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

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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.

(The Elements: Book $\text{V}$: Proposition $5$)


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

$\blacksquare$


Sources