reserve X for 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 Th19:
  for z,w st z,w are_commutative holds 1/ (n! ) *((z+w) #N n) =
  Partial_Sums(Expan_e(n,z,w)).n
proof
  let z,w;
  assume z,w are_commutative;
  hence 1/(n!)*((z+w) #N n)= 1/(n! )*(Partial_Sums(Expan(n,z,w)).n) by Th17
    .= ((1/(n! )) * (Partial_Sums(Expan(n,z,w)))).n by NORMSP_1:def 5
    .= Partial_Sums( (1/(n! )) * Expan(n,z,w)).n by LOPBAN_3:19
    .= Partial_Sums(Expan_e(n,z,w)).n by Th18;
end;
