End NormalHall.
Section Morphim.
Variables (aT rT : finGroupType) (D : {group aT}) (f : {morphism D >-> rT}).
Implicit Types (pi : nat_pred) (G H P : {group aT}).

(* Specifications: *)
Lemma morphim_pgroup pi G : pi.-group G -> pi.-group (f @* G).
Lemma morphim_p_index pi G H :
H \subset D -> pi.-nat #|G : H| -> pi.-nat #|f @* G : f @* H|.
Lemma morphim_pHall pi G H :
H \subset D -> pi.-Hall(G) H -> pi.-Hall(f @* G) (f @* H).
Lemma pmorphim_pHall pi G H :
G \subset D -> H \subset D -> pi.-subgroup(H :&: G) ('ker f) ->
pi.-Hall(f @* G) (f @* H) = pi.-Hall(G) H.
Lemma morphim_Hall G H : H \subset D -> Hall G H -> Hall (f @* G) (f @* H).
Lemma morphim_pSylow p G P :
P \subset D -> p.-Sylow(G) P -> p.-Sylow(f @* G) (f @* P).
Lemma morph_p_elt pi x : x \in D -> pi.-elt x -> pi.-elt (f x).
End Morphim.