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The same paper shows that the converse is true: starting from time-ordered Green functions satisfying axioms T1-T7 as in subsection I.1 one can get the Schwinger functions (Theorem 1, pp. 99, add Corollary 3 from same page to get full Euclidean covariance S5 of Schwinger functions from Poincaré covariance T8 of time-ordered Green functions). Since these will then satisfy a strengthened form S1-S4 (+S5 if T8 holds) of the Osterwalder-Schrader axioms (see subsection I.2) to allow for coinciding points, the Osterwalder-Schrader reconstruction theorem applies and from them one can recover the Wightman field operator-valued distributions and the corresponding Wightman functions.