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 Th20:
  0.X rExpSeq is norm_summable & Sum(0.X rExpSeq)=1.X
proof
  defpred X[set] means Partial_Sums( ||. 0.X rExpSeq.||).$1=1;
A1: for n being Nat st X[n] holds X[n+1]
  proof
    let n be Nat such that
A2: Partial_Sums(||.0.X rExpSeq .||).n=1;
    thus Partial_Sums(||.0.X rExpSeq .||).(n+1) =1 + ||.0.X rExpSeq .||.(n+1)
    by A2,SERIES_1:def 1
      .=1 + ||.(0.X rExpSeq ).(n+1).|| by NORMSP_0:def 4
      .=1 + ||.(1/(n+1)*0.X)*(0.X rExpSeq .n) .|| by Th14
      .=1 + ||.(0.X)*(0.X rExpSeq .n) .|| by LOPBAN_3:38
      .=1 + ||.0.X.|| by LOPBAN_3:38
      .=1 + 0 by LOPBAN_3:38
      .=1;
  end;
  Partial_Sums(||.(0.X rExpSeq).||).0 =||. (0.X rExpSeq) .||.0 by
SERIES_1:def 1
    .=||. (0.X rExpSeq).0 .|| by NORMSP_0:def 4
    .=||. 1.X .|| by Th14
    .=1 by LOPBAN_3:38;
  then
A3: X[0];
A4: for n being Nat holds X[n] from NAT_1:sch 2(A3,A1);
  Partial_Sums(||.0.X rExpSeq .||) is constant
  by A4;
  then
A5: ||.0.X rExpSeq .|| is summable by SERIES_1:def 2;
  defpred X[set] means Partial_Sums(0.X rExpSeq ).$1=1.X;
A6: for n st X[n] holds X[n+1]
  proof
    let n;
    assume Partial_Sums(0.X rExpSeq ).n=1.X;
    hence Partial_Sums(0.X rExpSeq ).(n+1) =1.X + (0.X rExpSeq ).(n+1) by
BHSP_4:def 1
      .=1.X + (1/(n+1) *0.X)*(0.X rExpSeq .n) by Th14
      .=1.X + (0.X)*(0.X rExpSeq .n) by LOPBAN_3:38
      .=1.X + 0.X by LOPBAN_3:38
      .=1.X by LOPBAN_3:38;
  end;
  Partial_Sums(0.X rExpSeq ).0 =(0.X rExpSeq ).0 by BHSP_4:def 1
    .=1.X by Th14;
  then
A7: X[0];
  for n holds X[n] from NAT_1:sch 2(A7,A6);
  hence thesis by A5,Th2;
end;
