For, if not, A is either equal to B or less.

Now A is not equal to B; for in that case each of the magnitudes A, B would have had the same ratio to C; [[V. 7](elem.5.7)] but they have not; therefore A is not equal to B.

Nor again is A less than B; for in that case A would have had to C a less ratio than B has to C; [[V. 8](elem.5.8)] but it has not; therefore A is not less than B.

But it was proved not to be equal either; therefore A is greater than B.

Again, let C have to B a greater ratio than C has to A; I say that B is less than A.

For, if not, it is either equal or greater.

Now B is not equal to A; for in that case C would have had the same ratio to each of the magnitudes A, B; [[V. 7](elem.5.7)] but it has not; therefore A is not equal to B.

Nor again is B greater than A; for in that case C would have had to B a less ratio than it has to A; [[V. 8](elem.5.8)] but it has not; therefore B is not greater than A.

But it was proved that it is not equal either; therefore B is less than A.

Therefore etc. Q. E. D.