The Bolzano-Poincaré Type Theorems

In 1883–1884, Henri Poincaré announced the result about the structure of the set of zeros of function f : I → R, or alternatively the existence of solutions of the equation f x 0. In the case n 1 the Poincaré Theorem is well known Bolzano Theorem. In 1940Miranda rediscovered the Poincaré Theorem. Except for few isolated results it is essentially a non-algorithmic theory. The aim of this article is to introduce an algorithmical proof of the Theorem “On the existence of a chain” and for n 3 an algorithmical proof of the Bolzano-Poincaré Theorem and to show the equivalence of Poincaré, Brouwer and “On the existence of a chain” theorems.