Ltac lg_instance1 l A B :=
  assert(tempo_sg:= ex_point_lg l);
  match goal with
    |H: Q_Cong l |-  _ => assert(tempo_H:=H); apply (tempo_sg A) in tempo_H; ex_elim tempo_H B; exists B
  end;
  clear tempo_sg.Ltac lg_instance2 l A P B :=
  assert(tempo_sg:= ex_point_lg_out l);
  match goal with
    |H: A <> P |-  _ =>
                        match goal with
                           |HP : Q_Cong l |-  _ =>
                                               match goal with
                                                 |HQ : ~ Q_Cong_Null l |-  _ => assert(tempo_HQ:=HQ);
                                                                           apply (tempo_sg A P H HP) in tempo_HQ;
                                                                           ex_and tempo_HQ B
                                               end
                        end
  end;
  clear tempo_sg.

Lemma ex_points_lg_not_col : forall l P, Q_Cong l -> ~ Q_Cong_Null l -> exists A, exists B, l A B /\ ~Col A B P.