reserve x,y for Real;
reserve z,z1,z2 for Complex;
reserve n for Element of NAT;

theorem
  for n being Element of NAT, z being Element of COMPLEX holds exp(-<i>*
  n*z) = (cos_C/.z - <i>*sin_C/.z) |^ n
proof
  let n be Element of NAT;
  let z be Element of COMPLEX;
  exp(-<i>*n*z) = exp(-<i>*(n*z)) .=cos_C/.(n*z) - <i>*sin_C/.(n*z) by Th37;
  hence thesis by Th51;
end;
