Product of Negative with Product Inverse

Theorem
Let $\struct {R, +, \circ}$ be a ring with unity.

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

Then:


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


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