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

theorem
  Niemytzki-plane is Tychonoff
proof
  set X = Niemytzki-plane;
  consider B being Neighborhood_System of X such that
A1: for x holds B.(|[x,0]|) = {Ball(|[x,r]|,r) \/ {|[x,0]|} where r is
  Real: r > 0} and
A2: for x,y st y > 0 holds B.(|[x,y]|) = {Ball(|[x,y]|,r) /\ y>=0-plane
  where r is Real: r > 0} by Def3;
  reconsider uB = Union B as prebasis of X by YELLOW_9:27;
A3: the carrier of X = y>=0-plane by Def3;
  now
    let x be Point of X;
    let V be Subset of X;
    assume that
A4: x in V and
A5: V in uB;
    V is open by A5,TOPS_2:def 1;
    then consider V9 being Subset of X such that
A6: V9 in B.x and
A7: V9 c= V by A4,YELLOW_8:def 1;
    x in y>=0-plane by A3;
    then reconsider x9 = x as Point of TOP-REAL 2;
A8: x = |[x9`1,x9`2]| by EUCLID:53;
    per cases by A3,A8,Th18;
    suppose
A9:   x9`2 = 0;
      then B.x = {Ball(|[x9`1,r]|,r) \/ {|[x9`1,0]|}
         where r is Real: r > 0} by A1,A8;
      then consider r being Real such that
A10:  V9 = Ball(|[x9`1,r]|,r)\/{|[x9`1,0]|} and
A11:  r > 0 by A6;
      consider f being continuous Function of Niemytzki-plane, I[01] such that
A12:  f.(|[x9`1,0]|) = 0 and
A13:  for a,b being Real holds (|[a,b]| in V9` implies f.(|[a,
b]|) = 1) & (|[a,b]| in V9\{|[x9`1,0]|} implies f.(|[a,b]|) = |.|[x9`1,0]|-|[a,
      b]|.| ^2/(2*r*b)) by A10,A11,Th76;
      take f;
      thus f.x = 0 by A9,A12,EUCLID:53;
      thus f.:V` c= {1}
      proof
        let u be object;
        assume u in f.:V`;
        then consider b being Point of X such that
A14:    b in V` and
A15:    u = f.b by FUNCT_2:65;
        b in y>=0-plane by A3;
        then reconsider c = b as Element of TOP-REAL 2;
A16:    V` c= V9` by A7,SUBSET_1:12;
        b = |[c`1,c`2]| by EUCLID:53;
        then u = 1 by A16,A14,A13,A15;
        hence thesis by TARSKI:def 1;
      end;
    end;
    suppose
A17:  x9`2 > 0;
      then B.x = {Ball(|[x9`1,x9`2]|,r) /\ y>=0-plane
      where r is Real: r > 0} by A2,A8;
      then consider r being Real such that
A18:  V9 = Ball(|[x9`1,x9`2]|,r)/\y>=0-plane and
A19:  r > 0 by A6;
      consider f being continuous Function of Niemytzki-plane, I[01] such that
A20:  f.(|[x9`1,x9`2]|) = 0 and
A21:  for a,b being Real holds (|[a,b]| in V9` implies f.(|[a,
b]|) = 1) & (|[a,b]| in V9 implies f.(|[a,b]|) = |.|[x9`1,x9`2]|-|[a,b]|.|/r)
      by A17,A18,A19,Th81;
      take f;
      thus f.x = 0 by A20,EUCLID:53;
      thus f.:V` c= {1}
      proof
        let u be object;
        assume u in f.:V`;
        then consider b being Point of X such that
A22:    b in V` and
A23:    u = f.b by FUNCT_2:65;
        b in y>=0-plane by A3;
        then reconsider c = b as Element of TOP-REAL 2;
A24:    V` c= V9` by A7,SUBSET_1:12;
        b = |[c`1,c`2]| by EUCLID:53;
        then u = 1 by A24,A22,A21,A23;
        hence thesis by TARSKI:def 1;
      end;
    end;
  end;
  hence thesis by Th52,Th82;
end;
