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
  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 rExpSeq)).n =z1*(z2 rExpSeq).n by LOPBAN_3:def 5
      .=z1*(1/(n! )*(z2 #N n)) by Def2
      .=(1/(n! ))*(z1*(z2 #N n)) by LOPBAN_3:38
      .=(1/(n! ))*((z2 #N n)*z1) by A1,Lm2
      .=(1/(n! )*(z2 #N n))*z1 by LOPBAN_3:38
      .=((z2 rExpSeq).n)*z1 by Def2
      .=((z2 rExpSeq)*z1).n by LOPBAN_3:def 6;
  end;
  then
A2: z1*(z2 rExpSeq) = (z2 rExpSeq) *z1 by FUNCT_2:63;
A3: Partial_Sums( z2 rExpSeq) is convergent by LOPBAN_3:def 1;
  thus z1*exp(z2) =z1*Sum(z2 rExpSeq) by Def10
    .=lim(z1*Partial_Sums(z2 rExpSeq)) by A3,Th6
    .=lim(Partial_Sums(z1*(z2 rExpSeq))) by Th9
    .=lim(Partial_Sums(z2 rExpSeq)*z1) by A2,Th9
    .=Sum(z2 rExpSeq) *z1 by A3,Th7
    .=exp(z2) *z1 by Def10;
end;
