
theorem Th2:
  for M being subset-closed SubsetFamilyStr holds M is non void iff
  {} in the_family_of M
proof
  let M be subset-closed SubsetFamilyStr;
  hereby
    assume M is non void;
    then reconsider M9 = M as non void subset-closed SubsetFamilyStr;
    set a = the independent Subset of M9;
    {} c= a;
    then {} is independent Subset of M9 by Th1;
    hence {} in the_family_of M by Def2;
  end;
  assume {} in the_family_of M;
  hence the topology of M is non empty;
end;
