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

theorem
  (y>=0-plane \ y=0-line) /\ product <*RAT,RAT*> is dense-in-itself
  Subset of Niemytzki-plane
proof
  (y>=0-plane \ y=0-line) /\ product <*RAT,RAT*> c= y>=0-plane \ y=0-line
  by XBOOLE_1:17;
  then reconsider
  A = (y>=0-plane \ y=0-line) /\ product <*RAT,RAT*> as Subset of
  Niemytzki-plane by Th25,XBOOLE_1:1;
  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 Th32;
    then x is_an_accumulation_point_of A;
    hence thesis by TOPGEN_1:def 3;
  end;
  hence thesis;
end;
