
theorem Th13:
  for T being up-complete Scott TopLattice, x being Element of T,
  A being upper Subset of T st not x in A
  holds (downarrow x)` is a_neighborhood of A
proof
  let T be up-complete Scott TopLattice, x be Element of T,
  A be upper Subset of T such that
A1: not x in A;
  (downarrow x)` is open by Th12;
  then
A2: Int (downarrow x)` = (downarrow x)` by TOPS_1:23;
  A misses downarrow x by A1,Th5;
  then A c= (downarrow x)` by SUBSET_1:23;
  hence thesis by A2,CONNSP_2:def 2;
end;
