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

theorem Th32:
  for A being Subset of Niemytzki-plane st A = (y>=0-plane \
  y=0-line) /\ product <*RAT,RAT*> for x being set holds Cl (A \ {x}) = [#]
  Niemytzki-plane
proof
  let A be Subset of Niemytzki-plane;
  assume
A1: A = (y>=0-plane \ y=0-line) /\ product <*RAT,RAT*>;
  let s be set;
  thus Cl (A\{s}) c= [#] Niemytzki-plane;
  let x be object;
  assume x in [#] Niemytzki-plane;
  then reconsider a = x as Element of Niemytzki-plane;
  reconsider b = a as Element of y>=0-plane by Def3;
  consider BB being Neighborhood_System of Niemytzki-plane such that
A2: for x holds BB.(|[x,0]|) = {Ball(|[x,q]|,q) \/ {|[x,0]|} where q is
  Real: q > 0} and
A3: for x,y st y > 0 holds BB.(|[x,y]|) = {Ball(|[x,y]|,q) /\ y>=0-plane
  where q is Real: q > 0} by Def3;
A4: a = |[b`1,b`2]| by EUCLID:53;
  for U being set st U in BB.a holds U meets A\{s}
  proof
    let U be set;
    assume
A5: U in BB.a;
    per cases by A4,Th18;
    suppose
A6:   b`2 = 0;
      then
      BB.a = {Ball(|[b`1,q]|,q) \/ {|[b`1,0]|} where q is Real
      : q > 0} by A2,A4;
      then consider q being Real such that
A7:   U = Ball(|[b`1,q]|,q) \/ {a} and
A8:   q > 0 by A4,A5,A6;
      reconsider q as positive Real by A8;
      consider w1,v1 being Rational such that
A9:   |[w1,v1]| in Ball(|[b`1,q]|,q) and
A10:  |[w1,v1]| <> |[b`1,q]| by Th31;
A11:  |[w1,v1]| in U by A7,A9,XBOOLE_0:def 3;
      set q2 = |.|[w1,v1]|-|[b`1,q]|.|;
      |[w1,v1]|-|[b`1,q]| <> 0.TOP-REAL 2 by A10,RLVECT_1:21;
      then q2 <> 0 by EUCLID_2:42;
      then reconsider q2 as positive Real;
A12:  q2 < q by A9,TOPREAL9:7;
      consider w2,v2 being Rational such that
A13:  |[w2,v2]| in Ball(|[b`1,q]|,q2) and
      |[w2,v2]| <> |[b`1,q]| by Th31;
      |.|[w2,v2]|-|[b`1,q]|.| < q2 by A13,TOPREAL9:7;
      then |.|[w2,v2]|-|[b`1,q]|.| < q by A12,XXREAL_0:2;
      then
A14:  |[w2,v2]| in Ball(|[b`1,q]|,q) by TOPREAL9:7;
      then
A15:  |[w2,v2]| in U by A7,XBOOLE_0:def 3;
A16:  Ball(|[b`1,q]|,q) misses y=0-line by Th21;
      Ball(|[b`1,q]|,q) c= y>=0-plane by Th20;
      then
A17:  Ball(|[b`1,q]|,q) c= y>=0-plane \ y=0-line by A16,XBOOLE_1:86;
A18:  v1 in RAT by RAT_1:def 2;
      w1 in RAT by RAT_1:def 2;
      then |[w1,v1]| in product <*RAT, RAT*> by A18,FINSEQ_3:124;
      then
A19:  |[w1,v1]| in A by A17,A9,A1,XBOOLE_0:def 4;
A20:  s = |[w1,v1]| or s <> |[w1,v1]|;
A21:  v2 in RAT by RAT_1:def 2;
      w2 in RAT by RAT_1:def 2;
      then |[w2,v2]| in product <*RAT,RAT*> by A21,FINSEQ_3:124;
      then
A22:  |[w2,v2]| in A by A17,A14,A1,XBOOLE_0:def 4;
      |[w2,v2]| <> |[w1,v1]| by A13,TOPREAL9:7;
      then |[w1,v1]| in A\{s} or |[w2,v2]| in A\{s} by A20,A19,A22,ZFMISC_1:56;
      hence thesis by A11,A15,XBOOLE_0:3;
    end;
    suppose
A23:  b`2 > 0;
      then BB.a = {Ball(|[b`1,b`2]|,q) /\ y>=0-plane
     where q is Real: q > 0} by A3,A4;
      then consider q being Real such that
A24:  U = Ball(b,q) /\ y>=0-plane and
A25:  q > 0 by A4,A5;
      reconsider q, b2 = b`2 as positive Real by A23,A25;
      reconsider q1 = min(q,b2) as positive Real by XXREAL_0:def 9;
      consider w1,v1 being Rational such that
A26:  |[w1,v1]| in Ball(b,q1) and
A27:  |[w1,v1]| <> b by A4,Th31;
A28:  v1 in RAT by RAT_1:def 2;
      set q2 = |.|[w1,v1]|-b.|;
      |[w1,v1]|-b <> 0.TOP-REAL 2 by A27,RLVECT_1:21;
      then q2 <> 0 by EUCLID_2:42;
      then reconsider q2 as positive Real;
A29:  q2 < q1 by A26,TOPREAL9:7;
A30:  q1 <= b`2 by XXREAL_0:17;
      then
A31:  Ball(b,q1) c= y>=0-plane by A4,Th20;
      Ball(b,q1) misses y=0-line by A30,A4,Th21;
      then
A32:  Ball(b,q1) c= y>=0-plane \ y=0-line by A31,XBOOLE_1:86;
      w1 in RAT by RAT_1:def 2;
      then |[w1,v1]| in product <*RAT, RAT*> by A28,FINSEQ_3:124;
      then
A33:  |[w1,v1]| in A by A32,A26,A1,XBOOLE_0:def 4;
A34:  s = |[w1,v1]| or s <> |[w1,v1]|;
      consider w2,v2 being Rational such that
A35:  |[w2,v2]| in Ball(b,q2) and
      |[w2,v2]| <> b by A4,Th31;
A36:  |[w2,v2]| <> |[w1,v1]| by A35,TOPREAL9:7;
      |.|[w2,v2]|-b.| < q2 by A35,TOPREAL9:7;
      then
A37:  |.|[w2,v2]|-b.| < q1 by A29,XXREAL_0:2;
      then
A38:  |[w2,v2]| in Ball(b,q1) by TOPREAL9:7;
A39:  q1 <= q by XXREAL_0:17;
      then |.|[w2,v2]|-b.| < q by A37,XXREAL_0:2;
      then |[w2,v2]| in Ball(b,q) by TOPREAL9:7;
      then
A40:  |[w2,v2]| in U by A24,A38,A31,XBOOLE_0:def 4;
A41:  v2 in RAT by RAT_1:def 2;
      w2 in RAT by RAT_1:def 2;
      then |[w2,v2]| in product <*RAT,RAT*> by A41,FINSEQ_3:124;
      then |[w2,v2]| in A by A32,A38,A1,XBOOLE_0:def 4;
      then
A42:  |[w1,v1]| in A\{s} or |[w2,v2]| in A\{s} by A34,A36,A33,ZFMISC_1:56;
      q2 < q by A39,A29,XXREAL_0:2;
      then |[w1,v1]| in Ball(b,q) by TOPREAL9:7;
      then |[w1,v1]| in U by A24,A26,A31,XBOOLE_0:def 4;
      hence thesis by A42,A40,XBOOLE_0:3;
    end;
  end;
  hence thesis by TOPGEN_2:10;
end;
