$\forall$$a$, $b$:$\mathbb{Q}$, $p$:($\mathbb{Q}$ List), $F$, $G$:(\{0..$\parallel$$p$$\parallel^{-}$\}$\rightarrow\mathbb{Q}$).
\\[0ex]weighted{-}sum($p$;$\lambda$$x$.($a$ $\ast$ ($F$($x$))) + ($b$ $\ast$ ($G$($x$))))
\\[0ex]=
\\[0ex](($a$ $\ast$ weighted{-}sum($p$;$F$)) + ($b$ $\ast$ weighted{-}sum($p$;$G$)))
\\[0ex]$\in$ $\mathbb{Q}$