
theorem Th14:
  for R being up-complete non empty reflexive transitive antisymmetric
  RelStr for T being Scott TopAugmentation of R, 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 R be up-complete non empty reflexive transitive antisymmetric RelStr,
  T be Scott TopAugmentation of R, x be Element of T,
  A be upper Subset of T such that
A1: not x in A;
  downarrow x is closed by WAYBEL11:11;
  then (downarrow x)` is open;
  then
A2: Int (downarrow x)` = (downarrow x)` by TOPS_1:23;
  A misses downarrow x by A1,WAYBEL11:5;
  then A c= (downarrow x)` by SUBSET_1:23;
  hence thesis by A2,CONNSP_2:def 2;
end;
