(** Lemma 16.2 in dimension 2. *)

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.