First let us recall the theorem 
:
Let M be a globally hyperbolic spacetime which satisfies the averaged null energy condition (ANEC) . Let the metric be de Sitter outside of a compact set S. Let S be such that 
is not empty. Then any topological structure in S surrounded by an inner trapped surface cannot be probed.
Proof of Theorem 1. Consider the universal covering space
and the corresponding
spacetime 
, with 
the pullback
of 
to 
by 
. By construction 
is
simply connected, and any point in M has a simply connected
neighborhood A whose inverse image 
is the disjoint
union of open simply connected sets in 
. Each of these
copies of A in 
corresponds to a homotopically distinct way
of reaching A from a fiducial point of M, and we can choose the
fiducial point to lie on 
. The projection 
, restricted
to any single copy of A, is an isometry.
Since the open neighborhood U of 
is chosen to be simply
connected and M itself is not simply connected, U will be covered
by multiple copies of itself in 
, which will therefore have
multiple asymptotic regions. Let 
be one of
these copies, an open connected neighborhood of a single asymptotic
region of 
. Construct a partial conformal completion
by adjoining a single copy
of 
to 
. Then 
, with
one asymptotic region singled out, satisfies the requirements of the
Lemmas.
Suppose the theorem is false. Then there is a causal curve 
in
M, from 
to 
that traverse the topological structure, which is not deformable to
relative to 
which do not traverse the topological structure. The curves 
and
can be lifted to curves 
and 
in 
that meet the same point of 
. Because the construction of
assigns distinct points to homotopically different ways of
reaching the same point of M, the curves 
and 
will
join 
to different copies of the asymptotic
region 
. Because 
lies in the simply connected
neighborhood U of 
, 
will lie in the
neighborhood 
of 
, while 
will join 
to another copy of U. In this
second asymptotic region, surfaces are outer trapped with respect to 
as it is surrounded by an inner trapped surface in the original space (M, not 
)
These large spheres appear outer
trapped as seen from the first asymptotic region, 
.
Let 
be the covering space of a
Cauchy surface 
of M and let 
be a sphere in an
asymptotic region of 
different from the one
containing 
. If we define outer-directed curves from any
sphere 
to be those that reach 
without
intersecting 
a second time, then the outer directed curves
from 
are curves from its concave surface --- curves that
would ordinarily be called inner directed by an observer in the
asymptotic region near 
. As 
is causal, this
implies that there are strongly outer trapped surfaces that intersect
. But this contradicts Lemma 2. Hence any
causal curve 
from 
to 
must be
deformable to 
.