theorem Th37:
  (superior_realsequence seq).n = upper_bound (seq ^\n)
proof
  reconsider Y = {seq.k: n <= k} as Subset of REAL by Th29;
  (superior_realsequence seq).n = upper_bound Y by Def5
    .= upper_bound rng (seq ^\ n) by Th30;
  hence thesis;
end;
