*We assume that all motions are translations only in
the same direction l and that each* *segment
is moved only once and one-at-a-time.*

Without any loss of generality, we can assume that the
direction ** l **in which the segments are to be separated is
the positive x-direction. By induction, if there exists one segment in

Let us illuminate all the segments from x=+ ,
as shown in the following Figure:

Is there always one segment completely illuminated?

**Lemma: **In any collection of disjoint line segments,
there is always at least one that is completely illuminated from x=+ .

*Proof: *Consider the subset ** U **of segments
whose upper endpoint is illuminated, that is, a horizontal rightward ray
from their upper endpoint does not hit any segment. There is at
least one segment in

As shown in the Figure, this rightmost highest segment
is not necessarily completely illuminated. It can be hidden by a segment
below. However, our claim is that the segment ** b** in