For since, as AB is to CD, so is AE to CF, alternately also, as BA is to AE, so is DC to CF. [V. 16]
And, since the magnitudes are proportional componendo, they will also be proportional separando, [V. 17]
that is, as BE is to EA, so is DF to CF,
and, alternately, as BE is to DF, so is EA to FC. [V. 16]
But, as AE is to CF, so by hypothesis is the whole AB to the whole CD.
Therefore also the remainder EB will be to the remainder FD as the whole AB is to the whole CD. [V. 11]