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).

(* Specifications: *)
Lemma l8_12 : forall A B C D X, Perp_in X A B C D -> Perp_in X C D A B.
Lemma l8_13_2 : forall A B C D X, A <> B -> C <> D -> Col X A B -> Col X C D -> (exists U, exists V :Point, Col U A B /\ Col V C D /\ U<>X /\ V<>X /\ Per U X V) -> Perp_in X A B C D.