simple induction 1; intros; auto with pts.
apply rt_step.
red in |- *.
apply (clos_rt_clos_rst term (R f) (subst_rec s x n)).
fold (R_rt R f) in |- *.
apply R_substs with (2 := H1); auto with pts.
elim H1 with (1 := H2); auto with pts; intros.
apply rt_trans with y0; auto with pts.
apply rt_trans with (subst_rec s y n); auto with pts.
Qed.
End Prop_R_subst.
Section Prop_R_stable_env.
Hypothesis R_stable : R_stable_env R.
Hypothesis R_stable' : R_stable_env R'.

(* Specifications: *)
Lemma reunion_stable : R_stable_env (reunion R R').
Lemma ctxt_stable : R_stable_env (ctxt R).
red in |- *.