Lemma square_distance_formula : forall O E E' S U1 U2 P Q PX PY QX QY PQ PQ2 PXMQX PYMQY PXMQX2 PYMQY2 F, Cd O E S U1 U2 P PX PY -> Cd O E S U1 U2 Q QX QY -> Is_length O E E' P Q PQ -> Prod O E E' PQ PQ PQ2 -> Diff O E E' PX QX PXMQX -> Prod O E E' PXMQX PXMQX PXMQX2 -> Diff O E E' PY QY PYMQY -> Prod O E E' PYMQY PYMQY PYMQY2 -> Sum O E E' PXMQX2 PYMQY2 F -> PQ2 = F.