split; try split; auto with algebra.
Qed.

(** - %\intoc{\bf Theorem 1.3}% literally (more-or-less; now easy to prove): *)
Lemma subspace_alt_characterization :
forall Ws : part_set V,
in_part (zero V) Ws /\
(forall x y : V, in_part x Ws -> in_part y Ws -> in_part (x +' y) Ws) /\
(forall (c : F) (x : V), in_part x Ws -> in_part (c mX x) Ws) <->
(exists W : subspace V, W =' Ws in part_set V).