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 st n <= m holds rseq1.n <= p * rseq2.n) implies Partial_Sums(
  rseq1).m <= p * Partial_Sums(rseq2).m
proof
  defpred P[Nat] means
$1 <= m implies Partial_Sums(rseq1).$1 <= p
  * Partial_Sums(rseq2).$1;
  assume
A1: for n st n <= m holds rseq1.n <= p * rseq2.n;
A2: now
    let n be Nat such that
A3: P[n];
    now
      assume
A4:   n + 1 <= m;
      then rseq1.(n+1) <= p * rseq2.(n+1) by A1;
      then
A5:   p * Partial_Sums(rseq2).(n) + rseq1.(n+1) <= p * Partial_Sums(rseq2
      ).(n) + p * rseq2.(n+1) by XREAL_1:6;
      n < n+1 & Partial_Sums(rseq1).(n+1) =Partial_Sums(rseq1).(n)+rseq1.
      (n+1) by SERIES_1:def 1,XREAL_1:29;
      then
A6:   Partial_Sums(rseq1).(n+1) <= p * Partial_Sums(rseq2).(n) + rseq1.(n
      +1) by A3,A4,XREAL_1:6,XXREAL_0:2;
      p * Partial_Sums(rseq2).(n) + p * rseq2.(n+1) =p * (Partial_Sums(
      rseq2).(n) +rseq2.(n+1))
        .=p * Partial_Sums(rseq2).(n+1) by SERIES_1:def 1;
      hence Partial_Sums(rseq1).(n+1) <= p * Partial_Sums(rseq2).(n+1) by A6,A5
,XXREAL_0:2;
    end;
    hence P[n+1];
  end;
A7: P[0]
  proof
    assume 0 <= m;
    Partial_Sums(rseq1).(0) =rseq1.0 & p * Partial_Sums(rseq2).(0) =p *
    rseq2.0 by SERIES_1:def 1;
    hence thesis by A1;
  end;
  for n be Nat holds P[n] from NAT_1:sch 2(A7,A2);
  hence thesis;
end;
