Section Grid.
Variable O E E' : Tpoint.
Variable grid_ok : ~ Col O E E'.
(** We are not faithful to Tarski's def because for uniqueness we do not need the assumption that A and B are on line OE as it is implied by the definition of sum. *)
End Grid.

Lemma pj_uniqueness : forall O E E' A A' A'', ~Col O E E' -> Col O E A -> Col O E' A' -> Col O E' A'' -> Pj E E' A A' -> Pj E E' A A'' -> A' = A''.