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 Th67:
  seq is non-decreasing bounded_above implies (
  superior_realsequence seq).n = upper_bound seq & (superior_realsequence seq)
   is
  constant
proof
  reconsider ubs = upper_bound seq as Element of REAL by XREAL_0:def 1;
  defpred P[Nat] means (superior_realsequence seq).$1 = ubs;
  assume
A1: seq is non-decreasing bounded_above;
A2: for k being Nat st P[k] holds P[k+1]
  by A1,Th66;
A3: P[0] by A1,Th39;
  for k being Nat holds P[k] from NAT_1:sch 2(A3,A2);
  hence thesis by VALUED_0:def 18;
end;
