Proof using.
move => l.
apply: always_map.
apply: eventually_map.
case => e s.
rewrite /= /occurred /=.
move => H_eq.
by rewrite H_eq.
Qed.

Lemma tot_map_onet_label_event_inf_often_occurred_conv :
forall l s,
inf_often (now (occurred (tot_map_label l))) (map tot_map_onet_event s) ->
inf_often (now (occurred l)) s.