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
  for z1,z2 st z1,z2 are_commutative holds z1 * exp(z2)=exp(z2) * z1
proof
  let z1,z2 such that
A1: z1,z2 are_commutative;
  now
    let n be Element of NAT;
    thus (z1*(z2 ExpSeq)).n =z1*(z2 ExpSeq).n by LOPBAN_3:def 5
      .=z1*(1r/(n!)*(z2 #N n)) by Def1
      .=(1r/(n!))*(z1*(z2 #N n)) by CLOPBAN3:38
      .=(1r/(n!))*((z2 #N n)*z1) by A1,Lm2
      .=(1r/(n!)*(z2 #N n))*z1 by CLOPBAN3:38
      .=((z2 ExpSeq).n)*z1 by Def1
      .=((z2 ExpSeq)*z1).n by LOPBAN_3:def 6;
  end;
  then
A2: z1*(z2 ExpSeq) = (z2 ExpSeq) *z1 by FUNCT_2:63;
A3: Partial_Sums( z2 ExpSeq) is convergent by CLOPBAN3:def 1;
  thus z1*exp(z2) =z1*Sum(z2 ExpSeq) by Def6
    .=z1*lim (Partial_Sums(z2 ExpSeq)) by CLOPBAN3:def 2
    .=lim(z1*Partial_Sums(z2 ExpSeq)) by A3,Th6
    .=lim(Partial_Sums(z1*(z2 ExpSeq))) by Th9
    .=lim(Partial_Sums(z2 ExpSeq)*z1) by A2,Th9
    .=lim(Partial_Sums(z2 ExpSeq))*z1 by A3,Th7
    .=Sum(z2 ExpSeq) *z1 by CLOPBAN3:def 2
    .=exp(z2) *z1 by Def6;
end;
