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 Th29:
  |.(Partial_Sums(||.z.|| rExpSeq)).n.| = Partial_Sums(||.z.||
rExpSeq).n & ( n <= m implies |.(Partial_Sums(||.z.|| rExpSeq).m-Partial_Sums
  (||.z.|| rExpSeq).n).|
    = Partial_Sums(||.z.|| rExpSeq).m-Partial_Sums(||.z.||
  rExpSeq).n )
proof
  for n holds 0 <= (||. z .|| rExpSeq).n by Th26;
  hence thesis by COMSEQ_3:9;
end;
