Binomial Coefficient involving Power of Prime/Proof 2

Proof
Lucas' Theorem states that for $n, k, p \in \Z$ and $p$ be a prime number, such that:
 * $n = a_r p^r + \cdots + a_1 p + a_0$
 * $k = b_r p^r + \cdots + b_1 p + b_0$

then:
 * $\displaystyle \binom n k \equiv \prod_{j \mathop = 0}^r \binom {a_j}{b_j} \pmod p$

Therefore: