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 Th10:
  for P be SubdivisionStr of Kr holds |.Kr.| = |.P.|
  proof
  let P be SubdivisionStr of Kr;
  thus|.Kr.|c=|.P.| by Def4;
  let x be object;
  assume x in |.P.|;
  then consider A be Subset of P such that
   A1: A is simplex-like and
   A2: x in conv@A by Def3;
  ex B be Subset of Kr st B is simplex-like & conv@A c=conv@B by A1,Def4;
  hence thesis by A2,Def3;
 end;
