---
book: 7
number: 7
id: "VII.7"
kind: "theorem"
uses: ["[[book-7/proposition-5]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.7"
license: "CC-BY-SA-4.0"
---

# VII.7

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

## Proof

For 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*.

For, whatever part *AE* is of *CF*, the same part also let *EB* be of *CG*.

Now since, whatever part *AE* is of *CF*, the same part also is *EB* of *CG*, therefore, whatever part *AE* is of *CF*, the same part also is *AB* of *GF*. [[book-7/proposition-5|VII. 5]]

But, whatever part *AE* is of *CF*, the same part also, by hypothesis, is *AB* of *CD*; therefore, whatever part *AB* is of *GF*, the same part is it of *CD* also; therefore *GF* is equal to *CD*.

Let *CF* be subtracted from each; therefore the remainder *GC* is equal to the remainder *FD*.

Now since, whatever part *AE* is of *CF*, the same part also is *EB* of *GC*, while *GC* is equal to *FD*, therefore, whatever part *AE* is of *CF*, the same part also is *EB* of *FD*.

But, whatever part *AE* is of *CF*, the same part also is *AB* of *CD*; therefore also the remainder *EB* is the same part of the remainder *FD* that the whole *AB* is of the whole *CD*. Q. E. D.