G^\sigma$ with cardinality $m(\sigma)$, $D=\{d_\eta \colon \eta
<m(\sigma)\}$, and two families of sets of coordinates $\{
\mathcal{S}_\theta \colon \theta \in [\alpha]^\omega\}, \{
\mathcal{N}_\eta\colon \eta< \alpha\}\subset \sigma$ such that:

1.  $|\mathcal{S}_\theta|=\sigma$.

2.  $\mathcal{S}_\theta\cap\mathcal{S}_{\theta^\prime}=\emptyset$, if $\theta\neq \theta^\prime$.

3.  ${\displaystyle \left|\mathcal{S_\theta}\setminus
    \bigcup_{\eta \in \theta}\mathcal{N}_\eta\right|=\sigma}$ for every $\theta\in [\alpha]^{ \omega}$.

4.  Every subset $\{g_\eta \colon \eta <\alpha\}$ of $G^\sigma$ with $\pi_{_{\mathcal{N}_\eta}}(g_\eta)=\pi_{_{\mathcal{N}_\eta}}(d_\eta)$, for all $\eta <\alpha$ is ${G_{\delta}}$-dense.

Let $\mathcal{A}_\beta=\{a_{\gamma}\colon \gamma<\sigma\}$ be a set with $|\mathcal{A}_\beta|=\sigma$ and consider the disjoint union $\mathcal{A}=\bigcup_{\beta<{\mathfrak{c}}} \mathcal{A}_\beta$. We identify $G^\sigma$ with $G^\mathcal{A}$ and $\alpha$ with $[{\mathfrak{c}}]^\omega
\times \alpha$. Since $\alpha^\omega \leq \sigma$, we can as well decompose each $\mathcal{A}_\beta$ as a disjoint union ${\displaystyle \mathcal{A}_\beta= \bigcup_{\widetilde{\theta}\in
[[{\mathfrak{c}}]^\omega\times \alpha]^\omega}
\mathcal{A}_{\beta,\widetilde{\theta}}}$ of sets of cardinality $|\mathcal{A}_{\beta,\widetilde{\theta}}|=\sigma$.

For each $N\in [{\mathfrak{c}}]^\omega$, let next $F_N=\{f_{(N,\eta)}\colon
\eta<\alpha\}$ be an independent ${G_{\delta}}$-dense subset of the product ${\displaystyle G^{\cup_{\gamma\in N} \mathcal{A}_\gamma}}$ (note that $m(\sigma)\leq \alpha$ and that $G$ is metrizable). Assume that each $f_{(N,\eta)}$ actually belongs to $G^\mathcal{A}$ by putting $\pi_{\mathcal{A}_\gamma}(f_{(N,\eta)})=0$ if $\gamma\notin N$.

We now order $\alpha=[{\mathfrak{c}}]^\omega \times \alpha$ lexicographically and define the sets $N_{\widetilde{\eta}}$, $\widetilde{\eta} \in
[{\mathfrak{c}}]^\omega \times \alpha$ and $\mathcal{S}_{\widetilde{\theta}}$, $\widetilde{\theta} \in [[{\mathfrak{c}}]^\omega \times \alpha]^\omega$. For $\widetilde{\eta}=(N,\eta)\in [{\mathfrak{c}}]^\omega \times \alpha$ define $\mathcal{N}_{(N,\eta)}=\bigcup_{\gamma\in N}
\mathcal{A}_{\gamma,\widetilde{\eta}}$ and given $\widetilde{\theta}=\{(N_k,\eta_k)\colon k<\omega,\;\:
(N_k,\eta_k)\in [{\mathfrak{c}}]^\omega \times\alpha\}$, we define 