Proof of Pascal's rule from the recursive definition

Proof

1. Let $n$ and $k$ be natural numbers.
2. The binomial coefficient is defined by the recursion $\binom{n}{0} = 1$, $\binom{0}{k+1} = 0$ and $\binom{n+1}{k+1} = \binom{n}{k} + \binom{n}{k+1}$.
3. The third clause of this definition, applied to $n$ and $k$, is the equation $\binom{n+1}{k+1} = \binom{n}{k} + \binom{n}{k+1}$.