Computational detail in derivation of Green-Kubo relation for diffusion coefficient

I read Kubos paper on the Fluctuation Dissipation Theorem

https://www.mrc-lmb.cam.ac.uk/genomes/madanm/balaji/kubo.pdf

and right at the beginning there is a calculation which seems a bit miraculous to me

$$D=lim_{ttoinfty}frac{1}{2t}langle{x(t)-x(0)}^2rangletag{2.4}$$
where the average $langlerangle$ is taken over an ensemble in thermal equilibrium. Since we have
$$x(t)-x(0)=int_0^t u(t’)dt’$$
equation $(2.4)$ is transformed as
begin{align}
D=&lim_{ttoinfty}frac{1}{2t}int_0^{t} dt_1int_0^t dt_2langle u(t_1)u(t_2)rangle
=&lim_{ttoinfty}frac{1}{t}int_0^{t} dt_1int_0^{t-t_1} dt’ langle u(t_1)u(t_1+t’)rangle
=&int_0^{infty} langle u(t_0)u(t_0+t’)rangle dt’tag{2.5}
end{align}
where we assumed that
$$lim_{ttoinfty}langle u(t_0)u(t_0+t)rangle=0tag{2.6}$$

Can someone give me a hint which steps in the calculation (lines 2 and 3 under segment 2.5) are explicitly left out or recommend a resource where a more detailed derivation can be found?

Edit: Following tbt’s hint a more detailed way to arrive at the solution is:

begin{align}
D=&lim_{ttoinfty}frac{1}{2t}int_0^{t} dt_1int_0^t dt_2langle u(t_1)u(t_2)rangle
=&lim_{ttoinfty}frac{1}{2t}left[2int_0^{t} dt_1int_0^{t_1} dt_2 langle u(t_1)u(t_2)rangleright]
=&lim_{ttoinfty}frac{1}{t}int_0^{t} dt_1int_0^{t_1} dt_2 langle u((t_1-t_2)+t_2)u(t_2)rangle
=&lim_{ttoinfty}frac{1}{t}int_0^{t} dt_1int_0^{t_1} dt_2 langle u((t_1-t_2)+t_0)u(t_0)rangle
=&lim_{ttoinfty}frac{1}{t}int_0^{t} dt_1int_0^{t_1} dt’ langle u(t’+t_0)u(t_0)rangle
=&int_0^{infty} langle u(t_0)u(t_0+t’)rangle dt
end{align}