VII.13

If four numbers be proportional, they will also be proportional alternately.

Proof

Let the four numbers A, B, C, D be proportional, so that, as A is to B, so is C to D; I say that they will also be proportional alternately, so that, as A is to C, so will B be to D.

For since, as A is to B, so is C to D, therefore, whatever part or parts A is of B, the same part or the same parts is C of D also. VII. Def. 20

Therefore, alternately, whatever part or parts A is of C, the same part or the same parts is B of D also. VII. 10

Therefore, as A is to C, so is B to D. VII. Def. 20 Q. E. D.