Subtraction of Multiples of Divisors obeys Distributive Law/Proof 2

Theorem

In the words of Euclid:

If a number be the same parts of a 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 $\text{VII}$: Proposition $8$)

In modern algebraic language:

$a = \dfrac m n b, c = \dfrac m n d \implies a - c = \dfrac m n \paren {b - d}$

Proof

A direct application of the Distributive Property:

\(\ds \frac m n b - \frac m n d\)

\(=\)

\(\ds \frac m n b + \frac m n \paren {-d}\)

\(\ds \)

\(=\)

\(\ds \frac m n \paren {b + \paren {-d} }\)

\(\ds \)

\(=\)

\(\ds \frac m n \paren {b - d}\)

$\blacksquare$