F_{k-1}$, be the form $(x,x^{'},y,y^{'})$, where $(x,x^{'}), (y,y^{'})\in E$ and $(y,x^{'}), (x,y^{'})\in E^{'}$.

First, we estimate the number of $\overline{C_4}$-type cycles $\gamma$ for which $(x,x^{'})$ is contained in a particular cycle $\gamma_i\in E$. We say that $\gamma$ is $rooted$ at $\gamma_i$. Assume, for a moment, that the vertices $x, x^{'}$ is chosen. There are at least $\alpha n-(n_i-1)$ ways to extend the path first to $y$ then to $y^{'}$ since the indegrees of $x^{'}$ is at least $\alpha n$. Denote $Y^{'}$ the set of all vertices $y^{'}$ reachable. Recall $N^{+}(x,G)$ is the set of neighbors $x$ points to. Thus the number of ways of completing a $\overline{C_4}$-type cycle $(x,x^{'},y,y^{'})$ is at least $$\begin{split}
|N^{+}(x,G)| + |Y^{'}|-n &\leq \alpha n + (\alpha n-(n_i-1))  -n\\
&=2\alpha n-n_i-n+1.
\end{split}$$ Hence the total number of $\overline{C_4}$-type cycles rooted at $\gamma_i$ is at least $n_i(2\alpha n-n_i-n+1)$.

We are now poised to bound $w^{+}(E)$. Each arc $(E,E^{'})$ defined by a $\overline{C_4}$-type $\gamma$ rooted at $\gamma_i$ has weight at least $n_i^{-1}f(k-1)$, which, by Lemma 8, bounded below by $(\beta\log n)(kn_i)^{-1}f(k)$, Thus\
$$\begin{split}
w^{+}(E)&=\sum\limits_{E^{'}:(E,E^{'})\in K}w(E,E^{'})\\
&\geq \sum\limits_{i=1}^kn_i(2\alpha n-n_i-n+1)n_i^{-1}f(k-1)\\
&= \sum\limits_{i=1}^k(2\alpha n-n_i-n)f(k-1)+ k f(k-1) \\
&\geq (2\alpha kn-n-kn)(\beta \log n)k^{-1}f(k) + k f(k-1) \\
&=(2\alpha -1/k-1)(\beta \log n)f(k)n + 