  We present a proof algorithm associated with the dipole splitting algorithm
(DSA). The proof algorithm (PRA) is a straightforward algorithm to prove that
the summation of all the subtraction terms created by the DSA vanishes. The
execution of the PRA provides a strong consistency check including all the
subtraction terms --the dipole, I, P, and K terms-- in an analytical way. Thus
we can obtain more reliable QCD NLO corrections. We clearly define the PRA with
all the necessary formulae and demonstrate it in the hadron collider processes
$pp \to \mu^{+}\mu^{-},\,2\,jets$, and $n\,jets$.
