induction x; simpl; intros.
- apply Rlt_trans with (1 + 0).
+ rewrite Rplus_comm.
apply Rlt_n_Sn.
+ apply Rplus_lt_compat_l.
rewrite <- (Rmul_0_l Rset Rext RTheory 2).
rewrite Rmult_comm.
apply Rmult_lt_compat_l.
* apply Rlt_0_2.
* trivial.
- rewrite <- (Rmul_0_l Rset Rext RTheory 2).
rewrite Rmult_comm.
apply Rmult_lt_compat_l.
+ apply Rlt_0_2.
+ trivial.
- replace 1 with (0 + 1).
+ apply Rlt_n_Sn.
+ apply Rplus_0_l.
Qed.

Lemma Rgen_phiPOS_not_0 :
forall x, InitialRing.gen_phiPOS1 1 Rplus Rmult x <> 0.