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 n be non zero Element of NAT, k be Element of NAT holds cos(2*PI*
  k/n) + sin(2*PI*k/n)*<i> is CRoot of n,1
proof
  let n be non zero Element of NAT,k be Element of NAT;
  (cos(2*PI*k/n)+sin(2*PI*k/n)*<i>)|^n =cos(n*((2*PI*k)/n))+sin(n*((2*PI*k
  )/n))*<i> by Th31
    .=cos(2*PI*k)+(sin(n*((2*PI*k)/n)))*<i> by XCMPLX_1:87
    .=cos(2*PI*k+0)+(sin(2*PI*k+0))*<i> by XCMPLX_1:87
    .=cos.(2*PI*k+0)+(sin(2*PI*k+0))*<i> by SIN_COS:def 19
    .=cos.(2*PI*k+0)+(sin.(2*PI*k+0))*<i> by SIN_COS:def 17
    .=cos.(2*PI*k+0)+(sin.0)*<i> by SIN_COS2:10
    .=1 by SIN_COS:30,SIN_COS2:11;
  hence thesis by COMPTRIG:def 2;
end;
