Proof:

To prove that the condition (18.1) is necessary, let us assume that a the lift exists. Then $p\circ \tilde{f} = f$ and $p_*\circ {\tilde f}_* = f_{*}$ whereby,

$${f}_{*}(\pi_1(Y, y_0)) = p_*\Big({\tilde f}_{*}(\pi_1(Y, y_0))\Big) \subset p_*(\pi_1({\tilde X}, {\tilde x}_0)).$$

We now turn to the proof of sufficiency of (18.1). To construct the lift ${\tilde f}$ let $y \in Y$ and $\gamma$ be a path in $Y$ joining $y_0$ and $y$. Take the lift of $f\circ \gamma : [0, 1]\longrightarrow X$ starting at ${\tilde x}_0$ and we declare

$${\tilde f}(y) = \widetilde{f\circ \gamma}(1).$$

To show that the function ${\tilde f}$ is well-defined, take two paths $\gamma_1$ and $\gamma_2$ joining $y_0$ and $y$ in $Y$ and form the closed loop $\gamma_1*\gamma^{-1}_2$ at $y_0$. Then $f\circ (\gamma_1*\gamma^{-1}_2)$ is a loop in $X$ based at $x_0$ and so

$$[f\circ (\gamma_1*\gamma^{-1}_2)] \in f_*(\pi_1(Y, y_0)) \subset p_*(\pi_1(\tilde{X}, {\tilde x}_0)).$$

Choose a loop $\sigma$ in ${\tilde X}$ based at ${\tilde x}_0$ such that $p_*([\sigma]) = [f\circ (\gamma_1*\gamma^{-1}_2)]$. In other words, the loop $(f\circ \gamma_1)*(f \circ\gamma^{-1}_2)$ is homotopic to $p\circ \sigma$. By the covering homotopy lemma, The lift of $(f\circ \gamma_1)*(f \circ\gamma^{-1}_2)$ starting at ${\tilde x}_0$ which will be denoted by $\tau$, is homotopic to $\sigma$. As a result, $\tau$ is also closed loop at ${\tilde x}_0$. Let $\widetilde{f\circ \gamma_1}$ be the lift of $f \circ \gamma_1$ starting at ${\tilde x}_0$ and $\widetilde{f\circ \gamma^{-1}_2}$ be the lift of $f \circ\gamma_2^{-1}$ starting at the terminal point $\widetilde{f\circ \gamma_1}(1)$. Observe that

$$\tau(t) = \left\{\begin{array}{lll} {\widetilde{f\circ \gamma_1}}... ...e{f\circ \gamma_2^{-1}}}(2t - 1) & & 1/2 \leq t \leq 1 \\ \end{array} \right.$$

We now look at the projection of the two paths $\tau(s/2)$ and $\tau\big(\frac{2-s}{2}\big)$ ( $0 \leq s \leq 1$):

$$p\circ \tau(s/2) = f \circ \gamma_1(s),\quad 0 \leq s \leq 1$$

and

$$p\circ \tau\big(\frac{2-s}{2}\big) = f \circ \gamma_2(s),\quad 0 \leq s \leq 1.$$

The paths $\tau(s/2)$ and $\tau\big(\frac{2-s}{2}\big)$ ( $0 \leq s \leq 1$) are thus lifts of $f \circ \gamma_1$ and $f \circ \gamma_2$, both starting at ${\tilde x}_0$ since $\tau$ is a closed loop. Hence

$${\widetilde{f\circ \gamma_1}}(1) = \tau(1/2) = {\widetilde{f\circ \gamma_2}}(1)$$

proving that ${\tilde f}(y)$ is well-defined.