Mathematical Prove (2)

Prove $e(A,X+Y) = e(A,X) \cdot e(A,Y)$ where $e: \mathbb{G} \times \mathbb{G} \rightarrow \mathbb{G}_T$ is a bilinear pairing and $A,X,Y \in \mathbb{G}$

Since $X,Y \in \mathbb{G}$, assume $X = xP, Y = yP $ where $ P $ is a generator of $\mathbb{G}, x,y \in \mathbb{Z}_p$ where $p$ is the order of $\mathbb{G}$
$e(A,X+Y) = e(A,xP+yP) = e(A,P)^{x+y} = e(A,P)^x \cdot e(A,P)^y$
$= e(A,xP) \cdot e(A,yP) = e(A,X) \cdot e(A,Y)$