Riemannian length is independent of piecewise c one subdivision

Given: Two admissible subdivisions of a piecewise $C^1$ curve.

Riemannian speed and length: The Riemannian speed on a $C^1$ piece is $\mathrm{\mid }\dot{\gamma }(t){\mathrm{\mid }}_g=\sqrt{g_{\gamma (t)}(\dot{\gamma }(t),\dot{\gamma }(t))}$. Its length is $L_g(\gamma )={\sum }_j{\int }_{t_{j-1}}^{t_j}\mathrm{\mid }\dot{\gamma }(t){\mathrm{\mid }}_{g}dt$. The curve convention is def-piecewise-c-one-curve-on-a-manifold and the norm is def-pointwise-norm-and-angle-from-a-riemannian-metric. Each integrand is continuous on its closed piece with the one-sided endpoint derivative, hence Riemann integrable and nonnegative. Values chosen at the finitely many corners do not change its integral. For a singleton interval the empty sum is zero; a constant curve also has zero length. Partition independence is established next.

The piecewise-C1 line-integral sums do not depend on the admissible partition: For a piecewise-$C^1$ path $\gamma$, the scalar and vector line-integral sums in def-scalar-and-vector-line-integrals-along-piecewise-c1-paths have the same value for every admissible partition. Thus both line integrals are well-defined.