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 KX be subset-closed SimplicialComplexStr of X
    for A be Subset of KX for S be finite-membered Subset-Family of A st
        S c= the topology of KX
      holds Complex_of S is strict SubSimplicialComplex of KX
 proof
  let KX be subset-closed SimplicialComplexStr of X;
  let A be Subset of KX;
  let S be finite-membered Subset-Family of A;
  A1: the_family_of KX is subset-closed;
  assume S c=the topology of KX;
  hence thesis by A1,Th3,Th28;
 end;
