End SubSocle.

(* Specifications: *)
Lemma subSocle_direct P : mxdirect (\sum_(W : sG | P W) W).
Definition Socle := (\sum_(W : sG) W)%MS.
Lemma simple_Socle M : mxsimple M -> (M <= Socle)%MS.
Lemma semisimple_Socle U : mxsemisimple U -> (U <= Socle)%MS.
Lemma reducible_Socle U :
mxmodule U -> mx_completely_reducible U -> (U <= Socle)%MS.
Lemma genmx_Socle : <<Socle>>%MS = Socle.
Lemma Socle_module : mxmodule Socle. Proof. exact: subSocle_module. Qed.
Lemma Socle_semisimple : mxsemisimple Socle.
End Socle.