proposition: Lemma ordinal_next : forall a x, next_compatible ID a -> inc x (U a) -> next a x = x.
proof: Proof. ir. uh H; ee. cp (H1 _ H0). ufi ID H2. wr H2. wr next_punctured_downward. sy; am. am. am. Qed.
imports: Arith transfinite