reserve x for Real;

theorem Th55:
  for x be Real, n,k be Nat 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 x be Real;
  let n,k be Nat;
  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 Th53
    .= 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;
