theorem
  ex L st not A on L & L on P
proof
  consider B,C such that
A1: {B,C} on P and
A2: not {A,B,C} is linear by Th46;
  consider L such that
A3: {B,C} on L by Def9;
  take L;
  A on L implies {B,C} \/ {A} on L by A3,Th8;
  then A on L implies {B,C,A} on L by ENUMSET1:3;
  then A on L implies {A,B,C} on L by ENUMSET1:59;
  hence not A on L by A2;
  not {B,C,A} is linear by A2,ENUMSET1:59;
  then B <> C by Th15;
  hence thesis by A1,A3,Def14;
end;
