Proving the Wiener-Khinchin theorem

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Homework Statement
The question is attached below.
Relevant Equations
correlation equations
Wiener-Khinchin theorem
My try:

I tried to take the expression for the decomposition coefficients and put it into the equation that I had to prove.
Then, I tried to work with the integral limits in order to get into Wiener-Khinchin theorem or maybe Fourier transform of delta function but I didn't see any success in those tries.

It would be nice to get a new point of view.

Thank you.

EDIT: I did solve this problem eventually, but I did not use the relation for z(t) - how is that?
Typeset problem statement: a stationary real process z(t) is spectrally decomposed as z(t) = integral from minus infinity to infinity of a(omega) e^(i omega t) d omega, with a(omega) = (1/2pi) times the integral of z(t) e^(i omega t) dt, and the task is to show <a(omega)a(omega')> = I(omega) delta(omega + omega').
 
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Watch my comment - I have solved this question but without the use of the relation for z(t).
I wanted to ask if it is hidden somewhere throughout the solution.
 
So you want someone to tell you if you assumed anything in a derivation that you refuse to share? I don’t think you understand how PF works.
 

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