Proper time is invariant under regular increasing weak piecewise changes of parameter

On each open piece of its partition, $\phi$ is $C^1$ with ${\phi }^{\mathrm{\prime }}>0$; the mean value theorem [L2] applied on each closed piece gives $\phi (\beta )-\phi (\alpha )={\phi }^{\mathrm{\prime }}(c)(\beta -\alpha )>0$ for its endpoints $\alpha <\beta$, so $\phi$ is strictly increasing on every piece and, concatenating the finitely many pieces, strictly increasing on $[u,v]$; hence $\phi$ is injective with $\phi (u)=a$, $\phi (v)=b$, and its inverse is continuous by [L3].

Let ${\rho }_0<\cdots <{\rho }_N$ be the increasing enumeration of the partition points of $\phi$ together with the preimages ${\phi }^{-1}(s_1),\dots ,{\phi }^{-1}(s_{m-1})$, which exist uniquely by [L3]; this is a finite partition of $[u,v]$ (the common refinement).

On each open subinterval $({\rho }_{k-1},{\rho }_k)$ the map $\phi$ is $C^1$ and $x$ is $C^1$ on the open piece of $x$ containing $\phi (({\rho }_{k-1},{\rho }_k))$, so $y=x\circ \phi$ is $C^1$ there with $y^{\mathrm{\prime }}={\phi }^{\mathrm{\prime }}(x^{\mathrm{\prime }}\circ \phi )$ in coordinates, by the chain rule [L4]; hence every open-piece tangent of $y$ is a positive multiple of a future timelike vector and is future timelike by [L5].

At each endpoint ${\rho }_k$, the one-sided difference quotients of $y$ exist: writing $k_h:=\phi ({\rho }_k+h)-\phi ({\rho }_k)$ for small $h$ on one side, $(y({\rho }_k+h)-y({\rho }_k))\mathrm{/}h=\left[\right(x(\phi ({\rho }_k)+k_h)-x(\phi ({\rho }_k))\left)\mathrm{/}{k}_h\right]\cdot (k_h\mathrm{/}h)$, where $k_h\to 0$ on the correct side, the first factor tends to the finite one-sided quotient tangent of $x$ at $\phi ({\rho }_k)$ (or to the tangent $x^{\mathrm{\prime }}$ when $\phi ({\rho }_k)$ is interior to an $x$-piece), and the second tends to the finite positive one-sided quotient derivative of $\phi$ at ${\rho }_k$; the product is finite and, being a positive multiple of a future timelike vector, future timelike by [L5].

Let ${\rho }_{k-1}<r<s<{\rho }_k$. Then $\phi$ is $C^1$ on $[r,s]$ with ${\phi }^{\mathrm{\prime }}$ continuous, hence integrable; the function $f:=\sqrt{-\eta (x^{\mathrm{\prime }},x^{\mathrm{\prime }})}$ is continuous on the order-convex open piece of $x$ containing $\phi ([r,s])$ by [L7] (it is the composition of the continuous tangent component functions with the square root); substitution [L6] gives ${\int }_r^s{\phi }^{\mathrm{\prime }}(w)f(\phi (w))dw={\int }_{\phi (r)}^{\phi (s)}f$.

By bilinearity [L8], $-\eta (y^{\mathrm{\prime }}(w),y^{\mathrm{\prime }}(w))={\phi }^{\mathrm{\prime }}(w{)}^2(-\eta (x^{\mathrm{\prime }}(\phi (w)),x^{\mathrm{\prime }}(\phi (w))))$, so $\sqrt{-\eta (y^{\mathrm{\prime }},y^{\mathrm{\prime }})}={\phi }^{\mathrm{\prime }}(w)f(\phi (w))$ on the open subinterval because ${\phi }^{\mathrm{\prime }}>0$; hence the left side of step 4.2 is the truncation integral of $y$'s length density over $[r,s]$.

Both $x$ and $y$ are weak piecewise future timelike curves (the first by hypothesis, the second by steps 3.1 and 4.1), so the improper integrals of $f$ over the pieces of $x$ and of $\sqrt{-\eta (y^{\mathrm{\prime }},y^{\mathrm{\prime }})}$ over the pieces of $y$ exist and are finite by [L9]. Letting $r\downarrow {\rho }_{k-1}$ and $s\uparrow {\rho }_k$ in step 4.2, the left side tends to ${\int }_{({\rho }_{k-1},{\rho }_k)}\sqrt{-\eta (y^{\mathrm{\prime }},y^{\mathrm{\prime }})}$ by definition of the improper integral, and the right side tends to the integral of $f$ over the interval $(\phi ({\rho }_{k-1}),\phi ({\rho }_k))$ contained in a single open piece of $x$, by continuity and strict monotonicity of $\phi$; the limits are equal.

Summing step 6.1 over the refined pieces and using the additivity of the improper integrals [L9], the intervals $(\phi ({\rho }_{k-1}),\phi ({\rho }_k))$ partition the pieces of $x$ up to finitely many cutting points, so $\mathrm{\Delta }{\tau }_{[u,v]}(y)=c^{-1}{\sum }_k{\int }_{({\rho }_{k-1},{\rho }_k)}\sqrt{-\eta (y^{\mathrm{\prime }},y^{\mathrm{\prime }})}=c^{-1}{\sum }_i{\int }_{(s_{i-1},s_i)}f=\mathrm{\Delta }{\tau }_{[a,b]}(x)$.

The same argument applied to the restrictions of $x$ and $\phi$ to corresponding restricted parameter intervals gives $\mathrm{\Delta }\tau$ over every pair of corresponding restricted intervals; consequently the based functions ${\tau }_y$ and ${\tau }_x\circ \phi$ differ by the base-point constant.

The hypotheses used are ${\phi }^{\mathrm{\prime }}>0$ at every interior smooth point and finite positive one-sided endpoint quotients; at a point where ${\phi }^{\mathrm{\prime }}=0$ the composed tangent is zero and the composition is not a worldline, so no total proper time is asserted for it, and no inverse of $\phi$ is used in the integral identity of step 7.1; no object is selected from a family, so no choice principle is used.