Theorem elim_py_on_line_d_left_right : forall (A P Q Y : Point) (f : F), on_line_d Y P Q f -> Py Y A Y = f* (f * Py Q A Q + (1 - f) * Py Q A P) + (1 - f) * (f * Py P A Q + (1 - f) * Py P A P) .