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_collinearity :
forall O E E' S U1 U2
A AX AY B BX BY C CX CY
AXMBX AYMBY BXMCX BYMCY XProd YProd,
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' AX BX AXMBX -> Diff O E E' AY BY AYMBY ->
Diff O E E' BX CX BXMCX -> Diff O E E' BY CY BYMCY ->
Prod O E E' AXMBX BYMCY XProd -> Prod O E E' AYMBY BXMCX YProd ->
(Col A B C <-> XProd = YProd).