Kantorovitch's theorem gives sufficient conditions for the convergence of Newton's method. We present a full formalization of this theorem in the case of a real function. The work is accomplished inside the Coq proof assistant and it is based on the Reals library provided by the theorem prover. For the general case of the theorem we first describe a way to represent concepts from multivariate analysis, as Coq does not offer such a library. We then discuss the proof of Kantorovitch's theorem based on this representation.