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 Th37:
  exp(z)*exp(-z)= 1.X & exp(-z)*exp(z)= 1.X
proof
  z *(-z) =z *((-1r)*z) by CLVECT_1:3
    .=(-1r)*(z*z) by CLOPBAN3:38
    .=(-1r)*z*z by CLOPBAN3:38
    .=(-z)*z by CLVECT_1:3;
  then
A1: z,(-z) are_commutative by LOPBAN_4:def 1;
  hence exp(z)*exp(-z) =exp(z+(-z)) by Th34
    .=exp(0.X) by RLVECT_1:5
    .=1.X by Th36;
  thus exp(-z)*exp(z) =exp((-z)+z) by A1,Th34
    .=exp(0.X) by RLVECT_1:5
    .=1.X by Th36;
end;
