Let X be a locally simply connected space. (X1, p1) is a cover for X. (X2,p2) is a covering space for X1. Then prove that (X2,p1*p2) is a covering space for X.

I thought of this. But I am quite sure that I am wrong.

Let x1 be an element X1 such that p(x1)=x. Then there is an admissible open set U1 is X1, such that p2^(-1) (U1) is a disjoint union of open sets each of which is homeomorphically mapped onto U1 by the map p2. I thought p1(U1) will be the required admissible open set about x. But I am not sure.