  we will show the existence and uniqueness of a real-time, time-sliced Feynman
path integral for quantum systems with vector potential. Our formulation of the
path integral will be derived on the $L^2$ transition probability amplitude via
improper Riemann integrals. Our formulation will hold for vector potential
Hamiltonian for which its potential and vector potential each carries at most a
finite number of singularities and discontinuities.
