Let $T_1, \ldots, T_N \in \text{Exponential}(\theta)$ each be an D214: Exponential random positive real number such that

(i)	$T_1, \ldots, T_N$ is an D3842: Uncorrelated random collection

Using results

(i)	R5250: Characteristic function factorizes for finite uncorrelated sum of random real numbers
(ii)	R2411: Characteristic function of exponential random positive real number

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. $\square$