{determined by $\Lambda$ together with a pair of conjugate points of $\gamma_{\Lambda}$}\}\end{aligned}$$ The function $f(\Lambda)$ is upper semicontinuous [@gaowald], when the *second requirement* and the *third requirement* are satisfied, the proof has been given in the reference [@gaowald]. Let $K\subset M$ be a compact set. Let $S_{K}=\{(p,k^{\mu}\in S \text{ with } p \in K)\}$, since the tangent bundle has the product topology, $K$ is compact and $k^{\mu}$ is of bounded norm, then $S_{K}$ is compact. Since $f$ is upper-semicontinuous, it must achieve a maximum in $S_{K}$, let’s denote it by $\bar R$. Let $K' = B_{\bar R}$. Let $p,q\notin K'$ and $q\in J_+(p)-I_+(p)$ and let $\gamma$ a causal curve joining $p$ with $q$, then $\gamma$ must be a null geodesic since $q\in J_+(p)-I_+(p)$. However, the Proposition \[temporal\] insures that $\gamma$ does not contain a pair of conjugate points between $p$ and $q$. If $\gamma\cap K\neq \emptyset$ then by the definition of $K'$, $\gamma$ must have a pair of conjugate points lying in $K'$ and in between $p$ and $q$. This contradiction completes the proof.

Note that the theorem given above is true if Properties 1 and 2 are replaced by the *first*, *second* and *third requirements* given in the introduction, as the proof would be unchanged.

In brief, in the present work it has been shown that the Gao-Wald theorem holds when the space time ($M$, $g_{\mu\nu}$) is null geodesically complete, every null geodesic posses at least a pair of conjugate points, and the curvature is such that a the quantity $R_{\mu\nu}k^\mu k^\nu\neq -e^{c\lambda}$ for large $\lambda$ values, when evaluated on a null geodesic. Therefore this result may apply to models which violate the Null Energy Condition [@averaged1]-[@averaged25] or to modified gravity theories such as the ones described in [@odintsov]-[@odintsov3]. This deserves further attention. In our opinion, the curvature condition introduced in the text may hold for several interesting solutions for these models. On the other hand, t