Subtraction of Multiples of Divisors obeys Distributive Law/Proof 2

Theorem

 * 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.

In modern algebraic language:
 * $a = \dfrac m n b, c = \dfrac m n d \implies a - c = \dfrac m n \left({b - d}\right)$

Proof
A direct application of the Distributive Property: