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 sub_im_coset (C : {set coset_of}) : C \subset coset @* 'N(A).