Let there be any number of magnitudes A, B, and C, and others D, E, and F equal to them in multitude, which taken two and two together are in the same ratio, so that A is to B as D is to E, and B is to C as E is to F.
I say that they are also in the same ratio ex aequali, that is, A is to C as D is to F.
Take equimultiples G and H of A and D, and take other, arbitrary, equimultiples K and L of B and E, and, further, take other, arbitrary, equimultiples M and N of C and F.
Then, since A is to B as D is to E, and of A and D equimultiples G and H have been taken, and of B and E other, arbitrary, equimultiples K and L, therefore G is to K as H is to L.
For the same reason also K is to M as L is to N.
Since, then, there are three magnitudes G, K, and M, and others H, L, and N equal to them in multitude, which taken two and two together are in the same ratio, therefore, ex aequali, if G is in excess of M, H is also in excess of N; if equal, equal; and if less, less.
And G and H are equimultiples of A and D, and M and N other, arbitrary, equimultiples of C and F.
Therefore A is to C as D is to F.
Therefore, if there are any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, then they are also in the same ratio ex aequali.
The proof builds on proposition V.20. Assume a : b = d : e, and b : c = e : f. To show a : c = d : f.
Let n, m, and k be three numbers. By V.4, na : mb = nd : me, and ma : kc = md : kf. By V.20,
This proposition can also be proved directly from the definition Def.V.5 very easily.
The analogous proposition for ratios of numbers is given in proposition VII.14. The proof given there works for magnitudes as well, but they all have to be of the same kind.
This proposition is used in V.24 and several propositions in Books VI, X, and XII.