Negative of Division Product

Theorem
Let $\left({R, +, \circ}\right)$ be a commutative ring with unity.

Let $\left({U_R, \circ}\right)$ be the group of units of $\left({R, +, \circ}\right)$.

Then:
 * $\displaystyle \forall x \in R: - \frac x z = \frac {- x} z = \frac x {- z}$

where $\dfrac x z$ is defined as $x \circ \left({z^{-1}}\right)$, that is, $x$ divided by $z$.

Proof
Follows directly from Product of Negative with Product Inverse and the definition of divided by.