intros.
assert (T:=lower_dim).
DecompEx T X.
DecompEx H0 Y.
DecompEx H1 Z.
(* *)
cases_col A B X.
2:exists X;auto.
(* *)
cases_col A B Y.
2:exists Y;auto.
(* *)
cases_col A B Z.
2:exists Z;auto.
(* *)
cases_equality A X.
smart_subst X.
assert (Col A Y Z).
eapply col_transitivity_1;try apply H;auto.
intuition.
(* *)
assert (Col A X Y).
eapply col_transitivity_1;try apply H;auto.
assert (Col A X Z).
eapply col_transitivity_1;try apply H;auto.
(* *)
assert (Col X Y Z).
eapply col_transitivity_2;try apply H4;auto.
intuition.
Qed.

Theorem t2_8 :
forall A B C D E : Point,
Bet A B C ->
Bet D B E -> Cong A B D B -> Cong B C B E -> Cong A E C D.