Section Class_Def.
Variables aT rT : FalgType K.
Definition ahom_in (U : {vspace aT}) (f : 'Hom(aT, rT)) :=
all2rel (fun x y : aT => f (x * y) == f x * f y) (vbasis U) && (f 1 == 1).
Structure ahom := AHom {ahval :> 'Hom(aT, rT); _ : ahom_in {:aT} ahval}.
Canonical ahom_subType := Eval hnf in [subType for ahval].
Definition ahom_eqMixin := [eqMixin of ahom by <:].
Canonical ahom_eqType := Eval hnf in EqType ahom ahom_eqMixin.
Definition ahom_choiceMixin := [choiceMixin of ahom by <:].
Canonical ahom_choiceType := Eval hnf in ChoiceType ahom ahom_choiceMixin.
Fact linfun_is_ahom (f : {lrmorphism aT -> rT}) : ahom_in {:aT} (linfun f).
Canonical comp_ahom (f : ahom rT sT) (g : ahom aT rT) :=
AHom (comp_is_ahom (valP f) (valP g)).

Lemma aimg_adjoin (f : ahom aT rT) U x : (f @: <<U; x>> = <<f @: U; f x>>)%VS.