(* Partial identity matrix (used in rank decomposition). *)
Fact pid_mx_key : unit. Proof. by []. Qed.
Definition pid_mx {m n} r : 'M[R]_(m, n) :=
\matrix[pid_mx_key]_(i, j) ((i == j :> nat) && (i < r))%:R.

Lemma mul_col_row m1 m2 n p1 p2 (Au : 'M_(m1, n)) (Ad : 'M_(m2, n))
(Bl : 'M_(n, p1)) (Br : 'M_(n, p2)) :
col_mx Au Ad *m row_mx Bl Br
= block_mx (Au *m Bl) (Au *m Br) (Ad *m Bl) (Ad *m Br).