Definition max_submod (U V : 'M_n) :=
(U < V)%MS /\ (forall W, ~ [/\ modG W, U < W & W < V])%MS.

Lemma max_submod_eqmx U1 U2 V1 V2 :
(U1 :=: U2)%MS -> (V1 :=: V2)%MS -> max_submod U1 V1 -> max_submod U2 V2.