and $D(t(G_0))$ divisible hulls of $F$ and $t(G_0)$, respectively. There is then a chain of group embeddings (here we use [@fuchs Lemmas 16.2 and 24.3])

$$\label{basicemb}
 F \overset{j_1}{\to} G_0 \overset{j_2}{\to} D(F)\oplus D(t(G_0))$$

Denote by $\chi$ the quotient homomorphism obtained as the dual map of the canonical embedding ${{\mathbb Z}}\to{{\mathbb Q}}$. Observe that identifying $F$ with $\oplus_{r_0(G)}{{\mathbb Z}}$ and $D(F)$ with $\oplus_{r_0(G)}{{\mathbb Q}}$, the dual map of $j_2\circ j_1$ is exactly $\chi^{r_0(G)}$.

Taking $\sigma=r_0(G)$, $G={{\mathbb Q}}_d^\wedge$ and $\alpha=\kappa^{\omega}$, we can apply Proposition \[prop:freetorsion\] to get a ${G_{\delta}}$-dense subgroup $H_1$ of $\left(D(F)_d\right)^\wedge=\Bigl({{\mathbb Q}}_d^\wedge\Bigr)^{r_0(G)}$ with $|H_1|=\kappa^\omega$ and such that $H_1$ and $\chi^{r_0(G)}(H_1)$ are isomorphic and have property ${{\mathbf \sharp}}$ (notice that $\kappa^\omega $ and $r_0(G)$ satisfy the hypothesis of that Proposition).

We now apply Corollary \[cor:comfgali\] to $\chi^{r_0(G)}(H_1)$ to obtain another free subgroup $H_2^\prime$ of ${{\mathbb T\,}}^{r_0(G)}$ with $\chi^{r_0(G)}(H_1)\cap H_2^\prime=\{0\}$, $|H_2^\prime|=2^\kappa$ and such that $\chi^{r_0(G)}(H_1)+H_2^\prime$ has property ${{\mathbf \sharp}}$. By lifting (through $\chi^{r_0(G)}$) the free generators of $H_2^\prime$ to $(D(F)_d)^\wedge$, we obtain a free subgroup $H_2$ of $(D(F)_d)^\wedge$ such that $H_1\cap H_ 2=\{0\}$ and $|H_2|=
2^\kappa$. Clearly $H_1+H_2$ is isomorphic to $\chi^{r_0(G)}(H_1)+H_2^\prime$ and therefore $H_1+H_2$ has property ${{\mathbf \sharp}}$ by Lemma \[lem:quotients\].

We finally apply Lemma \[enumer\]. The role of $G_1\times L$ is 