reserve X for Complex_Banach_Algebra,
  w,z,z1,z2 for Element of X,
  k,l,m,n,n1, n2 for Nat,
  seq,seq1,seq2,s,s9 for sequence of X,
  rseq for Real_Sequence;

theorem Th32:
  for seq st for k holds seq.k=Partial_Sums((Conj(k,z,w))).k holds
  seq is convergent & lim(seq)=0.X
proof
  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
A1: for k holds rseq.k=Partial_Sums(||.Conj(k,z,w).||).k;
  let seq such that
A2: for k holds seq.k=Partial_Sums((Conj(k,z,w))).k;
A3: now
    let k;
    ||.seq.k.||=||.Partial_Sums((Conj(k,z,w))).k.|| by A2;
    hence ||.seq.k.|| <= Partial_Sums(||.Conj(k,z,w).||).k by Th10;
  end;
A4: now
    let k;
    ||.seq.k.|| <= Partial_Sums(||.Conj(k,z,w).||).k by A3;
    hence ||.seq.k.|| <= rseq.k by A1;
  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 Th31;
    take n;
    now
      let k such that
A7:   n <= k;
      |.rseq.k-0 .|=|.Partial_Sums(||.Conj(k,z,w).||).k.| by A1;
      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 thesis by A4,A8,Th12;
end;
