Prequisites

This proof uses the following 3 facts:

$$$1.\quad -\frac{a}{b}=\frac{a}{-b}$$$

$$$2.\quad -0=0$$$

$$$3.\quad x=x$$$ is always true.

The Proof

Let $$$e$$$ be assigned the fraction $$$\frac{n}{0}$$$ for clarity.

Lemma: $$$e=-e$$$

Proof of Lemma:

$$$e=-e$$$

$$$\frac{n}{0}=-(\frac{n}{0})$$$

Move the $$$-$$$ in the denominator.

$$$\frac{n}{0}=\frac{n}{-0}$$$

And $$$-0=0$$$ so:

$$$\frac{n}{0}=\frac{n}{0}$$$

Hence $$$e=-e$$$

If $$$e=-e$$$ then:

$$$2e=0$$$

$$$e=0$$$

Thus, $$$\frac{n}{0}$$$ is $$$0$$$ for all $$$n$$$.

I would like to know about any mistakes in the comments. (Except for the fact that $$$\frac{n}{0}$$$ is illegal)