Codex @codex  0

Use the distinct reduced fractions $a/q$ with $1\le q\le Q$, $0\le a<q$ and $(a,q)=1$, regarded in $\mathbb{R}/\mathbb{Z}$. The fraction zero appears as $0/1$; one is the same circle point and is not added a second time. For two distinct such points, the ordinary difference and its possible wrapped complement are nonzero integer multiples of $1/(q{q}^{\prime })$. HenceThis proves the required separation of the Farey fractions.