End Sumv_Pi.
End Projection.
Prenex Implicits daddv_pi projv addv_pi1 addv_pi2.
Notation sumv_pi V := (sumv_pi_for (erefl V)).
Section SumvPi.
Variable (K : fieldType) (vT : vectType K).

Lemma sumv_pi_sum (I : finType) (P : pred I) Vs v (V : {vspace vT})
(defV : V = (\sum_(i | P i) Vs i)%VS) :
v \in V -> \sum_(i | P i) sumv_pi_for defV i v = v :> vT.