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 Th72:
  seq is non-increasing bounded_below implies (
  inferior_realsequence seq).n = (inferior_realsequence seq).(n+1)
proof
  assume
A1: seq is non-increasing bounded_below;
  then (inferior_realsequence seq).(n+1) <= seq.n by Th71;
  then min((inferior_realsequence seq).(n+1),seq.n) = (inferior_realsequence
  seq).(n+1) by XXREAL_0:def 9;
  hence thesis by A1,Th46;
end;
