# Statement(s) > [!NOTE] Statement 1 (....) > If $\Pi$ is a [[Lorentz Plane|Lorentz plane]], then $\Pi \cap \mathcal{H}^n \neq \emptyset$ where $\mathcal{H}^n$ is the [[Hyperbolic space|upper sheet of the n-hyperboloid]]. # Proof(s) ###### Proof of statement 1: By definition there exists $x\in \Pi$ such that $\langle x, x \rangle_{L}<0$. Let $\lambda = \sqrt{-\langle x,x \rangle_{L} }$ then $\underline{x}/\lambda\in \mathcal{H}^n$ since $\left\langle \frac{x}{\lambda} , \frac{x}{\lambda} \right\rangle= \frac{1}{\lambda^2} \langle x, x \rangle_{L} = -\frac{\lambda^2}{\lambda^2} = -1. $ $\blacksquare$ # Application(s) **Consequences**: **Examples**: # Reference(s)