We introduce horseshoe-type mappings which are geometrically similar to Smale's horseshoes. For such mappings we prove by means of the fixed point index the existence of chaotic dynamics - the semi-conjugacy to the shift on a finite number of symbols. Our theorem does not require any assumptions concerning derivatives, it is a purely topological result. The assumptions of our theorem are then rigorously verified by computer assisted computations for the classical Hénon map and for classical Rössler equations.