Section NaryDirect.
Context {I : finType} {P : pred I}.

Lemma directv_sum_independent {Us : I -> {vspace vT}} :
reflect (forall us,
(forall i, P i -> us i \in Us i) -> \sum_(i | P i) us i = 0 ->
(forall i, P i -> us i = 0))
(directv (\sum_(i | P i) Us i)).