Subtraction of Multiples of Divisors Obeys Distributive Law

Theorem

As Euclid defined it:

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

(The Elements: Book VII: Proposition $8$)

Proof

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

We need to show that $EB$ is also the same parts of the remainder $FD$ that the whole $AB$ is of the whole $CD$.

Let $GH = AB$.

Then whatever parts $GH$ is of $CD$, the same parts also is $AE$ of $CF$.

Let $GH$ be divided into the parts of $CD$, namely $GK + KH$, and $AE$ into the parts of $CF$, namely $AL + LE$.

Thus the multitude of $GK, KH$ will be equal to the multitude of $AL, LE$.

We have that whatever part $GK$ is of $CD$, the same part also is $AL$ of $CF$.

We also have that $CD > CF$.

Therefore $GK > AL$.

Now let $GM = AL$.

Then whatever part $GK$ is of $CD$, the same part also is $GM$ of $CF$.

Therefore from Subtraction of Divisors Obeys Distributive Law the remainder $MK$ is of the same part of the remainder $FD$ that the whole $GK$ is of the whole $CD$.

Again, we have that whatever part $KH$ is of $CD$, the same part also is $EL$ of $CF$.

We also have that $CD > CF$.

Therefore $HK > EL$.

Let $KN = EL$.

Then whatever part $KH$ is of $CD$, the same part also is $KN$ of $CF$.

Therefore from Subtraction of Divisors Obeys Distributive Law the remainder $NH$ is of the same part of the remainder $FD$ that the whole $KH$ is of the whole $CD$.

But $MK$ was proved to be the same part of $FD$ that $GK$ is of $CD$.

Therefore $MK + NH$ is the same parts of $DF$ that $HG$ is of $CD$.

But $MK + NH = EB$ and $HG = BA$.

Therefore $EB$ is the same parts of $FD$ that $AB$ is of $CD$.

$\blacksquare$

Historical Note

This is Proposition 8 of Book VII of Euclid's The Elements.