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

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'.