Let $X = (X_1, \ldots, X_D) \in \mathsf{MVGaussian}_D(\mu, \Sigma)$ be a D4856: Gaussian random euclidean real number such that

(i)	\begin{equation} \Sigma = \begin{pmatrix} \sigma^2_1 & \mathsf{Cov}(X_1, X_2) & \cdots & \mathsf{Cov}(X_1, X_D) \\ \mathsf{Cov}(X_2, X_1) & \sigma^2_2 & \vdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ \mathsf{Cov}(X_D, X_1) & \cdots & \cdots & \sigma^2_D \end{pmatrix} \end{equation}

Result R4806: Equivalent characterisations of gaussian random euclidean real numbers shows that every real-linear combination of the components of $X$ is a basic real gaussian. Thus, for example, the linear combination \begin{equation} e_d \cdot X = X_d \end{equation} is a basic real gaussian, where $e_d = (0, \ldots, 1, \ldots, 0)$ is the standard basis vector in $\mathbb{R}^D$ which has all components $0$ except the $d$th component, which equals $1$. The claim follows. $\square$