simpl in |- *.
intros.
generalize H; elim y.
intros yv yp Hy.
simpl in yp.
simpl in Hy.
simpl in |- *.
unfold not in Hy.
case yp; auto.
tauto.
Qed.

(* Specifications: *)
(** - %\intoc{\bf Theorem 1.8}%: If $S\subset V$ is linearly independent and $x\not\in S$ then $S\cup\{x\}$ is linearly dependent iff $x\in span(S)$  (By the by - this innocent-looking lemma was extremely hard to prove) *)
Lemma lin_dep_vs_span_lemma :
forall (F : field) (V : vectorspace F) (s : part_set V),
lin_indep s ->
forall x : V,
~ in_part x s -> (lin_dep (union s (single x)) <-> in_part x (span s)).