Lemma Upw_transitive (T : Tree) (x : pr11 T) : ∏ y z w, (Upw_E T x) y z → (Upw_E T x) z w → (Upw_E T x) y w. Proof. intros y z w P Q. unfold Upw_E in P. simpl in P. exact (TRRG_transitivity (pr1 T) (pr1 y) (pr1 z) (pr1 w) P Q). Qed.