Divisibility by 9/Proof 1

Theorem
A number expressed in decimal notation is divisible by $9$ iff the sum of its digits is divisible by $9$:

That is:
 * $N = [a_0 a_1 a_2 \ldots a_n]_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $9$

iff:
 * $a_0 + a_1 + \ldots + a_n$ is divisible by $9$.

Proof
If $N$ is divisible by 9, then