Since we define \begin{equation} \text{Var} \left( \sum_{n = 1}^N X_n \right) = \text{Cov} \left( \sum_{n = 1}^N X_n, \sum_{n = 1}^N X_n \right) \end{equation} then the first claim is a special case of R5113: Componentwise additivity of covariance for random real numbers. If $X_1, \ldots, X_N$ are uncorrelated, then result R3632: Uncorrelated iff covariances zero shows that $\text{Cov}(X_n, X_m) = 0$ whenever $n \neq m$. Since $\text{Cov}(X_n, X_n) = \text{Var} X_n$, the second claim follows. $\square$