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 Th73:
  seq is non-increasing & seq is bounded_below implies (
  inferior_realsequence seq).n = lower_bound seq &
  (inferior_realsequence seq) is
  constant
proof
  assume that
A1: seq is non-increasing and
A2: seq is bounded_below;
   reconsider lbs = lower_bound seq as Element of REAL by XREAL_0:def 1;
  defpred P[Nat] means (inferior_realsequence seq).$1 = lbs;
A3: for k being Nat st P[k] holds P[k+1]
  by A1,A2,Th72;
A4: P[0] by A2,Th38;
  for k being Nat holds P[k] from NAT_1:sch 2(A4,A3);
  hence thesis by VALUED_0:def 18;
end;
