Problem
Show that $1x = x$ for all $x \in \mathbb{F}^n$
Solution
Multiplication by scalar is done coordinatewise, and since $1a = a$ for all $a \in \mathbb{F}$ we must have that: $$1x \stackrel{D1.10}{=} 1(x_1, x_2, \dots, x_n) \stackrel{D1.17}{=} (1x_1, 1x_2, \dots, 1x_n) = (x_1, x_2, \dots, x_n) \stackrel{D1.10}{=} x$$ Proving that $1$ is a multiplicative identity for $\mathbb{F}^n$.