Casting Out 9s

Proof Technique
Casting out nines is a technique for checking that the result of an addition sum is correct.

Let $a$ and $b$ be two integers for which their sum:
 * $c = a + b$

is to be calculated.

For each of $a$ and $b$, expressed in conventional decimal notation, the digital root is extracted.

These are added together, and the digital root of the result is extracted.

During this process, any occurrences of the digit $9$ can be cast out, as they have no effect on the digital root.

If that digital root of the sum of the digital roots of $a$ and $b$ do not match the digital root of $c$, it means something must have gone wrong with the addition.

Proof
From Digital Root is Preserved by Addition:
 * $\dr c = \dr {\dr a + \dr b}$

where $\dr a$ denotes the digital root of $a$.

Hence the result.