Products of bounded subsets of paratopological groups

We prove that if Bi is a bounded subset of a totally ω-narrow paratopological group Gi, where i∈I, then ∏i∈IBi is bounded in ∏i∈IGi. The same conclusion remains valid in the case of products of bounded subsets of Hausdorff commutative paratopological groups with countable Hausdorff number or products of Lindelöf paratopological groups. In fact, we show that if B is a bounded subset of a paratopological group G satisfying one of the conditions (a)-(d) below, then B is strongly bounded in G: