Section UNION.
Variable f : nat -> ensemble (frm L).
Definition union_f : ensemble (frm L) :=
fun p => exists n, p \in f n.
Hypothesis incl : forall n1 : nat, forall n2 : nat, forall LE : n1 <= n2, f n1 \subseteq f n2.

Lemma union_f_proves_iff p
: union_f ⊢ p <-> (exists n, f n ⊢ p).