
theorem

:: Remark before 1.3., p. 143
  for T being lower TopSpace-like reflexive transitive non empty
  TopRelStr  for x being Point of T holds Cl {x} = uparrow x
proof
  let T be lower TopSpace-like reflexive transitive non empty TopRelStr;
  let x be Point of T;
  reconsider y = x as Element of T;
  y <= y;
  then x in uparrow x by WAYBEL_0:18;
  then {x} c= uparrow x by ZFMISC_1:31;
  hence Cl {x} c= uparrow x by Th4,TOPS_1:5;
  Cl {x} is upper by Th6;
  then
A1: uparrow Cl {x} c= Cl {x} by WAYBEL_0:24;
  uparrow {x} c= uparrow Cl {x} by PRE_TOPC:18,YELLOW_3:7;
  hence thesis by A1;
end;
