Non-Zero Real Numbers Closed under Multiplication/Proof 2

Proof
Let $x \times y = 0$.

, suppose that $x \ne 0$.

Then:

Thus:
 * $x \times y = 0, x \ne 0 \implies y = 0$

Mutatis mutandis
 * $x \times y = 0, y \ne 0 \implies x = 0$

and so:
 * $x \times y = 0 \implies y = 0 \lor x = 0$

So:

The result follows by the Rule of Transposition.