
theorem
  for X being finite subset-closed set holds Sub_of_Fin X = X
proof
  let X be finite subset-closed set;
  thus Sub_of_Fin X c= X;
  let x be object;
      reconsider xx=x as set by TARSKI:1;
  assume
A1: x in X;
  bool xx c= X
  by A1,CLASSES1:def 1;
  then xx is finite;
  hence thesis by A1,Def3;
end;
