Problem
Show that $(a + b)x = ax + bx$ for all $a,b \in \mathbb{F}$ and all $x \in \mathbb{F}^n$
Solution
Multiplication of elements in $\mathbb{F}^n$ by scalars is defined coordinate-wise and from problem 9 we know that multiplication of elements in $\mathbb{F}$ is distributive. Therefore: $$(a + b)x \stackrel{D1.10}{=} (a + b)(x_1, x_2, \dots, x_n) $$ $$\stackrel{D1.17}{=} ((a + b)x_1, (a+b)x_2, \dots, (a+b)x_n)$$ $$\stackrel{E1.4 \land P1A9}{=} (ax_1 + bx_1, ax_2 + bx_2, \dots, ax_n + bx_n)$$ $$ \stackrel{D1.12 \land D1.17}{=} a (x_1, x_2, \dots, x_n) + b (x_1, x_2, \dots, x_n) \stackrel{D1.10}{=} ax + bx$$ Proving that elements of $\mathbb{F}^n$ distribute onto sums of scalars.