Section T16.
Context `{T2D:Tarski_2D}.
Context `{TE:@Tarski_euclidean Tn TnEQD}.
(** Lemma 16.2 in dimension 2. *)
(** As we are in dimension 2, we skip 16.3 which is only needed to prove 16.4 in dimension n. *)

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).