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 Th18:
  for z,w st z,w are_commutative holds 1r/(n!) *((z+w) #N n) =
  Partial_Sums(Expan_e(n,z,w)).n
proof
  let z,w;
  assume z,w are_commutative;
  hence 1r/(n!)*((z+w) #N n) = 1r/(n!)*(Partial_Sums(Expan(n,z,w)).n) by Th16
    .= ((1r/(n!)) * (Partial_Sums(Expan(n,z,w)))).n by CLVECT_1:def 14
    .= Partial_Sums((1r/(n!)) * Expan(n,z,w)).n by CLOPBAN3:19
    .= Partial_Sums(Expan_e(n,z,w)).n by Th17;
end;
