second quantization - A question regarding to the one-body operators in N-particle Hilbert space

Recently, I've started studying the book "condensed matter field theory" by Alexander Altland and Ben Simons. At second chapter when he's trying to rationalize the one-body operators in Fock space, I can not get it. I googled it and found out this one. During this article we can start from equation 2.28 which it's been told that one-body operator can be written in the following form, $$F=\sum _{ \alpha }{ \sum _{ \beta }{\left| \alpha \right> \left< \alpha \right| F \left| \beta \right> \left< \beta \right| } }$$ and then it claims that the corresponding extension of this operator in N-particle space is simply, $$F_{N} = F(1) + F(2) + ... + F(N) = \sum_{i=1}^{N}{F(i)}$$ which each operator $F(i)$ acts only on particle $i$. Then he applys $F(i)$ on a product state. $$F(i) \left| \alpha_{1}\alpha_{2} ...\alpha_{N} \right) = \left| \alpha_{1} \right> \left| \alpha_{2} \right> ... \left| \alpha_{i-1} \right> \{ \sum_{\beta_{i}}{\left| \beta_{i} \right> \left< \beta_{i} \right| F \left| \alpha_{i} \right> \} \left| \alpha_{i+1} \right> ... \left| \alpha_{N} \right> }$$ $$= \sum_{\beta_{i}}{{\left< \beta_{i} \right| F \left| \alpha_{i} \right>}} \left| \alpha_{1} ... \alpha_{i-1} \beta_{i} \alpha_{i+1} ...\alpha_{N} \right)$$ I can not understand this part that why he does believe that the two following terms are same? $$\left< \beta_{i} \right| F(i) \left| \alpha_{i} \right> =\left< \beta_{i} \right| F \left| \alpha_{i} \right>$$ and it says, "Note that the matrix elements of $F$ do not depend on which particle is considered". And another question for me would be that what's the form for $F(i)$?