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.

Lemma kAutE K E f : kAut K E f = kHom K E f && (f @: E <= E)%VS.