(**
  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.

Theorem prod_inv_1_eq_1 : forall x : E, prod_is_inv x 1 -> x = 1.