Product of Division Products

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

Let $\struct {U_R, \circ}$ be the group of units of $\struct {R, +, \circ}$.

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 $x \circ \paren {z^{-1} }$, that is, $x$ divided by $z$.