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.
(** Given two angles a and b, a/2 = b/2 -> a = b *)
(** Given two angles a and b, a = b -> a/2 = b/2 *)
(** Unicity of the double angle *)
(** Unicity of the half angle *)
(** CongA preserves gHalfA *)
(** Out preserves gHalfA *)

Lemma ghalfa_out4__ghalfa : forall A B C D E F A' C' D' F',
Out B A' A -> Out B C' C -> Out E D' D -> Out E F' F -> gHalfA A B C D E F ->
gHalfA A' B C' D' E F'.