The lemma above ensures we maintain the correct order of intervals along the scanline. We emit a point IFF it is in front of the closest interval in our heap along a scanline. This is the same as emitting it IFF we can draw a line-segment from \(q\) to our point and not intersect an interval.
Furthermore if we have multiple points on the same scanline, we only consider the closest one as an event point. This ensures that our line-segment does not intersect any other points.
As such, we emit a point IFF we can draw a line-segment from \(q\) and hit our point without intersecting any other objects in the plane. This is equivalent to our point being visible from \(q\), and the proof is complete.