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