simple induction 1; intros; auto with pts.
apply rst_step.
apply R_lifts with (2 := H1); auto with pts.
apply rst_sym; auto with pts.
apply rst_trans with (lift_rec 1 y k); auto with pts.
Qed.
End Prop_R_lift.
Section Prop_R_subst.
Hypothesis R_substs : R_subst R.
Hypothesis R_substs' : R_subst R'.

Lemma reunion_subst : R_subst (reunion R R').