Canonical coset_baseGroupType :=
Eval hnf in BaseFinGroupType coset_of coset_of_groupMixin.
Canonical coset_groupType := FinGroupType coset_invP.
(* Projection of the initial group type over the cosets groupType.  *)
Definition coset x : coset_of := insubd (1 : coset_of) (H :* x).

Lemma rcoset_kercosetP x y :
x \in 'N(H) -> y \in 'N(H) -> reflect (coset H x = coset H y) (x \in H :* y).