reserve x,y for Real,
  u,v,w for set,
  r for positive Real;

theorem Th33:
  for A being Subset of Niemytzki-plane st A = y>=0-plane \
  y=0-line for x being set holds Cl (A \ {x}) = [#] Niemytzki-plane
proof
  let A be Subset of Niemytzki-plane;
  assume
A1: A = y>=0-plane \ y=0-line;
  let s be set;
  reconsider B = A /\ product <*RAT,RAT*> as Subset of Niemytzki-plane;
  thus Cl (A\{s}) c= [#] Niemytzki-plane;
  B\{s} c= A\{s} by XBOOLE_1:17,33;
  then Cl (B\{s}) c= Cl (A\{s}) by PRE_TOPC:19;
  hence thesis by A1,Th32;
end;
