Every rectifiable path factors through its arc-length function as a unit-speed path on $[0,L]$

By continuity and the endpoint values in [L1], $s([a,b])=[0,L]$. If $s(u)=s(v)$ with $u\le v$, [L1] makes the intervening length zero and [L2] gives $\gamma (u)=\gamma (v)$.

For $r\in [0,L]$, define $\bar{\gamma }(r)$ to be the unique common value $\gamma (t)$ of all $t$ with $s(t)=r$. Existence follows from surjectivity and well-definedness from step 1.1. This definition immediately gives $\gamma =\bar{\gamma }\circ s$ and uniqueness.

For $r<q$, take $u\le v$ with $s(u)=r$ and $s(v)=q$. The chord bound and [L1] give $\parallel \bar{\gamma }(q)-\bar{\gamma }(r){\parallel }_2\le L_{[u,v]}(\gamma )=q-r$. Hence $\bar{\gamma }$ is $1$-Lipschitz and continuous.

The restriction $s{\mathrm{\mid }}_{[u,v]}$ is a continuous surjective nondecreasing map onto $[r,q]$, and $\gamma {\mathrm{\mid }}_{[u,v]}=\bar{\gamma }{\mathrm{\mid }}_{[r,q]}\circ s{\mathrm{\mid }}_{[u,v]}$. By [L3], $L(\bar{\gamma }{\mathrm{\mid }}_{[r,q]})=L(\gamma {\mathrm{\mid }}_{[u,v]})=q-r$.

If $L=0$, [L2] makes $\gamma$ constant, $s$ has singleton image, and the construction gives the unique constant map on $[0,0]$; the subinterval formula is $0=0$.