Definition mx_subseries := all modG.
Definition mx_composition_series V :=
mx_subseries V /\ (forall i, i < size V -> max_submod (0 :: V)`_i V`_i).
Local Notation mx_series := mx_composition_series.
Fact mx_subseries_module V i : mx_subseries V -> mxmodule rG V`_i.
Fact mx_subseries_module' V i : mx_subseries V -> mxmodule rG (0 :: V)`_i.

Lemma mx_series_rcons U V :
mx_series (rcons U V) <-> [/\ mx_series U, modG V & max_submod (last 0 U) V].