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 Th34:
  for S1 be SubSimplicialComplex of SX st SX is maximal & S1 is maximal
    holds S1 is maximal SubSimplicialComplex of KX
 proof
  let S1 be SubSimplicialComplex of SX;
  reconsider s1=S1 as SubSimplicialComplex of KX by Th27;
  assume that
   A1: SX is maximal and
   A2: S1 is maximal;
  A3: [#]S1 c=[#]SX by Def13;
  now let A be Subset of s1;
   reconsider a=A as Subset of SX by A3,XBOOLE_1:1;
   assume A in the topology of KX;
   then a is simplex-like by A1;
   then a in the topology of SX;
   hence A is simplex-like by A2;
  end;
  hence S1 is maximal SubSimplicialComplex of KX by Def14;
 end;
