reserve n,m,k for Nat;
reserve a,p,r for Real;
reserve s,s1,s2,s3 for Real_Sequence;

theorem Th16:
  (for n holds 0<=s.n) implies Partial_Sums(s) is non-decreasing
proof
  assume
A1: for n holds 0<=s.n;
  now
    let n;
    0<=s.(n+1) by A1;
    then 0 + Partial_Sums(s).n <= s.(n+1) + Partial_Sums(s).n by XREAL_1:6;
    hence Partial_Sums(s).n <= Partial_Sums(s).(n+1) by Def1;
  end;
  hence thesis;
end;
