First of all, I was confused as to why the thread is called "Show Hilbert Space has a pre inner product". Certainly any Hilbert space has a pre-inner product, namely the inner product making the underlying vector space a Hilbert space in the first place.

Second of all, your notation is not really user-friendly, but I think I've deciphered it.

Let me show you how to prove additivity in the first variable. I want to show that .

The left hand side is just

On the right hand side of this equality, for each term in this (huge) sum, you just multiply into the parenthesis, giving

and hey, this is exactly the right hand side of the equality I was trying to prove.