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

theorem
  Bottom (Closed_Domains_Lattice T) = {}T & Top (Closed_Domains_Lattice
  T) = [#]T
proof
  thus Bottom (Closed_Domains_Lattice T) = {}T
  proof
    {}T is closed_condensed by TDLAT_1:18;
    then
A1: {}T in {A where A is Subset of T : A is closed_condensed};
    then reconsider E = {}T as Element of Closed_Domains_of T by TDLAT_1:def 5;
    {}T in Closed_Domains_of T by A1,TDLAT_1:def 5;
    then reconsider e = {}T as Element of Closed_Domains_Lattice T by Th93;
    for a being Element of Closed_Domains_Lattice T holds e "\/" a = a
    proof
      let a be Element of Closed_Domains_Lattice T;
      reconsider A = a as Element of Closed_Domains_of T by Th93;
      thus e "\/" a = E \/ A by Th94
        .= a;
    end;
    hence thesis by LATTICE2:14;
  end;
  thus Top (Closed_Domains_Lattice T) = [#]T
  proof
    [#]T is closed_condensed by TDLAT_1:19;
    then
A2: [#]T in {A where A is Subset of T : A is closed_condensed};
    then reconsider E = [#]T as Element of Closed_Domains_of T by TDLAT_1:def 5
;
    [#]T in Closed_Domains_of T by A2,TDLAT_1:def 5;
    then reconsider e = [#]T as Element of Closed_Domains_Lattice T by Th93;
    for a being Element of Closed_Domains_Lattice T holds e "/\" a = a
    proof
      let a be Element of Closed_Domains_Lattice T;
      reconsider A = a as Element of Closed_Domains_of T by Th93;
      A in Closed_Domains_of T;
      then A in {C where C is Subset of T : C is closed_condensed} by
TDLAT_1:def 5;
      then
A3:   ex D being Subset of T st D = A & D is closed_condensed;
      thus e "/\" a = Cl(Int(E /\ A)) by Th94
        .= Cl(Int(A)) by XBOOLE_1:28
        .= a by A3,TOPS_1:def 7;
    end;
    hence thesis by LATTICE2:16;
  end;
end;
