Abstract:   We present a method to prove hypergeometric double summation identities. Given a hypergeometric term F(n, i, j), we aim to find a difference operator L = a₀(n)N ⁰ + a₁(n)N ¹ + ··· + a_r(n)N ^r and rational functions R₁(n, i, j), R₂(n, i, j) such that LF = _i(R₁F) + _j(R₂F). Based on simple divisibility considerations, we show that the denominators of R₁ and R₂ must possess certain factors which can be computed from F(n, i, j). Using these factors as estimates, we may find the numerators of R₁ and R₂ by guessing the upper bounds of the degrees and solving systems of linear equations. Our method is valid for the Andrews-Paule identity, the Carlitz's identities, the Apéry- Schmidt-Strehl identity, the Graham-Knuth-Patashnik identity, and the Petkovšek-Wilf-Zeilberger identity.