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 Th38:
  exp(z)*exp(-z)= 1.X & exp(-z)*exp(z)= 1.X
proof
  reconsider jj=1 as Real;
  z *(-z) =z *((-jj)*z) by LOPBAN_3:38
    .=(-jj)*(z*z) by LOPBAN_3:38
    .=(-jj)*z*z by LOPBAN_3:38
    .=(-z)*z by LOPBAN_3:38;
  then
A1: z,(-z) are_commutative;
  hence exp(z)*exp(-z) =exp(z+(-z)) by Th35
    .=exp(0.X) by RLVECT_1:5
    .=1.X by Th37;
  thus exp(-z)*exp(z) =exp((-z)+z) by A1,Th35
    .=exp(0.X) by RLVECT_1:5
    .=1.X by Th37;
end;
