Conditional on the Poisson count, event times are order statistics of a uniform sample

Let $\{X(t),t\ge0\}$ be a time homogeneous Poisson process with rate $\lambda$. Let $X(T)=n>0$ and let $T_1<\dots<T_n<T$ be times at which these n events occur. Then, $E(T_1|X(T)=n)$ is given by: