For let A by multiplying C make G.
Since, then, A by multiplying C has made G, and by multiplying D has made E,
the number A by multiplying the two numbers C, D has made G, E.
Therefore, as C is to D, so is G to E. [VII. 17]
But, as C is to D, so is A to B;
therefore also, as A is to B, so is G to E.
Again, since A by multiplying C has made G,
but, further, B has also by multiplying C made F,
the two numbers A, B by multiplying a certain number C have made G, F.
Therefore, as A is to B, so is G to F. [VII. 18]
But further, as A is to B, so is G to E also; therefore also,
as G is to E, so is G to F.
Therefore G has to each of the numbers E, F the same ratio;
therefore E is equal to F. [cf. V. 9]