split either to the left or to the right. The picture on the left of Figure \[fig-ex2-simple-no-21-after-max\] shows that $\{a,b\}$ cannot be split solely to the right, as then taking $c$ to be the rightmost such separator we see that $\{a,b,c\}$ would lie in a proper interval. Similarly, $\{a,b\}$ cannot be split solely to the left. Thus $\{a,b\}$ must be split on both the left and the right. Now, taking $c$ to be the rightmost separator and $d$ the leftmost separator, we have two cases, depicted in the centre and right of Figure \[fig-ex2-simple-no-21-after-max\]. In both cases it is clear that $\{a,b,c,d\}$ is contained in a proper interval, and this contradiction completes the proof.

[ccccc]{}

(0,0)(50,40) (0,10)(10,10)(10,20)(0,20) (0,20)(10,20)(10,30)(0,30) (0,30)(10,30)(10,40)(0,40) (0,30)(10,40) (10,0)(20,0)(20,10)(10,10) (10,20)(20,20)(20,30)(10,30) (10,30)(20,30)(20,40)(10,40) (20,0)(30,0)(30,10)(20,10) (20,30)(30,30)(30,40)(20,40) (30,0)(40,0)(40,10)(30,10) (30,10)(40,10)(40,20)(30,20) (30,20)(40,20)(40,30)(30,30) (40,10)(50,10)(50,20)(40,20) (40,20)(50,20)(50,30)(40,30) (10,30)[2.0]{} (10,30)[$b$]{} (20,10)[2.0]{} (20,10)[$a$]{} (30,20)[2.0]{} (30,20)[$c$]{} (40,40)[2.0]{} (0,0)(0,40) (0,0)(50,0) (10,0)(10,40) (0,10)(50,10) (20,0)(20,40) (0,20)(50,20) (30,0)(30,40) (0,30)(50,30) (40,0)(40,40) (0,40)(50,40) (50,0)(50,40)

&

------------------------------------------------------------------------

&

(0,0)(60,50) (0,10)(10,10)(10,20)(0,20) (0,20)(10,20)(10,30)(0,30) (0,30)(10,30)(10,40)(0,40) (0,40)(10,40)(10,50)(0,50) (10,0)(20,0)(20,10)(10,10) (10,10)(20,10)(20,20)(10,20) (10,20)(20,20)(20,30)(10,30) (10,40)(20,40)(20,50)(10,50) (10,40)(20,50) (20,0)(30,0)(30,10)(20,10) (20,20)(30,20)(30,30)(20,30) (20,30)(30,30)(30,40)(20,40) (20,40)(30,40)(30,50)(20,50) (30,0)(40,0)(40,10)(30,10) (30,30)(40,30)(40,40)(30,40) (30,40)(40,40)(40,50)(30,50) (40,0)(50,0)(50,10)(40,10) (40,10)(50,10)(50,20)(40,20) (40,20)(50,20)(50,30)(40,30) (40,30)(50,30)(50,40)(40,40) (50,10)(60,10)(60,20)(50,20) (50,20)(60,20)(60,30)(50,30) (50,30)(60,30)(60,40)(50,40) (10,30)[2.0]{} (10,30)[$d$]{} (20,40)[2.0]{} (20,40)[$b$]{} (30,10)[2.0]{} (30,10)[$a$]{} (40,20)[2.0]{} (40,20)[$c$]{} (50,50)[2.0]{} (0,0)(0,50) (0,0)(60,0) (10,0)(10,50) (0,10)(60,10) (20,0)(20,50) (0,20)(60,20) (30,0)(30,50) (0,30)(60,30) (40,0)(40,50) (0,40)(60,40) (50,0)(50,50) (0,50)(60,50) (60,0)(60,50)

&

------------------------------------------------------------------------

&

