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

theorem
  y>=0-plane \ y=0-line is dense-in-itself Subset of Niemytzki-plane
proof
  the carrier of Niemytzki-plane = y>=0-plane by Def3;
  then reconsider A = y>=0-plane \ y=0-line as Subset of Niemytzki-plane by
XBOOLE_1:36;
  A is dense-in-itself
  proof
    let a be object;
    assume a in A;
    then reconsider x = a as Point of Niemytzki-plane;
    Cl (A\{x}) = the carrier of Niemytzki-plane by Th33;
    then x is_an_accumulation_point_of A;
    hence thesis by TOPGEN_1:def 3;
  end;
  hence thesis;
end;
