intros.
simpl.
rewrite fold_add.
assert (expnat 2 (lognat pf) <> 0). apply expnat_nz. auto.
assert (2 * (expnat 2 (lognat pf)) <> 0). omega.
eapply (lognat_ge_monotonic_mt pf H0).
rewrite <- (lognat_prod_sum H).
rewrite lognat_correct_eq.
omega.
Qed.

Lemma lognat_odd : forall(a : nat)(pf1 : (2 * a) <> 0)(pf2 : (S (2 * a)) <> 0),
((lognat pf2) = (lognat pf1)).