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 Th66:
  seq is non-decreasing bounded_above implies (
  superior_realsequence seq).n = (superior_realsequence seq).(n+1)
proof
  assume
A1: seq is non-decreasing bounded_above;
  then seq.n <= (superior_realsequence seq).(n+1) by Th65;
  then max((superior_realsequence seq).(n+1),seq.n) = (superior_realsequence
  seq).(n+1) by XXREAL_0:def 10;
  hence thesis by A1,Th47;
end;
