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 quotient_gen A : A \subset 'N(H) -> <<A>> / H = <<A / H>>.