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;
reserve d for Real;
reserve th,th1,th2 for Real;

theorem Th56:
  for z being Complex holds z*(Sum(z P_dt))=(Sum(z ExpSeq))-1r-z
proof
  let z be Complex;
A1: for z being Complex holds z (#) (z P_dt)=(z ExpSeq)^\ 2
  proof
    let z be Complex;
 for n being Element of NAT holds (z(#)(z P_dt)).n=((z ExpSeq)^\ 2).n
    proof
      let n be Element of NAT;
   (z(#)(z P_dt)).n=z*(z P_dt).(n) by VALUED_1:6
        .=z*((z|^(n+1))/((n+2)!)) by Def24;
then    (z(#)(z P_dt)).n= (z*(z|^(n+1)))/((n+2)!)
        .=(z|^(n+1+1))/((n+2)!) by NEWTON:6
        .=z ExpSeq.(n+2) by Def4
        .=((z ExpSeq)^\ 2).n by NAT_1:def 3;
      hence thesis;
    end;
    hence thesis;
  end;
   Sum
(z ExpSeq)= Partial_Sums(z ExpSeq).1 + Sum(z ExpSeq^\(1+1)) by COMSEQ_3:60
    .=Partial_Sums(z ExpSeq).1 + Sum(z ExpSeq^\(2));
then
A2: Sum(z (#) (z P_dt))=Sum(z ExpSeq) - Partial_Sums(z ExpSeq).(0+1) by A1
    .=Sum(z ExpSeq)-(Partial_Sums(z ExpSeq).0+z ExpSeq.1) by SERIES_1:def 1
    .=Sum(z ExpSeq)-(z ExpSeq.0+z ExpSeq.1) by SERIES_1:def 1
    .=Sum(z ExpSeq)-(1r+z ExpSeq.1) by Lm8
    .=Sum(z ExpSeq)-(1r+z) by Lm8
    .=(Sum(z ExpSeq))-1r-z;
  reconsider BBB=z P_dt as absolutely_summable Complex_Sequence by Th55;
 BBB is summable;
  hence thesis by A2,COMSEQ_3:56;
end;
