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-decreasing implies seq.n <= (inferior_realsequence seq).(n+ 1)
proof
  reconsider Y1 = {seq.k : n+1 <= k} as Subset of REAL by Th29;
A1: (inferior_realsequence seq).(n+1) = lower_bound Y1 by Def4;
  assume
A2: seq is non-decreasing;
  then lower_bound Y1 = seq.(n+1) by Th34;
  hence thesis by A2,A1;
end;
