reserve a,b,c for positive Real,
  m,x,y,z for Real,
  n for Nat,
  s,s1,s2,s3,s4,s5 for Real_Sequence;

theorem Th34:
  (for n holds s.n>=0) implies for n holds (Partial_Sums s).n>=0
proof
  defpred X[Nat] means (Partial_Sums(s)).$1>=0;
  assume
A1: for n holds s.n>=0;
A2: for n st X[n] holds X[n+1]
  proof
    let n;
    assume
A3: (Partial_Sums(s)).n>=0;
    (Partial_Sums(s)).(n+1)=(Partial_Sums(s)).n+s.(n+1) & s.(n+1)>=0 by A1,
SERIES_1:def 1;
    hence thesis by A3;
  end;
  (Partial_Sums(s)).0=s.0 by SERIES_1:def 1;
  then
A4: X[0] by A1;
  for n holds X[n] from NAT_1:sch 2(A4,A2);
  hence thesis;
end;
