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_conga_1 : forall A B C A' B' C' X Y Z X' Y' Z', gHalfA X Y Z A B C -> gHalfA X' Y' Z' A' B' C' -> CongA X Y Z X' Y' Z' -> CongA A B C A' B' C'.