ion (\[explota\]) is obtained.

Proof of the *third requirement* as a consequence of Properties 1 and 2
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The purpose of this section is to show that Property 1 and Property 2 given above imply the *third requirement*. This leads to the conclusion that the Gao-Wald theorem holds for space times satisfying these properties.

Before making formal statements, it is worth to give the intuition behind the proof of *third requirement* given below. One considers a reference null geodesic $\gamma_0(\lambda)$ which, by Property 1, has two conjugate points let’s say $p_0$ and $q_0$. Remark \[theta\_i\] shows that (\[tang2\]) is satisfied for some point in the middle $s_0$. That is, for this point $s_0=\gamma_0(\lambda_i)$ one has $$\theta_0(\lambda_i)+\frac{c_\Lambda}{2}< \lim_{\lambda\to\infty} \text{inf}\int_{\lambda_i}^\lambda e^{-c_\Lambda \xi} p_0(\xi) 
d\xi.$$ The strategy is to show that, for some geodesics emanating from points $p$ close to $p_0$, the condition (\[tang2\]) is also satisfied for the points $s=\gamma_\Lambda(\lambda_i)$, which are close to $s_0$. That is $$\theta_\Lambda(\lambda_i)+\frac{c_\Lambda}{2}< \lim_{\lambda\to\infty} \text{inf}\int_{\lambda_i}^\lambda e^{-c_\Lambda \xi} p_\Lambda(\xi) 
d\xi,$$ for all these geodesics. The Proposition \[prop1\] will allow to conclude that $\theta_\Lambda(\lambda)$ also is going to tend to $-\infty$ at a finite value of the affine parameter $\lambda$. Therefore, these points $p$ will have a conjugate point $q$ joined by the null geodesic $\gamma_\Lambda(\lambda)$. This leads to the first part of the *third requirement*.

The second part is more subtle. The point is that, by assumption, the reference geodesic $\gamma_0(\lambda)$ has two conjugate points $p_0$ and $q_0$. At these points the determinant $G_0(\lambda)$ defined in (\[smile2\]) vanishes. Thus $G_0(0)=0$ and $G_0(\lambda_e)=0$. By continuity in $S$, one may work by analogy with section 2.3 and try to show that $|G_\Lambda(\lambd