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

Let the number *AB* be the same parts of the number *CD* that *AE* subtracted is of *CF* subtracted.

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

Make *GH* equal to *AB.*

Therefore *AE* is the same parts of *CF* that *GH* is of *CD.*

Divide *GH* into the parts of *CD,* namely *GK* and *KH,* and divide *AE* into the parts of *CF,* namely *AL* and *LE.* Then the multitude of *GK* and *KH* equals the multitude of *AL* and *LE.*

Now since *AL* is the same part of *CF* that *GK* is of *CD,* and *CD* is greater than *CF,* therefore *GK* is also greater than *AL.*

Make *GM* equal to *AL.*

Then *GK* is the same part of *CD* that *GM* is of *CF.* Therefore the remainder *MK* is the same part of the remainder *FD* that the whole *GK* is of the whole *CD.*

Again, since *EL* is the same part of *CF* that *KH* is of *CD,* and *CD* is greater than *CF,* therefore *HK* is also greater than *EL.*

Make *KN* equal to *EL.*

Therefore *KN* is the same part of *CF* that *KH* is of *CD.* Therefore the remainder *NH* is the same part of the remainder *FD* that the whole *KH* is of the whole *CD.*

But the remainder *MK* was proved to be the same part of the remainder *FD* that the whole *GK* is of the whole *CD,* therefore the sum of *MK* and *NH* is the same parts of *DF* that the whole *HG* is of the whole *CD.*

But the sum of *MK* and *NH* equals *EB,* and *HG* equals *BA,* therefore the remainder *EB* is the same parts of the remainder *FD* that the whole *AB* is of the whole *CD.*

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

Q.E.D.

The sample value taken for *m/n* in the proof is 2/3.

This proposition is used in VII.11.