I'm trying to understand the proof of a mixing property of the Gauss map from the paper - 'Some metrical theorems in number theory' and I'm getting confused by the logic in a step.

The Gauss map $T$, defined in the interval $I$, is given as the fractional part of $\frac{1}{x}$ where $x\in(0,1]$. If you use Kuzmin's proof of convergence to the invariant measure $\frac{1}{\ln 2}\frac{1}{1+x}dx$, you get (after deducing it for cylindrical sets)

**Lemma 3:** If $F \subset I$ is measurable and if $E=(x,y) \subset I$ is such that $x, y$ are $k$th convergents (associated to some $k$ continued fraction digits), then
\begin{equation}
|E\cap T^{-(n+k)}F| = |E||F|\left(1 + O(q^{\sqrt{n}})\right)
\end{equation}
where $q<1$ and the constant implicit in the big-O notation are independent of the sets and $k$, and where $|\cdot|$ denotes the invariant measure described above.

From this I want to go to

**Lemma 4:** If $F\subset I$ is measurable and $E=(x,y) \subset I$ is *any* interval, then
\begin{equation}
|E \cap T^{-n}F| = |E||F| + |F|O\left(q^{\sqrt{n}} \right)
\end{equation}
with the same constant.

**Proof:** Fix $k$ to be the floor of $n/2$. Find continued fraction convergents (of $k$ digits) $p^i_k, q^i_k$ such that
$$
x \in \left[\frac{p^1_k}{q^1_k}, \frac{p^1_k + p^1_{k-1}}{q^1_k + p^1_{k-1}}\right] =: G_1 \text{ and }
y \in \left[\frac{p^2_k}{q^2_k}, \frac{p^2_k + p^2_{k-1}}{q^2_k + p^2_{k-1}}\right] =: G_2
$$
and decompose $E$ as the disjoint union
$$
E = E_1 \cup E_0 \cup E_2.
$$
Here $E_1 = E \cap G_1$, $E_2 = E \cap G_2$ and $E_0$ is the interval between the endpoints of $G_1$ and $G_2$.

Now supposedly we have **by lemma 3 and using that $|\cdot|$ is $T$-invariant** that
\begin{equation}
\left||E\cap T^{-n}F| - |E||F| \right| \leq C|E_0||F|q^{\sqrt{k}} + |F|\left(|E_1| + |E_2|\right).
\end{equation}

**Question:** How does this follow?

I think I understand where the first term comes from but not the second - \begin{equation} \begin{split} \left||E\cap T^{-n}F| - |E||F| \right| &\leq \sum^{2}_{i=0}\left| |E_i\cap T^{-n}F| - |E_i||F|\right| \\ &= C|E_0||F|q^{\sqrt{k}} + \sum^2_{i=1}||E_i\cap T^{-n}F| - |E_i||F||.\ \text{By Lemma 3.} \end{split} \end{equation} But how to estimate the second term(s)? Is it somethink trivial?

The factor $|F|$ in Lemma $4$ seems to be crucial when applying it elsewhere in the paper.