
theorem Th8:
  for T being up-complete non empty reflexive transitive antisymmetric
  TopRelStr, x being Element of T holds downarrow x is directly_closed
proof
  let T be up-complete
  non empty reflexive transitive antisymmetric TopRelStr, x be Element of T;
  downarrow x is directly_closed
  proof
    let D be non empty directed Subset of T;
    assume
A1: D c= downarrow x;
    ex a being Element of T st a is_>=_than D &
    for b being Element of T st b is_>=_than D holds a <= b
    by WAYBEL_0:def 39;
    then
A2: ex_sup_of D,T by YELLOW_0:15;
    x is_>=_than D
    by A1,WAYBEL_0:17;
    then sup D <= x by A2,YELLOW_0:30;
    hence thesis by WAYBEL_0:17;
  end;
  hence thesis;
end;
