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(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
(* This program is distributed in the hope that it will be useful,    *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of     *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
(** * subspaces.v *)
Section MAIN.
Set Implicit Arguments.
Unset Strict Implicit.
Require Export vecspaces_verybasic.
Require Export arb_intersections.
From Algebra Require Export Sub_module.
From Algebra Require Export Singleton.
Require Export algebra_omissions.
Variable F : field.
Variable V : vectorspace F.
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 *)

Lemma inj_subspace_injective : injective inj_subspace.
red in |- *.