Let $X \in \text{Gaussian}(\mu, \sigma)$ be a D210: Gaussian random real number.
Let $t \in \mathbb{R}$ be a D993: Real number.

Let $Z \in \text{Gaussian}(0, 1)$ be a D211: Standard gaussian random real number. Now $\sigma Z \in \text{Gaussian}(0, \sigma^2)$ and $\sigma Z + \mu \in \text{Gaussian}(\mu, \sigma^2)$. Result R3680: Characteristic function of standard gaussian random real number shows that \begin{equation} \mathbb{E} (e^{i t Z}) = e^{- t^2 / 2} \end{equation} Thus \begin{equation} \mathfrak{F}_{\sigma Z}(t) = \mathbb{E}(e^{i t a Z}) = \mathfrak{F}_Z(\sigma t) = e^{- (\sigma t)^2 / 2} = e^{- \sigma^2 t^2 / 2} \end{equation} Next, result R4639: Characteristic function for translated random real number shows that \begin{equation} \mathfrak{F}_{\sigma Z + \mu} (t) = e^{i t \mu} \mathfrak{F}_{\sigma Z} (t) = e^{i t \mu} e^{- \sigma^2 t^2 / 2} = e^{i t \mu - \sigma^2 t^2 / 2} \end{equation} To establish the result, it is only left to confirm that \begin{equation} \mathfrak{F}_{\sigma Z + \mu} (t) = \mathfrak{F}_X (t) \end{equation} for every real number $t \in \mathbb{R}$. Since $\sigma Z + \mu \overset{d}{=} X$, this is guaranteed by R2405: Characteristic function uniquely identifies the distribution of a random real number. $\square$