Section SomeHall.
Variable gT : finGroupType.
Implicit Types (p : nat) (pi : nat_pred) (G H K P R : {group gT}).

(* Specifications: *)
Lemma Hall_pJsub p pi G H P :
pi.-Hall(G) H -> p \in pi -> P \subset G -> p.-group P ->
exists2 x, x \in G & P :^ x \subset H.
Lemma Hall_psubJ p pi G H P :
pi.-Hall(G) H -> p \in pi -> P \subset G -> p.-group P ->
exists2 x, x \in G & P \subset H :^ x.
Lemma Hall_setI_normal pi G K H :
K <| G -> pi.-Hall(G) H -> pi.-Hall(K) (H :&: K).
Lemma coprime_mulG_setI_norm H G K R :
K * R = G -> G \subset 'N(H) -> coprime #|K| #|R| ->
(K :&: H) * (R :&: H) = G :&: H.