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 Th4:
  the topology of K1r c= the topology of K2r implies |.K1r.| c= |.K2r.|
 proof
  assume A1: the topology of K1r c=the topology of K2r;
  let x be object;
  assume x in |.K1r.|;
  then consider A be Subset of K1r such that
   A2: A is simplex-like and
   A3: x in conv@A by Def3;
  A4: A in the topology of K1r by A2;
  then A in the topology of K2r by A1;
  then reconsider A1=A as Subset of K2r;
  @A=@A1 & A1 is simplex-like by A1,A4;
  hence thesis by A3,Def3;
 end;
