$G^*$ be the graph obtained from $G$ by contracting $T_2$ to a vertex (which we will call $v^*$). Note that $G^*$ has fewer edges than $G$, and so satisfies Theorem \[thm:main\]. This means that $G^*$ either contains $k$ edge-disjoint long cycles or a small hitting set. We will now analyse these two cases separately.\
First suppose that $G^*$ contains a collection $\mathcal{C}$ of $k$ edge-disjoint long cycles. To obtain a contradiction, it suffices to derive that then $G$ also has $k$ edge-disjoint long cycles. To that end, we partition $\mathcal{C}$ into three subcollections $\mathcal{C}_1$, $\mathcal{C}_2$ and $\mathcal{C}_3$. By definition, each long cycle in $\mathcal{C}_1$ visits at least one edge of $T_1$ (and thus in particular visits $v^*$), while $\mathcal{C}_2$ contains the long cycles that are not in $\mathcal{C}_1$ but do visit $v^*$ and, finally, $\mathcal{C}_3$ contains the long cycles that do not visit $v^*$. Decontracting $v^*$ back to $T_2$ yields the following natural bijection $\sigma$ from $\mathcal{C}$ to edge-disjoint cycles in $G$. If $C\in \mathcal{C}_3$ then the decontraction does not affect $C$, so we just take $\sigma(C)=C$. If $C \in \mathcal{C}_2$, then $\sigma(C)$ is the cycle obtained from $C$ by identifying $v^*$ with $t_2$ (in particular, this ensures that $\sigma(C)$ 