Lemma characterization_of_betweenness_aux : forall O E E' S U1 U2 A AX AY B BX BY C CX CY BXMAX CXMAX AB AC IAC T, 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 -> ~ Col O E E' -> Col O E AX -> Col O E BX -> Col O E CX -> Col O E BXMAX -> Col O E CXMAX -> Col O E T -> Col O E AB -> Col O E AC -> Col O E IAC -> Diff O E E' BX AX BXMAX -> Diff O E E' CX AX CXMAX -> Length O E E' A B AB -> Length O E E' A C AC -> Prod O E E' T AC AB -> Prod O E E' IAC AC E -> Bet A B C -> A <> B -> A <> C -> B <> C -> Prod O E E' T CXMAX BXMAX.