let V be Z_Module; :: thesis: for a being Element of INT.Ring
for v, u, w being Vector of V holds a * (Sum <*v,u,w*>) = ((a * v) + (a * u)) + (a * w)
let a be Element of INT.Ring; :: thesis: for v, u, w being Vector of V holds a * (Sum <*v,u,w*>) = ((a * v) + (a * u)) + (a * w)
let v, u, w be Vector of V; :: thesis: a * (Sum <*v,u,w*>) = ((a * v) + (a * u)) + (a * w)
thus a * (Sum <*v,u,w*>) = a * ((v + u) + w) by RLVECT_1:46
.= (a * (v + u)) + (a * w) by VECTSP_1:def 14
.= ((a * v) + (a * u)) + (a * w) by VECTSP_1:def 14 ; :: thesis: verum