Proof Monoid.op_inv_r_uniq prod_monoid.
(** Proves that an element's inverse is unique.  Note: this theorem is defined as transparent to allow a theorem in the field module to compile. *)
Definition prod_inv_uniq
:  forall x y z : E, prod_is_inv x y -> prod_is_inv x z -> z = y
:= Monoid.op_inv_uniq prod_monoid.

(** Proves that the identity element is its own left inverse. *)
Theorem prod_inv_1_l
:  prod_is_inv_l 1 1.