  The Laplace transform is an algebraic method that is widely used for
analyzing physical systems by either solving the differential equations
modeling their dynamics or by evaluating their transfer function. The dynamics
of the given system are firstly modeled using differential equations and then
Laplace transform is applied to convert these differential equations to their
equivalent algebraic equations. These equations can further be simplified to
either obtain the transfer function of the system or to find out the solution
of the differential equations in frequency domain. Next, the uniqueness of the
Laplace transform provides the solution of these differential equations in the
time domain. The traditional Laplace transform based analysis techniques, i.e.,
paper-and-pencil proofs and computer simulation methods are error-prone due to
their inherent limitations and thus are not suitable for the analysis of the
systems. Higher-order-logic theorem proving can overcome these limitations of
these techniques and can ascertain accurate analysis of the systems. In this
paper, we extend our higher-order logic formalization of the Laplace transform,
which includes the formal definition of the Laplace transform and verification
of its various classical properties. One of the main contributions of the paper
is the formalization of Lerch's theorem, which describes the uniqueness of the
Laplace transform and thus plays a vital role in solving linear differential
equations in the frequency domain. For illustration, we present the formal
analysis of a $4$-$\pi$ soft error crosstalk model, which is widely used in
nanometer technologies, such as, Integrated Circuits (ICs).
