unfold eqo.
repeat split.
assumption.
assert(HH:=H0).
unfold Out in H0.
spliter.
apply out_col in HH.
intro.
apply H.
ColR.
intros.
assert(B=B2).
eapply l6_11_uniqueness.
3:apply H0.
unfold Out in H0.
spliter.
assumption.
2: apply H6.
unfold Out in H5.
spliter.
auto.
apply l6_6.
assumption.
apply cong_reflexivity.
subst B2.
apply l7_3 in H7.
subst M.
assert(Perp A B C1 A).
eapply perp_col.
intro.
subst B.
apply H.
Col.
apply H3.
apply out_col in H5.
assumption.
assert(A <> C).
apply perp_not_eq_2 in H1.
auto.
assert(A <> C1).
apply perp_not_eq_2 in H3.
auto.
assert(Col A C C1).
eapply perp2__col.
apply perp_sym.
apply perp_right_comm.
apply H1.
apply perp_sym.
Perp.
induction (eq_dec_points P C).
subst P.
apply l8_9 in H4.
induction H4.
subst C1.
unfold Midpoint in H9.
spliter.
left.
assumption.
subst C1.
apply perp_not_eq_2 in H3.
tauto.
Col.
assert(P <> C1).
intro.
subst P.
apply H14.
apply l8_9 in H2.
induction H2.
assumption.
subst C.
tauto.
Col.
assert(C1=C).
eapply per_id.
4:apply H4.
assumption.
auto.
auto.
assumption.
Col.
subst C1.
unfold Midpoint in H9.
spliter.
left.
assumption.
Qed.

Lemma out_preserves_eqo : forall A B P B' P', ~Col A B P -> Out A B B' -> Out A P P' -> eqo A B P A B' P'.
intros.