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