reserve x,y,X for set,
        r for Real,
        n,k for Nat;
reserve RLS for non empty RLSStruct,
        Kr,K1r,K2r for SimplicialComplexStr of RLS,
        V for RealLinearSpace,
        Kv for non void SimplicialComplex of V;
reserve Ks for simplex-join-closed SimplicialComplex of V,
        As,Bs for Subset of Ks,
        Ka for non void affinely-independent SimplicialComplex of V,
        Kas for non void affinely-independent simplex-join-closed
                 SimplicialComplex of V,
        K for non void affinely-independent simplex-join-closed total
                 SimplicialComplex of V;
reserve Aff for finite affinely-independent Subset of V,
        Af,Bf for finite Subset of V,
        B for Subset of V,
        S,T for finite Subset-Family of V,
        Sf for c=-linear finite finite-membered Subset-Family of V,
        Sk,Tk for finite simplex-like Subset-Family of K,
        Ak for Simplex of K;
reserve v for Vertex of BCS(k,Complex_of{Aff}),
        F for Function of Vertices BCS(k,Complex_of{Aff}),Aff;

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;
