Definition func_group (A : Setoid) (B : group) : GROUP :=
  Build_group
    (Build_group_on (group_inverse_map:=func_inv A B)
       (func_inv_is_r_inverse (A:=A) (B:=B))
       (func_inv_is_l_inverse (A:=A) (B:=B))).

Lemma func_group_is_abelian : forall (A : Setoid) (B : abelian_group), abelian_group_on (func_group A B).