reserve q,th,r for Real,
  a,b,p for Real,
  w,z for Complex,
  k,l,m,n,n1,n2 for Nat,
  seq,seq1,seq2,cq1 for Complex_Sequence,
  rseq,rseq1,rseq2 for Real_Sequence,
  rr for set,
  hy1 for 0-convergent non-zero Real_Sequence;

theorem Th9:
  ((0c qua Complex) ExpSeq is absolutely_summable)
  & Sum((0c qua Complex) ExpSeq)=1r
proof
  defpred X[set] means Partial_Sums(|.(0c qua Complex) ExpSeq.|).$1=jj;
 Partial_Sums(|.((0c qua Complex) ExpSeq).|).0
  =|.((0c qua Complex) ExpSeq).|.0 by SERIES_1:def 1
    .=|. ((0c qua Complex) ExpSeq).0 .| by VALUED_1:18
    .=1 by Th3,COMPLEX1:48;
then A1: X[0];
A2: for n being Nat st X[n] holds X[n+1]
  proof
    let n being Nat such that
A3: Partial_Sums(|.(0c qua Complex) ExpSeq.|).n=jj;
    thus Partial_Sums(|.(0c qua Complex) ExpSeq.|).(n+1)
    =1 + |.(0c qua Complex) ExpSeq.|.(n+1) by A3,SERIES_1:def 1
      .=1 + |.((0c qua Complex) ExpSeq).(n+1).| by VALUED_1:18
      .=1 + |.((0c qua Complex) ExpSeq.n) * 0c/(n+1+0*<i>) .| by Th3
      .=jj by COMPLEX1:44;
  end;
 for n being Nat holds X[n] from NAT_1:sch 2(A1,A2);
then  Partial_Sums(|.(0c qua Complex) ExpSeq.|)
  is constant by VALUED_0:def 18;
then A4: |.(0c qua Complex) ExpSeq.| is summable by SERIES_1:def 2;
  defpred X[set] means Partial_Sums((0c qua Complex) ExpSeq).$1=1;
 Partial_Sums((0c qua Complex) ExpSeq).0
  =((0c qua Complex) ExpSeq).0 by SERIES_1:def 1
    .=1 by Th3;
then A5: X[0];
A6: for n st X[n] holds X[n+1]
  proof
    let n such that
A7: Partial_Sums((0c qua Complex) ExpSeq).n=1;
    thus Partial_Sums((0c qua Complex) ExpSeq).(n+1)
    =1r + ((0c qua Complex) ExpSeq).(n+1) by A7,SERIES_1:def 1
      .=1r + ((0c qua Complex) ExpSeq.n) * 0c/(n+1+0*<i>) by Th3
      .=1;
  end;
 for n holds X[n] from NAT_1:sch 2(A5,A6);
  hence thesis by A4,COMSEQ_2:10;
end;
