
theorem Th6:
  for T being up-complete Scott non empty TopPoset holds
  T is order_consistent
proof
  let T be up-complete Scott non empty TopPoset;
  for x being Element of T holds downarrow x = Cl {x} &
  for N being eventually-directed net of T st x = sup N
  for V being a_neighborhood of x holds N is_eventually_in V
  proof
    let x be Element of T;
    for N being eventually-directed net of T st x = sup N
    for V being a_neighborhood of x holds N is_eventually_in V
    proof
      let N be eventually-directed net of T;
      assume x = sup N;
      then x = Sup the mapping of N by WAYBEL_2:def 1;
      then
A1:   x = sup rng the mapping of N by YELLOW_2:def 5;
      let V be a_neighborhood of x;
A2:   x in Int V by CONNSP_2:def 1;
      reconsider rngN = rng netmap (N,T) as Subset of T;
      rngN is directed by WAYBEL_2:18;
      then Int V meets rngN by A1,A2,WAYBEL11:def 1;
      then consider z being object such that
A3:   z in Int V and
A4:   z in rngN by XBOOLE_0:3;
      reconsider z9 = z as Element of T by A3;
      consider i being object such that
A5:   i in dom the mapping of N and
A6:   z9 = (the mapping of N).i by A4,FUNCT_1:def 3;
      reconsider i as Element of N by A5;
      consider j being Element of N such that
A7:   for k being Element of N st j <= k holds N.i <= N.k by WAYBEL_0:11;
      take j;
      let l be Element of N;
      assume j <= l;
      then N.i <= N.l by A7;
      then
A8:   N.l in Int V by A3,A6,WAYBEL_0:def 20;
      Int V c= V by TOPS_1:16;
      hence thesis by A8;
    end;
    hence thesis by Th5;
  end;
  hence thesis;
end;
