intros.
cases_col P U Q.
unfold parallel in H.
unfold S4 in H.
rewrite H0 in H.
assert (0 + S P Q V = S P Q V).
ring.
rewrite H1 in H.
right.
unfold Col in |- *.
trivial.
left.
trivial.
Qed.

Theorem co_side_main :
forall A B P Q M : Point,
~ Col Q A B ->
~ Col P A M ->
~ Col Q A M ->
Q <> M -> Col A B M -> Col P Q M -> P ** M / Q ** M = S P A B / S Q A B.