Loops over a two-set path-connected open cover factor through the covering sets

By [F5], ${\alpha }^{-1}(U)$ and ${\alpha }^{-1}(V)$ form an open cover of the compact metric interval $I$. Choose a Lebesgue number $\delta >0$ by [F1], then choose $q\ge 1$ with $1\mathrm{/}q<\delta$ by [F7] and use the subdivision $t_i=i\mathrm{/}q$. Each restricted path ${\alpha }_i:=\alpha {\mathrm{\mid }}_{[t_{i-1},t_i]}$, reparametrized to $I$, is continuous by [F8] and lies wholly in $U$ or wholly in $V$; assign it to $U$ if its image lies in $U$, and to $V$ otherwise. Merge adjacent pieces assigned to the same set. After merging, every interior subdivision value $\alpha (t_i)$ lies in $U\cap V$; the construction also admits the constant loop and the case of one retained piece.

Put ${\lambda }_0={\lambda }_m=c_{x_0}$. For each remaining interior vertex, [F2] makes the family of paths in $U\cap V$ from $x_0$ to $\alpha (t_i)$ nonempty, so [F4] supplies paths ${\lambda }_i$ for the finitely many vertices. If ${\alpha }_i$ lies in $A_i\in \{U,V\}$, then ${\beta }_i:={\lambda }_{i-1}\ast {\alpha }_i\ast {\bar{\lambda }}_i$ is a based loop in $A_i$.

In the product $[{\beta }_1]\cdots [{\beta }_m]$, every adjacent pair ${\bar{\lambda }}_i\ast {\lambda }_i$ cancels up to endpoint-fixed path homotopy, and the outside connectors are constant. Thus [F3] gives $[\alpha ]=(j_{A_1}{)}_{\ast }[{\beta }_1]\cdots (j_{A_m}{)}_{\ast }[{\beta }_m]$. For $m=1$ this is the single factor $[\alpha ]$, and for a constant loop it is the identity, so every loop class has the asserted factorization.