apply lt_times;apply lt_O_fact]
qed.
+theorem lt_O_bc: \forall n,m. m \le n \to O < bc n m.
+intro.elim n
+ [apply (le_n_O_elim ? H).
+ simplify.apply le_n
+ |elim (le_to_or_lt_eq ? ? H1)
+ [generalize in match H2.cases m;intro
+ [rewrite > bc_n_O.apply le_n
+ |rewrite > bc1
+ [apply (trans_le ? (bc n1 n2))
+ [apply H.apply le_S_S_to_le.apply lt_to_le.assumption
+ |apply le_plus_n_r
+ ]
+ |apply le_S_S_to_le.assumption
+ ]
+ ]
+ |rewrite > H2.
+ rewrite > bc_n_n.
+ apply le_n
+ ]
+ ]
+qed.
+
theorem exp_plus_sigma_p:\forall a,b,n.
exp (a+b) n = sigma_p (S n) (\lambda k.true)
(\lambda k.(bc n k)*(exp a (n-k))*(exp b k)).