For, if they are not prime to one another, some number will measure them.

Let some number measure them, and let it be C.

And, as many times as C measures A, so many units let there be in D, and, as many times as C measures B, so many units let there be in E

Since C measures A according to the units in D, therefore C by multiplying D has made A. [[VII. Def. 15](elem.7.def.15)]

For the same reason also C by multiplying E has made B.

Thus the number C by multiplying the two numbers D, E has made A, B; therefore, as D is to E, so is A to B; [[VII. 17](elem.7.17)] therefore D, E are in the same ratio with A, B, being less than they: which is impossible.

Therefore no number will measure the numbers A, B.

Therefore A, B are prime to one another. Q. E. D.