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 Th55:
  for z being Complex holds z P_dt is absolutely_summable
proof
  let z be Complex;
 ex r st for n holds Partial_Sums(|.z P_dt.|).n<r
  proof
A1: for n holds Partial_Sums(|.z P_dt.|).n<Partial_Sums(|.z ExpSeq.|).(n+1)
    proof
      let n;
   Partial_Sums(|.z P_dt.|).0=|.z P_dt.|.0 by SERIES_1:def 1
        .=|.z P_dt.0 .| by VALUED_1:18
        .=|.(z|^(0+1))/((0+2)!).| by Def24
        .=|.z/2.| by NEWTON:14;
then A2:   Partial_Sums(|.z P_dt.|).0=|.z.|/2 by Lm13;
   Partial_Sums(|.z ExpSeq.|).(0+1)
      =Partial_Sums(|.z ExpSeq.|).0+|.z ExpSeq.|.(0+1) by SERIES_1:def 1
        .=|.z ExpSeq.|.0+|.z ExpSeq.|.1 by SERIES_1:def 1
        .=|.z ExpSeq.|.0+|.z ExpSeq.1 .| by VALUED_1:18
        .=|.z ExpSeq.0 .|+|.z ExpSeq.1 .| by VALUED_1:18
        .=|.z ExpSeq.0 .|+|.z .| by Lm8
        .=1+|.z .| by Lm8,COMPLEX1:48;
then A3:   Partial_Sums(|.z ExpSeq.|).(0+1)-Partial_Sums(|.z P_dt.|). 0
      =1+(|.z .| - |.z.|/2) by A2;
      defpred X[Nat] means
      Partial_Sums(|.z P_dt.|).$1 <Partial_Sums(|.z ExpSeq.|).($1+1);
   0 <= |.z.| by COMPLEX1:46;
then A4:   X[0] by A3,XREAL_1:47;
A5:   for n st X[n] holds X[n+1]
      proof
        let n such that
A6:    Partial_Sums(|.z P_dt.|).n <Partial_Sums(|.z ExpSeq.|).(n+1);
    Partial_Sums(|.z P_dt.|).(n+1)
        =Partial_Sums(|.z P_dt.|).n+|.z P_dt.|.(n+1) by SERIES_1:def 1
          .=Partial_Sums(|.z P_dt.|).n+|.z P_dt.(n+1).| by VALUED_1:18
.=Partial_Sums(|.z P_dt.|).n+|.(z|^((n+1)+1))/(((n+1)+2)!).| by Def24
          .=Partial_Sums(|.z P_dt.|).n+|.(z|^(n+2))/((n+3)!).|;
then A7:    Partial_Sums(|.z P_dt.|).(n+1)
        =Partial_Sums(|.z P_dt.|).n+|.(z|^(n+2)).|/((n+3)!) by Lm13;
    Partial_Sums(|.z ExpSeq.|).((n+1)+1)
        = Partial_Sums(|.z ExpSeq.|).(n+1)+|.z ExpSeq.|.(n+(1+1)) by
SERIES_1:def 1
          .=Partial_Sums(|.z ExpSeq.|).(n+1)+|.z ExpSeq.(n+2).| by VALUED_1:18
          .=Partial_Sums(|.z ExpSeq.|).(n+1)+|.(z|^(n+2))/((n+2)!).| by Def4
          .=Partial_Sums(|.z ExpSeq.|).(n+1)+|.(z|^(n+2))/((n+2)!).|;
then A8:    Partial_Sums(|.z ExpSeq.|).((n+1)+1)
        =Partial_Sums(|.z ExpSeq.|).(n+1)+|.(z|^(n+2)).|/((n+2)!)
        by Lm13;
    (n+2)<(n+3) by XREAL_1:6;
then A9:    (n+2)!<=(n+3)! by Th38;
    |.(z|^(n+2)).|>=0 by COMPLEX1:46;
then     |.(z|^(n+2)).|/((n+2)!)>=|.(z|^(n+2)).|/((n+3)!) by A9,XREAL_1:118;
then A10:    Partial_Sums(|.z P_dt.|).n+|.(z|^(n+2)).|/((n+3)!)<=
        Partial_Sums(|.z P_dt.|).n+|.(z|^(n+2)).|/((n+2)!) by XREAL_1:6;
    Partial_Sums(|.z P_dt.|).n+|.(z|^(n+2)).|/((n+2)!)<
        Partial_Sums(|.z ExpSeq.|).(n+1)+|.(z|^(n+2)).|/((n+2)!)
        by A6,XREAL_1:6;
        hence thesis by A7,A8,A10,XXREAL_0:2;
      end;
  for n holds X[n] from NAT_1:sch 2(A4,A5);
      hence thesis;
    end;
    consider r be Real such that
A11: for n holds Partial_Sums(|.z ExpSeq.|).n<r by SEQ_2:def 3;
A12: for n holds Partial_Sums(|.z P_dt.|).n<r
    proof
      let n;
  Partial_Sums(|.z P_dt.|).n<Partial_Sums(|.z ExpSeq.|). (n+1) by A1;
      hence thesis by A11,XXREAL_0:2;
    end;
    take r;
    thus thesis by A12;
  end;
then  Partial_Sums(|.z P_dt.|) is bounded_above by SEQ_2:def 3;
  hence thesis by COMSEQ_3:61;
end;
