Uniform-spanning-tree path representation of loop-erased random walk

On a finite connected unweighted graph, the graph path between $x$ and $y$ in a uniform spanning tree has the law of loop-erased random walk from $x$ to $y$. Root Wilson's algorithm at $y$ and process $x$ first: the first attached graph path is exactly that loop erasure, and later attachments cannot change it. Changing the root to $x$ proves reversibility of loop-erased random walk, even though chronological loop erasure does not commute with reversing each individual random-walk trajectory.