$\mathcal{S}_{\widetilde{\theta}}=\mathcal{A}_{\beta_0,\widetilde{\theta}}$ where $\beta_0$ is such that $\beta \in N_k$ for some $k$, implies $\beta<\beta_0$ (recall that ${\mathfrak{c}}$ has uncountable cofinality). By construction of the sets $\mathcal{A}_{\beta,\widetilde{\theta}}$, we have $\mathcal{S}_{\widetilde{\theta}}\cap\mathcal{S}_{\widetilde{\theta}^\prime}=\emptyset$, when $\widetilde{\theta}\neq \widetilde{\theta}^\prime$. Condition (3) obviously holds, since $S_{\widetilde{\theta}}$ and $\bigcup_{\widetilde{\eta}\in\widetilde{\theta}}\mathcal{N}_{\widetilde{\eta}}$ are even disjoint.

Define finally $D=\{ f_{\widetilde{\eta}}\colon \widetilde{\eta}\in
[{\mathfrak{c}}]^\omega  \times \alpha\}=\cup_{N\in[{\mathfrak{c}}]^\omega}F_N$.

Suppose $\widetilde{D}=\{g_{\widetilde{\eta}}\colon
\widetilde{\eta}\in [{\mathfrak{c}}]^\omega \times \alpha\}$ is such that $\pi_{_{\mathcal{N}_{\widetilde{\eta}}}} (g_{\widetilde{\eta}})=
\pi_{_{\mathcal{N}_{\widetilde{\eta}} } } (f_{\widetilde{\eta}})$, for all $\widetilde{\eta} \in [{\mathfrak{c}}]^\omega \times \alpha$.

To check that $\widetilde{D}$ is indeed ${G_{\delta}}$-dense we choose a ${G_{\delta}}$-subset $U$ of $G^\mathcal{A}$ . There will be then $N=\{\alpha_n \colon n<\omega\}\in [{\mathfrak{c}}]^\omega$ and a ${G_{\delta}}$-set $V \subset G^{\cup\mathcal{A}_{\alpha_n}}$ such that $\{ \bar{x}\in G^{\mathcal{A}}\colon
\pi_{\cup_n\mathcal{A}_{\alpha_n}} (\bar{x})\in V \mbox{ for each
}n<\omega\}\subset U$. Since $F_N$ is ${G_{\delta}}$-dense in $G^{{\displaystyle \cup_{\gamma\in N}}
\mathcal{A}_\gamma}=G^{{\displaystyle \cup_n
\mathcal{A}_{\alpha_n}}}$, there will be an element $f_{(N,\eta)}\in F_N$ with $\pi_{\cup_n \mathcal{A}_{\alpha_n}}(f_{(N,\eta)})\in V$ for every $\alpha_n \in N$.

As $ g_{(N,\eta)}$ and $f_{(N,\eta)}$ have the same $\cup_{\gamma
\in N}\mathcal{A}_\gamma$-coordinates, we conclude that $g_{(N,\eta)}\in U\cap \widetilde{D}$.

If $\chi$ is a homomorphism between two groups $G_1$ and $G_2$ and $\sigma$ is a cardinal number, we denote by $\chi^{\sigma}$ the product homomorphism $\chi^\sigma \colon G_1^{\sigma}\to G_2^{\sigma}$ defined by $\chi^{\sigma}((g_\eta)_{\eta<\sigma})=(\chi(g_\eta))_{\eta<\sigma}$. It is easily verified that, for any $\mathcal{D}\subseteq\sigma$, the projections 