Proof of Theorem CCRM

Let $\alpha=a+bi$ and $\beta=r+si$. Then \begin{align*} \conjugate{\alpha\beta}&=\conjugate{(ar-bs)+(as+br)i}=(ar-bs)-(as+br)i\\ &=(ar-(-b)(-s))+(a(-s)+(-b)r)i=(a-bi)(r-si)=\conjugate{\alpha}\conjugate{\beta} \end{align*}