assert(A <> B /\ A <> B').
unfold Out in H0.
spliter.
split; auto.
spliter.
unfold eq_o.
repeat split.
assumption.
apply out_col in H0.
intro.
apply H.
ColR.
intros.
apply l7_3 in H9.
subst M.
assert(A <> C).
apply perp_not_eq_2 in H3.
auto.
assert(A <> C1).
apply perp_not_eq_2 in H5.
auto.
assert(Perp A B C1 A).
eapply perp_col.
assumption.
apply H5.
apply out_col in H0.
Col.
assert(Col A C C1).
eapply perp2__col.
apply perp_sym.
apply perp_comm.
apply H3.
apply perp_comm.
Perp.
assert(proj B A C1 A).
unfold proj.
split; auto.
left.
repeat split.
intro.
apply perp_not_col in H14.
apply H14.
Col.
apply perp_comm.
Perp.
Col.
assert(proj P A C C1).
eapply proj_col_proj.
apply H6.
assumption.
Col.
assert(C= C1).
eapply proj_uniqueness.
2:apply H17.
assumption.
subst C1.
clear H17 H6 H14.
left.
apply midpoint_bet.
assumption.
Qed.

Lemma cong_identity_inv : forall A B C, A <> B -> ~Cong A B C C.
intros.