proposition: Lemma exists_Forall A B : forall (P : A -> B -> Prop) l, (exists k, Forall (P k) l) -> Forall (fun x => exists k, P k x) l.
proof: Proof. intros P l; induction l as [|a l IHl]; intros [k HF]; constructor; inversion_clear HF. - now exists k. - now apply IHl; exists k. Qed.
imports: PeanoNat