unfold eq_o in H.
spliter.
assert(A <> B).
intro.
subst B.
apply H.
Col.
assert(HH:=ex_per_cong B A A X A B).
assert(exists P : Tpoint, Per P A B /\ Cong P A A B /\ OS B A P X).
apply HH;
auto.
Col.
intro.
apply H.
Col.
clear HH.
ex_and H3 T.
assert(A <> T).
intro.
subst T.
apply cong_symmetry in H4.
apply cong_identity in H4.
contradiction.
assert(Perp T A A B).
apply per_perp_in in H3.
apply perp_in_perp_bis in H3.
induction H3.
apply perp_not_eq_1 in H3.
tauto.
assumption.
auto.
assumption.
assert(HH:=proj_exists A T X H6).
ex_and HH PX.
assert(HH:=proj_exists A T Y H6).
ex_and HH PY.
prolong B A B' B A.
prolong PY A C' PY A.
assert(Col PX A T).
eapply proj_col.
apply H8.
assert(Col PY A T).
eapply proj_col.
apply H9.
assert(PX <> A).
intro.
subst PX.
apply proj_per in H8.
spliter.
apply H.
apply col_permutation_2.
eapply per2__col.
apply l8_2.
apply H3.
assumption.
assumption.
assumption.
assert(PY <> A).
intro.
subst PY.
apply proj_per in H9.
spliter.
apply H0.
apply col_permutation_2.
eapply per2__col.
apply l8_2.
apply H3.
assumption.
assumption.
assumption.
assert(~Col T A B).
apply per_not_col in H3.
assumption.
auto.
assumption.
assert(Col A PX PY).
ColR.
assert(~Col PX A B).
intro.
apply H18.
ColR.
assert(~Col PY A B).
intro.
apply H18.
ColR.
assert(A <> C').
intro.
subst C'.
apply cong_symmetry in H13.
apply cong_identity in H13.
contradiction.
assert(TS A B PY C').
unfold TS.
repeat split; auto.
intro.
apply H21.
apply bet_col in H12.
ColR.
exists A.
split.
Col.
assumption.
assert(HH:= H1 PX PY B A B' C' A).
assert(Bet PX A C' \/ OS A A PX C').
apply HH;
clear HH; clear H1.
apply per_perp_in in H3.
apply perp_in_comm in H3.
apply perp_in_perp_bis in H3.
induction H3.
apply perp_comm.
apply perp_sym.
eapply perp_col.
auto.
apply H1.
Col.
apply perp_not_eq_1 in H1.
tauto.
auto.
assumption.
eapply proj_col_proj.
apply H8.
auto.
Col.
apply per_perp_in in H3.
apply perp_in_comm in H3.
apply perp_in_perp_bis in H3.
induction H3.
apply perp_comm.
apply perp_sym.
eapply perp_col.
auto.
apply H1.
Col.
apply perp_not_eq_1 in H1.
tauto.
auto.
assumption.
eapply proj_col_proj.
apply H9.
auto.
Col.
apply out_trivial.
auto.
apply cong_reflexivity.
apply l7_3_2.
unfold Midpoint.
apply cong_symmetry in H11.
split;auto.
unfold Midpoint.
apply cong_symmetry in H13.
split;auto.
unfold Midpoint.
apply cong_symmetry in H11.
split;auto.
clear H1 HH.
assert(TS A B PX C').
unfold TS.
repeat split; auto.
intro.
apply H21.
apply bet_col in H12.
ColR.
exists A.
induction H24.
split.
Col.
assumption.
unfold OS in H1.
ex_and H1 U.
unfold TS in H1.
spliter.
assert_diffs; tauto.
unfold proj in H8.
spliter.
induction H25.
spliter.
assert(Par A B X PX).
apply l12_9_2D with T A.
apply perp_sym.
apply H7.
apply perp_right_comm.
Perp.
assert(Par_strict A B X PX).
induction H28.
assumption.
spliter.
apply False_ind.
apply H.
ColR.
unfold proj in H9.
spliter.
induction H30.
spliter.
assert(Par A B Y PY).
apply l12_9_2D with T A.
apply perp_sym.
apply H7.
apply perp_right_comm.
Perp.
assert(Par_strict A B Y PY).
induction H33.
assumption.
spliter.
apply False_ind.
apply H0.
ColR.
apply l12_6 in H29.
apply l12_6 in H34.
assert(TS A B X C').
eapply l9_8_2.
apply H1.
apply one_side_symmetry.
assumption.
assert(TS A B Y C').
eapply l9_8_2.
apply H23.
apply one_side_symmetry.
assumption.
eapply l9_8_1.
apply H35.
assumption.
spliter.
subst Y.
apply l12_6 in H29.
eapply one_side_transitivity.
apply H29.
eapply l9_8_1.
apply H1.
assumption.
spliter.
subst X.
unfold proj in H9.
spliter.
induction H26.
spliter.
assert(Par A B Y PY).
apply l12_9_2D with T A.
apply perp_sym.
apply H7.
apply perp_right_comm.
Perp.
assert(Par_strict A B Y PY).
induction H29.
assumption.
spliter.
apply False_ind.
apply H0.
ColR.
apply l12_6 in H30.
eapply one_side_transitivity.
eapply l9_8_1.
apply H1.
apply H23.
apply one_side_symmetry.
assumption.
spliter.
subst Y.
eapply l9_8_1.
apply H1.
assumption.
Qed.

Lemma eqo_one_side : forall A B X Y, eqo A B X A B Y -> OS A B X Y.
intros.