Rational Addition Identity is Zero

Theorem
The identity of rational number addition is $$0$$:
 * $$\exists 0 \in \Q: \forall a \in \Q: a + 0 = a = 0 + a$$

Proof
The identity for $$+$$ is $$\frac 0 k$$ where $$k \in \Z^*$$:

$$ $$ $$

Similarly for $$\frac 0 k + \frac a b$$.