Section AbelianGroup1.
Context {G : AbelianGroup}.
Definition zero := AbelianGroup.zero _ (AbelianGroup.class G).
Definition plus := AbelianGroup.plus _ (AbelianGroup.class G).
Definition opp := AbelianGroup.opp _ (AbelianGroup.class G).
Definition minus x y := (plus x (opp y)).

Lemma plus_reg_l :
forall r x y : G,
plus r x = plus r y -> x = y.