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
  exp(z) is invertible & (exp(z))" = exp(-z) & exp(-z) is invertible & (
  exp(-z))" = exp(z)
proof
A1: exp(-z)*exp(z)= 1.X by Th38;
A2: exp(z)*exp(-z)= 1.X by Th38;
  hence exp(z) is invertible by A1;
  hence (exp(z))" = exp(-z) by A2,A1,LOPBAN_3:def 8;
  thus exp(-z) is invertible by A2,A1;
  hence thesis by A2,A1,LOPBAN_3:def 8;
end;
