Product of Division Products

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

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

Let $a, b \in R, c, d \in U_R$. Then:


 * $\dfrac a c \circ \dfrac b d = \dfrac {a \circ b} {c \circ d}$

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