reserve X for Complex_Banach_Algebra,
  w,z,z1,z2 for Element of X,
  k,l,m,n,n1, n2 for Nat,
  seq,seq1,seq2,s,s9 for sequence of X,
  rseq for Real_Sequence;

theorem Th10:
  ||.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;
A3: ||. Partial_Sums(seq).k + (seq).(k+1) .|| <= ||. Partial_Sums(seq).k
    .|| + ||. (seq).(k+1) .|| by CLVECT_1:def 13;
    ||. Partial_Sums(seq).(k+1) .|| =||. Partial_Sums(seq).k + (seq).(k+1)
    .|| by BHSP_4:def 1;
    then
    ||. Partial_Sums(seq).(k+1) .|| <= Partial_Sums(||.seq.||).k + ||.seq.
    (k+1).|| by A3,A2,XXREAL_0:2;
    then
    ||. Partial_Sums(seq).(k+1) .|| <= Partial_Sums(||.seq.||).k+||.seq.||
    .(k+1) by NORMSP_0:def 4;
    hence P[k+1] by SERIES_1:def 1;
  end;
  Partial_Sums(||.seq.||).0 = (||.seq.||).0 by SERIES_1:def 1
    .= ||. seq.0 .|| by NORMSP_0:def 4;
  then
A4: P[0] by BHSP_4:def 1;
  for k holds P[k] from NAT_1:sch 2(A4,A1);
  hence thesis;
end;
