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(* as published by the Free Software Foundation; either version 2.1   *)
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(* GNU General Public License for more details.                       *)
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(* 02110-1301 USA                                                     *)
(** * bases_from_generating_sets.v *)
Set Implicit Arguments.
Unset Strict Implicit.
Require Export bases.
Require Export finite_subsets.
Require Export lin_dep_facts.
Section MAIN.
Variable F : field.
Variable V : vectorspace F.
Variable W0 : part_set V.
Variable H : is_finite_subset W0.
Variable H0 : generates W0 (full V).

(* Specifications: *)
(** - %\intoc{\bf Theorem 1.9}% in various flavours: *)
Lemma every_finite_generating_set_has_a_subset_that_is_a_basis :
exists W : part_set W0, is_basis (inject_subsets W).