reserve Y for TopStruct;

theorem
  for A being Subset of Y holds A is discrete implies for x being Point
  of Y st x in A ex F being Subset of Y st F is closed & A /\ F = {x}
proof
  let A be Subset of Y;
  assume
A1: A is discrete;
  let x be Point of Y;
  assume
A2: x in A;
  then reconsider Y9 = Y as non empty TopStruct;
  reconsider B = {x} as Subset of Y9 by ZFMISC_1:31;
  reconsider A9 = A as Subset of Y9;
  {x} c= A9 by A2,ZFMISC_1:31;
  then consider F being Subset of Y such that
A3: F is closed and
A4: A9 /\ F = B by A1;
  take F;
  thus thesis by A3,A4;
end;
