Proof

Inverting s(t) requires s'(t) 0. But, by the Fundamental theorem of Calculus and the definition of L(a, t),

for all parameter values t. In other words,

But this is the "never null" condition which we have assumed. Also,

For the converse, we are given a parameter t such that

In other words,

But now, with s defined to be arc-length from t = a, we have

(the signs cancel for time-like curves) so that

meaning of course that t = s + C.

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