Problem
Show that $(\alpha \beta) \lambda = \alpha (\beta \lambda)$ for all $\alpha, \beta, \lambda \in \mathbb{C}$
Solution
Let $\alpha = a + bi$, $\beta = c + di$ and $\lambda = e + fi$. Then by the associativity of real numbers: $$(\alpha \beta) \lambda = \left((a + bi)(c + di)\right)(e + fi) \stackrel{D1.1}{=} \left(ac - df + (bc + ad)i\right)(e + fi)$$ $$\stackrel{D1.1}{=} (ace - bdf - bcf - adf) + (acf - bdf + bce + ade)i$$
$$\stackrel{D1.1}{=} (a + bi)\left((ce - df) + (de + cf)i\right)$$ $$\stackrel{D1.1}{=} (a + bi)\left((c + di)(e + fi)\right) = \alpha (\beta \lambda)$$ Proving the associativity of complex numbers under multiplication.