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 Th8:
  ((z+w) |^ n) / (n!) = Partial_Sums(Expan_e(n,z,w)).n
proof
  thus ((z+w) |^ n)/(n! )
  = (Partial_Sums(Expan(n,z,w)).n) * (1r/(n! )) by Th6
    .= ((1r/(n! )) (#) (Partial_Sums(Expan(n,z,w)))).n by VALUED_1:6
    .= Partial_Sums( (1r/(n! )) (#) Expan(n,z,w)).n by COMSEQ_3:29
    .= Partial_Sums(Expan_e(n,z,w)).n by Th7;
end;
