intros a b0 b1 k forces_ab0 forces_ab1.
unfold forces_t in |- *; unfold forces_t2 in |- *; simpl in |- *.
intros k' suc_k' suc_k''.
change (forces_t2 k k' a -> forces_t2 k k' b0 /\ forces_t2 k k' b1) in |- *.
intros forces_a.
split.
apply (forces_ab0 k'); assumption.
apply (forces_ab1 k'); assumption.
Qed.

(* Specifications: *)
Lemma forces_a_imp_b__forces_a_imp_falsum_or_b :
forall (k : kripke_tree) (a b : form),
forces_t k (Imp a b) -> forces_t k (Imp a (OrF Falsum b)).