Section Subspace_def.
(** %\tocdef{subspace}% *)
Definition subspace (F : field) (V : vectorspace F) := submodule V.
(** - Some finger-flexing first *)
Variable W : subspace V.
Definition inj_subspace : Hom (W:VECSP F) V.
apply
(BUILD_HOM_MODULE (R:=F) (Mod:=W:VECSP F) (Mod':=V)
(ff:=fun x : W => subtype_elt x)); auto with algebra.
Defined.
(* This injects the subspace into its "parent" space *)
red in |- *.
auto with algebra.
Qed.

Lemma mult_inherited :
forall (c : F) (x : W), inj_subspace (c mX x) =' c mX inj_subspace x in _.