If a number is that part of a number which a subtracted number is of a subtracted number, then the remainder is also the same part of the remainder that the whole is of the whole.

Let the number *AB* be that part of the number *CD* which *AE* subtracted is of *CF* subtracted.

I say that the remainder *EB* is also the same part of the remainder *FD* that the whole *AB* is of the whole *CD.*

Let *EB* be the same part of *CG* that *AE* is of *CF.*

Now since *EB* is the same part of *CG* that *AE* is of *CF,* therefore *AB* is the same part of *GF* that *AE* is of *CF.*

But, by hypothesis, *AB* is the same part of *CD* that *AE* is of *CF,* therefore *AB* is the same part of *CD* that it is of *GF.* Therefore *GF* equals *CD.*

Subtract *CF* from each. Then the remainder *GC* equals the remainder *FD.*

Now since *EB* is the same part of *GC* that *AE* is of *CF,* and *GC* equals *FD,* therefore *EB* is the same part of *FD* that *AE* is of *CF.*

But *AB* is the same part of *CD* that *AE* is of *CF,* therefore the remainder *EB* is the same part of the remainder *FD* that the whole *AB* is of the whole *CD.*

Therefore, *if a number is that part of a number which a subtracted number is of a subtracted number, then the remainder is also the same part of the remainder that the whole is of the whole.*

Q.E.D.

This proposition is used in the next proposition and in VII.11.