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 Th14:
  for P1 be SubdivisionStr of Kr for P2 be SubdivisionStr of P1
    holds P2 is SubdivisionStr of Kr
 proof
  let P1 be SubdivisionStr of Kr;
  let P2 be SubdivisionStr of P1;
  |.P2.|=|.P1.| by Th10
   .=|.Kr.| by Th10;
  hence |.Kr.|c=|.P2.|;
  let A2 be Subset of P2;
  assume A2 is simplex-like;
  then consider A1 be Subset of P1 such that
   A1: A1 is simplex-like and
   A2: conv@A2 c=conv@A1 by Def4;
  ex A be Subset of Kr st A is simplex-like & conv@A1 c=conv@A by A1,Def4;
  hence thesis by A2,XBOOLE_1:1;
 end;
