If two numbers be prime to one another, and each by multiplying itself make a certain number, the products will be prime to one another; and, if the original numbers by multiplying the products make certain numbers, the latter will also be prime to one another [and this is always the case with the extremes].

Let A, B be two numbers prime to one another, let A by multiplying itself make C, and by multiplying C make D, and let B by multiplying itself make E, and by multiplying E make F; I say that both C, E and D, F are prime to one another.

For, since A, B are prime to one another, and A by multiplying itself has made C, therefore C, B are prime to one another. [VII. 25]

Since then C, B are prime to one another, and B by multiplying itself has made E, therefore C, E are prime to one another. [id.]

Again, since A, B are prime to one another, and B by multiplying itself has made E, therefore A, E are prime to one another. [id.]

Since then the two numbers A, C are prime to the two numbers B, E, both to each, therefore also the product of A, C is prime to the product of B, E. [VII. 26]

And the product of A, C is D, and the product of B, E is F.

Therefore D, F are prime to one another. Q. E. D.