Definition separable U W : bool :=
all (separable_element U) (vbasis W).
Definition purely_inseparable U W : bool :=
all (purely_inseparable_element U) (vbasis W).

Lemma separable_sum I r (P : pred I) (v_ : I -> L) K :
(forall i, P i -> separable_element K (v_ i)) ->
separable_element K (\sum_(i <- r | P i) v_ i).