Proof with sTarski.
intros.
cases_equality P A.
unfold is_midpoint in *.
intuition.
(* *)
treat_equalities...
(* *)
assert (exists X, Bet P' P X /\ Cong P X Q A)...
DecompExAnd H2 X.
assert (exists X', Bet X P' X' /\ Cong P' X' Q A)...
DecompExAnd H2 X'.
assert (exists Y, Bet Q' Q Y /\ Cong Q Y P A)...
DecompExAnd H2 Y.
assert (exists Y', Bet Y Q' Y' /\ Cong Q' Y' P A)...
DecompExAnd H2 Y'.
(* *)
unfold is_midpoint in *;intuition.
(* *)
assert (Cong A X Y A).
apply (l2_11 A P X Y Q A);Between...
(* *)
assert (Cong A Y' X' A).
apply (l2_11 A Q' Y' X' P' A);Between.
eapply cong_transitivity.
apply cong_commutativity.
apply H12.
sTarski.
eapply cong_transitivity;eauto...
(* *)
assert (Cong A Y A Y').
apply (l2_11 A Q Y A Q' Y');Between;eauto...
eapply cong_transitivity;eauto...
(* *)
assert (FSC X A X' Y' Y' A Y X).
unfold FSC;repeat split.
unfold Col;left;Between.
eapply cong_transitivity with (C:=Y) (D:=A)...
4:eapply cong_transitivity with (C:=Y) (D:=A)...
2:eapply cong_transitivity with (C:=Y') (D:=A)...
2:Tarski.
(* *)
eapply (l2_11 X A X' Y' A Y);Between.
eapply cong_transitivity with (C:=Y) (D:=A)...
eapply cong_transitivity with (C:=Y') (D:=A)...
(* *)
assert (X<>A).
unfold not;intro.
treat_equalities...
(* *)
assert (Cong X' Y' Y X).
eapply l4_16.
apply H15.
auto.
(* *)
assert (IFSC Y Q A X Y' Q' A X').
unfold IFSC,FSC in *;intuition;Between...
eapply cong_transitivity with (C:=Y') (D:=A)...
(* *)
assert (Cong Q X Q' X').
eapply l4_2.
apply H18.
(* *)
assert (IFSC X P A Q X' P' A Q').
unfold IFSC,FSC in *;intuition;Between...
(* *)
eapply l4_2.
apply H20.
Qed.
(** Symmetry preserves Bet. *)

Lemma l7_15 : forall P Q R P' Q' R' A,
is_midpoint A P P' -> is_midpoint A Q Q' -> is_midpoint A R R' ->
Bet P Q R -> Bet P' Q' R'.