End ExternalAction.
Section SylowSolvableAct.
Variables (gT : finGroupType) (p : nat).
Implicit Types A B G X : {group gT}.

Lemma sol_coprime_Sylow_subset A G X :
A \subset 'N(G) -> coprime #|G| #|A| -> solvable A ->
X \subset G -> p.-group X -> A \subset 'N(X) ->
exists P : {group gT}, [/\ p.-Sylow(G) P, A \subset 'N(P) & X \subset P].