Lemma pythagoras : forall O E E' A B C AC BC AB AC2 BC2 AB2, O <> E -> Per A C B -> Length O E E' A B AB -> Length O E E' A C AC -> Length O E E' B C BC -> Prod O E E' AC AC AC2 -> Prod O E E' BC BC BC2 -> Prod O E E' AB AB AB2 -> Sum O E E' AC2 BC2 AB2.