 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 PartialLeSum:
  s is summable & (for n holds 0 <= s.n) implies
  Partial_Sums(s).i <= Sum(s)
  proof
    assume A1: s is summable & for n holds 0<=s.n; then
A2: Sum(s) = Partial_Sums(s).i + Sum(s^\(i+1)) by SERIES_1:15;
A3: s^\(i+1) is summable by A1,SERIES_1:12;
    for n holds 0 <= (s^\(i+1)).n
    proof
      let n;
      0 <= s.((i+1)+n) by A1;
      hence thesis by NAT_1:def 3;
    end;
    hence thesis by A2,A3,SERIES_1:18,XREAL_1:31;
  end;
