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 Th92:
  for F being Subset-Family of T st F is domains-family for X
  being Subset of Domains_Lattice T st X = F holds (X <> {} implies "/\"(X,
  Domains_Lattice T) = (meet F) /\ (Cl Int(meet F))) & (X = {} implies "/\"(X,
  Domains_Lattice T) = [#]T)
proof
  let F be Subset-Family of T;
  assume
A1: F is domains-family;
  let X be Subset of Domains_Lattice T;
  assume
A2: X = F;
  thus X <> {} implies "/\"(X,Domains_Lattice T) = (meet F) /\ (Cl Int(meet F)
  )
  proof
    set A = (meet F) /\ (Cl Int(meet F));
    A is condensed by A1,Th69;
    then A in {C where C is Subset of T : C is condensed};
    then
A3: A in Domains_of T by TDLAT_1:def 1;
    then reconsider a = A as Element of Domains_Lattice T by Th85;
A4: a is_less_than X
    proof
      let b be Element of Domains_Lattice T;
      reconsider B = b as Element of Domains_of T by Th85;
      assume b in X;
      then A c= B by A2,Th70;
      hence thesis by A3,Th88;
    end;
    assume
A5: X <> {};
A6: for b being Element of Domains_Lattice T st b is_less_than X holds b [= a
    proof
      let b be Element of Domains_Lattice T;
      reconsider B = b as Element of Domains_of T by Th85;
      assume
A7:   b is_less_than X;
A8:   for C being Subset of T st C in F holds B c= C
      proof
        let C be Subset of T;
        reconsider C1 = C as Subset of T;
        assume
A9:     C in F;
        then C1 is condensed by A1;
        then C in {P where P is Subset of T : P is condensed};
        then
A10:    C in Domains_of T by TDLAT_1:def 1;
        then reconsider c = C as Element of Domains_Lattice T by Th85;
        b [= c by A2,A7,A9;
        hence thesis by A10,Th88;
      end;
      B in Domains_of T;
      then B in {C where C is Subset of T : C is condensed} by TDLAT_1:def 1;
      then ex C being Subset of T st C = B & C is condensed;
      then B c= A by A2,A5,A8,Th70;
      hence thesis by A3,Th88;
    end;
    Domains_Lattice T is complete by Th90;
    hence thesis by A4,A6,LATTICE3:34;
  end;
  thus X = {} implies "/\"(X,Domains_Lattice T) = [#]T
  proof
    set A = [#]T;
    A is condensed by TDLAT_1:15;
    then A in {C where C is Subset of T : C is condensed};
    then
A11: A in Domains_of T by TDLAT_1:def 1;
    then reconsider a = A as Element of Domains_Lattice T by Th85;
A12: for b being Element of Domains_Lattice T st b is_less_than X holds b [= a
    proof
      let b be Element of Domains_Lattice T;
      reconsider B = b as Element of Domains_of T by Th85;
      assume b is_less_than X;
      B c= A;
      hence thesis by A11,Th88;
    end;
    assume
A13: X = {};
A14: a is_less_than X
    by A13;
    Domains_Lattice T is complete by Th90;
    hence thesis by A14,A12,LATTICE3:34;
  end;
end;
