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 Th55:
  seq is bounded implies upper_bound(inferior_realsequence seq) <= lower_bound(
  superior_realsequence seq)
proof
  set seq1 = (inferior_realsequence seq);
  set r = lower_bound(superior_realsequence seq);
  assume seq is bounded;
  then seq1.n <= r by Th53;
  hence thesis by Th9;
end;
