Lemma eq_points_coordinates : forall O E S U1 U2 P1 X1 Y1 P2 X2 Y2, Cd O E S U1 U2 P1 X1 Y1 -> Cd O E S U1 U2 P2 X2 Y2 -> (P1 = P2 <-> (X1 = X2 /\ Y1 = Y2)).