Lemma characterization_of_betweenness : forall O E E' S U1 U2 A AX AY B BX BY C CX CY BXMAX BYMAY CXMAX CYMAY, Cd O E S U1 U2 A AX AY -> Cd O E S U1 U2 B BX BY -> Cd O E S U1 U2 C CX CY -> Diff O E E' BX AX BXMAX -> Diff O E E' BY AY BYMAY -> Diff O E E' CX AX CXMAX -> Diff O E E' CY AY CYMAY -> (Bet A B C <-> exists T, O <> E /\ Col O E T /\ LeP O E E' O T /\ LeP O E E' T E /\ Prod O E E' T CXMAX BXMAX /\ Prod O E E' T CYMAY BYMAY).