
theorem Th7:  :: p. 100, Remark 1.4 (i)
  for T being up-complete Scott non empty reflexive transitive TopRelStr,
  S being Subset of T holds S is closed iff S is directly_closed lower
proof
  let T be up-complete Scott non empty reflexive transitive TopRelStr,
  S be Subset of T;
  hereby
    assume S is closed;
    then S` is open;
    then reconsider A = S` as inaccessible upper Subset of T by Def4;
    A` is directly_closed lower;
    hence S is directly_closed lower;
  end;
  assume S is directly_closed lower;
  then reconsider S as directly_closed lower Subset of T;
  S` is open by Def4;
  hence thesis;
end;
