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:


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


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

Proof
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