Section HalfAngle.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.
Definition HalfA P A O B := ~ Bet A O B /\ InAngle P A O B /\ CongA A O P B O P.
(* Lemma lta_nbet__ncol : forall A B C X Y Z, ~ Bet X Y Z -> LtA A B C X Y Z -> ~ Col X Y Z. *)
Definition gHalfA A' O' B' A O B := exists P, HalfA P A O B /\ CongA A' O' B' A O P.

Lemma ghalfa_preserves_lta : forall A B C X Y Z A' B' C' X' Y' Z',
gHalfA A' B' C' A B C -> gHalfA X' Y' Z' X Y Z -> LtA A B C X Y Z -> LtA A' B' C' X' Y' Z'.