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The present paper, III, is the third part of a series of papers, under theglobal title "the non triviality of a $\Phi_4^4$ model". Parts I and II havebeen previously completed. In them thanks to the properties we dubbed"splitting -tree structure", and "alternating signs", which characterize ourconnected Green's functions, we have constructed a unique non trivial solutionto a $\Phi_4^4$ non linear renormalized system of equations of motion inEuclidean space.  In the present work, we show how, by application of these properties, thesolution of our $\Phi^4_4$ model verifies the Osterwalder-Schrader Positivityrequirement. This result complements those obtained in I and II where, apartfrom the Positivity, the Axiomatic Quantum Field theory properties have beenestablished.  The O.S. Positivity is verified under a condition on the physical couplingconstant relatively weaker than the one imposed in order to obtain theconvergence of the $\Phi^4_4$ mapping to the unique non trivial solution.

The "non triviality" of a $\Phi_4^4$ model, III, the "Osterwalder-Schrader Positivity"