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
  seq is bounded implies (inferior_realsequence seq).n <= (
  superior_realsequence seq).n
proof
  reconsider Y1 = {seq.k : n <= k} as Subset of REAL by Th29;
  assume
A1: seq is bounded;
A2: Y1 <> {} by SETLIM_1:1;
  (superior_realsequence seq).n = upper_bound Y1 &
  (inferior_realsequence seq).n =
  lower_bound Y1 by Def4,Def5;
  hence thesis by A1,A2,Th33,SEQ_4:11;
end;
