reserve T for TopSpace;

theorem
  for F being Subset-Family of T holds Cl F = Cl Cl F
proof
  let F be Subset-Family of T;
A1: Cl Cl F = {D where D is Subset of T : ex B being Subset of T st D = Cl B
  & B in Cl F} by Th7;
A2: Cl F = {A where A is Subset of T : ex B being Subset of T st A = Cl B &
  B in F} by Th7;
  for X being object holds X in Cl F iff X in Cl Cl F
  proof
    let X be object;
A3: now
      assume
A4:   X in Cl F;
      then reconsider C = X as Subset of T;
      consider B being Subset of T such that
A5:   C = Cl B and
A6:   B in F by A4,PCOMPS_1:def 2;
A7:   Cl B in Cl F by A6,PCOMPS_1:def 2;
      C = Cl Cl B by A5;
      hence X in Cl Cl F by A1,A7;
    end;
    now
      assume
A8:   X in Cl Cl F;
      then reconsider C = X as Subset of T;
      ex Q being Subset of T st Q = C & ex B being Subset of T st Q = Cl
      B & B in Cl F by A1,A8;
      then consider B being Subset of T such that
A9:   C = Cl B and
A10:  B in Cl F;
      ex Q being Subset of T st Q = B & ex R being Subset of T st Q = Cl
      R & R in F by A2,A10;
      hence X in Cl F by A9,PCOMPS_1:def 2;
    end;
    hence thesis by A3;
  end;
  hence thesis by TARSKI:2;
end;
