(**
  Accepts two elements, x and y, and
  asserts that y is x's left inverse.
*)
Definition prod_is_inv_l := Monoid.op_is_inv_l prod_monoid.(**
  Accepts two elements, x and y, and
  asserts that y is x's right inverse.
*)
Definition prod_is_inv_r := Monoid.op_is_inv_r prod_monoid.(**
  Accepts two elements, x and y, and
  asserts that y is x's inverse.
*)
Definition prod_is_inv := Monoid.op_is_inv prod_monoid.(**
  Accepts one argument, x, and asserts that
  x has a left inverse.
*)
Definition prod_has_inv_l := Monoid.has_inv_l prod_monoid.(**
  Accepts one argument, x, and asserts that
  x has a right inverse.
*)
Definition prod_has_inv_r := Monoid.has_inv_r prod_monoid.(**
  Accepts one argument, x, and asserts that
  x has an inverse.
*)
Definition prod_has_inv := Monoid.has_inv prod_monoid.

Theorem prod_inv_sym : forall x y : E, prod_is_inv x y <-> prod_is_inv y x.