Problem
Show that for every $\alpha \in \mathbb{C}$ with $\alpha \neq 0$, there exists a unique $\beta \in \mathbb{C}$ such that $\alpha \beta = 1$
Solution
Let $\alpha,\beta,\lambda \in \mathbb{C}$ with $\alpha\beta = \alpha\lambda = 1$, then: $$\beta = \beta \cdot 1 = \beta(\alpha\lambda) \stackrel{P1A6}{=} (\beta\alpha)\lambda \stackrel{E1.4}{=} (\alpha\beta)\lambda = 1 \cdot \lambda = \lambda$$ Proving uniqueness of the multiplicative inverse $\beta$, existence was shown in P1A1.