+theorem log_times_l: \forall p,n,m.O < n \to O < m \to S O < p \to
+log p n+log p m \le log p (n*m) .
+intros.
+unfold log in ⊢ (? ? (% ? ?)).
+apply f_m_to_le_max
+ [elim H
+ [rewrite > log_SO
+ [simplify.
+ rewrite < plus_n_O.
+ apply le_log_n_n.
+ assumption
+ |assumption
+ ]
+ |elim H1
+ [rewrite > log_SO
+ [rewrite < plus_n_O.
+ rewrite < times_n_SO.
+ apply le_log_n_n.
+ assumption
+ |assumption
+ ]
+ |apply (trans_le ? (S n1 + S n2))
+ [apply le_plus;apply le_log_n_n;assumption
+ |simplify.
+ apply le_S_S.
+ rewrite < plus_n_Sm.
+ change in ⊢ (? % ?) with ((S n1)+n2).
+ rewrite > sym_plus.
+ apply le_plus_r.
+ change with (n1 < n1*S n2).
+ rewrite > times_n_SO in ⊢ (? % ?).
+ apply lt_times_r1
+ [assumption
+ |apply le_S_S.assumption
+ ]
+ ]
+ ]
+ ]
+ |apply le_to_leb_true.
+ rewrite > exp_plus_times.
+ apply le_times;apply le_exp_log;assumption
+ ]
+qed.
+