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Proof: This is an induction on tex2html_wrap_inline11004 . For n=0 there is nothing to prove. So now let n>0 and suppose the result holds for all k<n. Let tex2html_wrap_inline11012 and let tex2html_wrap_inline11014 , tex2html_wrap_inline11016 and let tex2html_wrap_inline11018 . Then K' and B' satisfy the hypothesis of the theorem: In fact if tex2html_wrap_inline11024 is a simplex in K', then by the clique condition tex2html_wrap_inline11028 is a simplex in K and again via the clique condition


where the latter simplicial complex is contractible. Moreover tex2html_wrap_inline11034 is contractible by assumption. Now since tex2html_wrap_inline11036 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 tex2html_wrap_inline11046 and B satisfy the assumption of the theorem. Let tex2html_wrap_inline11050 be a simplex. Then tex2html_wrap_inline11052 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 tex2html_wrap_inline11122 be a simplex. Then the set tex2html_wrap_inline11124 has a center and is thus contractible. \