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 Th34:
  for z1,z2 st z1,z2 are_commutative holds exp(z1+z2)=exp(z1) *exp
  (z2) & exp(z2+z1)=exp(z2) *exp(z1) & exp(z1+z2)=exp(z2+z1) & exp(z1),exp(z2)
  are_commutative
proof
  let z1,z2 such that
A1: z1,z2 are_commutative;
A2: exp(z2+z1)=Sum((z2+z1) ExpSeq) by Def6
    .=Sum(z2 ExpSeq)*Sum(z1 ExpSeq) by A1,Lm3
    .=exp(z2)*Sum(z1 ExpSeq) by Def6
    .=exp(z2) *exp(z1) by Def6;
  exp(z1+z2)=Sum((z1+z2) ExpSeq) by Def6
    .=Sum(z1 ExpSeq)*Sum(z2 ExpSeq) by A1,Lm3
    .=exp(z1)*Sum(z2 ExpSeq) by Def6
    .=exp(z1) *exp(z2) by Def6;
  hence thesis by A2,LOPBAN_4:def 1;
end;
