Source — Euclid's Elements, encoded as an rrxiv paper

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V.22 Proposition V.22

If there be any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, they will also be in the same ratio ex aequali.

Proof

Apply V.20 inductively across the chain

a_{1} : a_{2} = b_{1} : b_{2}

,

a_{2} : a_{3} = b_{2} : b_{3}

,

\dots

,

a_{n - 1} : a_{n} = b_{n - 1} : b_{n}

to conclude

a_{1} : a_{n} = b_{1} : b_{n}

.

lines 74–74 in main.tex