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 Th20:
  |.Partial_Sums(|.Conj(k,z,w).|).n.|=Partial_Sums(|.Conj(k,z,w).|).n
proof
A1: now
    let n;
 |.Conj(k,z,w).|.n=|.(Conj(k,z,w)).n.| by VALUED_1:18;
    hence 0 <= |.Conj(k,z,w).|.n by COMPLEX1:46;
  end;
  A2: Partial_Sums(|.Conj(k,z,w).|).0 <= Partial_Sums(|.Conj(k,z,w).|).n &
  Partial_Sums(|.Conj(k,z,w).|).0=(|.Conj(k,z,w).|).0 by SEQM_3:6
,SERIES_1:def 1;
 0 <= (|.Conj(k,z,w).|).0 by A1;
  hence thesis by A2,ABSVALUE:def 1;
end;
