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

Theorem
Consider Pascal's triangle:

Let $\tuple {n, m}$ be the element in the $n$th row and $m$th column.

Then:
 * $\tuple {n, m} = \ds \sum_{k \mathop \ge 0} \tuple {n - k - 1, m - 1}$

and:
 * $\tuple {n, m} = \ds \sum_{k \mathop \ge 0} \tuple {n - k - 1, m - k - 1}$

Proof
We have $\tuple {n, m} = \dbinom n m$, and we have: