theorem Th18:
  A <> B & {A,B} on L & not C on L implies not {A,B,C} is linear
proof
  assume that
A1: A <> B & {A,B} on L and
A2: not C on L;
  given K such that
A3: {A,B,C} on K;
  {A,B} \/ {C} on K by A3,ENUMSET1:3;
  then {A,B} on K by Th10;
  then L = K by A1,Def10;
  hence contradiction by A2,A3,Th2;
end;
