... Poincar´e-Miranda theorem. In fact, by (2.14), f<0ontheleftsideofΞ, f>0 on the right side of Ξ and, by (2.11), g<0 on the lower side of Ξ and g>0 on the upper side of Ξ. Therefore, there ... gave a proof of Theorem 3.9 by an argument based on Lemma 2.3,which, in turn, was proved using the Poincar´e-Miranda theorem. On the other hand, the Poincar´e-Miranda theorem itself (Theorem ... obtain a proof of the Brouwer fixed point theorem for the rectangle via the Poincar´e-Miranda theorem. In fact,if φ : Ξ→ Ξ,thenI − φ satisfies the assumptions of Theorem 2.1. Remark 2.2. The Poincar´e-Miranda...