intros.
induction(eq_dec_points A A').
subst A'.
eapply (per_preserves_bet_aux1 P Q A B C B' C'); assumption.
induction(eq_dec_points B B').
subst B'.
eapply (per_preserves_bet_aux2 P Q A B C A' C'); assumption.
induction(eq_dec_points C C').
subst C'.
assert(HH:=(per_preserves_bet_aux1 P Q C B A B' A')).
apply between_symmetry.
apply HH; try assumption.
Between.
induction (eq_dec_points A B).
subst B.
assert(A'=B').
eapply (per_id A _ _ P); try auto.
ColR.
subst B'.
apply between_trivial2.
induction (eq_dec_points C B).
subst B.
assert(C'=B').
eapply (per_id C _ _ P); try auto.
ColR.
subst B'.
apply between_trivial.
assert(A <> C).
intro.
subst C.
apply between_identity in H0.
contradiction.
assert(Perp B' B' B' P \/ Perp B B' B' P).
eapply perp_in_perp_bis.
apply per_perp_in.
auto.
auto.
assumption.
induction H16.
apply perp_not_eq_1 in H16.
tauto.
assert(Perp C' C' C' P \/ Perp C C' C' P).
eapply perp_in_perp_bis.
apply per_perp_in.
auto.
auto.
assumption.
induction H17.
apply perp_not_eq_1 in H17.
tauto.
assert(Perp A' A' A' P \/ Perp A A' A' P).
eapply perp_in_perp_bis.
apply per_perp_in.
auto.
auto.
assumption.
induction H18.
apply perp_not_eq_1 in H18.
tauto.
assert(Par B B' C C').
eapply l12_9_2D.
apply H16.
apply perp_sym.
apply perp_left_comm.
eapply perp_col.
auto.
apply perp_left_comm.
apply perp_sym.
apply H17.
eapply (col_transitivity_1 _ Q);
auto.
assert(Par B B' A A').
eapply l12_9_2D.
apply H16.
apply perp_sym.
apply perp_left_comm.
eapply perp_col.
auto.
apply perp_left_comm.
apply perp_sym.
apply H18.
eapply (col_transitivity_1 _ Q);
auto.
(**************************************************************************************************)
(*induction(eq_dec_points A C). subst C. apply between_identity in H0. subst B. assert(A' = B'). eapply per_id. 4: apply H1. auto. auto. auto. auto. ColR. subst B'. apply between_trivial2.*)
assert(Col A B C).
apply bet_col.
assumption.
induction(eq_dec_points A' C').
subst C'.
assert(Col A C A').
eapply per2__col.
apply H1.
auto.
assumption.
assert(Col A B A').
eapply (col_transitivity_1 _ C); try auto.
Col.
assert(Per B A' P).
apply l8_2.
eapply (per_col _ _ A).
auto.
apply l8_2.
assumption.
Col.
assert(A' = B').
eapply (per_id B _ _ P); try auto.
intro.
subst A'.
apply par_right_comm in H20.
apply par_col in H20.
assert(Per P B B').
eapply (per_col _ _ A).
auto.
apply l8_2.
assumption.
Col.
apply H11.
eapply l8_7.
apply H25.
apply l8_2.
assumption.
ColR.
subst B'.
apply between_trivial.
(************************************* General case *********************************)
assert(Par A A' C C').
eapply par_trans.
apply par_symmetry.
apply H20.
assumption.
assert(A' <> B').
intro.
subst B'.
apply H22.
assert(Col A B A').
eapply per2__col.
apply H1.
auto.
assumption.
assert(Col C A A').
ColR.
assert(Per P A' C).
eapply (per_col _ _ A).
auto.
apply l8_2.
assumption.
Col.
apply l8_2 in H26.
eapply (per_id).
4: apply H26.
intro.
subst C.
apply perp_comm in H17.
apply perp_not_col in H17.
apply H17.
ColR.
auto.
auto.
assumption.
ColR.
assert(C' <> B').
intro.
subst B'.
apply H22.
assert(Col B C C').
eapply per2__col.
apply H3.
auto.
assumption.
assert(Col A C C').
ColR.
assert(Per P C' A).
eapply (per_col _ _ C).
auto.
apply l8_2.
assumption.
Col.
apply l8_2 in H27.
eapply (per_id).
5: apply H27.
auto.
auto.
auto.
auto.
ColR.
assert(Par_strict B B' C C').
induction H19.
assumption.
spliter.
apply False_ind.
apply H25.
eapply per_id.
5: apply H3.
intro.
subst B.
apply perp_comm in H16.
apply perp_not_col in H16.
apply H16.
ColR.
auto.
auto.
apply l8_2.
eapply (per_col _ _ C).
auto.
apply l8_2.
assumption.
Col.
ColR.
assert(Par_strict B B' A A').
induction H20.
assumption.
spliter.
apply False_ind.
apply H24.
eapply per_id.
5: apply H3.
intro.
subst B.
apply perp_comm in H16.
apply perp_not_col in H16.
apply H16.
ColR.
auto.
auto.
apply l8_2.
eapply (per_col _ _ A).
auto.
apply l8_2.
assumption.
Col.
ColR.
assert(OS B B' A A').
apply l12_6.
assumption.
assert(OS B B' C C').
apply l12_6.
assumption.
assert(TS B B' A C).
unfold TS.
repeat split; auto.
intro.
apply H24.
eapply per_id.
4:apply H1.
auto.
auto.
auto.
apply l8_2.
eapply per_col.
2: apply l8_2.
2:apply H3.
auto.
Col.
ColR.
intro.
apply H25.
eapply per_id.
4: apply H5.
auto.
auto.
auto.
apply l8_2.
eapply per_col.
2:apply l8_2.
2: apply H3.
auto.
Col.
ColR.
exists B.
split.
Col.
assumption.
assert(TS B B' A' C).
eapply l9_8_2.
apply H30.
assumption.
assert(TS B B' A' C').
apply l9_2.
eapply l9_8_2.
apply l9_2.
apply H31.
assumption.
unfold TS in H32.
spliter.
ex_and H34 BB.
assert(BB = B').
eapply l6_21.
apply not_col_permutation_1.
apply H33.
4:apply col_permutation_2.
4:apply bet_col.
4:apply H35.
auto.
Col.
Col.
ColR.
subst BB.
assumption.
Qed.

Lemma ex_col : forall A B C, A <> B /\ A <> C /\ B <> C -> Col A B C -> exists D, Col A B D /\ A <> D /\ B <> D /\ C <> D.
intros.