reserve rseq, rseq1, rseq2 for Real_Sequence;
reserve seq, seq1, seq2 for Complex_Sequence;
reserve k, n, n1, n2, m for Nat;
reserve p, r for Real;

theorem
  (for n holds rseq.n <= p) implies for n holds Partial_Sums(rseq).n <=
  p * (n+1)
proof
  defpred P[Nat] means Partial_Sums(rseq).$1 <= p * ($1+1);
  assume
A1: for n holds rseq.n <= p;
A2: now
    let n be Nat such that
A3: P[n];
    rseq.(n+1) <= p by A1;
    then
A4: p * (n+1)+ rseq.(n+1) <= p * (n+1) + p by XREAL_1:6;
    Partial_Sums(rseq).(n+1) =Partial_Sums(rseq).(n)+rseq.(n+1) by
SERIES_1:def 1;
    then Partial_Sums(rseq).(n+1) <= p * (n+1) + rseq.(n+1) by A3,XREAL_1:6;
    hence P[n+1] by A4,XXREAL_0:2;
  end;
  Partial_Sums(rseq).(0) =rseq.0 by SERIES_1:def 1;
  then
A5: P[0] by A1;
  thus for n be Nat holds P[n] from NAT_1:sch 2(A5,A2);
end;
