# Induced image of the fundamental group of a covering space

I have been reading Hatcher´s Algebraic Topology, and he wants to prove that if we have a covering space $$(E,p)$$, with $$p(e)=x_0$$ then $$p_*(\pi_1(E,e)$$) consists of the homotopy classes of loops in $$X$$ starting at $$x_0$$ such that their lifts are loops in $$E$$ starting at $$e$$.To do this in the proof he says that a loop representing an element of the image $$p_*$$ is homotopic to a loop having such a lift , and intuitively it seems right but i cant seem to see why this is true theoretically , so any help is apreciated, Thanks.

Let $$\gamma$$ be a loop on $$x_0$$ that represents $$p_*([\vartheta])\,\in\pi_1(X,x_0)$$ with $$[\vartheta] \in\pi_1(E,e)$$.
This means $$[\gamma] =p_*([\vartheta]) =[p\circ\vartheta]$$, that is, $$\gamma$$ is homotopic to $$p\circ\vartheta$$, which lifted to $$e$$ obviously gives $$\vartheta$$.