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 Th23:
  for Sv be non void SubSimplicialComplex of Kv st
      |.Kv.| c= [#]Kv & |.Sv.| c= [#]Sv
    holds BCS(n,Sv) is SubSimplicialComplex of BCS(n,Kv)
 proof
  let S be non void SubSimplicialComplex of Kv;
  assume|.Kv.|c=[#]Kv & |.S.|c=[#]S;
  then BCS(n,S)=subdivision(n,center_of_mass V,S) &
  BCS(n,Kv)=subdivision(n,center_of_mass V,Kv)
    by Def6;
  hence thesis by SIMPLEX0:65;
 end;
