Equality 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:


 * $\dfrac a c = \dfrac b d \iff a \circ d = b \circ c$

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

Proof
Alternatively, a proof can be built using Addition of Division Products.