Equality 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:


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

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

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