This paper solves the open problem on the sharp bound for the number of
isolated solutions in $\mathbf{R}_*^n$ to the real system of $n$ polynomial
equations in $n$ variables, i.e., the real $n$ by $n$ fewnomial system. For an
unmixed system of $n$ polynomial equations in $n$ variables, this paper shows
that the number of its positive solutions in $\mathbf{R}_*^n$ is sharply
bounded by that of the simplex configurations in the triangulation of its
support generically. The proof is based on a homotopic argument and an
inductive triangulation of the support of the system via a hierarchy of pyramid
configurations of different orders. For the mixed system of $n$ polynomial
equations in $n$ variables, this paper shows that the maximal number of
positive solutions in $\mathbf{R}_*^n$ to the systems with the same support is
a symmetric multilinear function of the support generically and hence can be
computed via the polarization identity.