# 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: $\R D$: 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

- 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $1 \ \text{(g)}$