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

theorem Th42:
  for A being Subset of y=0-line holds A is closed Subset of Niemytzki-plane
proof
  reconsider B = y=0-line as closed Subset of Niemytzki-plane by Th26;
  let A be Subset of y=0-line;
  A c= B;
  then reconsider A as Subset of Niemytzki-plane by XBOOLE_1:1;
  Der A c= Der B by TOPGEN_1:30;
  then Der A c= {} by Th41;
  then Der A = {};
  then Cl A = A \/ {} by TOPGEN_1:29;
  hence thesis;
end;
