# Looking for the proof of the prod check gadget referred to by Boneh in his PLONK video

I am going through Dan Boneh's video tutorial on PLONK Polynomial IOPs - https://www.youtube.com/watch?v=vxyoPM2m7Yg

He describes 3 type of proof gadgets he will use

He gives a proof of the Zero Test which I understood. However, he doesn't cover the proof for the Sum Check & Product Check in his video.

Prod Check

Prove that $$\prod_{a \in H} f(a) = c$$

He says that has Product Check covered in his slides - https://drive.google.com/file/d/1CMAoUSBl5vN8u88-ud_OJaCI9uBWABXG/view

But I don't find it covered even in the slides.

I found another set of Boneh slides which does have it - https://zk-learning.org/assets/lecture5-2023.pdf (Page 26 on)

This has the following 4 slides on Prod Check (it's a little different $$f(a) = 1$$ instead of $$f(a) = c$$)

But it still is very difficult to understand just from the slide without any explanation

For e.g. I don't understand how in the first of the 4 slides itself, he gets

$$t(\omega^{k-1}) = \prod_{a \in \Omega} f(a) = 1$$

And in the next line

$$t(\omega \cdot X) = t(x)\cdot f(\omega \cdot X)$$

Again, it's not even clear what the two different x's are

I googled & looked through several videos & documents on PLONK including the original PLONK document but I can't find this covered anywhere.

Does anyone know where I can read up more on this?