Section defs.
Variable F : field.
Variable V : vectorspace F.
(** %\tocdef{lin\_{\em[in]}dep}% *)
Definition lin_dep (X : part_set V) :=
exists n : Nat,
(exists a : seq (S n) F,
(exists v : seq (S n) X,
distinct v /\
~ a =' const_seq (S n) (zero F) in _ /\
sum (mult_by_scalars a (Map_embed v)) =' (zero V) in _)).
Definition lin_indep (X : part_set V) := ~ lin_dep X.
Definition lin_indep' (X : part_set V) :=
forall (n : Nat) (a : seq (S n) F) (v : seq (S n) X),
distinct v ->
~ a =' const_seq (S n) (zero F) in _ ->
~ sum (mult_by_scalars a (Map_embed v)) =' (zero V) in _.

Lemma lin_dep_comp :
forall X Y : part_set V, X =' Y in _ -> lin_dep X -> lin_dep Y.