  As a basic requirement of the renormalization group invariance, any physical
observable must be independent of the choice of both the renormalization scheme
and the initial renormalization scale. In this paper, we show that by using the
newly suggested $C$-scheme coupling, one can obtain a demonstration that the
{\it Principle of Maximum Conformality} prediction is scheme-independent to
all-orders for any renormalization schemes, thus satisfying all of the
conditions of the renormalization group invariance. We illustrate these
features for the non-singlet Adler function and for $\tau $ decay to $\nu +$
hadrons at the four-loop level.
