Proof with Geometry.
unfold parallel,S4 in |- *.
intros.
assert (S R A P = S R A Q + S R Q P + S Q A P)...
assert (Col R Q P)...
rewrite H4 in H3.
NormalizeRing H3...
assert (S A R Q = - S R A Q)...
rewrite H5 in H1.
RewriteVar (S R A Q) H1...
assert (S4 B A P Q = S B A P + S B P Q)...
assert (S B P Q = - S B Q P)...
rewrite H7 in H6.
RewriteVar (S B Q P) H2...
NormalizeRing H6...
assert (S C A P = S C A B + S C B P + S B A P)...
assert (Col C A B)...
rewrite H9 in H8...
assert (S C B P = S B P C)...
rewrite <- H10 in H6...
assert (S4 B A P Q = S C A P)...
rewrite H8.
rewrite H6.
ring_simplify...
cut (S A Q B + S Q A P = S4 B A P Q).
intro.
rewrite H12 in H3.
rewrite H11 in H3.
assert (S A P C = S C A P)...
rewrite H13.
rewrite <- H3.
assert (S A R P = - S R A P)...
rewrite H14.
ring.
assert (S A Q B = S B A Q)...
rewrite H12.
assert (S4 B A P Q = S B A Q - S P A Q)...
rewrite H13.
assert (S Q A P = - S P A Q)...
rewrite H14.
ring.
Qed.

Theorem l1_25_aux :
forall A B C X Y Z : Point,
~ Col Z B Y ->
~ Col C B Y ->
Col A B C ->
Col X Y Z ->
parallel A X B Y ->
parallel B Y C Z ->
B <> C -> Z <> Y ->
A ** B / C ** B = X ** Y / Z ** Y.