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 z for s,t be Real holds exp(s*z)*exp(t*z) = exp((s+t)*z) & exp(t*z
  )*exp(s*z) = exp((t+s)*z) & exp((s+t)*z) = exp((t+s)*z) & exp(s*z),exp(t*z)
  are_commutative
proof
  let z;
  let s, t be Real;
A1: s*z,t*z are_commutative by Th40;
  hence
A2: exp(s*z)*exp(t*z) = exp(s*z+t*z) by Th35
    .= exp((s+t)*z) by LOPBAN_3:38;
  thus
A3: exp(t*z)*exp(s*z) = exp(t*z+s*z) by A1,Th35
    .= exp((t+s)*z) by LOPBAN_3:38;
  thus exp((s+t)*z) = exp((t+s)*z);
  thus thesis by A2,A3;
end;
