For, otherwise, each of the magnitudes A, B would not have had the same ratio to C; [[V. 8](elem.5.8)] but it has; therefore A is equal to B.

Again, let C have the same ratio to each of the magnitudes A, B; I say that A is equal to B.

For, otherwise, C would not have had the same ratio to each of the magnitudes A, B; [[V. 8](elem.5.8)] but it has; therefore A is equal to B.

Therefore etc. Q. E. D.