(related to Proposition: Recursive Formula for Binomial Coefficients)
This proof is a simple recalculation of the formula, following the closed formula
$$\begin{align} \binom{n-1}{k-1}+\binom{n-1}{k}&=\frac{(n-1) !}{(n-1-k+1) !\cdot (k-1) !}+\frac{(n-1) !}{(n-1-k) !\cdot k !}\nonumber\\ &=\frac{(n-1) !}{(n-1) !\cdot (k-1) !}+\frac{(n-1) !}{(n-1-k) !\cdot k !}\nonumber\\ &=\frac{k(n-1) !}{(n-1) !\cdot k !}+\frac{(n-k)(n-1) !}{(n-k) !\cdot k !}\nonumber\\ &=\frac{k(n-1) ! + (n-k)(n-1) !}{(n-k) !\cdot k !}\nonumber\\ &=\frac{n !}{(n-k) !\cdot k !}\nonumber\\ &=\binom nk\nonumber\\ \end{align}$$
