
theorem Th11:
  for n, i, j being Element of NAT holds [** cos((2*PI*i)/n),sin((
  2*PI*i)/n) **]*[** cos((2*PI*j)/n),sin((2*PI*j)/n) **] = [** cos((2*PI*((i+j)
  mod n))/n), sin((2*PI*((i+j)mod n))/n)**]
proof
  let n, i, j be Element of NAT;
  (2*PI*i)/n + (2*PI*j)/n = (2*PI*i + 2*PI*j)/n by XCMPLX_1:62
    .= (2*PI*(i+j))/n;
  then
  cos((2*PI*i)/n) * cos((2*PI*j)/n) - sin((2*PI*i)/n) * sin((2*PI*j)/n ) =
  cos( (2*PI*(i+j))/n) & cos((2*PI*i)/n) * sin((2*PI*j)/n) + cos((2*PI*j)/n) *
  sin((2 *PI*i)/n) = sin((2*PI*(i+j))/n) by SIN_COS:75;
  then
  [** cos((2*PI*i)/n), sin((2*PI*i)/n) **] * [** cos((2*PI*j)/n), sin((2*
  PI*j)/n) **] = [** cos((2*PI*(i+j))/n), sin((2*PI*(i+j))/n) **]
    .= [** cos((2*PI*((i+j) mod n))/n), sin((2*PI*((i+j) mod n))/n)**] by Th10;
  hence thesis;
end;
