Let $\xi \in \mathbb{R}$ be a real number. Since the $X_1, \, \ldots, \, X_N$ are uncorrelated, applying results

(i)	R2340: Uncorrelated random real collection iff expectation multiplicative
(ii)	R3200: Characteristic function of Bernoulli random boolean number

yields \begin{equation} \begin{split} \mathfrak{F}_{\sum_{n = 1}^N X_n}(\xi) : = \mathbb{E} \left(e^{i \xi \sum_{n = 1}^N X_n} \right) & = \mathbb{E} \left( \prod_{n = 1}^N e^{i \xi X_n} \right) \\ & = \prod_{n = 1}^N \mathbb{E} \left( e^{i \xi X_n} \right) \\ & = \prod_{n = 1}^N \left( \theta (e^{i \xi} - 1) + 1 \right) \\ & = \left( \theta (e^{i \xi} - 1) + 1 \right)^N \\ \end{split} \end{equation} The claim is now a consequence of the results

(i)	R3201: Characteristic function of a binomial random natural number
(ii)	R2405: Characteristic function uniquely identifies the distribution of a random real number

$\square$