theorem
  ex K st not(ex P st L on P & K on P)
proof
  consider A,B such that
A1: A <> B and
A2: {A,B} on L by Def8;
  consider C,D such that
A3: not {A,B,C,D} is planar by A1,Th47;
  take K = Line(C,D);
  given P such that
A4: L on P and
A5: K on P;
  not {C,D,A,B} is planar by A3,ENUMSET1:73;
  then C <> D by Th16;
  then {C,D} on K by Def19;
  then
A6: {C,D} on P by A5,Th14;
  {A,B} on P by A2,A4,Th14;
  then {A,B} \/ {C,D} on P by A6,Th11;
  then {A,B,C,D} on P by ENUMSET1:5;
  hence thesis by A3;
end;
