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 Th16:
  |. Partial_Sums((z ExpSeq)).k .| <= Partial_Sums(|.z.| rExpSeq).k &
  Partial_Sums((|.z.| rExpSeq)).k <= Sum(|.z.| rExpSeq) &
  |. Partial_Sums(( z ExpSeq)).k .| <= Sum(|.z.| rExpSeq)
proof
  defpred X[Nat] means |. Partial_Sums(( z ExpSeq)).$1 .|
  <= Partial_Sums((|.z.| rExpSeq)).$1;
  A1: |.
 Partial_Sums(( z ExpSeq)).0 .| = |. ((z ExpSeq)).0 .| by SERIES_1:def 1
    .= |. (z |^ 0)/ (0 !).| by Def4
    .= 1 by Th1,COMPLEX1:48,COMSEQ_3:def 1;
 Partial_Sums((|.z.| rExpSeq)).0 = ((|.z.| rExpSeq)).0 by SERIES_1:def 1
    .= ((|.z .|) |^ 0)/ (0!) by Def5
    .= 1 by NEWTON:4,12;
then A2: X[0] by A1;
A3: for k st X[k] holds X[k+1]
  proof
    let k such that
A4: |. Partial_Sums(( z ExpSeq)).k .| <= Partial_Sums((|.z.| rExpSeq)).k;
     |.
 Partial_Sums(( z ExpSeq)).(k+1) .| =|. Partial_Sums(( z ExpSeq)).k + ((z
ExpSeq)).(k+1) .| & |. Partial_Sums(( z ExpSeq)).k + ((z ExpSeq)).(k+1) .| <=
    |. Partial_Sums(( z ExpSeq)).k.| + |. ((z ExpSeq)).(k+1) .| by COMPLEX1:56
,SERIES_1:def 1;
then A5: |. Partial_Sums(( z ExpSeq)).(k+1) .|
    <= |. Partial_Sums(( z ExpSeq)).k.| + (|.z.| rExpSeq).(k+1) by Th3;
A6: |. Partial_Sums(( z ExpSeq)).k.| + (|.z.| rExpSeq).(k+1)
    <= Partial_Sums((|.z.| rExpSeq)).k + (|.z.| rExpSeq).(k+1) by A4,XREAL_1:6;
 Partial_Sums((|.z.| rExpSeq)).k + (|.z.| rExpSeq).(k+1)
    =Partial_Sums((|.z.| rExpSeq)).(k+1) by SERIES_1:def 1;
    hence thesis by A5,A6,XXREAL_0:2;
  end;
A7: for k holds X[k] from NAT_1:sch 2(A2,A3);
  thus
  then
|. Partial_Sums(( z ExpSeq)).k .| <= Partial_Sums((|.z.| rExpSeq)).k;
 now
    let k be object such that
A8: k in NAT;
    thus |. (z ExpSeq).|.k = |. (z ExpSeq).k .| by VALUED_1:18
      .= (|.z.| rExpSeq).k by A8,Th3;
  end;
then A9: |. z ExpSeq .| = |.z.| rExpSeq;
then
 Partial_Sums((|.z.| rExpSeq)).k <= lim(Partial_Sums((|.z.| rExpSeq )))
  by SEQ_4:37;
  hence Partial_Sums((|.z.| rExpSeq)).k <= Sum(|.z.| rExpSeq)
  by SERIES_1:def 3;
A10: now
    let k;
     lim
(Partial_Sums((|.z.| rExpSeq)))=Sum(|.z.| rExpSeq) by SERIES_1:def 3;
    hence Partial_Sums((|.z.| rExpSeq)).k <= Sum(|.z.| rExpSeq)
    by A9,SEQ_4:37;
  end;
A11: |. Partial_Sums(( z ExpSeq)).k .|
  <= Partial_Sums((|.z.| rExpSeq)).k by A7;
 Partial_Sums((|.z.| rExpSeq)).k <= Sum(|.z.| rExpSeq) by A10;
  hence thesis by A11,XXREAL_0:2;
end;
