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A proof algorithm associated with the dipole splitting algorithm

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 $pp \to μ^{+}μ^{-},\,2\,jets$, and $n\,jets$.