Canonical capmx_monoid n :=
Monoid.Law (@capmxA n n n n) (@cap1mx n) (@capmx1 n).
Canonical capmx_comoid n := Monoid.ComLaw (@capmxC n n n).

Lemma addsmx_diff_cap_eq m1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A :\: B + A :&: B :=: A)%MS.