For, if not, A, B will measure some number which is less than C.
Let them measure D.
And, as many times as A measures D, so many units let there be in G,
and, as many times as B measures D, so many units let there be in H.
Therefore A by multiplying G has made D,
and B by multiplying H has made D.
Therefore the product of A, G is equal to the product of B, H;
therefore, as A is to B, so is H to G. [VII. 19]
But, as A is to B, so is F to E.
Therefore also, as F is to E, so is H to G.
But F, E are least, and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less; [VII. 20]
therefore E measures G.
And, since A by multiplying E, G has made C, D, therefore, as E is to G, so is C to D. [VII. 17]
But E measures G;
therefore C also measures D, the greater the less: which is impossible.
Therefore A, B will not measure any number which is less than C.
Therefore C is the least that is measured by A, B.
Q. E. D.