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

Lemma ghalfa_distincts : forall A' O' B' A O B, gHalfA A' O' B' A O B ->
O' <> A' /\ O' <> B' /\ O <> A /\ O <> B.