A proof of Yomdin-Gromov's Algebraic Lemma

Following the analysis of differentiable mappings of Y. Yomdin, M. Gromov has stated a very elegant "Algebraic Lemma" which says that the "differentiable size" of an algebraic subset may be bounded only in terms of its dimension, degree and diameter, regardless of the size and specific values of the underlying coefficients. We give a complete and elementary proof of Gromov's result.