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 Th22:
  (for k holds seq.k=Partial_Sums((Conj(k,z,w))).k) implies
  seq is convergent & lim seq = 0
proof
 now
    let seq such that
A1: for k holds seq.k=Partial_Sums((Conj(k,z,w))).k;
A2: now
      let k;
   |.seq.k.|=|.Partial_Sums((Conj(k,z,w))).k.| by A1;
      hence |.seq.k.| <= Partial_Sums(|.Conj(k,z,w).|).k by COMSEQ_3:30;
    end;
    deffunc U(Nat) = Partial_Sums(|.Conj($1,z,w).|).$1;
     ex rseq be Real_Sequence st for k holds rseq.k = U(k) from SEQ_1:sch 1;
    then consider rseq be Real_Sequence such that
A3: for k holds rseq.k=Partial_Sums(|.Conj(k,z,w).|).k;
A4: now
      let k;
   |.seq.k.| <= Partial_Sums(|.Conj(k,z,w).|).k by A2;
      hence |.seq.k.| <= rseq.k by A3;
    end;
A5: now
      let p be Real;
      assume p>0;
      then consider n such that
A6:  for k st n <= k
      holds |.Partial_Sums(|.Conj(k,z,w).|).k.| < p by Th21;
      take n;
  now
        let k such that
A7:    n <= k;
    |.rseq.k-0.|=|.Partial_Sums(|.Conj(k,z,w).|).k.| by A3;
        hence |.rseq.k-0.| < p by A6,A7;
      end;
      hence for k st n <= k holds |.rseq.k-0.| < p;
    end;
then A8: rseq is convergent by SEQ_2:def 6;
then  lim(rseq) =0 by A5,SEQ_2:def 7;
    hence seq is convergent & lim(seq)=0c by A4,A8,COMSEQ_3:48;
  end;
  hence thesis;
end;
