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;
reserve z for Complex;
reserve Nseq,Nseq1 for increasing sequence of NAT;

theorem
  |.Partial_Sums(seq).k.| <= (Partial_Sums |.seq.|).k
proof
  defpred P[Nat] means
|. Partial_Sums(seq).$1 .| <= Partial_Sums(|.seq.|).$1;
A1: now
    let k;
    assume P[k];
    then
A2: |. Partial_Sums(seq).k.| + |.(seq).(k+1).| <= Partial_Sums(|.seq.| ).k
    + |.(seq).(k+1).| by XREAL_1:6;
    |. Partial_Sums(seq).(k+1) .| =|. Partial_Sums(seq).k + (seq).(k+1) .|
& |. Partial_Sums(seq).k + (seq).(k+1) .| <= |. Partial_Sums(seq).k.| + |. (seq
    ).(k+ 1) .| by COMPLEX1:56,SERIES_1:def 1;
    then
    |. Partial_Sums(seq).(k+1) .| <= Partial_Sums(|.seq.|).k + |.seq.(k+1)
    .| by A2,XXREAL_0:2;
    then
    |. Partial_Sums(seq).(k+1) .| <= Partial_Sums(|.seq.|).k+|.seq.|.(k+1)
    by VALUED_1:18;
    hence P[k+1] by SERIES_1:def 1;
  end;
  Partial_Sums(|.seq.|).0 = (|.seq.|).0 by SERIES_1:def 1
    .= |. seq.0 .| by VALUED_1:18;
  then
A3: P[0] by SERIES_1:def 1;
  for k being Nat holds P[k] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
