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 Th90:
  Domains_Lattice T is complete
proof
  thus for X being set ex a being Element of Domains_Lattice T st X
  is_less_than a & for b being Element of Domains_Lattice T st X is_less_than b
  holds a [= b
  proof
    let X be set;
    set Y = { c where c is Element of Domains_Lattice T : c in X};
A1: for d being Element of Domains_Lattice T holds Y is_less_than d
    implies X is_less_than d
    proof
      let d be Element of Domains_Lattice T;
      assume
A2:   Y is_less_than d;
      thus for e being Element of Domains_Lattice T st e in X holds e [= d
      proof
        let e be Element of Domains_Lattice T;
        assume e in X;
        then e in Y;
        hence thesis by A2;
      end;
    end;
A3: for d being Element of Domains_Lattice T holds X is_less_than d
    implies Y is_less_than d
    proof
      let d be Element of Domains_Lattice T;
      assume
A4:   X is_less_than d;
      thus for e being Element of Domains_Lattice T st e in Y holds e [= d
      proof
        let e be Element of Domains_Lattice T;
        assume e in Y;
        then ex e0 being Element of Domains_Lattice T st e0 = e & e0 in X;
        hence thesis by A4;
      end;
    end;
    now
      let x be object;
      assume x in Y;
      then ex y being Element of Domains_Lattice T st y = x & y in X;
      hence x in the carrier of Domains_Lattice T;
    end;
    then reconsider Y as Subset of Domains_Lattice T by TARSKI:def 3;
    consider a being Element of Domains_Lattice T such that
A5: Y is_less_than a and
A6: for b being Element of Domains_Lattice T st Y is_less_than b
    holds a [= b by Th89;
    take a;
    for b being Element of Domains_Lattice T st X is_less_than b holds a
    [= b by A3,A6;
    hence thesis by A1,A5;
  end;
end;
