Suppose first that there are no circles at all (n = 0).
Then 
is all of 
,
and we can make all of our tubes vertical
(parallel to the z-axis).
The cross-section of the tubes grows exponentially as a function of the
length along the tube,
thanks to the factor of 
in the metric:
Let dx dy denote the cross-section of a tube at height 1,
that is,
where it passes through the surface z=1.
Then the cross-section at height h is
But moving from height 1 to height h corresponds to going a distance
along the tube, so the cross-section as a function of distance along the tube is
Applying the final proposition of section 2, we conclude that
In fact,
so by cutting we haven't thrown anything away.