Poisson Process Distribution of waiting time$W_{X(t)+2}$ in poisson process

Let's say $$N(t)$$ is a poisson process with parameter $$\lambda$$. Let $$W_n$$ to represent the waiting time of $$n$$th arrival. What is $$P(W_{N(t)+2}≤t+s)$$? I solved this question by the following way:$$P(W_{N(t)+2}≤t+s)=P(N(t,t+s] ≥2)$$$$=1-e^{\lambda s}-\lambda s e^{\lambda s}$$ so basically $$W_{N(t)+2}$$ has the same distribution as $$W_2$$ and the only difference is the time shifted to $$t$$. Is my proof right? Is this the rigorous way to reach the answer? It looks so straightforward.