Let $\widehat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$ be the extended complex plane, and let $\operatorname{Mob}(\widehat{\mathbb{C}})$ denote the set of maps $T:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ represented by coefficients $a,b,c,d\in\mathbb{C}$ with $ad-bc\ne 0$ through the formula
with the standard extended-value convention on $\widehat{\mathbb{C}}$. Let $z_1,z_2,z_3 \in \widehat{\mathbb{C}}$ be pairwise distinct, and let $w_1,w_2,w_3 \in \widehat{\mathbb{C}}$ be pairwise distinct. Then there exists a unique $T\in\operatorname{Mob}(\widehat{\mathbb{C}})$ such that
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\begin{align*}
T(z_i)=w_i \quad \text{for every } i \in \{1,2,3\}.
\end{align*}