Why are $T_2/T_1, ..., T_k/T_{k-1}, T_k$ independent conditional on $N((0,t])=k$, where $N((0,t]) = \sum_i \epsilon_{T_K}((0,t])$ is a point process?

Question

Why are $T_2/T_1, ..., T_k/T_{k-1}, T_k$ independent conditional on $N(t)=k$, where $N(t) = N((0,t]) = \sum_i \epsilon_{T_K}((0,t])$ is a point process, $0,<T_1<T_2<...$ and $N(t)$ has the order statistic property? It is not stated whether the point process is a Poisson process.

So far I concluded:

$ P(T_2/T_1=a_1, ..., T_k/T_{k-1}=a_{k-1}, T_k=a_k | N(t) = k) $

$ =P(T_1=a_k/\sum_{i=1}^{k-1}a_i, ..., T_{k-1}=a_k/a_{k-1}, T_k=a_k | N(t) =k) = \frac{k!}{t^k}, $

where the last equality follows from the order statistic property. But I don't see how this implies that:

$ P(T_2/T_1=a_1, ..., T_k/T_{k-1}=a_{k-1}, T_k=a_k | N(t) = k) $

$ =P(T_2/T_1=a_1 | N(t) = k) ... P(T_k/T_{k-1}=a_{k-1} | N(t) = k) P(T_k=a_k | N(t) = k), $

which should be the case for conditional independence to be true.

(This question is from Resnick's Adventures in Stochastic processes, exercise 4.14. If someone knows where to find a complete solutions manual, please let me know)