Complex line integrals change sign under reversal and add under concatenation

By [L1] the reversal is ${\gamma }^{-}(t)=\gamma (a+b-t)$. The map $t\mapsto a+b-t$ sends a partition $a=t_0<\cdots <t_n=b$ to the partition with points $a+b-t_n<\cdots <a+b-t_0$ and sends tags to tags, so it is a mesh-preserving bijection between tagged partitions of $[a,b]$ and tagged partitions of $[a,b]$. Under it each coordinate increment for ${\gamma }^{-}$ is the negative of the matching increment for $\gamma$, since the two subinterval endpoints are exchanged; each arc-length increment is instead unchanged, because by [L1] length is unaffected by monotone reparametrization. Hence every Riemann–Stieltjes sum in the coordinate integrators of [L4] for ${\gamma }^{-}$ is the negative of the corresponding sum for $\gamma$, and every sum in the arc-length integrator is equal to it. Passing to the limit over refinements yields the two reversal identities.

By [L1] the concatenation $\alpha \ast \beta$ is given on $[0,\frac{1}{2}]$ and $[\frac{1}{2},1]$ by the two standard affine pieces, each a strictly increasing continuous bijection onto its factor's parameter interval. Apply [L5] to each component Stieltjes integral in [L4] to transport it to the factor's own interval. The integrators are coordinates and arc length of a rectifiable path, hence of bounded variation, and the integrand is continuous, so the join $\frac{1}{2}$ satisfies the additivity hypotheses of [L2]; splitting there and recombining gives both additive identities.

On piecewise-$C^1$ contours these conclusions agree exactly with [L3], whose hypotheses and orientation distinction are preserved. Constant pieces contribute $0$.