Proposition 10.10 of Kanamori's The Higher Infinite states:
Suppose that $R$ is a [poset], $G$ is $R$-generic over $V$, and $N$ is a transitive $\in$-model of $\mathsf{ZFC}$ such that $V \subseteq N \subseteq V[G]$. Then there is a [poset] $P$ and a $P$-name $\dot{Q}$ such that $\Vdash_P \dot{Q}$ is a [poset] with $P * \dot{Q}$ isomorphic to $R$ so that, thus identifying $R$ with $P * \dot{Q}$ is as in 10.9(b), $V[G_0] = N$.
Proposition 10.9(b) of the same book states:
Conversely, suppose that $G$ is $P * \dot{Q}$-generic, and set: $$ G_0 = \{p \in P \mid \exists \dot{q}(\langle p,\dot{q} \rangle \in G)\}, \text{and} \\ G_1 = \{q^{G_0} \mid \exists p(\langle p,\dot{q} \rangle \in G)\} $$ Then $G_0$ is $P$-generic, $G_1$ is $\dot{Q}^{G_0}$-generic over $V[G_0]$, and $G = G_0 * G_1$.
Kanamori referred to Jech's Set Theory (2003) for the proof of Proposition 10.10, which uses a completely Boolean algebraic approach. I would like to ask for a proof of Proposition 10.10 without using any tools involving Boolean algebra (a quick search in other common set theory texts does not give any such proof).
Any help is appreciated.