
theorem Th9:  :: p. 100, Remark 1.4 (ii)
  for T being complete Scott TopLattice, x being Element of T
  holds Cl {x} = downarrow x
proof
  let T be complete Scott TopLattice, x be Element of T;
  downarrow x is directly_closed by Th8;
  then
A1: downarrow x is closed by Th7;
  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 T such that
A3: {x} c= C;
    reconsider D = C as Subset of T;
    assume C is closed;
    then
A4: D is lower by Th7;
    x in C by A3,ZFMISC_1:31;
    hence downarrow x c= C by A4,Th6;
  end;
  hence thesis by A1,A2,YELLOW_8:8;
end;
