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 Th19:
  the topology of K is finite & K is non finite-vertices implies
    K is non finite-membered
 proof
  defpred P[object,object] means ex D2 being set st D2 = $2 & $1 in D2;
  set V=Vertices K;
  assume that
   A1: the topology of K is finite and
   A2: K is non finite-vertices;
  A3: V is infinite by A2;
  A4: V=union the topology of K by Lm5;
  A5: for x being object st x in V
    ex y being object st y in the topology of K & P[x,y]
  proof
   let x be object;
   assume x in V;
   then ex y st x in y & y in the topology of K by A4,TARSKI:def 4;
   hence thesis;
  end;
  consider f be Function of V,the topology of K such that
A6: for x being object st x in V holds P[x,f.x] from FUNCT_2:sch 1(A5);
  the topology of K is non empty by A2,A4;
  then dom f=V by FUNCT_2:def 1;
  then consider x being object such that
   A7: x in rng f and
   A8: f"{x} is infinite by A1,A3,CARD_2:101;
  x in the topology of K by A7;
  then reconsider x as Subset of K;
  f"{x}c=x
  proof
   let y be object;
   assume
A9:  y in f"{x};
   then P[y,f.y] by A6;
   then f.y in {x} & y in f.y by A9,FUNCT_1:def 7;
   hence thesis by TARSKI:def 1;
  end;
  then x is non finite by A8;
  then the_family_of K is non finite-membered by A7;
  hence thesis;
 end;
