Section Projection.
Variables (K : fieldType) (vT : vectType K).
Implicit Types U V : {vspace vT}.
Definition daddv_pi U V := Vector.Hom (proj_mx (vs2mx U) (vs2mx V)).
Definition projv U := daddv_pi U U^C.
Definition addv_pi1 U V := daddv_pi (U :\: V) V.
Definition addv_pi2 U V := daddv_pi V (U :\: V).

Lemma sumv_pi_uniq_sum v :
uniq (filter P r0) -> v \in V ->
\sum_(i <- r0 | P i) sumv_pi_for defV i v = v.