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:

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

where $$\frac x z$$ 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.