
theorem Th5:  :: Remark 1.4 (ii)
  for T being Scott up-complete non empty reflexive transitive antisymmetric
  TopRelStr, x being Element of T holds Cl {x} = downarrow x
proof
  let T be Scott up-complete non empty reflexive transitive
  antisymmetric TopRelStr, x be Element of T;
  reconsider T9 = T as Scott TopAugmentation of T by YELLOW_9:44;
  reconsider dx = downarrow x as Subset of T9;
  reconsider A = {x} as Subset of T9;
A1: downarrow x is closed by WAYBEL11:11;
  x <= x;
  then x in downarrow x by WAYBEL_0:17;
  then
A2: {x} c= downarrow x by ZFMISC_1:31;
  now
    let C be Subset of T9 such that
A3: A c= C;
    reconsider D = C as Subset of T9;
    assume C is closed;
    then
A4: D is lower by WAYBEL11:7;
    x in C by A3,ZFMISC_1:31;
    hence dx c= C by A4,WAYBEL11:6;
  end;
  hence thesis by A1,A2,YELLOW_8:8;
end;
