Fact mul_polyA : associative mul_poly.
Fact mul_1poly : left_id 1%:P mul_poly.
Fact mul_poly1 : right_id 1%:P mul_poly.
Fact mul_polyDl : left_distributive mul_poly +%R.
Fact mul_polyDr : right_distributive mul_poly +%R.
Fact poly1_neq0 : 1%:P != 0 :> {poly R}.

Lemma coefMC c p i : (p * c%:P)`_i = p`_i * c.