(* Properties of the laws of internal composition. *)
Section Internal.
Variable S : Set.
Variable G : S -> Prop.
Variable Add : S -> S -> S.
Variable O I : S.
Variable Opp Inv : S -> S.
Definition intern := forall x y : S, G x -> G y -> G (Add x y).
Definition commutativity := forall x y : S, Add x y = Add y x.
Definition associativity :=
forall x y z : S, Add x (Add y z) = Add (Add x y) z :>S.
Definition neutral (S : Set) (G : S -> Prop) (Add : S -> S -> S)
(O : S) := G O /\ (forall x : S, G x -> Add x O = x /\ Add O x = x).

(* Specifications: *)
Lemma neutral_add : neutral S G Add O -> O = Add O O.
Definition is_opposite (x y : S) := G x /\ G y /\ Add x y = O /\ Add y x = O.
Lemma opp_com :
commutativity ->
forall x y : S, G x -> G y -> Add x y = O -> is_opposite x y.