Let $N^O$ and $N^U$ be an independent pair Poisson processes with intensities $\alpha_O$ and $\alpha_U$. We will model the incoming trains in the central station (trains that are arriving above the ground(O) and trains that are arriving in the underground(U)), this means that for every left-open, right closed Interval $I\subseteq(0,\infty)$ let $N{_I}^{\bullet}$ be the number of trains that have arrived within the interval $I$ in the part of the station $\bullet\in\text{{O,U}}$.
For every left open, right closed interval $I\subseteq (0,\infty)$ let $N_I$ be a random variable that denotes the number of trains that have arrived within a time interval $I$ in the central station. I want to show that in our model $N$ is a Poisson process with the intensity $\alpha := \alpha_O + \alpha_U$. My intuition tells me that the statement is correct, but I am struggling to show it. Any tips or solutions would be greatly appreciated.