we choose and fix a set of coordinates $\mathcal{D}_\theta \subseteq \sigma$ of cardinality $|\mathcal{D}_\theta|=\sigma$ in such a way that $$\mathcal{D}_\theta\subseteq \mathcal{S}_\theta
 \setminus {\displaystyle\bigcup_{\eta \in \theta}
 \mathcal{N}_\eta}$$ (recall that by Lemma \[room\], $\left|\mathcal{S_\theta}\setminus \bigcup_{\eta \in
\theta}\mathcal{N}_\eta\right|=\sigma$)

Given each $\theta\in[\alpha]^{\omega}$, we consider the free subgroup $\left\langle  \chi^\sigma(d_{\eta})
\colon \eta \in \theta \right\rangle$ and equip it with its maximal totally bounded topology. Denoting the resulting topological group as $\left\langle  \chi^\sigma( d_{\eta}) \colon \eta \in \theta
\right\rangle^\sharp$, and taking into account that it has weight ${\mathfrak{c}}$, we can find an embedding $$\label{(1)}
j_{\theta} \colon \left\langle
\chi^\sigma(d_{\eta}) \colon \eta \in \theta \right\rangle^{\sharp}
\hookrightarrow {{\mathbb T\,}}^{\mathcal{D}_\theta}.$$

For each $\theta \in [\alpha]^\omega$ and each $\eta\in \theta$, let $g_{\eta,\theta}$ denote an element of $G^\mathcal{D_\theta}$ with $\chi^{\mathcal{D}_\theta}(g_{\eta,\theta})=j_\theta(\chi^\sigma(d_\eta))$. Observe that the set $\{ g_{\eta,\theta}\colon \eta \in \theta\}$ is independent.

We finally define the elements $f_\eta$, $\eta<\alpha$, by the rules: $$\begin{aligned}
 &\pi_{\mathcal{D}_\theta}^G(f_\eta)=
g_{\eta,\theta} \mbox{, if } \theta \in [\alpha]^\omega \mbox{ is
such that } \eta \in \theta, \qquad \mbox{ and }\\ &
\pi_{\gamma}^G(f_\eta)=\pi_\gamma^G(d_\eta) \mbox{ if }  \gamma
\notin  \mathcal{D}_\theta \mbox{ for any } \theta \in
[\alpha]^{\omega} \mbox{ with } \eta \in \theta. \end{aligned}$$

Let us see that $F=\{f_\eta\colon\eta<\alpha\}$ satisfies the desired properties:

1.  *$F$ and $\chi^\sigma(F)$ are independent.* Suppose that $\sum_{k=1}^m 