proposition: Lemma submxMfree m1 m2 n p (A : 'M_(m1, n)) (B : 'M_(m2, n)) (C : 'M_(n, p)) : row_free C -> (A *m C <= B *m C)%MS = (A <= B)%MS.
proof: Proof. case/row_freeP=> C' C_C'_1; apply/idP/idP=> sAB; last exact: submxMr. by rewrite -[A]mulmx1 -[B]mulmx1 -C_C'_1 !mulmxA submxMr. Qed.
imports: HB(structures) mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq choice) mathcomp(fintype finfun bigop finset fingroup perm order) mathcomp(div prime binomial ssralg finalg zmodp matrix)