If a whole is to a whole as a subtracted number is to a subtracted number, then the remainder is to the remainder as the whole is to the whole.

Let the whole *AB* be to the whole *CD* as *AE* subtracted is to *CF* subtracted.

I say that the remainder *EB* is to the remainder *FD* as the whole *AB* is to the whole *CD.*

Since *AB* is to *CD* as *AE* is to *CF,* therefore *AE* is the same part or parts of *CF* as *AB* is of *CD.* Therefore the remainder *EB* is the same part or parts of *FD* that *AB* is of *CD.*

Therefore *EB* is to *FD* as *AB* is to *CD.*

Therefore, *if a whole is to a whole as a subtracted number is to a subtracted number, then the remainder is to the remainder as the whole is to the whole.*

Q.E.D.

This proposition is used in IX.35.