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 Y1,Y2 be Subset-Family of X st
      Y1 is finite-membered & Y1 is_finer_than Y2
    holds Complex_of Y1 is SubSimplicialComplex of Complex_of Y2
 proof
  let Y1,Y2 be Subset-Family of X such that
   A1: Y1 is finite-membered and
   A2: Y1 is_finer_than Y2;
  set C1=Complex_of Y1,C2=Complex_of Y2;
  A3: [#]C1=X & [#]C2=X;
  subset-closed_closure_of Y1 c=subset-closed_closure_of Y2 by A2,Th6;
  hence thesis by A1,A3,Def13;
 end;
