(* The converse seems to require much more work *)

Definition fstAB := fst (A:=A) (B:=B).

Lemma continuously_map : forall (f: A->B) (P: infseq A->Prop) (Q: infseq B->Prop), (forall s, P s -> Q (map f s)) -> forall (s: infseq A), continuously P s -> continuously Q (map f s).