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

theorem Th32:
  Cl Int Cl F = {A where A is Subset of T : ex B being Subset of T
  st A = Cl Int Cl B & B in F}
proof
  set P = {A where A is Subset of T : ex B being Subset of T st A = Cl Int Cl
  B & B in F};
  now
    let C be object;
    assume C in P;
    then
    ex A being Subset of T st C = A & ex B being Subset of T st A = Cl Int
    Cl B & B in F;
    hence C in bool the carrier of T;
  end;
  then reconsider P as Subset-Family of T by TARSKI:def 3;
  reconsider P as Subset-Family of T;
  for X being object holds X in Cl Int Cl F iff X in P
  proof
    let X be object;
A1: now
      assume
A2:   X in P;
      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 Cl B & B in F by A2;
      then consider B being Subset of T such that
A3:   C = Cl Int Cl B and
A4:   B in F;
      Cl B in Cl F by A4,PCOMPS_1:def 2;
      then Int Cl B in Int Cl F by Def1;
      hence X in Cl Int Cl F by A3,PCOMPS_1:def 2;
    end;
    now
      assume
A5:   X in Cl Int Cl F;
      then reconsider C = X as Subset of T;
      consider B being Subset of T such that
A6:   C = Cl B and
A7:   B in Int Cl F by A5,PCOMPS_1:def 2;
      consider D being Subset of T such that
A8:   B = Int D and
A9:   D in Cl F by A7,Def1;
      ex E being Subset of T st D = Cl E & E in F by A9,PCOMPS_1:def 2;
      hence X in P by A6,A8;
    end;
    hence thesis by A1;
  end;
  hence thesis by TARSKI:2;
end;
