Section T18.
Context `{T2D:Tarski_2D}.
Context `{TE:@Tarski_euclidean Tn TnEQD}.
Variable SS U1 U2 : Tpoint.
Variable orthonormal_grid : Cs O E SS U1 U2.
Definition F : Type := {P : Tpoint | Col O E P}.
Definition EqF (x y : F) := (proj1_sig x) = (proj1_sig y).
Instance eqF_Reflexive : Reflexive EqF.
Definition LeF (x y : F) := LeP O E E' (proj1_sig x) (proj1_sig y).
Instance leF_Antisymmetric : Antisymmetric F EqF LeF.
(* For 8.5 Instance leF_Antisymmetric : Antisymmetric eqF leF. *)
Definition MulF (x y : F) : F.
Add Ring GeometricRing : ringF.
Add Field GeometricField : fieldF.
Global Instance Fops: (@Ring_ops F OF OneF AddF MulF SubF OppF EqF) := {}.
Global Instance FRing : (Ring (Ro:=Fops)).
Global Instance Fintegral : (Integral_domain (Rcr:=Fcri)).
Definition sqrt3 := PythF 1 (PythF 1 1).

Lemma characterization_of_parallelism_F : forall A B C D,
Par A B C D <->
let (Ac, _) := coordinates_of_point_F A in
let (Ax, Ay) := Ac in
let (Bc, _) := coordinates_of_point_F B in
let (Bx, By) := Bc in
let (Cc, _) := coordinates_of_point_F C in
let (Cx, Cy) := Cc in
let (Dc, _) := coordinates_of_point_F D in
let (Dx, Dy) := Dc in
(Ax - Bx) * (Cy - Dy) - (Ay - By) * (Cx - Dx) =F= 0 /\
~ ((Ax - Bx) * (Ax - Bx) + (Ay - By) * (Ay - By) =F= OF) /\
~ ((Cx - Dx) * (Cx - Dx) + (Cy - Dy) * (Cy - Dy) =F= OF).