Path lifting lemma

Let $p:E\to X$ be a cover, $x\in X$, $e\in p^{-1}(x)$

1. Let $\gamma :I\to X$ be a path with $\gamma (0)=x$. There exists a unique path lift ($p\circ \tilde{\gamma }=\gamma$) $\tilde{\gamma }:I\to E$ such that $\tilde{\gamma }(0)=e$.

   Written another way, there is a unique path $\tilde{\gamma }:I\to E$ such that the following diagram commutes.

2. homotopies lift uniquely

Proof

todo