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

theorem
  Niemytzki-plane is separable
proof
  reconsider A = (y>=0-plane \ y=0-line) /\ product <*RAT,RAT*> as dense
  Subset of Niemytzki-plane by Th36;
A1: card A c= card product <*RAT, RAT*> by CARD_1:11,XBOOLE_1:17;
  density Niemytzki-plane c= card A by TOPGEN_1:def 12;
  then density Niemytzki-plane c= card product <*RAT,RAT*> by A1;
  hence density Niemytzki-plane c= omega by Th8,CARD_4:6,TOPGEN_3:17;
end;
