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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.             *)
(*                                                                    *)
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(* 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                                                     *)
(** * replacement_theorem.v *)
Set Implicit Arguments.
Unset Strict Implicit.
Require Export has_n_elements.
Require Export bases.
Require Export lin_dep_facts.

(* Specifications: *)
(** - %\intoc{\bf Theorem 1.10}% in this file, nothing else *)
Lemma replacement_theorem :
forall (F : field) (V : vectorspace F) (beta : basis V) (n : Nat),
has_n_elements n beta ->
forall Sset : part_set V,
lin_indep Sset ->
forall m : Nat,
m <= n ->
has_n_elements m Sset ->
exists S1 : part_set beta,
has_n_elements (n - m) S1 /\
generates (union Sset (inject_subsets S1)) (full V).