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;

theorem Th20:
  K is subset-closed & the topology of K is finite implies K is finite-vertices
  proof
  assume that
   A1: K is subset-closed and
   A2: the topology of K is finite;
  assume K is non finite-vertices;
  then K is non finite-membered by A2,Th19;
  then the_family_of K is non finite-membered;
  then consider x such that
   A3: x in the_family_of K and
   A4: x is non finite;
  {x}c=the_family_of K by A3,ZFMISC_1:31;
  then subset-closed_closure_of{x}c=the_family_of K by A1,Th3;
  then bool x c=the_family_of K by Th4;
  hence contradiction by A2,A4;
 end;
