Writing out the expression $(N - 1) S^2_N$, we have \begin{equation} \begin{split} (N - 1) S^2_N & = \sum_{n = 1}^N (X_n - \overline{X}_N)^2 \\ & = \sum_{n = 1}^N (X^2_n - 2 X_n \overline{X}_N + \overline{X}^2_N) \\ & = \sum_{n = 1}^N X^2_n - 2 \overline{X}_N \sum_{n = 1}^N X_n + N \overline{X}^2_N \\ & = \sum_{n = 1}^N X^2_n - 2 N \overline{X}^2_N + N \overline{X}^2_N \\ & = \sum_{n = 1}^N X^2_n - N \overline{X}^2_N \\ \end{split} \end{equation} Adding $N \overline{X}^2_N$ to both sides we then conclude \begin{equation} (N - 1) S^2_N + N \overline{X}^2_N = \sum_{n = 1}^N X^2_n \end{equation} $\square$