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

Question

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.

Answer 1

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.