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;

theorem Th17:
  |.Kv.| c= [#]Kv implies BCS(1,Kv) = BCS Kv
 proof
  assume A1: |.Kv.|c=[#]Kv;
  hence BCS(1,Kv)=subdivision(1,center_of_mass V,Kv) by Def6
   .=subdivision(center_of_mass V,Kv) by SIMPLEX0:62
   .=BCS Kv by A1,Def5;
 end;
