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 Th15:
  Partial_Sums(seq).k=Partial_Sums(Shift seq).k+seq.k
proof
  defpred X[Nat] means
Partial_Sums(seq).$1=Partial_Sums(Shift seq)
  .$1+seq.$1;
A1: for k st X[k] holds X[k+1]
  proof
    let k;
    assume Partial_Sums(seq).k=Partial_Sums(Shift(seq)).k+seq.k;
    hence
    Partial_Sums(seq).(k+1) = (Partial_Sums(Shift(seq)).k+seq.k) + seq.(k
    +1) by BHSP_4:def 1
      .=(Partial_Sums(Shift(seq)).k+(Shift(seq)).(k+1)) + seq.(k+1) by
LOPBAN_4:def 5
      .=Partial_Sums(Shift(seq)).(k+1)+seq.(k+1) by BHSP_4:def 1;
  end;
  Partial_Sums(seq).0 = seq.0 by BHSP_4:def 1
    .= 0.X + seq.0 by RLVECT_1:4
    .= (Shift(seq)).0 + seq.0 by LOPBAN_4:def 5
    .= Partial_Sums(Shift(seq)).0+seq.0 by BHSP_4:def 1;
  then
A2: X[0];
  for k holds X[k] from NAT_1:sch 2(A2,A1);
  hence thesis;
end;
