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

theorem Th1:
  (inferior_setsequence(A1)).n = Intersection (A1 ^\n)
proof
  reconsider Y = {A1.k: n <= k} as Subset-Family of X by SETLIM_1:28;
  (inferior_setsequence(A1)).n = meet Y by SETLIM_1:def 2
    .= meet rng (A1 ^\ n) by SETLIM_1:6;
  hence thesis by SETLIM_1:8;
end;
