Bolzano’s diagram-free proof of the IVT is an argument from what later became known as the Dedekind completeness of the real numbers: every non-empty set of reals bounded above (below) has a least upper bound (greatest lower bound). The value of Bolzano’s deduction of the IVT from the Dedekind completeness of the reals, according to Brown, is not that it proves the IVT but that it gives us confirmation of Dedekind completeness, just as an empirical hypothesis in empirical science gets confirmed by deducing some consequence of the hypothesis and observing those consequence to be true. This view assumes that we already know the IVT to be true by observing a diagram relevantly like Figure 5.