Section AlgC.
Variable (gT : finGroupType).
Definition trowb n1 m2 n2 B A :=  @trow n1 A m2 n2 B.
Canonical Structure trowb_linear n1 m2 n2 B :=
Linear (@trowb_is_linear n1 m2 n2 B).
Canonical Structure trow_linear n1 m2 n2 A :=
Linear (@trow_is_linear n1 m2 n2 A).
Fixpoint tprod  (m1 : nat) :
forall n1 (A : 'M[F]_(m1,n1)) m2 n2 (B : 'M[F]_(m2,n2)),
'M[F]_(m1 * m2,n1 * n2) :=
if m1 is m'1.+1
return forall n1 (A : 'M[F]_(m1,n1)) m2 n2 (B : 'M[F]_(m2,n2)),
'M[F]_(m1 * m2,n1 * n2)
then
fun n1 (A : 'M[F]_(1 + m'1,n1)) m2 n2 B =>
(col_mx (trow (usubmx A) B) (tprod (dsubmx A) B))
else (fun _ _ _ _ _ => 0).
Let trow_mul (m1 m2 n2 p2 : nat)
(A : 'rV_m1) (B1: 'M[F]_(m2,n2)) (B2 :'M[F]_(n2,p2)) :
trow A (B1 *m B2) = B1 *m trow A B2.
Let tprod_tr m1 n1 (A :'M[F]_(m1, 1 + n1)) m2 n2 (B :'M[F]_(m2, n2)) :
tprod A B =  row_mx (trow (lsubmx A)^T B^T)^T (tprod (rsubmx A) B).

Lemma mxtrace_prod m n (A :'M[F]_(m)) (B :'M[F]_(n)) :
\tr (tprod A B) =  \tr A * \tr B.