Finite sums and finite products in IP^ sets

In {key-4}, Hindman proved that for N=C₁∪ C₂∪…∪ Cᵣ, one cell contains an $AIP$ set, which was a conjecture of Graham and Rothschild. Hindman's method was combinatorial, and an algebraic proof of the Hindman finite sum theorem is followed from [Corollary 5.10]{key-5}, but it was originally due to Glazer and Galvin. In [Theorem 2.6]{key-2}, V. Bergelson and Hindman proved that for any AIP^ set A and sequence xₙₙ₌₁∞, there exists a sum subsystem yₙₙ₌₁∞ of xₙₙ₌₁∞ such that FS( yₙₙ₌₁∞)∪ FP( yₙₙ₌₁∞)⊆ A using algebra of . In this article, we will prove the same result combinatorially. As a consequence, we will prove that MIP^ sets contain two different exponential patterns.