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 Th28:
  for A be Subset of KX for S be finite-membered Subset-Family of A st
        subset-closed_closure_of S c=the topology of KX
      holds Complex_of S is strict SubSimplicialComplex of KX
 proof
  let A be Subset of KX;
  let S be finite-membered Subset-Family of A such that
   A1: subset-closed_closure_of S c=the topology of KX;
  set C=Complex_of S;
  [#]KX c=X by Def9;
  then [#]C c=X;
  then [#]C c=[#]KX & C is strict SimplicialComplexStr of X by Def9;
  hence thesis by A1,Def13;
 end;
