Using results
we have
\begin{equation}
\begin{split}
\mathbb{E}(e^{i t \sum_{n = 1}^N T_n})
= \prod_{n = 1}^N \mathbb{E}(e^{i t T_n})
& = \prod_{n = 1}^N \frac{\theta}{\theta - i t} \\
& = \prod_{n = 1}^N \frac{\theta}{\theta - i t} \frac{1 / \theta}{1 / \theta}
= \prod_{n = 1}^N \frac{1}{1 - i t / \theta}
= \frac{1}{(1 - i t / \theta)^N}
\end{split}
\end{equation}
By definition, a gamma random positive real number with parameters $N$ and $\theta$ has characteristic function $(1 - i t / \theta)^{- N}$, so we are done. $\square$