
theorem Th35:
  for T being Lawson complete TopLattice
  for D being directed non empty Subset of T holds sup D in Lim Net-Str D
proof
  let T be Lawson complete TopLattice;
  let D be directed non empty Subset of T;
  set N = Net-Str D;
A1: the mapping of N = id D by Th32;
A2: the carrier of N = D by Th32;
  set K = the prebasis of T;
  now
    let A be Subset of T;
    assume
A3: sup D in A;
A4: K c= the topology of T by TOPS_2:64;
    assume A in K;
    then A is open by A4,PRE_TOPC:def 2;
    then A is property(S) by WAYBEL19:36;
    then consider y being Element of T such that
A5: y in D and
A6: for x being Element of T st x in D & x >= y holds x in A
    by A3,WAYBEL11:def 3;
    reconsider i = y as Element of N by A5,Th32;
    thus N is_eventually_in A
    proof
      take i;
      let j be Element of N;
A7:   j = N.j by A1,A2;
A8:   y = N.i by A1,A2;
      assume j >= i;
      then N.j >= N.i by Th34;
      hence thesis by A2,A6,A7,A8;
    end;
  end;
  hence thesis by WAYBEL19:25;
end;
