Section KernelLemmas.
Variable K : fieldType.
(* convertible to kermx (horner_mx g p) when n = n.+1 *)
Definition kermxpoly n (g : 'M_n) (p : {poly K}) : 'M_n :=
kermx ((if n is n.+1 then horner_mx^~ p : 'M_n.+1 -> 'M_n.+1 else \0) g).

Lemma kermxpolyM n (g : 'M_n) (p q : {poly K}) : coprimep p q ->
(kermxpoly g (p * q) :=: kermxpoly g p + kermxpoly g q)%MS.