reserve T for TopSpace;

theorem Th18:
  for F being Subset-Family of T holds F = {} iff Int F = {}
proof
  let F be Subset-Family of T;
  thus F = {} implies Int F = {}
  proof
    set A = the Element of Int F;
    assume
A1: F = {};
    assume
A2: not thesis;
    then reconsider A as Subset of T by TARSKI:def 3;
    ex V being Subset of T st A = Int V & V in F by A2,Def1;
    hence contradiction by A1;
  end;
  thus Int F = {} implies F = {}
  proof
    set B = the Element of F;
    assume
A3: Int F = {};
    assume
A4: not thesis;
    then reconsider B as Subset of T by TARSKI:def 3;
    reconsider A = Int B as set;
    ex A be set st A in Int F
    proof
      take A;
      thus thesis by A4,Def1;
    end;
    hence contradiction by A3;
  end;
end;
