Multiples of Divisors obey Distributive Law/Proof 2

Theorem

In the words of Euclid:

If a number be parts of a number, and another be the same parts of another, the sum will also be the same parts of the sum that the one is of the one.

(The Elements: Book $\text{VII}$: Proposition $6$)

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:

$\dfrac m n b + \dfrac m n d = \dfrac m n \paren {b + d}$

$\blacksquare$