Zero is Zero Element for Natural Number Multiplication

Theorem
Let $\N$ be the natural numbers.

Then:
 * $\forall n \in \N: 0 \times n = 0 = n \times 0$

Proof
Proof by induction:

From Definition by Induction of Natural Number Addition‎, we have by definition that:

From the definition of natural number multiplication, we have by definition that:

For all $n \in \N^*$, let $P \left({n}\right)$ be the proposition:
 * $0 \times n = 0 = n \times 0$

Basis for the Induction
By definition, we have:
 * $0 \times 0 = 0 = 0 \times 0$

Thus $P \left({0}\right)$ is seen to be true.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 0$, then it logically follows that $P \left({k^+}\right)$ is true.

So this is our induction hypothesis $P \left({k}\right)$:
 * $0 \times k = 0 = k \times 0$

Then we need to show that $P \left({k^+}\right)$ follows directly from $P \left({k}\right)$:
 * $0 \times k^+ = 0 = k^+ \times 0$

Induction Step
This is our induction step:

By definition:
 * $k^+ \times 0 = 0$

So $P \left({k}\right) \implies P \left({k^+}\right)$ and the result follows by the Principle of Finite Induction.

Therefore:
 * $\forall n \in \N: 0 \times n = 0 = n \times 0$