Inverse for Rational Addition

Theorem
Each element $$x$$ of the set of rational numbers $$\Q$$ has an inverse element $$-x$$ under the operation of rational number addition:
 * $$\forall x \in \Q: \exists -x \in \Q: x + \left({-x}\right) = 0 = \left({-x}\right) + x$$

Proof
Let $$x = \frac a b$$ where $$b \ne 0$$.

We take the definition of rational numbers as the quotient field of the integral domain $$\left({\Z, +, \times}\right)$$ of integers.

From Existence of Quotient Field, we have that the inverse of $$\frac a b$$ for $$+$$ is $$\frac {-a} b$$:

$$ $$ $$ $$ $$

From Negative of Divided By, we have that:
 * $$-\frac a b = \frac {-a} b = \frac a {-b}$$

So $$\frac a b$$ has a unique and unambiguous inverse for $$+$$.