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;
reserve SC for SimplicialComplex of X;

theorem
  for A,B be Subset of SC st A c= B holds SC|A is SubSimplicialComplex of SC|B
 proof
  let A,B be Subset of SC;
  A1: bool A/\the topology of SC=the topology of SC|A & bool B/\the topology of
SC=the topology of SC|B by Th37;
  assume A2: A c=B;
  then bool A c=bool B by ZFMISC_1:67;
  then A3: bool A/\the topology of SC c=bool B/\the topology of SC by
XBOOLE_1:27;
  [#](SC|A)=A & [#](SC|B)=B by Def16;
  hence thesis by A2,A3,A1,Def13;
 end;
