Definition socle_eqMixin := Eval hnf in [eqMixin of sG by <:].
Canonical socle_eqType := Eval hnf in EqType sG socle_eqMixin.
Definition socle_choiceMixin := Eval hnf in [choiceMixin of sG by <:].
Canonical socle_choiceType := ChoiceType sG socle_choiceMixin.

Lemma subSocle_iso M :
mxsimple M -> (M <= S)%MS -> {W : sG | P W & mx_iso (socle_base W) M}.