Section MAIN.
Variable F : field.
Variable V : vectorspace F.
(** - A vectorspace being a direct sum is only defined as a predicate; the sum_set however is defined as a construction: %\tocdef{sum\_set}% *)
Definition sum_set (S1 S2 : part_set V) : part_set V.
intros.
simpl in |- *.
apply
Build_Predicate
with
(fun x : V =>
exists s1 : S1,
(exists s2 : S2, x =' subtype_elt s1 +' subtype_elt s2 in _)).
red in |- *.
intros.
inversion_clear H.
inversion_clear H1.
exists x0.
exists x1.
apply Trans with x; auto with algebra.
Defined.

Lemma subspace_sum_set :
forall W1 W2 : part_set V,
is_subspace W1 -> is_subspace W2 -> is_subspace (sum_set W1 W2).