1:6;
  hence thesis by CARD_3:def 4;
end;
