Section MatrixAlgebra.
Variables F : fieldType.
Local Notation "A \in R" := (@submx F _ _ _ (mxvec A) R).
Arguments memmx_subP {m1 m2 n R1 R2}.
Arguments memmx_eqP {m1 m2 n R1 R2}.
Arguments memmx_addsP {m1 m2 n A R1 R2}.
Arguments memmx_sumsP {I P n A R_}.
Definition mulsmx m1 m2 n (R1 : 'A[F]_(m1, n)) (R2 : 'A_(m2, n)) :=
(\sum_i <<R1 *m lin_mx (mulmxr (vec_mx (row i R2)))>>)%MS.
Arguments mulsmx {m1%N m2%N n%N} R1%MS R2%MS.
Local Notation "R1 * R2" := (mulsmx R1 R2) : matrix_set_scope.
Arguments mulsmx_subP {m1 m2 m n R1 R2 R}.

Lemma mulsmxS m1 m2 m3 m4 n (R1 : 'A_(m1, n)) (R2 : 'A_(m2, n))
(R3 : 'A_(m3, n)) (R4 : 'A_(m4, n)) :
(R1 <= R3 -> R2 <= R4 -> R1 * R2 <= R3 * R4)%MS.