Existence. By compactness of $[0, 1]$ and the Lebesgue covering lemma, there exists $\delta > 0$ such that every subinterval of length at most $\delta$ maps under $\gamma$ into an evenly covered neighborhood. Partition $[0, 1] = [t_0, t_1] \cup [t_1, t_2] \cup \cdots \cup [t_{N-1}, t_N]$ with $t_i - t_{i-1} < \delta$.

On $[t_0, t_1]$: $\gamma([t_0, t_1])$ lies in an evenly covered neighborhood $U$. Let $\tilde{U}$ be the sheet of $p^{-1}(U)$ containing $\tilde{s}_0$. Define $\tilde{\gamma}|_{[t_0, t_1]} = (p|_{\tilde{U}})^{-1} \circ \gamma|_{[t_0, t_1]}$.

Continue inductively: at step $i$, $\tilde{\gamma}(t_i)$ is defined, and $\gamma([t_i, t_{i+1}])$ lies in an evenly covered $U_i$. The unique sheet $\tilde{U}_i$ containing $\tilde{\gamma}(t_i)$ gives $\tilde{\gamma}|_{[t_i, t_{i+1}]}$.

By the pasting lemma, $\tilde{\gamma}$ is continuous on $[0, 1]$.

Uniqueness. If $\tilde{\gamma}_1, \tilde{\gamma}_2$ are two lifts, the set $\{t : \tilde{\gamma}_1(t) = \tilde{\gamma}_2(t)\}$ is closed (preimage of the diagonal in $\mathbb{R} \times \mathbb{R}$, since $\mathbb{R}$ is Hausdorff) and open (locally, $\tilde{\gamma}_1$ and $\tilde{\gamma}_2$ both lie in the same sheet, where $p$ is a homeomorphism, so equality at one point forces equality in a neighborhood). Since $[0,1]$ is connected and the set contains $0$, it equals $[0, 1]$.

The existence and uniqueness of $\tilde{H}$ follow from the same partition-and-lift argument applied to the square $[0,1]^2$ (subdivide into small squares, each mapping into an evenly covered neighborhood, and lift inductively).

The function $t \mapsto \tilde{H}(1, t)$ is continuous (as a restriction of $\tilde{H}$) and takes values in $p^{-1}(H(1, t))$. If $H$ is a path homotopy (rel endpoints), then $H(1, t) = \gamma_0(1) = \gamma_1(1)$ for all $t$, so $\tilde{H}(1, t) \in p^{-1}(\gamma_0(1)) = \mathbb{Z}$. A continuous function from $[0, 1]$ to $\mathbb{Z}$ (discrete) must be constant.

Well-definedness. If $[\gamma_0] = [\gamma_1]$, the homotopy lifting lemma gives $\tilde{\gamma}_0(1) = \tilde{\gamma}_1(1)$.

Homomorphism. Let $\gamma_1, \gamma_2$ be loops at $1$. The lift of $\gamma_1 * \gamma_2$ starting at $0$ is:

• On $[0, 1/2]$: the reparametrized lift $\tilde{\gamma}_1(2t)$, ending at $n_1 = \tilde{\gamma}_1(1)$.
• On $[1/2, 1]$: the reparametrized lift $\tilde{\gamma}_2(2t - 1) + n_1$, ending at $n_1 + n_2$.

So $\Phi([\gamma_1] \cdot [\gamma_2]) = n_1 + n_2 = \Phi([\gamma_1]) + \Phi([\gamma_2])$.

Surjectivity. For $n \in \mathbb{Z}$, the loop $\omega_n(t) = e^{2\pi i n t}$ has lift $\tilde{\omega}_n(t) = nt$, so $\Phi([\omega_n]) = n$.

Injectivity. If $\Phi([\gamma]) = 0$, then $\tilde{\gamma}$ is a loop in $\mathbb{R}$ based at $0$. Since $\mathbb{R}$ is contractible (simply connected), $\tilde{\gamma} \simeq \tilde{e}_0$ rel $\{0, 1\}$ via some homotopy $\tilde{H}$. Then $H = p \circ \tilde{H}$ gives $\gamma \simeq e_1$ rel $\{0, 1\}$ in $S^1$. So $[\gamma] = [e_1]$.