I'm studying Laver's proof of the consistency of Borel's conjecture and he says that the following is a standard fact about iterated forcing:
Define $\mathbb{P}^{\alpha \beta}$ as the set of all functions $f$ with domain $[\alpha, \beta)$ such that $1_{\alpha} \cup f \in \mathbb{P}_{\beta}$ and order $\mathbb{P}^{\alpha \beta}$ in $\mathfrak{M}[G_{\alpha}]$ with:
$f \le g \longleftrightarrow (\exists p \in G_{\alpha}) p \cup f \le p\cup g$
Let $G_{\beta}$ be generic over $\mathbb{P}_{\beta}$, then $\mathfrak{M}[G_{\beta}] = \mathfrak{M}[G_{\alpha}][G^{\alpha \beta}]$ where $G^{\alpha \beta}$ is $\mathfrak{M}[G_{\alpha}]\text{-generic}$ over $\mathbb{P}^{\alpha \beta}$.
My thoughts:
I tried to prove $G_{\beta} = G_{\alpha} * G^{\alpha \beta}$ and that $G^{\alpha \beta} = G^{\beta}|_{[\alpha, \beta)}$ but i failed because $G^{\alpha \beta}$ is so random. I would be really glad if someone pointed me in the right direction.
Edit I: The definitions that i forgot: $G_{\alpha} = G_{\beta} |_{\alpha}$ and $1_{\alpha}$ is the canonical name for the greatest element of $\mathbb{P}_{\alpha}$ and $\mathbb{P}_{\beta}$ is the $\beta\text{-th}$ stage iteration of Laver forcing.