= Solution
Choose a <Möbius transformation> of $\mathbb H$ that fixes $0$ and exchanges $x$ with $\infty$. By <Conformal invariance of SLE>, it transforms an $\operatorname{SLE}_6$ from $0$ to $x$ into one from $0$ to $\infty$. Until $x$ is disconnected from infinity, the discrepancy between the two target domains lies beyond the component visible from the growing tip. The <Locality property of SLE> therefore makes the two initial curve laws identical up to that disconnection time. Hence an $\operatorname{SLE}_6$ from $0$ to $x$, stopped at $\tau_x$, has the law of an $\operatorname{SLE}_6$ from $0$ to $\infty$ stopped when it disconnects $x$ from infinity.
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