Automorphism of $L|K$ mapping 3 distinct rational points of $S_{L|K}$ to other 3 distinct ones

Question

Let $K$ be a field and consider $L = K(x)$ the field of rational functions. Let $v_{1}, v_{2}, v_{3}$ rational points in the abstract Riemann surface $S_{L|K}$, distinct from each other, and $w_{1}, w_{2}, w_{3}$ also rationals points in $S_{L|K}$, distinct from each other. I have to prove that there is a unique automorphism $\sigma$ of the extension $L|K$ such that $w_{i} = v_{i} \circ \sigma$.