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 Th33:
  for n,k be Element of NAT,x be Real st n <> 0 holds (cos((x+2*PI
  *k)/n) + sin((x+2*PI*k)/n)*<i>)|^ n = cos x + (sin x)*<i>
proof
  let n,k be Element of NAT,x be Real;
  assume
A1: n <> 0;
  thus (cos((x+2*PI*k)/n)+ sin((x+2*PI*k)/n)*<i>)|^ n = cos (n*((x+2*PI*k)/n))
  +sin (n*((x+2*PI*k)/n))*<i> by Th31
    .= cos(x+2*PI*k)+sin(n*((x+2*PI*k)/n))*<i> by A1,XCMPLX_1:87
    .= cos(x+2*PI*k)+sin(x+2*PI*k)*<i> by A1,XCMPLX_1:87
    .= cos.(x+2*PI*k)+sin(x+2*PI*k)*<i> by SIN_COS:def 19
    .= cos.(x+2*PI*k)+(sin.(x+2*PI*k))*<i> by SIN_COS:def 17
    .= cos.(x+2*PI*k)+(sin.x)*<i> by SIN_COS2:10
    .= cos.x + (sin.x)*<i> by SIN_COS2:11
    .= cos.x + (sin x)*<i> by SIN_COS:def 17
    .= cos x + (sin x)*<i> by SIN_COS:def 19;
end;
