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 Th23:
  Partial_Sums((Alfa(k+1,z,w))).k = (Partial_Sums(( Alfa(k,z,w))))
  .k + (Partial_Sums(( Expan_e(k+1,z,w) ))).k
proof
  now
    let l be Nat;
    assume l <= k;
    hence (Alfa(k+1,z,w)).l = (Alfa(k,z,w)).l + Expan_e(k+1,z,w).l by Th22
      .= ( (Alfa(k,z,w)) + Expan_e(k+1,z,w)).l by NORMSP_1:def 2;
  end;
  hence
  Partial_Sums((Alfa(k+1,z,w))).k =Partial_Sums(((Alfa(k,z,w)) + Expan_e(
  k+1,z,w))).k by Th11
    .=(Partial_Sums(( Alfa(k,z,w))) +Partial_Sums(( Expan_e(k+1,z,w )))).k
  by LOPBAN_3:15
    .= (Partial_Sums(( Alfa(k,z,w)))).k + (Partial_Sums(( Expan_e(k+1,z,w) )
  )).k by NORMSP_1:def 2;
end;
