 reserve i,j,k,k0,m,n,N for Nat;
 reserve x,y for Real;
 reserve p for Prime;
 reserve s for Real_Sequence;

theorem PartialNonneg:
  (for n holds 0 <= s.n) implies 0 <= Partial_Sums(s).n
  proof
    assume A1: for n holds 0 <= s.n;
    Partial_Sums(s).0<=Partial_Sums(s).n & 0<=s.0 by A1,SERIES_1:16,SEQM_3:11;
    hence thesis by SERIES_1:def 1;
  end;
