For, if CA, AB are not prime to one another, some number will measure CA, AB.
Let a number measure them, and let it be D.
Since then D measures CA, AB, therefore it will also measure the remainder BC.
But it also measures BA; therefore D measures AB, BC which are prime to one another: which is impossible. [VII. Def. 12]
Therefore no number will measure the numbers CA, AB;
therefore CA, AB are prime to one another.
For the same reason AC, CB are also prime to one another.
Therefore CA is prime to each of the numbers AB, BC.