  The Berezin--Simon (BS) quantization is a rigorous version of the ``operator
formalism'' of quantization procedure. The goal of the paper is to present a
rigorous real-time (not imaginary-time) path-integral formalism corresponding
to the BS operator formalism of quantization; Here we consider the classical
systems whose phase space $M$ is a (possibly non-compact) K\"ahler manifold
which satisfies some conditions, with a Hamiltonian $H:M\rightarrow\mathbb{R}$.
For technical reasons, we consider only the cases where $H$ is smooth and
bounded. We use G\"uneysu's extended version of the Feynman--Kac theorem to
formulate the path-integral formula.
