I've just come across this proof of the following theorem that I can't convince myself is true. Any ideas whether it's correct?
Suppose $\gamma$ and $\lambda$ are homotopic paths starting at $x$ in a Riemann surface $X$ with the same endpoint. Let $Y$ be a covering space for $X$, with $Y$ a Riemann surface, $f$ the holomorphic covering map. Pick $y \in f^{-1}(x)$. Let $\tilde{\gamma},\tilde{\lambda}$ be the unique lifts of $\gamma,\lambda$ to $Y$. Then $\tilde{\gamma}(1)=\tilde{\lambda}(1)$.
The proof simply states that by the homotopy lifting theorem $\tilde{\gamma}$ is homotopic to $\tilde{\lambda}$ by a homotopy starting at $y$, so $\tilde{\gamma}(1)=\tilde{\lambda}(1)$. I don't see why this must be the case though. Surely we could have $y_1\neq y_2 \in Y$ with $f(y_1)=f(y_2)=\gamma(1)=\lambda(1)$ and $\tilde{\gamma}(1)=y_1$, $\tilde{\lambda}(1)=y_2$. What prevents this from happening? I assume that the proof is correct since it's a trick that the lecturer on the course used several times!
Many thanks!