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 non-increasing implies (superior_realsequence seq).(n+1) <= seq .n
proof
  reconsider Y1 = {seq.k : n+1 <= k} as Subset of REAL by Th29;
A1: (superior_realsequence seq).(n+1) = upper_bound Y1 by Def5;
  assume
A2: seq is non-increasing;
  then upper_bound Y1 = seq.(n+1) by Th35;
  hence thesis by A2,A1;
end;
