induction 1 as [|x y z0].
- auto.
- intros; right with y; eauto.
Qed.

(* Specifications: *)
Lemma clos_rst1n_sym : forall x y, clos_refl_sym_trans_1n R x y ->
clos_refl_sym_trans_1n R y x.
Lemma clos_rst_rst1n  : forall x y,
clos_refl_sym_trans R x y -> clos_refl_sym_trans_1n R x y.