Proof
fun x H
=> ex_intro
(op_is_inv ({-} x H))
x
(op_neg_inv x H).

(** Proves that negation is injective over the set of invertible elements - I.E. forall x and y if the negation of x equals the negation of y then x and y must be equal. *)
Theorem op_neg_inj
:  forall (x : M) (H : has_inv x) (y : M) (H0 : has_inv y),
{-} x H = {-} y H0 ->
x = y.