In the process sketched above, Bolzano's Theorem is inferred from V together with the two 'link' assumptions.

Therefore, coming to believe Bolzano's Theorem by visualizing in the manner sketched above cannot be a way of discovering the theorem.

One may feel that the visual route to the theorem can be defended against this argument in the following way: the counter-examples to the link assumption that we have described are curves of discontinuous functions; since Bolzano's theorem is about continuous functions only, we can ignore these curves.

So Bolzano's Theorem cannot be discovered by visualizing in the manner described above.

Consider the following route to the Intermediate Value Theorem (IVT) from Bolzano's Theorem.

Assuming that mere translation preserves continuity, the newly positioned curve is the curve of a function which satisfies the conditions of Bolzano's Theorem, and so its curve meets the x-axis at some point between a and b.

So we can apply Bolzano's Theorem to f: 0 = f(c) for some c in [a, b].

For one thing, if they are totally unrelated concepts then it would make no sense to even illustrate

Bolzano's theorems with the diagrams, nor would it make any sense to apply Bolzano's result to situations in geometry or mechanics, as is commonly done.