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

theorem
  s is summable & (for n holds 0<=s.n) implies 0<=Sum(s)
proof
  assume that
A1: s is summable and
A2: for n holds 0<=s.n;
A3: now
    let n;
A4: Partial_Sums(s).0 = s.0 by Def1;
    Partial_Sums(s).0<=Partial_Sums(s).n & 0<=s.0 by A2,Th16,SEQM_3:11;
    hence 0<=Partial_Sums(s).n by A4;
  end;
  Partial_Sums(s) is convergent by A1;
  hence thesis by A3,SEQ_2:17;
end;
