# Proof

In order to avoid complicated distinctions of cases, we can without loss of generality assume the integers $a,b,c,d,e,f\in Z$ to be all positive integers. We first show the trichotomy of the order relation "$<$" for rational numbers $$\frac ab,\frac cd, \in\mathbb Q$$:

• By the definition of the order relation for rational numbers,
• $\frac ab < \frac cd$ is equivalent to $\frac ab - \frac cd < 0$,
• $\frac ab > \frac cd$ is equivalent to $\frac ab - \frac cd > 0$, and
• $\frac ab = \frac cd$ is equivalent to $\frac ab - \frac cd = 0$.
• Therefore, the order relation of rational numbers "$<$" can be reduced to the order relation for integers "$<$".
• Now, the trichotomy of the order relation for rational numbers follows from the trichotomy of the order relation for integers.
• Therefore, only one of the cases $\frac ab = \frac cd$, or $\frac ab < \frac cd$, or $\frac ab > \frac cd$ can occur at once.

We now show the transitivity.

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