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

theorem
  Int Cl Int Cl F = Int Cl F
proof
  set H = {A where A is Subset of T : ex B being Subset of T st A = Cl Int Cl
  B & B in F};
  Cl Int Cl F = H by Th32;
  then reconsider H as Subset-Family of T;
A1: Int Cl Int Cl F = Int H by Th32;
A2: Int Cl F = {A where A is Subset of T : ex B being Subset of T st A = Int
  Cl B & B in F} by Th31;
  for X being object holds X in Int Cl Int Cl F iff X in Int Cl F
  proof
    let X be object;
A3: now
      assume
A4:   X in Int Cl 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 = Int
      Cl B & B in F by A2,A4;
      then consider B being Subset of T such that
A5:   C = Int Cl B and
A6:   B in F;
      Cl B in Cl F by A6,PCOMPS_1:def 2;
      then Int Cl B in Int Cl F by Def1;
      then
A7:   Cl Int Cl B in Cl Int Cl F by PCOMPS_1:def 2;
      C = Int Cl Int Cl B by A5,TDLAT_1:5;
      hence X in Int Cl Int Cl F by A7,Def1;
    end;
    now
      assume
A8:   X in Int Cl Int Cl F;
      then reconsider C = X as Subset of T;
      consider B being Subset of T such that
A9:   C = Int B and
A10:  B in {A where A is Subset of T : ex B being Subset of T st A =
      Cl Int Cl B & B in F} by A1,A8,Def1;
      ex S being Subset of T st S = B & ex R being Subset of T st S = Cl
      Int Cl R & R in F by A10;
      then consider D being Subset of T such that
A11:  B = Cl Int Cl D and
A12:  D in F;
A13:  Cl D in Cl F by A12,PCOMPS_1:def 2;
      C = Int Cl D by A9,A11,TDLAT_1:5;
      hence X in Int Cl F by A13,Def1;
    end;
    hence thesis by A3;
  end;
  hence thesis by TARSKI:2;
end;
