Product of Negative with Product Inverse

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

Let $z \in U_R$, where $U_R$ is the set of units.

Then:


 * $(1): \quad \forall x \in R: -\left({x \circ z^{-1}}\right) = \left({- x}\right) \circ z^{-1} = x \circ \left({\left({- z}\right)^{-1}}\right)$


 * $(2): \quad \forall x \in R: -\left({z^{-1} \circ x}\right) = z^{-1} \circ \left({- x}\right) = \left({\left({- z}\right)^{-1}}\right) \circ x$