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

(* Specifications: *)
Lemma separable_add K x y : separable_element K x -> separable_element K y -> separable_element K (x + y).
Lemma inseparable_add K x y : purely_inseparable_element K x -> purely_inseparable_element K y -> purely_inseparable_element K (x + y).