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

theorem Th41:
  for A being Subset of Niemytzki-plane st A = y=0-line holds Der A is empty
proof
  consider BB being Neighborhood_System of Niemytzki-plane such that
A1: for x holds BB.(|[x,0]|) = {Ball(|[x,r]|,r) \/ {|[x,0]|} where r is
  Real: r > 0} and
  for x,y st y > 0 holds BB.(|[x,y]|) = {Ball(|[x,y]|,r) /\ y>=0-plane
  where r is Real: r > 0} by Def3;
  let A be Subset of Niemytzki-plane;
  assume that
A2: A = y=0-line and
A3: Der A is not empty;
  set a = the Element of Der A;
  a in Der A by A3;
  then reconsider a as Point of Niemytzki-plane;
A4: a in Der A by A3;
  a is_an_accumulation_point_of A by A3,TOPGEN_1:def 3;
  then
A5: a in Cl (A \ {a});
  the carrier of Niemytzki-plane = y>=0-plane by Def3;
  then a in y>=0-plane;
  then reconsider b = a as Point of TOP-REAL 2;
A6: a = |[b`1,b`2]| by EUCLID:53;
A7: Der A c= Cl A by TOPGEN_1:28;
  Cl A = A by A2,Th35;
  then
A8: b`2 = 0 by A4,A7,A2,A6,Th15;
  then
  BB.a = {Ball(|[b`1,r]|,r) \/ {|[b`1,0]|} where r is Real:
    r > 0 } by A1,A6;
  then Ball(|[b`1,1]|,1) \/ {b} in BB.a by A6,A8;
  then Ball(|[b`1,1]|,1) \/ {b} meets A \ {a} by A5,TOPGEN_2:9;
  then consider z being object such that
A9: z in Ball(|[b`1,1]|,1) \/ {b} and
A10: z in A \ {a} by XBOOLE_0:3;
A11: z in A by A10,ZFMISC_1:56;
  z <> a by A10,ZFMISC_1:56;
  then
A12: z in Ball(|[b`1,1]|,1) by A9,ZFMISC_1:136;
  reconsider z as Point of TOP-REAL 2 by A9;
A13: z = |[z`1,z`2]| by EUCLID:53;
  then z`2 = 0 by A2,A11,Th15;
  then
A14: z-|[b`1,1]| = |[z`1-b`1,0-1]| by A13,EUCLID:62;
A15: |[z`1-b`1,0-1]|`2 = 0-1 by EUCLID:52;
  |[z`1-b`1,0-1]|`1 = z`1-b`1 by EUCLID:52;
  then |.z-|[b`1,1]|.| = sqrt((z`1-b`1)^2+(-1)^2) by A14,A15,JGRAPH_1:30
    .= sqrt((z`1-b`1)^2+1^2);
  then
A16: |.z-|[b`1,1]|.| >= |.1.| by COMPLEX1:79;
  |.z-|[b`1,1]|.| < 1 by A12,TOPREAL9:7;
  then |.1.| < 1 by A16,XXREAL_0:2;
  hence contradiction by ABSVALUE:4;
end;
