Problem:Let $p:X_2 \rightarrow X_1$ and $p':X_1 \rightarrow X$ be two covering maps.If $X$ is locally path connected and semilocally simply connected the $p\circ p'$ is a covering map.
My attempt: Let $x$ be a point of $X$. Then there is an open set $U$ that contains $x$ such that the induced homomorphism $π_1(U,x) \rightarrow π_1(X,x)$ is trivial. Then I took an open neighborhood of $x$, $V$,such as $(p')^{-1}(V)$ is a disjoint union of open sets of $X_1$ each of which is mapped homeomorphically onto $V$ by $p'$. From there I tried some basic ideas but I don't seem to make any progress.I'm self studying Algebraic Topology so any help would be much appreciated.