Lemma line_completeness_aux : line_completeness -> forall (Tm: Tarski_neutral_dimensionless) (Tm2 : Tarski_neutral_dimensionless_with_decidable_point_equality Tm) (f : @Tpoint Tn -> @Tpoint Tm), @archimedes_axiom Tm -> extension f -> forall A P Q R, ~ Col P Q R -> Coplanar (f P) (f Q) (f R) A -> exists B, Coplanar P Q R B /\ f B = A.