It may seem that we have gone as far as we can go
with vertical tubes, but this isn't quite true.
Suppose that there are two circles 
and 
that aren't tangent.
Pick a point on ,
move it to 
,
and normalize as in the one-circle case above.
is a bona fide circle in the half-plane
Now construct a circle 
between and
that is tangent to
and .
(See Figure 4.)
Figure 4: Introducing a third circle.
Decompose 
into the part
between 
and 
,
and the part
between 
and .
If we move
the point of tangency of and
to ,
we can cut
into vertical tubes.
Similarly,
if we move
the point of tangency of and
to ,
we can cut
into vertical tubes.
As for the boundary between
and
,
we can simply ignore it, since it has measure 0.
In this way, we get a cutting of into tubes,
some of which are vertical from one point of view and some from another.
Once again,
we conclude that
whereas in fact
Taking a second look at the argument we have just gone through,
we see that it can be simplified as follows:
Cut into the two pieces
and
.
We already know that
and
By the cutting law,
