Problem
Show that for every $\alpha \in \mathbb{C}$, there exists a unique $\beta \in \mathbb{C}$ such that $\alpha + \beta = 0$
Solution
Let $\alpha, \beta, \lambda \in \mathbb{F}$ and let $\alpha + \beta = 0$ and $\alpha + \lambda = 0$, then $$\beta = \beta + 0 = \beta + (\alpha + \lambda) \stackrel{P1A5}{=} (\beta + \alpha) + \lambda = 0 + \lambda = \lambda$$ Thus any additive inverse of $\alpha$ is unique, existence can be shown by seeing that if $\alpha = a + bi$, then $\beta = -a - bi$ is an additive inverse. Proof: $$\alpha + \beta = (a + bi) + (-a - bi) \stackrel{D1.1}{=} (a - a) + (b - b)i = 0 + 0i = 0$$ Proving existence of an additive inverse.