End SplittingFieldFor.
Section kHom.
Variables (F : fieldType) (L : fieldExtType F).
Implicit Types (U V : {vspace L}) (K E : {subfield L}) (f g : 'End(L)).
Definition kHom U V f := ahom_in V f && (U <= fixedSpace f)%VS.

(* Specifications: *)
Lemma kHomP {K V f} :
reflect [/\ {in V &, forall x y, f (x * y) = f x * f y}
& {in K, forall x, f x = x}]
(kHom K V f).
Lemma kAHomP {U V} {f : 'AEnd(L)} :
reflect {in U, forall x, f x = x} (kHom U V f).
Lemma kHomSl U1 U2 V f : (U1 <= U2)%VS -> kHom U2 V f -> kHom U1 V f.
Lemma kHom_inv K E f : kHom K E f -> {in E, {morph f : x / x^-1}}.
Lemma kHom_dim K E f : kHom K E f -> \dim (f @: E) = \dim E.
Lemma kHom_is_rmorphism K E f :
kHom K E f -> rmorphism (f \o vsval : subvs_of E -> L).