Lifts of circle-loop concatenations and reversals

Given: Based loops $\alpha ,\beta$, their lifts $\tilde{\alpha },\tilde{\beta }$ from zero, and terminal values $m=\tilde{\alpha }(1)$ and $n=\tilde{\beta }(1)$.

For a based circle loop $\gamma$ with lift $\tilde{\gamma }$ from zero, the terminal value $\tilde{\gamma }(1)$ is an integer and $\mathrm{deg}(\gamma )=\tilde{\gamma }(1)$ (The degree of a based circle loop).

The product $[\alpha ][\beta ]$ traverses $\alpha$ first and $\beta$ second, and is represented by $\alpha \ast \beta$ (Based loops and the fundamental group).

A path through a covering has a unique lift once its initial point is prescribed (Existence and uniqueness of path lifts through a covering map).

Functions continuous on each member of a finite closed cover, and agreeing where the pieces meet, paste to a continuous function (Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous).

For the quotient projection $p$, one has $p(x+k)=p(x)$ for every real $x$ and integer $k$ (The circle as $S^1=\mathbb{R}\mathrm{/}\mathbb{Z}$ with basepoint $[0]$).

Constant functions, the identity, finite sums, and scalar multiples are continuous on real intervals (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function).