intros A World le forces0 kripke k w a0 forces_a0 a1 b forces_a0a1_b.
simpl in |- *.
intros k' w' le_k_k' forces_a1.
apply (forces_a0a1_b k').
assumption.
assumption.
change (forces A World le forces0 k' (Imp a0 a1)) in |- *.
simpl in |- *.
intros k'' w'' le'' forces_a0''.
elim kripke.
intros refl trans mon.
apply forces_mon with (k := k'); assumption.
Qed.

Lemma forces_a0_imp_b__forces_a0_and_a1_imp_b :
forall (A : Set) (World : A -> Type) (le : A -> A -> Type)
(forces0 : A -> Int -> Prop) (k : A) (a0 b : form),
forces A World le forces0 k (Imp a0 b) ->
forall a1 : form, forces A World le forces0 k (Imp (AndF a0 a1) b).