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
  for S1 be SubSimplicialComplex of SX st
      S1 is maximal SubSimplicialComplex of KX
    holds S1 is maximal
 proof
  let S1 be SubSimplicialComplex of SX;
  assume S1 is maximal SubSimplicialComplex of KX;
  then reconsider s1=S1 as maximal SubSimplicialComplex of KX;
  the topology of SX c=the topology of KX by Def13;
  then A1: bool[#]s1/\the topology of SX c=bool[#]s1/\the topology of KX by
XBOOLE_1:26;
  bool[#]s1/\the topology of KX c=the topology of s1 by Th33;
  then bool[#]S1/\the topology of SX c=the topology of S1 by A1;
  hence thesis by Th33;
 end;
