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F.Treves proves, in [14], the following: Theorem. Let Ω be a non-empty open set of Rⁿ and let P = P(x,D) be a linear partial differential operator of order m, with analytic coefficients, of principal type in Ω. If P satisfies hypothesis (|) then P is analytic-hypoelliptic in Ω. Hypothesis (l) For any given x₀ ε &Omega, there exists an integer k₀ ≥ 0 such that the following holds: (*) for every ξ₀ ε Rⁿ ∣ {0} and every complex number z such that p(x₀, ξ₀) = 0, d_ξ Re(zp)(x₀,ξ₀) ≠ 0 the function Im(zp), restricted to the bicharacteristic strip of Re(zp) through (x₀, ξ₀), has a zero of even order less than or equal to 2k₀ at the point (x₀, ξ₀). Our purpose is to study the work mentioned above in a detailed fashion, including complete proofs of the main results in such a way as to make it accessible to non-specialists and more easily readable to students of Partial Differential Equations. An original contribution is the proof of some inequalities which are essential for the obtainment of the semi-regularity of the parametrix constructed for the operator P.