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 Th19:
  |.(Partial_Sums(|.z .| rExpSeq)).n.| = Partial_Sums(|.z .| rExpSeq).n &
  (n <= m implies
  |.(Partial_Sums(|.z .| rExpSeq).m-Partial_Sums(|.z .| rExpSeq).n).|
  = Partial_Sums(|.z .| rExpSeq).m-Partial_Sums(|.z .| rExpSeq).n)
proof
 for n holds 0 <= (|. z .| rExpSeq).n by Th18;
  hence thesis by COMSEQ_3:9;
end;
