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 Th21:
  for p being Real st p>0 ex n st for k st n <= k
  holds |.Partial_Sums(|.Conj(k,z,w).|).k.| < p
proof
  let p be Real such that
A1: p>0;
  reconsider pp = p as Real;
A2: 1 <= Sum(|.z.| rExpSeq) by Th17;
A3: 0 < Sum(|.w.| rExpSeq) by Th17;
  set p1=min((pp/(4 * Sum(|.z.| rExpSeq))),(pp/(4 * Sum(|.w.| rExpSeq))));
A4: p1 > 0 by A1,A2,A3,XXREAL_0:15;
 now
    let k be object such that
A5: k in NAT;
    thus |. (z ExpSeq).|.k = |. (z ExpSeq).k .| by VALUED_1:18
      .= (|.z.| rExpSeq).k by A5,Th3;
  end;
then  |. z ExpSeq .| = |.z.| rExpSeq;
  then consider n1 such that
A6: for k,l be Nat st n1 <= k & n1 <= l holds
  |.Partial_Sums(( |.z.| rExpSeq )).k - Partial_Sums(( |.z.| rExpSeq )).l.|
  < p1 by A4,COMSEQ_3:4;
  consider n2 being Nat such that
A7: for k,l be Nat st n2 <= k & n2 <= l holds
  |.Partial_Sums(( w ExpSeq )).k - Partial_Sums(( w ExpSeq )).l.|
  < p1 by A4,COMSEQ_3:47;
  set n3=n1+n2;
  take n4 = n3+n3;
A8: now
    let n;
    let k;
 now
      let l be Nat such that
A9:  l <= k;
      thus |.Conj(k,z,w).|.l =|.( Conj(k,z,w).l) .| by VALUED_1:18
        .=|.(z ExpSeq).l * (Partial_Sums(w ExpSeq).k
      -Partial_Sums(w ExpSeq).(k-'l)) .| by A9,Def13
        .=|.(z ExpSeq).l.| * |.(Partial_Sums(w ExpSeq).k
      -Partial_Sums(w ExpSeq).(k-'l)).| by COMPLEX1:65
        .=(|.z.| rExpSeq).l * |.(Partial_Sums(w ExpSeq).k
      -Partial_Sums(w ExpSeq).(k-'l)).| by Th3;
    end;
    hence for l be Nat st l <= k holds
    |.Conj(k,z,w).|.l =(|.z.| rExpSeq).l * |.(Partial_Sums(w ExpSeq).k
    -Partial_Sums(w ExpSeq).(k-'l)).|;
  end;
A10: now
    let k;
 now
      let l be Nat;
      assume l <= k;
      then
A11:  |.Conj(k,z,w).|.l =(|.z .| rExpSeq).l * |.Partial_Sums(w ExpSeq ).k
      -Partial_Sums(w ExpSeq).(k-'l).| by A8;
  |.Partial_Sums(w ExpSeq).k .| <= Sum(|.w.| rExpSeq) by Th16;
then A12:  |.Partial_Sums(w ExpSeq).k .| + |.Partial_Sums(w ExpSeq).(k-'l ).|
      <= Sum(|.w.| rExpSeq) + |.Partial_Sums(w ExpSeq).(k-'l).| by XREAL_1:6;
  |.Partial_Sums(w ExpSeq).(k-'l).| <= Sum(|.w.| rExpSeq) by Th16;
then   Sum(|.w.| rExpSeq) + |.Partial_Sums(w ExpSeq).(k-'l).|
      <= Sum(|.w.| rExpSeq)+ Sum(|.w.| rExpSeq) by XREAL_1:6;
      then   |.
Partial_Sums(w ExpSeq).k -Partial_Sums(w ExpSeq).(k-'l).| <= |.Partial_Sums
(w ExpSeq).k .| + |.Partial_Sums(w ExpSeq).(k-'l).| & |.Partial_Sums(w ExpSeq)
.k .| + |.Partial_Sums(w ExpSeq).(k-'l).| <= 2 * Sum(|.w.| rExpSeq) by A12,
COMPLEX1:57,XXREAL_0:2;
then A13:  |.Partial_Sums(w ExpSeq).k -Partial_Sums(w ExpSeq).(k-'l).|
      <= 2 * Sum(|.w.| rExpSeq) by XXREAL_0:2;
  0 <= (|.z .| rExpSeq).l by Th18;
      thus
      then |.Conj(k,z,w).|.l <= (|.z .| rExpSeq).l * (2 * Sum(|.w.| rExpSeq))
      by A11,A13,XREAL_1:64;
    end;
    hence for l be Nat st l <= k holds
    |.Conj(k,z,w).|.l <= (|.z .| rExpSeq).l * (2 * Sum(|.w.| rExpSeq));
  end;
 now
    let k such that
A14: n4 <= k;
A15: 0+n3 <= n3+n3 by XREAL_1:6;
then A16: n3 <= k by A14,XXREAL_0:2;
A17: n1+0 <= n1+n2 by XREAL_1:6;
then A18: n1 <= k by A16,XXREAL_0:2;
 now
      let l be Nat;
      assume
A19:  l <= n3;
then A20:  k-'l=k-l by A16,XREAL_1:233,XXREAL_0:2;
A21:  0+n2 <= n1+n2 by XREAL_1:6;
A22:  n4-l <= k-l by A14,XREAL_1:9;
  n3+n3-n3 <= n3+n3-l by A19,XREAL_1:10;
then   n3 <= k-l by A22,XXREAL_0:2;
then A23:  n2 <= k-'l by A20,A21,XXREAL_0:2;
  0+n3 <= n3+n3 by XREAL_1:6;
then   n2 <= n4 by A21,XXREAL_0:2;
then   n2 <= k by A14,XXREAL_0:2;
      then A24:  |.
Partial_Sums(( w ExpSeq )).k - Partial_Sums(( w ExpSeq )).(k -'l).|
      < p1 by A7,A23;
  0 <= (|.z .| rExpSeq).l by Th18;
then   (|.z .| rExpSeq).l * |.Partial_Sums(( w ExpSeq )).k
      - Partial_Sums(( w ExpSeq )).(k-'l).|
      <= (|.z .| rExpSeq).l * p1 by A24,XREAL_1:64;
      hence
|.Conj(k,z,w).|.l <= p1 * (|.z .| rExpSeq).l by A8,A16,A19,XXREAL_0:2;
    end;
then A25: Partial_Sums(|.Conj(k,z,w).|).n3
    <= Partial_Sums(|.z .| rExpSeq).n3 * p1 by COMSEQ_3:7;
 Partial_Sums(|.z .| rExpSeq).n3 * p1
    <= Sum(|.z.| rExpSeq) * p1 by A4,Th16,XREAL_1:64;
then A26: Partial_Sums(|.Conj(k,z,w).|).n3
    <= Sum(|.z.| rExpSeq) * p1 by A25,XXREAL_0:2;
A27: Sum(|.z.| rExpSeq) * p1 <= Sum(|.z.| rExpSeq) * (p/(4 * Sum
    (|.z.| rExpSeq))) by A2,XREAL_1:64,XXREAL_0:17;
A28: 0 <> Sum(|.z.| rExpSeq) by Th17;
 Sum(|.z.| rExpSeq) * (p/(4 * Sum(|.z.| rExpSeq)))
    =(Sum(|.z.| rExpSeq) * p)/(4 * Sum(|.z.| rExpSeq))
      .=p/4 by A28,XCMPLX_1:91;
then A29: Partial_Sums(|.Conj(k,z,w).|).n3 <= p/4 by A26,A27,XXREAL_0:2;
 0+p/4 < p/4 + p/4 by A1,XREAL_1:6;
then A30: Partial_Sums(|.Conj(k,z,w).|).n3 < p/2 by A29,XXREAL_0:2;
 k-'n3=k-n3 by A14,A15,XREAL_1:233,XXREAL_0:2;
then A31: k=(k-'n3)+n3;
 for l be Nat st l <= k holds
    |.Conj(k,z,w).|.l <= (2 * Sum(|.w.| rExpSeq)) * (|.z .| rExpSeq).l by A10;
then A32: Partial_Sums(|.Conj(k,z,w).|).(k) -Partial_Sums(|.Conj(k,z,w).|). n3
    <= (Partial_Sums(|.z .| rExpSeq).(k) -Partial_Sums(|.z .| rExpSeq).n3)
    * (2 * Sum(|.w.| rExpSeq)) by A31,COMSEQ_3:8;
 |.(Partial_Sums(|.z .| rExpSeq).k-Partial_Sums(|.z .| rExpSeq).n3).|
    <= p1 by A6,A17,A18;
then  Partial_Sums(|.z .| rExpSeq).k-Partial_Sums(|.z .| rExpSeq).n3
    <= p1 by A14,A15,Th19,XXREAL_0:2;
then  (Partial_Sums(|.z .| rExpSeq).k -Partial_Sums(|.z .| rExpSeq).n3)
    * (2 * Sum(|.w.| rExpSeq))
    <= p1 * (2 * Sum(|.w.| rExpSeq)) by A3,XREAL_1:64;
then A33: Partial_Sums(|.Conj(k,z,w).|).k-Partial_Sums(|.Conj(k,z,w).|).n3
    <= p1 * (2 * Sum(|.w.| rExpSeq)) by A32,XXREAL_0:2;
A34: (2 * Sum(|.w.| rExpSeq)) * p1
    <= (2 * Sum(|.w.| rExpSeq)) * (p/(4 * Sum(|.w.| rExpSeq)))
    by A3,XREAL_1:64,XXREAL_0:17;
A35: 0 <> Sum(|.w.| rExpSeq) by Th17;
 (2 * Sum(|.w.| rExpSeq) ) * (p/(4 * Sum(|.w.| rExpSeq)))
    =((2 * p ) * Sum(|.w.| rExpSeq) ) /(4 * Sum(|.w.| rExpSeq))
      .= (p +p )/4 by A35,XCMPLX_1:91
      .= p/2;
then  Partial_Sums(|.Conj(k,z,w).|).k-Partial_Sums(|.Conj(k,z,w).|).n3
    <= p/2 by A33,A34,XXREAL_0:2;
then  (Partial_Sums(|.Conj(k,z,w).|).k -Partial_Sums(|.Conj(k,z,w).|).n3)
    +Partial_Sums(|.Conj(k,z,w).|).n3 < (p/2)+(p/2) by A30,XREAL_1:8;
    hence |.Partial_Sums(|.Conj(k,z,w).|).k.| < p by Th20;
  end;
  hence thesis;
end;
