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in 1971, F. Treves stated the equivalence between the hypoellipticity of LPDO of principal type and order m > 0 and the properties (P) and (Q). He proved the sufficiency of (P) and (Q) and the necessity of (Q). By using the result of Moyer and Hõrmander ((P) is necessary for local solvability) one obtains a proof the above mentioned equivalence. Our purpose is to prove tnat (P) e (Q) are sufficient for the hypoellipticity of first order LPDO of principal type. We include a detailed construction of approximate solutions to Cauchy problems; this is necessary in order to obtain parametrices for our operators. One contribution is the statement and proof of the non-existence of hypoelliptic first-order LPDO of principal type with C^∞ coefficients in Ω ⊂ Rⁿ, n ≥ 3.