assert ({A : Point &  {B : Point &  {C : Point | ~ Col A B C}}}).
apply A4.
intros.
elim H;intro X;intros.
elim p;intro Y;intros.
elim p0;intro Z;intros.
cases_col A B X.
cases_col A B Y.
cases_col A B Z.
cases_equality Y A.
subst Y.
assert (Col A X Z).
eapply col_trans_1 with (B:=B)...
assert (Col X A Z)...
intuition.
assert (Col A Y Z).
eapply col_trans_1 with (B:=B)...
assert (Col A Y X).
eapply col_trans_1 with (B:=B)...
assert (Col Y X Z).
eapply col_trans_1 with (B:=A)...
assert (Col X Y Z)...
intuition.
exists Z...
exists Y...
exists X...
Qed.

(* Specifications: *)
Lemma build_point_not_collinear_2 : forall A, exists B, exists C, ~ Col A B C.
Proof with Geometry.