Proving a that the arrival times of a Poisson process is uniformly distributed

Question

I am stuck on this question for a very long time and I can't figure out the solution:

"Suppose $N_t$, $t\ge0$, is a Poisson process with rate $\lambda$ and $N_t = 0$. $T_0$ and $T_1$ are the first and second arrival times of this Poisson process.

1. Prove that given $N_t=1$, $T_0$ is uniformly distributed in $(0,t]$, i.e. $P(T_0 \le t_o|N_t = 1) = \frac{t_o}{t}$.
2. Hence or otherwise, state the distribution of $T_1$ given $T_0 = t_o$ and $N_t=2$.

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Edit: I can't seem to make what I type into MathJax so i'm typing here:

From $P(N_{t_0}=1∣N_t=1) = \frac{e^{-\lambda t_o} \lambda t_0}{e^{-\lambda t} \lambda t}$, of which there seems to be an extra $\frac{e^{t_o}}{e^{t}}$ in the solution.