Proof Monoid.op_intro_r prod_monoid.
(** 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.

(** Proves that the left and right inverses of an element must be equal. *)
Theorem prod_inv_l_r_eq
:  forall x y : E, prod_is_inv_l x y -> forall z : E, prod_is_inv_r x z -> y = z.