Inverse for Rational Multiplication

Theorem
Each element $$x$$ of the set of non-zero rational numbers $$\Q^*$$ has an inverse element $$\frac 1 x$$ under the operation of rational number multiplication:
 * $$\forall x \in \Q^*: \exists \frac 1 x \in \Q^*: x \times \frac 1 x = 1 = \frac 1 x \times x$$

Proof
From the definition, the field $$\left({\Q, +, \times}\right)$$ of rational numbers is the quotient field of the integral domain $$\left({\Z, +, \times}\right)$$ of integers.

From Rational Multiplication Identity is One, the identity for $$\left({\Q, \times}\right)$$ is $$1 = \frac 1 1 = \frac p p$$ where $$p \in \Z$$ and $$p \ne 0$$.

From the definition of product inverses in quotient field, we have that:
 * $$\left({\frac p q}\right)^{-1} = \frac q p$$

which can be demonstrated directly:

$$ $$ $$

Note that this is defined only when $$p \ne 0$$ and $$q \ne 0$$.

Now let $$x = \frac p q \in \Q$$.

We define $$\frac 1 x$$ as $$1 \times \left({\frac p q}\right)^{-1}$$.

Hence:

$$ $$ $$ $$