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;

theorem
  Bottom (Domains_Lattice T) = {}T & Top (Domains_Lattice T) = [#]T
proof
  thus Bottom (Domains_Lattice T) = {}T
  proof
    {}T is condensed by TDLAT_1:14;
    then
A1: {}T in {A where A is Subset of T : A is condensed};
    then reconsider E = {}T as Element of Domains_of T by TDLAT_1:def 1;
    {}T in Domains_of T by A1,TDLAT_1:def 1;
    then reconsider e = {}T as Element of Domains_Lattice T by Th85;
    for a being Element of Domains_Lattice T holds e "\/" a = a
    proof
      let a be Element of Domains_Lattice T;
      reconsider A = a as Element of Domains_of T by Th85;
      A in Domains_of T;
      then A in {C where C is Subset of T : C is condensed} by TDLAT_1:def 1;
      then ex D being Subset of T st D = A & D is condensed;
      then
A2:   Int(Cl A) c= A by TOPS_1:def 6;
      thus e "\/" a = Int(Cl(E \/ A)) \/ (E \/ A) by Th86
        .= a by A2,XBOOLE_1:12;
    end;
    hence thesis by LATTICE2:14;
  end;
  thus Top (Domains_Lattice T) = [#]T
  proof
    [#]T is condensed by TDLAT_1:15;
    then
A3: [#]T in {A where A is Subset of T : A is condensed};
    then reconsider E = [#]T as Element of Domains_of T by TDLAT_1:def 1;
    [#]T in Domains_of T by A3,TDLAT_1:def 1;
    then reconsider e = [#]T as Element of Domains_Lattice T by Th85;
    for a being Element of Domains_Lattice T holds e "/\" a = a
    proof
      let a be Element of Domains_Lattice T;
      reconsider A = a as Element of Domains_of T by Th85;
      A in Domains_of T;
      then A in {C where C is Subset of T : C is condensed} by TDLAT_1:def 1;
      then ex D being Subset of T st D = A & D is condensed;
      then
A4:   A c= Cl(Int A) by TOPS_1:def 6;
      thus e "/\" a = Cl(Int(E /\ A)) /\ (E /\ A) by Th86
        .= Cl(Int(A)) /\ ([#]T /\ A) by XBOOLE_1:28
        .= Cl(Int(A)) /\ A by XBOOLE_1:28
        .= a by A4,XBOOLE_1:28;
    end;
    hence thesis by LATTICE2:16;
  end;
end;
