reserve n,m for Nat;

theorem Th16:
  for R be real-valued FinSequence st len R = 0 or len R = 1 holds
  R is non-decreasing FinSequence of REAL
proof
  let R be real-valued FinSequence;
  assume
A1: len R = 0 or len R = 1;
  reconsider R as FinSequence of REAL by RVSUM_1:145;
  per cases by A1;
  suppose
    len R = 0;
    then R = <*>REAL;
    then for n st n in dom R & n+1 in dom R holds R.n<=R.(n+1);
    hence thesis by INTEGRA2:def 1;
  end;
  suppose
    len R = 1;
    then
A2: dom R = {1} by FINSEQ_1:2,def 3;
    now
      let n;
      assume that
A3:   n in dom R and
A4:   n+1 in dom R;
      n = 1 by A2,A3,TARSKI:def 1;
      hence R.n<=R.(n+1) by A2,A4,TARSKI:def 1;
    end;
    hence thesis by INTEGRA2:def 1;
  end;
end;
