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 bounded_below implies upper_bound inferior_realsequence
  seq = lower_bound seq
proof
  assume
A1: seq is non-increasing bounded_below;
  then inferior_realsequence seq is constant by Th73;
  then consider r1 being Element of REAL such that
A2: rng inferior_realsequence seq={r1} by FUNCT_2:111;
  r1 in rng inferior_realsequence seq by A2,TARSKI:def 1;
  then ex n being Element of NAT
st r1 = (inferior_realsequence seq).n by FUNCT_2:113;
  then upper_bound inferior_realsequence seq =
  upper_bound {lower_bound seq} by A1,A2,Th73
    .= lower_bound seq by SEQ_4:9;
  hence thesis;
end;
