theorem
  ex A st A on P & not A on L
proof
  consider A,B such that
A1: A <> B and
A2: {A,B} on L by Def8;
  consider C such that
A3: C on P and
A4: not {A,B,C} is linear by A1,Th44;
  take C;
  thus C on P by A3;
  C on L implies {A,B} \/ {C} on L by A2,Th8;
  then C on L implies {A,B,C} on L by ENUMSET1:3;
  hence thesis by A4;
end;
