  Hille in 1959 gave a stronger form for the classical implicit function
theorem of Cauchy for equations of the type $y = \psi(x,y)$, $x,y$ real or
complex, $\psi$ analytic with a condition which determines a radius of
convergence of the resulting series solution. Based on a geometrical
interpretation of Hille's condition, this report generalizes his result to the
case when x and y are elements of lattice-normed linear spaces in the sense of
Kantorovich. This formulation includes the special cases when x and y belong to
Banach or Riesz spaces.
