Using result R5721: Base-2 integer representation for minus one, we have \begin{equation} \begin{split} - (a) = 1 - (a) + (- 1) & = 1 - \left(- r_{N - 1} 2^{N - 1} + \sum_{n = 0}^{N - 2} r_n 2^n \right) + \left( - 2^{N - 1} + \sum_{n = 0}^{N - 2} 2^n \right) \\ & = 1 + \left(r_{N - 1} 2^{N - 1} + \sum_{n = 0}^{N - 2} - r_n 2^n \right) + \left( - 2^{N - 1} + \sum_{n = 0}^{N - 2} 2^n \right) \\ & = 1 + (r_{N - 1} - 1) 2^{N - 1} + \sum_{n = 0}^{N - 2} (1 - r_n) 2^n \\ & = 1 - (1 - r_{N - 1}) 2^{N - 1} + \sum_{n = 0}^{N - 2} (1 - r_n) 2^n \\ \end{split} \end{equation} $\square$