Proof with Perp.
intros.
cases_equality B C.
unfold Cong_3 in H0;intuition.
treat_equalities...
assert (B'<>C').
unfold Cong_3 in H0;intuition.
treat_equalities...
unfold Per in H.
DecompExAnd H D.
let_symmetric D' B' C'.
assert (OFSC C B D A C' B' D' A').
unfold OFSC,Cong_3,is_midpoint in *;intuition.
eapply cong_transitivity with (C:=B) (D:=C);sTarski.
eapply cong_transitivity with (C:=B') (D:=C');sTarski.
assert (Cong D A D' A').
eapply five_segments_with_def;eauto.
unfold OFSC,Cong_3,is_midpoint in *;intuition.
clean_duplicated_hyps.
assert (Cong A' C' A' D').
eapply cong_transitivity with (C:=A) (D:=C);sTarski.
eapply cong_transitivity with (C:=A) (D:=D);sTarski.
unfold Per.
exists D'.
intuition.
unfold is_midpoint...
Qed.
Definition Perp_in := fun X A B C D =>
A<>B /\ C<>D /\ Col X A B /\ Col X C D /\
(forall U V,  Col U A B -> Col V C D -> Per U X V).
Definition Perp := fun A B C D => exists X, Perp_in X A B C D.

Lemma l8_12 : forall A B C D X, Perp_in X A B C D ->  Perp_in X C D A B.
Proof with Perp.