Let ˜𝑥′ ∈˜𝑋′, and let ˜𝛼′ :𝕀 →˜𝑋′ be a path from ˜𝑥′0 to ˜𝑥′. Further let 𝛼 =𝑞 ∘˜𝛼′ and ˜𝛼 :𝕀 →˜𝑋 be the unique lift of 𝛼 with ˜𝛼(0) =˜𝑥0. Then

and thus both 𝑓 ∘˜𝛼 and ˜𝛼′ are lifts of 𝛼 over 𝑞 with 𝑓 ∘˜𝛼(0) =˜𝛼′(0) =˜𝑥′0, so 𝑓 ∘˜𝛼 =˜𝛼′ and in particular 𝑓(˜𝛼(1)) =˜𝑥′. Thus 𝑓 is surjective.

Let 𝑥1 ∈𝑋. Then 𝑥1 has a open neighbourhood 𝑈 that is evenly covered by both 𝑝 and 𝑞 (simply take the intersection of open neighbourhoods with respect to each covering) which we may assume to be connected without loss of generality (otherwise take the connected component containing 𝑥1). Now let {˜𝑈𝑖}𝑖∈𝐼 and {˜𝑈′𝑗}𝑗∈𝐽 denote the sheets over 𝑈 in ˜𝑋 and ˜𝑋′ respectively. By connectedness it follows that for each 𝑖 ∈𝐼, 𝑓(˜𝑈𝑖) ⊆˜𝑈′𝑗 for exactly one 𝑗 ∈𝐽. Fix some 𝑖 ∈𝐼 and let 𝑗 ∈𝐽 as above. It follows

since

hence (𝑓 ↾˜𝑈𝑖) :˜𝑈𝑖 →˜𝑈′𝑗 is a homeomorphism. Clearly 𝑓−1(˜𝑈′𝑗) ⊆𝑝−1(𝑈), and so from above it follows that the former is some disjoint union of ˜𝑈𝑖. Therefore 𝑓 is a locally path-connected, connected covering.