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 Th57:
  for p st p>0 holds ex q st q>0 &
  for z being Complex st |.z.|<q holds |.(Sum(z P_dt)).|<p
proof
A1: for z holds |.(Sum(z P_dt)).| <= Sum (|.z P_dt.|)
  proof
    let z;
A2: for k holds |.Partial_Sums(z P_dt).|.k <= Partial_Sums(|.z P_dt.|).k
    proof
      let k;
         |.
Partial_Sums(z P_dt).k.|=|.Partial_Sums(z P_dt).|.k by VALUED_1:18;
      hence thesis by COMSEQ_3:30;
    end;
A3: z P_dt is absolutely_summable by Th55;
A4: z P_dt is summable by Th55,COMSEQ_3:63;
A5: lim|.Partial_Sums(z P_dt).|=|.lim(Partial_Sums(z P_dt)).|
    by A4,SEQ_2:27;
 lim|.Partial_Sums(z P_dt).|<=lim(Partial_Sums(|.z P_dt.|)) by A2,A3,SEQ_2:18;
    hence thesis by A5,SERIES_1:def 3;
  end;
A6: for z,n holds |.z P_dt.|.n<=|.z.|*|.z.| GeoSeq.n
  proof
    let z,n;
 |.z P_dt.|.n=|.z P_dt.n.| by VALUED_1:18
      .=|.(z|^(n+1))/((n+2)!).| by Def24
      .=|.(z|^(n+1))/((n+2)!).|;
then A7: |.z P_dt.|.n=|.(z|^(n+1)).|/((n+2)!) by Lm13
      .=|.z.| |^(n+1)/((n+2)!) by Lm8;
A8: |.z.|*|.z.| GeoSeq.n=|.z.|* (|.z.| |^ n) by PREPOWER:def 1
      .=|.z.| |^(n+1) by NEWTON:6;
     (n+2)!>=1 & |.z.| |^(n+1) >=0 by Th38,COMPLEX1:46,NEWTON:12,POWER:3;
then  |.z.| |^(n+1)/1>=|.z.| |^(n+1)/((n+2)!) by XREAL_1:118;
    hence thesis by A7,A8;
  end;
  let p0 be Real;
  assume
A9: p0>0;
  reconsider p = p0 as Real;
  consider q such that
A10: q=p/(p+1);
 p+1>p by XREAL_1:29;
then A11: q <1 by A9,A10,XREAL_1:189;
A12: for z st |.z.|<q holds |.(Sum(z P_dt)).|<p
  proof
    let z;
    assume
A13: |.z.|<q;
then A14: |.z.|<1 by A11,XXREAL_0:2;
A15: |.|.z.|.|<1 by A11,A13,XXREAL_0:2;
then A16: |.z.| GeoSeq is summable by SERIES_1:24;
A17: Sum(|.z.| GeoSeq) = 1/(1- |.z.|) by A15,SERIES_1:24;
A18: |.z.|(#)|.z.| GeoSeq is summable by A16,SERIES_1:10;
A19: for n holds |.z P_dt.|.n<=(|.z.|(#)|.z.| GeoSeq).n
    proof
      let n;
  |.z P_dt.|.n<=|.z.|*|.z.| GeoSeq.n by A6;
      hence thesis by SEQ_1:9;
    end;
 for n holds 0<=|.z P_dt.|.n
    proof
      let n;
  |.z P_dt.|.n=|.z P_dt.n.| by VALUED_1:18;
      hence thesis by COMPLEX1:46;
    end;
then
A20: Sum(|.z P_dt.|)<=Sum(|.z.|(#)|.z.| GeoSeq) by A18,A19,SERIES_1:20;
A21: Sum(|.z.|(#)|.z.| GeoSeq) =|.z.|/(1- |.z.|) by A16,A17,SERIES_1:10;
A22: |.z.|*(p+1)<(p/(p+1))*(p+1) by A9,A10,A13,XREAL_1:68;
 (p/(p+1))*(p+1)=p by A9,XCMPLX_1:87;
then A23: p*|.z.|+|.z.|-p*|.z.|<p-p*|.z.| by A22,XREAL_1:9;
A24: 1- |.z.|>0 by A14,XREAL_1:50;
then  |.z.|/(1- |.z.|)< p*(1- |.z.|)/(1- |.z.|) by A23,XREAL_1:74;
then  |.z.|/(1- |.z.|)<p by A24,XCMPLX_1:89;
then A25: Sum(|.z P_dt.|)<p by A20,A21,XXREAL_0:2;
 |.(Sum(z P_dt)).|<=Sum (|.z P_dt.|) by A1;
    hence thesis by A25,XXREAL_0:2;
  end;
  take q;
  thus q>0 by A9,A10;
  let z being Complex;
  thus thesis by A12;
end;
