theorem
  for F st for v,B st B c= Aff & v in conv B holds F.v in B
    ex S be Simplex of card Aff-1,BCS(k,Complex_of{Aff}) st F.:S = Aff
 proof
  let F be Function of Vertices BCS(k,Complex_of{Aff}),Aff;
  set XX={S where S is Simplex of card Aff-1,BCS(k,Complex_of{Aff}):F.:S=Aff};
  assume for v being Vertex of BCS(k,Complex_of{Aff})for B st B c=Aff & v in
conv B holds F.v in B;
  then ex n st card XX=2*n+1 by Th46;
  then XX is non empty;
  then consider x being object such that
A1: x in XX;
  ex S be Simplex of card Aff-1,BCS(k,Complex_of{Aff}) st x=S & F.:S=Aff by A1;
  hence thesis;
 end;
