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 bounded_above implies
  lower_bound (superior_realsequence
  seq) = upper_bound seq
proof
  assume
A1: seq is non-decreasing bounded_above;
  then (superior_realsequence seq) is constant by Th67;
  then consider r1 being Element of REAL such that
A2: rng (superior_realsequence seq)={r1} by FUNCT_2:111;
  r1 in rng (superior_realsequence seq) by A2,TARSKI:def 1;
  then ex n being Element of NAT
st r1 = (superior_realsequence seq).n by FUNCT_2:113;
  then rng (superior_realsequence seq)= {upper_bound seq} by A1,A2,Th67;
  hence thesis by SEQ_4:9;
end;
