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 Th27:
  for S1 be SubSimplicialComplex of SX holds S1 is SubSimplicialComplex of KX
 proof
  let S1 be SubSimplicialComplex of SX;
  [#]SX c=[#]KX & [#]S1 c=[#]SX by Def13;
  then A1: [#]S1 c=[#]KX;
  the topology of SX c=the topology of KX & the topology of S1 c=the topology
of SX by Def13;
  then the topology of S1 c=the topology of KX;
  hence thesis by A1,Def13;
 end;
