Section Defs.
Variable a : A.
Canonical horner_alg_additive := Additive horner_alg_is_lrmorphism.
Canonical horner_alg_rmorphism := RMorphism horner_alg_is_lrmorphism.
Canonical horner_alg_linear := AddLinear horner_alg_is_lrmorphism.
Canonical horner_alg_lrmorphism := [lrmorphism of horner_alg].
End Defs.
Variable (pf : {lrmorphism {poly R} -> A}).
End HornerAlg.
Fact comp_poly_multiplicative q : multiplicative (comp_poly q).
Canonical comp_poly_rmorphism q := AddRMorphism (comp_poly_multiplicative q).
Canonical comp_poly_lrmorphism q := [lrmorphism of comp_poly q].

Lemma horner_comp p q x : (p \Po q).[x] = p.[q.[x]].