Suppose $K\in\mathbb{N}$ and the sets $\mathcal{S}\subseteq\{1,\dots,K\}$ and $\mathcal{S}^c=\{1,\dots,K\}\setminus\mathcal{S}$. Finally, let's use the notation for sets of random variable as $X_\mathcal{S}=\{X_i:i\in \mathcal{S}\}$ and $Y_{\mathcal{S}^c}=\{Y_i:i\in \mathcal{S}^c\}$. I am trying to lower bound the entropy $H(X_\mathcal{S}|Y_{\mathcal{S}^c})$ as: \begin{equation} H(X_\mathcal{S}|Y_{\mathcal{S}^c})\geq\frac{1}{|\mathcal{S}|-1}\sum\limits_{k\in\mathcal{S}}H(X_\mathcal{S}|X_k,Y_{\mathcal{S}^c})\label{a}\tag{1} \end{equation} So, I can show the following steps: \begin{eqnarray} H(X_\mathcal{S}|Y_{\mathcal{S}^c})&=&\frac{1}{|\mathcal{S}|}\sum\limits_{k\in\mathcal{S}}H(X_\mathcal{S},X_k|Y_{\mathcal{S}^c})\\ &=&\frac{1}{|\mathcal{S}|}\sum\limits_{k\in\mathcal{S}}\Big\{H(X_\mathcal{S}|X_k,Y_{\mathcal{S}^c})+H(X_k|Y_{\mathcal{S}^c})\Big\}\\ &\geq&\frac{1}{|\mathcal{S}|}\sum\limits_{k\in\mathcal{S}}\Big\{H(X_\mathcal{S}|X_k,Y_{\mathcal{S}^c})\Big\}+\frac{1}{|\mathcal{S}|}H(X_\mathcal{S}|Y_{\mathcal{S}^c}) \end{eqnarray} by using the generalizations of the properties \begin{equation} H(X,Y|Z)=H(X|Z)+H(Y|X,Z) \end{equation} and \begin{equation} H(X,Y)\leq H(X)+H(Y) \end{equation} but I cannot proceed to the next step to see that eq. (\ref{a}) holds.
EDIT
Initially, I forgot to mention the restriction of $|\mathcal{S}|>1$
You already got the right answer. Just shuffle around some terms:
\begin{align} H(X_\mathcal{S}|Y_{\mathcal{S}^c})&\geq\frac{1}{|\mathcal{S}|}\sum\limits_{k\in\mathcal{S}}\Big\{H(X_\mathcal{S}|X_k,Y_{\mathcal{S}^c})\Big\}+\frac{1}{|\mathcal{S}|}H(X_\mathcal{S}|Y_{\mathcal{S}^c}) \\ &\Updownarrow \\ \left(1 - \frac{1}{|\mathcal{S}|}\right)H(X_\mathcal{S}|Y_{\mathcal{S}^c}) &\geq \frac{1}{|\mathcal{S}|}\sum\limits_{k\in\mathcal{S}}\Big\{H(X_\mathcal{S}|X_k,Y_{\mathcal{S}^c})\Big\} \\ &\Updownarrow \\ H(X_\mathcal{S}|Y_{\mathcal{S}^c})&\geq\frac{1}{|\mathcal{S}|-1}\sum\limits_{k\in\mathcal{S}}H(X_\mathcal{S}|X_k,Y_{\mathcal{S}^c}) \end{align}