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:
 * $\forall x \in \Q: \exists -x \in \Q: x + \paren {-x} = 0 = \paren {-x} + x$

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

We take the definition of rational numbers as the field of quotients of the integral domain $\struct {\Z, +, \times}$ of integers.

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

From Negative of Division Product, we have that:
 * $\displaystyle -\frac a b = \frac {-a} b = \frac a {-b}$

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