(**
  Proves that an element's inverse is unique.

  Note: the field module depends on this proof
  being transparent.
*)
Definition prod_inv_uniq
  :  forall x y z : E, prod_is_inv x y -> prod_is_inv x z -> z = y
  := Ring.prod_inv_uniq ring.

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