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.

(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

\(\ds m \times \paren {a - b}\)

\(=\)

\(\ds m \times \paren {a + \paren {- b} }\)

Definition of Real Subtraction

\(\ds \)

\(=\)

\(\ds m \times a + m \times \paren {- b}\)

Real Number Axiom $\R \text D$: Distributivity of Multiplication over Addition

\(\ds \)

\(=\)

\(\ds m \times a + \paren {- m \times b}\)

Multiplication by Negative Real Number

\(\ds \)

\(=\)

\(\ds m \times a - m \times b\)

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 $1 \ \text{(g)}$