
theorem
  for R being non empty TopRelStr for A being Subset of R holds
  (for x being Element of R holds downarrow x = Cl {x}) implies
  for A being Subset of R st A is closed holds A is lower
proof
  let R be non empty TopRelStr, A be Subset of R;
  assume
A1: for x being Element of R holds downarrow x = Cl {x};
  let A be Subset of R such that
A2: A is closed;
  let x,y be Element of R such that
A3: x in A and
A4: y <= x;
  y in downarrow x by A4,WAYBEL_0:17;
  then
A5: y in Cl {x} by A1;
  {x} c= A
  by A3,TARSKI:def 1;
  hence thesis by A2,A5,PRE_TOPC:15;
end;
