theorem Th12:
  for n,seq st for m st m<=n holds seq.m<=r for m st m<=n holds
  Partial_Sums(seq).m <= r * (m+1)
proof
  let n,seq such that
A1: for m st m<=n holds seq.m<=r;
  defpred P[Nat] means
   $1<=n implies Partial_Sums(seq).$1<=r*($1+1);
A2: for m st P[m] holds P[m+1]
  proof
    let m such that
A3: P[m];
A4: Partial_Sums(seq).(m+1) = Partial_Sums(seq).m+seq.(m+1) by SERIES_1:def 1;
    assume
A5: m+1<=n;
    then seq.(m+1)<=r by A1;
    then Partial_Sums(seq).(m+1)<=(r*(m+1)) + r by A3,A5,A4,NAT_1:13,XREAL_1:7;
    hence thesis;
  end;
  Partial_Sums(seq).0=seq.0 by SERIES_1:def 1;
  then
A6: P[ 0 ] by A1;
  for m holds P[m] from NAT_1:sch 2(A6,A2);
  hence thesis;
end;
