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 Th3:
  for z being Complex holds
  z ExpSeq.(n+1) = z ExpSeq.n * z /(n+1+0*<i>) & z ExpSeq.0=1 &
  |.(z ExpSeq).n .| = (|.z.| rExpSeq ).n
proof
  let z be Complex;
A1: now
    let n be Nat;
    thus (z ExpSeq).(n+1)=z |^ (n+1) / ((n+1) ! ) by Def4
      .=(z GeoSeq).n * z /((n+1)! ) by COMSEQ_3:def 1
      .=z |^ n * z /(n! * (n+1+0*<i>)) by Th1
      .=(z |^ n / (n! ) )*( z / (n+1+0*<i>)) by XCMPLX_1:76
      .=(z |^ n / (n! ) * z) / (n+1+0*<i>)
      .=z ExpSeq.n * z /(n+1+0*<i>) by Def4;
  end;
A2: (z ExpSeq).0=z |^ 0 /(0! ) by Def4
    .=1 by Th1,COMSEQ_3:def 1;
  defpred X[Nat] means
|. ((z ExpSeq)).$1 .| =( |.z.| rExpSeq ).$1;
 ( |.z.| rExpSeq ).0 = ((|.z .|) |^ 0)/ (0!) by Def5
    .= 1/((Prod_real_n).0) by NEWTON:4
    .= 1/1 by Def2
    .= 1;
then A3: X[0] by A2,COMPLEX1:48;
A4: now
    let n such that
A5: X[n];
A6: |.(n+1+0*<i>).|=n+1 by ABSVALUE:def 1;
 |. ((z ExpSeq)).(n+1) .| =|. z ExpSeq.n * z /(n+1+0*<i>) .| by A1
      .=|. z ExpSeq.n * z .|/|.(n+1+0*<i>).| by COMPLEX1:67
      .=(( |.z.| rExpSeq ).n) * |.z .|/|.(n+1+0*<i>).| by A5,COMPLEX1:65
      .= (|.z.| |^ n)/(n!) * |.z .|/|.(n+1+0*<i>).| by Def5
      .=( (|.z.| |^ n) * |.z .| )/( (n!) * |.(n+1+0*<i>).| ) by XCMPLX_1:83
      .=( (|.z.| |^ n) * |.z .| )/( (n+1)! ) by A6,NEWTON:15
      .=( (|.z.| |^ (n+1)) )/( (n+1)! ) by NEWTON:6
      .=( |.z.| rExpSeq ).(n+1) by Def5;
    hence X[n+1];
  end;
 for n holds X[n] from NAT_1:sch 2(A3,A4);
  hence thesis by A1,A2;
end;
