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 daddv_pi_proj U V w (pi := daddv_pi U V) :
(U :&: V = 0)%VS -> pi (pi w) = pi w.