nonhomogeneous Poisson process elementary question

Question

Let $N_t$ is a nonhomogeneous Poisson process. Find $P(N_{t_2}=n|N_{t_1}=m)$ where $t_1<t_2$ and $n \geq m$. My solution: \begin{eqnarray} P(N_{t_2}=n\mid N_{t_1}=m)&=&\frac{P(N_{t_2}=n,N_{t_1}=m)}{P(N_{t_1}=m)}\\ &=&\frac{P(N_{t_2}-N_{t_1}=n-m,N_{t_1}=m)}{P(N_{t_1}=m)}\\ &=&\frac{P(N_{t_2}-N_{t_1}=n-m)P(N_{t_1}=m)}{P(N_{t_1}=m)}\\ &=&P(N_{t_2}-N_{t_1}=n-m)\\ &=&e^{-(\Lambda(t_2)-\Lambda(t_1))}\frac{(\Lambda(t_2)-\Lambda(t_1))^{n-m}}{(n-m)!}\\ \end{eqnarray} is it correct? Thank you