Definition muln_grepr rW k := \big[dadd_grepr/grepr0]_(i < k) rW.

(* Specifications: *)
Lemma mx_rsim_socle (sG : socleType rG) (W : sG) (rW : representation) :
let modW : mxmodule rG W := component_mx_module rG (socle_base W) in
mx_rsim (socle_repr W) rW ->
mx_rsim (submod_repr modW) (muln_grepr rW (socle_mult W)).