Subtraction of Divisors Obeys Distributive Law

Theorem

As Euclid defined it:

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

(The Elements: Book VII: Proposition $7$)

Proof

Let $AB$ be that part of the (natural) number $CD$ which $AE$ subtracted is of $CF$ subtracted.

We need to show that the remainder $EB$ is also the same part of the number $CD$ which $AE$ subtracted is of $CF$ subtracted.

Whatever part $AE$ is of $CF$, let the same part $EB$ be of $CG$.

Then from Book VII Proposition 5: Divisors Obey Distributive Law, whatever part $AE$ is of $CF$, the same part also is $AB$ of $GF$.

But whatever part $AE$ is of $CF$, the same part also is $AB$ of $CD$, by hypothesis.

Therefore, whatever part $AB$ is of $GF$, the same pat is it of $CD$ also.

Therefore $GF = CD$.

Let $CF$ be subtracted from each.

Therefore $GC = FD$.

We have that 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 $EB$ of $FD$.

But whatever part $AE$ is of $CF$, the same part also is $AB$ of $CD$.

Therefore the remainder $EB$ is the same part of the remainder $FD$ that the whole $AB$ is of the whole $CD$.

$\blacksquare$

Historical Note

This is Proposition 7 of Book VII of Euclid's The Elements.