intros A World le forces0 k a0 b forces_a0_b a1.
simpl in |- *.
intros k' w' le_k_k' forces_a0_a1.
apply (forces_a0_b k').
assumption.
assumption.
change (forces A World le forces0 k' a0) in |- *.
elim forces_a0_a1; trivial.
Qed.

Lemma forces_a1_imp_b__forces_a0_and_a1_imp_b :
forall (A : Set) (World : A -> Type) (le : A -> A -> Type)
(forces0 : A -> Int -> Prop) (k : A) (a1 b : form),
forces A World le forces0 k (Imp a1 b) ->
forall a0 : form, forces A World le forces0 k (Imp (AndF a0 a1) b).