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}).

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.