Applying R1831: Real arithmetic expression for binomial coefficient, we may directly compute \begin{equation} \begin{split} \binom{n - 1}{m} + \binom{n - 1}{m - 1} & = \frac{(n - 1)!}{(n - m - 1)! m!} + \frac{(n - 1)!}{(n - m)! (m - 1)!} \\ & = (n - 1)! \left( \frac{1}{(n - m - 1)! m!} + \frac{1}{(n - m)! (m - 1)!} \right) \\ & = (n - 1)! \left( \frac{1}{(n - m - 1)! m!} \frac{n - m}{n - m} + \frac{1}{(n - m)! (m - 1)!} \frac{m}{m} \right) \\ & = (n - 1)! \left( \frac{n - m}{(n - m)! m!} + \frac{m}{(n - m)! m!} \right) \\ & = (n - 1)! \frac{n}{(n - m)! m!} \\ & = \frac{n!}{(n - m)! m!} \\ & = \binom{n}{m} \\ \end{split} \end{equation} $\square$