reserve X for 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:
  1 <= Sum(||.z.|| rExpSeq)
proof
  ||. Partial_Sums(z rExpSeq).0 .|| =||. (z rExpSeq).0 .|| by BHSP_4:def 1
    .=||. 1.X.|| by Th14
    .=1 by LOPBAN_3:38;
  hence thesis by Th28;
end;
