# How does multiple integral change into terms multiplying each other in convolution theorem of Laplace?

In the steps of the proofs highlighted below, how does a multiple integral changes in to multiplication of two integral. This is only possible if V is independent of u, but as it turns out V = t - u, so they are not actually independent.

The substitution leads to a definite integral $$I = \int\limits_0^\infty e^{-sv} f(v) \, dv$$ which does not depend on $u$.
The prior integral $$I_0 = \int\limits_u^\infty e^{-s(t-u)} f(t-u) \, dt$$ seems to depend on $u$, but it is not as $I$ shows.
• $v$ is derived from $u$, but that $I$ is not dependent on $u$ anymore, $v$ is just an integration variable.