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For the past two days, I have been stuck on the last sentence of the proof below: enter image description here

I cannot think of any reason why a loop that is the image of $p_{*}$ must be homotopic to a loop whose lift is a loop as well.

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If $[f]\in\pi_1(X,x_0)$ is of the form $[f]=p_*[g]$ for some $[g]\in\pi_1(\tilde X,\tilde x_0)$, then $f$ is homotopic to $p\circ g$, which lifts to $g$, which is a loop at $\tilde x_0$.

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If $[\gamma] = p_*[\delta]$, then $\delta$ is (up to homotopy) a lift of $\gamma$, therefore it is the unique such lift, therefore any lift of $\gamma$ starting at $\tilde{x_0}$ is homotopic to $\delta$, which is a loop, therefore any such lift is also a loop.

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