---
book: 7
number: 11
id: "VII.11"
kind: "theorem"
uses: ["[[book-7/proposition-7]]", "[[book-7/proposition-8]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.11"
license: "CC-BY-SA-4.0"
---

# VII.11

*If, as whole is to whole, so is a number subtracted to a number subtracted, the remainder will also be to the remainder as whole to whole.*

## Proof

As the whole *AB* is to the whole *CD*, so let *AE* subtracted be to *CF* subtracted; I say that the remainder *EB* is also to the remainder *FD* as the whole *AB* to the whole *CD*.

Since, as *AB* is to *CD*, so is *AE* to *CF*, whatever part or parts *AB* is of *CD*, the same part or the same parts is *AE* of *CF* also; [[book-7/definitions#Definition 20|VII. Def. 20]]

Therefore also the remainder *EB* is the same part or parts of *FD* that *AB* is of *CD*. [[book-7/proposition-7|VII. 7, 8]]

Therefore, as *EB* is to *FD*, so is *AB* to *CD*. [[book-7/definitions#Definition 20|VII. Def. 20]] Q. E. D.