Section kHomExtend.
Variables (K E : {subfield L}) (f : 'End(L)) (x y : L).
Fact kHomExtend_subproof :
linear (fun z => (map_poly f (Fadjoin_poly E x z)).[y]).
Definition kHomExtend := linfun (Linear kHomExtend_subproof).
End kHomExtend.
Definition kAut U V f := kHom U V f && (f @: V == V)%VS.
Canonical inv_ahom (f : 'AEnd(L)) : 'AEnd(L) := AHom (inv_is_ahom f).
Notation "f ^-1" := (inv_ahom f) : lrfun_scope.

Lemma comp_kHom_img K E f g :
kHom K (g @: E) f -> kHom K E g -> kHom K E (f \o g).