reserve T for TopSpace;
reserve T for non empty TopSpace;
reserve F for Subset-Family of T;

theorem
  Cl Int Cl Int F = Cl Int F
proof
  set H = {A where A is Subset of T : ex B being Subset of T st A = Int Cl Int
  B & B in F};
  Int Cl Int F = H by Th33;
  then reconsider H as Subset-Family of T;
A1: Cl Int Cl Int F = Cl H by Th33;
A2: Cl Int F = {A where A is Subset of T : ex B being Subset of T st A = Cl
  Int B & B in F} by Th30;
  for X being object holds X in Cl Int Cl Int F iff X in Cl Int F
  proof
    let X be object;
A3: now
      assume
A4:   X in Cl Int F;
      then reconsider C = X as Subset of T;
      ex D being Subset of T st D = C & ex B being Subset of T st D = Cl
      Int B & B in F by A2,A4;
      then consider B being Subset of T such that
A5:   C = Cl Int B and
A6:   B in F;
      Int B in Int F by A6,Def1;
      then Cl Int B in Cl Int F by PCOMPS_1:def 2;
      then
A7:   Int Cl Int B in Int Cl Int F by Def1;
      C = Cl Int Cl Int B by A5,TOPS_1:26;
      hence X in Cl Int Cl Int F by A7,PCOMPS_1:def 2;
    end;
    now
      assume
A8:   X in Cl Int Cl Int F;
      then reconsider C = X as Subset of T;
      consider B being Subset of T such that
A9:   C = Cl B and
A10:  B in {A where A is Subset of T : ex B being Subset of T st A =
      Int Cl Int B & B in F} by A1,A8,PCOMPS_1:def 2;
      ex S being Subset of T st S = B & ex R being Subset of T st S = Int
      Cl Int R & R in F by A10;
      then consider D being Subset of T such that
A11:  B = Int Cl Int D and
A12:  D in F;
A13:  Int D in Int F by A12,Def1;
      C = Cl Int D by A9,A11,TOPS_1:26;
      hence X in Cl Int F by A13,PCOMPS_1:def 2;
    end;
    hence thesis by A3;
  end;
  hence thesis by TARSKI:2;
end;
