Lindenbaum's lemma and the hypotesis of deductively closure

Question

In my lectures, we define an Henkin set for a theory $T$ as a set $H$ of enunciates in a signature $L$ that is:

1. Deductively closed for $T$: if $T \vdash A_1,...,A_n \Rightarrow B$ and $A_1,...,A_n \in H $, then $ B \in H$;

2. Rich: if $\exists xB \in H$, then it exists a ground term $t$ s.t. $B(t/x) \in H$;

3. Maximal: for all $A$ enunciate, $A \in H$ or $\neg A \in H$;

4. Consistent: with the following equivalent definitions: it exists an enunciate $B$ s.t. $B \notin H$, or for all enunciate $B$, $B \in H$ and $\neg B \in H$ is not acceptable at the same time.

And then we try to prove the Lindebaum's Lemma:

Suppose $S$ a set of enunciates consistent, it exists a set $S^* \supseteq S$ s.t. $S^*$ is consistent and maximal.

My question is: shouldn't we have the hypotesis that $S$ is deductively closed for $T$ in Lindenbaum's lemma? And shouldn't we use the hypotesis of closure for deduction in order to demonstrate that the two definitions of consistency for a set of enunciates are equivalent?