Local Notation "phi /  'B'" := (cfQuo phi) (at level 40) : cfun_scope.
Local Notation "phi %%  'B'" := (cfMod phi) (at level 40) : cfun_scope.
(* We specialize the cfMorph lemmas to cfMod by strengthening the domain      *)
(* condition G \subset 'N(H) to H <| G; the cfMorph lemmas can be used if the *)
(* stronger results are needed.                                               *)

(* Specifications: *)
Lemma cfModE phi x : B <| G -> x \in G -> (phi %% B)%CF x = phi (coset B x).
(* Note that cfQuo is nondegenerate even when G does not normalize B.         *)
Lemma cfQuoEnorm (phi : 'CF(G)) x :
B \subset cfker phi -> x \in 'N_G(B) -> (phi / B)%CF (coset B x) = phi x.
Lemma cfQuoE (phi : 'CF(G)) x :
B <| G -> B \subset cfker phi -> x \in G -> (phi / B)%CF (coset B x) = phi x.