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_add :
is_group ->
commutativity S Add ->
forall x y : S, G x -> G y -> Opp (Add x y) = Add (Opp x) (Opp y).