proposition: Theorem Msigma_Hall : \sigma(M).-Hall(M) M`_\sigma.
proof: Proof. have [H hallH] := Hall_exists \sigma(M) solM; have [sHM sH _] := and3P hallH. rewrite /M`_\sigma (normal_Hall_pcore hallH) // -(quotientGK nsMaM). rewrite -(quotientGK (normalS _ sHM nsMaM)) ?cosetpre_normal //; last first. by rewrite (subset_trans sMaMs) ?pcore_sub_Hall. have hallHa: \sigma(M).-Hall(F) (H / M`_\alpha). apply: pHall_subl (subset_trans _ Malpha_quo_sub_Fitting) (Fitting_sub _) _. by rewrite quotientS ?sigma_Hall_sub_der1. exact: quotient_pHall (subset_trans sHM (normal_norm nsMaM)) hallH. rewrite (nilpotent_Hall_pcore (Fitting_nil _) hallHa) /=. exact: char_normal_trans (pcore_char _ _) nsFMa. Qed.
imports: mathcomp.ssreflect.ssreflect mathcomp(ssrbool ssrfun eqtype ssrnat seq div path fintype) mathcomp(bigop finset prime fingroup morphism perm automorphism quotient) mathcomp(action gproduct gfunctor pgroup cyclic center commutator) mathcomp(gseries nilpotent sylow abelian maximal hall) odd_order(BGsection1 BGsection3 BGsection4 BGsection5 BGsection6) odd_order(BGsection7 BGsection9)