If a number is parts of a number, and another is the same parts of another, then the sum is also the same parts of the sum that the one is of the one.

Let the number *AB* be parts of the number *C,* and another number *DE* be the same parts of another number *F* that *AB* is of *C.*

I say that the sum of *AB* and *DE* is also the same parts of the sum of *C* and *F* that *AB* is of *C.*

Since there are as many parts of *DE* in *F* as there are parts *AB* is of *C,* therefore there are as many parts of *F* in *DE* as there are parts of *C* in *AB.*

Divide *AB* into the parts of *C,* namely *AG* and *GB,* and divide *DE* into the parts of *F,* namely *DH,* and *HE.* Then the multitude of *AG* and *GB* equals the multitude of *DH* and *HE.*

And since *DH* is the same part of *F* that *AG* is of *C,* therefore the sum of *AG* and *DH* is the same part of the sum of *C* and *F* that *AG* is of *C.* For the same reason, the sum of *GB* and *HE* is the same parts of the sum of *C* and *F* that *GB* is of *C.*

Therefore the sum of *AB* and *DE* is the same parts of the sum of *C* and *F* that *AB* is of *C.*

Therefore, *if a number is parts of a number, and another is the same parts of another, then the sum is also the same parts of the sum that the one is of the one.*

Q.E.D.