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

theorem Th51:
  for n being Element of NAT, z being Complex holds (
  cos_C/.z - <i>*sin_C/.z) |^ n = cos_C/.(n*z) - <i>*sin_C/.(n*z)
proof
  let n be Element of NAT;
  let z be Complex;
  (cos_C/.z - <i>*sin_C/.z) |^ n = (cos_C/.z + <i>*-sin_C/.z) |^ n
    .=(cos_C/.z + <i>*sin_C/.-z) |^ n by Th2
    .=(cos_C/.-z + <i>*sin_C/.-z) |^ n by Th3
    .= cos_C/.(-n*z) + <i>*sin_C/.(n*-z) by Th50
    .= cos_C/.(n*z) + <i>*(sin_C/.(-n*z)) by Th3
    .= cos_C/.(n*z) + <i>*(-sin_C/.(n*z)) by Th2
    .= cos_C/.(n*z) + -<i>*sin_C/.(n*z);
  hence thesis;
end;
