unfold proj.
split; auto.
induction (col_dec A B T).
right.
split; auto.
assert(Col T P A).
ColR.
assert(Col T P B).
ColR.
induction(eq_dec_points A P).
subst P.
assert(T=A \/ B=A).
apply l8_9; auto.
induction H6.
auto.
subst B.
tauto.
assert(T=P \/ A=P).
apply l8_9; auto.
induction H7.
auto.
subst P.
tauto.
left.
induction (eq_dec_points A P).
subst P.
repeat split; auto.
apply per_perp_in in H1.
apply perp_in_comm in H1.
apply perp_in_perp_bis in H1.
induction H1.
Perp.
apply perp_not_eq_1 in H1.
tauto.
intro.
subst T.
contradiction.
assumption.
repeat split; auto.
apply per_perp_in in H0.
apply perp_in_comm in H0.
apply perp_in_perp_bis in H0.
induction H0.
eapply perp_col.
assumption.
eauto with perp.
Col.
apply perp_not_eq_1 in H0.
tauto.
intro.
subst T.
contradiction.
auto.
Qed.
Definition eqo := fun A B P A1 B1 P1 => ~Col A B P /\ ~Col A1 B1 P1 /\
forall C C1 B2 M B' C' K,
Perp A B C A  -> Per P C A -> Perp A1 B1 C1 A1 -> Per P1 C1 A1 ->
Out A1 B1 B2 -> Cong A B A1 B2 ->
Midpoint M A A1 -> Midpoint M B2 B' -> Midpoint M C1 C' -> Midpoint K B B' ->
Bet C A C' \/ OS A K C C'.
Definition eq_o := fun A B P A1 B1 P1 => ~Col A B P /\ ~Col A1 B1 P1 /\
forall C C1 B2 M B' C' K,
Perp A B C A -> proj P A C C -> Perp A1 B1 C1 A1 -> proj P1 A1 C1 C1 ->
Out A1 B1 B2 -> Cong A B A1 B2 ->
Midpoint M A A1 -> Midpoint M B2 B' -> Midpoint M C1 C' -> Midpoint K B B' ->
Bet C A C' \/ OS A K C C'.

Lemma eqo_eq_o : forall A B P A1 B1 P1, eqo A B P A1 B1 P1 -> eq_o A B P A1 B1 P1.
intros.