reserve x,y, X,Y,Z for set,
        D for non empty set,
        n,k for Nat,
        i,i1,i2 for Integer;
reserve K for SimplicialComplexStr;
reserve KX for SimplicialComplexStr of X,
        SX for SubSimplicialComplex of KX;

theorem
  Vertices SX c= Vertices KX
 proof
  let x be object;
  assume A1: x in Vertices SX;
  then reconsider v=x as Element of SX;
  v is vertex-like by A1,Def4;
  then consider S be Subset of SX such that
   A2: S is simplex-like and
   A3: v in S;
  A4: the topology of SX c=the topology of KX by Def13;
  A5: S in the topology of SX by A2;
  then S in the topology of KX by A4;
  then reconsider S as Subset of KX;
  v in S by A3;
  then reconsider v as Element of KX;
  S is simplex-like by A4,A5;
  then v is vertex-like by A3;
  hence thesis by Def4;
 end;
