Element of Pascal's Triangle is Sum of Diagonal or Column starting above it going Upwards

Theorem
Consider Pascal's triangle:

Let $\left({n, m}\right)$ be the element in the $n$th row and $m$th column.

Then:
 * $\left({n, m}\right) = \displaystyle \sum_{k \mathop \ge 0} \left({n - k - 1, m - 1}\right)$

and:
 * $\left({n, m}\right) = \displaystyle \sum_{k \mathop \ge 0} \left({n - k, m - k - 1}\right)$