Section groups.
Variable S : Set.
Variable G : S -> Prop.
Variable Add : S -> S -> S.
Variable O : S.
Variable Opp : S -> S.
Definition is_group :=
intern S G Add /\
associativity S Add /\ neutral S G Add O /\ opposite S G Add O Opp.

Lemma opp_opp : is_group -> forall x : S, G x -> x = Opp (Opp x).