Variations of $v$-change of time in an optimal control problem with state and mixed constraints

For a general optimal control problem with state and regular mixed constraints we propose a proof of the maximum principle based on the so-called $v$-change of time variable $t \mapsto \tau$, under which the original time becomes an additional state variable subject to the equation $dt/d\tau = v(\tau)$, while the additional control variable $v(\tau)\geqslant 0$ is piecewise constant, and its values become arguments of the new problem.