Two Coprime Integers have no Third Integer Proportional

Theorem
Let $a, b \in \Z_{>0}$ be integers such that $a$ and $b$ are coprime.

Then there is no integer $c \in \Z$ such that:
 * $\dfrac a b = \dfrac b c$

Proof
Suppose such a $c$ exists.

From Coprime Numbers form Fraction in Lowest Terms, $\dfrac a b$ is in canonical form.

From Ratios of Fractions in Lowest Terms:
 * $a \mathop \backslash b$

where $\backslash$ denotes divisibility.

This contradicts the fact that $a$ and $b$ are coprime.

Hence such a $c$ cannot exist.