generalize Rstar_contains_Rplus; intro T; red in T.
generalize Rstar_transitive; intro T1; red in T1.
intros U R x y z H'; elim H'.
intros x0 H'0; elim H'0.
intros; exists y0; auto.
intros; exists y0; auto.
intros; exists y0; auto.
split; [ try assumption | idtac ].
apply T1 with z0; auto.
Qed.

Theorem Lemma1 :
forall (U : Type) (R : Relation U),
Strongly_confluent U R ->
forall x b : U,
Rstar U R x b -> forall a : U, R x a -> exists z : U, Rstar U R a z /\ R b z.