In this work, we use a labelled deduction system based on the concept of computational paths (sequence of rewrites) as equalities between two terms of the same type. We also define a term rewriting system that is used to make computations between these computational paths, establishing equalities between equalities. We use a labelled deduction system based on the concept of computational paths (sequence of rewrites) to obtain some results of algebraic topology and with support of the Seifet-Van Kampen Theorem we will calculate, in a way less complex than the one made in mathematics \cite{ Munkres} and the technique of homotopy type theory \cite{ hott}, the fundamental group of Klein Blottle $\mathbb{ K}^2$, of the Torus $\mathbb{ T}^2$ and Two holed Torus $\mathbb{ M}_2=\mathbb{ T}^2\# \mathbb{ T}^2$ (the connected sum two torus).
Categories: cs.LO math.AT