Definition gHalfA A' O' B' A O B := exists P, HalfA P A O B /\ CongA A' O' B' A O P.

Lemma ghalfa2__conga_1 : forall A B C A' B' C' A'' B'' C'', gHalfA A B C A' B' C' -> gHalfA A B C A'' B'' C'' -> CongA A' B' C' A'' B'' C''.