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 Th65:
  seq is non-decreasing bounded_above implies seq.n <= (
  superior_realsequence seq).(n+1)
proof
  reconsider Y1 = {seq.k : n+1 <= k} as Subset of REAL by Th29;
A1: seq.(n+1) in Y1;
  assume
A2: seq is non-decreasing bounded_above;
  then Y1 is bounded_above by Th31;
  then
A3: seq.(n+1) <= upper_bound Y1 by A1,SEQ_4:def 1;
A4: (superior_realsequence seq).(n+1) = upper_bound Y1 by Def5;
  seq.n <= seq.(n+1) by A2;
  hence thesis by A4,A3,XXREAL_0:2;
end;
