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

A proof algorithm associated with the dipole splitting algorithm