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I cannot disprove anything they are saying.

That's why you should familiarize yourself with the operation of turbojet and turbofan engines; do your research from multiple sources, not just the ones promoting this "air compression combine theory".

Do you know the differences among the following types of aircraft engines?

1) Turbojet
2) Turbofan
3) Tubosprop

Can you distinguish between a turboshaft engine and a free turbine engine (hint: two types of Turboprop)?

I'm going to guess that the answer is "No" and "No". And that's OK; its not your field of expertise. But you can get a non-professional's working understanding of these topics by your own research on the internet. Do that, and you'll be ready to address the problems with that video you linked for me.

"Wikipedia" is not a bad place to start.