If two numbers be prime to two numbers, both to each, their products also will be prime to one another.

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

For, since each of the numbers A, B is prime to C, therefore the product of A, B will also be prime to C. [VII. 24]

But the product of A, B is E; therefore E, C are prime to one another.

For the same reason E, D are also prime to one another.

Therefore each of the numbers C, D is prime to E.

Therefore the product of C, D will also be prime to E. [VII. 24]

But the product of C, D is F.

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