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 Th19:
  0.X ExpSeq is norm_summable & Sum(0.X ExpSeq)=1.X
proof
 reconsider jj=1 as Element of REAL by XREAL_0:def 1;
  defpred X[set] means Partial_Sums( ||. 0.X ExpSeq.||).$1=jj;
A1: for n being Nat st X[n] holds X[n+1]
  proof
    let n be Nat such that
A2: Partial_Sums(||.0.X ExpSeq .||).n=jj;
    thus
    Partial_Sums(||.0.X ExpSeq .||).(n+1) =1 + ||.0.X ExpSeq .||.(n+1) by A2,
SERIES_1:def 1
      .=1 + ||.(0.X ExpSeq ).(n+1).|| by NORMSP_0:def 4
      .=1 + ||.(1r/(n+1) * 0.X)*(0.X ExpSeq .n) .|| by Th13
      .=1 + ||.(0.X)*(0.X ExpSeq .n) .|| by CLVECT_1:1
      .=1 + ||.0.X.|| by CLOPBAN3:38
      .=1 + 0 by CLOPBAN3:38
      .=jj;
  end;
  Partial_Sums(||.(0.X ExpSeq).||).0 =||. (0.X ExpSeq) .||.0 by SERIES_1:def 1
    .=||. (0.X ExpSeq).0 .|| by NORMSP_0:def 4
    .=||. 1.X .|| by Th13
    .=1 by CLOPBAN3:38;
  then
A3: X[0];
  for n being Nat holds X[n] from NAT_1:sch 2(A3,A1);
  then Partial_Sums(||.0.X ExpSeq .||) is constant by VALUED_0:def 18;
  then
A4: ||.0.X ExpSeq .|| is summable by SERIES_1:def 2;
  defpred X[set] means Partial_Sums(0.X ExpSeq ).$1=1.X;
A5: for n st X[n] holds X[n+1]
  proof
    let n;
    assume Partial_Sums(0.X ExpSeq ).n=1.X;
    hence Partial_Sums(0.X ExpSeq ).(n+1) =1.X + (0.X ExpSeq ).(n+1) by
BHSP_4:def 1
      .=1.X + (1r/(n+1) *0.X)*(0.X ExpSeq .n) by Th13
      .=1.X + (0.X)*(0.X ExpSeq .n) by CLVECT_1:1
      .=1.X + 0.X by CLOPBAN3:38
      .=1.X by RLVECT_1:def 4;
  end;
  Partial_Sums(0.X ExpSeq ).0 =(0.X ExpSeq ).0 by BHSP_4:def 1
    .=1.X by Th13;
  then
A6: X[0];
  for n holds X[n] from NAT_1:sch 2(A6,A5);
  then lim(Partial_Sums(0.X ExpSeq ))=1.X by Th2;
  hence thesis by A4,CLOPBAN3:def 2,def 3;
end;
