Proof
fun x y H
=> H || a = 1 @a by prod_is_comm x y.

(* Specifications: *)
(** 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.