Definition inversible (S : Set) (Mult : S -> S -> S) 
  (I x : S) := exists y : S, Mult x y = I /\ Mult y x = I.

(* Specifications: *)
Lemma inv_com : forall (S : Set) (Mult : S -> S -> S) (I x : S), commutativity S Mult -> (exists y : S, Mult x y = I) -> inversible S Mult I x.