= Solution
For increasing parameter $s$ along a future-directed <timelike curve>, the elapsed <proper time> is
$$
\boxed{\Delta\tau=\frac1c\int_{s_1}^{s_2}\sqrt{\eta_{ab}\frac{dx^a}{ds}\frac{dx^b}{ds}}\,ds}
$$
or $\int\sqrt{1-|\boldsymbol v|^2/c^2}\,dt$ when coordinate time is the parameter. A <Lorentz transformation> satisfies $L^T\eta L=\eta$, so replacing $\dot x$ by $L\dot{\tilde x}$ leaves the expression under the square root unchanged. Hence the integral is invariant. A monotone change of path parameter also leaves it unchanged by the ordinary substitution rule.
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