Equality of Division Products

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:


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

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

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