intros.
induction (eq_dec_points A A').
subst A'.
assert(Col A B C').
apply (col3 P Q); try auto.
assert(Col P A B).
eapply (col_transitivity_1 _ Q); try auto.
assert(Col Q A B).
eapply (col_transitivity_1 _ P).
auto.
Col.
Col.
induction(eq_dec_points A B).
subst B.
apply between_trivial2.
assert(Col P Q C).
eapply col3.
apply H12.
Col.
Col.
apply bet_col.
assumption.
assert(Col P C' C).
eapply col_transitivity_1.
apply H.
assumption.
assumption.
assert(P=C' \/ C=C').
eapply l8_9.
apply l8_2.
assumption.
assumption.
induction H15.
subst C'.
tauto.
subst C'.
assumption.
induction(eq_dec_points A C).
subst C.
apply between_identity in H0.
subst B.
apply False_ind.
apply H9.
apply l8_9 in H1.
induction H1.
assumption.
subst A'.
tauto.
ColR.
assert(~Col P Q A).
intro.
assert (Col A A' P).
ColR.
apply l8_9 in H1.
induction H1;tauto.
assumption.
assert(A <> B).
intro.
subst B.
contradiction.
induction (eq_dec_points C C').
subst C'.
assert(Col A' B C).
eapply (col3 P Q ); assumption.
assert(Col C A' P).
ColR.
assert(B=C).
eapply l6_21.
apply H11.
apply H12.
Col.
Col.
Col.
Col.
subst C.
apply between_trivial.
induction(eq_dec_points A' C').
subst C'.
assert(Per  A A' B).
eapply per_col.
2:apply H1.
auto.
ColR.
assert(Per C A' B).
eapply per_col.
2:apply H4.
auto.
ColR.
eapply l8_6 in H14.
subst B.
apply between_trivial.
apply H15.
Between.
assert(HH:=ex_per_cong P Q B A P Q).
assert(exists T, Per T B P /\ Cong T B P Q /\ OS P Q T A).
apply HH; auto.
ex_and H15 T.
clear HH.
assert(Perp B B B P \/ Perp T B B P).
eapply perp_in_perp_bis.
apply per_perp_in.
intro.
subst T.
apply cong_symmetry in H16.
apply cong_identity in H16.
contradiction.
auto.
auto.
induction H18.
apply perp_not_eq_1 in H18.
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 H19.
apply perp_not_eq_1 in H19.
tauto.
assert(Par T B C C').
eapply l12_9_2D.
apply H18.
apply perp_sym.
apply perp_left_comm.
eapply perp_col.
auto.
apply perp_left_comm.
apply perp_sym.
apply H19.
eapply (col_transitivity_1 _ Q);
auto.
assert(Perp A' A' A' P \/ Perp A A' A' P).
eapply perp_in_perp_bis.
apply per_perp_in.
auto.
auto.
assumption.
induction H21.
apply perp_not_eq_1 in H21.
tauto.
assert(Par T B A A').
eapply l12_9_2D.
apply H18.
apply perp_sym.
apply perp_left_comm.
eapply perp_col.
auto.
apply perp_left_comm.
apply perp_sym.
apply H21.
eapply (col_transitivity_1 _ Q);
auto.
assert(Par_strict T B C C').
induction H20.
assumption.
spliter.
apply False_ind.
apply H14.
eapply per_id.
4:apply H1.
assumption.
auto.
auto.
apply l8_2.
eapply per_col.
2: apply l8_2.
2:apply H4.
auto.
apply col_permutation_2.
eapply col_transitivity_1.
2:apply bet_col.
2:apply between_symmetry.
apply bet_col in H0.
2: apply H0.
assert(~Col P Q C).
intro.
assert (Col C C' P).
ColR.
apply l8_9 in H4.
induction H1;tauto.
assumption.
intro.
subst C.
contradiction.
Col.
ColR.
assert(Par_strict T B A A').
induction H22.
assumption.
spliter.
apply False_ind.
apply H14.
eapply sym_equal.
eapply per_id.
4:apply H4.
assumption.
auto.
auto.
apply l8_2.
eapply per_col.
2: apply l8_2.
2:apply H1.
auto.
apply col_permutation_2.
eapply col_transitivity_1.
2:apply bet_col.
apply bet_col in H0.
2: apply H0.
assumption.
Col.
ColR.
assert(OS T B C C').
apply l12_6.
assumption.
assert(OS T B A A').
apply l12_6.
assumption.
assert(TS T B A C).
unfold TS.
repeat split.
intro.
unfold Par_strict in H24.
spliter.
apply H30.
exists A.
split.
assumption.
Col.
intro.
unfold Par_strict in H23.
spliter.
apply H30.
exists C.
split.
assumption.
Col.
exists B.
split.
Col.
assumption.
assert(TS T B C' A).
eapply l9_8_2.
apply l9_2.
apply H27.
assumption.
assert(TS T B A' C').
eapply l9_8_2.
apply l9_2.
apply H28.
assumption.
unfold TS in H29.
spliter.
ex_and H31 BB.
assert(B=BB).
eapply (l6_21 T B A' C'); Col.
ColR.
subst BB.
assumption.
Qed.

Lemma par_col : forall A B C, Par A B A C -> Col A B C.
intros.