For let D be measured by any prime number E; I say that E measures A.

For suppose it does not; now E is prime, and any prime number is prime to any which it does not measure; [[VII. 29](elem.7.29)] therefore E, A are prime to one another.

And, since E measures D, let it measure it according to F, therefore E by multiplying F has made D.

Again, since A measures D according to the units in C, [[IX. 11 and Por.](elem.9.11 elem.9.11.p.1)] therefore A by multiplying C has made D.

But, further, E has also by multiplying F made D; therefore the product of A, C is equal to the product of E, F.

Therefore, as A is to E, so is F to C. [[VII. 19](elem.7.19)]

But A, E are prime, primes are also least, [[VII. 21](elem.7.21)] and the least measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent; [[VII. 20](elem.7.20)] therefore E measures C.

Let it measure it according to G; therefore E by multiplying G has made C.

But, further, by the theorem before this, A has also by multiplying B made C. [[IX. 11 and Por.](elem.9.11 elem.9.11.p.1)]

Therefore the product of A, B is equal to the product of E, G.

Therefore, as A is to E, so is G to B. [[VII. 19](elem.7.19)]

But A, E are prime, primes are also least, [[VII. 21](elem.7.21)] and the least numbers measure those which have the same ratio with them the same number of times, the antecedent the antecedent and the consequent the consequent: [[VII. 20](elem.7.20)] therefore E measures B.

Let it measure it according to H; therefore E by multiplying H has made B.

But further A has also by multiplying itself made B; [[IX. 8](elem.9.8)] therefore the product of E, H is equal to the square on A.

Therefore, as E is to A, so is A to H. [[VII. 19](elem.7.19)]

But A, E are prime, primes are also least, [[VII. 21](elem.7.21)] and the least measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent; [[VII. 20](elem.7.20)] therefore E measures A, as antecedent antecedent.

But, again, it also does not measure it: which is impossible.

Therefore E, A are not prime to one another.

Therefore they are composite to one another.

But numbers composite to one another are measured by some number. [[VII. Def. 14](elem.7.def.14)]

And, since E is by hypothesis prime, and the prime is not measured by any number other than itself, therefore E measures A, E, so that E measures A.

[But it also measures D; therefore E measures A, D.]

Similarly we can prove that, by however many prime numbers D is measured, A will also be measured by the same. Q. E. D.