intuition.
unfold ratInverse in *.
rattac.
destruct n0.
unfold posnatToNat, natToPosnat in *.
destruct p.
destruct p0.
inversion H0; clear H0; subst.
rewrite mult_1_r in e.
rewrite mult_1_r in e.
omega.
unfold posnatToNat, natToPosnat in *.
destruct p.
destruct p0.
inversion H0; clear H0; subst.
rewrite mult_1_r in e.
rewrite mult_0_l in e.
omega.
Qed.

Lemma ratInverse_1_swap : forall r,
~ r == 0 ->
r <= 1 ->
1 <= ratInverse r.