theorem
  not(ex P st K on P & L on P) implies K <> L
proof
  assume that
A1: not(ex P st K on P & L on P) and
A2: K = L;
  consider A,B such that
A3: A <> B and
A4: {A,B} on K by Def8;
A5: A on K & B on K by A4,Th1;
  consider C,D such that
A6: C <> D and
A7: {C,D} on L by Def8;
  C on K by A2,A7,Th1;
  then {A,B,C} on K by A5,Th2;
  then {A,B,C} is linear;
  then {A,B,C,D} is planar by Th17;
  then consider Q such that
A8: {A,B,C,D} on Q;
  C on Q & D on Q by A8,Th5;
  then {C,D} on Q by Th3;
  then
A9: L on Q by A6,A7,Def14;
  A on Q & B on Q by A8,Th5;
  then {A,B} on Q by Th3;
  then K on Q by A3,A4,Def14;
  hence contradiction by A1,A9;
end;
