Finite sums and finite products in IP^ sets

In key-4, Hindman proved that for N=C₁ C₂ Cₑ, 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 5. 10key-5, but it was originally due to Glazer and Galvin. In 2. 6key-2, V. Bergelson and Hindman proved that for any AIP^ set A and sequence x₍₍=₁^, there exists a sum subsystem y₍₍=₁^ of x₍₍=₁^ such that FS (y₍₍=₁^) FP (y₍₍=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.