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 Th30:
  |.(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 being Nat holds 0 <= (||. z .|| rExpSeq).n by Th27;
  hence thesis by COMSEQ_3:9;
end;
