**Proof:**
This is an induction on . For
*n*=0 there is nothing to prove.
So now let *n*>0 and suppose the result holds for
all *k*<*n*.
Let and let
,
and let
.
Then *K*' and *B*' satisfy the hypothesis of the theorem:
In fact if is a simplex in *K*', then
by the clique condition
is a simplex in *K* and
again via the clique condition

where the latter simplicial complex is contractible.
Moreover
is contractible by assumption.
Now since by induction hypothesis
*K*' is contractible and thus *x* is escamotable.

To prove the ``moreover" part we need to prove that
*K* is ``dismantlable via escamotable points" to
*K*[*B*]. To see this it is good enough to show that
and *B* satisfy the assumption of the theorem.
Let be a simplex.
Then
is contractible by assumption and we are done.
\

Clearly a set *B* as above must intersect every maximal
simplex, which is exactly the notion of a cutset.

**Proof:**
We will prove that the simplicial complex satisfies the
assumption
of Theorem 4.41.
Let be a simplex.
Then the set
has a center and is thus contractible. \

Bernd.S.W.Schroder