reserve n,m,k,k1,k2 for Nat;
reserve r,r1,r2,s,t,p for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve x,y for set;

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;
