# Multiplication of Real Numbers is Left Distributive over Subtraction/Proof 2

## Theorem

In the words of Euclid:

If a magnitude be the same multiple of a magnitude that a part subtracted is of a part subtracted, the remainder will also be the same multiple of the remainder that the whole is of the whole.

That is, for any numbers $a, b$ and for any integer $m$:

$m a - m b = m \paren {a - b}$

## Proof

 $\displaystyle m \times \paren {a - b}$ $=$ $\displaystyle m \times \paren {a + \paren {- b} }$ Definition of Real Subtraction $\displaystyle$ $=$ $\displaystyle m \times a + m \times \paren {- b}$ Real Number Axioms: $\R \text D$: Distributivity $\displaystyle$ $=$ $\displaystyle m \times a + \paren {- m \times b}$ Multiplication by Negative Real Number $\displaystyle$ $=$ $\displaystyle m \times a - m \times b$ Definition of Real Subtraction

$\blacksquare$