My question is how to deduce the red inequality from the Shannon-McMillan-Breiman theorem?

First I state some lemmas which will be used in the **QUESTION**:
the proceeding lemmas are in the setting where $X$ is a non-compact metric space and $T$ is a homeomorphism.

**Lemma:**:For any invariant measure $\mu$ and compact set $C$ with $\mu(C) > 1-\sigma/2$,
there is a compact set $K \subset C$ with $\mu(K) > 1 - \sigma$ and an $N$ so that for every
$n\ge N$ and $x\in K$
$$\frac{1}{n}\sum_{i=0}^{n-1}X_{A}(T^{i}x)\ge1-\sigma$$

**lemma:**
Let $P = \{P_1,.-. ,P_k\}$ be a Borel partition of $X$, let $C_i\subset P_i$ be
compact subsets and let $C = \{C_1, . . . , C_k\}$. Suppose that $K \subset C = \cup C_i$ is
compact and that there exists $\delta > 0$ so that for all $n > N$ and all$x\in K$
$$\frac{1}{n}\sum_{i=0}^{n-1}X_{A}(T^{i}x)\ge1-\sigma$$
then

$$ \limsup_{n\to \infty}\log r(n, C, K) \ge \limsup_{n\to \infty}\log r(n, P, K) -\epsilon(\sigma)$$ where $\epsilon(\sigma)\to 0$ as $\epsilon\to 0$.

where $r(n, C, K)$ is defined as below

Let $C = \{C_1,... , C_k\}$ be any finite collection of disjoint subsets of $X$; C need not be a cover of $X$. We say that a finite set $E \subset K$ is $(n, C)-$ separated if for any pair of distinct points $x, y \in E$, there exists $1<i<r\neq s<k $ and i\in {0,1,...,n-1}such that $T^i x \in C_r$ and $T^i y \in C_s$. Define $r(n, C, K)$ to be the maximal cardinality of a $(n, C)$-separated subset of $k$.

Now fix a metric $d$, an invariant Borel probability measure $\mu$, a finite Borel partition $P = \{ P_1 , . . . , P_k\}$ and $\sigma > 0$. Choose compact sets $k \subset C_r \subset P_r$ so that $C = \cup C_i$ satisfies it$\mu(C) > 1 - \sigma/2$. Choose $K$ and $N$ as in mentioned lemmas By the Shannon-McMillan-Breiman theorem

$$\color{red}{r(n,P, K) > (1 - 2\sigma)2 ^{(h_{\mu}(T'P)-\sigma)n}}.$$

This problems comes up from paper by Michael Handel, Bruce Kitchens and Daniel J. Rudolph on Entropy: http://link.springer.com/article/10.1007/BF02761650