reserve a,b,c,d,a9,b9,c9,d9,y,x1,u,v for Real,
  s,t,h,z,z1,z2,z3,s1,s2,s3 for Complex;

theorem
  for z being Element of COMPLEX for n being Element of NAT
  holds z|^ n = (|.z.| to_power n)*cos (n*Arg z) + (|.z.| to_power n)*sin (
  n*Arg z)*<i>
proof
  let z be Element of COMPLEX;
  let n be Element of NAT;
    z|^ n =(|.z.|*cos Arg z-0*sin Arg z+(|.z.|*sin Arg z+cos Arg z*0)*<i>)
    |^ n by COMPTRIG:62
      .=(|.z.|*(cos Arg z+sin Arg z*<i>))|^ n
      .= |.z.||^ n*(cos Arg z+sin Arg z*<i>)|^ n by NEWTON:7;
    hence z|^ n = (|.z.| to_power n)*(cos (n*Arg z)+sin (n*Arg z)*<i>) by Th31
      .= (|.z.| to_power n)*cos(n*Arg z)+(|.z.| to_power n)*sin(n*Arg z)*<i>;
end;
