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 Th36:
  s=s1(#)s1 implies for n holds (Partial_Sums s).n>=0
proof
  defpred X[Nat] means (Partial_Sums s).$1>=0;
  assume
A1: s=s1(#)s1;
A2: for n st X[n] holds X[n+1]
  proof
    let n;
    assume
A3: (Partial_Sums(s)).n>=0;
A4: (s1.(n+1))^2>=0 by XREAL_1:63;
    (Partial_Sums s).(n+1)=(Partial_Sums(s)).n+s.(n+1) by SERIES_1:def 1
      .=(Partial_Sums s).n+(s1.(n+1))^2 by A1,SEQ_1:8;
    hence thesis by A3,A4;
  end;
  (Partial_Sums s).0=s.0 by SERIES_1:def 1
    .=(s1.0)^2 by A1,SEQ_1:8;
  then
A5: X[0] by XREAL_1:63;
  for n holds X[n] from NAT_1:sch 2(A5,A2);
  hence thesis;
end;
