Definition lin_indep (X : part_set V) := ~ lin_dep X.

(* Specifications: *)
Lemma lin_dep_defs_eqv : forall X : part_set V, lin_indep X <-> lin_indep' X.
Lemma zero_imp_lin_dep : forall (F : field) (V : vectorspace F) (X : part_set V), in_part (zero V) X -> lin_dep X.
Lemma empty_lin_indep : forall (F : field) (V : vectorspace F), lin_indep (empty V).
Lemma lin_dep_include : forall S1 S2 : part_set V, included S1 S2 -> lin_dep S1 -> lin_dep S2.
Lemma lin_indep_include : forall S1 S2 : part_set V, included S1 S2 -> lin_indep S2 -> lin_indep S1.
Lemma lin_indep_doesn't_contain_zero : forall X : part_set V, lin_indep X -> ~ in_part (zero V) X.
Lemma inject_subsets_lin_dep : forall (W : subspace V) (X : part_set W), lin_dep X <-> lin_dep (inject_subsets X).